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EFFECTIVE MASS APPROXIMATION FOR SURFACE PHONON STATES
A. Fasolino, G. Santoro, E. Tosatti
To cite this version:
A. Fasolino, G. Santoro, E. Tosatti. EFFECTIVE MASS APPROXIMATION FOR SUR- FACE PHONON STATES. Journal de Physique Colloques, 1981, 42 (C6), pp.C6-840-C6-842.
�10.1051/jphyscol:19816248�. �jpa-00221335�
JOURNAL DE PHYSIQUE
Colloque CG, suppllment au n o 12, l'ome 42, dicembre 1981 page C6-840
E F F E C T I V E MASS A P P R O X I M A T I O N FOR SURFACE PHONON STATES
A .~asolino*, G. Santoro*" and E. Tosatti +*
*GNSM-CNR and SISSA, I s t i t u t o d i Fisica Teorica, Universitci d i T r i e s t e , 3401 4 Miramare-Grigmo, T r i e s t e , I t a l y
** I s t i t u t o d i Fisica and GNSM-CNR, Universitd d i Modena, 41700 Modem, I t a l y + ~ n t e r n a t i o n u ~ Centre f o r Theoretical Physics, 34014 Miramare-Grigmno, T r i e s t e ,
I t a l y
Abstract.- We show how effective mass concepts borrowed from the theory of in- purity states in semiconductors can usefully be applied to the calculation of surface phonon states.
1. Introduction.- Given an infinite crystalline solid, two independent surfaces can be created by adding a localized perturbation V, which consists of cutting all bonds that cross a plane. Historically the effect of localized perturbations on an infinite Bloch problem have been dealt with first in connection with impurity states in semi- conductors (lt2). When dealing in particular with the bound states that the impurity potential is able to split from the Bloch bands, the popular effective mass approxi- mation has been most useful in providing a qualitative understanding of the pro- blem(1f2).
Asurface state - electronic, vibrational, whatever its microscopic natu- re - is, in principle, a bound state of the same kind; only the perturbation has in- finite extension along (x,yl and is localized just along
z .This work represent an attempt to apply the concepts typical of the effective mass formulation to the study of a surface state problem. In order to make our study more definite and provide ourselves with accurate numerical reference calculations to use as a check, we shall concentrate in the following on a surface phonon problem, in particular the phonons of a bcc (100) surface.
2. Method.- We start by formally writing the surface phonon eigenvector as a linear
-+ -t
combination of bulk phonon states u with the same in-plane momentum q and variable normal component k:
Here R is an atomic site and + denotes the branch. The "envelope" function then satisfies:
AX'
-+ -+ -twhere Vkk ,=<uGkX [vI u;~, > is taken between bulk states. For a general q-direction and for a Bravais lattice, Eq.(2) constitutes a set of three coupled equations. For -.
particular q values, however, symmetry may allows a decoupling. Such is the case for
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19816248
example of q=(i,i)2n/a-~ + on the bcc (100) surface, that is our test case. ~t that
-+ - + -
point we have a non-degenerate v// z mode (M 1
)and a twofold degenerate vlz mode (M 5
).
The symmetry labels are those of Cqv, that is the two-dimensional part of the crystal symmetry group as reduced by the surface perturbation V. We now develop this particu- lar case as an instructive exemplifica tion. For comparison, we also perform accurate numerical surface state cal- culations with the usual slab techni- que using the parameters of ~ ( 1 0 0 ) (3) .
We use variable intra-surface force constants a and 6s(4) to control the actual surface state energy and eigen- vector.
The bulk bands w for GZM are
skA
shown in Fig.1. Here k spans twice the Brillouin zone (from -2n/a to 2n/a) as required by ha-ring two alternating Fig. 1
:2ulk phonon branches types of layers in the problem. We no- of W for qiM
te that this give rise to two equiva- lent minima at ko=O and ko=2n/a for both the M1 symmetry and the M5 symmetry and in fact for all other zone-border <-points as well. However, the two minima are orthogo- ~.
nal by symmetry (belonging to the two different rows (x-y) and (x+y) of the twofold degenerate irreducible representation M5) and do not mix in the M5 case, while they are mixed by V in the M case. The V-matrix is given by all those slab matrix elements that are removed by creating the surface, taken with negative sign. We obtain analyti cal expressions for vkk, at the M point as a function of a and Bs. However they are too long to be reported here(5); we shall just proceed to :iscus the results.
3. Bound state problem: solutions.- a) variational
:we try exponentially decaying solutions:
and
:where the "+" and "-" signs in Eq.(3) correspond to x+y and x-y respectively, the two terms in the M1 case are necessary in order to include intervalley mixing (that remo- ves the degeneracy between the "+" and "-" case) and the variational parameter is the penetration depth 1.
b) Effective mass approximation
:For shallow bound states, one may approximate
C$.,Xd+x2 (k-kg)
2 b b/m::. The Schroedinger-like problem (2) can be solved varia-
qk,A qk,X
tionally, with trial functions such as ( 3 ) and (4).
C6-842 JOURNAL DE PHYSIQUE
C) Koster-Slater approximation
:If we further replace Vkk, with Vk , where kois
0 0
the relevant extremum, the solution becanes similar to the well known Koster-Slater problem (except for a cutoff at +2n/a), that is straightforward to evaluate analyti- cally.
Fig.2 presents a comparison of the approximate solutions a), b) and c) with the exact numerical slab surface phonon "binding energy". This quantity, introduced in
Bulk States
O~(THZ')
0.8 0.4 0 -0A -Q8
W 6
-
+ ~ x a c t
Fig. 2
:Surface phonon "binding energies" calculated at theMpoint of W(100) for Bs=3.7 as+4.222
analogytothe impurity binding energy in the usual effective mass theory(lt2), is defi- ned as ~ ~ = ( w ' - ~ ~ ~ ) / 2 ~ The variational result is in perfect aggreement with
qkol'
the exact calculation over the whole range of parameters that have been tested. This means that the straight exponentials ( 3 ) and (4) in fact represent extremely well the behaviour of the surface phonon. We have checked that this is indeed the case.
The approximations b) and c) are clearly acceptable only for shallow surface states. However, the effective mass approximation b) remains fairly accurate in the whole range, while the Koster-Slater result c) fail appreciably already for E /band
+ B+
widthz0.1. We note that while for M5 the two states $- are degenerate, only bM
leads to a bound state in the M1 case, where $- yields E =O. M5 1
1
