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Computational Solid State Physics 計算物性学特論 5 回. 5.B and offset at hetero-interfaces and effective mass approximation. Energy gaps vs. lattice constants. Band alignment at hetero-interfaces. : conduction band edge. : valence band edge. crystal B. crystal A. - PowerPoint PPT Presentation
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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
BvE
AcE
BcE
crystal A crystal B
AvE
vE
cE
AgE B
gE
: conduction band edge
: valence band edge
None of the interface effects are considered.
χ:electron affinity
Anderson’s rule for the band alignment (1)
Anderson’s rule for the band alignment (2)
Ag
Bgvc EEEE
)( Ag
ABg
Bv
BAc
EEE
E
v
c
E
E
: conduction band offset
: valence band offset
type I type II
type III
Types of band alignment
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
)()(
)()(][
rrH
rErVH
nknknkcrys
crys
)()()2(
)()()(
)()()()(
)2()()()(
030
00
3
rrdk
ekrr
ereruerur
dkrkr
nnrik
nn
rikn
rikn
riknknk
mmkm
Effective mass approximation (1)
assume: conduction band n is minimum at k=0
)(rnk : Bloch function
V : external potential
303
3
)2()()(
)2()()(
)2()()()(
dkekr
dkrk
dkrkHrH
riknknnnknkn
nkncryscrys
m
rikmnmncrys
m
mmnk
dkekkarrH
ka
30 )2()()()(
)()(][ rErVH crys
Effective mass approximation(2)
)()()]()([ rErrVin
)()()( 0 rrr nn Schroedinger equation for envelope function χ(r)
Effective mass approximation(3)
)()()()()()()(
)()()(
00 rirriarrH
kikfdxexfikdxedx
xdf
nm
nm
mncrys
ikxikx
If 0)( f
)()()()](*2
[
*2)(
22
22
rErrVm
m
kk
c
cn
Effective mass approximation (4)
)()()( 0 rrr nn All the effects of crystal potential are included in εc and effective mass m*.
・ Schroedinger equation for an envelope function χ(r)
r
erV
s0
2
4)( :potential from a donor ion
Impurity
20
220
2
4 *
8
*
sc
s
c m
mRy
h
meE
Ry=13.6 eV: Rydber
g 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
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.
Crystal A Crystal B
Kronig-Penny model (1)
SbV
b
V
0
0
0 n
anxSxV )()(
)()()()](2
[2
22
xkExxVdx
d
m kk
Schroedinger equation in the effective mass approximation
)()( xeax kika
k
Bloch condition for superlattice
k: wave vector of Bloch function in the superlattice
Kronog-Penney model (2)
0)()()(2
)0()0(0
0
0
02
22
dxxxVdxxdx
d
m kk
kk
)cos()sin()( 11 xkAxkxk ax 0
Boundary condition at x=0
Solution of Schroedinger equation
0)0()]0()0([2 0
2
kkk Vm
(1)continuity of wavefunction
(2)connection condition for the 1st derivative of wavefunction
(2’)
21
2
2)( k
mkE
for
Kronig-Penney model (3)
0)]sin(cos1[2
)cos(sin
11
2
11
SAakAakeqm
akAakeA
ika
ika
1
121
sincoscos
k
akmSakka
(1)
(2’)
21
2
2)( k
mkE
Simultaneous equation for E(k)
Kronig-Penney model (4)
Sm
P2
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 Al0.35
Ga0.65As. 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.
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