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Klimeck – ECE606 Fall 2012 – notes adopted from Alam
ECE606: Solid State Devices
Lecture 6Gerhard Klimeck
gekco@purdue.edu
1
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Reference: Vol. 6, Ch. 3 & 4
Presentation Outline
•Reminder – Density of states»Possible states as a function of Energy
•Reality check - Measurements of Bandgaps•Reality check - Measurements of Effective Mass•Rules of filling electronic states •Derivation of Fermi-Dirac Statistics: three techniques
•Intrinsic carrier concentration•Conclusions
2
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
k
a
πa
π− 2k
Na
π∆ =
E E
DOS
( )3 3 0
2 3
2* *m m E EDOS
π−
=ℏ
Reminder: Momentum vs. DOS
Important things to remember:• Momentum k entered our thinking as a quantum number• Each quantum number is creating ONE state• Often “just” need the number of available states in an energy range
=> Density of States => appears to be solely determined by »1) band edge, »2) effective mass 3
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Measurement of Band Gap
k
E
1
2
3
4
Photons are only absorbed between bands that have filled and empty states
E23
abso
rptio
n
4
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Measurement of Energy Gap
k
hν
Iout
Iin
E23
I out -I in
E23+∆
23E E−∼
( )2
23 phE E E− −∼
k
E
1
2
3
4
5
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Temperature-dependent Band Gap
k
E
1
2
3
4
( ) ( )2
0G G
TE T E
T
αβ
= −+
6
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Reference: Vol. 6, Ch. 3 & 4
Presentation Outline
•Reminder – Density of states»Possible states as a function of Energy
•Reality check - Measurements of Bandgaps•Reality check - Measurements of Effective Mass•Rules of filling electronic states •Derivation of Fermi-Dirac Statistics: three techniques
•Intrinsic carrier concentration•Conclusions
7
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Measurements of Effective Masses
Important things to remember:• Full bands do not conduct –
electrons have no space to go• Empty bands to not conduct –
there are no electrons to go aroundQuestion:• We are interested in the top-most
valence band holes and the bottom-most electron states
• We want to figure out the slope of the bands
• How can we probe just one particular species of electrons/holes?
• We do not want to transfer them from one band to the next!
=> can we rotate the electrons around in a single band?
k
E
3
2
1
8
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
(kx,0)
(0,-ky)
(0,ky)
(-kx,0)
kx
Energy=constant.
kxky
kz
ky
Liquid He temperature …x
y
Motion in Real Space and Phase Space
9
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Derive the Cyclotron Formula
2
0
0 0
z z
m*q B q B
r
qB r
m*
υ υ υ
υ
= × =
=
B
For an particle in (x-y) plane with B-field in z-direction, the Lorentz force is …
0
0
00
00 0
2 2
12
2
r m*
qB
qB
m*qB
m*
π πτυ
ντ π
ω πν
= =
≡ =
= =
0
02
qm*
B
vπ=
10
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Measurement of Effective Mass
ν0=24 GHz(fixed)
IoutIin
B field variable …
00
0
0
2 2
q qm
B B*
vm*πν
π= =
I out -I in
BBcon
kxky
kz
Bval kxky
kz
11
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Effective mass in Ge
[ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ]
111 111 111 111
111 111 111 111
4 angles between B field and the ellipsoids …Recall the HW1
B
12
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Derivation for the Cyclotron Formula
B
Show that 2 2
2 2
1
c t l t
cos sin
m m m m
θ θ= +
[ ] dF q B M
dt
υυ= × =
The Lorentz force on electrons in a B-field
( )
( )
( )
* xx y z z y t
y*y z x x z t
* zz x y y x l
dF q B B m
dtd
F q B B mdt
dF q B B m
dt
υυ υ
υυ υ
υυ υ
= − =
= − =
= − =
In other words,
Given three mc and three θ,we will Find mt, and ml
13
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Continued …
22 2 2 2 2
2
0 0 00
0 with l
l
yy t t
t* * *c t l
dsin cos
dtqB qB qB
m m m
wω ω
ω
υυ
ω
ω ω θ θ
ω
+ = ≡ +
≡ ≡ ≡
Differentiate (vy) and use other equations to find …
Let (B) make an angle (θ) with longitudinal axis of the ellipsoid (ellipsoids oriented along kz)
( )2 2
2 2
1*
l t tc
sin cos
m m mm
θ θ= +so that …
B0
ky
kx
kz
( ) ( )0 00x y zB B cos , B , B B sin ,θ θ= = =
14
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Measurement of Effective Mass
Three peaks B1, B2, B3Three masses mc1,mc2,mc3Three unique angles: 7, 65, 73
2 2
2 2
1
tc l t
cos sin
m mm m
θ θ= +
Known θ and mc allows calculation of mt and ml.
[110]
B=[0.61, 0.61, 0.5]
1
02cmqB
vπ=
15
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Valence Band Effective Mass
Which peaks relate to valence band?Why are there two valence band peaks?
16
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Conclusions
1) Only a fraction of the available states are occupied. The
number of available states change with energy. DOS
captures this variation.
2) DOS is an important and useful characteristic of a
material that should be understood carefully.
3) Experimental measurements are key to making sure that
the theoretical calculations are correct. We will discuss
them in the next class.
17
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Reference: Vol. 6, Ch. 3 & 4
Presentation Outline
•Reminder – Density of states»Possible states as a function of Energy
•Reality check - Measurements of Bandgaps•Reality check - Measurements of Effective Mass•Rules of filling electronic states •Derivation of Fermi-Dirac Statistics: three techniques
•Intrinsic carrier concentration•Conclusions
18
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Carrier Density
Carrier number = Number of states x filling factor
Chapters 2-3 Chapter 4
19
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
E-k diagram and Electronic States
ka
πa
π−
E
g(E)
E2 3
2
2π−
ℏ
*
cm m*
E E
E1
E2
E3
Energy-Band Density of States
20
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Rules for filling up the States
� Pauli Principle: Only one electron per state
� Total number of electrons is conserved
� Total energy of the system is conserved
E2
E3
T iiN N=∑
T i iiE E N=∑
E1
21
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Presentation Outline
•Reminder – Density of states»Possible states as a function of Energy
•Reality check - Measurements of Bandgaps•Reality check - Measurements of Effective Mass•Rules of filling electronic states •Derivation of Fermi-Dirac Statistics: three techniques
•Intrinsic carrier concentration•Conclusions
In 1926, Fowler studied collapse of a star to white dwarf by F-D statistics, before Sommerfeld used the F-D statistics to develop a theory of electrons in metals in 1927. Wikipedia has a nice article on this topic. Difference between a trick and a method: A method is a trick used more than once!
22
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Illustrative Example: 3 Energy Levels
E=0
E=2
E=4
T iiN N=∑ T i ii
E E N=∑ET=12
203
5!
0!5!
2!
1!2
7!
3!5
4!3
!W = • •
=
122
5!
2!3!
72!
1!1
!
5!2!!420
W = • •
=
041
5!
4!1!
2!
0!2
7!
6!5
1!3
!W = • •
=
Particle conservation Energy conservation
NT=5
23
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Occupation Statistics
E=0
E=2
E=4
122 420W = 041 35W =203 35W =
W (E
)
2,0,3 1,2,2 0,4,1
Choose the most probable configuration.
24
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Occupation Statistics
E=0
E=2
E=4
122 420W =
*3
*2
*1
2
7
2
1
2
5
f
f
f
=
=
=W (E)
2,0,3 1,2,2 0,4,1
f(E)
E
Side note:So far everything shown here is EXACT!No approximations on the occupation probability!=> direct application to nano-scale electronics!
25
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
For N-states
( )!
! !i
ci i i i
SW
S N N=
−∏
lnW
2,0,3 1,2,2 0,4,1
[ ]ln ( ) ln( ) ( ) ln− − − − + − − +∑≃ i i i i i i i ii i i ii
NS S S N N NS NS N S
203
5!
0!5
2!
1 !
7!
3!4!!2!= • •WRecall.
Si
Ni
Stirling approx.
[ ]ln( ) ln ! ln( )! ln != − − −∑ i i i ii
W S S N N
[ ]ln ( ) ln( ) ln= − − − −∑ i i i i i i i ii
S S S N S N N N
( )ln( !) ln 10S S S S for S≈ − >
configurations
26
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Optimization with Lagrange-Multiplier
ln 1ii i
i i
SE dN
Nα β
= − − −
∑
T iiN N=∑
T i iiE E N=∑
lnW
configurations
lnln( ) i
i i
WW dN
Nδ ∂=
∂∑
ln 1ii i i i
i i ii
SdN dN E dN
Nα β
− − −
∑ ∑ ∑≃
Choose the most
probable
configuration.[ ]ln ln ( ) ln( ) lni i i i i i i i
i
W S S S N S N N N= − − − −∑
ln 1ii
i i
SdN
N
= −
∑
See additional notes on Lagrange multiplies on ece606 page and blackboard
Particle conservation
Energy conservation
Optimization with constraints!
0=
27
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Final steps …
1
1i
Ei
N
S eα β+=+
ln 1 0ii
i
SE
Nα β
− − − =
1At , ( ) 0
2FF FE f EE Eα β= ≡ ⇒ + =
f(E)
E
fmax(E)=1
1
Bk Tβ⇒ =
( )1
( )1 F
iE
iE
Nf E
S eβ −= =+
/At , ( ) BE k TBoltzmanE f E Ae−→ ∞ =
( ) /
1
1 BFE kE Te −=+
EF
ln ln 1 0ii i
i i
SW E dN
Nδ α β
= − − − =
∑
( )f E ≡
28
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Derivation by Detailed Balance
� Pauli Principle, energy, and number conservation all satisfied
0 3 0 4 0 1 0 2( ) ( )[1 ( )][1 ( )]f E f E f E f E− −
E1+ E
2= E
3+ E
4 Only solution is …. 0 ( )
1( )
1 β −=+ FE E
f Ee
E=0
E=2
E=4
1E
2E
3E 4E
0 1 0 2 0 3 0 4( ) ( )[1 ( )][1 ( )]f E f E f E f E− −=
Energy conversation
Pauli Exclusion
Detailed Balance in Equilibrium
29
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Derivation by Partition Function
( )
( )
0
1
000
11
0
1
β
β
− − ×
− − ×i
F
F
i i i
E
E E
state E N P
e Z
e Z
β β
β
− − − −
− −= ≡∑
F F
i i F
iiii( E ) ( E )
i ( E N E )
NE E
i
Ne eP
e Z
Ei
E=0
E=2E=4
1β = Bk T
30
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Derivation by Partition Function
( ) 1
0 1
=+P
f EP P
Probability that state is filled ….
( )
( )
0
1
000
11
0
1
β
β
− − ×
− − ×i
F
F
i i i
E
E E
state E N P
e Z
e Z
1
− −
− −=+
i F B
i F B
( E E ) / k T
( E E ) / k T
e / Z
/ Z e / Z
1
1 −=+ i F B( E E ) / k Te
1
f(E)
EF
1/2
E
1
f(E)
EF
1/2
E
1
31
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Few comments on Fermi-Dirac Statistics
� Applies to all spin-1/2 particles
� Information about spin is not explicit; multiply DOS by 2. May be more complicated for magnetic semiconductors.
� Coulomb-interaction among particles is neglected,Therefore it applies to extended solids, not to small molecules
Lx32
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Presentation Outline
•Reminder – Density of states»Possible states as a function of Energy
•Reality check - Measurements of Bandgaps•Reality check - Measurements of Effective Mass•Rules of filling electronic states •Derivation of Fermi-Dirac Statistics: three techniques
•Intrinsic carrier concentration•Conclusions
33
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Carrier Distribution
( )cg E
Eυ
FE( )g Eυ
cE
( )1 f E−
( )f E
( ) ( )cg E f E
( ) ( )1g E f Eυ −
( ) ( )top
c
E
cEn g E f E dE= ∫
( ) ( )1bot
E
Ep g E f E dE
υ
υ = − ∫
DOS F-D concentrationE E
1
34
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Electron Concentration in 3D solids
( ) ( )
( )( )2 3
2 12
2 1 βπ −
=
−= ×
+
∫
∫ℏ
top
c
top
Fc
E
cE
* *E n n C
E EE
n g E f E dE
m m E EdE
e
( )3 2
2 1 2
0 1
22
*n
C
dF
e
mN
h ξ ηπ β ξ ξη
∞
−=
≡ +∫
( )( ) ( )2 3
2 1
1 β βπ − −
∞
+
−∫≃
ℏ c Fc c
*C
E
n
E EE E
*nm m
dEe e
E E
( ) ( )1 2
2 η η βπ
= ≡ −c c FC CF EN E
Assume wide bands
Include spinfactor of 2
35
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Boltzmann vs. Fermi-Dirac Statistics
( ) ( )1 2
23ηη η β
π= → − ≡ − >c
C c C c C Fn N F N e if E E
( ) ( )cg E f E
( ) ( )1g E f Eυ − FE
ηce ( )1 2 ηcF
36
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Effective Density of States
( ) ( )1 2
23c FE E
C c C c Fn N F N e if E Eβη βπ
− −= → − >
( ) ( )cg E f E
( ) ( )1g E f Eυ −
CN
VN
FEFE
As if all states are at a single level EC
37
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Law of Mass-Action
( )
( )
β
β
− −
+ −
=
=
c F
v F
E EC
E EV
n N e
p N eFE
( )β
β
− −
−
× =
=
c v
g
E EC V
E
C V
n p N N e
N N e
Product is independent of the Fermi level!Very useful balance equation! Will use it often
38
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Fermi-Level for Intrinsic Semiconductors
2β−=≡
gE
i
F
V
i
Cn e
E
N
E
NFE
( ) ( )
1
2 2
β β
β
− − + −= ⇒ =
= +
c i v iE E E E
Gi
V
C
VCn p e e
EE ln
N
N
N
N
E
k
3
2
2 β−
= =
= g
V
i
E
i C N
p n
n eN
n
39
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Conclusions
• We discussed how electrons are distributed in electronic states
defined by the solution of Schrodinger equation.
• Since electrons are distributed according to their energy,
irrespective of their momentum states, the previously
developed concepts of constant energy surfaces, density of
states etc. turn out to be very useful.
=> will not discuss Schroedinger Eq. anymore
=> everything is captured in bandedges and effective masses
• We still do not know where EF is for general semiconductors … If
we did, we could calculate electron concentration.
40
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Presentation Outline
•Reminder – Density of states»Possible states as a function of Energy
•Reality check - Measurements of Bandgaps•Reality check - Measurements of Effective Mass•Rules of filling electronic states •Derivation of Fermi-Dirac Statistics: three techniques
•Intrinsic carrier concentration•Conclusions
41
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Illustrative Example: 3 Energy Levels
E=0
E=2
E=4
T iiN N=∑ T i ii
E E N=∑ET=12
203
5!
0!5!
2!
1!2
7!
3!5
4!3
!W = • •
=
122
5!
2!3!
72!
1!1
!
5!2!!420
W = • •
=
041
5!
4!1!
2!
0!2
7!
6!5
1!3
!W = • •
=
Particle conservation Energy conservation
NT=5
42
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Occupation Statistics
E=0
E=2
E=4
122 420W = 041 35W =203 35W =
W (E
)
2,0,3 1,2,2 0,4,1
Choose the most probable configuration.
43
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Occupation Statistics
E=0
E=2
E=4
122 420W =
*3
*2
*1
2
7
2
1
2
5
f
f
f
=
=
=
W (E)
2,0,3 1,2,2 0,4,1
f(E)
E
Side note:So far everything shown here is EXACT!No approximations on the occupation probability!=> direct application to nano-scale electronics!
44
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
For N-states
( )!
! !i
ci i i i
SW
S N N=
−∏
lnW
2,0,3 1,2,2 0,4,1
[ ]ln ( ) ln( ) ( ) ln− − − − + − − +∑≃ i i i i i i i ii i i ii
NS S S N N NS NS N S
203
5!
0!5
2!
1 !
7!
3!4!!2!= • •WRecall.
Si
Ni
Stirling approx.
[ ]ln( ) ln ! ln( )! ln != − − −∑ i i i ii
W S S N N
[ ]ln ( ) ln( ) ln= − − − −∑ i i i i i i i ii
S S S N S N N N
( )ln( !) ln 10S S S S for S≈ − >
configurations
45
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Optimization with Lagrange-Multiplier
ln 1ii i
i i
SE dN
Nα β
= − − −
∑
T iiN N=∑
T i iiE E N=∑
lnW
configurations
lnln( ) i
i i
WW dN
Nδ ∂=
∂∑
ln 1ii i i i
i i ii
SdN dN E dN
Nα β
− − −
∑ ∑ ∑≃
Choose the most
probable
configuration.[ ]ln ln ( ) ln( ) lni i i i i i i i
i
W S S S N S N N N= − − − −∑
ln 1ii
i i
SdN
N
= −
∑
See additional notes on Lagrange multiplies on ece606 page and blackboard
Particle conservation
Energy conservation
Optimization with constraints!
0=
46
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Final steps …
1
1i
Ei
N
S eα β+=+
ln 1 0ii
i
SE
Nα β
− − − =
1At , ( ) 0
2FF FE f EE Eα β= ≡ ⇒ + =
f(E)
E
fmax(E)=1
1
Bk Tβ⇒ =
( )1
( )1 F
iE
iE
Nf E
S eβ −= =+
/At , ( ) BE k TBoltzmanE f E Ae−→ ∞ =
( ) /
1
1 BFE kE Te −=+
EF
ln ln 1 0ii i
i i
SW E dN
Nδ α β
= − − − =
∑
( )f E ≡
47
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Reference: Vol. 6, Ch. 3 & 4
Presentation Outline
•Reminder - Rules of filling electronic states •Derivation of Fermi-Dirac Statistics:
»three techniques
• Intrinsic carrier concentration•Potential, field, and charge•E-k diagram vs. band-diagram•Basic concepts of donors and acceptors •Conclusions
48
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Derivation by Detailed Balance
� Pauli Principle, energy, and number conservation all satisfied
0 3 0 4 0 1 0 2( ) ( )[1 ( )][1 ( )]f E f E f E f E− −
E1+ E
2= E
3+ E
4 Only solution is …. 0 ( )
1( )
1 β −=+ FE E
f Ee
E=0
E=2
E=4
1E
2E
3E 4E
0 1 0 2 0 3 0 4( ) ( )[1 ( )][1 ( )]f E f E f E f E− −=
Energy conversation
Pauli Exclusion
Detailed Balance in Equilibrium
49
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Reference: Vol. 6, Ch. 3 & 4
Presentation Outline
•Reminder - Rules of filling electronic states •Derivation of Fermi-Dirac Statistics:
»three techniques
• Intrinsic carrier concentration•Potential, field, and charge•E-k diagram vs. band-diagram•Basic concepts of donors and acceptors •Conclusions
50
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Derivation by Partition Function
( )
( )
0
1
000
11
0
1
β
β
− − ×
− − ×i
F
F
i i i
E
E E
state E N P
e Z
e Z
β β
β
− − − −
− −= ≡∑
F F
i i F
iiii( E ) ( E )
i ( E N E )
NE E
i
Ne eP
e Z
Ei
E=0
E=2E=4
1β = Bk T
51
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Derivation by Partition Function
( ) 1
0 1
=+P
f EP P
Probability that state is filled ….
( )
( )
0
1
000
11
0
1
β
β
− − ×
− − ×i
F
F
i i i
E
E E
state E N P
e Z
e Z
1
− −
− −=+
i F B
i F B
( E E ) / k T
( E E ) / k T
e / Z
/ Z e / Z
1
1 −=+ i F B( E E ) / k Te
1
f(E)
EF
1/2
E
1
f(E)
EF
1/2
E1
52
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Few comments on Fermi-Dirac Statistics
� Applies to all spin-1/2 particles
� Information about spin is not explicit; multiply DOS by 2. May be more complicated for magnetic semiconductors.
� Coulomb-interaction among particles is neglected,Therefore it applies to extended solids, not to small molecules
Lx53
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Reference: Vol. 6, Ch. 3 & 4
Presentation Outline
•Reminder - Rules of filling electronic states •Derivation of Fermi-Dirac Statistics:
»three techniques
• Intrinsic carrier concentration•Potential, field, and charge•E-k diagram vs. band-diagram•Basic concepts of donors and acceptors •Conclusions
54
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Carrier Distribution
( )cg E
Eυ
FE
( )g Eυ
cE
( )1 f E−
( )f E
( ) ( )cg E f E
( ) ( )1g E f Eυ −
( ) ( )top
c
E
cEn g E f E dE= ∫
( ) ( )1bot
E
Ep g E f E dE
υ
υ = − ∫
DOS F-D concentrationE E
1
55
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Electron Concentration in 3D solids
( ) ( )
( )( )2 3
2 12
2 1 βπ −
=
−= ×
+
∫
∫ℏ
top
c
top
Fc
E
cE
* *E n n C
E EE
n g E f E dE
m m E EdE
e
( )3 2
2 1 2
0 1
22
*n
C
dF
e
mN
h ξ ηπ β ξ ξη
∞
−=
≡ +∫
( )( ) ( )2 3
2 1
1 β βπ − −
∞
+
−∫≃
ℏ c Fc c
*C
E
n
E EE E
*nm m
dEe e
E E
( ) ( )1 2
2 η η βπ
= ≡ −c c FC CF EN E
Assume wide bands
Include spinfactor of 2
56
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Boltzmann vs. Fermi-Dirac Statistics
( ) ( )1 2
23ηη η β
π= → − ≡ − >c
C c C c C Fn N F N e if E E
( ) ( )cg E f E
( ) ( )1g E f Eυ − FE
ηce ( )1 2 ηcF
57
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Effective Density of States
( ) ( )1 2
23c FE E
C c C c Fn N F N e if E Eβη βπ
− −= → − >
( ) ( )cg E f E
( ) ( )1g E f Eυ −
CN
VN
FEFE
As if all states are at a single level EC
58
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Law of Mass-Action
( )
( )
β
β
− −
+ −
=
=
c F
v F
E EC
E EV
n N e
p N eFE
( )β
β
− −
−
× =
=
c v
g
E EC V
E
C V
n p N N e
N N e
Product is independent of the Fermi level!Very useful balance equation! Will use it often
59
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Fermi-Level for Intrinsic Semiconductors
2β−=≡
gE
i
F
V
i
Cn e
E
N
E
NFE
( ) ( )
1
2 2
β β
β
− − + −= ⇒ =
= +
c i v iE E E E
Gi
V
C
VCn p e e
EE ln
N
N
N
N
E
k
3
2
2 β−
= =
= g
V
i
E
i C N
p n
n eN
n
60
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
• We discussed how electrons are distributed in electronic states
defined by the solution of Schrodinger equation.
• Since electrons are distributed according to their energy,
irrespective of their momentum states, the previously
developed concepts of constant energy surfaces, density of
states etc. turn out to be very useful.
=> will not discuss Schroedinger Eq. anymore
=> everything is captured in bandedges and effective masses
• We still do not know where EF is for general semiconductors … If
we did, we could calculate electron concentration.
Summary – DOS and Fermi Functions
61
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Reference: Vol. 6, Ch. 3
Presentation Outline
•Schrodinger equation in periodic U(x)•Bloch theorem•Band structure•Properties of electronic bands•E-k diagram and constant energy surfaces •Conclusions
62
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