EECS 105 Fall 2003, Lecture 10
Lecture 10:
PN Junction & MOS Capacitors
Prof. Niknejad
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 10
Prof. A. Niknejad
Lecture Outline



Review: PN Junctions Thermal
Equilibrium
PN Junctions with Reverse Bias (3.33.6)
MOS Capacitors (3.7-3.9):
–
–
–
Department of EECS
Accumulation, Depletion, Inversion
Threshold Voltage
CV Curve
University of California, Berkeley
EECS 105 Fall 2003, Lecture 10
Prof. A. Niknejad
Results of MT #1


Good Job!
AVG
74
MIN
34
MAX
99
STD DEV
13
This is only 17% of your grade
Homework
Laboratory
Midterm #1
Midterm #2
Final
Department of EECS
15%
20%
17%
18%
30%
University of California, Berkeley
EECS 105 Fall 2003, Lecture 10
Prof. A. Niknejad
PN Junction in Thermal Equilibrium





Contact potential develops between P and N region
Diffusion current balanced by drift current
Depletion region is a “space-charge” region where
the concentration of free carriers is low
The depletion region is charged due to the
immobile background ions (donors and acceptors)
Used the “Depletion Approximation” to estimate
the charge density  calculate the electric fields
and potential variation using electrostatics in 1D
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 10
Prof. A. Niknejad
Have we invented a battery?

Can we harness the PN junction and turn it into a
battery?
 ND
NA 
ND N A


bi  n   p  Vth  ln
 ln
 Vth ln
2

n
n
n
i
i 
i

?

Numerical example:
ND N A
10151015
bi  26mV ln
 60mV  log
 600mV
2
20
ni
10
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 10
Prof. A. Niknejad
Contact Potential




The contact between a PN junction creates a
potential difference
Likewise, the contact between two dissimilar
metals creates a potential difference (proportional
to the difference between the work functions)
When a metal semiconductor junction is formed, a
contact potential forms as well
If we short a PN junction, the sum of the voltages
around the loop must be zero:
+
bi
−
Department of EECS
mn
 pm
0  bi   pm  mn
n
p
bi  ( pm  mn )
University of California, Berkeley
EECS 105 Fall 2003, Lecture 10
Prof. A. Niknejad
PN Junction Capacitor




Under thermal equilibrium, the PN junction does
not draw any current
But notice that a PN junction stores charge in the
space charge region (transition region)
Since the device is storing charge, it’s acting like a
capacitor
Positive charge is stored in the n-region, and
negative charge is in the p-region:
qN a x po  qN d xno
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 10
Prof. A. Niknejad
Reverse Biased PN Junction

What happens if we “reverse-bias” the PN
junction?
+
 bi  VD
−



VD
VD  0
Since no current is flowing, the entire reverse
biased potential is dropped across the transition
region
To accommodate the extra potential, the charge in
these regions must increase
If no current is flowing, the only way for the charge
to increase is to grow (shrink) the depletion regions
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 10
Prof. A. Niknejad
Current Under Reverse Bias
n  VD
n
n
p
+
−
 VD
E0
p
Xd0




E0
p
X d (VD )
Under thermal equilibrium current is zero
If we apply a reverse bias, we are increasing the barrier
against diffusion current
Drift current is low since the field only moves minority
carriers across junction
In fact, current is not zero but very small since the minority
carrier concentration is low. Minority carriers within one
diffusion length of junction can contribute to a reverse bias
current. This is more or less independent of the applied bias
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 10
Prof. A. Niknejad
Voltage Dependence of Depletion Width

Can redo the math but in the end we realize that the
equations are the same except we replace the builtin potential with the effective reverse bias:
2 s (bi  VD )  N a

xn (VD ) 
qN d
 Na  Nd

V
  xn 0 1  D
bi

2 s (bi  VD )  N d

x p (VD ) 
qN a
 Na  Nd

VD
  x p 0 1 
bi

2 s (bi  VD )  1
1 


X d (VD )  x p (VD )  xn (VD ) 


q
 Na Nd 
VD
X d (VD )  X d 0 1 
bi
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 10
Prof. A. Niknejad
Charge Versus Bias

As we increase the reverse bias, the depletion
region grows to accommodate more charge
QJ (VD )  qN a x p (VD )  qN a 1 



VD
bi
Charge is not a linear function of voltage
This is a non-linear capacitor
We can define a small signal capacitance for small
signals by breaking up the charge into two terms
QJ (VD  vD )  QJ (VD )  q(vD )
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 10
Prof. A. Niknejad
Derivation of Small Signal Capacitance

From last lecture we found
dQD
QJ (VD  vD )  QJ (VD ) 
dV
C j  C j (VD ) 
dQ j
dV
Cj 
V VD
d

dV
qN a x p 0
2bi 1 

Notice that
C j0 
qN a x p 0
2bi
Department of EECS
qN a

2bi
VD
bi
vD  
VD


V
  qN a x p 0 1 




bi

V VR

C j0
1
 2 sbi  N d


 qN a  N a  N d
VD
bi

q s N a N d
 
2bi N a  N d

University of California, Berkeley
EECS 105 Fall 2003, Lecture 10
Prof. A. Niknejad
Physical Interpretation of Depletion Cap
q s N a N d
C j0 
2bi N a  N d

Notice that the expression on the right-hand-side is
just the depletion width in thermal equilibrium
C j0   s

 1
1 



2 sbi  N a N d 
q
1

s
Xd0
This looks like a parallel plate capacitor!
C j (VD ) 
Department of EECS
s
X d (VD )
University of California, Berkeley
EECS 105 Fall 2003, Lecture 10
Prof. A. Niknejad
A Variable Capacitor (Varactor)

Capacitance varies versus bias:
Cj
C j0

Application: Radio Tuner
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 10
Prof. A. Niknejad
“Diffusion” Resistor
N-type Diffusion Region
Oxide
P-type Si Substrate

Resistor is capacitively isolation from substrate
–
–
Must Reverse Bias PN Junction!
PN Junction creates a distributed capacitance with
substrate (RC transmission line)
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 10
Prof. A. Niknejad
MOS Capacitor
Oxide (SiO2)
 ox  3.9 0
Gate (n+ poly)
0
Very Thin!
tox ~ 1nm
Body (p-type substrate)
x
 s  11.7 0




MOS = Metal Oxide Silicon
Sandwich of conductors separated by an insulator
“Metal” is more commonly a heavily doped polysilicon
layer n+ or p+ layer
NMOS  p-type substrate, PMOS  n-type substrate
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 10
Prof. A. Niknejad
P-I-N Junction
Gate (n+ poly)
Body (p-type substrate)

Under thermal equilibrium, the n-type poly gate is at a
higher potential than the p-type substrate
kT N a
 p   ln
n  550mV
q
ni

No current can flow because of the insulator but this
potential difference is accompanied with an electric field
Fields terminate on charge!

Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 10
Prof. A. Niknejad
Fields and Charge at Equilibrium
+
Vox
−

++++++++++++++++++
+ − − − − − − − − −
VB − − − − − − − −
− − − − − − − − − −
Body (p-type substrate)
Eox
Xd0
At equilibrium there is an electric field from the
gate to the body. The charges on the gate are
positive. The negative charges in the body come
from a depletion region
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 10
Prof. A. Niknejad
Good Place to Sleep: Flat Band
QG (VGB  VFB )  0
VFB  0
+
−
Body (p-type substrate)

If we apply a bias, we can compensate for this
built-in potential
VFB  (n   p )


In this case the charge on the gate goes to zero and
the depletion region disappears
In solid-state physics lingo, the energy bands are
“flat” under this condition
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 10
Prof. A. Niknejad
Accumulation
QG  Cox (VGB  VFB )
VGB  VFB
−
+
−−−−−−−−−−−−−−−−−−
++++++++++++++++++
QB  QG
Body (p-type substrate)



If we further decrease the potential beyond the
“flat-band” condition, we essentially have a parallel
plate capacitor
Plenty of holes and electrons are available to charge
up the plates
Negative bias attracts holes under gate
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 10
Prof. A. Niknejad
Depletion
VGB  VFB
+
−
++ + + + + + + ++
− − − − − − − − −
− − − − − − − −
Body (p-type substrate)



QG (VGB )  QB
QB  qN a X d (VGB )
Similar to equilibrium, the potential in the gate is
higher than the body
Body charge is made up of the depletion region
ions
Potential drop across the body and depletion region
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 10
Prof. A. Niknejad
Inversion
s
VGB  VT
++ + + + + + + ++
− − − − − − − − −
− − − − − − − −
+
−
Body (p-type substrate)

As we further increase the gate voltage, eventually the
surface potential increases to a point where the electron
density at the surface equals the background ion density
ns  ni e

qs
kT
 Na
s   p
At this point, the depletion region stops growing and the
extra charge is provided by the inversion charge at surface
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 10
Prof. A. Niknejad
Threshold Voltage



The threshold voltage is defined as the gate-body
voltage that causes the surface to change from ptype to n-type
For this condition, the surface potential has to equal
the negative of the p-type potential
We’ll derive that this voltage is equal to:
1
VTn  VFB  2 p 
Cox
Department of EECS
2q s N a (2 p )
University of California, Berkeley
EECS 105 Fall 2003, Lecture 10
Prof. A. Niknejad
Inversion Stops Depletion



A simple approximation is to assume that once
inversion happens, the depletion region stops
growing
This is a good assumption since the inversion
charge is an exponential function of the surface
potential
Under this condition:
QG (VTn )  QB ,max
QG (VGB )  Cox (VGB  VTn )  QB ,max
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 10
Prof. A. Niknejad
Q-V Curve for MOS Capacitor
QG
QN (VGB )
QB ,max
VFB



VTn
VGB (V )
In accumulation, the charge is simply proportional
to the applies gate-body bias
In inversion, the same is true
In depletion, the charge grows slower since the
voltage is applied over a depletion region
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 10
Prof. A. Niknejad
Numerical Example

MOS Capacitor with p-type substrate:
tox  20nm

N a  5 1016 cm 3
Calculate flat-band:
VFB  (n   p )  (550  (402))  0.95V

Calculate threshold voltage:
 ox
3.45 1013 F/cm
Cox 

tox
2 10-6cm
1
VTn  VFB  2 p 
2q s N a (2 p )
Cox
2 1.6 1019 1.04 1012  5 1016  2  0.4
VTn  .95  2(0.4) 
 0.52V
Cox
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 10
Prof. A. Niknejad
Num Example: Electric Field in Oxide

Apply a gate-to-body voltage:
VGB  2.5  VFB


Device is in accumulation
The entire voltage drop is across the oxide:
Vox VGB  n   p 2.5  0.55  (0.4)
5 V
Eox 




8

10
tox
tox
2 106
cm

The charge in the substrate (body) consist of holes:
QB  Cox (VGB  VFB )  2.67 107 C/cm 2
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 10
Prof. A. Niknejad
Numerical Example: Depletion Region

In inversion, what’s the depletion region width and
charge?
VB ,max  s   p   p   p  2 p  0.8V
VB ,max
X d ,max 
1  qNa
 
2  s
 2
 X d ,max

2 sVB ,max
qN a
 144nm
QB,max  qNa X d ,max  1.15 107 C/cm2
Department of EECS
University of California, Berkeley