Lecture #6
OUTLINE
• Carrier scattering mechanisms
• Drift current
• Conductivity and resistivity
• Relationship between band diagrams & V, Read: Section 3.1
EE130 Lecture 6, Slide 2Spring 2007
Dominant scattering mechanisms:
1. Phonon scattering (lattice scattering)
2. Impurity (dopant) ion scattering
2/32/1
1
velocityermalcarrier thdensityphonon
1
TTTphononphonon
Phonon scattering mobility decreases when T increases:
= q / m
Mechanisms of Carrier Scattering
Tvth
EE130 Lecture 6, Slide 3Spring 2007
_+
- -Electron
Boron Ion Electron
Arsenic Ion
DADA
thimpurity NN
T
NN
v
2/33
There is less change in the electron’s direction of travel if the electron zips by the ion at a higher speed.
Impurity Ion Scattering
EE130 Lecture 6, Slide 4Spring 2007
Matthiessen's Rule
• The probability that a carrier will be scattered by
mechanism i within a time period dt is
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
impurityphonon
impurityphonon
111
111
i i
dt
i
dt
EE130 Lecture 6, Slide 5Spring 2007
1E14 1E15 1E16 1E17 1E18 1E19 1E20
0
200
400
600
800
1000
1200
1400
1600
Holes
Electrons
Mo
bili
ty (
cm2 V
-1 s
-1)
Total Impurity Concenration (atoms cm-3)Total Doping Concentration NA + ND (cm-3)
Mobility Dependence on Doping
EE130 Lecture 6, Slide 6Spring 2007
Temperature Effect on Mobility
impurityphonon
impurityphonon
111
111
EE130 Lecture 6, Slide 7Spring 2007
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
Drift Current
EE130 Lecture 6, Slide 8Spring 2007
Jp,drift = qpvdn = qppJn,drift = –qnvdn = qnn
Jdrift = Jn,drift + Jp,drift = =(qnn+qpp)
Conductivity of a semiconductor is qnn + qpp
Resistivity 1 /
Conductivity and Resistivity
(Unit: ohm-cm)
EE130 Lecture 6, Slide 9Spring 2007
n-type
p-type
Resistivity Dependence on Doping
For n-type material:
nqn 1
For p-type material:
pqp 1
Note: This plot does not apply for compensated material!
EE130 Lecture 6, Slide 10Spring 2007
Electrical Resistance
where is the resistivity
Resistance Wt
L
I
VR (Unit: ohms)
V
+_
L
tW
I
homogeneously doped sample
EE130 Lecture 6, Slide 11Spring 2007
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
Example
cm 4.1)450)(10)(106.1(
11
11619
ppn qpqpqn
EE130 Lecture 6, Slide 12Spring 2007
Example: Dopant Compensation
Consider the same Si sample, doped additionallywith 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
cm 12.0)600)(109)(106.1(
11
11619
npn qnqpqn
EE130 Lecture 6, Slide 13Spring 2007
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
Example: Temperature Dependence of
93.1400
770
EE130 Lecture 6, Slide 14Spring 2007
Potential vs. Kinetic Energy
electron kinetic energyin
crea
sing
ele
ctro
n en
ergy
Ec
Evhole kinetic energy
incr
easi
ng h
ole
ener
gy
referencecP.E. EE Ec represents the electron potential energy:
EE130 Lecture 6, Slide 15Spring 2007
N-+ –
0 .7 V
Si
E
x
Ec(x)
Ef(x)
Ev(x)
E
0.7 V
-
+
V(x)
0.7 V
x0
(a)
(b )
(c)
+ –
0 .7 V
Si
E
x
Ec(x)
Ef(x)
Ev(x)
E
0.7 V
-
+
V(x)
0.7 V
x0
(a)
(b )
(c)
Electrostatic Potential, V
• The potential energy of a particle with charge -q is related to the electrostatic potential V(x):
)(1
creference EEq
V
qVP.E.
EE130 Lecture 6, Slide 16Spring 2007
N-+ –
0 .7 V
Si
E
x
Ec(x)
Ef(x)
Ev(x)
E
0.7 V
-
+
V(x)
0.7 V
x0
(a)
(b )
(c)
+ –
0 .7 V
Si
E
x
Ec(x)
Ef(x)
Ev(x)
E
0.7 V
-
+
V(x)
0.7 V
x0
(a)
(b )
(c)
Electric Field,
dx
dE
qdx
dV c1
• Variation of Ec with position is called “band bending.”
EE130 Lecture 6, Slide 17Spring 2007
Carrier Drift (Band Diagram Visualization)
Ec
Ev
EE130 Lecture 6, Slide 18Spring 2007
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
• Ec represents the electron potential energy
Variation in Ec(x) variation in electric potential V
Electric field
• E - Ec represents the electron kinetic energy
= qnn + qpp
dx
dE
dx
dE vc
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