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Anomalous specific heat in La : Ce alloys
Solid State Communications, Vol. 15, pp. 1633—1638, 1974.
Pergamon Press:
Printed in Great Britain
ANOMALOUS SPECIFIC HEAT IN La : Ce ALLOYS* H.V. Culbert Argonne National Laboratory, Argonne, Illinois 60439, U.S.A. and A.S. Edeistein University of Illinois, Chicago, Illinois 60680, U.S.A. and Argonne ,National Laboratory, Argonne, Illinois 60439, U.S.A. (Received 26 July 1974 by H. Suhi)
The magnetic contribution to the specific heat in La : Ce alloys shows a broad anomaly at 1 .2°Kwhich is weakly dependent on the concentration. This anomaly, which accountsfor a substantial portion of the entropy of a doublet, cannot be attributed to either usual magnetic ordering or to a dilute alloy Kondo effect, but may represent the behavior of a concentrated Kondo system.
THE PROPERTiES of La : Ce alloys have been studied extensively in recent years. These alloys permit one to study the competition between the Kondo effect and superconductivity,13 the Kondo effect at high concentrations,4 and magnetic ordering.5’6 Earlier specific heat measurements2’6 were in two very different concentration ranges. Roberts and Lock’s measurements6 were at high concentrations where they found anomalies due to magnetic ordering. Presumably there should also be an anomaly at approximately 23 K due to crystal field effects,7 but their measurements do not extend up to this temperature. Often there are two anomalies at low temperatures. One of these can be attributed to magnetic ordering but it is not at all clear why there should be a second anomaly. One possible explanation8 concerns the fact that for the double hexagonal close packed (d.h.c.p.) phase there
anomalies might represent magnetic ordering on the two different sites. We think this explanation unlikely because of the small number of hexagonal sites in our samples and the appearance of only one ordering anomaly in susceptibility measurements.9 The other previous measurements2 have been made at low concentrations, i.e., for concentrations x ~ 0.01. These measurements showed the existence of 7T + aT~terms (1 ~ n ~ 2) in the specific heat while the samples were in the superconducting state. The magnitude of y, which was linear in x for 0.0025> x > 0.01 yields a value for the density of states at the Fermi level in good agreement with that determined by electron tunneling measurements.1 For La Ce alloys y = 0 for x < 0.001. This is in contrast with some recent results’°on (LaCe) Al 2, for which cxx down to their lowest concentrations. ,
are two types of sites, one with cubic symmetry and the other with hexagonal symmetry. The two *
The prime purpose of the present measurements is to examine the intermediate concentration region where long range magnetic ordering effects and
Based on work performed under the auspices of the U.S. Atomic Energy Commission, 1633
1634
ANOMALOUS SPECIFIC HEAT IN La : Ce ALLOYS
10,000
—
Vol. 15, No. 10
______
Ce~La,..~ Cp DATA 5,000
x~0.20 x~0.I0 x~0.07 • x~0.04 o x~0.0275 +
o
2,000 1,000-
+ * ~,
~
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-
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5
6
7
8
9
10
II
T(K) FIG.
1. Specific heat at constant pressure for Ce~La1_~ (x = 0.0275, 0.04,0.07,0.10,0.20) vs temperature.
superconductivity are not present. The concentrations employed were from 1 .25—20 at.%. At the lowest concentrations it was necessary to employ a magnetic field to quench superconductivity, while at the highest concentration, 20 at.%, long range ordering effects are present. It with was hoped that we might see interaction an anomaly associated the Kondo state and/or effects. What we have observed is an anomaly which does not fit the usual behavior observed either for the Kondo state or for interaction effects, measurements were made the conventionalThe heat pulse-drift method in anusing apparatus that has been previously described.11 The samples were as cast arc-melted alloys. They have both the face-centered cubic (f.c.c.) and the double hexagonal close packed (d.h.c.p.) phases present. The low concentration alloys are approximately 90 per cent f.c.c. This number is determined by the ratio of the magnitudes of the discontinuities in the specific heat at the superconducting transitions corresponding to the f.c.c. and d.h.c.p. phases. The higher concentration alloys may contain as much as 30 per cent of the d.h.c.p. phase, since neutron measurements on a 10 at.% sample showed this amount of the d.h.c.p. phase. Despite the obvious undesirability of mixed phase samples, we do
not believe our results would change qualitatively if we had single phase samples. In Fig. 1 the measured specific heat is shown as a function of temperature for several samples. From the specific heat we cornpute the incremental specific heat z~C by substracting 2’~.In computing these values ofA we ~ ~ T assume that the phase mixture of our samples is 90 per cent f.c.c. and 10 per cent d.h.c.p. The results of our analysis are only weakly dependent on this assumption. For T~4K,the values ofA 1 and A3 12 are obtained from of those of Johnson and are Finnemore for each phase La and higher Ar’s neglected. For T~’4K the appropriate linear combination of the data for each phase obtained by Johnson and Finnemore was fitted to a polynomial up to A 7. In Fig. 2 we plot iXC/x where x is the impurity concentration. One sees that there is a broad anomaly at 1 .2 K which is approximately concentration independent. The position and amplitude of this peak shifts weakly with increasing concentration. This description is appropriate for all of the samples with the exception of the 20 at.% Ce sample, which is the most concentrated that we have investigated. For the 20 at.% sample there is structure at approximately 1.4
Vol. 15, No. 10
ANOMALOUS SPECIFIC HEAT IN La : Ce ALLOYS
6
6
14
0~
2
••• • •
•
•
—
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0
:
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~0c
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I
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3.0
41.0
5.0
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0
0
2LEVELSCHOTKYCp
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0
FIG. 3. Comparison of the excess specific heat for Ce0,07 La0,93 vs temperature with that predicted for
o
X007
2 C 0
1635
c I
I
I
I
2
3
4
5
T(K)
FIG. 2. Excess specific heat after the lattice and conduction electron contribution have been subtracted for Ce~La 1_~ (x = 0.0275,0.04,0.07,0.10, 0.20) vs temperature. and 2.6 K. The structure at 1.4 K is due to magnetic 9 The origin of the added feature at 2.6 K is ordering. less clear, In the remainder of this paper we shall analyze the broad anomaly. The two most obvious possible explanations for it are that it is a short range magnetic ordering anomaly of the kind described by Klein, Brout, and Marshall~or that it is a Kondo anomaly. The above interaction effect theories, which are really only applicable at more dilute concentrations, predict that the temperature at which the maximum in the specific heat should occur is proportional to the impurity concentration. This has been observed in the case of Zn : Mn14 but clearly is in disagreement with the results of Fig. 2. Further, with the exception of the 20 at.% sample for which interaction effects are obviously present, the susceptibility does not show interaction effects for T> 1K. In fact the susceptibility’5 x is of the form x = xF(T, H) where the function F is independent of the concentration x. It may be, as discussed below, that interaction effects manifest themselves in a more subtle way in these concentrated alloys. There is a maximum in the resistivity4’16’17 which is strongly concentration dependent, but the unusual behavior of the magneto-
a two discrete levels (E = 3.16 K) and a Kondo anomaly. 16 and the behavior of the susceptibility resistance described above indicate that one should be cautious in applying a simple picture of interaction effects. In at least some of the cases where interactions are believed present, they manifest themselves in the 15 For susceptibility at very low temperatures. example, Ce shows magnetic at is 0.25 K. In 0,1La0,9 short, the evidence that ordering the anomaly due to interaction effects is at best ambiguous. The other obvious possibility, that the anomaly is due to the Kondo effect, also is not without its difficulties. In the Kondo effect one expects a broad anomaly of the type shown in Fig. 2. Such an anomaly has probably been observed in the related system (La,Ce)Al 1°In this system the Kondo 2. effect and superconductivity also are competing effects but the peak in the normal state specific heat occurs at a much lower temperature T~0.14. In this case, the anomaly either is partially suppressed or shifted to a lower temperature by the transition into the superconducting state and is only weakly affected by the presence of small magnetic fields. As discussed below, the anomaly in the present case is strongly attenuated in the superconducting state. In Fig. 3 we compare the specific heat of a 7 at.% sample with two theoretical predictions, one based on two discrete energy levels and the other that for a dilute ahoy Kondo anomaly. For the two level system the number of states and the level spacing = 3.16 K have been adjusted to fit the data at the
1636
ANOMALOUS SPECIFIC HEAT IN La : Ce ALLOYS
6000 R2n2~
~
4004-
~
3000
~
2OOO~-
—--- ---
--
Vol. 15, No. 10
T = 15 K show no sign of a CF anomaly. This suggests that the ~F splitting is larger than ~50K. The
-—--—-—---——--—-----~~-
entropy change ~S for 0.9< T< 15 decreases rapidly forx <0.02 where the alloys are superconducting. Presumably the extra entropy to make R ln 2 for all samples is found below 0.9 K. With the exception of the entropy change associated with the consensation into the crystal field ground state that occurs for kT ECF, at most a 5 per cent error is
C
made by not including data for T> 15. 000
~
I
In the case of the only superconducting alloy for which the change in Cwas measured on going into the
22
superconducting state, our measurements agree within 19 and experimental errorof with measurements hence give a value 1 .1 previous for TK. This value is higher
Clot. %Ce)
FIG. 4. Entropy change between 0.9 and 15 K associated with the excess specific heat for Ce~La 1_~ as a function of x.
peak. It is seen that the two level system prediction is too narrow to fit the data adequately. Taking three levels does not improve the fit. On the other18hand, is as seen in Fig. a dilute alloyanomaly. Kondo anomaly broader than3,the observed The theoretical
.
7
than previous values and is still so low as to make one further doubt that the anomaly shown in Fig. 2 is a dilute alloy Kondo anomaly.
specific heat curve shown in Fig. 3 was derived from a phenomenological model18 for the density of states. In the modelN(c) cx (E2 + ~2)_~ with j3 0.8. This theoretical prediction provides an adequate fit to data on Cu : Fe and is in agreement with the Bloomfield—Hamann calculation at high temperatures. Further, in contrast to the results shown in Fig. 2, if the peak were a dilute alloy Kondo anomaly one would expect that tIC ax.
Also as discussed earlier the anomaly is probably not associated with x. magnetic it also does not scale with It shiftsordering, to lower but temperatures and/or decreases in amplitude at low concentrations where the sample is superconducting. It is interesting to try to associate the b/T2 specific heat term seen in dilute superconducting alloys2 with the high temperature side of the anomaly after it has shifted below 1 K. Unfortunately, it is somewhat difficult to follow the anomaly continuously as a function of x since for x 0.02 the anomaly and the superconducting T~ occur at the same approximate temperature. If one suppresses superconductivity by applying a magnetic
In order to gain some insight into the nature of the anomaly we have computed the entropy change associated with the integral of tIC/xT from T°~° 0.9-15 K. For x = 0.0125, tIS was computed from the integral of (CA — CLa)/XT where CA is the superconducting alloy specific heat and CLa is the normal state specific heat of La. This entropy change LIS per Ce atom is shown in Fig. 4 as a function of concentration. One sees that as x —~ 0.2, LIS approaches R ln 2, i.e., the value for a doublet. This is consistent with the view that the anomaly is due to the Ce and that the cubic crystalline field has split the levels into a ground state doublet and a quartet which lies an energy LIECF above the doublet. Presumably the remainder of the spin entropy R ln 6 is recovered for kT> LIECF. Our measurements which extend up to
field, one also affects the anomaly. It may be possible to apply a weak field which will not affect the anomaly very much and yet be strong enough to destroy superconductivity. Our preliminary measurements of the magnetic field dependence of the specific heat for n = 0.0275 at a fixed temperature show that the specific heat is increased monotonically by the application of magnetic fields up to our highest field, 8 kG. At a fixed field the increase in the specific heat caused by the application of the field, i.e., C(H) — C(O), increases monotonically with decreasing temperature. This is consistent with the fact that at this concentration the magnetization shows no sign of magnetic ordering above 0.6K. The b/T2 term is a strong function of x. It depends onx’~where n = 2 or 3. This strong dependence on x certainly suggests that interaction effects are important. Distributions
Vol. 15, No. 10
ANOMALOUS SPECIFIC HEAT IN La : Ce ALLOYS
of local fields such as those considered by Kline, Brout and Marshall could be important. Because of the absence of interaction effects in the susceptibility at higher x, it is perhaps more likely that the anomaly is associated with an unusual sort of phase transition, The decreasing resistivity is another indication of the transition. A similar more dramatic effect has been seen in the resistivity of CeA132°The essential cornmon features of the transitions that are observed for La : Ce alloys and CeAl 3 are:in(1) fraction of the spin entropy is removed thea large transitions without magnetic ordering; (2) a gradual, approximately logarithmic, resistivity decrease occurs; (3) in the case of the alloys the transition depends on the concentration of Ce. This is also the case21 in La 1_~Ce~ Al3. These properties suggest that what has been observed is a new type of electronic transition. This transition might prevent reentrant superconducting 22 in (La, Ce)A1behavior such as has been observed 2 from 23 occurring in La : Ce alloys. A more important matter is to understand the nature of the transition. It is quite possible that it is a Kondo transition but —
.
1637
because the system is not dilute the specific heat anomaly is not the usual Kondo anomaly. If one attempts to assign the temperature at which the peak in the specific heat occurs as approximately one third of an effective Kondo temperature, then the effective Kondo temperature shifts to lower temperatures with decreasing concentration. This suggestion of a high, concentration dependent, effective Kondo temperature is not what one would conclude from 7 the susceptibility measurements. A gives theoretical fit to the susceptibility measurements a concentration independent value of Tk < 1K. One may not be able to describe concentrated Kondo systems by a single parameter such as TK. Further work at lower temperatures is planned to clarify the nature of this transition.
Acknowledgements The authors wish to thank D.K. Finnemore for supplying the specific heat data on both phases of pure La which we used in computing the excess specific heat and to acknowledge the helpful assistance of Mr. Z. Sungaila in the laboratory. —
REFERENCES I. 2.
EDELSTEIN A.S.,Phys. Rev. 180, 505 (1969);Phys. Rev. Lett. 19, 1184 (1967). CULBERT H.V. and EDELSTEIN A.S., Solid State Commun. 8,445 (1970).
3.
CHAIKIN P.M. and MIHALISIN T.W.,Phys. Rev. B6, 839 (1972);Solid State Commun. 10,465 (1972); SUGAWARA T. and EGUCHI H.,J. Phys. Soc. Japan 23,965 (1967).
4.
EDELSTEIN A.S.,Phys. Lett. 27A, 614 (1968). ELLIOTT R.O., HILL H.H. and MINER W.N.,Phys. Status Solidi 32, 609 (1969).
5.
6. 7.
ROBERTS L.M. and LOCK J.M.,Phil. Mag. 2,811(1957). DEGENNARO S. and BORCHI E. [Phys.Rev. Lett. 30, 377 (1973)] have fitted susceptibility measurements to their theory which includes both the Kondo effect and a cubic crystalline field to obtain a splitting of approximately 60 K between a doublet and a higher energy quarter.
8.
GSCHNEIDNER K.A., Jr. (private communication).
9.
EDELSTEIN A.S. (unpublished data).
10.
ARMBRUSTER H., LOHNEYSEN H.V., RIBLET G. and STEGLICH F.,Solfd State Commun. 14, 15 (1974); BADER S.D., PHILLIPS N.E. and MAPLE M.B. (private communication); LUENGO C.A., MAPLE M.B. and FERTIG W.A., Solid State Commun. 11, 1445 (1972).
11.
CULBERT H.V., FARRELL D.E. and CHANDRASEKHAR B.S.,Phys. Rev. B 3,794 (1971).
12.
JOHNSON D.L. and FINNEMORE D.K., Phys. Rev. 158, 376 (1967); FINNEMORE D.K. (private communication). MARSHALL W.,Phys. Rev. 118, 1519 (1960); KLEIN M.W. and BROUT R.,Phys. Rev. 132, 2412 (1963); KLEIN M.W.,Phys. Rev. 136, A1156 (1964); 173, 552 (1968); KLEIN M.W. and SHEN L.,Phys. Rev. B5, 1174 (1972).
13.
1638
ANOMALOUS SPECIFIC HEAT IN La: Ce ALLOYS
Vol. 15, No. 10
14.
SMITH F.W. (to be published in the Phys. Rev.).
15.
EDELSTEIN A.S.,Phys. Rev. Lett. 20, 1348 (1968); EDELSTEIN A.S., WINDMILLER L.R., KETTERSON J.B., CRABTREE G.W. and BOWEN S.P.,Phys. Rev. Lett. 26, 516 (1971).
16.
WOLLAN J.J. and FINNEMORE D.K.,Phys. Rev. 4, 2996 (1971).
17.
SUGAWARA T., YAMASE I. and SOGA R.,J. Phys. Soc. Japan 20, 618 (1965); SUGAWARA T. and EGUCHI H., ibid., 21, 725 (1966); 26, 1322 (1969). EDELSTEIN A.S.,Phys. Rev. Lett. 29, 1522 (1972); BLOOMFIELD P.E. and HAMANN D.R.,Phys. Rev. 164,856 (1967). LUENGO CA., HUBER J.G., MAPLE M.B. and ROTH M.,Phys. Rev. Lett. 32, 54(1974).
18. 19. 20. 21. 22. 23.
EDELSTEIN A.S., TRANCHITA C.J., McMASTERS O.D. and GSCHNEIDNER K.A., Jr. (to be published in Solid State Commun.). BUSCHOW K.H.S., VAN DAALH.J., MARANZANA F.E. and VAN AKEN P.B.,Phys. Rev. B 3, 1662 (1971). RIBLET G. and WINZER K., Solid State Commun. 9, 1663 (1971); MAPLE M.B., FERTIG W.A., MOTA A.C., DELONG L.E., WOHLLEBEN D. and FITZGERALD R.,Solid State Commun. 11,829(1972). UMLAUF E., SCHNEIDER J., MEIER R. and KREUZER H., J. Low Temp. Phys. 5, 191(1971).
La contribution magnétique a la chaleur specifique des alhiages La : Ce montre une anomalie d’une largeur considerable a une temperature de l.2°K;la dependence de la concentration est faible. Cette anomalie qui est responsable pour une, partie importante de l’entropie dun doublet, ne peut pas etre atribuée ni a un ordre magnétique ordinaire ni a l’effet Kondo des alliages diluées, mais II peut représenter les proprietés d’un système concentré de Kondo.
Pergamon Press:
Printed in Great Britain
ANOMALOUS SPECIFIC HEAT IN La : Ce ALLOYS* H.V. Culbert Argonne National Laboratory, Argonne, Illinois 60439, U.S.A. and A.S. Edeistein University of Illinois, Chicago, Illinois 60680, U.S.A. and Argonne ,National Laboratory, Argonne, Illinois 60439, U.S.A. (Received 26 July 1974 by H. Suhi)
The magnetic contribution to the specific heat in La : Ce alloys shows a broad anomaly at 1 .2°Kwhich is weakly dependent on the concentration. This anomaly, which accountsfor a substantial portion of the entropy of a doublet, cannot be attributed to either usual magnetic ordering or to a dilute alloy Kondo effect, but may represent the behavior of a concentrated Kondo system.
THE PROPERTiES of La : Ce alloys have been studied extensively in recent years. These alloys permit one to study the competition between the Kondo effect and superconductivity,13 the Kondo effect at high concentrations,4 and magnetic ordering.5’6 Earlier specific heat measurements2’6 were in two very different concentration ranges. Roberts and Lock’s measurements6 were at high concentrations where they found anomalies due to magnetic ordering. Presumably there should also be an anomaly at approximately 23 K due to crystal field effects,7 but their measurements do not extend up to this temperature. Often there are two anomalies at low temperatures. One of these can be attributed to magnetic ordering but it is not at all clear why there should be a second anomaly. One possible explanation8 concerns the fact that for the double hexagonal close packed (d.h.c.p.) phase there
anomalies might represent magnetic ordering on the two different sites. We think this explanation unlikely because of the small number of hexagonal sites in our samples and the appearance of only one ordering anomaly in susceptibility measurements.9 The other previous measurements2 have been made at low concentrations, i.e., for concentrations x ~ 0.01. These measurements showed the existence of 7T + aT~terms (1 ~ n ~ 2) in the specific heat while the samples were in the superconducting state. The magnitude of y, which was linear in x for 0.0025> x > 0.01 yields a value for the density of states at the Fermi level in good agreement with that determined by electron tunneling measurements.1 For La Ce alloys y = 0 for x < 0.001. This is in contrast with some recent results’°on (LaCe) Al 2, for which cxx down to their lowest concentrations. ,
are two types of sites, one with cubic symmetry and the other with hexagonal symmetry. The two *
The prime purpose of the present measurements is to examine the intermediate concentration region where long range magnetic ordering effects and
Based on work performed under the auspices of the U.S. Atomic Energy Commission, 1633
1634
ANOMALOUS SPECIFIC HEAT IN La : Ce ALLOYS
10,000
—
Vol. 15, No. 10
______
Ce~La,..~ Cp DATA 5,000
x~0.20 x~0.I0 x~0.07 • x~0.04 o x~0.0275 +
o
2,000 1,000-
+ * ~,
~
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-
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6
7
8
9
10
II
T(K) FIG.
1. Specific heat at constant pressure for Ce~La1_~ (x = 0.0275, 0.04,0.07,0.10,0.20) vs temperature.
superconductivity are not present. The concentrations employed were from 1 .25—20 at.%. At the lowest concentrations it was necessary to employ a magnetic field to quench superconductivity, while at the highest concentration, 20 at.%, long range ordering effects are present. It with was hoped that we might see interaction an anomaly associated the Kondo state and/or effects. What we have observed is an anomaly which does not fit the usual behavior observed either for the Kondo state or for interaction effects, measurements were made the conventionalThe heat pulse-drift method in anusing apparatus that has been previously described.11 The samples were as cast arc-melted alloys. They have both the face-centered cubic (f.c.c.) and the double hexagonal close packed (d.h.c.p.) phases present. The low concentration alloys are approximately 90 per cent f.c.c. This number is determined by the ratio of the magnitudes of the discontinuities in the specific heat at the superconducting transitions corresponding to the f.c.c. and d.h.c.p. phases. The higher concentration alloys may contain as much as 30 per cent of the d.h.c.p. phase, since neutron measurements on a 10 at.% sample showed this amount of the d.h.c.p. phase. Despite the obvious undesirability of mixed phase samples, we do
not believe our results would change qualitatively if we had single phase samples. In Fig. 1 the measured specific heat is shown as a function of temperature for several samples. From the specific heat we cornpute the incremental specific heat z~C by substracting 2’~.In computing these values ofA we ~ ~ T assume that the phase mixture of our samples is 90 per cent f.c.c. and 10 per cent d.h.c.p. The results of our analysis are only weakly dependent on this assumption. For T~4K,the values ofA 1 and A3 12 are obtained from of those of Johnson and are Finnemore for each phase La and higher Ar’s neglected. For T~’4K the appropriate linear combination of the data for each phase obtained by Johnson and Finnemore was fitted to a polynomial up to A 7. In Fig. 2 we plot iXC/x where x is the impurity concentration. One sees that there is a broad anomaly at 1 .2 K which is approximately concentration independent. The position and amplitude of this peak shifts weakly with increasing concentration. This description is appropriate for all of the samples with the exception of the 20 at.% Ce sample, which is the most concentrated that we have investigated. For the 20 at.% sample there is structure at approximately 1.4
Vol. 15, No. 10
ANOMALOUS SPECIFIC HEAT IN La : Ce ALLOYS
6
6
14
0~
2
••• • •
•
•
—
0 0
0
:
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3.0
41.0
5.0
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0
FIG. 3. Comparison of the excess specific heat for Ce0,07 La0,93 vs temperature with that predicted for
o
X007
2 C 0
1635
c I
I
I
I
2
3
4
5
T(K)
FIG. 2. Excess specific heat after the lattice and conduction electron contribution have been subtracted for Ce~La 1_~ (x = 0.0275,0.04,0.07,0.10, 0.20) vs temperature. and 2.6 K. The structure at 1.4 K is due to magnetic 9 The origin of the added feature at 2.6 K is ordering. less clear, In the remainder of this paper we shall analyze the broad anomaly. The two most obvious possible explanations for it are that it is a short range magnetic ordering anomaly of the kind described by Klein, Brout, and Marshall~or that it is a Kondo anomaly. The above interaction effect theories, which are really only applicable at more dilute concentrations, predict that the temperature at which the maximum in the specific heat should occur is proportional to the impurity concentration. This has been observed in the case of Zn : Mn14 but clearly is in disagreement with the results of Fig. 2. Further, with the exception of the 20 at.% sample for which interaction effects are obviously present, the susceptibility does not show interaction effects for T> 1K. In fact the susceptibility’5 x is of the form x = xF(T, H) where the function F is independent of the concentration x. It may be, as discussed below, that interaction effects manifest themselves in a more subtle way in these concentrated alloys. There is a maximum in the resistivity4’16’17 which is strongly concentration dependent, but the unusual behavior of the magneto-
a two discrete levels (E = 3.16 K) and a Kondo anomaly. 16 and the behavior of the susceptibility resistance described above indicate that one should be cautious in applying a simple picture of interaction effects. In at least some of the cases where interactions are believed present, they manifest themselves in the 15 For susceptibility at very low temperatures. example, Ce shows magnetic at is 0.25 K. In 0,1La0,9 short, the evidence that ordering the anomaly due to interaction effects is at best ambiguous. The other obvious possibility, that the anomaly is due to the Kondo effect, also is not without its difficulties. In the Kondo effect one expects a broad anomaly of the type shown in Fig. 2. Such an anomaly has probably been observed in the related system (La,Ce)Al 1°In this system the Kondo 2. effect and superconductivity also are competing effects but the peak in the normal state specific heat occurs at a much lower temperature T~0.14. In this case, the anomaly either is partially suppressed or shifted to a lower temperature by the transition into the superconducting state and is only weakly affected by the presence of small magnetic fields. As discussed below, the anomaly in the present case is strongly attenuated in the superconducting state. In Fig. 3 we compare the specific heat of a 7 at.% sample with two theoretical predictions, one based on two discrete energy levels and the other that for a dilute ahoy Kondo anomaly. For the two level system the number of states and the level spacing = 3.16 K have been adjusted to fit the data at the
1636
ANOMALOUS SPECIFIC HEAT IN La : Ce ALLOYS
6000 R2n2~
~
4004-
~
3000
~
2OOO~-
—--- ---
--
Vol. 15, No. 10
T = 15 K show no sign of a CF anomaly. This suggests that the ~F splitting is larger than ~50K. The
-—--—-—---——--—-----~~-
entropy change ~S for 0.9< T< 15 decreases rapidly forx <0.02 where the alloys are superconducting. Presumably the extra entropy to make R ln 2 for all samples is found below 0.9 K. With the exception of the entropy change associated with the consensation into the crystal field ground state that occurs for kT ECF, at most a 5 per cent error is
C
made by not including data for T> 15. 000
~
I
In the case of the only superconducting alloy for which the change in Cwas measured on going into the
22
superconducting state, our measurements agree within 19 and experimental errorof with measurements hence give a value 1 .1 previous for TK. This value is higher
Clot. %Ce)
FIG. 4. Entropy change between 0.9 and 15 K associated with the excess specific heat for Ce~La 1_~ as a function of x.
peak. It is seen that the two level system prediction is too narrow to fit the data adequately. Taking three levels does not improve the fit. On the other18hand, is as seen in Fig. a dilute alloyanomaly. Kondo anomaly broader than3,the observed The theoretical
.
7
than previous values and is still so low as to make one further doubt that the anomaly shown in Fig. 2 is a dilute alloy Kondo anomaly.
specific heat curve shown in Fig. 3 was derived from a phenomenological model18 for the density of states. In the modelN(c) cx (E2 + ~2)_~ with j3 0.8. This theoretical prediction provides an adequate fit to data on Cu : Fe and is in agreement with the Bloomfield—Hamann calculation at high temperatures. Further, in contrast to the results shown in Fig. 2, if the peak were a dilute alloy Kondo anomaly one would expect that tIC ax.
Also as discussed earlier the anomaly is probably not associated with x. magnetic it also does not scale with It shiftsordering, to lower but temperatures and/or decreases in amplitude at low concentrations where the sample is superconducting. It is interesting to try to associate the b/T2 specific heat term seen in dilute superconducting alloys2 with the high temperature side of the anomaly after it has shifted below 1 K. Unfortunately, it is somewhat difficult to follow the anomaly continuously as a function of x since for x 0.02 the anomaly and the superconducting T~ occur at the same approximate temperature. If one suppresses superconductivity by applying a magnetic
In order to gain some insight into the nature of the anomaly we have computed the entropy change associated with the integral of tIC/xT from T°~° 0.9-15 K. For x = 0.0125, tIS was computed from the integral of (CA — CLa)/XT where CA is the superconducting alloy specific heat and CLa is the normal state specific heat of La. This entropy change LIS per Ce atom is shown in Fig. 4 as a function of concentration. One sees that as x —~ 0.2, LIS approaches R ln 2, i.e., the value for a doublet. This is consistent with the view that the anomaly is due to the Ce and that the cubic crystalline field has split the levels into a ground state doublet and a quartet which lies an energy LIECF above the doublet. Presumably the remainder of the spin entropy R ln 6 is recovered for kT> LIECF. Our measurements which extend up to
field, one also affects the anomaly. It may be possible to apply a weak field which will not affect the anomaly very much and yet be strong enough to destroy superconductivity. Our preliminary measurements of the magnetic field dependence of the specific heat for n = 0.0275 at a fixed temperature show that the specific heat is increased monotonically by the application of magnetic fields up to our highest field, 8 kG. At a fixed field the increase in the specific heat caused by the application of the field, i.e., C(H) — C(O), increases monotonically with decreasing temperature. This is consistent with the fact that at this concentration the magnetization shows no sign of magnetic ordering above 0.6K. The b/T2 term is a strong function of x. It depends onx’~where n = 2 or 3. This strong dependence on x certainly suggests that interaction effects are important. Distributions
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ANOMALOUS SPECIFIC HEAT IN La : Ce ALLOYS
of local fields such as those considered by Kline, Brout and Marshall could be important. Because of the absence of interaction effects in the susceptibility at higher x, it is perhaps more likely that the anomaly is associated with an unusual sort of phase transition, The decreasing resistivity is another indication of the transition. A similar more dramatic effect has been seen in the resistivity of CeA132°The essential cornmon features of the transitions that are observed for La : Ce alloys and CeAl 3 are:in(1) fraction of the spin entropy is removed thea large transitions without magnetic ordering; (2) a gradual, approximately logarithmic, resistivity decrease occurs; (3) in the case of the alloys the transition depends on the concentration of Ce. This is also the case21 in La 1_~Ce~ Al3. These properties suggest that what has been observed is a new type of electronic transition. This transition might prevent reentrant superconducting 22 in (La, Ce)A1behavior such as has been observed 2 from 23 occurring in La : Ce alloys. A more important matter is to understand the nature of the transition. It is quite possible that it is a Kondo transition but —
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because the system is not dilute the specific heat anomaly is not the usual Kondo anomaly. If one attempts to assign the temperature at which the peak in the specific heat occurs as approximately one third of an effective Kondo temperature, then the effective Kondo temperature shifts to lower temperatures with decreasing concentration. This suggestion of a high, concentration dependent, effective Kondo temperature is not what one would conclude from 7 the susceptibility measurements. A gives theoretical fit to the susceptibility measurements a concentration independent value of Tk < 1K. One may not be able to describe concentrated Kondo systems by a single parameter such as TK. Further work at lower temperatures is planned to clarify the nature of this transition.
Acknowledgements The authors wish to thank D.K. Finnemore for supplying the specific heat data on both phases of pure La which we used in computing the excess specific heat and to acknowledge the helpful assistance of Mr. Z. Sungaila in the laboratory. —
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ANOMALOUS SPECIFIC HEAT IN La: Ce ALLOYS
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La contribution magnétique a la chaleur specifique des alhiages La : Ce montre une anomalie d’une largeur considerable a une temperature de l.2°K;la dependence de la concentration est faible. Cette anomalie qui est responsable pour une, partie importante de l’entropie dun doublet, ne peut pas etre atribuée ni a un ordre magnétique ordinaire ni a l’effet Kondo des alliages diluées, mais II peut représenter les proprietés d’un système concentré de Kondo.