Lecture #6
OUTLINE
• Carrier scattering mechanisms
• Drift current
• Conductivity and resistivity
• Relationship between band diagrams & V,
Read: Section 3.1
e
Mechanisms of Carrier Scattering
Dominant scattering mechanisms:
1. Phonon scattering (lattice scattering)
2. Impurity (dopant) ion scattering
Phonon scattering mobility decreases when T increases:
 phonon   phonon 
1
1
3 / 2


T
phonon density  carrier th ermal velocity T  T 1/ 2
 = q / m
Spring 2007
vth 
EE130 Lecture 6, Slide 2
T
Impurity Ion Scattering
-
Boron
_ Ion
Electron
-
+
Electron
Arsenic
Ion
There is less change in the electron’s direction of travel
if the electron zips by the ion at a higher speed.
3
vth
T 3/ 2
impurity 

N A  ND
N A  ND
Spring 2007
EE130 Lecture 6, Slide 3
Matthiessen's Rule
• The probability that a carrier will be scattered by
mechanism i within a time period dt is
dt
i
where i is the mean time between scattering events
due to mechanism i
 The probability that a carrier will be scattered within
a time period dt is
dt

i
1
i


Spring 2007
EE130 Lecture 6, Slide 4
1



1

1
 phonon  impurity
1
 phonon

1
impurity
Mobility Dependence on Doping
1600
1400
Electrons
1000
2
-1
-1
Mobility (cm V s )
1200
800
600
400
Holes
200
0
1E14
1E15
1E16
1E17
1E18
1E19
-3
Total
Impurity
ConcenrationN(atoms
cm -3))
Total
Doping
Concentration
A + ND (cm
Spring 2007
EE130 Lecture 6, Slide 5
1E20
Temperature Effect on Mobility
1

1

Spring 2007
EE130 Lecture 6, Slide 6


1
 phonon

1
 phonon
1
 impurity

1
 impurity
Drift Current
vd t A = volume from which all holes cross plane in time t
p vd t A = # of holes crossing plane in time t
q p vd t A = charge crossing plane in time t
q p vd A = charge crossing plane per unit time = hole current
 Hole current per unit area J = q p vd
Spring 2007
EE130 Lecture 6, Slide 7
Conductivity and Resistivity
Jn,drift = –qnvdn = qnne
Jp,drift = qpvdn = qppe
Jdrift = Jn,drift + Jp,drift = e =(qnn+qpp)e
Conductivity of a semiconductor is   qnn + qpp
Resistivity   1 / 
Spring 2007
(Unit: ohm-cm)
EE130 Lecture 6, Slide 8
Resistivity Dependence on Doping
For n-type material:
1

qn n
p-type
For p-type material:
1

qp p
n-type
Note: This plot does not apply
for compensated material!
Spring 2007
EE130 Lecture 6, Slide 9
Electrical Resistance
I
V
+
_
W
t
homogeneously doped sample
L
V
L
Resistance R   
I
Wt
where  is the resistivity
Spring 2007
EE130 Lecture 6, Slide 10
(Unit: ohms)
Example
Consider a Si sample doped with 1016/cm3 Boron.
What is its resistivity?
Answer:
NA = 1016/cm3 , ND = 0
(NA >> ND  p-type)
 p  1016/cm3 and n  104/cm3
1
1


qn n  qp p qp p

 (1.6 10
Spring 2007
19
16
)(10 )( 450)
EE130 Lecture 6, Slide 11

1
 1.4   cm
Example: Dopant Compensation
Consider the same Si sample, doped additionally
with 1017/cm3 Arsenic. What is its resistivity?
Answer:
NA = 1016/cm3, ND = 1017/cm3 (ND>>NA  n-type)
 n  9x1016/cm3 and p  1.1x103/cm3
1
1


qn n  qp p qn n

 (1.6  10
Spring 2007
19
)(9 10 )(600)
16
EE130 Lecture 6, Slide 12

1
 0.12   cm
Example: Temperature Dependence of 
Consider a Si sample doped with 1017cm-3 As.
How will its resistivity change when the temperature is
increased from T=300K to T=400K?
Solution:
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 300K to
400 at 400K. As a result,  increases by
770
 1.93
400
Spring 2007
EE130 Lecture 6, Slide 13
electron kinetic energy
Ec
hole kinetic energy
Ev
increasing hole energy
increasing electron energy
Potential vs. Kinetic Energy
Ec represents the electron potential energy:
P.E.  Ec  Ereference
Spring 2007
EE130 Lecture 6, Slide 14
E
+
–
Electrostatic Potential, V
Si
(a)
V(x)
0.7V
0.7V
E
+
–
N-Si
x
0
(b)
(a)
E
V(x)
-
Ec(x)
Ef(x)
• The
potential energy of a particle with charge -q is
0.7V
x
related
to the electrostatic
potential V(x):
0
(b)
E
-
P.E.  qV
Ec(x)
Ef(x)
x
1
V  ( Ereference  Ec )
q
Ev(x)
0.7V
+
x
Spring 2007
(c)
EE130 Lecture 6, Slide 15
Ev(x)
0.7V
+
(c)
0.7V
Electric Field,
+
e
E
–
Si
(a)
0.7V
V(x)
0.7V
E
+
–
N-Si
x
0
(b)
(a)
E
V(x)
-
e
0.7V
0
(b)
dV 1 dEc


dx q dx
Ec(x)
Ef(x)
Ev(x)
x
0.7V
+
x
E
-
Ec(x)
Ef(x)
(c)
• Variation of Ec with position is called “band bending.”
Ev(x)
0.7V
+
x
Spring 2007
(c)
EE130 Lecture 6, Slide 16
Carrier Drift (Band Diagram Visualization)
Ec
Ev
Spring 2007
EE130 Lecture 6, Slide 17
Summary
• Carrier mobility varies with doping
– decreases w/ increasing total concentration of ionized dopants
• Carrier mobility varies with temperature
– decreases w/ increasing T if lattice scattering is dominant
– decreases w/ decreasing T if impurity scattering is dominant
• The conductivity of a semiconductor is dependent on
the carrier concentrations and mobilities
 = qnn + qpp
• Ec represents the electron potential energy
Variation in Ec(x)  variation in electric potential V
dEc dEv
e  dx  dx
• E - Ec represents the electron kinetic energy
Electric field
Spring 2007
EE130 Lecture 6, Slide 18