Self-trapping of an electron by the surface modes

Solid State Comunicacions, Drinted in Great Britain.

VoL.53,Yo.2,

The

of

Physics

0038-1398/82/26009j-OJ+SO3.00/0

1932.

Pergamon

SELF-TRAPPING

Department

pp.95-98,

OF

AN

ELECTRON

THE

SURFACE

of

California,

Irvine,

Nelson Studart** Harvard University,

Laboratories,

MODES

Farias*

Gil A. University

Physics,

BY

Ltd.

Press

CA

Cambridge,

92717,

MASS

USA

02138,

USA

and

Departamento mica de Sao

de Fisica Carlos

e

(Received

Oscar Hipoli to CiGncia dos Materiais, I 3560 Sao Carlos-SP on

August

lnstituto, Brasi I

-

21thby

de

Fisica

e

Qui-

R.C.C.Leite)

A generalized form of the Feynman’s path is used in order to calculate the ground-state the effective mass of an electron interacting surface optical modes for the whole range of constant. A phase-transition-like behaviour free electron to a self-trapping electron as constant exceeds a certain critical value o

integral method energy and with the the coupling from nearly the coupling -3.8 is observed.

c

and effective mass of the surface polaron as functions of the couplinq constant. It will be shown that a phase transitionlike behaviour will occur at a critical value of the coupling constant. that is

The possibility that an electron may be trapped at the surface of a dielectric material has been recently The attracting much interest (I). attention has been directed mainly to the coupling between an electron and In the the surface optical phonons. interaction theories the weak coup1 ing electron behaves more or less like a free particle dressed with a few surface On the other hand the situation phonons. correspondinq to the sel f-trappinq state occurs when the electron-surface interaction becomes extremely phonons Quite recently two strong (2.3) different approaches namely the Feynman path integral (4) and the Huybrechts canonical transformation formalism (5) have been applied respectively by Huybrechts (6) and Hipolito (7) to study the surface polaron state problem for the whole range of the electron-surface phonon coupling constant c(. Nevertheless neither calculations have explicity shown the existence of a “phase-transirion” between nearly free and self-trapping electron

~~~-3.8. The electron interacting be written P

+

**

leave Federal

of

absence do Ceara,

On de

leave Federal

of

absence de Sao

from Departamento 60.000 Fortaleza, from Carlos,

Departamento 13.560 Sao

C

2m

the

So

fol

lo,,:ino

the surface

phonons

can

form

aq+Lnus(

a+

S5!

q

1 1 _ I--5 bq

/2

where p. coordina aq+ (aq) _

r

and

m are

5

and the

free mass crearion

of

a

surface

surface oscillarions, the surface and u constant characterizing between the electron According formalism the polaron

.-BE/ o -exp

=

de Fisica, Ceara,

95

(a

the

.

may

be

+ q

+a

_q)

(I)

momentum. of the electron. (annihilation)

phonon of to the surface the frequency

wave with of the

A

is

is the area dimensionless the couplinq and the phonon;

of

a

to the path qround-state written

inreqral energy

E.

of

as

TV- (e -BH, (-

; 3 H(t)dt)

2: Universida Brasil

de Fisica. Carlos, SP,

ejEi.1

te

operator

the

h)“4

_ m”S

. c+- jm,

i

On de

In

the

2 H=

The purpose of this work is to reinvestigate the problem and point out situations in which the electron could be trapped at the surface. Our approach is based on the generalized path-integral formalism recently employed to the bulk polaron problem by Luttinger and Lu (8). We discuss both the ground-state energy

l

Hamiltonian describing to motion parallel the surface with

Universida Brasil

-

-

D(path),

(7.)

95

SELF-TRQPISG

OF tU ELECTRON

0

where B= I/kRT and the limit 3 *= has to be taken. Since the path integral with the above Hami 1 ton i an is very difficult to be evaluated we wil I follow Luttinger and Lu approach Hamiltonian Hv between

the

particle

by

usinq

a

describing

electron

variational the

and

a

coupling

fictitious

as 2

.2

h=-!?.._+p v 2m

-+

”

(ir-

R)

(3)

2M M are the momentum, where ,D, R and coordinate and mass of the fictitious partitle and v(r-R) is (whitout any speci fit form) the electron-particle interaction potential. Then, after some manipulations and with the use of cx 1 Jensen’s inequality > e We for the groundobtain an upper bound E state energy of the sysyem with Hami ltonian HV,

BY THE SURFACE

It is interesting to point out that the present calculation tne expression the surface polaron energy, (viz Eq. identical to the three-dimensional 7) is polaron case (Eq.14 from Ref.3), except by a factor in the kinetic energy term where the dimensionality of the problem the dimensionality enters. In general, of the problem for different types of polarons changes also the interaction potential energy term. in for

By considering the first right hand side of Eq. (7) kinetic energy and through principle we can estimate the surface polaron radius Q-

Returninq ground-state mize it with W and 0, that

. ( d

+I)

”

“’

:I

-

;

where

j’\

to the energy, respect is

expression for the in Eq. (7). we minito both parameter S

From the first condition we direct that the best value of W is zero the ground-state energy reduces

Ev=

_ ,-Zc(AE

term of the as the the uncertainty the measure of Ri, as R = f

’ /2.

find Then,

,

Vol. $3, x0. 2

NODES

-

4.

r

!Y(l/Q) (1/2+1/R)

r

is

the

Euler

F-function.

(4)

IO'

where l/2 c

z

p

, AE

(MC!)

n

2 I/:! -

=

I/m

+

”

f

=E

-E

n

0

l/M,

2

I

-.. P

0)_ -1 L!.$i)=

In makinq this relative coordinate ~~~c~~~~e:h~ij~:::m’-

If we now v(r-R potential harmonic-interaction v

Enu.(i).

(5)

calculation the i and coordinate have been used

of to

_'

-2P

(i-!)

= 1

t_he upper energy

the

following

Eo4E

”

=;-

‘-

-Ev

choose for the variational ) a specific form of approximation. i.e.

K

(r-R)‘.

2 ron

IO’

(6)

--

of the bound E hiii in unl .Y s of

surface reduces

polato

5

form

_(n. T

j’h lo

e-’ iW2x+&

dx

o2-W2)

,n)

(,-,+q’2 (7)

l/2

are where W = (K/M) “2 and C=(K/p) variational parameters, from which the lowest upper bound for the ground-state It is interesting to is derived. ene rgv note that R represents the internal frequency of relative motion and S12/W2 the total mass of the trial model.

Fig.

1 -

The variational energy Ev of polaron of the

to (8)

1 1

En+

IY

the

ground-state surface

is plotted as a function coup1 ing consiant a.

vo .

ci3,

performed effective in Fig.

The other condition 6E /&a = g yields o + 3.8 when R + 0. A plgt of this expression as a function of D. for various values of a shows that there is no when a c 3.8. minimum for E miniWe have peyformed the numerical mization

of

97

SELF-TF4PPIXG OF AX ELECTRON BY THE SURFACE ??ODES

2

x0.

Eq.

(8)

with

respect

to

change constant

exceeds

With the also plotted

R

and the results for the surface polaron ground-state energy E (a) as a function in units of of the coupling constznt, of the surface phonon are energy )iws

and the mass in As i t 2. of m’ occurs

ron the

in Fig. I. As we can see, a order phase-transition hehaviour as a exceeds the critical value Although E (a) increases a 2 3.8. the slope ctintinuously as a inxreases, dE,,(a)da changes discontinuously of Ev, at the critical value of the coupling constant.

results for the units of m are is shown a very as the

optimum in Fiq.

radius

Rf

as

a

the

plotted rapid

coupling

critical

value.

values 3 the

of s1 we surface

function

of

UC

IO

have pola-

the

IO

plotted first occurs

Along the above lines the effective mass of the surface polaron can be found from the lowest energy of an electron with small velocity and can written in terms of the free electron mass as m*

=

I

dn3/2

m

2l-F

0

i

+

with n being the best ing the ground-state numerical calculations

(1

-

2

e e

-x

Rf be

dx

4x)

parameter energy, of Ea.

I(

(9)

3/2

minimizEq. (8). The (9) were

IO

IO‘

10. 0.1

1

lo0 4

IO' Fig.

3

-

The ron

ma

of

radius of the surface polaR is plotted as a function f the coupling constant a.

constant a. of the polaron point a .

coupling radius at the

As we shrinks

see

the abruptly

C

Let us discuss now the characteristic behaviours of the energy and the effective mass of the surface polaron in the limits of weak and strong coupling. In the weak couplinq theory we have small a and R -f 0. -Then in this limit the ground-state energy which actual ly is the interaction potential energy becomes Ev and

= the

m*=

Fig.

The effective mass of Feynman surface polaron plotted as a function coup1 ing constant a.

(IO)

effective

I +

mass

is

given

m*, the

by (11)

+4

In the strong coupling limit, for large values of a we have resulting then for the energy expression

the of

-4

that is, large Q Eq. (8) the

is E

= ; V

_ 4 (gj,/2(,+ II

2

,n2+C’ R

)

(121

SELF-TR%??IXC

3s

where =

C’

is

the

0.5772...

(2 In2 energy

+ C’) and becomes

iv= For

M’

The

the

=

I

*L 27

Euler best

then

Mascheroni value of

the

surface

OF .VY ELECTROZ

3

cons;ant = 2 -

BY TFIE SURFACE

shows 2

of

clearly

(2

In2

effective

+

Cl).

mass

we

Cl 4)

In both limits of small and large coup1 ing constant our results agree with those obtained from the canonical transformation method proposed by Huybrechts and applied to the surface polaron by one of us (OH) and E.L. Bodas (7.9). As we can note in the two limiting cases the principal contribution to the energy came from the self-induced potential. The surface polaron behaviour is nearly free at small couplinq constant but being trapped by the self-induced of the potential at the critical value coupling constant. the is

discontinuity an

the

change

the

surface

coupling

the at

of

the

polaron

constant

a phase-transition state changes

No.

phenomenon

abrupt

and

63,

Vol.

this

0 c-3.8. surface critical

value of c1 From a nearly free type to a self-tapping state in the same sense as suggested by Gross (IO) for the case of accoustic polaron. It is interesting to note that in our case the phase-transition is caused by the long range interaction between the electron and the surface polaron contrasting the proposition of Toyozawa (II) and Sumi and Toyozawa (12) which claimed that selftrapping is caused only by the short-raninteraction. ge Wheter or not the phase-transition is a common feature of variational methods and of different tyoes of polarons rather than an intrins’ic property Of the surface polaron itself, it is still an open quest ion. At the present we do not know any exoerimental observation that could conf’irm our theoretical predictions to the properties of the surface polaron, but we feel that the results presented here give a good picture of some qu,ll itative aspects of an electron self-trapped by the surface mode!

obtain

As it was pointed out in Ref.8. model calculation we have used here integral milar to the Feynman’s path formalism (4). but the present model

at

In such polaron

(13)

+014/r!2

and mass

radius

-

a

dEv(o)da

effective

polar2c?n

MODES

si -

REFERENCES

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