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Lecture 3
OUTLINE
• Semiconductor Basics (cont’d)
– Carrier drift and diffusion
• PN Junction Diodes
– Electrostatics
– Capacitance
Reading: Chapter 2.1-2.2
EE105 Fall 2011
Lecture 3, Slide 1
Prof. Salahuddin, UC Berkeley
Recap: Drift Current
• Drift current is proportional to the carrier velocity
and carrier concentration:
Total current Jp,drift= Q/t
Q= total charge contained in the
volume shown to the right
t= time taken by Q to cross the
volume
Q=qp(in cm3)X Volume=qpAL=qpAvht
 Hole current per unit area (i.e. current density) Jp,drift = q p vh
EE105 Fall 2011
Lecture 3, Slide 2
Prof. Salahuddin, UC Berkeley
Recap: Conductivity and Resistivity
• In a semiconductor, both electrons and holes
conduct current:
J p ,drift  qp p E
J n ,drift  qn(  n E )
J tot,drift  J p ,drift  J n ,drift  qp p E  qn n E
J tot,drift  q( p p  n n ) E  E
• The conductivity of a semiconductor is   qp p  qn n
– Unit: mho/cm
• The resistivity of a semiconductor is  
– Unit: ohm-cm
EE105 Fall 2011
Lecture 3, Slide 3
1

Prof. Salahuddin, UC Berkeley
Electrical Resistance
I
V
+
_
W
t
homogeneously doped sample
L
V
L
Resistance R   
I
Wt
(Unit: ohms)
where  is the resistivity
EE105 Fall 2011
Lecture 3, Slide 4
Prof. Salahuddin, UC Berkeley
Resistivity Example
EE105 Fall 2011
Lecture 3, Slide 5
Prof. Salahuddin, UC Berkeley
A Second Mechanism of Current Flow is Diffusion
EE105 Fall 2011
Lecture 3, Slide 6
Prof. Salahuddin, UC Berkeley
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
EE105 Fall 2011
Lecture 3, Slide 7
Prof. Salahuddin, UC Berkeley
Carrier Diffusion
• Current flow due to mobile charge diffusion is
proportional to the carrier concentration gradient.
– The proportionality constant is the diffusion constant.
dp
J p  qD p
dx
Notation:
Dp  hole diffusion constant (cm2/s)
Dn  electron diffusion constant (cm2/s)
EE105 Fall 2011
Lecture 3, Slide 8
Prof. Salahuddin, UC Berkeley
Diffusion Examples
• Linear concentration profile
 constant diffusion current
• Non-linear concentration profile
 varying diffusion current
 x
p  N 1  
 L
p  N exp
dp
J p ,diff  qD p
dx
N
 qD p
L
EE105 Fall 2011
J p ,diff
Lecture 3, Slide 9
x
Ld
dp
 qD p
dx
qD p N
x

exp
Ld
Ld
Prof. Salahuddin, UC Berkeley
Diffusion Current
• Diffusion current within a semiconductor consists of
hole and electron components:
dp
dn
J p ,diff  qD p
J n ,diff  qDn
dx
dx
dn
dp
J tot,diff  q ( Dn
 Dp )
dx
dx
• The total current flowing in a semiconductor is the
sum of drift current and diffusion current:
J tot  J p ,drift  J n ,drift  J p ,diff  J n ,diff
EE105 Fall 2011
Lecture 3, Slide 10
Prof. Salahuddin, UC Berkeley
The Einstein Relation
• The characteristic constants for drift and diffusion are
related:
D
kT

 q
kT
• Note that
 26mV at room temperature (300K)
q
– This is often referred to as the “thermal voltage”.
EE105 Fall 2011
Lecture 3, Slide 11
Prof. Salahuddin, UC Berkeley
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
EE105 Fall 2011
Lecture 3, Slide 12
Prof. Salahuddin, UC Berkeley
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
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VD < 0
Lecture 3, Slide 13
VD > 0
Prof. Salahuddin, UC Berkeley
Carrier Diffusion across the Junction
• Because of the differences 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)
EE105 Fall 2011
Lecture 3, Slide 14
Prof. Salahuddin, UC Berkeley
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.
quasiquasineutral
neutral
region width=Wdep region
EE105 Fall 2011
Lecture 3, Slide 15
Prof. Salahuddin, UC Berkeley
Some Important Relations
dE 

dx 

dV
E 
dx
Energy=-qV

EE105 Fall 2011
Lecture 3, Slide 16
Prof. Salahuddin, UC Berkeley
The Depletion Approximation
Because charge density ≠ 0 in the depletion region, a large E-field exists in this region:
In the depletion region on the N side:
dE  qN D


dx  si
 si
E
(x)
 si
x  b 
In the depletion region on the P side:
qND
a
-b
-qNA
qN D
x
dE   qN A


dx  si
 si
E
qN A
 si
a  x 
aN A  bN D
EE105 Fall 2011
Lecture 3, Slide 17
Prof. Salahuddin, UC Berkeley
Carrier Drift across the Junction
EE105 Fall 2011
Lecture 3, Slide 18
Prof. Salahuddin, UC Berkeley
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.
J p ,drift   J p ,diff
J n,drift   J n,diff
J tot  J p ,drift  J n,drift  J p ,diff  J n,diff  0
EE105 Fall 2011
Lecture 3, Slide 19
Prof. Salahuddin, UC Berkeley
Built-in Potential, V0
• Because there is a large electric field in the depletion region,
there is a significant potential drop across this region:
dp
qp p E  qDp
dx

 p


pp
x2
 dV D 
p
x1
 dV 
dp
p p 

D

p
 dx 
dx
pn
dp
p
Dp
p p kT
N
V (x 2 )  V (x1 ) 
ln

ln 2 A
 p pn
q n i /N D 
kT N A N D
V0 
ln
2
q
ni
EE105 Fall 2011
Lecture 3, Slide 20
(Unit: Volts)
Prof. Salahuddin, UC Berkeley
Built-In Potential Example
• Estimate the built-in potential for PN junction below.
– Note that
EE105 Fall 2011
kT
ln( 10)  26mV  2.3  60mV
q
N
P
ND = 1018 cm-3
NA = 1015 cm-3
Lecture 3, Slide 21
Prof. Salahuddin, UC Berkeley
PN Junction under Forward Bias
• A forward bias decreases the potential drop across the
junction. As a result, the magnitude of the electric field
decreases and the width of the depletion region narrows.
(x)
qND
a
-b
-qNA
x
ID
V(x)
V0
-b
EE105 Fall 2011
0
a
x
Lecture 3, Slide 22
Prof. Salahuddin, UC Berkeley