Advances in Condensed Matter Physics

Volume 2018, Article ID 1380986, 12 pages

https://doi.org/10.1155/2018/1380986

## Superconducting Properties of 3D Low-Density Translation-Invariant Bipolaron Gas

Keldysh Institute of Applied Mathematics, RAS, Miusskaya Sq. 4, Moscow 125047, Russia

Correspondence should be addressed to V. D. Lakhno; ur.bpmi@kal

Received 12 December 2017; Accepted 28 January 2018; Published 15 April 2018

Academic Editor: Jörg Fink

Copyright © 2018 V. D. Lakhno. 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

Consideration is given to thermodynamical properties of a three-dimensional Bose-condensate of translation-invariant bipolarons (TI-bipolarons). The critical temperature of transition, energy, heat capacity, and the transition heat of ideal TI-bipolaron gas are calculated. The results obtained are used to explain experiments on high-temperature superconductors.

#### 1. Introduction

The theory of superconductivity is one of the finest and oldest subject matters of condensed matter physics which involves both macroscopic and microscopic theories as well as derivation of macroscopic equations of the theory from microscopic description [1]. In this sense the theory was thought to be basically completed and its further development was to have been concerned with further details and consideration of various special cases.

The situation changed when the high-temperature superconductivity (HTSC) was discovered [2]. Surprisingly, it was found that, in oxide ceramics, the correlation length is some orders of magnitude less than that in traditional metal superconductors while the ratio of the energy gap to the temperature of superconducting transition is much greater [3]. The current status of research can be found in books and reviews [4–18].

Today the main problem in this field is to develop a microscopic theory capable of explaining experimental facts which cannot be accounted for by the standard BCS theory. One might expect that development of such a theory would not affect the macroscopic theory based on phenomenological approach.

With all the variety of modern versions of HTSC microscopic descriptions: phonon, plasmon, spin, exciton, and other mechanisms, the central point of constructing the microscopic theory is the effect of electron pairing (Cooper effect). In what follows such a bosonization of electrons provides the basis for the description of their superconducting condensate.

The phenomenon of pairing, in a broad sense, is considered as arising of bielectron states, while, in a narrow sense, if the description is based on phonon mechanism, it is treated as formation of bipolaron states [19]. For a long time this view was hindered by a large correlation length or the size of Cooper pairs in BCS theory. For the same reason, over a long period, the superconductivity was not viewed as a boson condensate (see footnote at p. 1177 in [20]). A significant reason of this lack of understanding was a standard idea that bipolarons are very compact particles.

The most dramatic illustration is the use of the small-radius bipolaron (SRB) theory to describe HTSC [10, 21, 22]. It implies that a stable bound bipolaron state is formed at one node of the lattice and subsequently such small-radius bipolarons are considered as a gas of charged bosons (as a variant individual SRP are formed and then are considered within BCS of creation of the bosonic states). Despite the elegance of such a picture, its actual realization for HTSC comes up against inextricable difficulties caused by impossibility to meet antagonistic requirements. On the one hand, the constant of electron-phonon interaction (EPI) should be large for bipolaron states of small radius to form. On the other hand, it should be small for the bipolaron mass (on which the superconducting temperature depends [23–28]) to be small too. Obviously, the HTSC theory based on SRP concept which uses any other (nonphonon) interaction mechanism mentioned above will run into the same problems.

Alternatively, in describing HTSC one can believe that the role of a fundamental charged boson particle can be played by large-radius bipolarons (LRB) [30–34]. Historically just this assumption was made by Ogg [30] and Schafroth [35] long before the development of the SRP theory. When viewing Cooper pairs as a peculiar kind of large-radius bipolaron states, one might expect that the LRP theory should be used to solve the HTSC problem.

As pointed out above, the main obstacle to consistent use of the LRP theory for explaining high-temperature superconductivity was an idea that electron pairs are localized in a small region, the constant of electron-phonon coupling should be large, and, as a consequence, the effective mass of electron pairs should be large.

In the light of the latest advances in the theory of LRP and LRB, namely, in view of development of an all-new concept of delocalized polaron and bipolaron states, translation-invariant polarons (TI-polarons) and bipolarons (TI-bipolarons) [36–42], it seems appropriate to consider their role in the HTSC theory in a new angle.

We recall the main results of the theory of TI-polarons and bipolarons obtained in [36–42]. Notice that consideration of just electron-phonon interaction is not essential for the theory and can be generalized to any type of interaction.

In what follows we will deal only with the main points of the theory important for the HTSC theory. The main result of papers [36–42] is construction of delocalized polaron and bipolaron states in the limit of strong electron-phonon interaction. The theory of TI-bipolarons is based on the theory of TI-polarons [36, 37] and retains the validity of basic statements proved for TI-polarons. The chief of them is the theorem of analytic properties of the ground state of a TI-polaron (accordingly TI-bipolaron) depending on the constant of electron-phonon interaction . The main implication of this statement is the absence of a critical value of the EPI constant , below which the bipolaron state becomes impossible since it decays into independent polaron states. In other words, if there exists a value of , at which the TI-state becomes energetically disadvantageous with respect to its decay into individual polarons, then nothing occurs at this point but for and the state becomes metastable. Hence, over the whole range of variation we can consider TI-polarons as charged bosons capable of forming a superconducting condensate.

Another important property of TI-bipolarons is the possibility of changing the correlation length over the whole range of depending on the Hamiltonian parameters [39]. Hence, it can be both much larger (as is the case in metals) and much less than the characteristic size between the electrons in an electron gas (as happens with ceramics).

A detailed description of the theory of TI-polarons and bipolarons and description of their various properties is given in review [42].

An outstandingly important property of TI-polarons and bipolarons is the availability of an energy gap between their ground and excited states (Section 3).

The above-indicated characteristics can be used to develop a microscopic HTSC theory on the basis of TI-bipolarons.

The paper is arranged as follows. In Section 2 we take Pekar-Froehlich Hamiltonian for a bipolaron as an initial Hamiltonian. The results of three canonical transformations, such as Heisenberg transformation, Lee-Low-Pines transformation, and that of Bogolyubov-Tyablikov are briefly outlined. Equations determining the TI-bipolaron spectrum are derived.

In Section 3 we analyze solutions of the equations for the TI-bipolaron spectrum. It is shown that the spectrum has a gap separating the ground state of a TI-bipolaron from its excited states which form a quasicontinuous spectrum. The concept of an ideal gas of TI-bipolarons is substantiated.

With the use of the spectrum obtained, in Section 4, we consider thermodynamic characteristics of an ideal gas of TI-bipolarons. For various values of the parameters, namely, phonon frequencies, we calculate the values of critical temperatures of Bose condensation, latent heat of transition into the condensed state, heat capacity, and heat capacity jumps at the point of transition.

In Section 5 we discuss the nature of current states in Bose-condensate of TI-bipolarons. It is shown that the transition from a currentless state to a current one is sharp.

In Section 6 the results obtained are compared with the experiment.

In Section 7 we consider the problems of expanding the theory which would enable one to make a more detailed comparison with experimental data on HTSC materials.

In Section 8 we sum up the results obtained.

#### 2. Pekar-Froehlich Hamiltonian: Canonical Transformations

Following [38–42], in describing bipolarons, we will proceed from Pekar-Froehlich Hamiltonian:where are coordinates of the first and second electrons, respectively; , are operators of the birth and annihilation of the field quanta with energy ; is the electron effective mass; the quantity describes Coulomb repulsion between the electrons; is the function of the wave vector :where is the electron charge; and are high-frequency and static dielectric permittivities; is the constant of electron-phonon interaction; is the systems volume.

In the system of the center of mass Hamiltonian (1) takes the form:In what follows in this section we will believe , , (accordingly ).

The coordinates of the center of mass can be excluded from Hamiltonian (3) using Heisenberg’s canonical transformation [43]:Accordingly, the transformed Hamiltonian will be written asFrom (5) it follows that the exact solution of the bipolaron function is determined by the wave function , which contains only relative coordinates and, therefore, is translation-invariant.

Averaging of over yields the Hamiltonian :Equation (6) suggests that the bipolaron Hamiltonian differs from the polaron one in that in the latter the quantity is replaced by and the constants , are added.

With the use of Lee-Low-Pines canonical transformation [44]:where are variational parameters having the sense of the distance by which the field oscillators are displaced from their equilibrium positions:for Hamiltonian :we get Hamiltonian contains linear, threefold, and fourfold terms in the birth and annihilation operators. Its explicit form is given in [36–38].

Then, as is shown in [36, 37], the use of Bogolyubov-Tyablikov canonical transformation [45] for passing on from operators , to new operators : (in which is a diagonal operator), makes mathematical expectation of equal to zero.

In the new operators , Hamiltonian (11) takes on the form : where is the so-called recoil energy. The general expression for was obtained in [37]. Actually, calculation of the ground state energy was performed in [41] by minimization of (14) in and in .

Notice that in the theory of a polaron with broken symmetry a diagonalized electron-phonon Hamiltonian has the form of (13)-(14) [46]. This Hamiltonian can be treated as a Hamiltonian of a polaron and a system of its associated renormalized real phonons or as a Hamiltonian whose quasiparticle excitations spectrum is determined by (13)-(14) [47]. In the latter case excited states of a polaron are Fermi quasiparticles.

In the case of a bipolaron the situation is qualitatively different since a bipolaron is a boson quasiparticle whose spectrum is determined by (13)-(14). Obviously, a gas of such quasiparticles can experience Bose-Einstein condensation (BEC). Treatment of (13)-(14) as a bipolaron and its associated renormalized phonons does not prevent their BEC since maintenance of the number of particles required in this case takes place automatically due to commutation of the total number of renormalized phonons with Hamiltonian (13)-(14).

Renormalized frequencies involved in (13)-(14), according to [36, 37], are determined by the equation for :solutions of which yield the spectrum of values.

#### 3. Energy Spectrum of a TI-Bipolaron

Hamiltonian (13)-(14) is conveniently presented in the form:where in the case of a three-dimensional ionic crystal iswhere is the number of atoms along the th crystallographic axis.

Let us prove the validity of the expression for the spectrum (16), (17). Since operators , obey Bose commutation relations:they can be considered to be operators of birth and annihilation of TI-bipolarons. The energy spectrum of TI-bipolarons, according to (15), is determined by the equationwhereIt is convenient to solve (20) graphically (Figure 1).