1 Material Model 2 Single Electron Band Structure in Bulk Semiconductors

1

Material Model 2Single Electron Band Structure in Bulk

Semiconductors

2

Single Electron in Bulk Semiconductors

The SchrÖdinger’s equation in its general form:

),(),( trHtrt

j

The Hamiltonian in semiconductors:

K

lklk lk

lkK

k

N

n kn

k

N

mnmn mn

K

k k

kN

nn

RRZZe

RrZe

rre

MmH

,1,0

2

1 10

2

,1,0

2

1

22

1

2

0

2

||421

||4

||1

421

22

1st term – electron kinetic energy, 2nd term – lattice ion kinetic energy, 3rd term – Coulomb potential among electrons, 4th term – Coulomb potential among electrons and lattice ions, 5th term – Coulomb potential among lattice ions

3


Since the Hamiltonian is not explicitly time dependent, we find:

)(),( retrtj h

)()( rrH h

Simplification of the Hamiltonian: step-1, the Born-Oppenheimerapproximation (i.e., to ignore lattice ion self-effects);

K

k

N

n kn

kN

mnmn mn

N

nn Rr

Zerr

em

H1 10

2

,1,0

2

1

2

0

2

||4||1

82

step-2, the Hartree self-consistent model (i.e., to count in theCoulomb effect successively rather than simultaneously);

N

nmmmmm

mnmm

K

k kn

knn rdr

rrre

RrZe

mH

,1

*

0

2

10

22

0

2

)(||

1)(8||42

4


If the n_th electron wave function is solved as:

What is the wave function for a collection of N electrons?

)( nn r

N

nnn rr

1

)()( Incorrect form

Nrrr

Nrrr

Nrrr

NNN

Nr

...............

...

...

!1)(

21

21

21

222

111

Correct form due to Slater

step-3, the Hartree’s model must be modified to take Hartree-Fock’s form (i.e., to count in the electron spin effect):

mnmmn

N

mnnmm

mnmm

nn

K

k kn

knnnn

rdrrrrrr

re

rRr

Zem

rH

)]()()()([||

1)(8

)(]||42

[)(

1

*

0

2

10

22

0

2

5

Single Electron in Bulk Semiconductors• Solution techniques

– The pseudo-potential method (Physicists’ method)– The tight-binding method (Chemists’ method)– The k-p method (for direct bandgap material at its band

structure extremes and neighborhoods)

• We will focus on the k-p method only in dealing with the material optoelectronic property for compound semiconductors

• Otherwise, we have to rely on, e.g., the pseudo-potential method

6


Following the k-p method, we can rewrite the Hamiltonian as:

K

k kn

nknn Rr

Zem

H10

22

0

2

||42

)(242 0

0

222

0

2

0 rVmp

re

mH

with a most important feature that describe the electron behaviorinside bulk semiconductors:

)]([)( 00 nRrVrV

3

1

)(i

iiannR

It is this feature that leads to Bloch’s theorem:

)()( ruer krkj

)]([)]([1)(3

1 1

)]([

321

nRrunRreNNN

ru ki

N

n

nRrkjk

i

i

�

2,1

2,...1,0,1,...1

2,

2iiii

iNNNN

n

7


Instead of the original equation (only one) for electron wavefunction which is defined in the entire crystal bulk, this equationfor the lattice wave function is defined in a primitive cell. Thereare, however, N1N2N3 such equations, for all possible k values,which is also consistent with the total number of primitive cellsinside the bulk crystal.

)()()]()(2

[ 02

0

2

rururVjkm khk

Why bother then?

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Conceptually, the electron E~k relation shows partially almostcontinuous, partially discrete, hence we have the famous electronband structure for semiconductors.

Why the eigen value (energy) shows the band structure?

Gkhi m

GGkGGk uuVumGkm

i

~~~~)]([

2

3

1)'('

2

0

2

'

It can be seen from the spectrum of the lattice wave function thatsatisfies:

3

1

)(i

iibmmG

,...3,2,1,0 im

3

1ii

i

i bNnk

|)(|

2kji

kji aaa

aab

9

k

G

G

0~V

GV~

1~V

1~

V

kGu~

0~

ku1

~ku)1(

~ku

k

0 b1/2

0

b1/N1

hk

0

b1/N1

-b1/2

b1/2-b1/2b1/2-b1/2

1st BZ

Energy band(continuous as N10)

Energy band gap

Energy band(continuous as N10)

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• Why discontinuities on E~k curve must happen at the Brillouin zone (BZ) edges?

• At discontinuities, E must take its extreme at least along one direction in k-space.

• Motion of electron in bulk semiconductors – partially free and partially bounded, the former reflects the property of the free-space occupied by the whole bulk, the latter reflects the property of the periodic structure (semiconductor lattice)

• The expanded BZ and the reduced BZ, the reason we adopt reduced BZ description.

11

E

k0

1st BZ 2nd BZ2nd BZ 3rd BZ3rd BZ 4th BZ4th BZ… …

Extended BZ

Reduced BZ Electron in 1D lattice

(energy bands appear)

Electron in free space

(continuous energy in parabolic shape)

Bands with

negative effective mass

Bands with positive effective mass

Valence bands (filled up with electrons at 0K)

Conduction bands (empty at 0K)

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Single Electron in Bulk Semiconductors• The free space – electrons have continuous E~k relation, hence

limit themselves to one type, multiple states; easiest for transport, not possible for discrete transitions; easy to access but limited useful property to offer; applications include: vacuum tube transistor, microwave traveling wave amplifier, etc.

• Atomic, molecular, or ionic systems – electrons have discrete E~k relation (discrete energy levels), hence limit themselves to multiple type, single states; easiest for excitations and transitions, hard for transport; useful property to offer but hard to access; application include: lasers, masers, etc.

• Bulk semiconductors – electrons have partially continuous partially discrete E~k relation (energy bands), hence they are in multiple type and multiple states; easy for both excitations or transitions and transport; useful property to offer and easy to access; a variety of applications

13

Single Electron in Bulk Semiconductors• Major drawback of bulk semiconductors compared to

the atomic, molecular, or ionic systems – intra-band electron interactions

• Ideal solution – low dimension semiconductor structures, i.e., quantum wells, quantum wires, and quantum dots; e.g., similar to the atomic system, QD has discrete energy levels, hence it has all the merits possessed by the atomic system; when combined with the bulk semiconductor, it also becomes accessible; it is this feature that makes the low dimension semiconductor structures (e.g., QD) so attractive!

14


Step-1 Kane’s model

nnr

em

nH kn

)

42(

2

0

22

0 YRn r

The radial dependence follows the Laguerre polynomials.The angular dependence follows:

)(cos)1(|)!|(|)!|(

412

||2

||

ml

jmmm

lm PemlmllY

lml

mlm

llm xdxdx

lxP )1()1(

!21)( 222

Spherical harmonic function

Legendre function

If k=0

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n=1,2,3,… (known as the total quantum number)l=0,1,2,…,n-1 (known as the angular quantum number, whichcorresponds to s, p, d, f,… orbits)m=-l,-l+1,…,-1,0,1,…l-1,l (known as the magnetic quantumnumber)ms=-1/2,1/2 (known as the spin quantum number)

Therefore, a bound electron by the lattice ion, similar to theorbiting electron in H-atom, is described by a set of quantumnumbers: (n, l, m, ms).

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The first few orbits (electron eigen states) are given as:

,

0l 0m SjY 0,04100

1l 0m ZrzY 0,1

43cos

4310

1l 1m2

1,183sin

8311 YjX

rjyxeY j

17


Under these states, the Hamiltonian is diagonalized as:

,

00

0,11,1

1,10,0

000000000000

H

EE

EE

H

p

p

p

s

Considering the spin of electrons, but ignoring the spin-orbitinginteraction, we have:

T]0,1,1,1,1,1,0,0,0,1,1,1,1,1,0,0[

880

00 0

0

HH

H

under:

18


However, the spin-orbiting interaction is not negligible incalculating the semiconductor band structure. We haveto include this interaction by modifying the Hamiltonian to:

,

,

)(

4 220

0 pVcm

H

Replacing p by p+ħk, we obtain the equation for the latticewave function again (with spin-orbiting interaction included):

knmk

knkVcm

pVcm

pkm

H

kn

)2

(

])(4

)(4

[

0

22

220

2200

0

19


It is easy to prove that, under the reordered base:

,

T]1,1,0,1,1,1,0,0,1,1,0,1,1,1,0,0[

T]0,1,1,1,1,1,0,0,0,1,1,1,1,1,0,0[

we have:880

0

H

H

3000

032

032

30

00

p

p

p

s

E

EkP

E

kPE

Hwhere: ZpS

mjP z

0

YpyVp

xVX

cmj xy

2204

3

20


Following a standard perturbation procedure to diagonalizethe Hamiltonian, we find that under the electron state:

,

the Hamiltonian becomes:

T]1,1,0,13

11,132,0,1

321,1

31,0,0[

0

22

22

0

22

22

0

22

22

0

22

2000

03

22

00

00)(32

0

000)(

)3/2(2

mk

EPk

mk

EPk

mk

EEEPk

mkE

H

g

g

gg

gg

d

21


The other half of the state is:

,

Hence, the full Hamiltonian becomes:

T]1,1,0,13

11,132,0,1

321,1

31,0,0[

880

0

d

d

HH

22


Final result of Kane’s model:

,

Conduction band gck E0 Sj0,0

Heavy hole band 00 hhk

)(2

11,123,

23

)(2

11,123,

23

YjX

YjX

Light hole band 00 lhk

ZYjX

ZYjX

32)(

610,1

321,1

31

21,

23

32)(

610,1

321,1

31

21,

23

Spin-orbit split band

sok 0

ZYjX

ZYjX

31)(

310,1

311,1

32

21,

21

31)(

310,1

311,1

32

21,

21

23


Step-2 Luttinger-Kohn’s model

Following the Lowdin’s renormalization theorem and takingthe Kane’s solution at k=0 for valence bands, we write:

Rewrite the equation for the lattice wave function as:

knknHmkpV

cmH kn

]

2)(

4[ '

0

22

220

0

Bv

B

v

Bv

Aj

A

j

Aj nkankakn )()(

21,

21

21,

21

23,

23

21,

23

21,

23

23,

23

654

321

AAA

AAA

nnn

nnn

24


Hence we obtain:

A

A

A

A

A

A

A

A

A

A

A

A

aaaaaa

E

aaaaaa

PSQSR

PRSQS

SRQPSR

QSSQPR

SQRQPS

RSRSQP

6

5

4

3

2

1

6

5

4

3

2

1

***

**

****

**

*

02

2232

02232

2

220

2230

2320

22

0

)(2

222

0

12

zyx kkkm

P

)2(2

222

0

22

zyx kkkm

Q

]2)([23

322

20

2

yxyx kkjkkm

R

zyx kjkkm

S )(3

0

32

25


Once the above eigen value problem is solved (a 6-orderpolynomial equation root-searching problem), we findvalence bands as:

Aj

j

Ajn nkakn

6

1

)(

subject to: 1|)(|6

1

2 j

Ajn ka

26


For conduction bands, we write:

Bv

B

v

Bv

AA nkankakn )()( 00

0,00

An

We obtain the solution in closed form:

0,00Ankn

eg

B

vBvg

zyxv

ava

g mkE

E

ppkk

mmkEE

22

22,,,

00

20

2

0

22

27


By ignoring spin-orbit split bands and the small anisotropyin x-y plane, we may reduce the 6 6 Hamiltonian to 4 4. Hence a closed form for valence bands is obtainable as:

hh

bhh m

kmkE

2)2(

2

22

210

22

lh

blh m

kmkE

2)2(

2

22

210

22

28

Single Electron in Bulk Semiconductors• As a summary of this section, we give following examples on the Hamiltonian

selection in k-p theory based on Luttinger-Kohn’s model.

• - Two bands in group A (heavy hole and light hole) with 44 Hamiltonian: InGaAs-AlGaAs-GaAs, InGaP-AlInGaP-GaAs, InGaAsP-InP and AlGaInAs-InP [12]

• - Three bands in group A (heavy hole, light hole, and spin-orbit split) with 66 Hamiltonian: InGaAsP-InP, AlGaInAs-InP and group-III nitrides with wurtzite structure such as InGaN-AlGaN [16]

• - Four bands (e.g., conduction, heavy hole, light hole, and spin-orbit split) with 88 Hamiltonian: wide bandgap II-VI compounds [17], group-III nitrides with wurtzite structure such as InGaN-AlGaN [18], group-III antimonides, and narrow bandgap II-VI compounds [19]

• - Five bands (e.g., N-resonant, conduction, heavy hole, light hole, and spin-

orbit split) with 1010 Hamiltonian: diluted nitrides such as GaInNAs-AlGaAs-GaAs [20]

Documents

1 Material Model 2 Single Electron Band Structure in Bulk Semiconductors