EE105 Fall 2011
Lecture 3
OUTLINE
• Semiconductor Basics (cont’d)
– Carrier drift and diffusion
• PN Junction Diodes
– Electrostatics
– Capacitance
Reading: Chapter 2.1-2.2
Lecture 3, Slide 1 Prof. Salahuddin, UC Berkeley

Recap: Drift Current
• Drift current is proportional to the carrier velocity and carrier concentration:
Total current J p,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 cm 3 )X Volume=qpAL=qpAv h t
 Hole current per unit area (i.e. current density) J p,drift
= q p v h
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
J tot , drift

 qp
 p
E J n , drift
J p , drift

J n , drift

 qp

 qn (
  n
E ) p
E
 qn
 n
E
J tot , drift
 q ( p
 p
 n
 n
) E
 
E
• The conductivity of a semiconductor is
– Unit: mho/cm
• The resistivity of a semiconductor is
– Unit: ohm-cm



1

 qp
 p
 qn
 n
EE105 Fall 2011 Lecture 3, Slide 3 Prof. Salahuddin, UC Berkeley

Electrical Resistance
V
I _
+
W t homogeneously doped sample
L
EE105 Fall 2011
Resistance R

V
I
 
L
Wt where
 is the resistivity
Lecture 3, Slide 4
(Unit: ohms)
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.
J p
  qD p dp dx
Notation:
D
D p n
 hole diffusion constant (cm 2 /s)
 electron diffusion constant (cm 2 /s)
EE105 Fall 2011 Lecture 3, Slide 8 Prof. Salahuddin, UC Berkeley

Diffusion Examples
• Linear concentration profile
 constant diffusion current p

N

1 x
L
• Non-linear concentration profile
 varying diffusion current p

N exp
 x
L d
EE105 Fall 2011
J p , diff
  qD p dp dx
 qD p
N
L
Lecture 3, Slide 9
J p , diff
  qD p
 dp qD p
N
L d dx exp
 x
L d
Prof. Salahuddin, UC Berkeley

Diffusion Current
• Diffusion current within a semiconductor consists of hole and electron components:
J p , diff
J tot , diff
  qD p dp
J dx n , diff
 q ( D n dn

D p dx dp
) dx
 qD n dn 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 q

26 mV
– 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.
–
V
D
+
I
D
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.
V
D
= 0 V
D
< 0 V
D
> 0
EE105 Fall 2011 Lecture 3, Slide 13 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:
EE105 Fall 2011
Notation: n n
 electron concentration on N-type side (cm -3 ) p p n n p p
 hole concentration on N-type side (cm -3 )
 hole concentration on P-type side (cm -3 )
 electron concentration on P-type side (cm -3 )
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.
Lecture 3, Slide 15 quasineutral region width=W dep quasineutral region
Prof. Salahuddin, UC Berkeley EE105 Fall 2011

Some Important Relations
EE105 Fall 2011

 dE dx



E   dV dx
Energy=-qV
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:
-b
-qN
A

(x) qN
D a
EE105 Fall 2011 x
In the depletion region on the N side : dE dx


 si
 qN
D
 si
E
 qN
D
 si
 x
 b

In the depletion region on the P side : dE dx


 si

 qN
A
 si
E
 qN
 si
A
 a
 x
 aN

A bN
D
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, V
0
• Because there is a large electric field in the depletion region, there is a significant potential drop across this region: qp  p
E  qD p dp dx
 p  p

 dV dx


D p dp dx
   p x
1 x
2  dV  D p
 p p p n
 V ( x
2
)  V ( x
1
)  dp
D p
 p p ln p p p n
 kT q ln  n i
2
N
A
/ N
D

EE105 Fall 2011
V
0
 kT ln q
N
A
N
D n i
2
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 kT ln( 10 )

26 mV

2 .
3

60 mV q
N P
N
D
= 10 18 cm -3 N
A
= 10 15 cm -3
EE105 Fall 2011 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) qN
D a
-b
-qN
A x
I
D
V
0
-b
EE105 Fall 2011
V(x)
0 a x
Lecture 3, Slide 22 Prof. Salahuddin, UC Berkeley