Semiconductor bands Effective mass Holes in …soktyabr/NNSE508/NNSE508_NENG452...NNSE508 / NENG452 Lecture #11 13 Energy bands in semiconductors • Bandgap dependence on temperature

NNSE508 / NENG452 Lecture #11

1

Lecture contents

• Semiconductor bands

• Effective mass

• Holes in semiconductors

• Semiconductor alloys


2 Band-structure of Si

Conduction band valleys in Si

VB in the BZ center

is degenerate: heavy

holes and light holes


3

Density of states in 3D and DOS effective mass

Effective mass density of states

21

032

23*2)( VE

mEN

3D density of states

**

cdos mm

Conduction band DOS mass in

G point:

1 32 3* * * *

1 2 3dos cm m m m

Conduction band DOS mass in

indirect gap semiconductors

(c– degeneracy of the valley):

Valence band DOS mass : 322/3*2/3**

lhhhdos mmm

GaAs


4 Free electrons and crystal electrons

Free electrons

m

kdr

m

iv

*

Kinetic energy: m

kE

2

22

Electrons in solid

*2

)( 20

2

m

kkE

Dispersion near band

extremum

(isotropic and parabolic):

Velocity or group velocity:

)()( ruer k

ikr

k Wave function: Wave function: ikr

k eV

r1

)(

*

)( 0

m

kkv

Group velocity: )(1

kEv k

Velocity at band extremum:

Dynamics (F – force): Dynamics in a band:

Fmdt

dv 1

2

2

2

2

1

*

1;

*

1

111

k

E

mF

mdt

dv

FEFvdt

dE

dt

dvkkkk

(if m* isotropic and parabolic) Force equation:

dt

dk

dt

dpF

Force equation:

dt

dkF Fv

dt

dkE

dt

kdEk

)(


5 Holes

• It is convenient to treat top of the

uppermost valence band as hole states

• Wavevector of a hole = total wavevector of

the valence band (=zero) minus

wavevector of removed electron:

• Energy of a hole. Energy of the system

increases as missing electron wavevector

increases:

• Mass of a hole. Positive! (Electron

effective mass is negative!)

• Group velocity of a hole is the same as of

the missing electron

• Charge of a hole. Positive!

eh kk 0

Hole energy:

Missing electron energy: )()( eehh kEkE

*

22

2)(

e

evee

m

kEkE

*

22

2)(

h

hvhh

m

kEkE

**eh mm

eee eh

eeekhhkh vkEkEv )(1

)(1

hh

e

edt

dk

edt

dk


6

Example: electron-hole pairs in semiconductors

Electron (e-)

Si atom

Hole (h+)

Eg

E

c

Ev

EHP generation : Minimum energy required to break

covalent bonding is Eg.


7

Charge carriers in a crystal

V

E Si atom

• Charge carriers in a crystal are

not completely free. Need to

use effective mass NOT REST

MASS !!!

• Electron and hole currents

have to be treated separately in

devices

+ -


8

Bandgap and lattice constants

Very close

lattice

matching


9

Alloys

Vegard’s law for lattice constant of alloy

Works for

• Random alloys

• With constant crystal structure

Average periodic potential of a “virtual

crystal”

Band extrema follow almost linear

dependence with slight bowing due to

alloy disorder

Effective mass for a given extremum

BAalloy axxaa )1( xxBA 1

From Singh, 2003

2cxbxaEalloy

g

***

11

BAalloy m

x

m

x

m

BAalloy VxxVV )1(


10

Bandgap in AlGaAs

Bandgap of AlGaAs


11

Bandgap in AlGaAs

Bandgap of AlGaAs related to the

vacuum level

X G

Valence band

• Knowledge of band edge

positions is important in junctions


12

Temperature dependence of the energy bandgap


13

Energy bands in semiconductors

• Bandgap dependence on temperature is mostly due to thermal expansion


Lecture recap

14

• In semiconductors extrema of CB and VB are most important

• VB in the BZ center is degenerate: heavy holes and light holes

• Holes as “independent” quasi-particles

• Vegards’ law

• Bandgap dependence on temperature

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Semiconductor bands Effective mass Holes in …soktyabr/NNSE508/NNSE508_NENG452...NNSE508 / NENG452 Lecture #11 13 Energy bands in semiconductors • Bandgap dependence on temperature