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EE 232: Lightwave Devices
Lecture #5 – Quantum well
Instructor: Seth A. Fortuna
Dept. of Electrical Engineering and Computer Sciences
University of California, Berkeley
2L− 2L0
z
Infinite potential well
( )V z
0 for 2
for (
2)
z LV
z Lz
=
Separation of variables
)( , , ) ( ( ) ( )x y zxx y y zz =
If ( , , ) ( )V x y z V z=
2
*
2
2
2
2
2
2
2
*
2
*
( ) ( ) ( )
2
,
2
( )
, )
2
(
y
y
z
x y
y
zz z
z
xx x
x y z
Em x
z
x y
Em y
m
z
V Ez
− =
− =
− + =
=
)
2
(
11
|
1
|
x y
t
i k
t
x k y
dxdy CA
eA
+
→ ==
=
( )
y
x
x y
x
y
y
x k y
t
ik
x
ik
i k
x y
Ae
Be
Ce
+
=
=
= =
22
*( , , ) ( , ,) )( , ,
2x y z Ey x y zV x z
m
− +
=
*
2
2
2
( )2
xx x xV x E
m x
− + =
*
2
2
2
( )2
zz z zV z E
m z
− + =
*
2
2
2
( )2
y
y y yV y Em y
− + =
2L− 2L0
z
Infinite potential well
( , , ) ( , ) ( )t zx y z x y z =
( )V z
0 for 2
for (
2)
z LV
z Lz
=
1n =
2n =
3n = )(
2cos 1,3
)
1
,5,...
(2
sin 2,4,6,...
x yx k yi k
t
z
nz n
L Lz
nz n
L L
eA
+
=
=
=
=
222 2
*2x y
w
nE k k
m L
= + +
2L− 2L0
z
Finite potential well
0
0 for 2
fo)
r 2(
z LV
Vz
z L
=
Barrier*( )bm
Barrier*( )bm
Well*( )wm
( )V z 0V
2L− 2L0
z
Finite potential well
1n =
2n =)(1
x yi k
t
x k ye
A
+=
( , , ) ( , ) ( )t zx y z x y z =
( /2)
2
( /2)
2
( )
z L L
z z L L
Ce zz
Ce z
− −
+
=
2 2( ) si )n( L L
z zz A k zz −=
2 2( ) co )s( L L
z zz B k zz −=
Well even solution
Well odd solution
Barrier solution
0
0 for 2
fo)
r 2(
z LV
Vz
z L
=
( )V z 0V
Finite potential well
Plug into Schrodinger’s Equation
*
0( )2 bV mE
−=
*2 w
z
m Ek =
Apply boundary conditions
( ) ( )
( ) ( )* *
2 2
1 12 2
w
z z
z z
b
L L
d dL L
m dz m dz
−
+ −
+ =
=
*
*tan
2
bz z
wm
Lmkk =
1
2
( /2)
2
( /2)
2
( )
z L L
z z L L
Ce zz
Ce z
− −
+
=
2 2
( ) si )n( L Lz zz A k zz −=
2 2( ) co )s( L L
z zz B k zz −=
Well even solution
Well odd solution
Barrier solution
*
*o
2c tb
z z
w
mk
m
Lk = −
(even)
(odd)
Finite potential well
After rearranging
*
*tan
2 2 2
bz z
w
mL L Lk
mk
=
32 2 2*
0
22
2
2 2
wz
mL Lk
V L
=
+
( /2)
2
( /2)
2
( )
z L L
z z L L
Ce zz
Ce z
− −
+
=
2 2( ) co )s( L L
z zz B k zz −=
Well even solution
Well odd solution
Barrier solution
2 2( ) si )n( L L
z zz A k zz −=
*
*c
2ot
2 2
bz z
w
mLk
L Lk
m
= −
*
*
w
b
m
m =
(even)
(odd)
where
Finite potential well
2
2z
Lk
2
L
*
02 2wm V L
1n = 2n = 3n =
Boundsolution
Solve graphically4
Semiconductor quantum well
2L− 2L0
z
cE
cE
vE
gEGE
InP
InG
aAs
InP
vE
Semiconductor quantum well
InP
InG
aAs
InP
2L− 2L0
z
𝐸𝑒1
𝐸𝑒2
𝐸ℎℎ2
𝐸ℎℎ1
𝐸ℎℎ3
Only heavy-holeshowed for clarity
Semiconductor quantum well
222 2
*2x y
w
nE k k
m L
= + +
C1C2C
HH LH HH1 LH1
0gE 0 1 1g g e hhE E E E= + +
222 2
*2x y
w
nE k k
m L
= + +
“Bulk” materialNo quantum confinement
Quantum well
Density of states (2D)
2
2
2
0
12
(2
22 )(
)
2
N d
dkk
−
=
=
k𝑘𝑥
𝑘𝑦d2𝑘 = 2𝜋𝑘𝑑𝑘
2
*
2
2n
e
c e
kE E E
m= + +
2
*
*
2
2 ( )
21 1
2
c en
c en
e
e
m E E Ek
mdk
dE E E E
− −=
=− −
*
2
0
2
*
2
*
( )
( ) ( )
c enE
c
e
en
e
E
e
ec n
mN dE
mH E E E dE
mg E H E E E
+
=
= − −
= − −
𝑑𝑘
(conduction band)
1 0( )
0 0
xH x
x
=
(Heavisidestep function)
Quantum well density of states
1eE 2eE 3eE
Energy
Den
sity
of
stat
es
*
*
*
1
1 22
2 32
3 42
0 0
( )2
3
c e
e c e
e c e
e c e
e
e
e
E E E
mE E E E
g E mE E E E
mE E E E
− −
= −
−
*
2)( ) (c c
een
n
mg E H E E E
= − −
(and so on…)
cE
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