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Anisotropic superconductivity in UPt3
Volume 117, number 8
PHYSICS LETTERS A
8 September 1986
A N I S O T R O P I C SUPERCONDUCTIVITY IN UPt 3 T. O G U C H I 1, A.J. F R E E M A N Department of Physics and Astronomy, Northwestern University, Evanston, IL 60201, USA
and G.W. CRABTREE Argonne National Laboratory, Argonne, IL 60439, USA Received 24 June 1986; accepted for publication 3 July 1986
A consistent picture of conventional superconductivity in UPt 3 is presented which explains qualitatively the gap anisotropy, the dependence of T¢ on crystal perfection, the low value of the specific heat discontinuity at Tc and the induced magnetic form factor. The model obtained is based on results of a local density band calculation for UPt 3 which are in excellent agreement with photoemission experiments and on estimates of the strength and anisotropy of the electron-phonon interaction. The key feature is anisotropy of the superconducting energy gap which arises from the variation of the electronic mass on the Fermi surface.
Since the discovery of superconductivity in by Steglich et al. [1], and the observation of superconductivity in the heavy fermion systems UBe13 and UPt3, there has been considerable controversy as to its origin [2]. The superconducting properties of UPt 3 are particularly interesting in several respects. The unusual behavior of its low temperature specific heat has led to the suggestion that superconductivity and spin fluctuations may coexist [3,4]. In contrast to UBet3 and C e C u E S i 2 , the specific heat jump at T~ is rather low - only 30-60% of the BCS value [3,4]. Ultrasonic attenuation experiments indicate there is strong anisotropy in the superconducting gap, which is often attributed to p- or d-wave pairing [5]. Neutron scattering measurements of the magnetic form factor of U show little or no change when the crystal enters the superconducting state [6]. Finally, strain induced by grinding the sample to a powder suppresses the superconductivity completely [4]. CeCu2Si 2
1 Present address: National Research Institute for Metals, 2-3-12, Nakameguro, Meguri-ku, Tokyo, Japan.
428
In this letter we provide a consistent picture for these unusual properties based on ordinary s-wave Cooper pairing induced by the electron-phonon interaction. The basis for our model is a fully relativistic, local density band calculation of the electronic structure of UPt 3 [7], and a crude estimate of the electron-phonon interaction based on a relativistic generalization [8] of the rigid ion approximation [9]. We find a band structure in surprisingly good agreement with photoemission experiments of Arko et al. [10] and a highly anisotropic electron-phonon interaction which is strong for electrons on the Pt atoms but quite weak for electrons on the U atoms. The corresponding anisotropy in )~, the electron-phonon coupling constant, on the Fermi surface is strongly correlated with the degree of f-character in the hybridized wavefunctions, and leads to strong gap anisotropy which follows the )~ anisotropy. The model obtained, which is independent of the quantitative computational results on which it is based, leads to a number of predictions and a consistent explanation of the observed properties of UPt 3 including the specific heat jump at Tc.
0375-9601/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 117, number 8
PHYSICS LETTERS A
The local density equations are solved using a fully relativistic formulation of the L M T O m e t h o d - a generalization developed by G o d r e c h e [11]. We employ the linear tetrahedron m e t h o d for the k-space integrations (up to 120 k-points in the irreducible Brillouin zone were used in the selfconsistent procedure) and the Hedin and L u n d qvist form of the exchange and correlation potential [12]. As discussed elsewhere, the calculated b a n d structure is in very good agreement with angle resolved photoemission experiments of A r k o et al. [10]. In the local site projected density of states (DOS) shown in fig. 1, there are two separate b a n d regions, Pt d and U f, with significant hybridization near E v between them. As shown, E v is located just above a very sharp peak in the D O S with a width of 7 m R y (0.1 eV) at the beginning of the U f band. Such a narrow peak has been observed in high-resolution photoemission measurements by A r k o et al. [10]. The U f b a n d has a s p i n - o r b i t splitting of 60 mRy, which corresponds to a s p i n - o r b i t parameter of 1900 c m - l . The b a n d width of the split f subbands with j = 5 / 2 and 7 / 2 is about 70 mRy. Since the s p i n - o r b i t splitting of the U f b a n d is comparable in magnitude to the b a n d width, a fully relativistic a p p r o a c h is crucial for a correct description for the electronic structure of UPt3. The total D O S at E F, N ( E v ) = 114.6 s t a t e s / R y
_J ~E rY 0 u_
8 September 1986
U P t 3, yields a bare specific heat coefficient Y0 = 19.8 m J / m o l e K z. If we introduce a total enhancement factor ~kt by y = (1 + ?~t)Y0, we obtain X t = 20.3 from the observed [3] y = 422 m J / m o l e K 2. This enhancement of the specific heat is larger than usually observed in other f-electron materials like CeSn3, CeA12 and U S n 3 where A t is about 5 - 9 [13]. Here it should be noted that, as discussed in several contexts recently, the enhancem e n t contains contributions from spin fluctuations and the e l e c t r o n - p h o n o n interactions, and that the e l e c t r o n - p h o n o n coupling can be strongly enhanced by the exchange interaction [14]. A n estimate of the (exchange unenhanced) elect r o n - p h o n o n coupling constant X [15], written for U P t 3 as 2~
~/v
+ 3
M U ( ¢..d2 )
~/Pt
(1)
Mpt(td2 ) '
m a y be made using a relativistic generalization [8] of the rigid ion approximation [9] to the electronic contribution ~/ (cf. eq. (9) of ref. [8]). Here M U and Met are the U and Pt masses and the averaged p h o n o n frequency (0~2) is approximated by 1 2 ~Oi~ with the D e b y e temperature ( 9 0 = 210 K [16]. In this crude approximation of ( ~ 2 ) , no account is taken of possible p h o n o n softening [14] or multiple p h o n o n modes [17]. Table 1 Calculated density of states (DOS) at the Fermi energy projected by atom type and angular moment j = l - 1/2 or l + 1/2, and the individual l ~ l + 1 scattering contributions to 71, the electronic part of k.
200 160
DOS (/eV/spin/atom) l l-1/2 l +1/2
~/t~ i+ 1 (eV/AZ)
s p d f
0.008 0.126 0.133
)k
Pr
w
c~ w
U
120 8O
total
40 >I----
0.004 0.041 2.475
Pt
0
0.4 0.6 OB 1.0 ENERGY (Ry) Fig. 1. Density of states projected by atom types. Dashed lines denote Pt and solid U. 0
2.908 s p d f
0.2
total Total
0.008 0.018 0.083 0.279
0.034 0.053 0.006
0.266 0.047 0.043 0.230 0.021
0.434 4.210(/eV/spin/UPt 3)
0.0284
0.002 0.062 0.701 0.765
0.0994 0.327 429
Volume 117, number 8
PHYSICS LETTERS A 0.46
0.42 /
H
A
0.28 ~
024 ~
~
O
0.27
0.28
028
0,28/ I--...IK 0.30
0.29 K
035
Fig. 2. Fermi surface of UPt 3 with values of X given at selected k-points.
Table 1 lists the calculated ~ and X values for UPt 3. Some values of X(k) at selected k-points on the different Fermi surface sheets are shown in fig. 2. Using the average X (averaged over all sheets of the Fermi surface shown in fig. 2) of 0.33 and an assumed Coulomb pseudopotential /**= 0.1, we obtain T~ = 0.2 K (Tc~xp = 0.5 K) using McMillan's formula [15]. (This value of Tc is of course a lower bound since it is governed by the largest X valued sheets on the Fermi surface.) Thus conventional superconductivity in UPt 3 is indeed possible. The most interesting feature in table 1 is that the dominant contribution to X comes from the Pt sites, especially the d - f channel. Hence, the superconductivity of UPt 3 is driven by the strong Pt electron-phonon interaction on those sheets of the Fermi surface for which the Pt component of the wavefunction is large (sheets 4 and 5 in fig. 2). Of course, these sheets also contain strongly hybridized U f-electron components which, while not contributing strongly to the pairing interaction, do participate in the superconductivity and contribute to the specific heat jump (as discussed more fully below). It is widely believed that large spin fluctuation effects in UPt 3 suppress s-wave pairing in favor of p-wave pairing. To investigate the role of spin fluctuations, we calculated the magnitude and anisotropy of the Stoner factor S = N(Ev)Ixc. The effective exchange-correlation integral, I×c [18], is evaluated with the local spin density functional of 430
8 September 1986
von Barth and Hedin [19]. The present band calculation gives S of 1.06 _+ 0.10, implying that the Stoner exchange enhancement factor (1 - S ) - 1 is large and that spin fluctuations are important [3,4]. However, we find that the Stoner factor is quite anisotropic and complements the anisotropy of the electron-phonon interaction: the U sites contribute - 0.99 and the Pt sites only 0.07 to S. These results imply that the large enhancement of the electronic specific heat, X t = 20.3, arises mostly from exchange interactions among the conduction electrons on the U site [20] nl. The band calculation provides a guide to how the electron-phonon interaction and Stoner factor vary over the Fermi surface. The former is associated with regions of large Pt d-character, while the latter is associated with regions of large U f-character. For the calculated Fermi surface shown in fig. 2, the three electron surfaces (labeled 1, 2 and 3) around F all have nearly pure f-character, the hole sheets (labeled 4 and 5) near H and L have regions where Pt d-character is stronger than U f-character and the electron sheet around K has intermediate U - P t hybridization. The formal connection between X anisotropy and gap anisotropy is implicit in the Eliashberg equations [21] and requires numerical solution for quantitative results. However, qualitative results can be obtained in the weak anisotropy limit and these illustrate the essential features of our model. Butler and Allen [22] have shown that a(k) A0
1+
1+/**
(1 +
X)(a-/**)
[ a ( k ) - )~1
1 x -/** [/**(t,) -/**],
(2)
where A 0 is the averaged superconducting gap. From this analysis and the results of table 1, it is clear that those parts of the Fermi surface which are largely f-like (sheets 1, 2 and 3 in fig. 2) will have a very small energy gap, while those with significant d-character (sheets 4 and 5 in fig. 2) will have a much larger gap. The order of magnitude difference in the ?t values for the U and Pt *~1 Note that Fay and Appel [20] have found interesting our results that spin fluctuations on the U sites do not suppress s-pairing on the Pt sites.
Volume 117, number 8
PHYSICS LETTERS A
sites (enhanced by the large difference in Stoner factors) implies a large anisotropy in the energy gap which is consistent with the unusual superconducting density of states observed in the ultrasonic attenuation experiments [5]. The correlation between the gap anisotropy and the effective mass anisotropy allows many unusual experimental observations to be explained. The induced form factor measurements [6] are sensitive mainly to the heavy (U f-)electrons which have a larger Stoner enhancement and a small energy gap. Because the gap is small, there is relatively little change in the electronic structure of these electrons in going below Tc and the form factor remains essentially constant. The sensitivity of T~ to strain produced by grinding [4] is explained by increased scattering of electrons from the high gap regions of the Fermi surface to the low gap regions (cf. fig. 2). Because most of the Fermi surface is f-like and unfavorable for Cooper pairing, the homogenizing effect of the increased scattering lowers the average attractive interaction and T~. The relatively small j u m p in the specific heat [3,4] at Tc is also naturally explained by gap anisotropy. The low gap sheets of the Fermi surface make a relatively smaller contributions to the specific heat discontinuity than do the high sheets, so that the total discontinuity in the specific heat is smaller than would be obtained without anisotropy. Following completion of this work, we have learned of two very recent experimental developments which support our calcul~ttions. Specific heat measurements below Tc in UPt 3 [23] show that C/T extrapolates to a finite value (3'N --- 260 m J / m o l e K 2) at T = 0, suggesting that there are normal electrons remaining on sheets of the Fermi surface where the gap is very small. Steglich et al. [23] further show that if one subtracts "YN to estimate 7s (the specific heat coefficient of those electrons which actually form Cooper pairs) and compares the magnitude of the superconducting j u m p with Ys one gets a value approaching the BCS value for s-wave pairing. A comparison of the extrapolated residual specific heat with extrapolated thermal conductivity suggests that the low gap regions of the Fermi surface hold heavy rather than light electrons. This result is a natural
8 September 1986
consequence of the correlation between gap anisotropy and mass anisotropy in our model. As for the j u m p in specific heat, by using the definition of
X=
f dk 8( EI, - Ev) we can divide 7 into contributions from different Fermi surface sheets. If we assume an averaged Xi within each FS sheet i then ~, = Y" N , ( E v ) ( 1 + )t,). i
F r o m our band results, we find that sheets 3 and 4 contribute essentially the entire DOS. Since sheet 3 has almost purely U f-character, we may consider the electrons on this sheet to remain normal below T~ over the temperature range measured [23]. Making a projection by atomic sites on each FS sheet, we can calculate the quasiparticle density of states from sheet 4 as Y4 -- [N4(U) + N4(Pt)]
N4(U))k(U) + N4(Pt)X(Pt ) ] × 1+
N4(U ) + N4(Pt)
.
(3)
Using the calculated N4(U ) = 40 and N4(Pt ) --- 35 ( s t a t e s / R y formula) unit and the values for X t(U ) and Xt(Pt ), we find Y4 = 210 m J / m o l e K z which is of the order of Steglich et al.'s Ys value. Further support for our results comes from the variation of T~ with substitution of Pd for Pt in UPt 3 [24]. With only 1% Pd, T~ drops from 0.5 K to less than 30 mK, a much faster reduction than occurs for substitution by Th on the U site. Such a difference in sensitivity between the two sites is predicted by our model, since Pt defects will be much more effective at scattering electrons from the high gap regions of the Fermi surface than will U defects. These two experimental results strongly support the fundamental correlation between gap anisotropy and electron mass which we propose. We are grateful to P.B. Allen, B.D. Dunlap, D.J. Kim, and I.K. Schuller for helpful discussions and criticism and to J. Mydosh and the authors of 431
Volume 117, number 8
PHYSICS LETTERS A
ref. [24] for p e r m i s s i o n to q u o t e their results p r i o r to p u b l i c a t i o n . W o r k s u p p o r t e d b y N S F ( D M R g r a n t N o . 82-16543) a n d D O E , a n d b y a g r a n t for computing time made available by DOE on the ER-CRAY.
References [1] [2] [3] [4] [5] [6] [7]
[8] [9]
432
F. Steglich et al., Phys. Rev. Lett. 43 (1979) 1892. G.R. Stewart, Rev. Mod. Phys. 56 (1984) 755. A. de Visser et al., J. Phys. F 14 (1984) L191. G.R. Stewart, Z. Fisk, J.O. Willis and J.L. Smith, Phys. Rev. Lett. 52 (1984) 679. D.J. Bishop et al., Phys. Rev. Lett. 53 (1984) 1009; B. Batlogg et al., preprint. C. Stassis et al., Bull. Am. Phys. Soc. 30 (1985) 356. T. Oguchi and A.J. Freeman, Bull. Am. Phys. Soc. 30 (1985) 408; T. Oguchi and A.J. Freeman, J. Magn. Magn. Mater. 52 (1985) 176. W. John and D. Hamann, Phys. Stat. Sol. (b) 93 (1979) K143. G.D. Gaspari and B.L. Gyorffy, Phys. Rev. Lett. 28 (1972) 801.
8 September 1986
[10] A.J. Arko et al., Phys. Rev. Lett. 53 (1984) 2050; A.J. Arko, Bull. Am. Phys. Soc. 30 (1985) 515. [11] C. Godreche, J. Magn. Magn. Mater. 29 (1982) 262. [12] L. Hedin and B.I. Lundqvist, J. Phys. C 4 (1971) 2064. [13] D.D. Koelling, private communication. [14] D.J. Kim, Phys. Rev. B 17 (1978) 468. [15] W.L. McMillan, Phys. Rev. 167 (1968) 331. [16] A. de Visser, J.J.M. Franse and A. Menovsky, J. Phys. F 15 (1985) 153. [17] B.M. Klein and D.A. Papaconstantopoulos, J. Phys. F 6 (1976) 1135. [18] J.F. Janak, Phys. Rev. B 16 (1977) 255; T. Jarlborg and A.J. Freeman, Phys. Rev. B 22 (1980) 2332. [19] U. von Barth and L. Hedin, J. Phys. C 5 (1972) 1629. [20] D. Fay and J. Appel, Phys. Rev. B 32 (1985) 6071. [21] G.M. Eliashberg, Sov. Phys. JETP 11 (1961) 6967; 12 (1961) 1000. [22] W.H. Butler and P.B. Allen, Superconductivity in d- and f-band metals, ed. D.H. Douglass (Plenum, New York, 1976) p. 73. [23] F. Steglich et al., J. Magn. Magn. Mater. 52 (1985) 54. [24] A. de Visser et al., J. Magn. Magn. Mater. 54-57 (1986) 375.
PHYSICS LETTERS A
8 September 1986
A N I S O T R O P I C SUPERCONDUCTIVITY IN UPt 3 T. O G U C H I 1, A.J. F R E E M A N Department of Physics and Astronomy, Northwestern University, Evanston, IL 60201, USA
and G.W. CRABTREE Argonne National Laboratory, Argonne, IL 60439, USA Received 24 June 1986; accepted for publication 3 July 1986
A consistent picture of conventional superconductivity in UPt 3 is presented which explains qualitatively the gap anisotropy, the dependence of T¢ on crystal perfection, the low value of the specific heat discontinuity at Tc and the induced magnetic form factor. The model obtained is based on results of a local density band calculation for UPt 3 which are in excellent agreement with photoemission experiments and on estimates of the strength and anisotropy of the electron-phonon interaction. The key feature is anisotropy of the superconducting energy gap which arises from the variation of the electronic mass on the Fermi surface.
Since the discovery of superconductivity in by Steglich et al. [1], and the observation of superconductivity in the heavy fermion systems UBe13 and UPt3, there has been considerable controversy as to its origin [2]. The superconducting properties of UPt 3 are particularly interesting in several respects. The unusual behavior of its low temperature specific heat has led to the suggestion that superconductivity and spin fluctuations may coexist [3,4]. In contrast to UBet3 and C e C u E S i 2 , the specific heat jump at T~ is rather low - only 30-60% of the BCS value [3,4]. Ultrasonic attenuation experiments indicate there is strong anisotropy in the superconducting gap, which is often attributed to p- or d-wave pairing [5]. Neutron scattering measurements of the magnetic form factor of U show little or no change when the crystal enters the superconducting state [6]. Finally, strain induced by grinding the sample to a powder suppresses the superconductivity completely [4]. CeCu2Si 2
1 Present address: National Research Institute for Metals, 2-3-12, Nakameguro, Meguri-ku, Tokyo, Japan.
428
In this letter we provide a consistent picture for these unusual properties based on ordinary s-wave Cooper pairing induced by the electron-phonon interaction. The basis for our model is a fully relativistic, local density band calculation of the electronic structure of UPt 3 [7], and a crude estimate of the electron-phonon interaction based on a relativistic generalization [8] of the rigid ion approximation [9]. We find a band structure in surprisingly good agreement with photoemission experiments of Arko et al. [10] and a highly anisotropic electron-phonon interaction which is strong for electrons on the Pt atoms but quite weak for electrons on the U atoms. The corresponding anisotropy in )~, the electron-phonon coupling constant, on the Fermi surface is strongly correlated with the degree of f-character in the hybridized wavefunctions, and leads to strong gap anisotropy which follows the )~ anisotropy. The model obtained, which is independent of the quantitative computational results on which it is based, leads to a number of predictions and a consistent explanation of the observed properties of UPt 3 including the specific heat jump at Tc.
0375-9601/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 117, number 8
PHYSICS LETTERS A
The local density equations are solved using a fully relativistic formulation of the L M T O m e t h o d - a generalization developed by G o d r e c h e [11]. We employ the linear tetrahedron m e t h o d for the k-space integrations (up to 120 k-points in the irreducible Brillouin zone were used in the selfconsistent procedure) and the Hedin and L u n d qvist form of the exchange and correlation potential [12]. As discussed elsewhere, the calculated b a n d structure is in very good agreement with angle resolved photoemission experiments of A r k o et al. [10]. In the local site projected density of states (DOS) shown in fig. 1, there are two separate b a n d regions, Pt d and U f, with significant hybridization near E v between them. As shown, E v is located just above a very sharp peak in the D O S with a width of 7 m R y (0.1 eV) at the beginning of the U f band. Such a narrow peak has been observed in high-resolution photoemission measurements by A r k o et al. [10]. The U f b a n d has a s p i n - o r b i t splitting of 60 mRy, which corresponds to a s p i n - o r b i t parameter of 1900 c m - l . The b a n d width of the split f subbands with j = 5 / 2 and 7 / 2 is about 70 mRy. Since the s p i n - o r b i t splitting of the U f b a n d is comparable in magnitude to the b a n d width, a fully relativistic a p p r o a c h is crucial for a correct description for the electronic structure of UPt3. The total D O S at E F, N ( E v ) = 114.6 s t a t e s / R y
_J ~E rY 0 u_
8 September 1986
U P t 3, yields a bare specific heat coefficient Y0 = 19.8 m J / m o l e K z. If we introduce a total enhancement factor ~kt by y = (1 + ?~t)Y0, we obtain X t = 20.3 from the observed [3] y = 422 m J / m o l e K 2. This enhancement of the specific heat is larger than usually observed in other f-electron materials like CeSn3, CeA12 and U S n 3 where A t is about 5 - 9 [13]. Here it should be noted that, as discussed in several contexts recently, the enhancem e n t contains contributions from spin fluctuations and the e l e c t r o n - p h o n o n interactions, and that the e l e c t r o n - p h o n o n coupling can be strongly enhanced by the exchange interaction [14]. A n estimate of the (exchange unenhanced) elect r o n - p h o n o n coupling constant X [15], written for U P t 3 as 2~
~/v
+ 3
M U ( ¢..d2 )
~/Pt
(1)
Mpt(td2 ) '
m a y be made using a relativistic generalization [8] of the rigid ion approximation [9] to the electronic contribution ~/ (cf. eq. (9) of ref. [8]). Here M U and Met are the U and Pt masses and the averaged p h o n o n frequency (0~2) is approximated by 1 2 ~Oi~ with the D e b y e temperature ( 9 0 = 210 K [16]. In this crude approximation of ( ~ 2 ) , no account is taken of possible p h o n o n softening [14] or multiple p h o n o n modes [17]. Table 1 Calculated density of states (DOS) at the Fermi energy projected by atom type and angular moment j = l - 1/2 or l + 1/2, and the individual l ~ l + 1 scattering contributions to 71, the electronic part of k.
200 160
DOS (/eV/spin/atom) l l-1/2 l +1/2
~/t~ i+ 1 (eV/AZ)
s p d f
0.008 0.126 0.133
)k
Pr
w
c~ w
U
120 8O
total
40 >I----
0.004 0.041 2.475
Pt
0
0.4 0.6 OB 1.0 ENERGY (Ry) Fig. 1. Density of states projected by atom types. Dashed lines denote Pt and solid U. 0
2.908 s p d f
0.2
total Total
0.008 0.018 0.083 0.279
0.034 0.053 0.006
0.266 0.047 0.043 0.230 0.021
0.434 4.210(/eV/spin/UPt 3)
0.0284
0.002 0.062 0.701 0.765
0.0994 0.327 429
Volume 117, number 8
PHYSICS LETTERS A 0.46
0.42 /
H
A
0.28 ~
024 ~
~
O
0.27
0.28
028
0,28/ I--...IK 0.30
0.29 K
035
Fig. 2. Fermi surface of UPt 3 with values of X given at selected k-points.
Table 1 lists the calculated ~ and X values for UPt 3. Some values of X(k) at selected k-points on the different Fermi surface sheets are shown in fig. 2. Using the average X (averaged over all sheets of the Fermi surface shown in fig. 2) of 0.33 and an assumed Coulomb pseudopotential /**= 0.1, we obtain T~ = 0.2 K (Tc~xp = 0.5 K) using McMillan's formula [15]. (This value of Tc is of course a lower bound since it is governed by the largest X valued sheets on the Fermi surface.) Thus conventional superconductivity in UPt 3 is indeed possible. The most interesting feature in table 1 is that the dominant contribution to X comes from the Pt sites, especially the d - f channel. Hence, the superconductivity of UPt 3 is driven by the strong Pt electron-phonon interaction on those sheets of the Fermi surface for which the Pt component of the wavefunction is large (sheets 4 and 5 in fig. 2). Of course, these sheets also contain strongly hybridized U f-electron components which, while not contributing strongly to the pairing interaction, do participate in the superconductivity and contribute to the specific heat jump (as discussed more fully below). It is widely believed that large spin fluctuation effects in UPt 3 suppress s-wave pairing in favor of p-wave pairing. To investigate the role of spin fluctuations, we calculated the magnitude and anisotropy of the Stoner factor S = N(Ev)Ixc. The effective exchange-correlation integral, I×c [18], is evaluated with the local spin density functional of 430
8 September 1986
von Barth and Hedin [19]. The present band calculation gives S of 1.06 _+ 0.10, implying that the Stoner exchange enhancement factor (1 - S ) - 1 is large and that spin fluctuations are important [3,4]. However, we find that the Stoner factor is quite anisotropic and complements the anisotropy of the electron-phonon interaction: the U sites contribute - 0.99 and the Pt sites only 0.07 to S. These results imply that the large enhancement of the electronic specific heat, X t = 20.3, arises mostly from exchange interactions among the conduction electrons on the U site [20] nl. The band calculation provides a guide to how the electron-phonon interaction and Stoner factor vary over the Fermi surface. The former is associated with regions of large Pt d-character, while the latter is associated with regions of large U f-character. For the calculated Fermi surface shown in fig. 2, the three electron surfaces (labeled 1, 2 and 3) around F all have nearly pure f-character, the hole sheets (labeled 4 and 5) near H and L have regions where Pt d-character is stronger than U f-character and the electron sheet around K has intermediate U - P t hybridization. The formal connection between X anisotropy and gap anisotropy is implicit in the Eliashberg equations [21] and requires numerical solution for quantitative results. However, qualitative results can be obtained in the weak anisotropy limit and these illustrate the essential features of our model. Butler and Allen [22] have shown that a(k) A0
1+
1+/**
(1 +
X)(a-/**)
[ a ( k ) - )~1
1 x -/** [/**(t,) -/**],
(2)
where A 0 is the averaged superconducting gap. From this analysis and the results of table 1, it is clear that those parts of the Fermi surface which are largely f-like (sheets 1, 2 and 3 in fig. 2) will have a very small energy gap, while those with significant d-character (sheets 4 and 5 in fig. 2) will have a much larger gap. The order of magnitude difference in the ?t values for the U and Pt *~1 Note that Fay and Appel [20] have found interesting our results that spin fluctuations on the U sites do not suppress s-pairing on the Pt sites.
Volume 117, number 8
PHYSICS LETTERS A
sites (enhanced by the large difference in Stoner factors) implies a large anisotropy in the energy gap which is consistent with the unusual superconducting density of states observed in the ultrasonic attenuation experiments [5]. The correlation between the gap anisotropy and the effective mass anisotropy allows many unusual experimental observations to be explained. The induced form factor measurements [6] are sensitive mainly to the heavy (U f-)electrons which have a larger Stoner enhancement and a small energy gap. Because the gap is small, there is relatively little change in the electronic structure of these electrons in going below Tc and the form factor remains essentially constant. The sensitivity of T~ to strain produced by grinding [4] is explained by increased scattering of electrons from the high gap regions of the Fermi surface to the low gap regions (cf. fig. 2). Because most of the Fermi surface is f-like and unfavorable for Cooper pairing, the homogenizing effect of the increased scattering lowers the average attractive interaction and T~. The relatively small j u m p in the specific heat [3,4] at Tc is also naturally explained by gap anisotropy. The low gap sheets of the Fermi surface make a relatively smaller contributions to the specific heat discontinuity than do the high sheets, so that the total discontinuity in the specific heat is smaller than would be obtained without anisotropy. Following completion of this work, we have learned of two very recent experimental developments which support our calcul~ttions. Specific heat measurements below Tc in UPt 3 [23] show that C/T extrapolates to a finite value (3'N --- 260 m J / m o l e K 2) at T = 0, suggesting that there are normal electrons remaining on sheets of the Fermi surface where the gap is very small. Steglich et al. [23] further show that if one subtracts "YN to estimate 7s (the specific heat coefficient of those electrons which actually form Cooper pairs) and compares the magnitude of the superconducting j u m p with Ys one gets a value approaching the BCS value for s-wave pairing. A comparison of the extrapolated residual specific heat with extrapolated thermal conductivity suggests that the low gap regions of the Fermi surface hold heavy rather than light electrons. This result is a natural
8 September 1986
consequence of the correlation between gap anisotropy and mass anisotropy in our model. As for the j u m p in specific heat, by using the definition of
X=
f dk 8( EI, - Ev) we can divide 7 into contributions from different Fermi surface sheets. If we assume an averaged Xi within each FS sheet i then ~, = Y" N , ( E v ) ( 1 + )t,). i
F r o m our band results, we find that sheets 3 and 4 contribute essentially the entire DOS. Since sheet 3 has almost purely U f-character, we may consider the electrons on this sheet to remain normal below T~ over the temperature range measured [23]. Making a projection by atomic sites on each FS sheet, we can calculate the quasiparticle density of states from sheet 4 as Y4 -- [N4(U) + N4(Pt)]
N4(U))k(U) + N4(Pt)X(Pt ) ] × 1+
N4(U ) + N4(Pt)
.
(3)
Using the calculated N4(U ) = 40 and N4(Pt ) --- 35 ( s t a t e s / R y formula) unit and the values for X t(U ) and Xt(Pt ), we find Y4 = 210 m J / m o l e K z which is of the order of Steglich et al.'s Ys value. Further support for our results comes from the variation of T~ with substitution of Pd for Pt in UPt 3 [24]. With only 1% Pd, T~ drops from 0.5 K to less than 30 mK, a much faster reduction than occurs for substitution by Th on the U site. Such a difference in sensitivity between the two sites is predicted by our model, since Pt defects will be much more effective at scattering electrons from the high gap regions of the Fermi surface than will U defects. These two experimental results strongly support the fundamental correlation between gap anisotropy and electron mass which we propose. We are grateful to P.B. Allen, B.D. Dunlap, D.J. Kim, and I.K. Schuller for helpful discussions and criticism and to J. Mydosh and the authors of 431
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PHYSICS LETTERS A
ref. [24] for p e r m i s s i o n to q u o t e their results p r i o r to p u b l i c a t i o n . W o r k s u p p o r t e d b y N S F ( D M R g r a n t N o . 82-16543) a n d D O E , a n d b y a g r a n t for computing time made available by DOE on the ER-CRAY.
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