GROUND-STATE ENERGY OF FRÖHLICH ELECTRON-PHONON SYSTEM

(1)

HAL Id: jpa-00221205

https://hal.archives-ouvertes.fr/jpa-00221205

Submitted on 1 Jan 1981

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

GROUND-STATE ENERGY OF FRÖHLICH

ELECTRON-PHONON SYSTEM

Chih-Yuan Lu

To cite this version:

(2)

JOURNAL DE PHYSIQUE

CoZloque C6, suppZlment au n o 12, Tome 42, de'cembre 1981 page c6-481

Chih-Yuan Lu

I n s t i t u t e of Electronics, National Chiao Tung University, Hsin-Chu, Ta'aiwan 300, Republic of Chinu.

Abstract

-

Recently Feynman's path-integral formalism of the Frahlich optical polaron problem is generalized, by which it is easy and natural to get the second-order perturbation result in the weak-coupling case and the Pekar's result in the strong- coupling case, even in the crudest ground-state approximation.

By using numerical method to the equations which are given by the ground-state approximation, the ground-state energy of the electron-optical phonon system can obtained for the overall range of coupling strength. In addition, a self-consistent set of equations is derived for a central optimized potential without using the ground-state approximation. This effective local, central potential is the solution of a linear integral equation which was similar to the extensive work of optimized-potential model (OPM) in atomic physics. By using OPM formalism, no par- ticular choice of interaction form is taken, therefore, except the approximation by Jensen's inequality, the ground-state energy of the electron-optical phonon interacting system can be obtained without any approximation.

The Hamiltonian of the idealized Frohlich optical polaron problem is given by

-+ 3

where we use the units fi=m=w=l. p, x are the momentum and coordinate operators of electron, pj, yj are those of phonons of mode j, and the

+

interaction terms W . (x) q are ( 8 n no/V) 'CCOS

( 2

):a /k .+sin (kj );- /k

.

lq

3 j j I I j.

a is the dimensionless coupling strength. The generalized model comes from an intuitive belief that in some sense the reaction of the

lattice system to the motion of an electron might be represented approximately by a fictitious particle coupled to the electron. We assume the variational Hamiltonian as

Feynman has used a specific form of interaction-harmonic interaction between the electron and the fictitious particle. We formulate the upper bound of the polaron energy for the general form of variational

(3)

C6-482 JOURNAL DE PHYSIQUE

,

potential v(x-R), then the result is given by

C

1

I

where C =

M/C

2 ( ~ + 1 )

'

1

( 4 ) and = E~

-

E

0 (5

-t

M is the reduced mass, p=M/(M+l), and ui(r) and si are the eigenstates and eigenvalues of the SchrGdinger equation of the system

From Eq.(3) it is obvious that if we take only the ground state term (i=O only) in the energy expression, because every term is posi- tive in the summation and of decreasing importance by the energy difference denominator of increasing magnitude of i, then the right hand side of Eq.(3) is still an upper bound of the polaron energy. Even withim this ground-state approximation, the second-order perturbation result in the weak-coupling case and Pekarls result in the strong-coupling case is obtained very naturally.

c

11 By using numerical direct integration or Ritz variational method the ground state energy can be obtained for the overall range of coupling strength, and the phase-transition-like behavior such as abrupt changes of slope of the ground state energy at some coupling strength ac is found

c

2

1

similar to that of Shoji and Tokuda. C3l

In order to calculate the Eq.(3) without using ground-state approximation, a self-consistent set of equations is derived for a central optimized potential, this effective local, central potential is the solution of a linear integral equation which is similar to the extensive work of optimized-potential model in atomic physics. C41C5l The v(;) above is to be varied to minimized the functional Ev, therefore a system of self-consistent equations is derived for this pro-

+

blem, in which v(r) is the solution of an integral equation. It can be seen that in our case the exchange terms in atomic physics will not appear. The variational problem to be solved is

(4)

+ u; (B~,(;)

I - ~ x ~ c - ~ c

(~+AE~)'

I;-:'

I

where W(rU)=

-

~ J d z -( As

+

1 (9) i,o Jrp i 6u. ( $ I ) 1

-

_-

_-Gi (;I + -+ and

,

r) ui (r) 6v

(z)

+ -f u*

( Z *

) u

.

)(; where G r r = C jfl (Ej

-

E ~ )

The results (8) and (10) can now be substituted into ( 7 ) , and if the Eq. (6) is used, it is found that

_,.

-+ -+ -+ -+ -+

where H(;', r) = u2(r1)Go(r1

,

r)uo(r), (13)

Therefore the variational problem has now been reduced to the problem of obtaining self-consistent solution of Eqs. (6) and (12).

In our problem there is another parameter p which should be determined by another variational equation. But it is suggested that we may choose some appropriate value which might come from ground state approximation or Feynman's harmonic model to the Eqs.(6) and

(12), and then substitute the solutions of the self-consistent equations (6) and (12) into the aEv/ap=O to check our choice, or we may plot the Ev versus p to determine it.

References

1 J. M. Luttinger and Chih-Yuan Lu, Phys. Rev.=, 4251(1980). 2 Chih-Yuan Lu and Chi-Kuang.Shen, to be published.

3 H. Shoji and N. Tokuda, J. Phys. C:14, 1231(1981).

4 J. D. Talman and W. F. Shadwick, ~ h 5 . Rev.=, 36(1976).