On the problem of the phase diagram of new superconductors

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On the problem of the phase diagram of new superconductors

L.P. Gor’Kov, A.V. Sokol

To cite this version:

L.P. Gor’Kov, A.V. Sokol. On the problem of the phase diagram of new superconductors. Journal de

Physique, 1989, 50 (18), pp.2823-2832. �10.1051/jphys:0198900500180282300�. �jpa-00211105�

2823

On the problem ^{of the} phase diagram ^{of new} superconductors

L. P. Gor’kov and A. V. Sokol

L. D. Landau Institute for Theoretical Physics, Academy of Sciences of the USSR, 142432

Chernogolovka, ^Moscow Region, ^U.S.S.R.

(Reçu ^{le 13 mars 1989,} accepté le 12 mai 1989)

Résumé.

Nous présentons ^{un modèle de} spectre électronique ^{avec une} large ^{bande de trous}

délocalisés et faiblement hybridée ^{avec un} réseau de niveaux localisés. Nous en dérivons un

hamiltonien effectif incluant les interactions spin-spin, spin-trou ^et trou-trou. Avec quelques hypothèses, l’intégrale d’échange peut ^{être à} longue portée. ^{Nous montrons} que, ^à faible concentration de dopant, ^les porteurs de la bande de conduction forment de grands polarons.

Nous donnons la susceptibilité magnétique ^du système ^{entier et sa} dépendance ^en concentration et température (dans la limite des grandes concentrations). ^{Nous montrons} que, à ^cause que la contribution compensatrice ^de l’interaction de type RKKY, ^le système antiferromagnétique peut présenter ^une instabilité ferromagnétique ^{avec une} énergie caractéristique de l’ordre de la

température ^{de Néel.} L’importance des effets quantiques ^est soulignée.

Abstract.

We present ^a model of the electron spectrum ^{with one} delocalized hole-like broad band weakly hybridized with the array of localized levels. Effective Hamiltonian including ^the spin-spin interaction, spin-hole interaction and hole-hole interaction is derived. At some

assumptions ^the exchange integral proves ^to be a long-range ^one. It is shown that at small dopant

concentrations carriers in the conduction band form the large ^size polarons. ^The magnetic susceptibility of the whole system ^{and its} dependence ^on concentration and temperature (in ^the

range of large concentrations) ^are found. It is shown that, ^{due to the} compensating contribution from the RKKY-interaction, ^an antiferromagnetic system may tend ^{to a} ferromagnetic instability

with the characteristic energy of the order of the Neel temperature. ^The importance ^{of the} quantum effects is emphasized.

J. Phys. ^{France 50} (1989) ^{2823-2832 15 SEPTEMBRE 1989,}

Classification

Physics ^Abstracts

74.20

74.30

74.70J

1. Introduction.

In this article within the framework of the model proposed previously [1, 2] ^an attempt ^is

made to interpret peculiarities ^{of the} phase diagram ^{of new} superconductors ^under doping.

For definiteness below we shall have in mind the properties ^of La2 - ,Sr,,CU04. The latter have been extensively discussed in the literature (see, e.g., [3]). ^{Of the} highest interest is naturally

the sharp dependence ^{of the nature of the} ground ^{state on} doping, e.g., transformation of the

antiferromagnetic (apparently, dielectric) ^{state at} small x into a metallic behavior at high

strontium concentrations. Although ^{in the} experimental picture ^{there are} still many obscure

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500180282300

problems, ^{a number of} qualitative considerations was given, ^{which we} think may have ^to do with the problem. ^{We mean, for} instance, ^{the comment made in} [4] ^{that the} exchange

interaction between the oxygen hole and the moment on Cu 2, is large ⁱⁿ comparison ^{with the} exchange integrals between the localized spins of copper ions. We aim ^at transferring ^this

discussion to a more microscopic ^{level of the band structure of the} compounds ^under study.

The fact that all the diversity ^{of the} properties ^{of the new} superconductors ^{cannot be}

described in terms of the Hubbard model of only ^Cu ions, as, for instance, has been assumed in [5], ^entailed already from the numerical calculations (see, e.g., [6]) which revealed that the

overlapping ^of d-copper shells and p-oxygen orbitals is ^not small. At the same time the presence of localized ^{moments on} Cu2+ is already ^an established fact [7, 8] and the role of interelectron correlations is undoubtful. Now there are two models whose major ^difference

lies in the physical ^{nature of how a localized state of the hole on} Cu ion is formed.

The model [9] ^{is based on the} picture of atomic copper d-orbitals and oxygen p-orbitals.

The latter, ^{as is} assumed, ^{in terms} of energy lies higher ^{than the one} hole-filled orbitals

Cu 2+ (configuration d9). ^{Cu2 + ions are Hubbard centers.} Doping ^is responsible ^{for the}

emergence of holes ^on 02 - ions and transport properties (the band) originate ^{from the} overlapping (hybridization) between d- and p-orbitals.

In the model [1] ^a sufficiently broad band for holes is introduced from the very beginning.

Such a band could, ^for instance, ^{be a result} of numerical calculations performed ^{in the} assumption that the self-consistent potential corresponds ^{to a} configuration ^{d10 for all}

Cu+ ions. Let the same calculation yield ^{a level £’ for one Cu+ ion. In} [1] ^an interaction of the hole on this level, V Q ^{Q, with any} phonon degree ^of freedom, Q, ^{is assumed to be} strong. ^As

a result of this interaction, the level goes down :

where Mil 2 is the appropriate rigidity of the lattice. The strong polaron ^effect (1) ^{for the state} Cu2 + originates from the Jahn-Teller instability of the octahedric configuration ⁱⁿ perovskites.

Localization of the states CU2 + and their orthogonality ^{to the band states} is, thus, ^ensured

and enhanced by the inclusion of slow lattice degrees of freedom. If the lattice times of relaxation are smaller than the lifetime of the state Cu2 + ,

^{these states} can, of course, be treated as Hubbard centers.

The position ^{of the level} (1) ^{relative to} the bottom of the conduction band determines the carrier population. ^{Since, for instance, the} rigidity ^{of the} system Mf2 2, ^no doubt, depends ^on doping ^{and on} other lattice parameters, the motion of the level £o ^with respect ^{to the} delocalized band could in principle describe the transition from the dielectric regime ^{to the}

metallic regime ^at strong doping. It is this aspect ^{of the} problem that has been discussed in [1].

Probably ^{it has} something ^to do with the problem in view of the discovery ^{of two} phases ⁱⁿ La2Cu04.13 [10] ^or of metallization in La2-xSrxCu04 ^{at x}

^0.17 [11,12]. In this article however we study only ^the region ^of relatively ^low concentrations. It is our assumption ^that

the ground ^{state is} magnetic and dielectric. Correspondingly, the relative position ^{of local}

levels and of the conduction band (for ^holes !) is chosen as is given ⁱⁿ figure ^1. With all this

taken into account, ^{we can} describe this case by ^the periodic ^Anderson Hamiltonian

2825

where V is weak hybridization ; repulsion ^on the Hubbard centers is assumed to be infinitely strong. The conduction band close to its bottom is quadratic, £(p)

p 2/2 mB. The width of the band, D, ^can arbitrarily be related to the position ^{of level} £0

à

0. Henceforth we

shall hold that the dispersion law 8. ^{is close to a 2D law} reflecting ^the layered character of new

superconducting materials. It is convenient to introduce the following ^{set of} parameters, characterizing the band and magnitude ^of hybridization :

where a is a lattice parameter, mT is the effective ^mass of the hole near the top ^{of the} band, ^at

p = Qo

(7r la, 7r la ). ^In general, ^{17 and v are not} equal ^to unity. Below it will be assumed that the parameter À is small and the value q - v - 1 for the band of ^an arbitrary ^{form. Note}

that in what follows only ^one conduction band is taken into account for the purposes of ^a

simplification.

2. Effective Hamiltonian.

The Hamiltonian (2) despite ^its seeming simplicity ^{describes a most} complicated problem ⁱⁿ general conditions. Yet, under the constraints formulated above, ^it permits ^this problem ^{to be}

reduced to the problem ^of Heisenberg spins. ^{Some main} parameters ^{of the} spin Hamiltonian have already been obtained in [2]. In this section we shall describe a regular ^{method of} deriving the effective Hamiltonian, H,,ff , ^{which is} equivalent ^to (2) ^{at T « A In} contrast to the derivation in [2], this method makes it possible ^not only ^{to obtain} higher ^{order terms with} respect ^to spin-spin interactions but also consistently ^{to take into account} the interaction between spins and carriers (holes) in the conduction band and carrier-carrier interaction.

Treating ^{the term with} hybridization ⁱⁿ (2) ^{as a} perturbation, ^{we can} standardly rewrite the

grand partition ^{function as}

where

The problem ^{is to reduce} expressions (3, 3’) ^{to the form}

employing à > T, ^i.e. the fact that transitions from local levels into the conduction band are

absent at low temperatures. ^The Hamiltonian Beff will, ^{as a} result, ^{be defined on} the manifold of wave functions CPh Et) CPs, where 0,

^{are wave functions of 2N-fold} degenerate ^{states of}

Hubbard centers and oh ^{are wave functions of} holes, ^{added to} the band under the condition that filling of the conduction band is not large (EF «’:,à)- (The derivation allows for

generalizations ^{which we shall not} dwell upon here.)

The first non-zero new contribution to Heff (4) ^comes from the second order term in the

expansion (3’) ^{in V :}

The order of operators d,,, (r2) dnu,(T1) (T2:> Tl) ^{is fixed} by ^the assumption ^of strong

repulsion ^{on the n-th center.} We rewrite the operators b (2 ) 4, ’ (1 ) in the interaction

representation ^as

Whereas the first term in (6) ^is c-number, ^{it is} possible ^to put -rl !-- T2 (A> T, EF) in ^the

second term and now it transforms into the operator of the number of particles ^{in the} secondary quantisation : the contribution to Heff ^is (« ^{s-d »} interaction)

where Sn ^{is a localized} (one-half) spin of the n-th center and the first term corresponds ^{to the}

renormalization of the energy of the localized level.

We now calculated S4({3) (the fourth order term over V in (3’». ^Unlike (5) ⁱⁿ S4({3) ^{there are} already ^two independent ^{sums over local centers} (say, ^{m and} n). ^{Let first}

n =A ^m. A relative order of times for the operators dm ( T4 ), dn ( T3 ), dn(7-2) ^and dm( Tl) (and, accordingly, ^{for the} operators 1/1 m ( T 4), 1/1 n ( T 3), 1/1:- ( T 2) ^and 1/1:’ ( Tl)) ^is again determined by

the requirement that the local center be occupied : T4::> Tl 1 and -r3 :::’ T2. The difference of

S4 (,S ) ^from

is in the presence in S4({3) ^{under the} integral ^{of the terms,} e.g. of the form

(at 7"4 >- T3:> ’r > T2). After commutation (i.e. ^{reduction to} independent integrals ^with respect ^to (T4’ T 1 ) ^and (7-3, T2)) there emerge ^new contributions (of the order of

V4) ^to the effective Hamiltonian of the form :

exchange interaction :

«

s-d » scattering by spins ^{of two centers}

The terms with n = m should be investigated separately. ^After extracting S4({3) ^from 54(Q ) there emerge ^a nontrivial local contribution

Apart ^from (8-10), ⁱⁿ S4 (/3 ), ^{like in} (7), ^{there are of course} corrections which correspond ^to

the general shift of the ground ^state energy. Henceforth these corrections ^are dropped. ^The

2827

expressions ^for J (R ) ^and t (R ) have been derived in [2] (for ^one conduction band). ^{The term} (10) corresponds ^{to the contact} interaction of carriers on one center. As we see, it has a rather

peculiar structure. Yet, ^{it is} comparatively small. Its estimate will be given ^below.

3. Antiferromagnetic ^state (mean ^field approximation).

Let doping ^{be absent} (stoechiometric composition). ^{Then the} problem ^{is reduced to the} spin

Hamiltonian (8). ^{Here we shall} give ^{a brief} description of the results of [2] ^{which we shall}

need further. The value of the exchange integral ^{on atomic distances is} of the order

At the same time we get [2] ^{that at D » A the} asymptotic ^behavior of J (R ) ^at large distances is

where

Thus, under these conditions the exchange interaction proves ^to be a long-range interaction.

This long-range behavior of the interaction gives ^{rise to the} so-called « frustrations » and therefore even if we neglect the fact that the Hamiltonian (8) ^{describes a 2D system of} Heisenberg spins, ^{where the} fluctuation contribution is large, the presence of such frustration

even more complicates the solution of the problem ^{of the nature of the} ground ^state. Yet, bearing in mind that in experiment La2Cu04 ^{is an} antiferromagnet, ^{let us} make certain estimates for the Neel state, making ^{use of the mean field method.} Generalized susceptibility

for the wave vector Qo, ^{which we assume to be} commensurate (Qo

(-ula, -ula», ^is

In conformity ^with [2], the magnon spectrum ^at small q « 11RO is ^described by

although

It is evident from these formulas that, ^at D > â, ^the slope of the linear part ^{of the} spectrum ^at small q is very steep. Thus the relation between the Neel temperature ^{and the} slope ^{of the}

linear part in the magnon spectrum largely ^{differs from what we obtain} in the model with only

nearest neighbor interaction. Formula (14) ^{is of course a} rough approximation. Calculation of

already ^second order bubble in the expansion ^{in J for} x (Qo) ^{testifies to} the presence of

significant deviations from (14) ^{at low} températures - TNf. ^Without giving details of the calculation we must point ^out that the actual temperature ^{of the} antiferromagnetic ^transition

may be in the range from - TNf/ln (Dlâ) ^{to ’"} TN . ^The lower-limit estimate results from the calculation of fluctuations in the low temperature region ^{due to} magnons. On the other hand,

the large interaction radius permits ^{us to calculate} exactly enough the static susceptibility ^of

the system :

where

In conclusion of this brief section, ^{note that the} slope of the magnon spectrum

d w Idq is estimated as 0.2-0.3 eV A [7]. ^This slope exceeds standard estimates (TN ⁱⁿ La2Cu04 amounts to 250-300 K). Although, ^{in a} layered La2Cu04, ^3D ordering, strictly speaking, results from coupling ^between separate layers, nevertheless, ^this large magnitude ^of

the slope d w Idq ^seems fairly unexpected. ^{In terms of the} model, adopted ^{in this} article, ^this result would testify in favor of the assumption D > 3.

4. Magnetic properties in the présence of carriers.

As long ^{as we} study ^{the bànd structure of} figure 1, ^we believe that even at a low dopant

concentration the system ^has conducting properties. ^An important ^{result obtained however is}

that the value of the

s-d

exchange interaction between a carrier and a localized center with respect ^{to the} parameter DA/V2_-l/À >1 exceeds the exchange integral (8). ^This

circumstance conforms to the assumptions ^of [4]. ^{Yet, in} contrast to [4] ^{where the} problem

has been discussed in terms of the ferromagnet ^{defects on the} background ^{of the} antiferromagnetic ground ^{state of Cu} ions, ^we shall first investigate what this fact gives ^{rise to}

in the framework of the band picture given ⁱⁿ figure ^{1, without} making any assumptions

relevant to the ground ^{state of the} system. ^In essence, the qualitative picture ^{of the} phenomena is well known (see, e.g., [13]). ^{Due to the} exchange interaction local spins ^near

the introduced carrier have the tendency ^to build up parallelly forming ^a ferromagnetic polaron. (In ^essence, the well known assertion made in [14] reflects the same effect). ^{It is}

known [15] that, ^{in the 2D} case, the formation of finite radius polarons ^is impossible ^without

some extra nonlinearity. ^{In our case this} nonlinearity is the condition that spin be finite. Thus, in the presence of ^a mobile carrier the system ^{has an} energy gain ^{from the} polarization ^of

localized spins ^{due to the}

^s-d

exchange ^{term but has an} energy loss from the kinetic ^term and from the term corresponding ^to the interaction of local spins because of the

antiferromagnetic character of the exchange interaction (8). ^It is easy ^to estimate the order of

magnitude of the radius of such polaron. The energy of the hole in the region ^{of local} spin

Fig. ^1. - Relative position of local levels and of the conduction band. Carriers are holes. Coulomb

repulsion ^{on local centers} is assumed to be infinitely strong.

2829

saturation decreases by the value -- - V 2/ i1. Comparing it with the energy loss due ^to the

antiferromagnetic character of the interaction of the order Or21, ^{we get}

It is seen that r pol:> Ro.

We have performed the calculation of the polaron energy by ^means of the variational

method, treating ^the quantum spin S ^{= 1/2 as a} classical. The equation ^{for the} polaron ^wave

function reads :

(in ^the ground state tp ^is axially symmetric). The value of tp at ^{which the} spin gets

^saturated

amounts to

We choose the approximate variational function in the form :

Minimization of the energy functional, corresponding ^to equation (19), yields

The area of the intemal polaron region ^{where the} spin achieves, say, one-half of its saturation

value, ^is equal (1)

Taking the estimate for the carrier concentration corresponding ^{on the} phase diagram ^of La2 -xSrxCu04 ^{to the} boundary ^{of the localized} and delocalized phases ne

0.06 [12], ^we

obtain À

0.2 - 0.3. (This estimate is grounded ^{on the} assumption ^{that a} separately ^taken heavy polaron ^is practically localized). ^As long ^as the carrier concentration is growing, polarons ^are becoming ^more overlapped. ^{If the}

^s-d

exchange Hamiltonian is arbitrary ^and sufficiently large ⁱⁿ comparison ^{with the} exchange integral in the local spin Heisenberg Hamiltonian, ^{one should} expect emergence of the ferromagnetic ground ^state where carriers

are magnetized antiparallelly ^{to local} spins. ^{In our concrete} model with only ^{one conduction}

band there is ^an unexpected compensation : effective molecular fields acting ^{on a local} spin

from both subsystems (- 0 ) ^are equal ^{and the} opposite by ^the sign with the accuracy up ^to the value - TNf « 0 (remember ^that D » A). To illustrate the latter assertion, ^{write out the} equation of motion for ^a local spin

(1) Note that the aforementioned mechanism makes possible the formation of clusters of larger ^size

(bipolarons etc.).

Inserting ^into (23) ^the magnetization ^{of local} spins and of free carriers in the molecular field

approximation, ^we get ^{that the} precession frequency ^{is ’"} TNf. ^Hence the energy of the above described structure is of the same order as for antiferromagnetic ordering. This result deserves a more detailed discussion and can be illustrated somewhat differently. ^{For this}

purpose ^at low temperatures ^{let us} pass ^{over to} the effective exchange Hamiltonian for the localized spins :

The exchange integral (24) ^{differs from} (8) by ^the fact that it includes the contribution of the RKKY-interaction coming from the finite level of doping. Figure ² schematically ^{shows the} dependencies of Jq and îq ^on q. It is clear from figure ² that the RKKY interaction strongly

modifies the form of the exchange integral only ^at momenta q . pp - n dop 1/2 ^{Qo. The latter}

comment in particular agrees with the experimentally observed fact that the correlation length

of antiferromagnetic fluctuations decreases with concentration as 3.8 Â In "2 ^{[16]. A straight-}

forward calculation reveals that the values of -iq ^{at q}

^{0 and q}

^{Qo prove to be of the same}

order. It is interesting ^to express this result ^as the dependence of the static susceptibility ^both

on concentration (at ndop

nc) ^and temperature :

Here it becomes evident, ⁱⁿ particular, that in the mean field theory approach ^we should have obtained a ferromagnetic transition at temperatures ^somewhat higher ^than TNf. ^However,

comparing ^the transition temperature ^{with the} expression for the characteristic spin precession frequencies ^from equation (23) ^{we see that, at such} temperatures, the excitation of

spin oscillations with the frequency of the order of the transition temperature will ^be sufficiently large. ^{In other words, the mean} field method is inapplicable ^{due to} large

fluctuations. Thus at this level it is impossible ^{to find out the} genuine ^{nature of the} ground

Fig. ^{2. - The} q-dependence of Jq ^and Jq, solid and broken curve, respectively. 1. differs from

Jq ^{only at} momenta q « P F - Qo n 1/2

2831

state in which, ^{as we} have first mentioned, quantum ^{effects are} important. ^{In the} general ^case,

where the compensation of the main contributions is not complete, ^the temperature ^and concent]-ation dependence ^of X is, nevertheless, ^described by ^{the first term in} 0 (T, ndop ) ^(25).

5. Conclusion.

We have shown that the model of the electron spectrum ^with localized and delocalized states has sufficiently many parameters ^{and can account for a number of} peculiarities ^{of the} phase diagram ^{of new} superconductors ^at doping. ^{At the same} time it is natural that in the mean

field approximation ^{one cannot} correctly ^{take into account} quantum effects associated with the small value of the spin S ^{= 1/2.} Therefore, ⁱⁿ contrast to [4] ^we find it difficult to make any conclusions concerning the contribution to the mechanism of superconductivity ^{due to the} exchange ^of spin fluctuations. The only undoubtful fact is that delocalized carriers are

strongly magnetized ^{at low} temperatures. ^The magnitude ^{of this} magnetization is of the order of 0 /À > TNf. ^Therefore it is doubtful that the spin ^{mechanism could be} responsible ^{for s-} pairing ^{in new} superconductors. ^{At the same time the} phenomenological analysis reveals that

new superconductors belong ^{rather to a} conventional type (see, e.g. [17]). ^For the sake of accuracy ^note that at the moment it is still unclear to what degree in the above described model the Fermi liquid approach ^is applicable for delocalized carriers in the presence of strong magnetic fluctuations. This problem will be discussed elsewhere.

In conclusion, ^a few words about the term in Hamiltonian responsible ^{for the contact}

carrier interaction (10). ^{Since the strongest} Coulomb correlations on the center are taken into account by ^the assumption that these levels are Hubbard centers, ^{this term} does need no

corrections due to Coulomb repulsion. ^The dimensionless combination which enters into the characterization of the hole-hole interaction V4/~3 D is here of the form À 2 D / Li. According

to the above estimate À - 0.2-0.3, this value is not that small. If the average spin ^{on one}

center is zero, this term would correspond ^to the attraction of two carriers with different

spins, i.e. would give ^a contribution to the s-pairing with the cutoff parameter of the order of the Fermi energy. In ^essence this term in our model describes an

exciton

mechanism (see,

e.g. [18]) ^at which the virtual process corresponds ^to the formation of an empty ^{local state and}

one more carrier in the conduction band.

Acknowledgements.

The authors appreciate the useful discussions with G. M. Eliashberg, ^{D. E.} Khmel’nitskii,

V. L. Pokrovskii and D. V. Khveshchenko. The authors are very pleased that this paper is ^to appear in the special issue of the Journal devoted to Professor J. Friedel. One of us (L.G.)

had benefited very much from the ^numerous discussions of the high- Tc problem ^{with him.}

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