Solid State C o m m u n i c a t i o n s ,
P r i n t e d in G r e a t Britain.
PRESSURE
S.
Vol.56,No.6,
DEPENDENCE
Ves +,
K.
OF
Stressner,
THE
pp.479-483,
LOWEST
N.E.
M.
1985.
DIRECT
0 0 3 8 - I 0 9 8 / 8 5 $3.00 + .00
P e r g a m o n Press Ltd.
ABSORPTION
Christensen*,
Cardona
Chul
Koo
EDGE
OF
Kim**,
ZnSe
and
Max-Planck-Institut
for F e s t k e r p e r f o r s c h u n g ,
Helsenbergstrasse
D-7000 Stuttgart
80, F e d e r a l R e p u b l i c of G e r m a n y
Received:
August
5,
1985,
by M a n u e l
I,
Cardona
T h e v a r i a t i o n of the l o w e s t d i r e c t a b s o r p t i o n
e d g e (go) of Z n S e w i t h
hydrostatic
p r e s s u r e h a s b e e n m e a s u r e d at r o o m t e m p e r a t u r e
with a diam o n d a n v i l c e l l for p r e s s u r e s
up to the p h a s e t r a n s i t i o n
(13.5±0.2
GPa). T h e g a p v a r i e s s u b l l n e a r l y
w i t h p r e s s u r e . H o w e v e r , w h e n the g a p
is p l o t t e d as a f u n c t i o n of the r e l a t i v e c h a n g e of the l a t t i c e c o n s t a n t
(-Aa/ao)
it s h o w s a s l i g h t s u p r a l l n e a r i t y
in c o n t r a s t to m o s t I I I - V
tetrahedral
semiconductors
which retain a slight sublinearity
e v e n as a
function of (-Aa/ao) . The experimental
r e s u l t s a r e c o m p a r e d to t h e o r e tical calculations
b a s e d (a) on the l o c a l d e n s i t y a p p r o x i m a t i o n
comb i n e d w i t h the s e l f - c o n s l s t e n t
band calculations
u s i n g the L M T O m e t h o d
( L D - L M T O ) and (b) on a l o c a l e m p i r i c a l
pseudopotential.
Both calculations predict a subllnear
dependence
of E 0 as a f u n c t i o n o f ( - A a / a 0 ) .
The semiempirlcal
Murnaghan's
e q u a t i o n of s t a t e t h a t w a s c a l c u l a t e d
w i t h the L D - L M T O m e t h o d , h a s a l s o b e e n t e s t e d by d e t e r m i n i n g
the l a t t i c e c o n s t a n t u n d e r p r e s s u r e w i t h an e n e r g y d i s p e r s i v e
x-ray diffractometer.
the w a y up to the p h a s e t r a n s i t i o n
to the N a C I
s t r u c t u r e * * , *s (13.5 GPa). We c o m p a r e the
experimental
r e s u l t s w i t h two t y p e s of calculations, a eemlempirical
o n e b a s e d on the
pseudopotential
m e t h o d (EPM) and an a b - i n i t i o
one b a s e d on the l o c a l d e n s i t y a p p r o x i m a t i o n
in c o n j u n c t i o n
w i t h the L i n e a r M u f f i n Tin
O r b i t a l m e t h o d ( L D ~ L M T O ) . As a b y - p r o d u c t
of
the l a t t e r we h a v e c a l c u l a t e d
the e q u a t i o n of
s t a t e for p r e s s u r e s up to 12 G P a and c o m p a r e d
it to e x p e r i m e n t a l
d a t a o b t a i n e d w i t h an e n e ~
gy d i s p e r s i v e
x-ray diffractometer.
We f i n d
considerable
discrepancies
at h i g h p r e s s u r e s .
The p r e d i c t i o n s
of M u r n a g h a n ' s
e q u a t i o n for Bo
- 62.4 GPa ,2 and B~ = 4 . 7 7 GPa, .2 h o w e v e r ,
a g r e e p e r f e c t l y w i t h the e x p e r i m e n t a l
data.
INTRODUCTION
The d e v e l o p m e n t
of the d i a m o n d a n v i l
cell* has made possible the measurement
o f the
pressure dependence
of the a b s o r p t i o n
edge of
tetrahedral
semiconductors
up to the p r e s s u r e s
where phase transitions
o c c u r ( I 0 - 2 0 GPa). 2,'
Direct absorption
e d g e s in Ge, ~ OaAs, s, InP, e
GaP,* and ZnTe 7 vary sublinearly
with pressure. H o w e v e r ,
w h e n p l o t t e d vs. the c h a n g e in
lattice constant
( - A a / a 0 ) by u s i n g s o m e tellable theoretical
or e x p e r i m e n t a l
e q u a t i o n of
s t a t e m o s t of the n o n - l i n e a r i t y
disappears
and
a nearly linear, sometimes slightly sublinear,
sometimes
slightly
superlinear,
variation with
(-Aa/ao) obtains. More recent work using modu lation techniques'
(photoreflectance),
and
l u m i n e s c e n c e , ° c o n f i r m the g e n e r a l i t y
of these
results.
In m o s t m a t e r i a l s
measured
(Ge, GaAs,
InP, GaP) the l o w e s t g a p is e i t h e r i n d i r e c t or
b e c o m e s i n d i r e c t at h i g h p r e s s u r e s .
This fact
l e a d s to s o m e b r o a d e n i n g
of the d i r e c t gap
(Eo, r,s ÷ r,) w i t h i n c r e a s i n g
pressure.
In
the I I - V I c o m p o u n d s
the r,s ~ r, d i r e c t gap
(Eo) is the l o w e s t gap, i n d i r e c t g a p s s u c h as
r,s ~ X I and r , s +
L, t a k e p l a c e at l e a s t at I
eV a b o v e E 0 a n d s t a y a l w a y s a b o v e Eo u n d e r
pressure.
H e n c e the o p t i c a l a b s o r p t i o n
edge
remains sharp with increasing
p r e s s u r e a n d its
pressure dependence
c a n be d e t e r m i n e d
rather
accurately.
T h u s w e h a v e c h o s e n to s t u d y the
pressure dependence
of the Eo g a p of Z n S e a l l
EXPERIMENT
O p t i c a l a b s o r p t i o n measurements under
p r e s s u r e s up t o 14 OPa have been performed f o r
ZnSe in a gasketed diamond a n v i l c e l l . ~3 The
d e t a i l s of the o p t i c a l a n d p r e s s u r e a p p a r a t u s
have been published
in Ref. 3. T h e s a m p l e w a s
a -5 ~m t h i c k p i e c e of Z n S e m e a s u r i n g
about
0.1 mm a c r o s s , p r e p a r e d by m e c h a n i c a l
polishing. X - r a y d i f f r a c t i o n
measurements
were also
c a r r i e d o u t on Z n S e in a g a e k e t e d a n v i l c e l l
in the e n e r g y - d i s p e r s i v e
x-ray diffraction
(EDXD) mode. T h e s a m p l e s ,
f r o m the s a m e i n g o t
u s e d for the a b s o r p t i o n
measurements,
were
g r o u n d to p o w d e r for t h i s work. As a p r e s s u r e
transmitting
m e d i u m a 4:1 m e t h a n o l e t h a n o l
m i x t u r e w a s u s e d and the p r e s s u r e w a s m e a s u r e d
by the r u b y f l u o r e s c e n c e
technique**
for b o t h
types of measurements.
+On l e a v e f r o m A r i s t o t l e
University
of Thessaloniki,
First Laboratory
of P h y s i c s , Thessaloniki,
Greece
*On l e a v e f r o m The T e c h n i c a l
University
of
Denmark, Physics Laboratory
I, D K ~ 2 8 0 0 Lyngby, D e n m a r k
**Alexander
y o n H u m b o l d t F e l l o w . On l e a v e f r o m
Yonsei University,
Dept. of P h y s i c s , S e o u l
120, K o r e a
RESULTS
AND
DISCUSSION
The d i r e c t a b s o r p t i o n
e d g e s of Z n S e obt a i n e d at s e v e r a l p r e s s u r e s
up to 12 GPa a r e
s h o w n in Fig. I. T h e d a t a w e r e t a k e n o v e r a
p e r i o d o f s e v e r a l d a y s a n d the p r e s s u r e s w e r e
m e a s u r e d b e f o r e a n d a f t e r e a c h run. The p r e s s u r e s g i v e n in Fig. I a r e m e a n v a i u e s o f i n i /,70
480
PRESSURE DEPENDENCE
-ZnSe
~
OF THE L O W E S T D I R E C T A B S O R P T I O N
EDGE OF Z n S e
Vol.
56, No.
6
:.~ ,.qm ~qm c~~ ~
ZnSe
T=3OOK
34
LI.I
32
A
>
oo
Pseudopotemmt
m
I
i
25
27
29
ENERGY
Fi~.
I
I
31
I
I
33
I
35
tUJ
[eV)
I: A b s o r p t i o n
e d g e s of Z n S e o b t a i n e d at
room temperature
for s e v e r a l h y d r o s t a tic p r e s s u r e s up to 12 GPa. The a r r o w s
i n d i c a t e the p o i n t t a k e n to be the
" e x c i t o n e d g e " at the g i v e n p r e s s u r e .
t i a l and f i n a l o n e s w i t h t y p i c a l d i f f e r e n c e s
b e i n g ~0.I GPa.
In Z n S e the E0 (Fys ÷ F~) a b s o r p t i o n
edge
is s t r o n g l y s t e e p e n e d by e x c l t o n i n t e r a c t i o n ;
t h i s e f f e c t m a k e s i t s e l f felt e v e n at 300 K. l~
The e d g e w a s d e f i n e d for e a c h p r e s s u r e by
f i n d i n g the e n e r g y at w h i c h the r i s i n g a b s o r p tion coefficient
sharply bends and becomes
flat. T h i s e n e r g y is f o u n d q u i t e e a s i l y and
accurately
a n d is i n d i c a t e d by the a r r o w s in
Fig. I.
B e c a u s e of the a b s e n c e of an i n d i r e c t g a p
the k i n k s a t t r i b u t e d
to the d i r e c t gap a r e
w e l l d e f i n e d up to p r e s s u r e s J u s t b e l o w the
phase transition
w h i c h o c c u r s at 1 3 . 5 ± 0 . 2
GPa. T h e k i n k a s s i g n e d to the gap d o e s not
correspond
e x a c t l y to the e x c l t o n e n e r g y but
is s l i g h t l y l o w e r e d as a r e s u l t of s c a t t e r e d
and spuriously
transmitted
light which determ i n e s the p l a t e a u in the t r a n s m i s s i o n .
However, s i n c e the a b s o r p t i o n
e d g e s of Fig. I
shift parallelly
w i t h p r e s s u r e the s o m e w h a t
incorrect definition
of the e d g e s h o u l d not
a f f e c t the p r e s s u r e d e p e n d e n c e
obtained
in
this way.
At the p r e s s u r e of 1 3 . 5 ± 0 . 2 GPa the
s a m p l e s b e c o m e s u d d e n l y o p a q u e in the e n e r g y
r e g i o n c o v e r e d by o u r d e t e c t o r
(hm ~ 1.5 eV).
T h u s we w e r e n o t a b l e to v e r i f y the t h e o r e t i cal prediction
of Ref. 15, a c c o r d i n g
to w h i c h
the h i g h p r e s s u r e p h a s e of Z n S e is of N a C I t y p e w i t h an i n d i r e c t gap a r o u n d 1.1 eV. T h e
transition
f o u n d w i t h our a b s o r p t i o n
measurements agrees well with that reported
in the
l i t e r a t u r e at 1 3 . 7 ± O . 3 GPa. *t ~
After lowering the p r e s s u r e at a b o u t 10 GPa the c r y s t a l
becomes again transparent
b u t not of the same
o p t i c a l q u a l i t y as b e f o r e the p h a s e t r a n s ~
tlon. U p o n i n c r e a s i n g
the p r e s s u r e a g a i n the
m a t e r i a l b e c o m e s o n c e m o r e o p a q u e w i t h o u t any
s h i f t in the t r a n s i t i o n
pressure.
The e n e r g y of the Eo gap o b t a i n e d in the
way mentioned
a b o v e is p l o t t e d in Fig. 2 as a
f u n c t i o n of p r e s s u r e . The s o l i d l l n e t h r o u g h
the e x p e r i m e n t a l
points represents
a quadratic
fit w h i l e the d a s h e d and the d a s h e d - d o t t e d
l i n e s g i v e the r e s u l t s of b a n d s t r u c t u r e c a l c u l a t i o n s p e r f o r m e d w i t h the L ~ L M T O met h o d ~" ]' and w i t h the EPM, 3 r e s p e c t i v e l y .
The
analytic expressions
for the e x p e r i m e n t a l
and
theoretical
pressure dependences
are:
28
4
6
s
lb
13
Pressure (GPa)
Fi~.
Lowest
direct
energy
gap (r[5
+ F~) of
Z n S e vs. p r e s s u r e at r o o m t e m p e r a t u r e
as o b t a i n e d f r o m a b s o r p t i o n
measurem e n t s (o). T h e s o l i d l i n e t h r o u g h the
experimental
data represents
a quadratic l e a s ~ s q u a r e fit. T h e d a s h e d line
represents
calculations
w i t h the L D L M T O m e t h o d , w h i l e the d a s h e d - d o t t e d
llne represents
calculations
b a s e d on
a local empirical pseudopotentlal.
The
dashed llne (LD-LMTO-method)
has been
s h i f t e d up by 1 . 5 2 eV in o r d e r to
m a t c h the e x p e r i m e n t
at z e r o p r e s sure.
2:
Exp.:
Eo =
(2.688±0.004)
LMTO:
Eo
=
1.170
EPM:
Eo
= 2.697
+ (7.2±0.2~x10-2p
-(15zl)x10-Wp
2
+ 6.0x10-2p
+ 6.57x10-2p
-
14.5x10-"p
-27.5x10-Wp
-
=
2,
(1)
w h e r e Eo i s g i v e n
i n eV a n d p i n G P a . T h e
theoretical
curves have been converted
from
the r e l a t i v e c h a n g e of l a t t i c e c o n s t a n t ~ Aa/
ao) to p r e s s u r e s by u s i n g M u r n a g h a n ' s
equation
of s t a t e : 2°
p
-
(~o)[(a~) 3B; -
1 ],
(2)
where a is the lattice
constant
at the pressure
p , Be = 6 2 . 4
GPa t h e b u l k
modulus ]2 and
B6 = 4 . 7 7
its
pressure
derivative,
lz The exp~
rlmental curve shows a small subllnearlty
w h i c h t u r n s into a s l i g h t s u p r a l l n e a r l t y
when
we p l o t E o as a f u n c t i o n o f ( - A a / a o ) , as can
be s e e n in Fig. 3. A s i m i l a r r e v e r s a l of the
c u r v a t u r e of E o has a l s o b e e n o b s e r v e d r e c e n ~
ly o n G a A e " by c o n v e r t i n g
the e x p e r i m e n t a l
d a t a o b t a i n e d by p h o t o r e f l e c t a n c e
spectroscopy
f r o m p r e s s u r e v a l u e s to r e l a t i v e
c h a n g e s of
lattice constant. Nevertheless
transmission
d a t a y i e l d for G a A s a s l i g h t l y
eubllnear de
Vol. 56, No. 6
P R E S S U R E D E P E N D E N C E OF THE L O W E S T D I R E C T A B S O R P T I O N EDGE OF ZnSe
TABLE
481
I: C o e f f i c i e n t s o b t a i n e d f r o m l e a s t s q u a r e s fits w i t h
E o ( p ) - Eo + bp + cp 2 to the e x p e r i m e n t a l p r e s s u r e
d e p e n d e n c e of the E o gap o f Z n S e at r o o m t e m p e r a ture. A l s o c a l c u l a t e d v a l u e s of the c o e f f i c i e n t s
o b t a i n e d w i t h two d i f f e r e n t m e t h o d s and c o m p a r i s o n
w i t h v a l u e s f o u n d in the l i t e r a t u r e .
I
!:k[!
Experiment|2.688~0.001
b(10-2eV/GPa)
~
12.6922.67 g
Theory:
Pseudopotentlal
and KKR
2.697 a
2.67 h
[. . . . . . . . . . . .
Theory:
LMTO
-ASA
1.17 a
.17
c ( 1 0 - W e V / G P a 2)
7.17+0.18a
7.5~.0.3 b
6.0 ~
-15±I a
6.5X a
7.1 ~
5.8 e
3h
7.5 r , 7.
-27.5 a
4.9 *a
6.0 a
- 9 . 5 *a
-I 4.5 a
a p r e s e n t w o r k . For c o n v e r s i o n f r o m (- ~ )
to p r e s s u r e s the
, M u r n a g h a n e q u a t i o n o f s t a t e w a s used. a°
aTheoretloal equation of state.
bp. J a s z c z y n - K o p e c ,
B. C a n n y , G. S y f o s s e , and H. H a m e l ,
S o l i d S t a t e C o m m u n . 49, 795 ( 1 9 8 4 ) .
CA.L. E d w a r d s , T.E. S l y k h o u s e , a n d H.G. D r l c k a m e r , J. Phys.
C h e m . S o l i d s 11, 140 ( 1 9 5 9 ) .
d D o n L. Campha~-sen, G.A. N e v i l l e C o n n e l l , a n d W. P a u l , Phys.
Bey. L e t t . 26, 184 (1981).
e y . F . T s a y an-d S.S. M i t r a , Phys. Rev. B 10, 1476 ( 1 9 7 4 ) .
fRef. 15.
aM. C a r d o n a , J. A p p l . Phys. 32, 2151 (1961).
hKKR calculations:
F. C e r d eira, J.S. D e W i t t , U. R ~ s s l e r , a n d
M. C a r d o n a , phys. stat. sol. 4_!I, 735 (1970).
p e n d e n o e o f E o on Aa/a. H e n c e m o r e m e a s u r e m e n t s are r e q u i r e d in the c a s e of G a A s in
o r d e r to c l a r i f y the s i t u a t i o n . In Fig. 3 we
a l s o c o m p a r e the c a l c u l a t i o n s by b o t h m e t h o d s
d i s c u s s e d a b o v e w i t h the e x p e r i m e n t a l r e s u l t s ,
c o n v e r t e d into d e p e n d e n c e s on - A a / a o w i t h the
use o f Eq. (2) (the LD L M T O c u r v e h a s b e e n
s h i f t e d u p w a r d s so as to m a t c h the e x p e r i m e ~
tal one at z e r o p r e s s u r e ) . For c h a n g e s of the
l a t t i c e c o n s t a n t up to 1 . 5 % the p s e u d o p o t e n tial c a l c u l a t i o n s a g r e e w e l l w i t h the exper i m e n t a l r e s u l t s . T h e y d e v i a t e m a r k e d l y for
larger changes, exhibiting a strong subline~
r i t y w h i c h r e s u l t s f r o m the f a s t e r i n c r e a s e of
the v a l e n c e b a n d FYs c o m p a r e d to t h a t o f F~, a
fact w h i c h is a l s o f o u n d in the L D ' L M T O c a l c u l a t i o n s . The L D - L M T O c a l c u l a t i o n s
come closer
to the e x p e r i m e n t a l c u r v e at l a r g e r d e f o r m ~
t l o n s b e c a u s e o f the s m a l l e r q u a d r a t i c t e r m
(see T a b l e II), in s p i t e of the l a r g e r d i s a g r e e m e n t for s m a l l -Aa/a0. B o t h t h e o r e t i c a l
c a l c u l a t i o n s p r e d i c t a s u b l i n e a r d e p e n d e n c e of
E0 as a f u n c t i o n o f -Aa/ao, c o n t r a r y to the
experimental results.
In Fig. 4 we s h o w M u r n a g h a n ' s e q u a t i o n o f
s t a t e (Eq. (2)), w h e r e the p a r a m e t e r s Bo and
B' h a v e the v a l u e s 6 2 . 4 GPa a n d 4 . 7 7 , r e s p e ~
t ~ v e l y , t a k e n f r o m Ref. 12. The d a s h e d line
r e p r e s e n t s the t h e o r e t i c a l e q u a t i o n o f s t a t e
o b t a i n e d w i t h the L D - L M T O m e t h o d : It d e v i a t e s
c o n s i d e r a b l y f r o m the p r e d i c t i o n s of M u r n a g h a n ' s e q u a t i o n . In o r d e r to c h e c k the c o r r e c t n e s s of t h e s e e q u a t i o n s of s t a t e we
me~
s u r e d the r e l a t i v e c h a n g e o f the l a t t i c e p a r ~
m e t e r at s e v e r a l p r e s s u r e s . Our e x p e r i m e n t a l
r e s u l t s are g i v e n by the s o l i d c i r c l e s in Fig.
4. T h e y e s t a b l i s h the c o r r e c t n e s s of M u r n a g h a n ' s e q u a t i o n and the d e f i c i e n c i e s of t h a t
o b t a i n e d "ab i n l t l o " w i t h the L D - L M T O m e t h o d .
AS a l r e a d y m e n t i o n e d we h a v e u s e d for our
c a l c u l a t i o n the e m p i r i c a l p s e u d o p o t e n t l a l
m e t h o d a n d the LD L M T O m e t h o d . The d e t a i l s of
the u s e d p s e u d o p o t e n t i a l
method have been
p u b l i s h e d in Ref. 3 ( a l s o r e f e r e n c e s t h e r e i n ) .
We g i v e b e l o w a few d e t a i l s of the L D - L M T O
calculation.
The t o t a l e n e r g y o f ZnSe as a f u n c t i o n of
v o l u m e w a s c a l c u l a t e d u s i n g the l o c a l - d e n s l t y
(LDA) a p p r o x i m a t i o n , .7 w h i c h h a s p r o v e d to be
a c c u r a t e for a l a r g e n u m b e r of s i m i l a r p r o blems. The o n e - e l e c t r o n S c h r 6 d l n g e r - l l k e
equ~
t l o n was s o l v e d s e l ~ c o n s i s t e n t l y w i t h the
L M T O m e t h o d . *e :9 As u s u a l , for the c a s e o f
z l n c b l e n d e s t r u c t u r e s , we i n t r o d u c e d two " e m ~
ty s p h e r e s " c e n t e r e d at E l and E2, i.e., the
s t r u c t u r e is fcc w i t h a b a s i s c o n s l s t l n ~ of
1 1 Y
f o u r atomic s i t e s : Zn ( 0 , 0 , 0 ) , Se ( = , ~ , : ) a , E:
444
-~ J a ,
and E 2 2 2 ~)a.
The valence
bands
(3,3,3.~
~
(1,!,1
o f ZnSe c o n s i s t
of a low lying
~ 1 3 . 2 e V ) Se
4s b a n d w h i c h is r a t h e r n a r r o w ( - 1 . 2 eV), and
a Zn 3d b a n d (at
- 6 . 7 eV) w h i c h is o n l y - 0 . 8
eV w i d e (1.1 eV w h e n s p l n = o r b l t c o u p l i n g is
i n c l u d e d ) . The b a n d p o s i t i o n s r e f e r r e d to h e r e
are m e a s u r e d f r o m the t o p of the v a l e n c e b a n d
at F (Se 4p). The L M T O b a n d s t r u c t u r e w a s
c a l c u l a t e d u s i n g t w o e n e r g y " p a n e l s " , ~" the
l o w e r c o v e r i n g the Se 4s b a n d a l o n e , a n d the
u p p e r c o n t a i n i n g the r e s t of the o c c u p i e d
482
PRESSURE D E P E N D E N C E OF THE LOWEST DIRECT A B S O R P T I O N EDGE OF Z n S e
ZnSe
3.4
Vol. 56, No. 6
T=300K
..EExperirner~
D
,/LD-LMTO
ZnSe
.-. 3.2
>
~
T-300 K
LD~-
X//"/L;TI_
(D
,..=..
,/'+
//
/ f
Equat~
~potential
/,,;;+z
I.,.I 3.0
~
6
Bo=62.4 C4~
8~:437
eExpe.met
~x
k~,,
28
i
2
~
~
~
6
(-~-~0) (%)
Fi~.
~-- (%)
3: D e p e n d e n c e of the Eo d i r e c t gap on
~ a / a o for Z n S e c a l c u l a t e d w i t h the
L D - L M T O m e t h o d ( d a s h e d llne) and with
a local empirloal pseudopotential
( d a s h e d - d o t t e d llne). The s o l i d line
t h r o u g h the o p e n c i r c l e s r e p r e s e n t s a
q u a d r a t i c fit to the e x p e r i m e n t a l
p o i n t s of Fig. 2 c o n v e r t e d to a f u n ~
± i o n of Aa/a0 by u s i n g EQ. (2).
TABLE
II:
Coefficients obtained from least
s q u a r e s f ~ t s w i t h E.o(a) = E o + b ( - a~o
a)
+ c(= ~/k~)~ to the m e a s u r e d and oalc u l a t e ~ Q v a l u e s of the Eo g a p s of
ZnSe.
Eo(eV)
I
Experiment
2.685
. . . .
I
4
Theory
2.695
I
b(eV)
c(eV)
14.4±0.4
6.2±0.4
14.9
-125.7
Pseudopotential
Theory
iii iI111: ' ......
-26.8
LMT~
ASA
(Zo
Fi~.
4: P r e s s u r e vs. v o l u m e d a t a for Z n S e at
r o o m t e m p e r a t u r e . The s o l i d line is a
p l o t of M u r n a g h a n ' s e q u a t i o n (Eq.
(2)). (Ref. 20). The s o l i d c i r c l e s are
our e x p e r i m e n t a l d a t a and the d a s h e d
llne our t h e o r e t i c a l e q u a t i o n of s t a t e
o b t a i n e d w i t h the L D - L M T O m e t h o d .
w i t h e x p e r i m e n t over a l a r g e - & a / a 0 r a n g e
u s i n g the s a m e m e t h o d o f c a l c u l a t i o n . We h a v e
not yet f o u n d the s o u r c e s for t h i s d i f f e r e n c e
in a c c u r a c y , and we do not feel it is J u s t
fled, at p r e s e n t , to a s c r i b e it to e r r o r s
i n t r o d u c e d by the LDA. == H o w e v e r , it c a n n o t be
e x c l u d e d that the d i f f e r e n c e s o c c u r as a o o ~
s e q u e n c e of the s p h e r i c a l a v e r a g i n g of the
Z n - 3 d e l e c t r o n d e n s i t y . T h e Ga 3d s t a t e s p l a y
a o o n s l d e r a b l y less i m p o r t a n t r o l e for the
b i n d i n g in OaAs, and t h e r e f o r e the n e g l e c t o f
the n o n - s p h e r i c i t y of t h e i r d e n s i t y s h o u l d be
a m u c h less s e r i o u s a p p r o x i m a t i o n .
We l i s t b e l o w , for c o m p l e t e n e s s , the
d e p e n d e n c e on l a t t i c e c o n s t a n t and p r e s s u r e
o b t a i n e d for the f i r s t i n d l r e o t g a p s of Z n S e
( b o t h a b o v e E0) o b t a i n e d f r o m o u r EPM c a l c u l ~
tions:
E(ry.+x~) - 4.366-1.376(- a1~a+)-1~2[-aa~=)2
= 4.371-O.014p'9.2~IO-~p
v a l e n c e s t a t e s . The t h e o r e t i c a l e q u a t i o n o f
s t a t e o b t a i n e d f r o m the t o t a l e n e r g y f u n c t i o nal y i e l d s an e q u i l i b r i u m v o l u m e a n d a b u l k
m o d u l u s Bo in g o o d a g r e e m e n t w i t h e x p e r i m e n t :
the t h e o r e t i c a l l a t t i c e c o n s t a n t is 0 . 2 ~ s m a ~
let t h a n o b s e r v e d and Bo is 60.5 GPa ( e x p e r i m e n t B o = 62.4 GPa). In v i e w of t h e s e r e s u l t s
it is s u r p r i s i n g t h a t the p r e s s u r e ( - A a / a , )
r e l a t i o n (Fig. 4) s h o w s l a r g e d i s c r e p a n c i e s
w h e n c o m p a r e d to e x p e r i m e n t . T h i s is f u r t h e r
p u z z l i n g s i n c e in the c a s e of G a A s (see FJE. 4
of Ref. 20) we f o u n d e x t r e m e l y g o o d a g r e e m e n t
r,, L,)
E[ V + O
=
2
(3)
,.283+5.7~ ~1-1oo( ~12
= q.289+O.O22~'15.3.10""p
=
(41
w i t h the e n e r g y in eV and the p r e s s u r e in OPa.
The E o gap s t a y s l o w e r t h a n t h o s e in Eqs.
(3) and (4) up to the p h a s e t r a n s i t i o n .
We h a v e a l s o c a l c u l a t e d w i t h the EPM-me± h o d the d e p e n d e n c e of the E, gap ( L ~ + F ~ ) on
( - d a / a o ) a n d on p, by t a k i n g for t h i s gap the
Vol. 56, No. 6
PRESSURE DEPENDENCE OF THE LOWEST DIRECT ABSORPTION EDGE OF ZnSe
average of 10 points equally spaced between
=(0.4, O.4,0.4~ and ~(I,1,1) along the A-direction of ~-space. ~e found:
Ez = 4 . 6 6 + 1 1 . O ( =
where
Aa
~'T
)'I°7.("
4.67+4.Tx10-Zp'20.5x10-~p
E,
is
eV and
pressure,
Aa] 2
ao,
=,
p is
483
ACKNOWLEDGEMENTS
We would llke to thank W. Dieterloh, H.
Hirt, M. Siemers, and P. Wurster for technical
a s s i s t a n c e and E. Kisela for the preparation
of the samples.
(5)
in
GPa.
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