Spin-orbit coupling effects in zinc blende structures

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.


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


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}.


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).


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}.


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).


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.

Documents

Spin-orbit coupling effects in zinc blende structures