Theoretical investigation of high-temperature superconductivity

PHYSICS REPORTS (Review Section of Physics Letters) 190, Nos. 4 & 5 (1990) 191—306. North-Holland

THEORETICAL INVESTIGATION OF HIGH-TEMPERATURE SUPERCONDUCTIVITY

AS. DAVYDOV Institute for Theoretical Physics, Academy of Sciences of Ukrainian SSR, Kiev, USSR Received August 1989

Contents: 1. Introduction 2. A brief survey of history preceding the discovery of high- T~ superconductivity 2.1. The discovery of superconductivity in metals and alloys 2.2. Phenomenological theory of superconductivity of metals and alloys 2.3. Superconductors of the second kind. Abrikosov vortex lattice 2.4. Early microscopic theories of superconduction in metals 3. Foundations of modern theory of superconductivity in metal compounds 3.1. BCS microscopic theory of superconductivity 3.2. The theory of superconductivity developed by Bogoliubov, Eliashberg and McMillan 3.3. Energy gap in the spectrum of quasiparticle states of a superconductor 4. Superconductivity of compounds that contain transition elements 4.1. The basic properties of intermetallic compounds such as A-iS 4.2. Theoretical models of A-iS superconductors 4.3. Superconductivity of transition metal compounds with NaCl(B1) structure 4.4. Superconductors with heavy fermions 5. Superconductivity of metal oxide compounds 5.1. First metal oxide compounds with small concentration of charge carriers

193 194 194 196 198 201 204 214 210 215 217 217 223 226 230 238

5.2. Discovery of high-temperature superconductivity and its peculiarities 6. Nonphonon models of high-temperature superconductivity 6.1. Early theoretical research in high-temperature superconductivity 7. General information on solitons and bisolitons in quasione-dimensional systems 7.1. One-component solitons 7.2. Two-component solitons 7.3. Three-component solitons. Bisolitons 7.4. Electric charge transport along protein molecules 8. Bisoliton model of high-temperature superconductivity 8.1. Bisoliton condensate in ceramic superconductors 8.2. Conditions for stability of hisohton condensate and critical superconducting current 8.3. The breaking of Cooper pairs in a constant magnetic field 8.4. Meissner effect in the bisoliton model of superconductivity 9. One-particle excitations in a bisoliton model of superconductivity of ceramic oxides 9.1. Introduction 9.2. Deformation field in a superconducting state 9.3. One-particle states in superconducting condensate 9.4. The energy gaps References Note added in proof

238

Single orders for this issue PHYSICS REPORTS (Review Section of Physics Letters) 190, Nos. 4 & 5 (1990) 191—306. Copies of this issue may be obtained at the price given below. All orders should be sent directly to the Publisher. Orders must be accompanied by check. Single issue price Dfl. 87.00, postage included. 0 370-1573/90/$40.60

© Elsevier

Science Publishers B.V. (North-Holland)

241 249 249 258 258 261 264 268 270 270 280 281 291 294 294 294 296 298 301 306

THEORETICAL INVESTIGATION OF HIGH-TEMPERATURE SUPERCONDUCTIVITY

A.S. DAVYDOV Institute for Theoretical Physics, Academy of Sciences of Ukrainian SSR, Kiev, USSR

NORTH-HOLLAND

A. S. Davydov, Theoretical investigation of high-temperature superconductivity

193

Abstract: A brief history of the development ofthe concepts of superconductivity since its discovery and to this day is given. The early phenomenological and microscopic theories of superconductivity of metals and alloys are discussed. The properties of nonmetallic superconductors that contain transition elements and intermetallic compounds having an A-15 structure are analyzed. These superconductors are in wide technological use now. The properties of the compounds of palladium and thorium with hydrogen and deutenum, as well as of compounds with heavy fermions are presented. The theoretical attempts to explain the specific features of nonmetallic superconductors are reviewed. The peculiarities of the new high-temperature superconductors and the different models of their theoretical description based on the concepts of a nonphonon superconductivity mechanism are discussed. A new bisoliton model of high-temperature superconductivity based on a system of nonlinear equations is introduced. An exact solution to these equations is characterized by a single wave function for the whole crystal that describes a condensate of bisolitons — pairs of quasiparticles surrounded by local deformations — that move as a single whole with constant velocity. The conditions under which the condensate is stable are investigated. The bisoliton model enables us to explain the following points without employing nonphonon pairing mechanisms: (i) the small correlation radius of paired particles as compared with BCS theory; (2) a very weak isotopic effect; (3) a nonmonotonic dependence of T, on the current carrier concentration; (4) different values of the energy gap in the spectrum of quasiparticle states determined from tunelling and spectroscopic measurements; (5) a weak dependence of T~on the concentration of magnetic impurities for concentrations lower than the critical one, and an abrupt destruction of superconductivity for concentrations higher than the critical one; (6) a nonmonotonic dependence of T, on the external magnetic field strength.

1. Introduction

In the spring of1986, Muller and Bednorz revealed that at temperatures higher than 30 K the ceramic compounds of lanthanum, barium and copper oxides go into a superconducting state. In February, 1987, Chu et a!. synthesized yttrium compounds which go over into a superconducting state at temperatures close to 90 K. Several months later new superconductors were synthesized almost simultaneously at different laboratories in the USA, Japan, the USSR and other countries. The previously known materials go into a superconducting state when they are cooled by liquid helium with a boiling temperature near 4 K. This is an extremely expensive cooler. Production of artificial materials going into a superconducting state at the liquid nitrogen temperature (77.4K) offers wide use of the superconductivity phenomenon in science and technology. Many specialists maintain the view that the discovery of new materials going into a superconducting state at the liquid nitrogen temperature, can be compared in its profoundness and importance with the discovery of quantum generators lasers. The significance of this discovery was emphasized by the fact that the authors of this discovery, K. Muller and J. Bednorz, were awarded the Nobel prize in 1988 two years after the discovery. Numerous experimental studies of new superconducting materials such as ceramics and monocrystals uncovered unique properties that distinguish them from the other superconducting materials. In this review we present a brief history of the evolution of the superconductivity concept from its discovery up to the present. The first phenomenological and microscopic theories of metal and alloy superconductivity by Bardeen, Cooper, Schrieffer, Bogoliubov and Eliashberg are described. The properties of nonmetal superconductors containing transition elements are discussed. The properties of the intensively applied superconducting intermetal compounds with the p3-tungsten structure are noted. The properties of superconducting compounds of palladium and thorium with hydrogen and deuterium and the superconductivity of compounds with heavy fermions are also discussed. We present a brief history of the discovery of new high-temperature superconductors and discuss the main properties that distinguish them from the old superconductors. In this survey we focus on clarifying the physical properties of different superconductors and on discussing the search for possible theoretical explanations of this extremely interesting and remarkable phenomenon. —

—

194

A .S. Davydoc, Theoretical investigation of high-temperature superconductivity

Today the theory of new superconductors is at the initial stage of its evolution. Many model descriptions, sometimes quite contradictory, are suggested. We think it reasonable to put forward our point of view concerning this fairly interesting problem. Therefore a substantial part of the review is devoted to a new bisoliton model of high-temperature superconductivity developed at the Institute for Theoretical Physics, of the Ukr.SSR Academy of Sciences, by the present author and his colleagues Brizhik, Ermakov, et a!. A bisoliton mode! of superconductivity in metal oxide materials uses the experimental data on their layered structure and the relatively small density of charge carriers. As is done in the BCS theory, it assumes that the superconducting properties are due to the relatively strong electron—phonon interaction. This interaction is determined by changes in the charge state of the complexes (Cu—O) ~ (Cu— O)~in copper—oxygen chains with varying oxygen composition (or under doping). Due to the layered structure of a crystal and relatively large electron—phonon interaction, the superconducting properties of such crystals are described by a system of nonlinear equations. The pairs of excess quasiparticles (holes) in a singlet spin state cause local deformations in a crystal which, in turn, keep them in a paired state (nonlinearity). The exact solution of coupled nonlinear equations for systems of quasiparticles is characterized by a unique wave function for the whole crystal showing a periodic (in space) distribution of bisolitons, i.e., pairs of quasiparticles surrounded by local deformations. The spinless bisoliton condensate formed moves in a crystal as a single unit with constant velocity. The stability of this condensate, which determines the critical velocity and critical current resulting in the breaking of pairs, is studied. The developed bisoliton model of superconductivity of layered metal oxide compounds allows us to explain the following points without involving nonphonon pairing mechanisms: (1) a correlation radius of paired quasiparticles which is small as compared to that in the BCS theory; (2) a very small isotopic effect (without invoking other pairing mechanisms); (3) the nonmonotonic dependence of the temperature of the superconducting transition on the concentration of excess quasiparticles; (4) different energy gap values in the spectrum of quasiparticle states determined from tunneling and spectroscopic measurements; (5) weak dependence of the critical temperature on the concentration of magnetic impurities when it does not exceed the critical value and abrupt destruction of superconductivity for concentrations that do exceed the critical value; (6) nonmonotonic dependence of T~on the external magnetic field strength. —

2. A brief survey of history preceding the discovery of high-Ta superconductivity 2.1. The discovery of superconductivity in metals and alloys The phenomenon of superconductivity was discovered in 1911 by Kamerlingh Onnes at the Leiden cryogenic laboratory [1]. He investigated mercury cooled by liquid helium and distilled to a record purity and found that it produced no resistance to direct current at temperatures below T = 4.3 K. The phenomenon observed was called superconductivity. The temperature T~at which superconductivity occurs was referred to as the critical temperature for a transition into a superconducting state. Shortly afterwards the superconductivity phenomenon was also discovered in other metals (tin, lead, indium, aluminium, niobium, etc.). File and Mills [2] measured the damping of superconducting current and showed that the current lifetime in a superconducting ring is about ~ years.

AS. Davydov, Theoretical investigation of high-temperature superconductivity

195

The superconductivity in metals was destroyed when a specimen was heated above the critical temperature T~ It vanished also at T < T~when a massive specimen was placed into a constant magnetic field of strength H~called the critical magnetic field. It was thought for a long time that the superconduction phenomenon might be observed only in pure metal elements. Intermetallic compounds began to be studied only near the end of the twenties. In Leiden in 1929 the discovery was made that an intermetallic compound, Au2Bi, formed by individually -

nonsuperconducting elements was superconductive at the critical temperature T~= 1.8 K. This triggered a wide-scale search for new superconducting materials, especially intermetallic compounds and alloys. Until 1930, the highest temperature in metals for a transition into a superconducting state had been displayed by lead (T~= 7.2 K). In 1930, niobium was found to be superconductive at T~= 9.2 K. It was established afterwards that some compounds had higher T~than their metallic constituents did individually. For example, titanium and niobium carbides had critical temperatures equal to 9.9 K and 10 K, respectively, whereas the critical temperature of pure titanium is 1.77 K, that of niobium is 9.2 K, and that of hydrogen is 0 K. In 1950—1970, Matthias was conducting a semiempirical investigation in high-temperature superconductors. He formulated a principle [3] relating the value of T~to the number of valence electrons per atom of an alloy or a compound. Yet, the progress in searching for compounds with high T~was slow. In 1941, a niobium hydride (NbH) was found to be superconductive at T~= 13 K, and a niobium nitride (NbN) at T~= 15 K. In 1954, intermetallic compounds V3Si were discovered to be superconductive at T~= 17 K, and Nb3Sn at T~= 18 K. In 1967, Matthias and collaborators [4] succeeded in producing solid solutions of Nb3Al and Nb3Se with superconducting transition temperatures 20 K. In 1973, a NbGe compound was synthesized and its T~,equal to 23.2 K, was then the highest ever recorded. So, within the sixty one years of intensive search for new superconducting materials their critical temperatures went up only from 4 K to 23.2 K. Many researchers thought that superconductivity was impossible above 25—30 K. Matthias, a prominent specialist in synthesizing new superconducting materials, shared this view until his death [5]. Some other investigators were of the same opinion [6]. In the many years after the detection of the superconduction phenomenon there was consensus that the phenomenon implied the total absence of electrical resistance when metal is cooled below the critical temperature. Only in 1933, twenty two years after Kamerlingh Onnes had discovered the superconduction phenomenon, Meissner and Ochsenfeld [7] established that a massive superconductor could be distinguished by ideal diamagnetism, in addition to ideal superconductivity. It was found that a weak magnetic field penetrated into a massive superconductor to a very small depth and was pushed out from the rest of its mass according to the formula H(x)=H(0)exp(—xIL),

x0.

(2.1)

At absolute zero, the depth of penetration, L(0), of a weak magnetic field into Al, Sn, Pb was equal to 100, 340 and 370 A, respectively. As the temperature rises, the penetration depth increases according to the law 112 (2.2) L(t) = L(0)[1 (TIT~)~] Thus it became clear that the behaviour of a superconductor in an external magnetic field is much different from that of an ideal conductor. Inside a massive specimen, the magnetic induction B at T < T~is always zero irrespective of the way the specimen goes into a superconducting state. If only ideal conductivity was present, the magnetic field would persist in a specimen even under cooling. -

—

196

A .S. Davydor. Theoretical investigation of high-temperature superconductivits’

The exclusion of a magnetic field from a massive superconductor was called the Meissner effect. In diamagnetics, the magnetic field induction B and the magnetic field strength H are related by B = where the proportionality coefficient ~.t is referred to as magnetic permeability. It is related to the magnetic susceptibility coefficient x by ~ = 1 + 41Tx. The magnetic susceptibility coefficient x characterises the magnetization M, arising under the influence of an external magnetic field H in a specimen M = XH. For the majority of diamagnetics, the magnetic susceptibility is negative and small, x _(105_106). Inside a specimen, B = 0, according to the Meissner effect, and outside it, B = H. Consequently, the magnetic susceptibility of a superconductor corresponds to x = (4~)1 It is therefore referred to as an ideal diamagnetic. Until 1945, the superconduction phenomenon had found restricted practical application in treating liquid helium and small critical magnetic fields. So, for the first superconductor discovered (mercury with T~= 4.3 K), the critical field even at absolute zero was equal to 400 0. Twenty years ago among the available superconductors there was a special class of compounds with their superconduction transition temperature being about 20 K. These are able to endure magnetic fields up to 20 T. These superconductors include the compounds of some transition metals in the periodic table and some nontransition ones. In April 1986, the Swiss physicists Bednorz and Muller found superconductivity in new artificial materials metal oxides with their transition temperature about 40 K. Shortly afterwards the research done in many laboratories all over the world confirmed the extraordinary advantages of metal oxides in producing high-temperature superconductors. This signaled a new era of research in high-temperature superconductivity. The ways these materials were discovered and their properties will be described in section 5. —

2.2. Phenomenological theory of superconductivity of metals and alloys One of the first phenomenological theories of superconductivity was advanced in 1934 by Gorter and Casimir [8]. Using the concept of two interpenetrating liquids (superconducting and normal ones), they described a second-order transition from a superconducting to a normal state when the magnetic field rises above the critical value H~.The critical field was determined from the condition that the magnetic energy H~I81Tshould be equal to the difference between the free energies of the metal in the normal and superconducting states. The first theory that succeeded in describing the electrodynamics of superconductors was proposed by the brothers F. and G. London in 1935 [9]. The theory was based on the experimentally observed diamagnetic property of metal superconductors the Meissner effect. Assuming that inside a massive superconductor the magnetic induction B is zero, they advanced a hypothesis about an unambiguous local relation between the density of a superconducting current, J, and the vector potential of a magnetic field, A. If the relation is represented as —

J(r)

=

—(cl4ir)A(r),

(2.3)

it follows from the Maxwell equation that the strength of the magnetic field, H = rot A, decreases exponentially from the surface toward the interior of a superconductor, H(x)

=

H(0) exp(—xIL),

x

0,

(2.4)

A.S. Davydov, Theoretical investigation of high-temperature superconductivity

197

Thus, the parameter L in (2.1) defined the London depth of penetration of a magnetic field into a specimen. The screening superconducting current that flows over the surface of a specimen penetrates it to the same depth J(x)

=

[cH(0)I4irL] exp(—xIL).

The phenomenological models proposed by Gorter and Casimir and the London brothers were generalized to a large extent by Ginzburg and Landau in 1950 [10].They treated the case of nonuniform distribution of superconducting electrons in a metal. This enabled them to study an intermediate state at the boundary between normal and superconducting states. The theory was based on the assumption that some part of conduction electrons in a superconducting state of metal forms a peculiar superfluid liquid (condensate), distributed with local density p~(r)over the whole volume of a crystal. The superfluid is capable of moving as a whole with local velocity v~(r). As the temperature rises, part of the electrons “evaporates” from the condensate and forms a “weakly excited gas” a normal fluid which is also distributed over the whole volume of a crystal with local density p~(r)and capable of moving with local velocity v~(r).In this case p,, (r) n(r) p~(r). Remaining within the framework of London’s local theory, Ginzburg and Landau made an important step towards constructing a quasi-microscopic (quantum) theory of superconductivity. They postulated the density p~(r)of superconducting electrons to be proportional to the squared modulus of some wave function i/s5(r) of a superconducting state, which is dependent only on one coordinate. So, a coherent —

—

state of all superconducting electrons was postulated. In the state of dynamic equilibrium, when T> T~, this function is equal to zero, and when T < T~it is different from zero. Ginzburg and Landau [10] investigated the one-dimensional case (superconducting half-space and plane plates). In this case the function ~!i5(r) and the vector potential, that determines the intensity of a magnetic field, are dependent only on the coordinates z, perpendicular to the boundary surface, and satisfy a set of nonlinear equations which is valid near the critical temperature, 2~/i~Idz2 = K2[(A2

—

2AIdz2

l)ç11

d

5

+

ç1i~],

d

=

Aç1i~.

(2.5)

It was assumed that the vector potential A(z) was directed along the x-axis, and the magnetic field intensity H~(z)= dA~Idzwas directed along the y-axis. The boundary conditions were chosen as follows: H~(z= 0)

=

H 0,

(d~!i~Iôz)50 = 0.

It is important that this set of equations involved only one dimensionless parameter K, equal to the ratio of the penetration depth L to the correlation length The authors of ref. [10] investigated an approximate solution to the set of nonlinear equations (2.5) for small K = L I~.In that case, setting ~.

~/s1~,

(p~l,

K
they obtained the following set of linear equations accurate up to terms of order ~ 2 and q’A: 2~/i 2= K2(2ç~+ A2), d2AIdz2 = A,

d

5Idz

(2.6)

198

AS. Davydor, Theoretical investigation of high-temperature superconductivity

and this set has a solution of the form =

1+ [,H~I(2

—

2z

K2)V~1I[(KIV~)e

—

(2.7)

e~V2].

It was mentioned that at K> 1 tV~there is instability in the normal phase of the metal. This point was investigated in greater detail by Abrikosov in 1957. The theory of Ginzburg and Landau is, strictly speaking, applicable to states near the critical temperature. This theory, however, enables one to describe many qualitative features of the behaviour of superconductors in a magnetic field. It stimulated the further development of a microscopic theory of superconductivity, relating the macroscopic properties to the microscopic wave function ~ of a superconductor. The theory is, therefore, referred to as a quasi-microscopic one. In 1959, Gor’kov [11] showed that the wave function qi 5(r) was proportional to the width ii(r) of the energy gap in the spectrum of quasiparticles. He also indicated that at some temperatures and for certain magnetic fields, the Ginzburg—Landau equations follow from a microscopic theory. The London local relation (2.3) is valid, if the current density J~and the vector potential A(r) are smooth functions of r. In the general case, the current i~ (r) at the point r depends on the value of the vector potential at neighbouring points. A phenomenological description of this effect was proposed by Pippard [12] who used the equality

(2.8)

Js(~fAfr’)exp[(T’)1~o]d3T’,

where the parameter ~ defines the effective region of a superconductor that influences a local current. The quantity ~ is called a coherence length. Under very smooth spatial variations in the vector potential the inequality L ~ is valid, the Pippard relation (2.8) reduces to the London one (2.3). The correctness of the phenomenological relation (2.8) has been proved brilliantly by numerous experiments. In terms of the density of superconducting electrons, i.~, the penetration depth is defined by 2/4qri-’~e2)’~2 (2.9) L = (mc where m and e are the mass and charge of an electron, and c is the speed of light. ~‘

,

2.3. Superconductors of the second kind. Abrikosov vortex lattice The Ginzburg—Landau quasimicroscopic theory [10]introduced the important dimensionless parameK = LI~ 0.For pure metals (tin, aluminium, mercury, etc.), the value of K is small. For example, K = 0.16 for mercury (L = 500 A, ~ = 3000 A). As a result, Ginzburg and Landau considered only the cases when KV~~ 1. In 1957, Abrikosov showed [13] that the Ginzburg—Landau theory implied that two groups of superconductors are possible. Superconductors with K\/~ < 1 were referred to as superconductors of the first kind. In these superconductors, when H < H~,the mean magnetic field inside the sample is B = 0. As an external magnetic field increases, there is an abrupt (within the limits of not more than one or two gauss) destruction of superconductivity. ter

A.S. Davydov, Theoretical investigation of high-temperature superconductivity

199

The second group of superconductors are those in which the field partially penetrates into a massive superconductor within some range of magnetic fields. This group includes superconductors with KV’~> 1. These values of K are, generally, characteristic of alloys, for example, lead—bismuth, lead—thallium, lead—cadmium, mercury—cadmium and some impure metals that have a small coherence length ~ The conductors with KV~>1 are called superconductors of the second kind. They are characterized by two critical fields H~1and H~2.In such superconductors an external field does not penetrate into a massive specimen, up to fields H = H~1.As an external field increases from H~1to H~2,the field partially penetrates the specimen so that the field induction increases and, at T~2,approaches a value characteristic of the normal metal. The electric resistance of the specimen, up to H~2,remains zero. In massive superconductors of the second kind, the upper critical field H~2is related to the lower H~1 by

H~2=

KV~HCl.

(2.10)

In these superconductors, the transitions at H~1and H~2are phase transitions of second order. They are not accompanied by release of latent heat, but characterized by a jump in heat capacity. The average field B is zero inside a long cylindrical superconductor if the external field H is smaller

than H~1.When the external field H obeys the inequality H~1 H~2the average field B inside the specimen is equal to the external field H and superconductivity disappears. Thus, superconductors of the second kind, with the values of external magnetic field H lying in the interval from H~1to H~2,are not ideal diamagnetics. At these values of the field, electron pairing is absent along some lines parallel to an external magnetic field. A consistent phenomenological theory of superconductivity of the second kind was developed in 1957 by Abrikosov [13]for K\/~>> 1, using the Ginzburg—Landau quasi-microscopic theory. London’s local approximation is valid in this case. In magnetic fields a little less than H~2,the wave function of a superconducting state is small. This enabled Abrikosov to take account of the influence of nonlinear terms in the Ginzburg—Landau nonlinear equations by the perturbation series method. It was shown that for fields H greater than H~1and slightly different from Hc2, the magnetic flux penetrates into a specimen as a regular structure of tubes, each carrying a quantum of magnetic flux equal to 2. (2.11) = !IcI2e = 2 x i0~G cm At the periphery of each tube there is a vortex of supercurrent that compresses a magneticflux in the central region equal to one quantum of the flux, 4i~.The existence of a quantum of magnetic flux was first indicated by F. London in 1950 [14].When Cooper pairing was disregarded, the quantum of magnetic flux became twice as large as Weak magnetic fields H < H~ 1do not penetrate into a specimen, i.e., the Meissner effect is observed. In this case the eigenenergy of the vortex is greater than the magnetic energy produced when one quantum of magnetic flux penetrates into a superconductor. These energies become comparable when the field is H = H~1.When H> H~1,the magnetic vortices begin to penetrate into the superconductor,

200

AS. Davydov, Theoretical investigation of high-temperature superconductivit

arranging themselves in a position parallel to the external magnetic field. The calculations show that the lines begin to form when the intensity of the field H> H~1,reaches the value 1 ln K (2.12) H~= (2K) As the field increases further, the magnetic flux penetrates into the specimen as distant vortex lines that form a structure similar to a lattice with a very large period. For fields close to H~ 2 at the 2,the field ~i in the lattice sites is zero, the magnetic field has its maximal value there, and it is practically absent space between the lines (superconducting phase). lithe lines are sufficiently separated from one another, they may be regarded as independent, and it is then possible to treat an individual line. Structurally, the vortex line consists mainly of two regions: the central cylindrical region with a diameter approximately equal to the coherence length In this region the density of superconducting electrons, ~/‘j2,rises from zero to one. This internal region is surrounded by an external cylindrical region with a radius of the order of the penetration depth L of a magnetic field. Undamped currents necessary to create one quantum c~of magnetic flux circulate in this region. The energy per unit arc length is given by -

~.

r~=21rK2lnK.

(2.13)

Consequently, if the interaction between the lines is not taken into account, the energy N of vortex lines that cross a unit area is equal to Nr~.The free energy of a superconductor is defined by F=N(r 5—4HK’). (2.14) When the external field is weak, the free energy F is positive and the vortex formation is not advantageous, but at HH~,, where H~,is defined by the equality (2.12), the free energy becomes negative and the vortex formation is advantageous energetically. If in a zero magnetic field, F~is the energy density of a normal state, and F~0is the energy density of a superconducting mixed state of a second-kind superconductor, their difference determines the critical thermomagnetic field as F~ F~0= H~mI8~T.

(2.15)

—

For first-kind superconductors, this relation defines the true critical field H~= Hcm. For second-kind superconductors, the value of Hem characterises only an auxiliary quantity. The condition for the mixed state of a second-kind superconductor to be in thermodynamic equilibrium requires that the field in its normal phase be equal to the critical thermodynamical field Hem. This field is expressed through the parameters L and ~ and the magnetic flux quantum 4~as 1 (2.16) Hcm = ~0(2V’~1TL~0) The second critical field H~ 2of a second-kind superconductor is related to the field Hcm by 1~KHCm= ~ (2.17) H~2= \ -

AS. Davydov, Theoretical investigation of high-temperature superconductivify

201

For materials with small coherence length ~, the superconductivity persists up to very large values of the field H~2.Thus, in V3Ga at T = 0, the second critical field H~2 3 x i05 G. For fields H greater than the second critical field, the magnetic field is not pushed out from a cylindrical specimen (the Meissner effect is absent). But in the region in which the field H satisfies the inequality H~ 3 A) on the surface of a cylinder remains superconductive. The field H~ 2
flow in opposite directions over the outer and inner surfaces of this superconducting layer. With the value of a magnetic field close to H~2,the mixed state in a homogeneous second-kind

superconductor is characterized by Abrikosov’s regular two-dimensional triangular vortex lattice (Kliner et al. [15]).The lattice period decreases with increasing external field. As the value of H reaches H~2,the period attains a value of the order of ~ (vortex lines “come into contact”) and a phase transition of second order from a mixed to a normal state occurs. If a second-kind superconductor is in a mixed state and the transport current induced by an external source flows in a direction perpendicular to the vortices, then the vortices are acted upon by the Lorentz force. This force is perpendicular to a current and the magnetic flux of a vortex. The magnetic vortices acted upon by the Lorentz force move across a transport current. The motion of a vortex magnetic field generates an electric field directed along the vortex and damping the electrons. In a perfectly homogeneous superconducting specimen, the vortex motion in the presence of even an infinitely small Lorentz force is accompanied by energy losses and vanishing superconductivity. In inhomogeneous second-kind superconductors there are always defects of different sorts (grain boundaries, pores, dislocations, etc.). The vortices get fastened at these inhomogeneities. The vortex fastening phenomenon is called a pinning. Superconductors exhibiting a strong pinning are referred to as rigid ones.

When pinning is present, a finite transport current is needed to break off and start moving the vortices. The current density at which the vortex lines begin to break off from pinning centers is called the critical current density. Different superconducting inclusions having a size about equal to the correlation length ~ are effective pinning centers. They are characterized by the “pinning force” defined as the Lorentz force that lets a magnetic vortex go off. Special mechanical and thermal treatment combined with inclusion of nonsuperconducting additions produces rigid superconductors with numerous pinning centers. While the critical magnetic fields of pure metals did not exceed 0.2 T, the rigid superconductors produced at the end of the fifties and the beginning of the sixties from Nb—Ti, Nb—Zr, Nb—Sn and some other alloys made it possible to manufacture small solenoids with their 2. critical magnetic fields up to 10 T and highly dense critical transport currents of about 105_106 Ai~~ 2.4. Early microscopic theories of superconduction in metals

The properties of normal metals are well described by the Bloch—Sommerfeld theory based on the concept that every electron contributing to conduction moves independently in some self-consistent field generated by other conduction electrons and periodically situated metal ions. The influence of this field may approximately be taken into account by replacing the free electron mass me with the effective mass m*. This approximation is, commonly, referred to as the effective mass approximation.

When this approximation is used, the electrons in a metal are substituted by quasiparticles that carry the charge and spin of a free electron, but have the inertial properties determined both by the mass of a

202

A .S. Davydov, Theoretical investigation of high-temperature superconductivity

free electron and the influence of a periodic ion field. Generally, the variation of the mass is 10%—15% of that of a free electron. In the effective mass approximation, the states of individual electrons are characterized by plane waves that describe the states with well-defined values of the wave vector k, spin 1/2 (in units of II), and the energy E(k) = 112k2/2m*. When the crystal is in the ground state at zero temperature, all one-electron states with energy E(k) lower than the Fermi energy EF are, according to the Pauli principle, occupied by electron pairs with opposite spins, while the states with energy E(k) greater than EF are free. In an isotropic crystal in k-space, the Fermi energy is represented as a sphere with radius kF, so that EF

h2k~I2m*.

=

(2.18)

In normal metals, EF has order of magnitude equal to 10 eV. The Bloch—Sommerfeld model does not take into account the correlations between electrons that are generated by the Coulomb interaction between electrons; these are neglected in the average periodic field that determines the effective mass of electrons. The interaction with ions displaced from their equilibrium periodic positions is regarded, in quantum terms, as the interaction with virtual phonons and, therefore, referred to as an electron—phonon interaction. Since the Bloch—Sommerfeld theory did not take the correlation between conduction electrons into account, it was unable to explain the superconductivity phenomenon. The physical foundations of the quantum microscopic theory of superconductivity were formulated by F. London [161.He introduced the concept of a very strong correlation between conduction electrons at low temperatures and assumed that the wave function describing the superfluid component applied to the whole crystal generally and was “rigid”, i.e., it remained unchanged under the action of small perturbations thermal vibrations and a weak external magnetic field. In an isolated superconductor in thermal equilibrium at T 0, the electric current j, according to London’s equation (2.3), appears only in the presence of an external magnetic field and is unambiguously defined by the vector potential A of this field. Furthermore, the superconducting state of a crystal is regarded as a one-quantum state extended over the whole crystal. Employing the concept that the electron—phonon interaction plays a major role in the superconductivity phenomenon, Fröhlich predicted in 1950 [17] a very important feature of superconductors the dependence of the critical temperature T~on the isotopic composition. The dependence was referred to as the isotopic effect. In the same year the isotopic effect was discovered independently in experiments by Maxwell [18] and Reynolds and Serin [19]. The discovery of the isotopic effect confirmed Fröhlich’s basic concept that the electron—phonon interaction is crucial for superconductivity. It became clear that a microscopic superconduction theory should be based on taking this interaction into account. The early attempts to construct such a theory ran across considerable mathematical difficulties experienced in trying to solve the problem by methods of perturbation series. Fröhlich’s works were the most important contributions among the first microscopic theories aimed at taking electron—phonon interaction into account [17,20]. As the main form for the Hamiltonian that describes the interaction of conduction band quasipartides with the field of longitudinal phonons, Fröhlich used the expression —

—

H=

~

[E(k)

—

~]a~akS + ~ hQ(q)b~bq+ H~ 5,

(2.19)

AS. Davydov, Theoretical investigation of high-temperature superconductivity

~ 11int

—

/ ( \\1/2 + 2~ Z.1 \ _____ ~ r 3) aksak+qsuq

~ ‘v ~ k,q,s

WI V

o’-~

203

~ Ilerm. conj.

where L is the volume of the system; a, a,~,b

,

are the creation and annihilation operators for the relevant quasiparticles and phonons; E(k) = h2k212m* is the energy of the free quasiparticles in the conduction band; j.~is the chemical potential that takes into account the constant value of an average number of quasiparticles. At low temperatures j.t = EF, where EF = h2k~I2m * is the Fermi level energy in the conduction band of free quasiparticles, hIl( q) the phonon energy, o the electron—phonon interaction energy, M the ion mass, V 0 the speed of sound. By using the canonical transformation that annihilates terms linear in a-, Fröhlich showed that in the second order of perturbation theory there is an effective interaction between states with oppositely directed quasi-momenta and spin projections. It has an attractive character and is maximal in a very thin layer near the Fermi surface. With this property of interelectronic interaction taken into account, it is possible to replace the Hamiltonian (2.19) by the effective Hamiltonian HF = 2

E e(k)a~a~1 L —

~

bq

Vk,kakfa_kfa.kfak~,

(2.21)

which includes only the attractive interaction between quasiparticles in a narrow region near the Fermi surface, s(k)=E(k)—EF,

e(k)I~EF.

(2.22)

The width of this region e(k) is approximately equal to the energy limit hQ( q) of virtual phonons that participate in the interaction. Vkk are the matrix elements of the inter-electron interaction generated by virtual phonons. Fröhlich was the first to substantiate theoretically [17,20] the isotopic effect and

introduce an important concept about the indirect interaction between electrons generated by their interaction with phonons. Fröhlich’s Hamiltonian (2.21) that involves the attractive interaction between electrons, with equal but oppositely directed quasimomenta and spins underlay all further microscopic theories of metal and alloy superconductivity. In 1954 H. Fröhlich produced a very interesting paper [21] in which he treated the very strong coupling of electrons with ion vibrations in the framework of a one-dimensional model of a metal. The interaction of electrons with lattice ions was chosen to be so strong that all electrons adiabatically followed lattice vibrations. The isotopic effect was, therefore, absent in the theory. In that paper, Fröhlich made an attempt for the first time to go beyond the framework of

perturbation theory. By studying the one-dimensional model of a metal which had continuously distributed ion charges and the conduction band half-occupied by electrons, he showed that taking into account the interaction of electrons with a field of lattice vibrations (which has a simple sinusoidal form)

resulted in an energy gap in the electron spectrum. In the absence of a current and at low temperatures the electrons occupied the energy states below the gap 4. The value of the gap was expressed nonanalytically in terms of the dimensionless parameter A of electron—phonon interaction using the formula 4—exp(—1/A).

(2.23)

204

A.S. Davydov, Theoretical investigation of high-temperature superconductivity

The relations derived demonstrated that they could not be obtained by means of perturbation theory in the small parameter A to any of its finite approximations. It is worth mentioning that the occurrence of an energy gap in the electron spectrum is not itself a sufficient reason for superconductivity. A similar gap is also characteristic of dielectrics. Fröhlich also determined the temperature dependence of the energy gap width 4. He established that it became zero at some critical temperature. He also concluded that the heat capacity of a crystal near zero temperature varied exponentially. However, the use of a simplified one-dimensional model with a very strong electron—phonon interaction that takes no account of the direct Coulomb interaction between electrons rendered Fröhlich’s study less convincing. That was the reason why, as J. Bardeen pointed out [22], Frählich’s work was for a long time treated as a mathematical analysis of a model that has no real physical meaning. The inapplicability of perturbation theory to describe the properties of superconductors was first indicated by Schafroth [23]. He showed that the theory could not explain the Meissner effect to any finite order of perturbation theory, if the state involving no bound pairs was chosen as the initial state. The result was later confirmed by Migdal [24]who concluded that the system of electrons had no energy gap within the framework of perturbation theory. Migdal also showed that the electron—phonon interaction increased the effective electron mass. He derived an expression for the eigenenergy of electrons (due to their interaction with acoustic phonons) as a power series of the small parameter \/mIM ~ Although each of the correlations is small, it is impossible to restrict oneself to taking a small number of such corrections into account, because some of their sums give divergent expressions. To remove this kind of divergence, it is necessary to give up perturbation theory. At nonzero temperatures there are thermal excitations in a conductor. When T < T~,the number of such excitations is still very small and the single quantum superconducting state ordered over large distances remains unaffected. When T = T~,the number of thermal excitations becomes so large that the correlations between electrons vanish and the conductor goes into a normal state. According to F. London [161, the ground state described by a single wave function should correspond to the superfluid component of a phenomenological two-liquid model. Attempts to advance a detailed microscopic theory of superconductivity had produced no major results for quite a long time. Some progress was made only in 1950, when Fröhlich [21] advanced a daring hypothesis that the superconductivity phenomenon is generated by the interaction of electrons with lattice phonons in a crystal. The idea was extravagant because when conductors are in a normal state, it is this interaction that makes them current-resistant. The relation between superconductivity and electron—phonon interaction gives a qualitative answer to the question stated earlier: why are superconductors bad conductors in a normal state? For example, lead is highly resistant at room temperature, and when cooled below 4.2 K, it goes into a superconducting state. Silver and copper are good conductors, but they do not go into a superconducting state at moderately low temperatures. ~

3. Foundations of modern theory of superconductivity in metal compounds 3.1. BCS microscopic theory of superconductivity In 1956, Cooper [25] used the results derived by Fröhlich from studies of the effective electron— phonon interaction caused by virtual phonons and showed that at absolute zero the ground state of a

AS. Davydov, Theoretical investigation of high-temperature superconductivity

205

metal (with all one-particle states being occupied up to the Fermi energy) was unstable in the presence of weak attraction between electrons. This kind of attraction results in the electrons making singlet spin pairs (Cooper effect). The Cooper pair is characterized by the total momentum p and described by the wave function i/i~(r)=çbexp[i(pr— st)Ih],

(3.1)

where the exponential factor defines the motion of the center-of-mass of the pair. The function 4 defines the phase coherence. It is different from zero in a region that has the dimensions of the order of the coherence length ~ ~ cm). According to Cooper, the pairing energy of two quasiparticles on the Fermi surface in a singlet spin state is defined by the formula 4Cooper =

2hflD exp(—2/N

0W),

(3.2)

where hQD is the Debye energy, N0 is the density of energy states (one spin orientation) on the Fermi surface, and W is the interaction energy of two quasiparticles. The result obtained by Cooper was generalized by him in collaboration with Bardeen and Schrieffer [26,27]. They advanced a microscopic theory of superconductivity that subsequently became generally recognized. The theory developed by Bardeen, Cooper and Schrieffer, now referred to as the BCS theory, employed a model Hamiltonian based on the assumption that all electrons in a superconducting

state near the Fermi surface are correlated in pairs with equal and oppositely directed momenta and spins. Cooper pairing was thus taken into account in the zero-order approximation. lithe energies s(k) and the wave vector k of quasiparticles are measured from their values on the Fermi surface, i.e., if we

choose the following definitions: 2F., s(k)=h2(k~—k~)I2m, (3.3) k= ±~k~— k where k 1 is the wave vector in the conduction band of free quasiparticles, the model Hamiltonian in BCS theory has the form, Hmod

=

2

~ k>O

r(k)BBk

2

+

E(k)IBBk

~ k
+

~ ~

(3.4)

k,k1

The matrix elements Wkk are Fourier components of the energy of the quasiparticle interaction caused by virtual phonons. The creation, B, and the annihilation, Bk, operators of the pairs are expressed through the creation, a~5,and the annihilation, akS, operators of free quasiparticles in ks states using the equalities, =

Bk =

akfak~ ,

(3.5)

a_k~akI .

They satisfy the commutation relations, [Bk, B~]= (1

—

~k,t

—

n.kt)ôkkl

2BkBk BkBk

+ BkBk

=

—

[Bk, B~]= 0 (3.6)

6kk

1(l

,

1)

206

AS. Davydov. Theoretical investigation of high-temperature superconductivity

with the values n,53 = a~akS, s = defining the number operators of free quasiparticles in the k, s state. Consequently, the operators B, Bk present in the model Hamiltonian of the BCS theory are different from Bose particle operators. Their commutation relations (3.6) involve the factors (1 nkl) and (1 ~k.k1) which appear because individual quasiparticles forming pairs obey Fermi statistics. As a result, the model Hamiltonian (3.4) in the BCS theory is not consistent with the simple picture of a gas of interacting Bose particles. Each pair of quasiparticles in the currentless state of a superconductor has zero total momentum and zero spin. It is to be noted that in this case there is no sense in speaking about momenta and spins of individual quasiparticles making up a pair. In the presence of a current in the system when paired particles have the drift velocity v0 = hk0/m, the wave vectors k and —k that appear in the operator (3.4) must be replaced with k + ~k0 and —k + ~k0, respectively. The ground state energy of a superconductor with N pairs of bound quasiparticles at absolute zero is calculated by minimizing the expression ~,

~,

—

—

—

=

~0, N~H,~051~N, 0)

with the following additional condition to ensure that the number of pairs is conserved:

Ko,N~B;Bk~N,o)—N. The condition for (~~(0) to be minimal implies that, in a superconducting state, in order to form a quasiparticle with momentum hk when a pair breaks up, it is necessary to spend an amount of energy given by

2(k).

~E(k) + 4 Thus, the quantity ~‘k characterizes the energy of quasiparticles in a superconductor, including the energy of its interaction with virtual phonons. According to (3.7), the spectrum of excited particle states at r(k) 0 begins with the value mm ~‘k = 4k(0)- The lowest level corresponds to = 4~(0). If we recall that with our choice of measuring energy, the minimal energy of free quasiparticles, r(O), on the Fermi surface is equal to zero, then the quantity ~ can be viewed as an energy gap in the spectrum of quasiparticle states that takes account of the electron—phonon interaction. The quantity 4k(0), appearing in (3.3), is defined by =

~

4k(0)

=

~ WkkGk(0),

(3.8)

Ic 1 4k(0)

where the function Gk(O) itself is related to Gk(0)~(0, N+21B,flN,0) = 4k(0)/2~k.

by (3.9)

It is impossible to solve the integral equation (3.8) in the general case. Therefore in BCS theory a substantial simplification was introduced: an isotropic metal was treated and it was assumed that the

AS. Davydov, Theoretical investigation of high-temperature superconductivity

matrix

elements of the interaction,

Wkk

1,

207

can be replaced by constant values,

Wkk= —W<0, if k(k)I

=0, otherwise. (3.10) 2IMV~,M is the ion mass, 1’~the speed of sound, hQD is the maximal energy of phonons Hereparticipate W = a- in the interaction. The ratio hQDIEF is, generally, of the order of 10-2. As a result of that the simplification (3.10), only the states within a layer of width hfl~,near the Fermi surface, are taken into account to calculate the gap width. The quasiparticles in states outside this layer are considered to follow the movement of ions in a lattice adiabatically. Accompanying the motion of lattice ions, they enhance their inertial properties. This results in a small change in the sound speed, but produces no appreciable effect on the inter-electron interaction energy. In the approximation (3.10), the gap width 4k(O) is independent of the wave vector k, and the sum in (3.8) should be restricted to e(k)I hOD.

Consequently, after changing from summation to integration, equation (3.8) transforms to

(N

1

=

0W)

J

[2+

42(O)]1/2

de,

(3.11)

where N

0 is the density of free quasiparticle states (per spin state) per unit energy in an infinitely narrow

layer near the Fermi surface. By solving the integral equation (3.11), we find

1~ ifIs(k)~hI?

k

Jo,

if~e(k)I>hu10.

Furthermore, the width of the energy gap 4~in the spectrum of quasiparticle states of a superconductor is defined by ~0 =

hQD/sinh(1!NOW) = 2/lf1~exp(—1/N0W).

(3.13)

The expression obtained is valid only for weak coupling, when the following inequality holds: N0W<0.5.

(3.14)

In metals, the value of N0W is much less than one. Consequently the inequality is well satisfied. 11~D that amounts to(3.14) several hundreds of

Thus, thedegrees. gap width is much less than the Debye energy absolute By comparing the energy gap width in the spectrum of quasiparticle states (3.13) with the two-quasiparticle pairing energy (3.1) obtained by Cooper, we see that the inequality 4cooper~4o

(3.15)

208

AS. Davydov, Theoretical investigation of high-temperature superconductivity

is valid. In a normal metal, all one-electron states are occupied by pairs of electrons with opposite spins. The density of these states is practically constant within the limit of several millivolts. In a superconducting state, all quasiparticles in the region of the Fermi energy that has the width of the order of the Debye energy hQD are bound in Cooper pairs and go to the energy region of one-particle states < If the state density near the Fermi surface is equal to !V~,the number of Cooper pairs, n, is defined by ~‘

—

n=NOhI2D.

(3.16)

The one-particle states in a Cooper pair have no definite value of the wave vector k. The uncertainty ~Xk is of the order of i0~kF. The uncertainty in wavenumber, ~k, corresponds to the size of a Cooper pair in ordinary space —hkFI2m/iO i0~cm. When the values of e(k) are negative, the energy ~‘k of one-particle states (occupied by electrons) decreases, beginning from —4~.The density of these states is defined by ~ ~)

=

dN(~~k~)/d~~

At T = 0 all states with P(~’k) =

~

NO~f’k(~ —

=

‘~~kk~k

2,

~Yl

-

~

4~.

(3.17)

4 are empty. Their density is defined by ~k > ~o

-

(3.18)

In metals, the gap width 4~has a value of about i03—i04 eV. The binding energy D 0 in a pair of particles that participate in a superconducting condensate, i.e., the energy needed for a pair of particles to break up and its components to go into free quasiparticle states is twice as great as the gap width, D0=240. At absolute zero, all quasiparticle states above 4~are free. It is possible to find the condensation energy under a transition to a superconducting state, by subtracting from the average energy ~ of a condensate the average energy ~ of a normal state that contains N pairs of quasiparticles. When the inequality 4~~ hIlD holds, the condensation energy is 2/[exp(2/NOW) 1] —2N 2e2~5W. (3.19) 2N0(tLQD) 0(hQ~) By dividing the condensation energy (3.19) by the number of condensed pairs, defined by (3.16), we find the pairing energy per Cooper pair in a superconducting condensate, —

—

(~ —

=

~)/n

=

211 ~D exp(—2/N 0W).

(3.20)

It is noteworthy that this energy is just the same as the binding energy of two quasiparticles (3.1), calculated by Cooper. The decrease in the energy when metal goes into a superconducting state (condensate generation) is caused both by the quasiparticle pairing energy and the strong correlation between particles which is due to the Pauli principle, but not to the dynamic interaction between pairs. As the temperature rises, a part of the quasiparticle states, free at T = 0, is occupied as a result of pair breaking. The decay accompanied by quasiparticles going into unoccupied states with momenta 11k1

A. S. Davydov, Theoretical investigation of high-temperature superconductivity

and 11k2 consumes an amount of energy ~(k1, k2)= \1e2(k1) + 4~+ ~e2(k2)

+

209

4~.One-particle states

~(k) which are stationary states at T ~ 0 are occupied according to the Fermi distribution, fkS = [exp(—/3~’(k)) 1]1, ~3= 1IkBT. —

(3.21)

The crystal stationary state at T ~ 0 is defined from the condition that the free energy is minimal F= ~5(T)

TS.

—

(3.22)

Here ~,(T) is the energy calculated by averaging the energy operator in a state with partially unbroken and individual quasiparticles characterized by the distribution functions at a given temperature. The entropy term TS as a function of the distribution, fk5’ is defined by TS = —2kBT ~ [f~5 ln fk5

+

(1 —fkS)ln(1 —fkS)].

From the condition that the free energy F is minimal we find that the energy gap width in the

quasiparticle spectrum is defined by 4k(T)

=

—

~ WkkGk(T),

(3.23)

k1

in which Gk(T)

=

4k(T)(1 —fk1 —f-Il )I2lfk.

When the simplifying assumptions (3.12) are introduced and (3.20) is taken into account, the gap width 4(T) in a narrow region near the Fermi energy is independent of k and is defined by the integral equation

(N0W)~=

I

2(T) tanh{~/3[e2 + 42(T)]112}, ~2

(3.24)

+4

I

for e~ IuIlD and is equal to zero for e~> 11 liD. From (3.24) it follows that the energy gap width in the

spectrum of quasiparticles decreases with increasing temperature. The critical temperature T~of a superconducting transition can be defined as a temperature at which

the gap width 4(T) decreases to zero and all pairs of quasiparticles are destroyed. The critical temperature can thus be defined by an equality htlD

(N

1 0W)

=

J

~

tanh(SI2kBTC).

In the case of weak coupling when N 0W < 1/2, this equation yields

(3.25)

210

A.S. Davydov, Theoretical investigation of high-temperature superconductivity

T~= 1.44@ exp(—1/N0W),

~9= hl11~/k~.

(3.26)

This is the basic equation in BCS theory. The value of 9 is determined from heat capacity measurements. In metals, it is in the interval 100—500 K. From eq. (3.21) it follows that high values of T~cannot be obtained in BCS theory, because the theory is valid only if the inequality N0W < 1/2 holds. So, even at 0 500 K, the critical temperature is not higher than 40 K. The gap width becomes zero at T= T~with an infinite derivative, this corresponding to a second-order phase transition (from a superconducting into a normal state). At T = 0, the ratio of the energy gap to kBTC for all the superconductors described by BCS theory is defined by the equality 24(0)!k~T~ = 3.52,

(3.27)

which is well satisfied for many metals. With T below 0.6 T~,the energy gap is slightly dependent on the temperature. 3 of electrons participates in pair formation If we intosurface, account the thatdistance a small part kBTC/EF near the take Fermi between these— i0 pairs will be ~~~.10_6cm. With the density of electrons equal to 1022 the volume occupied by a single pair (10~)~ will thus contain 106 other pairs. So, the wave function of a superconducting state describes coherence of about a million quasiparticle pairs among which it is difficult to isolate any individual pair. Bardeen and Schrieffer [26,27] assert, “the picture of isolated pairs makes no sense”, and the superconducting transition is, strictly speaking, not similar to Bose—Einstein condensation. Moreover, the pairs are not strictly Bose particles, because their creation operators do not satisfy the commutation relation for Bose-particles [see eq. (3.6)]. The concept of a Bose—Einstein condensation of pairs of charged particles was earlier introduced by Schafroth and collaborators [29] to explain superconductivity. They introduced the hypothesis that “pseudomolecular” pairs of electrons are generated with opposite spins and momenta. It was assumed that the extent of the pairs is less than the average distance between the pairs. Bardeen and Schrieffer think that the quasiparticle pairs in BCS theory have nothing to do with Schafroth’s pseudomolecules. Conceptually, a superconducting state can be thought of as a kind of a “macromolecule” that consists of many millions of quasiparticles. They are all correlated in pairs with equal and opposite momenta and spins. Such a “molecule” occupies a large volume of a crystal and is capable of moving as a whole if there are rigid correlations between the particles constituting the molecule. At T <0.6 T~the number of pairs decoupled in a superconductor is proportional to exp[—4(0)/ kBTJ. Consequently, the electron heat capacity at these temperatures is also characterized by an exponential dependence. At temperatures close to the critical one, the energy gap width decreases with increasing temperature, so that the heat capacity increases faster. There are no bound pairs of quasiparticles above the critical temperature. As a result, a sharp drop in heat capacity is observed when the critical temperature is exceeded. If we consider that the total energy of the electrons remains unchanged when the critical temperature is reached, because the gap width is zero at T = T~,the superconducting transition should be referred to a second-order phase transition. 3.2. The theory of superconductivity developed by Bogoliubov, Eliashberg and McMillan The BCS theory has made it possible to explain many experimental facts observed in studying the superconductivity phenomenon of simple metals and alloys. Yet, the theory incorporated some

AS. Davydov, Theoretical investigation of high-temperature superconductivity

211

insufficiently substantiated assumptions and approximations. Therefore, the theory required to be substantiated within a stricter context. For a weak electron—phonon coupling, the theory was substantiated in a stricter way by Bogoliubov [30,31] and Valatin [32]. Bogoliubov found the excitation spectrum of a superconductor using the Fröhlich effective Hamiltonian (3.2). The study employed the generalized method of canonical transformations proposed by N. Bogoliubov for constructing the superfluidity theory [33].

The ordinary theory of perturbations in powers of the small parameter of electron—phonon interaction is inapplicable to investigate superconductivity. The electron—phonon interaction, small though it may be, has turned out to be quite important in studying the states near the Fermi surface. Near the Fermi surface, electron pairs with equal, but oppositely directed momenta (k and k) and spins (t and ~.) exhibit a tendency to form pairs with components (k, ~) and k, This kind of pairing was called Cooper pairing. N. Bogoliubov suggested that the pairing effect should be taken into account even to the zeroth approximation. For this purpose he performed a canonical transformation from the Fermi operators a1t and a_11 of independent quasiparticles to the new Fermi operators a10 and akl that characterize the elementary excitations of paired particles. —

(—

.t~).

This transformation is as follows: =

u1a10 + v1a~

a,~1=

,

(3.28)

ukakl + vkakO.

It is performed using two functions

Uk

and

Uk,

symmetric under the transformation k

—*

—

k, which

satisfy the condition u~+u~=1.

(3.29)

The ground state energy, 3 ~ Wkkukukukuk, E0 = 2

~

k

e(k)

—

L

k,k 1

corresponds to the part of a complete transformation of the Hamiltonian that involves no new creation and annihilation operators. The explicit form of the transformation functions Uk, Cl is derived from the condition that the energy E0 is minimal. From the minimumand condition for E0 it follows that the functions Uk andthat Uk are expressed in termsthe of the energy of quasiparticle excitations, take into account the energy gap electron—phonon interaction, g’ 2(k) + 4~] 1/2 by the equalities 1 = [r u~=~(1+e(k)/~Zk),v~=~(1—r(k)/~ 4k

~

1).

4k

is a solution to the integral equation

It is established that the ground state energy is minimal, if =

—

~ ~

(3.30)

When the approximate values (3.10) for the matrix elements Wkk are used, the width of the energy gap 4k is independent of the wave vector k.

212

AS. Davydor, Theoretical investigation of high-temperature superconductivity

In the asymptotic approximation, the energy gap width at zero temperature is defined with accuracy to terms of the first order in W by

2hl1,~exp(—1/WN0)

~0 =

(3.31)

where N0 = (dNIdE)E of the number of electronic states perion unitmass, energy near the Fermi 2/MV~,a-isisthe thedensity electron—phonon interaction energy, M the l’~ the velocity of surface, W= alongitudinal sound, and ul~is the limit (Debye) frequency of the phonon spectrum. The superconducting transition temperature T~is defined as a temperature at which the energy gap 4T vanishes. This yields the relation kBTC = 1.4411QD exp(—1/WN

0).

(3.32)

By comparing (3.31) and (3.32), we find the universal relation of BCS theory that connects the energy gap width at zero temperature, 4~,and the superconducting transition temperature T~, 24OIkBTC

3.52.

(3.33)

The relation is well confirmed experimentally for many superconducting elements. Expression (3.31) is the same as the basic formula (3.13) in BCS theory. Thus, Bogoliubov proved the validity of the BCS theory in the asymptotic approximation, with accuracy to terms of the order of the square of the coupling parameter a-. Bogoliubov, Tolmachev and Shirokov [34] developed further the Bogoliubov method of superconductivity theory. In particular, the effect of the Coulomb interaction between electrons under their pairing was analyzed in detail. It was shown that taking account of the screened Coulomb interaction between quasiparticles using the method of perturbation theory results in an effective decrease in the interaction parameter A = WN0, involved in the eqs. (3.30) and (3.31). It was shown that the superconducting transition temperature must be defined not by (3.32), but by the equality kBT~=1.44hu1Dexp[_1/(A_,.L*)],

(3.34)

which, in addition to the parameter A that describes the electron—electron attraction, involves the Coulomb pseudopotential ~ expressed in terms of the matrix element of the Coulomb interaction V~ of electrons on the Fermi surface by the relation ~,

=

N~Vj1+ N0V~ln(EF/1lu1D)]~

-

(3.35)

The logarithmic weakening of the Coulomb repulsion results from the fact that the average distance of electrons in a Cooper pair, characterized by the correlation length 11VF/40

i0~cm,

(3.36)

(where CF is the electron velocity on Fermi surface), is much greater than the lattice constant a -~108 cm. Therefore, the Coulomb repulsion is strongly screened. Within the BCS theory, the pseudopotential p. * can be defined from experimental values of the isotopic effect. With the Coulomb interaction neglected, the dependence of T~on the ion mass M is

A.S. Davydov, Theoretical investigation of high-temperature superconductivity

defined, according to (3.32), by the Debye frequency

11D

-~

213

M -1 / 2, so that the following relation holds:

~T~IT~= —~MI2M.

(3.37)

With the Coulomb interaction taken into account, the value of T~is defined by the formula (3.34). So, it is also dependent on T in terms of p. ~ Equation (3.37) is therefore replaced by a relation such as &T~IT~ = —a ~MIM, a

=

(3.38)

~[1—p.*/(A_ p.*)].

(3.39)

Thus, if T~and the isotopic shift ~

are measured, we can define p.” using (3.39). The range of applicability of the BCS theory and of that developed by Bogoliubov was extended by

Eliashberg [35] and Nambu [36]. They applied the results obtained by Migdal [37]in a study of electron—phonon interaction in homogeneous and normal metals, to investigate the case of superconductivity. The Green’s function method developed by Gor’kov [38]was used to calculate the Green’s function for electrons that interact with isotropic crystal phonons. The calculations were made using the perturbation theory in which the dimensionless ratio ~j = A11I1DIEF was taken as the expansion parameter.

In the above expression, ~D is the maximal frequency of the phonons, EF the Fermi energy, and A the dimensionless parameter of eletron—phonon coupling. If hliDIE~ V~7M41, then ~ <1 for A> 1, too, so that the Eliashberg theory is referred to as a strong coupling theory for isotropic crystals. As a zero approximation the calculations used the states found in the Bogoliubov superconductivity theory and corresponding to the energy spectrum = [e2(k)+ 4~] 1 but not the states of noninteracting electrons. The main pairing effect was thus taken into account already to the zeroth approximation. These functions were used to construct the Green’s function of the zeroth approximation. The perturbation theory was then used to solve the equations for the Green’s functions of the first approximation in the parameter ij. The frequency dependence of the matrix elements of the theory was thus taken into account. The phonon state density function, F(w) = Eq ~(w 11q), expressed through the spectral distribution uI~ of phonon frequencies and the “weight” function a 2(w) that characterizes the intensity of electron—phonon interaction is crucial in taking into account the frequency dependence of the matrix elements. The spectral and tunneling measurements of the properties of superconductors enable us to determine directly the product a 2(w) F(w). Allen [39, 40] has shown that if this function is given, we can -~

~‘k

~,

—

use the expressions

A2Ja2(w)F(w)~,

(w2)2A~Jw2a(~)F(w)~,

to define the dimensionless constant A of electron—phonon interaction that corresponds to the value of WN(0) in BCS theory and the average value of (w2) of the square of a weighted phonon spectrum. At present this function is well known for many superconducting metals and alloys. The plots of the electron—phonon spectral function a 2(w) F(co) and the state densities of the phonons F(o) derived from

214

A.S. Davydov, Theoretical investigation of high-temperature superconductivity

data on neutron scattering differ little from each other. This indicates the weak dependence of a2(w) on w in the basic frequency region. The Eliashberg theory was generalized by Karakozov, Maksimov and Mashkov [41]. Using the pseudopotential treatment of screened Coulomb interelectron repulsion proposed by Morel and Anderson [42], McMillan [43] generalized the Eliashberg theory to include the Coulomb interaction. He showed that the basic formula (3.26) in the BCS theory that characterizes the superconducting transition temperature is replaced by k~T~=hl1~exp{_(1+A)/[A+p.*_Ap.*((w)/Q~)]},

(3.40)

where (cv) is the average value of phonon frequency, p. * is the Coulomb pseudopotential of Morel and Anderson, expressed in terms of the product of the average screened Coulomb interaction V~and the electron state density N 0 using the equation =

p.[1 + p. ln(EFIh[lD)]~

,

p.

N0V~1-.

(3.41)

The equation was derived under the assumption that the electron—phonon coupling is weak. It implies that the pseudopotential characterizing the Coulomb repulsion is much less than the constant p. itself. An explicit expression such as (3.41) has not yet been found for a strong coupling. A direct Coulomb interelectron repulsion, however, is suppressed in these cases too. To analyze the dependence of the dimensionless parameter on the properties of a metal, McMillan proposed a simple formula, 21a2)IM(cv2) (3.42) A=N0(q where N 0 is the energy density 2) of the electron states onsquare the Fermi the matrix of average of the of thesurface, phonon qia frequency, M theelement ion mass. electron—phonon interaction, (W Using experimental data, McMillan concluded that the numerator in (3.42) has the same value for many transition metals and their alloys. The value of A is more sensitive to M cv2). The superconducting transition temperature T~(3.40) is dependent on the ion mass M directly via the Debye frequency and indirectly via the dependence on 11D of the Coulomb pseudopotential p. * ,

(

11D~

due to Morel and Anderson. Using (3.40) and (3.41), we can find T~—M~

(3.43)

,

a

=

~[1 (1 —

+

A)(1

+

0.6A)p.*21[A

—

p.*(1 + 0.02A)]2].

(3.44)

In the weak coupling limit A 4 1, eq. (3.40) simplifies to kB T~

11I1D

exp[—1!(A

—

p. *)].

Comparing this equation with formula (3.26) in BCS theory, we see that the difference A

(3.45) —

p. * now

plays the role of N 0W. Assuming, for simplicity, the value of (1 + 0.62A)I(1 + A) to be equal to unity in eq. (3.44), we can express the Coulomb pseudopotential p. * in terms of the isotopic shift coefficient a, the superconduct-

A.S.

Davydov, Theoretical investigation of high-temperature superconductivity

215

ing transition temperature T~and the Debye temperature 0, using the equality =

(~.46)

(1 2a)v211n(0I1.45Tc). —

The specific electron heat capacity coefficient y (in units mJ/mol K2) is proportional to the density N 0 of electronic states on the Fermi surface and the factor (1 + A) due to electron—phonon interaction

2k~(1 + A)N

7= ~1T

(3.47)

0.

The theories of BCS, Eliashberg and McMillan are concerned with isotropic media. This restricts their range of applicability to nonmetallic superconductors. That will be treated in the subsequent sections. 3.3. Energy gap in the spectrum of quasiparticle states of a superconductor

All quasiparticle energy states with the energy ~ EF are occupied by electrons in theHere normal 2k212m is called the Fermi energy. m isstate the ofa metal at zero temperature. The limit energy EF = 11 effective mass of the quasiparticles. With the temperature different from zero some free quasiparticle levels with energy e(k)> EF are filled by quasiparticles with the probability defined by the Fermi statistical distribution ~‘k

f(r(k))

=

{exp[s(k)Ik~T]

—

1)~, s(k)0.

(3.48)

According to the theories of BCS and Bogoliubov the spectrum of quasiparticle states readjusts when

quasiparticle pairing (Cooper pair formation) is taken into account. When all energies are measured with respect to the Fermi level of free quasiparticles, the quasiparticle spectrum for a superconductor is defined by ~

±~r2(k)+4~.

(3.49)

As mentioned above, the quantity 24~that characterizes the discontinuity in the spectrum of free quasiparticles is called an energy gap. Near the edge of the gap the density of states with energy ~(k) is defined by ~

~

(3.50)

At T ~ 0, the quasiparticles and Cooper pairs are present simultaneously in a crystal. In a superconductor at temperature T, they form two subsystems which are in statistical equilibrium, having the same level of the chemical potential. At low temperature the chemical potential, p., in a metal coincides with the Fermi energy EF. At zero temperature all quasiparticle states with energy ~‘k ~ 4~are free. At nonzero temperatures some of the Cooper pairs are destroyed and the constituent quasiparticles occupy free quasiparticle levels with energy ~Ek 4~in accordance with the Fermi distribution. The occurrence of the energy gap 24~in the spectrum of unpaired quasiparticles of superconductors is an important characteristic of the superconductivity phenomenon. A number of papers were,

therefore, intended to improve the methods of its experimental determination.

216

AS. Davydov, Theoretical investigation of high-temperature superconductivity

Direct evidence of the existence of an energy gap in the spectrum of quasiparticle excitations was first produced from measurements of the absorption of far-infrared radiation. The use of electromagnetic radiation to determine the superconduction decay threshold was discussed earlier in the thirties. It was assumed that the radiation with energy kBTC -~i0~eV and with a wavelength of about 1 mm should be strongly absorbed and reflected from a metallic superconducting specimen. At that time the idea was not experimentally confirmed, because there was no technical means of generating and recording long-wavelength radiation of this kind. The possibility to do this was realized only near the end of the fifties. By that time Bardeen, Cooper and Schrieffer had explained that the superconduction phenomenon was concerned with the formation of Cooper pairs and the occurrence of an energy gap in the quasiparticle state spectrum. It was very important to verify experimentally the availability of an energy gap of this sort. Photons with energy 24~can break the Cooper pair when two quasiparticles go into unbound states with energies ~‘k 4~.The probability of the process is very large because of the high density of states in the region 4~,according to (3.50). The transfer of quasiparticles to the quasiparticle levels > 4~is less probable due to the smaller density of these states. So, at zero temperature the infrared radiation absorption spectrum should exhibit a sharp peak in the threshold frequency region, hw1~= 24g. According to BCS theory, Fiw~~ = 3.5kBTC at absolute zero. In superconductors with T~equal to several kelvins, the absorption threshold is in the far-infrared region. For example, with 24~— i0~eV, a sharp increase in the absorption is observed for the radiation with a wavelength of about 1 mm. As the temperature rises, the value of the gap decreases, 4(T) approaching zero for T—s’ T~. Therefore, with increasing temperature the frequency w1~decreases, going into the microwave region. The first measurements of the energy gap in the excitation spectrum of unpaired quasiparticles in superconductors at T < T~were made by Glover and Tinkhan in 1957 [44].They studied the passage of infrared radiation through thin superconducting films whose width is small compared to the radiation penetration depth. The measurements showed the gap width to be equal to 2.6 meV. The reflection and absorption spectra are often investigated along with the radiation transmission spectra [45,46]. In 1960, Giaever [47] suggested that the tunneling current passing through a thin dielectric layer separating the surface of the metal in normal and superconducting states should be used to determine the gap width in the spectrum of the quasiparticles of a superconductor. The Giaever tunnneling from a normal metal to a superconductor through a thin oxide film produces direct evidence of the existence of an energy gap in the spectrum of the quasiparticles of a superconductor. Giaever investigated the electron tunneling through a layer of aluminium oxide lying between an aluminium plate in the normal state and a lead plate in the normal and superconducting states. The oxide is a good insulator creating a potential barrier about 10 eV high that prevents free passage of electrons. In the presence of an oxide film a tunneling current is absent at zero temperature until the applied voltage exceeds the value 40/e (about several millivolts). The tunneling current caused by quantummechanical tunneling and flowing through a barrier formed by an insulating film is carried by separate electrons. Passing through the barrier, the electrons join to form Cooper pairs. The positions of the Fermi levels to the left and right of an insulating layer are equalized in an equilibrium state when there is no external potential difference V at the contact. The total contact current is absent in this case. When a potential difference V is applied to an insulating layer, the Fermi levels are shifted by the value eV. The electron energy is reduced on the side of the positive potential. I

~‘k

A.S. Davydov, Theoretical investigation of high-temperature superconductivity

217

At zero temperature when the potential difference is small, the tunnel current does not arise with the electron energy being conserved, because the metal electrons do not find free quasiparticle levels in a

superconductor. When the equality eV= ~ holds, a tunnel current is generated. Its value is proportional to the number of electrons that come to the barrier, the tunneling probability and the number of vacant sites on the opposite side of the barrier. The number of such states is large near the bottom of the conduction band of one-particle states in a superconductor and falls exponentially as the levels increase. Consequently, as the potential difference V rises, the voltage—current curve displays a sharp peak at eV 4~. At finite temperatures the Fermi energy level in a normal state of metal is smeared by the value kB T and the energy gap is narrowed. Therefore, tunneling is also observed in the range of voltages lower than 401e [47—49].

Tunneling experiments make it possible to determine the temperature dependence of an energy gap. Infrared (IR) and tunneling (tun.) measurements have produced numerous data on the energy gap value for simple metals. In many cases both methods give for ratio 24~ 1k8 T~the same values close to 3.5 the value predicted by BCS theory. An appreciable deviation is observed only for superconducting Pb, Hg. —

4. Superconductivity of compounds that contain transition elements Transition elements with unoccupied d- and f-electron shells form superconducting compounds that have properties much different from those of ordinary metal superconductors described by the BCS theory. Generally the compounds under consideration have great anisotropy and contain two types of electric charge carriers with heavy and light effective masses.

In sections 4.1 and 4.2 we consider intermetallic compounds with the structure of beta-tungsten (A-is). These are ordered systems of transition and nontransition elements in which the atomic bond can be characterized as lying in between the valence and the metallic bond. In section 4.3 we consider chemical compounds with a Bi structure similar to that of NaCl. In compounds of this kind, covalent and ionic bonds are very important, in addition to the metallic type of bonding. In section 4.4 we investigate the class of superconductors called superconductors with heavy fermions. In this class of superconductors, the conductivity is produced by quasiparticles with effective masses much greater than that of free electrons. 4.1. The basic properties of intermetallic compounds such as A-15

Persistent search for superconducting materials with the superconducting transition temperatures above 10 K was crowned with success only in the sixties. In 1953, Hardy and Hulm [50]discovered the superconductivity in an intermetallic compound V3Si with the superconducting transition temperature = 17 K. Shortly afterwards other intermetallic compounds were synthesized: Nb3Sn (T~ = 18 K); Nb3Si (T~= 17K); Nb3Al (T~= 16K); V3Si (T~= 17K). In 1957, Mattison synthesized Nb3Al08Ge02 with a superconducting transition temperature T = 20 K. Finally, in 1973 a Nb3Ge compound was synthesized, exhibiting the highest ever superconducting

transition temperature of 23.2 K that was above the hydrogen boiling temperature (20 K).

218

AS. Davydov. Theoretical investigation of high-temperature superconductivity

In addition to the high superconducting transition temperature, these compounds display at higher temperatures anomalies in the temperature dependence of the resistance, the heat capacity, the magnetic permeability and elastic constants. They have extraordinarily soft acoustic and optic modes of lattice vibrations [51]. The synthesis of intermetallic compounds with the superconducting transition temperatures of about 20 K and very high values of the critical magnetic field and currents has broadened appreciably the range of practical application of superconductivity. These superconductors can be manufactured as wires or films. Therefore, they are widely used in producing superconducting magnets with magnetic fields of about 10 T. They have found many other uses in science and technology. The new superconductors have been applied for the following purposes: in constructing a huge tevatron accelerator at the Fermi National Laboratory in the USA; for a nuclear fusion installation (T-15 tokomak) in the USSR; in designing medical diagnostic equipment that reproduces human internal organs by means of nuclear magnetic resonance. These devices NMRtomographs—need for their operation strong (1—2T) magnetic fields uniform in cameras with a diameter of about one meter. Many practical applications of the phenomenon have generated much interest in elucidating the physical reasons for the high T~and the high critical current densities. The studies have shown that A-iS superconductors have many properties that cannot be explained by the generally accepted superconductivity theories of BCS, Bogoliubov and Eliashberg. Intermetallic compounds of transition metals of vanadium (V) and niobium (Nb) such as Nb3B and —

V3B, in which B is one of the nontransition metals, have the structure of beta-tungsten (13-W) designated in crystallography by the symbol A-15. Superconducting compounds of this kind are, therefore, called “A-iS superconductors”. 28Nb) and vanadium (33V) are transition elements of the first group. Their 5 or 10 outer Niobium ( electrons have in the normal state electron configurations 3d84S2 and 3d34S2 with incompletely filled internal d-shells characterized by an orbital angular momentum equal to two. In intermetallic compounds of transition and nontransition elements the atomic bonds have a character intermediate between the valence and the metallic bond. These bonds are generated both by valent S-electrons and internal d-electrons. Since unoccupied d-states in these compounds can partially be filled by electrons of nontransition elements, the valence of atoms that form the bonds is a continuous function of the possible variable composition of a compound. These compounds are often unstable with the number of A and B atoms deviating on both sides from stoichiometric values. Generally, the deviations are realized by atom substitutions in a lattice. The structure of A-is compounds such as A 3B is represented in fig. 4.i. Atoms B form a body-centered cubic lattice, and atoms A are situated pairwise on the cube faces, parallel to the coordinate axes. The unit cell of an A-15 compound contains 8 atoms and is characterized by the spatial group The lattice constant is 4.72 A for V3S; and 5.29 A for an Nb3Sn compound. It is important that A-atoms °h

form three mutually perpendicular families situated along the [100],[010]and [001]axes. Under deviations from stoichiometry the linear chains of A-atoms remain unchanged. For example, in the structure A3+~B1 A atoms in excess occupy sites of B atoms without violating the structure of the ~,

linear chains. Atoms A are arranged along the chains at relatively small distances. For example, in the compound V3Si the distances between vanadium atoms along3.the chains are of 10% smallerelectrons than in ispure The number valence 2.4 vanadium. per atom. The of vanadium isatoms is The —5.7>< 1022 energy cm EF is —7 eV. The density Debye temperature 300 K. Fermi

AS. Davydov, Theoretical investigation of high-temperature superconductivity

219

A

x

Fig. 4.1. The unit cell of an A-15 compound such as A

5B.

Table 4.1 gives the critical temperatures and the values of the energy gaps 4~in superconductors with an A-iS lattice, measured using infrared spectroscopy (IS) and tunneling. The parameter y is also represented in table 4.1. It characterizes the electron contribution to the specific 2 by the formula C(T) = -yT + /3T2. heat capacity expressed at low temperatures in units of mJ/mol K The coherence length of superconducting A-15 compounds is relatively small. For example, = iOO A for a Nb 3Sn compound and 500 A for a V3Si compound, i.e., it is two orders of magnitude less than the coherence length in metals described by BCS theory. At the temperature Tm not much greater than T~,Nb3Sn and V3Si crystals undergo a weak transformation from a cubic to a tetragonal symmetry. For a V3Si compound (T~= 17 K and Debye temperature —500 K), the values of Tm are equal to 18—22 K and 25 K, respectively. For a Nb3Sn compound (T~= 18 K and Debye temperature 300 K) this value is 43. The symmetry transition is accompanied only by a rearrangement of the crystal lattice, with the

composition and volume of the crystal remaining unchanged. Bathennan and Burrett [52]called this transition a martensite one. At temperatures immediately preceding the transition, there is a softening of the elastic modulus of the lattice.

A3B compounds have relatively high values of electron heat capacity and paramagnetic susceptibility. Since electron heat capacity and paramagnetic susceptibility are proportional to the density of electronic states at the Fermi level, their large values indicate the high density of electron states at the Fermi level for these compounds. This may be one of the reasons for the high values of T~. These compounds have exhibited a strong temperature dependence of paramagnetic susceptibility. This could be explained by the fact that they have on the Fermi surface a narrow peak ofthe density of~ Table 4.1 Superconductors with A-15 lattice Compounds V 3Si Nb3AI Nb35n

Nb3Ge

T~ (K)

24,fk8T~ (IS)

240/k8T, (tunn.)

15.7 16.0 18.3 23.2

3.8 4.4 4.2—4.4 4.2

3.8 1.0—2.8 3.6 —

y 2) (mJ/mol K —

0.13 0.13 0.13

220

AS. Davydov, Theoretical investigation of high-temperature superconductivity

electron states with a width comparable with the thermal motion energy -—0.04 eV. Measurements of electron heat capacity and magnetic susceptibility have produced the values of electron density of states on the Fermi surface for V3Ga and V3Si, which are equal to, respectively, 7.1 and 5.6 per electron volt for one atom and one spin orientation. Study of X-ray spectra [53]has shown that, in all V3B compounds, 3d-states of vanadium atoms appear on the Fermi surface, and these are, probably, responsible for the narrow peak of the density of electron states. Relatively wide S-electron states are situated near the bottom of the valence band. In V3Si and Nb3Sb compounds, the maximal temperature of a superconducting transition is reached when the ratio of the components is close to a stoichiometric one. When nontransition elements are in excess, the temperature T~changes weakly. When the atoms of transition elements A are in excess (they replace the atoms in positions B), there is a sharp decrease in T~.The reason seems to be the fact that the interaction between the chains is enhanced by the atoms of transition elements that occupy the positions of atoms B. Such an interaction violates the “independence” of linear chains. All presently available superconductors of the A-iS type with T~— 18 K are metastable. It has been established that the more metastable the compound, the higher the T~.It may be that in nonequilibrium systems there is a stronger electron—phonon coupling, indicating an easier deformability of the crystal lattice. To elucidate the specific features of intermetallic superconductors of the A-iS type, numerous studies have been made using infrared spectroscopy and tunneling methods. By studying the infrared radiation transmission and reflection spectra [54, 55] it was possible to measure the energy gap width and obtain data on electron—phonon coupling. Since the critical temperature is high, the absorption maximum shifts to the region of higher frequencies as compared to the exceptionally low frequencies that are observed in IR studies of metallic superconductors. The tunneling studies for A-iS compounds are more complicated than similar studies for simple metallic superconductors. The difficulties are caused by the fact that natural oxides of many transition metals and compounds do not form good insulating tunneling barriers, so that it is necessary to cover the surface of A-iS compounds with thin layers (15—20 A) of other insulating materials oxides of amorphous silicon. As a result of the small coherence value for these compounds (-—100 A), only the surface layers of the samples are observed in tunneling. In 1972, Shen [56] carried out tunneling studies of A3B-type superconductors, producing good results. He investigated the tunneling process between Nb3Sn and Pb superconductors separated by an oxide film. The voltage—current curves obtained for4NbSn the Nb3Sn—Pb oxidemeV. tunneling weremeV, usedthe to + 4Ph = 4.45 When contact = 1.35 derive the value of the sum of the energy widths sought value of zl(Nb 3Sn) is equal to 3.10meV. Experimental data were analyzed to find the phonon spectrum of the superconducting state of Nb3Sn. The spectrum exhibited a strong low-energy peak at —9 meV and a small peak in the region of 25 meV. The phonon spectrum of transverse vibrations of Nb measured by neutron scattering has a peak in the region of 16.6 meV and that of longitudinal vibrations has a peak in the region of 235 meV. The occurrence of the 9 meV peak in the phonon spectrum of Nb3Sn indicates that the lattice vibration mode was “softened”. The softening may be caused by lattice instability. Of much interest are the tunneling studies made by Kihlstrom and Geballe [57] at the contact between Nb3Ge and Pb superconductors separated by a thin layer of amorphous barium oxide. A Nb3Ge superconductor belongs to a class of unstable superconductors. It can crystallize in composi—

4pb

A.S. Davydov, Theoretical investigation of high-temperature superconductivity

>

5.0

Ge%

17 20

2~Q

221

21 2223

1234

—~

L,~,meV

Fig. 4.2. The dependence of the gap width on the composition of the Nb

3Ge compound, from Kihistrom and Geballe (57].

tions much different from the stoichiometric ones. When the composition of Nb3Ge is stoichiometric (25 at% Ge), this compound has the highest T~among other A3B compounds. Studies of the relationship between the superconducting properties of compounds and the thickness of their layers have shown that very thin layers are not superconducting. As the thickness increases, the energy gap rises from zero, reaches its maximum value for thicknesses above 1000 and remains A

unchanged up to a thickness of 3000 The basic study was made with films 2500 thick. Studies of theincreases dependence gap close width to on 3.5 the (derived composition a compound havesamples shown that 24OIkBTC fromofathe value fromofBCS theory) for poorthe in ratio germanium atoms, to a value 4.35 when the number of germanium atoms approached a stoichiometric one (25%). This dependence is illustrated in fig. 4.2. Simultaneously, the temperature T~increases from A.

A

7.0 K to 20 K and the average frequency (w2) of the phonon spectrum decreases.

The experimental data obtained made it possible to produce important estimates of the parameters of Nb 3Ge superconductors: interaction A and the average phonon spectrum 2), depending the on the deviationconstant of the number of Ge atomsvalue from ofthethe stoichiometric value frequency (w to 25%. The data are given in table 4.2. This table shows that an increase in the coupling which is equal parameter A is accompanied by an increase in the energy gap width 4~,the superconducting transition temperature T~and by a softening of the spectrum of phonon vibrations (w2). Using the data obtained the authors draw the conclusion that the main factor in increasing the critical temperature is the “softening” of the spectrum of the phonon vibrations, i.e. the shift to lower phonon energies.

It is worth mentioning that the electron—phonon coupling parameter A has a value less than unity in superconductors poor in Ge and more than unity when the composition approaches the stoichiometric Table 4.2 The dependence of the values of T~,A and (wa) on the concentration of Ge atoms in NbGe compounds % content of Ge 16.7 20.9 22.9

~l(meV) 1.03 2.62 3.70

T~(K) 7.0 16.8 19.8

A 0.81 1.05 1.64

~w2) (meV)2 250 225 200

222

AS. Davydov, Theoretical investigation of high-temperature superconductivity

5.0

1~

4.0 BC5~

,----

~

//

~

3.0

1.0 I

18

I

I

22

26 B%

Fig. 4.3. The dependence of the ratio 2~1/kB T~on the composition of the NIi~Bcompound. For curve 1, B Geballe (58].

=

Al; for curve 2, B = Sn, from Kwo and

one. The same tendency was also observed when the other unstable superconductor Nb3A1 was investigated, The crystals of Nb3Sn and V3Si usually grow in a composition close to the stoichiometric one, producing stable crystals. Another class of compounds, Nb3Al, Nb3Ga, Nb3Ge is less stable. Crystals of this kind can grow with considerable deviations from the stoichiometric composition. Using the possibility to grow Nb3A1 and Nb3 Sn crystals with different contents of aluminium and tin, Kwo and Geballe [58] applied the tunneling method to measure, at 4K, the ratios 24IkBTC at the contacts of Nb3Al—SiO—Pb and Nb3Sn—SiO—Pb, depending on the content of aluminium and tin. The measured results are given in fig. 4.3. The other characteristics of these superconductors were also obtained. They are represented in table 4.3. This table illustrates how the properties of Nb3Al change with changing content of Al. The crystal is characterized by large values of T~and 4 when its composition approaches the stoichiometric one (25% Al). The average value of (cv2) is very sensitive to the composition. Crystals with a composition 2), large coupling close to the stoichiometric one are “softer”. They are characterized by small (w parameter A and high T~.When the composition is not stoichiometric, the crystal becomes more rigid its phonon spectrum shifts to greater frequencies, resulting in decreased A and T~. To elucidate the nature of high T~in the superconductors listed in table 4.3, the physical parameters of Nb 3Al and Nb3Sn are compared. For a composition close to the stoichiometric they have almost 2) one, in Nb the same values of A and ~ However, the average value of the square of (w 3Al is less than the one in Nb3Sn. At the same time the studies of electron heat capacity [57,58] and magnetic susceptibility [59] revealed a small density of electronic states near the Fermi surface. Table 4.3 shows that as the content of aluminium changes from 2), 21.5% to 22.8%, thefrom critical however, changes 226 temperature T~changes slightly from 14.0 K to 16.4 K. The value of (w —

Table 4.3 Microscopic parameters of Nb

5AI and Nb3Sn T~ (K)

~ (meV)

(meV)

21.5% Al 22.8% Al 25% Sn

14.0 16.4 17.7

3.56 4.95 4.26

226 181 226

2

Al and Sn

Content

A-15 Nb 3AI Nb5AI Nb35n

(2)

A

~io/kBT,

1.2 1.7 1.8

3.5 4.4 4.2

AS. Davydov, Theoretical investigation of high-temperature superconductivity

223

to 181. Consequently, the increase in the ratio 24OIkBTc is due to a considerable decrease in (cv2). Taking these data into account, the authors of ref. [58]conclude that the basic reason for high Tc is not the high electron densities near the Fermi surface, as asserted by the BCS and Bogoliubov theories, but the high lattice deformability that leads to a softened phonon spectrum. We can think that the high values of Tc in A 3B compounds are related to the fact that the chains of transition elements in a lattice are quasi-one-dimensional. The energy gap width is more24oIkB sensitive the to composition of a value Nb3Alofsuperconductor than the T~ istoclose the theoretical 3.5 in BCS theory when critical temperature T~. T he ratio the Al content is less than 21.5%, and rises sharply to 4.4 when that content is equal to 22.8%. This ratio characterizes the value of the interaction constant A. To evaluate it quantitatively, a simple equation was proposed in ref. [58], A = 1.i4(24OIkBTC) —3.5.

(4.1)

As shown by Geilikman and Kresin [61],the increase in the ratio 24

0/k8T~when the phonon spectrum

becomes smooth follows from the Eliashberg theory. They used the Einstein phonon spectrum to derive an analytic dependence of the form, 24/kBTC = 3.53 [1 + 5.3(k8T~Ihw0)]ln(hWOIkBTC),

(4.2)

where cv~is the effective frequency of phonons. 4.2. Theoretical models of A-15 superconductors

The great practical importance of superconductors with an A-iS lattice and their extraordinary properties have generated numerous attempts to explain them theoretically. Unfortunately, so far there is no substantiated viewpoint that could explain the properties of these compounds: high critical superconducting transition temperatures, anomalous temperature dependence of the resistance, heat capacity and magnetic susceptibility and the commonly observed softening of

acoustic and optical modes. Many explanations of these properties are based on the concept that with increasing temperature the average value of the density of states, which at low temperature has the form of a very narrow peak

near the Fermi surface, gets “smeared”. The magnetic susceptibility and heat capacity are proportional to the average density of states near the Fermi surface. While the width of the peak of the density of states at low temperature is about 4—5 K, the average value of the density of states decreases considerably as the temperature increases above 10 K, resulting in decreased susceptibility and heat capacity. Thus, the task of the theory is to

explain the appearance of sharp high peaks of the density of states on the Fermi surface at low temperature. It is generally accepted that the narrow peaks of the density of states are generated by quasi-onedimensional chains of transition atoms in A-15 compounds. Indeed, according to X-ray studies, the transition-element atoms A in A3B compounds are situated pairwise in unit cells (fig. 4.1) forming three mutually perpendicular linear systems. The distances

between neighbouring atoms A along the chains are much smaller than those between the atoms situated in different chains.

224

A .S. Davydov, Theoretical investigation of high-temperature superconductivits’

Considering the small overlap of d-shells of transition atoms situated on neighbouring chains, Weger [62] was one of the first to indicate the possibility of treating them independently to explain the anomalous properties of A-15 compounds. Three independent orthogonal chains of transition atoms were also discussed by Barisic and De Gennes [63]. A quasi-one-dimensional model of electronic states of A3B compounds was developed in the most consistent way by Labbe and Friedel in 1967 [64]. When the crystal symmetry is taken into account, five-fold degenerate atomic 3d-levels in every one-dimensional chain are divided into three sublevels. For a chain directed along the z-axis, these sublevels are as follows: d3z2_r2~(d~2,d%,.,) and (d~2,d~2,~2). Since the exchange interaction between neighbouring atoms in a chain is weak, these sublevels are smeared into very narrow energy bands. In the strong-coupling approximation, such bands are characterized by the formula E1(k)

=

(4.3)

E~1+ J. cos ka,

in which E01 is the d-level energy of a free atom A, k is the wave number, J~.the matrix element of exchange interaction between neighbouring atoms A in the ith chain. The density p.(E) of states in such bands has a root singularity at the band boundaries ~

ifE—E(>Jf,

=0

if E-~E)1
.

(4.4)

In vanadium compounds, there are three to four d-electrons per vanadium atom. Therefore, ten energy levels of d-electrons in the band are occupied by less than half. In this case the Fermi level may happen to be near the bottom of the upper (3drn,, 3dX2_%,2)-band slightly occupied by electrons. This was the basic postulate in the Labbe—Friedel theory. The contribution to the density of states coming from other d subbands and wide bands of S-states is very small. Consequently, the electron spectrum of compounds A and B near the Fermi surface is approximated by the dispersion law (4.3) with the state density (4.4) having a root singularity near the Fermi surface. Using this quasi-one-dimensional model, Labbe [65]explained the strong temperature dependence of the magnetic susceptibility. The Labbe—Friedel model based on the Weger—Labbe—Friedel hypothesis about the quasi-onedimensional character of the electron spectrum and the close position of the Fermi level to the bottom of the energy band involved many undetermined parameters fitted to experimental data. With a good choice of these parameters, the model describes quantitatively some anomalous properties of A-IS compounds: the temperature behaviour of elastic moduli and magnetic susceptibility, the high temperature of superconductivity and its relation to a martensite transformation. The fitting character of the theory, however, makes it less convincing. So, according to the Labbe—Friedel theory, the high values of T~and the softening of the phonon spectrum caused by lattice loosening prior to the martensite transition, are due to the singularity in the density of electronic states in quasi-one-dimensional chains near the Fermi surface, situated close to the bottom of the d-subband of electronic states. As known, one-dimensional models of superconductivity have been subjected to criticism. Hohenberg and Rice [66] and Ferrell [67] proved rigorously that superconductivity is impossible in a one-dimensional system, because of the destructive effect of the fluctuations of the electron density. This rigorous proof was concerned with ideal one-dimensional systems. Real systems can be only quasi-one-dimensional, i.e., those are systems in which the transverse dimension is much less than the longitudinal one. Such systems also include the chains of atoms in A-is structures.

AS.

Davydov, Theoretical investigation of high-temperature superconductivity

225

Dzyaloshinskii and Katz [68] have shown that in the model of a metal consisting of a parallel system of linear chains at a distance much greater than the interatomic distance inside the chain, the electron density fluctuations are suppressed due to the long-range character of the Coulomb interaction. In such a system, long-wavelength density fluctuations are the same as in the ordinary three-dimensional metal. To study the wave function fluctuations in a more accurate way, it is necessary to take into account the real transitions of electrons from one chain to a neighbouring one. This was done by Dzyaloshinskii and Katz [68] for the model of a quasi-one-dimensional metal with a dispersion law such as E(k) = k~5J2m+ a(cos ak5 + cos ak5).

(4.5)

With the small parameters a ~ k~/2m~ kB T~the formula for the superconducting transition temperature is practically the same as that obtained in a quasi-one-dimensional system disregarding the fluctuations. For large values of the parameter a, comparable with (k~I2m),the energy spectrum is the same as that of an isotropic crystal. In this case the superconductivity is described by ordinary BCS

theory. Labbe and Friedel [64], as well as Barisic, Labbe and Friedel [69] have shown that when taking account of the interaction of d-electrons with the lattice deformation, the chain model of A-15 superconductor systems offers the possibility to explain qualitatively the phase transition at temperatures Tm> T~.As a result, the free energy involves a term that provides for an appreciable decrease (“softening”) in the elasticity modulus (C11 C12), the vanishing of this modulus determines the temperature for lattice instability at which structural transformation occurs. It was shown that large zero vibrations, anharmonicity and lattice instability take place only for a sufficiently small number of d-electrons in the upper subband. For typical values of the parameters that correspond to a V3Si crystal, this number is only 0.01—0.1 electrons per atom. If these estimates are correct, we can conclude that small electron concentrations contribute to the superconductivity of A-iS —

compounds. The basic feature of the Labbe—Friedel model is the one-dimensionality of the electronic spectrum that results in a singularity of the density of electronic states near the Fermi surface. The role of a weak interchain interaction that violates the quasi-one-dimensionality of A-iS compounds was investigated by Izyumov [70]. It was shown that such an interaction leads to a “smoothening” of the root singularity of the electron spectrum. The “smoothening” results in a decreased superconducting transition temperature T~. These studies made it possible to elucidate how the T~changes depending on the stoichiometric composition of the A3B compounds. When the transition atoms A replace atoms of type B, an additional interchain interaction effected through these transition atoms occurs. The additional interchain interaction results in a sharply decreased T~. As indicated above, the upper d-subband in the Labbe quasi-one-dimensional model is occupied by a

small number of electrons near the bottom of the subband where a great density of electron states is observed. It is also assumed that the Fermi level of all electrons is in this region. The anomalous temperature dependences, i.e., the heat capacity, resistance and heat expansion coefficient for A-i5 compounds above the superconducting transition temperature were investigated by

Gor’kov [71,72]. He showed that the main properties of A-i5 compounds could be explained by employing the concepts of Fermi liquid theory, incorporating the alterations introduced by the one-dimensionality of

fibres. As distinct from the model of Labbe and Friedel who positioned the Fermi level at the bottom of the d-subband, Gor’kov thinks that in the linear chains of these compounds the Fermi level passes

226

AS. Davydov. Theoretical investigation of high-temperature superconductivity

through a flat section of the energy band at the Brillouin zone boundary (point X), where the electron levels are doubly degenerate and obey a linear law of dispersion. It is clear that many properties of superconductors with A-iS structure could be understood by taking into account the instability due to a linear dispersion law of the electron spectrum in this region, resulting from the interaction of electrons with the lattice and among themselves. So, rejecting the nonphonon mechanisms of electron pairing discussed in some papers (see, for example, ref. [73]), Gor’kov based his theory on a phonon mechanism the interaction of electrons with lattice deformation. It was shown that as the temperature decreases, such an interaction leads to logarithmically decreasing elasticity moduli C11 and C12. At some temperature Tm 25K close to that —

of a superconducting transition, the shift modulus may vanish. The vanishing of this modulus causes a weak tetragonal deformation in a cubic lattice, i.e., a martensite transformation. Under this transformation the atoms shift by less than 0.01 A, and this is an order of magnitude less than the amplitude of thermal and zero vibrations (-—0.1 A). The assumption that the Fermi surface contacts the Brillouin zone boundary at point X, resulting in the shift modulus in A-iS compounds, is consistent with the experimental results reported by Testradi and Bateman [74]. So, there are two different models that explain qualitatively a number of properties of A-iS superconductors. Both models regard the motion of d-electrons as being quasi-one-dimensional. Further studies will show which of the models is more realistic. To explain the properties of A3B compounds, Yu and Anderson [75] proposed an alternative quasi-one-dimensional model in which an A3B crystal is replaced with a model system that consists of an individual atom (atom A with one d-electron), strongly interacting with an electron gas (electrons 4s and 4p of atoms A and B). Only one mode of vibrations of atom A along the chain axis is studied. It is assumed that neighbouring atoms of the chain produce an elastic restoring action, screened by the cloud

of surrounding electrons. Such a screening loosens the lattice, reducing the frequency of vibrations. When further simplified, the system is replaced by a local harmonic oscillator immersed into an electronic gas. When the temperature is high and the electron—phonon interaction is strong enough, a simple harmonic potential transforms into a double potential well. When the temperature decreases, the double well gets smoothed, transforming into one planar well. Moreover, the zero vibration amplitude increases considerably in the process. This is equivalent to the lattice softening discussed in the models of Labbe and Gor’kov. We conclude by mentioning that a theoretical description of A-iS compounds has not been completed as yet. Much attention has been focussed on studying the properties of these compounds at temperatures above the critical temperature T~.As yet there is no clear understanding of the role of the interaction of two types of quasiparticles that form narrow d-bands and wide sp-bands. Which of these quasiparticles or their combinations form singlet pairs and what is the mechanism of their pairing? All these questions still need to be answered. 4.3. Superconductivity of transition metal compounds with NaCI(B1) structure 4.3.1. Superconductivity of nitrides and carbides A large group of superconducting materials is made up of chemical compounds of transition metals A and nonmetals B, described by the chemical formula AB. They include an important class of compounds crystallized in the NaCI structure denoted as B! in crystallography. Transition elements of the 3rd, 4th, 5th and 6th subgroups of the periodic system appear as atoms A, and nontransition elements appear as atoms B [76—80].

A.S. Davydov, Theoretical investigation of high-temperature superconductivity

227

Table 4.4 Critical temperature Compounds

T

NbN ZrN HfN VN TaN

17.3 10.7 8.8 8.5 6.5

T~for some

1(K)

nitrides and carbides

Compounds

T,fK)

MoC NbC TaC WC TIC

14.3 12.0 10.4 10.0 3.4

The highest-temperature superconductors are generated by transition metals of the 4th, 5th and 6th subgroups: Zr, Nb, Mo, Ta, W, which have incomplete 4d- and Sd- shells when they join nitrogen (nitrides) or carbon (carbides). Table 4.4 gives the values of critical superconducting transition temperatures for some nitrides and carbides. Nitrogen and carbon, included in nitrides and carbides, are not superconducting. The superconducting transition temperature for transition elements A is indicated in table 4.5. As follows from tables 4.4 and 4.5 the chemical compounds AB have, generally, much higher temperatures T~than the components involved in them. This points to a considerable rearrangement of

the electron states of A and B atoms, that form a chemical bond. In AB crystals, metallic type and ionic bonds are equally significant. In these crystals, atoms A of transition elements form a cubic face-centered lattice, and nontransition atoms B occupy all octahedral vacancies. As a result, every A atom is surrounded by an octahedron of B atoms, and vice versa.

The electronic states in compounds AB have been calculated theoretically to many approximations. The calculations performed with different choices of the initial configurations of atoms A and B in the same compounds produce energy spectra much different both in the distribution and in the values of the energy of electronic states. Without a comparison with experimental data, it is difficult to favour a certain approach, because all of them are quite approximate and based on a number of arbitrary assumptions. Ordinarily, such calculations refer to the normal state. They give no answer to the main question: which quasiparticles are paired? Do only heavy d-electrons participate in the pairing, or light s-electrons, or their hybridized states? What is the mechanism of pairing? Unfortunately, all the calculations of electronic spectra have concerned themselves with a stoichiometric composition. When there are no vacancies, such compounds are dielectrics. Ordinarily, there is a certain number of vacancies even in compounds of a stoichiometric composition. Atoms A and atoms B may be absent simultaneously. In some cases the vacancies are distributed in an ordered way and form superstructures. Even for compounds of a stoichiometric composition, the concentration of vacancies is quite high in every subgroup of crystals Bi. For example, the percentage concentration of vacancies in nitrides ZrN, NbN and VN, is equal to, respectively, 3.5, 1.3 and 1.0, and that in carbides HfC, ZrC, TaC is 4.0, 3.5 and 0.5, respectively. When Bi compounds are doped, the temperature T~usually rises. In 1953, Matthias [81]obtained for a NbC—NbN system a superconducting transition temperature equal to 17.8 K. During the fourteen Table 4.5 Critical temperature for some transition elements Element

Nb

V

Ta

TI

Re

Mo

Zr

Hf

W

T~(K)

9.3

5.4

4.5

2.4

1.7

0.92

0.53

0.10

0.015

228

A.S. Davydov, Theoretical investigation of high-temperature superconductivdv

years that followed this was one of the highest critical temperatures exhibited by the then available superconductors. Since B! compounds are dielectrics when the composition is stoichiometric and the electrons fully occupy both sublattices of a crystal, their superconductivity is likely to be caused by the presence of vacancies or doping. As distinct from A-iS compounds, the compounds with an NaC1 lattice are very stable and less sensitive to mechanical defects. Like A-is compounds, they have extraordinary properties in normal and superconducting states. Contrary to A-iS compounds, the large values of the superconducting transition temperatures for Bi compounds are not necessarily related to the large values of the linear coefficient y of heat capacity and the large values of the coefficient of magnetic permeability. So, there is no direct relationship between the density of energy states on the Fermi surface and the superconducting transition temperature. Zeller [82] and Van Maaren [83]made tunneling studies and concluded that the basic reason for Bi compounds being superconductive is an electron—phonon interaction and the dominant contribution to the electron—phonon interaction parameter comes from acoustic phonons. In contrast to isotropic metals in which the superconduction process is caused by Cooper pairs formed by one sort of quasiparticles, Bi compounds have two sorts of quasiparticles that can carry an electric charge: d-electrons of very narrow energy bands and sp-electrons of wide energy bands. A complete theory of the properties of such compounds has not yet been constructed. So in order to interpret experimental data, use is made of the formulas proposed by McMillan to describe the properties of isotropic superconductors with a strong electron—phonon coupling of one sort of quasiparticles. Thus Weber [84]used the McMillan formulas to calculate the electron—phonon interaction constants A for several carbides, employing data on the temperature T~,the Debye temperature and the density of electronic states, N0, on the Fermi surface, determined from heat capacity data. The values obtained are given in table 4.6.

The neutron scattering studies of the phonon spectra of stoichiometric compounds HfC (T~= 0.2S K) and TaC (T~= iO.3 K) made by Smith and Glasser [85] are noteworthy. The values of longitudinal acoustic frequencies Q(~)in units of 1012 Hz are given in fig. 4.4 as function of ~ = aql8ir. In a TaC compound with large T~there is an anomaly (decreased vibration frequency) near the band boundary. In a HfC compound with a very low T~,however, such an anomaly is not observed. The data confirm the dominant role of electron—phonon interaction in the superconductivity of Bi compounds. The decreased frequency of longitudinal vibrations indicate that the TaC crystal lattice can be more easily deformed. 4.3.2. Superconductivity of compounds of palladium and thorium with hydrogen and deuterium In the preceding section we have discussed the B i-type superconducting compounds of transition elements that have the property of being superconductive even in pure form. Table 4.6 The values of A for some carbides Compounds

ZrC

HfC

NbC

TaC

TJK) A

0.25 0.3

0.25 0.3

11.1 0.7

10.3 0.7

AS. Davydov, Theoretical investigation of high-temperature superconductivity

6-

HfC ~

~1

229

~

o’~

/°

o~~J

TciC

/

0.0

0.5

1.0

Fig. 4.4. The longitudinal acoustic frequency 11(f) as the function of f = aq/8ir for HfC and TaC, from Smith and Glasser (85].

In the seventies it was discovered that some metals and alloys not superconductive in pure form become relatively good superconductors when they form alloys or compounds involving hydrogen or deuterium. These metals include the transition elements palladium (Pd) and thorium (Th) that have

unoccupied 4d- and 5f-electron shells, respectively. In 1972, Skoskewitz [86]discovered that the transition element palladium, that has a small magnetic moment normally preventing the pairing of electrons, joins hydrogen and forms a PdH compound that goes into a superconducting state at T~= 9 K. This compound has a cubic structure such as NaC1 (Bi). It was established that adding noble metals to such a system raises the critical transition temperature up to 17K. A palladium—deuterium system (PdD) is also superconducting. It has a higher superconducting transition temperature T~equal to 11 K. So, as Hertel showed [87], the replacement of hydrogen with deuterium leads to an increased T~.The phenomenon was called an “inverse isotopic effect”.

The main task was to explain the superconduction mechanism of PdH and PdD compounds, considering that pure Pd is not superconductive. According to Bennemann and Garland [88],the role of hydrogen in PdH compounds is to suppress spin fluctuations that enhanced electron repulsion in pure Pd and destroyed Cooper pairs. On the other hand, Papaconstantopoulos and Klein [89]think that the superconductivity of PdH and PdD compounds is due to the electron pairing effect being enhanced by an additional interaction with high-frequency optical vibrations of hydrogen and deuterium atoms in

these compounds. The interaction with optical vibrations raises the electron—phonon interaction constant A from 0.45 for pure Pd to 0.72 for a PdD compound. It is assumed that the inverse isotopic effect is due to higher values of the frequencies of optical vibrations in PdH compounds than in PdD ones. It is worth noting that thorium hydrides and deuterides have close values of T~.The reason may be the higher atomic mass of thorium. The compounds of transition elements of palladium and thorium with hydrogen and deuterium, as well as Nb3X and V3X compounds have two conduction bands: the wide valence band of s- and p-electrons and the very narrow band generated by electrons in the inner 4d- and 5f-subshells

230

A.S. Davydov, Theoretical investigation of high-temperature superconductivity

incompletely occupied by electrons. The superconductivity of palladium and thorium hydrides and deuterides seems to be caused by the electrons going to free states from their inner 4d- and Sf-shells. The electron donors that occupy such states are hydrogen and deuterium and some other elements: aluminium, copper, silver, indium, etc. Since the exchange interaction with neighbouring transition elements is very weak, these electrons form very narrow energy conduction bands with large effective masses of charge carriers. Figure 4.5 represents the experimental studies [90] of the dependence of the superconducting transition temperature T~for Pd1_XAIXHY and Pd11In~H~ compounds on the concentration of electron donors Al and In. The study was performed under the most favourable hydrogen concentrations that correspond to the maximum value of T~with fixed x. We can see that as the concentration rises, the critical temperature initially increases, reaching a value equal to ---8.5 K and then drops sharply. The measurements made by Buckel and Stritzker [9i] reproduce the dependence of the critical temperature of a compound on the concentration of silver (fig. 4.6). The maximal value, T~= 13 K, is attained with a 20% silver content. The value of T~then drops sharply as the silver concentration approaches 40%. Figure 4.7 represents the dependence of the superconducting transition temperature for a Pd1_1Cu~Hcompound on the atomic concentration of copper reported by G. Hem, Stritzker and Buckel [92, 93]. The maximal T~= i6 K is reached with H0 .,Pd0 55Cu0 We can thus observe the following situation: as the electron concentration rises as a result of electron donors (copper and silver atoms) being added to Pd—H and Pd—D compounds the critical temperature also rises, passes through its maximal value and then drops sharply. 4.4. Superconductors with heavy fermions By the end of the seventies or the beginning of the eighties new original superconductors that contain the transitional elements of cerium and uranium had been synthesized. In 1979, Steglich et al. [94] obtained the CeCu2Si1 intermetal compound with the superconducting transition temperature T~= 0.65 K.

io 2

15

__

0

10

Hb0~~

20

30 x%

Fig. 4.5. The dependence of T~for the Pd1 ,Ph,H~and Pd1,ln,l-I, compounds, on the concentration of Pb (curve I) and In (curve 2); from Freidberg, Rex and Ruvalds [90].

Fig. 4.6. The dependence of T~for the Pd ,AgD, compound on the Ag concentration; from Buckel and Stritzker [91].

A.S. Davydov, Theoretical investigation of high-temperature superconductivity

231

18 12

/4

100 ~ -‘

50~~

I—

II

VflV1~ffi

0

I

O

20

40

60

I

80 100

o.o

0.2

0.4

0.6 0.8

1.0

Cu% Fig. 4.7. The dependence of T~for the Pd,_~Cu~D compound 5 on the Cu concentration; from HeIn, Stritzker and Buckel [92,931.

Fig. 4.8. The temperature dependence of the resistance of UBe13 at different magnetic fields. Curve 1,0; curve II, 10 kOe; curve III, 20 kOe; curve IV, 30 kOe; curve V, 40 kOe; curve VI, 50 kOe; curve VII, 7OkOe; from Maple et al. [1001.

A UBe13 compound with T~= 0.95 K was synthesized by Ott’s group in 1983 [95],and a VPt3 compound at T~ 0.54 K by the group of Stewart [96].Compounds of such a type involve also UFe with T~ 3.86 K, CeRu2 with T~ 6 K, CeA12 and some others. All these compounds at room temperature have a lattice of magnetic ions that usually prevent superconductivity. It was expected that with decreasing temperature such compounds should go into a

magnetic state. It turned out, however, that at low temperatures they are characterized by the superconducting state not the magnetic one. The Meissner effect has proved the bulk nature of this superconductivity. In spite of comparatively low temperatures of superconducting transitions, these superconductors

attracted the attention of researchers. It was established that they possess unique properties. Soon afterwards they were called superconductors with heavy fermions, since the effective mass m of heat and charge carriers, according to low-temperature heat capacity measurements, happened to be more than two orders of magnitude larger than the mass of free electrons m1. The compounds are also distinguished because superconductivity in them is accompanied by strong Pauli paramagnetism. The unusual properties of superconductors with heavy fermions are discussed in reviews by Stewart [97], Alexeevskii and Kchomski [98], Moshchalkov and Brandt [99].

Superconductors with heavy fermions are strong second-kind superconductors. Their lower critical magnetic field has a comparatively low value. In CeCuSi2 it equals only 23.3 Oe. But upper critical magnetic fields H~2and the derivatives dH~2/dT may obtain large values at T = Tc. In the CeCu2Si2 and VBe13 compounds these derivatives equal, respectively, 420 kOe/K and 230 kOe/K. Figure 4.8 shows the temperature dependence of the resistance of the single crystal UPt3 at different magnetic fields that was obtained by Maple et al. [100]. They also have shown that in hexagonal VB13 and tetragonal CeCu2Si2 crystals the higher critical magnetic field H~2exhibits great anisotropy. In CeCu2Si2 the distance between the nearest Ce atoms equals 4.1 A. In VBe13 the uranium atoms form a cubic lattice. The Be13 clusters are situated between them. These are almost regular polyhedrons, icosahedrons Be12, with one additional centered atom Be. The distance between the nearest uranium atoms equals 5.13 A. The crystal UPt3 has a hexagonal structure, the distance between uranium atoms is 4.1 58Ce) andA.uranium (92U) atoms have magnetic moments of the order of three Bohr Cerium (which are caused by incomplete f-electron shells. These atoms belong to transition elements magnetons,

232

AS. Davydov, Theoretical investigation of high-temperature superconductivity

of the third and fourth groups. Their external electron shells in the atomic state are characterized, respectively, by the configurations, 4ftSs2Sp6Sdt6s2 and Sf36s26p66dt7s2. Since the f-shells with a large orbital angular momentum, equal to 311, have 14 vacant sites in these atoms, they are only partially filled with electrons. In crystals the electron levels s, d and p are smeared into conductivity bands of width S—iO eV. The remaining electrons of cerium and uranium atoms are in localized ion states which are dependent on their environment. For instance, a cerium ion can be in a valence-three state (Ce3~)where there is one electron on the f-shell (configuration 4f1). In some compounds it is in a valence-four state (Ce4~)where the shell 4f is empty (configuration 4f°).The total angular momentum of the Ce4~ion equals zero. In this state the magnetic moment is absent. At close location of the energy levels 4f and Sd of the bands the ions can be characterized by intermediate valency. In this case the electron is partly in the 4f and partly in the Sd states and is described by the wave function ~1i= atlld + f3t~sf, a~2+ ~2 = Since cerium and uranium atoms are transitional elements possessing, apart from d- p- and s-electrons, also partially filled f-shells, their electron spectrum is characterized by two conductivity bands: the very wide band of valance d- and s-electrons and the very narrow band of f-electrons. The effective mass of the valence electrons m~in the conductivity band differs little from the mass of free electrons. The effective mass of f-electrons m~in the f-band is large. It is determined by the value of the exchange integral J of f-electrons between the nearest cerium (uranium) atoms by means of the following formula: ,

(4.6)

m~=h2/2a2J.

If m~= i03me, a 4 x iIJ~cm the exchange integral value following from eq. (4.6) is equal to ——4 x i0~eV. Such a small value, having as a consequence a large value for the effective mass, is due to the weak overlap of the wave functions of f-electrons of neighbouring atoms. In the range of critical temperature T~,the specific heat capacity C(T) undergoes an abrupt change (up to i5%) (see fig. 4.9). The enormous value of the coefficient y and the jump of the specific heat capacity in the transition to a superconducting state prove that heavy electrons participate in superconducting pairing. On the other hand, studies of some properties in the normal phase conductivity, thermal expansion, thermal electromotive force etc. reveal the substantial role played by conduction electrons with normal mass. There exist also direct experimental data confirming the presence of two groups of electrons with fairly different effective masses. —

—

0

0.4

0.8 1K

Fig. 4.9. The abrupt change of the specific heat capacity C(T) of a superconductor. CeCu 25i2, from Steglich et al. [101].

AS. Davydov, Theoretical investigation of high-temperature superconductivity

233

For instance, the presence of two groups of charge carriers was found when studying the de Haas—van Alphen effect in CeSr2 compounds [101] and the Hall effect in UBe13 compounds in strong

magnetic fields [102].Alexeevskii et a!. [103]showed that in weak fields the Hall effect is generated by a small concentration of a light component. In strong fields the process involves heavy electrons. Therefore, the total concentration of charge carriers increases, inducing the drop of the Hall constant.

The London depth of penetration into UBe13 is of the order of 3.6 x i0~cm which is characteristic of the usual superconductors. Apparently, this also indicates that the light electrons are important in

the screening of the electromagnetic field. As is known, the specific 2)heat capacity of metals in the normal state, is determined at low temperatures by (in mJ/mol K C(T)yT+f3T3,

(4.7)

T<0D150.

Here is the Debye temperature. The first term in this expression defines the heat capacity of electrons, and the second one the heat capacity induced by lattice vibrations. The plot of C(T) IT versus T2 allows one to define the coefficient y as the value of C( T) IT with T—~0. The parameter 13 is found °D

from the slope of the linear dependence in this plot. The coefficient y in ordinary metals, in mJ/mol K2 units, has a value close to unity. Table 4.6 presents the values -y, in mJlmol K2 units, for some simple metals and compounds with heavy fermions. In the simplest model of a metal where the latter is treated as the gas of free electrons, the parameters y and x are proportional to the energy density N(EF) of states, i.e., to the number of allowed states per unit energy interval near the Fermi surface. For electrons following the quadratic law

of dispersion, the density of states in isotropic space is determined by the space dimension. Onedimensional, two-dimensional and three-dimensional measurements correspond to the following forms for N(E) as a function of E: N 112, N 112. At temperatures lower than 1(E) E K) in metals, 2(E) const., N3(E) yEhas a value close to unity and is that of electron gas degeneration (-—10~ the parameter independent of the temperature. In compounds with heavy fermions, the values of y and x on the Fermi surface at T T~exceed the relevant values in ordinary metals by two or three orders of magnitude. In this case their strong -—

-—

-—

-—

dependence on the temperature is observed. They decrease by an order of magnitude as the temperature increases up to TF 3—iO K. As these parameters are proportional to the mean density of -—

states N(EF) on the Fermi surface, their large values and strong temperature dependence reveal that the energy state density on the Fermi surface N(EF) at T < TF has a giant sharp peak of width

i0”—iO~eV which is equivalent to several degrees on the temperature scale. Such a high narrow energy peak is usually called a resonance. The temperature TF is the temperature of heavy quasiparticle degeneration. Compounds with heavy fermions are characterized by a rather unusual temperature behaviour of the electric resistance. Unlike the metals where the resistance falls with decreasing temperature, in these compounds it first rises, attains its maximum value and then falls, vanishing for T ~ T~. Table 4.6 Values of coefficient y at zero temperature Compounds:

Cu

2) 0.7

Ty (mJ/molK 0 (K)

0

Li

Sn

UPt

1.6

1.73

450

0

3.7

0.53

3

CeCu2Si2 UBe13 1000 1100 0.50

0.85

CeAI2 1650

234

AS. Davydov. Theoretical investigation of high-temperature superconductivity

The rise of the resistance with decreasing temperature in metals containing isolated magnetic impurities was explained by Kondo in i964 [104]. This phenomenon was called the Kondo effect. Kondo explained theoretically the intricate appearance of a minimum on the resistance—temperature curve in weak solutions of Cu, Ag, Au, Mg, Zn with magnetic impurities Mn, Fe, Mo, Re, Os that have d-electrons. It turned out that the observed minimum with decreasing temperature in the region of the critical temperature T~and the subsequent logarithmic growth of the resistance can be attributed to an increase in the probability of sd-electrons in the conduction band of the metal to scatter with spin flip as the temperature decreases, and to the partial screening of the magnetic moments by impurities. The critical temperature TK was called the Kondo temperature. For compounds such as CeCu2Si2, CeA3 and UBe13 the temperature TK equals, respectively, 5,8 and 10K. The Kondo scattering is caused by the antiferromagnetic interaction of the f-electrons of the magnetic impurity with the sd-electrons of the conduction band. This interaction is characterized by a negative exchange energy of about —0.2 eV. Thus, the experimentally observed rise of the resistance with decreasing temperature (exceeding TK) leads to the conclusion that at T> TK the centres behave independently as Kondo-impurities. The fall of the resistance observed at temperatures less than TK (fig. 4.10) is, apparently, caused by the fact that in the region of these temperatures one cannot neglect the ordered locations of f-centers and their interaction. It is necessary to consider the interaction between the sd-electrons of the conduction band and the Kondo-lattice where the f-level degeneration in spin components is removed by the crystal field. This is discussed by Moshchalkov and Brandt [i08]. In systems with heavy fermions the magnetic susceptibility x at high temperatures increases with decreasing temperature according to the Curie law x — 1 / T, just as is the case in ordinary systems containing local magnetic moments. With a substantial decrease in temperature, however, the increase in magnetic susceptibility ceases without magnetic ordering. The magnetic susceptibility x takes on a very large constant value. In units CGSM/mol in CeCu7Si,, UBe13 and CeA13 compounds the value x equals, respectively, 8, iS and 36. In ordinary metals (copper) the value is x 0.008 and at T ~ T~these materials lose their magnetic properties and go over into a superconducting state. Changes in the magnetic properties of a system with heavy fermions can be qualitatively explained by the Kondo effect. It implies that the decreasing temperature of a compound in the range of values lower than the Kondo temperature TK leads to the screening of the isolated magnetic moments of sd-electrons of the conduction band. Such screening is accompanied by the lowering of the energy of the system. However, with ordering in the location of f-centres, when the temperature becomes lower than Tcog the result is magnetic ordering with a gain in energy. If the inequality TK> Tcog is satisfied, the screening of magnetic moments occurs earlier and magnetic ordering will not be manifest.

Tk

I

Fig. 4.10. The qualitative dependence of the resistance on temperature for an ordinary metal (curve 1) and for a superconductor with heavy fermions (curve 2); TK is the Kondo temperature.

AS. Davydov, Theoretical investigation of high-temperature superconductivity

235

It has by now been established that the low-temperature anomalies in superconductors with heavy fermions are due to the giant narrow energy peak of the quasiparticle state density in the region of the Fermi level. Many papers are concerned with clarifying the nature of the appearance of such a peak. In 1972 Gruner and Zawadowski [105] suggested a phenomenological mode! according to which this peak is generated by resonances in the scattering of conduction electrons by the system of f-centres (Kondo impurities). The possibility of the formation of a narrow resonance at low temperatures due to the scattering of Fermi energy electrons on Kondo-impurities was investigated theoretically in 1965 by Abrikosov [106] and by Suh! [107]. Therefore, the narrow peak of the density of states arising in superconductors with f-electrons was called by Gruner and Zawadowski an Abrikosov—Suhl resonance. In investigations by Aliev, Brandt, Moshchalkov and Chudinov [108—111]the narrow peak of the state density at the Fermi level in CeA12 and CeCu2Si2 is represented as an Abrikosov—Suhl resonance on Kondo-lattices and the f-band itself is assumed to lie much lower than EF, since cerium valence is defined by integer numbers. The alternative point of view was developed in 1979 by Kchomski [112]. Proceeding from the fact that 4f-shell valence is not an integer, the 4f-electrons were assumed to go directly on the Fermi level due to interconfigurational transitions such as 4f” ~ 1 + sd. In some papers it is ascertained that the narrow peak of the state density on the Fermi surface is generated by mixed (hybridized) states arising due to the spin—spin interaction of d-electrons. The resulting bound states lie on the Fermi surface and have the effective mass of heavy quasiparticles. The hybridization of sd- and f-electrons was investigated on the basis of Anderson’s Hamiltonian [113]in the papers of Lin [114]and Martin [115]. As is shown by Martin, with such a hybridization the f-electron is clouded by sd-electrons. The main peculiarities of low-temperature anomalies in systems with heavy fermions are interpreted by Eliashberg [116] on the basis of a strong renormalization at low temperature of the spectra of conduction electrons near the Fermi surface. By analogy with renormalization of the spectrum of conduction electrons in metals (studied by Migdal [117]) which is caused by the electron—phonon interaction, Eliashberg assumes that in systems with heavy fermions such a renormalization of the spectrum of conduction electrons is generated by their short-range action and magnetic terms in spin-dependent interaction with weakly bound, well localized, !anthanide and actinide f-atoms. The properties of compounds with heavy fermions at temperatures lower than the critical ones, also differ essentially from the properties of superconductors described by BCS theory. According to this theory, at T < T~the availability of the energy gap 4 in the one-particle spectrum results in the heat capacity, thermal conductivity, ultrasonic damping, etc. decreasing exponentially with decreasing temperature. Studying the temperature dependence in the region 0.1 T~~ T T~,Ott et al. [118] revealed the power 3Cn, formwhere dependence of the heat capacity C,(T) in the superconductivity state, C5(T) 2.8 C (TI T~) 11 is the heat capacity4Cn, in the state. In CeCu2Si2 superconductors, this withnormal /3 = 2.4—3. dependence is described by CS(T) (TITc) Some approaches to explain the anomalous temperature dependence of C~(T)and of other physical quantities in superconductors with heavy fermions at T < T~are suggested. For instance, it is assumed that such a nonexponential character of the temperature dependence is due to the presence of two independent gaps in quasiparticle states with light and heavy masses. On the other hand, Volovik and Gor’kov [119] explain the power form of the dependence of the specific heat capacity on the temperature by strongly anisotropic properties of the energy gap. They showed that C( T) is propor-

236

AS. Davydov, Theoretical investigation of high-temperature superconductivity

tional to T3, if the energy gap vanishes at some points of the Fermi surface. The functional behaviour is T2, if the energy gap vanishes along some of the lines. Studying the de Haas—van Alphen effect in a UPt 3 compound Taillefer and Lonzarich [120]obtained direct evidence of the presence of heavy electrons in cerium and uranium compounds. A large value was found for the cyclotron mass. If the magnetic field is oriented along the axis a in the principal crystallographic plane, the cyclotron mass is (25—90) me, and if the orientation is along the axis b, the value is m* = (iS—S0)me. Although the existence of heavy and light electrons in the superconductors UPt3, CeCu5Si2, etc., was proved experimentally, the nature of these quasiparticles and the way in which they are paired are not quite understood. The pairing of electrons in Cooper pairs in ordinary superconductors is realized in a singlet spin state with zero orbital angular momentum. Such a pairing is called the S-pairing. At the same time it is known that when a Fermi-liquid (helium-3) goes to a superfluid state, two atoms are paired with the total spin equal to one and a nonzero orbital angular momentum. This was called the triplet or P-pairing. Liquid helium-3, being a strongly interacting Fermi-liquid, with a large effective quasiparticle mass and enhanced magnetic susceptibility at temperature lower 3 mK, goes over to a superfluid state in which it can exist in two different phases. The repulsive character of the nuclear He—He interaction admits pairs of quasiparticles in the states representing a triplet in spin space and a p-wave in ordinary space (the orbital angular momentum of the pair equals one) to be formed in liquid helium-3. The A-phase of superfluid helium-3 corresponds to a triplet state in which the energy gap vanishes at two diametrically opposite points of the Fermi surface.

If we consider an abstraction of the real crystal structure and assume heavy fermions to form an isotropic Fermi liquid, we can use the analogy with superfluid helium-3 to describe the superconducting state of these systems. This point of view was put forward by Valls and Tesanovich [121], Volovik and Gor’kov [122], Varma [123]and Anderson [i24]. Anderson put forward the hypothesis that the electron states of the compounds CeAl3, CeCu2Si5, UBe13 and UPt3 can be considered as a heavy Fermi-liquid similar to liquid helium-3. Then the electron pairing in such an isotropic liquid may conventionally be called the triplet pairing (p-pairing). However, a strong spin—orbit interaction is found in real superconductors with heavy fermions. Therefore, the spin turns out to be a bad quantum number. It is necessary to consider the total angular momentum and to take into account the lattice symmetry. The parity and time reversal (“odd-parity”) prove to be more correct characteristics for this pairing. Since compounds with heavy fermions go over into a superconducting state with critical temperature much lower than the Debye temperature, Anderson assumes that in these compounds the specific nonphonon pairing mechanism is manifest, e.g., via virtual spin excitations. The specificity of the pairing mechanism in heavy fermion compounds explains why the antiferromagnetic compound CaPb3 with heavy fermions is not a superconductor in the absence of a magnetic field. It becomes a superconductor with a critical temperature T~— 0.6 K in a strong magnetic field. With S-pairing, the magnetic field tends to break up the pairs. With P-pairing, the magnetic field promotes pairing. It is not yet clear whether the notion of “triplet” pairing is sufficiently convincing. A number of experiments, e.g., on Josephson’s effect in CeCu2Si2, point to the usual singlet pairing. Therefore, attempts are made to explain experimental results (in particular, nonexponential dependence) within the traditional framework of singlet pairing taking account of spin—orbit interaction, the effect of spin fluctuations, etc. A theoretical investigation of the ordinary S-wave Cooper pairing caused by the electron—phonon mechanism in a UPt3 superconductor was carried out by Oguchi et al. in 1986 [125]. Using relativistic

A.S. Davydov, Theoretical investigation of high-temperature superconductivity

237

calculations, they managed to estimate roughly the value of the electron—phonon interaction. The latter is shown to be strong when the electrons interact with platinum atoms and weak when they interact with uranium atoms. This anisotropy of the interaction resulted in the anisotropy of the energy gap. On the other hand, Fenton [126],using the Eliashberg equations, showed that ordinary superconductivity based on S-pairing in a UPt3 superconductor is impossible, since it results in the value T~—— iO_23 K. This result would be important if one could show that the Eliashberg equations are applicable for systems with heavy fermions. A possible superconducting pairing interaction based on antiferromagnetic correlations in a UPt3 compound, which were obtained from neutron measurements, was shown by Hirsch [127] and Miyake [128]. The subject of pairing in superconductors with heavy fermions still remains debatable. It is not even clearly understood which electrons participate in pairing conductivity electrons (as in metals) or f-electrons. It seems to us that quasiparticles determining superconductivity in such compounds are the pairs made up of the f-electrons of the magnetic ions Ce and U and the conduction-band electrons. With small concentrations of magnetic ions such bound states are stationary because of the absence of exchange interaction between neighbouring ions. In superconducting crystals, however, the magnetic ions are present in each unit cell, and their density is large. The distance between cerium and uranium atoms is of the order of 4—S A. In this case even the weak exchange interaction J -~4 x i0~eV makes it possible to collectivize the energy states of f-electrons 3 me. in the form of very narrow energy bands in which quasiparticles have the effective mass m* i0 So, heavy quasiparticles represent, apparently, the bound pairs generated by f- and d-electrons. When such pairs are formed, the energy spectrum is completely rearranged. The energy band of heavy quasiparticles gets much narrower abruptly, giving to the density of states the form of a narrow peak coinciding with the Fermi energy. Heavy quasiparticles are Bose-particles. They have zero spin and a double electric charge. Their mass is of the order of i03 me. As a result of the exchange interaction they form a combined system a condensate. In pairing their group velocity falls down to i0~cmls as compared to that of d-electrons, 108 cm/s, in the conduction band. Indeed, the measured velocities in the CeA1 3, CeCu2Si2 and UBe13 compounds 5, 1.0 x i05 and 3.4 X 106 cmls. The measured coherence lengths in theequal, UPt respectively, 1.18 X i0 3, UBe13 and CeCu2Si2 compounds are equal, respectively, to 142, 170 and 190 A. Thus, the “radius” of paired f- and d-electrons is much lower than that of Cooper pairs in metals described by the BCS theory. It exceeds, however, the lattice constant of these crystals by a factor 40—50. Consequently, as in metals, inside a —

——

—

4~

single pair there are centres of many other pairs. As has been noted, the formation of individual pairs (“hybridization”) of f- and d-electrons was studied by Lin [114] and Martin [115] using the representation of spin—spin interaction. I want to note that the concept of spin—spin interaction is not necessarily understood in a direct sense. It is necessary to describe an exchange interaction on the basis of a Coulomb interaction. Such an interaction defines the covalence chemical bond between atoms in molecules. So the possible formation of a chemical bond between (hydrogen) atoms in a singlet spin state is determined by the character of correlations in the motion of electrons in these states. Although this correlation is dependent on the relative orientation of electron spins, it is not generated by the direct interaction of the magnetic moments of the electrons. The energy of this interaction is negligibly small

as compared to the exchange interaction. To produce a chemical bond, it is necessary for the wavefunction to be symmetric under the interchange of the space coordinates of the electrons. In this case there is a large probability that the

238

A. S. Davydov, Theoretical investigation of high-temperature superconductivity

electrons are located between the nuclei of atoms. The Coulomb attraction between electrons and nuclei leads to a bound state. This interaction is usually called the exchange interaction. So, the exchange interaction is a part of the Coulomb interaction of an electron with nuclei, which is determined by the symmetry of the wave function. The direct interaction between the spins of two electrons is practically unimportant in forming a bond. This is shown by the possibility for such a bond to be formed by only one electron, as in the case of the ion of the hydrogen molecule, H~. Certainly the above interpretation of heavy quasiparticles as bound pairs of f- and d-electrons is fairly qualitative. Its foundation needs serious theoretical study.

5. Superconductivity of metal oxide compounds 5.1. First metal oxide compounds with small concentration of charge carriers The superconductivity of some oxide compounds, e.g., SrTiO2 with the transition temperature T~= 0.3 K and Li1+XTi2_~O4with the temperature T~= i3 K, had been detected before i97S. However, these compounds did not attract researchers, because there were many difficulties in preparing and producing them as mixtures of multiphase compounds. In i97S, Sleight et al. [129] discovered the superconductivity of a metal (bismuth) oxide (ceramics) BaPb1~Bi~O3.This compound was easily prepared as single crystals in the form of a film. This stimulated studies of such bismuth ceramics in many laboratories. The properties of a new stable ceramic material BaPb~_~Bi~O3 which will in the sequel be denoted by BPBiXO, puzzled many physicists. Usually the electroconductivity of ceramics is much lower than that of normal metals. The system BPBiXO, however, with x = 0.25 and at a temperature below 13.7 K, 3. went over to a superconducting state with the carrier density equal to ~~~i021 cm Before 1975, this critical temperature had been the highest among compounds not involving transition elements. The critical temperature of the transition to a superconducting state in ordinary metals increases as the density of carriers on the Fermi surface increases. It seemed as though in the ceramics BPBi 2O T~could be increased by raising the concentration of bismuth ions. However, this attempt was a failure. Measuring the conductivity and Hall effects, Thanh, Koma and Tanaka [130] investigated the dependence of T~on the concentration of bismuth ions and showed that this dependence has a nonmonotonic character. The transition temperature has a maximum value equal to 13.7 K with concentration x = 0.25 and decreases both with increasing and decreasing bismuth concentration. When x grows larger than 0.4, the metal—dielectric transition accompanied by structural rearrangement takes place. Within the interval of values x less than 0. i2, the critical temperature and the density of states N(x) increase almost linearly attaining a maximum at x = 0.25 and then decrease. Such a dependence is illustrated in table 5.1. Table .5.1 The dependence of T~(x)on the concentration of bismuth x 3) x0 T(x)(K) N(x)(cm 0.46 I.42x 1020 0.12 0,2

4.0 10.0

1.6 x 2.0 x

102 102

A. S. Davydov, Theoretical investigation of high-temperature superconductivity

239

The low density of charge carriers in metal oxide compounds is significant. Besides, these compounds are to be distinguished from the superconductors known earlier by the spectrum of lattice vibrations and large anisotropic structure (see a review by Gabovich and Moiseev [131]). The metal oxide BPBiXO has been studied thoroughly. It has the structure of a perovskite. Materials called perovskites are hard (ceramics) consisting of metal and nonmetal elements. Oxygen ordinarily behaves as a nonmetal element. These materials have diverse ‘properties from insulators to semiconductors and metals. In the ideal case the perovskite structure is described by the formula ABX3 or by AB2X3. The perovskite contains three elements A, B, X in the proportions 1:1:3 or 1:2:3. Atoms A are metal cations, atoms X and B are nonmetal anions. Element X is often represented by oxygen. The compound BaPb1_5Bi~O3is a perovskite of type 1: 1:3 with a part of the lead atoms being replaced by bismuth atoms. The properties of ceramics BPBiXO are dependent on the concentration of bismuth ions. In the range of concentrations x <0.35 corresponding to a superconducting state, the layered lattice BPBi1O has rhombic symmetry.83Bi) with an electron configuration [Xe] 5d106s26p3, where [Xe] is the electron Atoms of bismuth ( may be found in two valence states in the compound BPBi~O:Bi3~and Bi5. configuration of xenon, It was hypothesized by Sleight [129]that the electron properties of BPBiXO are determined by the overlap of partially filled s-bands of lead and bismuth ions and of partially filled oxygen ion p-bands.

These ions generate a conduction band of width —14 eV. The presence of the wide conduction band in the interval of bismuth concentrations 0 s x 0.25 is proved by the low values of the effective mass of the band electrons, m = (0.5—0.8)me [132]. Calorimetric measurements of hard solutions with x = 0.25 carried out in ref. [133]on monocrystals revealed that the coefficient y determining the linear dependence of the electron heat capacity, has the value y = 1.65 mJ/mol K2. As the phonons responsible for the pairing in this superconducting compound, one can take the displacements of Pb and Bi ions and “breathing” oxygen modes. The results of tunneling experiments seem to prove the existence of a low-frequency mode with energy lower than 2 meV. In spite of numerous experimental data, there is still no common viewpoint on the superconductivity of compounds such as BPBiXO. The Bilbro and McMillan phenomenological model [134]often used to interpret the experimental data contains some phenomenological fitting parameters. This model, by analogy with BCS theory, uses the representation of a single simple conduction band. Therefore, calculating the parameter A by means of McMillan’s formula is not very reliable. It was assumed that [135]high T~values with a low density of current carriers in this oxide are generated by a plasmon mechanism. However, this superconductivity mechanism was questioned in a number of papers [136—140]. In some papers [141—144] it was stated that BPBiXO is a superconductor where the Bose condensation of bipolarons of small radius takes place. Pairs of electrons in a singlet spin state which are bound due to the interaction with virtual optical phonons in the region of one or two unit cells are called bipolarons. In a certain sense the bipolaron mechanism resembles the hypothesis put forward in 1957 by Schafroth, Butler and Blatt [29],asserting that superconductivity is realized by quasimolecules. We shall discuss in more detail the bipolaron mechanism of superconductivity in section 6.1.5. It is noteworthy

that this mechanism cannot explain the superconductivity of a metal oxide compound BPBiXO. According to the bipolaron mechanism, the superconducting current carriers (bipolarons) should have a mass hundreds and thousands of times as large as that of a free electron. This contradicts the above experimental data according to which the conduction band width —10—14 eV implies an effective mass on the Fermi surface m* ~O.Sme.

240

AS. Davydov, Theoretical investigation of high-temperature superconductivity

In the bipolaron model, superconductivity is described by analogy with superfluidity of helium-four atoms. The helium atoms, however, can move comparatively freely, but bipolarons are fixed on lattice sites in crystals. Their coherent motion is possible only at a very small group velocity due to an extremely narrow energy band. Localization of electron pairs on one site does not allow a strong screening of the Coulomb repulsion within a pair. To compensate for the Coulomb repulsion, it is necessary to employ very large values of A. An essential deficiency of the bipolaron model is also the assertion that paired electrons (bipolarons) should exist at temperatures far exceeding that of the condensation T~.This conclusion contradicts the experimental data. The general drawback of the above cited theoretical works consists also in that they do not take into account the dependence of the BPBi5O properties on the concentration of bismuth ions and the variation of their valency. Such a dependence was studied in refs. [145] and [146], Here, proceeding from the Anderson idea [147] on the interelectron attraction on one site at the expense of the electron—phonon renormalization of the Coulomb repulsion, the dependence of the transition temperature Tc on the bismuth concentration was alternation derived. in a BaBiO 3~and Bi5~ Using the assumption on Bi 3 compound, Rice and Sneddon [148] conjectured that the high temperature T~of a superconducting transition of this compound is due to the softening of the crystal lattice and the related increase in the parameter A. The properties of the compound BaPb1_~Bi~O3 as a function of the concentration of bismuth ions taking account of the valence transition were investigated by Ionov and Manakova 2~in a crystal are unstable. With small bismuth concentrations when[149,150]. x < x() Valence-two ions Bi 0.1—0.05, they give their sp-electrons to the conduction band forming the ions Bi5t When the bismuth ion concentration grows there occurs the reaction 2Bi2~ Bi3~+ Bi5~with the formation of complexes Bi—O—Bi in a singlet spin state or independent stable ion states Bi3~and Bi5~with one unpaired electron. In the case of complexes, because of the resonance interaction of the structures Bi3~—O~Bi~ O—Bi5~,generated by the low-frequency mode of optical vibrations, the energy of the complex goes below the Fermi level at a distance equal to the binding energy in the complex —0.5—1.0 eV. The electron exchange between neighbouring complexes induces smearing of their energy level. As a result, below the Fermi zone a narrow energy band of electrons is formed. The optical mode of intercomplex vibrations is directly connected with displacements in equilibrium positions of inertia centres of complexes (the acoustic bands of vibrations). This connection is manifest in the large dispersion of optical vibrations observed experimentally [151,152]. The latter comes to about 100 cm1. With growing concentration of bismuth ions in the interval x 0 ~ x ~ xm, the number of bound complexes increases, reaching the maximum at x = xm. For this value of x one observes the maximum values of T~,the second critical magnetic field H~2 44 kOe and the largest value of the electron— phonon interaction A parameter. The increase in T~with the growth of Bi ion concentration for x < Xm is accompanied by a decrease in the elastic modulus, i.e., the lattice softening. Collectivization of the energy states of complexes caused by the exchange interaction between them competes with the Coulomb interaction. The Coulomb correlations tend to localize the complexes. Their influence increases with x > Xm~When the concentration grows in the interval xm xcr the complexes are completely localized, in each cell there is a donor—acceptor complex with a local pair of electrons. For x 0.4, a structural transition takes place. This is a qualitative picture, that reflects the main properties of BaPb1_~Bi~O3 compounds. In 1988 new oxide materials which contain bismuth and get over to a superconducting state at higher temperatures were discovered. —~

-

AS.

Davydov, Theoretical investigation of high-temperature superconductivity

5.2. Discovery of high-temperature superconductivity and its

241

peculiarities

5.2.1. Discovery of superconductivity of metal oxides The discovery of the superconducting properties of the metal oxides BaPb1_~Bi~O3 having a low

transition temperature to the superconducting state was not met enthusiastically in many laboratories. These new superconducting materials, the metal oxides, were little known to the majority of the researchers studying the superconductivity phenomenon. Quite a different approach to this discovery was taken by the collaborators at the Zurich IBM

laboratory, Bednorz and Muller. They did not specialize in superconductivity. They studied the structures and phase transformations in the perovskite-type oxide compounds. Having examined the

discovery of superconductivity in a BaPb1_~Bi~O3 metal oxide the researchers investigated thoroughly other analogous materials hoping to find among them compounds with higher T~. The comparitively high temperature of the superconducting transition, T~ 13 K, in the metal oxides BaPb1 _5Bi~O3which have a low density of current carriers indicated that in this compound a strong electron—phonon interaction is manifested. The assumption was therefore justified that there could be other metal oxides with a higher density of charge carriers, and hence a higher temperature for the

superconducting transition. This reasoning was a guideline for Bednorz and Muller in their research on compounds with high-temperature superconductivity. They commenced their research by the end of the summer of 1983, investigating oxide materials containing copper and nickel ions and capable of being in several valence states. Finally they succeeded in synthesizing the oxygen-deficient mixed oxide Ba—La—Cu—O that consists of three different crystallographic phases. One of them is a layered perovskite structure such as K2NiF4 that goes over into a superconducting state at temperatures lower than 13 K. This transition started at a temperature of about 30 K. On April 17, 1986, the results of the studies were sent by the authors to the journal Zeitschrift für Physik B. The article was published in September [153]under the title, “Possible high-temperature

superconductivity in the Ba—La—Cu—O system”. The Ba—La—Cu—O superconductivity was independently confirmed by Chu et al. of Houston University (USA) [154] and Tanaka of Tokyo University [155] after they became aware of Bednorz and Muller’s paper. Soon Bednorz, Muller and their collaborator Takashige [156], while studying the Meissner effect were convinced of the true nature of a

superconducting state in Ba—La—Cu—O. The discovery of Bednorz and Muller was proved in many laboratories, and was followed by the discovery that the critical temperature in the oxides studied by Bednorz and Muller can be increased up to 40 K by replacing barium with strontium. Within half a year after the discovery of superconductivity in Ba—La—Cu—O, in the first half of 1987 in Houston University, Wu, Chu et al. [157,158] synthesized a YBa2Cu3O7_~metal oxide compound abbreviated as the compound 1:2:3. By measuring the resistance and magnetization it was shown that

the latter goes into a superconducting state in the interval 89—93 K. The transition to zero resistance terminates at 92 K. The new compound allowed one to demonstrate the Meissner effect without any

experimental complications using liquid nitrogen as a coolant. It was then shown by Chu et al. that the replacement of yttrium in YBa2Cu3O6÷~ compounds by the

rare earth elements Gd, Ho, Er, Lu, Nd, Sm, Eu and La preserves superconductivity in the interval 70—94 K. By the beginning of 1988, in laboratories in Europe, USA and Asia new bismuth and thallium

compounds with critical temperatures 100 and 125 K had been synthesized. Thus, by 1988 the critical temperature of the transition into a superconducting state increased by more than 100 K. The increase by 80 K happened in the last two years. The Bednorz and Muller discovery made it possible to study superconductivity at the temperature of an easily accessible coolant liquid nitrogen. Studies of

242

AS. Davydov, Theoretical investigation of high-temperature superconductivity

superconductivity in oxide compounds were commenced in many laboratories. The appearance of a great number of papers devoted to this phenomenon was promoted by the simple production of the materials. Superconducting ceramics can be produced in any physical and chemical laboratories. It is a much more difficult task to grow the single crystals necessary for scientific research. The revolutionary papers by Bednorz and Muller were honoured in 1987 with the Nobel prize in physics “for the discovery of superconductivity in ceramic materials”. Many papers and reviews [1S9—i63]are devoted to the history of this remarkable discovery. 5.2.2. Main properties of copper oxide superconductors The tetragonal unit cell of the lanthanum compound La1~~Sr~CuO4_5, x 0—0.4, 8 0—0.4 is shown in fig. 5.1. The hatched circles denote the positions of La ions which are partially (x) replaced by strontium or barium ions. The maximum value of T~is —33 K for x 0.15, 6 —0.04. The unit cell dimensions are given by a b 3.8 A, c 13.2 A. The copper ions are arranged in the form of two planes with CuO4 complexes in the cell centre, and in the form of mutually perpendicular chains of alternating copper and oxygen ions on the cell base. The orthorhombic unit cell of the yttrium ceramic structure Y—Ba2—Cu3—O7~,6 0.1—0.3 is shown in fig. 5.2. Its dimension is determined by the parameters a = 3.81 A. b = 3.89 A, c = 11.7 A. The copper and oxygen ions are centred as two planes in the cell between the barium ions and as infinite

~

00 •Cu

• Lci(Sr)

•

0

Cu

0

Fig. 5.1. The unit cell of the tetragonal structure of lanthanum compound.

0 Fig. 5.2. The unit cell of the orthorhombic structure of yttrium ceramic.

A.S. Davydov, Theoretical investigation of high-temperature superconductivity

243

chains of alternating Cu and 0 ions along the b axis in crystal. Oxygen vacancies usually arise in two central planes. Eachcopper ion in the linear chains is surrounded by a pyramid of five oxygen ions. This coordination is well preserved in a wide range of temperatures and oxygen contents. The unit cell of the bismuth ceramic is shown in fig. 5.3. Here also there are chains of alternating Cu and 0 ions along the b axis. Thus, all these ceramics are characterized by large unit cell volumes and a pronounced layered

anisotropy. To elucidate the mechanism determining superconductivity, it is necessary to understand the nature of charge carriers in these systems. The pure lanthanum ceramic La~Cu~O~with valence-two copper

ions is a dielectric. When a small number of valence-three lanthanum ions is replaced by valence-two strontium (Sr2~)or barium (Ba2~)ions, one may assume that in order to preserve the charge neutrality of the unit cell, the following transformation takes place, La3 + Cu2~ Sr2~Cu3 which replaces a copper of valence two by a copper of valence three. This transformation is accompanied by the appearance of an excess hole localized on the relevant copper ion. Due to the translational invariance —~

~,

of a crystal this state is collectivized, and the hole conduction band is generated, having a width induced by the exchange energy J, i.e., the energy of the hole transition between the ions Cu3~~± Cu2t These mobile collectivized hole—quasiparticles with positive charge and effective mass m * = 112/ 2a2J provided for the conductivity of ceramics during doping. The excess holes on copper, generating the conduction

bands of positively charged quasiparticles during collectivization, appear similarly in the doped yttrium ceramics YBa 2CuO7_~with oxygen-atom vacancies. According to recent concepts, the holes are localized not on copper ions but on a hybridized complex of copper and oxygen ions. This complex, generated by a hybridization of the oxygen 2p states and the copper 3d~2_~2state with oxygen deficiency, i.e., with a hole, is labelled, according to Bednorz and Muller [159], as [Cu—O]~. During collectivization they form the p—d conduction band due to the

exchange interaction.

,

-Ca

s-Sr

L7~

•Cu

a

Fig. 5.3. The unit cell of the bismuth ceramic.

244

AS. Davydov. Theoretical investigation of high-temperature superconductivity

A detailed study of the [Cu—O]~complex was performed by Gaididei and Loktev in 1988 [195,196]. In ordinary metals and alloys the critical temperature T~grows monotonically with increasing concentration of electrons on the Fermi surface. In contrast to this, in ceramic superconductors the dependence of Tc on the concentration of charge carriers ([Cu—O] + complexes) has a nonmonotonic character. Such a dependence on strontium concentration (x) for the compound La2_~Sr5Cu04is shown in fig. 5.4 based on measurements [163]. In this compound the temperature Tc increases with increasing x from 0.05 to 0.15. When x = 0.15, it attains its maximum and then decreases down to zero for x0.1S. This is confirmed by measurements of the Hall coefficient [164]in the compounds La2_5Sr~CuO4,0 (y 0.01). For x <0.01 the carriers are localized in the lattice and characterized by a local magnetic moment. For x 0.01 the superconductivity attaining its maximum value for x —0.15 is developed. The Hall coefficient is positive. This confirms that in this compound the hole superconductivity is realized. 2t cm TheMeasuring maximumthe concentration measured on by the Hall coefficient is iO of charge carriers in dependenceof of carriers superconductivity the change in concentration a YBa 2Cu307_~compound also exhibits its nonmonotonic character. The concentration of carriers in this compound depends on the oxygen deficiency (y). The maximum value T~-—93 K is attained with y~eaO.l—O.2[165]. Just after the discovery of high-temperature superconductivity attempts were made to clarify the nature of current carriers in these superconductors. Are they Cooper pairs with a double electric charge or not? Studies of the Josephson effect in the ceramics [166] La1 85Sr015Cu04, performed in 1987 by the Saclay and Orsay groups (France) showed that the charge of current carriers in this superconductor equals 2e. The experiments carried out in 1987 in Birmingham [167] (England) on the single crystals

Y~.2Ba02CuO4 also confirmed these results.

Large anisotropy due to the layered structure of new superconducting materials is manifest in studying the electric resistance and critical magnetic fields in single crystals. For instance, Dinger et al. [167]have proved experimentally the electric resistance anisotropy in the single crystals YBa2Cu307_~,.

Good conductivity was observed only in the plane perpendicular to the c-axis. Twinning of the crystal made it impossible to reveal anisotropy in the a, b plane.

The twinning structure arises inevitably, because the YBa2Cu3O7_~crystal, when cooled, goes from a tetragonal to an orthorhombic phase. The availability of a regular succession of twinning layers about

pvSlcm

(öO7~

1500 1000

~

T~ Fig. 5.4. The dependence of T~on the strontium concentration (x) for the 5rLa2_SrCuO4 compound; from Van Dover et at. [1631.

A.S. Davydov, Theoretical investigation of high-temperature superconductivity

245

500 A thick was observed by Osipyan et al. at the Institute of Solids, the USSR Academy of Sciences [168].They studied polarized light reflection by means of an optical microscope and established that the reflection phase delay was experienced only by the light component with polarization along the Cu—O chains lying on the b axis of the crystal. The authors assume this to be evidence in favour of one-dimensional conductivity in YBa2Cu3O7_~crystals. A thin twinning structure was observed in the electron-microscopic studies of YBa2Cu3O7_~single crystals by Worthington et al. [169].Anisotropy in the electroconductivity of these single crystals was

also observed by Tozer et al. [170]. The value of H~2differed between the magnetic field directions along and perpendicular to the c axis by factors of ten. The large anisotropy of the critical magnetic field H~2and the large difference in the value of the specific resistance between the a, b plane and along the c-axis indicate that the transport properties of electrons are strongly anisotropic.

The replacement of a yttrium ion in a YBa2Cu3O7...~compound by any rare-earth element with a local magnetic moment, e.g., Dy, Ho, Er, Tm, Yb, does not practically affect the temperature of

transition into a superconducting state. 3) of The carriers. classical The BCScritical theorytemperature referred to T~ isotropic large in concentrations (~~.~1022 cm charge and the metals energywith gap width such superconductors is small 4 1.7 kBTC. However, the Fermi energy is large EF 10—14 eV due to the large density of charge carriers. Since the inequality 4IE~41 is satisfied, only a very small part (—.10~)of electrons with energies close to the Fermi energy participate in superconductivity in such metals. These superconductors are characterized by the large coherence length ~ i0~cm. If a is the lattice constant, then the coherence length can be evaluated by employing the uncertainty condition ——

~O~aEF/4.

(5.1)

In copper oxide superconductors the concentration of charge carriers is small (of the order of ~~~1021cm3). According to Gor’kov and Kopnin’s estimates [173], the Fermi energy EF in an yttrium crystal is equal to —0.37 eV, and lanthanum is ——0.34 eV.

The occurrence of twins in single crystals of YBa

2Cu3O7_~,etc., arising in a structural transition from a tetragonal to an orthorhombic superconducting phase, did not allow one to investigate completely their anisotropic properties. Therefore, much interest was generated by the works of Verkin, Dmitriev, Seminozhenko et al. [171,172] who managed to develop a method of growing single crystals YBa2_~Sr~Cu3O7_~ without twins. The authors showed that when barium is replaced partially by strontium, orthorhombic superconducting crystals of the above composition with a concentration x in the interval —~0.5—0.8are free of the twin structure inherent only to undoped crystals (with x = 0). With decreasing temperature a strong anisotropy (of several orders of magnitude) is found along the crystallographic axes of twinless orthorhombic crystals. The resistance (p = 102 11 cm) along the c axis happened to be maximum. The electroresistance in the direction of the a and b axes was found to be quite different (by a factor of 100 and 1000). The highest electroconductivity (p 10_5_10_6 fi cm) was observed along the b axis. These studies justify the concept of a quasi-one-dimensional structure of the energy conduction band of yttrium crystals. Since in copper oxide superconductors the energy gap width proportional to kB T~ and the Fermi energy have comparable values, a substantial part of the charge carriers is involved in superconductivity in these superconductors. This distinguishes them from ordinary metal superconduc-

246

AS. Davydov, Theoretical investigation of high-temperature superconductivity

tors where only —104th part of the electrons participates in superconductivity. Since EF and 4 have comparable values, the coherence length in new superconductors should be small. The energy gap and coherence length in these layered compounds are anisotropic. Indeed, as was shown by recent measurements carried out at IBM [174] and by a group of Stanford [175]the coherence length ~ along the c axis in single crystals equals 3—4 A and has the value ~a,b = 20—30 A for the directions perpendicular to the c axis. The values and when extrapolated to T 0, characterize the dimension of the Cooper pairs in these directions. Large anisotropy of coherence lengths and energy gaps indicates weak correlation between Cu—O planes in the unit cell. Table 5.2 gives the values for the following parameters: Fermi momentum the penetration depth L, coherence length the Ginzburg—Landau parameter K = LIE, and the ratio between the masses McIMab of the current carriers in a superconducting state of La 2_~Sr1CuO4and YBa2CuO7 ~ single crystals. The large values of the Ginzburg—Landau parameter K indicate that these superconductors belong to the class of second kind superconductors. The small value of the coherence length ~ along the c axis shows that in these compounds the superconductivity is caused by weakly bound two-dimensional planes parallel to the a, b plane. This is confirmed by the large values of the ratio of effective masses Me/Ma6 of the superconducting current carriers.

~,

~

~,

~F’

One of the remarkable properties revealed in experiments with new copper oxide high-temperature superconductors was the different values for the energy gap obtained in measurements with tunneling and infrared spectroscopy methods. Direct measurements of the energy gap width in the one-particle spectrum of classical superconductors with these methods gave similar results. The task of performing experiments on reflection and penetration of infrared radiation to determine the energy gap in new superconductors is simplified by applying shorter wavelength radiation made possible by the large value of the gap width. However, the small coherence length and its anisotropy

enhance the requirements on the purity and perfection of the crystal surface. The interpretation of the results is hindered by the presence of lattice vibrations at optical frequencies. The anisotropy and small coherence length complicate the interpretation of experiments on tunneling. The surface can affect the binding energy of a pair. Great difficulties may arise in using point contacts. The above difficulties can explain the large dispersion in experimental values obtained in different laboratories when materials such as ceramics were used. More definite results are given by studies performed on single crystals. Table 5.3 lists experimental values of 24Ik8T~(extrapolated to T = 0) which were obtained in infrared and tunneling measurements [176—183].It follows from table 5.3 that in spite of the spread in experimental values the tunneling measurements produce, approximately, twice as large a value for the ratio 24 1kBT~. Naturally, there arises the question: what is the physical meaning of this difference? Table 5.2 Values for parameters in superconducting state of two single crystals T~ (K)

EF (eV)

‘~F

L0

~

Single crystal

(10 cm)

(A)

(A)

La2 SrCuO4

33

0.34

0.64

700

58

12

io~

93

0.37

0.50

335 525

30 20

11 26

2700 1740

K~,

Lr

5e

Kr

Mr/M0

4.3

232

180

3.8 6

711 240

65 11

(A) (A)

x0.15 YBa2CuO70

A. S. Davydov, Theoretical investigation of high-temperature superconductivity

247

Table 5.3 Experimental values obtained in infrared and tunneling measurements

21i,RIkBTC

2/.tT/kBT~

Method of measurements infrared tunneling

LaSrCuO

YBaCuO

T~(K)

1.3—2.7 3.5—6.0

1.6—3.4 3.4—5.3

36 87

The tunneling research of the energy gap width performed by Kirtley et al. [184]on a YBa 2Cu3O9_~

single crystal confirming its anisotropy, is of great interest. When tunneling is perpendicular to the Cu—O planes, the gap width turns out to be equal to —0.018 eV, and when tunneling is parallel to these planes, it is —0.015 eV. In the new superconductors La2_~Sr~CuO4 and YBa2Cu3O7, the linear temperature dependence of the electric resistance is observed in a wide temperature range from the critical T~to 500—1000 K. This

interesting property of the normal state of oxide superconductors has not yet been given a reliable explanation. As known, the isotopic effect discovered by Frählich, Maxwell and Reynolds and Serin was one of the most convincing indicators of the role played by the phonon mechanism in forming Cooper pairs in conventional superconductors. To elucidate the mechanism of Cooper pair formation in new hightemperature superconduct9rs the isotopic effect was studied again. The isotopic effect characterizes the dependence of the critical temperature T~of the superconducting transition on the masses of ions actively participating in pairing. In the BCS theory, such a dependence is determined by the relation T~-’M~

(5.2)

without taking into account the Coulomb interaction a —0.5. The isotopic effect measured initially by Batlogg et al. [185]and other scientists [186,187] revealed that when the oxygen ion 160 is replaced by the isotope 180 in a YBa2Cu3O7 superconductor, the value of a is equal to 0.0 ±0.3. This was interpreted as the absence of isotopic effect in new high-temperature superconductors. The theorists were confronted with the necessity to solve the problem of finding a new

electron pairing mechanism without applying the electron—phonon interaction. There are many approaches. Some of them will be considered in the next chapter. Further measurements carried out by Norris et al. [188]displayed a very weak isotopic effect in yttrium crystals with the value of a = 0.2 ±0.05. In all experiments, only 90% of the oxygen atoms were

substituted. Therefore, the question arose whether the oxygen atoms participating in pairing were the ones which were substituted. An unambiguous answer to this question was given in 1988 by Thomson et al. [189].They measured specimens in which 100% substitution took place. Simultaneously the shift in the Raman and optically active frequencies of atom vibrations was studied. These frequencies shifted by nearly 5% depending on the change in mass of the ions. At the same time the temperature T~shifted by a very small, although finite value i~ s 0.5 ±0.5 K. A small isotopic effect with the value a = 0.2 ±0.05 was observed by Bourne et al. [1871who measured the resistance and magnetic permeability in an La1 85Sr0 15CuO4 oxide. Table 5.4 presents the isotopic shifts of T~when 160 is replaced by 180 in four metal oxides obtained in measuring the temperature dependence of the magnetic permeability by zur Loye [190].The value of

248

AS. Dai’ydov, Theoretical investigation of high-temperature superconductivity Table 5.4 The isotopic shifts of Tr T.(K) Compound BaPb

075Bi1,~O5 11.0 La,11Ca0 rCuO4 20.6 La1 85Sr0 rCuO4 37.0 YBa,Cu307 91.1

1XT~(K) BCS ~T(K)

180(%)

0.6 1.6 1.0 0.9

60 75 75 67

0.63 1.14 2.10 5.21

AT~(BCS) corresponds to that predicted by the BCS theory with a 100% substitution of oxygen ions, when a = 0.5 and M is the oxygen mass. The isotopic shift in T~with 160 being replaced by t60 was studied in a new superconducting system Bi2Sr5Ca2Cu3O8 (discovered in 1988) for two phases with the temperatures T~equal to 110 K and 75 K [191].The isotopic shift ~ for these two phases turned out to be equal to, respectively, 0.35 ±0.3 K and 0.40±0.5K. Thus, in all oxide superconductors when toO was replaced by 180 a finite change much smaller than the BCS prediction was observed for the critical temperature.

So, the properties of new high-temperature metal oxide superconductors differ essentially from the properties of ordinary superconductors described by BCS theory. Briefly, these differences come down to the following points. (1) All new high-temperature superconductors are characterized by a very large anisotropy manifesting itself in their layered structure perpendicular to the principal crystallographic axis (c axis). (2) The large anisotropy becomes evident in the appreciable difference of coherence lengths ~ for the direction along the c axis and directions perpendicular to it. The largest anisotropy is found in the bismuth superconductor Bi—Sr—Ca—Cu—O where the coherence length equals ——1 A along the c axis, and —40 A in the transverse direction. (3) In ordinary superconductors the coherence length is i0~cm. In oxide materials it is thousands of times smaller and equal to ——10—40 A. The small value of the coherence length characterizing the space stretching of the wave function of a Cooper pair points toward a strong coupling of quasiparticles

in the pair. Such a coupling results in a high critical temperature. (4) The dependence of T~on the concentration of charge carriers has nonmonotonic character.3.The In

maximum value of T~is attained at a relatively smallmonotonically density of charge equal to ~~1021 cm At ordinary superconductors the temperature T~rises withcarriers the rise of concentration.

small concentrations of charge carriers their Fermi energy in the conduction band of the new superconductors has a small value 0.4 eV. (5)

Due to the large coherence length (10~cm) in classical superconductors only a 104th part of

the electrons placed near the Fermi surface participate in forming Cooper pairs. In the new superconductors the space distribution of the wave function of Cooper pairs is small, therefore many charge carriers, involved in the conduction band structure, participate in their formation.

(6) Energy gap measurements in the one-particle spectrum of new superconductors by the tunneling and infrared radiation absorption and reflection methods give different values. The gap measured by the tunneling method is larger. (7) In high-temperature superconductors a pronounced isotopic effect is observed. However, its magnitude is much lower than that predicted by the BCS theory based on the electron—phonon interaction. Therefore, various models of nonphonon pairing mechanisms were suggested. Some of these models will be considered in the next sections.

AS. Davydov, Theoretical investigation of high~temperaturesuperconductivity

249

6. Nonphonon models Of high-temperature superconductivity 6.1. Early theoretical research in high-temperature superconductivity 6.1.1. Exciton mechanism

Starting from the discovery of the superconduction phenomenon, the researches always aimed at finding a substance that goes into the superconducting state at as high a temperature as possible, in particular, at the temperature of liquid nitrogen. The theoretical search for materials with high-temperature superconductivity, i.e., materials with T~ exceeding 30 K was pioneered by Little’s paper published in 1964 [192].In this paper Little indicated the possibility to obtain the high-temperature superconductivity in long organic polymer-type molecules. It was assumed that the conduction electrons will be paired due to their interaction with electron excitations such as excitons (bound states of an electron and a hole) in the surrounding atoms involved in lateral chains of the polymer. Since the mass M of such exciton excitations is small, one could expect the high temperature T~ M~. In practice, this model failed to be realized. Because of their large energy of intraexciton excitations (at small mass M their energy is of the order of one or several electronvolts), they could not provide the pairing of electrons. In 1964, Gin.zburg [192]proposed the possibility of superconductivity at the metal—dielectric contact. In this case, instead of the conventional electron—phonon mechanism of electron pairing the electron— exciton interaction was proposed. As a model of a superconductor, the “sandwich” type system, i.e. the alternating layers of metal and dielectric (or semiconductor) was considered. The existence of excitons in dielectric interlayers in such systems is simplified. Theoretical studies of these systems however, met with great difficulties because there was no microscopic theory of surface phenomena. The book Problems of High-Temperature Superconductivity [136]edited by Ginzburg and Kirzhnits had summarized the state of theoretical studies in high-temperature superconductivity by the middle of the seventies. It was stated (Kirzhnits) that from a purely theoretical viewpoint there are no restrictions on the T~value. A possible rise in T~was also discussed in Golovashkin’s review [194]. Searching for the conditions at which the T~could have a comparatively large value, Alexander, Brite and Bardeen returned in 1973 to the excitonic pairing mechanism suggested earlier by Little [192] and Ginzburg [193]. A peculiar exciton mechanism of high-temperature superconductivity was suggested by Gaididei and

Loktev [195,196]. In 1988, they studied the specific form of the electron structure of ceramic compounds such as La2_5Sr~Cu04,YBa2Cu3O7, etc. It was shown that an extra hole is situated, mainly, on the oxygen site of theSuch unit cell. Movingcorrespond, in it, this hole with low-frequency electron 2~ions. transitions in interacts copper oxides, to quadrupole-type d—d transitions in Cu excitations. Behaving approximately as Bose particles, they provide, according to Gaididei and Loktev, the pairing between p-holes in analogy with the pairing by acoustic phonons in BCS theory. It should, however, be noted that a direct implantation of the BCS theory into the given case is not entirely justified, since for the known high-temperature superconductors the ratio of the exciton energy to the Fermi energy is large (of the order of 1/2—1/3). In other words, the quadrupole excitations have a comparatively large energy. The large energy of quadrupole excitations is useless for an increase in T~ according to the formula determined by the BCS theory. High values, even with a large preexponential

multiplier, can formally be obtained only in the case of a strong quasiparticle (hole) interaction with optical quadrupole vibrations.

250

AS. Davydov, Theoretical investigation of high-temperature superconductivity

The large energy of exciton excitations results in a more serious complication. Because of insufficient inertia, such vibrations cannot provide a superconducting pairing. Following adiabatically the electron motion, these vibrations are only important in changing the dielectric properties of the crystal. This basic difficulty concerns all the attempts to explain high-temperature superconductivity using the exciton mechanism. A similar difficulty is inherent also in a plasmon mechanism in a three-dimensional crystal which is discussed in the section 6.1.3.

In spite of the above limitations, the paper by Gaididei and Loktev is of great interest. It analyses in detail the quadrupole excitations of complexes [CuO]~ in oxide ceramics. The studies made are complete when account is taken of the coupling between quadrupole excitations of the neighbouring complexes in the layered structure of crystals. Such an account leads to a great dispersion of the phonon spectrum and to its transformation into a quasiacoustic one. In this case the bisoliton model, in which carriers are paired due to local displacements from the equilibrium positions of unit cells, can be more correct. The local displacements are described by virtual acoustic phonons. This model is developed in section 8. The interest in nonphonon superconductivity mechanisms was much enhanced in 1987 when in the first papers by Batlogg et al. (see section 5.2.2) no isotopic effect was observed in the oxide ceramic YBa2Cu3O4 with the oxygen atom 160 replaced by t80 This result was interpreted by many theorists as an indication that pairing in oxide superconductors is generated by electron and magnetic excitations, not by phonons. As it was noted in section 5.2.2 the subsequent more refined measurements showed the presence of isotopic displacement in all oxide ceramic superconductors. The isotopic shift value, however, was much lower than was predicted by the BCS theory. The isotopic replacement of one or several ions in a superconducting system and the observation of a related change in T~were important historically for extracting information on the phonon contribution to the superconductivity mechanism. Since the isotopic effect in ceramic oxide superconductors was not manifest completely, the interest

in nonphonon superconductivity mechanisms was justified. Below we consider some of the models to describe the high-temperature superconductivity phenomena in new materials. 6.1.2. Model of Anderson resonating valence bonds Among the theoretical models used to describe high-temperature ceramic superconductors the model of resonating valence bonds (RVB) suggested in 1987 by Anderson [197]and further developed in many subsequent papers [198—2031 is discussed most often. Taking into account that the planes perpendicular to the c axis where copper and oxygen ions are situated play a major role in superconductivity, Anderson used (as a model for the undoped lanthanum dielectric ceramic La2CuO4) the square lattice at the sites of which a copper ion (and two oxygen ions) are located. In the ground state each site of the square lattice in the plane a, b of a crystal is characterized (according to Anderson) by an uncompensated spin on the copper ion. These spins seem to form an antiferromagnetic lattice. However, Anderson intended to consider, as the ground state, the dimerized states of lattice points. In other words, it was assumed that all sites are joined in spinless pairs by “valence bonds”. Such pairs cannot move in an undoped crystal and transfer the electric charge since there are no free sites (knots). There exist several equivalent ways of pairing the knots (domains) in the square lattice. Two of them are shown in fig. 6.1 by means of the example of a one-dimensional structure. The doped crystal La2~(Ba,Sr)~Cu04admits excited states with several dimerized state domains divided by walls.

A. S. Davydov, Theoretical investigation of high-temperature superconductivity

251

Each domain wall is a topological soliton-kink. Such a domain wall dividing two domains is plotted in fig. 6.1. It has no electric charge, but it is characterized by an unpaired spin 1/2. Such neutral excitations with a spin were called spinons. The effective spinon mass equals, in the order of magnitude, the electron mass. The spinon is a neutral fermion. Since spinons are topological solitons, they can be generated and annihilated by pairs only. The neutral fermion concept was first introduced by Pomeranchuk [204]as early as in 1941, to describe the properties of quantum paramagnetics (see also ref. [205]).In more detail the neutral fermion type excitations were investigated by Su, Schrieffer and Heeger [206]to explain experiments on the electron-spin resonance in polyacetylene molecules containing alternating double and single chemical bonds between hydrogen atoms. The double bonds between hydrogen atoms in such a molecule are equivalent to bound pairs in Anderson’s model. The isolated domain walls in polymers were studied also by Brazovskii [207]. Spinons in Anderson’s model can move due to the resonance of valence bonds. Such a motion is illustrated in fig. 6.2 by the example of a one-dimensional model. The valence bond (wavy line) of the bound pair situated near the spinon in position 1 can be shifted to the left in position 2 due to the resonance. This process will correspond to the neutral spinon transfer to the right. The spinon motion due to the resonating valence bond gave its name to Anderson’s model. It was called the “resonating valence bonds” model abbreviated as the RVB model. The analogous notion of a resonance was introduced in 1960 by Pauling [208]to explain the enhanced stability of aromatic molecules containing the alternating single and double chemical bonds in zero approximation. Double bonds are generated by IT-electrons. Due to the “resonance” of two structures distinguished by the positioning of IT-electrons (e.g., Kekulé structure in benzene), these electrons are uniformly distributed along the molecule. A smaller amount of energy corresponds with such a state. Figuratively, the “resonance” of simplified structures strengthens the molecule. The excess hole—quasiparticles with positive charge and spin 1/2— arising when the La2Cu3O4 crystal is doped by atoms Ba or Sr, can give rise, together with spinons, to the formations bound in the singlet spin state. Such formations having single positive electric charge and zero spin were called holons by Anderson. To explain superconductivity, one should assume that holon pairs are bound to form spinless bosons with a double charge. Probably, this pairing is generated by the exchange of spinons. Paired holons at low temperature form a superconducting condensate. All these processes have not yet been studied in explicit detail.

~

~4f~~lla

Ta

/

SPINON

ila

Fig. 6.1. Two kinds of bound pairs (domains) in the one-dimensional lattice: Ia and ha. The domain wall (spinon) divides the two domains Ia and ha.

252

A. S. Davydov. Theoretical investigation of high-temperature superconductivity

H 4~

~4

2~i Fig. 6.2. Illustration of spinon motion (solid arrow). The valence bond (wavy line) is shifted from position 1 to position 2.

Within Anderson’s model the state of a superconducting crystal in a two-dimensional crystal lattice is described by the Hubbard Hamiltonian H=

~

—

tmna~uanu+

m,n,u

U ~

~

(6.1)

mu

1 ,u2

Here the summation n, m is carried out over the nearest neighbours of a square lattice, u is the spin index, Nnu = a~ganuis the operator of the number of particles on a site n. t,~, is the operator of antiferromagnetic energy of the exchange interaction between the nodes n and m. U is the energy of Coulomb repulsion of a quasiparticle pair on a single node. 2IU > i03 cm’) the operator (6.1), after a When the inequality tnrn U is satisfied (however, canonical transformation, can~ be reduced to the form t H=~ ±t~mSn+Sm+••~,

(6.2)

n ,m

where S~= are the Anderson pseudospin operators. In this case the exchange interaction is formally expressed through the spin interaction. To remove the main deficiency in Anderson’s model with a rigid lattice, which is an arbitrary postulate on the absence of antiferromagnetic ordering in the ground state, it was proposed by Kivelson et al. [199] to extend the Hamiltonian (6.2) taking into account small displacements u,~of lattice sites from equilibrium positions. In this case disregarding the kinetic energy of displacements of sites of the mass M, the Hamiltonian (6.2) is replaced by

H= ~

n.m

(~

(t~~X~Unmi2)+ ~Ku~),

(6.3)

where u,,,,~= u,~ Urn 1S the change in distance between sites n and m, and K is the elasticity coefficient of the lattice; x is the parameter that determines the interaction of a quasiparticle with displacements of sites from equilibrium positions, i.e., their interaction with virtual acoustic phonons. When there is no interaction between a quasiparticle and displacements (x = 0), the Hamiltonian (6.3) characterizes the Néel antiferromagnetic spin ordering in the lattice with energy —2t~U1per site. However, when the inequality —

9UX2IK>1

(6.4)

is satisfied, the state at which quasiparticles of neighbouring sites form singlet pairs, will be advantageous. Thus, to explain the nonmagnetic ground state of an undoped crystal, it is necessary to introduce

A. S. Davydov, Theoretical investigation of high-temperature superconductivity

253

sufficiently strong electron—phonon interaction. Consequently, Anderson’s initial idea to construct the superconductivity theory of new superconductors without taking into account the electron—phonon interaction failed to be realized. In the paper by Kivelson et al. [199],the main properties of possible excitations are estimated using the operator (6.3). Taking the following values of the leading parameters of the theory, t~= 0.5 eV, x = 3 eVIA, the distance between copper ions equal to 3.79 A, they concluded that in pairing the sites = t0/6~ 0.03 A are displaced. The spinon effective mass determined by the equality m* = (u0Ia)M 5 x i0~M was comparable with the electron mass. Anderson’s conclusion concerning the availability of a spinon in a superconductor is not yet proved experimentally. Probably, the presence of spinons can be revealed in studying the electron spin resonance on these crystals. Anderson’s model of resonating valence bonds made it possible to explain some peculiarities observed in the normal state of superconductors (at T> T~)[202].In particular, the linear dependence of the resistance observed experimentally for (T> T~)was explained as well as the peculiarities observed in tunneling measurements. However, there is no experimental evidence in favour of the presence of spinon-type excitations in these materials. The basic conclusion of the theory concerning holons is also questioned. 6.1.3. Magnon and plasmon pairing mechanisms

The magnon pairing mechanism in high-temperature superconductors was studied in many works. It was assumed that the pairing is realized due to the exchange of spin excitations magnons. We should specially emphasize here the pioneer work of A.I. Achiezer and I.A. Achiezer [209].Here account was taken of the interaction between the conduction electrons, which is generated by the exchange of acoustic phonons, and also by the additional interaction connected with the exchange of spin waves (magnons). It was shown that superconductivity and ferromagnetism can coexist in the same space regions. At rather small concentrations of a ferromagnetic component the rise in their concentration results in an increasing T~if there is triplet pairing. If the pairing caused by the phonon mechanism occurs in the singlet spin state, then the rise in the concentration of the ferromagnetic component results in a decreasing T~. The magnon pairing mechanism in high-temperature superconductors was studied in a number of papers [210—213]. Calculations were made using the Holstein—Primakoff representation for the spin wave operators within the strong coupling approximation. These studies showed that the usual magnons without the other excitations do not result in a sufficiently strong attraction. This was confirmed by numerical calculations on finite lattices [213]. Many papers are intended to explain the high-temperature superconductivity using the concept of pairing due to the exchange of longitudinal plasma wave quanta plasmons. The longitudinal plasma waves are formed in a solid in the range of frequencies at which the dielectric permeability of the medium vanishes. The characteristic frequency of plasma waves in three-dimensional crystals is determined by the expression 2N/m, (6.5) —

—

=

4i~e

where N is the concentration of electrons, e and m are their charge and mass. When the electron density N—~(1—3) X 1022 cm3, the plasma frequency ~ (1015_1016) ~ One could suppose that the —

254

AS. Davydov, Theoretical investigation of high-temperature superconductivity

plasmon, not the phonon, exchange would lead to the preexponential multiplier in the formula of the BCS theory, T~= (9exp[1/(A

—

~*)]

(6.6)

increasing by two—three orders of magnitude, if 0= h&P/kB. However, such an increase in (9does not result in a pronounced increase in T~,because with a plasmon frequency ~ comparable with that of electrons EFI/i, the plasmons are unable to provide superconducting pairing and play only an important role in changing the dielectric properties of the crystal. As was pointed out by Kresin [2141and Ruvalds [215],the role of magnons in superconductivity can be more significant at high-temperature in the layered metal oxides La2_v—(Ba, Sr)~—CuO4and Y—Ba—Cu304. In these superconductors the superconductivity is generated by plane layers of copper and oxygen ions. It was shown by Ando, Fowler and Stern [2161that in two-dimensional systems the plasmons are characterized by specific properties. Their dispersion contains no energy gap (for q—~0) and has the square root behaviour, 2 =

a\/~, a

-

(6.7)

(2ITNI6m*)I

Here N is the surface concentration of electrons proportional to the thickness of the layer, m* is their effective mass. Although the frequency w~is larger than that of acoustic phonons Qq~it is much lower than the frequency (6.5) of plasmons in a three-dimensional medium. For that reason such plasmons can participate in pairing together with acoustic phonons. This idea was advanced by Pashitski and Vinetskii [2171. They assume that the exchange of virtual quasi-one-dimensional low-frequency plasmons promotes the Cooper pairing of charge carriers. 6.1.4. Structural transformations and superconductivity It is known that the superconductivity phenomenon is often preceded by structural transformations in a crystal. To explain the structural transformations, the anharmonic model is ordinarily used. According to some authors [218,219], structural transformations preceding the onset of superconductivity lower significantly the phonon frequency, thus increasing the electron—phonon interaction parameter. Such softening of the phonon spectrum is due to large amplitudes of ion displacements in the double-well potential modulating the structural transformation. The influence of a structural transformation on superconductivity within the limit of weak pairing interaction and the isotropic gap were studied by Kopaev [220], and Kopaev and Rusinov [221]. The high-temperature superconductivity properties were investigated by Gorbachevich, Elesin and Kopaev [222]using the model where superconductivity is enhanced due to the singularity of the electron state density arising under structural and antiferromagnetic phase transitions. It was shown that this model allows one to explain the absence of the isotopic effect and large variations of the ratio 2~I TCkB. The weak point in models connecting superconductivity with structural phase transitions is the significant temperature difference between the known structural transitions and the temperature T~. Among models incorporating the nonphonon pairing mechanisms one should note refs. [223,2241 where the Hubbard Hamiltonian is used in systems with repulsive interaction only. It is assumed that the pairing effect is caused only by the kinematic interaction with incompletely filled Hubbard subzones. In this case there is no common understanding of the nature the attraction. It is possible that

A. S. Davydov, Theoretical investigation of high.temperature superconductivity

255

the bound state of quasiparticles is virtual. The bound state has a larger energy than the free quasiparticles. Consequently, their formation does not lower, but increases the energy of the system. 6.1.5. Bipolaron mechanism of superconductivity

One of the attempts to explain the high-temperature superconductivity was called the bipolaron theory. It is based on the representation concerning the electron self-localization in an ionic crystal caused by its interaction with longitudinal optical vibrations of the local polarization realized by the electron itself. The electron is kept in the local polarization potential well and supports the latter by its field. The idea of the electron self-localization in an ionic crystal was first put forward by Landau [225] and was then intensively worked out by Pekar [226],Fröhlich [227],Tyablikov [228]and others. When we study the self-localization process, it is necessary to take into account only the inertial part of the polarization that is unable to follow a fast moving electron. The noninertial part of the polarization is included into the definition of an average periodic field of a crystal. The efficiency of the interaction of an electron of mass m and charge e with longitudinal long-wave optical vibrations in a medium is characterized by the dimensionless parameter g = (e2I~)Vm/2h2li,

(6.8)

which was introduced by Frflhlich. Here 12 is the frequency of the optical vibration, g is the dielectric permeability of the inertial polarization. The interaction is weak if g ~ 1. Because of a large frequency lithe deformation field will be a rapid subsystem, so it has no time to follow the motion of an electron. The deformation field accompanies the electron motion in the form of a small cloud of phonons. In this case the energy of interaction between the field and the electron is proportional to the first degree of g. A strong interaction there corresponds to the value g> 1. In this case the electron is a fast subsystem. Therefore, it is connected with inertial deformation. Its effective mass increases. The bound state of an electron with a local polarization deformation is called a polaron. The binding energy and effective mass of the polaron are proportional to g2. Thus, the Landau—Pekar polaron can be treated as the bound state of an electron with the local polarization deformation of a crystal, moving freely as an entity in a crystal, with an effective mass that exceeds the mass of a free electron. The polaron theory was developed by Pekar in the continuum approximation when the dimension of the localization region exceeded the lattice constant. Since the dimension of the local deformation region is inversely proportional to the parameter g, the theory was concerned with the case of a not very strong interaction. In 1959, Holstein developed a theory [229]of polarons for the case of strong electron coupling with local optical vibrations of frequency 12 of diatomic molecules involved in the composition of the polymer chain. Under these conditions the polaron dimensions were comparable with the molecular ones, therefore, the model of a discrete one-dimensional lattice was studied. The bound states of an electron with local intramolecular deformation in the polymer chain were called polarons of small radius. A mathematical theory of small radius polarons was proposed by Tyablikov [228].The study of small radius polarons in a three-dimensional lattice was carried out by Firsov [230]and Klinger [231]. The concept of Landau—Pekar—Holstein—Tyablikov polarons was used by Alexandrov, Ranninger [232,233] and others to explain the high-temperature superconductivity. The papers were based on Holstein’s studies, where it was assumed that quasiparticles in the conduction band are paired due to their strong interaction with local optical vibrations in a discrete crystal lattice.

256

AS. Davydov. Theoretical investigation of high-temperature superconductivity

The concept of polarons and bipolarons in the approximation of strong coupling was also used by Brazovskii and Kirova [234] to explain the optical properties of cis- and trans-polyacetylene. In the BCS theory the conduction electrons are paired due to the interaction with acoustic phonons and are characterized by the dimensionless interaction constant A

=

wN(0),

(6.9)

where N(0) is the density of electron energy states on the Fermi surface and the quantity W is inversely

proportional to the elasticity coefficient of a crystal. It rises with an increase in lattice “softening”. In bipolaron theory the pairing is realized by virtual optical vibrations of frequency 12 that characterize local deformation in the oxygen environment of a transition metal in a crystal. In this case the role of the parameter A is played by the value =

2zg2h[IID

—

V~.

(6.10)

Here g is determined by eq. (6.8), Z is the number of nearest neighbours, D is the energy conduction band width of free quasiparticles. Taking into account eq. (6.8), it is clear that the value of ~ is independent of the frequency of optical vibrations 11. Since the interaction is nonlinear, there arises the binding energy of electrons in a singlet spin state which, according to bipolaron theory is determined by L1=2g2F1fl—V~,

(6.11)

where V~is the screening Coulomb interaction. The theory suggests that g2 1 and the electron binding energy in a bipolaron is much larger than the conduction band width of free quasiparticles, i.e., the following inequality D ~ 4 is satisfied. When this inequality is satisfied, the effective bipolaron mass is determined by the relation ~‘

=

m(41D) exp(g2).

(6.12)

It is much larger than that of a free quasiparticle in its conduction band. According to Alexandrov and Kabanov’s estimates [235],the bisoliton mass t~ is 100 and 1000 times larger than that of a conduction electron. The soliton band width D, as compared to the conduction band width D of a free

quasiparticle, decreases by the same factor, D~(mIrñ)D~D.

(6.13)

Such a narrowing of the energy conduction band was called by Alexandrov the “polaron effect”. Due to the polaron effect the energy polaron band may become so narrow that polarons are practically

localized on the single site in a crystal and the polaron moves basically from site to site through the jump mechanism stimulated at T = 0 by zero vibrations. This mechanism of polaron motion becomes more important with increasing temperature. The peculiarities of the motion of small radius polarons were studied by Firsov [230]. Due to the small radius of bipolarons, and considering their concentration which is lower than 0.2 per unit cell, the bisolitons in a crystal, unlike the overlapping Cooper pairs in BCS theory, are spatially

A. S. Davydov, Theoretical investigation of high-temperature superconductivity

257

separated. In this case the set of bisolitons can be regarded as a charged gas of Bose particles. Under its Bose condensation a superconductivity similar to the superfluidity of helium atoms will arise. To calculate the temperature of the superconducting transition, T~,corresponding to Bose condensation, the authors of refs. [232,233] used the expression for an ideal Bose gas kBTC = 3.31h2n213/ii

.

(6.14)

For ~ = lOOme and n = 1021 cm3 we find from this expression T~—28 K. One of the bipolaron mechanisms taking into account the layered structure of metal oxide ceramics and based on the Jahn—Teller character of valence-two copper ions was discussed by Gaididei and Loktev in ref. [236].In contrast to the above single-site bipolaron mechanisms, these authors proceed from the assumption that paired holes occupy neighbouring sites. Unfortunately, the possible temperature of a superconducting transition was estimated by the formulae derived from the isotropic Bose gas (6.14). Concerning the bipolaron high-temperature superconductivity theories the following critical remarks can be made. (1) Due to the enormous binding energy the bipolarons will decay at temperatures exceeding that of their condensation. However, the availability of bipolarons in the normal state of a crystal, i.e., at T> T~has not yet been observed. They would be manifest as paired particles with a double electric charge. (2) It is known that the spectrum of excited states of superfluid liquid has no gap. To explain

experimentally the observed gaps in tunneling and the optical measurements, it is necessary to introduce specific assumptions on the sharp change in the density of states of bipolarons near the temperature of a superconducting transition. (3) Described in terms ofvirtual optical phonons without dispersion (12q lie), the displacements of oxygen atoms that surround the atom of a transition metal are inevitably connected with displacements of equilibrium positions of the neighbouring cells. Such displacements are described in terms of virtual long-wave acoustic phonons with large dispersion (Wq qV~).Since the frequency of these phonons is small, the local deformations caused by them should interactintensely with electrons. This interaction is not taken into account in bipolaron theory. (4) The layered structure of superconducting ceramic compounds was not taken into account in bipolaron theory. However, it is just the layered structure that defines the distinctions between new

superconducting materials and traditional metal ones. (5) The very strong attractive interaction of two electrons on one site that overcomes their unscreened Coulomb repulsion, postulated in bipolaron theory, should be substantiated. In all theoretical models of high-temperature superconductivity which were treated in this section basic attention was given to the explanation of the pairing effect. In all these theories the formulae of the theory of a condensed Bose gas, generated by these pairs to determine T~,were used. Meanwhile, the pairing is only the necessary, but insufficient condition for the superconductivity to arise. Electron pairing is involved in each chemical bond but it has no relation to superconductivity. In the next chapter we shall present a new bisoliton model of high-temperature superconductivity of metal oxides that takes into account their complicated structure, small number of charge carriers and moderately strong interaction with acoustic branch phonons. An explicit expression for the function describing a bisoliton condensate will be given.

258

AS. Davydov, Theoretical investigation of high-temperature superconductivity

7. General information on solitons and bisolitons in quasi-one-dimensional systems In recent years the theory of nonlinear nonlocal excitations in quasi-one-dimensional systems called solitons and bisolitons was developed at the Institute for Theoretical Physics, the Ukr. SSR Academy of Sciences. It turned out that these excitations have great stability with respect to external actions and are able to transfer without energy loss the electric charge and momentum at distances far exceeding the atomic dimensions. This property of solitons and bisolitons was used by the author and his collaborators to explain some phenomena in biology which are connected with the transport of intramolecular energy and electrons along the a-helical protein molecules [237,238]. A bisoliton model of high-temperature superconductivity of metal oxide compounds was also proposed (see section 8). Here we will describe the main properties of solitons and bisolitons capable of transporting energy and electric charge along quasi-onedimensional molecular chains. 7.1.

One-component solitons

Some time ago it became clear that in nonlinear systems with dispersion, i.e., in media where the phase velocity of simple waves depends on the wave length and the wave amplitude, the ideal energy transfer is realized by solitary waves which were called solitons. Unlike the usual waves representing a periodic alternation of elevations and depressions on the water surface, or density compressions and rarefactions, or deviations from the mean value of other physical values, the solitary waves (solitons) represent single space-localized excitations (water elevations, etc.) moving, as a whole, with constant velocity. Solitary waves were first observed about 150 years ago by Scott Russell. Studying long solitary waves he found out that they move without changing their shape and velocity and have unique properties. For instance, initially large solitary waves leave behind small ones and pass through one another without any changes. For a long period of time solitary waves did not attract the attention of scientists. Only 50 years later, in 1895, Korteweg and de Vries derived a nonlinear mathematical equation named the KdV equation. This partial differential equation has the form cP=~P(x,t).

(7.1)

The function 1(x, t) determines the deviation from the mean value of density, velocity or any other physical quantity. If 1 has dimension of velocity, then the constant f3, taking into account the role of medium dispersion, has the dimension of velocity multiplied by length squared. The dimensionless quantity 6g characterizes the role of nonlinearity. In the reference frame ~ x Vt, moving with constant velocity V, eq. (7.1) has a localized solution in the form of bell-like elevation (g > 0) or depression (g <0) which is described by the function (see fig. 7.1) —

1(4) = V(2g)’ sech2(~\1Vi~).

(7.2)

A remarkable property of this solution is the fact that its maximum amplitude is proportional to the

AS. Davydov, Theoretical investigation of high-temperature superconductivity

259

g
Fig. 7.1. The amplitudes P(f) for the positive and negative coefficient g.

velocity V. Therefore, as was pointed out by Scott Russell, solitary waves of larger amplitudes leave behind waves of smaller amplitudes. The interest in solitary waves was enhanced only in the fifties in connection with studies in the physics of plasma and thermonuclear fusion. Special interest among the researchers was triggered by a paper published in 1955 by the scientists from Los Alamos laboratory (USA), Fermi, Pasta and Ulam. It studied the conditions of energy thermalization in nonlinear vibrational systems. According to the concepts available at that time, the atom vibrations in any condensed medium can be represented in the form of a superposition of monochromatic vibrations, i.e., vibrations with fixed frequencies. In a linear medium such vibrations are independent. Under weak nonlinearity forces between atoms depend on higher degrees of displacement. In this case interactions between monochromatic vibrations arise which stimulate a redistribution of the energy of vibrations over all degrees of freedom according to a Boltzmann distribution at the given temperature. This process is called the thermalization of vibrational excitations. Using a new computer in the Los Alamos scientific laboratory, Fermi, Pasta and Ulam made an attempt to elucidate the thermalization in a chain of periodically arranged particles between which the linear and quadratic forces act. To the great surprise of the researchers and the other scientists, thermalization in numerical experiments in the system turned out not to occur. For a long time this result seemed paradoxical. It was resolved only ten years later by the American scientists Zabusky and Kruskal. In 1965 they established that the long-wave excitations in a discrete chain studied by Fermi and his collaborators, are described by the nonlinear KdV equation. The solutions to this equation obtained by computer ‘have the appearance of stable bell-shaped excitations solitary waves moving in the chain. It also turned out that some solitary waves, when colliding, pass through one another without changing their shape and velocity, i.e., without exchanging energy. That is why thermalization was not observed in Fermi’s numerical calculations. This remarkable property of solitary waves merging together their behaviour with the behaviour of particles, allowed Zabuski and Kruskal to call them solitons the abbreviated name of a solitary wave. From 1965 onwards, the word “soliton” occupied a firm position in the scientific literature as the name for nonlinear solitary waves moving without energy loss and without changing their form. The extraordinary stability of solitons is due to the mutual compensation of two phenomena dispersion and nonlinearity. Dispersion results in a spatial spread of excitation, as monochromatic components of excitation move with different velocities. In linear systems this spread is not compensated, since the waves are independent. In nonlinear systems monochromatic components interact —

—

—

260

A. S. Davydov, Theoretical investigation of high-temperature superconductivity

intensely. If they interact so that the energy of rapidly moving waves is taken up and passed on to those that are left behind, the stable soliton-type excitation moving as a whole is formed. The great soliton stability stimulates numerous attempts to use them in studying nonlinear wave phenomena in water, plasma, solids and many other fields of physics. To describe the self-focusing phenomena in nonlinear optics, one-dimensional self-modulation of monochromatic waves, in nonlinear plasma, etc., the nonlinear Schrödinger equation is also used, (ih ôIc9t

(h2I2m) a2Iax2

+

+

G~ifr(x,t)~2)1I(x,t) =0.

(7.3)

This equation is written in the long-wave approximation when the excitation wavelength A is much larger than the characteristic dimension of discreteness in the system, i.e., under the condition ka = 2iraIA 1. It describes the complex field ~i(x, t) with self-interaction. The function ~2 determines the position of a quasiparticle of mass m. The coefficient G characterizes the intensity of the nonlinearity. The second term in this equation is responsible for the dispersion, and the third one for nonlinearity. When there is no nonlinearity (G = 0), eq. (7.3) has solutions in the form of plane waves, ~

~i(x,t) =

~

exp{i[ka

—

w(k)t]}

(7.4)

,

with the square dispersion law (7.5)

w(k) = lIk2I2m. With nonlinearity G

0 in the system having translational invariance, the excited states move with

constant velocity V. Therefore it is convenient to study solutions of the equation in the reference frame ~~x—Vt)Ia, (7.6) moving with constant velocity. In this reference frame the nonlinear equation (7.3) has solutions in the

form of a complex function ~fr(x,t) =

I’(~)exp[i(kx

—

k = mV/h,

wt)],

(7.7)

where the real function P( ~) satisfies the nonlinear equation [11w ~mV2 + J ~I9x2 —

+

G

2(~)]I(~) = 0,

(7.8)

0cP

with the value J = h212ma2.

Equation (7.8) has two types of solutions: nonlocalized and localized ones. The nonlocalized solution corresponds to a constant value of the amplitude cP( ~)= ~ and the dispersion law 11w = ~mV2

—

G

(7.9) 1 and in the limit L—1~ccthe second term in (7.9) tends to

0cP~.

If a particle is at a distance L, then P~= zero.

L

A.S. Davydov, Theoretical investigation of high-temperature superconductivity

261

The localized solution to eq. (7.8) normalized by the condition

J

~2(~)d~=1,

(7.10)

is represented as the bell-shaped function =

vT~sech(g~),

(7.11)

with the dimensionless nonlinearity parameter given by g=G 0/4J.

(7.12)

The function I(~) is nonzero on a segment I ~—2ITIg. The larger the nonlinearity parameter the smaller is the localization region. The energy of localized excitations is determined by the expression 2—Jg2.

(7.13)

hw= ~mV

The solutions of the nonlinear equation (7.3) in the form of the function (7.7) with the bell-shaped envelope (7.11) are called the soliton solutions solitons. Each function (7.7) describes the moving excited state of a nonlinear field, i.e., has one component. —

7.2.

Two-component solitons

In the preceding section we studied single solitons which are the solution to nonlinear equations containing one field function i/i(x, t). In these cases the nonlinearity was generated by the “self-action” of the field. The nonlinear properties are often manifest when two linear systems interact. As an example we consider the excess electron in a quasi-one-dimensional atomic (molecular) chain. If neutral atoms (molecules) are rigidly fixed in periodically arranged sites na of a one-dimensional chain, then, due to the translational invariance of the system, the lowest energy states of an excess electron are determined by the conduction band. The latter is caused by the electron state collectivization. In the continuum approximation, the influence of a periodic potential is taken into account by replacing the electron mass me by the effective mass m = 11 I2a2J which is inversely proportional to the exchange interaction energy that characterizes the electron jump from one node site onto another. In this approximation the electron motion along an ideal chain corresponds to the free motion of a quasiparticle with effective mass m and electron charge. Taking account of small displacements of molecules of mass M(~m)from their periodic equilibrium positions, there arises the short-range deformation interaction of quasiparticles with these displacements. When the deformation interaction is rather strong the quasiparticle is self-localized. Local displacement caused by a quasiparticle is manifest as a potential well that contains the particle. In turn, a quasiparticle deepens the well. A self-localized state can be described by two coupled differential equations for the field ifr(x, t) that determines the position of a quasiparticle, and the field p(x, t) that characterizes local deformation of the chains and determines the decrease in the relative distance a a p(x, t) between molecules of the chain, —~

—

262

AS. Davydov. Theoretical investigation of high-temperature superconductivity

[i/IoIôt + (112/2m) 82/ôx2 + up(x, t)]~i(x,t) (c~2/ôt2 V~l92/dx2)p(x, —

t)

—

=

0,

(a2u/M) 92h3x2 ~i(x, t)~2= 0

(7.14) .

(7.15)

When there is one quasiparticle in the chain, the function ~fr(x, t) is normalized by

~J

[~(x,t)]2dx=1.

(7.16)

Equations (7.14) and (7.15) are connected through the parameter a of the interaction between a quasiparticle and local deformation. V 0 = a\/~.7Mis the velocity of longitudinal sound in the chain with elasticity coefficient K. Equation (7.14) characterizes the motion of a quasiparticle in the local deformation potential U = ap(x, t). Equation (7.15) determines the field of the local deformation caused by a quasiparticle. In the reference frame 41 =9p/~.In (x Vt) Iathismoving with constant velocity V, the following is case the induced solution to eq. (7.15) has the equality form satisfied: ôp(x, t)/19t= —Va p(x, t) = [u/K(l s2)]k’(x, t)~2, s2 = V2IV~ 1. (7.17) —

—

—

-~

Having substituted the values p(x, t) into eq. (7.14), we transform it to a nonlinear equation for the

function qi(x,

t),

[illô19t + (112/2m) ô~2/9x2+ 2gJ~/i(x,t)~2]~!F(x, t) = 0,

(7.18)

where g

u212K(1

s2)J

—

(7.19)

is the dimensionless parameter of the interaction of a quasiparticle with a local deformation. Having substituted into eq. (7.18) the function ~i(x,t) = cI(~~)exp[i(kx wt)], k = mV/h, —

(7.20)

we get the equation [11w ~mV2 J ~~ioe + 2gJ~I~2(~)jcI(~) = 0, —

—

for the amplitude function cP(~)normalized by J

P2(e) d4 = 1.

(7.21)

The solution to this equation reads (see

fig. 7.2)

~P(~)= ~/~sech(g~/2),

(7.22)

with the value 11w

=

~mV2

—

D(s).

(7.23)

AS. Davydov, Theoretical investigation of high-temperature superconductivity

,

263

~

Fig. 7.2. The motion of a spatially localized two-component soliton with constant velocity.

The quantity D(s)

(7.24)

2J

= ~g

determines the binding energy of a particle and the chain deformation produced by the particle itself. According to eq. (7.22), a quasiparticle is localized in moving reference frame in the interval = 2ITaIg.

(7.25)

In this region the localization is characterized by the function p(~)= [ga/4i(1

—

s2)] sech2(g~/2).

(7.26)

The following energy is necessary for deformation: W

~K(1 + ~2)

J

p2(e)

d~= (24)~g2J(1+

~2)

(7.27)

Measured from the bottom ofthe conduction band of a free quasiparticle, the total energy (including that of the deformation) transferred by a soliton moving with velocity V is determined by the expression

E,(V) = W + 11w =

E

2,

(7.28)

5(0) + ~MSOIV

in which the energy of a soliton at rest

E 5(0)

and its effective mass M,01 are determined, respectively, by

the equalities 1g2J,

E,(O) = (12) M, 2J/3a2K). 01 = m(1 + g

(7.29) (7.30)

M

501 exceeds the effective mass of a quasiparticle m, as its motion is accompanied by the motion of a local deformation [238]. The effective potential well where quasiparticles are placed, is determined, in the reference frame by the expression 2Jsech2(~g4). (7.31) U= —op(x, t) = —g ~,

264

A.S. Davydov, Theoretical investigation of high-temperature superconductivity

Such “two-component” solitons formed when quasiparticles are bound by a local deformation will represent a single entity only if they move with velocity less than that of longitudinal sound in the chain. Otherwise, the local deformation will not be able to follow the quasiparticle. More rigid restrictions on the velocity V in a theory using eq. (7.15) for a harmonic description of molecular displacements from equilibrium positions follow from the requirement that the change in distance between molecules, as compared with the lattice constant, should be small. It is clear from eq. (7.17) that this requirement p(~)~ a is violated for s—~1. Anharmonic interactions in soliton theory were taken into account in papers by this author and Zolotaryuk [239—241]. The great stability of solitons can be attributed to the following facts. (1) Their energy lies lower than the bottom of the conduction band of free quasiparticles (7.29). (2) Solitons always move with a velocity less than that of sound; consequently they do not emit

phonons, i.e., their kinetic energy is not transformed into thermal energy. (3) Solitons have topological stability. After the soliton has passed, the equilibrium positions of molecules remain displaced but before a soliton occurs their position is unchanged. To annihilate soliton excitation, it is necessary to return molecules to their initial positions. Thus, the interaction of a quasiparticle with displacements of molecules (local deformation) stabilizes their motion. As the concept of a soliton is often confused with that of a polaron, we note their remarkable difference. Solitons (two-component) arise when quasiparticles (neutral or charged ones) interact with local deformations described by virtual acoustic phonons with large dispersion. They can be at rest and move with velocity V < V0, their great stability is due to the compensation of dispersion and nonlinearity. Polarons are self-localized states arising when a charged quasiparticle interacts with virtual optical vibrations_(intramolecular or polarization in ionic crystals) with a very small dispersion, wq = 2 w \/ w~+ aq 0. As was shown by Enolsky and this author [242], such polarons can move coherently only with velocity V < a when a > 0. At these small velocities the dispersion compensates nonlinearity. Hence polarons are practically at rest. Their motion is realized by the jump mechanism stimulated by thermal vibrations. This motion rapidly decreases. 7.3.

Three-component solitons. Bisolitons

If a quasi-one-dimensional soft chain is able to keep some excess electrons with charge e and spin 1/2, they can be paired in a singlet spin state due to the interaction with local chain deformation created by them. The potential well formed by a short-range deformation interaction of one electron attracts

another electron which, in turn, deepens the well. A simple theory of the pairing of two quasiparticles in a soft quasi-one-dimensional chain was developed by Brizhik and this author [243] without taking into account the relative motion of quasiparticles in a paired state. Here we study the pairing, rejecting this simplification. The equations of motion of two quasiparticles with effective mass m in the potential field U(x,

t) =

—op(x,

t),

created by local deformation p(x, t) of the chain can be written in the form 2/2m) ~2/öx,~+ U(x,, t)]tp~(x~, t) = 0, i, J = 1, 2, [i/l o/ot + (11

where ~(x

1, t) is the coordinate function of particle i in state j.

(7.32) (7.33)

A.S. Davydov, Theoretical investigation of high-temperature superconductivity

265

A local deformation p(x, t) is caused by two quasiparticles due to their interaction with displace-

ments from equilibrium positions. The function p(x, t) characterizing this local deformation of the chain, is determined by the equation /92

it

—

2812\

crt~2

0 —~)p(x,t) + /ix

—a-,ivi

V

t9 2

2

2

~ [I~’1(x, t)l + l~2(x,t)~] = 0,

(7.34)

where %7~= a~/~7M is the velocity of longitudinal sound in the chain with elasticity coefficient K and distance a between molecules of mass M; au is the energy of deformational interaction between a quasiparticle and the chain.

Due to the translational symmetry of an infinite chain, one can consider excitations propagating with constant velocity V
I

p(x, t) = 0 for all ~=(x—Vt)Ia, has x, thet for form,which i/i(x, p(~)= [uI,(1

—

52)][l

(~)I2

t)1

+ kI’

2( ~)II

2/V~ . ,

(7.35)

S2 = V

The local deformation energy (elastic and kinetic) is determined by W

~K(1+ ~2)

J

p2(e) d~.

(7.36)

Taking account of eqs. (7.32), (7.33) and (7.36), the equation for the function ~ ~ t) that characterizes the motion of a quasiparticle pair in the potential field (7.32) disregarding the Coulomb repulsion of electrons, takes the form, (7.37)

where the following notation is used, G = o-21K(1

J = 11212ma2,

—

(7.38)

~2)

We shall further consider the case of small velocity, 52 ~ 1. The symmetric coordinate function hle( ~ ~2’ t) of a singlet spin state for a quasiparticle pair is

~

~

t) =

~

+ ~(~2)~2(~1)] e_~~pt~

(7.39)

Here If 1, jg the energy of a quasi-particle pair in the potential field (7.34). To obtain the total wave function for a quasiparticle pair, it is necessary to multiply the coordinate function (7.39) by the spin function of the singlet state of these particles. The total spin in a paired state equals zero, but the spins of each quasi-particle have no definite value. If the function of single-particle states in a bisoliton (bound pair of quasiparticle and local deformation) is chosen in the form =

P(~)exp(ik1~1), i, j = 1,2,

(7.40)

266

AS. Davydov, Theoretical investigation of high-temperature superconductivity

with the value k.

~‘i~ ~

=

t) =

mV/h, the function of a bisoliton (7.39) reads \r2~(~1)~z(~2) exp{i[k(~1 +

~2)

—

~‘~t/h]} .

Here the amplitude functions ‘1( 4~)satisfy the equation, 2(4) e]P(~)= 0, [/i2//i~2 + 4gc1 —

(7.41) (7.42)

with the dimensionless parameter, g=

(7.43)

G/2J,

characterizing the nonlinear quasiparticle interaction via the field of local chain deformation caused by these quasiparticles. The energy of a quasiparticle pair in the potential well of local deformation is expressed through the eigenvalue r of eq. (7.42) by the equality =

mV2

+

eJ.

(7.44)

The local deformation field p(~),according to (7.40), is also expressed through the function I(~) p(~)=

[2u1K(1

—

s2)]cP2(~).

(7.45)

When only one bisoliton is present in the chain, the solution to eq. (7.42) is normalized by the condition

J

(~)d~=1,

(7.46)

and has the form of a bell-shaped function cP(~)=V~7~sech(g~),

(7.47)

with the value (7.48) The local deformation energy transferred by a bisoliton is defined by W(V)

= ~g2J(1 +

s2)/(1

—

s2)

(7.49)

,

so, the full energy transferred by a bisoliton (together with double electric charge) is defined by the equality E(V)

=

W + If~(V)= mV2

—

2g2J(1

—

5s2)13(1

—

s2).

(7.50)

A.S. Davydov, Theoretical investigation of high-temperature superconductivity

For small velocities EbS(V) =

267

~ 1), this expression can be written as

(~2

(7.51)

2.

EbS(0) + ~MbSV

Here MbS

=

2m(1

+

4g2JI3a2K)

(7.52)

is the bisoliton effective mass. The energy EbS(O) of a bisoliton at rest, measured relative to the bottom

of the quasiparticle conduction band, is determined by EbS(0) =

(7.53)

—2g2J/3.

It is essential that this value be independent of the mass of molecules M. The energy 2g2JI3 is necessary for a bisoliton to decay into free quasiparticles. Comparing the energy of two solitons at rest (7.29) with the energy of one bisoliton at rest (7.53), we can see that bisoliton decay into two solitons requires an energy equal to 2E

(7.54)

2J.

5(0) EbS(0) = ~g Both quasiparticles in a bisoliton move in the common effective potential —

U(~)= —2g2Jsech2(g~), whose radius is half and the depth is twice the corresponding values of the potential well (7.30) of one soliton. According to eqs. (7.41) and (7.47), the motion with velocity V of two quasi-particles in a bisoliton is described by the coordinate wave function ~

~

t) =

g2112 sech(g~

2 1)sech(g~2)exp{(i/h)[mV(~1 +

~2)

—

—

g2J)t]}

.

(7.55)

(mV

The quasiparticles pair in a singlet spin state in a soft chain due to the nonlinear coupling of quasiparticles with the inertia chain deformation. The solution to the nonlinear Schrödinger equation describing an inertialess field with self-interaction contains no bound states. Zakharov and Shabat [244], using the inverse scattering method showed that in this case the energy of two-soliton states equals the sum of the energies of one-soliton states. The preceding results did not take into account the Coulomb repulsion of particles. In neutral systems the electron charges are screened, therefore, the Coulomb repulsion is weakened. Without taking into account the Coulomb repulsion, according to (7.55), the most probable location of quasiparticles corresponds to the value ~ = ~2 = 0. With the repulsion taken into account the maximum probabilities of the quasiparticle positions will be displaced with respect to each other. As it is shown by Brizhik and this author [243], when account is taken of the Coulomb repulsion by means of the perturbation theory method, such a displacement is expressed by 2 1/2 I = (alg2 )(eefflaf)

268

AS. Davydov, Theoretical investigation of high-temperature superconductivity

where eeff is the effective screened charge. A bisoliton is stable if this displacement is less than the bisoliton “dimensions” 2ira/g. For this inequality to be valid, it is necessary that the dimensionless electron—phonon interaction parameter g exceeds the critical value ~ given by ~cr

Veeff/4a7rJ.

7.4. Electric charge transport along protein molecules The electron transport from donor to acceptor molecules along protein molecules is important in bioenergetics of living organisms. It has been established experimentally that the electron can move along protein molecules over distances 10 and 100 times as large as the interatomic ones. Protein molecules are dielectrics. The conduction band of proper electrons in such molecules is situated relatively high —.4—6 eV. The transport takes place by extra electrons which come into a protein molecule from the donor one. Usually electron transport is realized through the a-helical part of the

protein molecule. Alpha-helical parts of the protein molecule contain periodic peptide groups made up of four atoms H, N, C and 0. The electric charges in such a group are located asymmetrically. Therefore, in the ground state such a group possesses a constant electric dipole moment (near 3—4 debye) [245] (see fig. 7.3). As was shown by Turner, Anderson and Fox [246],in the field of such a dipole the electron can be in the bound state with binding energy —--0.9 eV. In this state the electron cloud is stretched across a large distance: the average distance of the electron from the positive and negative ends of the dipole equals, respectively, ——4 and 5 au. Peptide groups in an alpha-helical protein molecule form three chains of hydrogen bonds. In each chain they create periodic potential wells for an extra electron that got into the chain from a donor. The overlapping of wave functions of the electron ground states in neighbouring potential wells makes these states collectivized, i.e., the conduction band for excess electrons is formed. The overlapping of wave functions of the electrons located on peptide groups of different chains is not so important. Therefore, the motion of an electron that has come from a donor can be considered in each chain independently.

Fig. 7.3. The distribution of electronic charges on atoms of the peptide group.

Fig. 7.4. The arrangement of the permanent electric dipole moments in the peptide group of the a-helix protein molecule.

AS. Davydov, Theoretical investigation of high-temperature superconductivity

269

In the continuum approximation the electron motion in the conduction band of a chain of peptide groups, fixed rigidly at the sites na, is characterized by the effective mass m=/12I2a2J,

(7.56)

which is inversely proportional to the energy J of the exchange interaction between neighbouring peptide groups. Introducing the effective mass (7.56) allows one to consider the electron motion in the periodic field of constant dipoles as the free motion of a quasiparticle with this mass and electron charge. In the alpha-helical part of a protein molecule, the distance between peptide groups a = 5.4 A, the energy J = i0~eV. Hence the effective mass of a quasiparticle is m 120 me.

The extra electron that entered the protein molecule from a donor caused local deformation by displacing the equilibrium positions of peptide groups. The motion of a quasiparticle, taking into account such local deformation caused by the particle itself, is described by a system of coupled equations for the function characterizing the probability of the quasi-particle position ~/i(x,t) and the function p(x, t) that determines the relative decrease in the distances between peptide groups. These equations coincide with eqs. (7.14) and (7.15) considered by us in section 7.2, if the value M

involved in them corresponds to the mass of a peptide group, and V~is the velocity of longitudinal sound in the chain of the peptide groups with the elasticity coefficient K. Repeating the calculations performed in section 7.2, it becomes clear that the electron in a protein

molecule will move along the alpha-helical part of the chain as a two-component soliton, i.e., the bound state of a quasi-particle with local deformation without energy expenditure at constant velocity V. A free quasi-particle can propagate in the conduction band with a velocity not exceeding the maximum group velocity V 8~2aJ/h=1.8x108 cm/s.

(7.57)

The velocity of a free local deformation coincides with that of longitudinal sound in the chain

5 cm/s. (7.58) i0 A two-component soliton can move in the chain as a single entity if its velocity does not exceed the 3.5 X

lesser of these velocities, i.e., the velocity of longitudinal sound. In this case its energy transferred along

the protein molecule, is described by the expression E,(V)

= E

2,

(7.59)

5(0) + ~M,Ø1V

where E,(0)

= —g2J/12 =

~a4! 48i2J

(7.60)

is the energy of a soliton at rest which is measured from the bottom of the conduction band of a free quasiparticle, and M

2JI3K2) 501 = m(1 + g is the soliton effective mass.

(7.61)

270

AS. Davydov, Theoretical investigation of high-temperature superconductivity

According to (7.60) when a soliton is formed, i.e., when a quasiparticle and a deformation become bound, the amount of energy released is (7.62) 4—a4148KJ. Consequently, the binding energy is proportional to the fourth power of the deformation parameter a and inversely proportional to the square of the elasticity coefficient K. The larger the 4, the more stable is the soliton. Thus, soliton stabilization is especially pronounced when the coupling between a quasiparticle and the deformation is strong. These chains include the hydrogen-bonded chains of peptide groups in alpha-helical parts of the protein molecules. Therefore, these parts are ideal guides for electron transport from donor to acceptor molecules. As was shown in section 7.3, the interaction between a small number of electrons and an inertial local deformation can lead to pairing, i.e., the union of two electrons with oppositely directed spins with a local deformation. Such three-component solitons can transfer a double electric charge more effectively than two independent two-component solitons. The combination of three electrons with a local deformation is impossible due to the Pauli principle that does not admit two electrons to be at the same place in a similar state. The properties of three-component solitons were considered in section 7.3. It was noted there that they are more stable at small velocities than two-component ones. Therefore, in living organisms electrons are often transferred in pairs, not alone. For instance, in redox reactions pairs of electrons are transferred from one molecule to another. In the internal membranes of chloroplasts when ATP molecules are synthesized, the electrons move through the membrane in pairs, not individually [2471.

8. Bisoliton model of high-temperature superconductivity In this section we present a new nonlinear theory of high-temperature superconductivity of anisotropic nonmetallic oxide ceramics based on moderately strong electron—phonon coupling that does not admit application of perturbation theory. This theory uses the concept of bisolitons bound in a singlet spin state due to the crystal local deformation of quasiparticle pairs (holes). Bisolitons transporting the double electric charge are Bose particles. It is shown that when their concentration is higher or lower than some critical values, they produce a Bose condensate a collective state with periodically distributed bisolitons moving as a whole without resistance. The critical temperature of bisolitons at rest is weakly dependent on the mass of ions inducing pairing (very small isotopic effect). The weak isotopic effect is manifest only when the condensate motion is taken into account. The conditions for the formation of bisoliton condensate and its properties are discussed in the next section. —

—

8.1. Bisoliton condensate in ceramic superconductors In addition to the ceramic dielectric oxides La 2CuO4 and YBa2Cu3O7, other analogous compounds have also been observed which, being supplied with doping components, two-valent ions of strontium, barium, etc., or with oxygen ion vacancies being available in the lattice, carry the electric current. With decreasing temperature they went over into a superconducting state at comparatively high T~.All of them contained 2~.oxygen ions and copper ions which, without doping components were in the valencetwo state Cu

AS. Davydov, Theoretical investigation of high-temperature superconductivity

271

The doping causes positively charged complexes [Cu—O] + with a deficient electron, i.e., with a hole, to arise in the crystal. Due to the translation symmetry of the crystal these hole states are collectivized forming the hole conduction bands.

Since the crystal has a layered structure the conductivity is caused by the planes CuO4 perpendicular to the symmetry axis C. In lanthanum crystals (see fig. 5.1) the ion groups Cu04 are distributed in layers so that parallel chains of alternating copper and oxygen ions are formed in the mutually perpendicular a and b directions. In yttrium and bismuth (see figs. 5.2 and 5.3) crystals the chains of alternating copper and oxygen ions are directed along the axis b. In this case the exchange interaction between ions disposed along the filaments is much larger than the interaction between ions composing different filaments. The conduction band generated by the exchange interaction of the hole states of complexes (Cu—0)~should have the “tube” structure with an isolated symmetry axis directed along the axis b and the effective mass m of hole quasi-particles determined by the equality 2/2a2J,

(8.1)

m=11 where a is the period (unit cell dimension) along the axis b and J is the exchange interaction energy induced by the hole transfer between the neighbouring cells. Without a magnetic field one can consider the hole states of such a system of filaments with identical phases in the transverse cross-section. So the problem is reduced to a quasi-one-dimensional one. The quasi-one-dimensional model can be applied also to a plane square lattice (such as in La—Ba—Cu—O) if

one studies the particle transfer along certain directions with a constant phase in the transverse direction. Such a state is provided by the initial conditions. We remind the reader that a laser beam shaped as a plane wave propagates at enormous distances in the three-dimensional space. The opinion that a quasi-one-dimensional model should not lead to superconductivity, has sometimes been put forward. It proceeds from the papers by Rice and Hohenberg [66] who showed that in ideal one-dimensional systems the superconducting phase transition is impossible due to the thermal fluctuations. What is not taken into consideration however is that there are no ideal systems in nature. Very strong anisotropic systems with a pronounced symmetry axis can be regarded in some cases as

quasi-one-dimensional. The conclusions made by Rice and Hohenberg are inapplicable to such systems. As such a superconductor model we shall consider a chain of periodically distributed unit cells of mass M. Each cell, in the plane (a, b) consists of two copper ions with oppositely directed spins and six oxygen ions. Cells are joined by valence bonds of copper and oxygen ions. When the crystal is doped the p—d hybridized complexes [Cu—O]+ with an additional positive charge (holes on the complex) and spin 1/2 are formed in some cells. Due to the translational symmetry of the chain and the exchange interaction J between the neighbouring unit cells the quantum state of holes is collectivized. This state is characterized by the conduction band where the hole motion, in the effective mass approximation, is treated as the motion of a free quasi-particle with effective mass given by (8.1) in a chain with unit cells fixed in periodic equilibrium positions na. The formation of complexes [Cu—Ol+ inside the cell causes the “breathing” displacements of oxygen ions around the copper ions (due to the change of the electric interactions) and lowering of the symmetry of the octahedron of anions (the Jahn—Teller effect). Such a displacement of ions inside the cell generated by the appearance of a quasiparticle (hole) leads to the displacement of equilibrium positions of neighbouring cells. This is how the strong electron—phonon interaction between quasiparticle and local chain deformation comes into play.

272

AS. Davydov, Theoretical investigation of high-temperature superconductivity

The assumption on strong electron—phonon interaction in oxide ceramics made Bednorz and Muller search for high-temperature superconductivity in these materials. So, as a ceramic superconductor model we shall consider the quasi-one-dimensional chain containing N pairs of quasi-particles with spin 1/2 and positive electric charge formed under the crystal doping. These quasiparticles induce local chain deformations. Each local deformation can contain two quasipartides with opposite spins. The excited states in such chains were studied by Ermakov and this author [248—250]. In a singlet spin state in the reference frame ~ = (x Vt) Ia moving with velocity V a pair of quasiparticles of sorts 1 and 2 distinguished by oppositely directed spins, are described by the function ~ ~ t) symmetric with respect to a permutation of space coordinates which has the form —

~,

~k

1k2(~1,

~2’ t) =

~

+ Qk1(~)~k2(~l)] eISpt~.

(8.2)

Here is the energy measured from the bottom of the conduction band of paired quasiparticles in the potential field U( ~) created by the particles themselves 2/K(1 s2)][I ~k~( ~~)j2 + ~k2( ~.)j2] s2 = V2/V~~ 1. (8.3) U( ~) = —[a V~is the speed of longitudinal sound in the chain. The one-particle functions ~k( ~,)that characterize the probable quasiparticle distributions in the state k. are determined by —

~‘k~(~i)=

cI(~~)exp(ik 1x~), i, j=1,2.

In the chain with N0 unit cells (N0 wave numbers

~‘

(8.4)

1) the conductivity band involves N0 states distinguished by the

k=2i~s/aN0, ~t.=O, ±1,±2,... ,N0/2.

(8.5)

If it contains N (sN0) independent quasiparticle pairs with spin 1/2 then at low temperature they will occupy N lowest states with the wave numbers kF = irN/aN0.

(8.6)

The maximum value kF is called the wave number of the Fermi surface of energy 2k~I2m. (8.7) EF = h At low temperatures some part of the quasiparticles, due to the interaction with local deformations generated by them, pair in a singlet spin state producing Bose particles of zero spin and charge 2e. The ability to pair is inherent only to quasiparticles that lie near the Fermi surface. By the quasimomentum conservation law in forming a pair that moves with small velocity V= hk/m,

only particles with the wave numbers k 1 = 2k + kF,

k2 = —kF

(8.8)

AS. Davydov, Theoretical investigation of high-temperature superconductivity

273

can participate. Consequently, the coordinate wave function of the condensate consisting of N pairs of quasiparticles which is formed by quasiparticle pairs in a singlet spin state takes, according to eqs. (8.2) and (8.4), the form

~

~2’ t) =

\r2~(~1)~(4~2) cos[(k + kF)(~l

—

42)]

exp{i[k(~1 +

~2)

—

If~(V)zVh]} .

(8.9)

The cosine appears due to the symmetry condition. To obtain the total wave function the coordinate

wave function (8.9) should be multiplied by the spin function of a quasiparticle singlet state. The function (8.9) must satisfy the equation 2Iô~+ ~2/~~) + U(~ [i/l/iIdt+ J(/i 1)+ U(~2)]~1’(~1, 5~2’t) = 0, (8.10) where the deformation fields U(~~), taking into account (8.3) and ~ = P(~,) exp(ik1~~) are determined by 2’P2(flIK(1 s2). (8.11) U(4) = —2u In a state characterized by the function (8.9) the total momentum 11k and the total zero spin have definite values. The momenta and spins of isolated quasiparticles have no definite value. Having substituted the function (8.9) into eq. (8.10) one can get the equation which is satisfied by the amplitude functions, [/i2/t9~2 + 4gtI2(~)+ A]cP(~) = 0, (8.12) —

with the dimensionless parameter, g

GI2J = a2I2KJ(1

—

s2),

~2

~ 1,

(8.13)

that characterizes the coupling between quasiparticles and local deformation field. The energy If~(V)of quasiparticles trapped by this local field is expressed through the eigenvalue A of eq. (8.12) by means of If

2k~!m+ mV2

+

AJ

—

11VkF.

(8.14)

1,(V) = h

The energy of the local chain deformation is expressed via the amplitude function ck( ~) by the integral ~=

~

J

i’4(~)d~.

(8.15)

If the density of bisolitons is small and the Coulomb interaction between them can be neglected, their distribution in the chain and condensate properties will be defined by solutions to eq. (8.12). At low temperature bisolitons form a single entity the Bose condensate. The properties of this condensate are defined by periodic solutions to eq. (8.12). Periodic solutions of a nonlinear equation such as (8.12) were studied by Pestryakov and this author [251](see also Davydov in ref. [238],section 26). Equation (8.12) admits periodic solutions corresponding to a uniform distribution N < N 0 of pairs of coupled quasiparticles along the chain. The real functions P( ~) of these solutions should satisfy the periodicity condition, —

274

AS. Davydov, Theoretical investigation of high-temperature superconductivity

~

L=N0IN,

(8.16)

and the normalization condition, (8.16a)

J~2(~)d~1

so that in each space interval specified by the period aL there is one pair of coupled quasi-particles. The exact periodic solutions to eqs. (8.12) are expressed through the Jacobi elliptic functions dn(u, q) by the equality 2E~’(q)dn(u, q), u=g~/E(q). (8.17) ~q(~j)=(g/2)” The explicit form of the Jacobi functions dn(u, q) is dependent on the elliptic function modulus q taking on continuous values from zero to one. The eigenvalues A of eq. (8.12) are also expressed through the elliptic function modulus q by means of the equality A=

(8.18)

—g2qIE(q).

The expressions (8.17) and (8.18) involve the function which is a second kind elliptic integral determined by ~( q)

E(q)=

[

dn2(u,q)du,

(8.19)

via the total first kind elliptic integral ~(q)

=

J

[(1

—

t2)(1

—

q2t2)]”2 dt.

(8.20)

The dependence of the elliptic integrals E( q) and X( q) on the elliptic function modulus q is shown in fig. 8.1. The value of the elliptic function modulus q is determined through the product of the period L and the electron—phonon interaction dimensionless parameter g; the elliptic integrals of the first kind ~( q) and second kind E( q), with the help of the equality gL=2E(q)~{(q) ~ir2

(8.21)

This equality allows one to find the value of the modulus q for each product gL. The elliptic function modulus q defines completely its form. Below we shall give the explicit expressions of the Jacobi elliptic functions for the values q 1 and q ~ 1. The space distribution of both quasiparticles in a bisoliton is characterized by the square of the function in (8.17). According to eq. (8.11) the field of the local chain deformation is also the periodic function

AS. Davydov, Theoretical investigation of high-temperature superconductivity

275

2.5

2.0

1.57

-~

_

~

E(q)

D(q)

q

0.5

\\~~1.a

Fig. 8.1. The dependence of the first [X(q)] and the second [E(q)] total elliptic integrals on the modulus of the elliptic function, q; 2)E(q) —(1— q2)x(q)]. D(q) = 112(2 — q

U(~)= —[ga2/ic(1

—

s2)E2(q)] dn2[g~IE(q), q].

(8.22)

Substituting the values of eq. (8.17) into eq. (8.15) and calculating the integral, we find the energy of the local chain deformation in one period W(q) = [(1 + 2s2)!2E(q)]g2JD(q).

(8.23)

The above expression involves the function D( q) which is determined by D(q)

~[2(2 q2)E(q) —

—

(1

—

q2)~(q)].

(8.24)

The dependence D(q) on q is shown in fig. 8.1. The periodic local deformation field U( ~) causes an indirect interaction between bisolitons combin~ ing them in a single entity — the condensate. All bisolitons in this condensate move with the same velocity V.

The coordinate wave function of this condensate made up of bound quasiparticle pairs (in a singlet spin state), bisolitons, is determined for each period L by the expression

276

AS. Davydov, Theoretical investigation of high-temperature superconductivity 2(q)

~

t) = \/~E

~,

cos[(k

+

kF)(~l

-

x dn[g 5e~/E(q), q] dn[g~,/E(q), q] exp{i[~(k) + I~~(V)tIh]} ,

(8.25)

where the Jacobi function modulus q is determined through an electron—phonon interaction dimensionless constant, and the period L by means of eq. (8.21). The number of hisolitons concentrated in the chain N/N0 determines, according to eq. (8.16) the period L. The unique coherent phase 4(k) of the wave function (8.25) is defined by 4(k)=2k(R—Vt),

R_~(x1+x5).

(8.26)

It experiences a monotonic variation when the coordinate of the bisoliton centre of gravity changes. Due to their coherence, bisolitons move uniformly. An abrupt change in the velocity of a bisoliton should result in a change in the character of the motion of all bisolitons that compose a condensate. This is one of the principal causes providing superconductivity. The velocity of the condensate motion is expressed through the space phase derivative of its single wave function (8.25). Thus, V= (1112m) dçb(k)/dR

=

11k/rn.

Here 2m is the mass of the two free quasiparticles involved in a bisoliton. The coherence velocity of bisoliton condensate motion is limited from above. Since bisolitons are formed when two quasiparticles are bound with local chain deformation, their combined motion, as a single entity, is possible only under the condition that the velocity of motion of the composite entity does not exceed the lesser of the two velocities: (1) that of longitudinal sound V0 whose order of magnitude equals —10~cm/s; (2) the maximum group velocity Vg of free quasiparticles in their conduction band. The maximum group velocity is determined by the exchange integral J characterizing

the width of the conduction band of free quasi-particles =

2aJ/h

=

h/am,

(8.27) 7 cm, J ——0.05eV, m = 1.2 me. Therefore Vg

m the quasiparticle effective mass. With a i0 106 is cm/s. It is apparent that in superconducting ceramics a more important limitation of the condensate velocity is caused by a possible decay of the bisolitons involved in the condensate, into free quasi-particles. This restriction will be studied in the next section. Since the velocity V of the bisoliton condensate is always less than that V 3 of the longitudinal sound in the chain, the bisoliton does not lose energy to produce the acoustic phonons. Unlike the ideal Bose gas condensate that expresses the state of Bose particles with zero energy, the bisolitons in a condensate moving with velocity V much less than V3 and V8 transport energy. This energy is the sum of the bisoliton energy given by (8.14) in the deformation field moving with velocity V and the energy of the chain deformation given by (8.23). Taking into account (8.18) we find —

2k~Im

— 2Aq

~q(V)

IZ~(V)+ W(q) = 11

+

mV2 + hkFV + Wq(V).

(8.28)

AS. Davydov, Theoretical investigation of high-temperature superconductivity

277

Here 2Aq defines the binding energy of two quasi-particles in the deformation field of a motionless bisoliton 2Aq

(8.29)

= g2q2JIE(q).

The total energy (8.28) of a bisoliton moving with velocity V that involves also the deformation energy Wq(V) transforms, in the case of small velocities, as t~q(V) ~q(0) +

~MbSV2

+

11kFV, V2 ~ V~,

(8.30)

where ~q(0) is the energy of a bisoliton at rest determined by the equality

~q(0)= h2k~!m— 24(q).

(8.31)

Here 24(q) is the binding energy of two quasiparticles in a motionless bisoliton, that involves also the local deformation energy 24(q) = g2JF(q).

(8.32)

The function F( q) is determined by F(q)~2E2(q)[2— q2

—

D(q)!E(q)].

(8.33)

The bisoliton mass MbS in eq. (8.30) is determined by the expression Mb,

=

2m

+

4Jg2D(q)IV~E2(q), V~= a2K/M.

(8.34)

The function D(q) involved in eqs. (8.33) and (8.34) is determined by the equality (8.24). The value of 24(q) determines the binding energy of two quasi-particles at absolute zero, i.e., the energy released in generating a fixed bisoliton when account is taken of the energy expenditure to form

the local deformation. 4(q) characterizes the energy gap in the spectrum of quasiparticles which is manifest in the spectrum of quasiparticles when bisolitons are formed. The binding energy of a fixed bisoliton (8.31) is independent of the ion mass M, i.e., there is no isotopic effect although the electron—phonon interaction underlies the pairing. The isotropic effect is manifest only through the bisoliton kinetic energy. At small velocities (V2 ~ V~)the isotopic effect is also small. With increasing temperature the pairing energy 24r(q) decreases. The critical temperature T~of the thermal destruction of superconductivity is determined as the temperature at which 4~( q) vanishes. The decrease of 4,.(q) with increasing temperature is, to a great extent, connected with the decrease in the exchange integral J involved in eq. (8.32), due to the fluctuations of the distances between the unit cells. The BCS theory shows that the critical temperature T~is determined directly by the energy gap in the quasiparticle spectrum 4~c~with the help of the relation kBTC

= 3~54BCS

(8.35)

278

AS. Davydov, Theoretical investigation of high-temperature superconductivity

A bisoliton theory at nonzero temperature is not yet developed. One may hope that in such a theory the proportionality k~T~—24(q)

(8.36)

will hold. The properties of a bisoliton condensate in our model are dependent on the exchange interaction integral J of an electron—phonon interaction, on the dimensionless parameter g and on the ratio of the bisoliton number N to the general number N0 of unit cells in the chain. In terms of a constant lattice spacing the parameter L = N0/N characterizes the uniform distribution of the bisolitons along the chain when gL> ~ii-. In the following section we shall discuss the bisoliton properties in chains with a comparatively small number of bisolitons when the product gL has a large value. This situation is, apparently, realized in real superconducting ceramics characterized by a small number of charge carriers. 8.1.1. Bisoliton condensate at small density of charge carriers When the value of the elliptic function modulus q is close to unity, the total elliptic integrals ~( q) and E( q) have the asymptotic values 2~1, (8.37) X(q)=r— ~(r— 1)q~+”~, E(q)~1 + ~(r—1/2)q~+..., q~=~1—q where T~ln(4/q~)~—1 since q~-~1. In this case eq. (8.21) is replaced by the approximate equation gL=4ln(4/q~) or q~=16exp(—~gL). Consequently, the small values q 1 correspond to gL 1, i.e., to a relatively small ratio of the number of bisolitons to the general number of unit cells in the chain. In this case the functions F( q) and D( q) that determine, by means of eqs. (8.32) and (8.34), the energy and the effective mass of a bisoliton take the values ~-

F(q)

~[1 + 4gL exp(—gL)],

D(q)

~[1 4gL exp(—gL)]. —

Therefore, the total pairing energy 24(q) and bisoliton mass MbS are determined by 24(q) ~g2J[1 + 4gL exp(—gL)], Mb,~2m+8g2J(3V~)’[1 —4gL exp(—gL)].

(8.38) (8.39)

With increasing concentration the value gL decreases, therefore, the pairing energy increases and the effective mass decreases. Such changes are due to the mutual indirect influence of bisolitons in a condensate. When there is one bisoliton in the chain the square brackets in these expressions equal one. According to eq. (8.29) when the inequality gL 1 is satisfied the binding energy of two quasiparticles in the field of a local deformation of a fixed bisoliton is characterized by ~-

26q=Jg2.

(8.40)

A. S. Davydov, Theoretical investigation of high-temperature superconductivity

279

This energy should be expended when two quasiparticles “are ejected” from the potential deformation

well, not destroying it. These rapid processes should occur during a time much less than —llY~s. The energy necessary in thermal or any other slow destruction process of a bisoliton should be one and a half times as large as that of eq. (8.38). Different values of the bisoliton destruction energy under rapid or slow processes arise from the essential role played by the local deformation in forming them. At small quasiparticle densities the Jacobi elliptic functions determining the space distribution of bisolitons along the axis ~ in eq. (8.17) are approximated by hyperbolic functions in each period dn(u, ~)

[1 + ~q2(u + sinh u cosh u) tanh u] sech u.

(8.41)

As a result, the overlap wave functions ‘I’( ~) in (8.9) have the bell-shaped elevations of width = 2irlg, situated a distance L apart (fig. 8.2~.All of them characterize the distribution of bisolitons moving with the same velocity V along the axis The value z~x= 2 irag 1 determines the soliton dimension (the correlation length). The formation of a bisoliton involves quasiparticles with wave numbers in the range of values L~kwithin the kF region ~.

determined by Ak——gI2i~a.

(8.42)

The continuum approximation used by us is justified when g <21T is fulfilled. 8.1.2. Bisoliton condensate at large density of charge carriers

To large densities of quasiparticles in the chain there correspond small periods L. With fixed parameter g and small L when gL <9 the elliptic function modulus q is small. Satisfying the inequality q2 ~ 1 the total elliptic integrals have the asymptotic values ~flq)ir(1+

~q2+ ~q4),

E(q)~~1T(1— ~q2—~q4).

Hence, equality (8.21) determining the modulus q becomes gL= ~r2(1—q4I32)

or q2=\/2gLIIT2—1>0.

2~/g~

L~/g~

Fig. 8.2. The distribution of bisolitons moving with the same velocity along the axis ~.

(8.43)

280

A. S. Davydov. Theoretical investigation of high-temperature superconductivity

It follows from eq. (8.43) that periodic solutions to eq. (8.12) in the form of Jacobi elliptic functions exist only when gL> ~~2 Thus, the admitted maximum density of quasiparticles of the bisoliton condensate corresponds to the minimum period Lmjn =

rr2I2g.

(8.44)

When the inequality ~ir~
trigonometric functions ~I’q(~)=

~_1~f~[1

—

~q2sin2(2g~/ir)].

(8.45)

When the bisoliton density attains its critical value, L = rr2I2g, modulations of this function vanish. The ordered distribution of bisolitons is broken the condensate uniqueness is violated. —

With q2 41, the function (8.24) and E(q) are approximated by 12 D(q)~ir(1—

32

94

12

12

4q +~q),

E(q)~ir(1—4q). 2. In this case, according to (8.32), the binding energy Then with the help of (8.33) we find F(q) 16/ir takes the value 24(q)=2g2J(2/ir)3.

(8.46)

It should certainly be noted that at large bisoliton densities the Coulomb interaction will essentially decrease the value in (8.46). The dependence of the superconductivity of ceramic superconductors on the concentration of charge carriers obtained on the basis of a bisoliton model is proved apparently, by experimental studies. So, it was noted in Khurana’s review [252] that no group of researchers observed superconductivity of these compounds at concentrations less than 0.05 and higher 0.4. The maximum critical temperature is

attained at concentrations of carriers near 0.15. So, according to the bisoliton model the most favourable conditions for superconductivity of ceramic oxides corresponds to an optimal concentration of quasiparticles when the function F( q) entering (8.32) attains its maximum value. The bisolitons form a Bose condensate distributed at equal distances as in a one-dimensional crystal. When a superconductor moves between two metallic electrodes and the

electric field is switched on the whole condensate moves with a unique velocity, as an entity, not delayed by small barriers and defects. In this case the bisoliton disappearance on one electrode is compensated by their generation on another electrode. 8.2. Conditions for stability of bisoliton condensate and critical superconducting current

The availability of a condensate of charged bisolitons (Bose particles) is the necessary but insufficient condition for possible emergence of superconductivity. At small velocities the stability of a bisoliton condensate is provided by: (1) the absence of the other energy one-particle excitations lying below the

bisoliton energy band; (2) the fulfilment of the Landau superfluidity condition [d~

9(V)/dV]~.0>0. For kF ~ 0 this condition follows directly from eq. (8.30). The first condition in our model as well as in the BCS one, is satisfied because quasiparticle pairing is provided by the electron—phonon mechanism.

AS. Davydov, Theoretical investigation of high-temperature superconductivity

281

With increasing bisoliton condensate velocity there are restrictions on the upper values of the velocity. Since bisolitons are formed when two quasiparticles are coupled with a local chain deformation, their combined motion, as a single entity, is possible under the condition that this velocity does not exceed the smaller of the two velocities: (1) that of longitudinal sound V3 i0~cm/s; (2) the maximum group velocity V8 2aJIh of free quasi-particles in their conduction band. At a 1.8 X --.

—

i0~cm, J = 0.01 eV, the value

V8

1.5 X 106 cm/s.

The second restriction of the condensate velocity comes from the term containing the product of the velocity and kF, present in the energy given by (8.28). The presence of such a term results, for velocities exceeding the critical one V~,in the bisoliton decay into two free quasiparticles with the total

quasimomentum conserved. This process is of the type of rapid processes at which the local deformation has no time to change and quasiparticles from a bound state in a bisoliton moving with velocity V = 11k/rn go over into free states with quasi-momenta 11(2k + kF) and —hkF. This transformation is studied by Ermakov and this author [248]. With the transformation conserving the 211k and local Wq, the kinetic 2k2/m of two quasiparticles in aquasimomentum soliton is transformed to thedeformation sum of kinetic energies, energy 11 112(2k + kF)212m + h2k~/2m,of two free quasiparticles. Consequently, the energy of the system after the transformation is determined by the expression =

Wq + 112(2k

+

kF)212m + 112k~/2m 112k21m. —

Taking into account that in a bisoliton state it had the value (8.28) we find the change in the energy ~q(V) —

~dIS(~”)

= IIkFV —

2ôq~

If this difference in energies is negative the bisoliton decay is forbidden. Thus, the bisoliton decay is forbidden if its velocity V is less than the critical velocity V3r’ determined by = 2t5qIhkF•

(8.47)

With the bisoliton density v in the condensate moving with the velocity V, the density of the electric current transferred is determined by the equality j = 2evV. The restriction given by (8.47) on the condensate velocity points out that superconductivity of ceramic superconductors is conserved only when the current density is less than the critical one Icr =

4ev6q/hkF.

(8.48)

In ceramic superconductors the soliton density ii and the Fermi momentum are small, however, the critical current density can be large. Measurements of the current density made by Dinger et al. [253]in the most favourable directions of a YBa 2. 2Cu3O78 single crystal at T = 4.5 K and zero magnetic field showed the value ‘Cr = 3 x 108 A/cm 8.3. The breaking of Cooper pairs in a constant magnetic field

Ceramic high-temperature superconductors refer to the second kind of superconductors. Their total superconductivity vanishes only in a constant magnetic field whose strength exceeds the second critical

282

AS. Davydov, Theoretical investigation of high-temperature superconductivity

magnetic field B2cr. According to experimental data the second critical field in lanthanum superconductors has the value 50—140 T and in yttrium superconductors B2~~ is 80—140 T [254—256].Such magnetic fields exceed appreciably the second critical fields observed in ordinary superconductors, by several tesla. The destruction of superconductivity by a constant magnetic field is generated by the breaking of Cooper pairs due to the following two effects: (1) the paramagnetic effect the reorientation of electron magnetic moments along the magnetic field which makes the singlet spin state transform into a triplet one; (2) the Landau diamagnetic effect according to which the electrons, affected by a magnetic field, describe helical trajectories. The quantized intensive circuit motion is accompanied by depairing of electrons. —

It has been shown by Brizhik, Ermakov and this author [257]that, in addition to the large values of the second critical field, the superconducting ceramics also have other peculiarities distinguishing them qualitatively from ordinary superconductors. These peculiarities can be explained by the bisoliton model of superconductivity. Below it will be shown that, according to the bisoliton model, the paramagnetic effect, and not the diamagnetic one as in the BCS theory, proves to be crucial in suppressing the superconductivity by the magnetic field. It will also be shown that the effect of magnetic impurities on superconductivity has a step-like character. Also in this respect they diverge from the predictions of BCS theory. Paramagnetic effect To study the paramagnetic effect, use will be made of the bisoliton model (developed in this section) of superconductivity of ceramic superconductors. In accordance with this model, bisolitons in a superconducting state form a condensate that moves with constant velocity V along chains of periodically distributed copper and oxygen ion complexes (lattice axis b). Solitons are distributed along these chains in a regular way with a period aL determined by the linear density of pairs of quasiparticles, N. In each period the bisoliton involves two quasiparticles with the wave numbers 8.3.1.

kl=2k+kF,

k2=—k~,

(8.49)

where k = m~V/h,m, is2I2a2L, the effective motionbetween of a quasi-particle (axisunit z) determined wheremass J~is of thelongitudinal exchange energy neighbouring cells along by the equality rn, = h the z axis. In a singlet spin state x 0~the superconducting condensate moving with velocity V is described by the unique wave function normalized in a single period aL, = \dI)(~1)~(~2) cos[(k + kF)(~l —

~2)]x0 exp{i[k(~1 + ~2)

—

~~(V)t/h]}

,

(8.50)

where ~ = z~ is the energy of two particles in the potential well U(~),created by the particles themselves, that moves with velocity V. The well is determined by the equality —

Vt,

If~(V)

U(fl—ap(fl.

(8.51)

Here p( ~) is the local deformation caused by the relative decrease in distances between unit cells; a- is

the electron—phonon interaction parameter; 2)]’P2(fl. p(~) =

[2oiK(1

—

s

K

is the chain elasticity coefficient,

(8.52)

A.S. Davydov, Theoretical investigation of high-temperature superconductivity

283

The function (8.50) satisfies the equation + /i2//i~2) + U(~ [i/l/1/it + (h2I2m~)(82//i~~ 1)+ U(~2)]~t’j,,( ‘1’

~2’ t) =

0,

with the eigenvalue 2 + 11k~V+ h2k~/m~. IfbS(V) = If9(0) + m5V The amplitude functions cP( ~) involved in (8.50) are determined by the nonlinear equation

(8.53)

[d2!d~ +4gI2(~,)+ If

9(0)/2J~]~(~) =0,

(8.54)

where g is the dimensionless electron—phonon interaction parameter whose value is given by 2/2K(1 — s2)J g= a

5.

(8.55)

Equation (8.54) at small densities of quasiparticles (gL ~ 1) has a solution normalized in one period 1’( fl = V~7~sech( ge/a),

(8.56)

with the value (8.57)

2J~. = —g

To the state (8.56) corresponds a chain deformation energy, W= ~K(1+

s2)

J

p2(e)

~

= ~g2J

~

5

(8.58)

~.

So when there is no magnetic field, a bisoliton (together with the local deformation) transfers an amount of energy given by 2k~j2m E(V)

= If(V) + W=

h

2 + kFV11

—

5+ m5V

D

(8.59)

0,

where

(8.60)

2J~

D0 = ~g is the pairing energy of two quasiparticles with the formation of a bisoliton at rest. In a constant magnetic field of strength B, some bisolitons, at a certain field value, can decay into free quasiparticles in the triplet spin state x 1 with the wave numbers k1 = k + kF and k2 = k In these states they are described by the wave function —

hId =

(iv’2!al) sin[k(

~ ~2)]x1 exp{i[k(z1 —

normalized by the large period al (>>aL).

+

z2)

—

Ifdt/h]}

,

kF.

(8.61)

284

AS. Davydov, Theoretical investigation of high-temperature superconductivity

We assume the magnetic field directed along axis z of the chain to act on these particles. The operator of this interaction has the form H1~1= —~(a1~ + a25)B,

p.~= e11/cm~.

Then the wave function (8.61) should satisfy the equation 2/2m~)(/i2//i~ + /i2//i~~) ,.tB(crlZ + a [ill /1/it + (11 2Z)B]h/Id —

=

0.

(8.62)

The energy of two free quasiparticles in the magnetic field measured from the Fermi level has the value ~‘d=11k/mZ+2,~tBB+hkFImZ.

In the into magnetic field the total wave decayed free quasiparticles, can befunction writtenofinthe thesystem form in which some part (1 ~P’(B)ah,bS(B)+V1—a2hfrd(B),

(8.63) 2) of bisolitons,

—

a

a2~1.

Vi

(8.64)

Since a part of the bisolitons given by a2 decayed in the system, the bisoliton deformation potential well has decreased and eq. (8.54) is replaced by [d2/d~2+4ga2~~(~)

+

—

Ifp(a)12J

5]’1a(fl =0.

(8.65)

Thus, the presence of a magnetic field in the system will affect indirectly the bisolitons due to the charge in the local deformations. The solution (8.65) normalized to unity has the form 7~sech(a2g~), (8.66) = a\/~ with the value If~(a)= —2g2a2J 5. In this case the deformation energy (8.58) also reduces to 2a6(1+ s2)/(1 ~2) W= ~J~g Therefore, the bisoliton energy (8.53) is transformed as follows: —

2+ 2J 2a6 + W(a). 5+ m2V 5g The total energy of the system containing a part a2 of bisolitons, measured from the Fermi energy, has the value EbS(a)

112k~Im

11kFV

(8.67)

—

E~(V)=m~V2—DØa6+211k~Va2—(i—a2)2j~B.

Ea(V)

=

EbS(a)

+

(1

—

(8.68)

At zero velocity it is reduced to E(0)

=

2(a2

—

1)p,~B

—

D

6. 0a

(8.69)

A. S. Davydov, Theoretical investigation of high-temperature superconductivity

285

The dependence of the ratio En (0) ID0 on a2 is shown in fig. 8.3 by the curves 1, 2, 3 and 4 for values B/Bcr equal to 0,0.5, 1.0 and 1.5, respectively. The field B~~ =2D0I2~ =

m5cD0I2eh

(8.70)

is directed along the b axis of the crystal. It follows from fig. 8.3 that when B < Bcr2 (curves 1 and 2) the value 2 = 1 corresponds to stable states, provided there are no decayed Cooper pairs (bisolitons). For B> Bcr2 (curve 4) all Cooper pairs are broken. These stable states are separated by a weak energy barrier. So, due to the paramagnetic effect, the Cooper pairs are broken in the fields that exceed the critical value (8.70) at which the energy of particle interaction with a magnetic field is compared with the pairing energy, i.e., when (8.71)

2/.LBBcr2 = D0.

In ceramic superconductors with pairing energy D0 10 meV (this corresponds to T~—90 K), we get from (8.71) the value Bcr2 100 T which is close to the observed one in YBa2Cu3O7 [255,256, 258]. When solitons move with velocity V along the z axis according to eq. (8.68), the requirement of Cooper pair stability in a magnetic field itof follows strengththat B, parallel to the z axis,ofisareduced satisfying theis 2(11kFV + p~B) < D~,hence the critical velocity bisolitonto condensate condition determined by the equality —

1’cr

= (D 0

—

2~B)/211kF,

B

< Bcr = D0I2p~.

(8.72)

The velocity decreases linearly with increasing magnetic field strength and vanishes at the critical value of the magnetic field strength Bcr2• There is no superconducting current in magnetic fields B > Bcr2.

1.0 E ~ o)

Do 2. Curves 1, 2, 3 and 4 correspond to values of BIB~,equal to 0, 0.5, 1.0 and 1.5, respectively. Fig. 8.3. The E~(0)/D0dependence on a

286

AS. Davydov, Theoretical investigation of high-temperature superconductivity

8.3.2. The diamagnetic effect To study the diamagnetic effect, it is necessary to take into account the fact that the electron motion in a superconductor in a constant magnetic field is of helical character. Therefore when examining this motion one should give up the quasi-one-dimensional model. To take into account the large anisotropy of ceramic superconductors, we shall model them by a system of parallel quasi one-dimensional chains and consider the possible motion of quasiparticles, involved in bisolitons, transverse to a system of parallel chains. Due to the great anisotropy of crystals, the longitudinal and transverse motion of quasi-particles is characterized by different effective masses. For instance, according to experimental data [254,260], the effective mass of quasiparticles in an yttrium crystal when they move along the chains (axis z) is much less (by nearly 40—70 times) than their effective mass when they move transversely. Therefore it will further be assumed that the inequality m~,m~>>m~

(8.73)

is satisfied. We study only bisoliton states moving along the chains in a magnetic field of strength B, parallel to the z axis. Such a field affects only the transverse motion of the quasiparticles. Hence it suffices to consider the two-dimensional problem of the motion of quasiparticles in the (x, y) plane which is perpendicular to the magnetic field. A quantum theory of charged particle motion in a constant magnetic field was first developed by Landau in 1930 [262]. The projection of the charged particle trajectory onto the plane perpendicular to the magnetic field is a circle along which it moves with cyclic (Larmor) frequency ~B’ proportional to the magnetic field strength =

eB/Mc.

(8.74)

Here M is the cyclotron mass determined, in our case, through the effective transverse masses m~and m~by the equality M = yrn~m.~ m.

(8.75)

In accordance with quantum theory, a brief account of which will be given in section 8.3.4, this rotational movement is quantized and the minimum energy is determined by E~0,= ~11[2B= heB/2Mc.

(8.76)

It is proportional to the magnetic field strength. In weak magnetic fields B this rotational motion decreases the pairing energy D0 in the absence of a magnetic field, D0(B)

= D0—

heB/2Mc.

(8.77)

This energy equals zero when the critical field is B~~2 = 2McD0/eh.

(8.78)

A.S. Davydov, Theoretical investigation of high-temperature superconductivity

287

Comparing the above value with the critical magnetic field of the Cooper pair break-down by the paramagnetic effect (8.70), we see that B~~=2(4MIm~)B~2 ~ B~2.

(8.79)

Thus, the pairing energy and, hence, the critical temperature of a superconducting transition in a magnetic field directed along the conducting chains first slowly decreases with increasing magnetic field strength (diamagnetic effect), then falls sharply to zero (paramagnetic effect) when B =

B ~.

8.3.3. Influence of paramagnetic impurities on superconductivity

The occurrence of magnetic impurities in ordinary isotropic superconductors suppresses superconductivity. A different picture is observed in anisotropic superconducting oxide materials. For instance, the 3~(10.6 replacement of nonmagnetic yttrium~),ions in yttrium by the magnetic ions p.s), Mo3~(10.6p~), Er~(3.6 Tm3~(7.6 ~), superconductors Yb3~(4.5~) changes insignificantly (by Dy 5—10 K) the critical temperature T~,while maintaining the other superconducting properties. When substituting nonmagnetic yttrium ions by magnetic ones with spin s~(in units 11), an additional interaction between quasi-particles with spin ~a, (a, is the Pauli matrix) and the magnetic field which is generated by the impurity ions, arises in the system. The spin—spin interaction operator can be written as ~

=

J

~

—

0s1(a-1

+

cr2),

(8.80)

where 10 is the mean value of the exchange integral between magnetic impurities and unpaired (in triplet spin states) quasi-particles of the unit cells distributed along the conduction chains (axis z). Averaging Hint over the homogeneous distribution of magnetic impurities and spin states, one can introduce the effective magnetic field Beff as an approximation linear in the impurity concentration v, by

means of the equality Beff

= vA,

A

~

Jo(~

(8.81)

Having incorporated the effective magnetic field Beff, we reduce the problem of studying the influence of magnetic impurities on superconductivity to the problem of the influence of an external magnetic field which was considered already in the previous sections. Replacing in eq. (8.77) of the preceding section the value of the external magnetic field strength by Bett, we get the expression determining the pairing energy dependence (and consequently T~)on small concentrations of magnetic impurity ions due to the diamagnetic effect D0(v) =

D0

—

vheAI2Mc.

(8.82)

When the effective magnetic field strength attains, with increasing concentration of magnetic impurities, the critical value (Betf)Cr determined by eq. (8.70), i.e., when the equality =

D0(v)cm5/2e11

D0cm5!2e11

is valid, Cooper pairs decay due to the paramagnetic effect.

(8.83)

288

AS. Davydov, Theoretical investigation of high-temperature superconductivity

So with increasing concentration of magnetic impurities within the limits v

~ ~

D

0(v), and therefore the critical temperature T~of the superconducting transition, is weakly dependent on i.’. This dependence is confirmed by experimental data indicated at the beginning of this section. However, at concentrations ~‘> i~ the superconductivity vanishes stepwise. The dependence of T~[proportional to D0(i.’)] on the concentration v of magnetic impurities is illustrated qualitatively in fig. 8.4a. The results of experimental measurements of T~as a function of composition in the compound (Y1_~Se)0

(Ba1~Sr)06CuO~ [258]are given in fig. 8.4b. Comparing figs. 8.4a and 8.4b, we see that a bisoliton theory describes qualitatively the dependence observed experimentally. Figure 8.5a illustrates the dependence of the critical temperature on the concentration of Ce atoms in the superconductor Lai_rCeAl (T~= 3.3 K) belonging to the type of superconductors with heavy fermions. Here a sharp decrease in T~at the critical concentration is also observed [259]. Thus, the influence of magnetic impurities on superconductivity of high-temperature superconductors, according to the bisoliton model, differs essentially from the predictions of the BCS theory. As was shown by Abrikosov and Gor’kov [261], in the BCS theory, as the concentration of magnetic impurities increases, the critical temperature decreases monotonically vanishing at some value v = This dependence is expressed by the law T~(~) = T~(0) 1rJ~v/4kBEF, —

~~‘cr=

(8.84)

4kBTCEF/11~JO,

where J0 is the exchange integral between magnetic impurity and the spin of a conduction electron averaged by the atomic volume. The qualitatively anologous dependence of T~on the concentration of magnetic atoms U in a Thi_rUr compound was indicated in Maple’s review [259], see fig. 8.6. 8.3.4.

Magnetic flux quantization in layered superconductors

The quantum motion of an electron in a constant magnetic field has been studied by Landau [262]. Here we present a simple theory of magnetic flux quantization in layered superconductors. Let us

100 80

g~~

T~,K

Theory ‘

(a)

~cr

0.0

0.2

0.4

0.6

(b)

Fig. 8.4. (a) The theoretical dependence of the D0(v)1D0 ratio on v, the concentration of magnetic impurities. (b) The dependence of T~on the concentration of magnetic ions in the (Y1_~Se)04(Ba1_~5ç)06CuO~compounds 1258].

AS. Davydov, Theoretical investigation of high-temperature superconductivity

TC lx)

1.0

1.0

r1

~

Ic

CIX! T~

o__.__o___o..

0.5

T~=3.3K

°

0

289

1~0=1.36K 0.5

O\

Jo

00 0

0.4

0.2

0.6

0.8

0.0

O..~,

0 Fig. 8.5. The dependence of T~ on the concentration of the Ce atoms in the La, ~Ce~AIcompounds 1259].

0.1

0.2

0.3

Fig. 8.6. The dependence of T~on the concentration atoms U in the Th,_~U~ compounds [259].

—~-x

v of magnetic

assume that the magnetic field is directed along the direction b (axis z) in the superconducting plane (a, b) of a layered superconductor. We will propose that the effective masses of current-conducting quasiparticles satisfy the inequality (8.73). If the magnetic field B = V x A is described by the vector potential A the latter can be written as

(— ~yB, ~xB, 0).

A

(8.85)

Projection of the quasiparticle motion trajectory onto the plane (x, y) perpendicular to the field is characterized by a closed curve. It moves along this curve with cyclic (Larmor) frequency ~B proportional to the magnetic field strength QB=eB/Mc.

(8.86)

Here e is a quasiparticle charge, c is the light velocity, M is the effective cyclotron mass, M=ym~m~>>m~.

(8.87)

Taking account of the Coulomb repulsion, the two quasi-particles of a bisoliton move near two centres displaced relative to each other by a small value. In the magnetic field their transverse motion onto the plane (x, y) can be described by the Hamiltonian 2+

1 i~~1,2 [(P~

H1

= (2M)

1+ ~

A~)

(~,+ ~ Ar)].

(8.88)

Expressing the magnetic field B through the cyclic frequency 118 one can transform this Hamiltonian, for each quasi-particle, to the form H

2+ 1~= (2M)~’[(j3~,— ~My1f18)

(ji,,,

+

~Mx

2].

(8.89)

1118)

Both particles move synchronized in the magnetic field, therefore the index i drops out. The Hamiltonian (8.89) characterizes the state of motion of a two-dimensional harmonic oscillator. The eigenvalues and functions of this operator were investigated by Johnson and Lippman in 1949 [263](see also Davydov [264]).

290

.4 .S. Davydov, Theoretical investigation of high-temperature superconductivity

It has been shown that the motion of a quasiparticle is characterized by two integrals of motion: the energy E1 and projection of the angular momentum, L~,onto the axis z. Their values are determined by the quantum numbers ii and rn by the equalities p0,i

E111Q8(v+~),

L511m,

(8.90)

m=0, ±1

(8.91)

When a particle moves in a homogeneus magnetic field the absolute value of its velocity is conserved. Therefore, using a quasiclassical description one can replace, in the 3~ operator H1, the projections of the —+ M dx/dt, ji,, —+ M dyldt. With

momentum operators the projections of the classical momenta, j this transformation thebyoperator H +

2

E1 =

M[(dx/dt

—

1 corresponds to the energy, (dyldt + ~xQB)2].

(8.92)

~y~B)

Consideration of the quantum character of the motion is reduced to the requirement that the equality (dx/dt— ~yflB)2+(dy/dt+

~x11 2—hQ 8) 8(~+fl/M

(8.93)

be satisfied. This equality is satisfied by the values 2cos11 2sinui x=(h(2~+1)/M118)” 8t, y=(11(2v+1)/Mu18)” 8t. Thus, with a fixed value of

i’,

each quasi-particle involved in a bisoliton moves in a closed trajectory

with cyclic frequency on a circle radius 2, ofv0,i R~[h(2~+1)IM118]”

The minimum radius R 0 = V11/M128

(8.94)

corresponds to the value r’ = 0. Consequently, with ii = 0, the area described by two quasiparticles in the plane, perpendicular to the magnetic field, is determined by S=

rrR~= rrh/Mu18.

(8.95)

Having substituted in this equality the cyclic frequency value of (8.86) we find the explicit expressions for the magnetic flux permeating the closed orbit ~P0=BS= lThc/e.

(8.96)

The magnetic 2. flux (8.96) coincides with the universal unit of the magnetic flux I~ 2.07 x i0~Oe cm

A.S. Davydov, Theoretical investigation of high-temperature superconductivity

291

8.4. Meissner effect in the bisoliton model of superconductivity

The theory of the Meissner effect in high-temperature superconducting ceramics was developed by Brizhik [267]. Here we shall present the main results of these investigations. Just as was done in studying the diamagnetic effect in section 8.3.2, when describing Meissner effect it is necessary to take into account the helical character of the quasiparticle motion in a constant magnetic field. Because of very strong anisotropy, superconducting ceramics could be modeled by the system of parallel quasi-one-dimensional chains aligned parallel to the b-axis of the crystal with a possible movement of the quasiparticles bound in a bisoliton state in directions perpendicular to the system of parallel chains. Due to the strong crystal anisotropy, the longitudinal and transverse motion of the quasiparticles is characterized by different effective masses [see eq. (8.73)]. As it was shown in section 8.3, bisolitons in a strongly anisotropic crystal are not broken by the magnetic field if its intensity does not exceed the critical value, Bcr = g2J6 I 3~. Here .1,, is the exchange

interaction along the chain, g is a dimensionless nonlinearity parameter, p.1~is the Bohr magneton. In the presence of a constant external magnetic field less than the critical one, bisolitons move in the crystal along belical orbits perpendicular to the magnetic field, creating, at low temperatures, a nondamped circular current which in its turn creates a magnetic moment in the sample that com-

pensates the external magnetic field, i.e., the Meissner effect takes place. To describe this effect one needs to study bisoliton dynamics in the crystal in the external subcritical magnetic field. As was shown in section 8.3.1 a condensate of bisolitons moving with constant velocity V = hklmb in

a system of parallel chains aligned along the x-axis (b-axis of the crystal) is characterized by a single wave junction which in the frame of one bisoliton period, aL, has the form, ~Iç(x1,x2, t) =

V’2~~(~1, t)45(~2,t) cos[(k

Here the amplitude functions

+

kF)(~l--

~2)]

exp{i[k(~1 +

~2)

—

~E~tIh]}

.

(8.97)

4~,describe

the motion of quasiparticles with opposite spins, ~ is the energy of quasiparticles in the potential field 24,~(~)/K(1s2) ~2 41 U(~)= —2o a is the electron—phonon coupling constant, ~ = x is the quasiparticle coordinate in a moving frame, s = VIV, 3, 1’o is the sound velocity. The functions 4~(4,) are expressed in terms of periodic elliptic Jacobi functions, dn(u, q), —

— Vt

cb5(~,q) = E

(q)~/~7~dn[g~1/aE(q), q],

(8.98)

with modulus q determined by the space period aL~of bisolitons from the equation gL~= 25~(q)E( q), with X( q) and E( q) being total elliptic integrals of the first and the second kinds. The function (8.97) is normalized to unity per one period aLt. One can introduce variables R~=~(~1+~2), r=~1—~2,

(8.99)

292

A. S. Davydov, Theoretical investigation of high-temperature superconductivity

that characterize centre of mass (Ri) and relative (r) motion of quasiparticles in a bisoliton. For magnetic fields less than the critical one, we shall be interested in the centre of mass motion of the bisoliton only and replace wave function (8.97) by the function hi~(R~, t) =

cP(R~,t)

exp[iq(R5, t)].

(8.100)

The real function 1 and phase ~ are determined by the following expressions:

f

aL’, / 2

qi(R~,t)

=

~ X

~(R5, t)

=

4~(R+ ~r, t)q5~(R ~r, t) cos kFr dr, —

—aL~/2

2kR~ ~ —

The total wave function of the bisoliton condensate in a three-dimensional anisotropic crystal can be represented then in the form of a product, ~Ofld(R, t) = ~1c(R~,t)~P±(R~, R~,t) ,

(8.101)

with ~c(R~, t) describing bisoliton motion along chains, and 1111(R,

R~,t)

=

\/~iiexp{2i[k~R~ + k~R5]

— i~f’1tI11}

(8.102) =

11~(k~/m~ + k~Im~),

describing bisoliton motion in the (y, z) plane perpendicular to the chains. Here n1 is density of chains in this plane. Due to the crystal anisotropy, the quasiparticle effective masses satisfy the inequality, my, m~~ m1

(8.103)

.

It turns out to be useful to write down the function (8.101) in the form, ~Ofld(R,

t)

=

PCOfld(R, t) exp[i~(R, t)].

(8.104)

Here ~‘COfld(R, t) is a real amplitude and ~‘(R,t)=2k.R—(~~+

~‘1)t/11

(8.105)

is the bisoliton condensate coherent phase, its gradient projection onto the x-axis determines the velocity of the bisoliton as it moves along the chain, V~= (11/2m~)/icoI/R~= 11kIm~. The derivatives /icoI/iR~and /itpI/iR~determine the velocities of the bisoliton as it moves in the corresponding directions perpendicular to chains.

AS. Davydov, Theoretical investigation of high-temperature superconductivity

293

The superconducting current density in the crystal in the external magnetic field, B = V X A, is expressed by the condensate wave function (8.104) J~(R,t) = (heImP)P~Ofld(R,t)[/iq~(R, t)I/R~— (2eIhc)A~].

(8.106)

Taking into account the fact that the magnetic field in the volume of the sample changes only slightly over distances with period aL~and distances between the chains, one can substitute the function ~ COfld(R, t) by its mean value n, equal to the bisoliton volumetric density determined by the equality n = nbSnl, with ~b, being the bisoliton density in a chain. In this approximation expression (8.106) transforms to J~(R,t)

=

(heIm~)n[/itp(R, t)//iR~ (2e/hc)A~]. —

Next taking the curl of both sides of the above equality, one can get an equation rota J(R, t)

=

, —(2e2/cm~)nB~

which together with the Maxwell equation, rot B

(8.107) =

(4lTIc)J(R, t), gives London’s equation generalized

to the anisotropic case, (8.108)

= ~c2B~.

Here the value =

(8.109)

c2m~I81Te2n

stands for the square of the magnetic field penetration depth. Indeed, in the particular case when the magnetic field is directed parallel to the chains and the sample surface lies in the xy plane, it follows

from eq. (8.108) and the Maxwell equation, div B = 0, that the magnetic field damps exponentially into the superconductor, penetrating it to a depth —As only, B~(R 5) = B0

exp(—R5/A~).

(8.110)

At distances R5 > A~the magnetic field in the sample is absent (Meissner effect). According to eq. (8.109), the anisotropy of the magnetic field penetration depth is determined by the quasiparticle effective mass anisotropy =

Vm~Im~, , v, ~

=

1,2,3.

(8.111)

In an experimental study of the magnetic field penetration depth in ceramic superconductors the asymmetry of the penetration depths in the ab-plane of the crystal was not observed due to the twin phenomenon. For that reason the relation (8.111) should be replaced, for the sake of comparison with experimental results, by the relation = mclmab,

(8.112)

294

AS. Davydov, Theoretical investigation of high-temperature superconductivity

where A~band mab are mean values of the square of the magnetic field penetration depth and of the effective mass in the ab-plane, respectively. Experimental investigations of a YBa2Cu3O7 single crystal gave the values A,, 1740—2700 A, ‘~“ab= 335—525 A [268], A,, = 4200 A, Aab = 1460 A [269]. Experimental values for the magnetic field penetration depth in the crystal La2_~Sr~CuO4_~ (x = 0.15) are A,, = 2 x io~A, Aab = 1400 A [270]. These results coincide qualitatively with estimates of quasiparticle effective mass anisotropy, which, for the compounds YBa2Cu3O7 and La2_~Sr~CuO4_~, e.g., attains the value 10—65 [268,270].

9. One-particle excitations in a bisoliton model of superconductivity of ceramic oxides 9.1.

Introduction

The optical and tunnel properties of superconducting systems are generated by peculiarities of a one-particle energy spectrum of charge carriers. The creation of Cooper pairs is accompanied by the appearance of a dielectric gap in the spectrum of quasiparticle states and the change in their dispersion law. For low-temperature metal isotropic superconductors these peculiarities are well described by BCS theory (see section 3.1). The high-temperature superconductor models suggested previously which differ from BCS theory only by the nature of pairing fail to describe satisfactorily the one-particle spectrum. The change in the one-particle spectrum near the Fermi energy was treated on the basis of a bisoliton model by Ermakov and Kruchinin [271]. Here we study the peculiarities of the one-particle spectrum of all bands of the allowed states of quasiparticles using the nonlinear bisoliton model of high-temperature superconductivity of strongly anisotropic ceramic superconductors which are modelled by a system of parallel quasi-one-dimensional filaments. According to the bisoliton theory presented in sections 8.1—8.3, the Fermi particles in quasi-onedimensional chains in states with energy close to the Fermi energy with opposite wave numbers and spins are paired due to the energy of local chain deformation. Such bisolitons produce in a chain a periodic structure of local chain deformations a stable condensate described by a single wave function. Periodically distributed bisolitons quasiparticle pairs in a singlet spin state and local deformations related to them can move, under the influence of the external field, along the chain with the same velocity. Below we study a bisoliton model of superconductivity in terms of one-particle states which was developed by Davydov and Ermakov [272]. —

—

—

9.2. Deformation field in a superconducting state As was shown earlier, the transition to a superconducting state generated by the appearance of a bisoliton condensate is accompanied by the formation of a periodic field of local deformations. This field generated by quasiparticle pairs in a system of coordinates ~that move with velocity U is described by the expression 2(~), (9.1) U(~)= —4Jg~

AS. Davydov, Theoretical investigation of high-temperature superconductivity

295

where ~ = (x — Vt) Ia, a is the distance between equilibrium positions of molecules in an undeformed chain. The constants J and g are determined, respectively, by the expressions, J = h2I2ma~,

g=

u212JK(1 — ~2)

52 = V2IV~41.

Here g is a dimensionless parameter of local deformational interaction, K is the elasticity coefficient of the chain. V 0 is the sound velocity, m is the effective mass of Fermi particles in the conduction band of current carriers. The function cP( ~) describes the space distribution of bisolitons. It is periodic, with period aL (L is the linear concentration of bisolitons) and normalized in each period L by the condition )=~+L).

(9.2)

This function is expressed through the Jacobi elliptic functions dn(u, q) by means of the equalities 1(q)\I~7~dn(u, g), u = g~/E(q), =

(9.3)

E~

where the elliptic function modulus q is determined from the equation (9.4)

gL =2~’(q)E(q), containing the total elliptic integrals 5l~(q) and E( q) of the first and second order, respectively.

The condensate of Cooper pairs of quasiparticles in a singlet spin state is described in each period aL by the wave function ‘P 0(~ ~2) = v

I(~1)cP(~2) cos[akF(~l —

Here

~2)]

exp{i[Ka(~1 +

E~ is the energy of quasiparticle pairs in the potential field are represented by eq. (9.3) and satisfy the nonlinear equation 2/d~2+ U( ~) + E [J d 5] P( ~) = 0,

~2)

U(

— E~tI11]}.

~),

(9.5)

the amplitude functions P( ~) (9.6)

E5 is a bisoliton energy measured from the bottom of the condition band of free quasiparticles. The potential energy U( ~) has the form of eq. (9.1). The wave number K characterizes the motion of a

bisoliton centre. If N0 is the number of unit cells and N1 is the number of pairs of free quasiparticles with effective mass m in the conduction band, which arise in the crystal under doping, then at low energy they occupy all states with wave numbers k kF where kF is a wave number corresponding to the Fermi energy, kF=ITN1IaNO, N1
(9.7)

The energy E~(V)of two quasiparticles in the state (9.5) is expressed through the eigenvalue E5 of

296

A. S. Davydov, Theoretical investigation of high-temperature superconductivity

eq. (9.6) by the equality 2 + 112k~/m+ 11VkF. mV The amplitude functions cP( ~) characterize the space distribution of quasiparticles ~, inside each period aL. In metals, according to the BCS theory, only a small part of the quasiparticles joins to form Cooper

E~(V)=

E

5 +

pairs. We shall assume that in our model the number of bisolitons is less than (or equal to) the number of quasiparticle pairs in the conduction band, i.e., it will be assumed that L satisfies L

N 0/N1.

9.3.

One-particle states in superconducting condensate

The equation for the wave function h’( ~) of a quasiparticle with definite spin orientation, involved in the bisoliton and characterizing the bisoliton state in the coordinate system that moves together with a bisoliton condensate, proves to be the Schrödinger equation with a periodic potential U( ~) determined by eq. (9.1), 2Id~2+U(~)+E~)hi(~)=0. (9.8) (Jd Here

E

is the quasiparticle energy measured from the Fermi energy EF of free quasiparticles. For T = 0 the energy EF coincides with the level of the chemical potential To determine the one-particle spectrum at a fixed potential U( ~) we represent the function h’( ~) in eq. (9.8) in the form of expansion in plane waves 2

= E + EF,

E

~.

112

exp(ika~),

(9.9)

= N0

orthonormalized in a chain of length

=

1

N 1L

=

~

and, consequently,

(9.10)

u(k)p,,,(~). 2 = 1.

TheUsing one-particle function is normalized in space and therefore ~k~u(k)~ eq. (9.10) we represent eq. (9.8) in theN0, form [E+

EF—

e(k)]u(k)+~ U(k— k 1)u(k1)=0,

(9.11)

where 2J, e(k)=ak

f

1/2

U(k-k 1)~

(9.12)

(U~k(~)d~.

—1/2

Making use of the property of periodicity of the potential we write eq. (9.12) as

U(

~)

U(k

+

L) (see ref. [264], section 2.2)

AS. Davydov, Theoretical investigation of high-temperalure superconductivity

297

mN

0I2L

U(k — k1)

=

N~’

~

m—N012L

V,,(k — k1) exp[iLam(k

— k1)].

(9.13)

Here L12

V,,(k — ki)~V(Qn)

=

J

~

U(~)

exp(—iQ~~) d~.

—L12

The function (9.13) is nonzero if Q~=k—k1=21TnIaL, n=1,2

(9.14)

Therefore for the states with wave vectors k satisfying the approximate equality I k~ 1k — Q~~ i.e., at the values k~kn=~Qn=~nIaL,

(9.15)

corresponding to the band boundaries in k space, the system of equations (9.11) is approximated by the system of two equations for each value of n, [E—EF— s(k)]u(k)+V~u(k—Q~)0,

(9.16) 2(k)+u2(k—Q~)=1. Vflu(k)+[E—EF—e(k—Qfl)]u(k—Qfl)=0, u If all N 1 pairs of quasiparticles do form a bisoliton condensate then the period of the bisoliton distribution L and, hence, the potential field period U( ~) are determined by L

=

N01N1.

(9.17)

Taking into account the value (9.14) of the Fermi energy wave number kF we get from eq. (9.15) 2nkF.

Qn = 2irnlaL = Taking this value into

(9.18)

account the integrals V, involved in the system of equations (9.16) will be

determined by L/2

J

V~~V(Q~)~j U(~)exp(2ikFan~)d~.

(9.19)

—L/2

I kI

We find from eqs. (9.16) that the energy spectrum of quasiparticles near the band boundaries =

~

Q,,

=

nkF is determined by the expression

E~”~(k)= ~(s(k)+ s(k— Qfl)—2EF±{[s(k— Q,,)— s(k)]2+4IV~I2}”2).

(9.20)

298

A. S. Davydov, Theoretical investigation of high-temperature superconductivity

In particular, on the band boundaries =

~

e(nkF) ~

1VJ.

(9.21)

Thus, in the one-particle spectrum in a bisoliton superconductivity model near the states with = nkF there arise dielectric gaps of width equal to 2IV~(.States with k < kF are occupied; those with I kI > kF are free. It follows from eqs. (9.16) and (9.20) that the functions u(k) and v(k) u(k Q~)are determined by kJ

—

2 k — 1 ( u~( ) - ~1

{[r(k)

-

-

2kF) 2kF)]2 + 4IVI2}1/2)~

e(k) — e(k — e(k -

(9.22) 1 ( v~(k)= 2 ~ 2

+

{[e(k)

r(k)—s(k—2kF) — e(k — 2kF)]2 + 4IVI2}~2

For the energies close to that of Fermi following equality is valid: r(k) (9.20) and (9.22) take the form 2

u (k)

=

1 / 2 ~

—

{[e(k)

EF

—

EF

EF — r(k

I

at n = 1 and e(k) — EFI z.1 where ~1 IV~ ~ EF the — 2kF). Taking into account the above equality, eqs.

2 — 1 / v (k) — 2 ~

e(k) — EF — EF]2 + ~2}1/2)~

+

{[s(k)

e(k) — EF — EF]2 + g}1/2).

(9.23)

The wave function of the relevant one-particle states will have a form for the low bands given by =

N~”2exp(ika~)[u~(k) + v~(k)e2’~”~”],

(9.24a)

and for the upper bands 2 exp(ika~)[u~(k) 1” In particular on the band boundaries k h’n,upper(~)

=

=

N

N~”2cos(kFa~n),

—

u~(k)e2”~~”] .

= ~Q,, =

h”n,upper

=

kF

(9.24b)

we get

N~”2i sin(kFa4n).

(9.25)

The energy spectrum of quasiparticle states is illustrated in fig. 9.1. At n = 1 the low band is occupied, the upper one is empty. The bands with n> 1 are free. 9.4. The energy gaps

The upper and lower allowed bands are separated by dielectric gaps. Their values are determined by 4, = IV~I.Using eqs. (9.21) and (9.3) and performing the integration we obtain — “

8rrak~nJ LE(q) sinh[2ak~E(q)X 1(q)Ig]

(9.26)

A.S. Davydov, Theoretical investigation of high-temperature superconductivity

299

J ~

--~~

-2Kp —KF

0

K

I 2Kp

Fig. 9.1. One-particle energy spectrum in the bisoliton model of superconductivity for

where 5~( q) = ~I~(\I 1 — q2). At small concentrations of bisolitons (q asymptotic values: .‘K~( q) irI2 and E( q) 1. In this case

v

=

1.

1) the elliptic integrals have the

2

— 8lTkFanJ — L sinh(7rk~anIg)

(9. 7)

To the value n = 1 there corresponds the main energy gap separating, at T 0, the occupied quasiparticle states and the free ones. The values n = 2, 3,. . correspond to the forbidden bands of the

.

states in a quasiparticle spectrum of unoccupied states. Their availability reflects the periodic distribu-

tion of bisolitons in a condensate. Such states can be observed when studying the quantum transitions of quasiparticles from the occupied states to the free ones. The gap width (9.27) is dependent on the period L of the bisoliton distribution in a condensate. When writing all above-mentioned expressions it was assumed that all quasiparticles determining the Fermi level (9.7) participate in the pairing. In this case the period of the bisoliton distribution in a condensate is given by L = N 0IN1 = 1TIak~(see fig. 9.1). However, just as in BCS theory, it is possible for only some part N11v of the quasiparticles (v = 1, 2,. .) to participate in pairing. In this case the space period in the bisoliton condensate (at fixed N1)

. increases

L~= vN0IN1 = vlrIakF.

(9.28)

Consequently eq. (4.2) becomes 4,,,

=

8k~anJIirvsinh(1Tk~anIg).

(9.29)

With increasing n the gap width increases exponentially. To clarify the physical meaning of the number v = 1, 2,... one should consider that taking into account eq. (4.3), the cosine argument akF( ~ — ~2) in the wave function of the bisoliton condensate

300

AS. Davydov, Theoretical investigation of high-temperature superconductivity

(9.5) in the region of the bisoliton

~

—

~2I< L,, changes in the range ~

Thus, the number z.’ — 1 characterizes the number of nodes in the cosine function. Hence there follows the integral value of v. The widest gap (4.4) corresponds to the value r’ = 1 at which all quasiparticles participate in generating the bisoliton condensate. Using the value of energy state density in a quasi-one-dimensional system near the Fermi energy,

N(EF) = -~- ~

ô[e(k)

—

EFI

=

2

(9.30)

k

ITaF

we can transform the expression for the width of the main gap 4,,, as follows:

4, =2D~Isinh(—1IA),

(9.31)

where A

= 2gJN(E~),

D,,

=

(9.32)

4ak~JIi’ii~.

(9.33)

Expression (9.31) coincides formally with the formula that determines the energy gap width in the spectrum of quasiparticle states of BCS theory for A 40.5. The value D,, is equivalent to the Debye energy in BCS theory. In a bisoliton theory the preexponential factor (9.33) depends nonmonotonically on the concentration N1 of pairs of quasiparticles in4,p)max the crystal. As was shown Ermakovthis andvalue Kruchinin [271], atthea corresponds to kF =by 2gIa~r.To corresponds constant value i., the maximum gap ( optimal number of carrier pairs, N~~t = 2N

2. (9.34) 0gIir For a smaller or larger number of carriers the value of the gap width is less than the optimal one, which equals (~i,,,)max= l6g2JIir3v.

(9.35)

The dependence of the maximum width (9.35) on the number r’ = 1, 2,.. indicates the possibility to observe various widths 4,,, in the same sample at places having a different surface structure. This conclusion is proved by experimental measurements of gap widths carried out by Yanson and others [273,274] using the method of microscopic spectroscopy of superconductors La 1 Sr02CuO4. The experiments revealed two gaps with the values 24, = 13.3 meV and 24, = 26meV. The multiplicity of these values agrees with the conclusions of the theory if one assumes that the cases v = 2 and v = 1 are realizable. When studying the electron tunneling in thin films of the superconductor LaSrCuO Kapitulnik, Naito et a!. [275] discovered three energy gaps with values of 24 equal to 20, 30 and 66 meV. Probably, they are described by the formula (9.35) when v assumes the values v = 3. 2 and 1. .

AS. Davydov, Theoretical investigation of high-temperature superconductivity

References [1] H. Kamerlingh Onnes, Commun. Phys. Lab. Univ. Leiden Nos. 119, 120, 122 (1911). [2] J. File and R.G. Mills, Phys. Rev. Lett. 10 (1963) 93. [3] B.T. Matthias, Phys. Rev. 97 (1955) 74. [4] B.T. Matthias, Phys. Today 24 (1971) 23. [5] J.K. Huim, J.E. Kunzler and B.T. Matthias, Phys. Today 34 (1981) No. 1, 34. [6] D. Kelvin, Science 156 (1967) 645. [7] W. Meissner and R. Ochsenfeld, Naturwissenschaften 21(1933) 787. [8] Ci. Gorter and H.B.G. Casimir, Phys. Z. 35(1934) 963. [9] H. London and F. London, Physica 2 (1935) 341. [10]V.L. Ginzburg and L.D. Landau, Zh. Eksp. Teor. Fiz. 20 (1950) 1064. [11] L.P. Gor’kov, Zh. Eksp. Teor. Fiz. 34 (1959) 735. [12] A.B. Pippard, Proc. R. Soc. London Ser. A 216 (1953) 547. [13]A.A. Abnkosov, Zh. Eksp. Teor. Fiz. 32 (1957) 1442. [14] F. London, Superfluids, Vol. 1 (Wiley, New York, 1950). [15] W.H. Kilner, L.H. Roth and S.H. Yutler, Phys. Rev. A 133 (1964)1226. [16]F. London, Proc. R. Soc. London Ser. A 152 (1935) 24. [17] H. Fröhlich, Phys. Rev. 79 (1950) 845. [18] E. Maxwell, Phys. Rev. 78 (1950) 477. [19] C.A. Reynolds and B. Serin, Phys. Rev. 78(1950) 487. [20] H. Fröhlich, Proc. R. Soc. London Ser. A 215 (1952) 291. [21] H. Fröhiich, Proc. R. Soc. London Ser. A 223 (1954) 296. [22]J. Bardeen, Solid State Comniun. 13 (1973) 367. [23] MR. Schafroth, Helv. Phys. Acta 24 (1951) 655. [24] A.B. Migdal, Zh. Eksp. Teor. Fiz. 34 (1958) 1438; 37 (1960) 249. [25]L.N. Cooper, Phys. Rev. 104 (1956) 1189. [26]J. Bardeen, L.N. Cooper and JR. Schrieffer, Phys. Rev. 108 (1957) 1175. [27] JR. Schrieffer, Phys. Rev. 106 (1957) 162. [28]JR. Schrieffer, Theory of Superconductivity (Nauka, Moscow, 1970). [29] MR. Schafroth, ST. Butler and J.M. Blatt, Helv. Phys. Acta 30 (1957) 93. [30] N.N. Bogoliubov, Zh. Eksp. Teor. Phys. 34 (1958) 58. [31]N.N. Bogoliubov, Nuovo Cimento 7 (1958) 794. [32] J.G. Valatin, Nuovo Cimento 7 (1958) 843. [33] N.N. Bogoliubov, J. Phys. 9 (1947) 23. [34]N.N. Bogoliubov, VV. Tolmachev and D.V. Shirkov, A New Method in the Theory of Superconductivity (AN SSSR, Moscow, 1958). [35] G.M. Eliashberg, Zh. Eksp. Teor. Phys. 38 (1960) 966. [36] Y. Nambu, Phys. Rev. 117 (1960) 648. [37] A.B. Migdal, Zh. Eksp. Teor. Phys. 38 (1960) 966. [38] L.P. Gor’kov, Zh. Eksp. Teor. Phys. 34 (1958) 735. [39] Ph.B. Allen, Phys. Rev. B3 (1971) 306. [40] Ph.B. Allen, Phys. Rev. Bil (1975) 2693. [41] A.E. Karakozov, E.G. Maksimov and CA. Mashkov, Zh. Eksp. Teor. Phys. 688 (1975) 1937. [42] P. Morel and P.W. Anderson, Phys. Rev. 125 (1962) 1263. [43] W.L. McMillan, Phys. Rev. 167 (1968) 331. [44] R.E. Glover and M. Tinkhan, Phys. Rev. 108 (1957) 243. [451J. Giaever and K. Meyerle, Phys. Rev. 122 (1961) 1101. [46]P. Tikman, Phys. Rev. 165 (1968) 588. [47] I. Giaever, Phys. Rev. Lett. 5 (1960) 147. [48]I. Giaever, Phys. Rev. Lett. 5 (1960) 464. [49]J. Sutten and P. Townsend, Proc. LT8 (1963) 182. [50] G.F. Hardy and J.K. Huim, Phys. Rev. 89 (1953) 884; 93(1954) 1004. [51]L.R. Testradi, Rev. Mod. Phys. 47 (1975) 637. [52]B.W. Batherman and CS. Burrett, Phys. Rev. Lett. 13 (1964) 390. [53] CA. Nemnonov, E.Z. Kurmaev, B.A. Fomichev et al., Izv. Akad. Nauk SSSR Ser. Fiz. 36 (1972) 317. [54]SW. McKnight, B.L. Bean and S. Perkowitz, Phys. Rev. B 19 (1979)1437.

301

302

A. S. Davydov, Theoretical investigation of high-temperature superconductivity

[55] SW. McKnight, S. Perkowitz, D.B. Tanner and L.S. Testradi, Phys. Rev. B19 (1979) 5689. [56] L.Y.L. Shen, Phys. Rev. Lett. 29 (1972) 1082. [57] K.E. Kihlstrom and T.H. Geballe, Phys. Rev. B 24 (1981) 4101. [58]J. Kwo and T.H. Geballe, Phys. Rev. B 23(1981) 3230. [59] RH. Willen, Solid State Commun. 7(1969) 837. [60] E. Ehrenfreund, Phys. Rev. B 4 (1971) 2906. [61] B.T. Geilikman and V.Z. Kresin, Phys. Lett. A 40 (1972) 123; J. Low Temp. Phys. 18 (1975) 241. [62] W. Weger, Rev. Mod. Phys. 36 (1964)175. [63] S. Barisic and PG. De Gennes, Solid State Commun. 6 (1968) 281. [64]J. Labbe and J. Friedel, J. Phys. Radium 27 (1966) 153, 303. [65]J. Labbe, Phys. Rev. 158 (1967) 647. [66]P.C. Hohenberg, Phys. Rev. 158 (1967) 383; TN. Rice, Phys. Rev. 140 (1965) 1189. [67] R.A. Ferrell, Phys. Rev. Lett. 16 (1964) 330. [68] I.E. Dzyaloshinskii and El. Katz, Zh. Eksp. Teor. Phys. 55(1968) 238. [69]J. Labbe, S. Barisic and J. Friedel, Phys. Rev. Lett. 19 (1967) 1030. [70]Yu.A. Izyumov, Fiz. Met. Metalloved. 35(1973) 687. [71]L.P. Gor’kov, Pis’ma Zh. Eksp. Teor. Fiz. 17 (1973) 525. [72] L.P. Gor’kov, Zh. Eksp. Teor. Fiz. 65(1973)1558. [73] Al. Golomashkin and G.P. Motulevich, Pis’ma Zh. Eksp. Teor. Fiz. 17 (1973) 114. [74] L.R. Testradi and T.B. Bateman, Phys. Rev. 154 (1967) 402. [75] CC. Yu and P.W. Anderson, Phys. Rev. B 29 (1984) 6165. [76] R.G. Lye and EM. Logothetis, Phys. Rev. 147 (1966) 622. [77]J.B. Conclin and D.J. Silversmith, J. Quant. Chem. 11(1968) 243. [78] E. Clementi and AD. McLeon, Phys. Rev. 133 (1964) 419. [791l.A. Britov, E.Z. Kurmaev and CA. Nemnonov, Fiz. Met. Metalloved. 26 (1968) 366. [80] E.Z. Kurmaev, G.P. Shveikin and S.A. Nemnonov, Phys. Status Solidi b 60 (1973) K65. [81] B.T. Matthias, Phys. Rev. 92 (1953) 874. [82] HR. Zeller, Phys. Rev. B 5 (1972) 1813. [83]M.H. Van Maaren, G.M. Schaeffer and F.K. Lotgering, Phys. Lett. 25A (1967) 238. [84] w. Weber, Phys. Rev. B 8 (1973) 5082. [85]HG. Smith and W. Glasser, Phys. Rev. Lett. 25 (1970) 1611. [86]T. Skoskiewicz, Phys. Status Solidi a 11(1972) K123. [87] P. Hertel, Z. Phys. B 268 (1974) 111. [88] K.H. Bennemann and J.W. Garland, Z. Phys. B 260 (1974) 367. [89] D.A. Papaconstantopoulos and B.M. Klein, Phys. Rev. Lett. 35(1975)110. [90] C.B. Freidberg, A.F. Rex and J. Ruvalds, Phys. Rev. B 19 (1979) 5694. [91] w. Buckel and B. Stritzker, Phys. Lett. A 43(1973) 403. [92] G. HeIn and B. Stritzker, AppI. Phys. 7 (1975) 239. [93] W. Buckel and B. Stritzker, Phys. Lett. A 43 (1973) 403. [94] F. Steglich, J. Aarts, CD. Bredi et al., Phys. Rev. Lett. 43(1979)1892. [95] HR. Ott and H. Rudigier, Phys. Rev. Lett. 50 (1983) 1595. [96] G.R. Stewart, Z. Fisk, JO. Willis and J.L. Smith, Phys. Rev. Lett. 52 (1984) 679. [971G.R. Stewart, Rev. Mod. Phys. 56 (1984) 755. [98]I.E. Alexeevski and DI. Kchomski, Usp. Fiz. Nauk 147 (1985) 767. [99] VV. Moshchalkov and NB. Brandt, Usp. Fiz. Nauk 149 (1986) 585. [100] MB. Maple, J.W. Chen, SE. Lambert et al., Phys. Rev. Lett. 54 (1985) 477. [101] F. Steglich, J. Aarts, CD. Bredl et al., Phys. Rev. Lett. 43(1979)1892; J. Magn. Magn. Mater. 52 (1985) 54. [102] G.R. Stewart, Z. Fisk, JO. Willis and J.L. Smith, Phys. Rev. Lett. 52 (1984) 697. [103]HE. Alexeevski and B.N. Narozhnii, Pis’ma Zh. Eksp. Teor. Fiz. 40 (1984) 421. [104]J. Kondo, Prog. Theor. Phys. 32 (1964) 37. [105]G. Gruner and A. Zawadowski, Solid State Commun. 11(1972) 663. [106]A.A. Abrikosov, Zh. Eksp. Teor. Fiz. 48 (1965) 930; Physics 2 (1965)5. [107]H. Suhi, Phys. Rev. 515A (1965) 138. [108]NB. Brandt and VV. Moshchalkov, Adv. Phys. 33 (1984) 373. [109]F.G. Aliev, NB. Brandt, V.V. Moshchalkov and SM. Chudinov, Sol. State Commun. 47 (1983) 693. [110]V.V. Moshchalkov, J. Magn. and Magn. Mater. 47—48 (1985) 7. [111]V.V. Moshchalkov and F.G. Aliev, Z. Phys. B 58 (1985) 58.

A.S. Davydov, Theoretical investigation of high-temperature superconductivity

303

[112]DI. Kchomski, Usp. Fiz. Nauk 129 (1979) 443. [113]P.W. Anderson, Phys. Rev. 124 (1961) 41. [114]S.H. Lin, Phys. Rev. Lett. 58 (1987) 2706. [115]R.M. Martin, Phys. Rev. Lett. 48 (1982) 362. [116]G.M. Eliashberg, Pis’ma Zh. Eksp. Teor. Fiz. 45 (1988) 28. [117]A.B. Migdal, Zh. Eksp. Teor. Fiz. 34 (1958) 34. [118]HR. Ott, Phys. Rev. Lett. 52 (1984)1551. [119]G.E. Volovik and L.P. Gor’kov, Pis’ma Zh. Eksp. Teor. Fiz. 39 (1984)550. [120]L. Taillefer and G.G. Lonzarich, Phys. Rev. Lett. 60 (1988)1570. [121]O.T. Valls and Z. Tesanovic, Phys. Rev. Lett. 53 (1984)1497. [122]G.E. Volovik and L.P. Gor’kov, Pis’ma Zh. Eksp. Teor. Fiz. 39 (1984) 550. [123]CM. Varma, Bull. Am. Phys. Soc. 39 (1984)1497. [124]P.W. Anderson, Phys. Rev. B 30 (1984)1549. [125]T. Oguchi, A.J. Freeman and G.W. Crabtree, Phys. Lett. A 117 (1986) 428. [126]E.W. Fenton, Solid State Commun. 60 (1986) 351. [127]J.E. Hirsch, Phys. Rev. B 34 (1986) 8190. [128]K. Miyake, S. Schmitt-Rinku and C.M. Vanna, Phys. Rev. B 34 (1984) 6554. [129]A.W. Sleight, J.L. Gilison and P.E. Bierstedt, Solid State Commun. 17 (1975) 27. [130]T.D. Thanh, A. Koma and S. Tanaka, Appi. Phys. 22 (1980) 205. [131]A.M. Gabovich and D.P. Moiseev, Usp. Fiz. Nauk 150 (1986) 599. [132] 5. Tajima, K. Kitazawa and S. Tanaka, Solid State Commun. 47 (1983) 659. [133] B. Batlogg, Physica B 126 (1984) 275. [134]G. Bilbro and W.L. McMillan, Phys. Rev. B 14 (1976) 1887. [135]T. Tani, T. Itoh and S. Tanaka, J. Phys. Soc. Jpn. 49 (1980) Suppl. A. 309. [136] V.L. Ginzburg and D.A. Kirzhnits, eds, Problems in High-Temperature Superconductivity (Nauka, Moscow, 1977). [137] AM. Gabovich and D.P. Moiseev, Fiz. Tverd. Tela 137 (1975) 269. [138]AM. Gabovich, E.A. Pashitski and S.K. Uvarova, Fiz. Nizk. Temp. 1 (1975) 984. [139]J. Yhm, ML. Cohen and S.B. Tuan, Phys. Rev. B 23 (1981) 3258. [140]J. Ruvalds, Adv. Phys. 30 (1981) 677. [141]AS. Alexandrov, Doctoral dissertation, MIPhN, Moscow (1984). [142]1.0. Kulick and AG. Pedan, Fiz. Nizk. Temp. 9 (1983) 256. [143] 1.0. Kulick, Usp. Fiz. Nauk 145 (1985) 155. [144] AS. Alexandrov and J. Ranninger, Phys. Rev. B 23(1981)1796; 24 (1988)1164. [145]L.F. Mattheiss and DR. Hamann, Phys. Rev. B 26 (1982) 2686; 28 (1983) 4227. [146] D. Yoshioka and H. Fukuyama, J. Phys. Soc. Jpn. 54 (1985) 2996. [147]P.W. Anderson, Phys. Rev. Lett. 34 (1975) 953. [148]TM. Rice and L. Sneddon, Phys. Rev. Lett. 47 (1981) 680; Physica B 107 (1981) 661. [149] S.P. Ionov and L.A. Manakova, preprint IAE AN SSSR 4462/9 (Moscow, 1987). [150]S.P. Ionov, G.V. Ionova and L.A. Manakova, Izv. Akad. Nauk SSSR Ser. Fiz. 45 (1981) 589. [151]5. Sugai, Phys. Rev. B 35 (1987) 3621. [152] K. Kitazawa, Solid State Commun. 52 (1984) 459. [153]J.G. Bednorz and K.A. Muller, Z. Phys. B 64 (1986)189. [154]C.W. Chu, PH. Hor, R.L. Meng et al., Phys. Rev. Lett. 58 (1987) 405. [155]H. Takagi, SI. Uchida, K. Kitazawa and S. Tanaka, Jap. J. Appi. Phys. 26 (1987) L123. [156]J.G. Bednorz, M. Takashige and K.A. Muller, Europhys. Lett. 3 (1987) 379. [157] M.K. Wu, JR. Ashburn, C.J. Torng et al., Phys. Rev. Lett. 58 (1987) 908. [158]PH. Hor, R.L. Meng, Y.Q. Wang et al., Phys. Rev. Lett. 58 (1987) 1891. [159]J.G. Bednorz and K.A. Muller, Rev. Mod. Phys. 60 (1988) 585. [160] C.W. Chu, P.H. Hor, R.L. Meng et at., Science 235 (1987) 567. [161]H.S. Haberstein and L.F. Mattheiss, Phys. Rev. Lett. 60 (1988) 585, 1661. [162]A. Khurana, Phys. Today 40 (1987) No. 4, 17. [163]RB. Van Dover, Ri. Cava, B. Batlogg and E.A. Rietman, Phys. Rev. B 35 (1987) 5337. [164]NP. Ong, Phys. Rev. B 35 (1987) 3807. [165] D.J. Esteve, J.M. Martinis, C. Urbina et al., Europhys. Lett. 3 (1987) 1237. [166] CE. Gough, M.S. Colciough, EM. Forgan, R.G. Jordan, M. Keene, CM. Muirhead, A.I.M. Rae, N. Thomas, J.S. Abell and S. Sutton, Nature 326 (1987) 855. [167]T.R. Dinger, T.K. Worthington, W.J. Gallagher et al., Phys. Rev. Lett. 58 (1987) 2697. [168]V.K. Vlasko-Vlasov, MV. Indenbom, Yu.A. Osipyan, Pis’ma Zh. Eksp. Teor. Fiz. 47 (1987) 312.

304

AS. Davydov, Theoretical investigation of high-temperature superconductivity

[169]T.K. Worthington, W.J. Gallagher and T.R. Dinger, Phys. Rev. Lett. 59 (1987) 1160. [170] SW. Tozer, A.W. Kleinsasser, I. Renny et al., Phys. Rev. Lett. 59 (1987) 1768. (171] B.I. Verkin, V.M. Dmitriev, V.P. Seminozhenko et at., Fiz. Nizk. Temp. 14 (1988) 1676. [172]A.P. Voronov, V.M. Dmitriev, MB. Kosmyna, S.F. Prokopovich and V.P. Seminozhenko, Physica C 153—155 (1988) p. 1675. [173]L.P. Gor’kov and NB. Kopnin, Usp. Fiz. Nauk 156 (1988) 118. [174]A. Kapitulnic, MR. Beasley, C. Castellani and C. Di Castro, Phys. Rev. B 37 (1988) 537. [175]U. Walter, MS. Sherwin, A. Stacy, R.L. Richards and A. Zetti, Phys. Rev. B 35 (1987) 532. [176] P.E. Sulewski, A.J. Sievers, S.E. Russek et at., Phys. Rev. B 35 (1987) 5330. [177]ZR. Schlesinger, J.G. Bednorz, R.L. Greene and K.A. Muller, Phys. Rev. B35 (1987) 5334. [178]S. Pan, K.W. Ng, AL. Lozane et al., Phys. Rev. B 35(1987) 7220. [179]JR. Kirtley, CC. Tsuei, SI. Park et al., Phys. Rev. B 35(1987) 7216. [180] Z.R. Schlesinger, R.T. Collins, D.L. Kaiser and F. Holtzberg, Phys. Rev. Lett. 59 (1987) 1958. [181]ZR. Schlesinger, R.T. Collins, MW. Shafer and EM. Eagler, Phys. Rev. B 36 (1987) 5275. [182]J. Moreland, J.W. Tekin and L.F. Goodrider, Phys. Rev. B 35(1987) 8856. [183] P. Van Bentum, L. Van de Lempt, L. Schreurs et al., Phys. Rev. B 36 (1987) 843. [184]JR. Kirtley, R.T. Collins and Z. Schlesinger et al., Phys. Rev. B 35(1987) 8846. [185]B. Batlogg, R.J. Cava, A. Jayaraman et at., Phys. Rev. Lett. 58 (1987) 2333. [186] K.J. Leary, H.C. zur Loye and SW. Keller et at., Phys. Rev. Lett. 59 (1987) 1236. [187]L.C. Bourne, M.F. Crommie, A. Zetti et at., Phys. Rev. Lett. 58 (1987) 2337. [188] D.E. Morris, R.M. Kuroda, AG. Markeiz et at., Phys. Rev. B 37 (1988) 5936. [189] C. Thomson, H. Mattausch, M. Bauer, W. Banhofer, R. Liu, L. Genzel and M. Cardona, Solid State Commun. 67 (1988) 1069. [190] H.C. zur Loye, K.J. Leary, SW. Keller et at., Science 238 (1987) 4833. [191] H. Katayama-Yoshida, T. Hirooka, A. Oyamada et at., Physica C 156 (1988) 481. [192]WA. Little, Phys. Rev. A 134 (1964)1416. [193]V.L. Ginzburg, Phys. Lett. 13 (1964) 101. [194] Al. Golovashkin, Usp. Fiz. Nauk 148 (1986) 363. (195] Yu.B. Gaididei and V.M. Loktev, DokI. Akad. Nauk Ukr. SSR. N. 1 (1988) 56. [196]Yu.B. Gaididei and V.M. Loktev, Phys. Status Solidi b 147 (1988) 307. [197] P.W. Anderson, Science 235 (1987) 1196. [198] G. Baskaran, Z. Zou and P.W. Anderson, Solid State Commun. 63(1987) 973. [199] S. Kivelson, D. Rokhsar and J. Sethna, Phys. Rev. B 35 (1987) 8865. [200] P.W. Anderson, G. Baskaran, Z. Zou and T. Hsu, Phys. Rev. Lett. 58 (1987) 2790. [201]A. Rudsenstein, P. Hirschfeld and J. Appel, Phys. Rev. B 36 (1987) 887. [202]P.W. Anderson and Z. Zou, Phys. Rev. Lett. 60 (1988) 132. [203] F.C. Zhang, C. Gross, TM. Rice and H. Shiba, Sci. Technol. Ser. I (1988) 36. [204] I. Pomeranchuk, Zh. Eksp. Teor. Fiz. 11(1941) 226. [205] I.E. Dzyaloshinskii, AM. Polyakov and P.B. Wiegmann, Phys. Lett. A 127 (1988) 112. [206] W.P. Su, JR. Schrieffer and A.J. Heeger, Phys. Rev. Lett. 42 (1979) 1698; Phys. Rev. B 22 (1980) 2099. [207]S.A. Brazovskii, Zh. Eksp. Teor. Fiz. 18 (1980) 678. [208] L. Pauling, The Nature of the Chemical Bond (Cornell Univ. Press, Ithaca, N.Y., 1960). [209]Al. Achiezer and IA. Achiezer, Zh. Eksp. Teor. Fiz. 43 (1962) 2208. [210]Guo Y., J.M. Langlois and WA. Goddard, Science 239 (1988) 896. [211] K. Hida, J. Phys. Soc. Jpn. 57 (1988) 1544. [212] P. Prelovsek, Phys. Lett. A 126 (1988) 287. [213] E.Y. Loh, T. Martin and P. Pretovsek, Phys. Rev. B 38 (1988) 2494. [214] V.Z. Kresin, Phys. Rev. B 35(1987) 8716. [215]J. Ruvalds, Phys. Rev. B 35 (1987) 8872. [2161T. Ando, A. Fowter and F. Stern, Rev. Mod. Phys. 54 (1982) 237. [2171E.A. Pashitski and B.L. Vinetskii, Pis’ma Zh. Exp. Teor. Fiz. Suppl., 46 (1987) 124. [218] N.M. Plakida and V.L. Aksenov, preprint Joint Institute for Nuclear Research, P-17-87-498 (Dubna, 1987). [219]JR. Hardy and J.W. Flocken, Phys. Rev. Lett. 60 (1988) 2191. [220]Yu.V. Kopaev, Zh. Eksp. Teor. Fiz. 58 (1970) 1012. [221] Yu.V. Kopaev and Al. Rusinov, Phys. Lett. A 121 (1987) 300. [222]A.A. Gorbatsevich, V.Ph. Elesin and Yu.V. Kopaev, Phys. Lett. A 125 (1987) 149. [223]R.O. Zaitsev and V.A. Ivanov, Pis’ma Zh. Eksp. Teor. Fiz. Suppt. 46 (1987) 140. [224]N.M. Plakida and IV. Stasyuk, preprint Joint Institute for Nuclear Research, E-17-88-96 (Dubna, 1988). [225]L. Landau, Phys. Z. Sovjetunion 3 (1933) 664. [226]SI. Pekar, Studies in the Electron Theory of Crystals (Gostekhizdat, Moscow, 1951).

A.S. Davydov, Theoretical investigation of high-temperature superconductivity

305

[227]H. Fröhlich, Adv. Phys., 3 (1954) 325. [228]SV. Tyablikov, Zh. Eksp. Teor. Fiz. 21(1954) 377; 22 (1952) 512. [229]T. Holstein, Ann. Phys. 8 (1959) 343. [230]Yu.A. Firsov, Polarons (Nauka, Moscow, 1975); Zh. Eksp. Teor. Fiz. 32 (1980)30. [231]MI. Klinger, Phys. Status Solidi, b 2 (1962)1062; 11(1965) 499. [232]AS. Alexandrov, JR. Ranninger and S. Polashkewicz, Phys. Rev. B 33(1986) 4526. [233]AS. Alexandrov, Pis’ma Zh. Eksp. Teor. Fiz. Suppl. 46 (1987) 128; Zh. Eksp. Teor. Fiz. 95 (1989) 296; Physica C 158 (1989) 273. [234]S.A. Brazovskii and N.N. Kirova, Pis’ma Zh. Eksp. Teor. Fiz. 33 (1981) 6. [235]AS. Atexandrov and ‘A’. Kabanov, Fiz. Tverd. Tela, 28 (1986) 1129. [236]Yu.B. Gaididei and V.M. Loktev, DokI. Ukr. SSR, N. 11(1987) 40. [237]AS. Davydov, Biology and Quantum Mechanics (Pergamon, Oxford, 1982). [238]AS. Davydov, Solitons in Bioenergetics (Naukova Dumka, Kiev 1986). AS. Davydov, Solitons in Molecular Systems (Naukova Dumka, Kiev, 1984); (Reidel, Dordrecht, 1985) p. 319; 2nd Ed. (Naukova Dumka, Kiev, 1988). [239]AS. Davydov and A.V. Zototariuk, Phys. Status Solidi b 115 (1985) 115. [240]AS. Davydov and A.V. Zolotariuk, Phys. Lett. A 94 (1983) 49. [241]AS. Davydov and A.V. Zolotanuk, Phys. Scr. 30 (1984) 426. [242]A.S. Davydov and V.Z. Enolskii, Phys. Status Solidi b 143 (1987) 167. [243]L.S. Brizhik and AS. Davydov, Fiz. Nizk. Temp. 10 (1984) 748. [244]V.E. Zakharov and A.B. Shabat, Zh. Eksp. Teor. Fiz. 61(1971)118. [245]W.G.J. Hol, PT. Van Duijnen and U.J.C. Berendsen, Nature 273 (1978) 443. [246]I.E. Turner, YE. Anderson and K. Fox, Phys. Rev. 174 (1968) 81. [2471 P.C. Hinkle and R.E. McCarthy, Sci. Am. 238 (1978)104. [248]AS. Davydov and V.N. Ermakov, Phys. Status Solidi b 148 (1988)305. [249]A.S. Davydov and VN. Ermakov, Physica D 12 (1988) 218. [250]A.S. Davydov and VN. Ermakov, DokI. Akad. Nauk SSSR, Ser. Fiz. 302 (1988) 601. [251]A.S. Davydov and G.M. Pestryakov, preprint ITP-81-112R, Institute of Theoretical Physics, Kiev (1981). [252]A. Khurana, Phys. Today 40 (1987) No. 4, 17. [253]T.E. Dinger, Phys. Rev. Lett. 58 (1987) 2637. [254]T.P. Orlando, HA. Felin, S. Foner et at., Phys. Rev. B 35 (1987) 5347. [255]V. Hidaka, Y. Enomoto, M. Sazuki et at., Jpn. J. Appl. Phys. 26 (1987) L726. [256]V. lye, T. Tomagai, H. Takya and Takei, Jpn. J. Appl. Phys. 26 (1987) L1057. [257]L.S. Brizhik, AS. Davydov and V.N. Ermakov, Phys. Status Solidi b 157 (1990) 417. [258]A.K. Oota, Y. Kiyoshi and T. Hioki, Jpn. J. AppI. Phys. 26 (1987) L1356. [259]MB. Maple, Phys. Today 39 (1986) No. 3, 72. [260]A. Junod, A. Bezinge, T. Graf et al., Europhys. Lett. 4 (1987) 247. [261]A.A. Abrikosov and L.P. Gor’kov, Zh. Eksp. Teor. Phys. 39 (1960)1781. [262] L. Landau, Z. Phys. 64 (1930) 629. [263] M.H. Johnson and BA. Lippman, Phys. Rev. 79 (1949) 828. [264] AS. Davydov, Quantum Mechanics, 2nd Ed. (Pergamon Oxford, 1976) pp. 26—29. [265] V.L. Pokrovskii, Pis’ma Zh. Eksp. Teor. Fiz. 47 (1988) 539. [266] P.L. Gammel and D.J. Bishop, Phys. Rev. Lett. 59 (1987) 2592. [267] L.S. Brizhik, Meissner Effect in the Bisoliton Model of Superconductivity, Preprint ITP-89-31 E, Inst. Theor. Phys., Kiev (1989). [268] A. Junod, B. Bezinge and T. Craft, Europhys. Lett. 4 (1987) no. 1, 247. [269] V. Isikawa, K. Mon. R. Kabayashi, Jpn. J. AppI. Phys. 27 (1968) 403. [270]T.P. Orlando, K.A. Pelin and S. Foner, Phys. Rev. B 35(1987)5347. [271] V.N. Ermakov and S.P. Kruchinin, Phys. Stat. Solidi b 156 (1989) 333. [272] AS. Davydov and V.N. Ermakov, One-Particle Excitation in a Bisoliton Model of Superconductivity of Ceramic Oxides, Preprint ITP-89-68 E, Inst. Theor. Physics, Kiev (1989). [273]1K. Yanson, L.F. Rybalchenko, VV. Fisun, N.L. Bobrov, MA. Obolenskii, Yu.D. Tretyakov, AR. Kaul and I.E. Graboi, Fiz. Nizk. Temp. 12 (1987) 557. [274]I.K. Yanson, L.F. Rybalchenko, VV. Fisun, N.L. Bobrov, NA. Obolenskii, Yu.D. Tretyakov, A.P. Kaut and I.E. Graboi. Fiz. Nizk. Temp. 15 (1989) 803. [275]N. Naito, D.P.E. Smith, M.D. Kirk, B. Oh, MR. Hahn, K. Char, D.B. Mitzi, J.Z. Sun, D.J. Webb, M.R. Beasley, 0. Fisher, T.H. Geballe, RH. Hammond, A. Kapitulnik and CF. Quate, Phys. Rev. B 35 (1987) 7228. [276]A.I. Akimenko, N.M. Ponomarenko, V.A. Gudimenko, 1K. Yanson, P. Samuely and P. Kus, Fiz. Nizk. Temp. 15(1989)1242. [277]AS. Davydov, Foundation of quasi-one-dimensional model of oxides, to be published in Ukr. Fiz. Zh. and Phys. Status Solidi b.

306

AS. Davydov, Theoretical investigation of high-temperature superconductivity

Note added in proof 1. Akimenko et al. [276],using the method of contact spectroscopy for a study of quasiparticle states in SmBa2Cu3O7_~crystals (T~= 90K), observed a series of energy gaps at 4.2 K, as shown the table (in meV). exp.

23.7

20.9

17.8

16.7

14.6

13.2

theor. 144/v

24 6

20.6 7

18.1 8

18 9

14.4 10

13.1 11

The integral values of v determine, in our theory, the number (v — 1) of nodes of the cosine function in the wave function of the condensate of Cooper pairs (eq. 8.25). So, the discrete values of ii reflect the discreteness of the internal states of a bisoliton. For i.’ — 1 only a fraction 1 / v of all quasiparticles in the

conduction in the formation Cooper pairs. We recall that in the BCS theory only 4th part band of theparticipate quasiparticles form Cooper of pairs. 102. Most ceramic superconductors do not have a chain structure but a layered one. The relevance of the one-dimensional bisoliton model for those superconductors has been discussed by Davydov in ref. [277]. There it was shown that bisolitons created in a square (x, y) lattice have a quasi-one-dimensional nature. They are localized in small region of the variable x (or y) with constant phase along the perpendicular direction. The lowering of the dimensionality is stipulated by an increase of the role of nonlinearity in the system and a decrease of the role of zero vibration.