MSE 101
SEMICONDUCTOR MATERIALS
& PROCESSES
Motion and Recombination
of Electrons and Holes
Thermal Motion
3
2
1
2
Average electron or hole kinetic energy  kT  mvth2
vth 
3kT

mef f
3 1.38 10 23 JK 1  300K
0.26  9.110 31 kg
 2.3 105 m/s  2.3 107 cm/s
Thermal Motion
• Zig-zag motion is due to collisions or scattering
with imperfections in the crystal.
• Net thermal velocity is zero.
• Mean time between collisions is m ~ 0.1ps
Hot-point Probe can determine sample doing type
Hot-point Probe
distinguishes N
and P type
semiconductors.
Thermoelectric Generator
(from heat to electricity )
and Cooler (from
electricity to refrigeration)
Drift
Electron and Hole Mobilities
• Drift is the motion caused by an electric field.
Electron and Hole Mobilities
m p v  q E mp
v
v   pE
p 
q mp
mp
q E mp
mp
v    nE
q mn
n 
mn
• p is the hole mobility and n is the electron mobility
Electron and Hole Mobilities
v = E ;  has the dimensions of v/E  cm/s  cm2  .


 V/cm
V s 
Electron and hole mobilities of selected
semiconductors
 n (cm2/V∙s)
 p (cm2/V∙s)
Si
1400
Ge
3900
GaAs
8500
InAs
30000
470
1900
400
500
Based on the above table alone, which semiconductor and which carriers
(electrons or holes) are attractive for applications in high-speed devices?
Drift Velocity, Mean Free Time, Mean Free Path
EXAMPLE: Given p = 470 cm2/V·s, what is the hole drift velocity at
E = 103 V/cm? What is mp and what is the distance traveled between
collisions (called the mean free path)? Hint: When in doubt, use the
MKS system of units.
Mechanisms of Carrier Scattering
There are two main causes of carrier scattering:
1. Phonon Scattering
2. Ionized-Impurity (Coulombic) Scattering
Phonon scattering mobility decreases when temperature rises:
 phonon   phonon 
1
1
3 / 2


T
phonon density  carrier thermal velocity T  T 1 / 2
 = q/m
T
vth  T1/2
Impurity (Dopant)-Ion Scattering or Coulombic Scattering
Boron Ion
Electron
_
-
-
+
Electron
Arsenic
Ion
There is less change in the direction of travel if the electron zips by
the ion at a higher speed.
impurity
3
th
3/ 2
v
T


Na  Nd
Na  Nd
Total Mobility
1600
1
1400
2

 phonon
Electrons
1
1000

-1
-1
Mobility (cm V s )
1200

1
800


1
 phonon
1
 impurity

1
 impurity
600
400
Holes
200
0
1E14
1E15
1E16
1E17
1E18
1E19
-3
Na +Concenration
Nd (cm-3)
Total Impurity
(atoms cm )
1E20
Temperature Effect on Mobility
10 1 5
Question:
What Nd will make
dn/dT = 0 at room
temperature?
Velocity Saturation
• When the kinetic energy of a carrier exceeds a critical value, it
generates an optical phonon and loses the kinetic energy.
• Therefore, the kinetic energy is capped at large E, and the
velocity does not rise above a saturation velocity, vsat .
• Velocity saturation has a deleterious effect on device speed as
shown in Ch. 6.
Drift Current and Conductivity
E
unit
area
+
+
Hole current density
Jp
n
Jp = qpv
A/cm2 or C/cm2·sec
EXAMPLE: If p = 1015cm-3 and v = 104 cm/s, then
Jp = ?
2.2.3 Drift Current and Conductivity
Jp,drift = qpv = qppE
Jn,drift = –qnv = qnnE
Jdrift = Jn,drift + Jp,drift =  E =(qnn+qpp)E
 conductivity (1/ohm-cm) of a semiconductor is
 = qnn + qpp
1/ = is resistivity (ohm-cm)
DOPANT DENSITY cm-3
Relationship between Resistivity and Dopant Density
P-type
N-type
RESISTIVITY (cm)
 = 1/
EXAMPLE: Temperature Dependence of Resistance
(a) What is the resistivity () of silicon doped
with 1017cm-3 of arsenic?
(b) What is the resistance (R) of a piece of this
silicon material 1m long and 0.1 m2 in crosssectional area?
Diffusion Current
Particles diffuse from a higher-concentration location
to a lower-concentration location.
Diffusion Current
dn
J n ,diffusion  qDn
dx
dp
J p ,diffusion  qD p
dx
D is called the diffusion constant. Signs explained:
p
n
x
x
Total Current – Review of Four Current Components
JTOTAL = Jn + Jp
dn
Jn = Jn,drift + Jn,diffusion = qnnE + qDn
dx
Jp = Jp,drift + Jp,diffusion = qppE –
dp
qD p
dx
E
–
+
Si
Relation Between the Energy
Diagram
and V, E
(a)
V(x)
0.7eV
0.7V
+
N-
N type Si
–
x
0
(b)
E
Ec and Ev vary in the opposite
direction from the voltage. That
is, Ec and Ev are higher where
the voltage is lower.
dV 1 dEc 1 dEv



E(x)=
dx q dx q dx
E
-
Ec(x)
E cE(x)
f(x)
E f (x)
Ev(x)
E v (x)
0.7V
+
x
x
+
(c)
0.7V
E
Einstein Relationship between
D and 
f
Ec(x)
Consider a piece of non-uniformly doped semiconductor.
n  N ce
N-type
semiconductor
n-type semiconductor
Decreasing donor concentration
 ( Ec  E f ) / kT
dn
N ( E  E ) / kT dEc
 ce c f
dx
kT
dx
n dEc
Ec(x)

kT dx
Ef

Ev(x)
n
qE
kT
Einstein Relationship between D and 
dn
n

qE
dx
kT
dn
 0 at equilibrium.
dx
qDn


E
0 qn nE qn
kT
J n  qn nE  qDn
kT
Dn 
n
q
kT
p
Similarly, Dp 
q
These are known as the Einstein relationship.
EXAMPLE: Diffusion Constant
What is the hole diffusion constant in a piece of
silicon with p = 410 cm2 V-1s-1 ?
Solution:
 kT 
D p     p  (26 mV)  410 cm2 V 1s 1  11 cm2 /s
 q 
Remember: kT/q = 26 mV at room temperature.
Electron-Hole Recombination
•The equilibrium carrier concentrations are denoted with
n0 and p0.
•The total electron and hole concentrations can be different
from n0 and p0 .
•The differences are called the excess carrier
concentrations n’ and p’.
n  n0  n'
p  p0  p'
Charge Neutrality
•Charge neutrality is satisfied at equilibrium (n’=
p’= 0).
• When a non-zero n’ is present, an equal p’ may
be assumed to be present to maintain charge
equality and vice-versa.
•If charge neutrality is not satisfied, the net charge
will attract or repel the (majority) carriers through
the drift current until neutrality is restored.
n'  p'
Recombination Lifetime
•Assume light generates n’ and p’. If the light is
suddenly turned off, n’ and p’ decay with time
until they become zero.
•The process of decay is called recombination.
•The time constant of decay is the recombination
time or carrier lifetime,  .
•Recombination is nature’s way of restoring
equilibrium (n’= p’= 0).
Recombination Lifetime
• ranges from 1ns to 1ms in Si and depends on
the density of metal impurities (contaminants)
such as Au and Pt.
•These deep traps capture electrons and holes to
facilitate recombination and are called
recombination centers.
Ec
Direct
Recombination
is unfavorable in
silicon
Ev
Recombination
centers
Direct and Indirect Band Gap
Trap
Direct band gap
Example: GaAs
Indirect band gap
Example: Si
Direct recombination is efficient
as k conservation is satisfied.
Direct recombination is rare as k
conservation is not satisfied
Modern Semiconductor Devices for Integrated Circuits (C. Hu)
Rate of recombination (s-1cm-3)
dn
n

dt

n  p
dn
n
p dp
  
dt


dt
EXAMPLE: Photoconductors
A bar of Si is doped with boron at 1015cm-3. It is
exposed to light such that electron-hole pairs are
generated throughout the volume of the bar at the
rate of 1020/s·cm3. The recombination lifetime is
10s. What are (a) p0 , (b) n0 , (c) p’, (d) n’, (e) p ,
(f) n, and (g) the np product?
Chapter Summary
v p   pE
vn  -  nE
J p ,drift  qp pE
J n ,drift  qn nE
dn
J n ,diffusion  qDn
dx
dp
J p ,diffusion  qD p
dx
kT
Dn 
n
q
kT
Dp 
p
q
Chapter Summary
 is the recombination lifetime.
n’ and p’ are the excess carrier concentrations.
n = n0+ n’
p = p0+ p’
Charge neutrality requires n’= p’.
rate of recombination = n’/ = p’/
Efn and Efp are the quasi-Fermi levels of electrons and
holes.
 ( E  E ) / kT
n  N c e c fn
 ( E  E ) / kT
p  N v e fp v