drift and diffusion current
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EE105 Fall 2007 Lecture 2, Slide 1 Prof. Liu, UC Berkeley
Announcements
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Lecture 1, Slide 1
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EE105 Fall 2007 Lecture 2, Slide 2 Prof. Liu, UC Berkeley
Lecture 2
OUTLINE
Basic Semiconductor Physics (contd)
Carrier drift and diffusion
PN Junction Diodes
Electrostatics
Capacitance
Reading: Chapter 2.1-2.2
Lecture 1, Slide 2
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EE105 Fall 2007 Lecture 2, Slide 3 Prof. Liu, UC Berkeley
Dopant Compensation
An N-type semiconductor can be converted into P-type material by counter-doping it with acceptors
such that NA > ND.
A compensated semiconductor materialhas both
acceptors and donors.
P-type material
(NA > ND)
DA
i
DA
NN
nn
NNp
2
AD
i
AD
NN
np
NNn
2
N-type material
(ND > NA)
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EE105 Fall 2007 Lecture 2, Slide 4 Prof. Liu, UC Berkeley
Types of Charge in a Semiconductor
Negative charges: Conduction electrons (density = n)
Ionized acceptor atoms (density = NA)
Positive charges:
Holes (density =p)
Ionized donor atoms (density = ND)
The net charge density (C/cm3) in a semiconductor is
AD NNnpq
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EE105 Fall 2007 Lecture 2, Slide 5 Prof. Liu, UC Berkeley
Carrier Drift
The process in which charged particles move becauseof an electric field is called drift.
Charged particles within a semiconductor move with
an average velocity proportional to the electric field.
The proportionality constant is the carrier mobility.
Ev
Ev
ne
ph
Notation:
p hole mobility (cm2/Vs)n electron mobility (cm2/Vs)
Hole velocity
Electron velocity
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EE105 Fall 2007 Lecture 2, Slide 6 Prof. Liu, UC Berkeley
Velocity Saturation
In reality, carrier velocities saturate at an upper limit,called the saturation velocity(vsat).
E
vE
v
bv
bE
sat
sat
0
0
0
0
1
1
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EE105 Fall 2007 Lecture 2, Slide 7 Prof. Liu, UC Berkeley
Drift Current
Drift current is proportional to the carrier velocityand carrier concentration:
vhtA = volume from which all holes cross plane in time t
pvhtA = # of holes crossing plane in time t
q pvhtA = charge crossing plane in time t
qpvhA = charge crossing plane per unit time = hole current
Hole current per unit area (i.e. current density) Jp,drift = qpvh
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EE105 Fall 2007 Lecture 2, Slide 8 Prof. Liu, UC Berkeley
Conductivity and Resistivity
In a semiconductor, both electrons and holesconduct current:
The conductivityof a semiconductor is Unit: mho/cm
The resistivityof a semiconductor is Unit: ohm-cm
EEnpqJ
EqnEqpJJJ
EqnJEqpJ
npdrifttot
npdriftndriftpdrifttot
ndriftnpdriftp
)(
)(
,
,,,
,,
np qnqp
1
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EE105 Fall 2007 Lecture 2, Slide 9 Prof. Liu, UC Berkeley
Resistivity Example
Estimate the resistivity of a Si sample doped withphosphorus to a concentration of 1015 cm-3 andboron to a concentration of 1017 cm-3. The electronmobility and hole mobility are 700 cm2/Vs and 300
cm
2
/Vs, respectively.
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EE105 Fall 2007 Lecture 2, Slide 10 Prof. Liu, UC Berkeley
Electrical Resistance
where is the resistivity
ResistanceWt
L
I
VR (Unit: ohms)
V
+ _
L
tW
I
homogeneously doped sample
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EE105 Fall 2007 Lecture 2, Slide 11 Prof. Liu, UC Berkeley
Carrier Diffusion
Due to thermally induced random motion, mobileparticles tend to move from a region of highconcentration to a region of low concentration.
Analogy: ink droplet in water
Current flow due to mobile charge diffusion isproportional to the carrier concentration gradient.
The proportionality constant is the diffusion constant.
dx
dpqDJ pp
Notation:
Dp hole diffusion constant (cm2/s)
Dn electron diffusion constant (cm2/s)
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EE105 Fall 2007 Lecture 2, Slide 12 Prof. Liu, UC Berkeley
Diffusion Examples
Non-linear concentration profile varying diffusion current
L
NqD
dx
dp
qDJ
p
pdiffp
,
dd
p
pdiffp
L
x
L
NqD
dx
dp
qDJ
exp
,
Linear concentration profile constant diffusion current
dL
xNp
exp
L
xNp 1
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EE105 Fall 2007 Lecture 2, Slide 13 Prof. Liu, UC Berkeley
Diffusion Current
Diffusion current within a semiconductor consists ofhole and electron components:
The total current flowing in a semiconductor is the
sum of drift current and diffusion current:
)(
,
,,
dx
dpD
dx
dnDqJ
dx
dnqDJ
dx
dpqDJ
pndifftot
ndiffnpdiffp
diffndiffpdriftndriftptot JJJJJ ,,,,
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EE105 Fall 2007 Lecture 2, Slide 14 Prof. Liu, UC Berkeley
The Einstein Relation
The characteristic constants for drift and diffusion arerelated:
Note that at room temperature (300K)
This is often referred to as the thermal voltage.
q
kTD
mV26q
kT
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EE105 Fall 2007 Lecture 2, Slide 15 Prof. Liu, UC Berkeley
The PN Junction Diode
When a P-type semiconductor region and an N-typesemiconductor region are in contact, a PN junction
diode is formed.VD
ID
+
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EE105 Fall 2007 Lecture 2, Slide 16 Prof. Liu, UC Berkeley
Diode Operating Regions
In order to understand the operation of a diode, it isnecessary to study its behavior in three operation
regions: equilibrium, reverse bias, and forward bias.
VD = 0 VD > 0VD < 0
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EE105 Fall 2007 Lecture 2, Slide 17 Prof. Liu, UC Berkeley
Carrier Diffusion across the Junction
Because of the difference in hole and electronconcentrations on each side of the junction, carriersdiffuse across the junction:
Notation:
nn electron concentration on N-type side (cm-3)
pn hole concentration on N-type side (cm-3)
pp hole concentration on P-type side (cm-3)
np electron concentration on P-type side (cm-3)
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EE105 Fall 2007 Lecture 2, Slide 18 Prof. Liu, UC Berkeley
Depletion Region
As conduction electrons and holes diffuse across thejunction, they leave behind ionized dopants. Thus, a
region that is depleted of mobile carriers is formed.
The charge density in the depletion region is not zero.
The carriers which diffuse across the junction recombinewith majority carriers, i.e. they are annihilated.
width=Wdep
quasi-
neutral
region
quasi-
neutral
region
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EE105 Fall 2007 Lecture 2, Slide 19 Prof. Liu, UC Berkeley
Carrier Drift across the Junction
Because charge density 0 in the depletion region,an electric field exists, hence there is drift current.
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EE105 Fall 2007 Lecture 2, Slide 20 Prof. Liu, UC Berkeley
PN Junction in Equilibrium
In equilibrium, the drift and diffusion components ofcurrent are balanced; therefore the net current
flowing across the junction is zero.
diffndriftn
diffpdriftp
JJ
JJ
,,
,,
0,,,, diffndiffpdriftndriftptot JJJJJ
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EE105 Fall 2007 Lecture 2, Slide 21 Prof. Liu, UC Berkeley
Built-in Potential, V0
Because of the electric field in the depletion region,there exists a potential drop across the junction:
DiA
n
p
p
p
p
p
p
x
x
p
pppp
Nn
N
q
kT
p
pDxVxV
p
dpDdV
dx
dpD
dx
dVp
dx
dpqDEqp
p
n
/
lnln)()(221
2
1
ln20
i
DA
n
NN
q
kTV (Unit: Volts)
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EE105 Fall 2007 Lecture 2, Slide 22 Prof. Liu, UC Berkeley
Built-In Potential Example
Estimate the built-in potential for PN junction below.
Note that
N P
ND = 1018 cm-3 NA = 10
15 cm-3
mV603.2mV26)10ln( q
kT
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EE105 Fall 2007 Lecture 2, Slide 23 Prof. Liu, UC Berkeley
PN Junction under Reverse Bias
A reverse bias increases the potential drop across thejunction. As a result, the magnitude of the electric fieldincreases and the width of the depletion region widens.
112
0 R
DA
si
dep
VVNNq
W
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EE105 Fall 2007 Lecture 2, Slide 24 Prof. Liu, UC Berkeley
Diode Current under Reverse Bias
In equilibrium, the built-in potential effectivelyprevents carriers from diffusing across the junction.
Under reverse bias, the potential drop across the
junction increases; therefore, negligible diffusion
current flows. A very small drift current flows, limitedby the rate at which minority carriers diffuse from the
quasi-neutral regions into the depletion region.
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EE105 Fall 2007 Lecture 2, Slide 25 Prof. Liu, UC Berkeley
PN Junction Capacitance
A reverse-biased PN junction can be viewed as acapacitor. The depletion width (Wdep) and hence the
junction capacitance (Cj) varies with VR.
dep
sij
WC
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EE105 Fall 2007 Lecture 2, Slide 26 Prof. Liu, UC Berkeley
Voltage-Dependent Capacitance
si 10-12 F/cm is the permittivity of silicon.
0
0
0
0
1
2
1
VNN
NNqC
V
V
CC
DA
DAsij
R
j
j
VD
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EE105 Fall 2007 Lecture 2, Slide 27 Prof. Liu, UC Berkeley
Reverse-Biased Diode Application
A very important application of a reverse-biased PNjunction is in a voltage controlled oscillator (VCO),
which uses an LC tank. By changing VR, we can
change C, which changes the oscillation frequency.
LC
fres
1
2
1
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EE105 Fall 2007 Lecture 2, Slide 28 Prof. Liu, UC Berkeley
Summary
Current flowing in a semiconductor is comprised of drift
and diffusion components:
A region depleted of mobile charge exists at the junction
between P-type and N-type materials.
A built-in potential drop (V0) across this region is establishedby the charge density profile; it opposes diffusion of carriers
across the junction. A reverse bias voltage serves to enhance
the potential drop across the depletion region, resulting in
very little (drift) current flowing across the junction.
The width of the depletion region (Wdep) is a function of the
bias voltage (VD).
dxdpqD
dxdnqDEqnEqpJ pnnptot
ln20
i
DA
n
NN
q
kTV
112 0 D
DA
sidep VV
NNqW
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