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