GROUND-STATE ENERGY OF THE OPTICAL POLARON

PHYSICAL REVIEWB VOLUME 26, NUMBER 8 15 OCTOBER 1982

Ground-state

energy

of

the optical polaron

Chih-Yuan Lu and Chi-Kuang Shen

Institute ofElectronics, National Chiao Tung University, Hsin-Chu, Taiwan 300,Republic ofChina (Received 13May 1982)

The ground-state energy ofthe optical polaron iscalculated by the generalized path-integral formulation. The ground-state approximation ismade to simplify the complicated expression, and the Ritz variationa1 and direct integration methods are used to obtain the ground-state ener-gyin the whole range ofcoupling strength. The results agree with previous work, and it is found that there isatransition for coupling constant 0.

, =

9.2. The ground-state energy ob-tained by harmonic approximation, which isequivalent totaking Gaussian-like trial wave func-tions, is compared with those obtained by Pekar's trial wave function.

I.

INTRODUCTION

The problem

of

the motion

of

an electron in ionic crystals or polar semiconductors has been attracting the interest

of

many solid-state physicists for

de-cades.

'

The perturbation and the intermediate-coupling theories4 are valid when the interaction between the electron and longitudinal-optical pho-nons isrelatively weak and the electron behaves more or less like afree particle dressed with afew

phonons. On the other hand, the strong-coupling theory is valid when the interaction isstrong enough

to make the electron captured in a self-induced po-tential which is built up by the field

of

the correlated virtual phonons.

'

There are also some theories which interpolate between the weak- and the strong-coupling theories,

"

and itis well known that among all the methods, path-integral theory gives the best ground-state energy in the overall range

of

the coupling strength. '

"

The Feynman's path-integral formalism was restricted to the harmonic-interaction approximation and it isgeneralized" recently to un-specified general form

of

interaction potential. In this paper, we try to use the generalized formalism to

calculate the optical polaron energy by analytic nu-merical variational and direct integration methods through a ground-state approximation. It isfound, in

our ground-state approximation, that the result is

better than Feynman's result in an extremely strong-coupling case and agree with that

of

Pekar's; italso agrees with weak-coupling theories in small-coupling strength; but this result isstill inferior to the

Feynman's harmonic model in the intermediate range

because

of

the ground-state approximation and one

additional use

of

Jensen's inequality for the general-ized formalism.

"

If

within this approximation, the harmonic-interaction model, which uses the harmonic

I

uo (r

)u

Eo

~

E„=

(uolp'/2p lluo)

—

~

$

~~I

_J

&2p,

„0

r _r

wave function in the generalized formalism with ground-state approximation, will be inferior to the results obtained by analytic numerical variational method with Pekar's-type wave function and direct integration method. And the transition point

(n, =

9.2) of

the unspecified potential case is lower

than that obtained from harmonic-interaction approx-imation ('n,

=

9.

4).

II. CALCULATION OF THE OPTICAL POLARON ENERGY

Because the derivation

of

generalized path-integral formalism is very lengthy and tedious, in the present paper we only briefly write down the results from

Ref.

11.

The physical motivation

of

Feynman's

theory and Luttinger and Lu's variational method

comes from an intuitive belief that in some sense the reaction

of

the lattice (phonon) system to the motions

of

an electron might be represented approxi-mately by the reaction

of

a small number (hopefully,

one)

of

fictitions particles coupled in some simple way to the electron and to one another. In the most

simple case, the variational Hamiltonian is chosen as

where p, x and P, Rare the momentum and

coordi-nate

of

the electron and the fictitions particle,

respec-tively;

M

is the mass

of

the fictitions particle; and where we use the units

f

=

m,

=

coo

=

1,

m, is the ef-fective mass

of

electron in the conduction band, coois the frequency

of

the optical branch phonon which is taken to be independent

of

wave vector

k.

'

The optical polaron energy Eoisa lower bound

of

the variational energy

E„,

o( r

)u„'(

r

)u„(

r

)

1

—

exp[

—

2C(1+Do„)'

2~_r

—

_{r ~]}

1+De„

(2)

26 4707

4708

BRIEF

REPORTS 26

where p,

=M/(M+I),

be„=e„—

eo,and C

=M/[2(M+I)]',

and

u„(r)

and

e„are

eigenstate and eigenvalue

of

Schrodinger equation with the undetermined variational potential

v(

r

),

(4) Since each term in the summation

of Eq.

(2) is positive,

"

it is obvious that ifwe take only the ground-state term

(n

=0)

in the summation, then the right-hand side

of

Eq. (2) is still an upper bound

of

the polaron energy.

Therefore, in this ground-state approximation, we can write

ED~

E„—

=

'

uo (r

)

p'

uo( r )

—,

_~2p"'"

~

&

(uo(r)up(r

)('

[I

—

exp(

—

2C~ r

—

r

()]

~ '

Now, p, and up( r ) are varied tomake

Eo (p„uo)

as small as possible.

The above energy expression is equal to that

of

Pekar's theory plus a term

I

J

d d

'

'

'

(

—

2cir

—

r'i)

.

(5)

In the strong-coupling case, we can imagine that the mass

of

the fictitions particle

M

should be large,

hence C

~

to make this additional term extremely srna11. Since p,is less than or equa1 to one, therefore our result must be better than, at least equal

to,

that

of

Pekar's. In the weak-coupling case, p, must be small, and the exponent

of

expression

(5)

can be

ex-panded, and it is easily seen in this limit that the

po-I

laron energy is bounded above by

—

0., which agrees with that

of

the second-order perturbation

calcula-tions.

In order to calculate the polaron energy by Eq.

(4)

for the overall range

of

coupling strength, we first ap-ply the Ritz's variational principle with Pekar's-type

trial wave function which is shown to be extremely

accurate for optical polaron.

"

This trial function is given by

u(

r

)

=N

[I

+bpr +g

(bpr)2]e

where

(6)

N2= 2(b

p)

3/[w(14+

42a

+

45a2)1

Eo

_(p„,a,b

)

can be calculated analytically by using the following formulas: OO +1

exp(ikl

r

—

r

I)

₌

_k

_X'(k

)h, '

(k

)

_g

I;

(8,

$)

Y;(8',

$')

Jl tne—btdt e-bx

+

Jz

~

k)

y(n-k+1) fl

t"e b'dt=

'

—

e 40

The expression

of Eo

(p„,a, b) is very tedious and complicated. Itwill be included in the Appendix.

We use the direct-search method to find the

ex-tremes

of

the function

Eo

(p„a,

b) by adopting the Rosenbrock's rotating-axis algorithm, ' because it needs only the evaluation

of

the function. It isfound that there are four local minima forgiven coupling strength otthey are:

—

0.

108

504m,

—

o._,

—

0.

75o., and

—

_0.

10114o.

',

and the situations are found in the

Pekar theory. The convergence criterion

of

our computation is set equal to 10

".

Among these local minima, the smallest one is the absolute minimum which should be taken as the upper bound to the po-laron energy. Hence for o.&o.

,

=

9.21,

the upper-bound polaron energy is equal to

—

o._{, and} for

a

~

n„

the upper-bound polaron energy is equal to

—

_0.

108 504u2.

If

harmonic interaction is assumed, then the wave function is Hermite function and the energy

expres-(10)

I

sion in the ground-state approximation can be calcu-lated analytically, and isgiven by

0

0

1

0

1

E(Q,

ru)

=

—

₄

0

—a

—

exp

—

—

erfc

GJ m

0

0

(11)

where

0

= (k/p)'

',

cu

=

(k/M)

'

',

and kis the

Hook's constant

of

the harmonic potential

v(x-K)

=-,

'k(x-K)'.

The expression

(11)

has two local minima, one is

—

_{0., the other is}

—

_0.

_{10610. ,}the transition point will

be o.

,

'

=

9.

42.

Itis seen that the improvement

of

harmonic poten-tial by the optimum one is about 2% when

e

)

9.

21.

There is no improvement when u &

9.

21,

for in weak coupling the Hermite wave function is very similar to

that

of

Pekar's-type wave function in the limit

of

small k.

26

BRIEF

REPORTS 4709 Besides the Ritz s variational method, the direct-integration method isalso used to calculate the polaron

ener-gy. Since

E„

is afunction

of

p, and a functional

of

up( r

),

and the only constraint is that up is normalized, the stationary conditions for the best choice

of

interaction potential are equivalent to

r t

8

E„—

X

_J

up(r

)2dr

Sup(r)=0

5EP/Bp,

=0

Therefore we have to solve the self-consistent Hartree-type Schrodinger equation

p'

_nJ2

"

ap(r

)'

up( r

)

—

~ d

r,

[1

—

exp(

—

2Ci

r

—

r

i)]up(

r

)

=epup( r )

2p, i r

—r'i

(12)

(14)

for each p,

.

This is a prohibitively laborious numerical work, since for each given p,, the self-consistent Hartree-type equation needs many iterations to give the po-laron energy. Because the second term in the in-tegral

of

Eq.

(14),

although the ingenious integration

scheme

of

Miyake is used,

"'

the iterations are still very time consuming. According to our experience

in the Ritz's method, and it isalso shown in Miyake's work, Pekar's trial function is an excellent

approximation, hence we take p,

=

0when

a

&

n,

and p,

=

1when n &

0,

By this assumption we can easily find: when

a

&

a„Eq.

(14)

will be reduced to a par-ticle moving in constant potential

of

magnitute

—

_u, when n

)

a„Eq.

(14)

will just be reduced to Pekar's

model and can be solved by direct integration to ob-tain the exact value

of

the polaron energy.

"

There-fore, by direct integration, the upper-bound polaron energy is equal to

—

0.

108513m'

for cxlarger than

0,

III. DISCUSSION AND CONCLUSION

The optical polaron energy is calculated by the gen-eralized formalism under ground-state

approxima-tion, although the energy is higher than harmonic ap-proximation

of

Feynman's model and Luttinger and

Lu's"

work which include a11the excited states; it is shown that the result

of

the optimum potential ap-proach is better than that

of

harmonic approximation

ifthey are both under ground-state approximation.

We also find a phase-transition-like behavior at

u,

which also occurred, in the work

of

some other

au-thors,

e.

g.,Gross, Larsen,' Luttinger and Lu,

"

Manka, '~Lepine and Matz,' and Shoji and Tokuda. '

Within our approximation, the mass

of

the fictitions particle changes abruptly, as coupling increases, from

zero to infinity, whihc shows the abrupt change

of

I

the polaron state from nearly free type to

self-trapping type. However, from the work

of

Sumi and Toyozawa,

'

the conjecture

of

Peeters and

Devresse"

and the fact that Feynman's polaron theory, which gives a lower upper bound to the ground-state energy than the other approaches in most part range

of

cou-pling strength, did not predict a phase transition. Therefore it is still an unanswered theoretical question

—

whether this feature is a property

of

gen-eral type or ifitjust comes from approximation.

Ac-cording to the generalized path-integral formalism, we find that ifmore terms

of

excited states were in-cluded, the lower polaron energy results and the

smaller critical transition coupling strength will be,

e.

g.,in harmonic interaction,

if

only ground state (n

=

0)

is included,

a,

=

9.

42; ifall the excited states

are included, o.

,

will be

5.8.

"

We also conjecture the

possibility that, by the generalized formalism, ifthe optimum potential can be determined, and when all the excited states are included, would make the

abrupt change disappear, because, in principle, the

generalized formalism should give lower energy than

Feynman's model.

ACKNOWLEDGMENTS

We wish to thank Professor Satoru

J.

Miyake

of

Tokyo Institute

of

Technology for his useful corn-munications. And support from the Computer Center

of

National Chiao Tung University ismuch appreciated.

APPENDIX: EXPRESSION OF

_E„(p„a,

b

)

The trial function u(r )

=

N

[1+

b p,r

+a(bpr)2]e

~""and

_{N'=2(bp)3/[w(14+42a}

+45a')],

then EPcan be expressed as

b'pA

i(a)

₊

_~

[~

'(a)~

—

(a)

-~+'(a))

10243

2'

(a)

2

₂

(a)

1

4710

BRIEF

REPORTS 26

where C

=

_~

(1

—

p,

)

'

'

and

A~(a)

=874+2622a+4320a +4095a3+

1771.

875a~

A,

(a)

=A4(a)

A6(a)

= 616+2364a+ 3933a'+ 3690a'+1S75a~,

A7(a)

=228+1848a+3159a

+2880a

+1181.

25a~,

A8(a)

=912a+2136a +1800a3+675a~,

A

9(a)

=1140a'+720a3+ 225a~,

5 A&o(a)

=

$P

(a')b"

'l(b+C)"+'

n 1 5 A~p(a)

=

XP„(a)b"

'/(b

—

C) "+'

n 1

P~(a)

=

0.

25

P2(a)

=

0.

5,

P3(a)

=

0.

75a+

0.

37S

P4(a)

=1.

5a

P5(a)

=1.

875a

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4T.D.Lee,

F.

E.Low, and D.Pines, Phys. Rev. 90,297

(1953).

~S.Pekar,

J.

Phys. 10,347 (1946);Zh. Eksp.Teor. Fiz, 19, 796(1954).

"E.

P. Gross, Ann. Phys. (N.Y.)8,78 (1959).

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M.Matsuura, Can.

J.

Phys. 52, 1(1974).

~OW.

J.

Huybrechts,

J.

Phys. C9,L211(1976);10, 3761

(1977).

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M.Luttinger and C. Y.Lu,Phys. Rev. B 21,4251 (1980)~

'

_J.

_{T.Marshall and L.R.Mills, Phys. Rev. B 2, 3143(1970).} '3S. J,Miyake,

J.

Phys. Soc.Jpn. 38, 181(1975).

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J.

3,175(1960).

'5S.

_J.

Miyake (private communication).

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(1979).

'9H. Shoji and N.Tokuda,

J.

Phys. C14, 1231(1981).

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J.

Phys, Soc.Jpn. 35,137 (1973).

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M.Peeters and

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