Advances in Condensed Matter Physics

Volume 2015, Article ID 969564, 10 pages

http://dx.doi.org/10.1155/2015/969564

## The Correlation between the Energy Gap and the Pseudogap Temperature in Cuprates: The YCBCZO and LSHCO Case

Institute of Physics, Czȩstochowa University of Technology, Aleja Armii Krajowej 19, 42-200 Czȩstochowa, Poland

Received 12 February 2015; Accepted 20 May 2015

Academic Editor: Guoying Gao

Copyright © 2015 R. Szczȩśniak et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The paper analyzes the influence of the hole density, the out-of-plane or in-plane disorder, and the isotopic oxygen mass on the zero temperature energy gap (2Δ(0)) (YCBCZO) and (LSHCO) superconductors. It has been found that the energy gap is visibly correlated with the value of the pseudogap temperature (). On the other hand, no correlation between 2Δ(0) and the critical temperature () has been found. The above results mean that the value of the dimensionless ratio can vary very strongly together with the chemical composition, while the parameter does not change significantly. In the paper, the analytical formula which binds the zero temperature energy gap and the pseudogap temperature has been also presented.

The superconductivity in the compounds of copper oxides (cuprates) was discovered in 1986 by Bednorz and Müller [1]. It is now known that in the family of cuprates the compounds of the highest critical temperatures () exist. For example, in the (HBCCO) superconductor under the pressure at 31 GPa, the critical temperature equals about 164 K [2]. However, Takeshita et al. have reported recently that the correct maximum value of the critical temperature for HBCCO is a little bit lower, and it appears at much lower pressure (K at 15 GPa) [3].

The thermodynamics of the high-temperature superconducting state in cuprates differs significantly from the thermodynamics predicted in the framework of the classical BCS theory [4–7]. In addition to the too high value of the critical temperature, the most important difference seems to be in the existence of the second characteristic temperature, which is called the pseudogap temperature ().

Currently, it is believed that the critical temperature in cuprates sets the maximum value of , at which disappears the coherence of the superconducting state, while determines the temperature, in which the energy gap () ceases to exist at the Fermi level [8].

It should be noted that both temperatures are equal in the classical BCS theory, wherein the theory predicts the universal relationship between the value of the zero temperature energy gap and the critical temperature: , where is the Boltzmann constant [9].

In the presented paper, we have examined the impact of the various factors (the hole density (), the disorder, and the oxygen isotopic mass) on the energy gap in the cuprates (YCBCZO) and (LSHCO) [10–12].

The primary objective of the study was to determine the relationship between the zero temperature energy gap and or . The obtained results allowed then the determination of the values of the dimensionless ratios: and .

In the last step, we have derived the analytical formula which binds the energy gap and the pseudogap temperature.

All calculations have been performed in the framework of the theory, which assumes that the pairing mechanism in cuprates is induced by the electron-phonon interaction and the electron-electron correlations renormalized by the phonons. Additionally, through the appropriate selection of the electron band energy, the influence of the quasi-two-dimensionality of the electron system (the cooper-oxygen plane) on the physical properties of the studied compounds has been taken into account.

The reader may find in [13] the detailed description of the considered theory, together with the corresponding analysis leading to the fundamental thermodynamic equation. Additional information is also contained in the following works: [14] presents analysis of the ARPES method, [15, 16] present thermodynamics and ARPES for and , and [17] presents thermodynamics of the high-temperature superconductors with the maximum .

In the cases considered in the presented paper, the fundamental thermodynamic equation that determines the properties of the high-temperature superconducting state of -wave symmetry has the following form [13]:where the pairing potentials for the electron-phonon and electron-electron-phonon interaction have been denoted by and , respectively. The quantity is the amplitude of the order parameter for -wave symmetry: .

The symbol is defined by the following expression:where the function determines the electron band energy, , denotes the hopping integral, and .

The inverted temperature () is given by the expression .

The normalization constant is given by . The symbol represents the characteristic phonon frequency, which is of the order of Debye frequency.

Note that the sum over the momentums in (1) should be replaced with the integral in the following manner: , where is the Heaviside function.

In order to simplify the numerical calculations and perform the analytical calculations in the subsequent part of the work, (1) should be converted into a more convenient form. For this reason, we have introduced the designations , , and

Now, (1) can be rewritten in the following way:where

Using expression (3), we can make the following transformation of (4):

It turns out that (6) can be written in the compact form:

The equivalence of (6) and (7) can be most easily proven when determining the quantity from (7) and then reinserting the resulting formula in the square brackets in (7).

The input parameters for (7) are as follows: the hopping integral, the characteristic phonon frequency, and the pairing potentials.

The same values of and for the YCBCZO superconductor have been assumed as for the compound (YBCO): meV and meV [18, 19]. In the case of the LSHCO superconductor, we have based on the values of and obtained for (LSCO): meV and meV [20, 21].

The pairing potentials and have been chosen in such a way that the values of the critical temperature and the pseudogap temperature calculated on the basis of (7) would agree with the experimental values of and determined in works [10–12]. It should be noted that this can be done in a relatively simple way, because the electron-phonon potential is the unique function of the critical temperature (). Then, we have been able to determine the renormalized potential of the electron-electron interaction: [13]. The values of , and the corresponding results have been presented in Figure 1 and in Table 1.