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.