Tc and deltao in a phenomenological ''pseudogap'' model

Abstract

We study numerically superconductivity in a system characterized by the presence of a phenomenological ”pseudogap”, Eg, in the energy spectrum, for 0 < T < Ts.T* is a crossover temperature. As a simplification, the pseudogap and the superconducting gap have the same s-wave symmetry. We find that for<FONT FACE=Symbol>"</FONT>Eg <FONT FACE=Symbol>¹</FONT> 0 we require a critical value of the superconducting interaction, Vd, to produce a finite superconducting critical temperature, Tc and another one for deltao <FONT FACE=Symbol>¹</FONT> 0.

T_c and D_o in a phenomenological ''pseudogap'' model

D. Romero^I; L. Sánchez^I; J. J. Rodríguez-Núñez^I; H. Beck^II

^IDepartamento de Física-FACYT, Universidad de Carabobo, Venezuela

^IIInstitut de Physique, Université de Neuchâtel, Switzerland

ABSTRACT

We study numerically superconductivity in a system characterized by the presence of a phenomenological ”pseudogap”, E_g, in the energy spectrum, for 0 < T < T^s.T* is a crossover temperature. As a simplification, the pseudogap and the superconducting gap have the same s–wave symmetry. We find that for "E_g ¹ 0 we require a critical value of the superconducting interaction, V_d, to produce a finite superconducting critical temperature, T_c and another one for D_o ¹ 0.

1 Introduction

After their discovery by Bednorz and Muller[1] in 1986, the high-temperature superconductors (HTSC) are still attracting a lot of interest due to their unusual physical properties, both in the normal and in the superconducting phases. For example, the HTSC exhibit a pseudogap in the energy spectrum in the temperature range 0 < T < T^*. T^* is defined by Maier et al.[2] as the crossover temperature where the spin-susceptibility is a maximum.

There is experimental evidence by the group of Tallon and Loram[3] where the pseudogap persists below T_c, being independent of the superconducting gap. This interpretation is in agreement with the experiment of energy gap evolution in the tunneling spectra of Bi2Sr2CaCu2_8+d performed by Dipasupil et al.[4]. They find that the pseudogap smoothly develops into the superconducting state gap with no tendency to close at T_c. Another proof that the pseudogap and the superconducting gap are independent of each other is given in the experiments of Krasnov et al.[5] where they apply a magnetic field to their superconducting samples and they destroy the superconducting gap, but the pseudogap remains. They conclude that the pseudogap and the superconducting gap coexist in Bi-2212 using intrinsic tunneling spectroscopy.

Rubio Temprano, Trounov and Müller[6] have recently studied the isotope effects on the pseudogap in the high-temperature superconductor La_1.81Ho_0.04Sr_0.15CuO₄ by neutron crystal field spectroscopy. They have found evidence for the opening of an electronic pseudogap at T* » 60 K, above the superconducting critical temperature, T_c » 32 K.

We exploit the consequences of the psudogap on two macroscopic quantities in the superconducting state, namely, the superconducting critical temperature, T_c, and the superconducting order parameter at T = 0, D_o.

This paper is organized as follows. In Section 2, we present the pseudogap model, following the steps of ifrea, Grosu and Crisan[7]. In Section 3 we present our numerical results. In Section 4 we present our discussion and conclusions.

2 The ''pseudogap'' model

We assume[7] that the PG and the normal one-particle self-energy are given by

where G_o(,iw_n) is the free one-particle Green function. is the wave vector and w_n = 2pT (n + 1/2) is the odd Matsubara frequency, with n an integer. With this choice of self-energy (Eq. (1)) is easy to show that the ''PG'' Green function is given by

where we have chosen the pseudogap of pure s-symmetry, since we want to look for details overlooked in Ref. [7]. For example, the authors of Ref. [7] did not find critical pairing interactions to have T_c¹ 0 and D_o¹ 0. These considerations have been properly taken into account by Pistolesi and Nozières[8] in a similar model to ours. A word of caution is in order here. The model we are discussing here appears to be more of a semiconductor type, as recognized in Ref. [8]. To transform the present model into a real pseudogap model, we should include a damping factor, as it has been done by Andrenacci and Beck[9].

The superconducting state in the HTSC is obtained from the two two BCS equations as follows

where (,iw_n) and †(,iw_n) are the diagonal and off-diagonal BCS Green functions, respectively. This interpretation of the PG is equivalent to make the following choice in the T-matrix approximation[10,11,12] for the superconducting self-energy:

This assumption produces two gaps, one coming from the PG and the other one from D in Eq. (4). Our approach is completely different from the one in Refs. [13-16] where they have an effective gap, given by D_eff = . Their approach is equivalent to taking D = 0 in our Eq. (4) and substituting S(,iw_n) by the diagonal self-energy (Eq. (1)), with E_g ® .

Solving Eq. (3), we obtain

where

and Î() = - 2t(cos(k_x) + cos(k_y)) is the free tight binding band in two dimensions. We are not considering here the presence of the chemical potential, which we leave for a future publication [17]. In Section we choose t = 1, as our unit of energy. In Eqs. (5-6), the spectral weights, a_i() (normal ones) and b_i() (superconducting ones), with i = 1,2,3,4, are given by

We have to solve the T_c equation and the gap equation, D_o, at T º 0. They are given by

where k_x = 2n_xp/N_x and k_y = 2n_yp/N_y, with n_x = 0,1,...,N_x-1, and n_y = 0,1,...,N_y-1, since we are solving our discrete system in two dimensions. We have chosen N_x = N_y = 1000 in our numerical calculations. V_d is the absolute value of the pairing interaction. We have used a precision of 10^-5 to solve Eqs. (11-12). From these equations we conclude that = and = 0 when E_g = 0.

3 Numerical results

In Fig. 1 we present T_c vs V_d for several values of the pseudogap parameter, E_g, for the case of pure s-wave symmetry. We observe that there is a critical value of the interaction potential, , in order to have T_c. As we will see in the results for D_o vs V_d, there is also a critical value of the pairing potential below which D_o = 0. In the case of V_d,c coming from the D_o ® 0, these two critical pairing interactions are different. These critical pairing interactions were not discussed in Ref. [7]. However, they were considered in a similar model by Pistolesi and Nozières[8].

In Fig. 2 we present D_o vs V_d for several values of E_g, when the pseudogap and the superconducting order parameter, at T = 0, have the same symmetry, namely, pure s-wave. We need a critical interaction potential, ¹ 0, when E_g ¹ 0, in order to have D_o ¹ 0. From Fig. 2, for E_g = 0.50, » 3.00. Comparing Figs. 1 and 2, we see that for a fixed value of E_g, < . This result implies that the ratio 2D_o(V,E_g)/k_BT_c(V,E_g) is well defined only for V > , for a fixed value of E_g.

From Fig. 3 we plot D_o/t vs V_d/t for several values of the pseudogap parameter, E_g/t, when we adopt the aproximation of Ref. [7], namely, B_o/t º 0. This approximation does not produce a critical value of the interaction potential. In consequence, = 0, " E_g/t. As our Eq. (11) does not have the presence of the factor B_o/t, we cannot perform this approximation. Because of this, T_c/t always needs a critical value of the interaction, for " E_g/t ¹ 0.

4 Discussion and conclusions

In this paper we have considered that the pseudogap and the superconducting order parameter both have the same symmetry, namely, pure s-wave symmetry. This is not a crazy idea, because it has been shown that the observed symmetry of the order parameter cannot be fitted with only the lowest harmonics of the d-wave order parameter[18,19,20,21]. Furthermore, a recent experiment with twisted Josephson junctions in the Bi-cuprates[22], is in favor of an extended s-wave order parameter, and has shown the absence of a d-wave part in the order parameter. However, the inclusion of order parameters with another symmetry is not a difficult task in our working scheme. With this point of view in mind and following the model of ifrea, Grosu and Crisan[7] we have studied their model to treat delicate points like the critical interaction potential.

In summary, we have numerically implemented a model which has a pseudogap (really, it is a semiconductor gap, since damping effects have been neglected in our calculations) in the one-particle energy spectrum of quasi-particles in the temperature range 0 < T < T^*. We have investigated the effect of E_g on the two basic parameters of the BCS theory, T_c and D_o. We have found that for E_g ¹ 0 two critical pairing potentials emerge from our calculations. In consequence, in order to define the ratio R º 2D_o(V_d,E_g)/k_BT_c(V_d,E_g) we need to be above the bigger of the two critical pairing potentials. When E_g = 0, R » 3.5 in the BCS-approximation. We have briefly discussed the case when both order parameters, E_g and D_o, have the same symmetry, namely, pure s-wave symmetry. However, the d-wave symmetry is not difficult to study and we leave for a future publication. Another aspect we could study is the crossover phase diagram from the BCS limit to the Bose-Einstei regime[23].

Acknowledgments

We are very grateful to I. ifrea, J. A. Budagosky-Marcilla, P. Martín, E. V. L. de Mello, J. Ferreira, I. Bonalde, R. Medina, M. Valera, S. Aquino, M. Rodríguez, H. Mateu, E. Orozco, J. Konior, C. Chiang, O. Alvarez-Llamoza and M. A. Suárez, for interesting discussions. The numerical calculations were performed at SUPERCOMP (Departamento de Física-FACYT-UC). We thank C.D.C.H.-U.C.-Venezuela (Project 2001-013), FONACYT-Venezuela (Project S1-2002000448) and FAPERGS-Brazil for partial financial support. One of the authors (J.J.R.N.) is a Fellow of the Venezuelan Program of Scientific Research (PPI-II), a Senior Associate (ICTP-Italy (2003-2008)) and a Visiting Scientist at IVIC-Venezuela. We thank M. D. García-González for helping us with the preparation of the manuscript.

References

[1] J. G. Bednorz and K. A. Müller, Z. Phys. B 64, 189-193 (1986).

[2] Th. A. Maier, M. Jarrel, A. Macridin, and F.-C. Zhang, cond-mat/0208419 (submitted to Phys. Rev. Lett.).

[3] J. L. Tallon and J. W. Loram, Physica C 349, 53 (2001).

[4] R. M. Dipasupil, M. Oda, N. Momono and M. Ido, J. Phys. Soc. Jpn. 71, 1535-1540 (2002).

[5] V. M. Krasnov et al. Phys. Rev. Lett. 84, 5860 (2000)

[6] D. Rubio Temprano, V. Trounov and K. A. Müller, Phys. Rev. B 66, 184506 (2002). See also, P. Häfliger, A. Podlesnyak, K. Conder and A. Furrer, PSI-Zürich Internal Report (2003).

[7] I. ifrea, I. Grosu and M. Crisan, Physica C 371, 104 (2002).

[8] F. Pistolesi and Ph. Nozières, Phys. Rev. B 66, 054501 (2002).

[9] N. Andrenacci and H. Beck, cond-mat/0304084; ibidem, Physica C (to be published, Proceedings of the M2S-HTSC-VII.

[10] M. H. Pedersen, J. J. Rodríguez-Núñez, H. Beck, T. Schneider and S. Schafroth, Z. Phys. B 103, 21-28 (1997).

[11] S. Schafroth and J. J. Rodríguez-Núñez, Z. Phys. B 102, 493-499 (1997).

[12] S. Schafroth, J. J. Rodríguez-Núñez and H. Beck, J. Phys.: Condens. Matter 9, L111-L118 (1997).

[13] Y.-J. Kao, A. P. Iyengar, Q. Chen and K. Levin, Phys. Rev. B 140505(R) (2001).

[14] I. Kosztin, Q. Chen, B. Jankó and K. Levin, cond-mat/9805065.

[15] K. Levin, Q. Chen, I. Kosztin, B. Jankó and A. Iyngar, cond-mat/0107275.

[16] Y.-J. Kao, A. P. Iyengar, J. Stajic, and K. Levin, cond-mat/0207004 v2.

[17] J. J. Rodríguez-Núñez, L. Sánchez, D. Romero and H. Beck (submitted).

[18] V. M. Loktev, R. M. Quick and S. G. Sharapov, Physics Reports 349, 1-123 (2001).

[19] J. Mesot, M. R. Norman, H. Ding et al., Phys. Rev. Lett. 83, 840-843 (1999), ibidem., cond-mat/9812377.

[20] R. Gat, S. Christensen, B. Frazer et al., cond-mat/9906070.

[21] G. G. N. Angilella, A. Sudbo, and R. Pucci, Eur. Phys. J. B 15, 269 (2000).

[22] R. A. Klemm, G. Arnold, A. Bille et al, Int. Mod. Phys. B 13, 3449-3454 1999).

[23] M. B. Soares, F. Kokubun, J. J. Rodríguez-Núñez and O. Rondón, Phys. Rev. B 65, 174506 (2002); see also, A. Perali, P. Pieri, and G. C. Strinati, Phys. Rev. B 68, 066501 (2003); J. Quintanilla and B. L. Györffy, J. Phys. A: Math. Gen. 36, 9375 (2003).

[24] J. J. Rodríguez-Núñez, O. Alvarez, E. Orozco, O. Rondón, F. Kokubun and M. B. Soares Phys. Rev. B 68, 066502 (2003); O. Alvarez-Llamoza, E. Orozco and J. J. Rodríguez-Núñez (submitted).

Received on 23 May, 2003

Publication Dates

• Publication in this collection
  25 Nov 2005
• Date of issue
  Dec 2003

History

• Received
  23 May 2003