### Boltzmann Approximation of Fermi Function

```Semiconductor Device Physics
Lecture 3
Dr. Gaurav Trivedi,
EEE Department,
IIT Guwahati
Boltzmann Approximation of Fermi Function
Boltzmann Approximation of Fermi Function
 The Fermi Function that describes the probability that a state
at energy E is filled with an electron, under equilibrium
conditions, is already given as:
f (E) 
1
1  e( E  EF ) / kT
 Fermi Function can be approximated as:
f ( E)  e( E EF ) / kT
if E – EF > 3kT
1  f ( E)  e( EF E )/ kT
if EF – E > 3kT
Nondegenerately Doped Semiconductor
 The expressions for n and p will now be derived in the range
Boltzmann
Approximation
ofapplied:
Fermi Function
where the
Boltzmann approximation
can be
Ev  3kT  EF  Ec  3kT
3kT
Ec
EF in this range
3kT
Ev
 The semiconductor is said to be nondegenerately doped
(lightly doped) in this case.
Degenerately Doped Semiconductor
Degenerately Doped Semiconductor
Degenerately Doped Semiconductor
 If a semiconductor is very heavily doped, the Boltzmann
approximation is not valid.
 For Si at T = 300 K,
EcEF < 3kT if ND > 1.6  1018 cm–3
EFEv < 3kT if NA > 9.1  1017 cm–3
 The semiconductor is said to be degenerately doped (heavily
doped) in this case.
• ND = total number of donor atoms/cm3
• NA = total number of acceptor atoms/cm3
Equilibrium Carrier Concentrations
 Integrating
n(E) over all the
energies in the conduction
band
to
Boltzmann
Approximation
of
Fermi
Function
obtain n (conduction electron concentration):
Etop
n

g c ( E ) f ( E )dE
Ec
 By using the Boltzmann approximation, and extending the
integration limit to ,
3/ 2
n  NCe
( EF  Ec ) kT
 mn* kT 
where N C  2 
2
2

h


• NC = “effective” density of conduction band states
• For Si at 300 K, NC = 3.22  1019 cm–3
Equilibrium Carrier Concentrations
 Integrating
p(E) over all the
energies in the conduction
band
to
Boltzmann
Approximation
of
Fermi
Function
obtain p (hole concentration):
p
EV
g v ( E ) 1  f ( E )  dE

Ebottom
 By using the Boltzmann approximation, and extending the
integration limit to ,
3/ 2
p  N V e( Ev  EF ) kT
 mp* kT 
where N V  2 
2
 2 h 
• NV = “effective” density of valence band states
• For Si at 300 K, NV = 1.83  1019 cm–3
Intrinsic Carrier Concentration
 Relationship
between EFApproximation
and n, p :
Boltzmann
of
n  NCe( EF Ec ) kT
( Ev  EF ) kT
p  NVe
 For intrinsic semiconductors, where n = p = ni,
np  n
ni  NC N V e EG
2
i
2 kT
• EG : band gap energy
Fermi Function
Intrinsic Carrier Concentration
kT
npBoltzmann
 ( NCe( EF Ec )Approximation
)  ( NVe( Ev EF ) kT of
) Fermi Function
( Ev  Ec ) kT
 NC NVe
 NC NVe EG
ni  NC NV e
kT
 EG 2 kT
Alternative Expressions: n(ni, Ei) and p(ni, Ei)
 In an intrinsic semiconductor, n = p = ni and EF = Ei, where Ei
denotesBoltzmann
the intrinsic Fermi
level.
Approximation
of Fermi Function
n  NCe( EF Ec ) kT
p  NVe( Ev EF ) kT
( Ei  Ec ) kT
pi  NVe( Ev Ei ) kT
 NV  ni e( Ev Ei ) kT
ni  NCe
( Ei  Ec ) kT
 NC  nie
n  ni e( Ei Ec ) kT  e( EF Ec ) kT
n  ni e( EF Ei ) kT
n
EF  Ei  kT ln  
 ni 
p  nie( Ev Ei ) kT  e( Ev EF ) kT
p  nie( Ei EF ) kT
 p
EF  Ei  kT ln  
 ni 
Intrinsic Fermi Level, Ei
 To find EF for an intrinsic semiconductor, we use the fact that
n = p. Boltzmann Approximation of Fermi Function
NCe( Ei Ec ) kT  NVe( Ev Ei ) kT
Ei
EG = 1.12 eV
Ec
Ev
Ec  Ev kT  N V 
Ei 

ln 

2
2  NC 
*


m
Ec  Ev 3kT
p
Ei 

ln  * 
m 
2
4
 n
Ec  Ev
Ei 
• Ei lies (almost) in the middle
2
between Ec and Ev
Si
Example: Energy-Band Diagram
Boltzmann Approximation 17of Fermi
Function
–3
 For Silicon at 300 K, where is EF if n = 10
Silicon at 300 K, ni = 1010 cm–3
n
EF  Ei  kT ln  
 ni 
17


10
5
 0.56  8.62 10  300  ln  10  eV
 10 
 0.56  0.417 eV
 0.977 eV
cm ?
Charge Neutrality and Carrier Concentration
Boltzmann
Approximation
 ND: concentration
of ionized
donor (cm–3) of Fermi
 NA: concentration of ionized acceptor (cm–3)?
 Charge neutrality condition:
p  n  N D  N A  0,
2
i
n
 n  ND  NA  0
n
n2  n( ND  NA )  ni2  0
ni2
p
n
• Ei quadratic equation in n
Function
Charge-Carrier Concentrations
 The solution
of the previous
for n Function
is:
Boltzmann
Approximation
of Fermi
12

N D  N A  N D  N A 
2
n
 
  ni 
2
2



2
 New quadratic equation can be constructed and the solution
for p is:
12
2


NA  ND  NA  ND 
2
p
 
  ni 
2
2



• Carrier concentrations depend
on net dopant concentration
ND–NA or NA–ND
Dependence of EF on Temperature
Ec
Boltzmann Approximation of Fermi Function
Ei
Ev
1013
1014
1015
1016
1017
1018
1019
1020
n
EF  Ei  kT ln   , donor-doped
 ni 
 p
EF  Ei  kT ln   , acceptor-doped
 ni 
Net dopant
concentration
(cm–3)
Carrier Concentration vs. Temperature
Phosphorus-doped Si
–3
Boltzmann Approximation NofD =Fermi
Function
1015 cm
• n : number of majority
carrier
• ND : number of donor
electron
• ni : number of intrinsic
conductive electron
Carrier Action
Boltzmann Approximation of Fermi Function
 Three primary types of carrier action occur inside a
semiconductor:
 Drift: charged particle motion in response to an applied
electric field.
 Diffusion: charged particle motion due to concentration
 Recombination-Generation: a process where charge
carriers (electrons and holes) are annihilated (destroyed)
and created.
Carrier Scattering
 Mobile electrons and atoms in the Si lattice are always in
random thermal motion.
Boltzmann Approximation of Fermi Function
 Electrons make frequent collisions with the vibrating atoms.
 “Lattice scattering” or “phonon scattering” increases with increasing
temperature.
 Average velocity of thermal motion for electrons: ~1/1000 x speed of
light at 300 K (even under equilibrium conditions).
 Other scattering mechanisms:
 Deflection by ionized impurity atoms.
 Deflection due to Coulombic force between carriers or “carrier-carrier
scattering.”
 Only significant at high carrier concentrations.
 The net current in any direction is zero, if no electric field is
applied.
2
3
1
electron
4
5
Carrier Drift
 When an electric field (e.g. due to an externally applied
of Fermi
Function
voltage)Boltzmann
is applied to aApproximation
semiconductor, mobile
charge-carriers
will be accelerated by the electrostatic force.
 This force superimposes on the random motion of electrons.
3
F = –qE
2
1
4
electron
5
E
 Electrons drift in the direction opposite to the electric field
 Current flows.
• Due to scattering, electrons in a semiconductor do not achieve
constant velocity nor acceleration.
• However, they can be viewed as particles moving at a constant
average drift velocity vd.
Drift Current
Boltzmann Approximation of Fermi Function
vd t
All holes this distance back from the normal plane
vd t A All holes in this volume will cross the plane in a time t
p vd t A
Holes crossing the plane in a time t
q p vd t A
Charge crossing the plane in a time t
q p vd A
Charge crossing the plane per unit time I (Ampere)
 Hole drift current
q p vd Current density associated with hole drift current J (A/m2)
Hole and Electron Mobility
 For holes,
oftoFermi
Function
IP|driftBoltzmann
 qpvd A Approximation
• Hole current due
drift
J P|drift  qpvd
• Hole current density due to drift
 In low-field limit,
vd  pE
JP|drift  qp pE
• μp : hole mobility
 Similarly for electrons,
J N|drift  qnvd
vd  n E
J N|drift  qn nE
• Electron current density due to drift
• μn : electron mobility
Drift Velocity vs. Electric Field
Boltzmann Approximation of Fermi Function
vd  pE
vd  n E
• Linear relation holds in low field
intensity, ~5103 V/cm
Hole and Electron Mobility
Boltzmann Approximation of Fermi Function
 has the dimensions of v/E :
 cm/s cm2 



V/cm
V

s


Electron and hole mobility of selected
intrinsic semiconductors (T = 300 K)
 n (cm2/V·s)
 p (cm2/V·s)
Si
1400
Ge
3900
GaAs
8500
InAs
30000
470
1900
400
500
Temperature Effect on Mobility
RL
RI
Boltzmann Approximation of Fermi Function
Impedance to motion due
to lattice scattering:
•
No doping
dependence
•
Decreases with
decreasing
temperature
Impedance to motion due to
ionized impurity scattering:
•
increases with NA or
ND
•
increases with
decreasing
temperature
Temperature Effect on Mobility
Boltzmann Approximation of Fermi Function
 Carrier mobility varies with doping:
 Decrease with increasing total concentration of ionized
dopants.
 Carrier mobility varies with temperature:
 Decreases with increasing T if lattice scattering is
dominant.
 Decreases with decreasing T if impurity scattering is
dominant.
Conductivity and Resistivity
Boltzmann Approximation of Fermi Function
JN|drift = –qnvd = qnnE
JP|drift = qpvd = qppE
Jdrift = JN|drift + JP|drift =q(nn+pp)E = E
 Conductivity of a semiconductor:  = q(nn+pp)
 Resistivity of a semiconductor:  = 1 / 
Resistivity Dependence on Doping
ForFermi
n-typeFunction
material:
Boltzmann Approximationof
1

q n N D
 For p-type material:
1

q p N A
Example
 Consider a Si sample at 300 K doped with 1016/cm3 Boron.
What is Boltzmann
its resistivity? Approximation of Fermi Function
NA = 1016/cm3 , ND = 0
(NA >> ND  p-type)
p  1016/cm3, n  104/cm3
1

qn n  qp p
1

qp p
 (1.6 10
)(470)(10 ) 
 1.330  cm
19
16
1
Example
 Consider a Si sample doped with 1017cm–3 As. How will its
resistivity
change when
the temperature isof
increased
from
Boltzmann
Approximation
Fermi Function
T = 300 K to T = 400 K?
The temperature dependent factor
in  (and therefore ) is n.
From the mobility vs. temperature
curve for 1017cm–3, we find that n
decreases from 770 at 300 K to
400 at 400 K.
As a result,  increases by a
factor of: 770/400 = 1.93
Assignment
Boltzmann Approximation of Fermi Function
```