- Email: [email protected]
UPt3, heavy fermions and superconductivity
Physica 147B (1987) 81-160 North-Holland, Amsterdam
UPt,,
HEAVY FERMIONS
A. DE VJSSER*, Natuurkundig Received
A. MENOVSKY
Laboratorium
1 June
AND SUPERCONDUCTIVITY
der Universiteit
and J.J.M.
FRANSE
van Amsterdam,
Valckenierstraat
65, 1018 XE Amsterdam,
The Netherlands
1987
This review considers studies of the normal and superconducting state of the heavy-fermion superconductor UPt, most of them performed on good quality (y = 422 mJ/K’molU, T, = 0.5 K). We discuss a large variety of experiments, single crystals. Concerning the normal state we concentrate on the magnetic, thermal, elastic and transport properties. These properties can be fairly described in a spin-fluctuation model. Experiments concerning the superconducting phase reveal several unusual features. The possibility of an unconventional pairing mechanism is discussed. On alloying UPt, with Pd, additional anomalies are observed, suggesting that UPt, is close to an antiferromagnetic instability.
Table
ofcontents
1. General
2.6.3.
introduction
1.1. Introduction 1.2. Related compounds 1.2.1. 1.2.2. 1.2.3. 1.3. 1.3.1. 1.3.2. 1.3.3.
2.7.
1.4. Phase diagram and crystallography 1.5. Sample preparation
2.1. 2.2. 2.3. 2.3.1. 2.3.2.
4. UPt,
state
4.1. 4.2. 4.3. 4.4. 4.5. 4.6.
Introduction Magnetic properties Thermal properties Specific heat Thermal expansion
2.4. Elastic properties 2.5. Transport properties 2.5.1. 2.5.2. 2.5.3. 2.6. 2.6.1. 2.6.2.
conductivity
address:
Centre
state
with Pd
Introduction Specific heat and thermal expansion Susceptibility and magnetization Electrical resistivity Superconductivity Discussion remarks
5.1. Normal phase 5.2. Superconducting
and Hall
phase
Acknowledgements
studies
Magnetization Magnetoresistivity
* Present
alloyed
5. Concluding
Electrical resistivity Thermoelectric power, thermal effect Point-contact spectroscopy High-magnetic-field
the superconducting
3.1. Introduction 3.2. The Meissner effect 3.3. Specific heat 3.4. Critical field studies 3.5. Fermi liquid analysis 3.6. Miscellaneous
theories
the normal
studies resistivity susceptibility
3. UPt,:
Spin fluctuations Fermi liquid theory Odd-parity superconductivity
2. UPt,:
magnetostriction
Electrical Magnetic
2.7.1. 2.7.2.
and systematics
The uranium-platinum compounds The heavy-fermion compounds The UX, compounds Relevant
Forced
High-pressure
References
de Recherches
037%4363/87/$03.50 0 (North-Holland Physics
sur les Tr&
Basses
Temptratures,
Elsevier Science Publishers Publishing Division)
B.V.
CNRS,
BP 166X, 38042,
Grenoble,
France.
82
A. de Visser et al. I UPt,, heavy fermions
1. General Introduction 1.1.
Introduction
The recent discovery of a class of intermetallic compounds, the so-called heavy-fermion system, has caused a great deal of excitement among solid-state physicists. On cooling these materials to liquid helium temperatures the coefficient, y, of the electronic term in the specific heat, i.e. the term linear in temperature, attains a giant value. For an ordinary metal the y-value is of the order of l-10 mJ/K’mol. For a heavy-fermion system the y-value amounts to 400-1000 mJ/K*mol, i.e. roughly a factor 102-lo3 larger. Such a large y-value is usually explained by the existence of heavy electrons or quasiparticles at the Fermi surface. The effective mass of these heavy electrons amounts up to lo’-lo3 times the free electron mass. Hence the name heavy fermions. Of course we have to bear in mind that this definition of a heavy-fermion system is a somewhat arbitrary one. Systems with intermediate yvalues are found as well, and it is not precisely clear where to draw the border-line. After the discovery of the first heavy-fermion system CeAl, in 1975 by Andres, Graebner and Ott [l], extensive research programs have been carried out in order to look for equivalent systems. Fortunately, CeAl, was not a singularity in nature, and, up to date, about ten other heavyfermion compounds are known, among which the most prominent ones are CeCu,Si,, CeCu,, UBe,,, UPt,, U,Zn,, and UCd,,. Each system has its own exciting low-temperature properties and a variety of ground states are observed: superconductivity in CeCu,Si,, UBe,, and UPt,, antiferromagnetic order in U,Zn,, and UCd,,, and no order at all in CeAl, and CeCu,, at least not down to temperatures as low as - 10 mK. In particular, the discovery of superconductivity in some of the heavy-fermion systems was quite unexpected. Recent speculations about a nonconventional pairing mechanism (odd parity) have attracted much attention to this class of materials. At present, heavy-fermion behaviour is mainly found in intermetallic cerium and uranium-based
and superconductivity
compounds, which suggests the formation of a narrow 4f (Ce) or 5f (U) band near the Fermi surface, resulting in a strongly interacting electron system. In this paper we focus on one of the uranium-based heavy-fermion superconductors: UPt,. This material is of particular interest since it exhibits a coexistence of pronounced spinfluctuation phenomena [2] and superconductivity (at T, = 0.5 K [3]). This article is organized as follows. In the remainder of section 1 we present a rather general introduction to the physics of UPt,. Successively we present: (i) a review of related compounds and a discussion of the systematics in the occurrence of heavy-fermion behaviour, (ii) some elements of the theoretical models that might be applicable to UPt, and (iii) some metallurgical aspects. In section 2 we present and analyse the bulk of experiments performed on UPt, in its normal state: magnetic, thermal, elastic and transport properties will be discussed, as well as the outcome of high-magnetic field (35 tesla) and high-pressure studies. Section 3 deals with the superconducting state of UPt,. After a discussion whether superconductivity in UPt, is a bulk property (studied by the Meissner effect), we describe a number of experiments that have been performed in order to characterize the superconducting state. Another way to study the anomalous properties of UPt, is by performing alloying experiments. In section 4 the first results of such experiments on a series of U(Pt,_,Pd,), compounds are presented. In section 5 we review a series of remaining experiments and subsequently, present the concluding remarks concerning the normal and superconducting phase of upt,. Since the field of heavy fermions evolves rapidly, it will be impossible to present a fully up to date insight in the properties of UPt,. This paper compasses experimental and theoretical work performed before end 1986 and was written in partial fulfilment of a Ph.D. degree (A. de Visser, Thesis, University of Amsterdam, 1986). 1.2. Related compounds
and systematics
In order to present a background
for the un
A. de Visser et al. I UPt,, heavy fermions
100 mJ/K*molU. In the case of UPt, and UPt, this can also be ascribed to spin-fluctuation effects, although not as pronouned as in UPt,. Such a picture is supported by resistivity measurements [lo]. Curie-Weiss analyses of the high-temperature susceptibility data show in all cases small deviations from linearity in the x-l vs. T curves. Nevertheless, for UPt and UPt, an effective can be deduced. moment, lteff, of 3.5&/U-atom This is consistent with a 5f2 or 5f3-configuration in an LS-coupling scheme. The pefL,,,-valuesfor UPt,, 3.0&U-atom, and UPt,, 4.3h,lU-atom, can not easily be interpreted in the LS-coupling of scheme, which suggests the importance crystal-field effects at high temperatures or, perhaps, the need for a more elaborate coupling scheme.
usual properties of UPt, we shall in the following sections briefly review a number of compounds related to UPt,. Successively we shall consider the uranium-platinum compounds, the heavyfermion compounds and the intermetallic UX, compounds. The uranium-piat~num compounds Four uranium-platinum compounds do exist [4]: UPt, UPt,, UPt, and UPt,. A survey of their crystallographic and magnetic properties is listed in tables I.1 and 1.11, respectively. Since the nearest uranium-uranium distance, d,_, , varies from 3.61 A in UPt, up to 5.25 A in UPt,, one would expect, at least according to the Hill plot [9], an increasing localization with increasing Pt content. However, the systematics of the Hill plot are not followed in the U-Pt system. Ferromagnctism, possibly of itinerant nature, is observed for UPt, superconductivity for UPt,. UPt, and UPt, do not order. As we shall describe in detail in section 2, the compound UPt, shows pronounced spin-fluctuation phenomena. Evidence hereto mainly arises from a T3 ln(T/T*)-cont~bution to the specific heat. This low-temperature upturn results in the anomalously large y-value of 422 mJfK*molU. It is noteworthy that also for UPt, UPt, and UPt, are found: enhanced y-values rather l.2.1f
Table 1.1 Some crystallographic
parameters
of uranium-platinum
Compound
Structure
upt
orthorhombic
upt, upt,
hexagonal orthorhombic cubic
Table 1.11 Magnetic properties of uranium-platinum
83
and superconductivity
The heavy-fermion compounds As mentioned in section 1.1 it is not clear-cut which compounds belong to the class of the heavy-fermion systems. In table 1.111 a number of compounds with large y-values have been listed. Given a somewhat arbitrary choice of y > 400 mJ/K’mol, at present, 8 heavy-fermion systems are known. The y-values enumerated in table 1.111 have been derived from the low1.2.2.
alloys d u-u
Vlll
(A)
( 10m6m3/mol)
Ref.
CrB
3.61
26.61
Fl
MgCd, InN&(dist) AuBe,
4.12 3.81 5.25
42.43 33.62 61.53
it;
151
alloys. Data taken from ref. [2] and ref. [8]
Compound
Type
&ff (&U-atom)
w0 (pa/U-atom)
Y (mJ/K2molU)
~(4.2 K) ( 10m9m3/molU)
upt upt, upt, UR,
ferrom. param. superc. param.
3.5 3.5 2.8 4.3
-0.5 -
114 89 422 95
45 51 103 34
-
A. de Visser et al. I UPt,, heavy fermions
84
and superconductivity
Table I.111 Materials having y > 200 mJ/K*mol. For ordered materials, y has been obtained temperatures
just above
Ref.
1620 1600
no order no order
-
[II [ll,
4.10 5.13 5.13 6.56 4.39 4.12
1100 1100 900 840 535 # 422
superc. superc. antif. antif. antif. superc.
-0.6 0.9 3.4 5.0 9.7 0.5
1131 [I41
tetragonal hexagonal cubic
4.20 3.65 4.96
350 260 250
cubic cubic cubic
4.63 3.25 4.87
242 234 >200
$;
CeAI, CeCu,
hexagonal orthorhombic
4.43 4.83
CeCu&
upt,
tetragonal cubic cubic cubic rhombohedral hexagonal
CeRu,Si, YbCuAl UC% NpSn,
U,Zn,,
NC, CePb,
temperature
T order
Structure
UCd,,
to zero
Type
Compound
UBe,, NpBe 13
by extrapolation
from
Tordcr
TmJ/K’mol)
no order no order antif. antif. antif. antif.
15.0 9.5 6.6 1.1
121
1151
1161
1171 [2, 3. 181 [191 [201 [21,221
I231 (241 1251
* Per mole U.
limit of the term in the specific heat linear in temperature:
temperature
(1.1) For ordering materials y has been obtained by extrapolation to zero temperature from temperatures just above Torder. The resulting y-values are enhanced with a factor lo’-lo3 with respect to ordinary metals. The origin of the enhancement is one of the key questions in understanding the heavy-fermion systems. We shall return to this point in the next section. At present a substantial number of materials with intermediate y-values is known. Some of them having y > 200 mJ/K’mol are listed in table 1.111 as well. Since their properties are quite similar to those of the heavy-fermion systems, these materials may likely serve as a link between the ordinary metals and the heavy metals. It is interesting to note that diluted materials can have huge -y-values as well, albeit per mole impurity. For instance, dilute Ce in Y has a y-value of 3500 mJ/K*molCe [26]. However, we shall confine ourselves to (pseudo) intermetallic compounds. A more elaborate introduction to the heavy-
fermion systems can be found in the review paper by Stewart [27]. From table 1.111 it follows that heavy-fermion behaviour is not directly correlated with a particular crystal structure. The same holds for the various ground states: superconductivity, magnetic order or no order at all. A common feature in these materials is the large f-atom spacing. larger than the Hill limit (- 3.5 A). Several attempts have been made in order to find some systematics in the occurrence of heavy-fermion behaviour. Below we present an overview 01 these attempts, available to us at the moment. Meisner et al. [28] have plotted the y-values oj a number of compounds as function of the f. atom spacing, and find a smooth variation (fig 1.1). These authors discern two regimes: (i) al sufficiently short spacing direct f-atom interac tion prevails, leading to roughly an exponentia dependence of y on spacing, and (ii) at large] spacing indirect f-atom interaction prevails, wit1 delocalization of the f-electrons accomplished b! intervening atoms. A significant correlation factor is thought to b< the ratio of the measured susceptibility to thr one calculated from the observed (enhanced y-value:
A. de Visser et al. I IJPt,, heavy fermions
8.5
and superconductivity
in the same units as above, and p,rr in units of pr+, eq. (1.4) reads as: R,
= 1.74 x 10’ +
.
(1.5)
Yt-k!ff
Fig. 1.1. The specific heat coefficient y vs. f-atom spacing plotted on logarithmic scales, for superconducting (circles), magnetic (squares) and non-ordering (triangles) (nearly) heavy-fermion compounds (after Meisner et al. [28]).
(1.2) Note that since we use rationalized molar units a factor p0 appears in the denominator of eq. (1.2). With the molar susceptibility in units of m3/mol and y in J/K2mol eq. (1.2) reads as:
Stewart [27] has evaluated R, for the class of heavy-fermion systems. The trend noticed by Stewart suggests that for the superconducting compounds R, is roughly a factor 2 smaller than for the non-superconducting ones. Kadowaki and Woods [30] have stressed the importance of a universal relationship between the resistivity and the specific heat. They noted that the ratio of the coefficient, A, of the T2term in the resistivity and r*, Aly2, has a common value of 1.0 x 10-j p.RcmK2mo12/(mJ)2 (fig. 1.2). These studies have failed, however, to reveal any satisfying criterion for the occurrence of heavy-fermion behaviour. Besides, the meaning of these correlations is not always clear. Furthermore, we have to remind that it is not in all cases obvious how to distract the proper physical parameters from the data, especially when a strong sample dependence and/or anisotropies are present. 1(-4
R = 5.80 x lo6 ; .
(1.3)
R equals 1 if the density of states probed
by x and y have the same enhancement factor. The use of the ratio R has the advantage that it can be obtained directly from the experimental values of x and y. DeLong et al. [29] have evaluated R for a large number of heavy and “nearly” heavy-fermion compounds. The correlation they found formulates as follows: superconductors exhibit values of R 6 2.0, whereas other systems have R b 2.0. Another correlation factor put forward is the “Wilson ratio”: (1.4) Here perr = gpu(J(J + 1))“2 is the effective moment as follows from a Curie-Weiss analysis of the high-temperature susceptibility. With x and y
Fig. 1.2. The coefficient A of the T*-term in the resistivity plotted versus the y-value for a number of (nearly) heavyfermion compounds (after Kadowaki and Woods [30]).
86
1.2.3.
A. de Visser et al. I UPt,, heavy fermions
The UX, compounds
Most of the compounds UX,, with X a 4d or Sd-metal or an element from group III or group IV, crystallize in the cubic AuCu, structure. Therefore, a direct comparison with UPt, (hexagonal structure) is substantially hampered. Nevertheless, a qualitative consideration of the hybridization mechanisms in these compounds has revealed some interesting systematics (Koelling et al. [31]). For the UX, compounds, discussed here, the f-atom spacing ranges from - 4 A up to - 4.8 A. Since these values exceed the Hill limit (- 3.5 A), one would expect local f-electron behaviour, due to the absence of direct f-f overlap. However, Koelling et al. have pointed out that the degree of delocalization of the felectrons is mainly governed by the hybridization of the f-electrons with s, p and d-orbitals on neighbouring non f-atom sites. Assuming that the f-ligand hybridization decreases with increasing UX-distance, the systematics in the occurrence of the electronic and magnetic properties of the UX, compounds (see fig. 1.3) can be understood. For small X-atoms (strong hybridization) one then expects Pauli paramagnetism, and for large X-atoms local moment behaviour. Intermediate hybridization would result in a large density of states, hence heavy-fermion behaviour and/or spin-fluctuation phenomena. This trend is nicely followed for the group III and group IV
and superconductivity
elements, moving down along the column in the periodic table (fig. 1.3). For the 4d and 5dmetals the systematic is not so clear. UPd, is the only system that exhibits localized Sf-electrons, it has, however, a dhcp structure. Although URu, and UIr, are superconductors with transition temperatures of 0.2K [32] and 3.OK [33], respectively, these compounds do not show any sign of heavy-fermion behaviour, which is illustrated by the y-values of 13.5 and 17.0mJ/ K*molU, respectively. 1.3. Relevant theories From the experimental data it follows that the physics in the heavy-fermion systems is rather complex. The localized character of the f-electrons at high temperatures crosses over to a delocalized one at low temperatures. The high y-values point to pronounced f-density of states, and, hence, to a strongly interacting electron liquid, assuring the presence of many-body effects. Moreover, electrical resistivity measurements have revealed that most of the heavyfermion systems exhibit the Kondo effect. The latter feature is often taken as a starting point for theoretical models. The Kondo Hamiltonian [34] was originally written down to describe the exchange interaction of a single magnetic impurity, with spin S( = i ), with a conduction electron, having spin s.
Ye,= -2Js.S.
Fig. 1.3. Rough overview of the electronic and magnetic properties of UX, compounds, where X is indicated in the figure (after Koelling et al. [31]). Vertical bars give the y-value. Horizontal hatching indicates Pauli paramagnetism. Diagonal hatching indicates spin-fluctuation behaviour. Cross hatching indicates local-moment behaviour. Superconductivity is observed in UIr,, URu, and UPt,.
(1.6)
For a negative coupling parameter J, the impurity spin is completely compensated at low temperatures, giving rise to a Kondo singlet. This is accompanied by a logarithmic increase, with decreasing temperature, of the resistivity, proportional to ln( T/T,). It can be shown that the Kondo Hamiltonian can be generated from the more general Anderson Hamiltonian [35]. Most attempts to model the properties of the heavyfermion systems on microscopic grounds make use of a somewhat modified version of the Anderson Hamiltonian. Hereto, the solid is thought
A. de Visser et al. I UPt,, heavy fermions
to consist of a periodic array of magnetic impurities, the Kondo lattice. Although the formation of the Kondo lattice seems physically reasonable, its low-temperature properties differ rather drastically from the single-ion Kondo case. In the latter, the resistivity shows the whereas in the former characteristic upturn, coherence sets in at low temperatures, resulting in a T2-temperature dependence at approaching T = 0 K. At present, it is not clear how the coherence should be incorporated in the model. A further complication in describing, in particular the actinide heavy-fermion systems, arises from the uncertainty in the number of f-electrons per atom. Whereas a free U-atom has a f3configuration, this needs not to be the case in the solid. Apart from the possibility of a mixed or intermediate valence state, different configurations must be considered: f’ (J = 5/2), f* (J = 4) and f3 (J= 9/2). Furthermore, the groundstate might not be a pure Hund’s rule state. For a free atom, the separation between equal J-components of different LS-multiplets is of the order of 6000 K. A certain admixture might occur due to spin-orbit interaction. For the configurations f* and f3 this results in a groundstate that consists for only 85% of the Hund’s rule component [36]. In addition, in the solid, crystal-field splitting of the J-multiplet should be taken into account. Little is known about its strength in the heavyfermion systems. It would be illuminating if we could present a density of states picture for the heavy-fermion case. In order to do so, it is tempting to plot a narrow f-band near the Fermi level. However, it is more likely that we have to deal with an induced many-body band originating from the admixture of the relevant electron states. At low temperatures a very narrow many-body band of width - lo-100 K is present, causing the characteristic Fermi liquid behaviour. Due to the number of degrees of freedom in the liquid the entropy, and hence the electronic specific heat is strongly enhanced. In some of the heavy-fermion systems the high density of states is decreased by ordering phenomena, therewith decreasing the symmetry and degrees of freedom. This seems to be the case in the heavy-fermion superconduc-
87
and superconductivity
tors and magnets. When the temperature is raised, the many-body band disappears, and a broad Kondo resonance is formed. At high temperatures the characteristics of the system are determined by the broad f-peak below E,, hence a more localized picture results. XPS and BIS spectroscopy have been used to shed some more light on the f-density of states. The experimental resolution, in best cases about 300 K, is, however, not sufficient to resolve the expected narrow structures in the density of states. The discussion presented above can be summarized by stating that a microscopic picture of the heavy-fermion systems is far from being settled. For a more comprehensive overview of the recent developments in attacking the heavyfermion problem, we refer to the papers of Varma [37] and Lee et al. [38]. 1.3.1.
Spin JEuctuations
Elaborate theories exist to account for the effect of paramagnons or spin fluctuations on the temperature dependence of the susceptibility, specific heat and resistivity. BCal-Monod (ref. [39] and references therein) has calculated the temperature dependence of the paramagnon contribution to the susceptibility:
x (2 5
- 1.2 ( g)*)S2T2]
.
(1.7)
Here x0 is the Pauli susceptibility and S is the Stoner enhancement factor. N’ and N” are the first and second derivatives of the density of states, N = N(E,), with respect to the energy, evaluated at E, The sign of the characteristic T*-term depends on N, N’ and N”. If N"(E,)< 0, X(T) decreases continuously, when T increases. N"(E,)> 0 is a necessary condition for x to first increase with T, and thus to exhibit a maximum, since at higher temperatures x ought to reach a Curie-Weiss law. The paramagnon contribution to the specific heat has been evaluated by Doniach and Engelsberg [40], and Brinkman and Engelsberg [41],
88
A. de Visser et al. I UPt,, heavy fermions
who first applied it to “He. Its characteristics are two-fold: (i) an enhancement of the linear electronic term, which can be seen as an enhancement of the effective mass of the electrons, and (ii) a negative T3 lnT-correction to this linear term. The combination of these two terms results in a low-temperature upturn in c/T. The enhanced linear term is given by c = y,,
c m
T=
y,,(l + h,,)T,
(1.8)
where m* represents the effective electron mass, and A,, is called the spin-fluctuation enhancement factor. Here ‘yOrepresents the unenhanced electronic coefficient. The unenhanced electronic term in the molar specific heat is given by
c” = 3/oT =
2N,N(E,)rr2k2, 3
T >
with N(E,) per atom per spin direction. ing the T3 lnT-contribution, the ratio enhanced to the unenhanced electronic heat can be written as [41]:
(1.9) Includof the specific
C
7i= C
(1.10) where (1.11)
k,TE
= pci2m
,
(1.12)
and PI = PllP,
.
(1.13)
Here TF is the Fermi temperature, and p, a cut-off momentum in the model. The paramagnon contribution to the resistivity has been evaluated by several groups (see, for instance, ref. [42] and ref. [43]). Their models predict a T2-law at low temperatures,
and superconductivity
p = A T2 ,
(1.14)
where A m ( T5f)-2. A subsequent strong increase in resistivity near the spin-fluctuation temperaat rather ture, T,r, is followed by a saturation high values in the room-temperature region. Unfortunately, neither the T2-term in x, the T” lnT-term in c, nor the T2-term in p, can be taken as decisive evidence for spin fluctuations. In addition, information may be obtained from magnetic-field effects on x and c (“quenching” of spin fluctuations). However, the measured field effects are often small and difficult to interpret [39]. Inelastic neutron-scattering experiments enable one to measure the excitation spectrum, Imx( q, w), and might, therefore, be considered as a powerful tool, as well. To conclude, one has to consider a bulk of experiments, in order to decide whether one deals with a spin-fluctuation material, or not.
1.3.2. Fermi liquid theory In 1956 Landau put forward a phenomenological theory for the macroscopic properties of an interacting normal fermion system, at low temperature: the Fermi liquid theory [44]. As we shall see, Fermi liquid theory is also applicable to the highly correlated heavy-fermion state. The kernel of Landau’s theory consists of a one-toone correspondence between the eigenstates of the non-interacting electron gas and those of the interacting liquid. The interaction between the electrons leads to a dressing of the particles, thus increasing their effective mass, m*. This interaction can be expressed in terms of the dimensionless Landau parameters, F:, where A = s(a) refers to the spin symmetric and spin antisymmetric parts of the interaction (see for instance Pines and Nozieres [45]). The Landau parameters appear in the expressions for the susceptibility, the specific heat, the compressibility, etc. The temperature independent molar Pauli susceptibility of the interacting Fermi liquid is given by 2 m”k,V,,,
x = cLo/-%
Tr2h2
1 l+F;’
(1.15)
A. de Visser et al. i UPt,, heavy fermions and superconductivity
89
where k, is the Fermi wave number. x is enhanced with respect to the Pauli susceptibility of the non-interacting system:
where the enhancement
factor s” is given by
s”=_ m*/Fn
(1.17)
l+F;’
Just as in the paramagnon model, the specific heat of the interacting Fermi liquid can be written as the sum of an enhanced linear term and a T3 InT-term. The enhanced linear term accounts for the specific heat of the dressed quasiparticles. Its coefficient is given by kiV,m* k, y* z.z 3fi2 *
(1.18)
The existence of a T3 InT-term in the specific heat of an interacting Fermi liquid has been demonstrated by Pethick and Carneiro 1461. This contribution is due to small momentum (q) transfer processes between the quasiparticles. The corresponding contribution to the specific heat per unit volume is given by (1.19) where h2k; TF = ___ 2m*k,
(1.20)
is the Fermi system, n-2N_
V
+ (A;)‘(
(1.16)
x = ix0 >
temperature
_- k:.
2
37T
of the
interacting
(1.21)
3
is the particle number density, B” = 2N~N(~~)~‘/V~
,
and T, is a cut-off expressed as
temperature.
I-
;
A;)
-2A;A;]
,
(1.23)
where ws = 1 and o, = 3. The A: parameters are the Landau scattering amplitudes, which are related to the Landau parameters, FF, by A; =
F: 1+ F;/(21+
1) ’
(1.24)
In eq. (1.23) terms with I 2 2 have been neglected. Since, as we will see, B” is positive, the T3 lnT-term can be seen as a negative correction to the enhanced linear term. The physical process responsible for the T3 lnT-term is the reof quasiparticle-quasihole peated scattering pairs. As in the paramagnon model, this gives rise to spin fluctuations. Therefore, we can say that the T3 fnT-term accounts for the spin fluctuations in the liquid. Note, however, that the Fermi liquid theory does not speak about the nature of the interaction, it is merely based on its mathematical description. This expression for the specific heat has successfully been applied to the Fermi liquid 3He. Since we have allowed onfy small q scattering, we cannot rule out that other scattering processes in the liquid may give rise to additional terms. Such terms, however, have not been observed. The T3 fnT-contribution in eq. (1.19) comes from long-wavelength spin fluctuations. There is however an additional contribution to the T3term in eq. (1.19), from shorter wavelength spin fluctuations, that was not included by Pethick and Carneiro. Recently, Coffey and Pethick [47] have worked out the coefficient of this term in more detail. Below we show their approach how to proceed in order to determine the spin-fluctuation parameters from the specific heat. The full expression for the specific heat is given by
(1.22) c=y*T+&,T3+ST31n(T/T*).
(1.25)
B” may be Here y* is the enhanced linear coefficient, &, is
90
A. de Visser et al. I UPt,, heavy fermions
the usual phonon coefficient (T + O,), 6 is the coefficient of the pure T3 lnT-term and S lnT* = Psr is the coefficient of the spin-fluctuation contributions to the T3-term. In order to perform a computer fit to the data, the more suitable expression
and superconductivity
If one
considers
B’
= -
B”
T*
=
fluctuations
are
equal
and
opposite:
e-&f’8
(1.28)
T* is often denoted as the spin-fluctuation temperature. The expression for the T’ lnT-term and the T3-term, from the spin-fluctuation contribution to the specific heat per unit volume, is written by Coffey and Pethick as
AC = $
nk,B”
(:I’
+ q] ,
[lnc&i
which differs from eq. (1.19) ment of the logarithm. Here
only
in the argu-
4, = qc/P, Y
(1.30)
where pF is the Fermi cut-off momentum.
momentum
and
q, is a
(1.31) and
q,T, T*
We see that 33r”
S=Tnk,z=FY
.
(1.32) (per unit volume)
B‘
3~”
to
(A;,)’
(A;)* >
might
=
_
;
(1.34)
be further
simplified
(A;)3
+ ; (A:,)*
_
* B”
z,
(1.33)
to
$ +; ,
(1.35)
since Ai = 1 for electron systems [48]. In order to determine the spin-fluctuation parameters one may proceed as follows. Input parameters of the model are y*, p* and 6, determined by a fit to eq. (1.26) (actually we fitted to c/T), the phonon k,, coefficient Pph and the Fermi wave number which might, for instance, be obtained from configurational arguments (eq. 1.21). We now successively calculate m* from eq. (l.lS), T, from eq. (1.20), B” from eq_ (1.33), Ai from eq. (1.35), F: from eq. (1.24), S from eq. (1.17), & from eq. (1.27) and T* from eq. (1.28). The last important parameter, S,, may be calculated from the relationship [47]:
(1.29)
n=ln
_ ;
(1.27)
T* is a characteristic temperature, at which the T” 1nT and T’-contributions to the specific heat spin
+ ; (A;)*
where
P* = Pph+ P,r .
from
the I= 0 contributions
(1.26) which
may be used,
(A;)”
;
+ ; c = y*T + /3*T3 + 6T” 1nT
only
B”. one has
T*k, ” = q(F;)(l
+ F;)(u;)*m*
’
(1.36)
where W(F:) is a dimensionless function, which depends weakly on FP, [47]. The cut-off parameter 4, provides us with a test of the model, since we assumed q, < pF.
1.3.3. Odd-parity superconductivity The discovery of superconductivity in the heavy-fermion compounds CeCu,Si, , UBe ,7 and UPt, came as a total surprise. In these systems a strongly correlated electron liquid persists, down to the ordering temperature, as is indicated by the huge specific heat. Such a correlation between the electrons is, in general, thought to suppress the conventional phonon-mediated type of superconductivity. In the superconducting state the coefficient of the electronic term in the specific heat (nearly) vanishes, suggesting that
A. de Visser et al. I UPt,, heavy fermions
the f-electrons play an important role in the formation of the condensate. Some other unusual observations have been made concerning the superconducting state: (i) the electronic properties reveal a power law, rather than an exponential temperature dependence, at least in the case of UBe,, and UPt,, (ii) anisotropy effects are present, and (iii) an unusual stability of the Cooper pairs with respect to a magnetic field, given the low T,-values. These remarkable features have led to speculations upon an unconventional type of odd-parity superconductivity [49, 501. In this section we shall present some theoretical background for odd-parity superconductivity. However, we have to realize, that many aspects of the newly developed theories are subject of lively discussions. For a comprehensive overview of the theoretical and experimental aspects of conventional superconductivity we refer to the standard work of Parks [51]. The present theoretical models that deal with odd-parity superconductivity in the heavy fermions, are largely based on a close analogy to the triplet or p-wave superfluidity in 3He. However, in offering an explanation for the unusual behaviour in the superconducting state one should also consider higher orbital momentum states (1 Z 2), odd as well as even. A direct comparison with 3He is substantially hampered, since in the solid spin-orbit coupling and crystal fields lower the symmetry. Taking the crystal symmetry into account, several authors calculated the allowed odd-parity states from group theoretical arguments, for cubic (UBe ,3), tetragonal (CeCu,Si,) and hexagonal (UPt,) crystal structures [52-5.51. At present, a point of great interest is how to distinguish experimentally an odd-parity super-
and superconductivity
91
conductor. Varma [37] predicted the temperature dependence of a number of electronic properties, assuming odd-parity pairing. These properties depend on the form of the energy gap over the Fermi surface. He discerns three cases: 1) The superconducting gap exists everywhere on the Fermi surface, like in the conventional superconducting state of liquid 3He. The quasiparticle density of states has a gap. 2) The superconducting gap vanishes at points on the Fermi surface, near which it varies quadratically with the energy. This is like the ABM or axial state in superfluid “He. 3) The superconducting gap vanishes at lines on the Fermi surface, near which it varies linearly with the energy (this is like the polar state in an isotropic system). The results of his calculations for the specific heat, sound attenuation, nuclear magnetic relaxation time and thermal conductivity are listed in table l.IV. In contrast herewith, Rodriguez [56] predicts a sound attenuation proportional to T and T’, for the polar and axial states, respectively. Besides, Coppersmith and Klemm [57] pointed out that ultrasonic attenuation depends also on the relative orientation of the sound waves and the nodal structure, resulting in a more complex situation. Regarding group theoretical arguments Blount [54] stated that a polar state (case III) of odd parity is not possible in cubic or hexagonal crystals. However, his arguments have been contradicted by Varma [37] and Scharnberg and Klemm [58]. Several other experiments have been suggested to clarify the nature of the superconducting state. We mention (i) tunneling experiments [59, 601, and (ii) studies of the anisotropy of the upper critical field [58, 611.
Table l.IV Form of the temperature dependence of the specific heat, sound attenuation, nuclear magnetic relaxation conductivity in the superconducting state (T < T,), for various pairing mechanisms (after Varma [37]) Case: Gap: Specific
heat
o,~o, T,‘lT-’ I” K,
I Isotropic-like
II Axial-like
III Polar-like
- exp( - A/T) - exp( - A/ T) - exp( - A/T) -exp(-A/T)
- (T/A)’ - (T/A)’ - (T/A)’ -(T/A)’
-
(T/A)’ (T/A)’ (T/A)’ (TIP)’
time
and thermal
A. de Visser et al. I UPt,, heavy fermions
92
Although, as we shall see, the outcome of a number of experiments is consistent with oddparity superconductivity, one still should be cautious to take this as hard proof. On the one hand, gapless and anisotropic singlet superconductivity, or I= 2 (d-wave) pairing, might lead in special cases to similar unusual features in the low-temperature properties. On the other hand, the theory still needs refining. A consensus on what microscopic pairing mechanism is applicable to the heavy-fermion systems is lacking. Several authors have presented mechanisms in favour of odd-parity superconductivity. We mention Varma [37], Anderson [50], Beal-Monod [62], Li [63] and van der Mare1 and Sawatzky [64]. Others have presented models in favour of singlet superconductivity (Razafimandimby et al. [65], Tachiki and Meakawa [66] and Hirsch ]671). A different approach to shed light on the nature of the superconducting state can be found in the phenomenological Fermi liquid theory. Valls and TeSanovid [68] applied an almost localized Fermi liquid model to the heavy-fermion systems, in close analogy to “He. Bedell and Quader [69] used an induced interaction approach. Pethick et al. 1701 focused on a Fermiliquid analysis of the T3 lnT-contribution to the specific heat of UPt,. All these analyses favour triplet superconductivity. Central in their approach is an expression for the superconducting transition temperature, that was first applied to ‘He by Patton and Zaringhalam [71]: TL” = 1.13aT,e1’KI
and superconductivity
The expressions for g, have been derived in the s-p approximation, that is to say, only s and p-waves contribute to the scattering processes. Furthermore, the forward scattering limit has been applied, which means that the interaction involves only small momentum transfer processes (with the angle between the planes of the incoming and outgoing momentum vectors near zero). Employing the forward scattering sum rule, (1.39) and neglecting g,=
$(-A;I-3A;-2A;),
g, = ; (A; + A;)
g, = & i
(-
l)‘(A; l)‘(A;
- 3A;) + A;)
.
(1.40b)
Estimates for the values of A;( = l), A,“, and Al, from the various normal state properties of UPt,, lead to g,
(1.41)
Unfortunately, the model fails to predict T, since (Yis not known. Pethick et al. [70] suggest to test the model by calculating the relative pressure dependence of T,, assuming a pressure independent LY, and, subsequently, by comparing this value with the experimental one. From eq. (1.41) we obtain aln T,
(-
with 12 2, we have
(1.37)
.
Here CYis a numerical constant, T, is the Fermi temperature and g, represents a pseudopotential. If g, < 0, the potential is attractive; j = 1 indicates triplet pairing, j = 0 singlet pairing. The pseudopotential can be expressed as a function of the Landau scattering amplitudes, defined in eq. (1.24), g,, = $ c
components
,
aln T,
6
aA:
dP
(1 + A;)*
ap
ap
(1.42)
The right hand side of eq. (1.42) can be calculated either from the pressure dependence of the specific heat, or from a combination of specific heat and thermal expansion data at zero pressure. We shall return to this subject in section 3. 1.4.
(1.38b)
=--
Phase
The
phase
diagram
diagram
and crystallography
of the uranium-platinum
A. de Visser et al.
/ UPt,, heavy fermions and superconductivity
93
system has been constructed by Park and Fickle [4] It is shown in fig. 1.4. The system is characterized by four intermetallic compounds: UPt formed peritectoidally at 1234 K, UPt, formed peritectically at 1643 K, UPt, melting congruently at 1973 K and UPt, formed peritectically at 1733 K. The crystal structure of UPt, has been determined by Heal and Williams [7]. UPt, crystallizes in the MgCd,-type of structure (lowtemperature form of SnNi,). This closed-packed hexagonal structure, with 2 formula units per cell, belongs to the space group P6,immc. The atomic positions in the unit cell are given by: 2U at c 6m2 6Pt at h mm
&,$,a; x,2x, a ; x, 2x, ; ;
i,+,+, 2x,x, 3x,x,
$ ; $ ;
X, i, $ , x,x, f )
where the ideal value of x equals 2. The lattice parameters are given by: a = 5.764 A, c = 4.899 A and c/a = 0.850. These crystallographic data have been confirmed by X-ray (Loopstra, unpublished) and neutron (Helmholdt , unpublished) experiments. Up to pressures of 500 kbar no crystallographic phase transitions have been observed [72]. The stacking of the atomic layers is shown in fig. 1.5. The nearest U-U distance is measured between atoms in adjacent layers and amounts to 4.132 A. The unit-cell volume, 0, equals 140.96 A3, the mean atomic volume equals 17.62 A’, and the molar volume, V,,
1234K -y-w, 1000
-\’
978 K
,;‘UP)
t326K
@cl -+ u (0) 0’ ”
10
20
30
40 PLATINUM
50
60
70
80
(at %I
Fig. 1.4. Phase diagram of the uranium-platinum (after ref. [4]).
90
100 Pt
system
Fig. 1.5. Stacking of the atomic layers in the UPt, structure. Open circles: Pt; closed circles: U; c-axis not drawn to scale. Note that throughout this paper the a and b-axis hexagonal plane) are considered to be orthogonal.
(in the
equals 4.244 x 10m5m3/mol. With the molecular weight of M = 823.3 g the density amounts to 1.940 x lo4 kg/m3. Crystallographic parameters for UPt, UPt, and UPt, can be found in table 1.1. 1 S.
Sample preparation
In general, the preparation of actinide intermetallics asks for special techniques, due to high reactivity and high melting point of the compounds. Moreover, radioactivity and toxicity of actinides ask for additional restrictions in handling these materials. The investigated UPt, samples have been prepared at the metallurgy department of the Natuurkundig Laboratorium by A. Menovsky. Experiments have been performed on three types of samples: arc-melted polycrystalline samples, Czochralski-grown single-crystalline samples and single-crystalline whiskers. As starting materials served 2’SU-depleted uranium (nominal purity 99.8%), supplied by Koch-Light Ltd., and platinum (99.999%), supplied by Material Research Corporation. The RRR for the uranium amounted to 2.7. Polycrystalline samples were prepared by arc
94
Fig. 1.6.
A. de Vimer et ai.
(a) Scanning
microphotograph
I UPt,, heavy fermions and s~perco~d~ct~v~ty
of UPt,
whiskers
grown
melting the appropriate amounts of the constituents in a titanium-gettered argon atmosphere. In order to obtain a homogeneous compound the buttons were remelted several times. In some cases, the melt was cast into a watercooled copper crucible, in order to obtain a particular shape, e.g. a cylinder. A further homogenization of the samples was achieved by Table l.V Lattice parameters
and density
of UPt,
(hexagonal
out of the melt;
(b) whisker
with well developed
surface.
annealing in evacuated quartz ampoules at a temperature of - 1200 K, for a period of 7-10 days. During the annealing procedure the samples were wrapped in tantalum foil, to avoid contact with the quartz. A piece of uranium served as getter. Additional shaping of the sampies was done by means of spark erosion. In this manner spheres, cylinders and cubes (with plan-
structure)
a
c
Sample
(A)
(A)
Exp. density (lO’kg/m’)
Caic. density (104kgim’)
Ref
Polycrystalline Czochralski-grown single-crystal
5.764 f 1 5.7520 rt 12
4.899 2 1 4.8970 ‘- 33
1 .Y45
1.940 1.949
171 j731
Whiskers
5.7530 + 4
4.9010 t- 4
-
1.947
[741
A. de Visser et al. I UPt,, heavy fermions
parallel surfaces) could be made, as well as samples with holes. The crystal structure was checked on small amounts of powdered samples Lattice with a Guinier-de Wolff camera. parameters were determined from the X-ray patterns, with SiO, as standard. Single-crystalline UPt, was prepared with a modified tri-arc Czochralski method [73]. First, the melt is prepared on a water cooled copper hearth. Then the crystal is pulled out of the melt, with, in the case of UPt,, a pulling rate of about 5 mm/h. An as-cast cylinder (3 mm diameter, 20 mm length) served as a polycrystalline pulling tip. During the whole growth procedure, tip and hearth rotate in opposite direction, with rates of 15 and 50 rpm, respectively. With a fourth arc a piece of uranium is melted, in order to getter the -argon atmosphere (P = 300 Torr). In this way,
Fig. 1.7. Cross-section
of UPt,
whisker;
and superconductivity
95
large cylindrical crystals could be grown, with sizes limited to 8 mm diameter and 30 mm length. Laue pictures were taken to check the single-crystalline nature of the grown crystals. Lattice parameters, determined by Weissenberg X-ray measurements, are listed in table l.V. The experimental density is in good agreement with the calculated one (see table 1.V). Shaping of the samples was done by means of spark erosion. In some cases, the bulk crystal, or the cut pieces, were annealed at 1200 K, during 24 h, in order to reduce strains. A third type of sample can be found in tiny needle-like single-crystalline whiskers. The whiskers jump spontaneously out of the stoichiometric UPt, melt, at fast cooling on a water-cooled copper hearth [74]. A “liquid-solid” screw dislocation growth
orientation
of the crystal
axes as indicated.
96
A. de Visser et al. I UPt,, heavy fermions
mechanism has been suggested. An example of an arc-melted button with whiskers on its surface ’ is shown in fig. 1.6. Most of the whiskers have a regular shape, with a typical cross-section in the form of an equilateral trapezium (see fig. 1.7). The c-axis is always found to be parallel to the whisker axis. Typical whisker dimensions are (mass 15(a-axis) x 60(b-axis) x lOOO(c-axis)pm’ - 18 pg). Lattice parameters, determined from Weissenberg X-ray measurements, are listed in table l.V. Stewart et al. [3] have reported the growth of single-crystalline UPt, from a bismuth flux. No information on lattice parameters is available. The UPt, specimens are stable with respect to oxidation and can be stored under normal atmosphere. Loss of chemical purity, due to radioaccan be neglected. Natural tive (CY) decay, uranium consists of the isotopes 23xU (99.283%), 235U (0.7110%) and 234U (0.0054%), with (Ydecay rates of 4.51 x lo”, 7.1 X 10’ and 2.47 X lo5 year, respectively. In the case of depleted uranium the percentage of 235U is somewhat lower, 0.4% for the Koch-Light batch.
2. UPt,:
the normal state
2.1. Introduction In this section we present, analyse and discuss the experimental results, focusing on the normal state of UPt,. We begin by recalling susceptibility measurements [75, 21 that were made in a first attempt to characterize UPt,. We then successively concentrate on the thermal, elastic and transport properties. Next we deal with highstudies. high-pressure and magnetic-field Specific-heat and high-field magnetization measurements performed by Frings et al. [2] are recalled in the relevant sections. In addition, some experimental results on the compounds UPt, UPt,, UPt, and UAl, are presented. 2.2.
Magnetic properties
Schneider tibility data
and Laubschat [75] reported suscepon a polycrystalline UPt, sample in
and superconductivity
the temperature range 2.5-210 K. These authors found a weak maximum in the susceptibility curve at 15 K. By comparing the susceptibility with that of materials like NpPd, and PuPt,, these authors suggested antiferromagnetic ordering in UPt, at 15 K. The paramagnetic region could be described by a Curie-Weiss behaviour with pCL,rf = 2.85,uL,lU-atom, between 23 and llOK, and with pL,rf= 2.61&U-atom, between 110 and 210K. The presence of a maximum in the susceptibility has been confirmed by Frings et al. [2]. These authors measured the susceptibility of polycrystalline UPt, (unpublished data), in the temperature range 4.2-300 K, in fields up to 1.3 T, using a pendulum magnetometer, and in the temperature range 300-1000 K, using a Faraday balance. From the high-temperature data an effective moment of 3.0 ? O.l~,/U-atom was deduced. Their data taken on an unannealed Czochralski-grown single-crystalline sample, shaped into a sphere (diameter 3 mm), are shown in fig. 2.1. This figure shows that the susceptibility is strongly anisotropic. The maximum in the susceptibility is only present for field directions in the basal plane, for which an isotropic susceptibility results. The x-values at 4.2 K, 87 X lOmy and 99 x lo-’ m’/molU, and the x,,,-values, 91 x lo-” (at 15 K) and 111 x lo-” m’/molU (at 17 K), obtained on polycrystalline samples by Schneider and Laubschat [75] and Frings et al. (unpublished), respectively, fall well in between the data for the basal plane and c-axis. The differences between the absolute x-values for various polycrystalline samples can mainly be attributed to the presence of crystallites with preferred orientations. A Curie-Weiss analysis of the data, for a field direction in the basal plane and along the hexagonal axis, in the temperature range 50-250 K, results in a value for Barr of 2.6 ? 0:2p,/U-atom, for both field directions. The paramagnetic Curie temperature amounts to - 50 and - 170 K for the basal plane and hexagonal axis, respectively. The large uncertainty in ~l.,~rmakes it difficult to interpret. Applying Hund’s rules the value of 2.6p,/U-atom is close to the value for a full
A. de Visser et al. I UPt,, heavy fermions
I50
97
and superconductivity
5
125
UPt3
ah
I3
B B 0
75 50
0
-%++
c +++
A +
g
$2 a
+
0
+
+
25 #
0
Susceptibility
of single-crystalline
UPt,;
8
+
I
1
+
1
200
100
0
Fig. 2.1.
I
+
a-axis
(0),
b-axis
300
(A) and c-axis
(+)
(after
ref. [2]).
quenched f’( j_+ = 2.54j_&J-atom), f2(2.83&U-atom) or a full f4(1.68&U-atom) configuration. Besides, small deviations from the Curie-Weiss behaviour , possibly due to crystalfield effects, hamper the analysis. Nevertheless, the susceptibility data strongly suggest local moment behaviour at temperatures above 50 K. From the data in fig. 2.1. it is obvious that UPt, is not a simple two sublattice antiferromagnet. A complex type of order can not be excluded. However, in an ac-susceptibility experiment (H < 0.2 Oe) any sign of an anomaly near 17 K was lacking. As we shall show in the next section the thermal properties provide strong evidence for spin fluctuations. In that respect the maximum in x might be explained as a spinfluctuation phenomenon, according to eq. (1.7).
sample [2], are shown in a plot of c vs T, for T < 20 K, in fig. 2.2. In fig. 2.3 we plot the data as clT vs T2, which shows that a description with a linear and cubic term is far from appropriate. The large y-value of 422 mJ/K2molU, obtained from an extrapolation to 0 K, illustrates the heavy-fermion behaviour. The field-effect on the specific heat is small. For the polycrystalline sample it is in the order of 1 ? 1% at 5T (fig. 2.3). Data taken on the single-crystalline sample, with a field of 5 T applied along the a and c-axis, for temperatures below 5.5 K, are shown in fig. 2.4. At low temperatures the normal-state specific heat of UPt, obeys the relationship
2.3.
The presence of a T3 ln( T/ T*)-contribution to c strongly evidences the presence of spin-fluctuations (see section 1.3). However, before presenting details of such an analysis, other possible contributions to c need to be considered. First we stress that no sign of long-range antifer-
Thermal properties
2.3.1. Specific heat The normal-state specific heat of an unannealed Czochralski-grown single-crystalline sample [18, 761, and data taken on a polycrystalline
c = y*T
+ &T3
+ 6T3 ln(TIT*)
.
(2.1)
A. de Visser et al. I UPt,, heuvy fennions 12
t
*
”
’
,
and superconductivity
”
7
I,
”
” a
2-
I
0 0
s
I
L
,
I,,
5
*
1
L
I,
‘
I
15
20
T'&)
Fig. 2.2. Specific heat of single (0) and poly (Cl) crystalline UPt, (squares taken from ref. [Z]).
romagnetic order is found near 17K, the temperature at which the maximum in x is observed. Crystal-field contributions to c, in the investigated temperature range, can be excluded, since no Schottky-type of anomaly is observed. Local
magnetic moments (due to the presence of impurities) in an internal field can give rise to an upturn in c/T at low temperatures. However, in that case one would expect an external field of 5 T clearly to influence the specific heat, which is
0.6
0
100
200
300
400
T2 (K2) Fig. 2.3.
Specific
heat of single-crystalline
UPt, at OT (0), and polycrystalline UPt, at OT (0)
and
ST (+).
A. de Visser et al. i UPt,, heavy fermions
99
and superconductivity
0.50
9
+
0.45
H//a
A H//c
-$ -T a -t0
&+A gp4 00
0.4-O
+A 0
+A+d+
A+,
000
op Am A
A
0.35
I
0.30
I
10
0
I
I
20
30
T2 (K2) Fig. 2.4.
Specific
heat of single-crystalline
UPt,
at 0 field (0)
apparently not the case. The possibility that the low-temperature upturn is due to details of the band structure can be discarded as well. The anomalous increase in c/T below - 10 K suggests a huge narrow peak in the density of states. Of course, the effect of a magnetic field of 5 T on such a narrow structure would be enormous. If one, with a specific-heat experiment, would only probe the band structure, the data with and without field would largely differ, which is not the case. To conclude: contributions to c from long-range magnetic order, crystal-field effects, magnetic impurities and band-structure effects are absent. Given the appropriateness of eq. (2.1) to describe the low-temperature specific heat of UPt,, we analyse the data in a spinfluctuation model. Similar specific-heat curves have been observed for the compounds UAl, [77], TiBe, [78] and UCo, [79]. However, in these compounds the low-temperature upturns in c/T are less pronounced: the y-values amount to 143, 56 and 35 mJ/K2molU, respectively. Due to the presence of the T3 ln(TlT*)-term, in combination with anomalies in magnetization and susceptibility curves, these materials are labeled as spinfluctuation compounds. The specific heat of
and in a field of 5 T directed
along
the a( +) and c-axis
(A).
these materials has been analysed in the paramagnon model [40, 411. In the case of UPt,, we concentrate, in this chapter, on the Fermi liquid analysis [47], as outlined in section 1.3.2. Hereto, we have performed a least-squares fit to the specific-heat of single-crystalline UPt,, for several temperature intervals, employing the relation: c
- = y*+p*T2+ST21nT. T
(2.2)
The coefficients deduced from these fits are listed in table 2.1. The values for the coefficients in eq. (2.2) do not differ much for the different temperature intervals, indicating that eq. (2.2) gives a proper description of the low-temperature specific heat. The fit performed in the temperature range 1.2-20 K follows all 85 data points to within 1% (see fig. 2.5). The results agree fairly well with a similar fit on polycrystalline UPt, [79]. Coefficients derived from fits to the specific-heat data obtained by Stewart [27] and Brodale et al. [80] are listed in table 2.1 as well. In the following paragraphs we focus on the physical parameters that can be deduced from y*, p* and 6.
A. de Visser et al. I UPt,, heavy fermions
100 Table 2.1 Coefficients
derived
from
a 3-parameter
fit to the specific
Interval
heat of UPt,
(K)
&K’moIU)
P* (mJ/K’molU)
1.4-20 1.4-15 1.4-10 6-10 8-20 1.2-17
422 423 421 424 413 422
-
3-17 0.5-20 0.5-G 0.5-4 1.2-20 1.2-10
# #
# Fit to phonon
and superconductivity
according
to eq. (2.2)
(SmJ!K’molU)
Ref.
3.12 3.80 3.53 3.78 3.45 4.18
1.38 1.41 1.30 1.40 1.30
118, 761 118, 761 [lg, 761 [l& 761 118, 761 18, 791
452 422 429
- 6.42 - 4.24 - 5.04
2.25 1.60 1.90
1271
426
- 6.92
3.22
PO1
424 418
- 3.44 - 2.77
0.97 0.72
1811
corrected
specific
1.54
PO1 147, SO]
WI
heat.
The y *-value enables us to determine the effective mass, m*, of the interacting electrons (es. (1.18)), P rovided the Fermi wave number, k,, is known. We assume the Fermi surface to be spherical (this is probably a rather crude assumption). In that case the value for k, can be obtained from an analysis of the temperature derivative of the upper critical field (- 4.4 T/K) that will be presented in section 3, and amounts
to 1.06 A-‘. The Fermi wave number can also be deduced from configurational arguments: k, = (!$?j”‘,
(2.3)
where 2 is the number of heavy electrons per unit cell volume (LJ) that take part in the forma-
0.6
100
200
400
300
T2 (K2) Fig. 2.5. Measured (0), calculated phonon (A, after dotted lines represent S-parameter fits to the measured in table 2.11.
ref. [82]) and phonon-corrected (fit Ia) and phonon-corrected
(0)
specific
heat of UPt,.
The solid and
(fit Ha) specific heat, with coefficients
listed
A. de Visser et al. I UPt,, heavy fermions and superconductivity
tion of the electron liquid. With 2 = 6 (3 felectrons per U-atom, 2 U-atoms per unit cell), and R = 140.9 A3, we calculate the corresponding value for k, of 1.08A’. Notice that in the Fermi liquid description the electrons in the interacting and non-interacting system are described by one and the same Fermi wave number. With the y-value of 422mJ/K2molU and k, = 1.06 A-‘, we obtain the large effective mass m* lm, = 3fi2y* l(kik,V,m,) = 180, the Fermi temperature T, = h2k:/2k,m* = 275 K and the Fermi velocity ug = hk,/m* = 6800 m/s. The renormalization of the effective mass by a factor of - 200, leading to the low values for TF and UC, is a characteristic feature for the heavy-fermion compounds. The coefficient of the T3-term is given by p* = p,, + &, where &, represents the usual phonon coefficient, and & = - 6 In T* represents the contribution from spin fluctuation to the T3-term. T* = exp( - &r/S) is a characteristic temperature, which is often called the spinfluctuation temperature. Mathematically it is the temperature at which the 6 T3 In T-contribution and the &T3-contribution to the specific heat are equal, but opposite in sign. The phonon contribution to the specific heat, in a first approximation given by cph = &,,T3 (T G O,,), is commonly obtained by measuring a nonmagnetic analog of the compound one is interested in. Unfortunately, no information on a non-magnetic analog of UPt, is available. From the published phase diagrams the existence of, for instance, ThPt, is not clear. Nevertheless, two groups performed XPS-measurements on a ThPt, sample, but gave no details about its structure [75, 831. Frings [8] estimated &,,, at 0.85 mJ/K4molU (0, = 210 K), a value derived from a two-parameter fit to the specific heat of a polycrystalline UPt, sample, in the temperature range 9-15 K (with ‘y. = 225 mJ/K2molU). This value for p nicely agrees with the one we deduced from sound velocity measurements at 0.76 mJ/K4molU (0, = temperature: room 217K). With &, = 0.85mJ/K4molU and the coefficients p* and 6 from the fit in the temperature range 1.2-20 K (see table 2.1), we calculate & = - 4.57 mJ/K4molU and T* = 27 K.
101
The coefficient of the T3 ln( T/ T*)-term can (eq. Tg) be expressed as 6 = 3n2y*Bsl(10 (1.33)), where B” can be written as a function of the I = 0 antisymmetric Landau scattering amplitude A:. The parameter Ai will play an important role in discussing the nature of the superconducting state (section 3). Following the procedure outlined in section 1.3.2, we calculate successively B” = 83.74 (eq. (1.33)), Af: = - 3.71 (_eq. (1.35)), Fl = Ai (1 - A:)-’ = - 0.788 and S = (m* lm)( 1 + Fi)-’ = 849. The cut-off parameter, S,, can be calculated from eq. (1.36), where 9(Ft) can be determined from ref. [47]: 4, = 1.40. Finally, we calculate T, = 94K (eq. (1.31)) and n = 1.57 (eq. (1.32)). These results, together with rather similar values, deduced from a fit in the temperature range 1.2-10 K, are summarized in table 2.11. The cut-off parameter, 4, = qClpF, is rather large. This implies that scattering processes over a large part of the Fermi surface contribute-to the quasiparticle energy. The value for S = x/x0 = 849, as calculated from the specific heat, represents the susceptibility of the interacting system over that of the non-interacting system. With this value we calculate a temperature x = 342 x lo-” susceptibility independent m3/molU, using eq. (1.15). A direct comparison with experiment is hampered since the measured x is not Pauli-like. Furthermore, the observed anisotropy in x complicates the analysis. Yet, a mean value for x = 90 x lo-” m3/molU at 1 K, assuming x = $ xa,h + f x,, is considerably lower than the calculated value. However, Pethick and Pines [84] argue that, as a result of spin-orbit coupling, the effective moment peff of a heavy electron may not be equal to the free-electron Bohr magneton. In addition there can be nonquasiparticle contributions to the susceptibility, and consequently the measurements of x for heavy-fermion systems do not give direct information about the Landau contribution. In order to let the calculated and the measured x coincide, one needs to replace ps by p,rf = 0.51 E*.~in eq. (1.15). The value for this effective moment largely differs from the value derived from the Curie-Weiss analysis at high temperatures, peff = 2.6p,/U-atom. However, these effective
A. de Visser et al. I UPt,, heavy fermions
102 Table 2.11 Coefhcients of 3-parameter various parameters derived in units of mJ/K4molU Fit
Interval
Ia Ib IIa IIb
1.2-20 1.2-10 1.2-20 1.2-10
and superconductivity
fits to the measured (I) and phonon (II) corrected in a Fermi liquid analysis, employing k, = 1.06 w -I
K K K K
Y*
P*
422 421 424 418
-
3.72 3.53 3.44 2.77
specific
heat of UPt,,
employing eq. (2.2). and and p *, &, , p,, and 6
; y * in units of mJ/K2molU
P ph
P-r
0.85 0.85
-
‘4:
F;I
6 4.57 4.38 3.44 2.77
1.38 1.30 0.97 0.72
m* Fit
(m,)
Ia Ib IIa IIb
180 180 181 179
275 276 274 278
27 29 35 48
83.74 79.45 57.73 44.71
moments arise from different models and therefore their relation is not quite clear. At this point we want to stress that, in order to derive the various parameters in the Fermi liquid analysis, we assumed a spherical Fermi surface. As the physical properties of UPt, are strongly anisotropic, this is certainly a crude approximation. However, the effect of anisotropy on the Landau parameters is probably small [84]. For instance, since the deduced average value for F: is large and negative, one needs only a comparatively small anisotropy in Fi, in order to account for the large anisotropy in the susceptibility (eq. (1.15)). We also emphasize that this analysis is rather insensitive to the exact choice of k,. A considerable variation in the value for kr( 2: 1.1 A-‘), for instance 30%) does not qualitatively alter the results derived in the Fermi liquid analysis. Recently, inelastic neutron-scattering experiments enabled Renker et al. [82] to measure the phonon density of states and hence to calculate the lattice contribution to the specific heat in a more sophisticated way. At temperatures between 9 and approximately 20 K the Debye temperature amounts to 200 K (close to the value we quoted above), whereas below 9 K the lattice hardens and 0, gradually increases up to a value of 320 K at 1 K. The lattice contribution and the electronic contribution after correcting for the lattice are shown in fig. 2.5. This figure illustrates
-
3.71 3.64 3.24 2.95
~ ~ -
0.788 0.785 0.764 0.747
s
WF6)
4,
849 837 767 708
0.168 0.168 0.170 0.172
1.40 1.46 1.61 1.98
that the c/T-value for the electronic specific heat is only reduced by a factor of 2 at - 20 K, with respect to the value at 0 K. The entropy is large: St?, - R ln2 up to 20 K. The coefficients of a fit to the purely electronic specific heat [Sl], employing eq. (2.2) with p* = &, are listed in table 2.1, for two different temperature intervals. This fit has the same quality as the fit to the non-phonon corrected specific heat, as can be seen in fig. 2.5. An important difference is the 30% smaller 6value. The various parameters deduced in the Fermi liquid analysis, listed in table 2.11, yield, however, similar conclusions. Up to now we have neglected the possibility that the electron-phonon interaction may lead as well to a T3 lnT-contribution to the specific heat. Coffey and Pethick [47] pointed out that this contribution is of opposite sign compared to the from spin fluctuations. T3 In T-contribution Furthermore, they demonstrated that the ratio of the coefficients of the T3 lnT-contribution coming from the electron-phonon interaction to that coming from spin fluctuations is less than for UPt,. Since &,, is of the order of 0.03 &h 2 or less for normal metals, the electron-phonon contribution to the T” lnT-term may be neglected. A model that accounts for the field effect on the specific heat is at present not available. However, it is interesting to compare qualitatively the field effect on c/T (or y in the low-
A. de Visser et al. I UPt,, heavy fermions
temperature limit) with the susceptibility, ing the thermodynamic relationship:
apply-
(2.4) Given the susceptibility data on the single-crystalline sample (fig. 2.1), one expects a positive A(clT) = (clT)(, - (clT)I,=,, at low temperatures, for a field direction in the basal plane and a (small) negative A(clT) for a field along the c-axis. Our specific-heat data, taken in a field of 5 T, reveal hardly any field-effect on c/T (fig. 2.4). Stewart et al. [85] measured the specific heat of a single-crystalline sample, in the temperature range 2-17 K, with a field of 11 T applied in the basal plane. These authors observed a 10% increase in c/T at 2 K, a crossing of the field and zero-field curve at about 6 K, and a negative field effect on c/T of 6% at 17 K. Brodale et al. [SO] observed a nearly constant positive A(c/T) (2% at 1 K) in a field of 8 T in the temperature range 0.3-4 K. Applying eq. (2.4) this corresponds to x m T2, as expected in the low-temperature limit in the paramagnon model (eq. (1.7)). A precise analysis of the measured ,y, in order to look for a T*-term, is troublesome, due to the large anisotropy. Data taken below 1.2 K, on the same polycrystalline sample as described in fig. 2.3, reveal a small positive A(c/ T), in a field of 1 T ([86]; see fig. 3.6). However, within the experimental accuracy of 2 2%, data taken at 2, 4 and 8 T are identical with the 1 T results. In summary, we state that the field-effect on the specific heat is small, and hardly exceeds the experimental uncertainty. The observed (small) positive A(clT) for a field direction in the basal plane, consistent with the susceptibility data, can be explained as an enhancement of the spin-fluctuation phenomena in the low-temperature limit, for this particular field direction. 2.3.2. Thermal expansion Thermal expansion measurements in the temperature range 1.4-100 K have been performed on the same single-crystalline sample as the one used for the specific heat experiments, employ-
and superconductivity
103
capacitance ing a sensitive three-terminal method [87]. Hereto the sample was shaped into a 5 x 5 x 5 mm3 cube, with pairwise planparallel surfaces. Data were gathered stepwise: below 15 K each temperature step (~0.3 K) was made three times, i.e. up, down and up again, whereas above 15 K only steps upwards (up to 3 K) were made. The relative accuracy in the results decreased with increasing temperature, due to small temperature gradients over the cell, and is limited to 3 x lo-’ K-’ in (Y near 100 K. The thermal expansion results, obtained in several runs, are presented in fig. 2.6 and, in some more detail, in fig. 2.7, plotted as (Y= L-‘(dLldT) vs T. The thermal expansion of UPt, is strongly anisotropic. The volume expansion is obtained by summing the data for the three crystallographic directions: CY,= (Y,+ CX~ + (Y,. Hereto the curves for ff,, (Yeand (Y,have been approximated with a 10th grade polynomial. In figs 2.6 and 2.7 we plotted a,/3 as well. In fig. 2.8 CY,- a, is plotted, illustrating that the maximum anisotropy in CYis found near 14 K. The c/a-ratio as function of temperature, normalized to 1 at 1.4 K, is shown in fig. 2.9. Relative length changes, AL/ L, with respect to the length of the sample at 1.4 K, for the basal plane and the hexagonal axis, are shown in fig. 2.10. The length changes have been calculated from the areas below the appropriate (Yvs T curves. The relative change in volume was calculated from (AVIV) = (AL/L),
+ (AL/L),
+ (AL/L),.
In figs 2.11 and 2.12 we present data for polycrystalline UP&, UPt,, UPt, [88] and UAl, [87]. Comparing the results for single- and polycrystalline UPt, we conclude that preferred orientations were present in the latter sample. In analysing the results for UPt,, we first remark that the linear thermal expansion coefficient is strongly anisotropic. The maximum anisotropy is found at about 14 K, close to the temperature at which the maximum in the susceptibility is found (17 K). This strongly suggests a correlation between the magnetic and thermal properties: the onset of spin fluctuations, confined to the basal plane, causes a relatively large dilation in the basal plane, accompanied by a contraction along the hexagonal axis. In a plot of
A. de Visser et al. I UPt,, heavy fermions
104
and superconductivity
15
10
c
5 Y
P 0 c ci
O
UPt3 -5
-10 0
Fig. 2.6.
Coefficient
of linear
20
thermal
40
expansion
60
for single-crystalline
the volume expansion, (Y,, a maximum results at 10 K (fig. 2.7). At 100 K (Y reaches a value between 8 and 9 X lop6 K-‘, which points to a dominant lattice contribution in this temperature region. In analogy to the specific heat we plot
UPt,;
80
a(U),
b(x)
100
and c(A)
axis and q,/3(0).
the thermal expansion data as (Y/T versus T* in fig. 2.13. Notice that the maximum at 10 K is obscured in this plot. The anomalous behaviour at low temperatures becomes visible as a pronounced upturn.
C AA A A AA
A A A
A
A !
0
20
10
30
-r 09 Fig. 2.7.
Coefficient
of linear
thermal
expansion
for single-crystalline
UPt,
A. de Visser et al. I UPt,, heavy fermions
0
20
80
60
40
I
loo
0
T (K>
Fig. 2.8. Difference between the coefficients of linear mal expansion for the basal plane and for the hexagonal of upt,.
theraxis
In order to relate the volume expansion to the specific heat, usually, an analysis employing Gruneisen parameters is made. Physically meaningful Griineisen relations emerge when the entropy, or rather a part of the entropy Sj, can be written as Sj( TIT,(V)). Here T,.(V) is a (volume
105
and superconductivity
20
I
1
I
I
60
80
I
40
4
xx]
T (K> Fig. 2.10 Relative length change with respect to 1.4 K for single crystalline UPt, for the a(O), b(X) and c(A) axis, and l/3 of the relative volume change (0).
dependent) characteristic temperature of the entropy term, e.g. TF or On. The dimensionless Gruneisen parameter is defined as (see for instance ref. [89]):
or in terms of the free energy - (dF,lc?T),
Fj, with S, =
(2.6) Since
0.999’ n ’ 0
20
’
’
b
40
’ 60
’
’
80
T 0.9 Fig. 2.9.
The c/a-ratio
vs. T for UPt,.
’
’
where K = - (1 lV)(dV/~3p)~ compressibility, and
is the isothermal
100
(2.8)
A. de Visser et al. 1 UPt,, heavy fermions and superconductivity
106
UPtlj
A
\
0
20
lJPt9
4.0
60
80
100
J” M
Fig. 2.11. Coefficient of linear thermal expansion for polycrystalline
one may write, dropping the index j: r--
Vffv KC,
(2.9)
’
UPt,(O),
UPt,(O),
UPt,(A)
and UAl,( x).
be defined, for instance, for the linear (electronic) and cubic (phonon) terms in c and (Y,. Writing LY(= 43) as (2.10)
a:=yaT+p:hT3, Such so-called proper Griineisen
parameters
can
one obtains (in the single band uniform scaling model) : J-
e
=
3YcxVm aln T, -=__ 8lnV ’ KY
0
0
(2.11)
100
200 T2
20
40
60
80
100
-r Cc<> Fig. 2.12. Relative iengt~ change with respect to 1.4 K for polycrystalline UPt,(O), UPt,(Cl), UP&(A) and UAl,( X).
300
400
Cd:“)
Fig. 2.13. (1/3)ar,/T plotted vs. T” for single-crystalline UPt,. The solid and dashed lines present 3-parameter fits according to eq. (2.15) in the temperature intervals up to 10 and 20 K, respectively.
A. de Visser et al. I UPt,, heavy fermions
and (in the Debye model) (2.12) where y and Pph are given in the appropriate units per mole and V, represents the molar volume. r, and rph amount to - 2 for ordinary metals. In the case of UPt,, the expression for the specific heat (eq. (2.1)) may serve as a starting point to derive an expression for the thermal expansion. Applying thermodynamic relations we find a=-y;l*T+PflhT3+P~T3+S~T31nT,
(2.13)
with (2.14a)
(2.14b)
P:=-~(~)T+&(~)T7 m m
(2.14c)
and (2.14d) By defining p,* = PI” + p t and /3: = - S, In TE , we obtain the equivalent of eq. (2.1): (Y=~~T+~~~T~+~~T~I~(TIT~). A three-parameter
(2.15)
fit to the thermal expansion
Table 2.111 Coefficients from a 3-parameter fit to eq. (2.15) c(P) data [80], employing eq. (2.14)
for the thermal
Interval (K) Fit Fit Calc. Calc.
1.4-20 1.4-10 0.5-20 0.5-18
and superconductivity
data has not the same quality as the one for the specific heat, as is illustrated in fig. 2.13. The coefficients 72, p,* and S, have been listed in table 2.111 for temperature intervals up to 10 and 20 K. Given the relatively large variation in p,* and S,, we conclude that eq. (2.15) may only serve as a first approximate description of the thermal expansion data. Thermal expansion and specific heat as function of pressure yield the same information, as can be seen from eq. (2.14). Recently, the specific heat at pressures up to 8.9 kbar has been measured by Brodale et al. [80]. Values for r,” , pz and S,, calculated from their 3-parameter fits to the specific heat, have been listed in table 2.111, as well. In view of the variation of the coefficients with the temperature interval for which the fit is made, the internal consistency of the two experiments is not as excellent as has been suggested in ref. [70]. The Griineisen parameter for the enhanced linear term is given by
r; =
(2.16)
’
KY*
and amounts to the huge value of 68 at 1 K (K = 0.48 Mbar-’ ). This illustrates the sensitivity of TF with respect to the volume. Mutatis mutandis one may define Griineisen parameters for the p*T3 and S,T3 InT-term. However, these are improper Gruneisen parameters, without a significant physical interpretation. We calculate from the fit parameters in the temperature range up to 20 K: r, = 101 and r,, = 126. In order to calculate Tz we write pX = j3:” + pf = pp = - 6, In Tz , since /3Eh is at least two orders of magnitude smaller than pz. We derive Tz = 29 K.
expansion
(q/3)
of UPt,,
P:: (IO-‘K-‘) 1.08 1.16 0.87 1.06
107
-
1.77 2.90 1.2 1.9
and coefficients
calculated
a<, (lo-‘Km’) 5.27 9.88 3.8 6.4
from the
A. de Visser et al. I UPt,, heavy fermions
108
Luthi [90] stresses the importance of electronphonon coupling in the heavy-fermion systems; the electron states are influenced by a coupling of the band electrons to the elastic waves (strains). In the so-called deformation potential model, Liithi is able to interrelate the specific heat, thermal expansion and elastic constants. In this way the thermal expansion of UPt, can be understood as due to a certain density of electron states. In a two-band model, with band widths of - 30 and - 50 K, Liithi is able, using the experimental data for the specific heat and elastic constants, to account, at least qualitatively, for the thermal expansion data (see also ref. [91]). The model needs two bands in order to account for the extrema in the thermal expansion. Their physical meaning remains, however, somewhat unclear. In the last part of this section we compare the results for UPt, with thermal expansion measurements on UPt,, UPt, and UAl,. In a plot of (Y/T vs. T2 (fig. 2.14) deviations from a linear behaviour become visible at low temperatures. These anomalies are found in the specific heat as well, and are ascribed to spin-fluctuation effects [S]. In orthorhombic UPt, this contribution might be obscured due to crystallites with preferred orientations. The unenhanced coefficients Table 2.1V Coefficients from 2-parameter UPt,, UPt,, UPt, and UAlz
fits to 01 and c, enhanced
UPt”
Parameter
Units
Y‘,
10~W2
::
mJ/K’molU
Y,* *
lO_“K
and superconductivity
from two-parameter fits conform eq. (2.10) (see solid lines in fig. 2.14), for UPt, and UPt, in the temperature range 50 < T2 < 250 K2, and for UAl, in the temperature range 150 < T2 < 400 K*, have been collected in table 2.IV. The enhanced coefficients are listed as well. Data for UPt have been taken from ref. [92]. By comparing the thermal expansion data with specific heat data on the same samples (except in the case of UPt,) values for r,, rp,, and c have been derived. Hereto we estimated K from an interpolaI
I
100
0
200
I
300
400
T2 (K*) Fig. 2.14. a/T plotted vs. T* for UPt,(O), UAI,( X). The solid lines represent 2-parameter eq. (2.10).
linear coefficients,
and the corresponding
Griineisen
UPt,(&) and fits conform
parameters
upt,
upt,
5.6
11
-
1006.4
75 25
-
12.4
108
1.8
33
24 89
422 6X
957.3
133 20
* -
UPQ 0.8 24 13
UAI, 14.0 52 21
;f
mJ/K’molU
P, P r Ph
10 “‘Km4 mJiK4molU
0.55 1.4 0.4
0.71 1.03 1.2
0.32 # 0.85 2
2.3 4.2 2.1
0.72 0.65 0.9
K V,
lO~“m’/N 10~5m’/molU
0.69 + 2.66
0.58 + 3.36
0.48 4.24
0.48 + 6.15
1.35h 3.59
* Calculated + Estimated “Ref. [92]. hRef. [93].
with &,, = 2 value.
for UPt,
A. de Visser et al. I UPt,, heavy fermions
2.4. Elastic properties Crystals with hexagonal symmetry - such as UPt, - require five independent elastic constants to specify the relationships between stresses and strains. In terms of stiffness coefficients, they may be listed as cli, c,~, cr3, c33 and cd4. These five quantities can be determined by sending ultrasonic waves through properly oriented crystal specimens and measuring the wave velocities, which depend on the elastic constants and the density. Obviously, at least five measurements must be made for hexagonal crystals. Relations between wave velocities and elastic constants can be found in e.g. Mason [94]. In order to determine the five independent elastic constants at room temperature [87], we
Kl7=--
1 da -= a dP
Type mode
Direction of propagation
Direction of particle motion
Velocity
I I I I I II II II
long. shear long. shear shear quasi long. quasi shear shear
C
C
3860 1385 3993 2076 1388 3754 u, = 1827 u* = 1753
: b b b’ b’ b’
C
b’ Cl
a’ = a
C33 c33(C,,
+
-
Cl3
Ciz)
(2.17) -
2Cf3
c,~ = 1.421 Mbar as follows from a combination
Sample
in ab-plane b a
109
followed the approach of McSkimin [95]. For these measurements we used conventional ulwith quartz transducers trasonic equipment, tuned at 20 MHz (longitudinal waves) and 5 MHz (transversal waves). Two single-crystalline samples, from the same batch as prepared for the thermal expansion sample, were machined into the proper shape, by means of spark erosion. One specimen (I) served for propagation directions along the c-axis (1 = 7 mm) and b-axis (I = 5 mm). The other specimen (II) was shaped for a propagation direction (b’) at 45” between the b and c-axis, but perpendicular to the a-axis (I = 2 mm). The measured sound velocities are listed in table 2.V, together with the formulae for the elastic constants and the derived values. From the velocities u1 . . . u,, measured on sample I, cll, c,*, cX3and cl4 could be obtained. The other sample (II) served to derive c13. For the calculation of c13, from a combination of uI, u2, u3 and u6 or u,, or from a combination of u3, us, u6 and u, we refer to McSkimin [95]. us served as a cross check. The c,,-values are based on the calculated density p = 1.940 X lo4 kg/m3. The accuracy in the ciivalues is 1%) except for c,~, in which case it is 3%. The elastic constants enable us to calculate the compressibility, K. The linear compressibilities in the basal plane, K,, and along the hexagonal aXiS, K,, are given by
tion between the values for U (0.99 Mbar-‘) and Pt (0.38Mbarr’), encouraged by the estimated (0.53 Mbar-‘) and experimental (0.48 Mbar-‘) value for UPt, . The resulting &-values are larger than for ordinary metals, giving rise to the question whether the intersections of the solid lines in fig. 2.14 and the ordinate indeed represent the coefficients of the unenhanced linear terms. The large r z-values are indicative for the relatively high density of states in these compounds. The &,-values are relatively small. Unfortunately, for none of these compounds thermal expansion data on non-magnetic analogs are available. Therefore, a more thorough analysis of the various contributions to the thermal expansion is, at this moment, not possible.
Table 2.V Sound velocities and elastic constants of UPt, at room temperature. The c,,-values are based on the density p = 1.940 x lo4 kg/m’
and superconductivity
(m/s) u, =
UI = u3 = u, = v, = U6=
’
of c,, and c,,.
Elastic constant (Mbar) cjl = pu; = 2.891 CA1= P”2’ = 0.372 ‘C,, = pu; = 3.093 c,, = !(c,, - c,,) = PU; = 0.836 CA1= pu; = 0.374 cL, = 1.732 c,., = 1.695 ic,, - fC,? + qc,, = pu;
A. de Visser et al. I UPt,, heavy fermions and superconductivity
110
up13
Q
i :
“...
29.0 -
C
32.0 t
.a.
31.8 n
C33
a.
Cl1 Y
5
‘.
‘.
28.5 -
281’
5
*. *. *. a. *.
200
10”
40°
3.85 \
1
31.6 -
Y
31.4 0. *. .
T(K)
“E ‘O”
,” P
31.2 -
l.
‘x.
*. Q.
I
I
200
1
1
400
9.5 -
Q
l*..
I T(K)
L :
b
*. -.
C&L 380
:yyy,
-**...._
375 0
200
a
Fig. 2.15.
T(K)
Elastic
constants
and 1 dc KC=---=
c dP
The volume
- 2c,, + Cl1 + Cl2 %(C,l
+ Cl,)
compressibility
-
2cL
is given
(2.18) . by
1 dV K=--
-=2K
Vdp
u
+K..
(2.19)
’
A small anistropy is observed: K, = 0.164 Mbar-’ and K, = 0.151 Mbar-‘. Therefore, at applying a relative increase in the pressure on UPt,, c/u-ratio dln(c/a)ldp = K, - K, = 0.013 Mbar-’ is expected. The volume compressibility amounts to 0.479 Mbar-‘, which fits nicely in between the values for uranium, 0.99Mbar-‘, and platinum, Finally, the Debye temperature, 0.38 Mbar-‘. On, has been determined from the elastic con-
Table 2.W Elastic constants
of UPt,
in Mbar
(= 10”Nim’);
^
c,~ is calculated
200
0
600
of UPt,
(after
400
T (K)
600
ref. [91])
stants. This may be accomplished by integrating the sound velocities (or elastic constants) over all crystallographic directions (see e.g. Alers [96]). The resulting value equals 217 K. Yoshizawa et al. [91] have measured the temperature dependence of cS3, Cam,cl 1 and ce6 from 1 up to 600K. These data, shown in fig. 2.15, clearly reveal low-temperature anomalies of 0.03-1.1% in cii, for all models for T < 50 K. The longitudinal modes (c,, and Cam) show an upturn, whereas the transversal modes (c,, and ce6) pass through a maximum. These anomalies are related to the deformation potential coupling, as mentioned in the previous section. In table 2.VI we compare our results with the data of ref. [91] at 300 K and 1 K. The agreement is reasonable. Assuming that one of the elastic constants, c,~ (not measured by Yoshizawa et al.)
from a combination
of ce6 and c,, (see table
2.V)
Ref.
T(K)
cl!
Cl2
c13
c33
c44
Chh
[871 I911 [911
300 300 1
3.093 3.174 3.206
1.421 1.390 1.351
1.718 -
2.891 2.872 2.929
0.372 0.378 0.385
0.836 0.892 0.928
A. de Visser et al. I UP&, heavy fermions
Table 2.VII Compressibility
of UPt,
and superconductivity
111
in Mbar-’
Ref.
T(K)
K,
K,
K
[871 t911 t911
300 300 1
0.164 0.160 0.163
0.151 0.157 0.151
0.479 0.477 0.476 c
is temperature independent, we calculated K, K, and K, at 300 K and 1 K from the data of Yoshizawa et al. The results are listed in table 2.VII. No significant temperature dependence for the compressibility is found. Allowing for a 5% variation with temperature in cl3 yields the same conclusion. Recently, Benedict et al. [72] have determined the compressibility of UPt, from Xray patterns, taken in a diamond anvil cell at pressures up to 500 kbar. These authors obtained an initial slope, K -0.5 Mbar-‘, in good agreement with our result. 2.5. Transport properties 2.5.1.
Electrical resistivity
The electrical resistivity of Czochralski-grown single-crystalline UPt, [lo] has been measured on three cylindrical samples (diameter = 1.5 mm, length = 6 mm), that have been spark eroded along the a, b and c-axis out of one large crystal. The results, obtained on unannealed specimens, in the temperature range 1.3-300 K, are shown in fig. 2.16, as smooth lines through the data points. The resistivity is strongly anisotropic. Data taken along the a and b-axis show no significant difference, within the experimental accuracy (the absolute and relative accuracy amount to 7% and O.l%, respectively). All runs were taken with increasing and decreasing temperature; no hysteresis effects were observed. The pO-values, determined from an extrapolation to 0 K, are 7.2, 7.2 and 3.6pRcm for the a, b and c-axis, respectively; residual resistance ratios, RRR = ~(300 K)lp,, amount to 33, 33 and 37, respectively. After annealing, the pOvalues decreased to 6.2, 3.0 and 1.7 p&m, leading to RRR’s of 38, 79 and 78, respectively, The differences in p,-values, possibly due to small deviations from stoichiometry , might be intro-
100
0
I
0
I
100
I
I
I
200
J
300
Fig. 2.16. Electrical resistivity vs. temperature for unannealed Czochralski-grown UPt, for current directions as indicated.
duced by cutting the samples from different parts of the single-crystalline batch. In the lowtemperature regime, the resistivity is strongly temperature dependent, as can be seen in more detail in fig. 2.17. At 4.2 K the p-values amount to 28.2, 24.1 and 11.1 @cm for the a, b and c-axis, respectively. The superconducting transition temperature equals 0.48 K. In fig. 2.18 we
8o r-----l a 80 0
T c40
b
3
Q
Fig. 2.17. Electrical resistivity for annealed grown UPt, for T < 10 K; a(O), b(O) and c(A) lines represent data taken in a ‘He-cryostat).
Czochralskiaxis (dashed
A. de Visser et al. I UPt,, heavy fermions
112
and superconductivity
1oc
P-P, IpRcml SC
. upt,-a o UPt,-b oUPi,-c
8 b b
t
bb
20
__t
TIKI
Fig. 2.18. Double logarithmic plot of p - p0 for annealed UPt,. The solid lines denote a T*-temperature dependence.
present the low-temperature data on a double logarithmic scale. The resistivity curve for a polycrystalline UPt, sample nearly coincides with the curves along the a and b-axis, indicating that crystallites with preferred orientations are present. The crystallites are mainly oriented with the c-axis perpendicular to the cylinder axis. This was confirmed by magnetization measurements on this sample and further supported by the orientations of the grains studied under a light microscope. The p,-value of this sample amounts to 5 t&cm and the RRR equals 48. Resistivity data taken on an unannealed single-crystalline whisker [97], with the current along the c-axis, are shown in fig. 2.19. The whisker dimensions are approximately: 20 km(aaxis) x 40 pm(b-axis) x 1000 pm (c-axis). The pOvalue amounts to 2.2 @cm, the RRR equals 66. The uncertainty in the resistivity value due to the geometrical factor is, for this particular whisker, estimated at t 10%. The low-tern erature data P. are presented in a plot of p vs. T m fig. 2.20. The electrical resistivity of polycrystalline UPt , UPt, and UPt, is shown in fig. 2.21. In fig. 2.22
Fig. 2.19. Electrical nealed UPt, whisker
I
I
I
resistivity vs. temperature for an unan(current along the c-axis).
we present low-temperature data for polycrystalline UPt,, UPt, and UPt,. For UPt, UPt, and UPt, the p,-values amount to 34, 11.2 and 25.7@cm, and the RRR’s to 3.5, 14 and 5.6, respectively. In fig. 2.23 the low-temperature
‘r-----l 6
z4 3 Q3
0
1
2
3
4
5
6
T2 (K2) Fig. 2.20 Electrical whisker.
resistivity
vs. TZ for an unannealed
UPt,
A. de Visser et al. I UPt,, heavy fermions 2001
,
I
I
1t
p(~Rcml
150
I
IpRcml t
100
Fig. 2.21 Electrical resistivity line UPt, UPt, and UPt,.
vs. temperature
UPts
I, 0
Fig. 2.22. Electrical resistivity UPt,(O) and UPt,(A) (dashed a 3He cryostat).
49
Q *
2°
IJpt, upt,
. .. .
0 000"
for polycrystal-
1
a
0
0
5-
300
200
7-----l
F
0
P-P,
UPt2
0
40
10
I
I
113
and superconductivity
I
I
I Kl
of polycrystalline UPt,(O), lines represent data taken in
data for UPt, and UPt, are plotted on a double logarithmic scale. The resistivity of UPt, displays a type of curve that is characteristic for many actinide compounds [98]: a sharp rise at low temperatures is followed by a negative curvature towards the temperature axis (i.e. d2pldT2<0) and a tendency to saturate at large values in the roomtemperature region. In general, the overall large electrical resistivities in the actinides are ascribed to strong scattering of the conduction electrons
1
2
10
5
20
dTIK1 Fig. 2.23. Double logarithmic plot of p - p0 for UPt, and UPt,. The solid lines denote a TZ-temperature dependence.
by the spin fluctuations of the Sf-electrons. In this sense, raising temperature increases the amount of excitation in the spin-fluctuation spectrum and, hence, the effective scattering. Saturation occurs in the room-temperature region, where the mean free path of the electrons becomes of the order of the interatomic distance. It is not quite clear how far this situation holds for UPt,. The resistivities of all other heavyfermion systems reveal a Kondo-lattice behaviour: at low temperatures correlations between the Kondo-scattering centers cause a drastic drop of the resistivity (coherence). In the same way, one might think the drop in the resistivity of UPt, to be caused by the onset of correlations between the scattering centers. To what extent Kondo screening plays a role in UPt, is still an open question. Given the absence of a Kondo minimum below room temperature, the Kondo temperature will at least be several hundred Kelvin. The electron-phonon scattering term in the resistivity is assumed to be negligible at low temperatures, whereas this term will only play a minor role at high temperatures. This is supported by comparing data on UPt and non-
114
A. de Visser et al. I UPt,, heavy fermions
magnetic ThPt, for which compounds ~(300 K) amounts to 120 and 25 P&m, respectively [99]. As mentioned in section 1.3.1, according to the paramagnon model the resistivity of a spinfluctuation system varies as T2, at low temperatures. A T*-law is also expected in a Fermi liquid description, as follows directly from FermiDirac statistics. In a double logarithmic plot of the resistivity data (fig. 2.18), we observe that, possibly, in the low temperature limit, a T2-law is present. The coefficient A of this term is estimated at 1.6 and 0.7 @cm/K2, for the basalplane and hexagonal axis, respectively. A plot of p vs T2 for the whisker (fig. 2.20) reveals a T*-term below 2 K, yielding A = 0.58 @cm/K2 (c-axis). These A-values are at least two orders of magnitude larger than for ordinary metals. The characteristic change to a negative curvature of the resistivity curve towards the temperature axis, is clearly reflected by a peak in a plot of dpldT versus T (see fig 2.24). The temperature derivative has a maximum at 6.5 K for the basal plane directions and at 7.5 K for the hexagonal axis. The temperature at which the maximum occurs is assumed to be proportional to the spin-fluctuation temperature. At room temperature the resistivity starts to saturate. The maximum (or saturation) in the resistivity of the heavy-fermion compounds is often used to estimate the number of conduction electrons per atom, Z,, and their Fermi wave number. According to Friedel [loo], the maximum in the
5
0 0
Fig. 2.24. Temperature derivative of uranium-platinum compounds vs. T.
the
resistivity
of
and superconductivity
resistivity, due to incoherent scattering of the conduction electrons at the uranium atoms, can be written as: Pmax=
2(21+ 1)h.X e2kFZa
.
(2.20)
Here the orbital quantum number of f-electrons 1= 3, x( = l/4) is the concentration of scattering centers and k, = (3~2Z,/0n,)1’3, where L$ = 17.6 A’ is the mean volume per atom. Assuming a spherical Fermi surface and a mean value for Pmax= 200 @cm, we derive Z, = 2.7 and k, = 1.66 A-‘. However, this implies 10.8 conduction electrons per formula unit UPt,, which is a rather large number (assuming trivalent U one finds 2.6 conduction electrons per Pt). Therefore, the present estimate of k, must not be taken to literally; the order of magnitude is correct. Note that k, varies as Z”3, and thus depends only weakly on Z. The value for k, for the conduction electron gas must be compared with the value of k, = 1.1 A-’ at low temperatures, where the electron gas condenses into a liquid, due to the presence of the f-electrons. The electronic mean free path is given by 1, = U~T, where ur = fik,lm, the relaxation time r = ml(pne2), and it = Z,ln, is the number of conduction electrons per unit volume, hence 1, =
*
hi
(2.21)
With k, = 1.66 A- ’ and p = 200 l.&zrn we calculate 1, = 2.2 A, which shows that the electronic mean free path is indeed of the order of the interatomic distance. Resistivity measurements performed by others are in good agreement with the results presented above [lOl-1051. Some characteristic resistivity parameters have been collected in table 2.VIII. Data taken in Darmstadt on the same polycrystalline sample as described above are shown in figs. 2.25 and 2.26, in the temperature regions below 1000 K and 6 K, respectively [86]. Superconductivity is observed below 0.48 K, the width of the transition is about 0.1 K. The detailed behaviour of the polycrystalline specimen below
A. de Visser et al. I UP&, heavy fermions
Table 2.VIII Some characteristic
resistivity
parameters
and superconductivity
for UPt,
PO (@cm)
~(300 K) (@cm)
RRR
Specimen Czochralski a-axis Czochralski b-axis Czochralski c-axis Whisker c-axis* Polycrystalline Polycrystalline # Polycrystalline Polycrystalline Flux-grown c-axis
6.2 3.0 1.7 2.2 3.9 2.9 0.46 2.2 -0.5
238 238 132 147 200 200 336 236 150
38 79 78 66 51 69 730 107 300
* Unannealed. * After additional
A (@cm/K*) 1.6 1.6 0.7 0.58 1.38 1.7 -3 2.0
Ref.
[W [lOI [lOI [971
WI WJI [IO41 w51 [lo21
annealing.
1 K is shown in the inset of fig. 2.26. The residual resistivity equals 3.9 @cm (we obtained 5 t_&cm); after an additional annealing p0 = 2.9 t.LIcm. A quadratic temperature dependence holds up to 2 K, with a coefficient A =
upt,
OO
115
1
I
I
200
LOO
600
800
1000
T(K) Fig. 2.25. Resistivity vs. temperature polycrystalline UPt, sample.
Fig. 2.26. Temperature dependence polycrystalline UPt, sample below below 2 K ploted vs. T’.
up to lOOOK for
of the resistivity 6K. Inset shows
a
of a data
1.38 t.LIcmlKZ before annealing and 1.7 @cm/ K2 thereafter. The high-temperature data confirm the tendency to saturate as suggested by our results below 300 K. Wire et al. [loll performed resistivity measurements up to 1100 K with a transient-pulse-heating technique. Their data revealed a very weak maximum at -700 K (~(700 K)lp(llOO K) = 1.03). The resistivity of the compounds UPt, and UPt, display a similar behaviour as the resistivity of UPt,, although spin-fluctuation effects are less pronounced. A T2-law is followed over a somewhat larger temperature range, as follows from fig. 2.23. The coefficient A takes values of 0.09 and 0.01 t&cm/K*, respectively. The inflection point in the p vs. T curve (fig. 2.24) is found at 26 K for UPt, and at 30 K for UPt,, indicating a much higher spin-fluctuation temperature than for UPt,. The resistivity of UPt, displays a weak maximum, around 165 K. The room-temperature values equal 154 and 141 @cm for UPt, and UPt,, respectively. Equi-atomic UPt orders magnetically below 27 K. This compound reveals a complex magnetic behaviour, reflected in two magnetic transitions at 27 K and 19 K, leading to a ferromagnetic arrangement of the magnetic moments below 19 K. The large resistivity values in the paramagnetic region are ascribed to spindisorder scatter. Up to - 10 K a T2-law is obeyed. A more detailed description of the lowtemperature properties of UPt can be found in a paper by Frings and Franse [106].
A. de Visser et al. I UPt,, heavy fermions
116
and superconductivity
Thermoelectric power, thermal conductivity and Hall effect In order to look in some more detail into the
2.5.2.
transport properties of UPt,, we next discuss thermoelectric power and thermal conductivity measurements performed by the Darmstadtgroup on the polycrystalline specimen [86]. Thermoelectric power, S(T), measurements in the temperature range 0.2-350 K are shown in fig. 2.27. The main feature of S(T) is a pronounced positive peak at T = 8 K. The temperature at which the maximum occurs nearly coincides with T,,, as follows from the temperature derivative of the resistivity and can be taken to be proportional to the spin-fluctuation temperature. S(T) changes sign at T = 24 K and reaches a nearly constant value of - 11 ~.LV/K above T- 100 K, where also the resistivity starts to saturate. For normal metals, in which the resistivity is governed by elastic scattering processes, S(T) is proportional to TdlncTlaE (a is the
electrical conductivity) at low temperatures (see e.g. ref. [107]). As can be seen in fig. 2.27 there is a monotonous decrease of S(T) below T = 4 K, deviating from a linear behaviour, indicating the presence of inelastic scattering processes. Detailed measurements by Sulpice et al. [108] revealed S CCT2 below 1 K. A theoretical interpretation of this term is, however, not yet available. In fig. 2.28 the thermal conductivity, K, is given as a function of temperature up to 200 K, and compared with the electronic part as calculated from the measured electrical resistivity, assuming the validity of the Wiedemann-Franz
x
r
,
‘-_-------------___-A
/
I
50
0
100
150
T(K)
Fig. 2.28. (a) Thermal conductivity of polycrystalline UPt, for temperatures up to 200 K, as measured and as calculated from the electrical resistivity by applying the WiedemannFranz law; (b) the experimentally determined value for the Lorenz ratio, L(T), divided by the Sommerfeld value for this ratio, L,.
law. In fig. 2.28b we show the Lorenz ratio, L = ~plT, obtained from the measured K(T) and p(T) data. Depending on the temperature region the Lorenz ratio in pure simple metals is either smaller than, or comparable to, L, = 2.45 x lo-’ V*/K*, i.e. the Sommerfeld value. In systems with a small electronic mean free path, for example in the presence of strong electronparamagnon scattering, phonons can dominate the heat transport, giving rise to a substantial increase of L above the Sommerfeld value. Since the phonon conductivity usually has a maximum this leads to a corresponding at (O.l-0.2)0,, peak in L(T). In the case of UPt, a maximum value L,,, = 5.5L, is found at a temperature T,,,,, = 24 K, which is slightly larger than 0.1 0, (On = 217 K). B ecause the total heat conductivity is the sum of an electronic part, K,, and a phonon part, K~, the measured quantity L(T) can be expressed by L
=
p(W(T) =
(2.22)
L
T
L, = ~,plT is the electronic Lorenz ratio. Since at higher temperatures L, should approach
where L,,
0
1
2
3 T(K)
0
100
200
T(K)
Fig. 2.27. Thermoelectric power, S, of polycrystalline UPt, for T < 5 K (a), and for temperatures up to 350 K (b).
this
analysis
shows
that
K~ 3
4~,
at T,,,,,,
which is not unreasonable in view of the already rather short electronic mean free path, I, - 4 A, as inferred from the electrical resistivity of 120 @cm at this temperature. The phonon mean free path I, at T,,, can be calculated from:
A. de Visser et al. I UPt,, heavy fermions
i
X-1 (2%)
UPf3
. WF
5
ii
,.+-
.
.a
. I
y’# , , 1
2
3
1
,r2J L
5
117
range L is smaller than L, by about 8%. An even weaker ratio L/L, - 0.45is calculated from the data by Sulpice et al. [108], pointing again to the presence of inelastic scattering processes. Another transport property that has been investigated for UPt, is the Hall effect [109-1111. We concentrate here on the work of Schoenes and Franse [109]. These authors performed measurements on a single-crystalline specimen in the temperature range 2-300 K. The magnetic field (4T) was applied along the b-axis, the current along the a-axis, and the Hall voltage was determined in the c-direction. The data are shown in fig. 2.30. For temperatures above - 50 K the temperature dependence of the Hall coefficient, R,, resembles that of the basal-plane susceptibility, which is also shown in fig. 2.30. This makes it possible to separate the Hall effect in a normal part (R,), that allows for the carrier determination, and an anomalous part (R,), closely related to the magnetism in UPt,. To separate the normal from the anomalous Hall effect, one may write [ 1121:
r?
-
1
and superconductivity
TiKl
Fig. 2.29. Thermal conductivity of polycrystalline UPt, below 6 K; the solid curve represents the Wiedemann-Franz law. Kp = i,v,c,/3. By inserting the sound velocity, u, 14 x lo5 cm/s (table 2.V), a lattice s ecific -P heat cp of approximately 0.27 JK-’ cm , and Kp = 22 mWcm_’ K-‘, as calculated from the difference between the measured K(T) and the calculated K,, one derives I, = 60 8, + I,. In fig. 2.29 the low-temperature normal-state heatconduction data are displayed in more detail. At - 3 K the measured curve and the calculated curve cross, which shows that the additional phonon contribution (as discussed above) is present down to - 3 K. In the low-temperature
pu= R,B+pu,MR,.
(2.23)
10
I
50
100
150 Temperature
I
I
I
200
250
300
[KI
Fig. 2.30. Total Hall effect of UPt, as measured in the field of 4 T (after ref. [109]). The inset shows the geometrical arrangement of the electrodes. The solid lines are fits (see text). Also shown for comparison are the susceptibility data in the basal plane.
A. de Visser et af. i UPt,, heavy fermions
118
For T>SOK we have 8=&?I, M=xH, C/( 7’ - O), and the Hall coefficient equals
R~= PH =R, POH
+ &
R, .
x=
(2.24)
In fig. 2.31 a plot of R&T - S) vs, T- 0 is shown, with 0 = - 50 K, as derived from the x -l(T) curve for T 5 300 K. From the slope of the solid line, which represents a fit to the data, a value R, = - 4.0 x 1O-4 cm3A-‘s~” results. The negative sign shows that the ordinary Hall effect is dominated by electron contributions. The value for R, ( = - line) corresponds in a oneband model to 1.1 conduction electrons per formula unit [109], a value that differs considerably from the crude estimate of 2 = 2.7 per atom, that we deduced from the maximum in the resistivity. Marabelli et al. [113] performed farinfrared optical reflectivity measurements and derived in a Fermi Iiquid model 2 = 1 per formula unit, close to the value derived by Schoenes and Frame. Z = 1 corresponds to k, = 0.75 A-‘, which is in reasonable agreement with the value for k, of - 1.1 A-’ that we used in the Fermi liquid analysis of the specific heat (section 2.3.1). The intersection of the straight line in fig. CR, = gives ordinate the 2.31 with 0.244 cm3KA-‘s-l. With an effective moment of R, = compute 2+,/U-atom, we
and superconductivity
0,97cm3A-1s-*, which is - 2500 times larger than R, and of opposite sign. The anomalous Hall effect is attributed to skew (asymmetric) scattering at the uncorrelated scattering centers (single impurities). At low temperatures, below - 50 K, the anomalous Hall effect R, = (R, R,)/x decreases rapidly, which is seen as a clear sign of the formation of a coherent state. 2.5.3. Point-contact spectroscopy We conclude this section on transport properties with a discussion of point-contact experiments on the uranium-platinum compounds (Jansen et al. [114], and references therein). Small constrictions between metals have been used to analyse the interaction of the conduction electrons with elementary excitations in the metal. By measuring the resistance of a point contact as a function of the applied voltage, direct information is obtained about the energy dependent scattering of the conduction electrons. Important criterion for the applicability of this method is that the inelastic mean free path, fi, of the electrons is larger than the contact diameter d. In this ballistic regime the electronic system will be out of equilibrium with respect to the elementary excitations (e.g. phonons) in the presence of an applied voltage over the contact. With this technique excitations like phonons, magnons and crystal-field levels have been prob-
T-B [Kl Fig. 2.31. The decomposition
of the total Hall effect of UPt, into the normal and anomalous part for T > 50 K (after ref. (1091).
A. de Visser et al. I UPt,, heavy fermions
ed. In the impure limit of a contact (Ii < d) the electrons stay in equilibrium with the lattice, at an applied voltage, and in this thermal regime heating occurs in the contact region. For instance, it could be shown that in ferromagnetic materials the Curie temperature was reached in the contact region at a certain voltage over the contact. The point-contact experiments on the uranium-platinum compounds have been performed in Nijmegen and Grenoble. At liquid He temperature (1.5 K) the contacts were adjusted by pressing the sharp edges of two pieces of bulk material against each other. Prior to mounting, the polycrystalline bulk material was cleaved at room temperature. The contact resistances were of the order of 1 a. The technique for recording dV/dZ and d2V/dZ2 makes use of a modulation method. On top of the bias current, which is swept in order to vary the applied voltage over the point contact, a small ac-current is superimposed. The ac-voltage over the contact is measured by phase-sensitive detection at the fundamental frequency (If-mode), and at twice the fundamental frequency (2f-mode) of the modulator, to obtain signals proportional to the first and second current derivatives of the voltage. The change in the resistance, which is only a few percent over the investigated voltage range, is recorded using a bridge circuit. For these contacts, between the same materials, the dV/dZ characteristics were symmetric with respect to the applied voltage. The relative change AR/R(V=O) of the static resistance of the point contacts is presented in fig. 2.32. The reproducibility of the curves for different contacts might be estimated from the width in the minimum. For the case of UPt, and UPt, the width was equal within roughly 10%. For UPt and UPt, the width varied up to 30%. The strong resemblance between the voltage dependence in the contact resistance (fig. 2.32) and the temperature dependence of the bulk resistance (figs. 2.16 and 2.21) suggests the importance of heating effects in these compounds. In the dirty limit (Zi < d) the resistance of a circular contact between two semi-infinite metals is given by
119
and superconductivity
R(V=0) = 1.09 n a=0.09
R(V10)=9.0 n
0
20
40
60
60 voltage
CmV)
Fig. 2.32. Normalized change in the static resistance as a function of the applied voltage for point contacts with uranium-platinum compounds at liquid helium temperatures. The dashed curves have been calculated with the heating model, using temperature dependent resistivity data with the effective Lorenz ratio’s L, and multiplied by the factor (1.
R=$,
(2.25)
where p is the resistivity of the metal. For the U-Pt compounds with p = 10 l&cm at 1.5 K, and R 2:1 R the diameter d > 1000 A. The mean free path amounts to - 50 A ( < d). Clearly, we are in the thermal limit and Joule heating in the
120
A. de Visser
et al. 1 UPt,,
heavy fermions and superconductivitj
contact area has to be considered. Assuming equal flow lines for the electrical and thermal conduction near the contact, one obtains for an arbitrary contact geometry at an applied voltage V the relation T;=
v’ 4L ’
(2.26)
for the maximum temperature T,,,,, at the center of the contact in a bath at temperature T,. In this expression L = prc/T is the Lorenz number. For T,,, 9 T,) a numerical calculation gives Tmttx= aV, with a = 3.2 x lo3 KIV. Hence, using eq. (2.25) and eq. (2.26), a simple relation exists between the voltage dependent contact resistance and the temperature dependent resistivity, if the resistance of the constriction is mostly determined by the resistivity at the center of the contact where the temperature is maximal. Taking into account the temperature gradient near the contact, using the model of spherical spreading out for a constriction, the point-contact resistance as a function of the applied voltage can be calculated. For the conversion of the temperature scale to the voltage scale a constant Lorenz number over the whole temperature interval is taken. The results of these calculations with the used effective Lorenz numbers have been plotted in fig. 2.32 as dashed curves. To get a good agreement of the calculated curves with the experimental ones the calculated values of ARIR(V=O) are multiplied by a constant a, smaller than 1. An explanation for this constant CYcould be that part of the normal contact resistance is a Sharvin-like resistance R, ~pl,ld~, which applies for a clean contact (1, > d) and does not depend on the temperature. The constant cy is different for different contacts of one material, because of the relative change of contact resistance. For UPt and UPt, we mostly find small values for cy, meaning that the Sharvin resistance is much larger than the temperature dependent resistance in eq. (2.25). In this way we find a discrepancy, because no longer the limit I, < d will hold and the heating mode1 does not apply to our problem. Note however, the resistivity at the constriction is different from the
bulk resistivity, due to anisotropy, grain boundaries, strain, impurities etc. For UPt, we have taken in fig. 2.32 the Lorenz ratio as obtained from the measurements on the electrical and thermal conductivity, which yielded a constant value above 70 K. For the other curves we have taken the Lorenz number, that optimizes the agreement between experiment and theory. The accuracy of the Lorenz numbers must be seen in view of the, before mentioned, reproducibility of the width in the minimum of the resistance curves. Recently, point-contact masurements on UPt, were interpreted as a measure of the energy-dependent density of states of the electrons [115]. From these experiments strong evidence for a narrow band of heavy electrons (m* = 250m,) situated near the Fermi level was deduced, In detail it is not yet clear how the energy-dependent mass-renormalization enters the point-contact problem. We believe that heating plays a role in the description of pointcontact experiments with these metallic compounds. 2.6. High-magnetic-field
studies
2.61. magnetization The magnetization of single-crystailine UPt, has been measured by Frings et al. [2] at fields up to 35 tesla in the Amsterdam High-Field Installation. These data, taken with stepped pulses, at temperatures of 4.2 K (a, b and caxis), 20.3 K (a-axis) and 77K (a-axis), are shown in fig. 2.33 (see also refs. [8, 79, 116, 1171). The magnetization in the basal plane is isotropic, within the experimental accuracy. An anomalous increase of the magnetization takes place between - 1.5 and - 25 T for field directions in the basal plane at 4.2 K, resulting in a value of - lp,JU-atom at 35 T. At 20.3 and 77 K the basal-plane magnetization is linear in field, as is the magnetization along the c-axis at 4.2 K. The specimen investigated in these experiments was the same unannealed 3 mm sphere as used for the susceptibility measurements (section 2.2). In fig. 2.34 we show data, taken with a smooth pulse at 4.2 K, on an annealed Czochralski-grown single-crystalline sample, prepared
A. de Visser et al. I UPt,, heavy fermions
/
61
I
I
121
and superconductivity
r
I
/
I
% 8 8
UPt3
5
Y,
l-i
+
%
a
+
+
E
*
2 5
++
0 + 8
+
b
0
B Fig. 2.33. Magnetization of unanneaied Czochralski-grown 4.2 K(0) and c-axis at 4.2 K(X) (after ref. [8]).
xx
20
10
x
A
A
Y
x
30
40
1T1
UPt, for the a-axis at 4.2 K(Q), 20.3 K(+) and 77 K(a),
b-axis at
UPt3
b
0
20
10
B
30
40
CT)
Fig. 2.34. Magnetizatjon of unanneaied UPt, for the b-axis at 4.2 K(0) and for the a-axis at 77 K(A) (after ref. 1791); and of annealed UPt, for the a-axis at 4.2 K(X).
122
A. de Visser et ul. I UPt,,
heavy fermions
from a different batch, for a field along the a-axis, and on the unannealed sample (h-axis) as measured by Frings and Franse [79]. The field at which the anomaly occurs (20 T), reproduces well for different samples and is not sensitive to strains. The most interesting feature of the high-field magnetization measurements is the presence of the anomaly near 20 T, at 1.5 K (not shown) and at 4.2 K. This anomaly appears as a pronounced peak in a plot of the differential susceptibility, X(H) = &r/AH, versus H (see fig. 2.35). At 20.3 and 77 K, i.e. at temperatures above the temperature at which the maximum in x(T) occurs (17 K), the anomaly is absent. This suggests a close connection between the anomalies in a(H) and x( ?“), which is supported by the nearly equal and thermal energies: magnetic iu,B,,X = The field at which the anomaly occurs k, T,,,. does not depend on temperature: from 1.5 K (= 0.09 I’,,,) to 4.2K (= 0.25 T,,,) no variation of R,,, is observed, within the experimental accuracy (see also section 2.6.2). Besides, magnetization curves, taken at 1.5 K and 4.2 K, did not reveal any broadening of the transition. Hys-
Fig. 2.35. Differential susceptibility for UPt, for the b-axis at 4.2K(O) represents
and for the a-axis at 77K(A). The dashed curve a fit according to eq. (2.27) (after ref. [79]).
and superconductivity
teresis effects are absent. In order to perform a quantitative analysis, Frings and Franse [79] have fitted x(H) for p& < 20T to the expression: X(H) = x& + a( rU,W2 + b(E_LoW4+ c( &H)“] , (2.27) and obtained the parameters: x0 = 112 f. 1 x lo-” m”/molU, a = 135 -+ 25 x low5 T-‘, b = 0 -+ 16 x lo-’ T-” and c = 3.0 + 0.3 x lo-* T-? Thus x(H) - x0 starts as HZ. In the next section we first present magnetoresistance data in the same field region, and then discuss the possible interpretations of the anomaly near 20T. 2.6.2. Mag~et5res~st~v~ty The magnetoresistivity of single-crystalline UPt, has been measured in the Amsterdam High-Field Installation at fields up to 35 tesla [97, 118, 1191. Experiments have been performed on two annealed Czochralski-grown singlecrystalline specimens with a cylind~cal shape (length = 6 mm, diameter = 1.5 mm). One sample was measured with the current directed along the a-axis, the other with the current directed along the c-axis. In both cases the effect of a magnetic field parallel and perpendicular to the current direction has been studied. The temperature dependence of the resistivity of these samples has been presented in section 2.5.1. For these experiments two pulse shapes were used: (1) a stepped pulse in which fields are kept constant for approximately 50ms, i.e. long enough to neglect the effect of eddy currents, and (2) a smooth pulse in which the field, after having reached 35 T, decreases at a constant rate of 56T/s. Since both pulse-types offered nearly identical results, we here present only data taken with the latter pulse-type, which has the advantage that many data points can be collected during one field sweep. The relative accuracy is estimated to be 3%, whereas the absolute accuracy amounts to 7% due to the determination of the geometrical factor, S/I. The current through the samples amounted to 0.5 A. In fig. 2.36 we show the experimental results
A. de Visser et al. I UPt,, heavy fermions
123
and superconductivity 25
20
% HLC
’
‘0 4.2K 15
%:
10
3 5 aQ 0
-5
-10
Fig. 2.36. Magnetoresistance 4.2K(O), and for Hl(Zllc)
of UPt, at 4.2K(O)
for and
HII(Z[lc) at at 1.5K(A).
for ZZ]](Z]]c), at 4.2K, and for H~(z]]c), at 1.5 and 4.2 K. Since during the experiment only the relative change of the resistance with field is recorded, the absolute values, presented in fig. 2.36, have been calculated using the p( T)-values from fig. 2.17. The zero-field p-values for the c-axis equal 3.1 and 11.2 $km at 1.5 and 4.2 K, The magnetoresistance passes respectively. through a maximum at - 20 T, for a field direction in the basal plane. At B,,, a huge magnetoresistivity is observed: an increase in resistivity with field of 230% at 1.5 K and of 120% at 4.2 K. At 77 K Ap is slightly negative (not shown), and amounts to - 1.4 @cm at 35 T For a field direction (~(77 K) = 100 @km). parallel to the c-axis Ap saturates at a modest value. In fig. 2.37 we present the experimental results for H]](Zl]a), at 1.5, 4.2, 20.3 and 77K; zerofield p-values amount to 9.4, 28.2, 127 and 180 @km, respectively. At 1.5 K and 4.2 K Ap is positive and has a maximum at - 20 T. At B,,, the increase in resistivity equals 130% at 1.5 K and 70% at 4.2K. At 20.3 K and 77K Ap is negative. In fig. 2.38 we show once more the curve for ~]](~l]a) at 4.2K, but also data for HI(Z]] a), for several orientations of H in the
Fig. 2.37. Magnetoresistance peratures as indicated.
of UPt,
for Hll(ljla)
at tem-
k-plane. Clearly, B,,, shifts to higher values as the field is rotated more and more towards the c-axis in the k-plane. A simple analysis shows that the maximum in the resistivity is attained for a value of the component of the field in the basal plane equal to 20 T (B,,, cos 8 = 20 T, where 0 is the angle between field direction and basal plane). This evidences the two-dimensional nature of the field effects. Since the data in fig. 2.38 suggest that the absolute Ap-values for ZZl](Z]]a) and (~((b)l(Z]]a) are almost equal, we conclude that p(H) is mainly governed by the direction of the field with respect to the crystallographic axes; the orientation of the field with respect to the current is of minor importance. In fig. 2.39 we present the temperature dependence of the magnetoresistance in fields of 5 and 8 T for kZl](Z]]a) [Sl]. defined as The magnetoresistance is Ap(T, H) = p(T, H) - p(T, 0). In general, two contributions to the magnetoresistance exist. The first contribution is due to the ordinary magnetoresistance, which is caused by the Lorentz force on the electrons. This contribution is always positive, and varies. in general, as H’, in the low-field limit. The ordinary magnetoresist-
A. de Visser et al. I UPt,, heavy fermions
124 25
I
I
I
ence. In the case of UPt, the orientation of the field with respect to the current is of minor importance, therefore, we conclude that the ordinary magnetoresistance is small and, consequently, that Ap(H) is mainly determined by the field effect on the effective scattering processes.
r
20
T” 3 aQ
10
5
0 40
30
10
0
I&
Fig. 2.38. Magnetoresistance of UPt, at a temperature of 4.2 K for Hll(llia) (0) and for Hl(llla) with the orientation of H with respect to the b-axis given by (20 + 5)” (O), (40 2 5)” (A) and (70 + 5)” (0).
-2
j
”
0
”
“‘I
”
c
j
20
10 T
and superconductivity
At zero field the resistivity can be written as p( T, 0) = p,, + AT*, for T < 2 K. A first question that arises is whether p,,, which is determined by scattering at defects and impurities, is field dependent. From an extrapolation to zero temperature, from temperatures above 2 K, of Ap(T, H), at 5 and ST (fig. 2.39), we expect a field independent pO. Remenyi et al. [120] performed similar measurements, on a Czochralskigrown sample in the temperature range 0.53.5 K, at 10 and 20.7 T, for a field direction in the basal plane. These data are shown in a plot of p vs. T* in fig. 2.40. From this figure it follows that Ape = 0.5 p..Rcm at 10 T and Ap,, = 3.2 @cm at 20.7T. Sulpice et al. [108] performed magnetoresistivity measurements on a polycrystalline sample, in the temperature range 35-800 mK, at fields up to 7.5 T. These authors measured a linear field dependence of po: p,,(B) = p,, + aB, with a = 0.22 @cm/T. At 7.5 T these authors obtained Ape = 1.65 @cm, a value much larger than observed in the case of the Czochralskigrown samples near 7.5 T. This strong sample
' 30
(K>
Fig. 2.39. Magnetoresistance of single-crystalline UPt, for Hll(Zlla) as function of temperature in a field of S(0) and 8(O) tesla. Closed symbols are taken from fig. 2.37.
ance
is larger
for (transverse),
a field
perpendicular
to
the
than for a field parallel to the current (longitudinal magnetoresistance). The second contribution arises from the fieldeffect on the effective scattering processes. Since several scattering mechanisms might be present, this term can exhibit a complex field dependcurrent
Fig. 2.40. Magnetoresistance of UPt, plotted VS. T* at fields (directed in the basal plane) as indicated (after ref. [120]).
A. de Visser et al. I UPt,,
heavy fertnions
dependence might be due to the variation in the p,-values: 6.2 p fz cm (de Visser et al.), 1.6 FRcrn (Remenyi et al.) and 0.46 @cm (Sulpice et al.). Compared to the overall large magnetoresistance, the field effect on pI, is small and plays a minor role. From the data in fig. 2.40 it follows that the coefficient. A. of the T”-term increases as function of field. It attains values of 0.97, 1.10 and 2.25 t&cm/K’ at 0, 10 and 20.7T, respectively. The temperature region in which this term holds seems to decrease with field. The magnetoresistance strongly resembles x(H) (compare figs 2.35 and 2.37). We have analysed the Ap(W)-curves in a plot of logAp vs. logH, in order to look for a specific term H” in Ap. However, no field dependence with a fixed exponent over a large field range is observed. For ~l~(Z~~~) fz ==2 in the field region 3-10T at temperatures of 1.5 and 4.2 K (it is difficult to determine the initial field dependence due to the spread in the data at low fields). For H_L(I/]c) n = 2 between 3 and 10T at 4.2 K, but Ahp(H) varies linearly with field up to 10T at 1.5 K. Since it is difficult to distinguish the various contributions to Ap(H), we confine ourselves in the following to a qualitative analysis. We emphasize that the magnetoresistance is unusually large at low temperatures, which implies a very effective scattering mechanism up to the critical field. The temperature dependence of Ap (fig 2.39) shows that the scattering is most effective at - 7 K (at 8 T), i.e. the temperature at which the maximum in AplAT occurs. Near 14K Ap becomes negative. Above 17 K a small negative value remains. Next we discuss whether x(H) and p(H) can be interpreted in a spin-fluctuation model. As can be conjectured from the specific-heat data in field, the influence of a magnetic field on spinfluctuation systems is usually subtle. For instance, in the case of UAlz ]121] u(H) and p(H) reveal a slightly weaker field dependence above 15 T, which has been interpreted as a suppression of the spin-~uctuation phenomena. The presence of a H’-term in x(H) for UPt, is consistent with a spin-fluctuation model (BeAlMonod f39j). Unfortunately, at present a theoretical description of the steplike c(H)-
and supereonduct~v~t~
125
curve is lacking. Beil-Monod argues that if x(H) increases with field, p(H) should also increase. However, for ferromagnetic spin fluctuations one expects Ap(H) < 0, since these would be suppressed (or even quenched) by an external field. On the contrary, for antiferromagnetic spin fluctuations one expects Ap(H) > 0, due to an enhanced effective scattering mechanism caused by the instability of moments oriented antiparallel to the field. Above a certain critical field the antiferromagnetic correlations will be destroyed and a negative contribution to Ap(H) results. The positive Ap(H) for UPt, would, therefore, be in favour of an antiferromagnetic type of spin ~uctuations. It is, however, not clear to us whether this is compatible with the presence of the T” ln(T/T*)-term in the specific heat, which is ascribed to ferromagnetic spin fluctuations in the paramagnon model (in a Fermi liquid picture the only constraint is that of small momentum transfer processes). If there is any negative term in the low-temperature regime, arising from ferromagnetic spin fluctuations, it is completely immersed in the huge positive effect. At 20.3 and 77K Ap(H) is negative (fig. 2.37), with a quadratic field dependence at 20.3 K. From the increase of the coefficient of the T’-term with field (Remenyi et al. [120]) it follows that the spinfluctuation temperature (denoted by Ti) decreases, assuming A cx(T:)-“. This can be seen as an enhancement of the spin-~uctuation effects with increasing field. Several other explanations of the anomaly at 207: have been put forward, among which bandstructure effects, a type of antiferromagnetic transition and crystal-field effects. A peak in the density of states just above E, might be responsible for the maxima in x(H) and x(T) assuming that at B,,, and T,,,,, the Fermi level passes through the maximum of the peak. However, one then expects the anomaly in c/T at T,,, as well, which is apparently not the case. The a(N) and p(i-l)-curves remind one of a type of antiferromagnetic transition: the step in the magnetization might be caused by a (possibly spin-flop Iike) reorientation of magnetic moments in the basal plane, accompanied by a peak in the magnetoresistance. However, no sign of long-range
A. de Visser et al. I UPt,, heavy fermions
126
magnetic order has been observed. Therefore, the type of order should be short-range. For instance strong antiferromagnetic correlations in the basal plane below 17 K, likely associated with the spin-fluctuation phenomena. The possibility that the anomalies are due to a field induced crossing of a crystal field level and E,, induced by the external field, can be discarded, since no low energetic crystal-field levels have been observed in the specific heat. Rather similar anomalies have been observed in the magnetization and magnetoresistivity data taken on a single-crystalline sample of the tetragonal compound CeRu,Si, [122]. However, for this compound the anomaly takes place at a considerably lower field (7.5 T) and has a onedimensional character (it is only observed for a field along the tetragonal axis). A possible explanation for this phenomenon is sought in a field induced polarized phase of heavy-fermions (y = 350 mJ/ K2mol). A rather similar temperature dependence of Ap(H) has been observed in the heavy-fermion systems CeCu,Si, and CeAl, [123]. At high temperatures Ap(H) is negative, but changes sign at when the coherent state is low temperatures,
20-
0
10
30
Fig. 2.41. Magnetoresistance of UPt, for Hill and for HLZ at 4.2 K(0) and 1.5 K(O).
40
at 4.2 K(A)
and superconductivity
I
1
I
I
UPt5
Fig. 2.42. Magnetoresistance of UPt, for HII1 at 4.2K(A) and for HII at 4.2 K(0) and 1.5 K(O).
formed. It is remarkable that also in these compounds a maximum in Ap(H) is present at low temperatures: at 2 T (T < 0.5 K) for CeAl, and at 20 T (T < 1.5 K) for CeCu,Si,. This indicates the destruction of the coherent state by a field. For comparison we have measured the magnetoresistivity of polycrystalline UPt, and UPt,. Data on UPt, for H_iZ at 1.5 and 4.2 K, and for H]]Z at 4.2K, are shown in fig. 2.41. Data on UPt, for HII at 1.5 and 4.2K, and for HjlZ at 4.2 K, are shown in fig. 2.42. No anomalies have been observed, as expected from the linear magnetization curves [2]. At these low temperatures, the magnetoresistances of both compounds are positive. For UPt, the longitudinal magnetoresistance saturates, as in the case for UPt, for H )](I\] c). The transversal magnetoresistance is large: at 35 T Ap equals 15.4 and 13.2 @cm, at 1.5 and 4.2 K, respectively. For UPt, the nearly isotropic Ap is small and has a W-term with n = 1.6. For UPt, no fixed exponent is found. 2.6.3. Forced magnetostriction Forced magnetostriction measurements on single-crystalline UPt, have been formed at fields up to 8 T at a temperature of 4.3 K [124]. Hereto
A. de Visser et al. I UPt,, heavy fermions
the same sample as used for the thermal expansion measurements (section 2.3.2) was mounted in a capacitance cell, made of OFHC copper. Care was taken to keep the sample at constant temperature, since otherwise the large thermal expansion of the sample would obscure the forced magnetostriction, A’ = L-‘(dLldB). The accuracy decreases with increasing field and is limited at 1.5 x lo-’ T-’ in A’ at 8 T. The experimental set up allows a set of nine measurements: the field can be applied perpendicular (2 X ) and parallel to the three dilatation directions. As UPt, has a hexagonal structure we performed the whole set of nine measurements. Some anisotropy effects were observed in the basal plane, but were of the order of the spread in the data. Therefore, we neglect the basalplane anisotropy and concentrate on the overall behaviour. The experimental results are displayed in fig. 2.43 and fig. 2.44, for the field along the c and u-axis, respectively. Values for the basal-plane magnetostriction, AAb, were obtained after averaging AAand A;. The forced volume magnetostriction is calculated from A: = 2ALb + AS. For a field along the c-axis A’ is linear in field: positive
1 -
F 0 G a ? ,-d -J
0’
_, -
or in terms of the relative pressure dependence of the molar susceptibility
H,T
(2.29)
=
Here V is the molar volume. With the appropriate susceptibility data, x~,~ = 115 X lop9 and x, = 57 X 10m9m3/molU (fig. 2.1), we calculate
I5
l/3 Vol.
a\&% \, c
0.5
‘A ‘a \
t I 0
(2.28)
1
‘A \
-3
(gY,.,= - L-jgp,T~
\
-2
Fig. 2.43. UPt, for dilatation l/3 of the
”
0’ /--.’ &JIO___o_Oo-O-O--_olA - \*
for a dilatation in the basal plane, but negative along the c-axis (fig. 2.43). For a field applied along the a-axis deviations from linearity occur above 3 T (fig. 2.44). Note that these deviations occur at considerably lower fields than in the corresponding magnetization curves (at - 10 T, see fig. 2.33). The coefficients of these linear terms, or the u-values, writing X = AL/L = iaB*, have been listed in table 2.1X. For both field directions the c/u-ratio decreases with field. The forced magnetostriction data can be related to the hydrostatic pressure dependence of the molar magnetic moment by a thermodynamic Maxwell relation:
0’ 0’ a,b
B IJc
127
and superconductivity
i 0
2
4
8
0
l0
w06 The forced magnetostriction of single-crystalline a field direction along the c-axis at T= 4.3 K; in the basal plane (III), along the c-axis (A) and volume effect (0).
Fig. 2.44. UPt, for dilatation l/3 of the
2
4
6
8
10
The forced magnetostriction of single-crystalline a field applied in the basal plane at T = 4.3 K; in the basal plane (O), along the c-axis (A) and volume effect (0).
A. de Visser et al. I UPt,,
128 Table 2.1X Values for a (in lO~‘T~‘)
for field directions
Dilatation in basal plane Dilatation along hexagonal Volume effect
along
axis
heavy fermions
the a, b-and
High-pressure
80
I
’
I
n
I
= JaB’
at low fields
c-axis
2.00 0.57 4.57
2.02 0.62 4.66
1.61 -2.91 0.31
studies
I
A = AL/L
b-axis
2.7.1. Electrical resistivity The electrical resistivity of single-crystalline UPt, has been studied at pressures up to 5 kbar [lo, 1251. Measurements have been performed 100
assuming
Field along a-axis
from the data in table 2.1X, values for alnxlap of - 22 and - 3 Mbar~ ‘, for a field direction in the basal plane and along the hexagonal axis, respectively. Thus, for a field in the basal plane the relative pressure dependence of x (- 22 Mbar-‘) is expected to be nearly one order of magnitude larger than for a field direction along the hexagonal axis. In section 2.7.2 we shall compare the magnetostriction data with the results obtained from the pressure dependence of the susceptibility. 2.7.
c-axis,
and superconductivity
on unannealed Czochralski-grown samples, with the current along the three crystallographic axes. Zero-pressure data on these samples have been presented in section 2.5.1. The low-temperature data, plotted as p vs. T*, at zero pressure and at 4.2 kbar, are shown in fig. 2.45 for the a and b-axis, and in fig. 2.46 for the c-axis. In fig. 2.47 we have plotted the temperature derivative Api AT vs. T for the b-axis, at zero pressure and at 4.2 kbar. Applying pressure results in a shift of the temperature at which the maximum occurs towards hi her temperatures; a considerable inQ crease in a p I a T2 near the temperature at which the maximum in the susceptibility is found (17 K), can be observed as well. Above - 45 K the pressure effect for a current in the basal plane is temperature independent and amounts to - 1.4 @cm/kbar. For the c-axis the pressure 40
’
UPt3
30
0.0 a +,x b -
x
60
-
L
T c
2%
3
20
Q
Q40
l0 20
d 0 0
T2 (K2) Fig. 2.45. Electrical resistivity vs.. T2 for unannealed UPt, for the a-axis at 1 bar(O) and at 4.2 kbar(Ll), and for the b-axis at 1 bar(+) and at 4.2 kbar( X).
20
40
60
80
100
T2 (K’) Fig. 2.46. Electrical resistivity vs. T’ for unannealed for the c-axis at 1 bar(O) and at 4.2 kbar(O).
UPt
A. de Visser et al. I UPt,,
1c
heavy fermions
1
L
2
e C
1
I
TIKI I
I
I
20 30 0 10 Fig. 2.47. Temperature derivative of the electrical resistivity of unannealed UPt, for a current direction along the b-axis at 1 bar (0) and at 4.2 kbar( x).
effects are less pronounced; at room temperature a pressure independent resistivity results. In the low-temperature limit (T < 2 K) the resistivity of UPt, can be expressed as p = p0 + AT’. As can be seen in fig. 2.45 and fig. 2.46 pO is pressure independent. The coefficient A rapidly decreases as function of pressure. By estimating A from the initial slope of the data, plotted as p vs. T2, we observe a decrease of this coefficient of about 40% over 4.2 kbar, for both current directions. At 1 bar the temperature derivative of the resistivity has a maximum at a temperature, T;, of 6.5 K and 7.5 K, for the basal plane and the hexagonal axis, respectively. Under pressure this temperature increases at a rate of 0.35 + 0.1 K/kbar (data for the b-axis are shown in fig. 2.47). From the increase of Tz and the decrease of A, as function of pressure, we conclude that the spin-fluctuation phenomena are strongly suppressed. Due to a lack of the theoretical understanding of the low-temperature resistivity, there is no proper definition of the characteristic spin-fluctuation temperature. As
and superconductivity
129
mentioned before, however, T; , assuming and Tz can both be taken to be A 0~(T;)-2, proportional to the spin-fluctuation temperature. As a check of this assumption we can compare the relative pressure dependences of Tz and T:, assuming that the proportionality constants are pressure independent. From the data in fig. 2.45 and fig. 2.47, we calculate alnT:/ap = 54 + alnAlap=-95+35Mbar-’ 16 Mbar-‘, and that hence aln T;/ap = 48 ? 18 Mbaf ’ (note throughout this work we approximate the relative pressure dependence of a quantity Q, alnQ/ ap, by (Q,,))’ AQlAp, where Q, represents the 1 bar value of Q). The agreement between these deduced values is good, however, the uncertainty, in particular in the value of A as function of pressure, is large. of Resistivity measurements as function pressure performed by others [lOl, 103, 1261 yield qualitatively the same information as our results. Wire et al. [loll performed resistivity measurements on flux-grown crystals at pressures up to 18 kbar, in the temperature range l-300 K. These authors analysed the data in a so-called “parallel resistor model”, assuming R ’= Rim’ + R,‘, where R is the measured resistance, Ri is the ideal resistance of the specimen, and R, is a shunt resistance, whose magnitude is near the saturation value, i.e. - R(300 K). Applying this analysis, the authors conclude that Ri follows a T2-law up to - 8 K and - 12 K at 1 bar and at 17 kbar, respectively. From the coefficient of the T2-law, as function of pressure, values were deduced for aln T: lap of 36 and 30 Mbar- ‘, for the basal plane and c-axis, respectively. Willis et al. [103] performed resistivity measurements (Illc-axis) at pressures up to 19.1 kbar, in the temperature range 0.1-2 K, on a flux-grown specimen (see fig. 3.10). From the variation of the coefficient A with pressure, these authors derived a value for aln TT,lap of 25 Mbar - ‘. Ponchet et al. [126] have measured the resistivity of an UPt, whisker (Zllc-axis) in a diamond anvil cell, at pressures up to 202 kbar. Their results, normalized at room temperature at a zeropressure value of 132 @cm, are shown in fig. 2.48. Apparently, the resistivity curve gradually loses its anomalous low-temperature behaviour;
130
A. de Visser et al. I UPt,, heavy fermions ,507
5:
l:O
150
2j)O
2TO
“7’
i
E z
;
I 8
d 1
Fig. 2.48. Electrical resistivity of an unannealed UPt, ker at pressures up to 202 kbar (after ref. [126]).
whis-
a smooth variation with temperature is found at 202 kbar, but ~(300 K) remains large. The pOvalue is essentially constant below 40 kbar, but then increases from 1.9 to 6 @cm between 40 and 200 kbar. At ambient pressure A amounts to Under pressure A decreases 0.52 I.Dcm/K2. rapidly by one order of magnitude between 0 then more slowly to 8.8 X and 50 kbar, lop3 I&cm/K’ at 202 kbar. From these data a value for alnT;/ap of 30Mbar-’ is deduced. Averaging the values for aln Ti/dp as obtained by the various authors, we deduce a mean value of 33 Mbar-‘. Below we compare this value with the pressure dependence of the characteristic temperature, T,,,,, , as derived from the susceptibility.
and superconductivity
UPt, samples at pressures up to 4.5 kbar, in the temperature range 4.2-40K [103, 127, 1281. The magnetization in fields up to 6.5 T, for polycrystalline UPt, (cylinder, diameter = 6 mm, length = 8 mm), at a temperature of 4.2 K and at pressures of 1 bar and 4.5 kbar, is shown in fig. 2.49. The susceptibility measured in an applied field of 5.3 T is shown in fig. 2.50. The relative pressure dependence of the susceptibility, &xl ap, amounts to a value of - 24 Mbar-’ at 4.2 K. In order to measure the magnetization of singlecrystalline UPt,, two cylindrical samples (diameter = 6 mm, length = 6 mm) have been spark-eroded along the b and c-axis, out of large Czochralski-grown specimens. The magnetization curves for these two samples, at a temperature of 4.2 K and at pressures of 1 bar and 4.6 kbar, are shown in fig. 2.51. The corresponding susceptibility data, measured in a field of 5 T, are shown in figs 2.52 and 2.53. From the data in fig. 2.51 we calculate values from &rx/ap at 4.2K of -28+3 and -5 *3Mbar~‘, for the basal plane and hexagonal axis, respectively. Apparently, crystallities with preferred orientations are present in the polycrystalline sample, as was
4.2 K
0.4
3 i
c.7.
0.3
E
4
1 bar
b 0.2
4.5 kbar
ov 2.7.2. Magnetic susceptibility Magnetic susceptibility measurements have been performed on poly- and single-crystalline
0
’
I
I
1
I
/LJ-l
Fig. 2.49. Magnetization pressures as indicated.
I
6
2
I
I a
R
of polycrystalline
UPt,
at 4.2 K at
131
A. de Vzker et af. I UPt,, heavy ferrni~~~ and ~~~~r&onductivi~ 1257
,
I
I
I
17.6 K
I
5.3 T
'\
\
// 3
\
I/
11s
19.6 K ii
I/
D D 4.6 kbar
“I,,
,
',lbar
\
,'
1
'0
\
,,i,.
I I
1;
- b-axis
m3
i , fl-.
I'
$05-:
“I,
'\,
{f
\
I
\ \ \
I' I i / I
\ 4.5 kbar \
I 95-
'
r
-: m 0
I
I
I
I
-r
10
20
30
40
50
T (K) Fig. 2.50. Susceptibiii~y of polycrysta~li~e UPt, in a field of 5.3 T at pressures as indicated.
upt,
Fig. 2.52. Susceptibility of single-crystalline LJPt, in a field of 5 T directed along the b-axis at 1 bar(O) and 4.6 kbar(D), and directed along the c-axis at 1 bar(A) and 4.6 kbar(V).
b-axis 0
1 bar
CI 4.6 kbor 0 0
0
4
0.2 i
D
D
0
1
:
1
n
c-axis
t
+
A
0.1
1 bar
v 4.6 kbar
:: * 0.0
f
0
’ - ’ 1
2
L
’
I
’
I
’
’
t
’
’
’
345678
Fig. 2.51. Magnetization of single-crystalline UPt, at 4.2 K for field directions and pressures as indicated.
Fig. 2.53. Susceptibility of single-crystalline UPt, in a field of 5 T directed along the b-axis at 1 bar(O) and at 4.6 kbar(C!).
132
A. de Visser et al. 1 UPt,, heavy fermions
concluded previously from the thermal expansion data on this sample. Therefore, we shall, in analysing the results, restrict ourselves to the data for the single-crystalline samples. At applying a pressure of 4.6 kbar, the susceptibility for the basal plane strongly reduces, whereas the susceptibility for the c-axis hardly shows any pressure dependence. The rates of suppression of dlnxlap, at 4.2K, -28 ?3 and - 5 2 3 Mbarr’, for the basal plane and c-axis, respectively, are in good agreement with the values of - 22 and - 3 Mbaf ‘, for the basal plane and c-axis, respectively, that we predicted from the forced magnetostriction measurements (see section 2.6.3). When comparing the highpressure susceptibility data with the forced volume magnetostriction results we have to keep in mind that volume changes in the magnetostriction experiments in fields up to 8 T are at least two orders of magnitude smaller than the volume changes induced by a pressure of, say, 5 kbar (the compressibility of UPt, amounts to 0.48Mbar~‘). The nearly identical values for alnxlap deduced from these experiments indicate a linear pressure dependence of x up to, at least, 4.6 kbar. The maximum in the susceptibility curve along the b-axis is found at 17.9 K at 1 bar, and shifts under applied pressures of 4.6 kbar with 2.5 * 0.5 K to higher temperatures: ~lnT,,,/~p = 30 ? 6 Mbar-‘. Since the values for the relative pressure dependence of x(4.2 K) in the basal plane and of T,,, are nearly equal and opposite, we conclude that the product x. T,,, is nearly pressure independent, at low temperatures. The value for alnT,,,/ap is in good agreement with the mean value for alnTT,/ap as derived from the resistivity measurements. For the polycrystalline sample a value for alnT,,,/ap of 25 Mbar-' results. The relative pressure dependences of x(4.2 K) can be transformed to Griineisen and T,,, parameters, by using the compressibility K = TX = alnx(4.2 K)/alnV= 58 2 13 0.48 Mbar-‘: These = -~lnT,,,i~lnV= 63 + 6. and r, Griineise~aparameters are almost equal to the one derived from the enhanced linear term in the specific heat and thermal expansion, r: = 68 at
and superconductivity
1 K (section 2.3.2). The latter value is in good directly agreement with r: = 55, as obtained from the pressure dependence of the enhanced linear term in the specific heat [80]. Although no clear-cut definition of the characteristic temperature for the low-temperature anomalies in resistivity and susceptibility can be presented, we have shown that the pressure dependences of Tz, T; and T,,, all range between 25 and 54 Mbar~ ‘. Hence, the positive value for the pressure dependence of the characteristic temperature is firmly established; the corresponding Griineisen parameter is large. From these high-pressure data we conclude that by applying pressure the spin-fluctuation phenomena are strongly suppressed.
3. UPt,: the superconducting 3.1
state
Introduction
In this section we deal with the superconducting ground state of UP&. Superconductivity in UPt, has been discovered by Stewart et al. [3]. These authors performed resistivity, ac-susceptibility and specific heat experiments on fluxgrown single-crystalline specimens and found a drop in the resistivity, a diamagnetic signal in x,, and a discontinuity in the specific heat, with onset temperatures of 0.54 K, 0.53 K and 0.54 K, respectively. This was taken as evidence for bulk superconductivity below 0.54 K. As mentioned in section 1.3, the occurrence of superconductivity in some of the heavy-fermion systems was quite unexpected. Furthermore, unusual observations concerning the superconducting state have given rise to speculations upon an unconventional pairing mechanism. Especially, the coexistence of superconductivity and strong spinfluctuation phenomena in UPt,, made this compound a likely candidate for triplet superconductivity. In. our case, measurements concerning the superconducting state have been performed in a close cooperation with colleagues in Leiden (de Visser et al. [18, 761 and Palstra et al. [129]). Superconductivity was first observed on three
A. de Visser et al. I UPt,, heavy fermions
annealed Czochralski-grown samples in electrical resistivity measurements, along different crystallographic directions (see table 3.1: sample #l, #2 and #3). From previous experiments, it was evident that the resistivity only follows a T2-law at approaching the low-temperature limit of our 4He cryostat, 1.3 K (section 25.1). In a subsequent investigation of this term in a “He cryostat, down to 0.30 K, we found a drop to zero resistivity for all three samples between 0.47 and 0.53 K. A second indication of superconductivity was obtained by ac-susceptibility measurements (f = 10.9 Hz) on two unannealed samples (#4 and #5): large diamagnetic signals were observed below 0.40 and 0.43 K, respectively. The annealed samples #6 and #7 (table 3.1) showed a superconducting transition at 0.48 K, consistent with the values that were derived from the resistivity measurements on samples #l, #2 and #3. Apparently, annealing of the samples increases T,. Evidence for bulk superconductivity was derived from a specific heat experiment on an unannealed Czochralski-grown single-crystalline sample (#8), for which the onset of the superconductivity occurred at 0.43 K, and the maximum in the specific heat discontinuity at 0.26 K (see fig. 3.1). The value of AciyT at 0.26K amounted to 0.35, i.e. one quarter of the value expected from BCS theory: 1.43. Two other
133
and superconductivity
observations support the evidence for bulk superconductivity: (i) annealed polycrystalline samples of the neighbouring phases UPt, and UPt,, that possibly could be present in spurious amounts, showed no traces of superconductivity down to 0.30 K, and (ii) a single-crystalline sample (#6), an annealed polycrystalline sample (#9) and an annealed powdered sample (#lo) all gave a diamagnetic signal of the same magnitude as a Cd reference sample (T, = 0.52 K), after the necessary corrections for demagnetizing factors.
0.3 tt
$ 13 /mole K’l T IKI
0
1.0
0.5
I
1.5
Fig 3.1. Specific heat of unannealed Czochralski-grown UPt, below 1.3 K (sample #8). The onset of superconductivity is at 0.43 K. The broken line serves as a guide to the eye.
Table 3.1 Superconducting transition temperature, T,, and width of the superconducting transition, AT,, for Czochralski-grown UPt, (all samples #l till #8 out of one single-crystalline batch) and for arc-melted polycrystalline UPt, (samples #9 and #lo). T, and AT,-values, as determined from the ac-susceptibility experiments, are defined at the 50% point and between the 10 and 90% points, respectively
AT, Sample #l #2 #3 #4 #5 #6 #7 #8 #9 #lO
annealed annealed annealed unannealed unannealed annealed annealed unannealed annealed (arc-melted) annealed (powdered)
(K)
PO (10~‘Rm)
Experiment
6.2 3.0 1.7
pdL(a-axis) pdc(b-axis) pdc(c-axis)
0.47-0.53 0.47-0.53 0.47-0.53 0.40 0.43 0.48 0.48 0.43 0.50
0.05 0.05 0.05 0.10 0.10
-
0.45
>O.lO
-
X8, X,‘ X,,(a-axis) X,,(c-axis) specific heat X,,
xx
A. de Visser et al. 1 UPt,, heavy fermions
134
A hard-proof
to demonstrate the in UPt,, i.e. the Meissner effect, is presented in the following section. Thereafter we present, successively, specific heat and critical-field studies that enable us to determine the parameters that characterize the superconducting state. Subsequently, we apply the expression for T,, derived in the Fermi liquid theory, as outlined in section 1.3.3, to UPt,. To conclude this section, we discuss several other techniques that have been applied in order to elucidate the nature of the superconducting state.
occurrence
3.2.
experiment
of bulk superconductivity
The Meissner effect
Superconductivity in UPt, was deduced from resistivity, ac-susceptibility and specific heat experiments. However, from the first two experiments, the occurrence of superconductivity possibly caused by the presence of filamentary exclusions of superconducting non-stoichiometric UPt, can not be excluded. Furthermore, the rather small value for the specific heat discontinuity AclyT =0.35 at 0.26 K (compared to a value of 1.43 predicted by the BCS theory) arose the question whether the superconductivity in UPt, should be considered as a bulk property. A direct measure for the superconducting volume fraction of the sample can be obtained by the so-called Meissner effect. Hereto, the amount of flux expelled by a specimen, when it is cooled through the superconducting transition temperature in a small magnetic field, is measured. Meissner experiments [129] were performed on an annealed arc-melted polycrystalline sample (cylinder, diameter = 4.8 mm, length = 4.5 mm). Ultrapure Cd (T, = 0.52 K) and Sn (T, = 3.72 K) cylinders of the same dimensions served as reference samples. The flux expulsion of the sample was measured by means of a pick-up coil. It is useful to compare these results with dc-susceptibility measurement. Such experiments were performed by placing the sample in a (secondary) pick-up coil and recording the induced voltage while slowly ramping the magnetic field (produced by a primary coil), at a typical rate of 0.2 mT/s. The magnetization can be obtained by
and superconductivity
numerical or analog integration. Since, before ramping the field, the sample is cooled in zerofield, so-called virgin magnetization curves can be recorded. In addition, the magnetization was measured directly by means of a flux transformer method. For the Meissner effect, as well as for the magnetization experiments, the signals are calibrated by those of the Cd and Sn reference samples. The two methods give a lower and an upper limit for the superconducting volume fraction, as we shall discuss below. In fig. 3.2 we show a typical trace of the dc-susceptibility of the UPt, cylinder versus magnetic field. The data in this figure suggest a superconducting volume fraction of 80% at 353 mK. However, in this experiment it is impossible to discern normal regions embedded in superconducting regions. Therefore, this method will set an upper limit of the superconducting volume fraction. Fig. 3.2 also shows that the virgin and non-virgin curves largely differ. The magnetization curve at 353 mK, obtained with the flux transformer method, is shown in fig. 3.3 (actually we plotted -~($4). The lower critical field B,, is represented by the field at which the virgin magnetization curve has a maximum. This maximum is somewhat rounded, probably because of demagnetizing effects. The value for B,, of 2.2 mT perfectly agrees with the value of 2.2 mT (taken at ,ydc = 0) derived from fig. 3.2. In addition, we measured an annealed powdered UPt, sample. The superconducting volume frac-
upta
Fig. 3.2. DC-susceptibility (normalized with respect to Cd and Sn) vs. magnetic field for UPt, at 353 mK (TIT, = 0.73). The lower curve is the virgin curve. For decreasing field the curve is a mirror image with respect to the ordinate.
A. de Visser et al. I UPt,, heavy fermions 1 T,352
Fig. 3.3. Virgin magnetization of UPt, at 352mK. A minor x,, at 13 mT is also included.
mK
curve and magnetization loop hysteresis loop determined by
tion amounted to 50% at 357 mK, somewhat lower than observed for the cylinder. This difference can be ascribed to a different transition temperature, 7’,(bulk) = 0.50 K, T,(powder) = 0.45 K, and a different transition width, AT,(bulk) = 0.10 K and AY’,(powder) > 0.10 K. In fig. 3.4 the results of the Meissner experiments are presented, together with the virgin magnetization curve. The curves denoted by “field cooling” were drawn through the data
332
mK
135
and superconductivity
points obtained by measuring the flux expulsion, at constant magnetic fields, while lowering the temperature. The initial slopes of these curves at H = 0 yield the superconducting volume fractions, which amount up to 30%. The superconducting volume fractions, as obtained from both methods, are plotted in fig. 3.5 as function of temperature. The differences in results from both methods, can mainly be attributed to flux-pinning in the “field cooled” case. It is well-known that the superconducting transition propagates more quickly along the surface than into the bulk [51]. Furthermore, one can calculate that any superconducting surface in a small magnetic field forms an energy barrier impeding the passage of flux lines. Thus, when cooling the sample below T,, initially the surface becomes superconducting, forming this energy barrier. When the superconducting transition propagates farther into the bulk, the flux lines must be expelled from the inner part of the sample. However, if the pinning forces are large, the flux lines will be trapped at the surface barrier. Therefore, although the sample may be a good superconductor, it is well possible that very little flux is expelled. These pinning forces are especially large when the surface is oriented parallel to the magnetic field. The pinning forces can be decreased by more than two orders of magnitude when the surface is not parallel to the magnetic
ZERO-FIELD
u
PoH
(mT)
Y COOLED
Fig. 3.4. Magnetization vs. magnetic field of UPt, obtained by the two methods. Zero-field cooling denotes the virgin magnetization curve. The curves denoted by “field cooling” were drawn through the data points (not shown), obtained by measuring the flux expulsion at constant magnetic field as a function of temperature. The dashed line represents the full Meissner effect (M = H) and the initial slope of the 332 mK curve.
0.2
0.4
0.6
T(K) Fig. 3.5. Superconducting volume fraction of UPt, vs. temperature as obtained by the field cooling (x) and the zerofield cooling (0) methods.
A. de Visser et al. I VPt,, heavy fermions and superconductivity
136
field. This explains why powdered samples will always show a more complete expulsion of flux than cylinders oriented parallel to the field, as, for instance, was observed for CeCu,Si, [130]. Besides surface pinning, bulk pinning will also occur: the flux lines are trapped at defects in the crystal lattice. The hysteresis of the magnetization loop (fig. 3.3) shows the result of these enormous pinning effects. Nevertheless, the expelled flux will still set a lower limit of the superconducting volume fraction. At measuring the virgin magnetization curves, one avoids the problem of flux pinning. However, as argued above, this method will set an upper limit of the superconducting volume fraction. We believe that for UPt, it gives a more reliable estimate, because of the enormous flux pinning effects. We conclude from these Meissner effect and dc-susceptibility studies that superconductivity in UPt, is a bulk property. 3.3.
Specific
heat
The specific heat data taken on the unannealed Czochralski-grown sample reveal a broad anomaly below 0.43 K (see fig. 3.1). Specific heat experiments on a polycrystalline sample, in the temperature range 0.17-1.1 K, have been performed in Darmstadt [86], with an improved accuracy. An experimental difficulty, at performing specific heat experiments on UPt, employing an adiabatic method, is the self-heating of the sample due to radioactive U-decay. This limited the minimum sample temperature in the used set-up to 170mK. The specific heat of the annealed polycrystalline sample is shown in fig. 3.6 as cl T versus T, in different fields up to 8 T (the normal-state specific-heat of this sample is shown in fig. 2.3). In this experiment the broad anomaly, as found for the unannealed sample, is reproduced at zero field; Tz”“’ = 0.52 K. The height of the peak at 0.35 K is about 30% of the specific heat extrapolated from above T,. Similar values for AclyT of 0.35, 0.48 and 0.66 have been observed for the unannealed Czochralski-grown sample (fig. 3.1), an annealed flux-grown sample [3] and an annealed polycrystalline sample [108],
.-.......L.........- . . . . . .
-
1 ... .. . . . . . . ... .......lL..--1 0.5T
0
____c__
_'k........................ . .._ _OT
/
/ ! _/ / OO
UPi,
I 0.5
,
I 1 T(K)
Fig. 3.6. The specific heat of polycrystalline UPt, at temperatures below 1.1 K and in fields up to 8 T. The dashed line represents the specific heat according to eq. (2.2). with coefficients derived from a fit in the temperature interval 1.2-20 K (table 2.1). Schematic extrapolation of the zerofield data by the dash-dotted line satisfies entropy balance.
respectively. These values are much lower than the one expected from BCS theory: 1.43. The relative jump height is increased to about 60% if the broad transition is replaced in the usual way by a sharp one. This procedure yields a calorimetric transition temperature T, = 0.40 K, in contrast to TEnset = 0.52 K. Well below T,, c/T is almost linear in T, extrapolating to a finite y-value of about 260mJ/K2molU at T = 0 K. Such an extrapolation, however, violates the entropy balance, imposed by the extrapolation of the normal state specific heat to OK (dashed line). Therefore, the low temperature specific heat in the superconducting state should decrease faster than given by the solid line. In a schematic way this is indicated by the dashdotted line for which the entropy balance is met exactly. In a magnetic field of 0.5 T the anomaly is much weaker. At higher fields T, falls outside the covered temperature range. The field effect on the normal-state specific heat has been discussed in section 2.3.1. The large electronic normal-state specific heat is caused by the strongly interacting f-electrons. The large anomaly at the superconducting transition evidences that the f-electrons are also invol-
A. de Visser et al.
ved in the formation of the superconducting condensate. However, the extrapolation of c/T to zero temperature, taking into account entropy balance (dash-dotted line in fig. 3.6), suggests a small finite y-value of about 20 mJ/K’molU, instead of a vanishing one, as expected for a normal superconductor. A similar type of specific heat curve has been obtained by Sulpice et al. [108]. These authors measured a polycrystalline sample down to 146 mK, at which temperature c/T = 293 mJ/K’molU, i.e. a value well below the normal-state c/T-value just above T,. For this sample the specific heat varied as c/T = “/o+ aOT, with ‘yo= llOmJ/K*molU and (Y”= 1250 mJ/K3molU, over the temperature range 146-470 mK. The extrapolation to a finite yvalue of llOmJ/K*molU leads, however, to a small excess of entropy (5%) with respect to the normal phase. Therefore, the actual “residual” y-value might be somewhat lower. The peculiar features observed in the specific heat might be indicative for a non-conventional pairing mechanism in the superconducting state. In particular, the observation of a power law rather than an exponential temperature dependence of c, for T 4 T,, can be taken as evidence for a nonconventional pairing mechanism (table l.IV). Taking into account the entropy balance for the broad superconducting transition a finite y-value results, which suggests that parts of the Fermi surface remain normal. It is interesting to note that also in CeCu,Si, a nearly T2-dependence of c(T) is found, just below T, [131]. At lower temperatures this behaviour gradually changes towards a T3-law. Further comparison with the latter compound suggests that UPt, is in between those samples with pronounced specific heat jumps and other ones (“gapless superconductors”), with significant jumps, but with giant linear terms in c as T goes to zero [132]. In case of UBe,, a T3-law has been observed in the superconducting state. The presence of this term was taken as evidence for triplet superconductivity [ 1331. 3.4.
Critical field studies
The
temperature
dependence
of
the
137
i UPt,, heavy fermions and superconductivity
upper
critical field, B& = - dB,,ld TI =_=,, has been studied in applied magnetic fields up to 0.5 T by means of ac-susceptibility experiments [18, 761. The results, taken on an annealed Czochralskigrown sample (#6, table 3.1), for a field along the a-axis, are shown in fig. 3.7. For this sample BL2 amounts to 4.4T/K. In order to give a quantitative analysis of the upper critical field we follow the approach that successfully has been applied to Al5 superconductors [134] and to CeCu,Si, [135]. Hereto, some crude assumption will be made: (i) the Fermi surface, characterized by a Fermi wave ignoring any possible number k,, is spherical, anisotropy, and (ii) UPt, is not a strong-coupling superconductor. Under these assumptions the temperature derivative of the upper critical field can be written as: (3.1)
0.E 0.:
0.1
0.2
0.3
’
’
‘\
B,,ITeslal
0.1
0.L 0.5
I
,
I
0.7 C 3
I
i
UP+
0.:
a-axis
0.; 0.'
0.6
i
(b)
-
T[Kl
\
I
(a
L 0
I
0.1
I
0.2
I
I
I
I
I
03
OL
0.5
06
0.7
C1.: B
Fig. 3.7. (a) AC-susceptibility (in arbitrary units) in zero magnetic field and in fields of 0.1, 0.2, 0.3, 0.4 and 0.5T (field directed along the u-axis) for the sample #6; and (b) the upper critical field as function of temperature; Yf, is defined at the 50% point (indicated by arrows in (a)).
138
A. de Visser et al. I UPt,, heavy fermions
where y is given in J/K2m3, k, in m-l and the normal state resistivity at T,, approximated by the residual resistivity, pO, in LRm. Here the quantities a and b are given by 7.95 x 1O32Tm2K2J-2 and 4780 TKm2fi~‘JJ’, respectively. Eq. (3.1) h as its origin in an expression for B,, taken from the Ginzburg-Landau theory: B,, 0~t-‘(T). Here t(T) represents the GL coherence length, which signifies the characteristic length for variation of the order parameter. The GL coherence length depends on the purity of the superconductor, which can be presented by the ratio Z,/&,, where 1, is the electronic mean free path and &, the BCS coherence length. In the pure limit (I, + &,) p0 is small and Bf2 will be mainly determined by the first term on the right hand side of eq. (3.1). On the other hand, in the dirty limit (I, e 5,) p0 is large (T, is low) and Bs2 is approximately given by the second term. Eq. (3.1) enables us to determine the Fermi wave number directly from measurable quantities. Subsequently, we are able to calculate some parameters that characterize the superconducting state: the BCS coherence length, 5” - uF/Tc kgl(yT,), where uF is the Fermi velocity, the electronic mean free path, 1, - (p”kg)-‘, and the London penetration depth h, - (~7) lk:. Inserting the experimental results (sample #6) for Bi2 ( = 4.4 T/K), T,( = 0.48 K), the residual resistivity p0 (although not measured on sample #6, we may estimate it at - 3 x lo-’ am) and the (Sam le inde endent) enhanced y-value of 9.94 x 10!?J/K2m P (which corresponds to 422 mJ/ K2molU), we obtain a value for k, of 1.06 X 101Om-l. This Fermi wave number characterizes the strongly interacting electrons that form the Fermi liquid at low temperatures. With this value for k, we calculated (see section 2.3.1): the effective mass m* = 200m,, the effective Fermi velocity u$ = 6800 m/s and the Fermi temperature T, = 275 K. The dirty and clean limit contributions to eq. (3.1) amount to 1.4T/K and 3.0T/K, respectively. We realize that p. has not been measured on sample #6. However, allowing, for instance, a 100% variation in po, k, varies in between the values 0.96 X 10” rn-’ (for po=O) and 1.25 x 101om-’ (for p. =6X lop8 am). Taking into account the possibility of
and superconductivity
such a variation in k,, our Fermi liquid analysis yields, however, qualitatively the same results. Parameters for the superconducting state can now easily be calculated using the expressions given by Orlando et al. [134]: 4, = 2.0 x lo-’ m, 1,=3.6x 10d8m and &=3.6x 10-7m. The ratio 1,/t, amounts to 1.8, which signifies that our samples are neither in the dirty limit, nor in the clean limit, as could be expected from the p,-values. The Ginzburg-Landau parameter, ~~~ is found to be about 23. A similar value for ~~~ can be deduced from the ratio of the upper to the lower critical field: B,,IB,, 2: 2K~L/lnK,,. The temperature dependence of the lower critical field as obtained from the virgin magnetization curves, taken on the polycrystalline sample, (section 3.2) is shown in fig. 3.8. With B,, = 0.35 T and B,, = 0.0017 T at 0.4 K, we calculate ~~~ = 17. In table 3.11 we compare these parameters for UPt, with those for CeCu,Si, and UBe,,. Herewith we have to realize that some of these parameters can exhibit a strong sample in particular in the case of dependence, CeCu,Si,. Critical field studies performed by others are in good agreement with our results [102, 105, 108, 137-1391. The observed values for Bs2 range from -4.0 to - 6.3 T/K. The limiting critical field at zero temperature, Bc2(0), amounts to - 1.8T. In general, two different mechanisms
0.6 1-1
u Pt3
4
t
T(K)
Fig. 3.8. Temperature dependence of the critical fields B,, for a polycrystalline sample, and B,, for a single-crystalline sample along the a-axis (see also fig. 3.7).
A. de Visser et al. I UPt,, heavy fermions
139
and superconductivity
Table 3.11 Some important parameters characterizing the heavy-fermion superconductors references therein, sample No. 7) and UBe,, [136]. For the latter compound determine the parameters in the superconducting state
UPt,, CeCu,Si, (Rauchschwalbe et al. [135], and a dirty limit expression has been applied in order to
Parameter
Units
upt,
CeCu,Si,
UBe,,
Y T, B:2 PO kr m* u:
mJ/K’mol K T/K lo-Ylm 10” mm’
422 0.48 4.4 3.0 1.06 180 6800
1006 0.64 5.8 3.5 1.7 220 8700
-1100 0.85 42.0 125 0.87 # 300 3390
2
K lo-’
m
2.0 275
582 1.9
112 1.42
L fk %L
lOen m 1Om8m -
3.6 36 23
1.2 20 22
0.13
# Determined
m, m/s
from
configurational
arguments,
assuming
3 heavy
lead to the disappearance of superconductivity: the interaction of the external field with the electronic orbits and the interaction with the electronic spin. The first interaction gives rise to the definition of an orbital critical field, which is given by [140]: Bz2(0)= 0.7B;,T,.
(3.2)
The second interaction leads to a destruction of the superconducting state when the free spin energy becomes larger than the condensation energy (paramagnetic or Pauli limiting). Applying eq. (3.2) we calculate values for B:*(O) ranging between 1.4 and 2.2T/K, compared to the measured value of 1.8 T/K. This suggests that Bc2(0) is mainly determined by the orbital critical field; Pauli limiting is absent. The absence of Pauli limiting is expected for an equal spin-pairing superconducting state in a p-wave model. To conclude this section we briefly discuss anisotropy effects in the upper critical field. Anisotropy of the upper critical field is expected for some types of odd-parity superconductivity [58, 611. However, anisotropy might also be observed in case of a singlet superconductor with an anisotropic gap. Our data for a field direction along the u-axis reveal a linear temperature dependence of B,,.Another sample (#7) measured
electrons
per formula
unit UBe,,.
with the field applied along the c-axis, seemed to have an increasing slope of the upper critical field with decreasing temperature: Bfz= 3.6 T/K at T, and Bs2= 6.0T/K near 0.40 K. However, this sample exhibited a rather broad transition ( - 0.1 K). Similar anisotropy effects have also been observed by other authors. Willis et al. [137] and Chen et al. [102] measured the resistive superconducting transition for a current along the c-axis. For HIZ these authors found a linear temperature dependence of B,, over a relatively large temperature range, whereas for HI] Z the critical field curves showed a curvature near T,,leading to a considerably lower value for BL2.Similar anisotropy effects have been observed by Rauchschwalbe et al. [138]: for ZZ]](Z]]c) the initial value for Bi2is larger than for ZZ]](Z]]basal plane). Tholence et al. [139] performed ac-susceptibility measurements, and observed for a field direction along the u-axis, as well as for a field direction along the c-axis, non-linearities in B,,vs.T.Furthermore, Rauchschwalbe et al. observed a kink in the B,,-curve near T,,which is possibly attributed to a fieldinduced reorientation of an anisotropic pairing state, as suggested by Scharnberg and Klemm [58]. Up to now, none of the observed anisotropies in B,, has been taken as decisive evidence for triplet superconductivity.
140
3.5.
A. de Visser et al. I UPt,, heavy fermions
Fermi Liquid analysis
In this section we present the results of a Fermi liquid analysis of the superconducting transition temperature, employing the model developed by Pethick et al. 1701 described in section 1.3.3. In this model the superconducting transition temperature can be expressed by the relationship (eq. (1.37)): T(j) C = I 13oT F elis,
.
(3.3)
Here (Yis a numerical constant, TF is the Fermi temperature and gj represents a pseudopotential, given by eq. (1.40). If g; CO, the potential is attractive; j = 1 indicates triplet pairing, j = 0 singlet pairing. The analysis of the normal-state specific heat (section 2.3.1) enables us to calculate g, , which can be expressed as function of the antisymmetric I = 0 Landau scattering amplitude:
=
6
performed by considering its pressure derivative, and, subsequently, by comparing the calculated relative pressure derivative of T,, aln TJap, with the experimental value. Taking the relative pressure derivative of eq. (3.3) we find: calnT
3P
_
.
(3.4)
Using the values for Ai, ranging from -3.71 to - 2.95, derived from the various fits to the as measured and the phonon corrected specific heat (table 2.1), we calculate values for g, ranging from - 0,452 to - 0.325. Clearly, g, is negative. Next we need to investigate whether g, is the most negative pseudopotential. Employing the expressions for go and g, (eq. (1.40)), the necessary condition for g, < g, is given by Ai < - $ 2 As. For galilean-invariant systems Fi is related to the effective mass, m*. For a large effective mass Fi is large, and hence Ai amounts to 3 (eq. (1.24)). This would imply that for AZ< - 1115 triplet pairing occurs. However, in case of UPt, the lattice breaks up the galilean invariance of the electrons, which hampers the derivation of Ai from the effective mass. Nevertheless, estimating A”, to be of the order of 3 the Fermi liquid analysis favours triplet superconductivity (see also the discussion in the paper by Pethick and Pines [84]). Wnfortunately, it is within the Fermi liquid model not possible to predict T,, since cy is not known. However, a test of eq. (3.3) can be
6
8A;
(1 + A;)2
6p
a’n7-F 8P
(3.5)
’
where we substituted eq. (3.4) and assumed Q to be pressure independent. In the uniform scaling approximation ( y * TF 0: n) the relative pressure dependence of TF can be written as &?lnT,
iJlny *
-=K--
dP
dP
(3.6)
’
where K is the isothermal compressibility. The relative pressure dependence of T, then takes the form aln T,
1+ A; g1
and superconductivity
3P
alny”’ =K-;Ip-
6 (1 + A;)2
aA;? z
’
(3.7)
Hence, it appears that alnT,/ap can be calculated from parameters that follow directly from specific heat data as function of pressure. Such experiments have been performed by Brodale et al. [SO], at pressures up to 8.9 kbar. These authors observed a decrease of the coefficients y* and 6 (and thus an increase of A;) as function of pressure. From frts to the specific heat data in the temperature range 0.5-18 K at 1 bar and 3.8 kbar, we calculate K - alny” lap = 31 Mbar-[ and aA;/dp = 109 Mbar-‘. With the 1 bar value for A: of - 4.09, calculated from the data of Brodale et al., we derive atn T,/ 8p = - 37 Mbar-‘. Willis et al. [103] measured the pressure dependence of T, and observed a reduction of T, as function of pressure with a rate of - 12.6mKlkbar (see section 3.6). The experimental value for alnT,/ap amounts to 26Mbar-‘. Hence, a reasonable agreement between the calculated and experimental values for aln T,lap results. A second check on the validity of eq. (3.3) can be performed by calculating the relative pressure dependence of T, from a combination of thermal
A. de Visser et al. I UPt,, heavy fermions
expansion and specific heat measurements at 1 bar. Starting point is again eq. (3.7). The second term on the right hand side of eq. (3.7) can be related to the coefficient of the linear term in the thermal expansion, ~2, conform eq. (2.14a). The pressure dependence of Ai in eq. (3.7) can be related to the coefficient of the T31n(T/T*)-contribution to the thermal expaneq. (2.14d), eq. (1.33) and sion, 6, . Employing eq. (3.6), S, can be written as
(3.8) Rearranging eq. (2.14a), the pressure
terms in eq. (3.8) and employing we find the following expression for derivative of B”:
aB” ~ = 9V,B” ap The relation between the pressure of A: and B” can be found by taking derivative of eq. (1.34), assuming pressure independent: dA; _- dB” ap ap
1
alnT,=
(3.10)
(A;)2 + 3A;
- $
Subsequently, eq. (2.14a), tionship:
dependence the pressure that Ai is
we derive from eq. (3.7), by using eq. (3.9) and eq. (3.10), the rela-
6
.+3+Y*
ap
(1 + A;)2
9V,B‘
X -
y
(A;)2 + 3A; (3.11)
From the coefficients of the fits to the specific heat and thermal expansion in the temperature range 1.4-20 K (tables 2.1 and 2.111) we calcu-
and superconductivity
141
late, using eq. (3.11), alnT,/ap = - 19 Mbar-‘, in good agreement with the experimental value of aln T,/ap = - 26 Mbar-‘. Here, the second and third term on the right hand side of eq. (3.11) have opposite signs and amount to 33Mbarr’ and -52Mbar~’ (the term with K may be neglected). However, for the fits in the temperature range 1.4-10 K alnT,lap amounts This large negative value is to - 175 Mbar-‘. caused by the larger a,-value in this case (almost a factor of two larger than derived from the fit up to 20 K, see table 2.111). Similar large negative values result if we use a combination of the coefficients derived from fits to the phononcorrected specific heat (fits IIa and IIb, table 2.11), and the thermal expansion (table 2.111), for both temperature intervals. In this case the relatively small a-value is debit to a large negative second term in eq. (3.11). To summarize these results, we conclude that the analysis of T, in the Fermi liquid model, under the assumptions given above (see also section 1.3), suggest that UPt, is a triplet superconductor (g, < 0). However, a test of the expression for T, (eq. (3.3)), by means of a combination of specific heat and thermal expansion data, leads not to an unambiguous approval of this equation. This has partly its origin in the forementioned difficulties in describing the thermal expansion data for UPt, including a T31n(TI T*)-term. 3.6.
Miscellaneous
In this last section we discuss briefly the outcome of several other experiments that have been performed in order to elucidate the nature of the superconducting ground state. First we mention the sensitivity of UPt, to mechanical stress. As follows from table 3.1, T, for annealed samples is roughly 0.1 K higher than for the unannealed ones. This is also nicely illustrated by x,, experiments, performed by Tholence et al. [139], on a Czochralski-grown sample, before and after annealing (see Fig. 3.9). Stewart et al. [3] noticed the absence of superconductivity in a ground sample (T, < 0.050 K). On annealing the sample, T, restores.
142
A. de Visser et al. I UPt,,
heavy fermions
0
-5
3 d :
x-
10
Annealed 15 0.1
0.2
0.3
0.4K
Fig. 3.9. Diamagnetic susceptibility UPt, at different fields. As grown (circles). After ref. [139].
T(K)
for Czochralski-grown (squares) and annealed
Such a strong sensitivity of T, to mechanical stress (and thus p,,) is expected for a triplet superconductor [ 1411. Willis et al. [103] studied the superconducting transition temperature, by means of resistivity at applied pressures up to measurements, 19.1 kbar. These results, obtained on a fluxgrown sample (T, = 0.48 K), are shown in fig. 3.10. T, reduces as function of pressure, at a rate of - 12.6 mK/kbar. The relative depression of T, with pressure amounts to - 26 Mbar-‘. The initial slope of the upper critical field, BL2 is also depressed with pressure, at a relative rate of - 28 Mbar-‘. Both T, and BL2 extrapolate to zero at 37 kbar, corresponding to a volume change of AVlV,, of - 0.018. 6
c4 .c5 d z Q 2
0 0
0.5
1.0 T2
I.5
2.0
(K')
Fig. 3.10. Resistance vs. temperature crystalline UPt,. Pressures were applied 10.2 kbar, 19.1 kbar and 4.1 kbar. After
squared for singlein the order: 1 bar, ref. [103].
and superconductivity
The BCS expressions for T, [51] exhibit, it general, a complex pressure dependence causec by the competing pressure dependences of 8, A > P* and h,,. Therefore, the sign of the pF:ssure dependence of T, can not easily bt predicted. A collection of characteristic parame ters for the superconducting state, and for tht pressure dependence of T,, for several uraniun intermetallic superconductors can be found in ; recent paper by Franse [142]. For the heavy fermion superconductor UBe,, a value for aT,, dp of - 12 mK/kbar has been obtained [143] whereas for CeCu,Si, T, first increases wit1 pressure (for P<40 kbar) and then decrease: [144]. In case of UPt,, a matter of great interest i! whether the occurrence of superconductivity i! mediated by spin-fluctuations. The depression o superconductivity as well as spin fluctuation! with pressure suggests indeed a correlation be tween both phenomena, but cannot be taken a! proof for spin-fluctuation mediated superconduc tivity. The relative pressure dependence of T and the relative pressure dependence of the spin fluctuation temperature, T*, amount to - 2t and 33 Mbar- r, respectively (T* 0~TT, , section! 2.7.1 and 2.7.2). These relative pressure depend ences have an almost equal magnitude, but op posite signs. Within the presented Fermi liquic model, qualitatively a depression of T, wit1 pressure is expected for UPt,, as has been showr in the previous section. The quantitative agree ment between the calculated and experimenta values for alnT,/aT is, however, subject of dis cussion. Low-temperature data for the thermal conduc, tivity, K, as obtained by the Darmstadt group or a polycrystalline UPt, sample [86], are shown ir fig. 3.11. At temperatures below 0.5 K there is i difference between the superconducting-state K,, and the normal-state (superconductivity sup pressed by a magnetic field of 2.5 T), K,-values This difference increases towards lower tempera tures where K, follows a power law with ar exponent between 1 and 2. The temperaturt dependence of K, shows negative curvature foi T Z-O.2 K, when plotted as K,/T vs. T (fig 3.11b). As is indicated by the dashed line, the
A. de Visser et al.
I UPC,, heavy fermions and superconductivity
I t 0.1’.
a
ii’.1 ;(K; ii”
Fig. 3.11. Temperature tivity of polycrystalline and as KIT vs. T (b).
,#1’
0
’ -0 ‘.’
T(K)
’
dependence of the thermal UPt, in a double-logarithmic
conducplot (a),
data points at the very temperature end are consistent with K, = a T2, with (Y= 32 mW/K3cm. The quadratic temperature dependence of K, shows that the thermal conductivity below T, must be of electronic origin, despite the importance of the phonon conductivity as inferred in the normal state for T > 3 K(section 2.5.2). An upper limit of the phonon contribution to K, can be obtained by taking into account scattering of the phonons from grain boundaries only (“size effect”). Using the average grain size of 50 urn as determined for this polycrystalline sample, the phonon contribution to the heat transport in the superconducting state is estimated at only a few % of the total heat transport. Thermal conductivity measurements performed by Sulpice et al. [108] revealed, besides a T*-term below 200mK ((Y = 20mW/K3 cm), a term linear in temperature with a coefficient of 0.55 mW/K’cm. In section 3.3 we mentioned that also in the specific heat “residual” linear terms have been observed. A T2-term in K, has also been found for UBe,, ((Y= 0.38 mW/K3 cm [145]) and CeCu,Si, ((Y= 1.8 mW/K3 cm [131]). In the latter system an additional term linear in T exists. As in case of the specific heat, a T*-term in K, can be taken as evidence for strong gap anistropy (table 1 .IV). Bishop et al. [146] performed sound velocity experiments on UPt, in the superconducting state and observed that the ultrasonic attenuation varied as T2. These results were explained by assuming that UPt, is an anisotropic (triplet) superconductor in a polar-like state, where the
143
gap vanishes along a line on the Fermi surface (table l.IV). However, recently Muller et al. [147] observed a T3-term in the same temperature region, as well as an ultrasonic attenuation peak just below T, (only for longitudinal ultrasound), possibly due to collective modes as observed in 3He. Poppe [148] performed several tunneling experiments on single-crystalline UPt, specimens, using Al or Nb as counterelectrode. However, no Josephson current was observed as might be expected for a weak link between a triplet and singlet superconductor. However, a contact formed with two pieces of UPt, did not show any pair current either. One difficulty with UPt, is that the material .in the superconducting state is very sensitive to mechanical stress, as mentioned above. Therefore the specimen might not be superconducting in the vicinity of the contact. Besides, triplet superconductivity would be very sensitive to pairbreaking at surfaces and restrictions. Therefore, it would be difficult to observe the Josephson effect for a triplet superconductor.
4. UPt, alloyed with Pd 4.1.
Zntroduction
In the preceding sections we presented a large variety of experiments that have been performed in order to characterize and understand the normal and superconducting-state properties of UPt,. In order to further elucidate the lowtemperature properties of UPt,, useful information might be obtained from alloying experiments. Possible candidates for substitutions at the U-site are La, Lu and Th. Pt might be substituted, for instance, by its neighbours in the periodic table: Ir, Rh, Pd, Ag and Au. In general, little information is available in the literature on the solubility of other elements in UPt,. The compounds UIr, (cubic), URh, (cubic) and UPd, (hexagonal) differ in crystal structure from UPt,. UAg, does not exist and the crystal structure of UAu, is not known. Probably the area of stability of the UPt, phase is small for the forementioned substitutions at the Pt-site. This can also
A. de Visser et al. I UPt,, heavy fermions
144
be inferred from a systematic study of the occurrence of intermediate phases in the UX, compounds [ 1491. Since the low-temperature properties of UPt, strongly depend on the lattice parameters (large pressure effects, large Gruneisen parameters etc.), we expect a large influence from substitutions of only a few atom % U or Pt. In this section we present the results of a first series of alloying experiments in which we substituted Pt by isoelectronic Pd. The low-temperature properties of the compounds UPd, and UPt, are quite different, although the distances between neighbouring uranium atoms are nearly identical: 4.13 A in UPt, and 4.11 A in UPd,. The main crystallographic difference between these compounds is the stacking of the atomic layers: in UPt, the stacking is ABAB (hexagonal closed packed; hcp), in UPd, the stacking is ABAC (double hexagonal closed packed; dhcp). In the hcp structure only U-sites with hexagonal symmetry are present, whereas in the dhcp structure hexagonal sites (atoms in B layers) and so-called quasi-cubic sites (atoms in A layers) are present. In table 4.1 some crystallographic and magnetic parameters for UPt, and UPd, have been collected. Neutron scattering experiments have revealed that UPd, has well-localized fwith a 5f2-configuration in a LS electrons, groundstate ‘Hq [150]. The low value for the coefficient of the linear term in the specific heat, y < 10 mJ/K’ molU [151, 1521, illustrates the absence of a narrow band, and contrasts with the large value of 422 mJ/K’molU for UPt,. In UPd, two phase transitions have been observed,
and superconductivity
at 5 and 7 K, both non-magnetic of origin [151,153]. Evidence for crystal-field states comes from neutron-scattering experiments [150]: for both uranium sites a singlet ground state with excited doublet states, yielding excitation energies of 164 and 24 K, has been derived. Which energy splitting belongs to which site has, however, not firmly been established. Given the obvious differences between both compounds, we expect large effects on the low-temperature properties of UPt, by substituting Pd for Pt. In this section we report specific heat, thermal expansion, susceptibility, high-field magnetization and electrical resistivity measurements on a series of pseudobinary U(Pt, _xPd,), compounds (0.015 x 5 0.30). In addition, samples with x = 0.001, 0.002, 0.005 and 0.01 were checked for superconductivity. Polycrystalline compounds were prepared by arc melting the appropriate amounts of the pure elements, U (Koch-Light Ltd., purity 99.8%), Pt and Pd (MRC-Marz grade), in a titanium gettered argon atmosphere. All samples were annealed, in evacuated sealed silica tubes, at 1000°C for a period of 10 days. X-ray diffraction patterns taken on powdered samples at room temperature confirmed the hexagonal MgCd,type of structure. Samples with x Z 0.15 showed additional unresolved diffraction lines, pointing to at least one second unknown phase (phase boundary for the MgCd,-structure between 10 and 15 atom% Pd). Small needle-like singlecrystalline whiskers were obtained from the arc melted buttons, for all pseudobinary compounds,
Table 4.1 Lattice parameters (a and c), neighbouring uranium distance (d,,_,), molar volume (V,,,), susceptibility (x.,~ and x,.), effective moment (M_,,), and the electronic term in the specific heat (y), for hcp UPt, (MgCd,-type of structure) and dhcp UPd, (TiNi,-type of structure). Data for UPd, taken from ref. [150-1521
upt,
UP4
a
c
d,-,
(A)
(A)
(A)
V, (m’imol)
5.752 5.770
4.897 9.631
4.13 4.11
4.24 x 10 ’ 4.18 x lOmY
/*
~mJ!K2molU)
2.6 2.8
422 10
Xu.h (lo-” UPt, UPd,
115 300
m’/molU)
x, (lo-’ 57 160
m’/molU)
A. de Visser et al. I UPt,, heavy fermions
just as for pure UPt, (see section 1.5). Lattice parameter determinations, by X-ray diffraction patterns taken on the powdered samples and on the whiskers, yield a constant u-parameter within the experimental accuracy, a = 5.752 2 3 A, whereas the c-parameter decreases linearly with from 4.897 + 3 A for pure Pd concentration, UPt, down to 4.892 + 3 A for x = 0.15. The c/aratio decreases slightly with increasing Pd content (a reduction of 0.1% for the 15% Pd compound). The reduction of the volume, AVlV, amounts to 7.5 x 1O-4 per atom% Pd. The same value for AVIV can be achieved by applying an external pressure on pure UPt, of - 1.6 kbar (K = 0.48 Mbarr’). 4.2.
Specific heat and thermal expansion
Specific heat measurements on the series of U( Pt 1_xPd,), compounds have been performed in the temperature range 1.2-30 K [97, 154, 1551. Data were taken in zero field and in a field of 5 T. The zero-field data are shown in fig. 4.1
F
145
(for T ~20 K). In fig. 4.2 we show the field effect for some selected compounds. On alloying UPt,, two remarkable features can be observed: (1) for x 5 0.10 the y-value increases with respect to pure UPt,, and (2) an anomaly develops at low temperatures for the 2% and 5% buttons. The former observation points to an enhancement of the many-body effects at low temperatures (larger effective mass). Although it is not unambiguous how to extrapolate c/T in fig. 4.1 to 0 K, y might easily amount to 600 or 700 mJ/K*molU for the 5% and 10% compounds. This signifies a surprisingly large increase (almost 50%) of the y-value with respect to pure UPt,. In a magnetic field of 5 T the y-values are only slightly modified, as indicated by the c/T-values at 1.4K in fig. 4.3. The entropy difference, in the temperature interval 1.2-20 K, between the curve for pure UPt, and the curve for U(Pt 0.80Pd0.20)3 equals 2.4 J/ KmolU. On diluting by Pd, the corresponding
o.7rrl 0.6
and superconductivity
0.6 $ +
t+ :*
U(Pt,_xPd,13 0.5 9 a CJ
0.4 -
Y a
0.3 -
0.2 -
0.1 -
’ 0
Fig. 4.1. Specific heat of U(Pt,_,Pd,), of c/T vs. T, for x 5 0.30.
compounds in a plot
5
10
Fig. 4.2. c/T vs. T for U(Pt,_,Pd,), compounds for x= 0.00(O) and (O), x =0.02(A) and (*), x = 0.05(+) and (0) and x = 0.30(O) and (*), at zero field and an applied field of 5 T.
146
A. de Visser et al. I UPt,, heavy fermions
0.6 9 $
O5
-Y a
0.4
Y d\-
0.3
;
0.2 0.1
0.0
Fig. 4.3. c/T values at T = 1.4 K vs. Pd concentration for U(Pt,-,Pdr), compounds (X 5 0.30) in a magnetic field 01 5 T(A) and at zero field (0). The broken line serves as a guide to the eye.
entropy differences with the 20% compound amount to the constant value of 2.4 J/KmolU for compounds with x S 0.05. The entropy difference between the curves for x = 0.20 and x = 0.30 amounts to 1.0 J/KmolU. Next we discuss the nature of the peaks in c/T vs. T for the 2% and 5% Pd compounds. These anomalies remind one of the phase transitions in UPd, at 5 K and 7 K [151, 1533. From entropy considerations it follows, however, that the anomalies in these pseudobinary compounds can not be due to a second phase of UPd, (that might be overlooked in the X-ray patterns). For UPd, the excess entropy up to 15 K equals 3 Ji KmolU [151], whereas the entropy involved in the peaks of the 5% and 2% samples amounts to 0.8 J/KmolU and 0.2 J/KmolU, respectively. In a magnetic field of 5 T the temperature at which the maximum in c/T is observed shifts from 5.8 K to 5.4 K (5% Pd), and from 3.6K to 3.3 K (2% Pd), but the shape of both peaks remains essentially unchanged (fig. 4.2). The field effect on the anomaly in the specific heat of UPd, has an opposite sign [152]. In one of the papers we suggested that these
and superconductivity
anomalies possibly indicate the onset of antiferromagnetic order [154]. This was recently confirmed by neutron experiments on a single-crystalsample [156]. We return to line U(Pt”.,,Pd,.,,), this point in section 4.6. The specific heat data of UPt, have been analysed with a T31n( T/T*)-contribution, characteristic for spin-fluctuation effects. In case of the U(Pt,_,Pd,), compounds, the additional low-temperature anomalies hamper such an analysis. A computer fit to the data, for the 1% and 10% compounds, in the temperature interval 1.2-10 K, including such a term, yields an increase of 6 and a decrease of /3* with increasing Pd content. The characteristic temperature, T *, reduces from 29 K (pure UPt,) to 22 K (1% Pd) and 19 K (10% Pd). For x 2 0.15 the spin-fluctuation properties are rapidly lost. In figs 4.4 and 4.5 we present thermal expansion measurements on some selected compounds. The maximum observed for pure (polycrystalline) UPt, at 12K, shifts towards lower
WQ’dX)
3
0 x=0.00
I\ t r’ $
A q
00
00.01 .02 0.05 0.15
+ 0
-15 L 3 t
-20
1
(111111111(IIIJ1111 0 20 40
60
80
100
T (K> Fig. 4.4. Linear thermal expansion coefficient for polycrys talline U(Pt,+,Pd,), compounds, for x =0.00(O), 0.01(O). 0.02(A), 0.05(+) and 0.15(O).
A. de Visser
et al. I UPt,, heavy fermions and
pound, whereas the 2% compound shows a sharp drop (on decreasing the temperature) in cy near 4 K. The thermal expansion of a 15% compound resembles that of the c-axis of UPt,. Since the thermal expansion coefficient is strongly anisotropic and since crystallites with preferred orientations are present in these samples, a more quantitative analysis awaits data on single-crystalline samples.
9 8 7c\
6-
+
5-
147
su~erconductiv~~
g..
4-
4.3. ~~ce~ti~ility
s
3-
The results of the susceptibility measurements in a field of 1.3 T, in the temperature range 1 S-300 K, and high-field magnetization measurements, at a temperature of 4.2 K, on some selected U(Pt,_,Pd,), compounds [155] are shown in fig. 4.6 and fig. 4.7, respectively. Both experiments have been performed on cylindrical samples, with a diameter of 1.5 mm and a length of 5 mm. Because of the forementioned presence of crystallites with preferred orientations, the absolute x-values in fig. 4.6 are somewhat sample dependent. Obviously, the most interesting result of alloying UPt, with Pd as to the susceptibility and ma~etization, is the reduction of the characteristic temperature and field at which the anomalies are observed. The maximum in the
2-
P
I
ad
lo-
%
2
-1""""""""""' 5 0
10
20
15
Fig. 4.5. The same as fig. 4.4. More detailed account in the temperature range I-20 K.
temperatures with increasing Pd content. Additional anomalies for the 2% and 5% Pd compounds are also found in the thermal expansion: a small peak results at 5.8 K for the 5% com-
u(Pt,,Pd,)
and ~ug~etizfftiun
3
x= 0 0
0.00 0.02
4 0.05 0 0.15
0
’
I
I
0
100
200
Fig. 4.6. Magnetic susceptibility vs. temperature
T (K> for polycrystalline
1
300 U(Pt,_,Pd,),
compounds.
A. de Visser et al. f UPt,, heavy fermions and superconductivity
148
0
l
,*
,o
.<
,*
,’
,2 / ,0’ b'
A’
q
I-
’
P I
0 0
10
P,?T) /
i
30
40
Fig. 4.7. High-field magnetization normalized to 1 at 35 T for polycrystalline U(Pt,_,Pd,), compounds. The absolute values at 35 T amount to 0,806, 0.782, 0.882 and 0.832 &U-atom, for compounds with x = 0.00, 0.02, 0.05 and 0.15, respectively.
susceptibility, at approximately 17 IS for pure UPt,, shifts towards lower temperatures (11 K and 7 K for the 2% and 5% sample, respectively), and becomes more pronounced. It is not observed for a sample with x = 0.15 down to 1.4 K. From a Curie-Weiss analysis of the data in the temperature range 50-300 K, it follows that the effective moment remains constant on alloying ( peri. = 2.6 f 0.1&U-atom). The paramagnetic Curie-Weiss temperature increases from - 80 I: 10 K, for pure (poly~rystalline) UPt,, up to - 50 + 10 K for U(Pt, ssPd,, ,&. In the case of UPd, a value for wcff of 2.8p.,/Uatom has been deduced from an analysis of x in the temperature interval 70-300 K [152]. The anomaly in the high-field magnetization curve at 4.2 K, i.e. a maximum in the differential susceptibility at - 20 T, for pure UPt,, shifts towards lower fields (approximately 16 T and 11 T for the 2% and 5% sample, respectively). Again, it has not been observed for a sample with x = 0.15, at this temperature.
U(Pt,_,Pd,), compounds, as well as on small single-crystalline whiskers, in the temperature range 1.3-300 K 11571. The experimental results are shown in figs 4.8, 4.9 and 4.10. In order to
0.6
U(Pt,_,Pd,), 0.4
0.2
0
4.4 Electrical resistivity Electrical performed
resistivity measurements have been on a number of polycrystalline
I
J
200
Jo0
I
0
xl0
T (K> Fig. 4.8. Reduced resistivity, ~(~)/~(3~ K), of single-crystalline U(Pt,_,Pd,), whiskers, for x = O.~(O~, 0.05(A) and 0.10(V). The current is applied along the c-axis.
A. de Visser et al. I UPt,, heavy fermions
I
I
0
300
200
100
0
T (K> Fig. 4.9. Reduced line U(Pt,_,Pd,), 0.10(V).
resistivity, samples,
p(T) lp(300 K), of polycrystalfor x=0.00(0), 0.05(A) and
l.2
0.6
0.4
02
0 0
I
I
I
I
10
20
30
40
50
-I-(K> Fig. 4.10. Reduced resistivity, p(T)/&300 K), of polycrystalline U(Pt,_,Pd,), samples, for x=0.00(0), 0.01(O), 0.02(+), 0.05(A), 0.10(V) and 0.15(x).
and superconductivity
149
illustrate the relative temperature dependence of the resistivity, the data are plotted as p(T) / ~(300 K) vs. T. Absolute values for p,, and p(300K) are listed in table 4.11. From figs 4.8 and 4.9 it follows that the experimental data for the whiskers and the polycrystalline samples nearly coincide. This illustrates that the whiskers may be considered as representative for the bulk material. The difference between the absolute values for the two types of samples partly has its origin in anisotropy. As discussed in section 2.5.1 due to preferential orientations, a polycrystalline sample has a resistivity almost equal to the basal plane value (238 l&cm for pure UPt,), whereas a whisker represents the c-axis value (132 p.Rcm). Furthermore, cracks and an irregular texture enlarge the resistivity of the polycrystalline samples. This is in particular the case for the specimens with high Pd contents (x = 0.10, 0.15). The main result from our measurements is the gradual change from the spin-fluctuation-like behaviour observed for pure UPt, (section 2.5.1) to a Kondo-like resistivity for high Pd contents. In combination herewith we observe a reduction of ~(300 K) - p0 with increasing Pd concentration. Temperature derivatives, AplAT, are shown in fig. 4.11 for some selected polycrystalline compounds. The maximum in AplAT, at 6.5 K for UPt,, shifts towards 5.5 K for the 1% Pd sample. For the 2% and 5% Pd samples ApiA T yields an anomaly near the temperature where the maximum in the specific heat is observed. In particular a sharp minimum occurs at 5.6 K for the 5% Pd sample. The resistivity of the Kondo-like compounds (X = 0.10, 0.15) is shown in a plot of P(T) lp(300 K) vs. 1nT in fig. 4.12. At the lowtemperature side the resistivity of both compounds deviates from the 1nT behaviour; coherence sets in. A low-temperature maximum, as we observe for x = 0.15 near 5 K, has also been observed by Stewart and Giorgi [158] on an U(Pt,,,,Pd,.,,), sample near 20 K. A T2temperature dependence, as observed for UPt, below 2 K, is not observable in the samples with low Pd contents in the investigated temperature range. In searching an explanation for the Kondo-like resistivity curves for the samples with
A. de Visser et al. I UPt,, heavy fermions
and superconductivity
x equal to 0.10 and 0.15, we compare UPt, wnh UPd,. A concentration of - 10% Pd on the Pt sublattice could, in principle, be sufficient to destroy, at least partially, the hcp structure, possibly leading to quasi-cubic uranium sites that could act as Kondo impurities. However, analysing the X-ray patterns for x 5 0.15, any sign of a (long-range) dhcp structure was lacking and no substantial broadening of the diffraction lines was observed. Analysing the X-ray patterns for higher Pd concentrations, additional lines are found indicating the presence of a second phase, which is not the UPd, phase. Whether the Kondo-type of resistivity curves are related to this second phase remains a question to be answered. 4.5.
Fig. 4.11. Temperature derivative. of the resistivity, vs. T, for polycrystalline U(Pt,_,Pd,), samples, 0.00(O), 0.01(O), O.OZ(+) and 0.05(n).
U(Pt,_xPd,l,
ApiA T, for x =
I
0.8 0
1
2
3
4
5
6
Inl Fig. 4.12. Reduced resistivity, p(T)lp(SOO K), vs. InT for polycrystalline U(Pt,_,Pd,), samples, for x =0.10(V) and 0.15(x).
Superconductivity
As follows from the specific heat measurements on these pseudobinary compounds, the spin-fluctuation temperature decreases with increasing Pd content, leading to an enhancement of the spin-fluctuation effects. An important question is whether the occurrence of superconductivity and spin-fluctuation effects in UPt, are closely related or not. Assuming that such a close relation exists, one would not expect a large change in the superconducting properties for samples with small amounts of Pd substituted for Pt (x 5 O.Ol), since the spin-fluctuation effects do not change drastically in this Pd concentration range. However, no superconductivity has been observed in a U(Pt,,,,Pd,,,,), sample as follows from an ac-susceptibility experiment down to 40 mK [155]. The low-field magnetization experiments on these pseudo-binary compounds and chemical analysis of the pure U, prove that the concentration of magnetic impurities (probably Fe) is limited to 600 ppm, which is approximately a factor of 10 less than the amount of Pd in this 1% sample. An annealed single-crystalline UPt, sample, grown from this batch of U, was found to be superunannealed 0.48 K. An conducting at sample, made of a purer batch U(Pt0.99,Pd0.0”,)3 of uranium, was not superconducting either, whereas an unannealed polycrystalline sample
A. de Visser et al. I UPt,, heavy fermions
Table 4.11 Residual resistivity,
pO, and room-temperature Whiskers
Compound X 0.00 0.01 0.02 0.05 0.10 0.15
values ~(300 K) in kacrn
and superconductivity
for single-
and polycrystalline
Polycrystalline
(Illc-axis)
151
U(Pt,_,Pd,),
compounds
samples
PO
~(300 K)
PO
~(300 K)
2.2 4.5 6.4 24.6 46.6
147 126 81 75 48 -
6.1 18.6 36.4 101 633 *.# 355 #
240 264 238 251 539 334
* Irregular structure. # Kondo-type compound.
made from the same uranium, had a superconducting transition temperature of 0.38 K. Recently, samples with an even lower Pd content (0.1% and 0.2%) have been prepared. Superconductivity is observed in these samples at and at 0.357 K (0.2%) [159]. 0.460 K (0.1%) Apparently, the superconducting ground state is rapidly destroyed by alloying UPt, with Pd. From the resistivity data we conclude that the strong suppression of superconductivity cannot simply be due to an impurity effect. From table 4.11 it follows that the p,-value of, for instance, the 0.5% Pd specimen lies in between 3 and 10 l.kkm. Yet, an annealed single-crystalline UPt, sample with a similar p,-value (6.2 @km) became superconducting near 0.5 K. Also the presented Fermi-liquid model (section 3.5) is not able to account for such a rapid depression of T,. From the normal-state specific heat data we calculate only a minor change in the pseudopotential g,, for x 5 0.01, and thus T, would remain unchanged (eq. (3.3)), in contrast to the experimental findings. Although the spin-fluctuation effects are enhanced on alloying with Pd, superconductivity is suppressed. Apparently, the high y-value is not a sufficient condition for a superconducting ground state. Therefore, the present data are not in favour of a direct correlation between spin fluctuations and superconductivity. However, the occurrence of superconductivity might depend on a delicate balance between several factors, the presence of spin-fluctuations being one of
them. More information is needed further elucidate this problem.
in order
to
4.6 Discussion From the specific heat, susceptibility and highexperiments on the field magnetization U(Pt,_,Pd,), compounds we conclude that a close connection between the thermal and magnetic properties exists: the characteristic temperature as derived from the specific heat (T* = 29 K), the temperature at which the maximum in the susceptibility occurs (T,,, = 17 K), and the field at which the maximum in the differential susceptibility is found (B,,, = 20 T), all reduce on alloying UPt, with Pd. Accepting the analysis as presented in section 2, this implies an enhancement of the spin-fluctuation effects (lower excitation energy). For x 2 0.15 the spin-fluctuation phenomena are lost. As mentioned before, recently, neutron experiments revealed that the anomaly in the 5% Pd sample at 5.8 K is associated with the onset of antiferromagnetism (Frings et al. [156]). From these first neutron data it follows that the magnetic moments are aligned antiferromagnetically in the basal plane. The ordered moment amounts to 0.6 -+ 0.2&U-atom. Furthermore, Frings et al. argue that the step-wise increase of the magnetization near 11 T in the polycrystalline 5% Pd sample (fig. 4.7) is due to the destruction of the antiferromagnetic correlations in the basal plane, followed by a ferromagnetic
152
A. de Visser et al. I UPt,, heavy fermions
alignment at high fields. Magnetization data taken by Van Sprang et al. [159] on a singlespecimen at 4.2K crystalline U(Pt0,95 Pd,,,,), (below the ordering temperature), reveal the same anisotropy as observed for pure UPt,; the main difference being the occurrence of the stepwise increase in the basal plane magnetization at lower fields (- 11 T, compare fig. 4.7). At 35 T the magnetization in the basal plane exceeds the value for the c-axis by - O.S~,iU-atom. Assuming that the reorientation of the moments is superimposed on an isotropic magnetization, the ordered moment as derived from the magnetization data amounts to 0.5pu,lU-atom, close to the value for the ordered moment as reported by Frings et al. It is very likely that the high-field anomaly observed for pure UPt, is due to the same phenomenon. However, in case of UPt, no long-range antiferromagneti~ order has been observed, which implies that the order must be short-range. This offers the following picture. In pure UPt, short-range antiferromagnetic eorreiations, confined to the basal plane, set in near the temperature at which the maximum in the susceptibility occurs, T,,, = 17 K. This short-range order is destroyed by a field of - 20T. For the 2% Pd sample the characteristic temperature and field for the short-range correlation are slightly lower, besides long-range order (accomplished between the hexagonal planes) sets in at 3.6 K. In the 5% Pd compound the onset of short-range and long-range order almost coincide. In the 10% Pd compound the ordering phenomena have nearly disappeared (only a small anomaly in c is observed near 2 K). Besides, in this entire concentration range the large y-value remains, which shows that spin-fluctuations persist down to the lowest temperatures. At higher Pd contents a second phase appears in the diffraction patterns: the spin-fluctuation phenomena are lost, and a Kondo behaviour results. These observations suggest that the compound UPt, is close to an antiferromagnetic instability: antiferromagnetism can be induced by substituting small amounts of Pd for Pt. Similar ordering phenomena in the U( Pt , _XPdx)3 compounds have recently been observed by Stewart et al. [160].
and superconductivity
In fig. 4.13 we present the resulting magnetic phase diagram for the U(Pt,_,Pd,), system. For low Pd concentrations superconductivity is rapidly destroyed; antiferromagnetism is observed in the concentration range 1 or 2% Pd up to 10% Pd. A small concentration range remains where neither superconductivity, nor antiferromagnetism is observed (-0.4-l% Pd). This shows that the superconductivity is not directly killed by the onset of antiferromagnetism. The arrows in fig. 4.13 indicate the sign of the pressure effect on the superconducting transition temperature and the spin-fluctuation temperature are observed for pure UPt,: &rT,lap = -26MbaC’ and alnT*/ap =33Mbar-*, and on the NCel temperature as observed on a 5% Pd specimen: aln TN/ ap = - 55 Mbar ’ (Van Sprang et al. [lSl]). Applying pressure and alloying with Pd both yield a reduction of the volume. In this respect the suppression of superconductivity and antiferromagnetism might originate from a volume effect (note that this implies that TN for the 2% Pd compound would increase with pressure). However, this would also imply that at a sufficicntly high pressure pure UPt, would be an an-
Ts(
T 1
.’
0
0
,
i:.
0.5
‘
&at%
Pd
1
0
lo
Fig. 4.13. Phase diagram for the U(Pt,_,TPd,), compounds for xsO.10; Ntel temperature, TN, and superconducting transition temperature, T,, vs. Pd concentration; S= AF= fluctuations, SF = spin superconductivity, antiferromagnetism, K = Kondo behaviour. Arrows indicate the sign of the pressure effect on the characteristic temperature. Nate the expanded scale for x CC0.005. Data for the 7% Pd compound taken from ref. 11591.
A. de Visser et al. I UPt,, heavy fermions
tiferromagnet (albeit in a small pressure range), which is apparently not the case. Furthermore, the pressure effect on T,,, of x for pure UPt, and on T, for x = 0.05 (as follows from measurements of the resistive anomaly as function of pressure) are of opposite sign. Specific heat, susceptibility and resistivity measurements on a series of U,_,Th,Pt, compounds identical low-temperature revealed nearly anomalies as the Pd doped samples [160, 1611. In contrast to Pd doping, the volume increases, with respect to pure UPt,, when Th is substituted for U. Therefore, Ramirez et al. [161] suggested that not the volume but the c/u-ratio plays a dominant role: for both substitutions, Pd and Th, the c/u-ratio decreases. This was further supported by recent alloying studies with Ir and Au [162]. When, on alloying, the c/a-ratio decreases, antiferromagnetism is induced. At applying pressure the c/a-ratio increases again, as follows from the anisotropic compressibility (section 2.4), and TN decreases (note that this implies that TN for the 2% Pd compound would decrease with pressure as well). A numerical assuming a linear calculation shows that, pressure dependence of TN, antiferromagnetism in the 5% Pd specimen would be totally suppressed at a pressure of - 30 kbar. An explanation for the occurrence of antiferromagnetism in the doped UPt, samples can be offered by Fermi surface instabilities, due to doping induced strains: a gap opens over a part of the Fermi surface, resulting in a combination of spin density waves and antiferromagnetic order, like observed in Chromium [163]. A similar explanation has been offered for the occurrence of antiferromagnetism in the heavy fermion systems UCd,, [16] and U,Zn,, [17].
5. Concluding 5.1.
remarks
Normal phase
In section 2 we presented studies of the magnetic, thermal, elastic and transport properties of UPt, in the normal state. These studies were
and superconductivity
153
based on a large variety of experiments, among which specific heat, thermal expansion, susceptibility, electrical resistivity, magnetostriction etc. Besides, we studied the effect of high magnetic fields and high pressures. Most of these experiments have been performed on good-quality single-crystals. We have shown that the thermal properties can be fairly well described in the Fermi liquid model, including a T31n(TIT*)-contribution to account for spin fluctuations. The magnetic properties are strongly anisotropic: a two-dimensional character of the field effect (confined to the hexagonal plane) results. On the one hand, the observed anomalies in the susceptibility at T max= 17 K, and in the high-field susceptibility at &n,X = 20 T (at liquid helium temperatures), seem to originate from spin-fluctuation effects as well. On the other hand, these anomalies are reminiscent of a type of antiferromagnetic order. On alloying with Pd, pseudobinary compounds U(Pt,_,Pd,), order indeed antiferromagnetically (for a limited concentration range). Obviously, the compound UPt, is close to an antiferromagnetic instability, which can be triggered by substituting small amounts of Pd for Pt. Below we present a few remaining experimental and theoretical aspects, in order to complete the discussion about the normal phase of UPt,. Recently, Aeppli et al. [164] and Johnson et al. [165] performed inelastic neutron scattering experiments on UPt,. From these experiments direct evidence for spin fluctuations was deduced. The energy scale for the spin fluctuations was found to correspond to a large effective mass, in rough agreement with the low-temperature specific heat data. Diffraction patterns recorded for a powdered sample at 5,20 and 300 K were similar [165], once more excluding the possibility of long-range antiferromagnetic order below - 17 K. We want to emphasize that the low-temperature anomalies are not caused by crystal-field levels, as might be suggested by the resemblance between UPt, and, for instance, the hexagonal crystal-field compound PrNi,. The susceptibility of PrNi, peaks at 17 K [166], as is the case for UPt,. Measurements on a single-crystalline
154
A. de Vikser et al. / UPt,, heavy fermions
specimen revealed a nearly identical anisotropy in x, as found in UPt,. Also thermal expansion data for PrNi, [167] and UPt, are much alike (apart from the low-temperature upturn in (y/T, which is only observed for UPt,). Furthermore, non-linearities have been observed in the basalplane magnetization of PrNi, above - 8 T [ 1681. The ground state of PrNi, in the C&coupling scheme is ‘H4(Pr”‘). Due to the crystal field the J = 4 multiplet is split into 3 singlets and 3 doublets. The lowest level is a singlet (r4) and the first excited level, also a singlet (r’), lies at an energy distance of - 20 K. The low-temperature and high-field anomalies in the case of PrNi, are attributed to crystal-field effects. Assuming an identical ground state for UPt, (a crystal-field split 3H, state), we calculate from the observed anisotropy in x, a value for the crystal-field parameter Bi of 6.5 K, quite close to the value of 5.8 K reported for PrNi, [169]. Nevertheless, the corresponding anomalies in case of UPt, are not caused by crystal-field effects. Evidence hereto mainly arises from the absence of a Schottky anomaly in the specific heat, and is further supported by comparing magnetization and magnetoresistivity data for single-crystalline UPt, and PrNi,. In particular, the magnetoresistance of PrNi, at 4.2 K, for a current along the hexagonal axis and the field applied in the basal plane, increases monotonically as a function of the field, from 3.0 l&cm at zero field to 3.5 @cm at 35 T (1191, whereas for UPt, a large peak is observed near 20T. Besides, at room temperature p of PrNi, amounts to 27 @cm, much lower than the corresponding value for UPt, (132 @cm). Also magnetization curves of PrNi, at 4.2 K, measured for the different crystallographic axes in fields up to 35 T (de Visser et al., unpublished), have a quite different shape, when compared with the corresponding curves for UPt,. To summarize: we believe that no low-energetic crystal-field levels play a role in the low-temperature properties of UPt,. In discussing the thermal properties of UPt,, we focussed on a Fermi liquid analysis. Below we briefly compare a few aspects of this analysis with a similar analysis in the paramagnon model. In the Fermi liquid model the enhancement of
and superconductivity
the susceptibility of the interacting system which respect to x of the non-interacting system is expressed by S = (m*lm)(l + Ft)-‘. With the values m* im = 180 and Ft = - 0.788, as derived from the linea_r and T31n( TIT*)-terms in the specific heat, S amounts to 849 (table 2.11). In the paramagnon model the ratio of x for the exchange enhanced system and the Pauli susceptibility, x0, is given by the Stoner enhancement factor S. Usually, x0 is calculated from the unenhanced linear term in the specific heat, yO. In the case of UPt, ‘yOcan be obtained from a twoparameter fit (linear and cubic terms) to the specific heat in the temperature range 9-15 K, yielding Pph = -yO= 225 mJ/K*molU (and 0.85 mJ/K4molU) [8]. Assuming such a large y,-value, the spin-fluctuation enhancement factor A,, amounts to 0.88, which implies only a small mass enhancement rn* Im = 1.88 (we neglect the enhancement due to the electron-phonon interaction). With the experimental value for x of 90 x low9 m3/molU (averaged over the three crystallographic directions), and x0 as calculated directly from ‘y,,,we subsequently derive a rather low value for S: 2.32. A higher value for S results when the density of states that follows from band-structure calculations is used. Several groups performed band-structure calculations [170-1751 yielding y-values between 14.9 and 22.5 mJ/K*molU (values for h,, range from 18 till 27). By calculating x0 from these band-structure derived y-values, we obtain values for S between 23 and 35. The corresponding mass enhancement, m* Im, ranges from 17 till 26. The low value for the effective mass as derived in the applied paramagnon model (when compared to the mass enhancement obtained in the Fermi liquid model) is partly due to the fact that we did not take into account the mass renormalization that arises from the band structure. In the Fermi liquid model the renormalization of the electron mass has been calculated with respect to the free electron mass. In section 2.3.2 we showed that the experimental value for the enhanced linear terms in the specific heat and thermal expansion, rz, amounts to 68 at 1 K (eq. (2.16)). In the paramagnon model rz can be expressed as
A. de Visser et al. I UPt,, heavy fermions
1+ ASf r:=l+
r e,
(5.1)
enwhere AZ represents the spin-fluctuation hancement factor for the linear term in the thermal expansion: a,/3 = yz T = y,(l + h:)T (we neglect the enhancement due to the electronphonon interaction). r, represents the Griineisen parameter for the unenhanced linear terms in c and CY(eq. (2.11)). Assuming r, = 3 for f-electrons, we calculate using eq. (2.11) from (yO = ) y = 225 mJ/K’molU a value for ‘y, of 2 X lo-’ K-*. Since yz amounts to 1.08 X 10m6K-* (table 2.111), AZ is large. We calculate AZ = 53. The corresponding enhancement factor for the specific heat, hsf, equals 0.88 and the ratio AZ/A,, amounts to 60. Employing y = 20 mJ/K*molU, as follows from the band-structure calculations, we derive the much larger enhancement factors At = 613 and A,, = 20. Their ratio amounts to 31. The ratio AZ/A,, can also be calculated from the volume dependence of the exchange enhancement factor S, employing the relationship [176]: A
sf
-E_1+--
Asf
1 alnS T,lnS alnV ’
(5.2)
The relative volume dependence of S can be related to the pressure dependence of the susceptibility by alnS -=--_ alnV
(5.3)
With the averaged value of - 20 Mbar-’ for the relative pressure dependence of x at 4.2 K (section 2.7.2), and S = 2.32 we calculate AZ/A,, = 21, whereas for S = 26, as follows from the bandstructure calculations ( y = 20 mJ/ K*molU), a value for this ratio of only 6.2 results. Apparently, the ratio AZ/A,, as calculated from the volume dependence of the Stoner factor is lower than the value as calculated from the specific heat and thermal expansion data. This inconsistency puts some question marks as to the foregoing analysis in the paramagnon model. Nevertheless, several authors focussed on an analysis in the paramag-
and superconductivity
155
non model, when discussing the spin-fluctuation phenomena in UPt, [SO, 1771. As mentioned above band-structure calculations have been performed by several groups. The basic result of these calculations is a broad Pt 5d band below E, (width -7 eV) and a narrow spin-orbit split U 5f band above E, (width - 2 eV). Most of the calculations yield a sharp peak in the density of states with a width of 0.1 eV at the beginning of the U 5f band. E, is located near this peak. Hybridization between the two bands removes states from the top of Pt 5d band and places them in energy region of the U 5f band. The calculated y-value amounts to - 20 mJ/K’molU, implying that an enhancement factor of about 20 is needed in order to account for the large experimental y-value. This enhancement factor takes into account the spinfluctuation effects. The band-structure calculawith high-resolution tions are consistent (0.13 eV) photoemission studies [178], in which a narrow structure of - 0.15 eV has been observed at E,. We made several attempts to observe the De Haas-Van Alphen effect on a Czochralski-grown specimen (RRR = 80) in the Amsterdam High Field Installation (T = 1.5 K, B,,, = 40 T). However, these attempts were unsuccessful, which we attributed to the high effective electron mass. Recently, a major achievement in this field was made by Taillefer et al. [179], who performed dHvA experiments on a high-purity single-crystalline UPt, sample (RRR- 500), in the temperature range 20-150 mK. Preliminary estimates of the cyclotron masses range from 25 to 90 times the free electron mass, which undoubtedly proves the existence of heavy electrons in upt,. In spite of the tremendous effort to unravel the intriguing low-temperature properties of UPt,, some of them remain quite puzzling. In particular, the possible interrelation of spin fluctuations, antiferromagnetism and superconductivity is an intriguing question. Hopefully, in the near future, theorists will be able to develop models that can successfully account for such interrelations in the field of heavy fermions. From the experimentalists side, other alloying
156
A. de Visser et al. I UPt,, heavy fermions
studies, and the search systems, are requested underlying physics. 5.2.
Superconducting
for more heavy-fermion in order to grasp the
phase
In section 3 we discussed the superconducting properties of UPt,. Meissner effect experiments demonstrated that the occurrence of superconductivity is a bulk property. The uranium 5felectrons are involved in the formation of the superconducting condensate, as follows from the magnitude of the anomaly in the specific heat at T,. The large initial value for the temperature derivative of the upper critical field can be analysed in a rather conventional way, yielding the Fermi wave number of the heavy electron liquid before it enters the superconducting state. Several unconventional features have been observed in the superconducting phase. The electronic properties (specific heat, thermal conductivity and sound attenuation) follow a power law, rather than an exponential temperature dependence. Taking into account entropy balance, residual linear terms are found in the specific heat (and thermal conductivity), which shows that parts of the Fermi surface remain normal. The observed temperature variation of the electronic properties is highly suggestive for a polar like odd-parity superconducting phase. A strong dependence of T, on stress, and the absence of a Josephson current in a weak link between a conventional superconductor and UPt,, are consistent herewith. A Fermi liquid analysis of the superconducting transition temperature, using Landau parameters derived from the normalstate thermal properties, favours triplet superconductivity. On substituting Pd for Pt, superconductivity is rapidly destroyed: T, < 0.040 K for 0.5 atom% Pd. In view of the present state of the theory none of these unconventional properties can be taken as decisive proof for spin-fluctuation mediated odd-parity superconductivity. Recently, Oguchi et al. [180] could explain several of the observed properties, based on a model for conventional phonon mediated superconductivity, albeit with an anisotropic gap. From their band-structure
and superconductivity
calculations the electron-phonon interaction was deduced: strong for electrons on the Pt atoms, quite weak for electrons on the U atoms. The corresponding anisotropy in the electronphonon coupling constant, hph, leads via the hybridized wave functions to strong gap anisotropy. With a calculated average value for A,, of 0.33 and an assumed Coulomb pseudopotential of 0.1, Oguchi et al. calculate T, ==0.2 K (the experimental value amounts to 0.5 K) using McMillan’s formula. This would show that phonon mediated singlet superconductivity is possible in UPt,, the superconductivity being driven by the strong electron-phonon interaction at the Pt sites. The strong reduction of T, on substituting Pd for Pt has been taken to support this model: strains or Pt defects would largely affect the electron-phonon interaction, thus reducing T,. Weighing the pros and cons of odd-parity superconductivity a conclusive judgment cannot be given at present. Obviously, UPt, is a very likely candidate for unconventional spin-fluctuation mediated superconductivity. If UPt, proves to be an odd-parity superconductor a wealth of exotic physics may lie ahead.
Acknowledgements The authors are grateful to J.C.P. Klaasse, E. Louis, R. Verhoef, F.E. Kayzel, P.H. Frings, P.E. Brommer, M. van Sprang, K. Kadowaki, T.T.M. Palstra, P.R. Nugteren, A.J. Dirkmaat, P.H. Kes, J.A. Mydosh, U. Rauchschwalbe, C.D. Bredl, F. Steglich, J. Schoenes, B. Liithi, J. Flouquet, A.G.M. Jansen, A.M. Duif and many other colleagues in the field of the heavy fermions, for a stimulating cooperation. This work was part of the research program of the Dutch Foundation for Fundamental Research of Matter (F.O.M.).
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A. de Visser et al. I UPt,,
heavy fermions
[2] P.H. Frings, J.J.M. Frame, F.R. de Boer and A. Menovsky, J. Magn. Magn. Mat. 31-34 (1983) 240. [3] G.R. Stewart, Z. Fisk, J.O. Willis and J.L. Smith, Phys. Rev. Lett. 52 (1984) 679. [4] J.J. Park and D.P. Fickle, J. Research N.B.S. 64A (1960) 107. [5] VI. Kutaitsev, N.T. Chebotarev, LG. Lebedev, M.A. Andrianov, V.N. Konev and T.S. Menshikova, in: Plutonium 1965, A.E. Kay and M.B. Waldron, eds. (Chapman and Hall, Hampshire, 1965). [6] B.A. Hatt and G.I. Williams, Acta Cryst. 12 (1959) 655. [7] T.J. Heal and G.I. Williams, Acta Cryst. 8 (1955) 494. [8] P.H. Frings, Thesis, University of Amsterdam (1984). [9] H.H. Hill, in: Plutonium 1970 and other actinides, W.N. Miner, ed. (AIME, New York, 1970), p. 2. [lo] A. de Visser, J.J.M. Franse and A. Menovsky, J. Magn. Magn. Mat. 43 (1984) 43. [ll] G.R. Stewart, Z. Fisk, M.S. Wire, Phys. Rev. B 30 (1984) 482. [12] T. Fujita, K. Satoh, Y. Onuki and T. Komatsubara, J. Magn. Mag. Mat. 47-48 (1985) 66. [13] F. Steglich, J. Aarts, C.D. Bredl, W. Lieke, D. Meschede, W. Franz and H. Schafer, Phys. Rev. Lett. 43 (1979) 1892. [14] H.R. Ott, H. Rudigier, Z. Fisk and J.L. Smith, Phys. Rev. Lett. 50 (1983) 1595. [15] G.R. Stewart, Z. Fisk, J.L. Smith, J.O. Willis and M.S. Wire, Phys. Rev. B 30 (1984) 1249. [16] Z. Fisk, G.R. Stewart, J.O. Willis, H.R. Ott and F. Hulliger, Phys. Rev. B 30 (1984) 6360. [17] H.R. Ott, H. Rudigier, P. Delsing and Z. Fisk, Phys. Rev. Lett. 52 (1984) 1551. [18] A. de Visser, J. J.M. Franse, A. Menovsky and T.T.M. Palstra, Physica 127B (1984) 442. [19] F. Steglich, U. Rauchschwalbe, U. Gottwick, H.M. Mayer, G. Sparn, N. Grewe, U. Poppe and J.J.M. Franse, J. Appl. Phys. 57 (1985) 3054. [20] W.C.M. Mattens, R.A. Elenbaas and F.R. de Boer, Comm. Phys. 2 (1977) 937. (211 H.J. van Daal, K.H.J. Buschow, P.B. van Aken Andy M.H. van Maaren, Phys. Rev. Lett. 34 (1975) 1457. [22] H.R. Ott, H. Rudigier, E. Felder, Z. Fisk and B. Batlogg, Phys. Rev. Lett. 55 (1985) 1595. [23] R.J. Trainor, M.B. Brodsky, B.D. Dunlap, and G.K. Shenoy, Phys. Rev. Lett. 37 (1976) 1511. (241 M.B. Brodsky and R.J. Trainor, J. Physique Coll. C 6 (1978) 777. [25] C.L. Lin, J. Teter, J.E. Crow, T. Mihalisin, J. Brooks, A.I. Abou-Aly and G.R. Stewart, Phys. Rev. Lett. 54 (1985) 2541. [26] T. Sugawara and S. Yoshida, J. Low Temp. Phys. 4 (1971) 657. [27] G.R. Stewart, Rev. Mod. Phys. 56 (1984) 755. [28] G.P. Meisner, A.L. Giorgi, A.C. Lawson, G.R. Stewart, J.O. Willis, M.S. Wire and J.L. Smith, Phys. Rev. Lett. 53 (1984) 1829.
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UPt,,
HEAVY FERMIONS
A. DE VJSSER*, Natuurkundig Received
A. MENOVSKY
Laboratorium
1 June
AND SUPERCONDUCTIVITY
der Universiteit
and J.J.M.
FRANSE
van Amsterdam,
Valckenierstraat
65, 1018 XE Amsterdam,
The Netherlands
1987
This review considers studies of the normal and superconducting state of the heavy-fermion superconductor UPt, most of them performed on good quality (y = 422 mJ/K’molU, T, = 0.5 K). We discuss a large variety of experiments, single crystals. Concerning the normal state we concentrate on the magnetic, thermal, elastic and transport properties. These properties can be fairly described in a spin-fluctuation model. Experiments concerning the superconducting phase reveal several unusual features. The possibility of an unconventional pairing mechanism is discussed. On alloying UPt, with Pd, additional anomalies are observed, suggesting that UPt, is close to an antiferromagnetic instability.
Table
ofcontents
1. General
2.6.3.
introduction
1.1. Introduction 1.2. Related compounds 1.2.1. 1.2.2. 1.2.3. 1.3. 1.3.1. 1.3.2. 1.3.3.
2.7.
1.4. Phase diagram and crystallography 1.5. Sample preparation
2.1. 2.2. 2.3. 2.3.1. 2.3.2.
4. UPt,
state
4.1. 4.2. 4.3. 4.4. 4.5. 4.6.
Introduction Magnetic properties Thermal properties Specific heat Thermal expansion
2.4. Elastic properties 2.5. Transport properties 2.5.1. 2.5.2. 2.5.3. 2.6. 2.6.1. 2.6.2.
conductivity
address:
Centre
state
with Pd
Introduction Specific heat and thermal expansion Susceptibility and magnetization Electrical resistivity Superconductivity Discussion remarks
5.1. Normal phase 5.2. Superconducting
and Hall
phase
Acknowledgements
studies
Magnetization Magnetoresistivity
* Present
alloyed
5. Concluding
Electrical resistivity Thermoelectric power, thermal effect Point-contact spectroscopy High-magnetic-field
the superconducting
3.1. Introduction 3.2. The Meissner effect 3.3. Specific heat 3.4. Critical field studies 3.5. Fermi liquid analysis 3.6. Miscellaneous
theories
the normal
studies resistivity susceptibility
3. UPt,:
Spin fluctuations Fermi liquid theory Odd-parity superconductivity
2. UPt,:
magnetostriction
Electrical Magnetic
2.7.1. 2.7.2.
and systematics
The uranium-platinum compounds The heavy-fermion compounds The UX, compounds Relevant
Forced
High-pressure
References
de Recherches
037%4363/87/$03.50 0 (North-Holland Physics
sur les Tr&
Basses
Temptratures,
Elsevier Science Publishers Publishing Division)
B.V.
CNRS,
BP 166X, 38042,
Grenoble,
France.
82
A. de Visser et al. I UPt,, heavy fermions
1. General Introduction 1.1.
Introduction
The recent discovery of a class of intermetallic compounds, the so-called heavy-fermion system, has caused a great deal of excitement among solid-state physicists. On cooling these materials to liquid helium temperatures the coefficient, y, of the electronic term in the specific heat, i.e. the term linear in temperature, attains a giant value. For an ordinary metal the y-value is of the order of l-10 mJ/K’mol. For a heavy-fermion system the y-value amounts to 400-1000 mJ/K*mol, i.e. roughly a factor 102-lo3 larger. Such a large y-value is usually explained by the existence of heavy electrons or quasiparticles at the Fermi surface. The effective mass of these heavy electrons amounts up to lo’-lo3 times the free electron mass. Hence the name heavy fermions. Of course we have to bear in mind that this definition of a heavy-fermion system is a somewhat arbitrary one. Systems with intermediate yvalues are found as well, and it is not precisely clear where to draw the border-line. After the discovery of the first heavy-fermion system CeAl, in 1975 by Andres, Graebner and Ott [l], extensive research programs have been carried out in order to look for equivalent systems. Fortunately, CeAl, was not a singularity in nature, and, up to date, about ten other heavyfermion compounds are known, among which the most prominent ones are CeCu,Si,, CeCu,, UBe,,, UPt,, U,Zn,, and UCd,,. Each system has its own exciting low-temperature properties and a variety of ground states are observed: superconductivity in CeCu,Si,, UBe,, and UPt,, antiferromagnetic order in U,Zn,, and UCd,,, and no order at all in CeAl, and CeCu,, at least not down to temperatures as low as - 10 mK. In particular, the discovery of superconductivity in some of the heavy-fermion systems was quite unexpected. Recent speculations about a nonconventional pairing mechanism (odd parity) have attracted much attention to this class of materials. At present, heavy-fermion behaviour is mainly found in intermetallic cerium and uranium-based
and superconductivity
compounds, which suggests the formation of a narrow 4f (Ce) or 5f (U) band near the Fermi surface, resulting in a strongly interacting electron system. In this paper we focus on one of the uranium-based heavy-fermion superconductors: UPt,. This material is of particular interest since it exhibits a coexistence of pronounced spinfluctuation phenomena [2] and superconductivity (at T, = 0.5 K [3]). This article is organized as follows. In the remainder of section 1 we present a rather general introduction to the physics of UPt,. Successively we present: (i) a review of related compounds and a discussion of the systematics in the occurrence of heavy-fermion behaviour, (ii) some elements of the theoretical models that might be applicable to UPt, and (iii) some metallurgical aspects. In section 2 we present and analyse the bulk of experiments performed on UPt, in its normal state: magnetic, thermal, elastic and transport properties will be discussed, as well as the outcome of high-magnetic field (35 tesla) and high-pressure studies. Section 3 deals with the superconducting state of UPt,. After a discussion whether superconductivity in UPt, is a bulk property (studied by the Meissner effect), we describe a number of experiments that have been performed in order to characterize the superconducting state. Another way to study the anomalous properties of UPt, is by performing alloying experiments. In section 4 the first results of such experiments on a series of U(Pt,_,Pd,), compounds are presented. In section 5 we review a series of remaining experiments and subsequently, present the concluding remarks concerning the normal and superconducting phase of upt,. Since the field of heavy fermions evolves rapidly, it will be impossible to present a fully up to date insight in the properties of UPt,. This paper compasses experimental and theoretical work performed before end 1986 and was written in partial fulfilment of a Ph.D. degree (A. de Visser, Thesis, University of Amsterdam, 1986). 1.2. Related compounds
and systematics
In order to present a background
for the un
A. de Visser et al. I UPt,, heavy fermions
100 mJ/K*molU. In the case of UPt, and UPt, this can also be ascribed to spin-fluctuation effects, although not as pronouned as in UPt,. Such a picture is supported by resistivity measurements [lo]. Curie-Weiss analyses of the high-temperature susceptibility data show in all cases small deviations from linearity in the x-l vs. T curves. Nevertheless, for UPt and UPt, an effective can be deduced. moment, lteff, of 3.5&/U-atom This is consistent with a 5f2 or 5f3-configuration in an LS-coupling scheme. The pefL,,,-valuesfor UPt,, 3.0&U-atom, and UPt,, 4.3h,lU-atom, can not easily be interpreted in the LS-coupling of scheme, which suggests the importance crystal-field effects at high temperatures or, perhaps, the need for a more elaborate coupling scheme.
usual properties of UPt, we shall in the following sections briefly review a number of compounds related to UPt,. Successively we shall consider the uranium-platinum compounds, the heavyfermion compounds and the intermetallic UX, compounds. The uranium-piat~num compounds Four uranium-platinum compounds do exist [4]: UPt, UPt,, UPt, and UPt,. A survey of their crystallographic and magnetic properties is listed in tables I.1 and 1.11, respectively. Since the nearest uranium-uranium distance, d,_, , varies from 3.61 A in UPt, up to 5.25 A in UPt,, one would expect, at least according to the Hill plot [9], an increasing localization with increasing Pt content. However, the systematics of the Hill plot are not followed in the U-Pt system. Ferromagnctism, possibly of itinerant nature, is observed for UPt, superconductivity for UPt,. UPt, and UPt, do not order. As we shall describe in detail in section 2, the compound UPt, shows pronounced spin-fluctuation phenomena. Evidence hereto mainly arises from a T3 ln(T/T*)-cont~bution to the specific heat. This low-temperature upturn results in the anomalously large y-value of 422 mJfK*molU. It is noteworthy that also for UPt, UPt, and UPt, are found: enhanced y-values rather l.2.1f
Table 1.1 Some crystallographic
parameters
of uranium-platinum
Compound
Structure
upt
orthorhombic
upt, upt,
hexagonal orthorhombic cubic
Table 1.11 Magnetic properties of uranium-platinum
83
and superconductivity
The heavy-fermion compounds As mentioned in section 1.1 it is not clear-cut which compounds belong to the class of the heavy-fermion systems. In table 1.111 a number of compounds with large y-values have been listed. Given a somewhat arbitrary choice of y > 400 mJ/K’mol, at present, 8 heavy-fermion systems are known. The y-values enumerated in table 1.111 have been derived from the low1.2.2.
alloys d u-u
Vlll
(A)
( 10m6m3/mol)
Ref.
CrB
3.61
26.61
Fl
MgCd, InN&(dist) AuBe,
4.12 3.81 5.25
42.43 33.62 61.53
it;
151
alloys. Data taken from ref. [2] and ref. [8]
Compound
Type
&ff (&U-atom)
w0 (pa/U-atom)
Y (mJ/K2molU)
~(4.2 K) ( 10m9m3/molU)
upt upt, upt, UR,
ferrom. param. superc. param.
3.5 3.5 2.8 4.3
-0.5 -
114 89 422 95
45 51 103 34
-
A. de Visser et al. I UPt,, heavy fermions
84
and superconductivity
Table I.111 Materials having y > 200 mJ/K*mol. For ordered materials, y has been obtained temperatures
just above
Ref.
1620 1600
no order no order
-
[II [ll,
4.10 5.13 5.13 6.56 4.39 4.12
1100 1100 900 840 535 # 422
superc. superc. antif. antif. antif. superc.
-0.6 0.9 3.4 5.0 9.7 0.5
1131 [I41
tetragonal hexagonal cubic
4.20 3.65 4.96
350 260 250
cubic cubic cubic
4.63 3.25 4.87
242 234 >200
$;
CeAI, CeCu,
hexagonal orthorhombic
4.43 4.83
CeCu&
upt,
tetragonal cubic cubic cubic rhombohedral hexagonal
CeRu,Si, YbCuAl UC% NpSn,
U,Zn,,
NC, CePb,
temperature
T order
Structure
UCd,,
to zero
Type
Compound
UBe,, NpBe 13
by extrapolation
from
Tordcr
TmJ/K’mol)
no order no order antif. antif. antif. antif.
15.0 9.5 6.6 1.1
121
1151
1161
1171 [2, 3. 181 [191 [201 [21,221
I231 (241 1251
* Per mole U.
limit of the term in the specific heat linear in temperature:
temperature
(1.1) For ordering materials y has been obtained by extrapolation to zero temperature from temperatures just above Torder. The resulting y-values are enhanced with a factor lo’-lo3 with respect to ordinary metals. The origin of the enhancement is one of the key questions in understanding the heavy-fermion systems. We shall return to this point in the next section. At present a substantial number of materials with intermediate y-values is known. Some of them having y > 200 mJ/K’mol are listed in table 1.111 as well. Since their properties are quite similar to those of the heavy-fermion systems, these materials may likely serve as a link between the ordinary metals and the heavy metals. It is interesting to note that diluted materials can have huge -y-values as well, albeit per mole impurity. For instance, dilute Ce in Y has a y-value of 3500 mJ/K*molCe [26]. However, we shall confine ourselves to (pseudo) intermetallic compounds. A more elaborate introduction to the heavy-
fermion systems can be found in the review paper by Stewart [27]. From table 1.111 it follows that heavy-fermion behaviour is not directly correlated with a particular crystal structure. The same holds for the various ground states: superconductivity, magnetic order or no order at all. A common feature in these materials is the large f-atom spacing. larger than the Hill limit (- 3.5 A). Several attempts have been made in order to find some systematics in the occurrence of heavy-fermion behaviour. Below we present an overview 01 these attempts, available to us at the moment. Meisner et al. [28] have plotted the y-values oj a number of compounds as function of the f. atom spacing, and find a smooth variation (fig 1.1). These authors discern two regimes: (i) al sufficiently short spacing direct f-atom interac tion prevails, leading to roughly an exponentia dependence of y on spacing, and (ii) at large] spacing indirect f-atom interaction prevails, wit1 delocalization of the f-electrons accomplished b! intervening atoms. A significant correlation factor is thought to b< the ratio of the measured susceptibility to thr one calculated from the observed (enhanced y-value:
A. de Visser et al. I IJPt,, heavy fermions
8.5
and superconductivity
in the same units as above, and p,rr in units of pr+, eq. (1.4) reads as: R,
= 1.74 x 10’ +
.
(1.5)
Yt-k!ff
Fig. 1.1. The specific heat coefficient y vs. f-atom spacing plotted on logarithmic scales, for superconducting (circles), magnetic (squares) and non-ordering (triangles) (nearly) heavy-fermion compounds (after Meisner et al. [28]).
(1.2) Note that since we use rationalized molar units a factor p0 appears in the denominator of eq. (1.2). With the molar susceptibility in units of m3/mol and y in J/K2mol eq. (1.2) reads as:
Stewart [27] has evaluated R, for the class of heavy-fermion systems. The trend noticed by Stewart suggests that for the superconducting compounds R, is roughly a factor 2 smaller than for the non-superconducting ones. Kadowaki and Woods [30] have stressed the importance of a universal relationship between the resistivity and the specific heat. They noted that the ratio of the coefficient, A, of the T2term in the resistivity and r*, Aly2, has a common value of 1.0 x 10-j p.RcmK2mo12/(mJ)2 (fig. 1.2). These studies have failed, however, to reveal any satisfying criterion for the occurrence of heavy-fermion behaviour. Besides, the meaning of these correlations is not always clear. Furthermore, we have to remind that it is not in all cases obvious how to distract the proper physical parameters from the data, especially when a strong sample dependence and/or anisotropies are present. 1(-4
R = 5.80 x lo6 ; .
(1.3)
R equals 1 if the density of states probed
by x and y have the same enhancement factor. The use of the ratio R has the advantage that it can be obtained directly from the experimental values of x and y. DeLong et al. [29] have evaluated R for a large number of heavy and “nearly” heavy-fermion compounds. The correlation they found formulates as follows: superconductors exhibit values of R 6 2.0, whereas other systems have R b 2.0. Another correlation factor put forward is the “Wilson ratio”: (1.4) Here perr = gpu(J(J + 1))“2 is the effective moment as follows from a Curie-Weiss analysis of the high-temperature susceptibility. With x and y
Fig. 1.2. The coefficient A of the T*-term in the resistivity plotted versus the y-value for a number of (nearly) heavyfermion compounds (after Kadowaki and Woods [30]).
86
1.2.3.
A. de Visser et al. I UPt,, heavy fermions
The UX, compounds
Most of the compounds UX,, with X a 4d or Sd-metal or an element from group III or group IV, crystallize in the cubic AuCu, structure. Therefore, a direct comparison with UPt, (hexagonal structure) is substantially hampered. Nevertheless, a qualitative consideration of the hybridization mechanisms in these compounds has revealed some interesting systematics (Koelling et al. [31]). For the UX, compounds, discussed here, the f-atom spacing ranges from - 4 A up to - 4.8 A. Since these values exceed the Hill limit (- 3.5 A), one would expect local f-electron behaviour, due to the absence of direct f-f overlap. However, Koelling et al. have pointed out that the degree of delocalization of the felectrons is mainly governed by the hybridization of the f-electrons with s, p and d-orbitals on neighbouring non f-atom sites. Assuming that the f-ligand hybridization decreases with increasing UX-distance, the systematics in the occurrence of the electronic and magnetic properties of the UX, compounds (see fig. 1.3) can be understood. For small X-atoms (strong hybridization) one then expects Pauli paramagnetism, and for large X-atoms local moment behaviour. Intermediate hybridization would result in a large density of states, hence heavy-fermion behaviour and/or spin-fluctuation phenomena. This trend is nicely followed for the group III and group IV
and superconductivity
elements, moving down along the column in the periodic table (fig. 1.3). For the 4d and 5dmetals the systematic is not so clear. UPd, is the only system that exhibits localized Sf-electrons, it has, however, a dhcp structure. Although URu, and UIr, are superconductors with transition temperatures of 0.2K [32] and 3.OK [33], respectively, these compounds do not show any sign of heavy-fermion behaviour, which is illustrated by the y-values of 13.5 and 17.0mJ/ K*molU, respectively. 1.3. Relevant theories From the experimental data it follows that the physics in the heavy-fermion systems is rather complex. The localized character of the f-electrons at high temperatures crosses over to a delocalized one at low temperatures. The high y-values point to pronounced f-density of states, and, hence, to a strongly interacting electron liquid, assuring the presence of many-body effects. Moreover, electrical resistivity measurements have revealed that most of the heavyfermion systems exhibit the Kondo effect. The latter feature is often taken as a starting point for theoretical models. The Kondo Hamiltonian [34] was originally written down to describe the exchange interaction of a single magnetic impurity, with spin S( = i ), with a conduction electron, having spin s.
Ye,= -2Js.S.
Fig. 1.3. Rough overview of the electronic and magnetic properties of UX, compounds, where X is indicated in the figure (after Koelling et al. [31]). Vertical bars give the y-value. Horizontal hatching indicates Pauli paramagnetism. Diagonal hatching indicates spin-fluctuation behaviour. Cross hatching indicates local-moment behaviour. Superconductivity is observed in UIr,, URu, and UPt,.
(1.6)
For a negative coupling parameter J, the impurity spin is completely compensated at low temperatures, giving rise to a Kondo singlet. This is accompanied by a logarithmic increase, with decreasing temperature, of the resistivity, proportional to ln( T/T,). It can be shown that the Kondo Hamiltonian can be generated from the more general Anderson Hamiltonian [35]. Most attempts to model the properties of the heavyfermion systems on microscopic grounds make use of a somewhat modified version of the Anderson Hamiltonian. Hereto, the solid is thought
A. de Visser et al. I UPt,, heavy fermions
to consist of a periodic array of magnetic impurities, the Kondo lattice. Although the formation of the Kondo lattice seems physically reasonable, its low-temperature properties differ rather drastically from the single-ion Kondo case. In the latter, the resistivity shows the whereas in the former characteristic upturn, coherence sets in at low temperatures, resulting in a T2-temperature dependence at approaching T = 0 K. At present, it is not clear how the coherence should be incorporated in the model. A further complication in describing, in particular the actinide heavy-fermion systems, arises from the uncertainty in the number of f-electrons per atom. Whereas a free U-atom has a f3configuration, this needs not to be the case in the solid. Apart from the possibility of a mixed or intermediate valence state, different configurations must be considered: f’ (J = 5/2), f* (J = 4) and f3 (J= 9/2). Furthermore, the groundstate might not be a pure Hund’s rule state. For a free atom, the separation between equal J-components of different LS-multiplets is of the order of 6000 K. A certain admixture might occur due to spin-orbit interaction. For the configurations f* and f3 this results in a groundstate that consists for only 85% of the Hund’s rule component [36]. In addition, in the solid, crystal-field splitting of the J-multiplet should be taken into account. Little is known about its strength in the heavyfermion systems. It would be illuminating if we could present a density of states picture for the heavy-fermion case. In order to do so, it is tempting to plot a narrow f-band near the Fermi level. However, it is more likely that we have to deal with an induced many-body band originating from the admixture of the relevant electron states. At low temperatures a very narrow many-body band of width - lo-100 K is present, causing the characteristic Fermi liquid behaviour. Due to the number of degrees of freedom in the liquid the entropy, and hence the electronic specific heat is strongly enhanced. In some of the heavy-fermion systems the high density of states is decreased by ordering phenomena, therewith decreasing the symmetry and degrees of freedom. This seems to be the case in the heavy-fermion superconduc-
87
and superconductivity
tors and magnets. When the temperature is raised, the many-body band disappears, and a broad Kondo resonance is formed. At high temperatures the characteristics of the system are determined by the broad f-peak below E,, hence a more localized picture results. XPS and BIS spectroscopy have been used to shed some more light on the f-density of states. The experimental resolution, in best cases about 300 K, is, however, not sufficient to resolve the expected narrow structures in the density of states. The discussion presented above can be summarized by stating that a microscopic picture of the heavy-fermion systems is far from being settled. For a more comprehensive overview of the recent developments in attacking the heavyfermion problem, we refer to the papers of Varma [37] and Lee et al. [38]. 1.3.1.
Spin JEuctuations
Elaborate theories exist to account for the effect of paramagnons or spin fluctuations on the temperature dependence of the susceptibility, specific heat and resistivity. BCal-Monod (ref. [39] and references therein) has calculated the temperature dependence of the paramagnon contribution to the susceptibility:
x (2 5
- 1.2 ( g)*)S2T2]
.
(1.7)
Here x0 is the Pauli susceptibility and S is the Stoner enhancement factor. N’ and N” are the first and second derivatives of the density of states, N = N(E,), with respect to the energy, evaluated at E, The sign of the characteristic T*-term depends on N, N’ and N”. If N"(E,)< 0, X(T) decreases continuously, when T increases. N"(E,)> 0 is a necessary condition for x to first increase with T, and thus to exhibit a maximum, since at higher temperatures x ought to reach a Curie-Weiss law. The paramagnon contribution to the specific heat has been evaluated by Doniach and Engelsberg [40], and Brinkman and Engelsberg [41],
88
A. de Visser et al. I UPt,, heavy fermions
who first applied it to “He. Its characteristics are two-fold: (i) an enhancement of the linear electronic term, which can be seen as an enhancement of the effective mass of the electrons, and (ii) a negative T3 lnT-correction to this linear term. The combination of these two terms results in a low-temperature upturn in c/T. The enhanced linear term is given by c = y,,
c m
T=
y,,(l + h,,)T,
(1.8)
where m* represents the effective electron mass, and A,, is called the spin-fluctuation enhancement factor. Here ‘yOrepresents the unenhanced electronic coefficient. The unenhanced electronic term in the molar specific heat is given by
c” = 3/oT =
2N,N(E,)rr2k2, 3
T >
with N(E,) per atom per spin direction. ing the T3 lnT-contribution, the ratio enhanced to the unenhanced electronic heat can be written as [41]:
(1.9) Includof the specific
C
7i= C
(1.10) where (1.11)
k,TE
= pci2m
,
(1.12)
and PI = PllP,
.
(1.13)
Here TF is the Fermi temperature, and p, a cut-off momentum in the model. The paramagnon contribution to the resistivity has been evaluated by several groups (see, for instance, ref. [42] and ref. [43]). Their models predict a T2-law at low temperatures,
and superconductivity
p = A T2 ,
(1.14)
where A m ( T5f)-2. A subsequent strong increase in resistivity near the spin-fluctuation temperaat rather ture, T,r, is followed by a saturation high values in the room-temperature region. Unfortunately, neither the T2-term in x, the T” lnT-term in c, nor the T2-term in p, can be taken as decisive evidence for spin fluctuations. In addition, information may be obtained from magnetic-field effects on x and c (“quenching” of spin fluctuations). However, the measured field effects are often small and difficult to interpret [39]. Inelastic neutron-scattering experiments enable one to measure the excitation spectrum, Imx( q, w), and might, therefore, be considered as a powerful tool, as well. To conclude, one has to consider a bulk of experiments, in order to decide whether one deals with a spin-fluctuation material, or not.
1.3.2. Fermi liquid theory In 1956 Landau put forward a phenomenological theory for the macroscopic properties of an interacting normal fermion system, at low temperature: the Fermi liquid theory [44]. As we shall see, Fermi liquid theory is also applicable to the highly correlated heavy-fermion state. The kernel of Landau’s theory consists of a one-toone correspondence between the eigenstates of the non-interacting electron gas and those of the interacting liquid. The interaction between the electrons leads to a dressing of the particles, thus increasing their effective mass, m*. This interaction can be expressed in terms of the dimensionless Landau parameters, F:, where A = s(a) refers to the spin symmetric and spin antisymmetric parts of the interaction (see for instance Pines and Nozieres [45]). The Landau parameters appear in the expressions for the susceptibility, the specific heat, the compressibility, etc. The temperature independent molar Pauli susceptibility of the interacting Fermi liquid is given by 2 m”k,V,,,
x = cLo/-%
Tr2h2
1 l+F;’
(1.15)
A. de Visser et al. i UPt,, heavy fermions and superconductivity
89
where k, is the Fermi wave number. x is enhanced with respect to the Pauli susceptibility of the non-interacting system:
where the enhancement
factor s” is given by
s”=_ m*/Fn
(1.17)
l+F;’
Just as in the paramagnon model, the specific heat of the interacting Fermi liquid can be written as the sum of an enhanced linear term and a T3 InT-term. The enhanced linear term accounts for the specific heat of the dressed quasiparticles. Its coefficient is given by kiV,m* k, y* z.z 3fi2 *
(1.18)
The existence of a T3 InT-term in the specific heat of an interacting Fermi liquid has been demonstrated by Pethick and Carneiro 1461. This contribution is due to small momentum (q) transfer processes between the quasiparticles. The corresponding contribution to the specific heat per unit volume is given by (1.19) where h2k; TF = ___ 2m*k,
(1.20)
is the Fermi system, n-2N_
V
+ (A;)‘(
(1.16)
x = ix0 >
temperature
_- k:.
2
37T
of the
interacting
(1.21)
3
is the particle number density, B” = 2N~N(~~)~‘/V~
,
and T, is a cut-off expressed as
temperature.
I-
;
A;)
-2A;A;]
,
(1.23)
where ws = 1 and o, = 3. The A: parameters are the Landau scattering amplitudes, which are related to the Landau parameters, FF, by A; =
F: 1+ F;/(21+
1) ’
(1.24)
In eq. (1.23) terms with I 2 2 have been neglected. Since, as we will see, B” is positive, the T3 lnT-term can be seen as a negative correction to the enhanced linear term. The physical process responsible for the T3 lnT-term is the reof quasiparticle-quasihole peated scattering pairs. As in the paramagnon model, this gives rise to spin fluctuations. Therefore, we can say that the T3 fnT-term accounts for the spin fluctuations in the liquid. Note, however, that the Fermi liquid theory does not speak about the nature of the interaction, it is merely based on its mathematical description. This expression for the specific heat has successfully been applied to the Fermi liquid 3He. Since we have allowed onfy small q scattering, we cannot rule out that other scattering processes in the liquid may give rise to additional terms. Such terms, however, have not been observed. The T3 fnT-contribution in eq. (1.19) comes from long-wavelength spin fluctuations. There is however an additional contribution to the T3term in eq. (1.19), from shorter wavelength spin fluctuations, that was not included by Pethick and Carneiro. Recently, Coffey and Pethick [47] have worked out the coefficient of this term in more detail. Below we show their approach how to proceed in order to determine the spin-fluctuation parameters from the specific heat. The full expression for the specific heat is given by
(1.22) c=y*T+&,T3+ST31n(T/T*).
(1.25)
B” may be Here y* is the enhanced linear coefficient, &, is
90
A. de Visser et al. I UPt,, heavy fermions
the usual phonon coefficient (T + O,), 6 is the coefficient of the pure T3 lnT-term and S lnT* = Psr is the coefficient of the spin-fluctuation contributions to the T3-term. In order to perform a computer fit to the data, the more suitable expression
and superconductivity
If one
considers
B’
= -
B”
T*
=
fluctuations
are
equal
and
opposite:
e-&f’8
(1.28)
T* is often denoted as the spin-fluctuation temperature. The expression for the T’ lnT-term and the T3-term, from the spin-fluctuation contribution to the specific heat per unit volume, is written by Coffey and Pethick as
AC = $
nk,B”
(:I’
+ q] ,
[lnc&i
which differs from eq. (1.19) ment of the logarithm. Here
only
in the argu-
4, = qc/P, Y
(1.30)
where pF is the Fermi cut-off momentum.
momentum
and
q, is a
(1.31) and
q,T, T*
We see that 33r”
S=Tnk,z=FY
.
(1.32) (per unit volume)
B‘
3~”
to
(A;,)’
(A;)* >
might
=
_
;
(1.34)
be further
simplified
(A;)3
+ ; (A:,)*
_
* B”
z,
(1.33)
to
$ +; ,
(1.35)
since Ai = 1 for electron systems [48]. In order to determine the spin-fluctuation parameters one may proceed as follows. Input parameters of the model are y*, p* and 6, determined by a fit to eq. (1.26) (actually we fitted to c/T), the phonon k,, coefficient Pph and the Fermi wave number which might, for instance, be obtained from configurational arguments (eq. 1.21). We now successively calculate m* from eq. (l.lS), T, from eq. (1.20), B” from eq_ (1.33), Ai from eq. (1.35), F: from eq. (1.24), S from eq. (1.17), & from eq. (1.27) and T* from eq. (1.28). The last important parameter, S,, may be calculated from the relationship [47]:
(1.29)
n=ln
_ ;
(1.27)
T* is a characteristic temperature, at which the T” 1nT and T’-contributions to the specific heat spin
+ ; (A;)*
where
P* = Pph+ P,r .
from
the I= 0 contributions
(1.26) which
may be used,
(A;)”
;
+ ; c = y*T + /3*T3 + 6T” 1nT
only
B”. one has
T*k, ” = q(F;)(l
+ F;)(u;)*m*
’
(1.36)
where W(F:) is a dimensionless function, which depends weakly on FP, [47]. The cut-off parameter 4, provides us with a test of the model, since we assumed q, < pF.
1.3.3. Odd-parity superconductivity The discovery of superconductivity in the heavy-fermion compounds CeCu,Si, , UBe ,7 and UPt, came as a total surprise. In these systems a strongly correlated electron liquid persists, down to the ordering temperature, as is indicated by the huge specific heat. Such a correlation between the electrons is, in general, thought to suppress the conventional phonon-mediated type of superconductivity. In the superconducting state the coefficient of the electronic term in the specific heat (nearly) vanishes, suggesting that
A. de Visser et al. I UPt,, heavy fermions
the f-electrons play an important role in the formation of the condensate. Some other unusual observations have been made concerning the superconducting state: (i) the electronic properties reveal a power law, rather than an exponential temperature dependence, at least in the case of UBe,, and UPt,, (ii) anisotropy effects are present, and (iii) an unusual stability of the Cooper pairs with respect to a magnetic field, given the low T,-values. These remarkable features have led to speculations upon an unconventional type of odd-parity superconductivity [49, 501. In this section we shall present some theoretical background for odd-parity superconductivity. However, we have to realize, that many aspects of the newly developed theories are subject of lively discussions. For a comprehensive overview of the theoretical and experimental aspects of conventional superconductivity we refer to the standard work of Parks [51]. The present theoretical models that deal with odd-parity superconductivity in the heavy fermions, are largely based on a close analogy to the triplet or p-wave superfluidity in 3He. However, in offering an explanation for the unusual behaviour in the superconducting state one should also consider higher orbital momentum states (1 Z 2), odd as well as even. A direct comparison with 3He is substantially hampered, since in the solid spin-orbit coupling and crystal fields lower the symmetry. Taking the crystal symmetry into account, several authors calculated the allowed odd-parity states from group theoretical arguments, for cubic (UBe ,3), tetragonal (CeCu,Si,) and hexagonal (UPt,) crystal structures [52-5.51. At present, a point of great interest is how to distinguish experimentally an odd-parity super-
and superconductivity
91
conductor. Varma [37] predicted the temperature dependence of a number of electronic properties, assuming odd-parity pairing. These properties depend on the form of the energy gap over the Fermi surface. He discerns three cases: 1) The superconducting gap exists everywhere on the Fermi surface, like in the conventional superconducting state of liquid 3He. The quasiparticle density of states has a gap. 2) The superconducting gap vanishes at points on the Fermi surface, near which it varies quadratically with the energy. This is like the ABM or axial state in superfluid “He. 3) The superconducting gap vanishes at lines on the Fermi surface, near which it varies linearly with the energy (this is like the polar state in an isotropic system). The results of his calculations for the specific heat, sound attenuation, nuclear magnetic relaxation time and thermal conductivity are listed in table l.IV. In contrast herewith, Rodriguez [56] predicts a sound attenuation proportional to T and T’, for the polar and axial states, respectively. Besides, Coppersmith and Klemm [57] pointed out that ultrasonic attenuation depends also on the relative orientation of the sound waves and the nodal structure, resulting in a more complex situation. Regarding group theoretical arguments Blount [54] stated that a polar state (case III) of odd parity is not possible in cubic or hexagonal crystals. However, his arguments have been contradicted by Varma [37] and Scharnberg and Klemm [58]. Several other experiments have been suggested to clarify the nature of the superconducting state. We mention (i) tunneling experiments [59, 601, and (ii) studies of the anisotropy of the upper critical field [58, 611.
Table l.IV Form of the temperature dependence of the specific heat, sound attenuation, nuclear magnetic relaxation conductivity in the superconducting state (T < T,), for various pairing mechanisms (after Varma [37]) Case: Gap: Specific
heat
o,~o, T,‘lT-’ I” K,
I Isotropic-like
II Axial-like
III Polar-like
- exp( - A/T) - exp( - A/ T) - exp( - A/T) -exp(-A/T)
- (T/A)’ - (T/A)’ - (T/A)’ -(T/A)’
-
(T/A)’ (T/A)’ (T/A)’ (TIP)’
time
and thermal
A. de Visser et al. I UPt,, heavy fermions
92
Although, as we shall see, the outcome of a number of experiments is consistent with oddparity superconductivity, one still should be cautious to take this as hard proof. On the one hand, gapless and anisotropic singlet superconductivity, or I= 2 (d-wave) pairing, might lead in special cases to similar unusual features in the low-temperature properties. On the other hand, the theory still needs refining. A consensus on what microscopic pairing mechanism is applicable to the heavy-fermion systems is lacking. Several authors have presented mechanisms in favour of odd-parity superconductivity. We mention Varma [37], Anderson [50], Beal-Monod [62], Li [63] and van der Mare1 and Sawatzky [64]. Others have presented models in favour of singlet superconductivity (Razafimandimby et al. [65], Tachiki and Meakawa [66] and Hirsch ]671). A different approach to shed light on the nature of the superconducting state can be found in the phenomenological Fermi liquid theory. Valls and TeSanovid [68] applied an almost localized Fermi liquid model to the heavy-fermion systems, in close analogy to “He. Bedell and Quader [69] used an induced interaction approach. Pethick et al. 1701 focused on a Fermiliquid analysis of the T3 lnT-contribution to the specific heat of UPt,. All these analyses favour triplet superconductivity. Central in their approach is an expression for the superconducting transition temperature, that was first applied to ‘He by Patton and Zaringhalam [71]: TL” = 1.13aT,e1’KI
and superconductivity
The expressions for g, have been derived in the s-p approximation, that is to say, only s and p-waves contribute to the scattering processes. Furthermore, the forward scattering limit has been applied, which means that the interaction involves only small momentum transfer processes (with the angle between the planes of the incoming and outgoing momentum vectors near zero). Employing the forward scattering sum rule, (1.39) and neglecting g,=
$(-A;I-3A;-2A;),
g, = ; (A; + A;)
g, = & i
(-
l)‘(A; l)‘(A;
- 3A;) + A;)
.
(1.40b)
Estimates for the values of A;( = l), A,“, and Al, from the various normal state properties of UPt,, lead to g,
(1.41)
Unfortunately, the model fails to predict T, since (Yis not known. Pethick et al. [70] suggest to test the model by calculating the relative pressure dependence of T,, assuming a pressure independent LY, and, subsequently, by comparing this value with the experimental one. From eq. (1.41) we obtain aln T,
(-
with 12 2, we have
(1.37)
.
Here CYis a numerical constant, T, is the Fermi temperature and g, represents a pseudopotential. If g, < 0, the potential is attractive; j = 1 indicates triplet pairing, j = 0 singlet pairing. The pseudopotential can be expressed as a function of the Landau scattering amplitudes, defined in eq. (1.24), g,, = $ c
components
,
aln T,
6
aA:
dP
(1 + A;)*
ap
ap
(1.42)
The right hand side of eq. (1.42) can be calculated either from the pressure dependence of the specific heat, or from a combination of specific heat and thermal expansion data at zero pressure. We shall return to this subject in section 3. 1.4.
(1.38b)
=--
Phase
The
phase
diagram
diagram
and crystallography
of the uranium-platinum
A. de Visser et al.
/ UPt,, heavy fermions and superconductivity
93
system has been constructed by Park and Fickle [4] It is shown in fig. 1.4. The system is characterized by four intermetallic compounds: UPt formed peritectoidally at 1234 K, UPt, formed peritectically at 1643 K, UPt, melting congruently at 1973 K and UPt, formed peritectically at 1733 K. The crystal structure of UPt, has been determined by Heal and Williams [7]. UPt, crystallizes in the MgCd,-type of structure (lowtemperature form of SnNi,). This closed-packed hexagonal structure, with 2 formula units per cell, belongs to the space group P6,immc. The atomic positions in the unit cell are given by: 2U at c 6m2 6Pt at h mm
&,$,a; x,2x, a ; x, 2x, ; ;
i,+,+, 2x,x, 3x,x,
$ ; $ ;
X, i, $ , x,x, f )
where the ideal value of x equals 2. The lattice parameters are given by: a = 5.764 A, c = 4.899 A and c/a = 0.850. These crystallographic data have been confirmed by X-ray (Loopstra, unpublished) and neutron (Helmholdt , unpublished) experiments. Up to pressures of 500 kbar no crystallographic phase transitions have been observed [72]. The stacking of the atomic layers is shown in fig. 1.5. The nearest U-U distance is measured between atoms in adjacent layers and amounts to 4.132 A. The unit-cell volume, 0, equals 140.96 A3, the mean atomic volume equals 17.62 A’, and the molar volume, V,,
1234K -y-w, 1000
-\’
978 K
,;‘UP)
t326K
@cl -+ u (0) 0’ ”
10
20
30
40 PLATINUM
50
60
70
80
(at %I
Fig. 1.4. Phase diagram of the uranium-platinum (after ref. [4]).
90
100 Pt
system
Fig. 1.5. Stacking of the atomic layers in the UPt, structure. Open circles: Pt; closed circles: U; c-axis not drawn to scale. Note that throughout this paper the a and b-axis hexagonal plane) are considered to be orthogonal.
(in the
equals 4.244 x 10m5m3/mol. With the molecular weight of M = 823.3 g the density amounts to 1.940 x lo4 kg/m3. Crystallographic parameters for UPt, UPt, and UPt, can be found in table 1.1. 1 S.
Sample preparation
In general, the preparation of actinide intermetallics asks for special techniques, due to high reactivity and high melting point of the compounds. Moreover, radioactivity and toxicity of actinides ask for additional restrictions in handling these materials. The investigated UPt, samples have been prepared at the metallurgy department of the Natuurkundig Laboratorium by A. Menovsky. Experiments have been performed on three types of samples: arc-melted polycrystalline samples, Czochralski-grown single-crystalline samples and single-crystalline whiskers. As starting materials served 2’SU-depleted uranium (nominal purity 99.8%), supplied by Koch-Light Ltd., and platinum (99.999%), supplied by Material Research Corporation. The RRR for the uranium amounted to 2.7. Polycrystalline samples were prepared by arc
94
Fig. 1.6.
A. de Vimer et ai.
(a) Scanning
microphotograph
I UPt,, heavy fermions and s~perco~d~ct~v~ty
of UPt,
whiskers
grown
melting the appropriate amounts of the constituents in a titanium-gettered argon atmosphere. In order to obtain a homogeneous compound the buttons were remelted several times. In some cases, the melt was cast into a watercooled copper crucible, in order to obtain a particular shape, e.g. a cylinder. A further homogenization of the samples was achieved by Table l.V Lattice parameters
and density
of UPt,
(hexagonal
out of the melt;
(b) whisker
with well developed
surface.
annealing in evacuated quartz ampoules at a temperature of - 1200 K, for a period of 7-10 days. During the annealing procedure the samples were wrapped in tantalum foil, to avoid contact with the quartz. A piece of uranium served as getter. Additional shaping of the sampies was done by means of spark erosion. In this manner spheres, cylinders and cubes (with plan-
structure)
a
c
Sample
(A)
(A)
Exp. density (lO’kg/m’)
Caic. density (104kgim’)
Ref
Polycrystalline Czochralski-grown single-crystal
5.764 f 1 5.7520 rt 12
4.899 2 1 4.8970 ‘- 33
1 .Y45
1.940 1.949
171 j731
Whiskers
5.7530 + 4
4.9010 t- 4
-
1.947
[741
A. de Visser et al. I UPt,, heavy fermions
parallel surfaces) could be made, as well as samples with holes. The crystal structure was checked on small amounts of powdered samples Lattice with a Guinier-de Wolff camera. parameters were determined from the X-ray patterns, with SiO, as standard. Single-crystalline UPt, was prepared with a modified tri-arc Czochralski method [73]. First, the melt is prepared on a water cooled copper hearth. Then the crystal is pulled out of the melt, with, in the case of UPt,, a pulling rate of about 5 mm/h. An as-cast cylinder (3 mm diameter, 20 mm length) served as a polycrystalline pulling tip. During the whole growth procedure, tip and hearth rotate in opposite direction, with rates of 15 and 50 rpm, respectively. With a fourth arc a piece of uranium is melted, in order to getter the -argon atmosphere (P = 300 Torr). In this way,
Fig. 1.7. Cross-section
of UPt,
whisker;
and superconductivity
95
large cylindrical crystals could be grown, with sizes limited to 8 mm diameter and 30 mm length. Laue pictures were taken to check the single-crystalline nature of the grown crystals. Lattice parameters, determined by Weissenberg X-ray measurements, are listed in table l.V. The experimental density is in good agreement with the calculated one (see table 1.V). Shaping of the samples was done by means of spark erosion. In some cases, the bulk crystal, or the cut pieces, were annealed at 1200 K, during 24 h, in order to reduce strains. A third type of sample can be found in tiny needle-like single-crystalline whiskers. The whiskers jump spontaneously out of the stoichiometric UPt, melt, at fast cooling on a water-cooled copper hearth [74]. A “liquid-solid” screw dislocation growth
orientation
of the crystal
axes as indicated.
96
A. de Visser et al. I UPt,, heavy fermions
mechanism has been suggested. An example of an arc-melted button with whiskers on its surface ’ is shown in fig. 1.6. Most of the whiskers have a regular shape, with a typical cross-section in the form of an equilateral trapezium (see fig. 1.7). The c-axis is always found to be parallel to the whisker axis. Typical whisker dimensions are (mass 15(a-axis) x 60(b-axis) x lOOO(c-axis)pm’ - 18 pg). Lattice parameters, determined from Weissenberg X-ray measurements, are listed in table l.V. Stewart et al. [3] have reported the growth of single-crystalline UPt, from a bismuth flux. No information on lattice parameters is available. The UPt, specimens are stable with respect to oxidation and can be stored under normal atmosphere. Loss of chemical purity, due to radioaccan be neglected. Natural tive (CY) decay, uranium consists of the isotopes 23xU (99.283%), 235U (0.7110%) and 234U (0.0054%), with (Ydecay rates of 4.51 x lo”, 7.1 X 10’ and 2.47 X lo5 year, respectively. In the case of depleted uranium the percentage of 235U is somewhat lower, 0.4% for the Koch-Light batch.
2. UPt,:
the normal state
2.1. Introduction In this section we present, analyse and discuss the experimental results, focusing on the normal state of UPt,. We begin by recalling susceptibility measurements [75, 21 that were made in a first attempt to characterize UPt,. We then successively concentrate on the thermal, elastic and transport properties. Next we deal with highstudies. high-pressure and magnetic-field Specific-heat and high-field magnetization measurements performed by Frings et al. [2] are recalled in the relevant sections. In addition, some experimental results on the compounds UPt, UPt,, UPt, and UAl, are presented. 2.2.
Magnetic properties
Schneider tibility data
and Laubschat [75] reported suscepon a polycrystalline UPt, sample in
and superconductivity
the temperature range 2.5-210 K. These authors found a weak maximum in the susceptibility curve at 15 K. By comparing the susceptibility with that of materials like NpPd, and PuPt,, these authors suggested antiferromagnetic ordering in UPt, at 15 K. The paramagnetic region could be described by a Curie-Weiss behaviour with pCL,rf = 2.85,uL,lU-atom, between 23 and llOK, and with pL,rf= 2.61&U-atom, between 110 and 210K. The presence of a maximum in the susceptibility has been confirmed by Frings et al. [2]. These authors measured the susceptibility of polycrystalline UPt, (unpublished data), in the temperature range 4.2-300 K, in fields up to 1.3 T, using a pendulum magnetometer, and in the temperature range 300-1000 K, using a Faraday balance. From the high-temperature data an effective moment of 3.0 ? O.l~,/U-atom was deduced. Their data taken on an unannealed Czochralski-grown single-crystalline sample, shaped into a sphere (diameter 3 mm), are shown in fig. 2.1. This figure shows that the susceptibility is strongly anisotropic. The maximum in the susceptibility is only present for field directions in the basal plane, for which an isotropic susceptibility results. The x-values at 4.2 K, 87 X lOmy and 99 x lo-’ m’/molU, and the x,,,-values, 91 x lo-” (at 15 K) and 111 x lo-” m’/molU (at 17 K), obtained on polycrystalline samples by Schneider and Laubschat [75] and Frings et al. (unpublished), respectively, fall well in between the data for the basal plane and c-axis. The differences between the absolute x-values for various polycrystalline samples can mainly be attributed to the presence of crystallites with preferred orientations. A Curie-Weiss analysis of the data, for a field direction in the basal plane and along the hexagonal axis, in the temperature range 50-250 K, results in a value for Barr of 2.6 ? 0:2p,/U-atom, for both field directions. The paramagnetic Curie temperature amounts to - 50 and - 170 K for the basal plane and hexagonal axis, respectively. The large uncertainty in ~l.,~rmakes it difficult to interpret. Applying Hund’s rules the value of 2.6p,/U-atom is close to the value for a full
A. de Visser et al. I UPt,, heavy fermions
I50
97
and superconductivity
5
125
UPt3
ah
I3
B B 0
75 50
0
-%++
c +++
A +
g
$2 a
+
0
+
+
25 #
0
Susceptibility
of single-crystalline
UPt,;
8
+
I
1
+
1
200
100
0
Fig. 2.1.
I
+
a-axis
(0),
b-axis
300
(A) and c-axis
(+)
(after
ref. [2]).
quenched f’( j_+ = 2.54j_&J-atom), f2(2.83&U-atom) or a full f4(1.68&U-atom) configuration. Besides, small deviations from the Curie-Weiss behaviour , possibly due to crystalfield effects, hamper the analysis. Nevertheless, the susceptibility data strongly suggest local moment behaviour at temperatures above 50 K. From the data in fig. 2.1. it is obvious that UPt, is not a simple two sublattice antiferromagnet. A complex type of order can not be excluded. However, in an ac-susceptibility experiment (H < 0.2 Oe) any sign of an anomaly near 17 K was lacking. As we shall show in the next section the thermal properties provide strong evidence for spin fluctuations. In that respect the maximum in x might be explained as a spinfluctuation phenomenon, according to eq. (1.7).
sample [2], are shown in a plot of c vs T, for T < 20 K, in fig. 2.2. In fig. 2.3 we plot the data as clT vs T2, which shows that a description with a linear and cubic term is far from appropriate. The large y-value of 422 mJ/K2molU, obtained from an extrapolation to 0 K, illustrates the heavy-fermion behaviour. The field-effect on the specific heat is small. For the polycrystalline sample it is in the order of 1 ? 1% at 5T (fig. 2.3). Data taken on the single-crystalline sample, with a field of 5 T applied along the a and c-axis, for temperatures below 5.5 K, are shown in fig. 2.4. At low temperatures the normal-state specific heat of UPt, obeys the relationship
2.3.
The presence of a T3 ln( T/ T*)-contribution to c strongly evidences the presence of spin-fluctuations (see section 1.3). However, before presenting details of such an analysis, other possible contributions to c need to be considered. First we stress that no sign of long-range antifer-
Thermal properties
2.3.1. Specific heat The normal-state specific heat of an unannealed Czochralski-grown single-crystalline sample [18, 761, and data taken on a polycrystalline
c = y*T
+ &T3
+ 6T3 ln(TIT*)
.
(2.1)
A. de Visser et al. I UPt,, heuvy fennions 12
t
*
”
’
,
and superconductivity
”
7
I,
”
” a
2-
I
0 0
s
I
L
,
I,,
5
*
1
L
I,
‘
I
15
20
T'&)
Fig. 2.2. Specific heat of single (0) and poly (Cl) crystalline UPt, (squares taken from ref. [Z]).
romagnetic order is found near 17K, the temperature at which the maximum in x is observed. Crystal-field contributions to c, in the investigated temperature range, can be excluded, since no Schottky-type of anomaly is observed. Local
magnetic moments (due to the presence of impurities) in an internal field can give rise to an upturn in c/T at low temperatures. However, in that case one would expect an external field of 5 T clearly to influence the specific heat, which is
0.6
0
100
200
300
400
T2 (K2) Fig. 2.3.
Specific
heat of single-crystalline
UPt, at OT (0), and polycrystalline UPt, at OT (0)
and
ST (+).
A. de Visser et al. i UPt,, heavy fermions
99
and superconductivity
0.50
9
+
0.45
H//a
A H//c
-$ -T a -t0
&+A gp4 00
0.4-O
+A 0
+A+d+
A+,
000
op Am A
A
0.35
I
0.30
I
10
0
I
I
20
30
T2 (K2) Fig. 2.4.
Specific
heat of single-crystalline
UPt,
at 0 field (0)
apparently not the case. The possibility that the low-temperature upturn is due to details of the band structure can be discarded as well. The anomalous increase in c/T below - 10 K suggests a huge narrow peak in the density of states. Of course, the effect of a magnetic field of 5 T on such a narrow structure would be enormous. If one, with a specific-heat experiment, would only probe the band structure, the data with and without field would largely differ, which is not the case. To conclude: contributions to c from long-range magnetic order, crystal-field effects, magnetic impurities and band-structure effects are absent. Given the appropriateness of eq. (2.1) to describe the low-temperature specific heat of UPt,, we analyse the data in a spinfluctuation model. Similar specific-heat curves have been observed for the compounds UAl, [77], TiBe, [78] and UCo, [79]. However, in these compounds the low-temperature upturns in c/T are less pronounced: the y-values amount to 143, 56 and 35 mJ/K2molU, respectively. Due to the presence of the T3 ln(TlT*)-term, in combination with anomalies in magnetization and susceptibility curves, these materials are labeled as spinfluctuation compounds. The specific heat of
and in a field of 5 T directed
along
the a( +) and c-axis
(A).
these materials has been analysed in the paramagnon model [40, 411. In the case of UPt,, we concentrate, in this chapter, on the Fermi liquid analysis [47], as outlined in section 1.3.2. Hereto, we have performed a least-squares fit to the specific-heat of single-crystalline UPt,, for several temperature intervals, employing the relation: c
- = y*+p*T2+ST21nT. T
(2.2)
The coefficients deduced from these fits are listed in table 2.1. The values for the coefficients in eq. (2.2) do not differ much for the different temperature intervals, indicating that eq. (2.2) gives a proper description of the low-temperature specific heat. The fit performed in the temperature range 1.2-20 K follows all 85 data points to within 1% (see fig. 2.5). The results agree fairly well with a similar fit on polycrystalline UPt, [79]. Coefficients derived from fits to the specific-heat data obtained by Stewart [27] and Brodale et al. [80] are listed in table 2.1 as well. In the following paragraphs we focus on the physical parameters that can be deduced from y*, p* and 6.
A. de Visser et al. I UPt,, heavy fermions
100 Table 2.1 Coefficients
derived
from
a 3-parameter
fit to the specific
Interval
heat of UPt,
(K)
&K’moIU)
P* (mJ/K’molU)
1.4-20 1.4-15 1.4-10 6-10 8-20 1.2-17
422 423 421 424 413 422
-
3-17 0.5-20 0.5-G 0.5-4 1.2-20 1.2-10
# #
# Fit to phonon
and superconductivity
according
to eq. (2.2)
(SmJ!K’molU)
Ref.
3.12 3.80 3.53 3.78 3.45 4.18
1.38 1.41 1.30 1.40 1.30
118, 761 118, 761 [lg, 761 [l& 761 118, 761 18, 791
452 422 429
- 6.42 - 4.24 - 5.04
2.25 1.60 1.90
1271
426
- 6.92
3.22
PO1
424 418
- 3.44 - 2.77
0.97 0.72
1811
corrected
specific
1.54
PO1 147, SO]
WI
heat.
The y *-value enables us to determine the effective mass, m*, of the interacting electrons (es. (1.18)), P rovided the Fermi wave number, k,, is known. We assume the Fermi surface to be spherical (this is probably a rather crude assumption). In that case the value for k, can be obtained from an analysis of the temperature derivative of the upper critical field (- 4.4 T/K) that will be presented in section 3, and amounts
to 1.06 A-‘. The Fermi wave number can also be deduced from configurational arguments: k, = (!$?j”‘,
(2.3)
where 2 is the number of heavy electrons per unit cell volume (LJ) that take part in the forma-
0.6
100
200
400
300
T2 (K2) Fig. 2.5. Measured (0), calculated phonon (A, after dotted lines represent S-parameter fits to the measured in table 2.11.
ref. [82]) and phonon-corrected (fit Ia) and phonon-corrected
(0)
specific
heat of UPt,.
The solid and
(fit Ha) specific heat, with coefficients
listed
A. de Visser et al. I UPt,, heavy fermions and superconductivity
tion of the electron liquid. With 2 = 6 (3 felectrons per U-atom, 2 U-atoms per unit cell), and R = 140.9 A3, we calculate the corresponding value for k, of 1.08A’. Notice that in the Fermi liquid description the electrons in the interacting and non-interacting system are described by one and the same Fermi wave number. With the y-value of 422mJ/K2molU and k, = 1.06 A-‘, we obtain the large effective mass m* lm, = 3fi2y* l(kik,V,m,) = 180, the Fermi temperature T, = h2k:/2k,m* = 275 K and the Fermi velocity ug = hk,/m* = 6800 m/s. The renormalization of the effective mass by a factor of - 200, leading to the low values for TF and UC, is a characteristic feature for the heavy-fermion compounds. The coefficient of the T3-term is given by p* = p,, + &, where &, represents the usual phonon coefficient, and & = - 6 In T* represents the contribution from spin fluctuation to the T3-term. T* = exp( - &r/S) is a characteristic temperature, which is often called the spinfluctuation temperature. Mathematically it is the temperature at which the 6 T3 In T-contribution and the &T3-contribution to the specific heat are equal, but opposite in sign. The phonon contribution to the specific heat, in a first approximation given by cph = &,,T3 (T G O,,), is commonly obtained by measuring a nonmagnetic analog of the compound one is interested in. Unfortunately, no information on a non-magnetic analog of UPt, is available. From the published phase diagrams the existence of, for instance, ThPt, is not clear. Nevertheless, two groups performed XPS-measurements on a ThPt, sample, but gave no details about its structure [75, 831. Frings [8] estimated &,,, at 0.85 mJ/K4molU (0, = 210 K), a value derived from a two-parameter fit to the specific heat of a polycrystalline UPt, sample, in the temperature range 9-15 K (with ‘y. = 225 mJ/K2molU). This value for p nicely agrees with the one we deduced from sound velocity measurements at 0.76 mJ/K4molU (0, = temperature: room 217K). With &, = 0.85mJ/K4molU and the coefficients p* and 6 from the fit in the temperature range 1.2-20 K (see table 2.1), we calculate & = - 4.57 mJ/K4molU and T* = 27 K.
101
The coefficient of the T3 ln( T/ T*)-term can (eq. Tg) be expressed as 6 = 3n2y*Bsl(10 (1.33)), where B” can be written as a function of the I = 0 antisymmetric Landau scattering amplitude A:. The parameter Ai will play an important role in discussing the nature of the superconducting state (section 3). Following the procedure outlined in section 1.3.2, we calculate successively B” = 83.74 (eq. (1.33)), Af: = - 3.71 (_eq. (1.35)), Fl = Ai (1 - A:)-’ = - 0.788 and S = (m* lm)( 1 + Fi)-’ = 849. The cut-off parameter, S,, can be calculated from eq. (1.36), where 9(Ft) can be determined from ref. [47]: 4, = 1.40. Finally, we calculate T, = 94K (eq. (1.31)) and n = 1.57 (eq. (1.32)). These results, together with rather similar values, deduced from a fit in the temperature range 1.2-10 K, are summarized in table 2.11. The cut-off parameter, 4, = qClpF, is rather large. This implies that scattering processes over a large part of the Fermi surface contribute-to the quasiparticle energy. The value for S = x/x0 = 849, as calculated from the specific heat, represents the susceptibility of the interacting system over that of the non-interacting system. With this value we calculate a temperature x = 342 x lo-” susceptibility independent m3/molU, using eq. (1.15). A direct comparison with experiment is hampered since the measured x is not Pauli-like. Furthermore, the observed anisotropy in x complicates the analysis. Yet, a mean value for x = 90 x lo-” m3/molU at 1 K, assuming x = $ xa,h + f x,, is considerably lower than the calculated value. However, Pethick and Pines [84] argue that, as a result of spin-orbit coupling, the effective moment peff of a heavy electron may not be equal to the free-electron Bohr magneton. In addition there can be nonquasiparticle contributions to the susceptibility, and consequently the measurements of x for heavy-fermion systems do not give direct information about the Landau contribution. In order to let the calculated and the measured x coincide, one needs to replace ps by p,rf = 0.51 E*.~in eq. (1.15). The value for this effective moment largely differs from the value derived from the Curie-Weiss analysis at high temperatures, peff = 2.6p,/U-atom. However, these effective
A. de Visser et al. I UPt,, heavy fermions
102 Table 2.11 Coefhcients of 3-parameter various parameters derived in units of mJ/K4molU Fit
Interval
Ia Ib IIa IIb
1.2-20 1.2-10 1.2-20 1.2-10
and superconductivity
fits to the measured (I) and phonon (II) corrected in a Fermi liquid analysis, employing k, = 1.06 w -I
K K K K
Y*
P*
422 421 424 418
-
3.72 3.53 3.44 2.77
specific
heat of UPt,,
employing eq. (2.2). and and p *, &, , p,, and 6
; y * in units of mJ/K2molU
P ph
P-r
0.85 0.85
-
‘4:
F;I
6 4.57 4.38 3.44 2.77
1.38 1.30 0.97 0.72
m* Fit
(m,)
Ia Ib IIa IIb
180 180 181 179
275 276 274 278
27 29 35 48
83.74 79.45 57.73 44.71
moments arise from different models and therefore their relation is not quite clear. At this point we want to stress that, in order to derive the various parameters in the Fermi liquid analysis, we assumed a spherical Fermi surface. As the physical properties of UPt, are strongly anisotropic, this is certainly a crude approximation. However, the effect of anisotropy on the Landau parameters is probably small [84]. For instance, since the deduced average value for F: is large and negative, one needs only a comparatively small anisotropy in Fi, in order to account for the large anisotropy in the susceptibility (eq. (1.15)). We also emphasize that this analysis is rather insensitive to the exact choice of k,. A considerable variation in the value for kr( 2: 1.1 A-‘), for instance 30%) does not qualitatively alter the results derived in the Fermi liquid analysis. Recently, inelastic neutron-scattering experiments enabled Renker et al. [82] to measure the phonon density of states and hence to calculate the lattice contribution to the specific heat in a more sophisticated way. At temperatures between 9 and approximately 20 K the Debye temperature amounts to 200 K (close to the value we quoted above), whereas below 9 K the lattice hardens and 0, gradually increases up to a value of 320 K at 1 K. The lattice contribution and the electronic contribution after correcting for the lattice are shown in fig. 2.5. This figure illustrates
-
3.71 3.64 3.24 2.95
~ ~ -
0.788 0.785 0.764 0.747
s
WF6)
4,
849 837 767 708
0.168 0.168 0.170 0.172
1.40 1.46 1.61 1.98
that the c/T-value for the electronic specific heat is only reduced by a factor of 2 at - 20 K, with respect to the value at 0 K. The entropy is large: St?, - R ln2 up to 20 K. The coefficients of a fit to the purely electronic specific heat [Sl], employing eq. (2.2) with p* = &, are listed in table 2.1, for two different temperature intervals. This fit has the same quality as the fit to the non-phonon corrected specific heat, as can be seen in fig. 2.5. An important difference is the 30% smaller 6value. The various parameters deduced in the Fermi liquid analysis, listed in table 2.11, yield, however, similar conclusions. Up to now we have neglected the possibility that the electron-phonon interaction may lead as well to a T3 lnT-contribution to the specific heat. Coffey and Pethick [47] pointed out that this contribution is of opposite sign compared to the from spin fluctuations. T3 In T-contribution Furthermore, they demonstrated that the ratio of the coefficients of the T3 lnT-contribution coming from the electron-phonon interaction to that coming from spin fluctuations is less than for UPt,. Since &,, is of the order of 0.03 &h 2 or less for normal metals, the electron-phonon contribution to the T” lnT-term may be neglected. A model that accounts for the field effect on the specific heat is at present not available. However, it is interesting to compare qualitatively the field effect on c/T (or y in the low-
A. de Visser et al. I UPt,, heavy fermions
temperature limit) with the susceptibility, ing the thermodynamic relationship:
apply-
(2.4) Given the susceptibility data on the single-crystalline sample (fig. 2.1), one expects a positive A(clT) = (clT)(, - (clT)I,=,, at low temperatures, for a field direction in the basal plane and a (small) negative A(clT) for a field along the c-axis. Our specific-heat data, taken in a field of 5 T, reveal hardly any field-effect on c/T (fig. 2.4). Stewart et al. [85] measured the specific heat of a single-crystalline sample, in the temperature range 2-17 K, with a field of 11 T applied in the basal plane. These authors observed a 10% increase in c/T at 2 K, a crossing of the field and zero-field curve at about 6 K, and a negative field effect on c/T of 6% at 17 K. Brodale et al. [SO] observed a nearly constant positive A(c/T) (2% at 1 K) in a field of 8 T in the temperature range 0.3-4 K. Applying eq. (2.4) this corresponds to x m T2, as expected in the low-temperature limit in the paramagnon model (eq. (1.7)). A precise analysis of the measured ,y, in order to look for a T*-term, is troublesome, due to the large anisotropy. Data taken below 1.2 K, on the same polycrystalline sample as described in fig. 2.3, reveal a small positive A(c/ T), in a field of 1 T ([86]; see fig. 3.6). However, within the experimental accuracy of 2 2%, data taken at 2, 4 and 8 T are identical with the 1 T results. In summary, we state that the field-effect on the specific heat is small, and hardly exceeds the experimental uncertainty. The observed (small) positive A(clT) for a field direction in the basal plane, consistent with the susceptibility data, can be explained as an enhancement of the spin-fluctuation phenomena in the low-temperature limit, for this particular field direction. 2.3.2. Thermal expansion Thermal expansion measurements in the temperature range 1.4-100 K have been performed on the same single-crystalline sample as the one used for the specific heat experiments, employ-
and superconductivity
103
capacitance ing a sensitive three-terminal method [87]. Hereto the sample was shaped into a 5 x 5 x 5 mm3 cube, with pairwise planparallel surfaces. Data were gathered stepwise: below 15 K each temperature step (~0.3 K) was made three times, i.e. up, down and up again, whereas above 15 K only steps upwards (up to 3 K) were made. The relative accuracy in the results decreased with increasing temperature, due to small temperature gradients over the cell, and is limited to 3 x lo-’ K-’ in (Y near 100 K. The thermal expansion results, obtained in several runs, are presented in fig. 2.6 and, in some more detail, in fig. 2.7, plotted as (Y= L-‘(dLldT) vs T. The thermal expansion of UPt, is strongly anisotropic. The volume expansion is obtained by summing the data for the three crystallographic directions: CY,= (Y,+ CX~ + (Y,. Hereto the curves for ff,, (Yeand (Y,have been approximated with a 10th grade polynomial. In figs 2.6 and 2.7 we plotted a,/3 as well. In fig. 2.8 CY,- a, is plotted, illustrating that the maximum anisotropy in CYis found near 14 K. The c/a-ratio as function of temperature, normalized to 1 at 1.4 K, is shown in fig. 2.9. Relative length changes, AL/ L, with respect to the length of the sample at 1.4 K, for the basal plane and the hexagonal axis, are shown in fig. 2.10. The length changes have been calculated from the areas below the appropriate (Yvs T curves. The relative change in volume was calculated from (AVIV) = (AL/L),
+ (AL/L),
+ (AL/L),.
In figs 2.11 and 2.12 we present data for polycrystalline UP&, UPt,, UPt, [88] and UAl, [87]. Comparing the results for single- and polycrystalline UPt, we conclude that preferred orientations were present in the latter sample. In analysing the results for UPt,, we first remark that the linear thermal expansion coefficient is strongly anisotropic. The maximum anisotropy is found at about 14 K, close to the temperature at which the maximum in the susceptibility is found (17 K). This strongly suggests a correlation between the magnetic and thermal properties: the onset of spin fluctuations, confined to the basal plane, causes a relatively large dilation in the basal plane, accompanied by a contraction along the hexagonal axis. In a plot of
A. de Visser et al. I UPt,, heavy fermions
104
and superconductivity
15
10
c
5 Y
P 0 c ci
O
UPt3 -5
-10 0
Fig. 2.6.
Coefficient
of linear
20
thermal
40
expansion
60
for single-crystalline
the volume expansion, (Y,, a maximum results at 10 K (fig. 2.7). At 100 K (Y reaches a value between 8 and 9 X lop6 K-‘, which points to a dominant lattice contribution in this temperature region. In analogy to the specific heat we plot
UPt,;
80
a(U),
b(x)
100
and c(A)
axis and q,/3(0).
the thermal expansion data as (Y/T versus T* in fig. 2.13. Notice that the maximum at 10 K is obscured in this plot. The anomalous behaviour at low temperatures becomes visible as a pronounced upturn.
C AA A A AA
A A A
A
A !
0
20
10
30
-r 09 Fig. 2.7.
Coefficient
of linear
thermal
expansion
for single-crystalline
UPt,
A. de Visser et al. I UPt,, heavy fermions
0
20
80
60
40
I
loo
0
T (K>
Fig. 2.8. Difference between the coefficients of linear mal expansion for the basal plane and for the hexagonal of upt,.
theraxis
In order to relate the volume expansion to the specific heat, usually, an analysis employing Gruneisen parameters is made. Physically meaningful Griineisen relations emerge when the entropy, or rather a part of the entropy Sj, can be written as Sj( TIT,(V)). Here T,.(V) is a (volume
105
and superconductivity
20
I
1
I
I
60
80
I
40
4
xx]
T (K> Fig. 2.10 Relative length change with respect to 1.4 K for single crystalline UPt, for the a(O), b(X) and c(A) axis, and l/3 of the relative volume change (0).
dependent) characteristic temperature of the entropy term, e.g. TF or On. The dimensionless Gruneisen parameter is defined as (see for instance ref. [89]):
or in terms of the free energy - (dF,lc?T),
Fj, with S, =
(2.6) Since
0.999’ n ’ 0
20
’
’
b
40
’ 60
’
’
80
T 0.9 Fig. 2.9.
The c/a-ratio
vs. T for UPt,.
’
’
where K = - (1 lV)(dV/~3p)~ compressibility, and
is the isothermal
100
(2.8)
A. de Visser et al. 1 UPt,, heavy fermions and superconductivity
106
UPtlj
A
\
0
20
lJPt9
4.0
60
80
100
J” M
Fig. 2.11. Coefficient of linear thermal expansion for polycrystalline
one may write, dropping the index j: r--
Vffv KC,
(2.9)
’
UPt,(O),
UPt,(O),
UPt,(A)
and UAl,( x).
be defined, for instance, for the linear (electronic) and cubic (phonon) terms in c and (Y,. Writing LY(= 43) as (2.10)
a:=yaT+p:hT3, Such so-called proper Griineisen
parameters
can
one obtains (in the single band uniform scaling model) : J-
e
=
3YcxVm aln T, -=__ 8lnV ’ KY
0
0
(2.11)
100
200 T2
20
40
60
80
100
-r Cc<> Fig. 2.12. Relative iengt~ change with respect to 1.4 K for polycrystalline UPt,(O), UPt,(Cl), UP&(A) and UAl,( X).
300
400
Cd:“)
Fig. 2.13. (1/3)ar,/T plotted vs. T” for single-crystalline UPt,. The solid and dashed lines present 3-parameter fits according to eq. (2.15) in the temperature intervals up to 10 and 20 K, respectively.
A. de Visser et al. I UPt,, heavy fermions
and (in the Debye model) (2.12) where y and Pph are given in the appropriate units per mole and V, represents the molar volume. r, and rph amount to - 2 for ordinary metals. In the case of UPt,, the expression for the specific heat (eq. (2.1)) may serve as a starting point to derive an expression for the thermal expansion. Applying thermodynamic relations we find a=-y;l*T+PflhT3+P~T3+S~T31nT,
(2.13)
with (2.14a)
(2.14b)
P:=-~(~)T+&(~)T7 m m
(2.14c)
and (2.14d) By defining p,* = PI” + p t and /3: = - S, In TE , we obtain the equivalent of eq. (2.1): (Y=~~T+~~~T~+~~T~I~(TIT~). A three-parameter
(2.15)
fit to the thermal expansion
Table 2.111 Coefficients from a 3-parameter fit to eq. (2.15) c(P) data [80], employing eq. (2.14)
for the thermal
Interval (K) Fit Fit Calc. Calc.
1.4-20 1.4-10 0.5-20 0.5-18
and superconductivity
data has not the same quality as the one for the specific heat, as is illustrated in fig. 2.13. The coefficients 72, p,* and S, have been listed in table 2.111 for temperature intervals up to 10 and 20 K. Given the relatively large variation in p,* and S,, we conclude that eq. (2.15) may only serve as a first approximate description of the thermal expansion data. Thermal expansion and specific heat as function of pressure yield the same information, as can be seen from eq. (2.14). Recently, the specific heat at pressures up to 8.9 kbar has been measured by Brodale et al. [80]. Values for r,” , pz and S,, calculated from their 3-parameter fits to the specific heat, have been listed in table 2.111, as well. In view of the variation of the coefficients with the temperature interval for which the fit is made, the internal consistency of the two experiments is not as excellent as has been suggested in ref. [70]. The Griineisen parameter for the enhanced linear term is given by
r; =
(2.16)
’
KY*
and amounts to the huge value of 68 at 1 K (K = 0.48 Mbar-’ ). This illustrates the sensitivity of TF with respect to the volume. Mutatis mutandis one may define Griineisen parameters for the p*T3 and S,T3 InT-term. However, these are improper Gruneisen parameters, without a significant physical interpretation. We calculate from the fit parameters in the temperature range up to 20 K: r, = 101 and r,, = 126. In order to calculate Tz we write pX = j3:” + pf = pp = - 6, In Tz , since /3Eh is at least two orders of magnitude smaller than pz. We derive Tz = 29 K.
expansion
(q/3)
of UPt,,
P:: (IO-‘K-‘) 1.08 1.16 0.87 1.06
107
-
1.77 2.90 1.2 1.9
and coefficients
calculated
a<, (lo-‘Km’) 5.27 9.88 3.8 6.4
from the
A. de Visser et al. I UPt,, heavy fermions
108
Luthi [90] stresses the importance of electronphonon coupling in the heavy-fermion systems; the electron states are influenced by a coupling of the band electrons to the elastic waves (strains). In the so-called deformation potential model, Liithi is able to interrelate the specific heat, thermal expansion and elastic constants. In this way the thermal expansion of UPt, can be understood as due to a certain density of electron states. In a two-band model, with band widths of - 30 and - 50 K, Liithi is able, using the experimental data for the specific heat and elastic constants, to account, at least qualitatively, for the thermal expansion data (see also ref. [91]). The model needs two bands in order to account for the extrema in the thermal expansion. Their physical meaning remains, however, somewhat unclear. In the last part of this section we compare the results for UPt, with thermal expansion measurements on UPt,, UPt, and UAl,. In a plot of (Y/T vs. T2 (fig. 2.14) deviations from a linear behaviour become visible at low temperatures. These anomalies are found in the specific heat as well, and are ascribed to spin-fluctuation effects [S]. In orthorhombic UPt, this contribution might be obscured due to crystallites with preferred orientations. The unenhanced coefficients Table 2.1V Coefficients from 2-parameter UPt,, UPt,, UPt, and UAlz
fits to 01 and c, enhanced
UPt”
Parameter
Units
Y‘,
10~W2
::
mJ/K’molU
Y,* *
lO_“K
and superconductivity
from two-parameter fits conform eq. (2.10) (see solid lines in fig. 2.14), for UPt, and UPt, in the temperature range 50 < T2 < 250 K2, and for UAl, in the temperature range 150 < T2 < 400 K*, have been collected in table 2.IV. The enhanced coefficients are listed as well. Data for UPt have been taken from ref. [92]. By comparing the thermal expansion data with specific heat data on the same samples (except in the case of UPt,) values for r,, rp,, and c have been derived. Hereto we estimated K from an interpolaI
I
100
0
200
I
300
400
T2 (K*) Fig. 2.14. a/T plotted vs. T* for UPt,(O), UAI,( X). The solid lines represent 2-parameter eq. (2.10).
linear coefficients,
and the corresponding
Griineisen
UPt,(&) and fits conform
parameters
upt,
upt,
5.6
11
-
1006.4
75 25
-
12.4
108
1.8
33
24 89
422 6X
957.3
133 20
* -
UPQ 0.8 24 13
UAI, 14.0 52 21
;f
mJ/K’molU
P, P r Ph
10 “‘Km4 mJiK4molU
0.55 1.4 0.4
0.71 1.03 1.2
0.32 # 0.85 2
2.3 4.2 2.1
0.72 0.65 0.9
K V,
lO~“m’/N 10~5m’/molU
0.69 + 2.66
0.58 + 3.36
0.48 4.24
0.48 + 6.15
1.35h 3.59
* Calculated + Estimated “Ref. [92]. hRef. [93].
with &,, = 2 value.
for UPt,
A. de Visser et al. I UPt,, heavy fermions
2.4. Elastic properties Crystals with hexagonal symmetry - such as UPt, - require five independent elastic constants to specify the relationships between stresses and strains. In terms of stiffness coefficients, they may be listed as cli, c,~, cr3, c33 and cd4. These five quantities can be determined by sending ultrasonic waves through properly oriented crystal specimens and measuring the wave velocities, which depend on the elastic constants and the density. Obviously, at least five measurements must be made for hexagonal crystals. Relations between wave velocities and elastic constants can be found in e.g. Mason [94]. In order to determine the five independent elastic constants at room temperature [87], we
Kl7=--
1 da -= a dP
Type mode
Direction of propagation
Direction of particle motion
Velocity
I I I I I II II II
long. shear long. shear shear quasi long. quasi shear shear
C
C
3860 1385 3993 2076 1388 3754 u, = 1827 u* = 1753
: b b b’ b’ b’
C
b’ Cl
a’ = a
C33 c33(C,,
+
-
Cl3
Ciz)
(2.17) -
2Cf3
c,~ = 1.421 Mbar as follows from a combination
Sample
in ab-plane b a
109
followed the approach of McSkimin [95]. For these measurements we used conventional ulwith quartz transducers trasonic equipment, tuned at 20 MHz (longitudinal waves) and 5 MHz (transversal waves). Two single-crystalline samples, from the same batch as prepared for the thermal expansion sample, were machined into the proper shape, by means of spark erosion. One specimen (I) served for propagation directions along the c-axis (1 = 7 mm) and b-axis (I = 5 mm). The other specimen (II) was shaped for a propagation direction (b’) at 45” between the b and c-axis, but perpendicular to the a-axis (I = 2 mm). The measured sound velocities are listed in table 2.V, together with the formulae for the elastic constants and the derived values. From the velocities u1 . . . u,, measured on sample I, cll, c,*, cX3and cl4 could be obtained. The other sample (II) served to derive c13. For the calculation of c13, from a combination of uI, u2, u3 and u6 or u,, or from a combination of u3, us, u6 and u, we refer to McSkimin [95]. us served as a cross check. The c,,-values are based on the calculated density p = 1.940 X lo4 kg/m3. The accuracy in the ciivalues is 1%) except for c,~, in which case it is 3%. The elastic constants enable us to calculate the compressibility, K. The linear compressibilities in the basal plane, K,, and along the hexagonal aXiS, K,, are given by
tion between the values for U (0.99 Mbar-‘) and Pt (0.38Mbarr’), encouraged by the estimated (0.53 Mbar-‘) and experimental (0.48 Mbar-‘) value for UPt, . The resulting &-values are larger than for ordinary metals, giving rise to the question whether the intersections of the solid lines in fig. 2.14 and the ordinate indeed represent the coefficients of the unenhanced linear terms. The large r z-values are indicative for the relatively high density of states in these compounds. The &,-values are relatively small. Unfortunately, for none of these compounds thermal expansion data on non-magnetic analogs are available. Therefore, a more thorough analysis of the various contributions to the thermal expansion is, at this moment, not possible.
Table 2.V Sound velocities and elastic constants of UPt, at room temperature. The c,,-values are based on the density p = 1.940 x lo4 kg/m’
and superconductivity
(m/s) u, =
UI = u3 = u, = v, = U6=
’
of c,, and c,,.
Elastic constant (Mbar) cjl = pu; = 2.891 CA1= P”2’ = 0.372 ‘C,, = pu; = 3.093 c,, = !(c,, - c,,) = PU; = 0.836 CA1= pu; = 0.374 cL, = 1.732 c,., = 1.695 ic,, - fC,? + qc,, = pu;
A. de Visser et al. I UPt,, heavy fermions and superconductivity
110
up13
Q
i :
“...
29.0 -
C
32.0 t
.a.
31.8 n
C33
a.
Cl1 Y
5
‘.
‘.
28.5 -
281’
5
*. *. *. a. *.
200
10”
40°
3.85 \
1
31.6 -
Y
31.4 0. *. .
T(K)
“E ‘O”
,” P
31.2 -
l.
‘x.
*. Q.
I
I
200
1
1
400
9.5 -
Q
l*..
I T(K)
L :
b
*. -.
C&L 380
:yyy,
-**...._
375 0
200
a
Fig. 2.15.
T(K)
Elastic
constants
and 1 dc KC=---=
c dP
The volume
- 2c,, + Cl1 + Cl2 %(C,l
+ Cl,)
compressibility
-
2cL
is given
(2.18) . by
1 dV K=--
-=2K
Vdp
u
+K..
(2.19)
’
A small anistropy is observed: K, = 0.164 Mbar-’ and K, = 0.151 Mbar-‘. Therefore, at applying a relative increase in the pressure on UPt,, c/u-ratio dln(c/a)ldp = K, - K, = 0.013 Mbar-’ is expected. The volume compressibility amounts to 0.479 Mbar-‘, which fits nicely in between the values for uranium, 0.99Mbar-‘, and platinum, Finally, the Debye temperature, 0.38 Mbar-‘. On, has been determined from the elastic con-
Table 2.W Elastic constants
of UPt,
in Mbar
(= 10”Nim’);
^
c,~ is calculated
200
0
600
of UPt,
(after
400
T (K)
600
ref. [91])
stants. This may be accomplished by integrating the sound velocities (or elastic constants) over all crystallographic directions (see e.g. Alers [96]). The resulting value equals 217 K. Yoshizawa et al. [91] have measured the temperature dependence of cS3, Cam,cl 1 and ce6 from 1 up to 600K. These data, shown in fig. 2.15, clearly reveal low-temperature anomalies of 0.03-1.1% in cii, for all models for T < 50 K. The longitudinal modes (c,, and Cam) show an upturn, whereas the transversal modes (c,, and ce6) pass through a maximum. These anomalies are related to the deformation potential coupling, as mentioned in the previous section. In table 2.VI we compare our results with the data of ref. [91] at 300 K and 1 K. The agreement is reasonable. Assuming that one of the elastic constants, c,~ (not measured by Yoshizawa et al.)
from a combination
of ce6 and c,, (see table
2.V)
Ref.
T(K)
cl!
Cl2
c13
c33
c44
Chh
[871 I911 [911
300 300 1
3.093 3.174 3.206
1.421 1.390 1.351
1.718 -
2.891 2.872 2.929
0.372 0.378 0.385
0.836 0.892 0.928
A. de Visser et al. I UP&, heavy fermions
Table 2.VII Compressibility
of UPt,
and superconductivity
111
in Mbar-’
Ref.
T(K)
K,
K,
K
[871 t911 t911
300 300 1
0.164 0.160 0.163
0.151 0.157 0.151
0.479 0.477 0.476 c
is temperature independent, we calculated K, K, and K, at 300 K and 1 K from the data of Yoshizawa et al. The results are listed in table 2.VII. No significant temperature dependence for the compressibility is found. Allowing for a 5% variation with temperature in cl3 yields the same conclusion. Recently, Benedict et al. [72] have determined the compressibility of UPt, from Xray patterns, taken in a diamond anvil cell at pressures up to 500 kbar. These authors obtained an initial slope, K -0.5 Mbar-‘, in good agreement with our result. 2.5. Transport properties 2.5.1.
Electrical resistivity
The electrical resistivity of Czochralski-grown single-crystalline UPt, [lo] has been measured on three cylindrical samples (diameter = 1.5 mm, length = 6 mm), that have been spark eroded along the a, b and c-axis out of one large crystal. The results, obtained on unannealed specimens, in the temperature range 1.3-300 K, are shown in fig. 2.16, as smooth lines through the data points. The resistivity is strongly anisotropic. Data taken along the a and b-axis show no significant difference, within the experimental accuracy (the absolute and relative accuracy amount to 7% and O.l%, respectively). All runs were taken with increasing and decreasing temperature; no hysteresis effects were observed. The pO-values, determined from an extrapolation to 0 K, are 7.2, 7.2 and 3.6pRcm for the a, b and c-axis, respectively; residual resistance ratios, RRR = ~(300 K)lp,, amount to 33, 33 and 37, respectively. After annealing, the pOvalues decreased to 6.2, 3.0 and 1.7 p&m, leading to RRR’s of 38, 79 and 78, respectively, The differences in p,-values, possibly due to small deviations from stoichiometry , might be intro-
100
0
I
0
I
100
I
I
I
200
J
300
Fig. 2.16. Electrical resistivity vs. temperature for unannealed Czochralski-grown UPt, for current directions as indicated.
duced by cutting the samples from different parts of the single-crystalline batch. In the lowtemperature regime, the resistivity is strongly temperature dependent, as can be seen in more detail in fig. 2.17. At 4.2 K the p-values amount to 28.2, 24.1 and 11.1 @cm for the a, b and c-axis, respectively. The superconducting transition temperature equals 0.48 K. In fig. 2.18 we
8o r-----l a 80 0
T c40
b
3
Q
Fig. 2.17. Electrical resistivity for annealed grown UPt, for T < 10 K; a(O), b(O) and c(A) lines represent data taken in a ‘He-cryostat).
Czochralskiaxis (dashed
A. de Visser et al. I UPt,, heavy fermions
112
and superconductivity
1oc
P-P, IpRcml SC
. upt,-a o UPt,-b oUPi,-c
8 b b
t
bb
20
__t
TIKI
Fig. 2.18. Double logarithmic plot of p - p0 for annealed UPt,. The solid lines denote a T*-temperature dependence.
present the low-temperature data on a double logarithmic scale. The resistivity curve for a polycrystalline UPt, sample nearly coincides with the curves along the a and b-axis, indicating that crystallites with preferred orientations are present. The crystallites are mainly oriented with the c-axis perpendicular to the cylinder axis. This was confirmed by magnetization measurements on this sample and further supported by the orientations of the grains studied under a light microscope. The p,-value of this sample amounts to 5 t&cm and the RRR equals 48. Resistivity data taken on an unannealed single-crystalline whisker [97], with the current along the c-axis, are shown in fig. 2.19. The whisker dimensions are approximately: 20 km(aaxis) x 40 pm(b-axis) x 1000 pm (c-axis). The pOvalue amounts to 2.2 @cm, the RRR equals 66. The uncertainty in the resistivity value due to the geometrical factor is, for this particular whisker, estimated at t 10%. The low-tern erature data P. are presented in a plot of p vs. T m fig. 2.20. The electrical resistivity of polycrystalline UPt , UPt, and UPt, is shown in fig. 2.21. In fig. 2.22
Fig. 2.19. Electrical nealed UPt, whisker
I
I
I
resistivity vs. temperature for an unan(current along the c-axis).
we present low-temperature data for polycrystalline UPt,, UPt, and UPt,. For UPt, UPt, and UPt, the p,-values amount to 34, 11.2 and 25.7@cm, and the RRR’s to 3.5, 14 and 5.6, respectively. In fig. 2.23 the low-temperature
‘r-----l 6
z4 3 Q3
0
1
2
3
4
5
6
T2 (K2) Fig. 2.20 Electrical whisker.
resistivity
vs. TZ for an unannealed
UPt,
A. de Visser et al. I UPt,, heavy fermions 2001
,
I
I
1t
p(~Rcml
150
I
IpRcml t
100
Fig. 2.21 Electrical resistivity line UPt, UPt, and UPt,.
vs. temperature
UPts
I, 0
Fig. 2.22. Electrical resistivity UPt,(O) and UPt,(A) (dashed a 3He cryostat).
49
Q *
2°
IJpt, upt,
. .. .
0 000"
for polycrystal-
1
a
0
0
5-
300
200
7-----l
F
0
P-P,
UPt2
0
40
10
I
I
113
and superconductivity
I
I
I Kl
of polycrystalline UPt,(O), lines represent data taken in
data for UPt, and UPt, are plotted on a double logarithmic scale. The resistivity of UPt, displays a type of curve that is characteristic for many actinide compounds [98]: a sharp rise at low temperatures is followed by a negative curvature towards the temperature axis (i.e. d2pldT2<0) and a tendency to saturate at large values in the roomtemperature region. In general, the overall large electrical resistivities in the actinides are ascribed to strong scattering of the conduction electrons
1
2
10
5
20
dTIK1 Fig. 2.23. Double logarithmic plot of p - p0 for UPt, and UPt,. The solid lines denote a TZ-temperature dependence.
by the spin fluctuations of the Sf-electrons. In this sense, raising temperature increases the amount of excitation in the spin-fluctuation spectrum and, hence, the effective scattering. Saturation occurs in the room-temperature region, where the mean free path of the electrons becomes of the order of the interatomic distance. It is not quite clear how far this situation holds for UPt,. The resistivities of all other heavyfermion systems reveal a Kondo-lattice behaviour: at low temperatures correlations between the Kondo-scattering centers cause a drastic drop of the resistivity (coherence). In the same way, one might think the drop in the resistivity of UPt, to be caused by the onset of correlations between the scattering centers. To what extent Kondo screening plays a role in UPt, is still an open question. Given the absence of a Kondo minimum below room temperature, the Kondo temperature will at least be several hundred Kelvin. The electron-phonon scattering term in the resistivity is assumed to be negligible at low temperatures, whereas this term will only play a minor role at high temperatures. This is supported by comparing data on UPt and non-
114
A. de Visser et al. I UPt,, heavy fermions
magnetic ThPt, for which compounds ~(300 K) amounts to 120 and 25 P&m, respectively [99]. As mentioned in section 1.3.1, according to the paramagnon model the resistivity of a spinfluctuation system varies as T2, at low temperatures. A T*-law is also expected in a Fermi liquid description, as follows directly from FermiDirac statistics. In a double logarithmic plot of the resistivity data (fig. 2.18), we observe that, possibly, in the low temperature limit, a T2-law is present. The coefficient A of this term is estimated at 1.6 and 0.7 @cm/K2, for the basalplane and hexagonal axis, respectively. A plot of p vs T2 for the whisker (fig. 2.20) reveals a T*-term below 2 K, yielding A = 0.58 @cm/K2 (c-axis). These A-values are at least two orders of magnitude larger than for ordinary metals. The characteristic change to a negative curvature of the resistivity curve towards the temperature axis, is clearly reflected by a peak in a plot of dpldT versus T (see fig 2.24). The temperature derivative has a maximum at 6.5 K for the basal plane directions and at 7.5 K for the hexagonal axis. The temperature at which the maximum occurs is assumed to be proportional to the spin-fluctuation temperature. At room temperature the resistivity starts to saturate. The maximum (or saturation) in the resistivity of the heavy-fermion compounds is often used to estimate the number of conduction electrons per atom, Z,, and their Fermi wave number. According to Friedel [loo], the maximum in the
5
0 0
Fig. 2.24. Temperature derivative of uranium-platinum compounds vs. T.
the
resistivity
of
and superconductivity
resistivity, due to incoherent scattering of the conduction electrons at the uranium atoms, can be written as: Pmax=
2(21+ 1)h.X e2kFZa
.
(2.20)
Here the orbital quantum number of f-electrons 1= 3, x( = l/4) is the concentration of scattering centers and k, = (3~2Z,/0n,)1’3, where L$ = 17.6 A’ is the mean volume per atom. Assuming a spherical Fermi surface and a mean value for Pmax= 200 @cm, we derive Z, = 2.7 and k, = 1.66 A-‘. However, this implies 10.8 conduction electrons per formula unit UPt,, which is a rather large number (assuming trivalent U one finds 2.6 conduction electrons per Pt). Therefore, the present estimate of k, must not be taken to literally; the order of magnitude is correct. Note that k, varies as Z”3, and thus depends only weakly on Z. The value for k, for the conduction electron gas must be compared with the value of k, = 1.1 A-’ at low temperatures, where the electron gas condenses into a liquid, due to the presence of the f-electrons. The electronic mean free path is given by 1, = U~T, where ur = fik,lm, the relaxation time r = ml(pne2), and it = Z,ln, is the number of conduction electrons per unit volume, hence 1, =
*
hi
(2.21)
With k, = 1.66 A- ’ and p = 200 l.&zrn we calculate 1, = 2.2 A, which shows that the electronic mean free path is indeed of the order of the interatomic distance. Resistivity measurements performed by others are in good agreement with the results presented above [lOl-1051. Some characteristic resistivity parameters have been collected in table 2.VIII. Data taken in Darmstadt on the same polycrystalline sample as described above are shown in figs. 2.25 and 2.26, in the temperature regions below 1000 K and 6 K, respectively [86]. Superconductivity is observed below 0.48 K, the width of the transition is about 0.1 K. The detailed behaviour of the polycrystalline specimen below
A. de Visser et al. I UP&, heavy fermions
Table 2.VIII Some characteristic
resistivity
parameters
and superconductivity
for UPt,
PO (@cm)
~(300 K) (@cm)
RRR
Specimen Czochralski a-axis Czochralski b-axis Czochralski c-axis Whisker c-axis* Polycrystalline Polycrystalline # Polycrystalline Polycrystalline Flux-grown c-axis
6.2 3.0 1.7 2.2 3.9 2.9 0.46 2.2 -0.5
238 238 132 147 200 200 336 236 150
38 79 78 66 51 69 730 107 300
* Unannealed. * After additional
A (@cm/K*) 1.6 1.6 0.7 0.58 1.38 1.7 -3 2.0
Ref.
[W [lOI [lOI [971
WI WJI [IO41 w51 [lo21
annealing.
1 K is shown in the inset of fig. 2.26. The residual resistivity equals 3.9 @cm (we obtained 5 t_&cm); after an additional annealing p0 = 2.9 t.LIcm. A quadratic temperature dependence holds up to 2 K, with a coefficient A =
upt,
OO
115
1
I
I
200
LOO
600
800
1000
T(K) Fig. 2.25. Resistivity vs. temperature polycrystalline UPt, sample.
Fig. 2.26. Temperature dependence polycrystalline UPt, sample below below 2 K ploted vs. T’.
up to lOOOK for
of the resistivity 6K. Inset shows
a
of a data
1.38 t.LIcmlKZ before annealing and 1.7 @cm/ K2 thereafter. The high-temperature data confirm the tendency to saturate as suggested by our results below 300 K. Wire et al. [loll performed resistivity measurements up to 1100 K with a transient-pulse-heating technique. Their data revealed a very weak maximum at -700 K (~(700 K)lp(llOO K) = 1.03). The resistivity of the compounds UPt, and UPt, display a similar behaviour as the resistivity of UPt,, although spin-fluctuation effects are less pronounced. A T2-law is followed over a somewhat larger temperature range, as follows from fig. 2.23. The coefficient A takes values of 0.09 and 0.01 t&cm/K*, respectively. The inflection point in the p vs. T curve (fig. 2.24) is found at 26 K for UPt, and at 30 K for UPt,, indicating a much higher spin-fluctuation temperature than for UPt,. The resistivity of UPt, displays a weak maximum, around 165 K. The room-temperature values equal 154 and 141 @cm for UPt, and UPt,, respectively. Equi-atomic UPt orders magnetically below 27 K. This compound reveals a complex magnetic behaviour, reflected in two magnetic transitions at 27 K and 19 K, leading to a ferromagnetic arrangement of the magnetic moments below 19 K. The large resistivity values in the paramagnetic region are ascribed to spindisorder scatter. Up to - 10 K a T2-law is obeyed. A more detailed description of the lowtemperature properties of UPt can be found in a paper by Frings and Franse [106].
A. de Visser et al. I UPt,, heavy fermions
116
and superconductivity
Thermoelectric power, thermal conductivity and Hall effect In order to look in some more detail into the
2.5.2.
transport properties of UPt,, we next discuss thermoelectric power and thermal conductivity measurements performed by the Darmstadtgroup on the polycrystalline specimen [86]. Thermoelectric power, S(T), measurements in the temperature range 0.2-350 K are shown in fig. 2.27. The main feature of S(T) is a pronounced positive peak at T = 8 K. The temperature at which the maximum occurs nearly coincides with T,,, as follows from the temperature derivative of the resistivity and can be taken to be proportional to the spin-fluctuation temperature. S(T) changes sign at T = 24 K and reaches a nearly constant value of - 11 ~.LV/K above T- 100 K, where also the resistivity starts to saturate. For normal metals, in which the resistivity is governed by elastic scattering processes, S(T) is proportional to TdlncTlaE (a is the
electrical conductivity) at low temperatures (see e.g. ref. [107]). As can be seen in fig. 2.27 there is a monotonous decrease of S(T) below T = 4 K, deviating from a linear behaviour, indicating the presence of inelastic scattering processes. Detailed measurements by Sulpice et al. [108] revealed S CCT2 below 1 K. A theoretical interpretation of this term is, however, not yet available. In fig. 2.28 the thermal conductivity, K, is given as a function of temperature up to 200 K, and compared with the electronic part as calculated from the measured electrical resistivity, assuming the validity of the Wiedemann-Franz
x
r
,
‘-_-------------___-A
/
I
50
0
100
150
T(K)
Fig. 2.28. (a) Thermal conductivity of polycrystalline UPt, for temperatures up to 200 K, as measured and as calculated from the electrical resistivity by applying the WiedemannFranz law; (b) the experimentally determined value for the Lorenz ratio, L(T), divided by the Sommerfeld value for this ratio, L,.
law. In fig. 2.28b we show the Lorenz ratio, L = ~plT, obtained from the measured K(T) and p(T) data. Depending on the temperature region the Lorenz ratio in pure simple metals is either smaller than, or comparable to, L, = 2.45 x lo-’ V*/K*, i.e. the Sommerfeld value. In systems with a small electronic mean free path, for example in the presence of strong electronparamagnon scattering, phonons can dominate the heat transport, giving rise to a substantial increase of L above the Sommerfeld value. Since the phonon conductivity usually has a maximum this leads to a corresponding at (O.l-0.2)0,, peak in L(T). In the case of UPt, a maximum value L,,, = 5.5L, is found at a temperature T,,,,, = 24 K, which is slightly larger than 0.1 0, (On = 217 K). B ecause the total heat conductivity is the sum of an electronic part, K,, and a phonon part, K~, the measured quantity L(T) can be expressed by L
=
p(W(T) =
(2.22)
L
T
L, = ~,plT is the electronic Lorenz ratio. Since at higher temperatures L, should approach
where L,,
0
1
2
3 T(K)
0
100
200
T(K)
Fig. 2.27. Thermoelectric power, S, of polycrystalline UPt, for T < 5 K (a), and for temperatures up to 350 K (b).
this
analysis
shows
that
K~ 3
4~,
at T,,,,,,
which is not unreasonable in view of the already rather short electronic mean free path, I, - 4 A, as inferred from the electrical resistivity of 120 @cm at this temperature. The phonon mean free path I, at T,,, can be calculated from:
A. de Visser et al. I UPt,, heavy fermions
i
X-1 (2%)
UPf3
. WF
5
ii
,.+-
.
.a
. I
y’# , , 1
2
3
1
,r2J L
5
117
range L is smaller than L, by about 8%. An even weaker ratio L/L, - 0.45is calculated from the data by Sulpice et al. [108], pointing again to the presence of inelastic scattering processes. Another transport property that has been investigated for UPt, is the Hall effect [109-1111. We concentrate here on the work of Schoenes and Franse [109]. These authors performed measurements on a single-crystalline specimen in the temperature range 2-300 K. The magnetic field (4T) was applied along the b-axis, the current along the a-axis, and the Hall voltage was determined in the c-direction. The data are shown in fig. 2.30. For temperatures above - 50 K the temperature dependence of the Hall coefficient, R,, resembles that of the basal-plane susceptibility, which is also shown in fig. 2.30. This makes it possible to separate the Hall effect in a normal part (R,), that allows for the carrier determination, and an anomalous part (R,), closely related to the magnetism in UPt,. To separate the normal from the anomalous Hall effect, one may write [ 1121:
r?
-
1
and superconductivity
TiKl
Fig. 2.29. Thermal conductivity of polycrystalline UPt, below 6 K; the solid curve represents the Wiedemann-Franz law. Kp = i,v,c,/3. By inserting the sound velocity, u, 14 x lo5 cm/s (table 2.V), a lattice s ecific -P heat cp of approximately 0.27 JK-’ cm , and Kp = 22 mWcm_’ K-‘, as calculated from the difference between the measured K(T) and the calculated K,, one derives I, = 60 8, + I,. In fig. 2.29 the low-temperature normal-state heatconduction data are displayed in more detail. At - 3 K the measured curve and the calculated curve cross, which shows that the additional phonon contribution (as discussed above) is present down to - 3 K. In the low-temperature
pu= R,B+pu,MR,.
(2.23)
10
I
50
100
150 Temperature
I
I
I
200
250
300
[KI
Fig. 2.30. Total Hall effect of UPt, as measured in the field of 4 T (after ref. [109]). The inset shows the geometrical arrangement of the electrodes. The solid lines are fits (see text). Also shown for comparison are the susceptibility data in the basal plane.
A. de Visser et af. i UPt,, heavy fermions
118
For T>SOK we have 8=&?I, M=xH, C/( 7’ - O), and the Hall coefficient equals
R~= PH =R, POH
+ &
R, .
x=
(2.24)
In fig. 2.31 a plot of R&T - S) vs, T- 0 is shown, with 0 = - 50 K, as derived from the x -l(T) curve for T 5 300 K. From the slope of the solid line, which represents a fit to the data, a value R, = - 4.0 x 1O-4 cm3A-‘s~” results. The negative sign shows that the ordinary Hall effect is dominated by electron contributions. The value for R, ( = - line) corresponds in a oneband model to 1.1 conduction electrons per formula unit [109], a value that differs considerably from the crude estimate of 2 = 2.7 per atom, that we deduced from the maximum in the resistivity. Marabelli et al. [113] performed farinfrared optical reflectivity measurements and derived in a Fermi Iiquid model 2 = 1 per formula unit, close to the value derived by Schoenes and Frame. Z = 1 corresponds to k, = 0.75 A-‘, which is in reasonable agreement with the value for k, of - 1.1 A-’ that we used in the Fermi liquid analysis of the specific heat (section 2.3.1). The intersection of the straight line in fig. CR, = gives ordinate the 2.31 with 0.244 cm3KA-‘s-l. With an effective moment of R, = compute 2+,/U-atom, we
and superconductivity
0,97cm3A-1s-*, which is - 2500 times larger than R, and of opposite sign. The anomalous Hall effect is attributed to skew (asymmetric) scattering at the uncorrelated scattering centers (single impurities). At low temperatures, below - 50 K, the anomalous Hall effect R, = (R, R,)/x decreases rapidly, which is seen as a clear sign of the formation of a coherent state. 2.5.3. Point-contact spectroscopy We conclude this section on transport properties with a discussion of point-contact experiments on the uranium-platinum compounds (Jansen et al. [114], and references therein). Small constrictions between metals have been used to analyse the interaction of the conduction electrons with elementary excitations in the metal. By measuring the resistance of a point contact as a function of the applied voltage, direct information is obtained about the energy dependent scattering of the conduction electrons. Important criterion for the applicability of this method is that the inelastic mean free path, fi, of the electrons is larger than the contact diameter d. In this ballistic regime the electronic system will be out of equilibrium with respect to the elementary excitations (e.g. phonons) in the presence of an applied voltage over the contact. With this technique excitations like phonons, magnons and crystal-field levels have been prob-
T-B [Kl Fig. 2.31. The decomposition
of the total Hall effect of UPt, into the normal and anomalous part for T > 50 K (after ref. (1091).
A. de Visser et al. I UPt,, heavy fermions
ed. In the impure limit of a contact (Ii < d) the electrons stay in equilibrium with the lattice, at an applied voltage, and in this thermal regime heating occurs in the contact region. For instance, it could be shown that in ferromagnetic materials the Curie temperature was reached in the contact region at a certain voltage over the contact. The point-contact experiments on the uranium-platinum compounds have been performed in Nijmegen and Grenoble. At liquid He temperature (1.5 K) the contacts were adjusted by pressing the sharp edges of two pieces of bulk material against each other. Prior to mounting, the polycrystalline bulk material was cleaved at room temperature. The contact resistances were of the order of 1 a. The technique for recording dV/dZ and d2V/dZ2 makes use of a modulation method. On top of the bias current, which is swept in order to vary the applied voltage over the point contact, a small ac-current is superimposed. The ac-voltage over the contact is measured by phase-sensitive detection at the fundamental frequency (If-mode), and at twice the fundamental frequency (2f-mode) of the modulator, to obtain signals proportional to the first and second current derivatives of the voltage. The change in the resistance, which is only a few percent over the investigated voltage range, is recorded using a bridge circuit. For these contacts, between the same materials, the dV/dZ characteristics were symmetric with respect to the applied voltage. The relative change AR/R(V=O) of the static resistance of the point contacts is presented in fig. 2.32. The reproducibility of the curves for different contacts might be estimated from the width in the minimum. For the case of UPt, and UPt, the width was equal within roughly 10%. For UPt and UPt, the width varied up to 30%. The strong resemblance between the voltage dependence in the contact resistance (fig. 2.32) and the temperature dependence of the bulk resistance (figs. 2.16 and 2.21) suggests the importance of heating effects in these compounds. In the dirty limit (Zi < d) the resistance of a circular contact between two semi-infinite metals is given by
119
and superconductivity
R(V=0) = 1.09 n a=0.09
R(V10)=9.0 n
0
20
40
60
60 voltage
CmV)
Fig. 2.32. Normalized change in the static resistance as a function of the applied voltage for point contacts with uranium-platinum compounds at liquid helium temperatures. The dashed curves have been calculated with the heating model, using temperature dependent resistivity data with the effective Lorenz ratio’s L, and multiplied by the factor (1.
R=$,
(2.25)
where p is the resistivity of the metal. For the U-Pt compounds with p = 10 l&cm at 1.5 K, and R 2:1 R the diameter d > 1000 A. The mean free path amounts to - 50 A ( < d). Clearly, we are in the thermal limit and Joule heating in the
120
A. de Visser
et al. 1 UPt,,
heavy fermions and superconductivitj
contact area has to be considered. Assuming equal flow lines for the electrical and thermal conduction near the contact, one obtains for an arbitrary contact geometry at an applied voltage V the relation T;=
v’ 4L ’
(2.26)
for the maximum temperature T,,,,, at the center of the contact in a bath at temperature T,. In this expression L = prc/T is the Lorenz number. For T,,, 9 T,) a numerical calculation gives Tmttx= aV, with a = 3.2 x lo3 KIV. Hence, using eq. (2.25) and eq. (2.26), a simple relation exists between the voltage dependent contact resistance and the temperature dependent resistivity, if the resistance of the constriction is mostly determined by the resistivity at the center of the contact where the temperature is maximal. Taking into account the temperature gradient near the contact, using the model of spherical spreading out for a constriction, the point-contact resistance as a function of the applied voltage can be calculated. For the conversion of the temperature scale to the voltage scale a constant Lorenz number over the whole temperature interval is taken. The results of these calculations with the used effective Lorenz numbers have been plotted in fig. 2.32 as dashed curves. To get a good agreement of the calculated curves with the experimental ones the calculated values of ARIR(V=O) are multiplied by a constant a, smaller than 1. An explanation for this constant CYcould be that part of the normal contact resistance is a Sharvin-like resistance R, ~pl,ld~, which applies for a clean contact (1, > d) and does not depend on the temperature. The constant cy is different for different contacts of one material, because of the relative change of contact resistance. For UPt and UPt, we mostly find small values for cy, meaning that the Sharvin resistance is much larger than the temperature dependent resistance in eq. (2.25). In this way we find a discrepancy, because no longer the limit I, < d will hold and the heating mode1 does not apply to our problem. Note however, the resistivity at the constriction is different from the
bulk resistivity, due to anisotropy, grain boundaries, strain, impurities etc. For UPt, we have taken in fig. 2.32 the Lorenz ratio as obtained from the measurements on the electrical and thermal conductivity, which yielded a constant value above 70 K. For the other curves we have taken the Lorenz number, that optimizes the agreement between experiment and theory. The accuracy of the Lorenz numbers must be seen in view of the, before mentioned, reproducibility of the width in the minimum of the resistance curves. Recently, point-contact masurements on UPt, were interpreted as a measure of the energy-dependent density of states of the electrons [115]. From these experiments strong evidence for a narrow band of heavy electrons (m* = 250m,) situated near the Fermi level was deduced, In detail it is not yet clear how the energy-dependent mass-renormalization enters the point-contact problem. We believe that heating plays a role in the description of pointcontact experiments with these metallic compounds. 2.6. High-magnetic-field
studies
2.61. magnetization The magnetization of single-crystailine UPt, has been measured by Frings et al. [2] at fields up to 35 tesla in the Amsterdam High-Field Installation. These data, taken with stepped pulses, at temperatures of 4.2 K (a, b and caxis), 20.3 K (a-axis) and 77K (a-axis), are shown in fig. 2.33 (see also refs. [8, 79, 116, 1171). The magnetization in the basal plane is isotropic, within the experimental accuracy. An anomalous increase of the magnetization takes place between - 1.5 and - 25 T for field directions in the basal plane at 4.2 K, resulting in a value of - lp,JU-atom at 35 T. At 20.3 and 77 K the basal-plane magnetization is linear in field, as is the magnetization along the c-axis at 4.2 K. The specimen investigated in these experiments was the same unannealed 3 mm sphere as used for the susceptibility measurements (section 2.2). In fig. 2.34 we show data, taken with a smooth pulse at 4.2 K, on an annealed Czochralski-grown single-crystalline sample, prepared
A. de Visser et al. I UPt,, heavy fermions
/
61
I
I
121
and superconductivity
r
I
/
I
% 8 8
UPt3
5
Y,
l-i
+
%
a
+
+
E
*
2 5
++
0 + 8
+
b
0
B Fig. 2.33. Magnetization of unanneaied Czochralski-grown 4.2 K(0) and c-axis at 4.2 K(X) (after ref. [8]).
xx
20
10
x
A
A
Y
x
30
40
1T1
UPt, for the a-axis at 4.2 K(Q), 20.3 K(+) and 77 K(a),
b-axis at
UPt3
b
0
20
10
B
30
40
CT)
Fig. 2.34. Magnetizatjon of unanneaied UPt, for the b-axis at 4.2 K(0) and for the a-axis at 77 K(A) (after ref. 1791); and of annealed UPt, for the a-axis at 4.2 K(X).
122
A. de Visser et ul. I UPt,,
heavy fermions
from a different batch, for a field along the a-axis, and on the unannealed sample (h-axis) as measured by Frings and Franse [79]. The field at which the anomaly occurs (20 T), reproduces well for different samples and is not sensitive to strains. The most interesting feature of the high-field magnetization measurements is the presence of the anomaly near 20 T, at 1.5 K (not shown) and at 4.2 K. This anomaly appears as a pronounced peak in a plot of the differential susceptibility, X(H) = &r/AH, versus H (see fig. 2.35). At 20.3 and 77 K, i.e. at temperatures above the temperature at which the maximum in x(T) occurs (17 K), the anomaly is absent. This suggests a close connection between the anomalies in a(H) and x( ?“), which is supported by the nearly equal and thermal energies: magnetic iu,B,,X = The field at which the anomaly occurs k, T,,,. does not depend on temperature: from 1.5 K (= 0.09 I’,,,) to 4.2K (= 0.25 T,,,) no variation of R,,, is observed, within the experimental accuracy (see also section 2.6.2). Besides, magnetization curves, taken at 1.5 K and 4.2 K, did not reveal any broadening of the transition. Hys-
Fig. 2.35. Differential susceptibility for UPt, for the b-axis at 4.2K(O) represents
and for the a-axis at 77K(A). The dashed curve a fit according to eq. (2.27) (after ref. [79]).
and superconductivity
teresis effects are absent. In order to perform a quantitative analysis, Frings and Franse [79] have fitted x(H) for p& < 20T to the expression: X(H) = x& + a( rU,W2 + b(E_LoW4+ c( &H)“] , (2.27) and obtained the parameters: x0 = 112 f. 1 x lo-” m”/molU, a = 135 -+ 25 x low5 T-‘, b = 0 -+ 16 x lo-’ T-” and c = 3.0 + 0.3 x lo-* T-? Thus x(H) - x0 starts as HZ. In the next section we first present magnetoresistance data in the same field region, and then discuss the possible interpretations of the anomaly near 20T. 2.6.2. Mag~et5res~st~v~ty The magnetoresistivity of single-crystalline UPt, has been measured in the Amsterdam High-Field Installation at fields up to 35 tesla [97, 118, 1191. Experiments have been performed on two annealed Czochralski-grown singlecrystalline specimens with a cylind~cal shape (length = 6 mm, diameter = 1.5 mm). One sample was measured with the current directed along the a-axis, the other with the current directed along the c-axis. In both cases the effect of a magnetic field parallel and perpendicular to the current direction has been studied. The temperature dependence of the resistivity of these samples has been presented in section 2.5.1. For these experiments two pulse shapes were used: (1) a stepped pulse in which fields are kept constant for approximately 50ms, i.e. long enough to neglect the effect of eddy currents, and (2) a smooth pulse in which the field, after having reached 35 T, decreases at a constant rate of 56T/s. Since both pulse-types offered nearly identical results, we here present only data taken with the latter pulse-type, which has the advantage that many data points can be collected during one field sweep. The relative accuracy is estimated to be 3%, whereas the absolute accuracy amounts to 7% due to the determination of the geometrical factor, S/I. The current through the samples amounted to 0.5 A. In fig. 2.36 we show the experimental results
A. de Visser et al. I UPt,, heavy fermions
123
and superconductivity 25
20
% HLC
’
‘0 4.2K 15
%:
10
3 5 aQ 0
-5
-10
Fig. 2.36. Magnetoresistance 4.2K(O), and for Hl(Zllc)
of UPt, at 4.2K(O)
for and
HII(Z[lc) at at 1.5K(A).
for ZZ]](Z]]c), at 4.2K, and for H~(z]]c), at 1.5 and 4.2 K. Since during the experiment only the relative change of the resistance with field is recorded, the absolute values, presented in fig. 2.36, have been calculated using the p( T)-values from fig. 2.17. The zero-field p-values for the c-axis equal 3.1 and 11.2 $km at 1.5 and 4.2 K, The magnetoresistance passes respectively. through a maximum at - 20 T, for a field direction in the basal plane. At B,,, a huge magnetoresistivity is observed: an increase in resistivity with field of 230% at 1.5 K and of 120% at 4.2 K. At 77 K Ap is slightly negative (not shown), and amounts to - 1.4 @cm at 35 T For a field direction (~(77 K) = 100 @km). parallel to the c-axis Ap saturates at a modest value. In fig. 2.37 we present the experimental results for H]](Zl]a), at 1.5, 4.2, 20.3 and 77K; zerofield p-values amount to 9.4, 28.2, 127 and 180 @km, respectively. At 1.5 K and 4.2 K Ap is positive and has a maximum at - 20 T. At B,,, the increase in resistivity equals 130% at 1.5 K and 70% at 4.2K. At 20.3 K and 77K Ap is negative. In fig. 2.38 we show once more the curve for ~]](~l]a) at 4.2K, but also data for HI(Z]] a), for several orientations of H in the
Fig. 2.37. Magnetoresistance peratures as indicated.
of UPt,
for Hll(ljla)
at tem-
k-plane. Clearly, B,,, shifts to higher values as the field is rotated more and more towards the c-axis in the k-plane. A simple analysis shows that the maximum in the resistivity is attained for a value of the component of the field in the basal plane equal to 20 T (B,,, cos 8 = 20 T, where 0 is the angle between field direction and basal plane). This evidences the two-dimensional nature of the field effects. Since the data in fig. 2.38 suggest that the absolute Ap-values for ZZl](Z]]a) and (~((b)l(Z]]a) are almost equal, we conclude that p(H) is mainly governed by the direction of the field with respect to the crystallographic axes; the orientation of the field with respect to the current is of minor importance. In fig. 2.39 we present the temperature dependence of the magnetoresistance in fields of 5 and 8 T for kZl](Z]]a) [Sl]. defined as The magnetoresistance is Ap(T, H) = p(T, H) - p(T, 0). In general, two contributions to the magnetoresistance exist. The first contribution is due to the ordinary magnetoresistance, which is caused by the Lorentz force on the electrons. This contribution is always positive, and varies. in general, as H’, in the low-field limit. The ordinary magnetoresist-
A. de Visser et al. I UPt,, heavy fermions
124 25
I
I
I
ence. In the case of UPt, the orientation of the field with respect to the current is of minor importance, therefore, we conclude that the ordinary magnetoresistance is small and, consequently, that Ap(H) is mainly determined by the field effect on the effective scattering processes.
r
20
T” 3 aQ
10
5
0 40
30
10
0
I&
Fig. 2.38. Magnetoresistance of UPt, at a temperature of 4.2 K for Hll(llia) (0) and for Hl(llla) with the orientation of H with respect to the b-axis given by (20 + 5)” (O), (40 2 5)” (A) and (70 + 5)” (0).
-2
j
”
0
”
“‘I
”
c
j
20
10 T
and superconductivity
At zero field the resistivity can be written as p( T, 0) = p,, + AT*, for T < 2 K. A first question that arises is whether p,,, which is determined by scattering at defects and impurities, is field dependent. From an extrapolation to zero temperature, from temperatures above 2 K, of Ap(T, H), at 5 and ST (fig. 2.39), we expect a field independent pO. Remenyi et al. [120] performed similar measurements, on a Czochralskigrown sample in the temperature range 0.53.5 K, at 10 and 20.7 T, for a field direction in the basal plane. These data are shown in a plot of p vs. T* in fig. 2.40. From this figure it follows that Ape = 0.5 p..Rcm at 10 T and Ap,, = 3.2 @cm at 20.7T. Sulpice et al. [108] performed magnetoresistivity measurements on a polycrystalline sample, in the temperature range 35-800 mK, at fields up to 7.5 T. These authors measured a linear field dependence of po: p,,(B) = p,, + aB, with a = 0.22 @cm/T. At 7.5 T these authors obtained Ape = 1.65 @cm, a value much larger than observed in the case of the Czochralskigrown samples near 7.5 T. This strong sample
' 30
(K>
Fig. 2.39. Magnetoresistance of single-crystalline UPt, for Hll(Zlla) as function of temperature in a field of S(0) and 8(O) tesla. Closed symbols are taken from fig. 2.37.
ance
is larger
for (transverse),
a field
perpendicular
to
the
than for a field parallel to the current (longitudinal magnetoresistance). The second contribution arises from the fieldeffect on the effective scattering processes. Since several scattering mechanisms might be present, this term can exhibit a complex field dependcurrent
Fig. 2.40. Magnetoresistance of UPt, plotted VS. T* at fields (directed in the basal plane) as indicated (after ref. [120]).
A. de Visser et al. I UPt,,
heavy fertnions
dependence might be due to the variation in the p,-values: 6.2 p fz cm (de Visser et al.), 1.6 FRcrn (Remenyi et al.) and 0.46 @cm (Sulpice et al.). Compared to the overall large magnetoresistance, the field effect on pI, is small and plays a minor role. From the data in fig. 2.40 it follows that the coefficient. A. of the T”-term increases as function of field. It attains values of 0.97, 1.10 and 2.25 t&cm/K’ at 0, 10 and 20.7T, respectively. The temperature region in which this term holds seems to decrease with field. The magnetoresistance strongly resembles x(H) (compare figs 2.35 and 2.37). We have analysed the Ap(W)-curves in a plot of logAp vs. logH, in order to look for a specific term H” in Ap. However, no field dependence with a fixed exponent over a large field range is observed. For ~l~(Z~~~) fz ==2 in the field region 3-10T at temperatures of 1.5 and 4.2 K (it is difficult to determine the initial field dependence due to the spread in the data at low fields). For H_L(I/]c) n = 2 between 3 and 10T at 4.2 K, but Ahp(H) varies linearly with field up to 10T at 1.5 K. Since it is difficult to distinguish the various contributions to Ap(H), we confine ourselves in the following to a qualitative analysis. We emphasize that the magnetoresistance is unusually large at low temperatures, which implies a very effective scattering mechanism up to the critical field. The temperature dependence of Ap (fig 2.39) shows that the scattering is most effective at - 7 K (at 8 T), i.e. the temperature at which the maximum in AplAT occurs. Near 14K Ap becomes negative. Above 17 K a small negative value remains. Next we discuss whether x(H) and p(H) can be interpreted in a spin-fluctuation model. As can be conjectured from the specific-heat data in field, the influence of a magnetic field on spinfluctuation systems is usually subtle. For instance, in the case of UAlz ]121] u(H) and p(H) reveal a slightly weaker field dependence above 15 T, which has been interpreted as a suppression of the spin-~uctuation phenomena. The presence of a H’-term in x(H) for UPt, is consistent with a spin-fluctuation model (BeAlMonod f39j). Unfortunately, at present a theoretical description of the steplike c(H)-
and supereonduct~v~t~
125
curve is lacking. Beil-Monod argues that if x(H) increases with field, p(H) should also increase. However, for ferromagnetic spin fluctuations one expects Ap(H) < 0, since these would be suppressed (or even quenched) by an external field. On the contrary, for antiferromagnetic spin fluctuations one expects Ap(H) > 0, due to an enhanced effective scattering mechanism caused by the instability of moments oriented antiparallel to the field. Above a certain critical field the antiferromagnetic correlations will be destroyed and a negative contribution to Ap(H) results. The positive Ap(H) for UPt, would, therefore, be in favour of an antiferromagnetic type of spin ~uctuations. It is, however, not clear to us whether this is compatible with the presence of the T” ln(T/T*)-term in the specific heat, which is ascribed to ferromagnetic spin fluctuations in the paramagnon model (in a Fermi liquid picture the only constraint is that of small momentum transfer processes). If there is any negative term in the low-temperature regime, arising from ferromagnetic spin fluctuations, it is completely immersed in the huge positive effect. At 20.3 and 77K Ap(H) is negative (fig. 2.37), with a quadratic field dependence at 20.3 K. From the increase of the coefficient of the T’-term with field (Remenyi et al. [120]) it follows that the spinfluctuation temperature (denoted by Ti) decreases, assuming A cx(T:)-“. This can be seen as an enhancement of the spin-~uctuation effects with increasing field. Several other explanations of the anomaly at 207: have been put forward, among which bandstructure effects, a type of antiferromagnetic transition and crystal-field effects. A peak in the density of states just above E, might be responsible for the maxima in x(H) and x(T) assuming that at B,,, and T,,,,, the Fermi level passes through the maximum of the peak. However, one then expects the anomaly in c/T at T,,, as well, which is apparently not the case. The a(N) and p(i-l)-curves remind one of a type of antiferromagnetic transition: the step in the magnetization might be caused by a (possibly spin-flop Iike) reorientation of magnetic moments in the basal plane, accompanied by a peak in the magnetoresistance. However, no sign of long-range
A. de Visser et al. I UPt,, heavy fermions
126
magnetic order has been observed. Therefore, the type of order should be short-range. For instance strong antiferromagnetic correlations in the basal plane below 17 K, likely associated with the spin-fluctuation phenomena. The possibility that the anomalies are due to a field induced crossing of a crystal field level and E,, induced by the external field, can be discarded, since no low energetic crystal-field levels have been observed in the specific heat. Rather similar anomalies have been observed in the magnetization and magnetoresistivity data taken on a single-crystalline sample of the tetragonal compound CeRu,Si, [122]. However, for this compound the anomaly takes place at a considerably lower field (7.5 T) and has a onedimensional character (it is only observed for a field along the tetragonal axis). A possible explanation for this phenomenon is sought in a field induced polarized phase of heavy-fermions (y = 350 mJ/ K2mol). A rather similar temperature dependence of Ap(H) has been observed in the heavy-fermion systems CeCu,Si, and CeAl, [123]. At high temperatures Ap(H) is negative, but changes sign at when the coherent state is low temperatures,
20-
0
10
30
Fig. 2.41. Magnetoresistance of UPt, for Hill and for HLZ at 4.2 K(0) and 1.5 K(O).
40
at 4.2 K(A)
and superconductivity
I
1
I
I
UPt5
Fig. 2.42. Magnetoresistance of UPt, for HII1 at 4.2K(A) and for HII at 4.2 K(0) and 1.5 K(O).
formed. It is remarkable that also in these compounds a maximum in Ap(H) is present at low temperatures: at 2 T (T < 0.5 K) for CeAl, and at 20 T (T < 1.5 K) for CeCu,Si,. This indicates the destruction of the coherent state by a field. For comparison we have measured the magnetoresistivity of polycrystalline UPt, and UPt,. Data on UPt, for H_iZ at 1.5 and 4.2 K, and for H]]Z at 4.2K, are shown in fig. 2.41. Data on UPt, for HII at 1.5 and 4.2K, and for HjlZ at 4.2 K, are shown in fig. 2.42. No anomalies have been observed, as expected from the linear magnetization curves [2]. At these low temperatures, the magnetoresistances of both compounds are positive. For UPt, the longitudinal magnetoresistance saturates, as in the case for UPt, for H )](I\] c). The transversal magnetoresistance is large: at 35 T Ap equals 15.4 and 13.2 @cm, at 1.5 and 4.2 K, respectively. For UPt, the nearly isotropic Ap is small and has a W-term with n = 1.6. For UPt, no fixed exponent is found. 2.6.3. Forced magnetostriction Forced magnetostriction measurements on single-crystalline UPt, have been formed at fields up to 8 T at a temperature of 4.3 K [124]. Hereto
A. de Visser et al. I UPt,, heavy fermions
the same sample as used for the thermal expansion measurements (section 2.3.2) was mounted in a capacitance cell, made of OFHC copper. Care was taken to keep the sample at constant temperature, since otherwise the large thermal expansion of the sample would obscure the forced magnetostriction, A’ = L-‘(dLldB). The accuracy decreases with increasing field and is limited at 1.5 x lo-’ T-’ in A’ at 8 T. The experimental set up allows a set of nine measurements: the field can be applied perpendicular (2 X ) and parallel to the three dilatation directions. As UPt, has a hexagonal structure we performed the whole set of nine measurements. Some anisotropy effects were observed in the basal plane, but were of the order of the spread in the data. Therefore, we neglect the basalplane anisotropy and concentrate on the overall behaviour. The experimental results are displayed in fig. 2.43 and fig. 2.44, for the field along the c and u-axis, respectively. Values for the basal-plane magnetostriction, AAb, were obtained after averaging AAand A;. The forced volume magnetostriction is calculated from A: = 2ALb + AS. For a field along the c-axis A’ is linear in field: positive
1 -
F 0 G a ? ,-d -J
0’
_, -
or in terms of the relative pressure dependence of the molar susceptibility
H,T
(2.29)
=
Here V is the molar volume. With the appropriate susceptibility data, x~,~ = 115 X lop9 and x, = 57 X 10m9m3/molU (fig. 2.1), we calculate
I5
l/3 Vol.
a\&% \, c
0.5
‘A ‘a \
t I 0
(2.28)
1
‘A \
-3
(gY,.,= - L-jgp,T~
\
-2
Fig. 2.43. UPt, for dilatation l/3 of the
”
0’ /--.’ &JIO___o_Oo-O-O--_olA - \*
for a dilatation in the basal plane, but negative along the c-axis (fig. 2.43). For a field applied along the a-axis deviations from linearity occur above 3 T (fig. 2.44). Note that these deviations occur at considerably lower fields than in the corresponding magnetization curves (at - 10 T, see fig. 2.33). The coefficients of these linear terms, or the u-values, writing X = AL/L = iaB*, have been listed in table 2.1X. For both field directions the c/u-ratio decreases with field. The forced magnetostriction data can be related to the hydrostatic pressure dependence of the molar magnetic moment by a thermodynamic Maxwell relation:
0’ 0’ a,b
B IJc
127
and superconductivity
i 0
2
4
8
0
l0
w06 The forced magnetostriction of single-crystalline a field direction along the c-axis at T= 4.3 K; in the basal plane (III), along the c-axis (A) and volume effect (0).
Fig. 2.44. UPt, for dilatation l/3 of the
2
4
6
8
10
The forced magnetostriction of single-crystalline a field applied in the basal plane at T = 4.3 K; in the basal plane (O), along the c-axis (A) and volume effect (0).
A. de Visser et al. I UPt,,
128 Table 2.1X Values for a (in lO~‘T~‘)
for field directions
Dilatation in basal plane Dilatation along hexagonal Volume effect
along
axis
heavy fermions
the a, b-and
High-pressure
80
I
’
I
n
I
= JaB’
at low fields
c-axis
2.00 0.57 4.57
2.02 0.62 4.66
1.61 -2.91 0.31
studies
I
A = AL/L
b-axis
2.7.1. Electrical resistivity The electrical resistivity of single-crystalline UPt, has been studied at pressures up to 5 kbar [lo, 1251. Measurements have been performed 100
assuming
Field along a-axis
from the data in table 2.1X, values for alnxlap of - 22 and - 3 Mbar~ ‘, for a field direction in the basal plane and along the hexagonal axis, respectively. Thus, for a field in the basal plane the relative pressure dependence of x (- 22 Mbar-‘) is expected to be nearly one order of magnitude larger than for a field direction along the hexagonal axis. In section 2.7.2 we shall compare the magnetostriction data with the results obtained from the pressure dependence of the susceptibility. 2.7.
c-axis,
and superconductivity
on unannealed Czochralski-grown samples, with the current along the three crystallographic axes. Zero-pressure data on these samples have been presented in section 2.5.1. The low-temperature data, plotted as p vs. T*, at zero pressure and at 4.2 kbar, are shown in fig. 2.45 for the a and b-axis, and in fig. 2.46 for the c-axis. In fig. 2.47 we have plotted the temperature derivative Api AT vs. T for the b-axis, at zero pressure and at 4.2 kbar. Applying pressure results in a shift of the temperature at which the maximum occurs towards hi her temperatures; a considerable inQ crease in a p I a T2 near the temperature at which the maximum in the susceptibility is found (17 K), can be observed as well. Above - 45 K the pressure effect for a current in the basal plane is temperature independent and amounts to - 1.4 @cm/kbar. For the c-axis the pressure 40
’
UPt3
30
0.0 a +,x b -
x
60
-
L
T c
2%
3
20
Q
Q40
l0 20
d 0 0
T2 (K2) Fig. 2.45. Electrical resistivity vs.. T2 for unannealed UPt, for the a-axis at 1 bar(O) and at 4.2 kbar(Ll), and for the b-axis at 1 bar(+) and at 4.2 kbar( X).
20
40
60
80
100
T2 (K’) Fig. 2.46. Electrical resistivity vs. T’ for unannealed for the c-axis at 1 bar(O) and at 4.2 kbar(O).
UPt
A. de Visser et al. I UPt,,
1c
heavy fermions
1
L
2
e C
1
I
TIKI I
I
I
20 30 0 10 Fig. 2.47. Temperature derivative of the electrical resistivity of unannealed UPt, for a current direction along the b-axis at 1 bar (0) and at 4.2 kbar( x).
effects are less pronounced; at room temperature a pressure independent resistivity results. In the low-temperature limit (T < 2 K) the resistivity of UPt, can be expressed as p = p0 + AT’. As can be seen in fig. 2.45 and fig. 2.46 pO is pressure independent. The coefficient A rapidly decreases as function of pressure. By estimating A from the initial slope of the data, plotted as p vs. T2, we observe a decrease of this coefficient of about 40% over 4.2 kbar, for both current directions. At 1 bar the temperature derivative of the resistivity has a maximum at a temperature, T;, of 6.5 K and 7.5 K, for the basal plane and the hexagonal axis, respectively. Under pressure this temperature increases at a rate of 0.35 + 0.1 K/kbar (data for the b-axis are shown in fig. 2.47). From the increase of Tz and the decrease of A, as function of pressure, we conclude that the spin-fluctuation phenomena are strongly suppressed. Due to a lack of the theoretical understanding of the low-temperature resistivity, there is no proper definition of the characteristic spin-fluctuation temperature. As
and superconductivity
129
mentioned before, however, T; , assuming and Tz can both be taken to be A 0~(T;)-2, proportional to the spin-fluctuation temperature. As a check of this assumption we can compare the relative pressure dependences of Tz and T:, assuming that the proportionality constants are pressure independent. From the data in fig. 2.45 and fig. 2.47, we calculate alnT:/ap = 54 + alnAlap=-95+35Mbar-’ 16 Mbar-‘, and that hence aln T;/ap = 48 ? 18 Mbaf ’ (note throughout this work we approximate the relative pressure dependence of a quantity Q, alnQ/ ap, by (Q,,))’ AQlAp, where Q, represents the 1 bar value of Q). The agreement between these deduced values is good, however, the uncertainty, in particular in the value of A as function of pressure, is large. of Resistivity measurements as function pressure performed by others [lOl, 103, 1261 yield qualitatively the same information as our results. Wire et al. [loll performed resistivity measurements on flux-grown crystals at pressures up to 18 kbar, in the temperature range l-300 K. These authors analysed the data in a so-called “parallel resistor model”, assuming R ’= Rim’ + R,‘, where R is the measured resistance, Ri is the ideal resistance of the specimen, and R, is a shunt resistance, whose magnitude is near the saturation value, i.e. - R(300 K). Applying this analysis, the authors conclude that Ri follows a T2-law up to - 8 K and - 12 K at 1 bar and at 17 kbar, respectively. From the coefficient of the T2-law, as function of pressure, values were deduced for aln T: lap of 36 and 30 Mbar- ‘, for the basal plane and c-axis, respectively. Willis et al. [103] performed resistivity measurements (Illc-axis) at pressures up to 19.1 kbar, in the temperature range 0.1-2 K, on a flux-grown specimen (see fig. 3.10). From the variation of the coefficient A with pressure, these authors derived a value for aln TT,lap of 25 Mbar - ‘. Ponchet et al. [126] have measured the resistivity of an UPt, whisker (Zllc-axis) in a diamond anvil cell, at pressures up to 202 kbar. Their results, normalized at room temperature at a zeropressure value of 132 @cm, are shown in fig. 2.48. Apparently, the resistivity curve gradually loses its anomalous low-temperature behaviour;
130
A. de Visser et al. I UPt,, heavy fermions ,507
5:
l:O
150
2j)O
2TO
“7’
i
E z
;
I 8
d 1
Fig. 2.48. Electrical resistivity of an unannealed UPt, ker at pressures up to 202 kbar (after ref. [126]).
whis-
a smooth variation with temperature is found at 202 kbar, but ~(300 K) remains large. The pOvalue is essentially constant below 40 kbar, but then increases from 1.9 to 6 @cm between 40 and 200 kbar. At ambient pressure A amounts to Under pressure A decreases 0.52 I.Dcm/K2. rapidly by one order of magnitude between 0 then more slowly to 8.8 X and 50 kbar, lop3 I&cm/K’ at 202 kbar. From these data a value for alnT;/ap of 30Mbar-’ is deduced. Averaging the values for aln Ti/dp as obtained by the various authors, we deduce a mean value of 33 Mbar-‘. Below we compare this value with the pressure dependence of the characteristic temperature, T,,,,, , as derived from the susceptibility.
and superconductivity
UPt, samples at pressures up to 4.5 kbar, in the temperature range 4.2-40K [103, 127, 1281. The magnetization in fields up to 6.5 T, for polycrystalline UPt, (cylinder, diameter = 6 mm, length = 8 mm), at a temperature of 4.2 K and at pressures of 1 bar and 4.5 kbar, is shown in fig. 2.49. The susceptibility measured in an applied field of 5.3 T is shown in fig. 2.50. The relative pressure dependence of the susceptibility, &xl ap, amounts to a value of - 24 Mbar-’ at 4.2 K. In order to measure the magnetization of singlecrystalline UPt,, two cylindrical samples (diameter = 6 mm, length = 6 mm) have been spark-eroded along the b and c-axis, out of large Czochralski-grown specimens. The magnetization curves for these two samples, at a temperature of 4.2 K and at pressures of 1 bar and 4.6 kbar, are shown in fig. 2.51. The corresponding susceptibility data, measured in a field of 5 T, are shown in figs 2.52 and 2.53. From the data in fig. 2.51 we calculate values from &rx/ap at 4.2K of -28+3 and -5 *3Mbar~‘, for the basal plane and hexagonal axis, respectively. Apparently, crystallities with preferred orientations are present in the polycrystalline sample, as was
4.2 K
0.4
3 i
c.7.
0.3
E
4
1 bar
b 0.2
4.5 kbar
ov 2.7.2. Magnetic susceptibility Magnetic susceptibility measurements have been performed on poly- and single-crystalline
0
’
I
I
1
I
/LJ-l
Fig. 2.49. Magnetization pressures as indicated.
I
6
2
I
I a
R
of polycrystalline
UPt,
at 4.2 K at
131
A. de Vzker et af. I UPt,, heavy ferrni~~~ and ~~~~r&onductivi~ 1257
,
I
I
I
17.6 K
I
5.3 T
'\
\
// 3
\
I/
11s
19.6 K ii
I/
D D 4.6 kbar
“I,,
,
',lbar
\
,'
1
'0
\
,,i,.
I I
1;
- b-axis
m3
i , fl-.
I'
$05-:
“I,
'\,
{f
\
I
\ \ \
I' I i / I
\ 4.5 kbar \
I 95-
'
r
-: m 0
I
I
I
I
-r
10
20
30
40
50
T (K) Fig. 2.50. Susceptibiii~y of polycrysta~li~e UPt, in a field of 5.3 T at pressures as indicated.
upt,
Fig. 2.52. Susceptibility of single-crystalline LJPt, in a field of 5 T directed along the b-axis at 1 bar(O) and 4.6 kbar(D), and directed along the c-axis at 1 bar(A) and 4.6 kbar(V).
b-axis 0
1 bar
CI 4.6 kbor 0 0
0
4
0.2 i
D
D
0
1
:
1
n
c-axis
t
+
A
0.1
1 bar
v 4.6 kbar
:: * 0.0
f
0
’ - ’ 1
2
L
’
I
’
I
’
’
t
’
’
’
345678
Fig. 2.51. Magnetization of single-crystalline UPt, at 4.2 K for field directions and pressures as indicated.
Fig. 2.53. Susceptibility of single-crystalline UPt, in a field of 5 T directed along the b-axis at 1 bar(O) and at 4.6 kbar(C!).
132
A. de Visser et al. 1 UPt,, heavy fermions
concluded previously from the thermal expansion data on this sample. Therefore, we shall, in analysing the results, restrict ourselves to the data for the single-crystalline samples. At applying a pressure of 4.6 kbar, the susceptibility for the basal plane strongly reduces, whereas the susceptibility for the c-axis hardly shows any pressure dependence. The rates of suppression of dlnxlap, at 4.2K, -28 ?3 and - 5 2 3 Mbarr’, for the basal plane and c-axis, respectively, are in good agreement with the values of - 22 and - 3 Mbaf ‘, for the basal plane and c-axis, respectively, that we predicted from the forced magnetostriction measurements (see section 2.6.3). When comparing the highpressure susceptibility data with the forced volume magnetostriction results we have to keep in mind that volume changes in the magnetostriction experiments in fields up to 8 T are at least two orders of magnitude smaller than the volume changes induced by a pressure of, say, 5 kbar (the compressibility of UPt, amounts to 0.48Mbar~‘). The nearly identical values for alnxlap deduced from these experiments indicate a linear pressure dependence of x up to, at least, 4.6 kbar. The maximum in the susceptibility curve along the b-axis is found at 17.9 K at 1 bar, and shifts under applied pressures of 4.6 kbar with 2.5 * 0.5 K to higher temperatures: ~lnT,,,/~p = 30 ? 6 Mbar-‘. Since the values for the relative pressure dependence of x(4.2 K) in the basal plane and of T,,, are nearly equal and opposite, we conclude that the product x. T,,, is nearly pressure independent, at low temperatures. The value for alnT,,,/ap is in good agreement with the mean value for alnTT,/ap as derived from the resistivity measurements. For the polycrystalline sample a value for alnT,,,/ap of 25 Mbar-' results. The relative pressure dependences of x(4.2 K) can be transformed to Griineisen and T,,, parameters, by using the compressibility K = TX = alnx(4.2 K)/alnV= 58 2 13 0.48 Mbar-‘: These = -~lnT,,,i~lnV= 63 + 6. and r, Griineise~aparameters are almost equal to the one derived from the enhanced linear term in the specific heat and thermal expansion, r: = 68 at
and superconductivity
1 K (section 2.3.2). The latter value is in good directly agreement with r: = 55, as obtained from the pressure dependence of the enhanced linear term in the specific heat [80]. Although no clear-cut definition of the characteristic temperature for the low-temperature anomalies in resistivity and susceptibility can be presented, we have shown that the pressure dependences of Tz, T; and T,,, all range between 25 and 54 Mbar~ ‘. Hence, the positive value for the pressure dependence of the characteristic temperature is firmly established; the corresponding Griineisen parameter is large. From these high-pressure data we conclude that by applying pressure the spin-fluctuation phenomena are strongly suppressed.
3. UPt,: the superconducting 3.1
state
Introduction
In this section we deal with the superconducting ground state of UP&. Superconductivity in UPt, has been discovered by Stewart et al. [3]. These authors performed resistivity, ac-susceptibility and specific heat experiments on fluxgrown single-crystalline specimens and found a drop in the resistivity, a diamagnetic signal in x,, and a discontinuity in the specific heat, with onset temperatures of 0.54 K, 0.53 K and 0.54 K, respectively. This was taken as evidence for bulk superconductivity below 0.54 K. As mentioned in section 1.3, the occurrence of superconductivity in some of the heavy-fermion systems was quite unexpected. Furthermore, unusual observations concerning the superconducting state have given rise to speculations upon an unconventional pairing mechanism. Especially, the coexistence of superconductivity and strong spinfluctuation phenomena in UPt,, made this compound a likely candidate for triplet superconductivity. In. our case, measurements concerning the superconducting state have been performed in a close cooperation with colleagues in Leiden (de Visser et al. [18, 761 and Palstra et al. [129]). Superconductivity was first observed on three
A. de Visser et al. I UPt,, heavy fermions
annealed Czochralski-grown samples in electrical resistivity measurements, along different crystallographic directions (see table 3.1: sample #l, #2 and #3). From previous experiments, it was evident that the resistivity only follows a T2-law at approaching the low-temperature limit of our 4He cryostat, 1.3 K (section 25.1). In a subsequent investigation of this term in a “He cryostat, down to 0.30 K, we found a drop to zero resistivity for all three samples between 0.47 and 0.53 K. A second indication of superconductivity was obtained by ac-susceptibility measurements (f = 10.9 Hz) on two unannealed samples (#4 and #5): large diamagnetic signals were observed below 0.40 and 0.43 K, respectively. The annealed samples #6 and #7 (table 3.1) showed a superconducting transition at 0.48 K, consistent with the values that were derived from the resistivity measurements on samples #l, #2 and #3. Apparently, annealing of the samples increases T,. Evidence for bulk superconductivity was derived from a specific heat experiment on an unannealed Czochralski-grown single-crystalline sample (#8), for which the onset of the superconductivity occurred at 0.43 K, and the maximum in the specific heat discontinuity at 0.26 K (see fig. 3.1). The value of AciyT at 0.26K amounted to 0.35, i.e. one quarter of the value expected from BCS theory: 1.43. Two other
133
and superconductivity
observations support the evidence for bulk superconductivity: (i) annealed polycrystalline samples of the neighbouring phases UPt, and UPt,, that possibly could be present in spurious amounts, showed no traces of superconductivity down to 0.30 K, and (ii) a single-crystalline sample (#6), an annealed polycrystalline sample (#9) and an annealed powdered sample (#lo) all gave a diamagnetic signal of the same magnitude as a Cd reference sample (T, = 0.52 K), after the necessary corrections for demagnetizing factors.
0.3 tt
$ 13 /mole K’l T IKI
0
1.0
0.5
I
1.5
Fig 3.1. Specific heat of unannealed Czochralski-grown UPt, below 1.3 K (sample #8). The onset of superconductivity is at 0.43 K. The broken line serves as a guide to the eye.
Table 3.1 Superconducting transition temperature, T,, and width of the superconducting transition, AT,, for Czochralski-grown UPt, (all samples #l till #8 out of one single-crystalline batch) and for arc-melted polycrystalline UPt, (samples #9 and #lo). T, and AT,-values, as determined from the ac-susceptibility experiments, are defined at the 50% point and between the 10 and 90% points, respectively
AT, Sample #l #2 #3 #4 #5 #6 #7 #8 #9 #lO
annealed annealed annealed unannealed unannealed annealed annealed unannealed annealed (arc-melted) annealed (powdered)
(K)
PO (10~‘Rm)
Experiment
6.2 3.0 1.7
pdL(a-axis) pdc(b-axis) pdc(c-axis)
0.47-0.53 0.47-0.53 0.47-0.53 0.40 0.43 0.48 0.48 0.43 0.50
0.05 0.05 0.05 0.10 0.10
-
0.45
>O.lO
-
X8, X,‘ X,,(a-axis) X,,(c-axis) specific heat X,,
xx
A. de Visser et al. 1 UPt,, heavy fermions
134
A hard-proof
to demonstrate the in UPt,, i.e. the Meissner effect, is presented in the following section. Thereafter we present, successively, specific heat and critical-field studies that enable us to determine the parameters that characterize the superconducting state. Subsequently, we apply the expression for T,, derived in the Fermi liquid theory, as outlined in section 1.3.3, to UPt,. To conclude this section, we discuss several other techniques that have been applied in order to elucidate the nature of the superconducting state.
occurrence
3.2.
experiment
of bulk superconductivity
The Meissner effect
Superconductivity in UPt, was deduced from resistivity, ac-susceptibility and specific heat experiments. However, from the first two experiments, the occurrence of superconductivity possibly caused by the presence of filamentary exclusions of superconducting non-stoichiometric UPt, can not be excluded. Furthermore, the rather small value for the specific heat discontinuity AclyT =0.35 at 0.26 K (compared to a value of 1.43 predicted by the BCS theory) arose the question whether the superconductivity in UPt, should be considered as a bulk property. A direct measure for the superconducting volume fraction of the sample can be obtained by the so-called Meissner effect. Hereto, the amount of flux expelled by a specimen, when it is cooled through the superconducting transition temperature in a small magnetic field, is measured. Meissner experiments [129] were performed on an annealed arc-melted polycrystalline sample (cylinder, diameter = 4.8 mm, length = 4.5 mm). Ultrapure Cd (T, = 0.52 K) and Sn (T, = 3.72 K) cylinders of the same dimensions served as reference samples. The flux expulsion of the sample was measured by means of a pick-up coil. It is useful to compare these results with dc-susceptibility measurement. Such experiments were performed by placing the sample in a (secondary) pick-up coil and recording the induced voltage while slowly ramping the magnetic field (produced by a primary coil), at a typical rate of 0.2 mT/s. The magnetization can be obtained by
and superconductivity
numerical or analog integration. Since, before ramping the field, the sample is cooled in zerofield, so-called virgin magnetization curves can be recorded. In addition, the magnetization was measured directly by means of a flux transformer method. For the Meissner effect, as well as for the magnetization experiments, the signals are calibrated by those of the Cd and Sn reference samples. The two methods give a lower and an upper limit for the superconducting volume fraction, as we shall discuss below. In fig. 3.2 we show a typical trace of the dc-susceptibility of the UPt, cylinder versus magnetic field. The data in this figure suggest a superconducting volume fraction of 80% at 353 mK. However, in this experiment it is impossible to discern normal regions embedded in superconducting regions. Therefore, this method will set an upper limit of the superconducting volume fraction. Fig. 3.2 also shows that the virgin and non-virgin curves largely differ. The magnetization curve at 353 mK, obtained with the flux transformer method, is shown in fig. 3.3 (actually we plotted -~($4). The lower critical field B,, is represented by the field at which the virgin magnetization curve has a maximum. This maximum is somewhat rounded, probably because of demagnetizing effects. The value for B,, of 2.2 mT perfectly agrees with the value of 2.2 mT (taken at ,ydc = 0) derived from fig. 3.2. In addition, we measured an annealed powdered UPt, sample. The superconducting volume frac-
upta
Fig. 3.2. DC-susceptibility (normalized with respect to Cd and Sn) vs. magnetic field for UPt, at 353 mK (TIT, = 0.73). The lower curve is the virgin curve. For decreasing field the curve is a mirror image with respect to the ordinate.
A. de Visser et al. I UPt,, heavy fermions 1 T,352
Fig. 3.3. Virgin magnetization of UPt, at 352mK. A minor x,, at 13 mT is also included.
mK
curve and magnetization loop hysteresis loop determined by
tion amounted to 50% at 357 mK, somewhat lower than observed for the cylinder. This difference can be ascribed to a different transition temperature, 7’,(bulk) = 0.50 K, T,(powder) = 0.45 K, and a different transition width, AT,(bulk) = 0.10 K and AY’,(powder) > 0.10 K. In fig. 3.4 the results of the Meissner experiments are presented, together with the virgin magnetization curve. The curves denoted by “field cooling” were drawn through the data
332
mK
135
and superconductivity
points obtained by measuring the flux expulsion, at constant magnetic fields, while lowering the temperature. The initial slopes of these curves at H = 0 yield the superconducting volume fractions, which amount up to 30%. The superconducting volume fractions, as obtained from both methods, are plotted in fig. 3.5 as function of temperature. The differences in results from both methods, can mainly be attributed to flux-pinning in the “field cooled” case. It is well-known that the superconducting transition propagates more quickly along the surface than into the bulk [51]. Furthermore, one can calculate that any superconducting surface in a small magnetic field forms an energy barrier impeding the passage of flux lines. Thus, when cooling the sample below T,, initially the surface becomes superconducting, forming this energy barrier. When the superconducting transition propagates farther into the bulk, the flux lines must be expelled from the inner part of the sample. However, if the pinning forces are large, the flux lines will be trapped at the surface barrier. Therefore, although the sample may be a good superconductor, it is well possible that very little flux is expelled. These pinning forces are especially large when the surface is oriented parallel to the magnetic field. The pinning forces can be decreased by more than two orders of magnitude when the surface is not parallel to the magnetic
ZERO-FIELD
u
PoH
(mT)
Y COOLED
Fig. 3.4. Magnetization vs. magnetic field of UPt, obtained by the two methods. Zero-field cooling denotes the virgin magnetization curve. The curves denoted by “field cooling” were drawn through the data points (not shown), obtained by measuring the flux expulsion at constant magnetic field as a function of temperature. The dashed line represents the full Meissner effect (M = H) and the initial slope of the 332 mK curve.
0.2
0.4
0.6
T(K) Fig. 3.5. Superconducting volume fraction of UPt, vs. temperature as obtained by the field cooling (x) and the zerofield cooling (0) methods.
A. de Visser et al. I VPt,, heavy fermions and superconductivity
136
field. This explains why powdered samples will always show a more complete expulsion of flux than cylinders oriented parallel to the field, as, for instance, was observed for CeCu,Si, [130]. Besides surface pinning, bulk pinning will also occur: the flux lines are trapped at defects in the crystal lattice. The hysteresis of the magnetization loop (fig. 3.3) shows the result of these enormous pinning effects. Nevertheless, the expelled flux will still set a lower limit of the superconducting volume fraction. At measuring the virgin magnetization curves, one avoids the problem of flux pinning. However, as argued above, this method will set an upper limit of the superconducting volume fraction. We believe that for UPt, it gives a more reliable estimate, because of the enormous flux pinning effects. We conclude from these Meissner effect and dc-susceptibility studies that superconductivity in UPt, is a bulk property. 3.3.
Specific
heat
The specific heat data taken on the unannealed Czochralski-grown sample reveal a broad anomaly below 0.43 K (see fig. 3.1). Specific heat experiments on a polycrystalline sample, in the temperature range 0.17-1.1 K, have been performed in Darmstadt [86], with an improved accuracy. An experimental difficulty, at performing specific heat experiments on UPt, employing an adiabatic method, is the self-heating of the sample due to radioactive U-decay. This limited the minimum sample temperature in the used set-up to 170mK. The specific heat of the annealed polycrystalline sample is shown in fig. 3.6 as cl T versus T, in different fields up to 8 T (the normal-state specific-heat of this sample is shown in fig. 2.3). In this experiment the broad anomaly, as found for the unannealed sample, is reproduced at zero field; Tz”“’ = 0.52 K. The height of the peak at 0.35 K is about 30% of the specific heat extrapolated from above T,. Similar values for AclyT of 0.35, 0.48 and 0.66 have been observed for the unannealed Czochralski-grown sample (fig. 3.1), an annealed flux-grown sample [3] and an annealed polycrystalline sample [108],
.-.......L.........- . . . . . .
-
1 ... .. . . . . . . ... .......lL..--1 0.5T
0
____c__
_'k........................ . .._ _OT
/
/ ! _/ / OO
UPi,
I 0.5
,
I 1 T(K)
Fig. 3.6. The specific heat of polycrystalline UPt, at temperatures below 1.1 K and in fields up to 8 T. The dashed line represents the specific heat according to eq. (2.2). with coefficients derived from a fit in the temperature interval 1.2-20 K (table 2.1). Schematic extrapolation of the zerofield data by the dash-dotted line satisfies entropy balance.
respectively. These values are much lower than the one expected from BCS theory: 1.43. The relative jump height is increased to about 60% if the broad transition is replaced in the usual way by a sharp one. This procedure yields a calorimetric transition temperature T, = 0.40 K, in contrast to TEnset = 0.52 K. Well below T,, c/T is almost linear in T, extrapolating to a finite y-value of about 260mJ/K2molU at T = 0 K. Such an extrapolation, however, violates the entropy balance, imposed by the extrapolation of the normal state specific heat to OK (dashed line). Therefore, the low temperature specific heat in the superconducting state should decrease faster than given by the solid line. In a schematic way this is indicated by the dashdotted line for which the entropy balance is met exactly. In a magnetic field of 0.5 T the anomaly is much weaker. At higher fields T, falls outside the covered temperature range. The field effect on the normal-state specific heat has been discussed in section 2.3.1. The large electronic normal-state specific heat is caused by the strongly interacting f-electrons. The large anomaly at the superconducting transition evidences that the f-electrons are also invol-
A. de Visser et al.
ved in the formation of the superconducting condensate. However, the extrapolation of c/T to zero temperature, taking into account entropy balance (dash-dotted line in fig. 3.6), suggests a small finite y-value of about 20 mJ/K’molU, instead of a vanishing one, as expected for a normal superconductor. A similar type of specific heat curve has been obtained by Sulpice et al. [108]. These authors measured a polycrystalline sample down to 146 mK, at which temperature c/T = 293 mJ/K’molU, i.e. a value well below the normal-state c/T-value just above T,. For this sample the specific heat varied as c/T = “/o+ aOT, with ‘yo= llOmJ/K*molU and (Y”= 1250 mJ/K3molU, over the temperature range 146-470 mK. The extrapolation to a finite yvalue of llOmJ/K*molU leads, however, to a small excess of entropy (5%) with respect to the normal phase. Therefore, the actual “residual” y-value might be somewhat lower. The peculiar features observed in the specific heat might be indicative for a non-conventional pairing mechanism in the superconducting state. In particular, the observation of a power law rather than an exponential temperature dependence of c, for T 4 T,, can be taken as evidence for a nonconventional pairing mechanism (table l.IV). Taking into account the entropy balance for the broad superconducting transition a finite y-value results, which suggests that parts of the Fermi surface remain normal. It is interesting to note that also in CeCu,Si, a nearly T2-dependence of c(T) is found, just below T, [131]. At lower temperatures this behaviour gradually changes towards a T3-law. Further comparison with the latter compound suggests that UPt, is in between those samples with pronounced specific heat jumps and other ones (“gapless superconductors”), with significant jumps, but with giant linear terms in c as T goes to zero [132]. In case of UBe,, a T3-law has been observed in the superconducting state. The presence of this term was taken as evidence for triplet superconductivity [ 1331. 3.4.
Critical field studies
The
temperature
dependence
of
the
137
i UPt,, heavy fermions and superconductivity
upper
critical field, B& = - dB,,ld TI =_=,, has been studied in applied magnetic fields up to 0.5 T by means of ac-susceptibility experiments [18, 761. The results, taken on an annealed Czochralskigrown sample (#6, table 3.1), for a field along the a-axis, are shown in fig. 3.7. For this sample BL2 amounts to 4.4T/K. In order to give a quantitative analysis of the upper critical field we follow the approach that successfully has been applied to Al5 superconductors [134] and to CeCu,Si, [135]. Hereto, some crude assumption will be made: (i) the Fermi surface, characterized by a Fermi wave ignoring any possible number k,, is spherical, anisotropy, and (ii) UPt, is not a strong-coupling superconductor. Under these assumptions the temperature derivative of the upper critical field can be written as: (3.1)
0.E 0.:
0.1
0.2
0.3
’
’
‘\
B,,ITeslal
0.1
0.L 0.5
I
,
I
0.7 C 3
I
i
UP+
0.:
a-axis
0.; 0.'
0.6
i
(b)
-
T[Kl
\
I
(a
L 0
I
0.1
I
0.2
I
I
I
I
I
03
OL
0.5
06
0.7
C1.: B
Fig. 3.7. (a) AC-susceptibility (in arbitrary units) in zero magnetic field and in fields of 0.1, 0.2, 0.3, 0.4 and 0.5T (field directed along the u-axis) for the sample #6; and (b) the upper critical field as function of temperature; Yf, is defined at the 50% point (indicated by arrows in (a)).
138
A. de Visser et al. I UPt,, heavy fermions
where y is given in J/K2m3, k, in m-l and the normal state resistivity at T,, approximated by the residual resistivity, pO, in LRm. Here the quantities a and b are given by 7.95 x 1O32Tm2K2J-2 and 4780 TKm2fi~‘JJ’, respectively. Eq. (3.1) h as its origin in an expression for B,, taken from the Ginzburg-Landau theory: B,, 0~t-‘(T). Here t(T) represents the GL coherence length, which signifies the characteristic length for variation of the order parameter. The GL coherence length depends on the purity of the superconductor, which can be presented by the ratio Z,/&,, where 1, is the electronic mean free path and &, the BCS coherence length. In the pure limit (I, + &,) p0 is small and Bf2 will be mainly determined by the first term on the right hand side of eq. (3.1). On the other hand, in the dirty limit (I, e 5,) p0 is large (T, is low) and Bs2 is approximately given by the second term. Eq. (3.1) enables us to determine the Fermi wave number directly from measurable quantities. Subsequently, we are able to calculate some parameters that characterize the superconducting state: the BCS coherence length, 5” - uF/Tc kgl(yT,), where uF is the Fermi velocity, the electronic mean free path, 1, - (p”kg)-‘, and the London penetration depth h, - (~7) lk:. Inserting the experimental results (sample #6) for Bi2 ( = 4.4 T/K), T,( = 0.48 K), the residual resistivity p0 (although not measured on sample #6, we may estimate it at - 3 x lo-’ am) and the (Sam le inde endent) enhanced y-value of 9.94 x 10!?J/K2m P (which corresponds to 422 mJ/ K2molU), we obtain a value for k, of 1.06 X 101Om-l. This Fermi wave number characterizes the strongly interacting electrons that form the Fermi liquid at low temperatures. With this value for k, we calculated (see section 2.3.1): the effective mass m* = 200m,, the effective Fermi velocity u$ = 6800 m/s and the Fermi temperature T, = 275 K. The dirty and clean limit contributions to eq. (3.1) amount to 1.4T/K and 3.0T/K, respectively. We realize that p. has not been measured on sample #6. However, allowing, for instance, a 100% variation in po, k, varies in between the values 0.96 X 10” rn-’ (for po=O) and 1.25 x 101om-’ (for p. =6X lop8 am). Taking into account the possibility of
and superconductivity
such a variation in k,, our Fermi liquid analysis yields, however, qualitatively the same results. Parameters for the superconducting state can now easily be calculated using the expressions given by Orlando et al. [134]: 4, = 2.0 x lo-’ m, 1,=3.6x 10d8m and &=3.6x 10-7m. The ratio 1,/t, amounts to 1.8, which signifies that our samples are neither in the dirty limit, nor in the clean limit, as could be expected from the p,-values. The Ginzburg-Landau parameter, ~~~ is found to be about 23. A similar value for ~~~ can be deduced from the ratio of the upper to the lower critical field: B,,IB,, 2: 2K~L/lnK,,. The temperature dependence of the lower critical field as obtained from the virgin magnetization curves, taken on the polycrystalline sample, (section 3.2) is shown in fig. 3.8. With B,, = 0.35 T and B,, = 0.0017 T at 0.4 K, we calculate ~~~ = 17. In table 3.11 we compare these parameters for UPt, with those for CeCu,Si, and UBe,,. Herewith we have to realize that some of these parameters can exhibit a strong sample in particular in the case of dependence, CeCu,Si,. Critical field studies performed by others are in good agreement with our results [102, 105, 108, 137-1391. The observed values for Bs2 range from -4.0 to - 6.3 T/K. The limiting critical field at zero temperature, Bc2(0), amounts to - 1.8T. In general, two different mechanisms
0.6 1-1
u Pt3
4
t
T(K)
Fig. 3.8. Temperature dependence of the critical fields B,, for a polycrystalline sample, and B,, for a single-crystalline sample along the a-axis (see also fig. 3.7).
A. de Visser et al. I UPt,, heavy fermions
139
and superconductivity
Table 3.11 Some important parameters characterizing the heavy-fermion superconductors references therein, sample No. 7) and UBe,, [136]. For the latter compound determine the parameters in the superconducting state
UPt,, CeCu,Si, (Rauchschwalbe et al. [135], and a dirty limit expression has been applied in order to
Parameter
Units
upt,
CeCu,Si,
UBe,,
Y T, B:2 PO kr m* u:
mJ/K’mol K T/K lo-Ylm 10” mm’
422 0.48 4.4 3.0 1.06 180 6800
1006 0.64 5.8 3.5 1.7 220 8700
-1100 0.85 42.0 125 0.87 # 300 3390
2
K lo-’
m
2.0 275
582 1.9
112 1.42
L fk %L
lOen m 1Om8m -
3.6 36 23
1.2 20 22
0.13
# Determined
m, m/s
from
configurational
arguments,
assuming
3 heavy
lead to the disappearance of superconductivity: the interaction of the external field with the electronic orbits and the interaction with the electronic spin. The first interaction gives rise to the definition of an orbital critical field, which is given by [140]: Bz2(0)= 0.7B;,T,.
(3.2)
The second interaction leads to a destruction of the superconducting state when the free spin energy becomes larger than the condensation energy (paramagnetic or Pauli limiting). Applying eq. (3.2) we calculate values for B:*(O) ranging between 1.4 and 2.2T/K, compared to the measured value of 1.8 T/K. This suggests that Bc2(0) is mainly determined by the orbital critical field; Pauli limiting is absent. The absence of Pauli limiting is expected for an equal spin-pairing superconducting state in a p-wave model. To conclude this section we briefly discuss anisotropy effects in the upper critical field. Anisotropy of the upper critical field is expected for some types of odd-parity superconductivity [58, 611. However, anisotropy might also be observed in case of a singlet superconductor with an anisotropic gap. Our data for a field direction along the u-axis reveal a linear temperature dependence of B,,.Another sample (#7) measured
electrons
per formula
unit UBe,,.
with the field applied along the c-axis, seemed to have an increasing slope of the upper critical field with decreasing temperature: Bfz= 3.6 T/K at T, and Bs2= 6.0T/K near 0.40 K. However, this sample exhibited a rather broad transition ( - 0.1 K). Similar anisotropy effects have also been observed by other authors. Willis et al. [137] and Chen et al. [102] measured the resistive superconducting transition for a current along the c-axis. For HIZ these authors found a linear temperature dependence of B,, over a relatively large temperature range, whereas for HI] Z the critical field curves showed a curvature near T,,leading to a considerably lower value for BL2.Similar anisotropy effects have been observed by Rauchschwalbe et al. [138]: for ZZ]](Z]]c) the initial value for Bi2is larger than for ZZ]](Z]]basal plane). Tholence et al. [139] performed ac-susceptibility measurements, and observed for a field direction along the u-axis, as well as for a field direction along the c-axis, non-linearities in B,,vs.T.Furthermore, Rauchschwalbe et al. observed a kink in the B,,-curve near T,,which is possibly attributed to a fieldinduced reorientation of an anisotropic pairing state, as suggested by Scharnberg and Klemm [58]. Up to now, none of the observed anisotropies in B,, has been taken as decisive evidence for triplet superconductivity.
140
3.5.
A. de Visser et al. I UPt,, heavy fermions
Fermi Liquid analysis
In this section we present the results of a Fermi liquid analysis of the superconducting transition temperature, employing the model developed by Pethick et al. 1701 described in section 1.3.3. In this model the superconducting transition temperature can be expressed by the relationship (eq. (1.37)): T(j) C = I 13oT F elis,
.
(3.3)
Here (Yis a numerical constant, TF is the Fermi temperature and gj represents a pseudopotential, given by eq. (1.40). If g; CO, the potential is attractive; j = 1 indicates triplet pairing, j = 0 singlet pairing. The analysis of the normal-state specific heat (section 2.3.1) enables us to calculate g, , which can be expressed as function of the antisymmetric I = 0 Landau scattering amplitude:
=
6
performed by considering its pressure derivative, and, subsequently, by comparing the calculated relative pressure derivative of T,, aln TJap, with the experimental value. Taking the relative pressure derivative of eq. (3.3) we find: calnT
3P
_
.
(3.4)
Using the values for Ai, ranging from -3.71 to - 2.95, derived from the various fits to the as measured and the phonon corrected specific heat (table 2.1), we calculate values for g, ranging from - 0,452 to - 0.325. Clearly, g, is negative. Next we need to investigate whether g, is the most negative pseudopotential. Employing the expressions for go and g, (eq. (1.40)), the necessary condition for g, < g, is given by Ai < - $ 2 As. For galilean-invariant systems Fi is related to the effective mass, m*. For a large effective mass Fi is large, and hence Ai amounts to 3 (eq. (1.24)). This would imply that for AZ< - 1115 triplet pairing occurs. However, in case of UPt, the lattice breaks up the galilean invariance of the electrons, which hampers the derivation of Ai from the effective mass. Nevertheless, estimating A”, to be of the order of 3 the Fermi liquid analysis favours triplet superconductivity (see also the discussion in the paper by Pethick and Pines [84]). Wnfortunately, it is within the Fermi liquid model not possible to predict T,, since cy is not known. However, a test of eq. (3.3) can be
6
8A;
(1 + A;)2
6p
a’n7-F 8P
(3.5)
’
where we substituted eq. (3.4) and assumed Q to be pressure independent. In the uniform scaling approximation ( y * TF 0: n) the relative pressure dependence of TF can be written as &?lnT,
iJlny *
-=K--
dP
dP
(3.6)
’
where K is the isothermal compressibility. The relative pressure dependence of T, then takes the form aln T,
1+ A; g1
and superconductivity
3P
alny”’ =K-;Ip-
6 (1 + A;)2
aA;? z
’
(3.7)
Hence, it appears that alnT,/ap can be calculated from parameters that follow directly from specific heat data as function of pressure. Such experiments have been performed by Brodale et al. [SO], at pressures up to 8.9 kbar. These authors observed a decrease of the coefficients y* and 6 (and thus an increase of A;) as function of pressure. From frts to the specific heat data in the temperature range 0.5-18 K at 1 bar and 3.8 kbar, we calculate K - alny” lap = 31 Mbar-[ and aA;/dp = 109 Mbar-‘. With the 1 bar value for A: of - 4.09, calculated from the data of Brodale et al., we derive atn T,/ 8p = - 37 Mbar-‘. Willis et al. [103] measured the pressure dependence of T, and observed a reduction of T, as function of pressure with a rate of - 12.6mKlkbar (see section 3.6). The experimental value for alnT,/ap amounts to 26Mbar-‘. Hence, a reasonable agreement between the calculated and experimental values for aln T,lap results. A second check on the validity of eq. (3.3) can be performed by calculating the relative pressure dependence of T, from a combination of thermal
A. de Visser et al. I UPt,, heavy fermions
expansion and specific heat measurements at 1 bar. Starting point is again eq. (3.7). The second term on the right hand side of eq. (3.7) can be related to the coefficient of the linear term in the thermal expansion, ~2, conform eq. (2.14a). The pressure dependence of Ai in eq. (3.7) can be related to the coefficient of the T31n(T/T*)-contribution to the thermal expaneq. (2.14d), eq. (1.33) and sion, 6, . Employing eq. (3.6), S, can be written as
(3.8) Rearranging eq. (2.14a), the pressure
terms in eq. (3.8) and employing we find the following expression for derivative of B”:
aB” ~ = 9V,B” ap The relation between the pressure of A: and B” can be found by taking derivative of eq. (1.34), assuming pressure independent: dA; _- dB” ap ap
1
alnT,=
(3.10)
(A;)2 + 3A;
- $
Subsequently, eq. (2.14a), tionship:
dependence the pressure that Ai is
we derive from eq. (3.7), by using eq. (3.9) and eq. (3.10), the rela-
6
.+3+Y*
ap
(1 + A;)2
9V,B‘
X -
y
(A;)2 + 3A; (3.11)
From the coefficients of the fits to the specific heat and thermal expansion in the temperature range 1.4-20 K (tables 2.1 and 2.111) we calcu-
and superconductivity
141
late, using eq. (3.11), alnT,/ap = - 19 Mbar-‘, in good agreement with the experimental value of aln T,/ap = - 26 Mbar-‘. Here, the second and third term on the right hand side of eq. (3.11) have opposite signs and amount to 33Mbarr’ and -52Mbar~’ (the term with K may be neglected). However, for the fits in the temperature range 1.4-10 K alnT,lap amounts This large negative value is to - 175 Mbar-‘. caused by the larger a,-value in this case (almost a factor of two larger than derived from the fit up to 20 K, see table 2.111). Similar large negative values result if we use a combination of the coefficients derived from fits to the phononcorrected specific heat (fits IIa and IIb, table 2.11), and the thermal expansion (table 2.111), for both temperature intervals. In this case the relatively small a-value is debit to a large negative second term in eq. (3.11). To summarize these results, we conclude that the analysis of T, in the Fermi liquid model, under the assumptions given above (see also section 1.3), suggest that UPt, is a triplet superconductor (g, < 0). However, a test of the expression for T, (eq. (3.3)), by means of a combination of specific heat and thermal expansion data, leads not to an unambiguous approval of this equation. This has partly its origin in the forementioned difficulties in describing the thermal expansion data for UPt, including a T31n(TI T*)-term. 3.6.
Miscellaneous
In this last section we discuss briefly the outcome of several other experiments that have been performed in order to elucidate the nature of the superconducting ground state. First we mention the sensitivity of UPt, to mechanical stress. As follows from table 3.1, T, for annealed samples is roughly 0.1 K higher than for the unannealed ones. This is also nicely illustrated by x,, experiments, performed by Tholence et al. [139], on a Czochralski-grown sample, before and after annealing (see Fig. 3.9). Stewart et al. [3] noticed the absence of superconductivity in a ground sample (T, < 0.050 K). On annealing the sample, T, restores.
142
A. de Visser et al. I UPt,,
heavy fermions
0
-5
3 d :
x-
10
Annealed 15 0.1
0.2
0.3
0.4K
Fig. 3.9. Diamagnetic susceptibility UPt, at different fields. As grown (circles). After ref. [139].
T(K)
for Czochralski-grown (squares) and annealed
Such a strong sensitivity of T, to mechanical stress (and thus p,,) is expected for a triplet superconductor [ 1411. Willis et al. [103] studied the superconducting transition temperature, by means of resistivity at applied pressures up to measurements, 19.1 kbar. These results, obtained on a fluxgrown sample (T, = 0.48 K), are shown in fig. 3.10. T, reduces as function of pressure, at a rate of - 12.6 mK/kbar. The relative depression of T, with pressure amounts to - 26 Mbar-‘. The initial slope of the upper critical field, BL2 is also depressed with pressure, at a relative rate of - 28 Mbar-‘. Both T, and BL2 extrapolate to zero at 37 kbar, corresponding to a volume change of AVlV,, of - 0.018. 6
c4 .c5 d z Q 2
0 0
0.5
1.0 T2
I.5
2.0
(K')
Fig. 3.10. Resistance vs. temperature crystalline UPt,. Pressures were applied 10.2 kbar, 19.1 kbar and 4.1 kbar. After
squared for singlein the order: 1 bar, ref. [103].
and superconductivity
The BCS expressions for T, [51] exhibit, it general, a complex pressure dependence causec by the competing pressure dependences of 8, A > P* and h,,. Therefore, the sign of the pF:ssure dependence of T, can not easily bt predicted. A collection of characteristic parame ters for the superconducting state, and for tht pressure dependence of T,, for several uraniun intermetallic superconductors can be found in ; recent paper by Franse [142]. For the heavy fermion superconductor UBe,, a value for aT,, dp of - 12 mK/kbar has been obtained [143] whereas for CeCu,Si, T, first increases wit1 pressure (for P<40 kbar) and then decrease: [144]. In case of UPt,, a matter of great interest i! whether the occurrence of superconductivity i! mediated by spin-fluctuations. The depression o superconductivity as well as spin fluctuation! with pressure suggests indeed a correlation be tween both phenomena, but cannot be taken a! proof for spin-fluctuation mediated superconduc tivity. The relative pressure dependence of T and the relative pressure dependence of the spin fluctuation temperature, T*, amount to - 2t and 33 Mbar- r, respectively (T* 0~TT, , section! 2.7.1 and 2.7.2). These relative pressure depend ences have an almost equal magnitude, but op posite signs. Within the presented Fermi liquic model, qualitatively a depression of T, wit1 pressure is expected for UPt,, as has been showr in the previous section. The quantitative agree ment between the calculated and experimenta values for alnT,/aT is, however, subject of dis cussion. Low-temperature data for the thermal conduc, tivity, K, as obtained by the Darmstadt group or a polycrystalline UPt, sample [86], are shown ir fig. 3.11. At temperatures below 0.5 K there is i difference between the superconducting-state K,, and the normal-state (superconductivity sup pressed by a magnetic field of 2.5 T), K,-values This difference increases towards lower tempera tures where K, follows a power law with ar exponent between 1 and 2. The temperaturt dependence of K, shows negative curvature foi T Z-O.2 K, when plotted as K,/T vs. T (fig 3.11b). As is indicated by the dashed line, the
A. de Visser et al.
I UPC,, heavy fermions and superconductivity
I t 0.1’.
a
ii’.1 ;(K; ii”
Fig. 3.11. Temperature tivity of polycrystalline and as KIT vs. T (b).
,#1’
0
’ -0 ‘.’
T(K)
’
dependence of the thermal UPt, in a double-logarithmic
conducplot (a),
data points at the very temperature end are consistent with K, = a T2, with (Y= 32 mW/K3cm. The quadratic temperature dependence of K, shows that the thermal conductivity below T, must be of electronic origin, despite the importance of the phonon conductivity as inferred in the normal state for T > 3 K(section 2.5.2). An upper limit of the phonon contribution to K, can be obtained by taking into account scattering of the phonons from grain boundaries only (“size effect”). Using the average grain size of 50 urn as determined for this polycrystalline sample, the phonon contribution to the heat transport in the superconducting state is estimated at only a few % of the total heat transport. Thermal conductivity measurements performed by Sulpice et al. [108] revealed, besides a T*-term below 200mK ((Y = 20mW/K3 cm), a term linear in temperature with a coefficient of 0.55 mW/K’cm. In section 3.3 we mentioned that also in the specific heat “residual” linear terms have been observed. A T2-term in K, has also been found for UBe,, ((Y= 0.38 mW/K3 cm [145]) and CeCu,Si, ((Y= 1.8 mW/K3 cm [131]). In the latter system an additional term linear in T exists. As in case of the specific heat, a T*-term in K, can be taken as evidence for strong gap anistropy (table 1 .IV). Bishop et al. [146] performed sound velocity experiments on UPt, in the superconducting state and observed that the ultrasonic attenuation varied as T2. These results were explained by assuming that UPt, is an anisotropic (triplet) superconductor in a polar-like state, where the
143
gap vanishes along a line on the Fermi surface (table l.IV). However, recently Muller et al. [147] observed a T3-term in the same temperature region, as well as an ultrasonic attenuation peak just below T, (only for longitudinal ultrasound), possibly due to collective modes as observed in 3He. Poppe [148] performed several tunneling experiments on single-crystalline UPt, specimens, using Al or Nb as counterelectrode. However, no Josephson current was observed as might be expected for a weak link between a triplet and singlet superconductor. However, a contact formed with two pieces of UPt, did not show any pair current either. One difficulty with UPt, is that the material .in the superconducting state is very sensitive to mechanical stress, as mentioned above. Therefore the specimen might not be superconducting in the vicinity of the contact. Besides, triplet superconductivity would be very sensitive to pairbreaking at surfaces and restrictions. Therefore, it would be difficult to observe the Josephson effect for a triplet superconductor.
4. UPt, alloyed with Pd 4.1.
Zntroduction
In the preceding sections we presented a large variety of experiments that have been performed in order to characterize and understand the normal and superconducting-state properties of UPt,. In order to further elucidate the lowtemperature properties of UPt,, useful information might be obtained from alloying experiments. Possible candidates for substitutions at the U-site are La, Lu and Th. Pt might be substituted, for instance, by its neighbours in the periodic table: Ir, Rh, Pd, Ag and Au. In general, little information is available in the literature on the solubility of other elements in UPt,. The compounds UIr, (cubic), URh, (cubic) and UPd, (hexagonal) differ in crystal structure from UPt,. UAg, does not exist and the crystal structure of UAu, is not known. Probably the area of stability of the UPt, phase is small for the forementioned substitutions at the Pt-site. This can also
A. de Visser et al. I UPt,, heavy fermions
144
be inferred from a systematic study of the occurrence of intermediate phases in the UX, compounds [ 1491. Since the low-temperature properties of UPt, strongly depend on the lattice parameters (large pressure effects, large Gruneisen parameters etc.), we expect a large influence from substitutions of only a few atom % U or Pt. In this section we present the results of a first series of alloying experiments in which we substituted Pt by isoelectronic Pd. The low-temperature properties of the compounds UPd, and UPt, are quite different, although the distances between neighbouring uranium atoms are nearly identical: 4.13 A in UPt, and 4.11 A in UPd,. The main crystallographic difference between these compounds is the stacking of the atomic layers: in UPt, the stacking is ABAB (hexagonal closed packed; hcp), in UPd, the stacking is ABAC (double hexagonal closed packed; dhcp). In the hcp structure only U-sites with hexagonal symmetry are present, whereas in the dhcp structure hexagonal sites (atoms in B layers) and so-called quasi-cubic sites (atoms in A layers) are present. In table 4.1 some crystallographic and magnetic parameters for UPt, and UPd, have been collected. Neutron scattering experiments have revealed that UPd, has well-localized fwith a 5f2-configuration in a LS electrons, groundstate ‘Hq [150]. The low value for the coefficient of the linear term in the specific heat, y < 10 mJ/K’ molU [151, 1521, illustrates the absence of a narrow band, and contrasts with the large value of 422 mJ/K’molU for UPt,. In UPd, two phase transitions have been observed,
and superconductivity
at 5 and 7 K, both non-magnetic of origin [151,153]. Evidence for crystal-field states comes from neutron-scattering experiments [150]: for both uranium sites a singlet ground state with excited doublet states, yielding excitation energies of 164 and 24 K, has been derived. Which energy splitting belongs to which site has, however, not firmly been established. Given the obvious differences between both compounds, we expect large effects on the low-temperature properties of UPt, by substituting Pd for Pt. In this section we report specific heat, thermal expansion, susceptibility, high-field magnetization and electrical resistivity measurements on a series of pseudobinary U(Pt, _xPd,), compounds (0.015 x 5 0.30). In addition, samples with x = 0.001, 0.002, 0.005 and 0.01 were checked for superconductivity. Polycrystalline compounds were prepared by arc melting the appropriate amounts of the pure elements, U (Koch-Light Ltd., purity 99.8%), Pt and Pd (MRC-Marz grade), in a titanium gettered argon atmosphere. All samples were annealed, in evacuated sealed silica tubes, at 1000°C for a period of 10 days. X-ray diffraction patterns taken on powdered samples at room temperature confirmed the hexagonal MgCd,type of structure. Samples with x Z 0.15 showed additional unresolved diffraction lines, pointing to at least one second unknown phase (phase boundary for the MgCd,-structure between 10 and 15 atom% Pd). Small needle-like singlecrystalline whiskers were obtained from the arc melted buttons, for all pseudobinary compounds,
Table 4.1 Lattice parameters (a and c), neighbouring uranium distance (d,,_,), molar volume (V,,,), susceptibility (x.,~ and x,.), effective moment (M_,,), and the electronic term in the specific heat (y), for hcp UPt, (MgCd,-type of structure) and dhcp UPd, (TiNi,-type of structure). Data for UPd, taken from ref. [150-1521
upt,
UP4
a
c
d,-,
(A)
(A)
(A)
V, (m’imol)
5.752 5.770
4.897 9.631
4.13 4.11
4.24 x 10 ’ 4.18 x lOmY
/*
~mJ!K2molU)
2.6 2.8
422 10
Xu.h (lo-” UPt, UPd,
115 300
m’/molU)
x, (lo-’ 57 160
m’/molU)
A. de Visser et al. I UPt,, heavy fermions
just as for pure UPt, (see section 1.5). Lattice parameter determinations, by X-ray diffraction patterns taken on the powdered samples and on the whiskers, yield a constant u-parameter within the experimental accuracy, a = 5.752 2 3 A, whereas the c-parameter decreases linearly with from 4.897 + 3 A for pure Pd concentration, UPt, down to 4.892 + 3 A for x = 0.15. The c/aratio decreases slightly with increasing Pd content (a reduction of 0.1% for the 15% Pd compound). The reduction of the volume, AVlV, amounts to 7.5 x 1O-4 per atom% Pd. The same value for AVIV can be achieved by applying an external pressure on pure UPt, of - 1.6 kbar (K = 0.48 Mbarr’). 4.2.
Specific heat and thermal expansion
Specific heat measurements on the series of U( Pt 1_xPd,), compounds have been performed in the temperature range 1.2-30 K [97, 154, 1551. Data were taken in zero field and in a field of 5 T. The zero-field data are shown in fig. 4.1
F
145
(for T ~20 K). In fig. 4.2 we show the field effect for some selected compounds. On alloying UPt,, two remarkable features can be observed: (1) for x 5 0.10 the y-value increases with respect to pure UPt,, and (2) an anomaly develops at low temperatures for the 2% and 5% buttons. The former observation points to an enhancement of the many-body effects at low temperatures (larger effective mass). Although it is not unambiguous how to extrapolate c/T in fig. 4.1 to 0 K, y might easily amount to 600 or 700 mJ/K*molU for the 5% and 10% compounds. This signifies a surprisingly large increase (almost 50%) of the y-value with respect to pure UPt,. In a magnetic field of 5 T the y-values are only slightly modified, as indicated by the c/T-values at 1.4K in fig. 4.3. The entropy difference, in the temperature interval 1.2-20 K, between the curve for pure UPt, and the curve for U(Pt 0.80Pd0.20)3 equals 2.4 J/ KmolU. On diluting by Pd, the corresponding
o.7rrl 0.6
and superconductivity
0.6 $ +
t+ :*
U(Pt,_xPd,13 0.5 9 a CJ
0.4 -
Y a
0.3 -
0.2 -
0.1 -
’ 0
Fig. 4.1. Specific heat of U(Pt,_,Pd,), of c/T vs. T, for x 5 0.30.
compounds in a plot
5
10
Fig. 4.2. c/T vs. T for U(Pt,_,Pd,), compounds for x= 0.00(O) and (O), x =0.02(A) and (*), x = 0.05(+) and (0) and x = 0.30(O) and (*), at zero field and an applied field of 5 T.
146
A. de Visser et al. I UPt,, heavy fermions
0.6 9 $
O5
-Y a
0.4
Y d\-
0.3
;
0.2 0.1
0.0
Fig. 4.3. c/T values at T = 1.4 K vs. Pd concentration for U(Pt,-,Pdr), compounds (X 5 0.30) in a magnetic field 01 5 T(A) and at zero field (0). The broken line serves as a guide to the eye.
entropy differences with the 20% compound amount to the constant value of 2.4 J/KmolU for compounds with x S 0.05. The entropy difference between the curves for x = 0.20 and x = 0.30 amounts to 1.0 J/KmolU. Next we discuss the nature of the peaks in c/T vs. T for the 2% and 5% Pd compounds. These anomalies remind one of the phase transitions in UPd, at 5 K and 7 K [151, 1533. From entropy considerations it follows, however, that the anomalies in these pseudobinary compounds can not be due to a second phase of UPd, (that might be overlooked in the X-ray patterns). For UPd, the excess entropy up to 15 K equals 3 Ji KmolU [151], whereas the entropy involved in the peaks of the 5% and 2% samples amounts to 0.8 J/KmolU and 0.2 J/KmolU, respectively. In a magnetic field of 5 T the temperature at which the maximum in c/T is observed shifts from 5.8 K to 5.4 K (5% Pd), and from 3.6K to 3.3 K (2% Pd), but the shape of both peaks remains essentially unchanged (fig. 4.2). The field effect on the anomaly in the specific heat of UPd, has an opposite sign [152]. In one of the papers we suggested that these
and superconductivity
anomalies possibly indicate the onset of antiferromagnetic order [154]. This was recently confirmed by neutron experiments on a single-crystalsample [156]. We return to line U(Pt”.,,Pd,.,,), this point in section 4.6. The specific heat data of UPt, have been analysed with a T31n( T/T*)-contribution, characteristic for spin-fluctuation effects. In case of the U(Pt,_,Pd,), compounds, the additional low-temperature anomalies hamper such an analysis. A computer fit to the data, for the 1% and 10% compounds, in the temperature interval 1.2-10 K, including such a term, yields an increase of 6 and a decrease of /3* with increasing Pd content. The characteristic temperature, T *, reduces from 29 K (pure UPt,) to 22 K (1% Pd) and 19 K (10% Pd). For x 2 0.15 the spin-fluctuation properties are rapidly lost. In figs 4.4 and 4.5 we present thermal expansion measurements on some selected compounds. The maximum observed for pure (polycrystalline) UPt, at 12K, shifts towards lower
WQ’dX)
3
0 x=0.00
I\ t r’ $
A q
00
00.01 .02 0.05 0.15
+ 0
-15 L 3 t
-20
1
(111111111(IIIJ1111 0 20 40
60
80
100
T (K> Fig. 4.4. Linear thermal expansion coefficient for polycrys talline U(Pt,+,Pd,), compounds, for x =0.00(O), 0.01(O). 0.02(A), 0.05(+) and 0.15(O).
A. de Visser
et al. I UPt,, heavy fermions and
pound, whereas the 2% compound shows a sharp drop (on decreasing the temperature) in cy near 4 K. The thermal expansion of a 15% compound resembles that of the c-axis of UPt,. Since the thermal expansion coefficient is strongly anisotropic and since crystallites with preferred orientations are present in these samples, a more quantitative analysis awaits data on single-crystalline samples.
9 8 7c\
6-
+
5-
147
su~erconductiv~~
g..
4-
4.3. ~~ce~ti~ility
s
3-
The results of the susceptibility measurements in a field of 1.3 T, in the temperature range 1 S-300 K, and high-field magnetization measurements, at a temperature of 4.2 K, on some selected U(Pt,_,Pd,), compounds [155] are shown in fig. 4.6 and fig. 4.7, respectively. Both experiments have been performed on cylindrical samples, with a diameter of 1.5 mm and a length of 5 mm. Because of the forementioned presence of crystallites with preferred orientations, the absolute x-values in fig. 4.6 are somewhat sample dependent. Obviously, the most interesting result of alloying UPt, with Pd as to the susceptibility and ma~etization, is the reduction of the characteristic temperature and field at which the anomalies are observed. The maximum in the
2-
P
I
ad
lo-
%
2
-1""""""""""' 5 0
10
20
15
Fig. 4.5. The same as fig. 4.4. More detailed account in the temperature range I-20 K.
temperatures with increasing Pd content. Additional anomalies for the 2% and 5% Pd compounds are also found in the thermal expansion: a small peak results at 5.8 K for the 5% com-
u(Pt,,Pd,)
and ~ug~etizfftiun
3
x= 0 0
0.00 0.02
4 0.05 0 0.15
0
’
I
I
0
100
200
Fig. 4.6. Magnetic susceptibility vs. temperature
T (K> for polycrystalline
1
300 U(Pt,_,Pd,),
compounds.
A. de Visser et al. f UPt,, heavy fermions and superconductivity
148
0
l
,*
,o
.<
,*
,’
,2 / ,0’ b'
A’
q
I-
’
P I
0 0
10
P,?T) /
i
30
40
Fig. 4.7. High-field magnetization normalized to 1 at 35 T for polycrystalline U(Pt,_,Pd,), compounds. The absolute values at 35 T amount to 0,806, 0.782, 0.882 and 0.832 &U-atom, for compounds with x = 0.00, 0.02, 0.05 and 0.15, respectively.
susceptibility, at approximately 17 IS for pure UPt,, shifts towards lower temperatures (11 K and 7 K for the 2% and 5% sample, respectively), and becomes more pronounced. It is not observed for a sample with x = 0.15 down to 1.4 K. From a Curie-Weiss analysis of the data in the temperature range 50-300 K, it follows that the effective moment remains constant on alloying ( peri. = 2.6 f 0.1&U-atom). The paramagnetic Curie-Weiss temperature increases from - 80 I: 10 K, for pure (poly~rystalline) UPt,, up to - 50 + 10 K for U(Pt, ssPd,, ,&. In the case of UPd, a value for wcff of 2.8p.,/Uatom has been deduced from an analysis of x in the temperature interval 70-300 K [152]. The anomaly in the high-field magnetization curve at 4.2 K, i.e. a maximum in the differential susceptibility at - 20 T, for pure UPt,, shifts towards lower fields (approximately 16 T and 11 T for the 2% and 5% sample, respectively). Again, it has not been observed for a sample with x = 0.15, at this temperature.
U(Pt,_,Pd,), compounds, as well as on small single-crystalline whiskers, in the temperature range 1.3-300 K 11571. The experimental results are shown in figs 4.8, 4.9 and 4.10. In order to
0.6
U(Pt,_,Pd,), 0.4
0.2
0
4.4 Electrical resistivity Electrical performed
resistivity measurements have been on a number of polycrystalline
I
J
200
Jo0
I
0
xl0
T (K> Fig. 4.8. Reduced resistivity, ~(~)/~(3~ K), of single-crystalline U(Pt,_,Pd,), whiskers, for x = O.~(O~, 0.05(A) and 0.10(V). The current is applied along the c-axis.
A. de Visser et al. I UPt,, heavy fermions
I
I
0
300
200
100
0
T (K> Fig. 4.9. Reduced line U(Pt,_,Pd,), 0.10(V).
resistivity, samples,
p(T) lp(300 K), of polycrystalfor x=0.00(0), 0.05(A) and
l.2
0.6
0.4
02
0 0
I
I
I
I
10
20
30
40
50
-I-(K> Fig. 4.10. Reduced resistivity, p(T)/&300 K), of polycrystalline U(Pt,_,Pd,), samples, for x=0.00(0), 0.01(O), 0.02(+), 0.05(A), 0.10(V) and 0.15(x).
and superconductivity
149
illustrate the relative temperature dependence of the resistivity, the data are plotted as p(T) / ~(300 K) vs. T. Absolute values for p,, and p(300K) are listed in table 4.11. From figs 4.8 and 4.9 it follows that the experimental data for the whiskers and the polycrystalline samples nearly coincide. This illustrates that the whiskers may be considered as representative for the bulk material. The difference between the absolute values for the two types of samples partly has its origin in anisotropy. As discussed in section 2.5.1 due to preferential orientations, a polycrystalline sample has a resistivity almost equal to the basal plane value (238 l&cm for pure UPt,), whereas a whisker represents the c-axis value (132 p.Rcm). Furthermore, cracks and an irregular texture enlarge the resistivity of the polycrystalline samples. This is in particular the case for the specimens with high Pd contents (x = 0.10, 0.15). The main result from our measurements is the gradual change from the spin-fluctuation-like behaviour observed for pure UPt, (section 2.5.1) to a Kondo-like resistivity for high Pd contents. In combination herewith we observe a reduction of ~(300 K) - p0 with increasing Pd concentration. Temperature derivatives, AplAT, are shown in fig. 4.11 for some selected polycrystalline compounds. The maximum in AplAT, at 6.5 K for UPt,, shifts towards 5.5 K for the 1% Pd sample. For the 2% and 5% Pd samples ApiA T yields an anomaly near the temperature where the maximum in the specific heat is observed. In particular a sharp minimum occurs at 5.6 K for the 5% Pd sample. The resistivity of the Kondo-like compounds (X = 0.10, 0.15) is shown in a plot of P(T) lp(300 K) vs. 1nT in fig. 4.12. At the lowtemperature side the resistivity of both compounds deviates from the 1nT behaviour; coherence sets in. A low-temperature maximum, as we observe for x = 0.15 near 5 K, has also been observed by Stewart and Giorgi [158] on an U(Pt,,,,Pd,.,,), sample near 20 K. A T2temperature dependence, as observed for UPt, below 2 K, is not observable in the samples with low Pd contents in the investigated temperature range. In searching an explanation for the Kondo-like resistivity curves for the samples with
A. de Visser et al. I UPt,, heavy fermions
and superconductivity
x equal to 0.10 and 0.15, we compare UPt, wnh UPd,. A concentration of - 10% Pd on the Pt sublattice could, in principle, be sufficient to destroy, at least partially, the hcp structure, possibly leading to quasi-cubic uranium sites that could act as Kondo impurities. However, analysing the X-ray patterns for x 5 0.15, any sign of a (long-range) dhcp structure was lacking and no substantial broadening of the diffraction lines was observed. Analysing the X-ray patterns for higher Pd concentrations, additional lines are found indicating the presence of a second phase, which is not the UPd, phase. Whether the Kondo-type of resistivity curves are related to this second phase remains a question to be answered. 4.5.
Fig. 4.11. Temperature derivative. of the resistivity, vs. T, for polycrystalline U(Pt,_,Pd,), samples, 0.00(O), 0.01(O), O.OZ(+) and 0.05(n).
U(Pt,_xPd,l,
ApiA T, for x =
I
0.8 0
1
2
3
4
5
6
Inl Fig. 4.12. Reduced resistivity, p(T)lp(SOO K), vs. InT for polycrystalline U(Pt,_,Pd,), samples, for x =0.10(V) and 0.15(x).
Superconductivity
As follows from the specific heat measurements on these pseudobinary compounds, the spin-fluctuation temperature decreases with increasing Pd content, leading to an enhancement of the spin-fluctuation effects. An important question is whether the occurrence of superconductivity and spin-fluctuation effects in UPt, are closely related or not. Assuming that such a close relation exists, one would not expect a large change in the superconducting properties for samples with small amounts of Pd substituted for Pt (x 5 O.Ol), since the spin-fluctuation effects do not change drastically in this Pd concentration range. However, no superconductivity has been observed in a U(Pt,,,,Pd,,,,), sample as follows from an ac-susceptibility experiment down to 40 mK [155]. The low-field magnetization experiments on these pseudo-binary compounds and chemical analysis of the pure U, prove that the concentration of magnetic impurities (probably Fe) is limited to 600 ppm, which is approximately a factor of 10 less than the amount of Pd in this 1% sample. An annealed single-crystalline UPt, sample, grown from this batch of U, was found to be superunannealed 0.48 K. An conducting at sample, made of a purer batch U(Pt0.99,Pd0.0”,)3 of uranium, was not superconducting either, whereas an unannealed polycrystalline sample
A. de Visser et al. I UPt,, heavy fermions
Table 4.11 Residual resistivity,
pO, and room-temperature Whiskers
Compound X 0.00 0.01 0.02 0.05 0.10 0.15
values ~(300 K) in kacrn
and superconductivity
for single-
and polycrystalline
Polycrystalline
(Illc-axis)
151
U(Pt,_,Pd,),
compounds
samples
PO
~(300 K)
PO
~(300 K)
2.2 4.5 6.4 24.6 46.6
147 126 81 75 48 -
6.1 18.6 36.4 101 633 *.# 355 #
240 264 238 251 539 334
* Irregular structure. # Kondo-type compound.
made from the same uranium, had a superconducting transition temperature of 0.38 K. Recently, samples with an even lower Pd content (0.1% and 0.2%) have been prepared. Superconductivity is observed in these samples at and at 0.357 K (0.2%) [159]. 0.460 K (0.1%) Apparently, the superconducting ground state is rapidly destroyed by alloying UPt, with Pd. From the resistivity data we conclude that the strong suppression of superconductivity cannot simply be due to an impurity effect. From table 4.11 it follows that the p,-value of, for instance, the 0.5% Pd specimen lies in between 3 and 10 l.kkm. Yet, an annealed single-crystalline UPt, sample with a similar p,-value (6.2 @km) became superconducting near 0.5 K. Also the presented Fermi-liquid model (section 3.5) is not able to account for such a rapid depression of T,. From the normal-state specific heat data we calculate only a minor change in the pseudopotential g,, for x 5 0.01, and thus T, would remain unchanged (eq. (3.3)), in contrast to the experimental findings. Although the spin-fluctuation effects are enhanced on alloying with Pd, superconductivity is suppressed. Apparently, the high y-value is not a sufficient condition for a superconducting ground state. Therefore, the present data are not in favour of a direct correlation between spin fluctuations and superconductivity. However, the occurrence of superconductivity might depend on a delicate balance between several factors, the presence of spin-fluctuations being one of
them. More information is needed further elucidate this problem.
in order
to
4.6 Discussion From the specific heat, susceptibility and highexperiments on the field magnetization U(Pt,_,Pd,), compounds we conclude that a close connection between the thermal and magnetic properties exists: the characteristic temperature as derived from the specific heat (T* = 29 K), the temperature at which the maximum in the susceptibility occurs (T,,, = 17 K), and the field at which the maximum in the differential susceptibility is found (B,,, = 20 T), all reduce on alloying UPt, with Pd. Accepting the analysis as presented in section 2, this implies an enhancement of the spin-fluctuation effects (lower excitation energy). For x 2 0.15 the spin-fluctuation phenomena are lost. As mentioned before, recently, neutron experiments revealed that the anomaly in the 5% Pd sample at 5.8 K is associated with the onset of antiferromagnetism (Frings et al. [156]). From these first neutron data it follows that the magnetic moments are aligned antiferromagnetically in the basal plane. The ordered moment amounts to 0.6 -+ 0.2&U-atom. Furthermore, Frings et al. argue that the step-wise increase of the magnetization near 11 T in the polycrystalline 5% Pd sample (fig. 4.7) is due to the destruction of the antiferromagnetic correlations in the basal plane, followed by a ferromagnetic
152
A. de Visser et al. I UPt,, heavy fermions
alignment at high fields. Magnetization data taken by Van Sprang et al. [159] on a singlespecimen at 4.2K crystalline U(Pt0,95 Pd,,,,), (below the ordering temperature), reveal the same anisotropy as observed for pure UPt,; the main difference being the occurrence of the stepwise increase in the basal plane magnetization at lower fields (- 11 T, compare fig. 4.7). At 35 T the magnetization in the basal plane exceeds the value for the c-axis by - O.S~,iU-atom. Assuming that the reorientation of the moments is superimposed on an isotropic magnetization, the ordered moment as derived from the magnetization data amounts to 0.5pu,lU-atom, close to the value for the ordered moment as reported by Frings et al. It is very likely that the high-field anomaly observed for pure UPt, is due to the same phenomenon. However, in case of UPt, no long-range antiferromagneti~ order has been observed, which implies that the order must be short-range. This offers the following picture. In pure UPt, short-range antiferromagnetic eorreiations, confined to the basal plane, set in near the temperature at which the maximum in the susceptibility occurs, T,,, = 17 K. This short-range order is destroyed by a field of - 20T. For the 2% Pd sample the characteristic temperature and field for the short-range correlation are slightly lower, besides long-range order (accomplished between the hexagonal planes) sets in at 3.6 K. In the 5% Pd compound the onset of short-range and long-range order almost coincide. In the 10% Pd compound the ordering phenomena have nearly disappeared (only a small anomaly in c is observed near 2 K). Besides, in this entire concentration range the large y-value remains, which shows that spin-fluctuations persist down to the lowest temperatures. At higher Pd contents a second phase appears in the diffraction patterns: the spin-fluctuation phenomena are lost, and a Kondo behaviour results. These observations suggest that the compound UPt, is close to an antiferromagnetic instability: antiferromagnetism can be induced by substituting small amounts of Pd for Pt. Similar ordering phenomena in the U( Pt , _XPdx)3 compounds have recently been observed by Stewart et al. [160].
and superconductivity
In fig. 4.13 we present the resulting magnetic phase diagram for the U(Pt,_,Pd,), system. For low Pd concentrations superconductivity is rapidly destroyed; antiferromagnetism is observed in the concentration range 1 or 2% Pd up to 10% Pd. A small concentration range remains where neither superconductivity, nor antiferromagnetism is observed (-0.4-l% Pd). This shows that the superconductivity is not directly killed by the onset of antiferromagnetism. The arrows in fig. 4.13 indicate the sign of the pressure effect on the superconducting transition temperature and the spin-fluctuation temperature are observed for pure UPt,: &rT,lap = -26MbaC’ and alnT*/ap =33Mbar-*, and on the NCel temperature as observed on a 5% Pd specimen: aln TN/ ap = - 55 Mbar ’ (Van Sprang et al. [lSl]). Applying pressure and alloying with Pd both yield a reduction of the volume. In this respect the suppression of superconductivity and antiferromagnetism might originate from a volume effect (note that this implies that TN for the 2% Pd compound would increase with pressure). However, this would also imply that at a sufficicntly high pressure pure UPt, would be an an-
Ts(
T 1
.’
0
0
,
i:.
0.5
‘
&at%
Pd
1
0
lo
Fig. 4.13. Phase diagram for the U(Pt,_,TPd,), compounds for xsO.10; Ntel temperature, TN, and superconducting transition temperature, T,, vs. Pd concentration; S= AF= fluctuations, SF = spin superconductivity, antiferromagnetism, K = Kondo behaviour. Arrows indicate the sign of the pressure effect on the characteristic temperature. Nate the expanded scale for x CC0.005. Data for the 7% Pd compound taken from ref. 11591.
A. de Visser et al. I UPt,, heavy fermions
tiferromagnet (albeit in a small pressure range), which is apparently not the case. Furthermore, the pressure effect on T,,, of x for pure UPt, and on T, for x = 0.05 (as follows from measurements of the resistive anomaly as function of pressure) are of opposite sign. Specific heat, susceptibility and resistivity measurements on a series of U,_,Th,Pt, compounds identical low-temperature revealed nearly anomalies as the Pd doped samples [160, 1611. In contrast to Pd doping, the volume increases, with respect to pure UPt,, when Th is substituted for U. Therefore, Ramirez et al. [161] suggested that not the volume but the c/u-ratio plays a dominant role: for both substitutions, Pd and Th, the c/u-ratio decreases. This was further supported by recent alloying studies with Ir and Au [162]. When, on alloying, the c/a-ratio decreases, antiferromagnetism is induced. At applying pressure the c/a-ratio increases again, as follows from the anisotropic compressibility (section 2.4), and TN decreases (note that this implies that TN for the 2% Pd compound would decrease with pressure as well). A numerical assuming a linear calculation shows that, pressure dependence of TN, antiferromagnetism in the 5% Pd specimen would be totally suppressed at a pressure of - 30 kbar. An explanation for the occurrence of antiferromagnetism in the doped UPt, samples can be offered by Fermi surface instabilities, due to doping induced strains: a gap opens over a part of the Fermi surface, resulting in a combination of spin density waves and antiferromagnetic order, like observed in Chromium [163]. A similar explanation has been offered for the occurrence of antiferromagnetism in the heavy fermion systems UCd,, [16] and U,Zn,, [17].
5. Concluding 5.1.
remarks
Normal phase
In section 2 we presented studies of the magnetic, thermal, elastic and transport properties of UPt, in the normal state. These studies were
and superconductivity
153
based on a large variety of experiments, among which specific heat, thermal expansion, susceptibility, electrical resistivity, magnetostriction etc. Besides, we studied the effect of high magnetic fields and high pressures. Most of these experiments have been performed on good-quality single-crystals. We have shown that the thermal properties can be fairly well described in the Fermi liquid model, including a T31n(TIT*)-contribution to account for spin fluctuations. The magnetic properties are strongly anisotropic: a two-dimensional character of the field effect (confined to the hexagonal plane) results. On the one hand, the observed anomalies in the susceptibility at T max= 17 K, and in the high-field susceptibility at &n,X = 20 T (at liquid helium temperatures), seem to originate from spin-fluctuation effects as well. On the other hand, these anomalies are reminiscent of a type of antiferromagnetic order. On alloying with Pd, pseudobinary compounds U(Pt,_,Pd,), order indeed antiferromagnetically (for a limited concentration range). Obviously, the compound UPt, is close to an antiferromagnetic instability, which can be triggered by substituting small amounts of Pd for Pt. Below we present a few remaining experimental and theoretical aspects, in order to complete the discussion about the normal phase of UPt,. Recently, Aeppli et al. [164] and Johnson et al. [165] performed inelastic neutron scattering experiments on UPt,. From these experiments direct evidence for spin fluctuations was deduced. The energy scale for the spin fluctuations was found to correspond to a large effective mass, in rough agreement with the low-temperature specific heat data. Diffraction patterns recorded for a powdered sample at 5,20 and 300 K were similar [165], once more excluding the possibility of long-range antiferromagnetic order below - 17 K. We want to emphasize that the low-temperature anomalies are not caused by crystal-field levels, as might be suggested by the resemblance between UPt, and, for instance, the hexagonal crystal-field compound PrNi,. The susceptibility of PrNi, peaks at 17 K [166], as is the case for UPt,. Measurements on a single-crystalline
154
A. de Vikser et al. / UPt,, heavy fermions
specimen revealed a nearly identical anisotropy in x, as found in UPt,. Also thermal expansion data for PrNi, [167] and UPt, are much alike (apart from the low-temperature upturn in (y/T, which is only observed for UPt,). Furthermore, non-linearities have been observed in the basalplane magnetization of PrNi, above - 8 T [ 1681. The ground state of PrNi, in the C&coupling scheme is ‘H4(Pr”‘). Due to the crystal field the J = 4 multiplet is split into 3 singlets and 3 doublets. The lowest level is a singlet (r4) and the first excited level, also a singlet (r’), lies at an energy distance of - 20 K. The low-temperature and high-field anomalies in the case of PrNi, are attributed to crystal-field effects. Assuming an identical ground state for UPt, (a crystal-field split 3H, state), we calculate from the observed anisotropy in x, a value for the crystal-field parameter Bi of 6.5 K, quite close to the value of 5.8 K reported for PrNi, [169]. Nevertheless, the corresponding anomalies in case of UPt, are not caused by crystal-field effects. Evidence hereto mainly arises from the absence of a Schottky anomaly in the specific heat, and is further supported by comparing magnetization and magnetoresistivity data for single-crystalline UPt, and PrNi,. In particular, the magnetoresistance of PrNi, at 4.2 K, for a current along the hexagonal axis and the field applied in the basal plane, increases monotonically as a function of the field, from 3.0 l&cm at zero field to 3.5 @cm at 35 T (1191, whereas for UPt, a large peak is observed near 20T. Besides, at room temperature p of PrNi, amounts to 27 @cm, much lower than the corresponding value for UPt, (132 @cm). Also magnetization curves of PrNi, at 4.2 K, measured for the different crystallographic axes in fields up to 35 T (de Visser et al., unpublished), have a quite different shape, when compared with the corresponding curves for UPt,. To summarize: we believe that no low-energetic crystal-field levels play a role in the low-temperature properties of UPt,. In discussing the thermal properties of UPt,, we focussed on a Fermi liquid analysis. Below we briefly compare a few aspects of this analysis with a similar analysis in the paramagnon model. In the Fermi liquid model the enhancement of
and superconductivity
the susceptibility of the interacting system which respect to x of the non-interacting system is expressed by S = (m*lm)(l + Ft)-‘. With the values m* im = 180 and Ft = - 0.788, as derived from the linea_r and T31n( TIT*)-terms in the specific heat, S amounts to 849 (table 2.11). In the paramagnon model the ratio of x for the exchange enhanced system and the Pauli susceptibility, x0, is given by the Stoner enhancement factor S. Usually, x0 is calculated from the unenhanced linear term in the specific heat, yO. In the case of UPt, ‘yOcan be obtained from a twoparameter fit (linear and cubic terms) to the specific heat in the temperature range 9-15 K, yielding Pph = -yO= 225 mJ/K*molU (and 0.85 mJ/K4molU) [8]. Assuming such a large y,-value, the spin-fluctuation enhancement factor A,, amounts to 0.88, which implies only a small mass enhancement rn* Im = 1.88 (we neglect the enhancement due to the electron-phonon interaction). With the experimental value for x of 90 x low9 m3/molU (averaged over the three crystallographic directions), and x0 as calculated directly from ‘y,,,we subsequently derive a rather low value for S: 2.32. A higher value for S results when the density of states that follows from band-structure calculations is used. Several groups performed band-structure calculations [170-1751 yielding y-values between 14.9 and 22.5 mJ/K*molU (values for h,, range from 18 till 27). By calculating x0 from these band-structure derived y-values, we obtain values for S between 23 and 35. The corresponding mass enhancement, m* Im, ranges from 17 till 26. The low value for the effective mass as derived in the applied paramagnon model (when compared to the mass enhancement obtained in the Fermi liquid model) is partly due to the fact that we did not take into account the mass renormalization that arises from the band structure. In the Fermi liquid model the renormalization of the electron mass has been calculated with respect to the free electron mass. In section 2.3.2 we showed that the experimental value for the enhanced linear terms in the specific heat and thermal expansion, rz, amounts to 68 at 1 K (eq. (2.16)). In the paramagnon model rz can be expressed as
A. de Visser et al. I UPt,, heavy fermions
1+ ASf r:=l+
r e,
(5.1)
enwhere AZ represents the spin-fluctuation hancement factor for the linear term in the thermal expansion: a,/3 = yz T = y,(l + h:)T (we neglect the enhancement due to the electronphonon interaction). r, represents the Griineisen parameter for the unenhanced linear terms in c and CY(eq. (2.11)). Assuming r, = 3 for f-electrons, we calculate using eq. (2.11) from (yO = ) y = 225 mJ/K’molU a value for ‘y, of 2 X lo-’ K-*. Since yz amounts to 1.08 X 10m6K-* (table 2.111), AZ is large. We calculate AZ = 53. The corresponding enhancement factor for the specific heat, hsf, equals 0.88 and the ratio AZ/A,, amounts to 60. Employing y = 20 mJ/K*molU, as follows from the band-structure calculations, we derive the much larger enhancement factors At = 613 and A,, = 20. Their ratio amounts to 31. The ratio AZ/A,, can also be calculated from the volume dependence of the exchange enhancement factor S, employing the relationship [176]: A
sf
-E_1+--
Asf
1 alnS T,lnS alnV ’
(5.2)
The relative volume dependence of S can be related to the pressure dependence of the susceptibility by alnS -=--_ alnV
(5.3)
With the averaged value of - 20 Mbar-’ for the relative pressure dependence of x at 4.2 K (section 2.7.2), and S = 2.32 we calculate AZ/A,, = 21, whereas for S = 26, as follows from the bandstructure calculations ( y = 20 mJ/ K*molU), a value for this ratio of only 6.2 results. Apparently, the ratio AZ/A,, as calculated from the volume dependence of the Stoner factor is lower than the value as calculated from the specific heat and thermal expansion data. This inconsistency puts some question marks as to the foregoing analysis in the paramagnon model. Nevertheless, several authors focussed on an analysis in the paramag-
and superconductivity
155
non model, when discussing the spin-fluctuation phenomena in UPt, [SO, 1771. As mentioned above band-structure calculations have been performed by several groups. The basic result of these calculations is a broad Pt 5d band below E, (width -7 eV) and a narrow spin-orbit split U 5f band above E, (width - 2 eV). Most of the calculations yield a sharp peak in the density of states with a width of 0.1 eV at the beginning of the U 5f band. E, is located near this peak. Hybridization between the two bands removes states from the top of Pt 5d band and places them in energy region of the U 5f band. The calculated y-value amounts to - 20 mJ/K’molU, implying that an enhancement factor of about 20 is needed in order to account for the large experimental y-value. This enhancement factor takes into account the spinfluctuation effects. The band-structure calculawith high-resolution tions are consistent (0.13 eV) photoemission studies [178], in which a narrow structure of - 0.15 eV has been observed at E,. We made several attempts to observe the De Haas-Van Alphen effect on a Czochralski-grown specimen (RRR = 80) in the Amsterdam High Field Installation (T = 1.5 K, B,,, = 40 T). However, these attempts were unsuccessful, which we attributed to the high effective electron mass. Recently, a major achievement in this field was made by Taillefer et al. [179], who performed dHvA experiments on a high-purity single-crystalline UPt, sample (RRR- 500), in the temperature range 20-150 mK. Preliminary estimates of the cyclotron masses range from 25 to 90 times the free electron mass, which undoubtedly proves the existence of heavy electrons in upt,. In spite of the tremendous effort to unravel the intriguing low-temperature properties of UPt,, some of them remain quite puzzling. In particular, the possible interrelation of spin fluctuations, antiferromagnetism and superconductivity is an intriguing question. Hopefully, in the near future, theorists will be able to develop models that can successfully account for such interrelations in the field of heavy fermions. From the experimentalists side, other alloying
156
A. de Visser et al. I UPt,, heavy fermions
studies, and the search systems, are requested underlying physics. 5.2.
Superconducting
for more heavy-fermion in order to grasp the
phase
In section 3 we discussed the superconducting properties of UPt,. Meissner effect experiments demonstrated that the occurrence of superconductivity is a bulk property. The uranium 5felectrons are involved in the formation of the superconducting condensate, as follows from the magnitude of the anomaly in the specific heat at T,. The large initial value for the temperature derivative of the upper critical field can be analysed in a rather conventional way, yielding the Fermi wave number of the heavy electron liquid before it enters the superconducting state. Several unconventional features have been observed in the superconducting phase. The electronic properties (specific heat, thermal conductivity and sound attenuation) follow a power law, rather than an exponential temperature dependence. Taking into account entropy balance, residual linear terms are found in the specific heat (and thermal conductivity), which shows that parts of the Fermi surface remain normal. The observed temperature variation of the electronic properties is highly suggestive for a polar like odd-parity superconducting phase. A strong dependence of T, on stress, and the absence of a Josephson current in a weak link between a conventional superconductor and UPt,, are consistent herewith. A Fermi liquid analysis of the superconducting transition temperature, using Landau parameters derived from the normalstate thermal properties, favours triplet superconductivity. On substituting Pd for Pt, superconductivity is rapidly destroyed: T, < 0.040 K for 0.5 atom% Pd. In view of the present state of the theory none of these unconventional properties can be taken as decisive proof for spin-fluctuation mediated odd-parity superconductivity. Recently, Oguchi et al. [180] could explain several of the observed properties, based on a model for conventional phonon mediated superconductivity, albeit with an anisotropic gap. From their band-structure
and superconductivity
calculations the electron-phonon interaction was deduced: strong for electrons on the Pt atoms, quite weak for electrons on the U atoms. The corresponding anisotropy in the electronphonon coupling constant, hph, leads via the hybridized wave functions to strong gap anisotropy. With a calculated average value for A,, of 0.33 and an assumed Coulomb pseudopotential of 0.1, Oguchi et al. calculate T, ==0.2 K (the experimental value amounts to 0.5 K) using McMillan’s formula. This would show that phonon mediated singlet superconductivity is possible in UPt,, the superconductivity being driven by the strong electron-phonon interaction at the Pt sites. The strong reduction of T, on substituting Pd for Pt has been taken to support this model: strains or Pt defects would largely affect the electron-phonon interaction, thus reducing T,. Weighing the pros and cons of odd-parity superconductivity a conclusive judgment cannot be given at present. Obviously, UPt, is a very likely candidate for unconventional spin-fluctuation mediated superconductivity. If UPt, proves to be an odd-parity superconductor a wealth of exotic physics may lie ahead.
Acknowledgements The authors are grateful to J.C.P. Klaasse, E. Louis, R. Verhoef, F.E. Kayzel, P.H. Frings, P.E. Brommer, M. van Sprang, K. Kadowaki, T.T.M. Palstra, P.R. Nugteren, A.J. Dirkmaat, P.H. Kes, J.A. Mydosh, U. Rauchschwalbe, C.D. Bredl, F. Steglich, J. Schoenes, B. Liithi, J. Flouquet, A.G.M. Jansen, A.M. Duif and many other colleagues in the field of the heavy fermions, for a stimulating cooperation. This work was part of the research program of the Dutch Foundation for Fundamental Research of Matter (F.O.M.).
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