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

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Superconducting Properties of 3D Low-Density Translation-Invariant Bipolaron Gas Hindawi Advances in Condensed Matter Physics Volume 2018, Article ID 1380986, 12 pages https://doi.org/10.1155/2018/1380986 Research Article Superconducting Properties of 3D Low-Density Translation-Invariant Bipolaron Gas V. D. Lakhno Keldysh Institute of Applied Mathematics, RAS, Miusskaya Sq. 4, Moscow 125047, Russia Correspondence should be addressed to V. D. Lakhno; [email protected] Received 12 December 2017; Accepted 28 January 2018; Published 15 April 2018 Academic Editor: Jorg¨ Fink Copyright © 2018 V.D. Lakhno. Tis 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. Consideration is given to thermodynamical properties of a three-dimensional Bose-condensate of translation-invariant bipolarons (TI-bipolarons). Te critical temperature of transition, energy, heat capacity, and the transition heat of ideal TI-bipolaron gas are calculated. Te results obtained are used to explain experiments on high-temperature superconductors. 1. Introduction efect). In what follows such a bosonization of electrons pro- vides the basis for the description of their superconducting Te theory of superconductivity is one of the fnest and oldest condensate. subject matters of condensed matter physics which involves Te phenomenon of pairing, in a broad sense, is consid- both macroscopic and microscopic theories as well as deriva- ered as arising of bielectron states, while, in a narrow sense, if tion of macroscopic equations of the theory from microscopic the description is based on phonon mechanism, it is treated description [1]. In this sense the theory was thought to be as formation of bipolaron states [19]. For a long time this basically completed and its further development was to have view was hindered by a large correlation length or the size been concerned with further details and consideration of of Cooper pairs in BCS theory. For the same reason, over a various special cases. long period, the superconductivity was not viewed as a boson Te situation changed when the high-temperature super- condensate (see footnote at p. 1177 in [20]). A signifcant conductivity (HTSC) was discovered [2]. Surprisingly, it reason of this lack of understanding was a standard idea that was found that, in oxide ceramics, the correlation length is bipolarons are very compact particles. some orders of magnitude less than that in traditional metal Te most dramatic illustration is the use of the small- superconductors while the ratio of the energy gap to the radius bipolaron (SRB) theory to describe HTSC [10, 21, 22]. temperature of superconducting transition is much greater It implies that a stable bound bipolaron state is formed at [3]. Te current status of research can be found in books and onenodeofthelatticeandsubsequentlysuchsmall-radius reviews [4–18]. bipolarons are considered as a gas of charged bosons (as a Today the main problem in this feld is to develop a micro- variant individual SRP are formed and then are considered scopic theory capable of explaining experimental facts which within BCS of creation of the bosonic states). Despite the cannot be accounted for by the standard BCS theory. One elegance of such a picture, its actual realization for HTSC might expect that development of such a theory would not comes up against inextricable difculties caused by impos- afect the macroscopic theory based on phenomenological sibility to meet antagonistic requirements. On the one hand, approach. the constant of electron-phonon interaction (EPI) should be With all the variety of modern versions of HTSC micro- large for bipolaron states of small radius to form. On the other scopic descriptions: phonon, plasmon, spin, exciton, and hand,itshouldbesmallforthebipolaronmass(onwhich other mechanisms, the central point of constructing the the superconducting temperature depends [23–28]) to be microscopic theory is the efect of electron pairing (Cooper small too. Obviously, the HTSC theory based on SRP concept 2 Advances in Condensed Matter Physics which uses any other (nonphonon) interaction mechanism Te above-indicated characteristics can be used to mentioned above will run into the same problems. develop a microscopic HTSC theory on the basis of TI- Alternatively, in describing HTSC one can believe that the bipolarons. roleofafundamentalchargedbosonparticlecanbeplayedby Tepaperisarrangedasfollows.InSection2wetake large-radius bipolarons (LRB) [30–34]. Historically just this Pekar-Froehlich Hamiltonian for a bipolaron as an initial assumptionwasmadebyOgg[30]andSchafroth[35]long Hamiltonian. Te results of three canonical transformations, before the development of the SRP theory. When viewing such as Heisenberg transformation, Lee-Low-Pines trans- Cooper pairs as a peculiar kind of large-radius bipolaron formation, and that of Bogolyubov-Tyablikov are briefy states, one might expect that the LRP theory should be used outlined. Equations determining the TI-bipolaron spectrum to solve the HTSC problem. are derived. Aspointedoutabove,themainobstacletoconsistent InSection3weanalyzesolutionsoftheequationsforthe use of the LRP theory for explaining high-temperature TI-bipolaron spectrum. It is shown that the spectrum has a superconductivity was an idea that electron pairs are localized gap separating the ground state of a TI-bipolaron from its in a small region, the constant of electron-phonon coupling excited states which form a quasicontinuous spectrum. Te should be large, and, as a consequence, the efective mass of concept of an ideal gas of TI-bipolarons is substantiated. electron pairs should be large. With the use of the spectrum obtained, in Section 4, In the light of the latest advances in the theory of we consider thermodynamic characteristics of an ideal gas LRP and LRB, namely, in view of development of an all- of TI-bipolarons. For various values of the parameters, new concept of delocalized polaron and bipolaron states, namely,phononfrequencies,wecalculatethevaluesofcritical translation-invariant polarons (TI-polarons) and bipolarons temperatures of Bose condensation, latent heat of transition (TI-bipolarons) [36–42], it seems appropriate to consider into the condensed state, heat capacity, and heat capacity theirroleintheHTSCtheoryinanewangle. jumps at the point of transition. WerecallthemainresultsofthetheoryofTI-polarons In Section 5 we discuss the nature of current states and bipolarons obtained in [36–42]. Notice that considera- in Bose-condensate of TI-bipolarons. It is shown that the tion of just electron-phonon interaction is not essential for transition from a currentless state to a current one is sharp. thetheoryandcanbegeneralizedtoanytypeofinteraction. In Section 6 the results obtained are compared with the In what follows we will deal only with the main points of experiment. the theory important for the HTSC theory. Te main result In Section 7 we consider the problems of expanding the of papers [36–42] is construction of delocalized polaron theory which would enable one to make a more detailed and bipolaron states in the limit of strong electron-phonon comparison with experimental data on HTSC materials. interaction. Te theory of TI-bipolarons is based on the In Section 8 we sum up the results obtained. theory of TI-polarons [36, 37] and retains the validity of basic statements proved for TI-polarons. Te chief of them is the theorem of analytic properties of the ground state 2. Pekar-Froehlich Hamiltonian: of a TI-polaron (accordingly TI-bipolaron) depending on Canonical Transformations the constant of electron-phonon interaction �.Temain implication of this statement is the absence of a critical value Following [38–42], in describing bipolarons, we will proceed of the EPI constant ��,belowwhichthebipolaronstate from Pekar-Froehlich Hamiltonian: 2 2 becomes impossible since it decays into independent polaron ℏ ℏ 0 + � �=− Δ � − Δ � + ∑ℏ� � �� states. In other words, if there exists a value of �,atwhichthe 2�∗ 1 2�∗ 2 � � TI-state becomes energetically disadvantageous with respect � to its decay into individual polarons, then nothing occurs ��→ �→ � +�(� � 1 − � 2�) at this point but for �<�� and the state becomes � � � metastable. Hence, over the whole range of variation we can �→�→ �→�→ (1) � � � 1 � � � 2 consider TI-polarons as charged bosons capable of forming a + ∑ (��� �� +��� �� + �.�.) , superconducting condensate. � Another important property of TI-bipolarons is the � � �2 ��→ �→ � possibility of changing the correlation length over the whole �(� � 1 − � 2�)= � �, � � � ��→� − �→� � range of [0, ∞] depending on the Hamiltonian parameters ∞ � 1 2� [39].Hence,itcanbebothmuchlarger(asisthecasein �→ �→ where � 1 � 2 are coordinates of the frst and second electrons, metals) and much less than the characteristic size between + respectively; �� , �� areoperatorsofthebirthandannihilation the electrons in an electron gas (as happens with ceramics). 0 ∗ A detailed description of the theory of TI-polarons and of the feld quanta with energy ℏ�� =ℏ�0; � is the electron bipolarons and description of their various properties is given efective mass; the quantity � describes Coulomb repulsion in review [42]. between the electrons; �� is the function of the wave vector �: An outstandingly important property of TI-polarons and � 2�ℏ� ℏ� 4�� 1/2 bipolarons is the availability of an energy gap between their � = √ 0 = 0 ( ) , ground and excited states (Section 3). � � ��̃ ��1/2 � Advances in Condensed Matter
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