Computational Solid
State Physics
計算物性学特論 5回
5.Band offset at hetero-interfaces
and effective mass approximation
Energy gaps vs. lattice constants
Band alignment at
hetero-interfaces
B
Ec
Ec
A
E
Ev
A
g
Ec : conduction band edge
EgB
A
Ev
B
Ev : valence band edge
crystal A
crystal B
Anderson’s rule for the
band alignment (1)
χ:electron affinity
None of the interface effects are considered.
Anderson’s rule for the
band alignment (2)
Ec    
A
B
Ev    E  (   E )
B
B
g
A
Ec : conduction band offset
Ev : valence band offset
Ec  Ev  E  E
B
g
A
g
A
g
Types of band alignment
type I
type III
type II
Band bending in a doped
hetero-junction (1)
Band bending in a doped
hetero-junction (2)
Effective mass approximation
・Suppose that a perturbation
is added to a perfect crystal.
・How is the electronic state?
Examples of perturbations
an impurity, a quantum well, barrier, superlattice,
potential from a patterned gate, space charge potential
Effective mass approximation (1)
[ H crys  V ] (r )  E (r )
H crysnk (r )   nknk (r )
V : external potential
nk (r ) : Bloch function
dk
 (r )     m (k )mk (r )
3
(
2

)
m
assume: conduction band
n is minimum at k=0
nk (r )  unk (r )eikr  un 0 (r )eikr  n 0 (r )eikr
 ( r )  n 0 ( r )   n ( k ) e
ik r
dk
 n 0 ( r )  n ( r )
3
(2 )
Effective mass approximation(2)
[ Hcrys  V ] (r )  E (r )
dk
H crys (r )  H crys   n (k )nk (r )
(2 )3
dk
dk
ik r
   n (k ) nknk (r )
 n 0 (r )   n (k ) nk e
3
(2 )
(2 )3
 nk   am k m
m
H crys (r )  n 0 (r ) am   n (k )k e
m ikr
m
dk
(2 )3
Effective mass approximation(3)
df ( x) ikx
ikx
e
dx

ik
f
(
x
)
e
dx  ikf (k )
 dx


If f ()  0
H crys (r )  n 0 (r ) am (i) m  (r )  n 0 (r ) n (i)  (r )
m
Schroedinger equation for envelope function χ(r)
 (r )  n0 (r ) n (r )
[ n (i)  V (r )] (r )  E (r )
Effective mass approximation (4)
・Schroedinger equation for an envelope
function χ(r)
2
2
 k
 n (k )   c 
2m *
2 2
[
  V (r )] (r )  ( E   c )  (r )
2m *
 (r )  n0 (r ) n (r )
All the effects of crystal potential are included
in εc and effective mass m*.
Impurity
V (r )  
e2
4 0 s r
:potential from a donor ion
e4 m *
m*
E   c  2 2 2   c  Ry
m0 s2
8h  0  s
Ry=13.6 eV:
Rydberg constant
Quantum well
Quantum corral
HEMT
2D-electron confinement
in HEMT
The sub-band structure at the interface of the GaAs active channel in a HEMT structure. E1
and E2 are the confined levels. The approximate positions of E1 and E2 as well as the shape
of the wave functions are indicated in the lower part of the diagram. In the uper part, an
approximate form of the potential profile is shown, including contributions of the conduction
band offset and of the space charge potential.
Superlattice
Crystal A Crystal B
The Kronig-Penney model, a simple superlattice, showing wells of width w alternating with
barriers of thickness b and height V0. The (super)lattice constant is a=b+w.
Kronig-Penny model (1)
V0  
b0
V0b  S
V ( x)   S ( x  an)
n
Schroedinger equation in the effective mass approximation
2 d 2
[
 V ( x)] k ( x)  E (k ) k ( x)
2
2m dx
Bloch condition for superlattice
k ( x  a)  e k ( x)
ika
k: wave vector of Bloch
function in the superlattice
Kronog-Penney model (2)
Boundary condition at x=0
 k (0  )   k (0  )
0
0

2  d 2

 k ( x)dx   V ( x) k ( x)dx  0
2

2m 0 dx
0
2

[ k (0 )   k (0 )]  V0 k (0)  0
2m
(1)continuity of
wavefunction
(2)connection condition
for the 1st derivative of
wavefunction
(2’)
Solution of Schroedinger equation
k ( x)  sin(k1 x)  A cos(k1 x)
2 2
E (k ) 
k1
2m
for 0  x  a
Kronig-Penney model (3)
A  e ika (sin k1a  A cos k1a)
(1)
2

q[1  e ika (cosk1a  A sin k1a)]  SA  0
2m
(2’)
Simultaneous equation for E(k)
m S sin k1a
coska  cosk1a  2

k1
2 2
E (k ) 
k1
2m
Kronig-Penney model (4)
m
P 2S

allowed range
of cos(ka)
Kronig-Penney model (5)
Conduction band of crystal A is split into mini-bands with
mini-gaps by the Bragg reflection of the superlattice.
Problems 5
 Calculate the lowest energy level for
electrons and light and heavy holes in a
GaAs well 6 nm wide sandwiched between
layers of Al0.35Ga0.65As. Calculate the
photoluminescence energy of the optical
transition.
 Calculate the two-dimensional Schroedinger
equation for free electrons confined in a
cylindrical well with infinitely high walls for r>a.