8/11/2019 Spin-orbit coupling effects in zinc blende structures
1/7
PH
YSI
CAL
REVIEW
VOLUM E
100,
NUM B
ER 2
OCT OB
ER
15,
1955
Spin-Orbit
Coupling
Effects
in Zinc
Blende
Structures*
G.
DRESSELHAUst
Departmesss
of
Physics,
Umioersity
of
California, Berkeley,
Calcforrsia
(Received
June
30,
1955)
Character
tables
for
the
group
of
the wave
vector
at certain
points
of
symmetry
in the Brillouin
zone
are given.
The
additional
degeneracies
due
to
time reversal
symmetry
are indicated.
The form
of
energy
vz
wave vector at these
points
of
symmetry
is derived.
A possible
reason for
the
complications which
may
make
a
simple
effective
mass
concept
invalid
for
some
crystals
of
this
type
structure will be
presented.
HE
e6'ect
of
symmetry
on
the
energy
band
structures
of
crystals
of
the
zinc blende
type
can
be readily
derived
using
the
machinery developed
by
Bouck.
aert,
Smoluchowski,
and
Wigner'
and
Elliott.
'
Recent
extensive
studies
of
the
semiconductor
proper-
ties'
of
InSb,
which
has
the zinc blende structure,
and
preliminary
cyclotron
resonance
investigations'
have
indicated a
need for
a
more
thorough
understanding
of
the
possible energy
band structures
of a
zinc blende
type
crystal.
A
zinc
blende structure consists
of
two
interpene-
trating
face
centered
cubic
lattices;
each
f.
c.c.
may
be
considered
a
sublattice. The two
sublattices
are
dis-
placed
by
one quarter
of a
body
diagonal
and
each
consists entirely
of
one
species
of
atom.
If
the
two
sub-
lattices
are
identical,
one has a
diamond
structure. The
symmetry
properties
of
diamond
are
fully
discussed
in
reference
2.
FxG.
1.
The
erst
Brillouin
zone
for a
face
centered
cubic,
diamond,
and zinc blende
structure.
Points
and lines
of
symmetry
are
indicated
using
the notation
of
reference 1.
*This
work has
been
supported
in
part
by
the
Office
of
Naval
Research
and
the
U.
S.
Signal
Corps.
t
Now
at the Institute
for
the
Study
of
Metals,
University
of
Chicago,
Chicago,
Illinois.
'Bouckaert,
Smoluchowski,
and
Wigner,
Phys.
Rev.
50,
58
(1936).
s
R.
J.
Elliott,
Phys.
Rev.
96,
280 (1954).
e
H.
Welker,
Z.
Naturforsch.
7a,
744
(1952);
8a,
248
(1953);
M.
Tanenbautn
and H. B.
Briggs,
Phys.
Rev.
91,
1561
(1953);
G.
L.
Pearson
and
M.
Tanenbaum,
Phys.
Rev.
90,
153
1953);
M.
Tanenbaum
and
J.
P.
Maita,
Phys.
Rev.
91,
1009
1953);
H.
Weiss,
Z.
Naturforsch.
8a,
463
(1953);
O.
Madelung
and
H.
Weiss,
Z.
Naturforsch.
9a,
527 (1954).
4Dresselhaus,
Kip,
Kittel,
and
Wagoner,
Phys.
Rev.
98,
556
(1955).
The zinc blende
structure
has the
space
group
sym-
metry
F43m
or
T&'.
There
are
no
glide planes
or screw
axes,
so the
group
of
any
wave
vector
k
has
only
simple
operations. The
6rst Brillouin
zone is the
well-known
truncated
octahedron
shown
in
Fig.
1. The
character
tables
for
the
group
of
the wave vector k for certain
points
of
symmetry
in
the Brillouin zone are
given
in
Tables
I
through VI. When
the
spin
is
included in
the
problem
only
the
double
representations
occur
(i.
e.
,
representations
for which
a
360'
rotation,
E,
changes
the
sign
of
the wave
function).
The effect
of
including
spin
in
the
problem
is
to form wave functions
of a
spatial function
times
a
spin
function
which will
transform'
as
D;.
The
total
wave function will
then
transform
as
the
direct
product
of
a
single
group
repre-
sentation
with
D;.
This direct
product
then
can be
decomposed
in
terms
of
representations of
the double
group.
If more
than one representation
of
the
double
group
occurs
in
the
decomposition
of the direct
product,
a
spin-orbit
splitting
of the level
is indicated.
A
table
of
the
direct
products
of the
single
group
representa-
tions
with
Dg
is
included with
each
character
table.
The
compatibility
relations
for
certain
lines
of
sym-
metry
are
given
in
Table VII. These relations
give
the
splitting
of the
degeneracies
as one
proceeds
along
the
symmetry
axes. The
extra
degeneracies
due to time
reversal
symmetry
can
be found
using
the
standard
test
due to
Herring'
and Eliott.
'
The
extra
degeneracies
are
indicated in each table.
The
principal
diGerence
from
the diamond
structure
is
the
lack
of
inversion
symmetry
for
the
point
groups
in
the
zinc blende
structure. Without inversion
sym-
metry
one
still has the
result from
Kramers' theorem'
that
E(k)
=8(
k),
but now
the
periodic
part
of the
Bloch
functions no
longer
satisfies
the condition
I
q(r)
=N~(
r),
and
hence
a
twofold
degeneracy
throughout
the Brillouin
zone
is not
required.
The
one
electron Schrodinger
equation
for
the
'
E.
Wigner,
Grlppeatheorie
U.
W.
Edwards,
Michigan,
1944),
p.
245.
'
C. Herring, Phys.
Rev.
52,
361
(1937).
~
This
theorem
states
that in the
absence
of
magnetic
6elds
+lr
and
i0.
%'1,
*
are solutions of
the Hamiltonian for the same
energy.
The second
solution
belongs
to wave vector
,
and hence
we
have two
solutions at k
and
with
the
same
energy.
8/11/2019 Spin-orbit coupling effects in zinc blende structures
2/7
SPIN
ORB IT EFFECTS IN Zn
BLENDE
STRUCTURES
TAsLz I.
Character
table of the
double
group
of
r;
k=
(000).
r,
Fs
r4(x,
y,
z)
F5
r6
rv
rs
1
1
2
3
3
6C4B
1
1
2
0
0
0
1
1
0
0
1
1
8CB
1
1
0
0
1
6I
XC4
1
0
1
2
0
6I
XC4
1
0
1
2
v2
0
j.
2I
XCB
1
0
1
0
0
0
F;
F;XD)
Fy
F6
r,
Fv
Fs
rs
r,
Fv+Fs
r,
F6+Fs
Selection
rules
r;
r;xr
r,
F4
F2
r,
F3
r,
+r,
F4
r,
+r,+r,+r,
r,
r,
+r,+F,
yr,
F6
Fv+Fs
r,
F6+Fs
Fs
F6+Fv+2F
problem
with
spin-orbit
coupling
is
p'
+V+
(vVXp)
~
p~=Ea+~
2m
4m'c'
The translational
symmetry
of
the lattice
requires
that
the wave
functions
be
of
the
Bloch
form,
i.
e.
,
%g
Ng(r)e'~'
Degenerate levels
are
treated
by
solving the
customary
secular
determinants.
In
working
out the
matrix
elements
for
perturbation
theory,
it
is
helpful
to
use
group
theoretical selection
rules.
Due to the
scalar
character of
the
Hamiltonian,
TABLE
II.
Character
table of
the
double
group
of
6;
k=
kL100].
where
N(r)
is
periodic
and satisfies the
equation
P
+V+
(vVXp)
~
2m
4m'c'
ag(x)
~g
(y+z)
a4(y
)
1 1
1
1
1
1
1
1
2
2C4B
1
0
2I
XC2
1
1
0
2I
XC2'
1
1
0
f
p
5
l
p
fez/z'l
+
I
+
~xvv
I=I
E.
(m
4~'c'
)
E
2~)
The
equation
for
k+K
is
p2
+V+
(vVXp)
o+K
.
2m
4m'c'
f
p
+fzk
I
+
Em
4m'c'
&;XD)=~5,
i=
1, 2,
3,
4
53
and
A4
are
degenerate
by
time reversal
Selection
rules
6;XAy
~sX~B
~3
b.
;X
h4
A4
A4
b,
4
TABLE III. Character table of
the double
group
of A
or
L;
Ir=
(k/v3)(111]
or
(z-/a)(111).
3I
XC2
3I
XCB
,
L
ply'
+5K
I
+
~xvv
I+K
&m
4~zc'
)
A&(x+y+z)
A2
(x+cuy+aPz)
A 3
(x+~'y+cuz)
(~'=1)
I+K
(3)
~4
2m
)
+s
A6
I
E~+K
Treating
the
term
L4
and
L~
are
(A4
and
A;
AsXDy
Select&on
rules
A;
AI
Ag
A;XAz
A,XA&
t'
p
K'=5K
I
axvv
4m'c'
)
as a perturbation,
the
energy
at
k+K
for a
nonde-
generate
level
is
52)I|2
Ei+K
Eg+
.
+(pgI
K
I
g)+
2m
(4)
E
2CB
2CB
1 1
1
1 1 1
2
1
1
1 z
1
z
1
0
0
degenerate
by
time reversal
A5
are
nondegenerate)
Ag
A2 AB
A6
As A4+A5+A6
AB
A4
As A4
Ay+Ay+As
A6
A5
As
A.
5
A6
As
A4+A.
g+A6
8/11/2019 Spin-orbit coupling effects in zinc blende structures
3/7
G.
DRESSELHAUS
Z,
K
(s)
(*+y)
Zs(x
y)
Zs
Z4
I
XC&
ZsXD(=Zs+Z4,
i=
1,
2
Selection rules
Z;
Zg
ZsXZz
Zz
ZsXZ2
Zg
Zg Zs
Z4
Z2
Zs
Z4
Z1
Z4
Zs
I
XC2
1
2
TABLE
V. Character
table of the
double
group
of
X;
h=
(2s/a)(100).
TABLE
IV.
Character
table
of
the
double
group
of
Z
or
E;
lt
=(k/V2) 110j
or
(34r/2a)(110).
is
the
irreducible
representation
of
wave
vector
k
according
to which
4'k'
transforms
and
FR
is
the
repre-
sentation
according
to
which
a vector
transforms.
Another
way
of
viewing the
selection
rules
is
that the
only
representations
that
mix with
F;
under the
per-
turbation
BC'
are those contained
in
the
decomposition
of
the
direct
product
F;)&FR.
Tables
I
to VII
also
give
the
decomposition of
these
direct
products
for
the
points
of
symmetry
in
the
Brillouin
zone.
For
conveni-
ence
the
combinations
of
vector
components
which
transform
as
a
given
irreducible
representation
are
indicated
in
the
character
tables.
In
order
to
give
a
more
complete treatment
of
the
point
F,
the
bases
shown in
Table
VIII
may
be
selected
for
the
irreducible
representations.
'
In
this
notation
the
spin-orbit
splittings at
k=0
are,
for F41
4'*,
3)z
/'
BV
BV
AEso
zl
B,
p
p.
5,
l,
4m'c'
4 Bx
By
)
4C4~~
2C4~1)
2I
&(C4))
2I
XC411
4I
XC2
1
1
0
0
0
1
1 1
1
1
1
1
1
1
1
0
-2
0
0
o
o m
2
0
0
2 V2
Xg
X2
Xz(x)
X4
Xz(y,
s)
X6
X7
1
1
1
1
1
1 1
2
2
2
2
-2
and for
I 5&
5&,
35
BV
BV
++Be=
zl
el
pw
p~
es
4m'c'
Bx
By
g Xp
Xs
X4
Xs
Xs
X7
X'7
Xs
Xs+X7
X;
KXD)
Selection
rules
X; Xg
X2
Xs
X4
XXX,
X,
X4
X1
X~
X;XX5
X5
X5 X5 X5
Xs
X6
X7
X5
X7
Xs
X1+X2+X3+X4
Xs+X7
Xs+X7
TABLE
Vl. Character
table of
the double
group
of
5';
h
=
(24r/a)
(0
,
1).
I
XC4'
1
z
/i
C42
C4 I
XC4
I
XC4 I
XC4'
1
i+i-
i
iv'i
Qi
1
z
z
4/i
i/i-
+i
1
1
1 1
1
1
1
1
1
1
1
1
W1
Wz(x)
Wz(y+is}
W4(y
s)
8'5
W7
Ws
z
gi
iv'i
z
Z
b
z
z
TABLE
VII
Compatibility
relations
Selection
rules
suggest
that
the
representations
F4
and
F5
have
first-order
matrix
elements
with
3
and
hence finite
slopes.
Actually
all
6rst-order
matrix
elements
with
p
vanish
due
to time
reversal
symmetry.
For
example,
(Bzl
pl
Bz)
=
(8sl
plBt)
by
a
reflection
in
the
(101)
plane,
but
by
partial
integration
(8t
l
p
l
8s)
=
Bs
l
pl
Bz),
as the
8,
's
are
real
if
the
Hamiltonian
has
time-inversion
symmetry;
hence
all
such
matrix
ele-
ments
vanish.
This
argument
only
holds
if
all
5
s are
from
the
same
degenerate
F4
level.
When
spin-orbit
interaction
is
included
in the
per-
turbation
(i.
e.
,
the
oXV V term is
not
neglected),
the
energy
to
erst
order
in
k
for
a
F8
level is
given
by
the
W;
W'XD-;
Selection
rules
W;
W'X W2
W
XWs
W;X
W4
the
term
Ws+W6
Wg
W7+
Ws
Ws
W6+
W7
W4
Ws+
Ws
Ws W7
Ws W5
W5
Ws
W7
Ws
5(
fi
R=
l
p+
4rxv'U
m
(
4mc'
Wg Wp
Ws W4
Ws
Ws
W2
W1
W4
Ws
W7 W6
W4 W~
Wr Ws
W7
W4
Ws
W1
8'g
W5
F1~
Q1
F2
Q2
Fs
~
~1+~2
P4
~
411+
(413+~4)
pz
~
as+(&4+&4)
F6,
F7,
Fs~
~5
F1
Ai
F2
A.
g
Fs
As
F4
41+As
Fs
A~+A.
s
FF7~X,
Fs
~
~4+&5+As
X1~
A1
(L4+Lz)
~
A4+Az
X2~
h.
~
Xs~h1
X4
h2
X,
(~,+~,
}
Xs,
X7~
a5
Time reversal
degenerate
representations
are
indicated
by
paren-
theses.
transforms
as a
vector. Matrix elements of
the
type
(+
I&-le )
will
vanish
unless
the
direct
product
4
See
p
v& r
Laze
end
H.
A.
Bethe,
Phys. Rev.
71,
612
F,
XFaXF;
contains the
unit
representation, where
F;
(1942}.
8/11/2019 Spin-orbit coupling effects in zinc blende structures
4/7
SPI N
ORBIT
EFFECTS
IN
Zn
BLEN DE
STRUCTURES
secular determinant
(i/2)Ck+
i/2)Ck
Ck,
(iV3/2)Ck
(i'/2)Ck+
Ck,
Ck,
iV3/2)Ck+
(i/2)Ck
i'/2)Ck
Ck,
(i/2)Ck,
where X=E'
h'ks/2m),
which
has solutions
X=
~C(k'yL3(k
'k
'yk
'k
'+k
'k
')]&)-*'
'=
~C(ks
L3(k
sk
syk
sk
s+k
sk
s)]-:):
where
1 5s)
BU
for
r,
&,
2~pm'c'E
By
)
1ks( BU
2%3m'cs
E
By
C=O,
for
F3.
The
first-order
energies
are
shown
in
Fig.
2 for the
L100],
[111],
and
1-110]
directions.
In
polar
coordinates,
k,
=ksin8cosp, k=ksin8sing,
k,
=kcos8,
Eqs.
(8)
become
X=
~CkL1~(3)&
sin8(cosi8+r~
sin'8
sins2&)&]&.
(10)
Figure
3
shows the four
6rst-order
energies given
by
Eq.
(10)
for
constant
k plotted
against
the
angle
8
for
wave
vectors
in
a
(110)
plane (@=sr/4).
It
is
interesting
to
note
that
the
two-dimensional A.
6
level is
the
inter-
section of
surfaces
which
arise
from
two
diferent
repre-
sentations in
a
[100]
direction.
Any
large
separation
of the
energy
levels which transform as
F8
at the center
ENERGY
of
the
zone,
as one
proceeds
toward
the
edge
of
the
zone,
then
would
entail
a
rather
complex
band
structure
with
highly
deformed
energy
surfaces and accidental
de-
generacies.
'
Second-order
perturbation
theory
for the
F8
level
is
tractable
on
the
approximation
that
only
first-order
terms
in
e&VV and
second-order
terms
in
y
be
con-
sidered. The fourth-order
equation
which
results from
the
4)&4
secular
determinant
is
(L
M
l'
s
(L
M
'
y'
y'
I
3
)('(k
sk
2+k
sk
2+k
sk
2)+Csks
L
Mq'
/4yCsg(ksksyksks+ksks)+
~
~
k4
3
E'
(L,
M)'
+ (k.
'kys+k'k, '+k,
'k,
')
3
+C'Ek' 3(k'k'+k'k'+k'k')]
L
M's-
+2~
~
C'(k,
'+k
'+k ')
)
/Ll'
2
3~
~
+
S'
C'k'(k
'k
'+k
'kg'+kg'k-')
3 ) 3
pL
M'
+21~
~
C'k'k'k'=0,
(11)
)
where
I,
,
3f,
and
S
are real numbers
and
can be
ex-
pressed
in terms of sums over the
squares
of
the
Fn.
2.
Plot
of
energy
es
wave vector
showing
the
first-order
energy
for
the
spin-orbit
split
F4
or
I's
level in
100 ,
110),
and
(111
direc-
tions.
The circled
num-
bers indicate the
dimen-
sion
of the
representa-
tion.
P7 Qi
Qi
g4
{oooj
(ooo)
k[IIO]
k
goo]
(ooo)
Fxo.
3.
Cylindrical
cross
section
around
the
point
F of
a
plot
of
energy
ss wave
vector for wave vectors
in a
(110)
plane.
C.
Herring, Phys.
Rev.
52,
365
(1937).
8/11/2019 Spin-orbit coupling effects in zinc blende structures
5/7
G.
D RESSEL
HA
US
TAsI.E VIII.
Basis
functions for
the
irreducible
representations'
of the double
group
for
the
point
l
.
r~=1
rs,
dS
=
Lx'(ys
')
+yc(z'
')
+z'(*'
')7
Fg,
'y1=x
+Glg+GPS
y2
=
x'+any'+coo
cd=
1
rs,
n
}
0.
+)
r
Pl-)
lil+)
1
r.
(r),
'v+
~)l+)
v2
(~
'Ys
~71)
I )
V2
~'vs
~v~)
I+&
v2
Pro.
4. A
cylindrical cross
section
around
the
point
I
of
a
plot
of
energy
vs
wave
vector
for wave vectors
in
a
(110) plane
showing
the third order
splitting
of the
F6
or
I'7
surface.
14,
bg=x
82=y
53=8
~'vs+~v
~)
I
)
v2
1
rv('),
:
(si
os)
I+&+Ssl
&7
V3
1
I
~(s~
ss.
)
I
)
+
7
VS
r,
(r,&},
s,
Q,
)
I
)
W2
1
I
i(a,
b,
)
I+&+2s3I
&7
6
1
--
E~(s~+sss)
I
&+I
+&7
6
it=
%I
1007
the
energy
eigenvalues are
)L+2Mq
'
(L
Mq
'
+k[100l
=
u'+'
I
I
u'+
I
I
fs4+c
2m
E
3
J,
k
3
where
each
root
is
double;
and
for
k=
(k/V3)I
1117
(L+2M
)
~[sy~sl
[irtl
=
&
+
I
Ik'
-I~vl
(double)
(12)
r,
,
e,
=x(ys
')
es
=
y(z*-x')
es
(x'
')
z
~i+fss)
I+&
V2
1
rs(re),
p s(eq
ts)
+)+
3I
)7
&3
s(ei+ses)l
)
s3I+)
7
v3
rs(rat),
e&
s)
I
)
v2
b(e~
~s)l+)+2eal
&7
6
Ls(e~+s)
I
&+2esl+&7
6
~~+s)
I+)
V2
+
lP+%2Ck
(13)
3
O'
V2Ck.
An
estimate
of
the
magnitude
of
the
constant
C
may
be
obtained
if
one
considers the
zinc blende structure
as a
deformed
diamond
type
structure.
Here
we
will consider
U'=
V
V;
as a
perturbation,
where
Vi
is a
diamond
type
potential with inversion
symmetry
and
V
is
the
actual
zinc
blende
type
potential.
The
correct
6rst-order
spatial
wave
functions for
a zinc
blende structure are then
given
by
the
following
equations.
Fol
Py)
absolute
value of
matrix
elements,
and
(~'+I
1
l~+)
(&'
I
1
l~+)
n=u++
Q
n,
++
Q
P',
'
=I,
+
~~0
~;
' =~2
&O
&'
fi'
t'L+2M
~
Iu+y.
3
)
+0 ~i
i
=~2
+0
(14)
For
a
general
k,
Eq.
(11)
has
four real
roots;
for
io
Dresselhaus,
Kip,
and
Kittel, Phys.
Rev.
98,
368
(1955).
'
For
a
more
detailed
treatment see the
paper
by
F.
Herman
(to
be
published).
R.
H.
Parrnenter,
Phys.
Rev.
100,
573
(1955}.
8/11/2019 Spin-orbit coupling effects in zinc blende structures
6/7
SPIN
ORBIT EFFECTS
IN
ZII
BLEND E STRUCTURES
For
I'2,
p
i
=r2+
(-;-I
l
I=)
n;
+
g
lattice
constant
and
A~o
is
the spin-orbit
splitting
at
k=0.
The
first-order
energy
at
the
zone
edge
in
a
[100]
direction
is
(-;-I
I
IW)
e=~'+
Z;-+
E
jap
jv.
i
=r2+
For
I'3,
(v-;
II lv-
)
vx=vx++
2
QKi
(vx;
I
&'Ivx+)
+
2
+xi
)
jvp
jv.
=r12
(v;
II'lv
-)
vIc vK
+
VKi
(vx;
I
&'I
vx-)
+
2
%xi
~
(sx;+I
&'I
ex+)
&z
Ep
8
For
I'~,
(~-;-II
l~
-)
4c=&x
+ 4c
;
=r15-
EP
E;
(
'+Iv'ls
)
+
2
ex'+,
i
=rg5+
Ep
E.
(&x; I&
le++)
5K=+
p
.
~Z4
(17)
2m
~so
E
C
2s.
0.
02
ev (in
InSb).
8
Z
Hence the
splitting
due
to
the
first-order
terms would
be
only
of
the
order of
0.02 ev
if the
slope
were linear
all
the
way
to
the
edge
of the
zone.
Actually
the
second-
order
terms
which
should be
large
due
to the
small
energy
gap
will
turn the
surfaces
down
very rapidly.
Using
this
value
for
C
and
the
values of
I.
,
M,
and
X
from
cyclotron
resonance experiments on
Ge,
'
it
would
seem
from
Eq.
(11)
that near
the center
of
the Brillouin
zone the
removal of the
twofold Kramers
degeneracy
is
at most 10
4
ev
for holes with thermal
energies.
Under these
circumstances
it
seems
quite likely
that a
perturbation
expansion
about the
extremum
should
contain
several
orders
of
perturbation theory, and
a
simple
energy
surface with
effective
mass tensor
com-
ponents
independent
of
wave vector
would
be a
very
poor
approximation.
For the
I 6
and
F7
level
the
energy
to
third order in
k
is
given
by
E=
C
k'~C
[k'(k
'k
'+k
'k
'+k
'k
')
k
'k
'k
']'
(20)
In
third
order
the
levels are
split
in
all
but
the
[100]
and
[111]
directions.
In
polar
coordinates
Eq. (20)
reads
E=
Cek'+Cik'
sin8[1
in'8
(1+2
sin'2p)
+
(9/4)
sin'2p
sin48]l.
(21)
For
F5,
(&x;+I
&'I
&x+)
err
=~K
+
~z;+
gp
g
4
(;-I
v'l~
+)
&Xi
)
p
(&rr;+I
&'I
ere
)
ex=ex
+
~z'+
i =rIfi
jvp
jv.
(ex'
I
I
I
)
where
the cubic
harmonics
are
as
defined
in reference
8
except
that
Vt+
=
Vs+*=
x'+4ey'+to's', (oi'=
1),
and
the
&
superscript
is
used to denote
the
parity
of the
functions.
For the
I'4
level
arising
from
a
diamond
type
F+
level
fz'
C=-
&3
tS
C i
=r&4
(et+I
~l'*/~yl34'
)(~i'
I
l
I
et+)
(19)
(Eo
E')
To
a
very
rough
approximation
V'
P';/Z
where Z
is the atomic
number.
C
(1/Z)hsoa,
where
a
is
the
Figure
4
shows
a
plot
of
energy
es
8
for
constant k and
&=4r/4
[i.
e.
,
k in
a
(110)
plane].
The
surfaces
have
their maximum
separation
along
&il0) axes.
In the
III
V
class
of
semiconducting
compounds
like
InSb,
the
high
mobility
electrons
are
presumably
in
a
spheri-
cally
symmetric
I'6
state.
'
Higher
orders
in
k
should
not enter
until
the thermal
or Fermi
energy
for
the
electrons is
of
the
order
of the
band
splittings
at
k=0.
In
impure
e-type
InSb
with electron
concentrations
10 /cm'
and
a
Fermi
energy of
0.
2 ev one
may
be
entering
into a
region
where the
splitting
of
the
degeneracy
for the
electrons
should
be considered.
Perturbation
expansions
for
other
points
in
the
Brillouin
zone
are
facilitated by
writing
the
perturbation
in
a
form such
that
the vector combinations
indicated
in the character tables
appear;
these combinations
are,
for
the
points
I',
(000),
and
X,
(24r/a) (100),
K'=
k~
+kR+k&. (22)
for the
point
W,
(24r/a)
(O,
s,
1),
3C'
=
K+,
+p[(K+i
K,
)
(RE,
)
+(KsK,
)
(Z+4m.
)];
(23)
'4
Hrostowslti,
Wheatley,
and
Flood,
Phys,
Rev.
95,
1683
(1954).
8/11/2019 Spin-orbit coupling effects in zinc blende structures
7/7
G.
DRESSELHAUS
TABLE
IX. Possible
energy
extrema
for
the
representations
of
the
double
group
in
zinc
blende
structures.
X6
x
-0
7
f22
Cl
12
I
X
c9
CL
X7
-8
z
p-.
X6
LLI
0-
r
-0
6
rs
0
r
-0
7
Is
r,
L6
L6
(
L4&
Lg)
L6
L6
Extrema at
(000)
(k/VS)
L111]
h.
Representations
of
double
group
r,
F7
Fa
(from
Fq)
Constant
energy
surfaces
sphere
sphere
2
warped
energy
surfaces as
for
holes
in Si
and
Ge
8
spheroids;
&11'1)
axes
8
spheroids;
&111)
axes
0-
I
6
(2s.
/a)(1,
xs
0)
W
Ws,
Wo,
W7,
Ws
6
spheroids;
(100)
axes
2
/,
(IOO)
(ooo)
(ooo)
/p
(I
I
I)
REDUCF D
WAVE
VECTQR
K
General
point
48 general
ellipsoids
FIG.
5. A
schematic
drawing
of
the
energy
levels for a
zinc
blende
type
structure
modification of
boron
nitr ide based
on
Herman's
calculation
for
diamond
PF.
Herman,
Phys.
Rev.
88,
1210
(1952);
thesis,
Columbia
Vniversity,
1953
(unpublished)].
The
spin-orbit
splittings
are
highly
exaggerated
for
the
purpose
of
illustration.
The levels
marked with an 0 have
zero
slope
along
that axis.
This
6gure
should be
compared
with
Elliott's
Fig.
2.
L'R.
J.
Elliott,
Phys.
Rev.
96, 266
(1954).
g
for
the
points
A, (k/~3(111),
and
L,
(m/a)(111),
x'=
',
(E'
+K
+E,
)
(R,
+R+R,
)
+
(K,
+toK+cesK,
)
(R,
+aPR,
+(oR.
)
+
(E,
+tosE+toE,
)
(
R,
+
toR+c,
)i,
(24)
and
for
the
points
Z, (k/v2) (110),
and
6,
k(001)
5C'=
K&,
+ts[(K,+-K)
(R,
+R)
+
(K,
K)
(R,
R
)
).
(25)
Using
these
perturbations the
Grst-order
energies
can be
written down at
sight
and are
of the
following
form: for
hs,
k(100),
Z=CsK
+Cs(Ks+K,
)'*;
(26)
for
A4
and
As,
(k/~3
(111),
E=
C,
(E,+E+E,
);
(2&)
for
h6,
E=
Cs(E,+E+E,
+CsLEs
(EgK+KE,
+E,
K.
))&) (28)
for
Z4
or
Zs,
(k/~2(110),
Z=Cr(E,
+E
)+CsK,
;'
(29)
and
for
Xs
or
Xr, (2m/a)
(100),
Z=
&C
(Es+Es,
)
&.
The second-order energies
for
the
point
W,
(2s.
/u) (0,
ts,
1),
are of
the
form
Z=CtoE
'+C
(Ett'+E',
).
s(31)
If
the
energy
extremum is
at
the
point
8',
the constant
energy
surfaces will
be
spheroids
with
&
100& axes.
The
irreducible
representations
A4
and
A5
only
have
slopes
along
a
&111&
axis,
hence
if the constant
C4
is
zero at
some
point along
the axis
as
it
presumably
is
g.par
the
ct:nter of
the
zone
in InSb
due
to
the
perturbing
influence of
the lowest
conduction states
(see
Fig.
5),
then
the
constant
energy
surfaces
at
these
extrema will
be
spheroids
with
&111&
axes.
However,
because
of
the
smallness
of the
first-order terms
compared
to
the
second-order
terms,
it
is
not expected
that
these
extrema
will
be of
any
significance in
the band structure
if
they
occur near
the
center
of
the
zone.
The
representation
I'8
at the
center of the
zone
can
have zero
slope
if
it
arises from
the
two-dimensional
representation
F3
of the
single
group.
In this
case
the
energy
to
second order
in
k is
given
by
E=Ct2ks&LCtssk'+Ct4s(k,
'ks+k'k,s+k,
'k
')]l,
(32)
in
the limit
that
all
spin-orbit
splittings
are
negligible
compared
with the
spacing
between
levels
at k=0,
then
C~4'
C~3'. In
atomic Sn
it
is
known that
the 4d
levels
overlap
the Ss atomic
levels,
hence
in
grey
Sn
or
InSb
it is conceivable
that a
I 3
level,
which
can
be
represented
in
a
tight
binding approximation
by
d-orbitals,
could
be
the
uppermost valence
band.
Table
IX
gives
a
tabulation
of the
types
of
energy
surfaces
which
might
be
expected
at
certain
points
in
the
Brillouin
zone. It
should
be
emphasized
in all these
considerations
that
if
the
spin-orbit
interaction is
small
or
the
difference
in
the crystal
potential
from the
diamondlike
potential
is
only slight,
then the
region
of
convergence
of
the
energy
expressions
will be
small
compared
to
kT and one
should
then
consider
only
the
single
group
representations
in
the
6rst
case,
or the
diamond structure
double
group
representations
in the
latter
case.
If
the
region
of
convergence
of the per-
turbation
expansion
is ~kT,
then
one is
not
justi6ed
in
keeping only
the lowest
nonvanishing
term and
a
simple
effective mass
approximation
would
seem unjus-
ti6ed.
From the order
of magnitude
estimate
of
the
spin-orbit
splitting,
it
seems
this
might
be
the case
for
holes in InSb.
I should
like
to
thank
Professor
C.
Kittel
for
sug-
gesting
this
problem
and
for
guidance
throughout
the
course
of the
investigation.
I am
indebted
to
Dr. F.
Herman
and
Dr. R.
Paramenter
of
R.
C.A.
Laboratories
for
communicating ,
their
results
to
me
prior
to pub-
lication.
I also
wish to
thank
Mr. R,
K,
Qehringer
for
checking
the manuscript.