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Lecture 4
OUTLINE
• PN Junction Diodes
– Electrostatics
– Capacitance
– I/V
– Reverse Breakdown
– Large and Small signal models
Reading: Chapter 2.2-2.3,3.2-3.4
EE105 Fall 2010
Lecture 4, Slide 1
Prof. Salahuddin, UC Berkeley
Energy Band Description
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Lecture 4, Slide 2
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PN Junction under Reverse Bias
• A reverse bias increases the potential drop across the
junction. As a result, the magnitude of the electric field
in the depletion region increases and the width of the
depletion region widens.
EE105 Fall 2011
Lecture 4, Slide 3
Prof. Salahuddin, UC Berkeley
Energy Band Description
EE105 Fall 2011
Lecture 4, Slide 4
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I-V characteristic from energy
band description
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Lecture 4, Slide 5
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Mathematical description of current flow in a p-n
junction diode
EE105 Fall 2011
Lecture 4, Slide 6
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Minority Carrier Injection under Forward Bias
• The potential barrier to carrier diffusion is decreased by
a forward bias; thus, carriers diffuse across the junction.
– The carriers which diffuse across the junction become minority
carriers in the quasi-neutral regions; they recombine with
majority carriers, “dying out” with distance.
np(x)
np0
edge of depletion region
x'
0
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x'
Equilbrium concentration n
of electrons on the P side: p 0
Lecture 4, Slide 7
ni2

NA
Prof. Salahuddin, UC Berkeley
Minority Carrier Concentrations
at the Edges of the Depletion Region
• The minority-carrier concentrations at the edges of
qV / kT
V
e

e
the depletion region are changed by the factor
D
D
/ VT
– There is an excess concentration (Dpn, Dnp) of minority
carriers in the quasi-neutral regions, under forward bias.
• Within the quasi-neutral regions, the excess minoritycarrier concentrations decay exponentially with
distance from the depletion region, to zero:
n p ( x)  n p 0  Dn p ( x)
Dn p ( x) 
2
i

VD / VT
n e
NA
e
1
Notation:
Ln  electron diffusion length (cm)
 x / Ln
J n,diff


dn p qDn ni2 VD /VT
 qDn

e
 1 e  x / Ln
dx
N A Ln
x'
EE105 Fall 2011
Lecture 4, Slide 8
Prof. Salahuddin, UC Berkeley
Diode Current under Forward Bias
• The current flowing across the junction is comprised
of hole diffusion and electron diffusion components:
J tot  J p,drift
x 0
 J n,drift
x 0
 J p,diff
x 0
 J n,diff
x 0
• Assuming that the diffusion current components are
constant within the depletion region (i.e. no
recombination occurs in the depletion region):
J n,diff
x 0


J tot  J S e
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
qDn ni2 VD /VT

e
1
N A Ln
VD / VT
J p ,diff
x 0

qDp ni2
N D Lp
e
VD / VT

1
 Dn
Dp 

 1 where J S  qn 

N L N L 
D p 
 A n

2
i
Lecture 4, Slide 9
Prof. Salahuddin, UC Berkeley
Current Components under Forward Bias
• For a fixed bias voltage, Jtot is constant throughout
the diode, but Jn(x) and Jp(x) vary with position.
Jtot
-b
EE105 Fall 2011
Lecture 4, Slide 10
0
a
x
Prof. Salahuddin, UC Berkeley
I-V Characteristic of a PN Junction
• Current increases exponentially with applied forward
bias voltage, and “saturates” at a relatively small
negative current level for reverse bias voltages.
“Ideal diode” equation:


I D  I S eVD / VT  1
 Dn
Dp 

I S  AJ S  Aqn 

N L N L 
D p 
 A n
2
i
EE105 Fall 2011
Lecture 4, Slide 11
Prof. Salahuddin, UC Berkeley
Practical PN Junctions
• Typically, pn junctions in IC devices are formed by
counter-doping. The equations provided in class (and
in the textbook) can be readily applied to such diodes if
– NA  net acceptor doping on p-side (NA-ND)p-side
– ND  net donor doping on n-side (ND-NA)n-side
I D  I S (e qVD
kT
 1)
ID (A)
 Dn
Dp 

I S  Aqni 

L N

L
N
n
A
p
D


2
VD (V)
EE105 Fall 2011
Lecture 4, Slide 12
Prof. Salahuddin, UC Berkeley
Parallel PN Junctions
• Since the current flowing across a PN junction is
proportional to its cross-sectional area, two identical
PN junctions connected in parallel act effectively as a
single PN junction with twice the cross-sectional
area, hence twice the current.
EE105 Fall 2011
Lecture 4, Slide 13
Prof. Salahuddin, UC Berkeley
Diode Saturation Current IS
 Dn
Dp 

I S  Aqni 

L N

L
N
n
A
p
D


2
• IS can vary by orders of magnitude, depending on the diode
area, semiconductor material, and net dopant concentrations.
– typical range of values for Si PN diodes: 10-14 to 10-17 A/mm2
• In an asymmetrically doped PN junction, the term associated
with the more heavily doped side is negligible:
 Dp 

– If the P side is much more heavily doped, I S  Aqni 

L
N
 p D
2
 Dn 

– If the N side is much more heavily doped, I S  Aqni 
 Ln N A 
2
EE105 Fall 2011
Lecture 4, Slide 14
Prof. Salahuddin, 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
in the depletion region increases and the width of the
depletion region widens.
Wdep
2 si  1
1 

V0  VR 


q  N A ND 
EE105 Fall 2011
Lecture 4, Slide 15
Prof. Salahuddin, UC Berkeley
PN Junction Small-Signal Capacitance
• A reverse-biased PN junction can be viewed as a
capacitor, for incremental changes in applied voltage.
Cj 
EE105 Fall 2011
Lecture 4, Slide 16
 si
Wdep
Prof. Salahuddin, UC Berkeley
Voltage-Dependent Capacitance
• The depletion width (Wdep) and hence the junction
capacitance (Cj) varies with VR.
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 2011
Lecture 4, Slide 17
Prof. Salahuddin, 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 2011
Lecture 4, Slide 18
1

2
1
LC
Prof. Salahuddin, UC Berkeley
Reverse Breakdown
• As the reverse bias voltage increases, the electric
field in the depletion region increases. Eventually, it
can become large enough to cause the junction to
break down so that a large reverse current flows:
breakdown voltage
EE105 Fall 2011
Lecture 4, Slide 19
Prof. Salahuddin, UC Berkeley
Reverse Breakdown Mechanisms
a) Zener breakdown occurs when the electric field is
sufficiently high to pull an electron out of a covalent
bond (to generate an electron-hole pair).
b) Avalanche breakdown occurs when electrons and holes
gain sufficient kinetic energy (due to acceleration by the
E-field) in-between scattering events to cause electronhole pair generation upon colliding with the lattice.
EE105 Fall 2011
Lecture 4, Slide 20
Prof. Salahuddin, UC Berkeley
Constant-Voltage Diode Model
for Large-Signal Analysis
• If VD < VD,on: The diode operates as an open circuit.
• If VD  VD,on: The diode operates as a constant voltage
source with value VD,on.
EE105 Fall 2011
Lecture 4, Slide 21
Prof. Salahuddin, UC Berkeley
Example: Diode DC Bias Calculations
IX
VX  I X R1  VD  I X R1  VT ln
IS
I X  2.2mA for VX  3V
I X  0.2mA for VX  1V
• This example shows the simplicity provided by a
constant-voltage model over an exponential model.
• Using an exponential model, iteration is needed to
solve for current. Using a constant-voltage model,
only linear equations need to be solved.
EE105 Fall 2011
Lecture 4, Slide 22
Prof. Salahuddin, UC Berkeley
Small-Signal Analysis
• Small-signal analysis is performed at a DC bias point by
perturbing the voltage by a small amount and
observing the resulting linear current perturbation.
– If two points on the I-V curve are very close, the curve inbetween these points is well approximated by a straight line:
DI D
dI D

DVD dVD
2
3
x
x
ex  1 x 

 
2! 3!
EE105 Fall 2011
Lecture 4, Slide 23
VD VD1
I s VD1 / VT I D1
 e

VT
VT
Prof. Salahuddin, UC Berkeley
Diode Model for Small-Signal Analysis
• Since there is a linear relationship between the
small-signal current and small-signal voltage of a
diode, the diode can be viewed as a linear resistor
when only small changes in voltage are of interest.
Small-Signal Resistance
(or Dynamic Resistance)
EE105 Fall 2011
Lecture 4, Slide 24
VT
rd 
ID
Prof. Salahuddin, UC Berkeley
Small Sinusoidal Analysis
• If a sinusoidal voltage with small amplitude is applied
in addition to a DC bias voltage, the current is also a
sinusoid that varies about the DC bias current value.
V D(t )  V0  V p cos t
 V0
I D (t )  I 0  I p cos t  I s exp 
 VT
EE105 Fall 2011
Lecture 4, Slide 25
 V p cos t
 
 VT / I 0 
Prof. Salahuddin, UC Berkeley