Semiconductor Device Physics
Lecture 4
Dr.-Ing. Erwin Sitompul
President University
http://zitompul.wordpress.com
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Chapter 3
Carrier Action
Electron kinetic energy
Ec
Ev
Hole kinetic energy
Increasing hole energy
Increasing electron energy
Potential vs. Kinetic Energy
 Ec represents the electron potential energy:
P.E.  Ec  Ereference
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Chapter 3
Carrier Action
Band Bending
 Until now, Ec and Ev have always been drawn to be
independent of the position.
 When an electric field E exists inside a material, the band
energies become a function of position.
E
Ec
Ev
x
• Variation of Ec with position is
called “band bending”
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Chapter 3
Carrier Action
Band Bending
 The potential energy of a particle with charge –q is related to
the electrostatic potential V(x):
P.E.  qV
1
V   ( Ec  Ereference )
q
E  V
dV

dx
1 dEc 1 dEv 1 dEi
E


q dx q dx q dx
• Since Ec, Ev, and Ei differ
only by an additive constant
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Chapter 3
Carrier Action
Diffusion
 Particles diffuse from regions of higher concentration to
regions of lower concentration region, due to random thermal
motion (Brownian Motion).
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Chapter 3
Carrier Action
1-D Diffusion Example
 Thermal motion causes particles to
move into an adjacent compartment
every τ seconds.
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Chapter 3
J N|diff
Carrier Action
dn
 qDN
dx
e
n
Diffusion Currents
dp
J P|diff  qDP
dx
p
x
Electron flow
Current flow
h+
x
Hole flow
Current flow
• D is the diffusion coefficient
[cm2/sec]
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Chapter 3
Carrier Action
Total Currents
J  J N + JP
J N  J N|drift + J N|diff
J P  J P|drift + J P|diff
dn
 qn nE + qDN
dx
dp
 q p pE  qDP
dx
 Drift current flows when an electric field is applied.
 Diffusion current flows when a gradient of carrier concentration
exist.
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Chapter 3
Carrier Action
Current Flow Under Equilibrium Conditions
 In equilibrium, there is no net flow of electrons or :
J N  0, J P  0
 The drift and diffusion current components must balance each
other exactly.
 A built-in electric field of ionized atoms exists, such that the
drift current exactly cancels out the diffusion current due to the
concentration gradient.
dn
J N  qn nE + qDN
0
dx
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Chapter 3
Carrier Action
Current Flow Under Equilibrium Conditions
 Consider a piece of non-uniformly doped semiconductor:
 EF  Ec 
n-type semiconductor
Decreasing donor
concentration
Ec(x)
EF
Ev(x)
n  NCe
N C  EF  Ec  kT dEc
dn

e
dx
kT
dx
n dEc

kT dx
kT
dn
q

nE
dx
kT
• Under equilibrium, EF inside
a material or a group of
materials in intimate contact
is not a function of position
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Chapter 3
Carrier Action
Einstein Relationship between D and 
 But, under equilibrium conditions, JN = 0 and JP = 0
dn
J N  qn nE + qDN
0
dx
q
qnEn  qnE
DN  0
kT
Similarly,
DN
kT

n
q
DP
kT

p
q
• Einstein Relationship
 Further proof can show that the Einstein Relationship is valid
for a non-degenerate semiconductor, both in equilibrium and
non-equilibrium conditions.
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Chapter 3
Carrier Action
Example: Diffusion Coefficient
 What is the hole diffusion coefficient in a sample of silicon at
300 K with p = 410 cm2 / V.s ?
 kT 
DP  
 p
 q 
25.86 meV

 410 cm 2 V 1s 1
1e
cm 2
 25.86 mV  410
V s
1 eV
1 V
1e
 10.603 cm2 /s
1 eV  1.602 1019 J
• Remark: kT/q = 25.86 mV
at room temperature
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Chapter 3
Carrier Action
Recombination–Generation
 Recombination: a process by which conduction electrons and
holes are annihilated in pairs.
 Generation: a process by which conduction electrons and
holes are created in pairs.
 Generation and recombination processes act to change the
carrier concentrations, and thereby indirectly affect current
flow.
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Chapter 3
Carrier Action
Generation Processes
Band-to-Band
R–G Center
Impact Ionization
1 dEc
E
q dx
Release of
energy
ET: trap energy level
• Due to lattice defects or
unintentional impurities
• Also called indirect
generation
EG
• Only occurs in the
presence of large E
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Chapter 3
Carrier Action
Recombination Processes
Band-to-Band
R–G Center
Auger
Collision
• Rate is limited by
minority carrier trapping
• Primary recombination
way for Si
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• Occurs in heavily
doped material
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Chapter 3
Carrier Action
Direct and Indirect Semiconductors
E-k Diagrams
Ec
Ec
Phonon
Photon
GaAs, GaN
Photon
Ev
Si, Ge
(direct semiconductors)
(indirect semiconductors)
• Little change in momentum is
required for recombination
• Momentum is conserved by
photon (light) emission
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Ev
• Large change in momentum is
required for recombination
• Momentum is conserved by
mainly phonon (vibration)
emission + photon emission
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Chapter 3
Carrier Action
Excess Carrier Concentrations
Deviation from
equilibrium values
Values under
arbitrary conditions
Equilibrium values
n  n  n0
p  p  p0
n, p  0
 Positive deviation corresponds to a carrier excess, while
negative deviations corresponds to a carrier deficit.
n, p  0
 Charge neutrality condition:
n  p
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Chapter 3
Carrier Action
“Low-Level Injection”
 Often, the disturbance from equilibrium is small, such that the
majority carrier concentration is not affected significantly:
 For an n-type material p  n0 ,
 For a p-type material
n  p0 ,
n n0
p  p0
p
n  n0
p0
• Low-level injection condition
 However, the minority carrier concentration can be significantly
affected.
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Chapter 3
Carrier Action
Indirect Recombination Rate
 Suppose excess carriers are introduced into an n-type Si
sample by shining light onto it. At time t = 0, the light is turned
off. How does p vary with time t > 0?
 Consider the rate of hole recombination:
p
 cp NT p
t R
NT : number of R–G centers/cm3
Cp : hole capture coefficient
 In the midst of relaxing back to the equilibrium condition, the
hole generation rate is small and is taken to be approximately
equal to its equilibrium value:
p
p
p
 cp NT p0


t G t G-equilibrium
t R-equilibrium
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Chapter 3
Carrier Action
Indirect Recombination Rate
 The net rate of change in p is therefore:
p
p
p
 cp NT p + cp NT p0  cp NT  p  p0 

+
t R G t R t G
p
p
 cp NT p  
t R G
p
1
where  p 
cp N T
• For holes in
n-type material
 Similarly,
n
n
 cn NT n  
t R G
n
where  n 
1
cn NT
• For electrons
in p-type material
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Chapter 3
Carrier Action
Minority Carrier Lifetime
1
p 
cp NT
1
n 
cn NT
 The minority carrier lifetime τ is the average time for excess
minority carriers to “survive” in a sea of majority carriers.
 The value of τ ranges from 1 ns to 1 ms in Si and depends on
the density of metallic impurities and the density of
crystalline defects.
 The deep traps originated from impurity and defects capture
electrons or holes to facilitate recombination and are called
recombination-generation centers.
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Chapter 3
Carrier Action
Example: Photoconductor
 Consider a sample of Si doped with 1016 cm–3 Boron, with
recombination lifetime 1 μs. It is exposed continuously to light,
such that electron-hole pairs are generated throughout the
sample at the rate of 1020 per cm3 per second, i.e. the
generation rate GL = 1020/cm3/s
a) What are p0 and n0?
p0  1016 cm3
10 2
2
10 

ni
4
3

10
cm

n0 
1016
p0
b) What are Δn and Δp?
p  n  GL   1020 106  1014 cm3
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• Hint: In steady-state,
generation rate equals
recombination rate
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Chapter 3
Carrier Action
Example: Photoconductor
 Consider a sample of Si at 300 K doped with 1016 cm–3 Boron,
with recombination lifetime 1 μs. It is exposed continuously to
light, such that electron-hole pairs are generated throughout
the sample at the rate of 1020 per cm3 per second, i.e. the
generation rate GL = 1020/cm3/s.
c) What are p and n?
p  p0 + p  1016 + 1014  1016 cm 3
n  n0 + n  104 + 1014  1014 cm 3
d) What are np product?
30
3
np  1016 1014  10 cm  ni2 • Note: The np product can be
very different from ni2 in case
of perturbed/agitated
semiconductor
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Chapter 3
Carrier Action
Net Recombination Rate (General Case)
 For arbitrary injection levels and both carrier types in a nondegenerate semiconductor, the net rate of carrier
recombination is:
ni2  np
p
n


t R G
t R G  p (n + n1 ) +  n ( p + p1 )
where n1  ni e( ET Ei ) kT
p1  ni e( Ei ET ) kT
• ET : energy level of R–G center
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Chapter 2
Carrier Action
Homework 3
 1.
(4.27)
Problem 3.12, from (a) until (f), for Figure P3.12(a) and Figure P3.12(f),
Pierret’s “Semiconductor Device Fundamentals”.
 2.
(5.28)
The electron concentration in silicon at T = 300 K is given by
 x 
n( x)  1016 exp   cm3
 18 
where x is measured in μm and is limited to 0 ≤ x ≤ 25 μm. The electron
diffusion coefficient is DN = 25 cm2/s and the electron mobility is μn = 960
cm2/(Vs). The total electron current density through the semiconductor is
constant and equal to JN = –40 A/cm2. The electron current has both
diffusion and drift current components.
Determine the electric field as a function of x which must exist in the
semiconductor. Sketch the function.
 Deadline: 10 February 2011, at 07:30.
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