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Drift and Diffusion Current
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OUTLINE
Carrier drift and diffusion PN Junction Diodes
Electrostatics
Reading: Chapter 2.1-2.2
Lecture 1, Slide 2
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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 material has 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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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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Carrier Drift
The process in which charged particles move because of 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/V·s)n electron mobility (cm2/V·s)
Hole velocity
Electron velocity
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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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Drift Current Drift current is proportional to the carrier velocity and
carrier concentration:
vh t A = volume from which all holes cross plane in time t
p vh t A = # of holes crossing plane in time t
q p vh t A = charge crossing plane in time t
q p vh A = charge crossing plane per unit time = hole current
Hole current per unit area (i.e. current density) Jp,drift = q p vh
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Conductivity and Resistivity
In a semiconductor, both electrons and holes conduct current:
The conductivity of a semiconductor is Unit: mho/cm
The resistivity of a semiconductor is Unit: ohm-cm
EEnpqJ
EqnEqpJJJ
EqnJEqpJ
npdrifttot
npdriftndriftpdrifttot
ndriftnpdriftp
)(
)(
,
,,,
,,
np qnqp
1
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Resistivity Example Estimate the resistivity of a Si sample doped with phosphorus to
a concentration of 1015 cm-3 and boron to a concentration of 1017 cm-3. The electron mobility and hole mobility are 800 cm2/Vs and 300 cm2/Vs, respectively.
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Electrical Resistance
where is the resistivity
Resistance Wt
L
I
VR (Unit: ohms)
V
+_
L
tW
I
homogeneously doped sample
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Carrier Diffusion
Due to thermally induced random motion, mobile particles tend to move from a region of high concentration to a region of low concentration. Analogy: ink droplet in water
Current flow due to mobile charge diffusion is proportional 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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Diffusion Examples
Non-linear concentration profile varying diffusion current
L
NqD
dx
dpqDJ
p
pdiffp
,
dd
p
pdiffp
L
x
L
NqDdx
dpqDJ
exp
,
• Linear concentration profile constant diffusion current
dL
xNp
exp
L
xNp 1
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Diffusion Current
Diffusion current within a semiconductor consists of hole 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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The Einstein Relation
The characteristic constants for drift and diffusion are related:
Note that at room temperature (300K)
This is often referred to as the “thermal voltage”.
q
kTD
mV26q
kT
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The PN Junction Diode
When a P-type semiconductor region and an N-type semiconductor region are in contact, a PN junction diode is formed.
VD
ID
+–
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Diode Operating Regions
In order to understand the operation of a diode, it is necessary to study its behavior in three operation regions: equilibrium, reverse bias, and forward bias.
VD = 0 VD > 0VD < 0
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Carrier Diffusion across the Junction
Because of the difference in hole and electron concentrations on each side of the junction, carriers diffuse 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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Depletion Region As conduction electrons and holes diffuse across the
junction, 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 recombine with
majority carriers, i.e. they are annihilated.
width=Wdep
quasi-neutral region
quasi-neutral region
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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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PN Junction in Equilibrium
In equilibrium, the drift and diffusion components of current are balanced; therefore the net current flowing across the junction is zero.
diffndriftn
diffpdriftp
JJ
JJ
,,
,,
0,,,, diffndiffpdriftndriftptot JJJJJ
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Built-in Potential, V0
Because of the electric field in the depletion region, there exists a potential drop across the junction:
Di
A
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
ln 20
i
DA
n
NN
q
kTV (Unit: Volts)
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Built-In Potential Example
Estimate the built-in potential for PN junction below.
Note that
N P
ND = 1018 cm-3 NA = 1015 cm-3
mV603.2mV26)10ln( q
kT