the effective mass

7/31/2019 The Effective Mass

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The effective mass

Conductivity effective mass determinesmobility.

Density of states effective massdetermines NC

Cyclotron effective mass can bemeasured directly


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Electron in a periodic potential

Why does the semiconductor industry usesingle crystal material (when possible) ?


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Electrons are not scatteredby a periodic potential

move with a constantvelocity as in vacuum !


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Electron in vacuum

0)( xV

)](exp[)( tkxiAx

m

pE

m

k

22

)(22


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Dispersion

)(k

)](exp[)( tkxiAx


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Electron in vacuum

0)( xV

)](exp[)( tkxiAx

m

pE

m

k

22

)(22


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Electron in a periodic potential

)()2

(

)()(

)](exp[)()(

)()(

kEa

kE

xuaxu

tkxixux

xVaxV


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Electron in a 3D periodic potential

vectorlatticeReciprocal

)()(

)()()](exp[)()(

vectorlatticeBravais

)()(

K

kEKkE

ruRrutrkirur

R

rVRrV


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

zyxji

kkkkkk

EEkE jjiiji ji

,,,

))((21)( 00

,

2

0


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

tensormasseffectiveinverse1

where

,,2

)(

or

,,,

))((2

1)(

0

2

2

1

0

0

0

1

000

2

0

00

,

2

0

k

kk

EM

kk

kk

kk

MkkkkkkEkE

zyxji

kkkkkk

EEkE

ji

ij

zz

yy

xx

zzyyxx

jjii

ji ji


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eddiagonalizbecan

-matrixlsymmetricaais

tensormasseffectiveinversethe1

0

2

2

1

kkk

EM

ji

ij

I th di t t i hi h


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In the coordinate system in which

the effective mass tensor is

diagonal

0

0

0

1

3

1

2

1

1

3

2

1

1

zz

yy

xx

zyx

kk

kk

kk

m

m

m

zk

Ey

k

Ex

k

Ev

m

m

m

M


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Acceleration due to an electric filed (F)

F-qaM

)modelcalsemiclassi(the-

1

0

0

0

1

3

1

2

1

1

Fqdt

kd

dt

kdM

dt

vda

kk

kk

kk

m

m

m

v

zz

yy

xx


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Effective mass tensor valid nearE(k) minima and maxima only

F-qaM


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Constant energy surfaces in crystalmomentum space cookies


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Constant energy surfaces near aminimum are ellipsoids

1

3

1

2

1

1

1

0

0

0

1

000

2

0 ,,

2

)(

m

m

m

M

kk

kk

kk

MkkkkkkEkE

zz

yy

xx

zzyyxx

Constant energy surfaces in Si and


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Constant energy surfaces in Si and

Ge near a minimum are ellipsoids

of revolution

1

1

1

1

0

0

0

1

000

2

0 ,,2

)(

t

t

l

zz

yy

xx

zzyyxx

m

m

m

M

kk

kk

kk

MkkkkkkEkE


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Acceleration of an electron near anenergy minimum in silicon

6

i1

6-1

i

1

x

*

- electric field (main coordinate system)

-qF Ma -in each valley

1

-qF M a -average6

1a M F

6

1 2a ( )3 3

1 1 2

3 3

x y Z

i

i

x

l t

l t

F F x F y F z

q

qFm m

m m m


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Electron transport effective mass insilicon and germanium

tl mmm 3

2

3

11

*


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Homework competition find a shorter wayto prove this equation for germanium than

given in last years home exam

tl mmm 3

2

3

11

*


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The spherical case -

Electrons in GaAs

Holes in Si, Ge, GaAs

m

m

m

M


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Cyclotron resonance effective mass

qH

m


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Cyclotron resonance effective masstobe shown in the tutorial

1 2 3

2 2 2

1 1 2 2 3 3

2 2 2 2 22 2

1 1 2 2 3 3 31 2

1 2 3 2 3 1 3 1 2

1

m m mm

H m H m H m

H m H m H m HH H

m m m m m m m m m m


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Cyclotron resonance effective masstobe shown in the tutorial


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Density of states effective mass

3 2*

2

22

c fE E kT

c

c

c

n N e

m kTN

h


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Density of states

number of allowed energy states( )

dEdV

V- Volume

E- Energy

g E


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Example: density of states ofhydrogen gas

( ) ( )

- allowed energy level

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

n - density of hydrogen atoms

i i

i

i

i

g E n g E E

E

g

g(E)

E


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Density of states of solids

g(E)

E


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Density of electrons in an energyband

g(E)

E

Ef

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)


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Density of holes in an energy band

g(E)

E

Ef

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)


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Approximation for the Fermi Diracdistribution for E-Ef>3KT

1

1

F

F

E E kT

FD MBE E kTf E e f Ee


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Density of electrons in an energyband

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


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Density of states of solids in Kspace

3

x y z

number of allowed energy states 12

dk dk dk dV (2 )

V- Volume

The density of states in an energy interval is


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

constant energy surfaces

2

1

x y z312 dk dk dk (2 )

E

E


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Constant energy surfaces in crystalmomentum space cookies


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Volume of an ellipsoid

2 2 2

2 2 21

x y z

a b c

4

Volume= 3 abc


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Volume of a constant energy (E)

ellipsoid

22 22

31 2,

1 2 3

22 2

31 2

1 , 2 , 3 ,

2 2 2

'2

1

2 ' 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

3c vE m m m E E

Density of states near a conduction band minimum


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

the density of states effective mass

, 1 2 3 ,2 3

3 2*

, , ,2 3

2

12

elipso

c 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 m m m

C


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Conduction band density of stateseffective 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

V l b d d i f


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Valence band density of stateseffective 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


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Summary

Conductivity effectivemass determinesmobility.

Density of stateseffective massdetermines NC

Cyclotron effective

mass can bemeasured directly

2 3 1 3*

1 2 3c elipsom N m m m

tl mmm 3

2

3

11

*

1 2 3

2 2 2

1 1 2 2 3 3

m m mmH m H m H m


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Elective home exercise

Derive conductivity and density of stateseffective mass for holes.

k

E

hh

lh

the effective mass

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