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 Physics 3

2�∗� 1/2 1 �2� �→ �→ �=( 0 ) ,�= , �̃−1 =�−1 −�−1, � = {−�∑ ��+� } �, ℏ 2 ℏ� �̃ ∞ 0 1 exp � � 0 �

(2) �→�→ −1 −� � � (4) �1 ��1 =�� , where � is the electron charge; �∞ and �0 are high-frequency andstaticdielectricpermittivities;� is the constant of �→�→ �−1�+� =�+�� . electron-phonon interaction; � is the systems volume. 1 � 1 � In the system of the center of mass Hamiltonian (1) takes the form: Accordingly, the transformed Hamiltonian will be written as 2 2 ℏ ℏ 0 + ��→� � � −1 �=− Δ � − Δ � + ∑ℏ�� +�(� � �) ̃ 2� 2� � � �=�1 ��1 � � � �→ ��→� + �→ �→�→ =−2Δ +�(� � �)+∑� � � � � � � � � � � � + ∑2� (� � + �.�.) , (3) � � cos 2 � (5) � �→�→ 2 �→ �→ ∗ � � + 1 �→ + �→ ( � 1 + � 2) � + ∑2� (� +� )+ (∑ �� ) . �= , �→�=�→� − �→� ,�=2�∗,�= . � cos 2 � � 2 � � 2 1 2 � � 2 � � ℏ=1 �0 =1 Inwhatfollowsinthissectionwewillbelieve , � , From (5) it follows that the exact solution of the bipolaron �� =1(accordingly �� =1/4). �(�) �→ function is determined by the wave function ,whichcon- Te coordinates of the center of mass � can be excluded tains only relative coordinates � and, therefore, is translation- from Hamiltonian (3) using Heisenberg’s canonical transfor- invariant. mation [43]: Averaging of �̃ over �(�) yields the Hamiltonian �:

2 1 �→ �= (∑ ��+� ) + ∑�+� + ∑� (� +�+)+�+�, 2 � � � � � � � � � � � � (6) � �→�→� � � � � � � � =2� ⟨Ψ � � �⟩ , �=⟨Ψ |� (�)| Ψ⟩ , �=−2⟨Ψ�Δ � Ψ⟩ . � � �cos 2 � � ��

−1 Equation (6) suggests that the bipolaron Hamiltonian difers �2 �� =�� +��, from the polaron one in that in the latter the quantity �� is (8) � � � −1 + + replaced by � and the constants , are added. �2 �� =�� +��, With the use of Lee-Low-Pines canonical transformation [44]: ̃ for Hamiltonian �̃: + �2 = exp {∑� (�) (�� −��)} , (7) � ̃ −1 �=�2 ��2, (9) where �� are variational parameters having the sense of the distance by which the feld oscillators are displaced from their equilibrium positions: we get

̃ �=�0 +�1, (10)

2 1 �→ � =2∑� � + ∑�2 + (∑ ��2) + H + �+�, 0 � � � 2 � 0 � � � (11) �→ 2 � + 1 �→ � + + + + � �→ �→ 2 H = ∑� � � + ∑ � � � � � (� � � +� � +� � � +� � ), � =1+ + � ∑ � � . 0 � � � 2 � � � � � �� 2 �� ,�� 4 Advances in Condensed Matter Physics

� � ,�=0; Hamiltonian 1 contains linear, threefold, and fourfold terms { �� in the birth and annihilation operators. Its explicit form is � = 2 � { � (17) given in [36–38]. ] =� +� + � ,�=0,̸ { � �� 0 Ten,asisshownin[36,37],theuseofBogolyubov- 2 Tyablikov canonical transformation [45] for passing on from where in the case of a three-dimensional ionic crystal is �+ � �+,� operators � , � to new operators � �: 2� (� −1) � =± � , ∗ + �� = ∑� � � � + ∑� � � � 1�� 2�� (18) � (12) �� + ∗ + �� =1,2,..., +1, �=�,�,�, �� = ∑�1�� + ∑�2�� 2 �� where �� isthenumberofatomsalongthe th crystallo- (in which H0 is a diagonal operator), makes mathematical graphic axis. expectation of �1 equal to zero. Letusprovethevalidityoftheexpressionforthespec- + �+ � In the new operators �� , �� Hamiltonian (11) takes on the trum (16), (17). Since operators � , � obey Bose commuta- ̃ tion relations: �̃ form : + + + [� ,� ]=�� −� � =� � , ̃ � �� ,� (19) �=�̃ + ∑] �+� , �� (13) they can be considered to be operators of birth and annihila- � tion of TI-bipolarons. Te energy spectrum of TI-bipolarons, � =Δ� +2∑� � + ∑�2 + �+�, according to (15), is determined by the equation �� (14) � � � (�) =1, (20) where Δ�� is the so-called recoil energy. Te general expres- where sion for Δ�� =Δ��{��} was obtained in [37]. Actually, 2 2 2 2 �� = ∑ � � . calculation of the ground state energy �� was performed in ( ) 2 (21) 3 � �−� [41] by minimization of (14) in �� and in �. �� Notice that in the theory of a polaron with broken It is convenient to solve (20) graphically (Figure 1). symmetry a diagonalized electron-phonon Hamiltonian has Figure 1 suggests that the frequencies ]� (index � is the form of (13)-(14) [46]. Tis Hamiltonian can be treated � omitted) lie between the frequencies �� and �� .Hence,the as a Hamiltonian of a polaron and a system of its associated ] � � �+1 spectrum �� as well as the spectrum �� are quasicontinuous: renormalized real phonons or as a Hamiltonian whose −1 ]� −�� =�(� ), which just proves the validity of (16), (17). quasiparticle excitations spectrum is determined by (13)-(14) � � [47]. In the latter case excited states of a polaron are Fermi It follows that the spectrum of a TI-bipolaron has a gap between the ground state �� and the quasicontinuous quasiparticles. � In the case of a bipolaron the situation is qualitatively spectrum, equal to 0. diferent since a bipolaron is a boson quasiparticle whose Below we will consider the case of low concentration spectrum is determined by (13)-(14). Obviously, a gas of such of TI-bipolarons in a crystal. Ten they can adequately quasiparticles can experience Bose-Einstein condensation be considered as an ideal Bose gas, whose properties are (BEC). Treatment of (13)-(14) as a bipolaron and its associated determined by Hamiltonian (16). renormalized phonons does not prevent their BEC since maintenance of the number of particles required in this case 4. Statistical Thermodynamics of Low-Density takes place automatically due to commutation of the total TI-Bipolaron Gas number of renormalized phonons with Hamiltonian (13)- (14). Let us consider an ideal Bose gas of TI-bipolarons which � Renormalized frequencies ]� involved in (13)-(14), represents a system of particles occurring in some volume � �.Letuswrite�0 for the number of particles in the lower according to [36, 37], are determined by the equation for : � one-particle state and � for the number of particles in higher 2 �2�2� 1= ∑ � � , states. Ten 3 �−�2 (15) 1 � � �= ∑ �� = ∑ , �(��−�)/� −1 (22) 2 �=0,1,2,... � solutions of which yield the spectrum of �={]�} values. � �=�0 +�, 3. Energy Spectrum of a TI-Bipolaron 1 � = , 0 (� −�)/� Hamiltonian (13)-(14) is conveniently presented in the form: � �� −1 (23) ̃ � 1 ̃ + � = ∑ . �= ∑ �� , (��−�)/� (16) �=0 ̸ � −1 �=0,1,2,... Advances in Condensed Matter Physics 5

F(s)

2 2 2 2 ]k1 ]k2 ]k3 ]k4

s 2 2 2 2 k1 =1 k2 k3 k4

Figure 1: Graphical solution of (20).

� In expression � (23), we will perform integration over 350 quasicontinuous spectrum (instead of summation) (16), (17) 300 and assume �=�� . As a result, from (22), (23) we get an equation for determining the critical temperature of Bose 250 condensation ��: 200  

̃ f �� =��̃ (��), (24) 150

̃ ̃3/2 ̃ 100 ��̃ (��)=�� 3/2 (�/̃ ��), 50 2 ∞ �1/2�� (�) = ∫ , 3/2 �+� √� 0 � −1 0 10 20 30 40 50  3/2 T �2/32�ℏ2 c � =( ) , �� ∗ (25) 1 2 3 4 5 6 �� C＜Ｊ � �=̃ 0 , �∗

̃ ��  �� = ,  �∗ f where �=�/�. Figure 2 shows a graphical solution of (24) ∗ for the values of parameters �� =2� =2�0,where�0 is ∗ the mass of a free electron in vacuum, � =5meV (≈58 K), 21 −3 �=10 cm ,andthevalues�̃1 =0,2; �̃2 =1; �̃3 =2; �̃ =10 �̃ =15 �̃ =20 T T T T T T 4 ; 5 ; 6 . c1 c2 c3 c4 c5 c6  It is seen from Figure 2 that the critical temperature grows Tc with increasing phonon frequency �0. Te relations of critical Figure 2: Solutions of (24) with �� = 331, 3 and �̃� ={0, 2; 1; 2; 10; temperatures ��/�0� corresponding to the chosen parameter 15; 20} �̃ �̃ =27,3�̃ =30�̃ =32 values are given in Table 1. Table 1 suggests that the critical , which correspond to �� : �1 ; �2 ; �3 ; �̃ =42 �̃ = 46, 2 �̃ =50 temperature of a TI-bipolaron gas is always higher than that �4 ; �5 ; �6 . of ideal Bose gas (IBG). It is also evident from Figure 2 that an increase in the concentration of TI-bipolarons � will lead ∗ to an increase in the critical temperature, while a gain in the It should be stressed, however, that (26) involves �� =2� , �∗ �=0̃ electron mass to its decrease. For the results go over rather than the bipolaron mass. Tis resolves the problem of �=0̃ into the limit of IBG. In particular, (24), for ,yieldsthe the low temperature of condensation which arises both in expression for the critical temperature of IBG: the SRP theory and in the LRP theory in which expression 3, 31ℏ2�2/3 (26) involves the bipolaron mass [31–34]. Another important � = . (26) result is that the critical temperature �� for the parameter � � � values considerably exceeds the gap energy �0. 6 Advances in Condensed Matter Physics

21 −3 Table 1: Calculated characteristics of Bose gas of TI-bipolarons with concentration �=10 cm .

� 0123456

�̃� 0 0,2 1 2 10 15 20

��/�� ∞ 136,6 30 16 4,2 3 2,5

��/�� 1,3 1,44 1,64 1,8 2,5 2,8 3 ̃ −Δ(��V,�/��) 0,11 0,12 0,12 0,13 0,14 0,15 0,15

�V,�(�� −0) 1,9 2,16 2,46 2,7 3,74 4,2 1,6

(�� −��)/�� 0 0,16 0,36 0,52 1,23 1,53 1,8 � ⋅ 3 16 ⋅ 1019 9 4⋅1018 4 2⋅1018 2 0⋅1018 1 2⋅1017 5 2⋅1014 2 3⋅1013 �� cm , , , , , , �̃ =�/�∗ �∗ =5 � � � � � , meV, � is the energy of an optical phonon; �� is the critical temperature of the transition, � is the latent heat of the transition from ̃ ̃ ̃ ̃ ∗ condensate to supracondensate state; −Δ(��V,�/��) = ��V,�/��|�=̃ �̃ +0 −��V,�/��|�=̃ �̃ −0 isthejumpinheatcapacityduringSCtransition,�=�/�; �� V,�(�� −0)is the heat capacity in the SC phase at the critical point; �� =�V(�� −0), �� =�V(�� +0). Te calculations are carried out for the concentration 21 −3 ∗ of TI-bipolarons �=10 cm and the efective mass of a band electron � =�0. Te table also lists the values of concentrations of TI-bipolarons �� for HTSC ��2��3�7, based on the experimental value of the transition temperature �� =93K(Section6).

From (22), (23), it follows that 1 2 3 4 5 6 �� (�̃) �̃3/2 �̃ 1 = �3/2 ( ), � �� ̃ (27) �

� (�̃) � (�̃) /N 0 =1−  � � N Figure 3 shows temperature dependencies of the number of � supracondensate particles � and the number of particles �0 occurring in the condensate for the above-indicated �̃ parameter values �. T T T T T T c1 c2 c3 c4 c5 c6 Figure3suggeststhat,ascouldbeexpected,thenumber T of particles in the condensate grows as the gap �� increases. Te energy of a TI-bipolaron gas � is determined by the expression 1 �= ∑ � � =� � + ∑� � . � � �� 0 � � (28) �=0,1,2,... �=0 ̸ /N  With the use of (16), (17), and (28) the specifc energy (i.e., the ̃ ̃ ∗ ̃ ∗ 1−N energy per one TI-bipolaron) �(�) = �/�� , �� =�� /� will be 1 2 3 4 5 6 �(̃ �)̃ T T T T T T = �̃ c1 c2 c3 c4 c5 c6 �� T (29) ̃5/2 ̃ ̃ � ((�−̃ �)/̃ �)̃ Figure 3: Temperature dependencies of the relative number of � �−� �̃ 5/2 � + �3/2 ( )[ + ], supracondensate particles � /� and the particles occurring in the �� ̃ �̃ � ((�−̃ �)/̃ �)̃ � 3/2 condensate �0/� = 1 − � /� for the parameter values �̃�,givenin Figure 2. 2 ∞ �3/2�� (�) = ∫ , 5/2 �+� (30) √� 0 � −1 Relation of �̃ with the chemical potential of the system � �̃ ∗ where is determined from the equation is given by the formula �=(�−�̃ �� )/� . From (29)-(31) expressions for the free energy Δ� = −(2/3)Δ�, Δ� = � − ̃3/2 �−̃ �̃ � �3/2 ( )=�� , (31) �� ,Δ�=�−�� and entropy � = −��/�� also follow. �̃ Figure 4 illustrates temperature dependencies Δ� = �−̃ ̃ ̃ �̃ � {0, �≤��; �� for the above-indicated parameter values �. Break points �=̃ { (32) Δ� (�)̃ �(̃ �)̃ , �≥̃ �̃ . on the curves � correspond to the values of critical { � � temperatures �� . Advances in Condensed Matter Physics 7

30  with Hamiltonian (1) and therefore is a constant value, that 3 is, c-number.  25 2 For this reason, to consider nonequilibrium properties and, particularly current states, we can use generalization of  E  Heisenberg transformation (4) such that Δ 1 20 �→ �→ + �→ �1 (P) = exp {� (P − ∑ �� )} �. (37) 15 �

26 28 30 32 34 36 38 Te general expression for the functional of the total energy T �→ of a TI-bipolaron for P =0̸ isgivenin[36].TI-bipolarons ̃ ̃ ̃ ̃ Figure 4: Temperature dependencies Δ�(�) = �(�) − �� for the occurring in condensed state have a common wave function �̃ parameter values � presented in Figures 2 and 3. forthewholecondensateanddonotthermalizeinthestate�→ when P =0̸ . In the supracondensate part, phonons whose �→ wave vectors contribute to P (36) are thermalized and their Te dependencies obtained enable us to fnd the heat ̃ ̃ ̃ totalmomentumwillbeequaltozero. capacity of a TI-bipolaron gas: �V(�) = ��/��.Withtheuse ̃ ̃ ̃ Hence, all the changes arising in considering the case of of (29) for �≤��, we express �V(�) as �→ P =0̸ concern only the expression for the ground state �̃3/2 �̃2 �̃ �̃ �̃ �→ � (�)̃ = [ � ( )+6( )� ( ) energy which becomes dependent on P, while the spectrum V ̃2 1/2 ̃ ̃ 3/2 ̃ 2�� of excited states (16), (17) remains unchanged. It follows (33) that, in passing over the critical point, the currentless state �̃ suddenly becomes current which is in agreement with the +5�5/2 ( )] , �̃ experiment. 2 ∞ 1 �� (�) = ∫ . 1/2 �+� (34) √� 0 √� � −1 6. Comparison with the Experiment ̃ Expression (33) yields a well-known exponential dependence Figure 4 shows typical dependencies of �(�).Teysuggest of the heat capacity at low temperatures �V ∽ exp(−�0/�), that at the point of transition the energy is a continuous ̃ caused by the availability of the energy gap �0. function of �. Tis means that the transition per se occurs Figure 5 shows temperature dependencies of the heat without energy expenditure being a phase transition of the ̃ capacity �V(�) for the above-indicated parameter values �̃�. second kind in complete agreement with the experiment. At Table 1 lists the values of the heat capacity jumps: the same time transition of Bose particles from a condensate � � state to a supracondensate one occurs with consumption �� (�)̃ �� (�)̃ � �� (�)̃ � V V � V � of energy which is determined by the value � (Section 4, Δ = � − � (35) ��̃ ��̃ � ��̃ � Table 1), determining the latent heat of transition of a Bose ��=̃ �̃ +0 ��=̃ �̃ −0 � � gas which makes it a phase transition of the 1st kind. ̃ at the transition points for the parameter values ��. By way of example let us consider HTSC ��2��3�7 with Te dependencies obtained will enable one to fnd the the temperature of transition 90÷93K, volume of the unit cell −21 −3 21 −3 latent heat of transition �=��,where� is the entropy 0, 1734⋅10 cm , and concentration of holes �≈10 cm . of supracondensate particles. At the point of transition this According to estimates [48], Fermi energy is equal to �� value is �=2��V(�� − 0)/3,where�V(�) is determined by = 0,37 eV. Concentration of TI-bipolarons in ��2��3�7 is formula (33). For the above-indicated parameter values ��,it found from (24): is given in Table 1. � �� =� (�̃ ) � �� ̃ � (38) 5. Current States of a TI-Bipolaron Gas ̃ with �� =1,6.Table1liststhevaluesof��,� for the values of In the foregoing we have considered equilibrium properties parameters �̃� given in Section 4. It follows from Table 1 that of a TI-bipolaron gas. Te formation of Bose-condensate per ��,� ≪�. Hence, only a small part of charge carriers is in se does not mean that it has superconducting properties. a bipolaron state which justifes the approximation of a low- To demonstrate such a possibility let us consider the total density TI-bipolaron gas used by us. Tis fact correlates with momentum of a TI-bipolaron: recent experiments [49] where it was shown that only a small �→ �→̂ �→̂ �→ part of electrons are in SC state. Te energy levels of such TI- P = � + � + ∑ ��+� , (36) 1 2 � � bipolaronslienearFermisurfaceandaredescribedbythe �→̂ �→̂ wave function: � � �→ where 1 and 2 are the momenta of the frst and second �→ � � �→� �→ �→ Ψ(�)=� � �(�), (39) electron, respectively. It is easy to check that P commutes 8 Advances in Condensed Matter Physics

0

1.5

 1.0 C

0.5

5 10 15 20 25 30  Tc

1 2 1.4 1.5 1.2 1.0 0.8 1.0   C 0.6 C 0.4 0.5

0.2 TC TC 5 10 1520 25 30 5 1015 20 25 30 35 T T

3

1.5

 1.0 C

0.5

10 20 30 T � � =0 �̃ = 25, 2 � (�̃ )=2 � =0,2 Figure 5: Temperature dependencies of the heat capacity for various values of the parameters �: 0 ; �0 ; V �1 ; 1 ; �̃ = 27, 3 � (�̃ −0) = 2, 16 � (�̃ +0) = 1,9 � =1�̃ =30� (�̃ −0) = 2, 46 � (�̃ +0) = 1,8 � =2�̃ = 32, 1 � (�̃ −0) = 2,7 �1 ; V �1 ; V �1 ; 2 ; �2 ; V �2 ; V �2 ; 3 ; �3 ; V �3 ; ̃ �V(��3 + 0) = 1, 78.

which leads to replacement of �,involvedin(6),by For Δ 1 ≫�0 the value of a pseudogap can greatly exceed � � 2 both �� and the energy of the gap (i.e., �0). Te expression �=−2⟨��Δ �� ⟩ + 2��, (40) for the spectrums �� and �� (16)-(17) suggests that the that is, reckoning of the energy from Fermi level (the last term angular dependence of superconducting gap is completely in the right-hand side of (40) in dimensional units is equal to determined by the symmetry of the isoenergetic surface 2 2 ∗ �→ 2��,where�� =ℏ��/2� ). of the �0( �). Earlier this conclusion was made by Bennet According to our approach, superconductivity arises [50], who proved that the main source of anisotropy of whencoupledstatesareformed.Teconditionforthe formation of such states has the form superconducting properties is the angular dependence of the phonon spectrum, though some contribution is also made by � <0, �� (41) the anisotropy of Fermi surface. where �� is the energy of a TI-polaron [41]. Condition (41) It follows from what has been said that formation of determines the value of a pseudogap: apseudogapisaphasetransitionprecedingthephase � � Δ = �� . transition to the superconducting state. Recent experiments 1 � �� (42) [51] also testify in favor of this statement. Advances in Condensed Matter Physics 9

� =� =� =� � =� 2,02 this case � �� ‖, �� ⊥,we ̃ will get instead of �� , determined by (24), the value �� = 2 2,00 �� /�; � =�⊥/�‖ is the parameter of anisotropy. Hence ) 2 anisotropy of efective masses gives for the concentration �� the value �̃�� =�� . Terefore taking account of anisotropy

(mJ/gK 1,98 can, in an order of magnitude, enhance the estimate of the /T

 concentration of TI-bipolarons. If for ��2��3�7 we take

C 2 1,96 the estimate � =30[52], then for the concentration of TI- 19 −3 bipolarons we will get �̃�� =1,4⋅10 cm , which holds valid the general conclusion: in the case under consideration 20 40 60 80 100 only a small number of charge carriers are in TI-bipolaron T (K) state. Te situation can change if the anisotropy parameter is very large. Tus, for example, in layered HTSC Bi-Sr-Ca-Cu- Figure 6: Comparison of the theoretical (solid line) and experimen- Otheanisotropyparameteris� > 100; accordingly, the con- tal [29] (broken line) dependencies in the region of the heat capacity centration of TI-bipolarons in these compounds can have the jump. same order of magnitude as the total concentration of charge carriers. In paper [39] correlation length for TI-bipolarons was Another important conclusion emerging from taking account of the anisotropy of efective masses is that the tem- calculated. According to [39], in HTSC materials its value � � � can vary from several angstroms to several tens of angstroms, perature of the transition � depends not on �� and ‖ indi- which is also in agreement with the experiment. vidually, but on their relation which straightforwardly follows from (24). Of special interest is to determine the characteristic energy of phonons responsible for the formation of TI- bipolarons and superconducting properties of oxide ceram- 7. Essential Generalizations of the Theory ics. To do this let us compare the calculated values of the heat capacity jumps with experimental data. In the foregoing we considered the case of an ideal TI- As is known, in BCS theory, a jump in the heat capacity is bipolaron gas. At small concentration of TI-bipolarons their equal to Coulomb interaction will be greatly screened which justifes the use of the model of an ideal gas. � �� −�� = � =1,43, If the concentration of TI-bipolarons is large (e.g., � � (43) 1021 −3 � �� cm , as was believed in Section 4), then such Bose gas can no longer be considered to be ideal. Taking account of where �� is the heat capacity in the superconducting phase � Coulomb interaction between bipolarons becomes necessary. and � is the heat capacity in the normal phase and is For this purpose we can use Bogolyubov theory [53] for independent of the parameters of the model Hamiltonian. As weakly imperfect Bose gas, which implies that the spectrum it follows from numerical calculations shown in Figure 5 and of elementary excitations with the momentum � will be inTable1,asdistinctfromtheBCStheory,thevalueofthe determined by the expression: jump depends on the phonon frequency. Hence, the approach presented predicts the existence of an isotopy efect for the �2 2 heat capacity jump. � (�) = √�2�2 (�) +( ) , (44) As it is seen from Figure 6, the heat capacity jump 2�� calculated theoretically (Section 4) coincides with the exper- ��2��3�7 �=1,5̃ imental value in [29], for ,thatis,for where �(�) = √��(�)/��, �(�) is the Fourier component of �=7,5meV. Tis corresponds to the concentration of TI- 18 −3 the potential of pairwise interaction between charged bosons: bipolarons equal to �� =2,6⋅10 cm .Hence,incontrast 2 2 �(�) = 4��/�0� , �� and �� are the charge and mass of a to the widespread notion that in oxide ceramics supercon- boson, and �� is the concentration of bosons. ductivityisdeterminedbyhigh-energyphonons(withenergy 70 ÷ 80 meV [52]), actually, the superconductivity in HTSC Hence it follows that materials should be determined by sof phonon modes. 2 2 2 2 � 4�� Notice that in calculations of the temperature of tran- � (�) = √(ℏ� ) +( ) ,�= √ � . (45) � � � sition it was believed that the efective mass � in (24) is 2�� 0�� independent of the direction of the wave vector; that is, isotropic case was dealt with. �� is the plasma frequency. According to (45), the spec- In the anisotropic case, choosing principal axes of trum of excitations of quasiparticles of an imperfect gas vector �,ascoordinateaxes,wewillgetthequantity ischaracterizedbyafniteenergygapwhichinthelong- 1/3 (�� ) instead of the efective mass ��.Incomplex wavelength limit is equal to ��.Tesameresultcanbe HTSC materials the values of efective masses lying in the arrived at if the Coulomb interaction between the electrons plane of layers ��, �� are close in value. Assuming in is considered as a result of electron-plasmon interaction. 10 Advances in Condensed Matter Physics

According to [54], Hamiltonian of electron-plasmon interac- implies that the temperature of the transition is related to tion coincides in structure with Froehlich Hamiltonian which the concentration of charge carriers divided by its efective involves plasmon frequency instead of phonon frequency. mass [55, 56]. Te TI-bipolaron theory of superconductivity Tis straightforwardly leads to the energy gap equal to the developed in this paper gives an answer to the question of ∗ plasmon frequency ��,ifin(45)weput�� =2�, �� =2� , paper [49], namely, where most of electrons disappear in �� =�/2,where� is the electron concentration. Hence, for the superconductors analyzed. It lies in the fact that only a �� <�0, the energy gap will be determined by ��, while for small part of electrons are paired. Te results obtained suggest �� >�0 it will still be determined by �0. that in order to raise the critical temperature �� one should Actually, in real HTSC, there are not only phonon and increase the concentration of bipolarons. plasmon branches, but also some other elementary excita- Application of the theory to 1D and 2D systems leads to tions which can take part in electron pairing. An example is qualitatively new results since the occurrence of a gap in the spin fuctuations. In generalization of the theory to the case TI-bipolaron spectrum automatically removes divergences at of interaction with various brunches of excitations which will small momenta, inherent in the theory of ideal Bose gas. contribute to the ground state energy, the value of the gap and Tese problems were considered by the author in [57]. Te thedispersionlawofaTI-bipolaronareatopicalproblem. case of 1D-polaron was discussed in [58]. Obviously, taking account of this interaction will lead to an increase in the coupling energy of both TI-bipolarons Conflicts of Interest and TI-polarons. Terefore, a priori, without any particular calculations one cannot say anything of how the condition Te author declares that there are no conficts of interest of stability of TI-bipolaron states determined by (41) will regarding the publication of this paper. change. Another important problem is generalization of the the- Acknowledgments ory to the case of intermediate coupling of electron-phonon interaction. Formally, the expression for the functional of Te work was done with the support from the Russian the ground state energy of a TI-bipolaron (14) is valid for Foundation for Basic Research (RFBR Project no. 16-07- any value of the electron-phonon coupling constant. For this 00305). reason such a calculation will not change the spectrum of a TI-bipolaron; however it will alter the criteria of fulfllment References of the conditions of a TI-bipolaron stability (41). [1] E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics, Part 2: 8. Conclusive Remarks Teory of the Condensed State, Butterworth-Heinemann, 1st edition, 1980. In this paper we have presented conclusions emerging [2]J.G.BednorzandK.A.Muller,¨ “Possible high T� superconduc- from consistent translation-invariant consideration of EPI. It tivity in the Ba-La-Cu-O system,” Zeitschrif fur¨ Physik B,vol. implies that pairing of electrons, for any coupling constant, 64, pp. 189–193, 1986. leads to a concept of TI-polarons and TI-bipolarons. Being [3]V.L.Ginzburg,“Superconductivity:thedaybeforeyesterday— bosons, TI-bipolarons can experience Bose condensation yesterday — today — tomorrow,” Physics-Uspekhi, vol. 43, p. 573, leading to superconductivity. Let us list the main results 2000. following from this approach. First and foremost the theory [4] S. L. Kakani and S. Kakani, Superconductivity, Anshan, China, 2009. resolves the problem of the great value of the bipolaron efective mass (Section 4). As a consequence, formal limitations [5] T. Tohyama, “Recent progress in physics of high-temperature superconductors,” Japanese Journal of Applied Physics,vol.51, on the value of the critical temperature of the transition are Article ID 010004, 2012. eliminated too. Te theory quantitatively explains such ther- [6] S. Kruchinin, H. Nagao, and S. Aono, Modern Aspects of modynamic properties of HTSC-conductors as availability superconductivity. Teory of superconductivity, World Scientifc (Section 4) and value (Section 6) of the jump in the heat Publishing Co. Pte. Ltd., River Edge, NJ, USA, 2011. capacity lacking in the theory of Bose condensation of an [7] K. P. Sinha and S. L. Kakani, “Fermion local charged boson ideal gas. Te theory also gives an insight into the occurrence � model and cuprate superconductors,” Proceedings – National of a great ratio between the width of the pseudogap and � Academy of Sciences, India. Section A, Physical Sciences,vol.72, (Section 6). It accounts for the small value of the correlation p. 153, 2002. length [39] and explains the availability of a gap (Section 3) [8] K.H.BennemanandJ.B.Ketterson,Superconductivity: Conven- and a pseudogap (Section 6) in HTSC materials. Te angular tional and Unconventional Superconductors 1-2, Springer, New dependence of the gap and pseudogap gets a natural explana- York,NY,UK,2008. tion (Section 6). [9] J. R. Schriefer, Teory of Superconductivity, Westview Press, Accordingly, isotopic efect automatically follows from Oxford, UK, 1999. expression (24), where the phonon frequency �0 acts as a [10] A. S. Alexandrov, Teory of Superconductivity from weak to gap. Te conclusion of the dependence of the temperature of strong oupling,IOP,publishing,Bristol,UK,2003. the transition �� on the relation �� /�‖ (Section 6) correlates [11] N. M. Plakida, “High temperature cuprate superconductors: with Uemura law universal for all HTSC materials which experiment,” in Teory and Applications,vol.166ofSpringer Advances in Condensed Matter Physics 11

Series in Solid-State Sciences, pp. 479–494, Springer, Berlin, [33] D. Emin, “Formation, motion, and high-temperature supercon- Germany, 2010. ductivity of large bipolarons,” Physical Review Letters,vol.62, [12] I. Askerzade, “Physical Properties of Unconventional Supercon- no.13,pp.1544–1547,1989. ductors,”in Unconventional Superconductors,vol.153ofSpringer [34] V. L. Vinetskii, N. I. Kashirina, and E. A. Pashitskii, “Bipolaron Series in Materials Science, pp. 1–26, Springer Berlin Heidelberg, states in ion crystals and the problem of high temperature Berlin, Germany, 2012. superconductivity,” Ukrainian Journal of Physics,vol.37,no.76, [13] O. Gunnarsson and O. Rosch, “Interplay between electron– 1992. phonon and Coulomb interactions in cuprates,” Journal of [35] M. R. Schafroth, “Superconductivity of a charged boson gas,” Physics, vol. 20, Article ID 043201, 2008. Physical Review,vol.96,no.4,p.1149,1954. [14] T. Moriya and K. Ueda, “Spin fuctuations and high temperature [36] A. V.Tulub, “Recoil accounting in nonrelativistic quantum feld superconductivity,” Advances in Physics,vol.49,no.5,pp.556– theory,” Vestnik Leningrad University,vol.15,no.22,p.104,1960. 606, 2000. [37] A. V. Tulub, “Slow electrons in polar crystals,” Soviet Physics— [15] D. Manske, Teory of Unconventional Superconductors, JETP,vol.14,no.1301,1962. Springer, Heidelberg, Germany, 2004. [38] V.D.Lakhno,“Energyandcriticalionic-bondparameterofa3D [16] R. Szcze¸´sniak and A. P. Durajski, “Anisotropy of the gap large-radius bipolaron,” Journal of Experimental and Teoretical parameter in the hole-doped cuprates,” Superconductor Science Physics,vol.110,no.5,pp.811–815,2010. and Technology,vol.27,no.12,ArticleID125004,2014. [39] V. D. Lakhno, “Translation-invariant bipolarons and the prob- [17] A. P. Durajski, “Anisotropic evolution of energy gap in Bi2212 lem of high-temperature superconductivity,” Solid State Com- superconductor,” Frontiers of Physics, vol. 11, no. 5, Article ID munications,vol.152,p.621,2012. 117408, 2016. [40] N. I. Kashirina, V. D. Lakhno, and A. V. Tulub, “Te virial [18] R. Szcze¸´sniak, A. P. Durajski, and A. M. Duda, “Pseudogap in theorem and the ground state problem in polaron theory,” Eliashberg approach based on electron-phonon and electron- Journal of Experimental and Teoretical Physics,vol.114,no.5, electron-phonon interaction,” Annalen der Physik,vol.529, pp. 867–869, 2012. Article ID 1600254, 2017. [41] V. D. Lakhno, “Translation invariant theory of polaron (bipo- [19] V. D. Lakhno, “Phonon interaction of electrons in the laron) and the problem of quantizing near the classical solu- translation-invariant strong-coupling theory,” Modern Physics tion,” Journal of Experimental and Teoretical Physics,vol.116, Letters B,vol.30,ArticleID1650031,2016. no. 6, pp. 892–896, 2013. [20] J. Bardeen, L. N. Cooper, and J. R. Schriefer, “Teory of [42] V. D. Lakhno, “Pekar’s ansatz and the strong coupling problem superconductivity,” Physical Review Letters,vol.108,pp.1175– in polaron theory,” Physics-Uspekhi, vol. 58, p. 295, 2015. 1204, 1957. [43] W.Heisenberg, “Die Selbstenergie des Elektrons,” Zeitschrif fur¨ [21] A. S. Alexandrov and N. Mott, Polarons Bipolarons, World Sci, Physik,vol.65,no.1-2,pp.4–13,1930. Pub. CO Inc, Singapore, Singapore, 1996. [44] T. D. Lee, F. E. Low, and D. Pines, “Te motion of slow electrons [22] A.S.AlexandrovandA.B.Krebs,“Reviewsoftopicalproblems,” in a polar crystal,” Physical Review,vol.90,no.2,pp.297–302, Soviet Physics—Uspekhi, vol. 35, no. 5, pp. 345–383, 1992. 1953. [23] P. W. Anderson, “Condensed matter: the continuous revolu- [45] S. V. Tyablikov, Methods of Quantum Teory of Magnetizm, tion,” Physics World,vol.8,no.12,p.37,1995. Moscow, Russia, 1975. [24] N. F. Mott, “High-temperature superconductivity debate heats [46] S. J. Miyake, in Polarons and Applications,V.D.Lakhno,Ed., up,” Physics World, vol. 9, no. 1, p. 16, 1996. Wiley, Chichester, 1994. [25] P. W. Anderson, Physics World,vol.9,no.1,p.16,1996. [47] I. B. Levinson and E.´ I. Rashba, “Treshold phenomena and [26] E. V.de Mello and J. Ranninger, “Dynamical properties of small boundstatesinthepolaronproblem,”Soviet Physics—Uspekhi, polarons,” Physical Review B: Condensed Matter and Materials vol. 16, no. 6, pp. 892–912, 1974. Physics,vol.55,no.22,pp.14872–14885,1997. [48] L. P.Gor’kov and N. B. Kopnin, “High-Tcsuperconductors from [27] Y. A. Firsov, V.V.Kabanov, E. K. Kudinov, and A. S. Alexandrov, the experimental point of view,” Soviet Physics—Uspekhi,vol.31, “Comment on “Dynamical properties of small polarons”,” no. 9, pp. 850–860, 1988. Physical Review B: Condensed Matter and Materials Physics,vol. 59,p.12132,1999. [49] I. Bozoviˇ c,X.He,J.Wu,andA.T.Bollinger,“Dependenceofthe´ critical temperature in overdoped copper oxides on superfuid [28] E. V. de Mello and J. Ranninger, “Reply to “Comment on density,” Nature,vol.536,no.7616,pp.309–311,2016. ‘Dynamical properties of small polarons’ ”,” Physical Review B: Condensed Matter and Materials Physics,vol.59,no.18,pp. [50] A. J. Bennett, “Teory of the anisotropic energy gap in super- 12135-12136, 1999. conducting lead,” Physical Review,vol.140,p.A1902,1965. [29] N. Overend, M. A. Howson, and I. D. Lawrie, “3D X-Y scaling [51] R.-H. He, M. Hashimoto, H. Karapetyan et al., “From a single- band metal to a high-temperature superconductor via two of the specifc heat of YBa2Cu3O7−� single crystals,” Physical Review Letters,vol.72,p.3238,1994. thermal phase transitions,” Science,vol.331,p.1579,2011. [30] R. A. Ogg, “Superconductivity in solid metal-ammonia solu- [52] A. Marouchkine, Room-Temperature Superconductivity,vol.7, tions,” Physical Review,vol.70,p.93,1946. Cambridge Int. Sci. Publ., Cambridge, UK, 2004. [31] V. L. Vinetskii and E. A. Pashitskii, “Superfuidity of charged [53] N. N. Bogolyubov, “On the theory of superfuidity,” Journal of Bose-gas and bipolaron mechanism of superconductivity,” Physics USSR,vol.11,p.77,1947. Ukrainian Journal of Physics, vol. 20, p. 338, 1975. [54] V. D. Lakhno, In Polarons and Applications,V.D.Lakhno,Ed., [32] E. A. Pashitskii and V. L. Vinetskii, “Plasmon and bipolaron Wiley, Chichester, 1994. mechanisms of high-temperature superconductivity,” JETP Let- [55] Y.J.Uemura,G.M.Luke,B.J.Sternliebetal.,“Universalcorrela- * ters, vol. 46, p. S104, 1987. tions between T� and n�/� (carrier density over efective mass) 12 Advances in Condensed Matter Physics

in High-T� cuprate superconductors,” Physical Review Letters, vol. 62, p. 2317, 1989. [56] Y. J. Uemura, L. P. Le, G. M. Luke et al., “Basic similarities among cuprate, bismuthate, organic, Chevrel-phase, and heavy- fermion superconductors shown by penetration-depth mea- surements,” Physical Review Letters,vol.66,p.2665,1991. [57] V.D. Lakhno, “Atranslation invariant bipolaron in the Holstein model and superconductivity,” SpringerPlus,vol.5,p.1277,2016. [58] V.D. Lakhno, “Large-radius Holstein polaron and the problem of spontaneous symmetry breaking,” Progress of Teoretical and Experimental Physics,vol.2014,no.7,ArticleID073I01,2014. Journal of International Journal of Advances in The Scientifc Applied Bionics Chemistry Engineering World Journal Geophysics and Biomechanics Hindawi Hindawi Hindawi Publishing Corporation Hindawi Hindawi www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 http://www.hindawi.comwww.hindawi.com Volume 20182013 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018

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