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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