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VIBRATIONAL PROPERTIES OF VACANCIES IN
HOMOPOLAR SEMICONDUCTORS
K. Suzuki, D. Schmeltzer, A. Maradudin
To cite this version:
JOURNAL DE PHYSIQUE
CoZZoque C6, suppZ6ment au n o 12, Tome 42, de'cembre 1981 page C6-640
VIBRATIONAL PROPERTIES OF VACANCIES IN HOMOPOLAR SEMICONDUCTORS
K. Suzuki, D. Schmeltzer and A.A. Maradudin
Max-PZanek-Institut fiir Festk6'rp~rforsehungg Heisenbergstrasse 1, 7000 Stuttgart 80, F.R. G.
Abstract. Spectral densities of phonons i n the immediate v i c i n i t y of a vacan- cy i n Si and Ge a r e calculated in the continued fraction/recursion method. The r e s u l t s indicate t h a t both in Si and Ge the presence of vacancy gives r i s e to a sharp decrease in the COS i n t h e optical frequency range and a r a t h e r d i f f u s e increase i n t h e acoustic frequency range.
1. Introduction.
-
A t moderately high concentrations of hydrogen i n aworphous S i , four hydrogen atoms tend t o c l u s t e r as they s a t u r a t e qroups of four dangling bonds pointing towards a counterpart of t h e c r y s t a l l i n e vacancy.' Infrared a b s o r p t i o n l y 2 and %man s c a t t e r i n g 3 experiments on amorphous and c r y s t a l l i n e Si or Ge vith H ,O,
or F a s impurities show t h a t the frequency of t h e resonant vibration i n the acoustic range depends only very sl ightly on inpuri ty snecies. This suogests t h a t such vi bra- tional modes involve a large number of a t o m of the host and leads one t o suspect t h a t the vacancy i n whose v i c i n i t y the clustering occurs nay i t s e l f qive r i s e t o resonant modes i n the frequency range where they have been observed.
As a step i n the direction of elucidating the nature of the resonant rrodes we have studied vibrational properties of crystal1 i ne Si and Ge containing an isolated vacancy. Use has been made of the real space version of the continued f r a c t i o n l r e - cursion method4, which does not require a periodic arrangement of atoms. The method has been applied successfully
to
several types of l a t t i c e dynalrical defect problems involving a lowering of symnetry5, although not to point defects a s f a r as we a r e aware.The local density p m ( R . R ; w ) of vibrational nodes of polarization u a t s i t e L and frequency w i s d i r e c t l y related
to
the diagonal element of the Green's functionC
of the dynamical matrix D u B ( R , R ' )
.
In the recursion methodS the l a t t e r i s expressed in a continued f r a c t i o nL
where the c o e f f i c i e n t s ( a o , a l , . ;.) and (bo, bl,
..
.) can be obtained a s functions of R and ci algebraically once D u B ( Z , R 1 ) of the systen! i s given.2 . Perfect Crystal
.
-
To examine the a p p l i c a b i l i t y of the recursion rethod we f i r s t
c a l c u l a t e t h e phonon spectrum of Si crystal without vacancy. On t h e one hand the
phonon spectrurr, i s obtained in the recursion method by considering a f i n i t e crystal
containing up t o 18 000 atoms, a t t h e surface of which a l l bonds a r e terminated. The
dynamical matrix i s such t h a t includes only the f i r s t and second neighbor forces.
7This requires s i x independent force constants, of which f i v e have been chosen t o
give a best least-squares f i t to the experimental phonon frequencies
a t
r,
X,and
Lpoints.8 The l a s t one, the second neighbor force constant
6,which a f f e c t s none of
t h e above phonons, has been e i t h e r fixed on the basis of the valence force f i e l d
9
illode1
( 6 =(P-v-A)/2)
o r equated t o zero. The f i r s t 21 coefficients (ao,
.
.
,
azO)
and
(bo,
..,
bZ0) in Eq. ( 1 ) were d i r e c t l y calculated and t h e remining part of the
continued f r a c t i o n was replaced by an asymptotic form
where am and ba a r e chosen e i t h e r from an extrapolation of the calculated coeffi-
c i e n t s o r fro111
the requirement t h a t the obtained phonon spectrum extends from zero
t o the experimental highest frequency (namely that of the optical phonons a t
r ) .
The
phonon LEOS paa(R, R;w) calculated from E q . ( 1 ) i s independent of
a
and
a
(provided
Ri s close t o the center of c l u s t e r ) and i s represented by the upper curve i n Fig.
1.On
the other hand t h e phonon spectrum
i s calculated f o r an i n f i n i t e l a t t i c e us-
ing the same force constants in the usual
k-space sampl ing method. With about
80 000 sample points in
1/48of the B r i l -
louin zone we obtain a phonon 30s which i s
shown by the lower curve in Fig.
1.-
u. 0?
2
t--
Hi
t honly up t o second neighbor for-
0
ces neither of the DOS reproduces t h e
sharp peak in the acoustic region which
0 ,. .. ... ,' : ,---..1 \
_.I'
._
....
,I.
., ,would be present i f the substantial f l a t -
O
6'
5 10 15FREQUENCY 'ITHz)
tening of acoustic phonons in t h e v i c i n i t y
Fig.
1.Lattice vibration spectrum of
OfBZ
edge
were
properly
taken into ac-
m g
f i r s t and second neighbor
count.1° Except f o r t h i s , both of t h e
forces
i n t h e and i nthe
curves reproduce t h e general features of
k-space sampling method. The f o r c e
constants a r e those defined by Her-
the known phonon spectrum f a i r 1 y we1 1 . The
man.
two curves a r e very s i m i l a r , the only d i f -
ference being t h a t t h e recursion method general 1 y smoothes sharply edged van Hove
s i n g u l a r i t i e s c h a r a c t e r i s t i c of i n f i n i t e crystal s . 11
DOS FOR PHONONS IN SI
C6-642 JOURNAL DE PHYSIQUE
3. Vacancy.
-
To model a vacancy i n the s i m p l e s t way we remove t h e c e n t r a l S i atom a t R=o and equate t o zero a l l forces connected t o t h i s atom. I n c o n t r a s t t o t h e preceding case, t h e LDOS does depend on t h e l a t t i c e s i t e R a n d we evaluate i t a t one o f t h e four nearest neighbors o f t h e vacancy. The r e c u r s i o n c a l c u l a t i o n pro- ceeds i n a s i m i l a r way as before u s i n g t h e same s e t o f f o r c e constants, except t h a t we equate 6=0, which i s r e q u i r e d if t h e t r a n s l a t i o n a l i n v a r i a n c e c o n d i t i o n 6 i s t o be s a t i s f i e d w i t h o u t i n t r o d u c i n g new f o r c e constants. No r e l a x a t i o n o f atomic p o s i t i o n i s assumed. The s p e c t r a l d e n s i t y f o r t h e p e r f e c t c r y s t a l (6=0), t h a t o f t h e vacancy and t h e i r d i f f e r e n c e a r e shown i n F i g . 2.DOS FOR PHONONS IN SI
I
RECURSION METHODNCYC = 21 ,
I
a=5 6206 x
lo4
dynlcrnpc3.7417
A=-0.7221 (a1 PERFECT ~ ~ 0 . 3 8 2 1 CRYSTAL v.0.4695
1 6A
$
-0.1 IC 1 DIFFERENCE Z W lb)-
la) a -0.2 0 5 10 15 FREQUENCY (THz) Fig. 2 . V i b r a t i o n spectrum o f t h e s i n g l e v a c a n c y i n Si c a l c u l a t e d i n t h e r e c u r s i o n method. 4 . Discussion.-
I n F i g . 2 we see a d i f f u s e i n - crease o f DOS i n the lower acoustic frequency r e g i o n and a decrease i n t h e o p t i c a l r e g i o n . T h i s i s expected i f we consider t h e presence of a vacancy as a p a r t i a l s o f t e n i n g o f t h e force a c t i n g on i t s neighbors. The i n t r o d u c t i o r o f H or F, on t h e o t h e r hand, corresponds t o a hardening (decreased i o n i c mass) and gives r i s e t o a r a t h e r sharp resonant mode a t t h e upper edge o f t h e a c o u s t i c spectrum, as has been ob- served. A f u r t h e r d i s c u s s i o n on t h e change o f DOS f o r various p o i n t d e f e c t s can be made on the b a s i s o f the f u n c t i o n ( 1 ) f o r t h e p e r f e c t c r y s t a l.
E s s e n t i a l l y s i m i l a r r e s u l t s a r e ob- t a i n e d f o r a vacancy i n Ge.5. Acknowledgement.
-
We a r e indebted t oM. Cardona, H. B i l z , and W . Kress f o r discus- sions and t o H.J. StXrke f o r advice on numeri- c a l work.
References
1. S.C. Shen, C.J. Fang, W. Cardona, and L. Genzel, Phys. Eev.
B
22, 2913 (1980).2. H. Shanks, C.J. Fang, L. Ley,
M.
Cardom, F.J. Cernon, and S. K g b i t z e r , Phys. S t a t . Sol. ( b )100,
43 (1980);S.C. Shen, C.J. Fang, and M. Cardona, Phys. S t a t . Sol. ( b ) 101, 451 (1980). 3. D. Bermejo and
M.
Cardona, J . Pioncryst. S o l i d s 32, 405 ( 1 5 7 v4. R. Haydock, S o l i d S t a t e Phys
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K e l l y,
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.,
S o l i d R a t e Phys. Suppl. 3 (2nd ed.), 1 (1971).7. Our n o t a t i o n o f f o r c e constants i s t h e same as Fy H e r ~ a n , J . Phys. Chem. Sol i d s 8, 405 (1959).
8.
ST.
B i l z andW.
Kress, Phonon D i s p e r s i o n R e l a t i o n s i n I n s u l a t o r s , Springer, 1979.9. K. Kunc, PI. Balkanski, and ivl. Nusimovici, Phys. Rev. B
-
12, 4346 (1975). 10. IJ. Ueber, Phys. Rev. B 15, 4789 (1977).11. S i m i l a r conclusions h a v f i e e n obtained f o r a simpler case by P.E. Meek, P h i l . Mag. 33, 897 (1976). See a l s o C. H e r s c o v i c i and M. F i b i c h , J . Phys. C - 13, 1635
