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– If a section is over-subscribed, then priority will be given to
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EE105 Fall 2007
Lecture 1,
2, Slide 1
Prof. Liu, UC Berkeley
Lecture 2
OUTLINE
• Basic Semiconductor Physics (cont’d)
– Carrier drift and diffusion
• PN Junction Diodes
– Electrostatics
– Capacitance
Reading: Chapter 2.1-2.2
EE105 Fall 2007
Lecture 1,
2, Slide 2
Prof. Liu, UC Berkeley
Dopant Compensation
• An N-type semiconductor can be converted into Ptype material by counter-doping it with acceptors
such that NA > ND.
• A compensated semiconductor material has both
acceptors and donors.
N-type material
(ND > NA)
P-type material
(NA > ND)
n  ND  N A
p  N A  ND
2
2
ni
p
ND  N A
EE105 Fall 2007
ni
n
N A  ND
Lecture 2, Slide 3
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
  q p  n  N D  N A 
EE105 Fall 2007
Lecture 2, Slide 4
Prof. Liu, UC Berkeley
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.

Hole velocity
vh   p E

Electron velocity


ve    n E
Notation:
p  hole mobility (cm2/V·s)
n  electron mobility (cm2/V·s)
EE105 Fall 2007
Lecture 2, Slide 5
Prof. Liu, UC Berkeley
Velocity Saturation
• In reality, carrier velocities saturate at an upper limit,
called the saturation velocity (vsat).

0
1  bE
vsat 
0
b
v 
1
EE105 Fall 2007
Lecture 2, Slide 6
0
E
0 E
vsat
Prof. Liu, UC Berkeley
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
EE105 Fall 2007
Lecture 2, Slide 7
Prof. Liu, UC Berkeley
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 2007
Lecture 2, Slide 8
1

Prof. Liu, UC Berkeley
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 700 cm2/Vs and 300
cm2/Vs, respectively.
EE105 Fall 2007
Lecture 2, Slide 9
Prof. Liu, 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 2007
Lecture 2, Slide 10
Prof. Liu, 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
• 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 2007
Lecture 2, Slide 11
Prof. Liu, 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 2007
J p ,diff
Lecture 2, Slide 12
x
Ld
dp
 qD p
dx
qD p N
x

exp
Ld
Ld
Prof. Liu, 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 2007
Lecture 2, Slide 13
Prof. Liu, 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 2007
Lecture 2, Slide 14
Prof. Liu, 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 2007
Lecture 2, Slide 15
Prof. Liu, 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
EE105 Fall 2007
VD < 0
Lecture 2, Slide 16
VD > 0
Prof. Liu, UC Berkeley
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)
EE105 Fall 2007
Lecture 2, Slide 17
Prof. Liu, 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 2007
Lecture 2, Slide 18
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.
EE105 Fall 2007
Lecture 2, Slide 19
Prof. Liu, 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 2007
Lecture 2, Slide 20
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:
qp p E  qD p
x2
dp
dx
   p  dV D p
x1
 dV
p p  
 dx

dp


D

p
dx

pp
dp
p p
n
 V ( x1 )  V ( x2 ) 
Dp
p
ln
pp
pn

kT
N
ln 2 A
q
ni / N D

kT N A N D
V0 
ln
2
q
ni
EE105 Fall 2007
Lecture 2, Slide 21

(Unit: Volts)
Prof. Liu, UC Berkeley
Built-In Potential Example
• Estimate the built-in potential for PN junction below.
– Note that
EE105 Fall 2007
kT
ln( 10)  26mV  2.3  60mV
q
N
P
ND = 1018 cm-3
NA = 1015 cm-3
Lecture 2, Slide 22
Prof. Liu, UC Berkeley
PN Junction under Reverse Bias
• A reverse bias increases the potential drop across the
junction. As a result, the magnitude of the electric field
increases and the width of the depletion region widens.
Wdep
2 si  1
1 

V0  VR 


q  N A ND 
EE105 Fall 2007
Lecture 2, Slide 23
Prof. Liu, UC Berkeley
Diode Current under Reverse Bias
• In equilibrium, the built-in potential effectively
prevents 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, limited
by the rate at which minority carriers diffuse from the
quasi-neutral regions into the depletion region.
EE105 Fall 2007
Lecture 2, Slide 24
Prof. Liu, UC Berkeley
PN Junction Capacitance
• A reverse-biased PN junction can be viewed as a
capacitor. The depletion width (Wdep) and hence the
junction capacitance (Cj) varies with VR.
Cj 
EE105 Fall 2007
Lecture 2, Slide 25
 si
Wdep
Prof. Liu, UC Berkeley
Voltage-Dependent Capacitance
Cj 
VD
C j0 
C j0
VR
1
V0
 si q N A N D
1
2 N A  N D V0
si  10-12 F/cm is the permittivity of silicon.
EE105 Fall 2007
Lecture 2, Slide 26
Prof. Liu, UC Berkeley
Reverse-Biased Diode Application
• A very important application of a reverse-biased PN
junction is in a voltage controlled oscillator (VCO),
which uses an LC tank. By changing VR, we can
change C, which changes the oscillation frequency.
f res
EE105 Fall 2007
Lecture 2, Slide 27
1

2
1
LC
Prof. Liu, UC Berkeley
Summary
• Current flowing in a semiconductor is comprised of drift
dn
dp
J

qp

E

qn

E

qD

qD
and diffusion components: tot
p
n
n
p
dx
dx
• 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 established
by 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).
Wdep
EE105 Fall 2007
2 si  1
1 

V0  VD 



q  N A ND 
Lecture 2, Slide 28
V0 
kT N A N D
ln
2
q
ni
Prof. Liu, UC Berkeley