Handout 14 Statistics of Electrons in Energy Bands...1 ECE 407 – Spring 2009 – Farhan Rana –...
Transcript of Handout 14 Statistics of Electrons in Energy Bands...1 ECE 407 – Spring 2009 – Farhan Rana –...
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ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Handout 14
Statistics of Electrons in Energy Bands
In this lecture you will learn:
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Example: Electron Statistics in GaAs - Conduction Band
Consider the conduction band of GaAs near the band bottom at the -point:
e
e
e
m
m
m
M
100
010
0011
This implies the energy dispersion relation near the band bottom is:
e
ce
zyxcc m
kE
m
kkkEkE
22
222222
Suppose we want to find the total number of electrons in the conduction band:
We can write the following summation:
FBZin
2k
c kfN
2
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
fc
KTEkEc EkEfkffc
exp1
1
Where the Fermi-Dirac distribution function is:
We convert the summation into an integral:
KTEkE
kdVkfN
fckc
exp1
1
222
FBZ3
3
FBZ in
Then we convert the k-space integral into an integral over energy:
?
?FBZ3
3
exp1
1
22 fc
fc
EEfEgdE
KTEkE
kdVN
We need to find the density of states function gc(E) for the conduction band and need to find the limits of integration
Example: Electron Statistics in GaAs - Conduction Band
FBZin
2k
c kfN
Another way of writing itEf
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Density of States in Energy BandsEnergy
xk
a
a
sE ssV4 akVEkE xsssx cos2
sss VE 2
sss VE 2
Consider the 1D energy band that results from tight binding:
We need to find the density of states function g1D(E):
dEEgL
dEdEdk
Ldk
Ldk
L
sss
sss
sss
sssx
VE
VED
VE
VE
xa
xa
a
x
k
2
21
2
20FBZ in
22
42
22
akaVdkdE
xssx
sin2
2212
12
sssD
EEVaEg
E
Eg D1
sEsss VE 2 sss VE 2
3
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
?
?FBZ3
3
exp1
1
22 fc
fc
EEfEgdE
KT
EkE
kdVN
Example: Electron Statistics in GaAs - Conduction Band
e
ce
zyxcc m
kE
m
kkkEkE
22
222222
Energy dispersion near the band bottom is:
Electrons will only be present near the band bottom
(parabolic and isotropic)
Since the electrons are likely present near the band bottom, we can limit the integral over the entire FBZ to an integral in a spherical region right close to the -point:
point3
2
FBZ3
3
8
42
22 fcc EkEfdk
kVkf
kdVN
Ef
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Example: Electron Statistics in GaAs - Conduction Band
point
3
2
8
42 fc EkEfdk
kVN
Since the Fermi-Dirac distribution will be non-zero only for small values of k, one can safely extend the upper limit of the integration to infinity:
03
2
8
42 fc EkEfdk
kVN
emdkk
dkdE 2
ce EEm
k 22
and
We have finally:
e
ce
zyxcc m
kE
m
kkkEkE
22
222222
cEfcfc EEfEgdEVEkEfdk
kVN
03
2
8
42
We know that:
Ef
4
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
We have finally:
Where the conduction band density of states function is:
ce
c EEm
Eg
23
222
2
1
cEfcfc EEfEgdEVEkEfdk
kVN
03
2
8
42
E
Egc
cE
Example: Electron Statistics in GaAs - Conduction Band
The density of states function looks like that of a 3D free electron gas except that the mass is the effective mass and the density of states go to zero at the band edge energy
emcE
Ef
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
ce
c EEm
Eg
23
222
2
1
cEfc EEfEgdEn
fEEf
EfE
Egc Ef
KT
fEE
eEEf f
If then one may approximate the Fermi-Dirac function as an exponential:
KTEE fc
KTEE
KTEE
EEf f
ff exp
exp1
1
KTEE
NEEfEgdEn fcc
Efc
c
exp
cE
Where:
23
222
KTm
N ec
Maxwell-Boltzman approximation
Example: Electron Statistics in GaAs - Conduction Band
Effective density of states (units: #/cm3)
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ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Example: Electron Statistics in GaAs - Valence Band and Holes
Ef
• At zero temperature, the valence band is completely filled and the conduction band is completely empty
• At any finite temperature, some electrons near the top of the valence band will get thermally excited from the valence band and occupy the conduction band and their density will be given by:
• The question we ask here is how many empty states are left in the valence band as a result of the electrons being thermally excited. The answer is (assuming the heavy-hole valence band):
• We call this the number of “holes” left behind in the valence band and the number of these holes is P:
KTEE
Nn fcc exp
FBZ in
12k
fhh EkEf
fhhk
fhh EkEfkd
VEkEfP
12
212FBZ
3
3
FBZ in
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Example: Electron Statistics in GaAs - Valence Band and Holes
parabolic approx.
fhh EkEf
kdVP
12
2FBZ
3
3
hh
vhh
zyxhh m
kE
m
kkkEvkE
22
222222
Energy dispersion near the top of the valence band is:
Holes will only be present near the top of the valence band
Since the holes are likely present near the band maximum, we can limit the integral over the entire FBZ to an integral in a spherical region right close to the -point:
point
3
2
18
42 fhh EkEfdk
kVP
Ef
6
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Example: Electron Statistics in GaAs - Valence Band and Holes
03
2
18
42 fhh EkEfdk
kVP
Since the Fermi-Dirac distribution will be non-zero only for small values of k, one can safely extend the upper limit of the integration to infinity:
point
3
2
18
42 fhh EkEfdk
kVP
hh
vhh
zyxhh m
kE
m
kkkEvkE
22
222222
We know that:
hhmdkk
dkdE 2
EEm
k vhh 2
2
and
We have finally:
vE
fhh
fhh
EEfEgdEV
EkEfdkk
VP
1
18
42
03
2
Ef
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Example: Electron Statistics in GaAs - Valence Band and Holes
Where the heavy hole band density of states function is:
EEm
Eg vhh
hh
23
222
2
1
E
Eghh
vE
We have finally:
vE
fhh
fhh
EEfEgdEV
EkEfdkk
VP
1
18
42
03
2
Note that the mass that comes in the density of states is the heavy hole effective mass and the density of states go to zero at the band edge energy , and the density of states increase for smaller energies
hhm vE
Ef
7
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
EEm
Eg vhh
hh
23
222
2
1
vE
fhh EEfEgdEp 1
fEEf
EfE
Egc
Ef
If then one may approximate the Fermi-Dirac function as an exponential:
KTEE vf
KTEE
KTEE
EEf f
ff exp
exp1
11
KTEE
NEEfEgdEp vfhh
E
fhh
vexp1
cE
Where:
23
222
KTm
N hhhh
Maxwell-Boltzman approximation for holes
Example: Electron Statistics in GaAs - Valence Band and Holes
vE
Eghh
Effective density of states (units: #/cm3)
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Example: Electron Statistics in GaAs - Valence Band and HolesIn most semiconductors, the light-hole band is degenerate with the heavy hole band at the -point. So one always needs to include the holes in the light-hole valence band as well:
Ef
fEEf
EfE
Egc
cEvE
Eghh Eg h
v
v
vv
E
fv
E
fhhh
E
fh
E
fhh
EEfEgdE
EEfEgEgdE
EEfEgdEEEfEgdEp
1
1
11
KTEE
Np vfv exp
Where:
23
222
KTm
N hv
and 322323hhhh mmm
EEm
EgEgEg
vh
hhhv
23
222
2
1
Density of states effective mass
8
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Example: Electron Statistics in GaAs – Electrons and Holes
Ef
At any temperature, the total number of electrons and holes (including both heavy and light holes) must be equal:
c
vvcf
vcf
c
v
fcc
vfv
NNKTEE
E
KTEEE
NN
KTEE
NKTEE
N
np
log22
2exp
expexp
Because the effective density of states for electrons and holes are not the same (i.e. Nv ≠ Nc), the Fermi level at any finite temperature is not right in the middle of the bandgap.
But at zero temperature, the Fermi-level is exactly in the middle of the bandgap
fEEf E
fE
Egc
cEvE
Egv
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Example: Electron Statistics in GaAs – Electrons and Holes
Ef
At any temperature, the total number of electrons and holes (including both heavy and light holes) must be equal:
where ni is called the intrinsic electron (or hole) densityinnp
Note that the smaller the bandgap the larger than intrinsic electron (or hole) density
KT
ENNn
nKTEE
NN
nKTEE
NKTEE
N
nnp
nnp
gcvi
ivc
cv
ifc
cvf
v
i
i
2exp
exp
expexp
2
2
2
9
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Electron and Hole Pockets in GaAs
Ef
• At any non-zero temperature, electrons occupy states in k-space that are located in a spherically symmetric distribution around the -point
• This distrbution is referred to as the “electron pocket” at the -point
• At any non-zero temperature, the holes (heavy and light) also occupy states in k-space that are located in a spherically symmetric distribution around the -point
• This distribution is referred to as the “hole pocket” at the -point
Hole pocket
Electron pocket
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Shape of Fermi Surface/Contour and Mass Tensor: 2D Example
xkyk
Energy
xkyk
Energy
EF EF
kF
yy
y
xx
xcc m
k
mk
EkE22
2222
m
k
mk
EkE yxcc 22
2222
xxyy mm
When the energy dispersion relation is anisotropic, the distribution of carriers in k-space, and the Fermi surface/contour, are not spherical/circular but become ellipsoidal/elliptical
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ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Constant Energy Surfaces
e
cc mk
EkE2
22
Constant energy surfaces are in the reciprocal space and are such that the energy of every point on the surface is the same.
For example, the conduction band energy dispersion:
All points in k-space that are equidistant from the origin (-point) have the same energy. Constant energy surfaces in 3D are spherical shells, and in 2D are circles, with the origin as their center.
xk
yk
zk
Equation of a Constant Energy Surface with Energy Eo:
cozyxoe
c EEm
kkkEmk
E 2
22222 2
2
Equation of a sphere in k-space of radius = co EE
m
22
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Constant Energy Surfaces
Now consider the energy band dispersion: zz
z
yy
y
xx
xcc m
km
k
mk
EkE222
222222
Now the equation of a constant energy surface with energy Eo is:
Equation of an ellipsoid in k-space with semi-major axes given by:
coxx EE
m
22
cozz
z
yy
y
xx
xo
zz
z
yy
y
xx
xc EE
mk
m
k
mk
Emk
m
k
mk
E 2
222222222 2222
coyy EE
m
2
2
co
zz EEm
2
2
xk
yk
zk
Fermi-Surfaces are Examples of Constant Energy Surfaces:
xk
yk
zk
Fczz
z
yy
y
xx
xc EE
mk
m
k
mk
E 222
222222 Ec+EF
11
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Silicon: Electrons in the Conduction Band
coc EkE
t
t
m
m
m
M
100
010
0011
In Silicon there are six conduction band minima that occur along the six -X directions. These are also referred to as the six valleys. For the one that occurs along the -X(2/a,0,0) direction:
0,0,
285.0
ako
Not isotropic!
mℓ = 0.92 mmt = 0.19 m
This implies:
t
ozz
t
oyyoxxcc m
kkm
kk
mkk
EkE222
222222
fc
kkEkEf
kd
o
near
3
3
22
Expression for the electron density in the valley located at along the -X(2/a,0,0) direction can be written as:
Ef
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Silicon: Electrons in the Conduction Band
Define:
ozzt
z
oyyt
yoxxx
kkmm
q
kkmm
qkkmm
q
mq
EqE
mkk
m
kk
mkk
EkE
cc
t
ozz
t
oyyoxxcc
2
22222
222222
This implies:
fc
q
tt
fckk
EqEfqd
m
mmm
EkEfkd
o
0 near 3
3
3
near 3
3
22
22
Therefore, expression for the electron density in the valley located at along the -X(2/a,0,0) direction can be written as:
Ef
Dispersion is isotropic in q-space
12
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Silicon: Electrons in the Conduction Band
fc
q
tt EqEfqd
m
mmm
0 near 3
3
3 22
03
2
3 8
42 fc
tt EqEfdqq
m
mmm
Where: ce
c EEm
Eg
23
222
2
1
Total electron density in the conduction band consists of contributions from electron density sitting in all the six valleys:
03
2
3 8
426 fc
tt EqEfdqq
m
mmmn
cEfc EEfEgdEn
31326 tte mmmm and:Density of states effective mass
Ef
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Silicon: Electrons in the Conduction Band
KTEE
NEEfEgdEn fcc
Efc
c
exp
Where:
23
222
KTm
N ec
31326 tte mmmm And:
Six electron pockets in FBZ:
There are six electron pockets in Silicon - one at each of the valleys (conduction band minima)
The electron distribution in k-space in each pocket is not spherical but ellipsoidal since the electron masses in different directions are not the same
Ef
13
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Germanium: Electrons in the Conduction Band
In germanium there are eight conduction band minima that occur at the L-points
The L-point is at the edge of the FBZ, so one-half of each electron pocket is not in the FBZ and therefore one-half of the electron distribution in each L-valley should not be counted in the sum for calculating the number of electrons:
FBZin
2k
fc EkEfN
The other way to look at the problem is to realize that the other-half of each pocket is also located in the FBZ on the opposite side – so in reality there are four complete pockets of electrons in the FBZ
FBZ
ECE 407 – Spring 2009 – Farhan Rana – Cornell University