Semiconductor Device Physics
Lecture 4
Dr. Gaurav Trivedi,
EEE Department,
IIT Guwahati
Electron kinetic energy
Ec
Ev
Hole kinetic energy
 Ec represents the electron potential energy:
P.E.  Ec  Ereference
Increasing hole energy
Increasing electron energy
Potential vs. Kinetic Energy
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
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”
Diffusion
 Particles diffuse from regions of higher concentration to regions of lower concentration
region, due to random thermal motion (Brownian Motion).
1-D Diffusion Example
 Thermal motion causes particles to
move into an adjacent compartment
every τ seconds.
Diffusion Currents
J N|diff
dn
 qDN
dx
n
J P|diff
dp
 qDP
dx
p
x
Electron flow
Current flow
x
Hole flow
• D is the diffusion coefficient
[cm2/sec]
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.
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
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)
• Under equilibrium, EF inside a
material or a group of materials
in intimate contact is not a
function of position
n  NCe
N C  EF  Ec  kT dEc
dn

e
dx
kT
dx
n dEc

kT dx
dn
q

nE
dx
kT
kT
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.
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
 10.603 cm2 /s
1 eV
1 V
1e
1 eV  1.602 1019 J
• Remark: kT/q = 25.86 mV at
room temperature
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.
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
Recombination Processes
Band-to-Band
R–G Center
Auger
Collision
• Rate is limited by minority
carrier trapping
• Primary recombination way
for Si
• Occurs in heavily
doped material
Direct and Indirect Semiconductors
Ec
Ec
Phonon
Photon
GaAs, GaN
Ev
Photon
Si, Ge
Ev
(direct semiconductors)
(indirect semiconductors)
• Little change in momentum is
required for recombination
• Momentum is conserved by photon
(light) emission
• Large change in momentum is
required for recombination
• Momentum is conserved by mainly
phonon (vibration) emission +
photon emission
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
“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  p0
 For a p-type material
n  p0 ,
p
p0
• Low-level injection condition
 However, the minority carrier concentration can be significantly affected.
n  n0
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
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
n
n
 cn NT n  
t R G
n
where  n 
• For holes in
n-type material
 Similarly,
1
cn NT
• For electrons
in p-type material
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.
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
Net Recombination Rate (General Case)
 For arbitrary injection levels and both carrier types in a non-degenerate 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