A model with electron–phonon interaction for high-Tc d-wave cuprates

Physica C 478 (2012) 38–41

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A model with electron–phonon interaction for high-Tc d-wave cuprates S. Orozco ⇑, R.M. Méndez-Moreno, M.A. Ortiz Departamento de Fı´sica, Facultad de Ciencias, Universidad Nacional Autónoma de México, Av. Universidad 3000, Col. Ciudad Universitaria, C.P. 04000, Ap. Postal 21-092, 04021 México, D.F., Mexico

a r t i c l e

i n f o

Article history: Received 10 December 2011 Received in revised form 17 February 2012 Accepted 21 March 2012 Available online 29 March 2012 Keywords: Cuprate d-Wave High-Tc Electron–phonon interaction

a b s t r a c t Within the BCS framework a model with d-wave symmetry is introduced to describe high-Tc superconductor cuprates. The electron–phonon interaction is considered as the dominant cause of superconductivity. In order to increase the electronic density of states at the Fermi level, an anomalous occupation, has been proposed. The phonon available energy is introduced by the half-breathing phonons. Numerical results for the coupling parameter k, the energy gap D0, and parameters of the model, as a function of doping, in the (0.105, 0.22) range, are obtained for the cuprate La2xSrx CuO4. The k values obtained are in the intermediate and the weak coupling region and the experimental results for the gap D0 are reproduced with our model. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction Experimental data accumulated so far for high-Tc copper-oxide superconductors have given some useful clues to unravel the fundamental ingredients responsible for the high transition temperature Tc. However, the underlying physical process remains unknown [1]. In this context, it seems crucial to study new ideas that consider the most relevant experimental evidence and use simplified schematic models to isolate the mechanism(s) that generate high-Tc superconductivity (HTSC). The advances in theoretical methods [2] and the improvement of experimental techniques show the importance of the electron– phonon interaction (EPI) in HTSC. Measurements of ARPES [3–5], as well as tunneling [6] provided enough evidence for the relevant role of phonons in HTSC. The isotope shift of the kink energy gives direct proof of the EPI in the origin of the kink. However, the most obvious and undebatable evidence for the EPI can be found in phonon spectra where the EPI is manifested in the softening and broadening of particular phonons [5,7]. Some of these modes are experimentally found to be strongly coupled, in particular at low doping concentrations [8]. High precision experiments show that sudden changes in electron dispersion curves (kinks) occur from electronic coupling to optical phonon modes like oxygen bondstretching modes or buckling modes [9]. On the other hand, doping dependence of the kink in ARPES suggests that EPI strength decreases when the doping increase, reaching the intermediate coupling regime at optimal doping [5].

⇑ Corresponding author. Tel.: +52 55 56224964; fax: +52 55 56160326. E-mail address: [email protected] (S. Orozco). 0921-4534/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physc.2012.03.048

Various estimates for the EPI strength give the value k = 1 for undoped compounds [3,5]. Recently there has been interest in the introduction of several superconducting components [10] which may arise from multiband superconductors, or from two different electronic components [7,11,12]. ARPES has provided key experimental insights that unveil the electronic structure of high-Tc superconductors, these experimental results show that holes tend to separate into hole-rich and hole-poor regions. The so-called two component electronic structure, can be attributed to these two regions in a nanoscale phase-separated system. The combined results of energy distribution curves and momentum distribution curves suggest that the system has two electronic components and each has similar k-space dispersion [7,12]. In two-band superconductors the superconducting components may arise from Cooper pairing in different bands [13], from two groups of charge carriers associated with free or localized states, or from some lattice (rotational and/or translational) symmetry breaking [14]. All these additional ingredients in the band structure go beyond the single-band picture. Pairing symmetry is an important element toward understanding the mechanism of high-Tc superconductivity. For many cuprate superconductors it is generally accepted that the pairing symmetry is d-wave for hole-doped cuprate superconductors as for electron doped cuprates [15,16]. ARPES experiments have shown that the gap structure on high-Tc cuprate superconductors, as a function of the angle, is similar to a d-wave gap [17]. On the other hand, low temperature specific heat has proved to be a helpful tool in identifying the d-wave symmetry of the superconducting gap [18]. The small but non-vanishing isotope effects in high- Tc cuprates have been shown compatible with d-wave superconductivity. Nevertheless, controversial results have been reported that show

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a different scenario: two-gap, s-wave or p + s wave symmetries. Therefore, the symmetry and the evolution of the gap are questions being debated in the cuprates [19]. The BCS theory considers superconductivity mediated by the electron–phonon coupling. In this scheme a phonon-mediated BCS like model has been presented to describe layered cuprate superconductors. In this framework, a two bands model with s-wave and anomalous occupation, has been proposed in order to increase the density of states (DOS) at the Fermi level [20], to take into account the idea that the tendency towards superconductivity can be enhanced when the Fermi level lies at or close to the energy of a singularity in the DOS. In this model two superconducting components with similar k-space dispersion are considered. Even in the BCS framework, this model that can be taken as a minimal singularity in the DOS, can lead to higher Tc values than those expected from the traditional phonon barrier. The anomalous occupation modifies the DOS near the Fermi level allowing the high Tc values observed. A similar effect can be obtained with other mechanisms such as a van Hove singularity in the DOS [21]. Within the d-wave BCS framework, a two dimensional (2D) model, which takes into account two superconducting components with the same dispersion relation, is proposed in this work to describe high- Tc cuprate superconductors. With this model the DOS at the Fermi level is modified and electron–phonon coupling values in the intermediate coupling region are allowed. The EPI is considered as the dominant cause of superconductivity and dwave pairing is introduced. In order to obtain numerical results, the model here proposed will be used to describe properties of the cuprate superconductor La2x Srx CuO4 in terms of the doping and parameters of the model. Gap values, in agreement with the experimental results in the intermediate and weak coupling regimes are calculated. 2. The model To describe high-Tc cuprate superconductors within the BCS framework, we consider a 2D system with two superconducting components associated with two overlapping bands at the Fermi level. Our starting point is the gap equation, 0

Dðk Þ ¼

P

0

Vðk; k ÞDðkÞ

k

tanhðEk =2kB TÞ ; 2Ek

ð1Þ

with V(k,k0 ) the pairing interaction, E2k ¼ 2k þ D2k , where k ¼ h 2 k2 =2m are the self-consistent single-particle energies, and kB is the Boltzmann constant. For the electron–phonon interaction considered as the dominant cause of superconductivity, we have proposed V(k,k0 ) = V0w(k) w (k0 ), with w(k) = cos(2/k) for dx2 y2 pairing. Then Vðk; k0 Þ ¼ V 0 cosð2/k Þ cos 2/0k , with V0 a constant, when jkj and jk0 j 6 Eph and 0 elsewhere. Eph is the phonon available energy. As usual the attractive BCS interaction is nonzero only for unoccupied orbitals in the neighborhood of the Fermi level EF. Here /k = tan1 (ky/kx) is the angular direction of the momentum in the ab plane. The superconducting gap, D(k) = D(T)w(k). With these considerations we propose the next particular distribution with anomalous occupancy in momentum space, to describe a system with two superconducting components, with the same dispersion relation.

nk ¼ HðckF  kÞ þ HðckF  kÞHðk  bkF Þ;

ð2Þ

with kF the Fermi momentum and 0 < b < c < 1. In order to keep the average number of electron states constant, the parameters are related in a 2D system by the equation

2c2  b2 ¼ 1;

ð3Þ

then only one parameter is independent. The distribution in momentum induces one in energy, Eb < Ec where Eb = b2 EF and Ec = c2 EF. For physical consistency, an important requirement of the model is that the energy associated with the anomalous occupancy parameter (AOP) c2 is not larger than the phonon available energy, Eph, which means (1  c2) EF 6 Eph, i.e., Eph restricts the energy scale of the anomaly (1  c2) EF in this model. The minimum of c2 from this condition can be obtained from

ð1  c2 ÞEF ¼ Eph :

ð4Þ

Eqs. (3) and (4) together will give the minimum c2 value consistent with the model. In the last framework the summation in Eq. (1) is changed to an integration which is done over the two overlapping bands defined above. One gets

k 1¼ 2 þ

Z

Ec þEph

Ec Eph

k 2

Z

EF

Eb

Z 2p

2

d/ cos ð2/Þ tanh

0

Z 2p 0

d/ cos2 ð2/Þ tanh

 pffiffiffiffiffiffi 

Nk

2kB T  pffiffiffiffiffiffi 

Nk

2kB T

dk pffiffiffiffiffi ffi

Nk

dk pffiffiffiffiffi ffi:

Nk

ð5Þ

In this equation Nk = (k  EF)2 + D(T)2 cos2(2/), the coupling parameter is k = V0D(E), with D(E) the electronic density of states, which will be taken as a constant for the 2D system in the integra2 tion range. EF ¼ hmp n2D , with n2D the carriers density per CuO2 layer. The two integrals correspond to the model proposed by Eq. (2). The integration over the surface at Ec for the first component, is restricted to states in the interval Ec  Eph 6 Ek 6 Ec þ Eph . For the second component, in order to conserve the particle number, the integration is restricted to the interval Eb 6 Ek 6 EF, if Ec + Eph > EF, with Eb = (2c2  1)EF, according to Eq. (3). While EF  Ec 6 Eph, implies that the energy difference between the anomalously occupied states must be provided by the material itself. Finally D(T) w(k) = D(T) cos (2/) at the two integrals. At T = 0 K, Eq. (5) will be evaluated and c2 values consistent with the model and the experimental data will be obtained:

1¼

Z 2p k d/ cos2 ð2/Þ 4p 0  2 2 1 Eph  ð1  c ÞEF 1 ð1  c ÞEF þ Eph  sinh þ sinh D0 j cosð2/Þj D0 j cosð2/Þj  2 1 2EF ð1  c Þ þsinh ; D0 j cosð2/Þj

ð6Þ

with D(T = 0) = D0. The model presented in this section can be used to describe high-Tc cuprate superconductors, the AOP c2 and other relevant parameters are determined. In any case a specific material must be selected to introduce the available experimental data. As an application, the relationship between the characteristic parameters will be obtained for La-based compounds at several doping concentrations x. The single layer cuprate superconductor La2xSrx CuO4 has one of the simplest crystal structures among the high-Tc superconductors. This fact makes this cuprate very attractive for both theoretical and experimental studies. High quality single crystals of this material are available with several doping concentrations which are required for experimental studies. Even the determination of charge carrier concentration in the cuprate superconductors is quite difficult; the La2xSrx CuO4 is a system where the carrier concentration is nearly unambiguously determined. For this material, the hole concentration for the CuO2 plane, n2D, is equal to the x value, i.e. to the Sr concentration, as long as the oxygen is stoichiometric [22]. The phonon available energy provided by the material itself is introduced by the half-breathing phonons

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Numerical results are obtained for the cuprate La2xSrx CuO4. Experimental data for Eph, and D0 for several x values are introduced. Neutron scattering experiments [23], the EPI in the photoemission spectra of La2x Srx CuO4 [8] and first-principles phonon calculations of the role of phonons indicate a few phonon modes that are likely to be most relevant for the electron–phonon coupling. Among these, the 70 meV half-breathing mode [5,23] was selected as Eph for our calculations. The energy scale of the anomaly (1  c2) EF must be lower or equal to these phonon modes, because they provide an energy scale accessible to the lattice. There are experimental evidences of band structures beyond the single band picture [14]. For La2 x SrxCuO4 an apparent twocomponent behavior in the ARPES line shapes can be associated to an increase of the electron–phonon coupling [24]. In our approach the two-components system corresponds to the two overlapped bands. Shi et al. [17] measured the coherent d-wave superconducting gap and its momentum dependence in underdoped La2x Srx CuO4 by ARPES. The samples studied correspond to x = 0.105 with Tc = 30 K, and x = 0.145 with Tc = 36 K. They observe sharp spectral peaks along the entire underlying Fermi surface, which trace out a simple d-wave superconducting gap at the antinode, with an amplitude of 14 meV for a x = 0.145 sample, as well as a maximal gap of 26 meV with a x = 0.105 sample. Using these data it is possible to obtain the coupling parameter k at T = 0 as function of the AOP c2 from Eq. (6), in which k is function of c2, D0, Eph and x through EF. The integrals are numerically evaluated to avoid the use of approximations that limits the result to the weak coupling case, as in BCS (k < 0.5). This fact together with the increase of the DOS due to two bands, allows k > 0.5 values in the intermediate coupling region. In Fig. 1 the k values in terms of c2 are shown. To compare the system behavior at different doping values, the results for underdoped La2x SrxCuO4 with x = 0.145 (lower curve) and for x = 0.105 (upper curve) are shown. To show the behavior of k in terms of c2 in a larger range, the inset displays the parabolic behavior for x = 0.105, with a minimum near c2 = 0.746 in the intermediate coupling range. In the allowed c2 range, the underdoped sample with x = 0.105 requires larger coupling parameter k than the moderately underdoped sample with x = 0.145. We can see in these two curves that as x increases the k minimum occurs at greater c2 value. We

assume that this behavior is also valid for x values in the overdoped region. At each curve in Fig. 1, to the right of the minimum k, the coupling parameter increases when the overlapping decreases. When c2 = 1 this model goes back to the BCS model (one band system), then in the underdoped and the nearly optimal doped regions the calculated k values are larger than 1, out of the intermediate coupling region. There is experimental evidence of the superconducting gap behavior with doping in the x range (0.10, 0.18) [25–27]: as x increases the gap D0 decreases. In the overdoped region, for x P 0.19, experimental results are consistent with the d-wave BCS prediction of D0 = 2.14kB Tc [26], for these doping values we can obtain the coupling parameter k as function of c2, using the experimental values of the gap. The curves for k as functions of c2 for x P 0.19 have similar behavior to the ones reported in Fig. 1 with x = 0.105 and x = 0.145. In this overdoped region the minimum k values are obtained at greater c2 values than in the underdoped region. ARPES results show that the kink effect is stronger in the underdoped region and gets weaker with increasing doping [3]. Experimental values of the coupling parameter can be estimated from ARPES data [28], these experimental values are higher than the electron–phonon k because they included the electron– electron interaction. Nevertheless the decreasing behavior of the experimental coupling parameter with doping is assumed to be similar to the k behavior [28]. To determine the superconducting gap D0 for any x in the range 0.10 6 x 6 0.22 we consider, as first approximation, that k is a lineal decreasing function of the doping x. Among all the allowed c2 values for each x it is possible to choose the c2 consistent with this assumption, i.e. one point (c2,k) for each x. Two points were selected to construct k(x): the minimum of k at c2 = 0.746 for x = 0.105 shown in the curve in Fig. 1; and the value of k, at c2 = 0.99 for x = 0.22, i.e., an AOP close to c2 = 1, in order obtain a k value consistent with BCS. Other c2 values for x = 0.145,0.19,0.20 and 0.21 were adjusted to maintain this linear behavior of k with x. With these (x,c2) points the interpolated function c2(x) constructed is shown in Fig. 2. The curve shows that the overlapping decreases when the doping increases and an asymptotic behavior toward BCS is observed for the overlapping parameter in the overdoped region. The corresponding function to the linear fit of the points is k(x) = 0.939  2.118x. This k(x) function is shown in the inset of Fig. 2. In both figures the points designate the values of c2 and k at x = 0.105,0.145,0.19,0.20 0.21, and 0.22. For each x there are only one D0 and one c2 values in Eq. (6) which satisfy the lineal dependence k(x). Then we obtain the prediction for D0 at each x.

Fig. 1. Coupling parameter as a function of the anomalous occupation parameter c2. The upper curve is for x = 0.105 and the lower corresponds to x = 0.145. The inset shows a larger c2 range for the x = 0.105 curve.

Fig. 2. The AOP c2 as a function of x. The lineal fit k(x) is shown in the inset. The points corresponding to values of c2 and k at x = 0.105 and 0.145 were calculated using gap experimental values obtained from ARPES [17]. The points at x = 0.19, 0.20 0.21, and 0.22 were calculated using the d-wave BCS gap.

found in experiments [5,7]. There are also reliable data for the gap D0 for several samples in the superconducting region [17].

3. Results

S. Orozco et al. / Physica C 478 (2012) 38–41

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References

Fig. 3. The continuous curve show the prediction of the model in this work for D0 as a function of doping. The dotted curve shows the weak coupling d-wave BCS results. The rhombs show the experimental values from ARPES [17]; the stars show the derived values from low-temperature specific heat measures [27,26]; the triangle and the squares are reported by Nohara et al. [29] and Nakano et al. [30] respectively.

In Fig. 3 the predictions of this model for the superconducting gap D0 as a function of x in the range 0.105 6 x 6 0.22, are reported and compared with experimental values obtained with different methods. The dotted curve shows the well known weak coupling d-wave BCS result, the rhombs show the experimental values reported by Shi [17] from ARPES; the stars show the derived values from low-temperature specific heat measures by Wen et al. [27] and by Wang et al. [26]; the triangle is reported by Nohara et al. [29]; and the squares, the experimental results obtained by Nakano et al. [30]. In the overdoped region x > 0.19, the D0 curve converges to weak coupling d-wave BCS. This result is obtained for c2 ? 1, in these conditions our model goes to a one component system. From the graphic it is possible to conclude that the model in this work, with two components and the lineal approximation k(x) allows the gap evaluation in all the doped range studied, while the coupling parameter values were found in the range from intermediate to the weak coupling regimens. In contrast, the BCS result (one component) agrees with the experimental D0 values only in the overdoped region, for x > 0.19. In conclusion, we presented a model with anomalous occupancy which takes into account two superconducting components with the same dispersion relation to describe high-Tc cuprate superconductors. Within the BCS framework, for a 2D system, a d-wave pairing symmetry is considered. The enhancing of the DOS with these two superconducting components is associated with two overlapping bands at the Fermi level. The energy provided by the half-breathing phonons introduced as the phonon available energy allows to describe these systems in the intermediate and weak coupling regimes. The lineal approximation k(x) allows to predict the gap values and the d-wave character of the superconducting gap in the range 0.10 6 x 6 0.22. This model simulates quite well intermediate coupling corrections to the BCS framework.

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