NNSE 618 Lecture #4

1

Lecture contents

• Spin-orbit coupling

• kp method

• Valence band

• Band structures of semiconductors

NNSE 618 Lecture #4

2

Degenerate valence band: Spin-orbit coupling

SLrHso )(Hamiltonian of spin-orbit interaction includes

orbital momentum L and spin S operators

• SO-coupling is a relativistic effect

• Responsible for fine structure of atomic levels

• Interaction significant close to nucleus

(smaller than inner Bohr radius)

• Treated as a perturbation: need to find an

average over the unperturbed state

• Probability to find electron inside the inner

shell~ 1/Z2

• For the state with quantum numbers l and s,

the SO coupling will be determined by total

angular momentum j

dr

dV

rcm

1

2

122

)1()1()1(2

2

1

2

222

sslljj

SLJSL

2

S

prL

22

2

eZmr

r

ZerV

2

)(

4

422

2

meZ

c

e

Fine structure constant

NNSE 618 Lecture #4

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Degenerate valence band: Spin-orbit coupling

• States with different j will have different energies !

• In the G-point (k=0) valence band wavefunctions are

constructed from atomic p-states: px , py, pz (6 states

total)

• Each of these states has angular momentum l=1 and

spin s = 1/2

• We can find linear combinations of these states with

total angular momentum j in the range from |l - s| =

1/2 to |l + s| = 3/2

• Using the rule for summation of angular momentum :

j = 1/2 (2 states), j = 3/2 (4 states)

• Spin-orbit splitting:

2

1,

2

3,

2

)1()1()1(2

2

2

2

jfor

jfor

sslljj

ESO

2

3 2SO

NNSE 618 Lecture #4

4 kp method

nknknk uEurVm

k

m

pk

m

p

22

222

Perturbation

One-electron Schrödinger equation has

Bloch function solutions (n- band):

When k is substitiuted to Schrödinger

equation:

And can be treated as perturbation

near some (in general any) point of

Brillouin zone. For k0=(000) it reduces

to:

If En0 , un0 are known, they are used as

unperturbed values to calculate Enk ,

unk in the vicinity of k0 :

( ) ( )ikr

k nkr e u r

)()(2

2

rErrVm

p

2

0 0 02

n n n

pV r u E u

m

2 2 20 '0 0 '0

, ,0 2' 0 '0

| | | |

2

n n n n

n k n

n n n n

u p u u p ukE E k k

m m E E

0 '0

, ,0 '0

' 0 '0

| |n n

n k n n

n n n n

u k p uu u u

m E E

NNSE 618 Lecture #4

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kp method: Degenerate band extremum (valence band)

nknknk uEurVm

k

m

pk

m

p

22

222

Perturbation

One-electron Schrödinger equation

in the vicinity of band extremum :

nn nn

nnnn

ijEE

upkuupku

mH

ji

' 0'0

00'0'0

2

2 ||||Non-diagonal second order

perturbation matrix elements :

The perturbed energies E(k), from

secular equation:

with

0

333231

232221

131211

HHH

HHH

HHH

m

kkE

2)(

22

Band-structure close to G-point :

NNSE 618 Lecture #4

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Degenerate band extremum: valence band

Kohn-Lutttinger equation for valence band (heavy and light holes):

From Balkanski and Wallis, 2000

C 2.5 -0.1 0.63

Si 4.28 0.339 1.446

Ge 13.38 4.24 5.69

NNSE 618 Lecture #4

7 kp method at k=(000): Effective mass and Eg

Dispersion and effective

mass in nondegenerate

extremum:

Conduction band effective mass in

Gpoint (4-bands):

nn nn

nnnn

n EE

upuupu

mm

m

' 0'0

0'00'0 ||||2

*

22 2 1

1* 3

sp

e g g SO

pm

m m E E

2 2 20 '0 0 '0

, ,0 2' 0 '0

| | | |

2

n n n n

n k n

n n n n

u p u u p ukE E k k

m m E E

0 '0

'0

' 0 '0

| |n n

n

n n n n

u p uk u

E E

0 '0

, ,0 '0

' 0 '0

| |n n

n k n n

n n n n

u k p uu u u

m E E

14

g

eV

E

Contribution from

a band is inversly

proportional to the

energy difference!

NNSE 618 Lecture #4

8 General features of a bandstructure of semiconductors

with zinc-blende and diamond lattice

p-symmetry

NNSE 618 Lecture #4

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Effective masses at G-point

From Yu and Cordona, 2003

NNSE 618 Lecture #4

10 Example: Kane model

4-bands Kane model:

Two parabolic bands at k0 = 0 :

Conduction band : s-type

valence band: p-type with SO

coupling

(good for small bandgap semiconductors:

InSb, InAs)

2

0 0

0 0

| |21

*

c x v

c c v

u p um

m m E E

SOgg

sp

gcEEm

P

m

kEkE

12

3

)0(21

2)(

222

2 2

( )2

vh

kE k

m

g

sp

vlmE

P

m

kkE

3

)0(41

2)(

222

SOg

sp

SOSOEm

P

m

kkE

3

)0(21

2)(

222

Becomes negative as a result of inteaction with outher bands

NNSE 618 Lecture #4

11 Band-structures of Si and Ge

NNSE 618 Lecture #4

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Bandstructure of GaAs

1

22

0

22

67.0

*2)1(

067.0*

*2)(

eV

m

kEE

mm

m

kkE

cc

c

With nonparabolicity:

NNSE 618 Lecture #4

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Bandgaps

From Balkanski and Wallis, 2000

NNSE 618 Lecture #4

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Band structures of group III-nitrides

From Singh, 2003

Symmetry Lattice

parameters (Å)

Band gap (eV) TE Coef.

(106/K)

GaN (wurtzite) hex. a = 3.189,

c = 5.185

3.39-3.50 5.59

3.17

GaN (zinc-blende) cubic a = 4.531 3.30-3.45

AlN (wurtzite) hex. a = 3.112,

c = 4.982

6.20-6.28 4.2

5.3

AlN (zinc-blende) cubic a = 4.33 5.11 (Indirect)

InN (wurtzite) hex. a = 3.548,

c = 5.760

1.89 3.8

2.9

InN (zinc-blende) cubic a = 4.98 2.20

6H-SiC hex. a = 3.08,

c = 15.12

4.36 4.2

4.7

-Al2O3 hex. a = 4.758,

c = 12.991

7.5

8.5

NNSE 618 Lecture #4

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Temperature dependence of the energy bandgap

Bandgap change is mainly due to thermal expansion (through deformation potential)

NNSE 618 Lecture #4

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Energy bands in solids

NNSE 618 Lecture #4

17 Temperature dependence of the energy bandgap