MSE 310-ECE 340
Elec Props of Matls
Intrinsic S/C Prop.: Free Carrier
Statistics
MSE 310-ECE 340
Elec Props of Matls
 Let n be defined as the concentration of
electrons (#/volume) in the conduction band.
 Free Carrier Statistics:
 Carrier concentrations as a function of
temperature in intrinsic S/C’s.
n
o n = f(T)
o p = f(T)
E  ECT
E  ECB
 We will find that:
ni2  N c N v e
 Eg
g cb ( E )  f ( E )dE
[1]
 gcb(E) is the density of states in 3D. It is
obtained by solving the particle in the box in
3 dimensions. Hence, it is obtained using
quantum mechanics.
kT
 We want to determine how many free e-’s and
h+’s exist near their respective band edges as a
function of:
3
 meff  2
g cb ( E )  8 2  2  E  Ec  N1 E  Ec ;
 h 


o Temperature
m
where N1  8 2  eff2
 h

 Called “Free Carrier Statistics”, this is again
performed using:
o Statistical Mechanics
o Quantum Mechanics
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Intrinsic S/C Prop.: Free Carrier
Statistics
3
2


 The Fermi-Dirac distribution function, F(E) is
given:
1
F (E) 
(EE f )
kT
1 e
1
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2
1
MSE 310-ECE 340
Elec Props of Matls
Intrinsic S/C Prop.: Free Carrier
Statistics
MSE 310-ECE 340
Elec Props of Matls
 Let n be defined as the concentration of
electrons (#/volume) in the conduction band.
E  ECT
E  Ec
E  ECB
(EE f )
n  N1 
1 e
Intrinsic S/C Prop.: Free Carrier
Statistics
 Substituting, we have:
n  N1 kT 
E  ECB
dE
kT

E 
1 e
 
dE
kT
 Need to:
 Transform dE to dε
 Let: EcT  
 Multiply by:
n  N1
 Let:
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kT
kT
kT


E  Ec  Ec  E f
kT
E  Ec
kT
 d 
dE
dE
0 
; So: dE  kTd 
kT
kT
 Determine the limits for ε :
E  Ec
E 
kT
kT 
dE
(EE f )
E  ECB
kT
1 e
E  Ef
 Where:
 E 
 E 
E  Ec
kT
   E  Ec  
3
 Thus: n  N1  kT  2
Ec  Ec
  Ec
 0; And:   E    

kT
kT

 

0
1  e 
3
 N 2    F1/2  
2
  
E f  Ec
E  Ec
and  
kT
kT
3
d
[2]
N 2  N1  kT 
 Fj(η) is the Fermi-Dirac Integral of Order j:
x 
1
xj
Fj   
dx
  j  1 x 0 1  e x 
 Γ(j+1) is the Gamma Function of order j+1
x 
  j  1   x j e  x dx
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x 0
3
2
4
2
MSE 310-ECE 340
Elec Props of Matls
Intrinsic S/C Prop.: Free Carrier
Statistics
MSE 310-ECE 340
Elec Props of Matls
 Here: Fj(η) = F1/2(η)
 And: Γ(j+1) = Γ(3/2)
 Fj(η) was not analytically solved until 1995.
 Can now use Mathematica to solve.
 Gamma functions, which are less complicated
than Fj(η), can be solved using integral tables
or Mathematica.
 This leads to:
1
e
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E  Ef
kT
 
1

1
e
 
 e    e e 
[3]
 Note that the Fermi-Dirac distribution
function becomes the Maxwell-Boltzmann
distribution function.
 Thus, we have Arrhenius behavior resulting in
thermal activation of electron concentration in
the conduction band.
 Substituting equation [3] into [2], we have:
 BOLZMANN’S APPROXIMATION:
 Fj(η) was not analytically solved until 1995,
so historically the Boltzmann’s
Approximation was used.
 Boltzmann’s Approximation: Ef is well within
the energy band gap and not near the band
edges of EC and EV.
  
Intrinsic S/C Prop.: Free Carrier
Statistics
n  N2 
 
 N2 
 
e 
 0
 1
 0

 N 2e
5
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
1
2  
 e e d
 

d
0
1
2 
 e d
6
3
MSE 310-ECE 340
Elec Props of Matls
Intrinsic S/C Prop.: Free Carrier
Statistics
MSE 310-ECE 340
Elec Props of Matls
 Note that we have: Γ(j+1) = Γ(3/2)

n  N 2e
 

0

 N2
 NC e
 Hence:
 Compare before the Boltzmann Approximation:
3
n  N1  kT  2 
1
2 
 e d
kT
where: N C  N 2
n  Nce

 Ec  E f

 2 m kT 
where N c  2 

 h



n  N 2e
2
 N2
[2]
 


2
0
1
2 
3
 e d   N 2   e
2
e

e
3
2
 Hence, compare these two functions:
n  F1/2  
 NC is known as the Effective Density of States
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d
 To after the Boltzmann Approximation:
kT
*
e
2
1  e 
3
 N 2    F1/2  
2
 F1/2  
e
2
 Ec  E f 

 
 0

3
Note:    
2
2
3
 N 2    e
2
Intrinsic S/C Prop.: Free Carrier
Statistics
7
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n  e
8
4
MSE 310-ECE 340
Elec Props of Matls
Intrinsic S/C Prop.: Free Carrier
Statistics
MSE 310-ECE 340
Elec Props of Matls
 Hence, compare these two functions:
Intrinsic S/C Prop.: Free Carrier
Statistics
Density of States function:
n  F1/2  
n  e
Robert F. Pierret, Advanced Semiconductor
Fundamentals, Modular Series on Solid State Devices,
2nd Ed., Volume VI (Prentice Hall, 2003) p. 112-116.
 Note: eη diverges from F1/2(η) when η, or Ef- Ec or
9
Knowlton
Ev-Ef ~ 1kT.
g (E)  E
J.P. McKelvey, Solid State Physics for Engineering & Materials Science (Kreiger Publishing Company,
1993).
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10
5
MSE 310-ECE 340
Elec Props of Matls
Intrinsic S/C Prop.: Free Carrier
Statistics
1 e
Knowlton
kbT
 What happens when you combine the Fermi level
distribution with the density of states distribution?
Fermi-Dirac distribution & Density of States functions:
John P. McKelvey, Solid State Physics for
Engineering & Materials Science (Kreiger
Publishing Company, 1993).
Density of States function:
(EE f )
11
Intrinsic S/C Prop.: Free Carrier
Statistics
John P. McKelvey, Solid State Physics for Engineering & Materials Science (Kreiger
Publishing Company, 1993).
1
f E 
Robert F. Pierret, Advanced Semiconductor
Fundamentals, Modular Series on Solid State Devices,
2nd Ed., Volume VI (Prentice Hall, 2003) p. 112-116.
Fermi-Dirac distribution function:
MSE 310-ECE 340
Elec Props of Matls
Knowlton
12
6
MSE 310-ECE 340
Elec Props of Matls
Intrinsic S/C Prop.: Free Carrier
Statistics
MSE 310-ECE 340
Elec Props of Matls
 A similar approach is used to find the
concentration of holes, p, in the valence band.
 Let p be defined as the concentration of holes
(#/volume) in the valence band.
 What happens when you combine the Fermi level
distribution with the density of states distribution?
E g(E)
E c+ 
(E-Ec)1/2
CB
E
E
[1- f(E)]
For electrons
Ec
p
Area = n
Ec
nE(E)
Ev
Ev
For holes
VB
0
(a)
g(E)
(b)
f(E)
(c)

nE(E) or pE(E)
(d)
Fig 5.7
(a) Energy band diagram. (b) Density of states (number of states per unit energy
per unit volume). (c) Fermi-Dirac probability function (probability of occupancy
of a state). (d) The product of g(E) and f(E) is the energy density of electrons in
the CB (number of electrons per unit energy per unit volume). The area under
nE(E) vs. E is the electron concentration in the conduction band.

E  EVT

E  EVT
pE(E)
Area = p
E  EVT
E  EVT
EF
EF
Intrinsic S/C Prop.: Free Carrier
Statistics
g vb  E  f h  E dE
g vb  E  1  f e  E  dE
 gvb(E) is the density of states in 3D in the
valance band.
 gvb(E) is obtained by solving the particle in
the box in 3 dimensions.
 Hence, gvb(E) is obtained using quantum
mechanics.
 mh
g vb ( E )  8 2  eff2
 h


3
2
 Ev  E  P1 Ev  E ;


 mh
where P1  8 2  eff2
 h


From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Knowlton
13
[1]
Knowlton
3
2



14
7
MSE 310-ECE 340
Elec Props of Matls
Intrinsic S/C Prop.: Free Carrier
Statistics
MSE 310-ECE 340
Elec Props of Matls
 Evaluation of p eventually results in the
product of a Fermi-Dirac integral of order ½
and Gamma function of order 3/2.
3
 Applying the BOLZMANN’S
APPROXIMATION:
 Boltzmann’s Approximation: Ef is well within
the energy band gap and not near the band edges
of EC and EV.
E E
 That is:
   f
 1
1
*
1  2m kT
p  2  eff2
2  
2   2
d
 0
1  e 

*
1  2meff kT
 2
2   2
2  3 
    F1  
 2 2
Intrinsic S/C Prop.: Free Carrier
Statistics
3
kT
[2]
 Hence:
1
1
  

 e    e e 
1  e  e 
 Substituting into equation [2], we have:
Note - This may help the derivation:
0


0
  f ( x)dx   f ( x)dx
3
1
*
1  2m kT
p  2  eff2
2  
2   2
d
 0
1  e 

*
1  2m kT
 2  eff2
2  
 2   12 
 e 0  e d 

3
 Now we have a Gamma Function that is easily
3
found in integral tables:
*
1  2m kT  2
3
p
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15
Knowlton

2 2 
 p  Nve
eff
2

  
    e
 2
E f  Ev
kT
16
8
MSE 310-ECE 340
Elec Props of Matls
Intrinsic S/C Prop.: Free Carrier
Statistics
MSE 310-ECE 340
Elec Props of Matls
 n: concentration of electrons in the Conduction
Band (CB) per unit volume.
Intrinsic S/C Prop.: Free Carrier
Statistics
 i.e., Electron density
n
E 
E  Ec
g cb ( E )  f ( E )dE
where g cb ( E )  E  Ec
n  Nce

 Ec  E f

kT
3
 2 me*kT  2
where N c  2 

2
 h

 p: concentration of holes in the Valence Band
(VB) per unit volume.
 i.e., hole density
p
E  E
E 0
g ( E )  1  f ( E )  dE
where gb ( E )  E  E
p  N e

 E f  E

kbT
3
 2 mh* kT  2
where N  2 

2
 h

Knowlton
17
Intrinsic carrier concentration as a function of
1/T (Arrhenius plot). After Prof. E.H. Haller, UC Berkeley Lecture notes
Knowlton
18
9
MSE 310-ECE 340
Elec Props of Matls
Intrinsic S/C Prop.: Free Carrier
Statistics
MSE 310-ECE 340
Elec Props of Matls
 n: concentration of
electrons in the Conduction
Band (CB) per unit
volume.
 i.e., Electron density
 References:
 J.S. Blakemore, Semiconductor Statistics,
(Dover, 1987) p. 75-84.
 John P. McKelvey, Solid State Physics for
Engineering & Materials Science (Kreiger
Publishing Company, 1993) Ch. 9.
 Robert F. Pierret, Advanced Semiconductor
Fundamentals, Modular Series on Solid State
Devices, 2nd Ed., Volume VI (Prentice Hall,
2003) p. 112-116.
 David K. Ferry & Jonathan P. Bird, Electronic
Materials and Devices, (Academic Press, 2001)
p. 168-175.
 S.O. Kasap, Principles of Electronic Materials
and Devices, 3rd Ed. (McGraw-Hill, 2005) p.
380-387.
np  ni2
 N c N e
 N c N e
 N c N e

 Ec  E f

  Ec  E 
 Eg

 E f  E
kT
e

kbT
kT
kT
3
3
 2 me*kT  2
 2 mh* kT  2
where N c  2 
 & N  2 

2
2
 h

 h

ni  N c N e
 Eg
 N c N e
Ef 
Intrinsic S/C Prop.: Free Carrier
Statistics
 Eg
kT
2 kT
m 
3
 kT ln eff ,h
2 4
meff ,e
Eg
19
20
10