Enhancement of superconducting critical temperature due to metal-semiconductor transition

Volume 121, number 6

PHYSICS LETTERS A

4 May 1987

ENHANCEMENT OF SUPERCONDUCTING CRITICAL TEMPERATURE DUE TO METAL-SEMICONDUCTOR TRANSITION Yu.V. KOPAEV and A.!. RUSINOV P.N. Lebedev Physical Institute, Academy ofSciences ofthe USSR, Lenin Prospect 53, Moscow B-333, USSR Received 19 March 1987; accepted forpublication 20 March 1987

It is pointed out that there exists a qualitative agreement between the occurrence of high-temperature superconductivity in oxide systems (perovskite and layered perovskite structures) and the author’s earlier theoretical predictions of superconductivity in the model of metal—insulator transition (partly contained in the book: High-temperature superconductivity). Here a brief account ofthe published results is given and attention is paid to some properties of such superconductors.

Beginning with the well-known paper ofLittle [1], in the theory arises the problem of the connection between superconductivity and the structural transition of metal into insulator state for quasi-onedimensional systems. Also, a many superconducting layered compound exhibits lattice instabilities at a temperature TD (above the superconducting temperature T5) which cause the formation of an electronic energy gap at some parts of the Fermi surface. The source ofsuch transitions originates, as a rule, in a nesting of constant-energy surfaces. Structural instabilities with simultaneous formation of a dielectric gap at a part of the Fermi surface have also been observed some A-15 type transisuperconductors. Usually, the in influence of such tions on the superconducting critical temperature T~, and its increase, are discussed on the basis of the changes of phonon characteristics of the metal. It is a rather wide-spread opinion that such changes (dielectrisation) of the electronic spectrum reduce the critical temperature T~because of reducing the portion of electrons taking part in Cooper pairing, However, if the nesting ofisoenergysurfaces takes place near the Fermi energy, it turned out that a certam amount of filled electronic states is situated above the dielectric gap A~,which is formed on congruent parts of energy surfaces. In that case, the Fermi level~uis located in the conduction band, and the system behaves as a degenerate semiconductor (or 300

semimetal). It is not improbable that a transition temperature TD from a semiconducting to metallic phase exceeds the melting point ofthe crystal lattice, i.e. within the whole temperature range the system remains in a semiconducting state in which a manifestation of congruent parts of the energy surface in the high-temperature metallic phase (not observable experimentally) consists in a very weak dependence of the electron energy on its momentum along original congruent surfaces [2]. For example, the semiconducting superconductors SrTiO3, Pb~Sn1_~Teand Li~Ti2_~O4 have rather high T5 values versus very low 3), concentrations of the (n 10’ 9_ 1021 but just in that rangecarriers of concentrations the I/cm above-mentioned congruent states are filled. It has been experimentally observed that in the perovskite-structure compound BaPb, _~Bi~O [33] and in the layered-perovskite structures La2 _~M~CuO4 (M = Ba, Sr, Ca) [4—61there are metal—semiconductor transitions depending on the temperature and composition x, accompanied by a characteristic semiconducting behaviour ofthe resistivity above the superconducting critical temperature T’~. The formation of a dielectric gap on congruent parts ofan energy surface results in disappearance of a component of the electron velocity in the directions perpendicular to these parts of the surfaces. It means that the electron density ofstates N(E) has an

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E 1/2 type singularity near the bottom of the conduction band, exactly as in the case of a one-dimensional metal. The case of superconductivity in the situations when the Fermi level lies inside the conduction band with such a singularity of the density of states was investigated theoretically in refs. [7,81. It was found that there exists a real possibility for significant enhancement of the superconducting critical ternperatures T~in the region of low concentration of carriers as compared with ordinary metals with constant density of states near the Fermi level, N(E) = N( 0) = const. In order to demonstrate the physical mechanism of such an increase in T~,we turn back to the Cooper problem [9] and calculate the energy of the bound state oftwo electrons due to an attractive interaction A <0 in the presence of a Fermi sea. The bound state energy W is a solution of the equation 1 =:A~: k W— 2c(k)

,

(1)

where the attraction energy A was assumed constant in the energy range 0< e(k)
f E -4N(0) j de,

Cooper found a solution of eq. (1) in the form 2o I. W= ex [+ 2/N(0) IA I’ —1 (2) p Thus, in the weak-coupling limit Al N( 0) ‘~<1 the bound state energy W is exponentially small exp [ 2/N( 0)1 Al], as in the case of a two-dimensional well [10]. In the BCS model [11], the Fermi sea instability due to formation of a finite density of Cooper electron pairs gives a gap in the one-particle spectrum of the system, A 5(T=0)—~ exp[—l/N(0)l2l], which equalsthe superconducting critical temperature (A5( T=0) = 1 .76T5) in order ofmagnitude. In the case ofa metal with congruent parts ofenergy surfaces, due to the electron—hole pairing the electron spectrum__has a BCS-type [12]: 2 +A~,where ~(k) is the form spectrum in E(k)metallic = ,/c(k) parent phase. This spectrum correthe —

—

4 May 1987

sponds to an electron_density of states in the form N(E) = N( 0 )E/~,/E~ —A To obtain the energy of a Cooper pair in a dielectric phase, it is necessary to make the following change in eq. (1): e(k) —~E(k)—au, where jL is the chemical potential ofthe pair electrons in the conduction band (it=AD). In the weak coupling limit (1-+0 and W—~’0) the main contribution to the sum (1) is given by a range of small e. Writing the series expansion E(k) A~+ e 2/24D, one easily obtains the following result: ~ )~ 0 ~12A (3) ~.

~

—

—

~

L

~.

/J

D•

A more precise calculation gives for the numerical coefficient the value K( l/,.J~)= 1.85 instead of ~x = 1.57. Thus, the bound stateenergy Wofan electronic pair near the edge of the conduction band is proportional to ,~,2, as in the case of a one-dimensional potential well [10]. Comparison ofthe results (3) and (2) indicates that the formation of a dielectric gap may in principle be more favourable for Cooper pairing than in a normal metal. At finite concentration n ofelectrons (holes) in an n-type degenerate semiconductor, the problem of pairing ofthese carriers can be solved by analogy with the conventional BCS theory. However, for small n there is the question ofusing the BCS approximation when the interaction energy AN(E) is not small as compared with the Fermi energy of a degenerate semiconductor. Furthermore, if the cut-off energy Wph of the attractive interaction is larger than the dielectric gap AD, it is necessary to include in the description of a pairing state the contribution of electrons from the filled valence band. These difficulties can be easily avoided in a model that describes on an equal footing the dielectric and superconducting pairing in the original metallic state [7,8]. In the two-band approximation it was assumed that the energy surfaces e.(k) of the first (1) and second (2) bands are connected by the equation ~k’— ~k’+~ 4 ~ / I~. ( ) The parameter ~,u describes a small deviation (8~u<< I ci) of the energy surfaces from the nesting condition: ~1 = e2. In the isotropic case this parameter is proportional to the concentration of metallic electrons (holes) in the semiconducting state. the one-band model the role of bands 1 and 2 is For played by the parts of the Fermi surface that “nest” under —

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displacement by a constant wave vector Q. For a mathematical description of the phase transition from a metallic to a semiconducting state (due to structural transformation) and/or to a supercon-

tions were found in a simple analytic form. In the limit ofsmall concentration n (TF << 4~),at a temperature T= 0, the superconducting order parameter A5 is independent of n and coincides with

ducting state, the interband functions G12 and G21 (electron—hole pairing) and Gorkov functions F1, and F22 (Cooper pairing) are introduced along with the normal intraband functions G,1 and G22. These anomalous functions result from logarithmic singularities in the amplitudes of scattering electrons and holes from different bands (G12) and electrons in any of the two bands (F11 and F22). At the same time the theory contains, from the requirement of self-consistency, the interband anomalous functions F,2 and F21, in spite of the absence of a corresponding singularity in a scattering channel ofelectrons from different bands. The solution of the problem of combined dielectric and superconducting exact inin thethe same sense as that in the BCSpairings model is [11], and model of electron—hole pairing [12] in the limit ofweak coupling. A detailed theory ofthis problem is contained in refs. [13,7,8]. The theory permits taking into account fine details of mutual influences ofdielectric and superconducting phase transitions. In particular, the theory may be applied for a description of superconductivity in a semiconducting state with a

the two-electron bound state energy Wat the edge of the dielectric gap obtained above (eq. (3)). At finite concentration of carriers it is important to take into account in a self-consistent manner the interactions of Cooper pairs and the spread of the electrons in conduction bands in a finite energy range near the singularity of the density of states. The important case of interest corresponds to the range of concentration in which the superconducting order parameter A5( T= 0) or the critical temperature T~is less than the degeneration temperature TF in the semiconducting state (the latter is much less than the dielectric gap AD). The symmetric solution for A5 at T= 0 has 2 the/ form ~ 4ñ ~ nfb)~ (5)

very low density ofcarriers (the values the degen2/ADofmay be as eration temperature TF [nN(0)] 4s~or small as the superconducting order parameter temperature 7’s). A general criterion for the validity of the theory consists in the smallness of the dielectric (or superconducting) gap as compared with the Fermi energy of the original metallic phase. The contribution of valence-band electrons to superconducting pairing in this model gives rise to a renormalisation of the superconducting coupling constant due to spectrum dielectrisation. Depending on the phase relations between the superconducting order parameters A~,and A~2for different bands, only two cases are possible: A~,=A52 (symmetric case) and A5, = —A~2 (antisymmetric case j~2 0). A complete system of coupled equations at finite temperature for ofthe order parame4D~ A 4s,determination I = I A~ ter 5= I 2I, 4~2 and the “chemical potential” 8~uat a fixed value of carriers n was formulated in ref. [81. In some interesting cases solu302

ID

where ñ = 4N( 0) n, fl~,= ln(Af~/A~°~), A f~is the dielectric gap in the absence of superconductivity, and the superconducting order parameter A~°~ in the metallic phase is given by the BCS equation (2). In the derivation of eq. (5) it has been assumed that

p

0>> 1. critical temperature n) is proportional to theThe superconducting orderT5(parameter, A 5( T= 0, n) = 1.76 T~ ( n), as in the case of the BCS theory. The function T~ ( n) (or A 5( n)) at fixed values of 4ff) and A ~O) has a sharp maximum for concentration 4L0) 1 + IR 1n1/2(p + 8)1/2 (6) —

~

—

1)

4t’O

4t0

.“O

In the limit $0>> I we have / T \ / 4 \2 A~°~ ~~k)max ~e ln(A ~ /A~°~)) ~ ~ 1,

(7)

for ~ ~2Aco/lnALo/A1o U

k

U

Hence, the enhancement of the superconducting critical temperature T5 (and A~)may be rather large, which is caused by the specific behaviour of the elecironic density of states near the edge of the conduction band of the insulator with a small energy gap on the scale ofattractive interaction, that is, of the order of the Debye energy.

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It is necessary to mention that the antisymmetric solution for the superconducting state corresponds to lower values ofA5 and T5 than A ~O) and T~°~ i.e. dielectrisation leads to a reduction of T5. The region of enhancement of superconductivity in a plane (Ac”, A~°~) due to a weak dielectrisation of spectra was described in ref. [81. On the other hand, in the situation where a Fermi level happens to lie inside a dielectric gap of a transformed metal, the critical temperature T5 is reduced [81. This result was confirmed later in other works (see, e.g. ref. [14]), in which the authors assumed that after a structural transition associated with congruent parts of the energy surfaces, the Fermi level would lie inside the dielectric gap. Dielectrisation of these parts of the Fermi surface unambiguously causes a lowering of T5. For the BaPb1 _~Bi~O3 system the fact of the existence of nesting parts on the Fermi surface in the metallic phase results from calculation of the band structure [151. An important question is then whether or not a nesting exists in high-temperature superconductors of the type La2_~Sr (Ba, Ca)Cu04 We have not yet a positive answer to this question and can only guess that similarity ofcrystal structures ofthese two compounds is an indication of a similarity of their electronic spectra. It is well known that the tight-binding approximation gives congruent parts ofenergy surfaces for many crystal structures. The validity ofthis approximation follows from by theoxygen fact that octahedral of metal atoms atoms weakenssurrounding (loosens) the degree of overlapping cation orbitals. So, by variation of compositions x andy a fitting of the Fermi ~

4 May 1987

teristic phonon energy. From this point of view the excitonic mechanism of superconductivity [21 with large value of co. may be very effective in the model under consideration in spite of the fact that its role in the conventional BCS theory is less effective than the phonon mechanism. The mechanism ofenhancement ofT5 in the model ofrefs. [7,8] is sensitive to scattering of electrons on impurities due to spreading of the singularity of the density of states N(E). This scattering is however not important in the perovskite type system because of the high value of the dielectric permittivity ~ 10 2 1 0~.Moreover the substitution of one element for another (x 0) in perovskite structures favors the long-range order even with substantially different ionic radii of substituting and substituted metals. It also suppresses a spreading of the density of states. Such suppressing effect also takes place under oxygen vacancies ordering [161. The thermodynamics of the superconducting state under considerationis very much like that of the BCS state [11] (e.g. the ratio ofA5(T=0)/T5 is retained) except the case of a slightly doped semiconductor mentioned above. Landau expansion for the energy near the transition temperature T5 ( n) in zero magnetic field has the usual form: 2

~

T) =

,3~,,—

4

a( T)A5 + ~P4s

T~T5( n) The general formulas forthe coefficients a and $ the are 2/AD given in ref. a[19]. the be limit T5( n) =analytically: (2ñ) coefficients and 13Incan calculated a(T)=~~N(0) ln[T 5(n)/T]

surface to congruent parts of energy surfaces can be achieved. Replacement of La atoms by atoms with a smaller number of valence electrons leads to variation of band filling, but the latter is hampered evidently by the formation of a bound state of the electron (hole) on oxygen vacancies (y#0) below the Fermi level. The oxygen vacancies in Magneli’s vanadium oxide series play a similar role in the metal—semiconductor transition. As follows from (7) the values of A5 and T5 are proportional to 4~0) if the inequality Wph >> 40) is valid. Thus, the value of A6°~ in the phonon mechanism of superconductivity is limited to the charac-

n Ab°~

—

—

N= N( 0) 40)

13(T) =

7~(3)

_~

N(0)0 fl

07t

2~1-~ ~ ‘~2

(8)

5~fl/

This result will turn into the corresponding expressions in the BCS theory if we put the effective density of states N in the presence of an insulating gap instead of the density of states of the metallic parent phase. Hence the expression for the jump ofspecific 303

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heat coincides with that of the BCS theory with the density of states N, t~C-~N. In particular, the value ofA~does not depend on N. The relative value ofthe jump of specific heat L~C/C~ = 1.43 exactly coincides withthe BCS theory taking into account that at T? T5 the specific heat of doped semiconductors contains the renormalized 2NT, density of states N, C~(T) = ~x as well. We would like to point out some features of the superconducting state under consideration, which can be observed in BaPb 1 _~Bi~O3 and La2_~MeCuO4. The temperature and frequency dependence of a number of kinetic coefficients [17] may differ considerably from the corresponding ones in the BCS model, because the renormalization of the matrix elements of interaction with ultrasonics, electromagnetic field, nuclear spin, etc. is due to both superconducting and dielectric coherent factors but not only due to the first one. This is experimentally observed in a number of superconductors of A-l 5 type substances undergoing structural transition. The interaction ofinsulating and superconducting order parameter results in the existence of a low-frequency optically active collective mode [18] which was observed by means ofRaman scttering in NbSe2. An elementary excitation spectrum [13,81 differs considerably from the one in the model of ref. [11]. This fact can result in qualitative changes in tunneling characteristics of such superconductors. A Ginzburg—Landau type equation for A~~fl an electromagnetic field couples with the one for AD [19], thus making the electrodynamics ofthe superconductors under consideration rather specific. The peculiarities of the magnetic properties ofthe superconducting phase will be published elsewhere.

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References [1] W.A. Little, Phys. Rev. A 134 (1964) 1416. [2] B.A. Volkov and O.A. Pankratov, Zh. Eksp. Teor. Fiz. 75 (1978)Sleight, 1362. J.L. Gillson and P.E. Bierstedt, Solid State [3] A.W. Commun. 17 (1975) 27. [4] C.W. [5] J.G. Bednorz Chu, P.H. andHor, K.A.R.L. Muller, Meng, Z. Phys. L. Cao, B 64 Z.J.(1986) Huang 189. and Y.Q. Wang, Phys. Rev. Lett. 58 (1987) 405. [6] R.J. Cava, R.B. van Dover, B. Batlogg and E.A. Rietman, Phys. Rev. Lett. 58 (1987) 408. [7] Yu.V. Kopaev, Zh.W.D. Eksp. Teor. Fiz. (1970) D.C. Mattis and Langer, Phys.58Rev. Lett.1012; 25 (1970) 376. [8] A.I. Rusinov, Do Chan Kat and Yu.V. Kopaev, Zh. Eksp. Teor. Fiz. 65 (1973) 1984. [9] [10] [11]

L.N. Cooper, Phys. Rev. 104 (1956) 1189. L.D. Landau and E.M. Lifshitz, Quantum mechanics (Fizmatgiz, Mosow, 1963) p.192 (in Russian). J. Bardeen, L.N. Cooperand J.R. Sehrieffer, Phys.Rev. 106 (1957) 162. [12] L.V. Keldysh and Yu.V. Kopaev, Fiz. Tverd. Tela 6 (1964) 2791; P.A. Fedders and P.C. Martin, Phys. Rev. 143 (1966) 245. [13] V.L. Ginzburg and D.A. Kirzhnits, eds., High-temperature superconductivity (Consultants Bureau, New York, 1982) ch.5. [14] 0. Bilbro and W.L. McMillan, Phys. Rev. B 14(1976)1887. [15] L.F. Mattheise and D.R. Haman, Phys. Rev. B 28 (1983) 4227. [16] Yu.V. Kopaev and V.G. Mokerov, Dokl. Akad. Nauk SSSR 264 (1982) 1370. [17] Yu.V. Kopaev, V.V. Menyailenko and S.N. Molotkov, Zh. Eksp. Teor. Fiz. 77 (1979) 352. [18] A.A. Gorbatsevich, Yu.V. Kopaev and SN. Molotkov, Solid State Commun. 44 (1982) 193. [191 Yu,V. Kopaev and S.N. Molotkov, Fiz. Tverd. Tela 21 (1979) 1195.