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Very low temperature properties of heavy fermion materials
PHYSICAl] Physica B 199&200 (19941 70 75
ELSEVIER
Very low temperature properties of heavy fermion materials J.P. Brison"'*, N. Keller", P. Lejay a, A. I-[uxley b, L. Schmidt b, A. Buzdin b'l, N.R. Bernhoeft b, I. Mineev b'2, A.N. Stepanov b'3, J. Flouquet b, D. Jaccard ¢, S.R. Julian a, G.G. Lonzarich d " Centre de Recherches sur les Trbs Basses Tempbratures, Laboratoire associb h I' Universitb Joseph Fourier, CNRS, BP 166, 38042 Grenoble-Cbdex 9, France b DRFMC-SPSMS, CENG, BP 85X, 38041 Grenoble-C&tex, France DPMC, Gen&'e, Switzerland a Cavemlish Laboratory. Cambridge, UK
Abstract A review is given of recent very low temperature ,experiments in the heavy fermion superconductors UPt3, URu2Si2, UBe~ 3 and CeCu2Si2, Attention is focused on the temperature dependence of the specific heats in all four systems, the anisotropy of the upper critical fields in URu2Si2 and UPt 3, and the pressure dependence of the antiferromagnetic and superconducting transition temperatures in URu2Si2.
1. Introduction The heavy fermion superconductors URu2Si2 and UPt3 differ from the related materials CeCu2Sb. and UBet ~ in exhibiting clearly defined transitions to weakly polarised antiferromagnetic states at temperatures TN well above the superconducting critical points T,. The values of T¢ for these four systems are 1.4, 0.5, 0.6 and 1.0 K. The corresponding values of TN for the first two compounds, which will be the main focus of attentior~, are 17.5 and 5 K and hence ;an order of magnitude above
7",. * Corresponding2 author. 1 Present address: Argonne National Laboratory, Argonne. ll.. USA. ? Pnc~cn(address: [.andau Institute. Mrs,;cow. Ru'~l:m g~?d,,i~lion. Present address: im,tltute for High Pressure Studies .~,imc,m. Ru,~sian [:cdcratmn.
URu2Si2 and UPt3 share in common not only the condition TN >> T,, but also weak and comparable static moments of the order of 1O-2ltn in the low T limit. The effects of the magnetic transition on the specific heat and resistivity appear in the form of pronounced anomalies near TN in URu2Si 2 [1]. The corresponding signatures are, however, much weaker in UPt3 for which evidence of magnetic order has been provided mainly via direct neutron scattering experiments [2]. I I I x. t l l L O . a t . a l L ? l l t * l l t ? S Uv,d S b |
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{)921-4526 94 S07.00 { !994 Elsevier '~cicn¢c BA'. All rights re~er~cd SSDt 0921-4526(931F0406-7
J.P. Brison et al. / Physica B 199&200 (1994) 70-.75 2
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Fig. !. Pressure dependence of TNand T~ in URu:Si2 traced by their resistive anomaly. Full lines are guides to the eye. Note the coincidence of the collapse of 7", and enhancement of the (dTN/dP) slope at P ~ 12 kbars.
Fig. 2. Comparison of the ratio C/T of specific heat by the temperature of four different heavy fermion superconductors in reduced units. T, determined for an ideal transition by the usual equal entropy construction. C/T normalized also with respect to the entropy balance at 7",.
2. Experiments in URu2Si 2 Samples were grown by a tri-arc method and characterised by sp~cific heat and elastic neutron scattering measurements I3] before and after annealing at 1000°C for one week. This heat treatment had little effect on the specific heat anomaly near Try, but greatly altered the temperature dependence of the sublattice magnetisation and the form of the specific heat near 7",. The behaviour of the elastic peak intensity in the annealed state is similar to that reported previously [4]. The possible connection between the magnetic and superconducting transitions has been studied by means of measurements of the pressure and magnetic field dependences of TN and T¢. The transition points were inferred from anomalies in the temperature dependence of the resistivity. Measurements up to 17 kbar were made in CuBe clamped cells with N-pentane plus isoamylic alcohol as a transmitter medium. Higher pressures up to 70 kbar co,~!d be reached in a CW anvil with pyrophilite. The pressure variations of T,~ and T, are shown in Fig. 1. 7", ,decreases monotonically and tends towards zero near a pressure of F, = 12 ,:bar. TN. o~ the other hand, increases with pressure wi)h a slope which appears to change markedly near P,. The correlation in the behaviour of TN and 7", is not yct understood in detail and merits further investigation. These findings are augmented by measurements of the magnetic field deper~dence of 7",. They reveal that not only the magnitude but also the amsotropy of the upper critical fiekl diminishes with pressure. At ambient pressure in the low T limit H'~2 "-- 14 T and H:,,, --- 3 T for
a field H parallel to the a- and c-axes, respectively, of the tetragonal structure. At 8.5 kbar, on the other hand, in the low T limit H~,2 ,,- 2 T while H~,2 -- 0.6 T. Also worth noting is a pronounced pressure dependence of the quadratic (T:) coefficient A of the resistivity just above T, which is found to fall from 0.12 0.04 laf~cmK -2 in the range 0 20kbar. At higher temperatures approaching TN the resistivity increases much more rapidly than expected from the above T 2 forms. An interpretation of this behaviour has been considered on the basis of a model of the temperature and pressure dependences of the spin-wave spectrum whose gap collapses on approaching TN [5]. The effective gap in this analysis increases slowly from 78 to 110 K in the range 0 - 1 2 k b a r and is then almost constant up to 70 kbar in agreement with Ref. [6]. At ambient pressure this gap lies approximately at the centre of gravity of the measured spin-wave spectrum [7]. Turning now to the superconducting state, we contrast first the behaviour of the specific heat in the four heavy fermion systems URu2Si 2. UPt 3, CeCu2Si2 and UBe~3. An examination of the normalised data for all four materials shown in Fig. 2 reveals significant differences between URuzSi2 and UPt3 on the one hand (which, as we have r:tressed, exhibit ordered moments with T,~ ~ T,) and CeCu,Si2 and UBe~3 on the other. For T > 0.27", the curves for URu2Siz and UPt3 are quite comparable, except for the well defined splitting near T, in ploperly annealed samples of U Pt 3- In particular, the specific heat jump A C C at T~ is in both cases well below the usual
72
J.P. Brison et al,/ Physica B 199&200 (1994) 70-75
BCS value of 1.43. Furthermore, well within the range 0,2 To to T~ the curves are roughly linear, i.e. C is roughly quadratic in temperature. For T < 0.2To marked departures from linearity are observed in both materials. The strong upturn in UPts is comparable to that reported in [8]. A weaker but similar ano,naly is also present in URu2Si2. This behaviour is in sharp contrast to that observed in CeCu,Si2 and UBe~s in which a well defined antiferromagnetic order (with TN >> T~) has not been established. Here AC/C is comparable to or in excess of the BCS value, and particularly in UBets, C / T is markedly nonlinear over the entire range below T~. The difference in the form of C / T in URu2Si2 and UPts on the one hand and CeCuzSi2 and UBe~s on the other may plausibly be connected with the existence of ordered moments in the first pair and, for instance, with the difference in the magnitude of the linear coefficient of the specific heat in the normal state which are substantially higher in the latter than in the former pair. The effect of antiferromagnetic alignment, in particular, has been reconsidered within a simple BCS s-wave pair state for a system with TN ,3, T~ [9]. This contrasts, for example, with the Chevrel phases for which TN < T~. The finding in Ref. [9] is that even in this s-wave singlet model a line of zeroes of the superconducting gap may appear for states on the Fermi surface with wave vectors k nornml to the antiferromagnetic wave vector Q. This result is interpreted as arising from the pair breaking effect of the exchange field which is most effective for motion along directions of constant phase of the static magnetisation and tends to be cancelled out for propagation along directions of varying phase, i.e. along Q. {For URu2Si2 both Q and the orientations of the ordered moments are along the c-axis.) The model employed is based on a single new parameter ~, which measures the ratio of an exchange field 2h (in units of frequency) to vv Q, where vr is an effective Fermi velocity. It is interesting that the value of 7 required to fit the data for T > 0.2 T~ Isee Fig. 3) is of the orde- of magnitude to be expected from ," ~s*;,,-~,* ......... s of ,t.~ .,,. abtve parameters fo; URu,Si:. These fi~ding~ support :he assumption that antiferromagnetic order could have appreciable effects on the gap structure in this system and suggest that an extension of the above analysis to the more general cases of p and d pairing is warranted. Other properties of the superconducting state investigated include the temperature dependence of the upper critical field along the a- and c-axes, and the orientation dependence of H,2 between these axes at very low temperatures. The measurements, carried out in fields up to 18 T and temperatures down to 10 mK in Cambridge,
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Fig. 3. Experimental results for the specific heat in URu2Si2 together with the best fit obtained in our model of antiferromagnetic superconductors. The parameter measuring the influence of the AF ground state 7 = (2h/Qv) ~ 2T~/"f~/~Fis of the order of 1. In the inset, the dependence upon ~,of the specific heat jump AC/C at Tc (for "; = 0 notice the classical BCS 1.4 value). have also revealed the presence of quantum oscillations in the magnetoresistance at fields as low as He., itself, at least near the a-axis [10]. The field dependence of the amplitude of the osdllations confirms that in our samples the carrier mean free path is in excess of 900 A and hence much larger than the superconducting coherence length ~o ~ lOO L As shown in Fig. 4, He2 exhibits a pronounced orientation dependence in the a - c plane which at I0 m K is expected to be govcrncd by the Pauli paramagnctic limit. A phenomcnological model for a uniaxial anisotropy in lowest order provides an adequate description of the data over the entire angular range a - c (solid line and inset of Fig. 4). For a quantitative microscopic interpretation of this result it would be neccssary to consider the roles of the spin orbit interaction and anisotropic exchange molecular fields.Elementary modcls including such effects have been considered and iii lo.,;c~t, order they yield the same quaiitative form for the anisotropy of H~2 as presented in 7 , 4. The temperatt, re variation of H,2 along the two principal axes {Fig. 5} has an apparently conventional form, except at very low temperatures where a deviation from the expected T 2 saturation is observed. The interpretation of this and the much more dramatic departures from saturation of It,: observed at low temperatures in some oxide and organic low dimensional superconductors remains unclear. It is stressed that the width of the resistive transitions in all of these systems tends to be large and for URu2Si:
J.P. Brison ctal./Physica B 199&200 : 1994) 70-75 16
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T (K) Fig. 5. Upper critical field in URu.,Siz, measured down to 10 mK on two different samples. The same criterium i50% of the resistive IranshionJ was used for both measurements. The striking feature is the apparent lack of saturation of lt~2 for it--, 0.
at l0 mK it is as high as 0.5 T and 4 T along the c- and a-axes, respectively. We may also note in passing that no evidence is found in our measurements in URu2Si2 for the state of the kind described in [1 !].
3. Experiments on UPt 3 We n o w lurn to our measurements of the specific beat and H : in UPt3 and in particular to a discussion of the anomalous upturn of C f at very low temperature, the
73
double peak structure of C~ T n e a r To, and the anisotropy of the upper critical fields. The sharp upturn of C~T shown in Fig. 2 at low T has been investigated in the same sample under different annealing conditions (see below). The anomaly remains of comparable size even in cases where substantial changes in the form and position of the specific heat structure near Tc are observed. This finding is consistent with studies above H~2 I'8] which suggest that the upturn in C/T at low T is not correlated strongly with the superconducting state itself. Further support for this conclusion is found in an analysis of C/Tin terms of a regular component, equal to the total above 0.2 K plus that below 0.2 K defined via a linear extrapolation of C/T from data in the range 0.2 K to T,, and the remainder displaying the anomalous upturn at low T. The entropy associated with the regular component agrees at T, with that expected in the conventional description of the superconducting and normal states. The anomalous component (in the range from 15 to 100 mK) represents 7% of the total entropy of 0.255 J/mol at T,. We note that, in contrast to what is reported in Ref. [6], no maximum io C/T is seen down to 15 mK. It should also be pointed out that a similar anomalous contribution to the specific heat is observed in Pr at very low temperatures. Here the effect has been explained in terms of a coupling or nearly antiferromagnetic electrons and nuclear moments [12]. We next turn to a discussion of the double transition at T¢ in UPt3 for which, despite intensive study, a completely satisfactory interpretation is still lacking. Systematic measurements show that an unambiguous double peak structure can be obtained reversibly for a given range of annealing temperatures around 1200~C. In the Czochralski crystals as grown and when the annealing temperature is above about 1500~C, the width of the transition hinders any conclusion. Samples have been prepared which display two sharp peaks in the specific heat (Fig. 2~ and a very close correspondence between the position of the upper peak and the resistive transition t'he critical sensitivity to annealing has also been nmcd in the study of structura; modulations [13]. Further measurements of both the magnetic and structural properties and their relation to the double transition are in progress to fully clarify this problem, iNeu~ron experiments on the Orphee reactor at the Leon Brillouin Laboratory have confirmed the existence of antiferromagnetic order at low temperatures in our samples, but our search via neutron and X-ray scattering for the s~ructural modulations reported in Ref. [13] is still inconclusive.) Proposals for the origin of the double peak structure in
74
J.P. Brison et al./ Physica B 199&200 (1994) 70-75
the transition region include those based on effects of inhomogeneities or, more commonly, on the possible multi.component character of the superconducting order parameter. (For a discussion, for example, of the role of inhomogeneities in the high T~ materials see Ref. [14]. It seems unlikely, however, that models of this kind can account for all of the superconducting properties of UPh, including for example such features as the tricritical point in the H - T phase diagram.) It is also noted that, as for URuzSi~ discussed earlier, the effect of antiferromagnetic order on the superconducting gap structure can probably not be ignored in a detailed analysis of UPt3. In a treatment analogous to that given in the last section a splitting of T¢ can arise in principle via the effect of a change of the temperature dependence of the molecular field below T~ (as may be expected to occur on the basis of the neutron scattering measurements of Ref. [15]). This conjecture for the double peak structure, however, is not supported by quantitative estimates of the magnitude of the effect for UPt 3. Finally, we turn to a discussion of our studies of the upper critical field in the neighbourhood of T~, the anisotropy of which may also help to distinguish models of the superconducting state. Systematic measurements with H in the basal plane of the hexagonal structure of UPt 3 reveal a finite anisotropy of H¢2 with a six fold symmetry, in agreement with an earlier preliminary study [16]. This is in contrast with the prediction of the simplest models of anisotropic superconductivity [17] in which an orientation dependence of H~z ~s expected for a hexagonal system only for rotations away from the basal plane. The anisotropy observed may be understood in terms of surface superconductivity (note that the whiskers have cross-sectional areas with six-fold symmetry) or via a coupling in different domains between multicomponent superconducting and antiferromagnetic order parameters [18]. As the latter may be suppressed with a tiny compression of the lattice without destroying the superconducting order, the above two possible mechanisms for the anisotropy in the basal plane (as well as the role of the ordered moment on the gap structure discussed earlier) may be distinguished with further measurements under hydrostatic pressure. The angular dependen~ o~"11~2 {{~)i,, the a - c plane ha~ also been studied near T,. As shown in Fig. 6, a phenomenological model formally analogous to that discussed in the last section for URu:Si, but now (close to T,) in the orbital rather than Pauli limit, fails to describe the data satisfactorily. The observed sharp peak in H,2 10) near the c-axis in particular is more readily understood in terms of the effect of surface superconductivity [Fig. 61.
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4. Conclusion
The heavy Fermion superconductors with antiferromagnetic ordered moments, URu2Si2 and UPta, exhibit markedly different forms of C~ T versus T than CeCuzSi2 and UBet3. It is suggested that a key to understanding part of the difference may lie in the effects of the static exchange field on the superconducting gap structure. A simple model indicates that a line of zeroes of the gap on the Fermi surface may arise from this mechanism even in the case of an s-wave singlet state. The experimental results for UR,'zSi2 do not rule out a description in which a mechanism of this kind, but perhaps generalised iml
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References [1] W. Schlabitz. J. Baumann. B. Polht. U. Rauchschwalbe, H.M. Mayer. U. Ahlheim and C.D. Bredl. Z. Phys. B 62 (1986) 171.
J.P. Brison et aL/ P@sica B 199&200 (I994) 70-75
[2] G. Aeppli, E. Bucher, C. Broholm, J.K. Kjems, J. Baumann and J. Hufnagl, Phys. Rev, Lett. 60 {1988) 615. [3] B. F~lk, C, Vettier, R.A. Fisher, N. Phillips, P, Lejay, A. Vemi6re, ,I.P. Brison and !. Flouquet, to be published. [4] T.E. Mason, B.D. Gaulin, J.D. Garett, A. Tun, WJ.L Buyers and E.D. lsaacs, Phys. Rev, Lett. 65 (1990) 3189. [5! L. Schmidt, Thesis, Grenoble (1993), unpublished. [6] K. Iki, G. Oomi, Y. Uwatoko, H. Takahashi, N. Moil, Y. Onuki and T. Komatsubara, J. Alloys Compounds 181 (1992) 71. [7] C. Broholm, H. Lin, P.T, Matthews, T.E. Mason, W.J.L. Buyers, M.F. Collins, A.A. Menovsky, J.A. Mydosh and J.K. Kjems, Phys. Rev. B 43 (1991) 12809. [8] E.A. Schuberth, B. Strickler and K. Andres, Phys. Rev. Lett. 68 (1992) 117; E.A. Schuberth and M. Fischer, Physica B 194-196 (1994) 1983. [9] A. Buzdin, J.P. Brison, P. Lejay and J. Flouquct, to be published. [10] S.R. Julian and G.G. Lonzarich, Physica B 199&200{1994) 63.
75
[11] A.I Larkin and Yu.N. Ovchinnikov, Soy. Phys. JETP 20 (1965) 762; P. Fulde and R.A. Ferre!, Phys. Rev. 135 (1964) A550. [I 2] H.B. Moller,J.Z.Jensen, M, Wulff, A.R. Mackintosh, O.D. McMasters and K.A. Gschneidner Jr.,Phys. Rev. Latt.49 (1982) 482. [13] P.A. Midgley, S.M. Hayden, L. T~illefer,B. Bogenberger and H.V. L6hneysen, Phys. Rev. Lett.70 (1993) 678. [14] A.A. Abilkosov, A.I. Buzdin, M.L. Kulic' and D.A. Kuptsov, Soy. Phys. JETP 68 {1989) 210. ['15] G. Aeppli, D. Bishop, C. Broholm, E. Bucher, K. Siemensmeyer, M. Steiner and N. Stiisser,Phys. Rev. Lett. 63 0989) 676. [16] L. Taillefer,K. Behnia, K. Hasselbach, J. Flouquet, S.M Hayden and C. Vettier, J. Magn. Magn. Mater. 90&91 (I 9901 623. [17] G.E. Volovik and L.P. Gorkov, Soy. Phys. JETP 61 ~1n85) 843. [18] K. Machida, M. Ozaki and T. Ohmi, J. Phys. Soc. Japan 58 (1989) 4116.
ELSEVIER
Very low temperature properties of heavy fermion materials J.P. Brison"'*, N. Keller", P. Lejay a, A. I-[uxley b, L. Schmidt b, A. Buzdin b'l, N.R. Bernhoeft b, I. Mineev b'2, A.N. Stepanov b'3, J. Flouquet b, D. Jaccard ¢, S.R. Julian a, G.G. Lonzarich d " Centre de Recherches sur les Trbs Basses Tempbratures, Laboratoire associb h I' Universitb Joseph Fourier, CNRS, BP 166, 38042 Grenoble-Cbdex 9, France b DRFMC-SPSMS, CENG, BP 85X, 38041 Grenoble-C&tex, France DPMC, Gen&'e, Switzerland a Cavemlish Laboratory. Cambridge, UK
Abstract A review is given of recent very low temperature ,experiments in the heavy fermion superconductors UPt3, URu2Si2, UBe~ 3 and CeCu2Si2, Attention is focused on the temperature dependence of the specific heats in all four systems, the anisotropy of the upper critical fields in URu2Si2 and UPt 3, and the pressure dependence of the antiferromagnetic and superconducting transition temperatures in URu2Si2.
1. Introduction The heavy fermion superconductors URu2Si2 and UPt3 differ from the related materials CeCu2Sb. and UBet ~ in exhibiting clearly defined transitions to weakly polarised antiferromagnetic states at temperatures TN well above the superconducting critical points T,. The values of T¢ for these four systems are 1.4, 0.5, 0.6 and 1.0 K. The corresponding values of TN for the first two compounds, which will be the main focus of attentior~, are 17.5 and 5 K and hence ;an order of magnitude above
7",. * Corresponding2 author. 1 Present address: Argonne National Laboratory, Argonne. ll.. USA. ? Pnc~cn(address: [.andau Institute. Mrs,;cow. Ru'~l:m g~?d,,i~lion. Present address: im,tltute for High Pressure Studies .~,imc,m. Ru,~sian [:cdcratmn.
URu2Si2 and UPt3 share in common not only the condition TN >> T,, but also weak and comparable static moments of the order of 1O-2ltn in the low T limit. The effects of the magnetic transition on the specific heat and resistivity appear in the form of pronounced anomalies near TN in URu2Si 2 [1]. The corresponding signatures are, however, much weaker in UPt3 for which evidence of magnetic order has been provided mainly via direct neutron scattering experiments [2]. I I I x. t l l L O . a t . a l L ? l l t * l l t ? S Uv,d S b |
tly~U
~.,P~ILg"~
ave
een
CdllleU
out to help shed light on the possible effects of the weak antiferr:*m.~gnctic polarisation on the nature of the superconducting states of these two systems charactensed by the unusual conmtion TN ~ T,. It is suggested in parvcular that this magnetm order could contribute significantly to the anisotropy and nodal structure of the ,:nergy gap in either the convention,d or unconventiomd models of the superconducting state
{)921-4526 94 S07.00 { !994 Elsevier '~cicn¢c BA'. All rights re~er~cd SSDt 0921-4526(931F0406-7
J.P. Brison et al. / Physica B 199&200 (1994) 70-.75 2
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Fig. !. Pressure dependence of TNand T~ in URu:Si2 traced by their resistive anomaly. Full lines are guides to the eye. Note the coincidence of the collapse of 7", and enhancement of the (dTN/dP) slope at P ~ 12 kbars.
Fig. 2. Comparison of the ratio C/T of specific heat by the temperature of four different heavy fermion superconductors in reduced units. T, determined for an ideal transition by the usual equal entropy construction. C/T normalized also with respect to the entropy balance at 7",.
2. Experiments in URu2Si 2 Samples were grown by a tri-arc method and characterised by sp~cific heat and elastic neutron scattering measurements I3] before and after annealing at 1000°C for one week. This heat treatment had little effect on the specific heat anomaly near Try, but greatly altered the temperature dependence of the sublattice magnetisation and the form of the specific heat near 7",. The behaviour of the elastic peak intensity in the annealed state is similar to that reported previously [4]. The possible connection between the magnetic and superconducting transitions has been studied by means of measurements of the pressure and magnetic field dependences of TN and T¢. The transition points were inferred from anomalies in the temperature dependence of the resistivity. Measurements up to 17 kbar were made in CuBe clamped cells with N-pentane plus isoamylic alcohol as a transmitter medium. Higher pressures up to 70 kbar co,~!d be reached in a CW anvil with pyrophilite. The pressure variations of T,~ and T, are shown in Fig. 1. 7", ,decreases monotonically and tends towards zero near a pressure of F, = 12 ,:bar. TN. o~ the other hand, increases with pressure wi)h a slope which appears to change markedly near P,. The correlation in the behaviour of TN and 7", is not yct understood in detail and merits further investigation. These findings are augmented by measurements of the magnetic field deper~dence of 7",. They reveal that not only the magnitude but also the amsotropy of the upper critical fiekl diminishes with pressure. At ambient pressure in the low T limit H'~2 "-- 14 T and H:,,, --- 3 T for
a field H parallel to the a- and c-axes, respectively, of the tetragonal structure. At 8.5 kbar, on the other hand, in the low T limit H~,2 ,,- 2 T while H~,2 -- 0.6 T. Also worth noting is a pronounced pressure dependence of the quadratic (T:) coefficient A of the resistivity just above T, which is found to fall from 0.12 0.04 laf~cmK -2 in the range 0 20kbar. At higher temperatures approaching TN the resistivity increases much more rapidly than expected from the above T 2 forms. An interpretation of this behaviour has been considered on the basis of a model of the temperature and pressure dependences of the spin-wave spectrum whose gap collapses on approaching TN [5]. The effective gap in this analysis increases slowly from 78 to 110 K in the range 0 - 1 2 k b a r and is then almost constant up to 70 kbar in agreement with Ref. [6]. At ambient pressure this gap lies approximately at the centre of gravity of the measured spin-wave spectrum [7]. Turning now to the superconducting state, we contrast first the behaviour of the specific heat in the four heavy fermion systems URu2Si 2. UPt 3, CeCu2Si2 and UBe~3. An examination of the normalised data for all four materials shown in Fig. 2 reveals significant differences between URuzSi2 and UPt3 on the one hand (which, as we have r:tressed, exhibit ordered moments with T,~ ~ T,) and CeCu,Si2 and UBe~3 on the other. For T > 0.27", the curves for URu2Siz and UPt3 are quite comparable, except for the well defined splitting near T, in ploperly annealed samples of U Pt 3- In particular, the specific heat jump A C C at T~ is in both cases well below the usual
72
J.P. Brison et al,/ Physica B 199&200 (1994) 70-75
BCS value of 1.43. Furthermore, well within the range 0,2 To to T~ the curves are roughly linear, i.e. C is roughly quadratic in temperature. For T < 0.2To marked departures from linearity are observed in both materials. The strong upturn in UPts is comparable to that reported in [8]. A weaker but similar ano,naly is also present in URu2Si2. This behaviour is in sharp contrast to that observed in CeCu,Si2 and UBe~s in which a well defined antiferromagnetic order (with TN >> T~) has not been established. Here AC/C is comparable to or in excess of the BCS value, and particularly in UBets, C / T is markedly nonlinear over the entire range below T~. The difference in the form of C / T in URu2Si2 and UPts on the one hand and CeCuzSi2 and UBe~s on the other may plausibly be connected with the existence of ordered moments in the first pair and, for instance, with the difference in the magnitude of the linear coefficient of the specific heat in the normal state which are substantially higher in the latter than in the former pair. The effect of antiferromagnetic alignment, in particular, has been reconsidered within a simple BCS s-wave pair state for a system with TN ,3, T~ [9]. This contrasts, for example, with the Chevrel phases for which TN < T~. The finding in Ref. [9] is that even in this s-wave singlet model a line of zeroes of the superconducting gap may appear for states on the Fermi surface with wave vectors k nornml to the antiferromagnetic wave vector Q. This result is interpreted as arising from the pair breaking effect of the exchange field which is most effective for motion along directions of constant phase of the static magnetisation and tends to be cancelled out for propagation along directions of varying phase, i.e. along Q. {For URu2Si2 both Q and the orientations of the ordered moments are along the c-axis.) The model employed is based on a single new parameter ~, which measures the ratio of an exchange field 2h (in units of frequency) to vv Q, where vr is an effective Fermi velocity. It is interesting that the value of 7 required to fit the data for T > 0.2 T~ Isee Fig. 3) is of the orde- of magnitude to be expected from ," ~s*;,,-~,* ......... s of ,t.~ .,,. abtve parameters fo; URu,Si:. These fi~ding~ support :he assumption that antiferromagnetic order could have appreciable effects on the gap structure in this system and suggest that an extension of the above analysis to the more general cases of p and d pairing is warranted. Other properties of the superconducting state investigated include the temperature dependence of the upper critical field along the a- and c-axes, and the orientation dependence of H,2 between these axes at very low temperatures. The measurements, carried out in fields up to 18 T and temperatures down to 10 mK in Cambridge,
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Fig. 3. Experimental results for the specific heat in URu2Si2 together with the best fit obtained in our model of antiferromagnetic superconductors. The parameter measuring the influence of the AF ground state 7 = (2h/Qv) ~ 2T~/"f~/~Fis of the order of 1. In the inset, the dependence upon ~,of the specific heat jump AC/C at Tc (for "; = 0 notice the classical BCS 1.4 value). have also revealed the presence of quantum oscillations in the magnetoresistance at fields as low as He., itself, at least near the a-axis [10]. The field dependence of the amplitude of the osdllations confirms that in our samples the carrier mean free path is in excess of 900 A and hence much larger than the superconducting coherence length ~o ~ lOO L As shown in Fig. 4, He2 exhibits a pronounced orientation dependence in the a - c plane which at I0 m K is expected to be govcrncd by the Pauli paramagnctic limit. A phenomcnological model for a uniaxial anisotropy in lowest order provides an adequate description of the data over the entire angular range a - c (solid line and inset of Fig. 4). For a quantitative microscopic interpretation of this result it would be neccssary to consider the roles of the spin orbit interaction and anisotropic exchange molecular fields.Elementary modcls including such effects have been considered and iii lo.,;c~t, order they yield the same quaiitative form for the anisotropy of H~2 as presented in 7 , 4. The temperatt, re variation of H,2 along the two principal axes {Fig. 5} has an apparently conventional form, except at very low temperatures where a deviation from the expected T 2 saturation is observed. The interpretation of this and the much more dramatic departures from saturation of It,: observed at low temperatures in some oxide and organic low dimensional superconductors remains unclear. It is stressed that the width of the resistive transitions in all of these systems tends to be large and for URu2Si:
J.P. Brison ctal./Physica B 199&200 : 1994) 70-75 16
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T (K) Fig. 5. Upper critical field in URu.,Siz, measured down to 10 mK on two different samples. The same criterium i50% of the resistive IranshionJ was used for both measurements. The striking feature is the apparent lack of saturation of lt~2 for it--, 0.
at l0 mK it is as high as 0.5 T and 4 T along the c- and a-axes, respectively. We may also note in passing that no evidence is found in our measurements in URu2Si2 for the state of the kind described in [1 !].
3. Experiments on UPt 3 We n o w lurn to our measurements of the specific beat and H : in UPt3 and in particular to a discussion of the anomalous upturn of C f at very low temperature, the
73
double peak structure of C~ T n e a r To, and the anisotropy of the upper critical fields. The sharp upturn of C~T shown in Fig. 2 at low T has been investigated in the same sample under different annealing conditions (see below). The anomaly remains of comparable size even in cases where substantial changes in the form and position of the specific heat structure near Tc are observed. This finding is consistent with studies above H~2 I'8] which suggest that the upturn in C/T at low T is not correlated strongly with the superconducting state itself. Further support for this conclusion is found in an analysis of C/Tin terms of a regular component, equal to the total above 0.2 K plus that below 0.2 K defined via a linear extrapolation of C/T from data in the range 0.2 K to T,, and the remainder displaying the anomalous upturn at low T. The entropy associated with the regular component agrees at T, with that expected in the conventional description of the superconducting and normal states. The anomalous component (in the range from 15 to 100 mK) represents 7% of the total entropy of 0.255 J/mol at T,. We note that, in contrast to what is reported in Ref. [6], no maximum io C/T is seen down to 15 mK. It should also be pointed out that a similar anomalous contribution to the specific heat is observed in Pr at very low temperatures. Here the effect has been explained in terms of a coupling or nearly antiferromagnetic electrons and nuclear moments [12]. We next turn to a discussion of the double transition at T¢ in UPt3 for which, despite intensive study, a completely satisfactory interpretation is still lacking. Systematic measurements show that an unambiguous double peak structure can be obtained reversibly for a given range of annealing temperatures around 1200~C. In the Czochralski crystals as grown and when the annealing temperature is above about 1500~C, the width of the transition hinders any conclusion. Samples have been prepared which display two sharp peaks in the specific heat (Fig. 2~ and a very close correspondence between the position of the upper peak and the resistive transition t'he critical sensitivity to annealing has also been nmcd in the study of structura; modulations [13]. Further measurements of both the magnetic and structural properties and their relation to the double transition are in progress to fully clarify this problem, iNeu~ron experiments on the Orphee reactor at the Leon Brillouin Laboratory have confirmed the existence of antiferromagnetic order at low temperatures in our samples, but our search via neutron and X-ray scattering for the s~ructural modulations reported in Ref. [13] is still inconclusive.) Proposals for the origin of the double peak structure in
74
J.P. Brison et al./ Physica B 199&200 (1994) 70-75
the transition region include those based on effects of inhomogeneities or, more commonly, on the possible multi.component character of the superconducting order parameter. (For a discussion, for example, of the role of inhomogeneities in the high T~ materials see Ref. [14]. It seems unlikely, however, that models of this kind can account for all of the superconducting properties of UPh, including for example such features as the tricritical point in the H - T phase diagram.) It is also noted that, as for URuzSi~ discussed earlier, the effect of antiferromagnetic order on the superconducting gap structure can probably not be ignored in a detailed analysis of UPt3. In a treatment analogous to that given in the last section a splitting of T¢ can arise in principle via the effect of a change of the temperature dependence of the molecular field below T~ (as may be expected to occur on the basis of the neutron scattering measurements of Ref. [15]). This conjecture for the double peak structure, however, is not supported by quantitative estimates of the magnitude of the effect for UPt 3. Finally, we turn to a discussion of our studies of the upper critical field in the neighbourhood of T~, the anisotropy of which may also help to distinguish models of the superconducting state. Systematic measurements with H in the basal plane of the hexagonal structure of UPt 3 reveal a finite anisotropy of H¢2 with a six fold symmetry, in agreement with an earlier preliminary study [16]. This is in contrast with the prediction of the simplest models of anisotropic superconductivity [17] in which an orientation dependence of H~z ~s expected for a hexagonal system only for rotations away from the basal plane. The anisotropy observed may be understood in terms of surface superconductivity (note that the whiskers have cross-sectional areas with six-fold symmetry) or via a coupling in different domains between multicomponent superconducting and antiferromagnetic order parameters [18]. As the latter may be suppressed with a tiny compression of the lattice without destroying the superconducting order, the above two possible mechanisms for the anisotropy in the basal plane (as well as the role of the ordered moment on the gap structure discussed earlier) may be distinguished with further measurements under hydrostatic pressure. The angular dependen~ o~"11~2 {{~)i,, the a - c plane ha~ also been studied near T,. As shown in Fig. 6, a phenomenological model formally analogous to that discussed in the last section for URu:Si, but now (close to T,) in the orbital rather than Pauli limit, fails to describe the data satisfactorily. The observed sharp peak in H,2 10) near the c-axis in particular is more readily understood in terms of the effect of surface superconductivity [Fig. 61.
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Fig. 6. Angulardependence of H+, in the a-c plane, with fits in a model of anisotropie effective masses and surface superconductivity. Both effects~re probably necessaryto understand the data. As neither model, however, provides a detailed overall account of the results, a more elaborate analysis which also takes explicit account of effects of the multicomponent nature of the superconducting order parameter is being pursued.
4. Conclusion
The heavy Fermion superconductors with antiferromagnetic ordered moments, URu2Si2 and UPta, exhibit markedly different forms of C~ T versus T than CeCuzSi2 and UBet3. It is suggested that a key to understanding part of the difference may lie in the effects of the static exchange field on the superconducting gap structure. A simple model indicates that a line of zeroes of the gap on the Fermi surface may arise from this mechanism even in the case of an s-wave singlet state. The experimental results for UR,'zSi2 do not rule out a description in which a mechanism of this kind, but perhaps generalised iml
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References [1] W. Schlabitz. J. Baumann. B. Polht. U. Rauchschwalbe, H.M. Mayer. U. Ahlheim and C.D. Bredl. Z. Phys. B 62 (1986) 171.
J.P. Brison et aL/ P@sica B 199&200 (I994) 70-75
[2] G. Aeppli, E. Bucher, C. Broholm, J.K. Kjems, J. Baumann and J. Hufnagl, Phys. Rev, Lett. 60 {1988) 615. [3] B. F~lk, C, Vettier, R.A. Fisher, N. Phillips, P, Lejay, A. Vemi6re, ,I.P. Brison and !. Flouquet, to be published. [4] T.E. Mason, B.D. Gaulin, J.D. Garett, A. Tun, WJ.L Buyers and E.D. lsaacs, Phys. Rev, Lett. 65 (1990) 3189. [5! L. Schmidt, Thesis, Grenoble (1993), unpublished. [6] K. Iki, G. Oomi, Y. Uwatoko, H. Takahashi, N. Moil, Y. Onuki and T. Komatsubara, J. Alloys Compounds 181 (1992) 71. [7] C. Broholm, H. Lin, P.T, Matthews, T.E. Mason, W.J.L. Buyers, M.F. Collins, A.A. Menovsky, J.A. Mydosh and J.K. Kjems, Phys. Rev. B 43 (1991) 12809. [8] E.A. Schuberth, B. Strickler and K. Andres, Phys. Rev. Lett. 68 (1992) 117; E.A. Schuberth and M. Fischer, Physica B 194-196 (1994) 1983. [9] A. Buzdin, J.P. Brison, P. Lejay and J. Flouquct, to be published. [10] S.R. Julian and G.G. Lonzarich, Physica B 199&200{1994) 63.
75
[11] A.I Larkin and Yu.N. Ovchinnikov, Soy. Phys. JETP 20 (1965) 762; P. Fulde and R.A. Ferre!, Phys. Rev. 135 (1964) A550. [I 2] H.B. Moller,J.Z.Jensen, M, Wulff, A.R. Mackintosh, O.D. McMasters and K.A. Gschneidner Jr.,Phys. Rev. Latt.49 (1982) 482. [13] P.A. Midgley, S.M. Hayden, L. T~illefer,B. Bogenberger and H.V. L6hneysen, Phys. Rev. Lett.70 (1993) 678. [14] A.A. Abilkosov, A.I. Buzdin, M.L. Kulic' and D.A. Kuptsov, Soy. Phys. JETP 68 {1989) 210. ['15] G. Aeppli, D. Bishop, C. Broholm, E. Bucher, K. Siemensmeyer, M. Steiner and N. Stiisser,Phys. Rev. Lett. 63 0989) 676. [16] L. Taillefer,K. Behnia, K. Hasselbach, J. Flouquet, S.M Hayden and C. Vettier, J. Magn. Magn. Mater. 90&91 (I 9901 623. [17] G.E. Volovik and L.P. Gorkov, Soy. Phys. JETP 61 ~1n85) 843. [18] K. Machida, M. Ozaki and T. Ohmi, J. Phys. Soc. Japan 58 (1989) 4116.