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*3 Electrons and Holes in Semiconductors 3 Electrons and Holes in Semiconductors 3.1. Introduction...*

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3 Electrons and Holes in Semiconductors

3.1. Introduction

Semiconductors : optimum bandgap

Excited carriers → thermalisation

Chap. 3 : Density of states, electron distribution function, doping, quasi thermal equilibrium

electron and hole currents

Chap. 4 : Charge carrier generation, recombination, transport equation

3.2. Basic Concepts

3.2.1. Bonds and bands in crystals

Free electron approximation to band

Ref. CLASSIC “Bonds and Bands in Semiconductors” J. C. Phillips, Academic 1973

Atomic orbitals → molecular orbitals → bands

Valence band : HOMO

Conduction band : LUMO

2

Semiconductors : eVEg 35.0

Semi-metals : eVEg 5.00

Si : sp3 orbital, diamond-like structure

3.2.2. Electrons, holes and conductivity

Semiconductors

At KT 0 , all electrons are in valence band – no conductivity At elevated temperature,

some electrons in conduction band, and some holes in valence band

3

3.3. Electron States in Semiconductors

3.3.1. Band structure

Electronic states in crystalline

Solving Schrödinger equation in periodic potential (infinite)

Bloch wavefunction

rkk rrk ii eu, (3.1)

for crystal band i and a wavevector k, which is a good quantum number.

The energy E against wavevector k is called by crystal band structure.

Typical band diagram plotted along the 3 major directions in the reciprocal space.

111,011,001,000 LMX

a)

Conduction Band

Valence Band

Electrons

a)’

Electron in CB

b) b)’

Hole in VB

Fig. 3.4. Electron in CB and Hole in VB

4

Point a

k

is called the Brillouin zone boundary.

For zinc blende structure,

aaa L

aa K

a X

,0

2

3

2

3 ,00

2 ,000

3.3.2. Conduction band

At the vicinity of the minimum kE in the conduction band 0ckk , approximation

*

2

0

2

0 2 c

c

c m

EE kk

k

(3.2)

where 00 cc EE k and *

cm is a parameter with dimensions of mass, and defined by

2

2

2*

11

k

kE

m

c

c

(3.3)

is called the parabolic band approximation.

eg. *

cm for GaAs 0 * 067.0 mmc , for Si 0

*

// 19.0 mmc 0 * 92.0 mmc ,

where 0m is the static electron mass.

*

cm is determined by the atomic potentials, should actually be a tensor,

but is usually treated as a scalar, and is called the effective mass of electron.

Momentum of an electron and the applied force are related as

dt

d

dt

d kp F (3.4)

Following eq. (3.2), the velocity and momentum can be obtained as

*

01

c

c

m E

kk kv k

(3.5)

0 *

ccm kkvp (3.6)

3.3.3. Valence band

At the vicinity of the maximum kE in the valence band 0vkk

*

2

0

2

0 2 v

v

v m

EE kk

k

(3.7)

5

where 00 vv EE k and *

vm is the effective mass of hole.

*

01

v

v

m E

kk kv k

(3.8)

0 *

vvm kkvp (3.9)

2

2

2*

11

k

kE

m

v

v

(3.10)

eg. *

vm for GaAs 0 * 5.0 mmv , for Si 0

* 54.0 mmv ,

*

cm is the curvature of the band.

3.3.4. Direct and indirect band gaps

Whether or not minimum of conduction band and maximum of valence band occur at the same

wavevector 00 vc kkk

Reciprocal lattices for (a) fcc and (b) bcc crystals

6

When the band gap is indirect, transition requires involvement of phonons for conservation of

momentum.

(Sze)

3.3.5. Density of states

According to Pauli’s exclusion principle, each quantum state only supports one electron (two,

considering spins). Also, from Heisenberg’s uncertainty principle,

hxp ~ ∴ hxk ~ ∴ 2~xk

Therefore, for a crystal of volume LLL

Lx

k 22

~

∴

3

2

L states in this unit volume.

Hence, the density of electron states per unit crystal volume considering spin is

kkk 3

3

3

2

2 ddg

(3.11)

If the band structure kE is spherically symmetric and isotropic around 0k

dkkkgdg 23 4kk (3.12)

Therefore,

dE dE

dk kkgdEEg 24 (3.13)

∴ dE

dk kEg 2

3 4

2

2

(3.14)

7

From eq. (3.2) *

2

0

2

0 2 c

c

c m

EE kk

k

∴ 02 *

2 2 c

c EE m

k

(3.15)

Similarly, for the valence band (hole)

∴ EE m

k v v 02

*

2 2

(3.15’)

Differentiating (3.15)

210

21

2

*2 c

c EE m

k

∴ 210

21

2

*2

2

1

c

c EE m

dE

dk

Therefore, from (3.14)

210

23

2

*

2

21

0

23

2

*

3

21

0

21

2

*

02

*

3

21

0

21

2

*

2

3

2

3

2

2

12

2

4

2

2

12 4

2

2

2

2

1 4

2

2 4

2

2

c

c

c

c

c c

c c

c c

c

EE m

EE m

EE m

EE m

EE m

k dE

dk kEg

(3.16)

Similarly,

210

23

2

*

2

2

2

1 EE

m Eg v

v v

(3.17)

Question : Derive the DOS for 2D and 1D materials. (Box 3.3)

Excitons

A bound state of a pair of electron and hole of same k .

The wavefunction can be described as

rkrk ,, heex

Satisfying a hydrogenic effective mass equation

exexex

s

ex E q

rr

42

2 2

*

2

* : reduced effective mass of e-h pair

exE : binding energy of the exciton

8

22

2

0

0

*

l

Ryd

m E

s

ex

3.3.6 Electron distribution function

Fermi function in semiconductors

1exp

1

Tk

EE Ef

B

F

This can be expressed in terms of rk,f although the energy is only dependent on the wavevector.

krkkkr 33 , dfgdn

The total electron density is given by

k krkkr CB c dfgn 3, (3.20)

k krkkr VB v dfgp 3,1 (3.21)