1

EE232 Lecture 4-1 Prof. Ming Wu

EE  232  Lightwave  DevicesLecture  4:  Density  of  States,  Quantum  Wells  

and  Wires

Instructor:  Ming  C.  Wu

University  of  California,  BerkeleyElectrical  Engineering  and  Computer  Sciences  Dept.

EE232 Lecture 4-2 Prof. Ming Wu

Review  of  Quantum  Mechanics

Schrodinger Equation

HΨ(r!,t) = i! ∂

∂tΨ(r!,t)

H =P2

2m+V (r!,t) Hamiltonian = Kinetic + Potential Energy

Ψ(r!,t) Wavefunction

Ψ(r!,t)

2

=Ψ(r!,t) ⋅Ψ(r

!,t)* Probability of finding particle at r

!

P!"= −i!∇ Momentum operator

P!"

= −i! Ψ(r!,t)*∫ ∇Ψ(r

!,t)dr! Average Momentum

r!= Ψ(r

!,t)*∫ r!Ψ(r!,t)dr! Average Position

2

EE232 Lecture 4-3 Prof. Ming Wu

Electron  Plane  Wave

Ψ(r!,t) = eik

!⋅r!−iωt

LHS: HΨ(r!,t) = !

2k 2

2mΨ(r!,t)+V (r

!,t)Ψ(r

!,t)

P = !k

RHS: i! ∂∂tΨ(r!,t) = !ωΨ(r

!,t)

⇒!2k 2

2m+V (r!,t) = !ω

EE232 Lecture 4-4 Prof. Ming Wu

Example:  Infinite  Potential  Well

2 2

2

2

2 2

22

( , ) ( ) Solve Eigenvalue in

( ) ( ) ( ) ( )2

2For 0 , ( ) ( ) 0

sin( )( )

cos( )B.C. ( 0) ( ) 0

2( ) sin

2

i t

n

n

z t z eE

d z V z z E zm dz

d mEz L z zdz

kzz

kzz z L

nz zL L

nEm L

wfw

f f f

f f

f

f f

pf

p

-Y ==

- + =

< < + =

ì= íî

= = = =

æ ö= ç ÷è ø

æ ö= ç ÷è ø

h

h

h

h

0V =¥

0x = x L=Time Independent Potential

0 for 0( )

for 0 or z L

V zz z L

< <ì= í¥ < >î

1E

2E

3

EE232 Lecture 4-5 Prof. Ming Wu

Typical  Examples

*0

1

2 1

In GaAs, 0.067For a 10-nm-wide potential well ( 10 )

56 meV4 224 meV

em mL nm

EE E

=

=== =

EE232 Lecture 4-6 Prof. Ming Wu

Complete  Wavefunction for  Infinite  Potential  Well

22 2

( , ) '( , ) ( ) Electron confined in , but free in ,

Plane wave in x and y1'( , )

A: area (normalization const)

2 1( , ) sin

2

x y

x y

i t

ik x ik y

ik x ik y

n x y

r t x y z ez x y

x y eA

nr t e zL LA

nE k km L

wf f

f

p

p

-

+

+

Y =

Þ

=

æ öY = ç ÷è ø

æ ö= + + çè ø

r

r

h 2

Energy quantized only in directionzk

é ùê ú÷ê úë ûxk

yk

E

1n =

2n =

4

EE232 Lecture 4-7 Prof. Ming Wu

2-­d  Density  of  States

xk

yk

k

k k+D0V =¥

0x = x L=

1E

2E

xk

yk

E

1n =

2n =

EE232 Lecture 4-8 Prof. Ming Wu

2-­d  Density  of  States

222

*

2 2

Consider the lowest band first (n=1):Number of electron states between and per unit volume

2 2 2( ) 2 2 2

( )2

2 1( ) ( )2

kz

x y

e

d kz

k k k

kdk kk dk dkV L

L L

E k km L

dk kE dE k dEdE L

m

pr p p p

p

r rp

+ D

= × =

é ùæ ö= +ê úç ÷è øê úë û

æ ö= = ç ÷

è ø

h

h*

*

2 2( )

e

ed

z

dEk

mEL

rp

=h

5

EE232 Lecture 4-9 Prof. Ming Wu

2-­d  DOS  for  Multiple  Energy  Levels

1E 2E 3E

1 2

*

1 2 2 2

*

2 3 2 2

*

3 4 2 2

*

2 21

0 ( ) 0

( )

2( )

3( )

In general

( ) ( )

d

ed

z

ed

z

ed

z

ed n

nz

E E E

mE E E ELmE E E ELmE E E EL

mE H E EL

r

rp

rp

rp

rp

¥

=

ì< < =ï

ïï

< < =ïïíï < < =ïïï < < =ïî

= -å

h

h

h

h

2 ( )d Er

Step  Function

EE232 Lecture 4-10 Prof. Ming Wu

2-­d  Electron/Hole  Concentration

( )

( )

,2

,2

1 2

1 ,2 1 2

*

1 2

Electron and hole concentrations:

( ) ( )

( ) ( )

At T = 0K, and for ( )

C

V

n e dE

E

p h d

n e d

en

z

n f E E dE

p f E E dE

E E En F E E E E

mn F EL

r

r

r

p

¥

-¥

=

=

< <

= - × < <

= -

ò

ò

h

1

Example: 10-nm-wide GaAs quantum well quasi-Fermi energy is 100 meV above E2-­d  electron  concentration

m_e 0.067 m0×:= m_e 6.104 10 32-´ kg=

Lz 10nm:= Lz 1 10 8-´ m=

ρ2dm_e

π h_bar2× Lz×:= ρ2d 1.747 1044´

s2

kg m5×=

n 100meV ρ2d×:= n 2.795 1018´1

cm3×=

n_s n Lz×:= n_s 2.795 1012´1

cm2×=

6

EE232 Lecture 4-11 Prof. Ming Wu

1-­d  Density  of  States

yx

z2 22

2, *

22

, * *

, , 0

*

2, 0

*

2, 0

*

1 2

( )2

( ) 222 2

2

2

21 1

21 1( )

m n z ze

zm n z z z z

e e

zz

m n m nx y

z

e

m nx y z

e

m nx y mx ny

eD

x y

m nE k km L L

kdE k k dk dkm m

dkn dkV L L

Lm dE

L L k

m dEL L E E E

mEL L E

p p

pp

p

p

rp

¥ ¥

-¥

¥

¥

æ öæ ö æ ö= + +ç ÷ç ÷ ç ÷ç ÷è ø è øè ø

= × =

= =æ öç ÷è ø

=

=- -

=

å åò ò

åò

åò

h

hh

h

h

h ,m n mx nyE E- -å