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SUPERCONDUCTIVITY FROM NON-PHONON MECHANISMS

L.J. SHAM Department of Physics, University of California, San Diego, La Jolla, CA 92093, USA Our studies of the determination of the transition temperature from the non-phonon mechanisms is reviewed. Raising T c by replacing phonons with high frequency excitations does appear possible though the usual estimates are found to be too high.

1. Introduction

Several interesting proposals have been made for superconductivity by unconventional mechanisms, whereby the phonons responsible for the attraction in a Cooper pair are replaced by excitons [1, 2], acoustic plasmons [3], and plasmons [4]. The common feature in these mechanisms is the high energy of these mediating bosons, which may be comparable to the Fermi energy. The problem which concerns us here is the determination of the superconducting transition temperature given this feature of the non-phonon mechanism. Superconductivity of the so-called heavy fermion systems [5], which is the object of much experimental and theoretical research today, may also share such an attribute. The unusually large low-temperature specific heat and the comparable jump in specific heat at the superconducting transition suggest the presence of a narrow band of heavy fermions. According to the theory of the mixed valent compounds or the Kondo lattice [6, 7], the Fermi temperature of these heavy fermions is of the order of the Kondo temperature, i.e. of the order 10 to 100K. Quantitative calculations of temperature dependent normal state properties of these compounds [8] show that this is a reasonable picture. In this case, no matter whether the attraction between the electrons is due to the conventional phonons [7] or due to some sort of fluctuations [6, 9], the energy of the mediating bosons is of the same order as the Fermi energy of the heavy fermions. In a series of studies in collaboration with

Schuh [10], Rietschel [11] and Grabowski [12], we address the problem of careful determination of the superconducting transition temperature due to the attraction between electrons by exchanging high frequency bosons. Three main conclusions may be drawn from our studies. (1) The usual estimates of the transition temperature for the nonphonon mechanisms are too high. (2) Processes beyond those in the Migdal theorem tend to lower Tc. (3) The approximation of the Eliashberg equation suggested by D.A. Kirzhnits, E.G. Maksmov and D.I. Khomskii (KMK) [13] is valid only to the first order in the coupling constant, i.e. when the weak coupling BCS formula [14] for T c is applicable. The use of the Fermi liquid parameters to estimate Tc [15] has been recently applied to the heavy fermion compounds [16, 17]. The Fermi liquid interaction parameters are applicable only to low frequency excitations and cannot be used to estimate Tc if the mediating bosons have energies comparable to the Fermi energy. The argument is substantiated by examining the case of the electron gas. In the following sections, I shall discuss in more details our studies of the transition temperature from the high frequency attractive mechanisms and the use of the Fermi liquid parameters.

2. The plasmons

It is an interesting question whether the exchange of plasmons between two electrons can

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L.J. Sham / Superconductivity from non-phonon mechanisms

cause sufficient attraction to bind them into a Cooper pair. If the electron pair stays close to the Fermi surface, then the frequency of the retarded interaction is very small and the plasmons only provide the static dielectric screening. The screened Coulomb interaction averaged over the Fermi surface gives a positive/~, which tends to suppress T c. If, in addition, there are low frequency bosons, such as phonons, which bind the Cooper pairs, the effective repulsion due to the Coulomb interaction is scaled down to p~* in the usual way [18]. On the other hand, it can be argued that the frequency structure of the dielectric function due to the plasmon contribution is very similar to phonons or excitons. So, if we just extend the order parameter to a large enough frequency range, the electron pair may be able to take advantage of the attractiveness of the plasmons and form a Cooper pair [4]. Rietschel and I [11] carried out a numerical study of the possibility of superconducting transition in an electron gas in order to resolve the contradiction. A rigid positive background is taken so that there are no phonons. The screening of the interaction is taken in the random phase approximation. This gives us the essential plasmon and e l e c t r o n - h o l e pair contributions and includes a balance of the repulsive direct Coulomb interaction and the attractive interaction due to boson mediation. The RPA gives a clean cut approximation. We seek to solve the Eliashberg equation [19] without further approximation. In particular, both the frequency and momentum range of the pairing function are chosen appropriate to the electron gas, and not restricted to the neighborhood of the Fermi surface. Takada followed the K M K approximation [13] by converting the frequency dependence of the pairing function to be m o m e n t u m dependent. We remove the uncertainty of his conclusion by solving the Eliashberg equation without such an approximation. As a way of determining the effective p~ our procedure avoids the usual one of treating the Coulomb part of pairing function without regard to the frequency structure of the screened electron-electron interaction. Our calculation of the normal state properties

of the electron gas differs from the previous ones in that the frequency is taken along the imaginary axis. We treat the electron self-energy in two ways. In one, the electron Green's function which appears in the self-energy is the unperturbed one. This is usually done [20] and we shall term the results from this way the RPA results. In the second way, the Green's function is the exact one, obtained by successive iterations. Results from this will be termed the SRPA results. The normal state properties from our RPA, such as the effective mass, the wave function renormalization and the correlation part of the chemical potential, as functions of r s, are in good agreement with the RPA result of Hedin. The SRPA results differ by about ten percent. The numerical solution of the Eliashberg equation shows that the gap function, at a fixed frequency, has a slow momentum dependence within the Fermi sea and decreases slowly as the momentum increases beyond the Fermi wave vector. The frequency dependence of the gap function is more important. The gap changes sign once as the imaginary frequency varies at a fixed momentum. The results of our calculation can be understood roughly in terms of an attractive square well model as a function of the imaginary frequency after averaging the effective interaction over the m o m e n t u m space. Our SRPA results for /~ range from being positive at small rs, e.g. 0.045 at r = 1, to being negative for large r S, e.g. - 0 . 1 2 at r s = 5. t~ is defined here with the cut-off energy at 1000 K. The effective interaction is attractive for r Sgreater than about 2.5. This means that with the addition of the p h o n o n contribution, sodium is superconducting below 2 K. Potassium and cesium would have much higher T c. These results are contrary to what is known about the alkali metals. Since we consider our solution of the Eliashberg equation reliable, we conclude that RPA is inadequate. Because the plasmon energy is comparable to the Fermi, Migdal's theorem [21] cannot be used. Thus, higher order processes beyond RPA not only modify the dielectric function but also the pairing in the Eliashberg equation. Given that the RPA over-estimates Tc in the electron gas, the higher

L.J. Sham / Superconductivity from non-phonon mechanisms

order plasmon exchange processes in the Cooper pair must tend to reduce the pairing. Since the frequency structure of the excitons parallels that of the plasmons in RPA, we infer that the usual estimates of the room temperature superconductivity from the exciton mechanism are over-optimistic. Advocates of the higher order processes being negligible [22] or tending to increase Tc [2] have to explain the electron gas behavior. To use the electron gas as a model for simple metals to obtain the positive/x *, which empirically should be about 0.1 [23], we need clearly to go beyond the RPA. We have found that the static approximation to the vertex correction to RPA [24] increases /x and, therefore, depresses T~ drastically. However, the static corrections persist in the limit of infinite frequency and do not provide the correct frequency behavior for the gap function. They may, therefore, have over-estimated the effects of the higher order processes. What is needed is a theory of the vertex correction and other higher order terms with the correct asymptotic frequency behavior.

3. Non-Migdai terms In the Eliashberg equation, the binding of the two time-reversal symmetry related electrons is due to simple exchange of phonons. In the strong coupling case, the phonon contribution to the self-energies of these electrons has to be included. Higher processes are negligible because the phonon energy is much smaller than the Fermi energy. This is known as the Migdal theorem [21]. When bosons with energy comparable to the Fermi energy take the place of the phonons, the higher order processes may no longer be negligible. Grabowski and I [12] have studied the effects of some of the higher order terms. Because of the RPA study of the electron gas which shows that the frequency dependence is more important than the momentum dependence, we concentrate only on the frequency dependence. We choose a model interaction of the form

I

N(O)V(p,k;iw)=/z

1-o

2]

453

co b

2--- 2 w +~b

x O(X/2kf-p)O(X/-2kf- k ) . (1) The dimensionless interaction parameter/x is the interaction potential times the density of states at the Fermi level, N(0). The first term on the right represents the repulsion between the electrons. oJb is the frequency of the dispersionless bosons which mediate the interaction, o- represents the strength of the boson-electron coupling. To avoid structure instability, o- is chosen to be less than unity. This interaction is taken as the unified model for the non-phonon mechanisms for superconductivity. It does not describe well the one dimensional system [22] because of the pronounced momentum dependence. For the electron gas, ~ob is taken to be the plasma frequency and the parameters Iz and o-are obtained as functions of r s by averaging the dynamically screened interaction in RPA over the momentum space. As a check of the reliability of the model interaction, we calculate T c from the Eliashberg equation with these values of parameters of the interaction. The agreement with the RPA results [11] with or without the self-energy correction is good. For the more general non-phonon model superconductor, transition temperatures are first calculated by solving the Eliashberg equation including terms of the model interaction (1) only within the dictate of the Migdal theorem, for selected values of p~ and or as functions of the boson energy in units of the Fermi energy, oJb/Ef. Even though T c is a small fraction of the Fermi temperature which now also serves as the cut-off temperature, the system cannot be considered as a weak coupling superconductor because the self-energy terms reduce T c by an order of magnitude. For fixed/z and o-, Tc first rises with increasing Wb/E f and then drops, cresting when the boson energy is about a quarter of the Fermi energy. For /x = 0.9 and obetween 0.8 and 1, a system with a Fermi energy typical of a metal would have a maximum T c around room temperatures. We then include in the Eliashberg equation the

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L.J. Sham / Superconductivity from non-phonon mechanisms

lowest order corrections to the Migdal terms, namely, the vertex terms in the boson exchange and the repeated boson exchange processes in the particle-hole channel which are analogous to the spin fluctuation terms [25]. These terms are found to reduce T c drastically. For the electron gas, tx* now rises slightly between 0.1 and 0.2 as r s increases instead of the RPA behavior of decreasing from positive to negative values. Thus, nonMigdal terms of our model interaction suppress superconductivity in the electron gas entirely. For the exciton system with IX = 0.9 and o- between 0.8 and 1, T c now peaks a t OOb/E~ about 0.1 with values around 50 K for a Fermi temperature of 105 K. The conclusion drawn from our model calculation is that it is possible to raise the superconducting transition temperature by increasing the mediating boson energy from the phonon range. The point of diminishing returns comes when the boson energy reaches about 1/10 of the Fermi energy. Maximum T c is likely to be in tens of degree Kelvin rather than around room temperatures.

unreliable. For example, in the electron gas with phonons, where A = i x [29] so that A - i x * is of second order in tx, KMK is incorrect even in the weak coupling and Migdal limit. For RPA in the electron gas, the KMK result for T c obtained by Takada [4] is several orders of magnitude smaller than the numerical solution of the Eliashberg equation [11]. We conclude that KMK is unreliable in cases where the BCS formula cannot be used.

5. T c from the Fermi liquid parameters

In Landau's theory of the Fermi liquid, the interaction parameters characterize the forward scattering amplitude of two quasi-particles on the Fermi surface [30]. Patton and Zaringhalam [15] give an extrapolation to the interaction between two quasi-particles and thus obtain the superfluid transition temperature for liquid 3He. For charged systems, the scattering amplitude contains an additional screened Coulomb interaction [30]. The method of ref. 15 is extended to include this contribution to the superconducting transition temperature, given by

4. The K M K approximation

The pairing function in the Eliashberg equation is a function of momentum and frequency. In the weak coupling limit, K M K simplify the Eliashberg equation to one for the gap function dependent on the momentum only by taking the spectral density of the pairing function to be a delta function in frequency. The KMK equation has been used to calculate T c in a number of papers [22, 4, 26, 271. Schuh and I [10] have found, in the weak coupling expansion, a correction term to the KMK kernel, which is governed by another integral equation. Kirzhnits et al. [2] have further given corrections including lowest order nonMigdal terms. It is then clear that the original K M K approximation is correct only to the first order in the coupling constant. This shows up clearly in the comparative numerical study by Khan and Allen [28]. If the leading term in T c is of second order in the coupling, then KMK is

T(,J ) = 1.13TF exp(~Q).

(2)

For the singlet pairing, the effective coupling constant is given by ,,,(2l + I)A,] P~o = ~ 1 + / = 1

4

,(-1)'

I+B,

J"

(3)

l[t~ ( 2 I + 1)B,] _ (-1) / 1+~- 7 j"

(4)

For the triplet pairing, /x1 =

1 +l=1 t - l )

+~

~ + Az

A l and B l are the interaction parameters which satisfy the sum rule [31].

L.J. Sham / Superconductivity from non-phonon mechanisms

1

o1 + A

+ , : 0 ( 2 / + 1)

[AI_ A' +

=0.

(5) C o n s i d e r the R P A in the electron gas as a test case. F o r r s = 2, 4, 5, /% o b t a i n e d f r o m the interaction p a r a m e t e r s of H e d i n [20] is respectively 0.36, 0.36, 0.35 c o m p a r e d with the values of 0.02, - 0 . 0 5 , - 0 . 0 9 f r o m the solution of the Eliashberg e q u a t i o n [11]. T h e discrepancy is easy to understand. T h e f o r m e r c o r r e s p o n d s to a static average over the F e r m i surface and the latter includes the f r e q u e n c y d e p e n d e n t contribution. T o apply the m e t h o d to the h e a v y f e r m i o n superc o n d u c t o r s , let us follow Vails and Tesanovic [16] and assume that only l = 0 and 1 p a r a m e t e r s are non-zero. A 0 and A 1 are t a k e n to be large because of the h e a v y mass. In ref. 16, B 0 is taken to be - 3 / 4 because the h e a v y f e r m i o n system is considered to be an almost local F e r m i liquid [32]. We shall e x a m i n e T c as a function of B 0. B 1 is d e t e r m i n e d f r o m the sum rule. Fig. 1 shows the resulting effective coupling constants as functions of the susceptibility enh a n c e m e n t factor 1/(1 + B0). A n e n h a n c e m e n t factor o f 4 c o r r e s p o n d s to the results o f Valls and

I/(I.-Bo) I

I

2

3

4

I

I

-I-

-2-

Pj

glet -4-

-5

Fig. 1. The effective coupling constants for singlet and triplet pairing versus the susceptibility enhancement factor.

455

Tesanovic which show the triplet pairing to be s u p e r c o n d u c t i n g and the singlet pair to be unbound. However, experiment [5] indicates the e n h a n c e m e n t factor to be b e t w e e n 1 and 2, in which case, fig. 1 shows that the singlet dominates. B e c a u s e of the uncertainty of the frequency d e p e n d e n t contribution discussed above, this conclusion does not d e m o n s t r a t e so firmly that h e a v y f e r m i o n s u p e r c o n d u c t o r s are singlets as that the use of the F e r m i liquid t h e o r y to explain the h e a v y f e r m i o n superconductivity [16, 17] is o p e n to doubt.

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[22] D. Davis, H. Gutfreund and W.A. Little, Phys. Rev. B13 (1976) 4766. [23] W.L. McMillan and J.M. RoweU, in: Superconductivity, R.D. Parks, ed. (Marcel Dekker, New York, 1969) p. 561. [24] P. Vashishta and K.S. Singwi, Phys. Rev. B36 (1972) 875. [25] N.F. Berk and J.R. Schrieffer, Phys. Rev. Lett. 17 (1966) 433. [26] W. Hanke and M.J. Kelly, Phys. Rev. B23 (1981) 112. [27] G. Vignale and K.S. Singwi, Phys. Rev. B31 (1985) 2729.

[28] F.S. Khan and P.B. Allen, Solid State Commun. 36 (1980) 481. [29] C.M. Varma, in: Superconductivity in d- and f-Band Metals 1982, W. Buckel and W. Weber, eds. (Kernforschungszentrum, Karlsruhe, 1982) p. 603. [30] D. Pines and P. Nozieres, Theory of Interacting Fermi Systems (Benjamin, New York, 1964). [31] W. Brinkman, P.M. Platzman and T.M. Rice, Phys. Rev. 174 (1968) 495. [32] D. Vollhardt, Rev. Mod. Phys. 56 (1984) 99.

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