Discontinuity of the surface polaron

~

Solid State Communications, Printed in Great Britain.

Vol.44,No. II, pp.1471-]475,

]982.

0038-I098/82/47]47]-05503.00/0 Pergamon Press Ltd.

DISCONTINUITY OF THE SURFACE POLARON M. Matsuura Department of Applied Science, Faculty of Engineering, Yamaguchi University, Tokiwadai, Ube, Japan 755* and The institute for Solid State Physics, The University of Tokyo, Roppongi, Minatoku, Tokyo, Japan 106

(P~eeeived

21

September

1982

by

Y.

Toyozawa)

Using the Feynman's path integral method, energies and masses are calculated for various types of a surface polaron. It is shown that, when the electron-surface phonon interaction becomes strong, surface optical phonon yields continuous change from free to self-trapped states. On the other hand surface acoustic phonon yields discontinuous change for the deformation type interaction and continuous change for the piezoelectric interaction. All these results reflect the character of the force range of the electron-surface phonon interaction.

criterion. Let us consider a Surface electron interacting with phonons. As a characteristic problem of the surface polaron, an electron motion perpendicular to the surface is assumed to be completely localized at the surface. Then, we have two dimensional polaron problem, whose hamiltonian is written as

Recently the interaction of an electron with lattice vibrations at solid surfaces has been drawing much theoretical attention. The problem involves not only electronic states outside or inside of crystals I-I0, but also those in inversion layers and heterojunctions II-13. In these surface systems the electron motion may be affected very much by surface phonons. So far most of previous works are concearned with the surface optical (SO) phonon. Very recently, using the path-integral method of Luttinger and Lu 14, Farias et al I0 found the phase transition type behaviour of an SO polaron between the neary free (F) state and the self-trapped (S) state, when the electron-SO phonon coupling exceeds a certain critical value. On this context Toyozawa and Shinozuka (TS) 15 gave the criterion for the stability of an electron in the phonon field, using the stability index ~=~-21-2, where ~ and I are dimensionality and force range index, respectively. If this criterion is applied to the SO polaron problem, the change from F to S is expected to be continuous. This contradicts with the result of Farias et al. In the present work, using the Feynman's path integral method 16, we will discuss the behaviour of the polaron from F to S states for various types of the surface polaron. Changes of energies and effective masses from weak to strong electron-surface phonon interactions will clearly show (i) the continuous variation for the long range interaction such as electron-SO phonon interaction and electron-surface acoustic (SA) phonon piezoelectric interaction and (ii) the distinct discontinuous behaviour for the short range interaction such as electron-SA phonon deformation type interaction. All these results are in accordance with those of TS

H

= p2 ~m

+ ~

x a%~

h~ ~ a + ~

+ h.c. ],

a ~ + ~

[ V ~ i~'~

(i)

where p, r and m are momentum, position and mass of an electron, respectively. Operator a +~k -+ (avE~) is a creation (annhilation) operator of the phonon with the mode ~ and the wave vector ~, whose energy is h ~ . The electron-phonon coupling is described by V ~ . We assume the ~-dependence of the phonon energy and the coupling constant as h~

= h~

(k/k0)~V

and V~

= V~

(k/k0)ng,

(2)

where k 0 is the (average) wave vector of the edge of the first Brillouin zone. The strength of the electron-phonon coupling is conveniently characterized by the nondimensional quantity S~ ~ ELR / h ~ , where ELR is the lattice relaxation energy when the electron is completely localized at one point and is given by ~ k l V ~ k ] 2 / ~v~. Then V9 is written as V9 / h ~ = [Sv(2ngC9 + 2) /2N]I/2, where N is the mumber of the unit cell at the surface plane. For convenience we assume h~v is independent of the mode 9 and ~ v = h~ (= hmop, where h~op is the SO phonon energv.

* Permanent address 1471

DISCONTINUITY

1472

OF THE SURFACE POLARON

For the SO phonon I ~ = 0 and ~ = -0.5. In order to describe the strength of the electron-SO phonon interaction, another nondimensional quantity a s = e 2 (E - E~) (m/2Tl3~op) i/2 is commaonly used, where Ei 0= (ei-l) / (ei+l) for i = 0 and ~ and e0(e ~) is the low (high) frequency dielectric constant. The quantity a s is related with Sop by S O = as(Bm)1/2, where B m = 5kf12/2m~ is the band w i d t h of the electron in units-of the phonon energy h~. In p a s s i n g it should be noted that some literatures 5,10 have used another definition of as, i.e., as' , w h i c h is related with the above e s by a s ' = w as/2. In the case of SA phonon we have ~ = 1 and n~ = i for the deformation type interactionlT, 7 and q~ = 0 for the piezoelectric interactionl7. Using these values for ~9 and q~, we may apply the TS criterion for the stability of the surface polaron. The force range index ~ in TS is given by I = - ~ + ~v / 2. Then the stability index ~ = 6-2~-2 is equal to-I for SO phonon and the p i e z o e l e c t r i c interaction of the SA phonon and I for the deformation type interaction of the SA phonon 18 According to TS, a(0 (a)0) yields the continuous (discontinuous) change between the F and S states, w h i c h reflects the long (short) range character of the electron-phonon interaction. Then it is expected that, while the interaction of the SO phonon and the piezoelectric type interaction of the SA phonon yield the continuous behaviour, the deformation type interaction of the SA phonon has the discontinuous behaviour. Now the Feynman's path integral m e t h o d is used to calculate the energy and effective mass of the polaron. Adopting units of h = m = • = i, the ground state energy E T may be written as e -SET = / exp

(S) DX(t),

(3)

where

Vol.

44. No.

6 = 2, V 2= W2+ 4C/W, and

F(u) =

W2 [~

W2_V 2 V3 (I - e-VU)].

(7)

Variational parameters V and W instead of C and W are d e t e r m i n e d by the m i n i m i z a t i o n of the energy E. Also, with the use of Feynman's procedure, effective mass of the p o l a r o n is obtained as

m* ~-=

2 1 + ~BmS~(2n x I ~ d u u2exp 0

1 x2n~+6+l ~ - ~v + 6) f 0 d x [ - x ~ u - X2Bm F(u)].

(8)

The above results of energy and effective mass, i.e., Eqs. (6) and (8), m a y be applied to various cases of the surface polaron problem. Firstly let us consider an SO phonon only. In this case it is usual to take k 0 to be infinite. In our expression this approximation corresponds to take the upper limit of the x integration in Eqs. (6) and (8), I, to be infinite. If this is done, we obtain E = I (V-W)2 2 V

~i/2 2 as I 0 d u e-U

[F(u)]-i/2'

(9) w h i c h agrees with the result of Hubrechts 5 , if we notice his a is as'(=2as/Z) in the p r e s e n t notation. Also the effective mass is calculated to be

m* ~-=

~i/2 1 + --~-- a s /~du u2e -u O

[F(u)] -3/2.

(i0)

where the action S is given by "d~(t) ) 2

x e i~.[ f (t) - ~

/ ~dt~ ~ d s ~

(s) ] e - h ~

Jt-sJ

(4)

Here 8 = (KT) -I is taken to be infinity in the calculation. For the trial action St, the following form is chosen, .dX (t)) 2 C 8 S S t = -]~dt ½ ~d-~--- ~/0dt/0ds

J~(t) -

X(s) J2 e -Wlt-sj,

(5)

where C and W are variational parameters to be determined later. Then, using the inequality ~ e and calculating the path integral, upper bound of the ground state energy E is obtained as 6 (V-W) 2 ET_<

E = ~-- V -

- ~ Sv(2~

× x 2n9+6-! / o d U exp

- ~

1 + 6) I 0 d x

[-x~u-X2Bm

F(u)], (6)

If we compare Eqs. (9) and (i0) with the corresp o n d i n g Eqs. in Farias et al, we m a y see that the only difference is V/2 instead of (V-W) 2/2V in the expression of the energy in (9). However, this difference is very important. As noted in Farias et al, minimization of their energy yields W=0. If we take W=O, the present results completely agree with those of Farias et al. However, the m i n i m i z a t i o n of the present energy is attained by the nonzero value of W. As noted in Feynman's work 16, in the three dimensional polaron problem, W=0 in the trial action yields the abrupt change between weak and strong coupling cases around the value of the Frohlich constant e ~ 6 . The similar abrupt change occurs in two dimensional case, which is the origin of the discontinuous behaviour of Farias el al. Using (9) and (i0) with W=0 and w i t h o u t W=0, we calculate numerically the ground state energy and the effective mass. The results are compared in Figs la and b. It is seen explicitly that energies of the calculation w i t h o u t W=0, w h i c h is lower than those with W=0 (, i.e., those of Farias et al), changes continuously from the result of the weak coupling limit, i.e., E ~-~as~op/2 , to that of the strong coupling limit, i.e., E ~ - n a s 2 h W o p / 8 . The continuous change is more clearly seen in the variation of the effective mass in Fig. lb in contrast to the discontinuous behaviour of Farias et a119. At present the Qffective mass changes c o n t i n u o u s l y from I + ~ a s / 8 in the weak coupling limit to

I]

Vol. 44. No.

]]

DISCONTINUITY

i(a)

z

a~ 4

'

"

OF THE SURFACE POLARON

6

z

~

a~ 4

]473

6

lib)

Z -z Bo'

/./z ×

% io

2

3

4

~s

Q5

Fig.l E n e r g y E (la) a n d e f f e c t i v e m a s s m /m (Ib) of a s u r f a c e optical polaron of t h e p r e s e n t (solid l i n e ) a n d t h e F a r i a s et a l ' s ( b r o k e n l i n e w i t h a dot) c a l c u l a t i o n s as a f u n c t i o n of t h e e l e c t r o n -surface optical phonon coupling constant e s a n d =~ (= n~s/2).

~2Ss4/16 in the strong coupling limit. After having seen there is no discontinuity for the usual SO polaron problem, we now turn to the properties of various surface polarons. Here we consider the deformation type and the piezoelectric type interactions of SA phonon as well as interaction of SO phonon, keeping the value of the maximum wave vector of phonons as k 0. Numerical calculations of energy and effective mass of (6) and (8) for these types of polarons have been performed. Before showing results, calculated analytic behaviours of energies and effective masses of polarons in the weak and strong coupling limits are shown in Table i. It is noted that in the strong coupling limit there are two behaviours depending upon V~B m except for the deformation type interaction of SA phonon 20. Now calculated results of energies and effective masses for the band width Bm=50 are shown in Fig. 2a and b. In these figures it is

Table i.

seen that, when the strength of the electronphonon coupling $9 becomes large, the interaction of an electron with SO phonon and the piezoelectric interaction with SA phonon have the continuous variation for energies and masses from F to S states. However, the deformation interaction of an electron with SA phonon yields the distinct discontinuous behaviour. Especially the effective mass changes by three orders of magnitude at the transition point. All these continuous and discontinuous natures are in accordance with TS. Let us briefly discuss the discontinuous transition for the deformation type interaction of SA phonon. The magnitude of the discontinuous change and the value of S~ at the transition are sensitive to the value of Bm. For small Bm such as Bm=10, the discontinuous nature is much smaller and the effective mass jumps by only one order of magnitude. Actually, near the transit-

Energy E and effective mass m*/m of the surface polaron in various limits.

Strong coupling

Weak coupling Limit

(w-~v) SV

-E

( w<
< Bin)

tan-I/~

( W, B m ~ V

$92

S9

s% Optical phonon

m*

S~

1 +4

-E Deformation type of acoustic phonon

(l+Bm) 2

~2

1

tan-i/~

}

+~

16Bm 2 $94

3 S_a { ! _ i_ + i___ log Ii+~I } Bm

2

Bm

2 ~

BmS~ S~

Bm2

m__* 6S~ Bm(2+3B m) m 1 + Bm2{ + log ll+Bml } 2 (l+Bm) 2

-E Piezoelectric type of acoustic phonon

( - I + B m)

{

S__X~ logl I+%1

2BmS~

~--~S~2

S~

-~ Sv 2

2~s~

Bm

m* m

1 + S9 { 1 - (l+Bm)-2}

2

)

1474

DISCONTINUITY OF THE SURFACE POLARON

2co> /. I//'1

o~

7" //I./

i../" [/

]/yi

E

I/.i I

Vol. 44. No.

°4 /

I:

I/ I/ o

/

0

20

S,

/

/i

i

40

60

80

S~

Fig.2 E n e r g y E (2a) a n d e f f e c t i v e m a s s m /m (2b) of a p o l a r o n for the i n t e r a c t i o n of the s u r f a c e optical phonon ( s o l i d l i n e ) , the d e f o r m a t i o n type interaction ( b r o k e n l i n e w i t h a d o t ) a n d the p i e z o electric type interaction ( b r o k e n l i n e w i t h two d o t s ) of the s u r f a c e a c o u s t i c p h o n o n as a f u n c t i o n of the e l e c t r o n - surface phonon coupling strength S . The band width B is t a k e n to be 50. m

ion region, F(or S) state may have a metastable state S(or F), whose energy is higher than true stable state F(or S). Fot this case the notation F(S) (or S(F)) has been used to specify the state more accurately 15. The coexistence region of stable and metastable states, i.e., the region of F(S) or S(F) is hardly seen for Bm =i0, but the region can be seen clearly for Bm=50, though not shown in Fig.2. The value of S~ at the discontinuous change varies from $9= 36.3 for Bm=50 to Sv=7.6 for Bm=10. This change is expected because the larger band width means the more mobile nature of an electron and then the stronger electron-phonon interaction is needed for an electron to be self-trapped. It is noted that the adiabatic treatment of TS yields the phase boundary from F to F(S) at $9= 2Bm/3 and from F(S) to S(F) at Sg=B m. The present nonadiabatic treatment yields the deviation from this result as see above. Also it should

be noted that above results of the discontinuity are qualitatively similar to those for polarons in bulk( three dimensional) case of Sumi and Toyozawa 21. Two dimensional character of the smaller kinetic energy of an electron brings the smaller $9 at the discontinuous change in comparison with the three dimensional case( $9=55 for Bm=50 in three dimensinal case). With the inclusion of the electron motion perpendicular to the surface, the present method of the path integral can be easily extended to study many situations of polarons at surfaces, such as polaron effects of a surface electron interacting with surface phonons (as well as bulk optical phonons) in an electronically intrinsic surface state, in an image potential and in inversion layers. Calculations of some of these problems as well as the phase diagram of a polaron in both SO and SA phonon fields are in progress.

References

I) J. Sak, Phys. Rev. B6, 3981 (1972) 2) E. Evans and D. L. Mills, Solid State Commun. Ii, ].093 (1972) : Phys. Rev. B8, 4004 (1973) 3) M. Matsuura, J. Phys. Soc. Jpn. 41, 394 (1976) 4) T. D. Clark, Solid State Commun 16, 861 (]976) 5) W. J. Huybrechts, Solid State CoFm~un. 28, 95 (1978) • ! . 6) O. Hlpollto, J. Phys. C 12, 4667 (1979) : Solid State Com~un. 32, 515 (1979) 7) H. Ueba, Phys. Stat. Sol. (b) I00, 705 (1980) : Surface Science 97, 564 (1980) 8) T. S. Rahman and D. L. Mills, Phys. Rev. B 21, 1432 (1980) 9) N. Tokuda, J. Phys. C 15, 1353 (1982)

i0) G. A. Farias, N. Studart and O. Hip61ito, Solid State Commun. 43, 95 (1982) ii) K. Hess and P. Vog!, Solid State Cormnun. 30, 807 (1979 j 12) S. Das Sarma and A. Madhukar, Phys. Rev. B 22, 2823 (1980) : A. Madhukar and S. Das Sarma, Surface Science 98, 135 (1980) 13) T. S. Rahman, D. L. Mills and P. S. Risebor ough, Phys. Rev. B 23, 4081 (1980) 14) J. M. Luttinger and C. Y. Lu, Phys. Rev. B 21, 4251 (1980) 15) Y. Toyozawa and Y. Shinozuka, J. Phys. Soc. Jpn. 48, 472 (1980) 16) R. P. Feynman, Phys. Rev. 97, 660 (1955) 17) S. Tamura and T. Sakuma Phys. Rev. B 15, 4948 (1977)

l|

Vol. 44. No.

II

DISCONTINUITY OF THE SURFACE POLARON

18) It is interesting that these values of the stability index ~ are the same as those for the corresponding electron-phonon interactions in the three dimensional case. 19) It is noted that the abrupt change of Farias et al should be considered to occur at ~s' = 3.9 insted of ~s' = 3.7. This is because, between 3.7 and 3.9, strong coupling solution is metastable and has the higher energy than the weak coupling solution.

1475

20) These two behaviours of the strong coupling limits may be seen for lager Bm. For example the SO polaron for Bm=50 changes from the weak coupling result to the strong coupling result of V)B m through the strong coupling result of V