The effective mass

• Conductivity effective mass – determines mobility.

• Density of states effective mass – determines NC

• Cyclotron effective mass – can be measured directly

Electron in a periodic potential

• Why does the semiconductor industry use single crystal material (when possible) ?

Electrons are not scattered by a periodic potential – move with a constant velocity as in vacuum !

Electron in vacuum

0)( xV

)](exp[)( tkxiAx

m

pE

m

k

22

)( 22

Dispersion

)(k

)](exp[)( tkxiAx

Electron in vacuum

0)( xV

)](exp[)( tkxiAx

m

pE

m

k

22

)( 22

Electron in a periodic potential

)()2

(

)()(

)](exp[)()(

)()(

kEa

kE

xuaxu

tkxixux

xVaxV

 

Electron in a 3D periodic potential

vectorlatticeReciprocal

)()(

)()(

)](exp[)()(

vectorlatticeBravais

)()(

K

kEKkE

ruRru

trkirur

R

rVRrV

 

Expansion of E(kx,ky,kz) near a minimum value E0= E(kx0,ky0,kz0)

zyxji

kkkkkk

EEkE jjii

ji ji

,,,

))((2

1)( 00

,

2

0

Expansion of E(kx,ky,kz) near a minimum value E0= E(kx0,ky0,kz0)

tensormass effective inverse1

where

,,2

)(

or

,,,

))((2

1)(

0

2

21

0

0

01

000

2

0

00,

2

0

kkk

EM

kk

kk

kk

MkkkkkkEkE

zyxji

kkkkkk

EEkE

jiij

zz

yy

xx

zzyyxx

jjiiji ji

eddiagonaliz becan

-matrix lsymmetrica a is

tensor mass effective inverse the1

0

2

21

kkk

EM

jiij

In the coordinate system in which the effective mass tensor is

diagonal

0

0

0

13

12

11

3

2

1

ˆˆˆ1

zz

yy

xx

zyx

kk

kk

kk

m

m

m

zk

Ey

k

Ex

k

Ev

m

m

m

M

Acceleration due to an electric filed (F)

F-qaM

) model calsemiclassi (the -

1

0

0

0

13

12

11

Fqdt

kd

dt

kdM

dt

vda

kk

kk

kk

m

m

m

v

zz

yy

xx

Effective mass tensor – valid near E(k) minima and maxima only

F-qaM

Constant energy surfaces in crystal momentum space cookies

Constant energy surfaces near a minimum are ellipsoids

13

12

11

1

0

0

01

000

2

0 ,,2

)(

m

m

m

M

kk

kk

kk

MkkkkkkEkE

zz

yy

xx

zzyyxx

Constant energy surfaces in Si and Ge near a minimum are ellipsoids

of revolution

1

1

1

1

0

0

01

000

2

0 ,,2

)(

t

t

l

zz

yy

xx

zzyyxx

m

m

m

M

kk

kk

kk

MkkkkkkEkE

Acceleration of an electron near an energy minimum in silicon

6

i1

6-1

i1

x

*

ˆ ˆ ˆ - electric field (main coordinate system)

-qF Ma -in each valley

1-qF M a -average

6

1a M F

6

1 2a ( )

3 3

1 1 2

3 3

x y Z

i

i

xl t

l t

F F x F y F z

q

qFm m

m m m

Electron transport effective mass in silicon and germanium

tl mmm 3

2

3

11

*

Homework competition – find a shorter way to prove this equation for germanium than

given in last year’s home exam

tl mmm 3

2

3

11

*

The spherical case -

• Electrons in GaAs• Holes in Si, Ge, GaAs

m

m

m

M

Cyclotron resonance effective mass

qH

m

Cyclotron resonance effective mass –to be shown in the tutorial

1 2 32 2 2

1 1 2 2 3 3

2 2 2 2 22 21 1 2 2 3 3 31 2

1 2 3 2 3 1 3 1 2

ˆ ˆ ˆ

ˆ ˆ ˆ ˆˆ ˆ1

mm mm

H m H m H m

H m H m H m HH H

m mm m m m mm mm

Cyclotron resonance effective mass –to be shown in the tutorial

Density of states effective mass

3 2*

2

22

c fE E kT

c

cc

n N e

m kTN

h

Density of states

number of allowed energy states( )

dEdV

V- Volume

E- Energy

g E

Example: density of states of hydrogen gas

( ) ( )

- allowed energy level

- number of allowed states in energy level i (degeneracy)

n - density of hydrogen atoms

i ii

i

i

g E n g E E

E

g

g(E)

E

Density of states of solids

g(E)

E

Density of electrons in an energy band

g(E)

E

E f

Band

n g E f E dE

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2

E/Ef

f(E

)

Density of holes in an energy band

g(E)

E

E f

Band

[1 ]p g E f E dE

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2

E/Ef

1-f

(E)

Approximation for the Fermi Dirac distribution for E-Ef>3KT

1

1F

F

E E kTFD MBE E kTf E e f E

e

Density of electrons in an energy band

Band

Band

Band

exp[ ( ) / )]

exp[ ( ) / ]

exp[ ( ) / )]

f

C C f

C C

n g E f E dE

n g E E E KT dE

n N E E KT

N g E E E KT dE

g(E)

E

E f

Density of states of solids in K space

3x y z

number of allowed energy states 12

dk dk dk dV (2 )

V- Volume

The density of states in an energy interval is proportional to the volume in K space between two

constant energy surfaces

2

1

x y z3

12 dk dk dk

(2 )

E

E

Constant energy surfaces in crystal momentum space cookies

Volume of an ellipsoid

2 2 2

2 2 21

x y z

a b c

4Volume=

3abc

Volume of a constant energy (E’) ellipsoid

22 22

31 2,

1 2 3

22 231 2

1 , 2 , 3 ,

2 2 2

'2

12 ' 2 ' 2 '

c v

c v c v c v

kk kE k E

m m m

kk k

m E E m E E m E E

3

1 2 3 ,3

4' 8

3 c vE m m m E E

Density of states near a conduction band minimum or valence band maximum, and the definition of

the density of states effective mass

, 1 2 3 ,2 3

3 2*, , ,2 3

2

12

elipsoc v c v

c v c v c v

Ng E m m m E E

g E E E m

2 3 1 3*, 1 2 3c v elipsom N mm m

Conduction band density of states effective mass

Band

Band

3 2*

2

exp[( ) / ]

exp[ ( ) / )]

22

C C f

C C

cc

n g E f E dE

n N E E KT

N g E E E KT dE

m kTN

h

Valence band density of states effective mass

Band

Band

3 2*

2

[1 ]

exp[( ) / ]

exp[ ( ) / )]

22

V f V

V V

VV

p g E f E dE

p N E E KT

N g E E E KT dE

m kTN

h

Summary

• Conductivity effective mass – determines mobility.

• Density of states effective mass – determines NC

• Cyclotron effective mass – can be measured directly

2 3 1 3*1 2 3c elipsom N mm m

tl mmm 3

2

3

11

*

1 2 32 2 2

1 1 2 2 3 3ˆ ˆ ˆ

mm mm

H m H m H m

Elective home exercise

• Derive conductivity and density of states effective mass for holes.

k

E

hh

lh