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Temperature dependence of strongly coupled surface polaron in potassium iodide semi-infinite polar crystal
PHYSICA ELSEVIER
Physica B 229 (1997) 361-368
Temperature dependence of strongly coupled surface polaron in potassium iodide semi-infinite polar crystal C . M . L e e a, S.W. G u b, C . C L a m a,* aDepartment of Physics and Materials Science, City University of Hong Kong, 83, Tat Chee Avenue, Kowloon, Hong Kong bApplied Physics Department and Institute of Condensed Matter Physics, Shanghai Jiao Tong University, Shanghai 200030, China
Received24 May 1996;revised 7 October 1996
Abstract
Temperature dependences of the effective Hamiltonian and the renormalized mass of surface polaron, for which the electron is strongly coupled with the surface-optical (SO) phonons but weakly or intermediately coupled with the bulk longitudinal-optical (LO) phonons, are derived. Temperature dependences of the self-energy and the renormalized mass of the polaron for the polar crystal of potassium iodide are evaluated as a function of the depth from the crystal surface. Results of our calculation indicate that the self-trapping energy increases for strong electron-SO-phonon coupling but decreases for weak electron-LO-phonon coupling with temperature. Keywords: Surface polarons; KI; Strong coupling
1. Introduction
In 1970s, Ibach [1] found the existence of optical surface phonons on the crystal surface of zinc oxide by using a high-resolution electron-impact spectrometer. The surface polaron in crystals has then been of considerable interest for theorists. Evan [-2] and Sak [-3] studied the properties of weak and intermediate coupling surface polaron by using L L P [4] variational approach and a perturbation regime [3], respectively. Based on the strong-coupling regime, Sak [3] and Hipolito [5] also studied the same polaron system. Recently, the properties of the polaron at finite temperature have been investigated theoretically by several scientists [-6-8]. However, they arrived at two opposite conclusions [9]. Arisawa and Saitoh
* Corresponding author.
[-6] found that the polaron mass decreases with temperature. On the other hand, Peeters et al. [7], based on their theory, predicted the opposite results. The polaron mass can be determined experimentally by the cyclotron resonance experiments in which the stated opposite results can be found for different materials. In Ref. [11], the experimental data for silver halide showed that the polaron mass increases as the temperature raises up. However, the cyclotron resonance measurement [12] on heterojunctions of G a A s - G a l - xAlxAs showed that the effective mass anomalously increases as the temperature increases up to about 100 K and then decreases at higher temperature range. Ref. [13] argued that the temperature dependence of the polaron mass can be explained by the energy band nonparabolicity and the electron-LO-phonon interaction. At low temperature, the increase of the polaron mass with temperature is attributed to the nonparabolicity of the polaron energy-momentum
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C M . Lee et al. / Physica B 229 (1997) 361 368
spectrum [9, 10]. The temperature dependence of the polaron mass has been a controversial topic in the past [9]. The reasons are basically laid on different existing definitions for the effective mass, and this may lead to different temperature behaviours. So far, the results obtained from the cyclotron resonance were referred to the bulk materials for which the effect of surface phonon can be neglected. For the case of semi-infinite polar crystal, the surface phonon plays crucial role for the effective mass of polaron near the surface and the situation may then be greatly different from those cases of which the influence of the surface phonon can be neglected. In this paper, we investigate the interaction between the electron and the LO phonon as well as the SO phonon. As were known by scientists, the weak and intermediate coupling theories [14] for the bulk polaron are applicable for electron-bulk LO-phonon interaction with coupling constant ~f < 6. For the electron-SO-phonon interaction with coupling constant as > 2.5 [15], the strong coupling theory must be applied. Furthermore, one of the author has studied the behaviours of the weak or intermediate coupling polaron at nonzero temperature [16]. However, the research for the temperature dependence of the strong electron-phonon coupling is very scarce. In Ref. [17], one of the authors has studied the strong coupling of electron with the SO phonon, as well as the weak or intermediate coupling with the LO phonon in the semi-infinite crystal at zero temperature and has derived the effective Hamiltonian for surface polaron based on the method of the linear combination operator [18] and a Lagrangian multiplier [19]. The analytic results for the limiting case of z = 0 and z = ov have also been obtained. In this paper, we further study the effects of the LO phonon as well as the SO phonon on the polaron renormalized mass at different depths from the crystal surface at nonzero temperature. In Section 2, we first derive the temperature dependence of the effective Hamiltonian and the polaron renormalized mass. In Section 3, discussion and conclusions will be given. We choose one of the ionic compounds, namely potassium iodide, to discuss the temperature dependences of the self-energy and
the polaron renormalized mass in terms of the depth from the crystal surface.
2. T h e o r y
Consider a semi-infinite polar crystal with vaccum when z < 0, the Hamiltonian for an electron phonon interacting system is given by [20, 21]
/4 = HH + ti, Hit :
~m
(1)
+
^+^ w
Q
1 , + e iw[l'p + h.c.) + _~-sin(wzz)(V,~g~w w W
V / / C ~ f)~e-aZe ie.p + h.c.)
+ 7 t,,/-d /~±
p~
e2(goo - -
1)
(3)
+ 4zeo~ (e~ + 1)'
= ~
(2)
in which • /4rce2hcot V *w = 1 4
~
'
ca='x/
,
% = ~(~oe + 1
1
1
1
eo - - 1
e ~ -- 1
eoo
So'
e*
So + 1
soo + 1'
where/qll and/4± are the Hamiltonians for electron moving in the xy-plane and in the z-direction, respectively. Note that/-)ll involves the kinetic energy of electron moving in xy-plane, i.e. the first term in Eq. (2), the energies of the LO phonon and the SO phonon, which are represented in the second and the third terms of Eq. (2), and the interaction energies of electron-LO phonon and electron-SO phonon are described by the fourth and the fifth terms in Eq. (2) r~pectively. The other part of the Hamiltonian, i.e. H±, involves the kinetic energy of electron along z-direction and the image potential, which are represented by the first and the second
363
C.M. Lee et al. / P h y s i c a B 229 (1997) 3 6 1 - 3 6 8
terms of Eq. (3) respectively. The symbols COsand COe are the angular frequencies of the SO phonon and LO phonon, respectively. The creation (annihilation) operators fi~ (~iw) and b~ (DQ) are, respectively, for the LO phonon and the SO phonon, eo and e~ are the static and optical dielectric constants of the crystal, respectively, coe is the angular frequency of long-wavelength limit for transverse optical phonon. S and V are the surface area and the volume of the crystal, respectively. In order to solve the Hamiltonian in Eq. (1), first, we treat the wave function as Gaussian-type for the strong electron-SO-phonon coupling, this is equivalent to the treatment which introduces a transformation of linear-combination operators for the momentum operator PIIt and coordinate operator Pt of the electron along the j-direction [18, 19], P!,t = ~ ( b 7
pt=i
+ bj + Pot),
/~-=:-7~(bt-bf)
wherej=x,y,
(4)
(5)
in which Pot is a displacement of the momentum operator w.r.t, the old coordinate system and 2 is a variational parameter. Next we carry out the double unitary transformations U~ and U2, which are expressed by [14] ~', = exp( - i ~
fi+
awW,,'p),
(6)
w
/ U2 = exp{~" (fi~+fv - fiwfw) *
The transformed ground state of the electronphonon system at a certain temperature T can be assumed as
(8)
= 4,(z)lUt, N.NO),
where cb(z) is a specific orthonormal wave function describing the electron motion along z-direction. In Eq. (8), INt, Nw, N o) = INt)IN~)INQ), in which the components of the wave function are the conventional states in the particle number representation. INt) stands for the wave function of the polaron, while INI) and INQ) stand for the wave functions of the bulk and surface phonons, respectively• The product of all these wave functions represents the ground state of the whole system at nonzero temperature. At a finite temperature, the properties of the polaron are determined by the statistical average of the corresponding properties w.r.t, the various states in the electron-phonon system. As shown in Ref. [12], the phonon frequency decreases with the increasing temperature, but if the temperature is restricted to the range lower than the room temperature (T < 300 K), the relative changes of the frequncies, coe and COs, are only 1%. So we take the optical-phonon frequencies to be approximately constant, and according to the quantum statistical mechanics, the expectation values of b+ bj, aw ^+ aw, ^ ^__^ bQ bQ in the state I~b} at temperature T are determined by Ref. [161. Note that it is reasonable to take ~r = £rt for j = x and y for the symmetrical motion of the electron in potassium iodide which has a cubic structure. Assume the total momentum of the system, which contains phonons and electron that is moving in the xy-plane, is given by
\ ^ + + ~(bQ gQ _ _ begS) ~ , / O
(7)
• * ge, gQ * are displacement amplitudes in whmhf~,fw, that are treated as variation parameters. Note that ~-~1 transformation is used to eliminate the electron's coordinate p in the Hamiltonian of Eq. (2), especially the phase factor exp(-iwl~'p), for weak or intermediate electron-LO-phonon coupling; and (J2 is used to displace the phonons' coordinates.
~ol~= P,, + ~ hwd~+aw + y' hQTg~ bQ; w
(9)
Q
and introduce a vector ~'l which is defined as a Lagrangian multiplier and behaves as the velocity of the polaron, we may then evaluate the expectation value o f F(f~,ge, Po,2,alli, z , T ) = (J21(7; 1 (/~, -- ~ql" ,o H)U1 U2 with respect to the ground state INj, Nw, NQ), at a certain temperature. By performing the variation of F(fw, gQ,Po,2,qlll, z,T ) with respect tOfw and gQ, we then obtain the expression
C.M. Lee et al. /Physica B 229 (1997) 361 368
364
of F(Po, 2, ~il!l, z, T)
a<.ha~ { 7 _ [C,(2u,tz)sin(2u)z) x/ZN~ + 1
F(Po,2,~ll,,,z, T) = (/V + ½)fi2 + ~/~.fioe + ~ hCOsNQ ~,
Q
- Si(2u)z) cos(2uez)]},
+ ¼hPE2 -- 4 - - ~ P o ' ~gli -- Zw 4 ~i w -'
(15)
o" WllNw in which
_ ~12
IVwl2 sin2(WzZ) h~ol + [h2w~/(2mO] (2.~, + 1)
E(
1 mile2 ad --
)2]
h(/)g, - -
as -- g, h2us '
_ ~ ICQIZB2(N) exp( - hQZ/(2m!l2))e- 2Q,
, /-gl--
QhCOs
h2u2 , 2roll
hoJs h2u2
1 mile2
x 1 + fitat + [h2w~/(2mO] (2-Nw + 1) Q
h2ut
,
-
2mpl '
U~,
x/ZN, o + 1' (10)
Ci(2u, z) = - (|~ c~o(2zx)dx, j,~
x
where
B(N) = 1 -- R hQ2
Si(2ulz) = - i '° oi~(2zx) dx.
/ / h Q 2 \2~
).
+otto)
J, 4
(11)
in which O(x z) is an infinitesimal function that consists of x 2 and higher order terms. Further variation with respect to Po gives
~/(2ml!/(h2))alil Po = 1 - (2h2/mO f (z, T)'
(12)
where at and as are the coupling constants for LO phonon and SO phonon with electron, respectively. Obviously, the variation of F(2,z, T) with respect to 2 may determine the value of 2. Furthermore, the average value of the momentum fill for polaron is defined by
in which
1
f(z, T)
.g~i4UI U:INj, Nw,NQ) = mr(z, T)Ugi, IV~l 2 sinZ(~=z)
I--. 'W)
(13) After simplifying Eq. (10) by neglecting all higher order terms of ~ for slow electron, we obtain the expected value of F(Po,2,~lll, z, T) with respect to the ground state:
F(2, z, r) = ~Nwhoc + ~'.No.h¢os + G(2,z, T), (14) ,~ Q where
G(2,z, T) = (N + ½)h2 -- a~h(o)~2)1/2 x f~(1-
Nil (}21 ~i_ 1~ I ^
COS 2 (e, v ~'~(hcor~+ [h2co~/(2mll)] (2/~., + 1))3
-
~
2Nx )exp(-x
2
f2-
-24~u~xz)dx
(16)
X
^
(17)
where 1
m~l (z, T) = ml, -1 -- (2h2/mll)f(z, T) i/ ,~ "~3 / 2
+2aSt~ss )
f~x2(1-2~x2)
×exp(--x 2- 2 f~uszx)dx}
(18)
From the above, we can obviously see that ~j is the average velocity of polaron moving in the xyplane. In Eq. (18), there is a multiple factor for the renormalized mass mr(z, T), which consists of two terms. The first term is due to the LO phonon and the other term is due to the SO phonon. Note that,
CM. Lee et al. /Physica B 229 (1997) 361-368
in this paper, we do not consider the conventional band nonparabolicity, so the band mass mll is assumed to be temperature independent. The effective Hamiltonian is then given by Heft(Z , T) = H i + M i n ( N e, Nw, Nil (J~- 101 lff/ll
× (g~f_JelN2,N,~,Ne) p2
e2(eoo -- 1)
-- 2m± + 4ze~o(e~o +
+Z
+ Z Qho)s
w
+
1)
Q
P~ + (N + ½)h2 2m* (z, T)
- ~sh(m~,t)a/2 •
x exp( - x 2 -
- 2Nx 2)
2~u~zx)dx
~
- [Ci(2u'~z)sin(2u)z)
- Si(2u)z) cos(2u)z)]}
(19)
In the following section, we shall discuss the numerical calculation for relevant quantities of G(z, T) and m r (z, r ) .
3. Discussion and conclusions To illustrate our theoretical results in a quantitative manner, we choose an ionic compound, namely potassium iodide, as an example with a large coupling constant for electron-SO-phonon interaction having the value of es = 3.1 as well as an intermediate coupling constant for electronLO-phonon interaction, ~ = 2.5. All the parameters used in the numerical calculation are given as follows [22]:- e o = 4 . 6 8 , e~ =2.68, hco~= 18.0meV, ho)s= 16.9 meV, m~i = 2 . 1 × 1 0 -3 rob, where mb is the bare electron mass. For simplicity, the depth from the crystal surface is expressed in the unit of u; - 1 = 3 . 1 × 1 0 - v m . There are two points worth for discussion:
365
3.1. Self-trapping energy The self-energy of the polaron consists of two parts. One comes from the interaction between the electron and the LO phonon, while the other comes from electron-SO-phonon interaction. It is worth noting that, the negative value of the self-energy of the polaron represents a bound state of the self-trapped electron which is formed by an asymmetric local deformation in the lattice. As was known in the past, the magnitude of the self-energy G(z, T) is defined as the self-trapping energy. Our calculation results show that the self-trapping energy IG(z,Z)[ increases for strong electron-SOphonon coupling, i.e. the first and second terms in Eq. (15), and decreases for weak electronLO-phonon coupling, i.e. the third term, with temperature. From Fig. l(a), in general, the overall self-trapping energy IG(z,T)I monotonically decreases with temperature. When the temperature increases, the random motion of the ions in the crystal will become violent. The number of phonons then increases but the interaction between electron and phonon will be weakened [9]. It is consistent with the results of Ref. [6]. However, the strong electron-SO-phonon coupling increases with increasing temperature. Peeters et al. had also obtained the same results as stated previously [7]. Fig. l(a) also shows the overall self-trapping energy IG(z, T)I increases with increasing depth z from the crystal surface for a temperature T < 100 K, but decreases for T > 100 K. In Fig. l(b), the self-trapping energy [G(z, T)I decreases drastically with the depth near the crystal surface. It is because the polarised field produced by SO mode is dominant only within a very narrow depth from the crystal surface, z < 0.3u7 a, for lower temperature ranging from 0 K to 50 K. When z > 0.3ui 1, the self-trapping energy increases and tends to a constant value. However, for high temperature range, T > 100 K, the self-trapping energy gradually decreases. The effect of the bulk LO mode is dominant when the polaron is far from the crystal surface and more sensitive than the SO mode to temperature. Moreover, there is some interesting physical significance involved in Fig. l(b). Within a relatively low temperature range, T < 50 K, there is a minimum for the self-trapping energy IG(z, T)I. This minimum
366
C.M. Lee et al. / Physica B 229 (1997) 361-368 r
70.00 ~
>
N
1
I
,
Z ---- 0 . 5 1 1 1 - 1 .........
Z =
........
z = 1.5ul-1
1.0nl-1
50.00
30.00
10.00
I
i
I
I
I
0.00
50.00
100.00
150.00
200.00
(a)
250.00
T (K)
70.00
'
'
T=0K
~,=...5,o,K
65.00
. . . . . . . . . . . . . . . . . . . . . . . . .
60.00 v m
[..., t,4 v
1\ I ~,
55.00
T = lOOK
50.00 45.00 ""...... - . . , . 40.00
i
I
0.00
1.00
....
• ...........................
(b)
T = 150K
• ..........
I
i
2.00
3.00
Z (Ul
..=.
4.00
"1)
Fig. l(a). A plot of the self-trapping energy IG(z, T)[ against the temperature T for KI compound with different depths from the crystal surface, i.e. z = 0.5u[ 1, 1.0u71, 1.5u[l and 2.0uZ 1. (b) A plot of the self-trapping energy IG(z, T)I against the depth z from the crystal surface for KI compound with different temperatures, T = 0, 50, 100 and 150 K.
represents the m o s t u n s t a b l e p o l a r o n t h a t m a y occur at the valley of the [G(z, r ) l versus the d e p t h z. As the t e m p e r a t u r e increases, say T > 50 K, the t e n d e n c y for the p o l a r o n to m o v e to the i n t e r i o r is reduced. At high t e m p e r a t u r e , T > 100 K, the p o l a r o n tends to m o v e to the surface to increase the self-trapping energy Ia(z, r ) l . At zero t e m p e r a t u r e , the self-energies G(z, T ) of the p o l a r o n at the limiting cases of depths, z = 0 a n d z = o% for K I crystal are - 6 3 . 6 a n d - 7 0 . 7 meV, respectively.
3.2. Renormalized mass The c o n s e q u e n c e of the e l e c t r o n - p h o n o n interaction is to increase the r e n o r m a l i z e d m a s s of the p o l a r o n . This is because the electron d r a g s the h e a v y ion cores a l o n g with it. In Fig. 2(a), the p o l a r o n r e n o r m a l i z e d mass, in general, decreases with t e m p e r a t u r e as the d e p t h from the crystal surface is fixed. T h e e x p l a n a t i o n is the same as t h a t for the self-energy, because the higher the
367
C.M. Lee et al. / Physica B 229 (1997) 361 368
12.00
\
z=2.0ul'l
z = 1.5ux-l".
g
8.00 \ :
[.. ~f
Z ---- 1 . 0 U l -1 ~ %: "~
,,., 4.00
z = 0.5Ul-1
'..
"%.'$ ~ #
0.00
~
J
0.00
50.00
'
'
100.00 150.00 T (K)
(a)
200.00
250.00
Fig. 2a. A plot of the renormalized mass of polaron, rn~'(z,T), against the temperature T for K I with different depths from the crystal surface, i.e. z = 0.5u[ 1, 1.0u[ 1, 1.5u71 and 2.0u[ 1.
60.00 50.00 40.00 v [.., ~4
30.00 20.00 10.00
t
:,0K...............
0.00
T = 150K I
0.00
1.00
(b)
I
2.00 z (ui -1)
3.00
4.00
Fig. 2b. A plot of the renormalized mass of the polaron, m*(z, T), against the depth from the crystal surface, z, for KI compound with different temperatures, T = 0, 50, 100 and 150 K.
s e l f - t r a p p i n g e n e r g y the h i g h e r the r e n o r m a l i z e d m a s s is. As we c a n see in Fig. 2(a), the p o l a r o n r e n o r m a l i z e d m a s s i n c r e a s e s for a t e m p e r a t u r e T < 100 K a n d decreases for T > 100 K with inc r e a s i n g the d e p t h s f r o m the crystal surface. I n Fig. 2(b), the d r a s t i c decrease in the r e n o r m a l i z e d m a s s n e a r the surface is m o r e or less the s a m e for differ-
e n t t e m p e r a t u r e s , b u t w h e n the d e p t h i n c r e a s e s f r o m 0 . 5 u [ 1 o n w a r d s , the r e n o r m a l i z e d m a s s will be s i g n i f i c a n t l y affected b y the t e m p e r a t u r e . T h e h i g h e r the t e m p e r a t u r e , the l o w e r the r e n o r m a l i z e d m a s s will be. I n the i n t e r i o r of the b u l k s a m p l e , the p o l a r o n r e n o r m a l i z e d m a s s decreases w i t h i n c r e a s i n g t e m p e r a t u r e . It is o b v i o u s t h a t the effect of S O
368
CM. Lee et al. / Physica B 229 (1997) 361 368
mode is less sensitive to temperature than that of LO mode. Note that at zero temperature, the polaron renormalized masses at the limiting cases of depths, z = 0 and z = ~ , for KI crystal are 57.7mll and 54.8mll, respectively.
Acknowledgements We would like to cordially thank the Research Degree Committee, City University of Hong Kong, for finanica] support of this investigation. One of the authors, Gu, also thanks Lab. of Excited State Processes, Chang Chun Physics Institute, Academic Sinica of China, for financial support.
References [1] H. Ibach, Phys. Rev. Lett. 24 (1970) 1416. 1-2] E. Evans and D.L. Mills, Phys. Rev. B 5 (1972) 4126; E. Evans and D. L. Mills, Phys. Rev. B 8 (1973) 4004. 1-3] J. Sak, Phys. Rev. B 6 (1972) 3981. 1-4] T.D. Lee, F.E. Low and D. Pines, Phys. Rev. 90 (1953) 297. 1-5] W.J. Huybrechts, J. Phys. C 9 (1976) L211; O. Hipolito, J. Phys. C 12 (1979) 4667. [6] K. Arisawa and M. Saitoh, Phys. Star. Sol. B 120 (1983) 361.
I-7] F.M. Peeters and J.T. Devreese, Phys. Rev. B 25 (1982) 7302; F.M. Peeters and J.T. Devreese, Phys. Rev. B 31 (1985) 5500. [8] D.C. Khandekar, K.V. Bhagwat and S.V. Lawande, Phys. Rev. B 37 (1988) 3085. 1-9] F.M. Peeters and J. T. Devreese, Solid State Phys 38 (1984) 81. [10] K. Okamoto and S. Takeda, J. Phys. Soc. Japan 37 (1974) 333. 1-11] T. Masumi, Polarons and Excitons in Semiconductor and Ionic Crystals, eds. J.T. Devreese and F.M. Peeters (Plenum Press, New York, 1984) p. 99. [12] M.A. Brummell, R.J. Nicholas, M.A. Hopkins, J.J. Harris and C.T. Foxon, Phys. Rev. Lett. 58 (1987) 77. [13] W. Xiaoguang, F.M. Peeters and J.T. Devreese, Phys. Rev. B 36 (1987) 9765. [14] E. Haga, Progr. Theor. Phys. 11 (1954) 449. [15] P. Jin-Sheng, Phys. Stat. Sol. B 127 (1985) 307. [16] Y.-C. Li and S.W. Gu, J. Phys.: Condens. Matter 1 (1989) 3201. B.H. Wei and S.W. Gu, Phys. Stat. Sol. B 163 (1991) 161. Y.T. Wang, G.L. Gong and S.W. Gu, Phys. Stat. Sol. B, 179 (1993) K61. [17] DJ. Yang and S.W. Gu, Acta Sci. Natur. Univ. Intramongolicae 18 (1987) 2 (in Chinese). [18] W.J. Huybrechts, J. Phys. C 9 (1976) L211. [19] N. Tokuda, J. Phys. C 13 (1980) 173. [20] X.X. Liang and S.W. Gu, Solid State Commun. 50(1984) 505. [21] J.J. Licari and R. Evrard, Phys. Rev. B 15 (1977) 2254. [22] E. Kartheuser, Polarons in Ionic Crystals and Polar Semiconductors (North-Holland, New York, 1972).
Physica B 229 (1997) 361-368
Temperature dependence of strongly coupled surface polaron in potassium iodide semi-infinite polar crystal C . M . L e e a, S.W. G u b, C . C L a m a,* aDepartment of Physics and Materials Science, City University of Hong Kong, 83, Tat Chee Avenue, Kowloon, Hong Kong bApplied Physics Department and Institute of Condensed Matter Physics, Shanghai Jiao Tong University, Shanghai 200030, China
Received24 May 1996;revised 7 October 1996
Abstract
Temperature dependences of the effective Hamiltonian and the renormalized mass of surface polaron, for which the electron is strongly coupled with the surface-optical (SO) phonons but weakly or intermediately coupled with the bulk longitudinal-optical (LO) phonons, are derived. Temperature dependences of the self-energy and the renormalized mass of the polaron for the polar crystal of potassium iodide are evaluated as a function of the depth from the crystal surface. Results of our calculation indicate that the self-trapping energy increases for strong electron-SO-phonon coupling but decreases for weak electron-LO-phonon coupling with temperature. Keywords: Surface polarons; KI; Strong coupling
1. Introduction
In 1970s, Ibach [1] found the existence of optical surface phonons on the crystal surface of zinc oxide by using a high-resolution electron-impact spectrometer. The surface polaron in crystals has then been of considerable interest for theorists. Evan [-2] and Sak [-3] studied the properties of weak and intermediate coupling surface polaron by using L L P [4] variational approach and a perturbation regime [3], respectively. Based on the strong-coupling regime, Sak [3] and Hipolito [5] also studied the same polaron system. Recently, the properties of the polaron at finite temperature have been investigated theoretically by several scientists [-6-8]. However, they arrived at two opposite conclusions [9]. Arisawa and Saitoh
* Corresponding author.
[-6] found that the polaron mass decreases with temperature. On the other hand, Peeters et al. [7], based on their theory, predicted the opposite results. The polaron mass can be determined experimentally by the cyclotron resonance experiments in which the stated opposite results can be found for different materials. In Ref. [11], the experimental data for silver halide showed that the polaron mass increases as the temperature raises up. However, the cyclotron resonance measurement [12] on heterojunctions of G a A s - G a l - xAlxAs showed that the effective mass anomalously increases as the temperature increases up to about 100 K and then decreases at higher temperature range. Ref. [13] argued that the temperature dependence of the polaron mass can be explained by the energy band nonparabolicity and the electron-LO-phonon interaction. At low temperature, the increase of the polaron mass with temperature is attributed to the nonparabolicity of the polaron energy-momentum
0921-4526/97/$17.00 Copyright © 1997 ElsevierScienceB.V. All rights reserved Pll S092 1 -4526(96)00852-6
362
C M . Lee et al. / Physica B 229 (1997) 361 368
spectrum [9, 10]. The temperature dependence of the polaron mass has been a controversial topic in the past [9]. The reasons are basically laid on different existing definitions for the effective mass, and this may lead to different temperature behaviours. So far, the results obtained from the cyclotron resonance were referred to the bulk materials for which the effect of surface phonon can be neglected. For the case of semi-infinite polar crystal, the surface phonon plays crucial role for the effective mass of polaron near the surface and the situation may then be greatly different from those cases of which the influence of the surface phonon can be neglected. In this paper, we investigate the interaction between the electron and the LO phonon as well as the SO phonon. As were known by scientists, the weak and intermediate coupling theories [14] for the bulk polaron are applicable for electron-bulk LO-phonon interaction with coupling constant ~f < 6. For the electron-SO-phonon interaction with coupling constant as > 2.5 [15], the strong coupling theory must be applied. Furthermore, one of the author has studied the behaviours of the weak or intermediate coupling polaron at nonzero temperature [16]. However, the research for the temperature dependence of the strong electron-phonon coupling is very scarce. In Ref. [17], one of the authors has studied the strong coupling of electron with the SO phonon, as well as the weak or intermediate coupling with the LO phonon in the semi-infinite crystal at zero temperature and has derived the effective Hamiltonian for surface polaron based on the method of the linear combination operator [18] and a Lagrangian multiplier [19]. The analytic results for the limiting case of z = 0 and z = ov have also been obtained. In this paper, we further study the effects of the LO phonon as well as the SO phonon on the polaron renormalized mass at different depths from the crystal surface at nonzero temperature. In Section 2, we first derive the temperature dependence of the effective Hamiltonian and the polaron renormalized mass. In Section 3, discussion and conclusions will be given. We choose one of the ionic compounds, namely potassium iodide, to discuss the temperature dependences of the self-energy and
the polaron renormalized mass in terms of the depth from the crystal surface.
2. T h e o r y
Consider a semi-infinite polar crystal with vaccum when z < 0, the Hamiltonian for an electron phonon interacting system is given by [20, 21]
/4 = HH + ti, Hit :
~m
(1)
+
^+^ w
Q
1 , + e iw[l'p + h.c.) + _~-sin(wzz)(V,~g~w w W
V / / C ~ f)~e-aZe ie.p + h.c.)
+ 7 t,,/-d /~±
p~
e2(goo - -
1)
(3)
+ 4zeo~ (e~ + 1)'
= ~
(2)
in which • /4rce2hcot V *w = 1 4
~
'
ca='x/
,
% = ~(~oe + 1
1
1
1
eo - - 1
e ~ -- 1
eoo
So'
e*
So + 1
soo + 1'
where/qll and/4± are the Hamiltonians for electron moving in the xy-plane and in the z-direction, respectively. Note that/-)ll involves the kinetic energy of electron moving in xy-plane, i.e. the first term in Eq. (2), the energies of the LO phonon and the SO phonon, which are represented in the second and the third terms of Eq. (2), and the interaction energies of electron-LO phonon and electron-SO phonon are described by the fourth and the fifth terms in Eq. (2) r~pectively. The other part of the Hamiltonian, i.e. H±, involves the kinetic energy of electron along z-direction and the image potential, which are represented by the first and the second
363
C.M. Lee et al. / P h y s i c a B 229 (1997) 3 6 1 - 3 6 8
terms of Eq. (3) respectively. The symbols COsand COe are the angular frequencies of the SO phonon and LO phonon, respectively. The creation (annihilation) operators fi~ (~iw) and b~ (DQ) are, respectively, for the LO phonon and the SO phonon, eo and e~ are the static and optical dielectric constants of the crystal, respectively, coe is the angular frequency of long-wavelength limit for transverse optical phonon. S and V are the surface area and the volume of the crystal, respectively. In order to solve the Hamiltonian in Eq. (1), first, we treat the wave function as Gaussian-type for the strong electron-SO-phonon coupling, this is equivalent to the treatment which introduces a transformation of linear-combination operators for the momentum operator PIIt and coordinate operator Pt of the electron along the j-direction [18, 19], P!,t = ~ ( b 7
pt=i
+ bj + Pot),
/~-=:-7~(bt-bf)
wherej=x,y,
(4)
(5)
in which Pot is a displacement of the momentum operator w.r.t, the old coordinate system and 2 is a variational parameter. Next we carry out the double unitary transformations U~ and U2, which are expressed by [14] ~', = exp( - i ~
fi+
awW,,'p),
(6)
w
/ U2 = exp{~" (fi~+fv - fiwfw) *
The transformed ground state of the electronphonon system at a certain temperature T can be assumed as
(8)
= 4,(z)lUt, N.NO),
where cb(z) is a specific orthonormal wave function describing the electron motion along z-direction. In Eq. (8), INt, Nw, N o) = INt)IN~)INQ), in which the components of the wave function are the conventional states in the particle number representation. INt) stands for the wave function of the polaron, while INI) and INQ) stand for the wave functions of the bulk and surface phonons, respectively• The product of all these wave functions represents the ground state of the whole system at nonzero temperature. At a finite temperature, the properties of the polaron are determined by the statistical average of the corresponding properties w.r.t, the various states in the electron-phonon system. As shown in Ref. [12], the phonon frequency decreases with the increasing temperature, but if the temperature is restricted to the range lower than the room temperature (T < 300 K), the relative changes of the frequncies, coe and COs, are only 1%. So we take the optical-phonon frequencies to be approximately constant, and according to the quantum statistical mechanics, the expectation values of b+ bj, aw ^+ aw, ^ ^__^ bQ bQ in the state I~b} at temperature T are determined by Ref. [161. Note that it is reasonable to take ~r = £rt for j = x and y for the symmetrical motion of the electron in potassium iodide which has a cubic structure. Assume the total momentum of the system, which contains phonons and electron that is moving in the xy-plane, is given by
\ ^ + + ~(bQ gQ _ _ begS) ~ , / O
(7)
• * ge, gQ * are displacement amplitudes in whmhf~,fw, that are treated as variation parameters. Note that ~-~1 transformation is used to eliminate the electron's coordinate p in the Hamiltonian of Eq. (2), especially the phase factor exp(-iwl~'p), for weak or intermediate electron-LO-phonon coupling; and (J2 is used to displace the phonons' coordinates.
~ol~= P,, + ~ hwd~+aw + y' hQTg~ bQ; w
(9)
Q
and introduce a vector ~'l which is defined as a Lagrangian multiplier and behaves as the velocity of the polaron, we may then evaluate the expectation value o f F(f~,ge, Po,2,alli, z , T ) = (J21(7; 1 (/~, -- ~ql" ,o H)U1 U2 with respect to the ground state INj, Nw, NQ), at a certain temperature. By performing the variation of F(fw, gQ,Po,2,qlll, z,T ) with respect tOfw and gQ, we then obtain the expression
C.M. Lee et al. /Physica B 229 (1997) 361 368
364
of F(Po, 2, ~il!l, z, T)
a<.ha~ { 7 _ [C,(2u,tz)sin(2u)z) x/ZN~ + 1
F(Po,2,~ll,,,z, T) = (/V + ½)fi2 + ~/~.fioe + ~ hCOsNQ ~,
Q
- Si(2u)z) cos(2uez)]},
+ ¼hPE2 -- 4 - - ~ P o ' ~gli -- Zw 4 ~i w -'
(15)
o" WllNw in which
_ ~12
IVwl2 sin2(WzZ) h~ol + [h2w~/(2mO] (2.~, + 1)
E(
1 mile2 ad --
)2]
h(/)g, - -
as -- g, h2us '
_ ~ ICQIZB2(N) exp( - hQZ/(2m!l2))e- 2Q,
, /-gl--
QhCOs
h2u2 , 2roll
hoJs h2u2
1 mile2
x 1 + fitat + [h2w~/(2mO] (2-Nw + 1) Q
h2ut
,
-
2mpl '
U~,
x/ZN, o + 1' (10)
Ci(2u, z) = - (|~ c~o(2zx)dx, j,~
x
where
B(N) = 1 -- R hQ2
Si(2ulz) = - i '° oi~(2zx) dx.
/ / h Q 2 \2~
).
+otto)
J, 4
(11)
in which O(x z) is an infinitesimal function that consists of x 2 and higher order terms. Further variation with respect to Po gives
~/(2ml!/(h2))alil Po = 1 - (2h2/mO f (z, T)'
(12)
where at and as are the coupling constants for LO phonon and SO phonon with electron, respectively. Obviously, the variation of F(2,z, T) with respect to 2 may determine the value of 2. Furthermore, the average value of the momentum fill for polaron is defined by
in which
1
f(z, T)
.g~i4UI U:INj, Nw,NQ) = mr(z, T)Ugi, IV~l 2 sinZ(~=z)
I--. 'W)
(13) After simplifying Eq. (10) by neglecting all higher order terms of ~ for slow electron, we obtain the expected value of F(Po,2,~lll, z, T) with respect to the ground state:
F(2, z, r) = ~Nwhoc + ~'.No.h¢os + G(2,z, T), (14) ,~ Q where
G(2,z, T) = (N + ½)h2 -- a~h(o)~2)1/2 x f~(1-
Nil (}21 ~i_ 1~ I ^
COS 2 (e, v ~'~(hcor~+ [h2co~/(2mll)] (2/~., + 1))3
-
~
2Nx )exp(-x
2
f2-
-24~u~xz)dx
(16)
X
^
(17)
where 1
m~l (z, T) = ml, -1 -- (2h2/mll)f(z, T) i/ ,~ "~3 / 2
+2aSt~ss )
f~x2(1-2~x2)
×exp(--x 2- 2 f~uszx)dx}
(18)
From the above, we can obviously see that ~j is the average velocity of polaron moving in the xyplane. In Eq. (18), there is a multiple factor for the renormalized mass mr(z, T), which consists of two terms. The first term is due to the LO phonon and the other term is due to the SO phonon. Note that,
CM. Lee et al. /Physica B 229 (1997) 361-368
in this paper, we do not consider the conventional band nonparabolicity, so the band mass mll is assumed to be temperature independent. The effective Hamiltonian is then given by Heft(Z , T) = H i + M i n ( N e, Nw, Nil (J~- 101 lff/ll
× (g~f_JelN2,N,~,Ne) p2
e2(eoo -- 1)
-- 2m± + 4ze~o(e~o +
+Z
+ Z Qho)s
w
+
1)
Q
P~ + (N + ½)h2 2m* (z, T)
- ~sh(m~,t)a/2 •
x exp( - x 2 -
- 2Nx 2)
2~u~zx)dx
~
- [Ci(2u'~z)sin(2u)z)
- Si(2u)z) cos(2u)z)]}
(19)
In the following section, we shall discuss the numerical calculation for relevant quantities of G(z, T) and m r (z, r ) .
3. Discussion and conclusions To illustrate our theoretical results in a quantitative manner, we choose an ionic compound, namely potassium iodide, as an example with a large coupling constant for electron-SO-phonon interaction having the value of es = 3.1 as well as an intermediate coupling constant for electronLO-phonon interaction, ~ = 2.5. All the parameters used in the numerical calculation are given as follows [22]:- e o = 4 . 6 8 , e~ =2.68, hco~= 18.0meV, ho)s= 16.9 meV, m~i = 2 . 1 × 1 0 -3 rob, where mb is the bare electron mass. For simplicity, the depth from the crystal surface is expressed in the unit of u; - 1 = 3 . 1 × 1 0 - v m . There are two points worth for discussion:
365
3.1. Self-trapping energy The self-energy of the polaron consists of two parts. One comes from the interaction between the electron and the LO phonon, while the other comes from electron-SO-phonon interaction. It is worth noting that, the negative value of the self-energy of the polaron represents a bound state of the self-trapped electron which is formed by an asymmetric local deformation in the lattice. As was known in the past, the magnitude of the self-energy G(z, T) is defined as the self-trapping energy. Our calculation results show that the self-trapping energy IG(z,Z)[ increases for strong electron-SOphonon coupling, i.e. the first and second terms in Eq. (15), and decreases for weak electronLO-phonon coupling, i.e. the third term, with temperature. From Fig. l(a), in general, the overall self-trapping energy IG(z,T)I monotonically decreases with temperature. When the temperature increases, the random motion of the ions in the crystal will become violent. The number of phonons then increases but the interaction between electron and phonon will be weakened [9]. It is consistent with the results of Ref. [6]. However, the strong electron-SO-phonon coupling increases with increasing temperature. Peeters et al. had also obtained the same results as stated previously [7]. Fig. l(a) also shows the overall self-trapping energy IG(z, T)I increases with increasing depth z from the crystal surface for a temperature T < 100 K, but decreases for T > 100 K. In Fig. l(b), the self-trapping energy [G(z, T)I decreases drastically with the depth near the crystal surface. It is because the polarised field produced by SO mode is dominant only within a very narrow depth from the crystal surface, z < 0.3u7 a, for lower temperature ranging from 0 K to 50 K. When z > 0.3ui 1, the self-trapping energy increases and tends to a constant value. However, for high temperature range, T > 100 K, the self-trapping energy gradually decreases. The effect of the bulk LO mode is dominant when the polaron is far from the crystal surface and more sensitive than the SO mode to temperature. Moreover, there is some interesting physical significance involved in Fig. l(b). Within a relatively low temperature range, T < 50 K, there is a minimum for the self-trapping energy IG(z, T)I. This minimum
366
C.M. Lee et al. / Physica B 229 (1997) 361-368 r
70.00 ~
>
N
1
I
,
Z ---- 0 . 5 1 1 1 - 1 .........
Z =
........
z = 1.5ul-1
1.0nl-1
50.00
30.00
10.00
I
i
I
I
I
0.00
50.00
100.00
150.00
200.00
(a)
250.00
T (K)
70.00
'
'
T=0K
~,=...5,o,K
65.00
. . . . . . . . . . . . . . . . . . . . . . . . .
60.00 v m
[..., t,4 v
1\ I ~,
55.00
T = lOOK
50.00 45.00 ""...... - . . , . 40.00
i
I
0.00
1.00
....
• ...........................
(b)
T = 150K
• ..........
I
i
2.00
3.00
Z (Ul
..=.
4.00
"1)
Fig. l(a). A plot of the self-trapping energy IG(z, T)[ against the temperature T for KI compound with different depths from the crystal surface, i.e. z = 0.5u[ 1, 1.0u71, 1.5u[l and 2.0uZ 1. (b) A plot of the self-trapping energy IG(z, T)I against the depth z from the crystal surface for KI compound with different temperatures, T = 0, 50, 100 and 150 K.
represents the m o s t u n s t a b l e p o l a r o n t h a t m a y occur at the valley of the [G(z, r ) l versus the d e p t h z. As the t e m p e r a t u r e increases, say T > 50 K, the t e n d e n c y for the p o l a r o n to m o v e to the i n t e r i o r is reduced. At high t e m p e r a t u r e , T > 100 K, the p o l a r o n tends to m o v e to the surface to increase the self-trapping energy Ia(z, r ) l . At zero t e m p e r a t u r e , the self-energies G(z, T ) of the p o l a r o n at the limiting cases of depths, z = 0 a n d z = o% for K I crystal are - 6 3 . 6 a n d - 7 0 . 7 meV, respectively.
3.2. Renormalized mass The c o n s e q u e n c e of the e l e c t r o n - p h o n o n interaction is to increase the r e n o r m a l i z e d m a s s of the p o l a r o n . This is because the electron d r a g s the h e a v y ion cores a l o n g with it. In Fig. 2(a), the p o l a r o n r e n o r m a l i z e d mass, in general, decreases with t e m p e r a t u r e as the d e p t h from the crystal surface is fixed. T h e e x p l a n a t i o n is the same as t h a t for the self-energy, because the higher the
367
C.M. Lee et al. / Physica B 229 (1997) 361 368
12.00
\
z=2.0ul'l
z = 1.5ux-l".
g
8.00 \ :
[.. ~f
Z ---- 1 . 0 U l -1 ~ %: "~
,,., 4.00
z = 0.5Ul-1
'..
"%.'$ ~ #
0.00
~
J
0.00
50.00
'
'
100.00 150.00 T (K)
(a)
200.00
250.00
Fig. 2a. A plot of the renormalized mass of polaron, rn~'(z,T), against the temperature T for K I with different depths from the crystal surface, i.e. z = 0.5u[ 1, 1.0u[ 1, 1.5u71 and 2.0u[ 1.
60.00 50.00 40.00 v [.., ~4
30.00 20.00 10.00
t
:,0K...............
0.00
T = 150K I
0.00
1.00
(b)
I
2.00 z (ui -1)
3.00
4.00
Fig. 2b. A plot of the renormalized mass of the polaron, m*(z, T), against the depth from the crystal surface, z, for KI compound with different temperatures, T = 0, 50, 100 and 150 K.
s e l f - t r a p p i n g e n e r g y the h i g h e r the r e n o r m a l i z e d m a s s is. As we c a n see in Fig. 2(a), the p o l a r o n r e n o r m a l i z e d m a s s i n c r e a s e s for a t e m p e r a t u r e T < 100 K a n d decreases for T > 100 K with inc r e a s i n g the d e p t h s f r o m the crystal surface. I n Fig. 2(b), the d r a s t i c decrease in the r e n o r m a l i z e d m a s s n e a r the surface is m o r e or less the s a m e for differ-
e n t t e m p e r a t u r e s , b u t w h e n the d e p t h i n c r e a s e s f r o m 0 . 5 u [ 1 o n w a r d s , the r e n o r m a l i z e d m a s s will be s i g n i f i c a n t l y affected b y the t e m p e r a t u r e . T h e h i g h e r the t e m p e r a t u r e , the l o w e r the r e n o r m a l i z e d m a s s will be. I n the i n t e r i o r of the b u l k s a m p l e , the p o l a r o n r e n o r m a l i z e d m a s s decreases w i t h i n c r e a s i n g t e m p e r a t u r e . It is o b v i o u s t h a t the effect of S O
368
CM. Lee et al. / Physica B 229 (1997) 361 368
mode is less sensitive to temperature than that of LO mode. Note that at zero temperature, the polaron renormalized masses at the limiting cases of depths, z = 0 and z = ~ , for KI crystal are 57.7mll and 54.8mll, respectively.
Acknowledgements We would like to cordially thank the Research Degree Committee, City University of Hong Kong, for finanica] support of this investigation. One of the authors, Gu, also thanks Lab. of Excited State Processes, Chang Chun Physics Institute, Academic Sinica of China, for financial support.
References [1] H. Ibach, Phys. Rev. Lett. 24 (1970) 1416. 1-2] E. Evans and D.L. Mills, Phys. Rev. B 5 (1972) 4126; E. Evans and D. L. Mills, Phys. Rev. B 8 (1973) 4004. 1-3] J. Sak, Phys. Rev. B 6 (1972) 3981. 1-4] T.D. Lee, F.E. Low and D. Pines, Phys. Rev. 90 (1953) 297. 1-5] W.J. Huybrechts, J. Phys. C 9 (1976) L211; O. Hipolito, J. Phys. C 12 (1979) 4667. [6] K. Arisawa and M. Saitoh, Phys. Star. Sol. B 120 (1983) 361.
I-7] F.M. Peeters and J.T. Devreese, Phys. Rev. B 25 (1982) 7302; F.M. Peeters and J.T. Devreese, Phys. Rev. B 31 (1985) 5500. [8] D.C. Khandekar, K.V. Bhagwat and S.V. Lawande, Phys. Rev. B 37 (1988) 3085. 1-9] F.M. Peeters and J. T. Devreese, Solid State Phys 38 (1984) 81. [10] K. Okamoto and S. Takeda, J. Phys. Soc. Japan 37 (1974) 333. 1-11] T. Masumi, Polarons and Excitons in Semiconductor and Ionic Crystals, eds. J.T. Devreese and F.M. Peeters (Plenum Press, New York, 1984) p. 99. [12] M.A. Brummell, R.J. Nicholas, M.A. Hopkins, J.J. Harris and C.T. Foxon, Phys. Rev. Lett. 58 (1987) 77. [13] W. Xiaoguang, F.M. Peeters and J.T. Devreese, Phys. Rev. B 36 (1987) 9765. [14] E. Haga, Progr. Theor. Phys. 11 (1954) 449. [15] P. Jin-Sheng, Phys. Stat. Sol. B 127 (1985) 307. [16] Y.-C. Li and S.W. Gu, J. Phys.: Condens. Matter 1 (1989) 3201. B.H. Wei and S.W. Gu, Phys. Stat. Sol. B 163 (1991) 161. Y.T. Wang, G.L. Gong and S.W. Gu, Phys. Stat. Sol. B, 179 (1993) K61. [17] DJ. Yang and S.W. Gu, Acta Sci. Natur. Univ. Intramongolicae 18 (1987) 2 (in Chinese). [18] W.J. Huybrechts, J. Phys. C 9 (1976) L211. [19] N. Tokuda, J. Phys. C 13 (1980) 173. [20] X.X. Liang and S.W. Gu, Solid State Commun. 50(1984) 505. [21] J.J. Licari and R. Evrard, Phys. Rev. B 15 (1977) 2254. [22] E. Kartheuser, Polarons in Ionic Crystals and Polar Semiconductors (North-Holland, New York, 1972).