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Superconducting alloys with paramagnetic impurities (part II) magnetization and upper critical fields of LaAl2 with dilute Gd impurities
PhysicaC 167 (1990) North-Holland
198-211
SUPERCONDUCTING MAGNETIZATION IMPURITIES
ALLOYS
WITH PARAMAGNETIC
IMPURITIES
(PART II)
AND UPPER CRITICAL FIELDS OF LaA12 WITH DILUTE Gd
B. VLCEK, E. SEIDL and H.W. WEBER Atominstitut der tisterreichischen Universitiiten, A-1020 Wien, Austria
E. SCHACHINGER
and W. PINT
Institut ftir Theoretische Physik Technische UniversitSit Graz, A-8010 Graz, Austria Received 30 November 1989 Revised manuscript received 26 February
1990
Magnetization and AC susceptibility were made on single- and polycrystalline samples of LaAIZ doped with Cd in the concentration range 0 In, S 0.7 at%. The depression of T, with Cd content is found to be in excellent agreement with previous experimental work. Concerning the temperature dependence of H,,, peaks at low temperatures ( = 400 mK) and downward curvatures towards T=O have been observed. The upper critical field of the clean material can be explained quantitatively in terms of anisotropic Eliashberg theory using (a’) ~0.01 for the electron-phonon coupling anisotropy parameter and (b’) =O. 16 for the anisotropy of the Fermi velocity. With these values and the generalized phonon density of states reported by Yeh et al. excellent agreement between theory and experiment was achieved. Furthermore, results on the Cd doped materials can be explained quantitatively on the basis of strong coupling theory, if spin orientation effects of the Cd atoms are included in the theoretical treatment.
1. Introduction
Magnetic impurities in superconductors have been studied intensively, because small concentrations are sufficient to change the properties of the host material in a significant way. This is in clear contrast to normal conducting materials, where very high concentrations must be dissolved in order to change the normal state properties, leading almost certainly to an interaction between the impurity atoms. Therefore, magnetic impurities can be used either to study the influence of the host material on the magnetic state of the impurity, or to study strong coupling and anisotropy effects on the superconducting properties as a function of impurity concentration. As outlined by Matthias et al. [ 1 ] who studied rare earth solutions ( 1 atW) in lanthanum, normal (non magnetic) impurities have little or no effect on the transition temperature of the host material, a result which is in agreement with the theoretical argument given by Anderson [2] later on. Gadolinium im0921-4534/90/$03.50 (North-Holland )
0 Elsevier Science Publishers
B.V.
purities, on the other hand, which have the largest spin of the rare earths are most effective in lowering the critical temperature. This is due to the spin flip interaction of the conduction electrons with the spin of the gadolinium atoms which is pairbreaking. In contrast, the momentum exchange interaction which is typical for scattering processes of conduction electrons at normal impurity sites is not pairbreaking and has therefore no effect on the superconducting properties of isotopic superconductors. The first theoretical description of a superconductor containing paramagnetic impurities was given by Abrikosov and Gor’kov (AG-theory) [3,4] on the basis of BCS theory. They treated the magnetic scattering in first order Born approximation assuming a weak interaction between impurities and conduction electrons. In the case of a strong interaction, terms of higher order have to be included in order to describe the scattering of the conduction electrons at the impurity sites [ 5-7 1. Experimentally, the predictions of the AG-theory
B. Vlcek et al. /Superconducting
on the T, depression have been confirmed in the system LaAl,-Gd [8] except near the critical concentration. This small deviation is presumably related to strong coupling effects. Skalsky et al. [ 91 have extended the AG-theory to other thermodynamic properties of a superconductor containing paramagnetic impurities. For example, they have shown that the thermodynamic critical field deviation function,
=Ec
D(t)
(l-t’),
t=T/T,,
becomes more negative with increasing impurity concentration. These calculations have been confirmed by measurements on the weak coupling superconductor thorium [ 10 1. Schachinger et al. [ 111 calculated the deviation function D(t) for the strong coupling superconductor lead containing paramagnetic impurities. In contrast to the AG-theory, these calculations were based on the Eliashberg theory. It was shown in this work that the AG-theory overestimated the effect of paramagnetic impurities on the critical temperature quite substantially and pronounced differences were reported for the deviation function D(t). Additional deviations from the standard isotropic BCS-theory are based on anisotropy effects. They were studied first by Markovitz and Kadanoff [ 121 who introduced a separable pairing potential V(k, k’ ):
V(k,k’)=(l+a,)I’~,,(l+~~~)
(2)
to describe the anisotropy of the electron-phonon pairing interaction. Here Us is a temperature independent anisotropy parameter and V,,, is the constant and isotropic pairing potential of the BCS-theory. An extension of this work was given by Clem [ 13 ] who calculated the influence of anisotropy on the thermodynamic properties of superconductors, in particular the thermodynamic critical field H, ( T) and its deviation function D(t). All these calculations were limited to weak coupling superconductors. Strong coupling effects were first included in a phenomenological way by introducing a second fit parameter 6 [ 14,15 1. A full theoretical treatment of anisotropic superconductors including strong coupling effects became available later on [ 16- 19 1, and was confirmed experimentally on the superconductor indium doped with thallium using a mean square
alloys with paramagnetic impurities
199
electron-phonon interaction anisotropy of < a2) = 0.04 in the clean limit [ 20 1. It was also shown that the anisotropy was “washed out” by non-magnetic impurities. Since the theory was also suited to calculate the influence of paramagnetic impurities on the deviation function of strong coupling anisotropic superconductors it was tempting to investigate this thermodynamic aspect experimentally. However, in order to determine the deviation function as a function of magnetic impurity content, the thermodynamic critical field H,(T) has to be measured with high accuracy, a task which can be met only by type-1 superconductors. Based on calculations by Werthamer et al. [ 2 11, Eliashberg equations for the upper critical field H,, ( T) could be established by Rainer and Bergman [ 221 and Schossmann and Schachinger [ 23 1. Further work by Mars&ho et al. [24] showed that the upper critical field of type-II superconductors was very sensitive to strong coupling effects. This theory was then extended by Prohammer and Schachinger [27] to describe the upper critical field of anisotropic polycrystals. Experimental data on Hc2 in pure and nitrogen doped niobium [25,26] were explained on the basis of this extended theory. Finally, in part I of this series of two papers [ 28 1, two of us presented a full strong coupling theory of the upper critical field in isotropic superconductors containing paramagnetic impurities. An application of this theory, extended to the case of anisotropic polycrystals, will be presented in section 5 of this paper. For the experimental investigations the superconductor LaAl, which is well known in the literature [ 829-361, has been selected because various magnetic impurities, such as Gd, Ce, etc., can be dissolved homogeneously. Gadolinium was chosen as the magnetic impurity. Its large spin (S=i) ensures that the AG-theory which treats the atomic spins of the impurities as classical vectors can be applied to this system. Except for some work on the pure material [ 35,361, no information on the magnetic properties of the system LaAlz-Gd has been available so far. Thus, we will present some aspects of sample preparation and experimental techniques used for the measurement of superconducting parameters in sections 2 and 3, summarize the experimental results and evaluations in section 4 and devote section 5 to a comparison of experimental upper
200
B. Vlcek et al. /Superconducting alloys with paramagnetic impurities
critical field data with theory. Our conclusions drawn in section 6.
are
2. Materials In the present study a single crystal of pure LaAl, and six polycrystals were investigated. The single crystal which had been studied previously [ 341 was provided by F. Steglich (Darmstadt). It was annealed at 900°C for 50 h to remove internal stress and electropolished to clean the surface. The polycrystals were prepared using the cold boat technique under an atmosphere of 800 mbar Ar. As the starting materials, pure lanthanum (purchased from the Ames Laboratory and containing only minor amounts of magnetic impurity atoms, such as 15 at.ppm Fe, 0.96 at.ppm Ni and 2.2 at.ppm Ce, with all the other impurities falling below 1 ppm each) and 99.99999% Al were used. The samples were prepared by carefully weighing (five times) the desired amounts of material and melting them six times, together with Gd, to ensure a homogeneous distribution of Gd within the matrix. Then the polycrystals were annealed at 900°C for 50 h, electropolished and cast to their final cylindrical form (3 mm diameter, 20 mm length). A summary of material and characteristic superconductive parameters is presented in table I.
3. Experimental Conventional four point measurements were made at room temperature and at 4.2 K to evaluate the re-
sistivity p,, and the residual resistivity ratios RRR. The accuracy of this data is determined mainly by the geometry of the sample. While the error associated with the residual resistivity ratio is less than l%, an uncertainty of about 10% must be taken into account for the absolute values of p,,. In order to determine the magnetic properties in the superconducting state, two different methods were employed. Firstly, in the temperature range between T, and 1.6 K magnetization measurements using the differential technique [ 371 with a low temperature chopper were made. Very low field sweeping rates were used (0.6 mT/s) to achieve true equilibrium conditions in the magnetization cycle. All the data were stored on a computer and analyzed numerically. The resolution for determining the upper critical field H,, from the differential curve is better than f0.5 mT. Secondly, AC-susceptibility measurements were made at all temperatures by superimposing a small AC-ripple field ( lo-100 nT) onto a transverse DC field. This field was provided either by a superconducting magnet in a conventional bath cryostat or by an electromagnet or a normal conducting coil in the dilution refrigerator. The sample was placed into a pick-up coil and the signals were detected by a lockin amplifier while the external field was swept slowly at a fixed temperature. In order to find the optimal sweep time, the data in increasing and decreasing field were compared at constant temperature and the sweep time was decreased until both branches gave the same response indicating thermodynamic equilibrium conditions. Then the data were taken from increasing field measurements. This experimental
Table I Summary of material parameters Polycrystals
Sample
Single crystal
n,, at% Gd RRR ~“(4.2 K), CnRm) T,(K)
0.0 830 0.7385 3.31
0.0 343 1.478 3.29
0.1 217 2.19 2.91
92.04 0.75 26.112
92.48 0.757 26.68
72.07 24.53
k&CO), (mT) (-/@&(T)IdT)I~=r
0.3 105 4.11 2.084
0.41 72 5.671 (1.42)
0.5 64 7.881 1.032
28.41 18.82
15.06
9.54 14.36
0.55 58 1.896 0.705 5.096 9.46
0.58 45 11.31 <20mK
B. Vlcek et al. /Superconducting
technique was used throughout the entire study to evaluate the upper critical fields HC2. The field was produced by three different systems, a superconducting Helmholtz coil (O-4 T), a normal conducting Helmholtz coil (O-20 mT) and an electromagnet (0- 1 T). The field achieved by the Helmholtz coils is measured through the current producing the field. The error in this measurement can be disregarded, but some uncertainties concerning the absolute field error remain. The electromagnet was calibrated using a high resolution Hall probe at the position of the sample. Due to this calibration the accuracy of the field measurement is better than 0.1%. Through a comparison of several measurements in different magnets we conclude that the absolute error of H,, determined by the differential technique is less than 0.5%. The accuracy of HC2evaluated from the AC-susceptibility measurements was found to be f50 PT. In the temperature range from T, to 25 mK, calibrated germanium resistors were used whose maximum calibration errors, according to their specilications, are about 4 mK near 4 K. The accuracy of the temperature measurement below 1 K was better than 5 mK. From one T, measurement made in the overlapping temperature range between the conventional bath cryostat and the 3He-4He dilution refrigerator, which is not easily accessible, we believe that the T, values are accurate to better than 5 mK on an absolute scale. As mentioned above, two different techniques were available for the measurements, a differential magnetization and an AC-susceptibility technique. For a few samples which were investigated in the hightemperature range by both techniques, a test on the compatibility of the H,,-evaluations was made. In both cases the transition from the superconducting to the normal state is slightly rounded and was therefore determined by two tangents. A comparison of the values of H,, evaluated from both techniques is shown in fig. 1. As no noticable difference between the data could be found, we conclude that the procedures adopted to determine HC2 from experiment are correct. A further consistency check was made for the critical temperature. The transition temperature of the sample containing 0.1 and 0.3 at% Gd was measured using an AC-device. The transition width at T, in-
alloys with paramagnetic impurities
201
45
0
40
0 magnetization
0
.
3
10 5-
OyJ 2.0
I
I
I
I
2.2
2.4
2.6
2.8
0. , 3.0
o*%+_ 3.2
T(K) Fig. 1. Comparison of H,, data evaluated from AC-susceptibility (. ) and differential magnetization ( o ) measurements.
dicates that the impurities are dissolved homogeneously in the material. Compared to the value of T, extrapolated from the H,,(T) data, the agreement was found to be within the range of experimental error. The DC-magnetization measurement was used to determine HCI, K,, and JC~in the pure materials as well as the Bo-jump at H,, in the case of the LaA12 single crystal.
4. Results 4.1. Superconducting properties of pure LaA12 As mentioned in section 1, intensive studies of this material have been reported in the literature. It is interesting to note, however, that a basic characterization of LaA12 in terms of its magnetization behaviour and intrinsic superconducting properties has not been reported so far. The only short note available to us [ 351 quotes data which are off by a factor of four from the “clean limit” results to be presented below (cf. also [ 361). The residual resistivity ratio measured following the annealing treatment, was found to be 830 and, therefore, well exceeding all previously published data. The normal state resistivity P,, of the sample was found to be p,=O.7385 nQm, in reasonable agreement with the trend expected from less pure samples. Results evaluated from the magnetization measurements are shown in fig. 2. From this data T, was extrapolated to be
B. Vlcek et al. /Superconducting
202
I
alloys with paramagnetic impurities
Evidence for an attractive flux line interaction leading to a first order transition at H,, has been found [ 391. This first order phase transition disappears at a certain conversion temperature T* < T, and the usual higher order phase transition at H,, prevails for temperatures T* I TI: T,. From the differential magnetization curves we find T* = 3.262 K. A further consistency check for K was made by evaluating the slope of Hc2 and H, at T, and by employing the standard relations
6
3 20
Hc2 = fi
-1
ub
0
0
b
”
0.5
1’
1.0
’ 1.5
j T lu\ ’ \“I
” 2.0
” 2.5
I\
3.0
3.!5
LaA12
0.6 -
0
1
0.5
1.0
1.5
2.0
I
2.5
3.0
3.5
T(K)
(3) poW(t)
kdffc2(t)
dt
dt
04
0.6 -
KH,
1.
(4)
,+=I
With~odH,2(t)/dt],=I=-86.2mTand~odH~(t)/ dt If= 1= - 84.3 mT, we obtain ~=0.723, in reasonable agreement with the data quoted above. All these values and the magnitude of Hc2 at T=O, obtained directly from experiments in the dilution refrigerator (mc2(0) =92.04 mT), differ considerably from the data reported in ref. [ 351 (~=3.0, tic,(O)=250 mT, h cW,,(t)/dt(,=,=-359 mT). The discrepancy is attributed to material problems in this older study. Using the BCS relation hdH,(t) dt
I=,
= _ 1 74 &H,(O) T,
(5)
Fig. 2. (a) Temperature dependence of H,, and HC2in the LaAlz single crystal. (b ) Temperature dependence of the GLparameters K, and ic2in the LaAl, single crystal.
and
3.310+0.002 K, slightly higher than the value reported so far [ 8 1, as expected for the higher purity of the sample. The magnetization curve was nearly reversible and, hence, very well suited for an evaluation of superconducting parameters. The Ginzburg-Landau parameters icr and ~~ extrapolate to ~(7’,)=~=0.75+0.01 (fig. 2(b)), K, the low temperature resistivity pn and the zero temperature specific heat coefficient y [ 3 1 ] determine the clean limit value of the GLparameter K,, via the Gor’kovGoodman relation [ 381. We find rco= 0.745. which establishes that LaAl, is a type-II superconductor in the clean limit with a very low Ginzburg-Landau parameter.
we get the value h Hc2( 0) = 83 mT, which differs considerably from the experimental result. This discrepancy is attributed to strong coupling corrections to K, ( T) which become important for T-*0. Using standard evaluation procedures, a number of other basic superconductive and normal state parameters can be calculated and used for additional consistency checks. For instance, the slope of Hc2 at T, together with the appropriate value of the Gor’kov function [ 381 give for the average normalized Fermi the value 2.01 x lo5 m/s velocity ( & ) ‘I2 (l+d) (,J is the mass enhancement factor due to the electron-phonon interaction). This value and the measured T, give the clean limit coherence length
(6)
203
B. Vlceket al. /Superconducting alloyswithparamagneticimpurities
&,= 83.5 nm. It is also possible to calculate the London penetration depth at T= 0, A,( 0) from the slope of H, at T, using
0’
0.6 0.7 -
0.6 -
~,(0)=(~~~~,=,)i’2=~4.2nm.
(7)
P
Here &, is the flux quantum. n,(O) together with the clean limit GL-parameter K~ determine again the clean limit coherence length &, which is now found to be 82.8 nm which is in reasonable agreement with the value of 83.5 nm quoted above. In yet another check we can use one of the two values for &, together with the experimental resultfortheenergygapat T=O (2d(0)/(kr,TC)=3.7, ref. [ 301) to calculate again the Fermi velocity. We find (z$ > ‘I’/ ( 1 +A) ~2.08 x IO5 m/s, in excellent agreement with the previous value. Finally, the electronic mean free path 1 and the impurity parameter cz are evaluated from K~, A,(O) and &,. The corresponding values are 9.3 pm and 0.008, respectively. Magnetization measurements were also made on the pure LaA12 polycrystal. From the resistivity (po= 1.478 nQm), the Sommerfeld constant y and K~, the Gor’kov-Goodman [ 381 relation yields ~~0.755, in good agreement with the experimental result, K= 0.757 f 0.02. A summary of the clean limit data for LaAlz is presented in table II. 4.2. Influence of Gd impurities on pn and T, Results on the dependence of the normal state resistivity on Gd content are shown in fig. 3 and demonstrate that the transport relaxation time of the conduction electrons is affected by the normal scattering part of the impurity potential in the expected
&.4 -
a=
0.3 0
0
0.2 0.1 --
/ 0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
x (at%)
Fig. 3. Normal state resistivity concentration.
p. as a function
of Gd
(first order Born’s approximation) way. The transition temperature T, is lowered dramatically with increasing concentration of paramagnetic impurities. The experimental results on the change of T, with Gd content are presented in fig. 4 where T, was determined by HC2( T,) = 0. It will be noted that the present results (open circles) are in close agreement with the data reported by Maple [ 81. The sample containing 0.585 at% Gd did not show superconductivity at a temperature of 25 mK. Therefore, the critical concentration ncr at which superconductivity is totally destroyed is about 0.59 at%, in agreement with previous work [ 8 1. 4.3. The upper critical field HC2 The experimental results on the upper critical fields from T, down to about 40 mK are presented in fig.
Table II Summary of clean limit parameters for L.aA&
TC WI
P&L(O) (mT)
(-Icod&(T)/dT)Ir=Tc (mT/K)
RRR
3.31 f0.01
92.04 k 0.02
26.112kO.02
830+ 8.0
K
Ko
I
co
(rm)
(urn)
9.3
83.5
0.75f0.01
0.754
~~(4.2 K)
( - AdK ( T) /dT) I T=rc
(nfim)
(mT/K)
0.7385f0.07
AL(O) (nm) 64.2
25.54+0.01
v, ( lo5 m/s)
3.24
B. Vlcek et al. /Superconducting
204
alloys with paramagnetic impurities
5. As expected, the critical fields are lowered with increasing Gd concentration. An interesting effect is observed at low temperatures ( N 400 mK) where the upper critical fields start to decrease with decreasing temperature. This effect is most pronounced at intermediate concentrations and disappears again for n, 2 0.5 at%.
5. Comparison
between theory and experiment
The theoretical treatment of the upper critical field in superconducting alloys which is applied here to analyse the experimental data is based on a number of basic assumptions: - The spin of the impurity atom is treated classically which is certainly valid for Gd atoms with a spin of s=;. - The impurity concentration is very dilute and the impurities distributed are homogeneously throughout the sample. We can, therefore, neglect correlations between impurity sites and this allows us to assume a random impurity distribution. - The interaction of the conduction electrons is as-
Fig. 4. Critical temperature T, as a function of Gd content, (. ) data of ref. [ 81, ( o ) present results. (The solid line is drawn to guide the eye).
a0
60
-0
0.4
0.8
1.2
1.6
2.0
2.4
2.6
3.2
T(K) Fig. 5. Upper critical fields HE2 of LaAlz containing 0.0, eye).
0.1,
0.3, 0.41, 0.5 and 0.55 at% Gd. (The dashed lines are drawn
to
guide the
B. Vlceket al. /Superconducting alloyswithparamagneticimpurities
sumed to be weak and the scattering by the impurity atoms is treated in first order Born’s approximation. - Pauli spin paramagnetism is not included because of the very low upper critical fields ( S 100 mT). These points were discussed in detail in the first part of this series [ 281. Nevertheless, the theory presented therein has to be extended to include the anisotropy of the electron-phonon interaction and of the Fermi velocity. This necessity follows from the pronounced upward curvature of H,, ( T) near T, observed in clean LaA12 (fig. 7 (a) ) [ 421. This extension is analytically very tedious but straightforward and follows earlier work by Prohammer [ 43 ] closely. Hence, the theory contains quite a number of parameters which have to be determined unambiguously in order to give a conclusive interpretation. These parameters are: ,u*, the Coulomb pseudo-potential, a2F( o), the electron-phonon interaction spectral function, ( ur) , the Fermi surface average of the Fermi velocity, (a’), the mean square anisotropy of the electron-phonon interaction as a consequence of the separable ansatz of eq. (2), and ( b2), the mean square anisotropy of the Fermi velocity as a consequence of a similar separable ansatz [ 27 1. The impurities are described by three more parameters, t,, t_ and l’, (0) two of which are proportional to the impurity concentration n,. t+ = l/ (27rr,,), where the transport relaxation time T,, is related to the residual resistivity p,, by
(8)
205
ever, the generalized phonon density of states G(o) measured by Yeh et al. [ 411 is available and it is usually safe to assume that a’F(w) is proportional to G(o). The factor of proportionality is then chosen to give a certain value for the mass enhancement factor 1 co
(9) 0
If the sample were isotropic, the choice of il together with the experimental value of TC( = 3.29 K) would result in a unique value for ,u*, the Coulomb pseudopotential, as a result of the linearized Eliashberg equations, (As a rule of thumb ,u*z 0.1 which helps in making the “proper” choice for 2.) Our sample is clearly anisotropic and the choice for 2 for to be supplemented by a choice for (a’) in order to determine p* from the experimental value of T, and for a value of t, calculated from the experimental value of p,, using eq. (8) and S2p=$(z$)N(0)e2
(10)
and y=fxZN(0)(l+l)
(11)
where y is the experimentally determined zero temperature electronic specific heat (Sommerfeld constant) [ 3 11. The value ( V$) has already been evaluated in the previous chapter. Having determined p* from these assumptions, the full strong coupling theory of the upper critical field
and L$ is the “dressed” Drude plasma frequency. The parameter t_ = 1/ ( ~KZ,,,) is related to the magnetic relaxation time r, of the spin flip interaction between conduction electrons and impurity spins. Finally, V, (0) is the microscopic spin flip scattering potential which becomes important in the discussion of spin orientation effects. In the following, we will discuss the procedures employed to determine these eight parameters and the consequences for the interpretation of the results. -.__
0.00
5. I. The clean polycrystalline sample
The electron-phonon interaction spectral function, a’F(o), is not known for this material. How-
0.0
0.1
0.2
0.3
0.4
0.5
0.0
x (at%) Fig. 6. Inverse normal relaxation time t, and inverse magnetic relaxation time I_ as a function of Cd concentration.
206
B. Vlcek et al. /Superconducting
for anisotropic polycrystals is employed to fit the experimental Hc2 data over the whole temperature range (20 mKS TIT,). The parameter (b’) is used to model the curvature of HC2( T) at T, with all the other parameters already fixed by the first step. Of course, the first choice for L will not be the one which results in an optimal fit to the experimental data and several recursive steps are necessary, adjusting 5 (z+), (a’) and ( b2) until good agree-
alloys with paramagnetic impurities
ment between theory and experiment is established. The result of this procedure is as follows: LzO.55, p*=O.l, (vr)=3.23x105 m/s, (a2)=0.01 and ( b2) =0.16. As shown in fig. 7(a), excellent agreement between experiment and theory is achieved. These parameters characterize LaAlz as a rather weak coupling superconductor with little electron-phonon interaction anisotropy. The upward curvature of Hc2 ( T) at T, is mainly caused by the pronounced an-
(b) x- 0.1 at’/,Gd c
50-
E w
40-
‘.
‘\
r: I 9
‘.
‘\
30 20-
00. %%! %%
lo0.0
0.8
0.4
1.2
1.5
2.0
2.4
2.8
0.4
3.2
0.8
1.2
0.0
0.5
1.0
1.6
2.0
2.4
2.8
T (K)
T(K)
1.5
2.0
2.5
T(K)
Fig. 7. Upper critical dashed lines: theory.
fields He2 or ~a(x at’% Gd)A12 compared
to strong couplmg
calculations
including
anisotropy.
(o ) experiment,
B. Vlceket al. /Superconducting alloyswithparamagneticimpurities
isotropy of the Fermi velocity. Using these parameters, anisotropic Eliashberg theory predicts a zero temperature gap ratio 2A( 0) / (k,T,) of 3.6, which is in excellent agreement with the experimental value of 3.7. On the other hand, the analysis of section 4 would result in (0;) ‘I2 = 3.12 x 1O5 m/s, which is close enough to the value found by the full strong coupling analysis. (Note that ‘I2 has to be different from (L+) in an anisotropic system. ) 5.2. T, depression as a function of Gd content As LaAIZ is an anisotropic superconductor, the critical temperature of a sample with Gd content is affected in a twofold way: (a) T, decreases with decreasing transport relaxation time rt,, (increasing pn, increasing t+) because of “smearing out” the electron-phonon interaction anisotropy by normal momentum scattering processes [ 12 1, and (b ) T, also decreases rapidly with decreasing magnetic relaxation times (increasing t_ ). Using eqs. (8, 10) and (1 1 ), t, is calculated for each particular sample from its experimental p,, value. By keeping 1, ,u* and ( a2) fixed, the parameter t_ is fitted for each sample using anisotropic Eliashberg theory to give the experimental T, value [44]. Figure 6 shows that the values for t+ and t_ found using this procedure are proportional to the impurity concentration n, as required by theory (see eqs. ( 17) and (18) of ref. [28]). Standard weak coupling isotropic AG-theory describes the decrease in critical temperature with increasing impurity concentration by
ln+
c =
tr,(l+A)+
7 1--u/(4)
(12)
where T,,, is the critical temperature of the clean sample and v(x) is the digamma function. The critical concentration at which superconductivity is destroyed, is found for a value of t- (fig. 6) corresponding to a critical concentration of 0.57 atoh Gd, which is just a bit smaller than the experimentally found critical concentration of about 0.59 at%. We can therefore conclude that the depression of the critical temperature with increasing concentrations of paramagnetic impurities is described reasonably
207
well by the weak coupling limit AG-theory. Strong coupling effects beyond the ( 1+A) renormalization are very diflicult to be distinguished by experiment except at concentrations very close to the critical one (cf. fig. 1 of ref. [28]). 5.3. The upper critical field As a first step, spin orientation is ignored and the full strong coupling theory of HC2 is applied to reproduce the experimental HC2( T) data for five La (x at% Gd)A12 sample with x=0.1, 0.3, 0.41, 0.5 and 0.55. All the parameters are already determined and there are no degrees of freedom left. The dashed lines in figs. 7(b-f ) represent the theoretical predictions and they prove that indeed the theory reproduces the experimental data exceptionally well in the temperature range 0.4 KI TI T,. The upward curvature of HC2( T) near T, for the clean polycrystals is as well reproduced as its smearing out with increasing Gd concentration. Nevertheless, there are also a few inconsistencies: (a) The pronounced deviation of the experimental data from the theoretical predictions for the sample with x= 0.1 at temperatures < 1 K is not understood (fig. 7(b)). (b) The sample withx=0.55 (fig. 7(f)) shows definite deviations to smaller values in the temperature range 0.2 KI T40.6 K, while the low and high temperature parts of the HC2( T) curve are perfectly reproduced by theory. In view of the otherwise excellent agreement between experiment and theory, further investigations of the upper critical field in samples with Gd concentrations close to the critical one could result in interesting new results. At temperatures below 0.4 K the samples with x=0.1, 0.3, 0.41 and also with 0.5 show a pronounced deviation from the theoretical predictions. The first three show a maximum in HCz( T) at about 0.4 K and a decrease in HC2 with further decreasing temperatures. This effect is most pronounced in the sample with x=0.3. The sample with x=0.5 does not really develop a maximum but, nevertheless, HC2( T) levels off at temperatures ~0.25 K. Similar behaviour was predicted by the theoretical analysis presented in part I of this series [ 28 ] if spin orientation effects at low temperatures are included. The simplest model describes the spin orientation as a function of the applied magnetic field B and the
208
B. Vlcek et al. /Superconducting
temperature T by a Boltzmann distribution for the average spin (S) “seen” by the conduction electrons:
(S)
=a
s dQS(Q)exp
{ -,,,.,I
(13)
N=
(14)
This effect introduces a new parameter, Vi (0), the microscopic scattering potential of the paramagnetic impurities (see eq. (29 ) of ref. [ 28 ] ). This parameter is used to fit the low temperature &( T) data of the sample with x=0.3 as is shown in fig. 8. We find I’, (0) = 0.47 meV. HC2( T) data of the two other samples with x= 0.1 and 0.4 1 are then calculated with this potential. (No other fitting parameter is avail-
alloys with paramagnetic impurities
able for this additional calculation! ) It becomes obvious from the results presented in fig. 8, that spin orientation is indeed the explanation for the low temperature behaviour of H,,( T) in these samples. But a comparison with the x=0.41 sample also reveals that our simple model for the spin orientation obviously overestimates the amount of spin orientation at very low temperatures ( < 100 mK) while the onset of spin orientation is predicted properly in all three cases. With higher impurity concentrations, the external magnetic fields are already so small that spin orientation effects can only be observed at very low temperatures, if at all. The sample with x=0.5 could be indicative for this behaviour. (A theoretical analysis of these data was not feasible because of prohibitively long computing times. ) 5.4. Strong coupling effects
0
L CDoao 0
a**.
60
0 0 0
0
l
0 0
-
l
l
experiment strong-anirotropic + spin
orientation
2.0
1.0
T (K)
Fig. 8. Upper critical fields of La(x at% Gd)A12, x=0.1,0.3 and 0.4 1 compared to strong coupling theory including spin orientation. (o ) experiment, (. ) strong coupling theory without spin orientation term, and solid lines strong coupling theory with spin orientation.
The theoretical analysis of ref. [ 281 predicted rather pronounced strong coupling effects for the upper critical field, especially in samples with rather high concentrations of paramagnetic impurities and at low temperatures. Thus, in this section we will compare the predictions of the full strong coupling theory with results found from a ( 1 +A) renormalized weak coupling theory (eqs. (33-37) of ref. [ 281). In this comparison T,, t+, A and (z+) are the same; only t_ has to be adjusted using eq. ( 12) to give the correct T,. Figure 9(a) presents the results of such a comparison for the clean (x = 0 ) sample and for the sample with x=0.3. Whereas the full strong coupling theory (solid lines) reproduces the experimental data (open circles) almost perfectly, the predictions of the weak coupling theory (dashed lines) are well below the experimental data. This discrepancy becomes less pronounced for the sample with x=0.3. Finally, fig. 9(b) shows that for the sample with x=0.5 the opposite holds, namely that the weak coupling theory predicts upper critical fields which are above the experimental data and the results of the full strong coupling theory. This behaviour is in complete agreement with theoretical expectations formulated in ref. [ 28 ] and is based on the fact that paramagnetic impurities become more effective in their pair breaking abilities with decreasing temperatures. Since the magnetic re-
B. Vlceket al. /Superconducting alloyswithparamagneticimpurities
-
full
-----
weak coupling
*trong
-.---- weak
1.6
209
(4
coupling (1+X)
coupling( x=0)
2.0
2.4
2.8
3.;
T(K)
T (K)
laxation time is overestimated by the weak coupling theory (smaller values for t_ for the same Tc’s), it results, with increasing impurity concentrations, in higher upper critical fields relative to the predictions of the full strong coupling theory and therefore a “crossover” has to be observed. This effect is clearly demonstrated in fig. 9 and is an indication of strong coupling effects which go beyond the (1 +A) renormalization even for the rather weak coupling material LaA12. In order to emphasize the importance of the ( 1+,I) renormalization in the weak coupling theory, results for A= 0 (dashed dotted lines) are also included in fig. 9(a).
Fig. 9. (a) HCzdata of La(x at% Gd)A12 (x=0.0 and 0.3) compared to weak coupling calculations (dashed lines), strong coupling calculations including anisotropy (solid lines) and weak coupling calculations for A= 0 (dash-dotted lines ) (b ) H,, data of La(x at% Gd)A12 (x=0.5 at%) compared to weak coupling calculations (dashed line) and strong coupling calculations (solid line).
6. Conclusions Magnetization and AC-susceptibility measurements were made on single- and polycrystalline samples of LaAlz doped with Gd in the concentration range 0 I nr IO.6 at%. The results may be summarized as follows. 1) The resistivity measurements show that Gd is dissolved homogeneously in the material. 2) The T, values extrapolated from the Hc2( T) measurements are found to be in reasonable agreement with data published previously. From our measurements we conclude that the
210
B. Vlcek et al. /Superconducting
critical concentration is about 0.59 at% Gd. 3 ) From an evaluation of superconducting parameters we conclude that LaA12 is a rather weak coupling type-II superconductor. Due to the low GGparameter, a phase transition between typeII/ 1 and type-1112 superconductivity was found at T*=3.262 K. 4) The upper critical fields Hc2 are strongly depressed by the presence of magnetic impurities. At low temperatures ( N 400 mK) and intermediate impurity concentrations, peaks of H,, followed by a downward curvature towards T=O are observed. 5 ) The clean limit data on H,, could be explained quantitatively in terms of anisotropic Eliashberg theory using the anisotropy parameters (a*) = 0.0 1 for the electron-phonon coupling and (b’) =O. 16 for the Fermi velocity. Excellent agreement between the strong coupling theory and experiment was also achieved for the samples containing Gd impurities if spin orientation of the impurity atoms was included to explain the peaks of Hc2 at intermediate temperatures and Gd concentrations. 6) While the effects of strong coupling and anisotropy on the depression of T, are rather small, significant effects of strong coupling have been found for the upper critical field. As predicted by theory and confirmed by our measurements, the weak coupling theory underestimates the influence of Gd on Hc2 up to concentrations of about tncr. At higher concentrations the upper critical field is overestimated by the weak coupling theory.
Acknowledgements We wish to thank Mr. H. Niedermaier for his continuous and dedicated help with the operation of the dilution refrigerator, Prof. F. Steglich ( Darmstadt ) for providing the LaAlz single crystal and Mr. H. Beyss and co-workers (Jtilich ) for their help with sample preparation. This work was supported in part by Fonds zur Fijrderung der Wissenschaftlichen Forschung, Wien, under contract No. 5353 and by the Office of Naval Research under grant No. NO0 14-89J-1088.
alloys with paramagnetic impurities
References [ 1] B.T. Matthias,
H. Suhl and E. Corenzwit, Phys. Rev. Lett. 1 (1958) 92. [2] P.W. Anderson, J. Phys. Chem. Solids 1I (1959) 26. [ 31 A.A. Abrikosov and L.P. Gor’kov, Sov. Phys.-JEPT 9 ( 1958) 220. [4] A.A. Abrikosov and L.P. Gor’kov, Sov. Phys.-JETP 12 (1961) 1243. [ 51H. Shiba, Progr. Theor. Phys. 40 ( 1968) 435. [6] A.I. Rusinov, Sov. Phys.-JETP Lett. 9 ( 1969) 85. [7] A.I. Rusinov, Sov. Phys.-JETP 29 (1969) 1101. [8] M.B. Maple, Ph.D. Thesis, University of California, San Diego (1969). [ 91 S. Skalsky, 0. Betbeder-Matibet and P.R. Weiss, Phys. Rev. 136 (1964) A1500. [ lo] W.R. Decker and D.K. Finnemore, Phys. Rev. 172 ( 1968 ) 430. [ 111 E. Schachinger, J.M. Daams and J.P. Carbotte, Phys. Rev. B22 (1980) 3194. [ 121 D. Markovitz and L.P. Kadanoff, Phys. Rev. 131 ( 1963) 563. [ 131 J.R. Clem, Ann. Phys. (NY) 40 (1966) 268. [ 141 D.K. Finnemore and D.E. Mapother, Phys. Rev. 140 ( 1965) A507. [ 151 D. Gubser, Phys. Rev. B6 (1972) 827. [ 161 A.J. Bennet, Phys. Rev. 140 (1965) A1902. [ 171 C.R. Leavens and J.P. Carbotte, Ann. Phys. (NY) 70 (1965) 338. [ 181 J.P. Carbotte, in: Anisotropy Effects in Superconductors, ed. H.W. Weber (Plenum, New York, 1977) p. 183. [ 191 J.M. Daams and J.P. Carbotte, J. Low Temp. Phys. 43 (1981) 263. [ 201 L. Niel, N. Giesinger, H.W. Weber and E. Schachinger, Phys. Rev. B32 (1985) 2976. [ 211 N.R. Werthamer, E. Helfand and P.C. Hohenberg, Phys. Rev. 158 (1967) 415. [22] D. Rainer and G. Bergmann, J. Low Temp. Phys. 14 ( 1974) 501. [23] M. Schossmann and E. Schachinger, Phys. Rev. B33 (1986) 6123. [24] F. Marsiglio, M. Schossmann, E. Schachinger and J.P. Carbotte, Phys. Rev. B35 (1987) 3226. [25] E. Schachinger, M. Prohammer, E. Seidl and H.W. Weber, Physica C 153 (1988) 247. [ 26 ] H.W. Weber and E. Schachinger, Helv. Phys. Acta 6 1 ( 1988 ) 478.
[ 271 M. Prohammer
and E. Schachinger,
Phys. Rev. B36 ( 1987)
8353. [ 281 W. Pint and E. Schachinger, Physica C 159 ( 1989) 33. [29] J.H. Moeser, Ph.D. Thesis, Universitat Kijln, 1976, unpublished. [30] R.E. Hungsberg
and K.A. Gschneidner
Jr., J. Phys. Chem.
Solids 33 (1972) 401. [31]F.Steglich,Z.Phys.B23 (1976) 331. [ 321 J.H. Moeser, F. Steglich, C. Leser and E. Mtiller-Hartmann. Z. Phys. B 25 (1976) 339.
B. Vlcek et al. /Superconducting [33] G. Beuermann, Z. Phys. B 28 (1977) 207. [ 341 H. v. tihneysen, J.L. Tholence and F. Steglich, Z. Phys. B 29 (1978) 319. [35] G. Riblet and K. Winzer, Solid State Commun. 11 ( 1972) 175. [ 361 B. Vlcek, E. Seidl and H.W. Weber, Jpn. J. Appl. Phys. 263 (1987) 967. [37] P. Hahn and H.W. Weber, Cryogenics 23 (1983) 87. [38] L.P. Gor’kov, Sov. Phys.-JETP 10 (1960) 998; B.B. Goodman, IBM J. Res. Dev. 6 ( 1962) 63. [ 39 ] H.W. Weber, E. Seidl, M. Botlo, C. Laa, E. Mayerhofer, EM. Sauerzopf, R. Schalk, H.P. Wiesinger and J. Rammer, PhysicaC 161 (1989) 272.
alloys with paramagnetic impurities
211
[40] K. Maki, Phys. Rev. Lett. 23 (1969) 1223. [41] C.T. Yeh, W. Reichart, B. Renker, N. Nticker and M. Loewenhaupt, J. Phys. (Paris) C6 (1981) 371. [42] P.C. Hohenberg and N.R. Werthamer, Phys. Rev. 153 (1967) 493. [43] M. Prohammer, Solid State Commun. 67 (1988) 293. [ 441 J.M. Daams, E. Schachinger and J.P. Carbotte, J. Low Temp. Phys. 42 ( 198 I ) 69.
198-211
SUPERCONDUCTING MAGNETIZATION IMPURITIES
ALLOYS
WITH PARAMAGNETIC
IMPURITIES
(PART II)
AND UPPER CRITICAL FIELDS OF LaA12 WITH DILUTE Gd
B. VLCEK, E. SEIDL and H.W. WEBER Atominstitut der tisterreichischen Universitiiten, A-1020 Wien, Austria
E. SCHACHINGER
and W. PINT
Institut ftir Theoretische Physik Technische UniversitSit Graz, A-8010 Graz, Austria Received 30 November 1989 Revised manuscript received 26 February
1990
Magnetization and AC susceptibility were made on single- and polycrystalline samples of LaAIZ doped with Cd in the concentration range 0 In, S 0.7 at%. The depression of T, with Cd content is found to be in excellent agreement with previous experimental work. Concerning the temperature dependence of H,,, peaks at low temperatures ( = 400 mK) and downward curvatures towards T=O have been observed. The upper critical field of the clean material can be explained quantitatively in terms of anisotropic Eliashberg theory using (a’) ~0.01 for the electron-phonon coupling anisotropy parameter and (b’) =O. 16 for the anisotropy of the Fermi velocity. With these values and the generalized phonon density of states reported by Yeh et al. excellent agreement between theory and experiment was achieved. Furthermore, results on the Cd doped materials can be explained quantitatively on the basis of strong coupling theory, if spin orientation effects of the Cd atoms are included in the theoretical treatment.
1. Introduction
Magnetic impurities in superconductors have been studied intensively, because small concentrations are sufficient to change the properties of the host material in a significant way. This is in clear contrast to normal conducting materials, where very high concentrations must be dissolved in order to change the normal state properties, leading almost certainly to an interaction between the impurity atoms. Therefore, magnetic impurities can be used either to study the influence of the host material on the magnetic state of the impurity, or to study strong coupling and anisotropy effects on the superconducting properties as a function of impurity concentration. As outlined by Matthias et al. [ 1 ] who studied rare earth solutions ( 1 atW) in lanthanum, normal (non magnetic) impurities have little or no effect on the transition temperature of the host material, a result which is in agreement with the theoretical argument given by Anderson [2] later on. Gadolinium im0921-4534/90/$03.50 (North-Holland )
0 Elsevier Science Publishers
B.V.
purities, on the other hand, which have the largest spin of the rare earths are most effective in lowering the critical temperature. This is due to the spin flip interaction of the conduction electrons with the spin of the gadolinium atoms which is pairbreaking. In contrast, the momentum exchange interaction which is typical for scattering processes of conduction electrons at normal impurity sites is not pairbreaking and has therefore no effect on the superconducting properties of isotopic superconductors. The first theoretical description of a superconductor containing paramagnetic impurities was given by Abrikosov and Gor’kov (AG-theory) [3,4] on the basis of BCS theory. They treated the magnetic scattering in first order Born approximation assuming a weak interaction between impurities and conduction electrons. In the case of a strong interaction, terms of higher order have to be included in order to describe the scattering of the conduction electrons at the impurity sites [ 5-7 1. Experimentally, the predictions of the AG-theory
B. Vlcek et al. /Superconducting
on the T, depression have been confirmed in the system LaAl,-Gd [8] except near the critical concentration. This small deviation is presumably related to strong coupling effects. Skalsky et al. [ 91 have extended the AG-theory to other thermodynamic properties of a superconductor containing paramagnetic impurities. For example, they have shown that the thermodynamic critical field deviation function,
=Ec
D(t)
(l-t’),
t=T/T,,
becomes more negative with increasing impurity concentration. These calculations have been confirmed by measurements on the weak coupling superconductor thorium [ 10 1. Schachinger et al. [ 111 calculated the deviation function D(t) for the strong coupling superconductor lead containing paramagnetic impurities. In contrast to the AG-theory, these calculations were based on the Eliashberg theory. It was shown in this work that the AG-theory overestimated the effect of paramagnetic impurities on the critical temperature quite substantially and pronounced differences were reported for the deviation function D(t). Additional deviations from the standard isotropic BCS-theory are based on anisotropy effects. They were studied first by Markovitz and Kadanoff [ 121 who introduced a separable pairing potential V(k, k’ ):
V(k,k’)=(l+a,)I’~,,(l+~~~)
(2)
to describe the anisotropy of the electron-phonon pairing interaction. Here Us is a temperature independent anisotropy parameter and V,,, is the constant and isotropic pairing potential of the BCS-theory. An extension of this work was given by Clem [ 13 ] who calculated the influence of anisotropy on the thermodynamic properties of superconductors, in particular the thermodynamic critical field H, ( T) and its deviation function D(t). All these calculations were limited to weak coupling superconductors. Strong coupling effects were first included in a phenomenological way by introducing a second fit parameter 6 [ 14,15 1. A full theoretical treatment of anisotropic superconductors including strong coupling effects became available later on [ 16- 19 1, and was confirmed experimentally on the superconductor indium doped with thallium using a mean square
alloys with paramagnetic impurities
199
electron-phonon interaction anisotropy of < a2) = 0.04 in the clean limit [ 20 1. It was also shown that the anisotropy was “washed out” by non-magnetic impurities. Since the theory was also suited to calculate the influence of paramagnetic impurities on the deviation function of strong coupling anisotropic superconductors it was tempting to investigate this thermodynamic aspect experimentally. However, in order to determine the deviation function as a function of magnetic impurity content, the thermodynamic critical field H,(T) has to be measured with high accuracy, a task which can be met only by type-1 superconductors. Based on calculations by Werthamer et al. [ 2 11, Eliashberg equations for the upper critical field H,, ( T) could be established by Rainer and Bergman [ 221 and Schossmann and Schachinger [ 23 1. Further work by Mars&ho et al. [24] showed that the upper critical field of type-II superconductors was very sensitive to strong coupling effects. This theory was then extended by Prohammer and Schachinger [27] to describe the upper critical field of anisotropic polycrystals. Experimental data on Hc2 in pure and nitrogen doped niobium [25,26] were explained on the basis of this extended theory. Finally, in part I of this series of two papers [ 28 1, two of us presented a full strong coupling theory of the upper critical field in isotropic superconductors containing paramagnetic impurities. An application of this theory, extended to the case of anisotropic polycrystals, will be presented in section 5 of this paper. For the experimental investigations the superconductor LaAl, which is well known in the literature [ 829-361, has been selected because various magnetic impurities, such as Gd, Ce, etc., can be dissolved homogeneously. Gadolinium was chosen as the magnetic impurity. Its large spin (S=i) ensures that the AG-theory which treats the atomic spins of the impurities as classical vectors can be applied to this system. Except for some work on the pure material [ 35,361, no information on the magnetic properties of the system LaAlz-Gd has been available so far. Thus, we will present some aspects of sample preparation and experimental techniques used for the measurement of superconducting parameters in sections 2 and 3, summarize the experimental results and evaluations in section 4 and devote section 5 to a comparison of experimental upper
200
B. Vlcek et al. /Superconducting alloys with paramagnetic impurities
critical field data with theory. Our conclusions drawn in section 6.
are
2. Materials In the present study a single crystal of pure LaAl, and six polycrystals were investigated. The single crystal which had been studied previously [ 341 was provided by F. Steglich (Darmstadt). It was annealed at 900°C for 50 h to remove internal stress and electropolished to clean the surface. The polycrystals were prepared using the cold boat technique under an atmosphere of 800 mbar Ar. As the starting materials, pure lanthanum (purchased from the Ames Laboratory and containing only minor amounts of magnetic impurity atoms, such as 15 at.ppm Fe, 0.96 at.ppm Ni and 2.2 at.ppm Ce, with all the other impurities falling below 1 ppm each) and 99.99999% Al were used. The samples were prepared by carefully weighing (five times) the desired amounts of material and melting them six times, together with Gd, to ensure a homogeneous distribution of Gd within the matrix. Then the polycrystals were annealed at 900°C for 50 h, electropolished and cast to their final cylindrical form (3 mm diameter, 20 mm length). A summary of material and characteristic superconductive parameters is presented in table I.
3. Experimental Conventional four point measurements were made at room temperature and at 4.2 K to evaluate the re-
sistivity p,, and the residual resistivity ratios RRR. The accuracy of this data is determined mainly by the geometry of the sample. While the error associated with the residual resistivity ratio is less than l%, an uncertainty of about 10% must be taken into account for the absolute values of p,,. In order to determine the magnetic properties in the superconducting state, two different methods were employed. Firstly, in the temperature range between T, and 1.6 K magnetization measurements using the differential technique [ 371 with a low temperature chopper were made. Very low field sweeping rates were used (0.6 mT/s) to achieve true equilibrium conditions in the magnetization cycle. All the data were stored on a computer and analyzed numerically. The resolution for determining the upper critical field H,, from the differential curve is better than f0.5 mT. Secondly, AC-susceptibility measurements were made at all temperatures by superimposing a small AC-ripple field ( lo-100 nT) onto a transverse DC field. This field was provided either by a superconducting magnet in a conventional bath cryostat or by an electromagnet or a normal conducting coil in the dilution refrigerator. The sample was placed into a pick-up coil and the signals were detected by a lockin amplifier while the external field was swept slowly at a fixed temperature. In order to find the optimal sweep time, the data in increasing and decreasing field were compared at constant temperature and the sweep time was decreased until both branches gave the same response indicating thermodynamic equilibrium conditions. Then the data were taken from increasing field measurements. This experimental
Table I Summary of material parameters Polycrystals
Sample
Single crystal
n,, at% Gd RRR ~“(4.2 K), CnRm) T,(K)
0.0 830 0.7385 3.31
0.0 343 1.478 3.29
0.1 217 2.19 2.91
92.04 0.75 26.112
92.48 0.757 26.68
72.07 24.53
k&CO), (mT) (-/@&(T)IdT)I~=r
0.3 105 4.11 2.084
0.41 72 5.671 (1.42)
0.5 64 7.881 1.032
28.41 18.82
15.06
9.54 14.36
0.55 58 1.896 0.705 5.096 9.46
0.58 45 11.31 <20mK
B. Vlcek et al. /Superconducting
technique was used throughout the entire study to evaluate the upper critical fields HC2. The field was produced by three different systems, a superconducting Helmholtz coil (O-4 T), a normal conducting Helmholtz coil (O-20 mT) and an electromagnet (0- 1 T). The field achieved by the Helmholtz coils is measured through the current producing the field. The error in this measurement can be disregarded, but some uncertainties concerning the absolute field error remain. The electromagnet was calibrated using a high resolution Hall probe at the position of the sample. Due to this calibration the accuracy of the field measurement is better than 0.1%. Through a comparison of several measurements in different magnets we conclude that the absolute error of H,, determined by the differential technique is less than 0.5%. The accuracy of HC2evaluated from the AC-susceptibility measurements was found to be f50 PT. In the temperature range from T, to 25 mK, calibrated germanium resistors were used whose maximum calibration errors, according to their specilications, are about 4 mK near 4 K. The accuracy of the temperature measurement below 1 K was better than 5 mK. From one T, measurement made in the overlapping temperature range between the conventional bath cryostat and the 3He-4He dilution refrigerator, which is not easily accessible, we believe that the T, values are accurate to better than 5 mK on an absolute scale. As mentioned above, two different techniques were available for the measurements, a differential magnetization and an AC-susceptibility technique. For a few samples which were investigated in the hightemperature range by both techniques, a test on the compatibility of the H,,-evaluations was made. In both cases the transition from the superconducting to the normal state is slightly rounded and was therefore determined by two tangents. A comparison of the values of H,, evaluated from both techniques is shown in fig. 1. As no noticable difference between the data could be found, we conclude that the procedures adopted to determine HC2 from experiment are correct. A further consistency check was made for the critical temperature. The transition temperature of the sample containing 0.1 and 0.3 at% Gd was measured using an AC-device. The transition width at T, in-
alloys with paramagnetic impurities
201
45
0
40
0 magnetization
0
.
3
10 5-
OyJ 2.0
I
I
I
I
2.2
2.4
2.6
2.8
0. , 3.0
o*%+_ 3.2
T(K) Fig. 1. Comparison of H,, data evaluated from AC-susceptibility (. ) and differential magnetization ( o ) measurements.
dicates that the impurities are dissolved homogeneously in the material. Compared to the value of T, extrapolated from the H,,(T) data, the agreement was found to be within the range of experimental error. The DC-magnetization measurement was used to determine HCI, K,, and JC~in the pure materials as well as the Bo-jump at H,, in the case of the LaA12 single crystal.
4. Results 4.1. Superconducting properties of pure LaA12 As mentioned in section 1, intensive studies of this material have been reported in the literature. It is interesting to note, however, that a basic characterization of LaA12 in terms of its magnetization behaviour and intrinsic superconducting properties has not been reported so far. The only short note available to us [ 351 quotes data which are off by a factor of four from the “clean limit” results to be presented below (cf. also [ 361). The residual resistivity ratio measured following the annealing treatment, was found to be 830 and, therefore, well exceeding all previously published data. The normal state resistivity P,, of the sample was found to be p,=O.7385 nQm, in reasonable agreement with the trend expected from less pure samples. Results evaluated from the magnetization measurements are shown in fig. 2. From this data T, was extrapolated to be
B. Vlcek et al. /Superconducting
202
I
alloys with paramagnetic impurities
Evidence for an attractive flux line interaction leading to a first order transition at H,, has been found [ 391. This first order phase transition disappears at a certain conversion temperature T* < T, and the usual higher order phase transition at H,, prevails for temperatures T* I TI: T,. From the differential magnetization curves we find T* = 3.262 K. A further consistency check for K was made by evaluating the slope of Hc2 and H, at T, and by employing the standard relations
6
3 20
Hc2 = fi
-1
ub
0
0
b
”
0.5
1’
1.0
’ 1.5
j T lu\ ’ \“I
” 2.0
” 2.5
I\
3.0
3.!5
LaA12
0.6 -
0
1
0.5
1.0
1.5
2.0
I
2.5
3.0
3.5
T(K)
(3) poW(t)
kdffc2(t)
dt
dt
04
0.6 -
KH,
1.
(4)
,+=I
With~odH,2(t)/dt],=I=-86.2mTand~odH~(t)/ dt If= 1= - 84.3 mT, we obtain ~=0.723, in reasonable agreement with the data quoted above. All these values and the magnitude of Hc2 at T=O, obtained directly from experiments in the dilution refrigerator (mc2(0) =92.04 mT), differ considerably from the data reported in ref. [ 351 (~=3.0, tic,(O)=250 mT, h cW,,(t)/dt(,=,=-359 mT). The discrepancy is attributed to material problems in this older study. Using the BCS relation hdH,(t) dt
I=,
= _ 1 74 &H,(O) T,
(5)
Fig. 2. (a) Temperature dependence of H,, and HC2in the LaAlz single crystal. (b ) Temperature dependence of the GLparameters K, and ic2in the LaAl, single crystal.
and
3.310+0.002 K, slightly higher than the value reported so far [ 8 1, as expected for the higher purity of the sample. The magnetization curve was nearly reversible and, hence, very well suited for an evaluation of superconducting parameters. The Ginzburg-Landau parameters icr and ~~ extrapolate to ~(7’,)=~=0.75+0.01 (fig. 2(b)), K, the low temperature resistivity pn and the zero temperature specific heat coefficient y [ 3 1 ] determine the clean limit value of the GLparameter K,, via the Gor’kovGoodman relation [ 381. We find rco= 0.745. which establishes that LaAl, is a type-II superconductor in the clean limit with a very low Ginzburg-Landau parameter.
we get the value h Hc2( 0) = 83 mT, which differs considerably from the experimental result. This discrepancy is attributed to strong coupling corrections to K, ( T) which become important for T-*0. Using standard evaluation procedures, a number of other basic superconductive and normal state parameters can be calculated and used for additional consistency checks. For instance, the slope of Hc2 at T, together with the appropriate value of the Gor’kov function [ 381 give for the average normalized Fermi the value 2.01 x lo5 m/s velocity ( & ) ‘I2 (l+d) (,J is the mass enhancement factor due to the electron-phonon interaction). This value and the measured T, give the clean limit coherence length
(6)
203
B. Vlceket al. /Superconducting alloyswithparamagneticimpurities
&,= 83.5 nm. It is also possible to calculate the London penetration depth at T= 0, A,( 0) from the slope of H, at T, using
0’
0.6 0.7 -
0.6 -
~,(0)=(~~~~,=,)i’2=~4.2nm.
(7)
P
Here &, is the flux quantum. n,(O) together with the clean limit GL-parameter K~ determine again the clean limit coherence length &, which is now found to be 82.8 nm which is in reasonable agreement with the value of 83.5 nm quoted above. In yet another check we can use one of the two values for &, together with the experimental resultfortheenergygapat T=O (2d(0)/(kr,TC)=3.7, ref. [ 301) to calculate again the Fermi velocity. We find (z$ > ‘I’/ ( 1 +A) ~2.08 x IO5 m/s, in excellent agreement with the previous value. Finally, the electronic mean free path 1 and the impurity parameter cz are evaluated from K~, A,(O) and &,. The corresponding values are 9.3 pm and 0.008, respectively. Magnetization measurements were also made on the pure LaA12 polycrystal. From the resistivity (po= 1.478 nQm), the Sommerfeld constant y and K~, the Gor’kov-Goodman [ 381 relation yields ~~0.755, in good agreement with the experimental result, K= 0.757 f 0.02. A summary of the clean limit data for LaAlz is presented in table II. 4.2. Influence of Gd impurities on pn and T, Results on the dependence of the normal state resistivity on Gd content are shown in fig. 3 and demonstrate that the transport relaxation time of the conduction electrons is affected by the normal scattering part of the impurity potential in the expected
&.4 -
a=
0.3 0
0
0.2 0.1 --
/ 0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
x (at%)
Fig. 3. Normal state resistivity concentration.
p. as a function
of Gd
(first order Born’s approximation) way. The transition temperature T, is lowered dramatically with increasing concentration of paramagnetic impurities. The experimental results on the change of T, with Gd content are presented in fig. 4 where T, was determined by HC2( T,) = 0. It will be noted that the present results (open circles) are in close agreement with the data reported by Maple [ 81. The sample containing 0.585 at% Gd did not show superconductivity at a temperature of 25 mK. Therefore, the critical concentration ncr at which superconductivity is totally destroyed is about 0.59 at%, in agreement with previous work [ 8 1. 4.3. The upper critical field HC2 The experimental results on the upper critical fields from T, down to about 40 mK are presented in fig.
Table II Summary of clean limit parameters for L.aA&
TC WI
P&L(O) (mT)
(-Icod&(T)/dT)Ir=Tc (mT/K)
RRR
3.31 f0.01
92.04 k 0.02
26.112kO.02
830+ 8.0
K
Ko
I
co
(rm)
(urn)
9.3
83.5
0.75f0.01
0.754
~~(4.2 K)
( - AdK ( T) /dT) I T=rc
(nfim)
(mT/K)
0.7385f0.07
AL(O) (nm) 64.2
25.54+0.01
v, ( lo5 m/s)
3.24
B. Vlcek et al. /Superconducting
204
alloys with paramagnetic impurities
5. As expected, the critical fields are lowered with increasing Gd concentration. An interesting effect is observed at low temperatures ( N 400 mK) where the upper critical fields start to decrease with decreasing temperature. This effect is most pronounced at intermediate concentrations and disappears again for n, 2 0.5 at%.
5. Comparison
between theory and experiment
The theoretical treatment of the upper critical field in superconducting alloys which is applied here to analyse the experimental data is based on a number of basic assumptions: - The spin of the impurity atom is treated classically which is certainly valid for Gd atoms with a spin of s=;. - The impurity concentration is very dilute and the impurities distributed are homogeneously throughout the sample. We can, therefore, neglect correlations between impurity sites and this allows us to assume a random impurity distribution. - The interaction of the conduction electrons is as-
Fig. 4. Critical temperature T, as a function of Gd content, (. ) data of ref. [ 81, ( o ) present results. (The solid line is drawn to guide the eye).
a0
60
-0
0.4
0.8
1.2
1.6
2.0
2.4
2.6
3.2
T(K) Fig. 5. Upper critical fields HE2 of LaAlz containing 0.0, eye).
0.1,
0.3, 0.41, 0.5 and 0.55 at% Gd. (The dashed lines are drawn
to
guide the
B. Vlceket al. /Superconducting alloyswithparamagneticimpurities
sumed to be weak and the scattering by the impurity atoms is treated in first order Born’s approximation. - Pauli spin paramagnetism is not included because of the very low upper critical fields ( S 100 mT). These points were discussed in detail in the first part of this series [ 281. Nevertheless, the theory presented therein has to be extended to include the anisotropy of the electron-phonon interaction and of the Fermi velocity. This necessity follows from the pronounced upward curvature of H,, ( T) near T, observed in clean LaA12 (fig. 7 (a) ) [ 421. This extension is analytically very tedious but straightforward and follows earlier work by Prohammer [ 43 ] closely. Hence, the theory contains quite a number of parameters which have to be determined unambiguously in order to give a conclusive interpretation. These parameters are: ,u*, the Coulomb pseudo-potential, a2F( o), the electron-phonon interaction spectral function, ( ur) , the Fermi surface average of the Fermi velocity, (a’), the mean square anisotropy of the electron-phonon interaction as a consequence of the separable ansatz of eq. (2), and ( b2), the mean square anisotropy of the Fermi velocity as a consequence of a similar separable ansatz [ 27 1. The impurities are described by three more parameters, t,, t_ and l’, (0) two of which are proportional to the impurity concentration n,. t+ = l/ (27rr,,), where the transport relaxation time T,, is related to the residual resistivity p,, by
(8)
205
ever, the generalized phonon density of states G(o) measured by Yeh et al. [ 411 is available and it is usually safe to assume that a’F(w) is proportional to G(o). The factor of proportionality is then chosen to give a certain value for the mass enhancement factor 1 co
(9) 0
If the sample were isotropic, the choice of il together with the experimental value of TC( = 3.29 K) would result in a unique value for ,u*, the Coulomb pseudopotential, as a result of the linearized Eliashberg equations, (As a rule of thumb ,u*z 0.1 which helps in making the “proper” choice for 2.) Our sample is clearly anisotropic and the choice for 2 for to be supplemented by a choice for (a’) in order to determine p* from the experimental value of T, and for a value of t, calculated from the experimental value of p,, using eq. (8) and S2p=$(z$)N(0)e2
(10)
and y=fxZN(0)(l+l)
(11)
where y is the experimentally determined zero temperature electronic specific heat (Sommerfeld constant) [ 3 11. The value ( V$) has already been evaluated in the previous chapter. Having determined p* from these assumptions, the full strong coupling theory of the upper critical field
and L$ is the “dressed” Drude plasma frequency. The parameter t_ = 1/ ( ~KZ,,,) is related to the magnetic relaxation time r, of the spin flip interaction between conduction electrons and impurity spins. Finally, V, (0) is the microscopic spin flip scattering potential which becomes important in the discussion of spin orientation effects. In the following, we will discuss the procedures employed to determine these eight parameters and the consequences for the interpretation of the results. -.__
0.00
5. I. The clean polycrystalline sample
The electron-phonon interaction spectral function, a’F(o), is not known for this material. How-
0.0
0.1
0.2
0.3
0.4
0.5
0.0
x (at%) Fig. 6. Inverse normal relaxation time t, and inverse magnetic relaxation time I_ as a function of Cd concentration.
206
B. Vlcek et al. /Superconducting
for anisotropic polycrystals is employed to fit the experimental Hc2 data over the whole temperature range (20 mKS TIT,). The parameter (b’) is used to model the curvature of HC2( T) at T, with all the other parameters already fixed by the first step. Of course, the first choice for L will not be the one which results in an optimal fit to the experimental data and several recursive steps are necessary, adjusting 5 (z+), (a’) and ( b2) until good agree-
alloys with paramagnetic impurities
ment between theory and experiment is established. The result of this procedure is as follows: LzO.55, p*=O.l, (vr)=3.23x105 m/s, (a2)=0.01 and ( b2) =0.16. As shown in fig. 7(a), excellent agreement between experiment and theory is achieved. These parameters characterize LaAlz as a rather weak coupling superconductor with little electron-phonon interaction anisotropy. The upward curvature of Hc2 ( T) at T, is mainly caused by the pronounced an-
(b) x- 0.1 at’/,Gd c
50-
E w
40-
‘.
‘\
r: I 9
‘.
‘\
30 20-
00. %%! %%
lo0.0
0.8
0.4
1.2
1.5
2.0
2.4
2.8
0.4
3.2
0.8
1.2
0.0
0.5
1.0
1.6
2.0
2.4
2.8
T (K)
T(K)
1.5
2.0
2.5
T(K)
Fig. 7. Upper critical dashed lines: theory.
fields He2 or ~a(x at’% Gd)A12 compared
to strong couplmg
calculations
including
anisotropy.
(o ) experiment,
B. Vlceket al. /Superconducting alloyswithparamagneticimpurities
isotropy of the Fermi velocity. Using these parameters, anisotropic Eliashberg theory predicts a zero temperature gap ratio 2A( 0) / (k,T,) of 3.6, which is in excellent agreement with the experimental value of 3.7. On the other hand, the analysis of section 4 would result in (0;) ‘I2 = 3.12 x 1O5 m/s, which is close enough to the value found by the full strong coupling analysis. (Note that
ln+
c =
tr,(l+A)+
7 1--u/(4)
(12)
where T,,, is the critical temperature of the clean sample and v(x) is the digamma function. The critical concentration at which superconductivity is destroyed, is found for a value of t- (fig. 6) corresponding to a critical concentration of 0.57 atoh Gd, which is just a bit smaller than the experimentally found critical concentration of about 0.59 at%. We can therefore conclude that the depression of the critical temperature with increasing concentrations of paramagnetic impurities is described reasonably
207
well by the weak coupling limit AG-theory. Strong coupling effects beyond the ( 1+A) renormalization are very diflicult to be distinguished by experiment except at concentrations very close to the critical one (cf. fig. 1 of ref. [28]). 5.3. The upper critical field As a first step, spin orientation is ignored and the full strong coupling theory of HC2 is applied to reproduce the experimental HC2( T) data for five La (x at% Gd)A12 sample with x=0.1, 0.3, 0.41, 0.5 and 0.55. All the parameters are already determined and there are no degrees of freedom left. The dashed lines in figs. 7(b-f ) represent the theoretical predictions and they prove that indeed the theory reproduces the experimental data exceptionally well in the temperature range 0.4 KI TI T,. The upward curvature of HC2( T) near T, for the clean polycrystals is as well reproduced as its smearing out with increasing Gd concentration. Nevertheless, there are also a few inconsistencies: (a) The pronounced deviation of the experimental data from the theoretical predictions for the sample with x= 0.1 at temperatures < 1 K is not understood (fig. 7(b)). (b) The sample withx=0.55 (fig. 7(f)) shows definite deviations to smaller values in the temperature range 0.2 KI T40.6 K, while the low and high temperature parts of the HC2( T) curve are perfectly reproduced by theory. In view of the otherwise excellent agreement between experiment and theory, further investigations of the upper critical field in samples with Gd concentrations close to the critical one could result in interesting new results. At temperatures below 0.4 K the samples with x=0.1, 0.3, 0.41 and also with 0.5 show a pronounced deviation from the theoretical predictions. The first three show a maximum in HCz( T) at about 0.4 K and a decrease in HC2 with further decreasing temperatures. This effect is most pronounced in the sample with x=0.3. The sample with x=0.5 does not really develop a maximum but, nevertheless, HC2( T) levels off at temperatures ~0.25 K. Similar behaviour was predicted by the theoretical analysis presented in part I of this series [ 28 ] if spin orientation effects at low temperatures are included. The simplest model describes the spin orientation as a function of the applied magnetic field B and the
208
B. Vlcek et al. /Superconducting
temperature T by a Boltzmann distribution for the average spin (S) “seen” by the conduction electrons:
(S)
=a
s dQS(Q)exp
{ -,,,.,I
(13)
N=
(14)
This effect introduces a new parameter, Vi (0), the microscopic scattering potential of the paramagnetic impurities (see eq. (29 ) of ref. [ 28 ] ). This parameter is used to fit the low temperature &( T) data of the sample with x=0.3 as is shown in fig. 8. We find I’, (0) = 0.47 meV. HC2( T) data of the two other samples with x= 0.1 and 0.4 1 are then calculated with this potential. (No other fitting parameter is avail-
alloys with paramagnetic impurities
able for this additional calculation! ) It becomes obvious from the results presented in fig. 8, that spin orientation is indeed the explanation for the low temperature behaviour of H,,( T) in these samples. But a comparison with the x=0.41 sample also reveals that our simple model for the spin orientation obviously overestimates the amount of spin orientation at very low temperatures ( < 100 mK) while the onset of spin orientation is predicted properly in all three cases. With higher impurity concentrations, the external magnetic fields are already so small that spin orientation effects can only be observed at very low temperatures, if at all. The sample with x=0.5 could be indicative for this behaviour. (A theoretical analysis of these data was not feasible because of prohibitively long computing times. ) 5.4. Strong coupling effects
0
L CDoao 0
a**.
60
0 0 0
0
l
0 0
-
l
l
experiment strong-anirotropic + spin
orientation
2.0
1.0
T (K)
Fig. 8. Upper critical fields of La(x at% Gd)A12, x=0.1,0.3 and 0.4 1 compared to strong coupling theory including spin orientation. (o ) experiment, (. ) strong coupling theory without spin orientation term, and solid lines strong coupling theory with spin orientation.
The theoretical analysis of ref. [ 281 predicted rather pronounced strong coupling effects for the upper critical field, especially in samples with rather high concentrations of paramagnetic impurities and at low temperatures. Thus, in this section we will compare the predictions of the full strong coupling theory with results found from a ( 1 +A) renormalized weak coupling theory (eqs. (33-37) of ref. [ 281). In this comparison T,, t+, A and (z+) are the same; only t_ has to be adjusted using eq. ( 12) to give the correct T,. Figure 9(a) presents the results of such a comparison for the clean (x = 0 ) sample and for the sample with x=0.3. Whereas the full strong coupling theory (solid lines) reproduces the experimental data (open circles) almost perfectly, the predictions of the weak coupling theory (dashed lines) are well below the experimental data. This discrepancy becomes less pronounced for the sample with x=0.3. Finally, fig. 9(b) shows that for the sample with x=0.5 the opposite holds, namely that the weak coupling theory predicts upper critical fields which are above the experimental data and the results of the full strong coupling theory. This behaviour is in complete agreement with theoretical expectations formulated in ref. [ 28 ] and is based on the fact that paramagnetic impurities become more effective in their pair breaking abilities with decreasing temperatures. Since the magnetic re-
B. Vlceket al. /Superconducting alloyswithparamagneticimpurities
-
full
-----
weak coupling
*trong
-.---- weak
1.6
209
(4
coupling (1+X)
coupling( x=0)
2.0
2.4
2.8
3.;
T(K)
T (K)
laxation time is overestimated by the weak coupling theory (smaller values for t_ for the same Tc’s), it results, with increasing impurity concentrations, in higher upper critical fields relative to the predictions of the full strong coupling theory and therefore a “crossover” has to be observed. This effect is clearly demonstrated in fig. 9 and is an indication of strong coupling effects which go beyond the (1 +A) renormalization even for the rather weak coupling material LaA12. In order to emphasize the importance of the ( 1+,I) renormalization in the weak coupling theory, results for A= 0 (dashed dotted lines) are also included in fig. 9(a).
Fig. 9. (a) HCzdata of La(x at% Gd)A12 (x=0.0 and 0.3) compared to weak coupling calculations (dashed lines), strong coupling calculations including anisotropy (solid lines) and weak coupling calculations for A= 0 (dash-dotted lines ) (b ) H,, data of La(x at% Gd)A12 (x=0.5 at%) compared to weak coupling calculations (dashed line) and strong coupling calculations (solid line).
6. Conclusions Magnetization and AC-susceptibility measurements were made on single- and polycrystalline samples of LaAlz doped with Gd in the concentration range 0 I nr IO.6 at%. The results may be summarized as follows. 1) The resistivity measurements show that Gd is dissolved homogeneously in the material. 2) The T, values extrapolated from the Hc2( T) measurements are found to be in reasonable agreement with data published previously. From our measurements we conclude that the
210
B. Vlcek et al. /Superconducting
critical concentration is about 0.59 at% Gd. 3 ) From an evaluation of superconducting parameters we conclude that LaA12 is a rather weak coupling type-II superconductor. Due to the low GGparameter, a phase transition between typeII/ 1 and type-1112 superconductivity was found at T*=3.262 K. 4) The upper critical fields Hc2 are strongly depressed by the presence of magnetic impurities. At low temperatures ( N 400 mK) and intermediate impurity concentrations, peaks of H,, followed by a downward curvature towards T=O are observed. 5 ) The clean limit data on H,, could be explained quantitatively in terms of anisotropic Eliashberg theory using the anisotropy parameters (a*) = 0.0 1 for the electron-phonon coupling and (b’) =O. 16 for the Fermi velocity. Excellent agreement between the strong coupling theory and experiment was also achieved for the samples containing Gd impurities if spin orientation of the impurity atoms was included to explain the peaks of Hc2 at intermediate temperatures and Gd concentrations. 6) While the effects of strong coupling and anisotropy on the depression of T, are rather small, significant effects of strong coupling have been found for the upper critical field. As predicted by theory and confirmed by our measurements, the weak coupling theory underestimates the influence of Gd on Hc2 up to concentrations of about tncr. At higher concentrations the upper critical field is overestimated by the weak coupling theory.
Acknowledgements We wish to thank Mr. H. Niedermaier for his continuous and dedicated help with the operation of the dilution refrigerator, Prof. F. Steglich ( Darmstadt ) for providing the LaAlz single crystal and Mr. H. Beyss and co-workers (Jtilich ) for their help with sample preparation. This work was supported in part by Fonds zur Fijrderung der Wissenschaftlichen Forschung, Wien, under contract No. 5353 and by the Office of Naval Research under grant No. NO0 14-89J-1088.
alloys with paramagnetic impurities
References [ 1] B.T. Matthias,
H. Suhl and E. Corenzwit, Phys. Rev. Lett. 1 (1958) 92. [2] P.W. Anderson, J. Phys. Chem. Solids 1I (1959) 26. [ 31 A.A. Abrikosov and L.P. Gor’kov, Sov. Phys.-JEPT 9 ( 1958) 220. [4] A.A. Abrikosov and L.P. Gor’kov, Sov. Phys.-JETP 12 (1961) 1243. [ 51H. Shiba, Progr. Theor. Phys. 40 ( 1968) 435. [6] A.I. Rusinov, Sov. Phys.-JETP Lett. 9 ( 1969) 85. [7] A.I. Rusinov, Sov. Phys.-JETP 29 (1969) 1101. [8] M.B. Maple, Ph.D. Thesis, University of California, San Diego (1969). [ 91 S. Skalsky, 0. Betbeder-Matibet and P.R. Weiss, Phys. Rev. 136 (1964) A1500. [ lo] W.R. Decker and D.K. Finnemore, Phys. Rev. 172 ( 1968 ) 430. [ 111 E. Schachinger, J.M. Daams and J.P. Carbotte, Phys. Rev. B22 (1980) 3194. [ 121 D. Markovitz and L.P. Kadanoff, Phys. Rev. 131 ( 1963) 563. [ 131 J.R. Clem, Ann. Phys. (NY) 40 (1966) 268. [ 141 D.K. Finnemore and D.E. Mapother, Phys. Rev. 140 ( 1965) A507. [ 151 D. Gubser, Phys. Rev. B6 (1972) 827. [ 161 A.J. Bennet, Phys. Rev. 140 (1965) A1902. [ 171 C.R. Leavens and J.P. Carbotte, Ann. Phys. (NY) 70 (1965) 338. [ 181 J.P. Carbotte, in: Anisotropy Effects in Superconductors, ed. H.W. Weber (Plenum, New York, 1977) p. 183. [ 191 J.M. Daams and J.P. Carbotte, J. Low Temp. Phys. 43 (1981) 263. [ 201 L. Niel, N. Giesinger, H.W. Weber and E. Schachinger, Phys. Rev. B32 (1985) 2976. [ 211 N.R. Werthamer, E. Helfand and P.C. Hohenberg, Phys. Rev. 158 (1967) 415. [22] D. Rainer and G. Bergmann, J. Low Temp. Phys. 14 ( 1974) 501. [23] M. Schossmann and E. Schachinger, Phys. Rev. B33 (1986) 6123. [24] F. Marsiglio, M. Schossmann, E. Schachinger and J.P. Carbotte, Phys. Rev. B35 (1987) 3226. [25] E. Schachinger, M. Prohammer, E. Seidl and H.W. Weber, Physica C 153 (1988) 247. [ 26 ] H.W. Weber and E. Schachinger, Helv. Phys. Acta 6 1 ( 1988 ) 478.
[ 271 M. Prohammer
and E. Schachinger,
Phys. Rev. B36 ( 1987)
8353. [ 281 W. Pint and E. Schachinger, Physica C 159 ( 1989) 33. [29] J.H. Moeser, Ph.D. Thesis, Universitat Kijln, 1976, unpublished. [30] R.E. Hungsberg
and K.A. Gschneidner
Jr., J. Phys. Chem.
Solids 33 (1972) 401. [31]F.Steglich,Z.Phys.B23 (1976) 331. [ 321 J.H. Moeser, F. Steglich, C. Leser and E. Mtiller-Hartmann. Z. Phys. B 25 (1976) 339.
B. Vlcek et al. /Superconducting [33] G. Beuermann, Z. Phys. B 28 (1977) 207. [ 341 H. v. tihneysen, J.L. Tholence and F. Steglich, Z. Phys. B 29 (1978) 319. [35] G. Riblet and K. Winzer, Solid State Commun. 11 ( 1972) 175. [ 361 B. Vlcek, E. Seidl and H.W. Weber, Jpn. J. Appl. Phys. 263 (1987) 967. [37] P. Hahn and H.W. Weber, Cryogenics 23 (1983) 87. [38] L.P. Gor’kov, Sov. Phys.-JETP 10 (1960) 998; B.B. Goodman, IBM J. Res. Dev. 6 ( 1962) 63. [ 39 ] H.W. Weber, E. Seidl, M. Botlo, C. Laa, E. Mayerhofer, EM. Sauerzopf, R. Schalk, H.P. Wiesinger and J. Rammer, PhysicaC 161 (1989) 272.
alloys with paramagnetic impurities
211
[40] K. Maki, Phys. Rev. Lett. 23 (1969) 1223. [41] C.T. Yeh, W. Reichart, B. Renker, N. Nticker and M. Loewenhaupt, J. Phys. (Paris) C6 (1981) 371. [42] P.C. Hohenberg and N.R. Werthamer, Phys. Rev. 153 (1967) 493. [43] M. Prohammer, Solid State Commun. 67 (1988) 293. [ 441 J.M. Daams, E. Schachinger and J.P. Carbotte, J. Low Temp. Phys. 42 ( 198 I ) 69.