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29 February 1988

HIGH-T. SUPERCONDUCTIVITY EXPLAINED BY HOLE POLARIZATION E.V. KHOLOPOV The Institute of Inorganic Chemistry of the Siberian Division ofthe Academy of Sciences 0/the USSR, 630090 Novosihirsk, USSR Received 11 August 1987; revised manuscript received 8 January 1988; accepted for publication 11 January 1988 Communicatedby A.A. Maradudin

A model in which the hole conductivity in a valence band iscaused by random impurities is considered. Moreover, the spatially disordered distribution of those impurity levels leads to an interaction between holes. As a result, the superconducting coupling ofholes is possible at high temperatures. The relation between our results and experiments in ceramics is discussed.

1. Introduction In the last months the interest in the problem of high-temperature superconductivity has grown after the revealing of superconducting properties at about 40 K in the La—Ba--Cu—O system [1]. Further investigations [2—4]have resulted in swift progress in preparing related materials with significantly higher values of the critical temperature 7’, of superconductivity [5,6]. The theoretical explanations of that effect are mainly based on the conventional mechanisms [7]. Inasmuch as the phase with the K2NiF4 structureis typical for those compounds [21, the metallic character of their initial conductivity is assumed to be connected with the fact that the Peierls instability does not arise after adding bivalent metals (Ba, Sr). On the other hand, the high-temperature superconductivity is supposed to originate in the high-frequency breathing-type distortions of oxygen octahedra It is also pointed out that some frustrations[8—11]. accompanied by the spin-glass-like state can be important for high-temperature superconductivity [121. Apart from the Cu2~ions, the cxistence of the Cu3 + ions gives evidence of mixed valence in the systems of interest, so that appropriate possibilities [13,14] are taken into account as well. However, the high-temperature superconductivity effect was revealed rather unexpectedly. This is a Senous reason for the verisimilar conjecture that the 292

nature of this phenomenon is beyond the scope of the conventional mechanisms of superconductivity; some novel mechanism seems to be engaged accidentally. In particular, such a possibility is discussed in the present paper. A unique mechanism is proposed which reflects the peculiar features of the Ceramics at hand. Corresponding preliminary results are contained in the brief communication [15] (see also ref. [161). Note that although the region of applicability of the mechanism proposed seems to be rather wide, so far the ceramics of interest are the only objects where it is realized. This mechanism is not associated with phonons. Moreover, holes form the only dynamical system there. in contrast to the conventional mixed-valencesystems [171, there is a single conduction band in our system. If the La2CuO4 structure is immediately concerned, then the hole conductivity is, on our assumption, realized in the valence oxygen band due to the virtual exchange of 3 ± ions. The same holes between that band and the Cu mechanism is responsible for the superconducting coupling of holes as well. The treatment has been performed in terms of temperature Green functions [181. The difference of our mechanism from the nonphonon mechanisms [19—21] will be discussed as well.

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2. Hole band and chaotically arranged localized donors

29 February 1988

2cg tan 20k= cB—w~~

Let a regular valence band in which holes are mitially absent exist. Furthermore, a hole donor can cxist in every unit cell of a regular lattice. The hamiltonian of the whole system can be written in the form

(6)

On substituting expressions (5) into hamiltonian (3), the latter takes the form ~ (k~ k~+~*~(k~) (7) ,

k

where ~ k

+~

[Bb~b,~.+g(b~aj~ +a,~bia)] ,

(1)

where a~ and a*,. are respectively the creation and annihilation operators of a hole with momentum k, energy 03k. and spin ~r,a~and aja are the same operators in the site representation; the summation over repeated spin indices is supposed. The parameter i7~ describes the presence of a donor in the ith unit cell if ~,= 1, otherwise ~,=0. b,~and b~are the appropriate creation and annihilation operators specifying a hole located at the ith donor site and characterized by the energy value B. We assume B>> 03k for all k. g is the matrix element describing the hole transfer between the band and a donor. The sum over i is taken over all N units cells of the regular lattice. It is convenient for what follows to rewrite expression (1) in the form

~

+4Q2] 1/2} [(1 __&~)2 +4Q21 1/2}

~cB{1+&k~ [(1 —03k)

~k=

=

~cB{l

+&k+

,

(8)

wk=a)k/cB, Q=g/B. The Fermi level lies in the c~band and is determined by the condition ~ = 0, which reduces to the relationship (9) Note that according to expressions (5), the probability that a hole belonging to the c~-bandis located at donors is equal to sifl2Ok. In the practically interesting case of ceramics sin2O~ ~ 1. So Q << 1, i.e., Q is the small parameter of the task. Inasmuch as thk ~ 1, the c-band is described by rather high energies and therefore is effectively empty. Thus, the band is not important for what follows. According to expressions (4) and (7), the hamiltonian H describing the ~-band is of the form WkF =

.

~-

H=H 0+H,

(10)

,

(2) ~

H0=~[oka~ak~+cBb~bk~

(11)

(~k~F)~ka,

k

k

~ ~ ~

(3)

H,=

(4)

Here F=cg

where ~,, = c, Ho corresponds to the total diagonal part of the hamiltonian H, c is the donor concentration defined as the mean value of The canonical transformation reducing hamiltonian (3) to diagonat form is given by the equations

—

~Nk~2

~

(12)

212

2/B.

—

3~Thermodynamical pair interaction

~,.

a~=cos8k+sinOk~.~ b~~=—sinOk~ 3i)+cosOkCJ~) ,

(5)

where ~ and ~ are the annihilation (creation) 0k operators ofby thethe appropriate is specified conditionnew states. The value

Let us consider the thermodynamic potential Q specifying the canonical ensemble where the total number of hole donors is not fixed, but the mean concentration of those donors is c. Q can be written in the form Q=

—

[

Tin Tr{exP _fl(H_~~ ~ rp,

)] }.

(13)

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Here T= 11/3 is the temperature measured in energy units, ji is the chemical potential of donors, the trace is taken over all configurations of ç~,and holes in the ~-band. It is useful to separate the contribution of the non-interacting degrees of freedom, which is Q

0=

—

[

TIn Tr{ exp _fl(Ho

—~

~

~,)] }.

(14)

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arise in expression (17). Furthermore, it is obvious that the averages of the form (18) connect holes with opposing spins. According to this conclusion, we restrict ourselves to only even n on the right-hand side of relationship (17). Taking account of all possible configurations of pairs ofH~(r),which are then coupled via averages (18), formula (17) may be represented in the form

~In! ç dr, dr... dr~dt~ Ii

Due to the fact that H0 and H, are non-commuting operators, formula (13) can be rewritten as [17]

1=

.i

,,...~

/3

Q=Q0— Tln(exp(_

$

111(r) dx)),

(15)

0

where H~(r)=exp(rH0)H1exp(—rH0),

(16)

the angular brackets with the subscript t denote the mean value of the expression contained, which is ordered with respect to t, so that every array of factors as functions of r is arranged there in such a manner that r decreases upon moving along the array from left to right; averaging is carried out with the partition function corresponding to Q0. For further calculations we make use of the wellknown theorem for the irreducible averages, which is of the form I~ln(exp(_

J

H1(r) dr)~

0

o (~l)~z n~I n! 0 dr ~

=

J

...

dr~<

Here the double angular brackets denote the irreducible average. Itover is important for whatcontribution follows that upon averaging c°k the nonzero corresponds only to the pair average of the form

X ~ L/(t1, t~)...U(r~,r~)>>r, (19) where the pair interaction specifying the hole ther-

modynamics is of the form 4 U(t, ~ C(1NB2 c)g kik 2 k3k4 +k2,k3+&4

(20)

4. Critical temperature of superconductivity As usual [181, the instability of the system of holes which gives rise to their superconducting coupling is driven theoretically by the singularity of the whole vertex of the two-particle interaction at zero energy and momentum exchange. In the conventional manner, interaction (20) leads to the ladder diagrams as the predominant set, the sum over which reduces to a geometric progression. The appropriate calculations are omitted, as they are rather tedious, but have not any practical difficulty. As a result, the condition that the pole of interest arises in the two-particle temperature takes the form 4 Green function 1 c( 1NB2 c)g (~ T~)2+ (Wk EF) = —

—

(21)

where T~is the critical temperature of superconduc<49kQ~k 2>=c(l

—c)ö_kk,

(18)

tivity. Further calculations require the concrete form

where ö..kk2 is the Kronecker symbol. Indeed, according to definition,

of the spectrum ~ axial anisotropy, 2 03k = k~ + k

exclusion principle forbids the presence ofmore than two holes at the same site, so that such terms do not

where k~and k1 are the momentum components

294

~

2m

In the simplest case of the uni(22)

1,

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along and normal to the anisotropy axis, respectively, m and m1 are the appropriate effectivehole masses. After replacing the sum over k in formula (21) by the corresponding integral and carrying out the integration, we obtain ~ ~tan’(~~ \. (z+p)2+qj q \ q I _________

L

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appropriate curve are shown in fig. 1. According to relationship (23), in the limit it T~ ~ ~F the explicit expression for T~may be derived in the form [151 2~F [~ (~)1/2 1/2 + ~2~ (27) F Ii T~ = The vanishing of T~at ~F=03 is a peculiarity of de—

pendences (23) and (27).

_i/z+p~s1 ~

+

tan

~—)j=~

(23) 5. Discussion

where (24)

D— c(l_c)g4vm.J~

—

4ir2h3B2

‘

p=[(7c2T~+~)”2+EF1”2 ,

(25)

2T~ +f~)h/2—~F]112, (26) q=[(it v is the volume of the unit cell, z= (203)1/2, w is the width of the spectrum 03k~ Formula (23) specifies the dependence of T~on c, with the definitions of D and ~ taken into account. The typical shapes of the

The model involved will be applied to ceramics which are described by the chemical formula L 2_~M~CuO4, where Land Mare a trivalent and a bivalent metal, Inasmuch as cell the gives presence ofone atomrespectively. of the M type in the unit rise to the possibility of the appearance of the Cu3~ ion, which is the hole donor, we apparently have the equality x=c. On the other hand, as mentioned above, the ratio Cu3~/Cu2~ is specified by the parameter Q2=0.01. It is also instructive to write the density of states of holes on the Fermi surface, which is equal to

6

NF=

vm2h3 (2mr~cB)112. ir 1 Q

(28)

According to papers [8,10], for the La I

2

o

2

~2

0.’i

0.6

OZ

C 4/B)

Fig. I. Critical versus donor concentration temperature 7~ cat of superconductivity Q=O.1, D= bc(1 _c)QBL’2. (7= T~I 0 The different possible types of this dependence are represented by curves 1 and 2 which are described by w/B=O.l and 0.004, respectively,

1 85Ba01 5CuO4 compound we have T~=33 K, NF= 15 state/Ry cell. From formulae (27) and (28) we obtain the esti‘nate B 20 eV which is of a reasonable order of i~agnitude. On discussing the features of the obtained solution, we must point out that there is the non-exponential character of dependence (27). The latter is a direct consequence of the factthat interaction (20) is neither instantaneous nor time-retarded, but connects states divided by an arbitrary time interval. Such a peculiarity reflects the fact that the indirect interaction between holes is mediated by the static degrees of freedom ~. The condition that the superconducting state exists in such a system reduces to the absence any correlation between positions in theoflattice. The latter is also the in donor agreement with experiments. As far as the physical nature of the phenomenon at hand is concerned, it is worth noticing that this 295

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phenomenon distinguishes itself from the traditional ones, which are based on indirect interaction via some branch of boson excitations organized regularly. In our case we deal with the virtual excitations of chemical bonds distributed randomly. The appropriate energies are of the order of the ionization energy, so that they lie far beyond the upper threshold of excitation energies typical for the usual solid state. This seems to be the origin of the fact that the proximity to the instability with respect to the chemical composition is a peculiar feature of the corresponding real objects. Note that, as a rule, the appearance of the unusually high energies is compensated somewhat by the statistic of the system [22]. Finally we point out that the ratio between the energy gap and the critical temperature may be described experimentally by different values which is a consequence of the present model and will be discussed elsewhere. It is clear that our model differs drastically from those described in papers [19—21].First of all, superconductivity is not associated with a departure from nesting here. Conflicting with Emery’s model, the Cu3~ions generate holes in our model, but the Cu2~ions do not. Moreover, our interaction leading to the hole hybridization is localized only in those unit cells where the Ba (or Sr) ions are contained. Our indirect interaction is independent of time because it is mediated by the static degrees of freedom due to an arbitrary distribution of the Cu3~ions which in turn yields the unusual dependence of T~. Furthermore, we suppose that the electronic properties are anisotropic though it is not crucial for our mechanism. As far as the hole coupling is concerned, we believe that the picture of its spatial distribution is centred at unit cells mentioned above without any exchange of spin fluctuationsdiscussed in paper [21]. The condensation of hole pairs is not typical for our superconducting phase transition as well. In our case another nature is responsible for a maximum value of T~at an intermediate concentration of the Ba (or Sr) ions.

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References

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[4J R.J.

Cava, R.B. van Dover, B. Batlogg and E.A. Rietman. Phys. Rev. Lett. 58 (1987) 408. [5] M.K. Wu, J.R. Ashburn, C.J. Torng, P.H. Hor, R.L. Meng. L. Gao, Z.J. Huang, Y.Q. Wang and C.W. Chu, Phys. Rev. Lett. 58 (1987) 908. [6] P.H. Hor, L. Gao, R.L. Meng, Z.J. Huang, Y.Q. Wang. K. Forster, J. Vassilious, C.W. Chu, M.K. Wu, J.R. Ashburn and C.J. Torng, Phys. Rev. Leit. 58 (1987) 911. [7] V.L. Ginzburg and D.A. Kirzhnitz, eds., Problem of hightemperature superconductivity (Nauka, Moscow, 1977). [8] iD. Jorgensen, H.-B. Schuttler, D.G. Hinks, D.W. Capone it. K. Zhang, M.B. Brodsky and D.J. Scalapino, Phys. Rev. Lett. 58 (1987) 1024. [9] L.F. Mattheiss, Phys. Rev. Lett. 58 (1987) 1028. [10] J. Yu, A.J. Freeman and J.-H. Xu, Phys. Rev. Lett. 58 (1987) 1035. [11] N.M. Plakida, V.L. Aksenov and S.L. Drechsler, Preprint JINR, El7-87-198 (Dubna, 1987). [12] K.A. MUller, M. Takashlge and J.G. Bednorz, Phys. Rev. Lelt. 58 (1987) 1143. [13] E.P. Fetisov and DI. Khomskii, Zh. Eksp. Teor. Fiz. 92 (1987) lOS. [14] ChenChang-fengandZhangLi-yuan,J. Phys. Chem. Solids 47 (1986) 547. [15] E.V. Kholopov, Fiz. Nizk. Temp., to be published. [161E.V. Kholopov, in: Abstracts of the IV All-Union Conference on Physics and chemistry of rare-earth semiconductors (Novosibirsk, USSR, 1987). [17]Dl. Khomskii, Usp. Fiz. Nauk 129 (1979) 443. [18] A.A. Abrikosov, L.P. Gor’kov and I.E. Dzyaloshinksi, Quantum field theory in statistical physics (Prentice-Hall. Englewood Cliffs, 1963). [19] P.W. Anderson, Science 235 (1987) 1196 [20] V.J. Emery, Phys. Rev, Lett. 58 (1987) 2794. [21] J.E. Hirsch, Phys. Rev. Lett. 59 (1987) 228. [22]E.V. Kholopov, in: Abstracts of the Xi All-Union Conference on Calorimetry and chemical thermodynamics (Novosibirsk. USSR, 1986) Part 2, p. 95.