Competition between superconductivity and charge-density waves in the electron–phonon system

SSC 5010

PERGAMON

Solid State Communications 113 (2000) 327–330 www.elsevier.com/locate/ssc

Competition between superconductivity and charge-density waves in the electron–phonon system R. Szcze˛s`niak*, M. Mierzejewski Institute of Physics, University of Silesia, 40-007 Katowice, Poland Received 6 August 1999; accepted 13 October 1999 by H. Eschrig

Abstract We have considered the phonon-induced superconductivity and the charge-density waves. In order to find the leading contributions to the corresponding Green functions we made use of the decoupling on the second stage of the equations of motion. This procedure allows one to discuss both instabilities starting directly from the electron–phonon Hamiltonian. In contradistinction to the standard BCS-type approach we have taken into account modification of the normal-state properties which occur because of the electron–phonon interaction. It has been found that these many-body effects can contribute effectively to the stabilization of the superconducting phase. q 1999 Elsevier Science Ltd. All rights reserved. Keywords: A. Superconductors; D. Electron–phonon interactions

Anomalous normal-state properties of high-temperature superconductors have recently attracted a considerable attention. In particular, a gap in the charge excitation spectrum, which exists above the superconducting transition temperature, has been reported in several papers. The normal-state pseudogap in the underdoped superconductors can be inferred from heat capacity [1,2], Knight shift [3], and transport properties [4,5]. In particular, the angleresolved photoemission spectroscopy (ARPES) [6,7] and NMR [8] measurements indicate the presence of the pseudogap which has the dx2 2y2 symmetry. A mechanism, which is responsible for the formation of the normal-state pseudogap, is still unknown. The magnitude of the pseudogap depends on temperature, in particular, on the concentration of holes, vanishing above the optimal doping. Here, we refer to Refs. [9,10] for recent results and discussion. The coexistence of charge-density wave (CDW) and superconductivity (SC) can be considered as a possible scenario, which accounts for this anomalous property of high-Tc superconductors [11,12]. A competition between the charge-density wave and the superconducting gap has been discussed within

* Corresponding author.

the scheme where both the gaps originate either from the electron–phonon interaction [13] or from the exchange of antiferromagnetic spin fluctuations [11]. In Ref. [13] the electron–phonon coupling has been considered within a BCS-type approach, which neglects the impact of this interaction on the energy spectrum in a normal paramagnetic state. However, the relative stability of CDW and SC state can strongly depend on temperature, the electron–phonon interaction-strength and the electronic spectrum [13]. Therefore, the many-body effects which, in particular, lead to the modification of the normal-state properties, can play a nonnegligible role. Within the Eliashberg scheme [14] one accounts for these effects by introducing the wave function renormalization factor and the energy shift. The simultaneous consideration of SC and CDW order parameters within the Eliashberg formalism is a highly nontrivial problem. In the present paper we investigate the competition between superconducting and charge-density wave order parameter with the electron–phonon interaction being explicitly taken into account. Similarly to Ref. [15] we decouple electron and phonon operators on the second stage of equations of motion. Within this approach we stimate a modification of the normal-state properties due to the electron–phonon coupling and its impact on CDW and SC phases.

0038-1098/00/$ - see front matter q 1999 Elsevier Science Ltd. All rights reserved. PII: S0038-109 8(99)00474-3

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Our starting point is the usual form of the electron– phonon Hamiltonian: X 1 X 1 Hˆ ek ck;s ck;s 1 gkk1q c1 k1q;s ck;s …bq 1 b2q † k;s

1

leads to Rck" uc2k# S ˆ

D ; v2 2 Ek2

k;q;s

X

vq b1 q bq :

…1†

Rck" uc1 k1Q# S ˆ

q

2G0 2 G1 ; v2 2 Ek2

…7† …8†

where the quasiparticle energies are given by For the sake of simplicity we neglect the momentum dependence of the electron–phonon interaction, gkk1q ! g; and assume that phonons can be modeled by an Einstein oscillator of frequency v 0. With the help of equations of motion 1 we calculate the Green functions Rck" uc1 k" S; Rck" uck1Q" S and Rck" uc2k# S which determine normal-state, CDW and superconducting properties, respectively. Here, Q ˆ …p; p†: On the second stage of equations of motion we apply a decoupling scheme, which accounts for all possible oneparticle Green functions as represented by the above propagators. In order to obtain analytical results we assume that ek1Q ˆ 2ek [13]. In the case of two-dimensional lattice this corresponds to the dispersion which originates from the nearest-neighbor hopping. Finally, one ends up with the system of equations 2

6 v 1 ek 1 Z1 1 x1 6 6 G1 …v† 2 2N…v; ek †G0 6 4 2Dp …v†

G1 …v† 2 2N…v; 2ek †G0

v ˆ ek 1 Z2 1 x2 0

2g2 X 1 kc c l; v0 k k1Q" k"

G1 …v† ˆ 2

2g2 X N…v; 2ek †kc1 k1Q" ck" l; v0 k

…3†

…4†

and SC state 2g2 X D…v† ˆ 2 N…v; 2ek †kc2k# ck" l; v0 k

…5†

where N…v; ek † ˆ 2

v20 : …v 1 ek †2 2 v20

…6†

Z1, Z2, Z3 and x 1,x 2, x 3 are of the order of g 2. They determine the modification of the energy spectrum in the normal paramagnetic state and correspond, to some extend, to the wave-function renormalization factor and the energy shift in the Eliashberg formalism [14]. It is a difficult problem to account for Z and x values in a self-consistent way. In the BCS-type approach [13] these quantities are neglected as well as the frequency and momentum dependence of the pairing potential …22g2 =v0 †N…v; ek †: This simplification

…9†

Assuming a Lorentzian form of the density of states one can obtain analytical results for the CDW and SC transition temperatures, providing that these instabilities are discussed independently on each other. In particular, one can derive the BCS gap equation for the superconducting order parameter [15], whereas the CDW transition temperature is very close to that derived by Balseiro and Falicov [13] kTCDM ˆ 1:13v0 e21=…leff † ;

…10†

where

  1 1 1 v20 v0 ˆ 1 ˆ ln ; 2 2 2l 2 W 1 v0 leff W 32 3 2 3 Rc uc1 S 2D…v† 76 k1Q" k1Q" 7 6 1 7 76 7 6 7 76 Rck" uc1 7 6 7 0 k1Q" S 7 ˆ 6 0 7: 76 54 5 4 5 1 0 v 2 ek 1 Z3 2 x3 Rc1 2k2Q# uck1Q" S

Here, we have introduced order parameters corresponding to CDW: G0 ˆ 2

Ek2 ˆ e2k 1 ‰2G0 2 G1 Š2 1 uDu2 :

…11†

…2†

and W stands for the bandwidth. In the weak-coupling regime v0 p W and one can assume, with a very good accuracy, that the effective CDW coupling function l eff is twice larger than the superconducting one l ˆ 2g2 …pv20 †21 : Then, the CDW gap function at zero temperature is determined by 1 ˆ 2l

Z1 ∞ 2∞

v20 1 de; e2 1 v20 2…e2 1 G20 †1=2

…12†

As both the order parameters interfere, destructively each to the other, the CDW correlations tend to suppress the SC ones and lead to the lowering of the superconducting transition temperature. Making use of Eq. (7), one can estimate the critical value of the CDW gap function for which the superconducting phase transition takes place at zero temperature 1ˆl

Z1 ∞ 2∞

e2

v20 1 de: 1 v20 2…e2 1 G2cr †1=2

…13†

One can see that Gcr is smaller than the CDW gap function at zero temperature. This result is a consequence of the simplified band structure, where ek1Q ˆ 2ek ; and can not be considered as a general feature of the electron–phonon system [13]. In particular, the many-body effects, which are responsible for the modification of the normal-state properties, can effectively stabilize the superconducting

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we take xi . x and neglect Zi. It leads to the modification of the CDW and SC correlation functions (Eqs. (7) and (8)) which take on the form Rck" uc2k# S ˆ

D ; …v 2 ek 1 x†…v 1 ek 2 x†

Rck" uc1 k1Q" S ˆ

2G0 2 G1 : …v 2 ek 1 x†…v 2 ek 1 x†

…14†

…15†

The many-body effects show up in the equations that determine the transition temperatures for SC and CDW states. Taking the Lorentzian density of states and making use of the digamma function C [16] one gets " ! v2 1 b 1 SC …v0 2 ix† 1 ˆ l 2 0 2 Re C 2 2p x 1 v0 !# ! x 1 bSC 1 1 …16† 2 v0 ImC …v 2 ix† 2 C 2 2 2p 0 Fig. 1. The superconducting transition temperature as a function of x . x is the parameter which, to some extent, accounts for manybody effects. TSC(x ) (TSC(0)) stands for the critical temperature evaluated with (without) the modification of the normal-state properties. The phonon frequency v 0 has been taken as an energy unit.

state. To see the qualitative impact of these effects on the competition between SC and CDW phases, one should take into account the quantities Zi and x i which enter Eq. (2). For the sake of simplicity we neglect their frequency and momentum dependence and consider them as parameters. Within this approximation x 1, x 2, x 3 are of the same order of magnitude and are much larger than Zi. Therefore, to get the first insight into the actual significance of this quantities

for SC transition temperature, kTSC ˆ b21 SC and " ! 5 1 b 1 ˆ 2lRe 1 CDW …v0 2 ix† C 4 2 2p !# 1 ib x 2 CDW 2C 2 2p " ! 1 1 ibCDW 1 …v0 1 x† 2 l ReC 4 2 2p !# 1 ibCDW 1 1 ReC …v0 2 x† 2 2p

Fig. 2. The same as in Fig. 1 but for the CDW transition temperature.

for the CDW transition temperature, kTCDW ˆ b21 CDW : For the sake of brevity, the above equations are presented for v0 p W: Figs. 1 and 2 show the transition temperature as a function of x for superconductivity and charge-density waves, respectively. On one hand, the modification of the normalstate properties acts to the detriment of superconducting transition temperature. However, even for relatively large values of x one obtains a finite value of the critical temperature. This remains in agreement with results, which can be obtained within the Eliashberg-type of approach, where the inclusion of the wave-function renormalization factor results in a reduction of Tc. On the other hand, strong many-body effects lead to a pronounced reduction of the critical temperature for CDW instability. In particular, for sufficiently high values of x there is no solution of the CDW gap equation. These results indicate that many-body effects can play an important role in the stabilization of the superconducting phase. To summarize, we have adopted the method developed in [15] to discuss a competition between superconductivity and charge-density waves, when both these phases originate

…17†

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R. Szcze˛s`niak, M. Mierzejewski / Solid State Communications 113 (2000) 327–330

from the electron–phonon interaction. For the Lorentzian density of states, we have obtained an analytical form of the gap equations for SC and CDW instabilities. When neglecting the modification of the normal-state properties the CDW transition temperature is very close to that obtained by Balseiro and Falicov [13], and is much larger than the critical temperature for the superconducting transition. Although, the many-body effects lead to a reduction of Tc in both SC and CDW channels, these effects can also stabilize the superconducting state. In the present paper we have neglected the electronic correlations and dynamical effects. Therefore, our results should be confirmed within a more reliable method, e.g. Eliashberg formalism, which allows for a straightforward inclusion of these important properties. This problem is under our present investigation.

Acknowledgement We are grateful to Janusz Zielin´ski for a fruitful discussion.

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