Strong electron-phonon interaction in multiband superconductors

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Strong electron-phonon interaction in multiband superconductors

O. V. Dolgov1_{and A. A. Golubov}2

1_{Max-Planck-Institut für Festkörperphysik, Heisenbergstrasse 1, 70569 Stuttgart, Germany} 2_{Faculty of Science and Technology and MESA}_{⫹ Institute for Nanotechnology, University of Twente,}

7500 AE Enschede, The Netherlands

共Received 7 March 2008; revised manuscript received 15 May 2008; published 26 June 2008兲 We discuss the effects of anisotropy on superconducting critical temperature and order parameter in a strongly coupled regime. The multiband representation is used as a model for anisotropy. We show that strong-coupling effects in multiband superconductors lead to pair breaking due to interband coupling because soft phonon modes play the same role as static impurities. This effect makes the order parameters in different bands equal to each other and limits the upper bound on critical temperature.

DOI:10.1103/PhysRevB.77.214526 PACS number共s兲: 74.20.Mn, 74.62.⫺c, 74.70.⫺b

I. INTRODUCTION

Effects of anisotropy on superconducting critical tempera-ture and energy gap become of primary importance by ap-proaching the strong-coupling regime when transition tem-perature Tc becomes of the order or even larger than the

characteristic energy⍀ of a boson mode, which mediate su-perconductivity. This issue received little attention up to now. In the weak-coupling limit, the effects of anisotropy were investigated shortly after the Bardeen–Cooper– Schrieffer共BCS兲 theory 共see, e.g., Ref.1兲, and for multiband

systems 共Refs.2and3兲. Following the paper by Markowitz

and Kadanoff,4_{different authors}_{共references can be found in}

the review5_{兲 introduced the so-called separable interaction} Vkk⬘=共1 + ak兲V共1 + ak⬘兲, 共1兲

where akis an anisotropy parameter with the Fermi surface

averaging具ak典 being equal to zero. The result is the

enhance-ment of the effective coupling constant

␭eff=具N共0兲Vkk⬘典 = N共0兲V共1 + 具ak2典兲 ⬎ N共0兲V,

and corresponding rising of the Tcaccording to the standard

BCS expression

Tc= 1.14␪Dexp共− 1/␭eff兲, 共2兲

where␪Dis the phonon cutoff.

For multiband clean systems in the weak-coupling limit, the effective coupling constant in Eq. 共2兲 is determined by

the maximum eigenvalue of the matrix ␭_␣␤, where␣ and␤ are band indices. Intraband impurity scattering does not af-fect superconducting properties共Anderson’s theorem兲 while the interband one averages out the order parameters⌬_␣ and ␭eff共and Tc兲, and corresponds to the average value

具␭典 =

兺␣␤

N␣共0兲␭␣␤

兺␣

N_␣共0兲 , 共3兲

共see, e.g., Refs. 6and7兲. For positively defined matrix ␭_␣␤, the maximum eigenvalue is bigger than具␭典, and we have the enhancement of Tc for multiband systems in comparison

with the averaged value. This result does not depend on the sign of the nondiagonal matrix elements which determine the anisotropic contribution.8

Recent theoretical studies of superconductivity in the two-band superconductor MgB2 共Ref.9兲, and calculations of

co-valent metals as the hypothetical hexagonal LiB and boron-doped diamond renewed the interest on the problem of an upper bound on superconducting critical temperature in strongly coupled anisotropic systems. Some estimates pro-vide values of ␭ in anisotropic superconductors as large as four共Ref. 10兲 or even 25 共Ref.11兲.

Let us first recall the result for the strong-coupling ap-proach to isotropic systems. In the present paper we do not discuss effects of the strong electron–phonon interaction on phonon frequencies共see Ref. 12兲. For the case of ⍀Ⰶ2␲Tc

共which can occur for large ␭兲, real phonons give the pair breaking contributions to the superconducting pairing, as well as to the quasiparticle renormalization. The largest terms corresponding to pair breaking and quasiparticle damping 共see Appendix A兲 cancel each other out 共Refs. 13

and 14兲, and, as the result, one arrives at the following

strong-coupling expression共see Ref. 15兲

Tc= const

冑

␭⍀2, 共4兲

where in the simplest approximation const=共2␲兲−1_⯝0.15

共numerical calculations give 0.1827兲. There are interpolation expressions connecting strong- and weak-coupling limits 共see reviews5,16,17_兲.

Moussa and Cohen10 _{have imposed two possible upper}

bounds on a maximal critical temperature of multiband su-perconductors: the lower one is determined by the averaged coupling constant 关Eq. 共3兲兴 while the upper one is governed

by the maximal 共positive兲 eigenvalue of the matrix for the first momentum of the Eliashberg functions␣_␣␤2 共␻兲F_␣␤共␻兲,

关␭⍀2_兴

␣␤= M␣␤共1兲 = 2

冕

0 ⬁

d␻␻␣_␣␤2 共␻兲F_␣␤共␻兲共5兲共for the Einstein spectrum, this value is equal to ␭␣␤⍀2兲.

The purpose of this work is to analyze self-consistently the effects of anisotropy on the upper bound on Tc. We show

that the low-frequency phonons play a role similar to intra-band and interintra-band static impurities. The latter can lead to the suppression of the anisotropy and, as a result, the upper bound on Tc is determined by the averaged coupling

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stant. We consider in more detail the applications to the multiband systems.

II. GENERAL DESCRIPTION OF MULTIBAND SYSTEMS

The gap functions ⌬_␣共␻n兲 can be calculated within an

extension of the Eliashberg formalism to multiple bands;

=␲T共2n−1兲, standard Eliashberg functions define the super-conducting properties, and thermodynamical properties are

␣␣␤2 共␻兲F␣␤共␻兲 = 1 N_␣共0兲

兺

k,k⬘,␯ 兩gk,k⬘ ␣␤,␯_兩2_␦共␧ k ␣_兲␦共␧ k⬘ ␤_兲 ⫻␦共␻−␻_k−k␯ _⬘兲, 共8兲

where ␣,␤=兵1,2,...其, N_␣共0兲 is the partial density of states per spin at the Fermi energy, g_k,k␣␤_⬘ is the electron–phonon interaction 共EPI兲 matrix element, ␮_␣␤ⴱ is the renormalized Coulomb pseudopotential matrix element, and ␧k␣ and

␧k␤ are the quasiparticle energies. Defining ␭␣␤

= 2兰␻−1_␣ ␣␤ 2 _共␻兲F

␣␤共␻兲d␻, we obtain the partial EPI

con-stants. The values of⌬_␣共␻n兲 enter to the expression for the

superconducting density of states18

N共␻兲 =

兺

␣ N_␣共0兲Re

冋

冏

=⌬_␣共−␻n兲兴 B_␣␤共n,n

⬘

兲 =␭␣␤共n − n

_冑

册

, 共13兲

⌬˜␣共n兲=⌬␣共n兲/

冑

2n − 1, and␳共Tc兲 is the maximum eigenvalue

of the matrix B_␣␤共n,n

⬘

兲.

Here for the simplest Einstein spectrum with the fre-quency ⍀, we have ␭_␣␤共m兲=␭_␣␤⍀2/共⍀2+ 4␲2Tc

2

m2兲. The value of Tcis determined by the equation

␳共Tc兲 = 1. 共14兲

III. STRONG COUPLING

The simplest way to estimate ␳共Tc兲 for superstrong

cou-pling is to put in Eq. 共11兲 n=n

⬘

= 1. In this case, we have ⌬˜␣

冋

␳共Tc兲 +

兺

␥ ␭␣␥

册

=

兺

␤ ␭␣␤关⍀ 2_/共2␲_T c兲2+ 1兴⌬˜␤. 共15兲 In the isotropic system␭_␣␤=␭␦_␣␤, the last two terms in both sides of the equation cancel each other out and we have a standard expression for superstrong coupling 共see Refs. 5

and17兲,

Tc,iso=

⍀

2␲

冑

␭. 共16兲

For the nondiagonal matrix␭_␣␤, this cancellation does not occur and the large␭_␣␤terms play the role of pair breaking 共see Appendix A兲.

Let us consider, for the sake of simplicity, the two-band system. The solution of Eq. 共14兲 has a form

T_c,2b= ⍀ 2␲

冑

A +

冑

B2+ 4CD 2C with the eigenvector

兵⌬˜1,⌬˜2其 =

再

A

⬘

+

冑

B2+ 4CD 2␭21C ,1

冎

, 共17兲 where A =␭21␭11+␭11+ 2␭12␭21+␭12␭22+␭22, A

⬘

=␭11共1 +␭21兲−␭22共1+␭12兲 B=␭21␭11+␭11+ 2␭12␭21+␭12␭22+␭22, C = 1 +␭₁₂+␭₂₁, and D =␭₁₂␭₂₁−␭₁₁␭₂₂.

In this case, in the order of O共1/␭兲共we suppose ␭11

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T_c,2b⯝ ⍀

2␲

冑

具␭典, 共18兲

where具␭典 means averaging over both bands, and 具␭典 =共␭11+␭12兲N1共0兲 + 共␭22+␭21兲N2共0兲

N1共0兲 + N2共0兲

,

and N_␣共0兲 are the partial densities of states. In the case of multiple band system, we recover the general Eq.共3兲. For the

non-Einstein spectrum, we have 具␭典⍀2_{⇒具关␭⍀}2_兴 ␣␤典

=具M_␣␤共1兲典=具2兰₀⬁d␻␻关␣2_{共␻兲F共␻兲兴}

␣␤典. This means that

strong coupling leads to washing out the effects of aniso-tropy.

Similar statements were made in Refs.19 and20 where the authors have considered the momentum dependent interaction. In the former paper, the separable interaction similar to Eq. 共1兲关␣2共␻兲F共␻兲兴pp⬘=␣2共␻兲F共␻兲g共p兲g共p

⬘

兲

with 具g共p兲典=1 was used. They got the result that the expression for Tcin the superstrong limit reduces to the

iso-tropic one while the “pairing potential” is proportional to g共p兲. This contradicts the more general statement in the latter article where the positive 共attractive兲 interactions ␣2_共k,k

_⬘

_,_{␻兲F共k,k}

_⬘

_,_{␻兲 for all k,k}

_⬘

_{was investigated and it}

was shown that the gap function becomes k independent, which leads to the isotropic expression for Tc. The detail

inspection of the situation in Ref.19also shows that the real order parameter, which enters to the density of states关see Eq. 共9兲兴, is isotropic for large ␭.

The exact behavior of Tcdepends on the structure of the

matrix␣_␣␤2 共␻兲F_␣␤共␻兲 but qualitative results remain. We have investigated numerically the evolution of Tcand eigenvectors

⌬␣ as functions of the coupling strength␭␣␤ for the model

matrix of the Eliashberg functions

␣␣␤2 共␻兲F␣␤共␻兲 =␣2共␻兲F共␻兲

冉

1 1/5

1/10 0

冊

. 共19兲 We suppose, for simplicity, 2N1共0兲=N2共0兲 and ␭22= 0

共i.e., no intrinsic superconductivity in the second band兲. The average EPI constant 具␭典 is equal to 0.467␭. Results for Tc

are presented in Fig.1. We see that for weak and intermedi-ate couplings, there is an enhancement of Tc due to

aniso-tropic effects, in comparison to the averaged value. For small EPI, the result coincide with the weak-coupling expression for␭_eff=␭_max= 1.02␭, where ␭_maxis a maximal eigenvalue of the matrix关Eq. 共19兲兴. This enhancement, however, vanishes

for large values of␭ when phonons lead to isotropization of the superconducting order parameter.

We have to note that the result 关Eq. 共18兲兴 is obtained

under the condition of nonvanishing 具␭典 and in the Born approximation21_{for the spin-independent interaction.}

Recently the model for the system with strong-coupling anisotropic interaction was considered in Ref.22. It was sup-posed that the difference between the interaction in the qua-siparticle channel具␣2共k,k

⬘

,␻兲F共k,k

⬘

,␻兲典FSand in the

Coo-per channel 具⌬共k兲␣2共k,k

⬘

,␻兲F共k,k

⬘

,␻兲⌬共k

⬘

兲典FS/具⌬2共k兲典FS

is independent of the coupling strength. The above analyses 共as well as Ref.19and20兲 show that this difference vanishes

for strong coupling. This removes unphysical results for Tc

obtained in this limit in the mentioned paper.

In Appendix B, the sensitivity of Tc to different phonon

modes is considered by calculating the variational deriva-tives. It is shown that the negative 共divergent at small fre-quencies兲 contribution to the nondiagonal variational deriva-tive of Tcvanishes in the strongly coupled regime.

IV. CONCLUSIONS

We have shown that strong-coupling effects in the multi-band superconductors lead to the appearance of strong damp-ing, which results from pair breaking due to interband cou-pling.

For systems with attractive interaction, this effect leads to averaging of order parameters in different bands. As a result, asymptotic behavior of Tc is described by the well-known

single-band expression Tc⬀

冑

具␭⍀2典=

冑

2兰0⬁d␻␻具␣2共␻兲F共␻兲典.

This means that the upper bound on Tc in the superstrong

coupling regime is determined by the averaged coupling con-stant while the higher upper bound corresponding to the maximal eigenvalue of the matrix关␭⍀2_兴

␣␤is never reached. ACKNOWLEDGMENTS

The authors acknowledge many stimulating discussions with I. I. Mazin. The work is partially supported by NanoNed program Grant No. TCS7029.

APPENDIX A

We extend the results of Ref. 14 for effects of low-frequency intermediate boson modes共⍀ⱗ2␲Tc兲 on the

criti-FIG. 1. 共Color online兲 Critical temperature 共a兲 and the gap ratio 共c兲 in the two-band case as a function of intraband coupling con-stant␭ in the first band. The panel 共b兲 shows that Tcin the strongly

coupled regime is determined by the average coupling constant. The panel共c兲 shows the ratio of the order parameters in the two bands. The gap function becomes isotropic in the strongly coupled regime. The numerically calculated ratio⌬1/⌬2is very accurately described by the expression Eq.共17兲 in a broad range of ␭.

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cal temperature of the multiband superconductors. On the real frequency axis, the equations for the complex order pa-rameter⌬_␣共␻兲 and the renormalization function Z_␣共␻兲 have forms共neglecting the Coulomb contribution兲

. Now let us separate the functions ␣_␣␤2 共⍀兲F_␣␤共⍀兲 on to low-energy part关␣_␣␤2 共⍀兲F_␣␤共⍀兲兴⬍and the high-energy one

␣␣␤2 共⍀兲F␣␤共⍀兲 = 关␣␣␤2 共⍀兲F␣␤共⍀兲兴⬍⌰共2␲Tc−⍀兲

+关␣_␣␤2 共⍀兲F_␣␤共⍀兲兴⬎⌰共⍀ − 2␲Tc兲.

The same procedure can be done for the kernel K_␣␤共z

⬘

,␻兲, K_␣␤共z

⬘

,␻兲 = K_␣␤⬍共z

⬘

,␻兲 + K_␣␤⬎共z

⬘

,␻兲. 共A3兲 In the first term on the right-hand side of Eq. 共A3兲, we can

⬘

. We see that the low-frequency excitations play a role of

in-traband and interband static impurities. Inin-traband ⌫_␣␣⬍ ones drop out from the above equation 共so-called Anderson’s theorem兲. It is interesting to note that the famous cancella-tion of the largest terms proporcancella-tional to␭⬍共see, e.g., Ref.5兲

comes not from the strong renormalization of the quasiparti-cle energy共Re Z兲 but from the damping ⌫⬍⬃␲␭⬍T.

APPENDIX B

In Ref.23, the sensitivity of Tcto different phonon modes

was considered by calculating the variational derivatives ␦Tc/␦关␣2共⍀兲F共⍀兲兴␣␤. For the diagonal elements共␣=␤兲, the

result for small ⍀共⍀Ⰶ2␲Tc兲 coincides with the one

ob-tained by Bergmann and Rainer,24 ␦_T

c/␦关␣2共⍀兲F共⍀兲兴⬃⍀,

for the isotropic single-band system. This corresponds to the enhancement of Tc by adding low-frequency phonons

共bosons兲.

In the multiband case, the interband derivative has the following form ␦Tc ␦关␣2_{共⍀兲F共⍀兲兴} ␣⫽␤⬃ N_␣共0兲 ⍀ n

兺

ⱖ1 ⌬␣共␻n兲关⌬␤共␻n兲 − ⌬␣共␻n兲兴 ␻n 2 + O共⍀兲,

and ␦Tc/␦关␣2共⍀兲F共⍀兲兴12 and ␦Tc/␦关␣2共⍀兲F共⍀兲兴21 have

different signs. This contradicts to the symmetry relation 关Eq. 共10兲兴. If we change the function 关␣2共⍀兲F共⍀兲兴12,

the counterpart 关␣2共⍀兲F共⍀兲兴21 has to be changed

automati-cally. Only the symmetrized off-diagonal combination ␦Tc/␦␣2共⍀兲F共⍀兲o.d.has physical meaning. Here

␣2_{共⍀兲F共⍀兲} o.d. =N1共0兲关␣ 2_{共⍀兲F共⍀兲兴} 12+ N2共0兲关␣2共⍀兲F共⍀兲兴21 N₁共0兲 + N₂共0兲 . As a result, we obtain ␦Tc ␦␣2_{共⍀兲F共⍀兲} o.d. ⬃ −N␣共0兲 ⍀ n

兺

ⱖ1 关⌬2共␻n兲 − ⌬1共␻n兲兴2 ␻n 2 + O共⍀兲. 共B1兲 In contrast to the single-band case 共see Ref. 17兲, the

off-diagonal derivative has different behavior in the weak-coupling and strong-coupling regimes. For the former case, one can suppose ⌬_␣共␻n兲=⌬␣共⌰−兩␻n兩兲 and

␦Tc/␦␣2共⍀兲F共⍀兲o.d.⬃− 共⌬2−⌬1兲2

⍀ . This means that the addition

of nondiagonal interaction with low-frequency phonons leads to strong suppression of the critical temperature in weak-coupling anisotropic superconductors. This result was obtained in Ref.25for the anisotropic separable interaction. In the strong-coupling limit, as it was shown above,⌬1⇒⌬2,

then the first term in Eq. 共B1兲 vanishes and

␦Tc/␦␣2共⍀兲F共⍀兲o.d.⬃⍀⬎0, similar to the intraband

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