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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
3
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
5
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
6
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
9
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
12
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
13
Bandgaps
From Balkanski and Wallis, 2000
NNSE 618 Lecture #4
14
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
15
Temperature dependence of the energy bandgap
Bandgap change is mainly due to thermal expansion (through deformation potential)
NNSE 618 Lecture #4
16
Energy bands in solids
NNSE 618 Lecture #4
17 Temperature dependence of the energy bandgap