The influence of structural instabilities and non-linear electron-phonon coupling on the isotope effect

PHYSICA

Physica C 221 (1994) 363-386 North-Holland

The influence of structural instabilities and non-linear electron-phonon coupling on the isotope effect R a f a e l G u t i e r r e z , Jiirgen H e s s e a n d Jiirgen Schreiber Institut J~r Theoretische Physik, Technische Universitdt Dresden, Mommsenstr. 13, 01062 Dresden, Germany

Received 13 August 1993 Revised manuscript received 10 November 1993

In the framework of a two-component ~4-1attice model structural phase transitions are considered. It was found that both longrange ordered phases and glassy-likestates can be described in this model. On the basis of the standard Eliashberg theory the phonon-driven superconducting transition of electrons in a single band is investigated. To do this, the electron-phonon coupling is taken linear and quadratic in the fluctuations of the lattice order parameters. The dependence of the superconducting transition temperature and the isotope effect on the defect concentration is discussed for various parameter sets. Different qualitative features are obtained. The La2_~SrxCuO4is taken as a reference system.

1. Introduction The role of the lattice dynamics in the high-temperature cuprate superconductors ( H T S C ' s ) is still an open question. It is not definitively clear whether the conventional electron-phonon interaction, as treated in the framework o f the BCS theory [ 1 ] or even in the more sophisticated strong-coupling Eliashberg formalism [2], is the decisive mechanism in determining the superconducting transition. Several attempts have been made in order to relate lattice instabilities to the superconducting transition (cf. refs. [ 3 - 8 ] ) . Indications about lattice effects can be obtained, for instance, from measurements of the parameter ct u of the isotope effect, which is related to the superconducting transition temperature T~ by the relation Oln T~ °tu = - Oln M u ' with M u as the mass of the/~th ion. Crawford et al. [9 ] found a strong concentration dependence of the oxygen isotope effect Oto in La2_~SrxCuOa. The superconducting transition temperature and the measured a o do not correlate in a clear form. The authors suggest that some features of

a o could be interpreted by assuming an incipient structural instability, in analogy with the low-temperature orthorhombic to the low-temperature tetragonal structural phase transition in the L a - B a - C u O system [ 10 ]. On the other hand, there does not seem to be an appreciable oxygen isotope effect in YBa2Cu307_~ [11 ]. This is, however, no proof that an electronphonon mechanism has to be excluded for explaining the superconductivity in HTSC's. Hegenbarth et al. [12] found in Y - B a - C u - O a second change in the slope of the thermal conductivity at a temperature Tk about 10 K below To. In a wide temperature interval (Tk< T < To+ 30 K) they detected long time relaxation effects o f the thermal conductivity. The authors propose a freezing in of the microscopic dynamics of the oxygen ions, which could lead to a sort of structural glassy state, as a possible explanation of this fact. Anomalies in the ultrasound attenuation in La2_xSrxCuO, [ 13 ] have been interpreted as originating in a glassy-like state. A strong reduction of the mean square fluctuations of the Cu and O out-of-plane displacements near Tc in Y - B a - C u - O indicating a possible freezing in of the dynamics was also reported by Haga [ 14 ]. Both lattice instabilities, the structural phase transition and the transition to a structural glassy state,

0921-4534/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved. SSD1 0 9 2 1 - 4 5 3 4 ( 9 3 ) E 0 7 9 7 - 5

364

R. Gutierrez et al. /lnfluence Q/'sm~ctural mstabilitle.~

will be treated in this article for a model with a twocomponents order parameter and linear and quadratic coupling of electrons to the lattice deformations. The relevance of the quadratic coupling has been pointed out in several publications (cf. refs. [15-201 ). We concentrate our attention on the oxygen dynamics on the CuO planes and present calculations of the c% and Tc for this model system. As a reference point we use the La2_ ~SrxCuO4. It shows a structural phase transition from a high-temperature tetragonal to a low-temperature orthorhombic phase at a temperature To = 530 K for x = 0 [ 21 ]. T o ( x ) is a decreasing function of the Sr concentration; the transition is suppressed at x about 0.24. The phase transition is caused by the softening of the tilting mode, an optical rotational mode at the X point of the Brillouin zone, which can be described as rigid rotations of CuO6 octahedra. The tilting mode can be decomposed into two orthogonal rotations around the axis pointing to the [ 110] and [ l i 0 ] directions ( [ 6 - 2 2 ] ) . According to the frozen-phonon calculations of Cohen et al. [23 ] the potential in terms of the local normal coordinates of the tilting mode is strongly anharmonic, which enables one to understand the structural instabilities. Anharmonicity is also supported by neutron-scattering experiments, which show, for instance, large thermal ellipsoids for oxygen ions corresponding to the tilting mode [24]. Concerning the electronic structure of HTSC's there is no doubt that electron-electron correlations play a significant role in the origin of the details of the electronic band structure. However, in the metallic phase at low enough temperatures the conduction electrons can be described as a Fermi liquid [25]. Furthermore, band-structure calculations in the local-density approximation (LDA) show no significant changes of the density of states at the transition point from the tetragonal to the orthorhombic phase [23 ]. Hence the aim of this paper is to study the renormalization of the electron-phonon interaction due to anharmonicity and nonlinear coupling, including doping effects. For this purpose we consider a simplified model hamiltonian and certain approximations. In the next section the lattice hamiltonian, which describes structural instabilities, is presented. Using a meanfield-like approach a set of equations for the lattice

properties is derived. For the sake of simplicity the intersite coupling is chosen to be harmonic and of infinite range. Disorder is introduced by a dopingdependent harmonic part of the on-site potentials and the intersite coupling is taken as a eonfigurational averaged but concentration-dependent parameter. The dynamics is treated by the Green function's method in the frame of the first-order self-consistent phonon approximation; in this way we get expressions for the eigenfrequencies and mean square displacements. In section 3 the hamiltonian for the electronic system and the electron-phonon coupling is introduced. We assume a configurational averaged electronic structure, in order to avoid disorder effects in the electronic system, which go beyond the scope of this paper. Further, it is suggested that due to strong local perturbations around the defects only the host places yield a relevant contribution to the superconductivity. We show that the existence of a phase transition in the lattice system forbids a linear local coupling. Additionally, terms in the electronic self-energy that are not invariant on changing the sign of the order parameter vanish identically. Section 4 is devoted to the Eliashberg-Migdal formalism for the electron-phonon coupling. As an approximate solution of the Eliashberg equations the Krezin formula [26] is used to calculate To. In section 5 results for the superconducting temperature and the isotope effect are discussed for both a structural transition and a mixed phase, where a glassy and a long-range ordered state coexist. The main results can be stated as follows: C~o and Tc are very, sensitive to the characteristic parameters of the lattice dynamics. Especially, lattice instabilities strongly influence the superconducting transition. The mixed phase yields a considerable reduction of the isotope effect. The nonlinear electron-phonon coupling plays a considerable role when the lattice system becomes unstable. The essential effect is a renormalization of the linear coupling proportional to the squared order parameter. Last, we find that a concentration dependence of the electronic density of states, for which a phenomenological Ansatz was introduced, yields considerable changes in Tc and C~o; therefore, disorder effects in the electronic system have to be considered in a more sophisticated theory.

R. Gutierrezet aL / Influence of structural instabilities

365

2. Lattice model

with

2. I. Hamiltonian

V~' = - Az Bt (~of)"+ ~'~o~', ~- (~oy)2+ ~-

In this paper we consider a lattice model with two degenerated modes responsible for different lattice instabilities. Let us consider two local generalized coordinates (a) and ~ , which describe the motion of structural units in each unit cell labelled by I. In this paper we consider them to be the rotation angles of the CuO6 octahedra around the [110] and [ l i 0 ] axis, respectively. In the pure system the on-site potentials as a function of these coordinates are strongly anharmonic and we will approximate them by corresponding double well ~a potentials. The motion of the structural units in different unit cells will be coupled by harmonic forces Cfm~ (l, m are the numbers of the unit cells, #, v the numbers of the modes) [ 27 ]. In order to obtain degenerated modes in the hightemperature phase (according to the experimental facts in the La2_xSrxCuO4 [ 21 ] ) only diagonal contributions of this coupling with respect to the mode indices have to be considered. These additionally 22 have to satisfy the symmetry relations Ct~ = Ct,, = Ct,,. Disorder is included in the model in the form of frozen defects caused by a random substitution of certain constituents of the lattice, e.g. S r ~ L a in the case L a 2 _ x S r x C u O 4. In the defect cells the structure of the ion-site potentials will be modified. In accordance with the situation in LaE_xSrxCuO4 we assume that the defects cause a stiff potential locally, i.e. the motion will take place in single-well anharmonic potentials. Furthermore, we take into account that due to the relaxation of the local structure in the vicinity of a defect atom random local stress fields ~f will be induced. Finally we incorporate the fact that the harmonic interaction forces Ctm will be renormalized if the carrier concentration is changed and therefore the screening effect is altered. We will describe this circumstance by concentration-dependent configuration-averaged force constants. The model hamiltonian for the lattice reads

n,~,= ~ ((P~)~ \-Y~m + v ~')

+~ ~

Y Ctm(~'-~P+Hint,

lz l,m

(1)

H~.t=~ ~ ~EU"(¢f)z(¢Y)2, /~#v

l

and

(Ah >0, l~h Al-- ~[Ad< O, l~d

host, defect.

We assume approximately B(~B. pJ' are the canonically conjugate momenta of the corresponding normal coordinates. The interaction between the two modes is described by a local biquadratic coupling of strength EuL The sum over the mode indices can be restricted to/~ # v, since diagonal contributions only cause renormalization of the fourth-order term in the on-site potentials. We can consider the terms contained in Hint, generally speaking, as an expansion in powers of ( ¢ ~ ) p ( ~ ) n , with p, n integer. The symmetry properties of this model demand the lattice hamiltonian to be form-invariant by a transformation into new coordinates (a3 which are defined by 1

and correspond to rotations on n/4 of the ~',2-axis. By such a transformation only the coefficients in Hlat are renormalized, i.e.

At-~.4t =At,

B/'

E'~£= -~ Ctm - ' Ctm = C,,.,

(2)

where by symmetry considerations we have put E 12= E 2~. These symmetry features determine the structure of the terms contained in Him. In leading order we get the biquadratic coupling. A more realistic treatment of the lattice dynamics would require to include the coupling of the soft mode to the elastic strains. Such investigations have been carried out for a microscopic lattice model by

R. Gutierrez et al. / Influence of structural instabilities

366

Plakida et al. (cf. refs. [6] and [ 2 2 ] ) and in the frame of the Landau theory by Axe [28], and Heid and Rietschel [29 ]. Using a unitary transformation one can try to map the optoelastic coupling term onto a @2~02 interaction in the lattice hamiltonian. However, this cannot be fully accomplished for the phonon dynamics the way it is possible for electron or pseudo-spin systems [ 30 ]. Therefore, we assume, for the sake of simplicity that the anharmonic interactions are dominant and the influence of the elastic strains on the soft mode can be considered as a small perturbation. The local coordinate ~0~ can be decomposed into a thermal average r/f describing displacements from the high-temperature equilibrium positions, and dynamical fluctuations u f (t), ~o/(t)=rlf +uf(t)

g~('],,, = ~

(1 -t-(7, A) .

The free parameters g and 6t serve to control the phase transition temperature as a function of the defect concentration x. In correspondence to the MFA only local disorder fluctuations are taken into account and nonlocal fluctuations will be considered to be completely uncorrelated, i.e. (~CIm(~Ci,m, 5~ 0 only for l = / ' , m = m ' and ~C~,, St/,, = 0 otherwise. As far as we are interested in second-order phase transitions (see the situation in La2_,SrxCuO4 [ 3 1 ] ) we restrict our investigations to the case C o / A h > 2. For this parameter region the pseudo-harmonic MFA [ 27,32 ] works quite well and a second-order phase transition can be described.

,

~ E q u i l i b r i u m condition

t / / = ((0t') •

(3)

(...) means a thermal average using the lattice hamiltonian ( 1 ). The local order parameters qf show structural fluctuations caused by the stochastic strain fields ef and by the r a n d o m distribution o f the defects. As we are not interested in details of the disorder, but only consider general features of the model, we restrict ourselves to a simplified symmetric distribution function of the qt"s, which is characterized by the following relations:

In order to determine the order parameters of the model, we have to write the equilibrium condition for the lattice, namely, vanishing of the thermal averages of all forces,

This yields ql'F(~ll') = Z (l,,,q,. ~ " - 2/:,,2 { u l ~u ~ ) q/+.. +el' , m

m

+El2((q~.)2+((l.t~,~s,)2/~

i-8. f) ~+ ~ ~ O, 8r/~' = t/f - V/~ .

(4)

r / / i s a configurational averaged quantity according to the stochastic distribution of impurities. The same relations are assumed to hold for the local fields {f, too. The lattice dynamics will be treated in the frame of the first order of the self-consistent phonon approximation (pseudo-harmonic phonons). For the harmonic intercell coupling we exploit a mean-field like approach ( M F A ) , where the interaction terms are proposed to be o f infinite range, i.e.

..... h

I

q~,,

G,. =Gin + 6 G i n , N

--

Co ( 1 "AF6X N

--d

) ,

) .

(7)

We have used the pseudo-harmonic approach for decoupling higher-order correlation functions. The above equation allows us to calculate the averaged order parameter q / o f the phase transition after performing the configurational average over the defect distribution. In the following we will distinguish between host and defect places introducing the averaged values of the order parameter for both sites separately, i.e. -

Co(X)

(6)

( d p f / dl ) = (OH~a,/O{ot') =-0.

F{,I/) = -,4,+3,9{ {uI')~) +B(q/)2+ Z G,,,

- - k

(8~f) 2k ~ (8~/) 2k

Gm--

(5)

~

1

-~% ,~Z~,

367

R. Gutierrez et al. / Influence of structural instabilities

where Nh(Nd) are the number of host (defect) places. The averaged local stress fields ~' vanish and have no influence on the structural phase transition. Then, the equation for t/~' takes the following form:

help of the Green function method (see section 2.4). In the cases when (&/j,)2 vanishes, the eq. (10) has two different solutions for a phase with long-range order:

--h--

. ~ h,d

h

- - h

- - h

meh

~/t

reed

[ ' ~ h,d 2

--

2E12 \/,,1 2\,~#u u l Ul /rll

h

(8)

and a similar expression holds for the defect sites. Due to the assumed uncorrelated fluctuations of the coupling term Ctm and the order parameter, the averages on the r.h.s, of eq. (8) can be truncated. Avlid erages like 0/J')2< (u~') 2> ' in the function F ( q f ) can be decomposed, by neglecting structural fluctuations of the mean square fluctuations, into (t/f) 2 < (uf)2> + (Sqf):< (u~') 2 >. The second part of this the expression can be further truncated in (&/~,)2 < (uy)= >, since the mean square fluctuations are implicit functions of the order parameter fluctuations (&/f)z and according to the used distribution function of the disorder the following holds in general (mode indices are omitted):

X-"

&7~"(8,~)" dk< u~(8~) > < u ~ ( ~ ) > k!

d(@,~) ~

2" ~ ¢ k

dk

= ~k ~'' k ~ d(&/2) k 8r#i"

(9)

2

The fluctuations ( & I f ) 2 = ( q f ) E - q f define a new kind of order parameter, which describes frozen-in structural disorder and will be treated in the next section. Using eqs. (4) and (5) we find in this way -- h,d--h,d

--

rl~' F(q~')

=(1-x)Co(x)q

h

--

d

u +xCo(X)~l u

_2E12< Uli Ul2 > /~#,u

F(,~')

h,d

-- h'd)2

(,1~ ) =(,7~

s0.

With our choice of the coordinate system the first case corresponds to the orthorhombic phase in the LaSrCuO, while the second one belongs to a lowtemperature tetragonal state. Although the two solutions are different in a physical sense, they can be mapped onto each other by changing the reference axis from qy to ~F with corresponding new coupling parameters (eq. (2) ). 2. 3. Structural glassy state

The next step in the treatment of the model is to consider the existence of structural disorder (a structural glassy state), characterized by the so called structural glass order parameter q~A=(~/7~) 2 - - 2 (&/~') in analogy with the Edward-Anderson order parameter in the theory of spin glasses [ 33 ]. q~k describes a physical state in which some local atomic displacements are frozen-in stochastically. Investigations [ 34,35,36 ] have shown that for single-component models qEAS 0 and qS 0 cannot exist simultaneously. In the model with two order parameters we have the possibility to study the coexistence of both phases, i.e. to_find stable solutions of the kind

q~A=n~ =0, q~A, nl so. An equation for the structural glass order parameter can be found from the equilibrium condition (7) using a procedure taken from refs. [35] and [36]. Squaring eq. (7) and averaging over the host (defect) sublattice we find

= (t/~)2j,u

-- h,d ) 2 + ( ~ / ~ # / - t ) 2 h ' d

( (/,/~u

with h,d) •

=0,

X Gm'7~C'.'7~ "'d +~i~li'"

+3B < (U~:)2>h'd'4"B~h'd)2

+ < (U~g,u)2 >

h,d

ra,.

= (Co(x)-A~,a)

+ 3B(~/TJ*) 2h'd +EI2

~

#0, t/l

(10)

The quantities < (u f) 2 > will be calculated with the

h,d

-

-

- h,d

+ (t/~)2j2u

- -

+ (t/Y) 6

h,dB2

--

+Jag

h,d

, (11)

R. Gutierrezet al. / Influenceof structuralinstabilitie.~

368

h,d

.Jlp

j~u= (Co(x) - A t ) 2 + 9B2 ((u;~) 2 ) + 6 (CO(x) - A t ) B ( ( u t * ) 2 )

+6(Co(X)-A,.d)B(

+ 6 B E ' 2 ( ( ~ 0 f # " ) 2 ) ((b{l')

2 )

,

............

h.d

•

J2,

+2B(CO(x) --Al) + 6 B 2 ( ( u ~ ' ) 2 ) .

+4El2r/~,r/~,.l,( ul1ut2 ) (Co(x)

( (~f)2>

3B((bl~) = (~)2_{_

((Uf):)

h,d

J3.

2) + ((~#/*)2))

-

+ 6 B 2 ( ( u ~ ) 2)

u ~ ) (r/~)3q~ *"

-]- ( 8 7 / y ) 2 ,

a0~ = (qf)2 h,d = (r/y --h,d )2+q~;~.o,

a~u = (r/~S~a~'d= ( ~ h'd)4 + 6 (~/~ h'd)Zq~:~,d + (q~:~,d) 2 a2u = (r/y)6 h,d = (q~, --h,d )6+15(~//~ --h,d )4q~x,. h~

a3~ = (~t)3

) 2 ( q ~ xh d' ) h'd

2

qy

--h,d)

=Co(x)2((l-x)r/~

-

¢0.

2

t + qEA qEA --~qEA = ~--C

2 -+ qEA qEA ~

d_gl2 ,

--d

qEA ---%12 •

2

--d

-k-X~/~ )2 According to the value of the correlation function {re the following solutions can be found:

+ C2( (I -x)(q{~ + ( ~ h) 2) ) 2 ) ) .q_ (ff~t)2

The concentration factors arise from terms like N,,d/ N after separating the initial lattice sums into contributions arising from defect and host places. We find finally h,d

h,d

q-aluJ2u q-a2uB2Wj3.a - - h,d

= R ( q ~ ' d , rl~

2

=~qz

2

~ Clml'l~rnCln?l~n --h

h,d

2

q7

m,n

aouJlu

(Co(x)

h,d

~

+x(q~,dA+(~l~/

- b,d

Using the topological properties of the model it ~s possible to find symmetry-related solutions of the set of equations (10) and (12). In order to exemplify this idea, let us consider a possible solution (A): ~/] # 0 . r/2 = 0 and q~A = 0 , q~:A ~0. Transforming it into the rp~-representation we get

+(q~,d)3

The 1.h.s. of eq. ( 1 1 ) yields then

R(q~'nA'd, rlt'

(u)u~)

h,d

--Ah,d + 3 B ( ( u ~ ' ) 2 ) h,d+ ( (~0y#u)2 h,d) .

2

- - h,d h'd ) 3..[_ 3 ? i f q~;~,d .

(~

.

+ 2B(Co(X) --.4h,d )

a3.qlu¢ '

- - h , d - - h,d - -

+4El2qI'

--h,d + 1 5(r/~

h.d

h.d

=4(E12)2((U)U2~h'd) 2

+4E12B(u] u 2 )

where Z,,,Ctm~Z,,~CI,,,=CO(x). Performing the configurational averages we treat the fluctuations of the order parameter on the basis of the relations (4), i.e. we decouple powers of it in the following way:

- -

( ( u l ' ) 2)

b,d

- - h , d

,

h.d

= 2EI2B( (¢fly#,,)2 -

J3u = 4 ( E ' 2 ) 2(U) U 2 > 2 + 4 E ' 2 B ( u ]

--A/+

( u ~ ' ) 2 > " ' d + E 2( (0,~,~,~,)2 1,~

+6BE"(({0y#')23

j2,, = 2 E I i B ( ({OY*u) 2 )

2)

.... ~ h,d _j_ 2/L] 12 ( C o ( ~ ( ) __Ah,d ) ( ((ff~,,," / x ) . >

+ 2E12( Co(x) -AI) ( ((py.~,)2 ~

q_El2(((p~¢-/1)2)

h,d = (C0(X)--Ah.dJ2q-gB2((H~)

),

(12)

(1)

¢12 = 0 ~ q E A + = q E A_ =

(2)

¢,2=--+

(3)

q~a G2
qEa ~ -

v

q~-A 7 ~q+A=q~A(0),

qgA=0(q~A).

v

~+

+ qEA¢qgA"

In this way solution (A) and solutions ( 1 ) - ( 3 ) can be mapped onto each other and are, accordingly,

R. Gutierrezet al. /Influence ofstructural instabilities symmetry related. We can proceed analogously with --2

--2

a solution (B)r/~ = ~ 0. This yields

--2

=rh ~0,

2 =qEg q E~A = q E A

369

mto2D~'~(co) = h3uP~lm + ( - A t +3( (~/f)2 + ( ( u f ) 2 ) ) )D~'~(to) + E C,nD~'m#(°9)- E C'.D~m(w)

2 -2

n

=

n

+E'2( (~y*u)2)Dy,,,#(co)

~ - ~ , - =0; (1) (2) (3)

+ 2E12(~: ~2 )D,im~/,#(co).

( 15 )

~z =O--'q+A =q~A =qEA , +

+

~,2=__q~A--'qeA=2q~A(O),

--h,d

q~-A=0(2qUA)

+

~ 2 < Iq~AI--'qEA#qEA.

All other possibilities can always be mapped onto the solutions (A) and (B) presented. In this paper_we restrict the study to the solution qJ~0, q~= 0, qlA=0, q~A#0 with ~12=0, SO that eqs. (10), (12) and (20) (see the next section) yield a selfconsistent set for determining the lattice properties.

Let us n o w introduce the averaged,G F Dfm#(Og) , where the conditional average (...) means that the

lattice site l is fixed to either the host (h) or the defect (d) places. According to the assumptions in section 2.1, eqs. (4) and (5), the equation of motion takes the form 2

moo Df~

h,d

2 h,d - - h , d

=h~aJt,~+mI2ou

+ ~-,ClnD~f('O)h'd- E n

Dy.,#(~)

ClnD~n~m(°)) h'd

n h,d--h,d

2.4. Green functions

+E12((~y'.)2)

For treating the problem of the lattice dynamics we use the method of the equation of motion for the commutator Green function (GF), defined by

+2ElZ (~] ~2 ) h'dD~ u.#(CO)h,d,

=-iO(t)([uy(t),U~m(O)]-) .

(13)

The equation of motion in the frequency representation reads, using the hamiltonian ( 1 ),

n

(14)

D~"(to) Higher-order correlation functions will be decoupled in the pseudo-harmonic approximation, e.g.

Hence it follows that

~ =(n~')

h,d

,

+((uf) 2h'd

h,d

~

=~h~

?]l

h,d

. -/ 7 ~ T ~

+(u~u~)

h,d

- - h , d . q _ ( Ul1 Ul2 )

Since the averaged GF depends only on R~m=Rt-R~-Rm (Rt is the coordinate of the lattice site l) we can introduce q-dependent quantities,

+ E Ct,,D~'~(w)- E CtnD~(oJ)

(((u/')2, u~)),,,~O.

h,d

h,d

+ 8rl: 8~l~h,~.

+ B ( ( ~ f ) 3, U#m))eo

( ((u~') 3, u~) ),0--,3 ((u~') z ) ( (uf, u#~) )~,

-

= ( ~ h'~)~q~,~ + ((u~,)~---~~,d,

~--'/~1

moo2D fmP(o9) =hcJu#~,,a -AtD~m~(09)

+ ~ EU'((~'(q~Y) 2, u~)),o.

-

_Ah,d+3B ((~)2)

h,d-~ h,d

+ ( ( [ Hl~,, Pf] _, u~) )o,,

n

((#))2

(~1~)

mto2O~mP(to) = ( [P~, U#m]_ )

(16)

with m . Q 2 u h'd =

Of~(t) = ( (uf , u~) )

O~'~(co)

h,d

1

= ~t~ D~,~(o))

h,d e

-i,R,m

In the sense of the M F A the fourth term in eq. (16) will not contribute to the averaged G F at q ¢ 0 due to the structure of the inter-cellcoupling (infinite long-range interactions)and theiruncorrelated fluctuations to the lattice order parameter. But at q = 0

this term gives a non-vanishing contribution so that

370

R. Gutierrez et al. /In,fluence yf structural ~nstahdrric~.r

defect and host sites are coupled to each other. This situation is just of interest for calculating the softmode frequencies, since the soft mode in this model condensates at this critical point. This problem will be treated separately in appendix D. For calculating other quantities, like the mean square fluctuations. the q= 0 contribution goes with weight 0( 1/IV) in the q sums and therefore can be disregarded. After some algebra we find a linear system of equations for the matrix phonon Green function for q#O (notice that the GF is now local due to the

~~ImDg”(w+iO)

h.d

(Ml’lL;‘)

= -

A

1

,J

j&J

@3ew_,

h.rl --

From this results

B

+

FOL

-

v

h.d I’ )h,d

coth

MFA).

~-~-

h.d

= [m(d-r;2;,

KPW

--

h,d

)-cb(.y)

Bh,”

The condition Det [ K~” ] = 0 yields the eigenfrequenties of the problem. We find after diagonalizing this matrix equation

Bh.d = I’

____

(O)=

h,d

C02-fl:

sz:, h.d

sz:,

h3d

(18)

-hh.d w2-R:

(~5

)“.d

(20)

---ph.d =-Ah.d+3B(((4f”))

h,d

=2E’275i5h’d.

(19)

Using the spectral theorem [ 371 the mean square h,d ( uf u/ can be expressed by the corGreen functions

fluctuations responding

3. Electron-phonon

interaction

In this section we consider the interaction between electrons and phonons. The electrons will be described by an effective one-band hamiltonian. The electron-electron correlations are not considered. but can in principle be incorporated via an effective coupling constant in the Eliashberg theory (i+i.-11’ [ 2 ] ). At the same time we restrict the investigations to systems where the structural instabilities do not change the electronic structure abruptly. This means the usual perturbational approach to the electronphonon interaction is assumed. Hence the following model hamiltonian seems to be a good starting point to discuss the influence of certain lattice effects on superconductivity:

and

_

(O’,)“.“-

>

with

mG2

,h,d_$“d

(w2_(o~)h.d)(w2_(w’)h.dj

-

X

tw<

coth

film

h,d

Of;”

(CfJ_ )h’d

R. Gutierrezet al. / Influenceofstructural instabilities

371

q v, ( / 6u u _ q ~ N n ) = w ,u ,

He = H.l + H~l-oh ,

w , - ~ ~ (g~ N l , - g , y N , t ) + g , N , , #,1

--~)C?

#

lzv I~ra

"[3Cm+ 1 ~.,

l

j0u, = F ( q ~ ) - ~ C, mrlUm

qU(~oy)2c{- "c3cl,

m

,u,l

_2E12(

H~l = ~ tb~Cf" ' ~ 3 C m , I,m

{ CIT

c,=~c~),ci~=(c ~

c,~),

(21)

where the Nambu formalism was adopted [38] and z~ are the Pauli matrices. The ct+~) and cl~(~) + are annihilation and creation operators for electrons in Wannier states labelled by the index l and the spin eigenvalue +(?). A full treatment of disorder effects in the electron system and its influence on the superconducting state is a very complex problem and will not be considered in this paper. Two simplifying assumptions will be done for treating approximately the influence of the disorder on the electron-phonon interaction: ( 1 ) Due to strong perturbation of the electronic and lattice states at defect places, the pairing interaction is negligible small at these places; therefore, they do not give any contribution to the superconductivity. (2) Disorder fluctuations in the electronic system will be averaged over for small defect concentrations in the frame of a virtual crystal approximation. In this way we recover the translational invariance of the electronic system. We adopt the view that the average electron system couples to the lattice displacements at the host places in the same manner as it would do in an ordered system. Local (g~') and nonlocal (g~'m ~) linear coupling terms have been taken into account as well as a local coupling q~' quadratic in the lattice displacements. As we will show below the local linear coupling has to vanish in order to obtain the trivial qf = 0 solution when considering the equilibrium condition for the whole electron-phonon system. Otherwise this coupling produces always a qf ~ 0 solution, as long as the average electron density does not vanish. The existence of a phase transition in the lattice system will also impose some symmetry conditions on the nonlocal linear coupling g~'~. The equilibrium condition reads

U n1U n22qn

,,'u

"~

~

,

(22)

Here is N,l=(c+~z3ct) and N , = ( c , ~ c , t ) + (c,~ c,+ ), the average electron density at site n. The thermal average is taken with the whole hamiltonian H=H~.t+He and the function F(q~) has the same meaning as in section2.2, eq. (7). As follows from eq. (22), a solution q~ = 0 is in general not allowed; therefore we have to impose some conditions on the coupling terms. It is possible to transform w~ into the q representation, in the sense of a virtual crystal approximation for the electronic system, which gives w ~, -- ~

(g,u~ - g , ~u) N _, + g ,~N, ,

where

gu~= l ~ gyn~e-i,R,.. The existence of a phase transition requires the condition w~ = 0. Considering the electronic density as a free parameter, it follows that

gU, =g~U , g,~ = 0 ,

(23)

i.e. a local linear coupling is forbidden. At this point we use a general symmetry o f g ~ which one obtains for a hermitic hamiltonian. Namely, gym ~ = - (g,~§)*

gU~ = _ (g~U). Inserting this last expression into (23) we find gU~ = _ ( g ~ ) * ,

(24)

i.e. the linear nonlocal coupling is a pure imaginary quantity in the q-space. Additional symmetries are discussed in appendix A. We wish to add some remarks on the linear electron-phonon interaction. Some authors have worked with local [15,16] or local and nonlocal [7,39] linear electron-phonon coupling (EPC) in lattice

372

R. Gutierrez et al. / Influence <~fstructural instabilitie.s

models which show structural phase transitions, without imposing any restrictions on the structure of such coupling terms. So far as we do not consider higher-order coupling terms, the only difficulties arise from the non-existence of the r/t' = 0 solution in the equilibrium condition for the whole electron-phonon system. The relations (23) allow one to remove these. The inclusion of a nonlinear EPC brings a new problem into the formalism for systems with structural instabilities. It can be shown that the nonlinear coupling always yields a contribution to the electronic self-energy which is proportional to the order parameter: ~4qUq: ' [ 15]. Since in the low-temperature phase two symmetric equilibrium positions _+q:' exist, we see that this term is not uniquely determined physically, because it is not invariant under the transformation r/-~-q. However, using the conditions (23) this problem can be solved, then by the formulation of the Eliashberg theory, contributions which do not satisfy the foregoing symmetry vanish identically (the proof is contained in appendix B). This is a consequence of demanding a phase transition to exist. In this context, some results of previous papers require a revision.

low this approach too. For lattices with structural phase transitions the Migdal theorem can be violated in regions where the superconducting transition temperature lies near the phase-transition point, since critical fluctuations may yield large vertex corrections. In the same sense the quasi-harmonic approximation can break down in this region, as pointed out in ref. [20]. Therefore, any assertions near the critical concentration xo must be considered with caution. Let us introduce the electron matrix Green function G.,,(t) = ( (c:. ~'+, , , ) ) =

- i O ( t ) ( [ C l ( t ) , c,~(0)]+ ) ,

(25)

or .

//((el, c,+,) )

( (c/, C,n~ ) )))

a"'(t~=~,((c:~c,,+,,)) ((c:~c,,,~) " The non-diagonal terms corresponds to anomalous averages, which do not vanish in the superconducting state. Using the equation of motion method for this Green function, we get a Dyson equation for Gt,,, in the q-representation [43 ];

Gq(~o)=G°(oJ)+G°(o))Mq(o~)Gq(o~) . 4. Eliashberg formalism In this section we consider the conventional Eliashberg theory [2] to treat linear and quadratic electron-phonon interaction. The Eliashberg formalism with linear EPC has been widely used for investigating properties of the HTSC. Concerning lattice effects, solutions of the Eliashberg equations for uncoupled double-well (cf. refs. [ 5 ] and [40 ] ) and sixth-order anharmonic potentials [41] have been presented. In ref. [4] the Eliashberg equations were solved for a two-dimensional mode and sixth-order anharmonicity, obtaining a quite good fit to the measured isotope effect in the La system. Extensions of the formalism to include a nonlinear EPC has been also proposed (cf. refs. [1517,19,20] ). Such effects are considered to be relevant for systems with lattice instabilities. In all quoted papers the Migdal theorem [42] is assumed to be valid; consequently, vertex corrections are neglected. For the sake of simplicity we fol-

(26)

The self-energy operator contains different contributions. In appendix B it is shown that only the following terms remain after imposing some symmetry restrictions: M:,,(oJ)= Z Z Z g/~::~3 l~,e

p.K lt.I"

×((c:,,(ut'-u:;,).

( u f -u,,)c:,~

+4q"q"ql'q~,,*3((Gul', +qUq,t3((Cl(JUl,)2

+ ) ~,,,~i'"t3g:t,,,e"

" + ) ),o irr % U,,,Cm

, 2 Cm , + )),,, i~ *3. (JUm)

(27)

Next, the Green functions in the mass operator will be expressed by means of correlation functions according to the spectral theorem. These will be factorized into products of electron and lattice correlators (Migdal theorem). The further development of the theory follows the conventional approach [2,43]. In this way, an expression for the so called Eliashberg coupling function can be found averaging over the Fermi surface

373

R. Gutierrez et al. / Influence of structural instabilities

0~2 ( t o ) F ( t o ) = --1 --

- -

- -

/.0.'

nh(s~) vv ( ) vk \(~i) uv /

u,~

/tv ~ k,uv ( t o ) --- (Tpk +4qUq"rl~'qY) ImDU"-k (to)

+qUq~ Im HU~_k(to) , //V 7pk = ~ (g~-p-g~-k)(gv/¢t) -g~,") ;

(28)

pJ¢

vk is the velocity at the Fermi surface and V the volume of the first Brillouin zone. H~'p (to) is the Fourier transform of the two-particle Green function, given by

us to use approximate solutions of the Eliashberg equations as long as we are only interested in qualitative properties of superconducting systems with a nonlinear EPC and a phase transition in the phonon system. Of course, more elaborated approaches are necessary to get a complete solution of the investigated problem; but let us do the first step by considering the superconducting transition temperature with the help of the Krezin formula [26] given by

kBTc =0.25h

(32)

(o92) and 2e~-phare two moments of the Eliashberg function defined by

H~'g=( ( (au~') z, (au~)2))

oo

2 el-ph = 2 j dtoto- la 2F(to) ,

-iO(t) ([(6uf)2(t), (5u~)2(0)]_),

(Su~')2(t) = (uf)2(t) - ((u~') 2) ,

x/(to2) x/e 2/a°l-ph- 1 "

0

(29)

oo

which can be expressed in the framework of the pseudo-harmonic approximation by the one particle propagator D~U~(to). Truncating the twophonons correlator (auZ (0), a u 2 ( t ) ) into 2(um(O)ul(t) ) (um(O)ul(t) ) one gets 2

f dto2

e p'°~- 1 (to-to1 )(e t~°2- 1 )(e ~(°''-'°2)- 1 )

×

up

x I m D ~ ~ (o~2) Im D~_, (to1 -to2) •

(to2) =22~l_ph -, dtotoa2F(¢o) .

By its use it is possible to find a quite good interpolation of T¢ for 2 values between the weak- and the strong-coupling limit [ 5,44 ]. Using eqs. ( 18 ), (28 ), (30) and (33) a straightforward calculation yields the following expressions for 2~,-ph and (to2): el-ph

(30)

Within the MFA for the phonon system there is no wavevector dependence in the phonon Green functions. Therefore we can take the p- and k-integrations in eq. (28) which lead to averaged coupling parameters

.2 +to_

N(O)

( ( y u u ) z + 4 ( q U ) 2 ( ? l ~ ) 2 ) 0)2

m

~2 2

_ ((7u~) 2+ 4qUq.q},rff) to2+toz_

+

\to:+ + to-T__ - - 2

v~

(33)

0

16~Q22

vv /

)

tolto2V/~ (+ -- - ) 2 2

-p

up ~p , 2g_pgp

(31)

since one has --6p

qUq"(ff22---72((u2>+ +

6~

=g~g~V=0

in order to guarantee a symmetric coupling of the electrons to the two components of the soft mode. The knowledge of the Eliashberg function enables

~

-- 4 ( ( u 2 )

\to~+

+

to---U-_

+-, 2"T77~. -- 2(091092 U2) - )X/'~)] '

N(O) (0)2)-

(u:)_

m2el-ph

~ ((7.u)2+4(qU)2(rl~,) 2 uv~v

374

R. Gutierrezet al. / Influenceof structural instabilities 2

+

(qU)2(/~_

4(~,2)

.+_fl~

-k -

-

2 )(U2>+

--
E 2

8~22 qZ,qV +

( (/~2>

+ _[_ { / d 2 > _ ) )

,

(34)

with 4 ( u 2 ) + (0~2 _ Q 2 ) 2

N(x)=N0(l+

(7;v) 2-2- ~ °Kvo-pop ~'up ~cp

(u2)

,~

+- - mco+

/z~o+

coth~.

(35)

The density of states at the Fermi level is given by V ~ d2p N(O) = ~ t,',

perconducting properties. In a crude way we take also into account a concentration dependence of the electronic density of states at the Fermi level. Motivated by some investigations on the energy and concentration dependence of the density of states [45-48 ] we propose a simple Ansatz:

(36)

(Sf )

All quantities are averaged over the host sublattice. We omit for simplifying the notation the index (h).

5. Results

5.1. Lattice properties In the foregoing sections a self-consistent set of equations was derived for discussing the lattice and electronic properties quantitatively. At the beginning we should state that due to the used simplifications in the model and because of the applied approximations, we do not pretend to explain the properties of real HTSC materials. Here we are dealing with the analysis o f typical properties of systems with a nonlinear electron-phonon coupling and lattice instabilities. To have a starting point we use the lattice parameters estimated in ref. [22] for La2_~SrxCuO4:lt2Ah/m~.60 meV 2, Ah/B,~O.09 ~2, Co/Ah,~2.44, rnm2.6mo~y, N ( 0 ) = I eV -~. The parameter 6 and the ratio A~/Ah were chosen to get the phase transition point at x ~ 0.25. The parameter a~ was set equal to zero and the field fluctuations (ey)2 were considered as vanishingly small, so that we do not treat its influence on the lattice and su-

;\~ , ], ( x - x k ) 2 + F ~j

(37)

where No, N , Xk and F are free parameters. (Van Hove )-singularities in the density of states due to the strong 2D nature of the high-T~, compounds are considered to be smeared out by the disorder. The temperature and concentration dependence of the order parameters q, qEA and the soft-mode frequencies of the model are depicted in figs. 1-4 (the dimensionless temperature t is defined by kB(B/ A~ )T). Lowering the temperature or the impurity concentration different phases with long- a n d / o r short-range order can appear. Let us start with the long-range ordered phase 1 ~1) ¢ 0 , q~ = 0 , qEa = q E2 A = 0 . For these no real solutions ( u ) u~ ) # 0 of the non-diagonal fluctuations exist, so that the frequency' £222 vanishes and oJ+ =Q~, (o ~_ =~25 holds. As we see from figs. 1 and 2, an increase of the dimensionless inter-mode coupling E ' 2 / B causes a reduction of the stability region where the low-temperature phase exists. This effect

5:

F.0~

~'rr:-,=~= ! , r

~',~

,~ :~ ,-

;- : , - - -

,

~

1

Fig. 1. Temperature dependence of the order parameter of the structural phase transition, with Co/Ah=2.44 p=0.14, v=0. a=2.6; (A) E'2/B= 1.2. (B) EI2/B=3.2 (dashed line).

R. Gutierrez et al. / Influence of structural instabilities 2.00-

• 50-

1 . 50

00

\ ~ \ x

\\\\\\

,

X 1.00

0.50

x

375

--g

C I

t

, "

~

x:\\,.

\ \

-0.50

0.00

0.20

0.00

0.40

0.6~

0.80

...... 0.00

8.05

0.18

t

0.50

Fig. 4. Concentration dependence of the soft-mode eigenfrequencies (mO2+_(q=O-Co(x=O))/Ab for the mixed phase (dashed lines). For comparison the long-range ordered phase. We have Co/Ah=2.44, C2/Ah=9,p=0.3, t=0.05, EI2/B=5.12.

critical one, pc, which is determined by the condition 0) 2- ( q = 0 , x, T = 0 ) = 0 at P=Pc. For our model one finds in the special case x = 0

2

/Co(x=O)

P c - 3 + E 12 N/

~hh

"

(38)

B

0.40

X0.30 (]Y

X0,20 E--

0.10

0.00

0.20

X

Fig. 2. Temperature dependence of the s o , m o d e eigenfrequencies (toO2._(q=O-Co(x=O))/Ah for the long-range ordered phase and different inter-mode coupling terms E~2/B. Notice that for E ~2/B< 1 the lower frequency branch becomes negative• We have Co/Ah=2.44, p=0.14 x = 0 , J=2.6; (A) E12/B=0.88 (dashed line), (B) E12/B=2.6 (solid line).

0.00 ~

0.15

~ 0.05

-

~

0.10

0.75

0.~0

0.25

x

Fig. 3. Concentration dependence of the order parameters t/(upper solid line) and q ~ for the mixed phase. Both phases become unstable nearly above the same concentration. We have Co/ Ah= 2.44, C2/Ah=9.12, p=0.3, EI2/B=5.12.

is similar to that of quantum fluctuations, whose strength is characterized by the dimensionless parameter p=h(BA~3/2m-1/2). These fluctuations suppress long-range order for values larger than a

From this expression one sees that for increasing E 12/ B the critical value p diminishes and the long-range ordered phase becomes more and more unstable. The situation is clearly seen in fig. 2, where the q= 0 frequencies are presented. Both eigenfrequencies are degenerate in the hightemperature phase. Their splitting in the low-temperature phase depends strongly on E12/B. While the upper frequency branch 0)2 is not appreciably affected by changing the coupling between the modes, the lower branch becomes negative for low temperatures when E 12/B < 1. Free-energy calculations (see appendix C) for this model show that for this parameter region the symmetric solution of the equi--2

252

librium condition r/] = ~/t becomes the stable one. Solutions for a phase with only short-range order, i.e. r/~'=0, q~A # 0 were also found; free-energy calculations have shown, however, that these phases are not thermodynamically stable when comparing with the high-temperature disordered or low-temperature

376

R. Gutierrez el al. /Influence of structural instabilities

ordered phases. It is possible this is an artifact of our a p p r o x i m a t i o n scheme. In the next step we treat a m i x e d phase with both long- and short-range order (figs. 3 and 4). The solution r/) ~:0, r/2 = 0 , qEA = 0 , q2 A ~a0 will be considered later on. F o r this we get again a vanishing nondi_agonal frequency ,Q22. Contrary to the phase q] ~ 0, r/2 = 0 , q[/,----q2A ----0 we have found now a crossing of the eigenfrequencies below the phase-transition point. This is a result o f the used a p p r o x i m a t i o n s . Therefore, we have i m p r o v e d the a p p r o a c h for determining the Green functions for this special solution. The essential point is to include disorder fluctuations up to second order in the equation o f motion. As a consequence o f this, the new non-vanishing quantity (g2~2) 2 appears in the resulting expressions (details of the calculation are collected in a p p e n d i x D ) . It is interesting to notice that this solution was only found in a very small region o f the p a r a m e t e r space spanned by p, EI2/B, C2/An, & F o r instance, solutions with a critical concentration larger than 0.15 could not be found. Small changes in the m o d e l parameters, mainly E~2/B and C2/Ah, cause the disa p p e a r a n c e o f the m i x e d solution and we get either the r/]:/=0, r/~=0 or q~A=0, q2EA:fi0 solutions separately. F o r the m i x e d phase, cases where the glassy state exists above the structural phase-transition point could not be found in the investigated p a r a m e t e r region. The concentration dependence o f the soft-mode frequencies is shown in fig. 4. F o r comparison, the long-range ordered phase is also shown. This is, however, the unstable phase for these parameters, as freeenergy calculations show. While the lower m o d e has a regular x dependence, the o92 m o d e changes the curvature. We think this is connected with the used distribution function for the d i s o r d e r fluctuations. W i t h i n a justifiable effort it makes no sense to extend the calculations in the present approach to more realistic cases, e.g. to a G a u s s i a n d i s t r i b u t i o n of the fluctuations. This could be a future task in an improved a p p r o x i m a t i o n scheme. W h e n c o m p a r i n g with our reference system, La2_xSrxCuO4 [21], the results which have been presented hitherto show that in the absence o f structural disorder the temperature dependence of the softm o d e frequencies is quite well described. The ex-

perimental facts suggest a value of E12/B near I. The lack of experimental data on the concentration dependence o f the eigenfrequencies does not allow one to draw conclusions on the influence o f structural disorder. The strong correlation between the appearance o f the long-range ordered phase and the structural glassy state makes it difficult to detect the existence o f the latter for a mixed solution.

5.2. Superconducting properties Let us now discuss the influence of lattice instabilities on the superconducting transition temperature and the isotope effect. The results are presented in figs. 5-9. Due to the self-consistency the calculations are p e r f o r m e d for T= T~.(x). In these calculations we assume first a constant density o f states at the F e r m i level. The qualitative b e h a v i o u r o f C~o and T¢ for the two-modes system resembles in m a n y features that of single-component models (see, for instance, refs. [ 15] and [49] ). F o r 7, q < I, but ~/q> 1 (~, plays the role o f an effective linear coupling, see a p p e n d i x A ) , Tc shows a p r o n o u n c e d peak and ~xo a j u m p at the critical concentration x0, where the phase transition takes place. The height of the j u m p can be strongly influenced by changing the strength o f q u a n t u m fluctuations and the m o d e - m o d e cou-

". , S '

N

Fig. 5. Concentration dependence of the oxygen-isotopeeffect %r two different values of the inter-mode coupling E~Z in the case q~A= 0,/,t = 1, 2 (see section 6 ). We have Co/A, = 2.44, p = 0.15. 8=1.6; (A) EI2/B=l.08, (B) EI2/B=5.12 (dashed line), 7=0.45 eV//~, q=0.43 eV/.J~2.

R. Gutierrez et al. / Influence of structural instabilities

377

4. @@

@. B@

3.5@ @.

6@

".o "... !

\ x

I

3.

\

"''o......

@@

/

f

/

~ @ . 4@

~2.5@

I I

X

I o

I

5"2.@@

I

/ /

/

/

/

@.2@-

i

/ /

1 , 5@

./

@.@@

.09 @.@9 i

@.@@

@,

1@

9.2@

@. 3@

@.48

/

. . . .

/

/

/

I I [ l l l l l [ l l ] l l l l l r l l l l ' l l

@.@s

X

Fig. 6. Concentration dependence of the superconducting transition temperature for the case of fig. 5 for the same model parameters; (A) EI2/B= 1.08, (B) EI2/B= 5.12 (dashed line).

I I 1 ~ 1 [

@. 10 x

@. !5

Fig. 8. Superconducting transition temperature ters of fig. 7. (A) q--0.05 e V / i 2 (solid line dashed line (long-range ordered phase), (B) (dotted line (mixed phase), long-dashed line dered phase) ).

@.20

for the parame(mixed phase), q=0.85 e V / i 2 (long-range or-

i . 60

//I

1 .2@

5

~

~

~

1

0.55

I L

j

o@. B@-

/

'1/

~-

J~

/

tI

\ \

0.50

20.00

I

I,I

@.

4@

~

0.45

,

/

I

S

0"40 / @. @@

. . . . . . . . .

@.@@

]

. . . . . . . .

@,@B

I

. . . . . . . . .

@. I@ X

\~

"'', ~!~

/

, . . . . . . . . .

@. 15

,

@.2@

Fig. 7. Influence of the nonlinear electron-phonon coupling on the isotope effect for the mixed phase (solid and dotted lines). For comparison we show the long-range ordered phase (dashed and long-dashed lines). We have Co/Ah=2.44, C2/Ah=8.5, p=0.27, EI2/B=5.11, 6=1.6, 7=0.55 eV/A; (A) q=0.05 eV/ A 2 (solid line, dashed line), (B) q= 0.85 eV/A 2 (dotted line, longdashed line).

piing term_ El2/B (see figs. 5 and 6 for the phase r]) # 0, ~ / = 0, q~A = 0 ). Especially, values of a@ quite larger than the BCS result ( a o ~ 0.5 ) can be reached. Changes of E~2/B influence stronger the superconducting transition temperature in the low-temperature phase than for X>Xo (fig. 6, dashed line). The

\ -15.00

3

\ C\

0.35

~Z

x\

l

\ x\ \

0.30

025 . 0.00

.

.

. 0.10

0.20

\

5.00

,~ . . . . 0.30

X Fig. 9. Isotope effect and superconducting transition temperature using a phenomenological Ansatz for the density of states at the Fermi level. We have Co/Ah=2.44, p=0.35, E12/B= 1.2, 6=1.6, 7=0.22 eV/A, q=0.1 eV/]~ 2, N ( x = 0 ) = 0 . 5 eV -l, N~ =0.4, Xk=0.1, F2=0.01; (A) q=0.1 e V / A 2 (solid lines), (B) q=0.1 e V / A 2 (dashed lines).

378

R. Gutierrez et al. / Influence of structural instabilitie~

dependence of the isotope effect on the different model parameters near the structural instability has, however, to be taken with care because of the strong changes in the electron-phonon interaction which can occur in this region due to critical fluctuations. The quadratic coupling can affect drastically the behaviour C~o and To. Its influence is presented in figs. 7 and 8 for the investigated phases: r/) # 0, q7 = 0, q~A=0, q~A # 0 and the long-range ordered phase, which we show for comparison. At first it is to be noticed that the existence of a glassy state strongly reduces the isotope effect. We believe these effects can be influenced by the choice of other distribution functions for disorder fluctuations. For both phases the nonlinear electron-phonon coupling causes a reduction of the isotope effect in the low-temperature phase. By an appropriate choice of the model parameters we could get a zero isotope effect or even an inverse one. Contrary to the long-ranged phase, a strong quadratic coupling seems to have a quite weaker influence on C~o in the mixed phase. The changes of Tc in this phase are also clearly not so relevant. The strongest effect of the nonlinear coupling on T~ is found in the low-temperature phase (fig, 8 ). This means the pure nonlinear electron-phonon interaction term in the Eliashberg function, which is proportional to (qi')2<(Ut~)2), yields only small corrections to the linear contribution. This result is in agreement with general arguments concerning the effect of higher-order electron-phonon coupling and model calculations [15-17,20] Nevertheless, if a phase transition takes place, the linear coupling term is renormalized by a term 4(qa)2(r/~) 2. This contribution can drastically influence the superconducting properties. For very strong electron-phonon coupling, 7, q>> 1, we get naturally quite large transition temperatures, but the features in the isotope effect are washed out. Above and below the transition point we get nearly the BCS result and at x0 there is only a very small jump. Generally speaking, large transition temperatures and strong effects in C~o do not seem to be compatible in the frame of this theory'. Next, we consider a phenomenological concentration dependence in the density of states according to eq. (37). The results are presented in fig. 9 for the phase with long-range order. The results were calculated for a dominant linear (solid lines) and qua-

dratic (dashed lines) coupling. We see now that the superconducting transition temperature is considerably enhanced and reaches its maximum below the phase transition point .vo. Above the transition point it decreases rapidly to zero. The isotope effect, in contrast to the first calculations with a constant density of states, decreases in the high-temperature phase and shows a pronounced curvature below Xo. The main anomaly remains, however, at the phase transition point.

6. Conclusions We have investigated in this paper the influence of lattice instabilities on the superconducting transition temperature and the oxygen-isotope effect for a model with a two-components order parameter and electron-phonon coupling linear and quadratic in the order-parameter fluctuations. Due to the used approximation scheme and the simplifications contained in the model, a direct comparison with the experimental findings in the La system [9] is not possible. Especially, we have not dealt with the question: how do changes in the structural symmetry alter the electronic structure of strongly correlated electrons in the Cu-O units responsible for superconductivity in the majority of high-T~, compounds. This has to be the next step in a theory' only stated in its contours in this paper. We have merely investigated renormalization ell fects of the electron-phonon interaction in systems undergoing structural instabilities. The results show that this influence on the superconducting state is substantial. The isotope effect is strongly susceptible to changes in the characteristic parameters of the phonon dynamics. The general picture is, first of all, enriched by incorporating the (theoretical) existence of a structural glassy state. A progress towards better understanding of the concentration dependence of the isotope effect would be possible by a more realistic treatment of disorder effects, e.g. by an appropriate choice of the distribution function of structural fluctuating quantities. Then the following scenario is thinkable: starting with small defect concentrations. a pure long-range ordered state will be stable. Increasing x a transition may take place into a mixed

R. Gutierrez et al. / Influence of structural instabilities

state (at a concentration x¢ < x0) with both long- and short-range order causing a strong decrease in the isotope effect below the transition point to the hightemperature phase. We have looked for a crossover by investigating the free energies of the different phases; however, no indications for its existence were found in the model presented; the mixed state is thermodynamically stable for all concentrations in the investigated parameter region. We think this can be the result of the peculiar order-parameter distribution that we have used. Concerning the electron-phonon coupling, we have found out that the nonlinear electron-phonon interaction can be relevant in the presence of structural instabilities in agreement with early results (see, for instance, refs. [ 16,20,49 ] ). The strong changes in Tc as a consequence of a concentration-dependent density of states at the Fermi level show that the consideration of disorder effects in connection with a realistic description of the electronic structure is of interest for understanding the T¢(x) behaviour in HTSC's. In all calculations we find a non-vanishing Tc in the high-temperature phase. Recently, Takagashi et al. [50] reported that superconductivity was only detected in the orthorhombic phase of La2_ ~rxCuO4. This would support the idea that structural instabilities play a major role in determining the superconducting transition. Experiments on pressure effects in La2_xMxCuO4 ( M = B a , Sr) carried out by Yamada and Ido [ 51 ], do not seem to support those results. It may be that the strong decrease of T~ in the vicinity of Xo, found in the calculations with a concentration-dependent density of states are related to these experimental facts. Finally let us point out other lattice anomalies, which can have an influence on the superconducting properties of HTSC's. Neutron-scattering experiments [ 52 ] lead to indications that in LaSrCuO precursor clusters of the low-temperature phase are present in the high-temperature one. This is consistent with the observation of a central peak in the dynamical structure factor for this system [ 53 ]. In general, this behaviour seems to be characteristic of perovskite systems [27]. A possible theoretical description was given in ref. [ 54 ], suggesting a dynamical freezing transition above the structural phasetransition point. In the framework of this theory one

379

could suggest that the incoherent fluctuations of the correlated clusters completely suppress the superconductivity; this would only set in after the transition into a long-range ordered state. We are looking now for improvements of the theory in this direction.

Acknowledgements The authors would like to thank E.I. Kornilov for the technical support by solving the self-consistent set of equations, S.L. Drechsler for useful comments and N.M. Plakida, E. Olbrich and S. Flach for interesting discussions. One of us ( J H ) thanks the Bundesministerium f'tir Forschung und Technik for the financial support (Project Nr. 0117/010).

Appendix A Additional properties o f the E P C m a t r i x elements

Let us consider some additional properties of electron-phonon coupling terms. ( 1 ) According to the symmetries of the problem the following equations have to hold: Igt~l = Igt:2 [ , [gt~ l = Ig t ,21 ~ I

,

Iq11=lq~[.

From this follows 22

..1- ~ 1 1

g lm =

- - ,~ l m ,

gj2 = + gl~ , q) =q~ = q , or 22 glm

t ' ~ l 1 "~* =

-+

I,e~lm

]

gt~ = + t,,2L ~* -- k,Slm)

ql = q 2 = q ,

,

,

(39)

where g~'m ~ is in general a complex function and q~ is a real quantity. Due to the mode's degeneracy in the high-temperature phase, both components have to couple in the same form to the electrons independent of the coordinate system used. This sym-

380

R. Gutierrez et al. / Influence of structural instabilities

merry helps to select from all possible combinations in eq. (39) those which are physically relevant. There remain (A)

22

II

glm = --glm 12

21

lm (gy,,~) = l m (g,~5'})

have to hold. (3) Finally, let us consider the quantity y~'" introduced at the end of section 3 (see eq. ( 3 5 ) ) . It is ( 7 ~ ) 2 = 2 ( g ~ ' ( g ~ l )~v. p +gv-i5 =(grJ2

g lm = g l,n

q~=q2=q

g]~,= + -

q~ = q 2

11 *

(g,r.)

gll2m = g ' / m21

).v

l~, -i-U~.P+7 i i , . i2 p +gv ( g v ) tgv ) g~ ) .

or

(B)

(43)

(y22) 2 = 2 ( g 22 ,gvt~T~* + , P gv'21,gv(~U]; p, ,

+g~ (g~2). P + ( d =' ) *--g /:~

p

).

,

q.

(40)

For the first case (A), for instance, the corresponding interaction term for the linear coupling looks like H°m = E [g,m((a,-(a~)+g,,.(#-~,,,): 11 l 22 l.m

( 7 ' z ) 2 = ( ~ ' 2 J ) 2 = 2 ( g ~ ' ( g ~1=*~' ) +gp~"~(gu~)*P ,

gp (g~

)

(44)

Because g~': is an imaginary quantity, we can write it as g~,'~ =i0~ ~ and define .

.

.

.

p

P

gv/x/~ (gp#/z ) * -- -O p,l~z Op#IL =(g;,~)2

12 1 2 1 +g#.(~, -(a~) + g l 21 m ( ~ l --(am ) ]c? T'3Cn,l , H ° m = ~: [g,m " ((a,, -(aL _ # + (a2,)

g ~ " ( g ~ ' " ~ v - t- ~" ,7 ~ V~v - - ( g /--' " ) 2 = ( g m ' )

.

(45)

l,m

In the next step we have to impose the condition, evident from symmetry considerations, that ( 7 ~~) 2 = (~,22) 2 = (7 J2 ) 2 = ( ~,2~ ) 2. Then follows, using eq. (44)

12 1 +g+r.((a,. -(am2 +(a~ --~O'm) ]c? ~,Cm ,,

l,m

-

[glm( (al --(am)

~z + --(am+ ) ]eft- ~3Cm . + glm((al

(41)

Here is (az-+ = ½x/2((a) _+(a~). One has further to demand Ig+~ I = Ig+~ I. For the same symmetry reasons we find the only possible condition is 12 \ * g]~ = + tg+m } • In this way we guarantee that both modes couple symmetrically to the electronic system. A similar treatment can be done for the case (B). (2) The condition eq. (23) allows us to find an additional symmetry of the nonlocal linear coupling in the real space. Using Re(g~ ~) = 0 , one obtains --

[

(),11 ) _

"

(~,22)

2=:~. ( 0 1 I

- 0 ~ -vv) 0 vI '- = O .

We then have either 0 ~ = O~2 or 0~2=0. If we consider a general case with a non-vanishing non-diagonal coupling we have to choose the first condition. Inserting it into eq. (44) one finally gets an effective linear coupling ~,= (711)_~= (~,2:)2= (7 i_') : = (~,2J)2=2 ( ( g l , ) 2 + (gJ2)2+2p2) , p2 = 0~ 0p,2v.

( 46 )

[1 i~, ~ R e l - - ~ e ' "g~m~ / =

k,N +,,,

J

Appendix B

1

N /~ [c°s(qR+~)Re(gf"~) +sin(qR+m)Im(g~m ~) ] = 0 ,

(42)

Absence o f terms linear in the order p a r a m e t e r in the Eliashberg f u n c t i o n

so that the conditions Re (gym~) = -- Re ( g ~ ) ,

Using the imposed symmetry conditions on the linear coupling (eq. ( 2 3 ) ) we will show that all con-

R. Gutierrez et aL / Influence of structural instabilities tributions in the Eliashberg theory which are proportional to t/and, therefore, not invariant with respect to the transformation r/--.- r/, have to vanish identically. We treat only the nonlocal linear EPC, since terms which contain contributions from the local linear EPC vanish in any case. In order to simplify the following calculations, we consider only a perfect lattice. The results are not affected by this assumption. The equation of motion for the electron Green function (already omitting the local coupling) reads

381

with G O (o9) = (o9'o - % * a ) - l , 1

~,= ~ E e""t°', l,l'

and

Pu,(og) =

E E Z gy,e',3((c,,,ay,~',~yc?,

) ) w,~r* 3 g l ,~, l'

Itit" ~1,,' 11,l"

+ 4 ~ qf qyr/fr/y *3( (ctuCu~/.c~ ))o,i,~ *3

o9.o( (c,c? ) )0, =

Itu

+ ~. qUq~((ctS(uf) z, ~(uy,)2c¢-, ) ) ~ . 3 ,

&vto + ~ tw,((ct,,cit, ) )0,*3 Ire

It,t)

+ Y~ Y g~e'((cv,[~-~,;], c/t, ))0,,3

Zw(o9) =

Itit, l"

+ Y, qit((cl(~') 2, c~))0,,3 •

(47)

2

~ 2 Z g~t#"qvrl~'*3( It/z ~ P

/"

V~ ~ ~

h

( e l "A~t'u'', U~'C+ )

)~,~'3

It

We divide the coordinate ~ ' into a thermal average and fluctuations,

+ ~ ~ Y~g~e,'q%((c~.AW, O(Uy,)2C?,))~r*3 Nit' v l n

~of( t ) = t l f + uf( t) (uf)2(t) = ( ( u f ) 2) + cS(u~')E(t) .

+ Y, ~ ~, g~Vl;qU*3((Ct.6(U~) 2, A~;C~ ) )~r*3 , l,l:' It I1

Inserting this into eq. (47) it yields F.,(w)

o9*o( (c, cit, ) )0, = %6w + ~ to,,( (ce, c~, ) )0,*3

2 ~ q~qVr/~,*3((ClU~,t~(U~,)2C? ))co irr "3

+ E Eg~,~,;((c,,,afy,;,c?))0,,~

+2 if? qitqVt/~*3((Clt~(U~) 2, uvcv v + ))0,irr *3.

/tit' I"

It

+ Y, qit((c~6(ut') ~, c~))0,.3,

(48)

It

with

a~#'=uf - u ~ ' ,

The terms in Fu, (o9) are, in general, non-zero. In the framework of the pseudo-harmonic approach, however, they do not give any contribution, as one sees after decoupling the corresponding correlation functions with the Migdal theorem,

( c l ~ ( u f ) ~, uy,c + 5i~--,

to,, =tu, + ~ g~e,'(q~'--rl~:) +ate ~, qit( (q~,)2) .

( c~ c~) ( u~,~(uf) ~) = 0 .

It

In the next step one writes the equation of motion for the Green functions ((ce, A~e,'cit, ) ), ((c~uf, c ~ ) ), ( ( c:8 ( u ~') ~, c ~ ) ) and substitutes these in eq. (48). After some manipulations we obtain a Dyson equation for the electron Green function [ 49 ]: G,(og) = G o (o9) + G o (o9) (Z,(og) + P,(og) + F,(og) )G,(og),

(50)

Itv

+ 2 ~ qitqf( (ctu~', c~, ) )0,*3

/*it'

=

(49)

If they do not vanish, these terms in Fu, (o9) are invariant against the transformation q ~ -~/, then the corresponding phonon Green function ( ( 8 ( u f ) 2 , uY, )) is proportional to r/(as one can see by writing the equation of motion) so that Fu, (o9) ~ r/2. In what follows, we show that Zw (o9) has to vanish identically as a consequence of the symmetry

382

R. Gutierrez et al. / Influence of structural instabilities

properties of the nonlocal linear coupling (notice that Ztt, (co) is not invariant by a change in the sign of q). We limit ourselves to the first two terms (denoting them by Z~l!)(o))) in the sum; the others can be treated in a similar way. Exploiting the decoupling

((ct(ul'-ul;'), u;,c; ) ) oi~~
expression in parentheses vanishes identically. By the same procedure one can show that the other terms in Zq(~o) vanish too. Therefore 7q(~O)=0 holds and we have no linear terms in q in the electronic selfenergy operator. The further treatment of Pq (co) yields in the frame of the usual Eliashberg theory an expression for o<2F(o~), as given in the text.

~-
~

Appendix C

~ g~,U,'q~qY,gt,t,,(d~Y-dY,;[,) Free energy .lot" the two-components lattice model

+ ~

~ gt.t,~"' ~f'gt,~, ( d~," - df~Y' ) ,

( 51 )

IZPPP ll

where gtt, = %z3 and d~,~ = ( ufuy, ). Transforming the expression (51 ) into the q representation this yields

Z~'>~ o-q'

+ ~ ~p'p

ge

For getting an estimation for the free energy of the lattice model (eq. ( l ) , section 2) we use the Bogolyubow inequality [ 55 ] F<~ Fi = Fo + ( Hl~t -- t t o

dq_q,)~/{,q"

( ( p ) ) 2 + (p}):

/

(

Z o~""'a q t q~'- - q '

'

\p#'

~v'

( 53 )

which gives an upper bound for the exact free energy of the system. The index o refers to an appropriate trial hamiltonian; in our case we use a quasi-harmonic hamiltonian with the form with the form

tc "~v'rtu~ ,e~q ~q-¢,-g~,~'dUo~-q, , )rl~q u :

, Z gq' ~ gU¢e'~g-q' -

)..

H o = ~ \ 2rn

2t~

-~- ~e~l I 2 ( H ] ) e _~_ 1 0 221~

2I ) e - ~ 2 2 / g )

U/

+¼ Z c , , . [ ( u ) - u l , , ) e + ( u ~ - u ~ ) gq

q - q ' - Z ~.,

,-.').

(52)

with Mv v u o'aU_~,= Y~ dq-~,~/t q • p

Let us consider the first two terms in eq. (52). Interchanging the indices v*--*v' gives Z gay ,u ' ff~_¢, # v . -- ~ gq,v,v(~q--q'

Now we rename the sum indices in the second term v, v'*-,#/t' and find

Zq(,)

1

~ ¢2 g,, Z (g~/~'

- - g q eu',,

)O'q__q, ~ .

Using the symmetry relation o f section 3 derived from the equilibrium condition we see that the

[,t~l

2] .

I54)

Usually, the quasi-harmonic frequencies ~2~, are used as free variational parameters in order to find a local m i n i m u m of the free energy. If we employ the frequencies obtained by the equation of motion method (20), we get the same expression for the frequencies as with the variational method, i.e. they minimize the free energy, too. Introducing ~0f=~/~'+u~ one finds ( H,a, - Ho ) =

.,t - 2

((q~'):+((uf)2) +~_((qf)4

+ 6 ( q y ) Z ( ( u ~ ) 2 ) + 3 ( ( u y ) Z ) 2)] -}- ½ 2

I

El2( < (~1)2) < ((/92)2)

383

R. Gutierrez et al. / Influence of structural instabilities

+4ql r/2 (ul u 2 ) +2(u~ u 2)2)

host places• The averages over the corresponding sites yields an expression of the form

+¼ Z G~(,l~')-n~) =+ ~ , 7 ~ ' Lm,,u

Ida

F1

f~ = ~

--h

= (1-x)F,o

--d

+XFlo

(56)

,

1 + 1#a=2 (

(u~)~> + va~=( ul u] > ],

where

((~o~')=) = (nf)=+ ((u~') =) .

--h

After diagonalizing Ho we readily find the quasi-harmonic part of the free energy Fo

F,o

=

sin ~,2-kBT))

Fo = - k~ T ln Ho •

+ln(2 sinh(2~Bh-T))]

ha)+

=k~ T ~ (In(2 smh(2-kaT) ) +In(2 sinh(2~a-T))),

Bi

with a)~

a~ +a~ + / ( n ~ - a ~ ) ~

- -

2

-N/

g

- IB( ( < (uJ)5> ")2+ ( ( (u2)~ > h)~)

+(E22~)1"

+ ½E12((~)2(q])~ h

Then the free energy reads F1 •

-2(ulu]

ha)+

kBT ~t [In(2 slnh(2--kBT) )

2(h))

+ ½( 1 - x ) C o ( x )

+In(2 sinh(2~-B-T))]

--d Flo

+ 7 ( (~1)4+ (rl])4)

_ A~

- IB( < (ul)~>~+ < (u])~> 2)

(~+~0)

2 Bt ((~3~"+~3 ~) + ~-

+ ½E'2( (nJ)2(q2)2- ((ul)=) ((u]) 2) -2=)] +*4 Z Z C',~(~'--~m) =

- IB( ( ( (ul)~> ~)2+ ( < (u~)= > ~)=)

t~ l,m 1

=

kBT[ln(2 smh(2--ka ' hwa+T))+ln(2 slnh(2--ka " ha)a-T))]

Bl

u

(57)

for the host places and

+ E [- T ((hi)=+

+ Y'. Z ~r~'.

( q ~ +q2E~)

(55)

According to the average procedure discussed in the main text (section 2) we divide the sums over the cell indices into contributions arising from defect and

_2 ~ ) ) 1

ld

2d

+ ~xCo ( x ) ( q E A + q E A )

( 58 )

R. Gutierrez et al. / Influence of'structural mszabilitie.s

384

In this way we find on combining these expressions an 8 × 8 matrix equation coupling host and defect sites. The coefficient matrix has the form

for the defect sites.

Appendix

D

Eigenfrequencies at q = 0

(60)

The soft m o d e in the lattice model considered condenses at the critical wavevector q = 0 . In order to find the soft-mode frequencies we should r e m e m b e r (see remarks in section 2.4) that for this point in the M F A host and defect sites are coupled to each other, so that one has to reconsider the equation o f m o t i o n for the G r e e n function. In the first part of this appendix we get a set o f equations for d e t e r m i n i n g the soft-mode frequencies in a general manner. In the second part the frequencies for the special solution tl', q2A # 0 , q2, qlA= 0 are treated separately (see section 5), iterating the equation o f m o t i o n up to second o r d e r in the d i s o r d e r fluctuations. ( 1 ) The equation o f m o t i o n reads (section 2.4, eq. ( 1 5 ) )

m(o) 2 - ~ , )2D l , ,SiP = h S / ' V (~lrrt +

E

~ g ' ) 2ul'L* FIP[d__ ¢1t~, lnl E

(59)

Cln o ,u,8 ....

The frequencies have the same meaning as in section 2.4. F o r the averaged G F the last term on the r.h.s, at q = 0 is non-zero. D i v i d i n g the lattice sums in this term into host and defect contributions, one gets for the host places h-~-

C+

~7.~ ~

where

~7~<'= (.c2,2) 2 ~,<' r , , C =xC'o(X)*o. C

+ = ( 1-.v)Co(_V)% ,

and

The diagonalization o f the ~ matrix was done numerically. The condition that £22( T = To) = min(co2 ( T = To) ) (e)~ are the roots of the characteristic p o l y n o m e ) has to vanish at the phase-transition point was used to determine the softm o d e frequencies. (2) Let us now consider the special solution ~/~, q~A @ 0, q~, qlA = 0. Introducing m(o2~--Q~) = h ( D o ) , we get from eq. (59) a sort o f Dyson equation Df~,~=

h

m

p

(Ss"aimD(¢ + ~ Dr{ Y. --s,,*-'t,,, 02 r,,,l* ~----h

-

1,7~,t

d

ha *'~- (1 -x)Co(x)D~'"=o -xCo(x)DD'~=o ---

h

where D¢=o v, _- Z , , D z m u, -- d - - u B

K/'v D q f l o

d

111

D Dt;Y~

h

n

and similarly

: ----h

ha s'B- ( 1 -x)Co(x)Dg~=o -XCo(x)D~'~=o ~

for defect places, with K/t~'=

------h

--h

Dg~=o =du"O(~ //IH~2

u h

+[g)

E E

lJ~Jl

mD 2 --"('~I,) 6 ,ul, + £ 2 1 22 ( 1 - d " " ) .

"~

Iterating up to second order in the non-diagonal frequency (in the disorder fluctuations) and averaging under the condition l = h ( d ) one finds (with £222 = 0 )

-- D (0)2

'

l ) v4

h 0

(Cb(X)(I

h ~2

O2

t'

--~")Dq=o ~#

h

h

R. Gutierrez et al. / Influence of structural instabilities

+ Co(x)xD~o d) , and a similar expression for the defect places. Averages like ~,a c~2Uu.t.." nup q = 0 can be truncated due to the special form of the distribution of disorder fluctuations (see eq. ( 9 ) ) . The diagonalization of the obtained set of equations can be done in a similar manner to the foregoing case. It has only to be noticed that in-2 stead of 022 , which vanishes, the new quantity (022) 2 appears. It takes the value (022)2 = 4 (E~2) 2( 0/I )2(q2)2 +(ul

,2 u,2 ) 2 +2r/,rh (uJu~))

,

or approximating (-l-t" / ~ ( u l ul we finally find (022)2

h,d

--

=4(E~2)2(n ~

h,d

)2q2~,d.

_

22 = 0 ,

(61)

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