The stability region of bipolaron formation in two and three dimensions in relation to high-Tc superconductivity

Solid State Communications, Vol. Printed in Great Britain.

76, No. 8, pp. 1005-1007,

The Stability

1990.

0038-1098/90$3.00+.00

Pergamon Press plc

Region of Bipolaron

Formation

in Two and Three Dimensions in Relation

to High-T,

Superconductivity.

G. kierbist, F.M. Peeters and J.T. Devreese*

University of Antwerp (WA), Department of Physics, B-2610 Antyerpen, Belgium. 2 aug,ust 1990 by S. Zundqvist)

(Received The

groundstate

formalism.

of a bipolaron

A scaling relation

two and three dimensions.

is obtained

We found that

in two than in three dimensions. of Emin’s

theory,

as a mechanism

Bednorz perature

and Miiller’s

superconductors

within

is derived between

for high-?:

the bipolaron

The relevance

which describes

the Feynman

the free energies stability

of these results

Bose-Einstein

path

integral

of a bipolaron

is discussed

condensation

in

region is larger in view

of large bipolarons

superconductivity.

discovery1

of the high tem-

stimulated

both

Early Ritz

experimen-

studies4

variational

of this system

principle

mation

proposed Bose-Einstein condcnsa.tion of als 2. Emin large two-dimensional bipolarons into a superfluid state

with (twice) the single polaron groundstate energy, calculated within the same formalism. Since the latter

as a possible

quantity

mechanism

in these

responsible

for the bipolaron

for superconduc-

materials.

In this paper

a path integral

ditions

study

exist in the copper oxides. Some experimental difficulties in determining material parameters, like the bandmass, are also pointed out.

The

electron

together

with

a polaron.

Two

different

tion.

and with

possible

bipolaron

is a state

but with a larger internal

degrees

of two polarons

to each other through the mediating teraction. The distinction between bipolarons polarons.

is analogeous

(effective)

mass

of freedom.

The

which

very

for the existence

He finds a narrow

freedom

e.g.

favourable

from cool-

of bipolarons.

energy using a singlet an unitary transforma-

stability

region in three dimen-

within the path integral formalism alelimination of the phonon degrees of

just like for single polarons5.

Because

the free

energy, instead of the groundstate energy, is calculated, the generalization to non-zero tern erat,ures is straightforward. Hiramoto and Toyozawa K included two types

are bound

polarization insmall and la_rge

on large bipolarons,

were compared

to known results,

formalism516,

were obtained

Calculations low for the exact

of electron-phonon cal phonons,

to the one made for (single)

We will concentrate

energy

estimat,ions

3D.

are dis-

tinguished: a) the small polaron, where the polaron is localized, and b) the large polaron. which behaves much like a free electron,

poorly

The

sions. Recently, Bassani et aL8 have generalized this approach and they also reported results in two dimensions, where a larger stability region was found than in

its polarization cases

groundstate

imalization of the groundstate wavefunction after performing

Bipolarons are closely related to (single) polarons. The latter term is used to describe an electron in interaction with the lattice vibrations of the surroundfield is called

integral

cloud.

Adamowski7 resolved this difficulty by comparing with path integral results for the single polaron energy. His approach to the bipolaron consists of a numerical min-

of large bipolarons in both two and three dimensions. We will discuss conditions under which bipola.rons can

ing solid.

compared

the path

we will present

polarization

approxi-

tal and theoretical efforts to determine the mechanism responsible for superconductivity in these new materi-

tivity

for the

used the Rayleigh-

and the adiabatic

They

in

and

considered

interaction:

a) longitudinal

b) longitudinal

acoustical

only the three

dimensional

opti-

phonons. case and

which the electrons interact with the longitudinal optical (LO) phonons and the inter-electronic potential will be taken Coulombic.

emphasized the role of the acoustical phonons for the self-localization of the bipolaron. In a superconducting state, however, the bipolarons should bc mobile and therefore self-localization should be prevented. Follow-

*

ing Emin our aim is to study mobile bipolarons and thus we will concentrate on the LO-phonon interaction.

Also at: University of Antwerp (RUCA), B-2020 Antwerpen (Belgium); and Eindhovcn University of Technology, NL-5600 MB Eindhoven (The Netherlands).

Kochetov et al.1° reported results for the two-polaron 1005

analytical problem

strong-colll)lillg in thrre tlimcll-

1006

BIPOLARON FORMATION

sions within the path integral formalism, taking only LO-phonons into account. Here we wiU consider the whole coupling range and present our results in two and three dimensions. The system under study has the following Hamiltonian

~=~ 2-~-~+Z.., ~ V't~ LO at,, [. £

+

v ~

+

j=l,2

15

I

The first term describes two free electrons with broadmass rob and the second term represents the free LO phonons, with frequency WLO. The last terms are the interactions: the Coulomb repulsion between the electrons and the Fr6hlich interaction 11 between the electrons and the LO-phonons. In what follows we will use a unit system such that h = m b = WLo -= 1. The problem is characterized by two parameters: i) the electronphonon coupling constant a, which is hidden in the interaction coefficients V~ ( tV~p = 2 v ~ a / ~ ~- in 3D and IVI] 2 = v / 2 r a / k in 2D), and ii) the strength of the Coulombic repulsion U. Both parameters are related to the static (e0) and high-frequency ( e ~ ) dielectric constants of the crystal: a = e 2 ( e ~ ' -- %')/V/22 and U = e2/eoo. Since the dielectric constants are positive quantities only the region U _> v'~a of the (a,U)-plane is realisable in a physical system. After the elimination of the phonons, we use the Feynman-Jensen inequality to determine variationally an upper bound to the free energy. As a model system for the bipolaron we introduce two oscillators to represent the phonon clouds around each electron. All interactions are then replaced by harmonic interactions, which results in a quadratic trial-action after the elimination of the extra oscillators. The proposed trialaction is generalization of the one of ref. [9]. An important property of our trial-action is its translational invariance, which ensures the conservation of total linear momentum. This is necessary in order to have mobile bipolarons. Within this formalism, a scaling relation is obtained between our estimations of the free energy F in two (2D) and three (3D) dimensions

=

2 ~ .37r 37r U 5/'3D(-'~O', 4 ) .

I

(2)

Details of our calculations will be presented elsewhere 12. Here we will concentrate on the numerical results for the stability region of bipolarons at zero temperature. Figs. 1 and 2 show the phase diagram for bipolaron formation in three, respectively two dimensions. Our results indicate that in both cases the non-physical region (U < v'~a) is entirely bipolaronic. Below a critical value ac the bipolaron/single polaron transition is continuous and occurs at U~ = x/~2a. For a _> ac the transition is discontinuous and 1)ipolarons do exist for physically acceptable values of o. and U. The stability region for bipolarons (shaded area) lies below the full curve, while the dotted line (U = f i n )

I

¢- 10 o

8

I

Bipoloron,,

/

~./%=4/5o%-:..'"

" ~

o

~,6 8

"

/•/.*

f:k • Of

~. . 5 " " " - V = " J ' f f a

5

/.. .¢'.

F2D(~ , U)

Vol. 76, NO.

{D

(1)

+ E Z,

IN TWO AND THREE DIMENSIONS

s*"

.x.

--"

3D PHASE DIAGRAM I

I

I

!

2

4

6

8

0 0

/..

10

Coupling constant a FIG. 1 The stability region for bipolaron formation in 3D. The dotted line U = v ~ a separates the physic a / r e g i o n (U >_ ~/2(~) from the non-physical one (U <: v ~ a ) . The stability region lies below the full curve• The shaded area is the stability region in physical space. The dashed line is determined by U = x/~c~/(1 - eo~/e0) where we took the experimental values 14 coo = 4 and e0 = 50. The critical point a= = 6.8 is represented by a full dot.

15

I

I

I

2D PHASE DIAGRAM ¢O (n

10

3olaron Region

CL Of

5

point ac~,2.9 .4;"

,~ / " - U = x / 2 a i"

0

0

J

i

i

i

2

4

6

8

10

Coupling constant a FIG. 2

The same as Fig. 1, but now for 2D, where the critical point is ac = 2.9.

separates the physical region from the non-physical one. We found numerically ac ,-~ 2.9 in two dimensions and ac ~ 6.8 in three dimensions, these l)oints are 1)resented as full dots on the figures. Note that the stability region is larger in two than in three dimensions and, maybe more importantly, in two dimensions bipolarons exist at smaller and more realistic values of the coupling constant a. In order to discuss the relevance of bipolaron formotion in the copper oxides, we have to estimate o and U. First we rewrite these quantities as

Vol. 76, No. 8

~;~) and u (e~mb/2h2)/(~,Lo) is

= v~ (e2 where ~ =

BIPOLARON FORMATION IN TWO AND THREE DIMENSIONS

= v~/,~,

(3)

the ratio of the e/-

fective Rydberg e4mb/2h 2 to the LO-phonon energy hWLO. Note that mb is the band mass, not the free electron mass. As in single polaron physics 13, accurate values of mb are hard to obtain since the electronphonon interaction will increase the bsndmass mb to the effective polaron mass m*, which is the one measured in e.g. a cyclotron-resonance experiment. We can, however, eliminate the mass dependence by eliminating A between a and U. In doing so we obtain the relation U = V'2a/(1 - eoo/eo) which is shown as a dashed line in Figs. 1 and 2, where we have taken Coo/e0 = 4/50 = 0.08 which is the relevant ratio in La2CuO414 at zero temperature. With these numbers we find U = 1.53a as the equation of the dashed line. From Fig. 1 we can decide that three dimensional bipolaron formation is not expected in La2CuO4. At low temperatures e0 is found to increase with increasing temperature 14 and the dielectric constant is very anisotropic (e0 = 50 in the a, b-plane and 23 in the perpendicular c-direction). Therefore two dimensional bipolaron formation (Fig. 2) seems quite possible. For

1007

example, in order to have bipolarons at a = 4 (5) the ratio coo/e0 should be around 0.04 (0.055). If we assume an LO-phonon energy of 70 meV and coo = 4, then the bandmass mb (in units of the free electron mass) should equal 1.4 (2.3) in order to have a coupling constant a = 4 (5) at e~/e0 -- 0.04 (0.055}. It is hard to speculate whether such conditions are realized in other high temperature superconductors. Recent experiments 15 on YBa2CusO6+x report only polycrystalline measurements, where tile anisotropy effects, which lead to such large values of e0 in La2CuO4, are averaged out. The present study indicates that bipolaron formation in the copper oxides is not unrealistic. Here we considered only the interaction with the LO-phonons while it is expected that the inclusion of short-range phonons like the optic deformation 3 and/or acoustic deformation 9 phonons will enlarge the stability region for bipolaron formation. Acknowledgements - One of us (FMP) is supported by the Belgian National Science Foundation. Financial support was provided by "Krediet aml navorsers, nr. 4.0006.90" of the N.F.W.O. and I.U.A.P.-ll.

References

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