EECS 105 Fall 2003, Lecture 8
Lecture 8:
Capacitors and PN Junctions
Prof. Niknejad
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 8
Prof. A. Niknejad
Lecture Outline
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Department of EECS
Review of Electrostatics
IC MIM Capacitors
Non-Linear Capacitors
PN Junctions Thermal Equilibrium
University of California, Berkeley
EECS 105 Fall 2003, Lecture 8
Prof. A. Niknejad
Electrostatics Review (1)

Electric field go from positive charge to negative
charge (by convention)
+++++++++++++++++++++
−−−−−−−−−−−−−−−

Electric field lines diverge on charge

E 



In words, if the electric field changes magnitude,
there has to be charge involved!
Result: In a charge free region, the electric field
must be constant!
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 8
Prof. A. Niknejad
Electrostatics Review (2)

Gauss’ Law equivalently says that if there is a net
electric field leaving a region, there has to be
positive charge in that region:
+++++++++++++++++++++
−−−−−−−−−−−−−−−
Electric Fields are Leaving This Box!
 E  dS 
Recall:

V   E dV V  dV  Q / 
Department of EECS
Q

Q
   E dV  E  dS  
V
S
University of California, Berkeley
EECS 105 Fall 2003, Lecture 8
Prof. A. Niknejad
Electrostatics in 1D

Everything simplifies in 1-D
dE 
E 

dx 
dE 

dx

 ( x' )
E ( x)  E ( x0 )  
dx'

x
x
0

Consider a uniform charge distribution E (x)
Zero field
boundary
condition
 (x )
0
x1
Department of EECS
x x

 ( x' )
E ( x)  
dx'  0 x


0
0
x1

x1
University of California, Berkeley
EECS 105 Fall 2003, Lecture 8
Prof. A. Niknejad
Electrostatic Potential

The electric field (force) is related to the potential
(energy):
d
E
dx


Negative sign says that field lines go from high
potential points to lower potential points (negative
slope)
Note: An electron should “float” to a high potential
point:
d

d
Fe  qE  e
dx
1
Fe   e
e
dx
2
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 8
Prof. A. Niknejad
More Potential

Integrating this basic relation, we have that the
 (x)
potential is the integral of the field:

 ( x)   ( x0 )   E  dl

dl
C

In 1D, this is a simple integral:
x
 ( x)   ( x0 )    E ( x' )dx'
 ( x0 )
E
x0

Going the other way, we have Poisson’s equation in
1D:
d 2 ( x)
 ( x)

2
dx

Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 8
Prof. A. Niknejad
Boundary Conditions


Potential must be a continuous function. If not, the fields
(forces) would be infinite
Electric fields need not be continuous. We have already
seen that the electric fields diverge on charges. In fact,
across an interface we have:
x
  E  dS   E S  
1
E1 (1 )
1
2
E2 S  Qinside
Qinside x

0  0
 1E1S   2 E2 S  0
E2 ( 2 )

S
E1  2

E2  1
Field discontiuity implies charge density at surface!
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 8
Prof. A. Niknejad
IC MIM Capacitor
Bottom Plate
Top Plate
Bottom Plate
Contacts
Thin Oxide
Q  CV



By forming a thin oxide and metal (or polysilicon) plates, a
capacitor is formed
Contacts are made to top and bottom plate
Parasitic capacitance exists between bottom plate and
substrate
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 8
Prof. A. Niknejad
Review of Capacitors
 E  dS 
+
−
Vs
 E  dl  E0tox  Vs
Q
 E  dS  E A  
0


+++++++++++++++++++++
−−−−−−−−−−−−−−−

Q
Vs
E0 
tox
Vs
Q
A
tox

Q
 E  dS   
Q  CVs
A
C
tox
For an ideal metal, all charge must be at surface
Gauss’ law: Surface integral of electric field over
closed surface equals charge inside volume
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 8
Prof. A. Niknejad
Capacitor Q-V Relation
+++++++++++++++++++++
Q
y
−−−−−−−−−−−−−−−
Vs
Q( y )
y
Q  CVs


Total charge is linearly related to voltage
Charge density is a delta function at surface (for
perfect metals)
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 8
Prof. A. Niknejad
A Non-Linear Capacitor
+++++++++++++++++++++
y
Q
−−−−−−−−−−−−−−−
Vs
Q( y )
y
Q  f (Vs )


We’ll soon meet capacitors that have a non-linear Q-V
relationship
If plates are not ideal metal, the charge density can penetrate
into surface
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 8
Prof. A. Niknejad
What’s the Capacitance?

For a non-linear capacitor, we have
Q  f (Vs )  CVs


We can’t identify a capacitance
Imagine we apply a small signal on top of a bias
voltage:
Q  f (Vs  vs )  f (Vs ) 
df (V )
vs
dV V Vs
Constant charge

The incremental charge is therefore:
df (V )
Q  Q0  q  f (Vs ) 
vs
dV V Vs
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 8
Prof. A. Niknejad
Small Signal Capacitance

Break the equation for total charge into two terms:
Incremental
Charge
df (V )
Q  Q0  q  f (Vs ) 
vs
dV V Vs
Constant
Charge
df (V )
q
vs  C vs
dV V Vs
df (V )
C
dV V Vs
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 8
Prof. A. Niknejad
Example of Non-Linear Capacitor

Next lecture we’ll see that for a PN junction, the
charge is a function of the reverse bias:
Q j (V )  qN a x p 1 
Charge At N Side of Junction

V
Voltage Across NP
Junction
b
Constants
Small signal capacitance:
C j (V ) 
Department of EECS
dQ j
dV

qN a x p
2b
1
1
V
b

C j0
1
V
b
University of California, Berkeley
EECS 105 Fall 2003, Lecture 8
Prof. A. Niknejad
Carrier Concentration and Potential

In thermal equilibrium, there are no external fields
and we thus expect the electron and hole current
densities to be zero:
dno
J n  0  qn0  n E0  qDn
dx
 n 
dno
 q  d
  no E0   no 0
dx
 kT  dx
 Dn 
 kT  dno
dn0
d0   
 Vth
n0
 q  n0
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 8
Prof. A. Niknejad
Carrier Concentration and Potential (2)

We have an equation relating the potential to the
carrier concentration
 kT  dno
dn
d0   
 Vth 0
n0
 q  n0

If we integrate the above equation we have
n0 ( x)
0 ( x)  0 ( x0 )  Vth ln
n0 ( x0 )

We define the potential reference to be intrinsic Si:
0 ( x0 )  0
Department of EECS
n0 ( x0 )  ni
University of California, Berkeley
EECS 105 Fall 2003, Lecture 8
Prof. A. Niknejad
Carrier Concentration Versus Potential

The carrier concentration is thus a function of
potential
n0 ( x)  ni e0 ( x ) / Vth


Check that for zero potential, we have intrinsic
carrier concentration (reference).
If we do a similar calculation for holes, we arrive at
a similar equation
p0 ( x)  ni e 0 ( x ) / Vth

Note that the law of mass action is upheld
n0 ( x) p0 ( x)  ni2 e 0 ( x ) / Vth e0 ( x ) / Vth  ni2
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 8
Prof. A. Niknejad
The Doping Changes Potential

Due to the log nature of the potential, the potential changes
linearly for exponential increase in doping:
n0 ( x)
n0 ( x)
n0 ( x)
0 ( x)  Vth ln
 26mV ln
 26mV ln 10 log 10
ni ( x0 )
ni ( x0 )
10
n0 ( x)
0 ( x)  60mV log 10
10
p0 ( x )
0 ( x)  60mV log 10
10


Quick calculation aid: For a p-type concentration of 1016
cm-3, the potential is -360 mV
N-type materials have a positive potential with respect to
intrinsic Si
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 8
Prof. A. Niknejad
PN Junctions: Overview
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The most important device is a junction
between a p-type region and an n-type region
When the junction is first formed, due to the
concentration gradient, mobile charges
transfer near junction
Electrons leave n-type region and holes leave
p-type region
These mobile carriers become minority
carriers in new region (can’t penetrate far due
to recombination)
Due to charge transfer, a voltage difference
occurs between regions
This creates a field at the junction that causes
drift currents to oppose the diffusion current
In thermal equilibrium, drift current and
diffusion must balance
Department of EECS
p-type
NA
−+−+−+−+−+− −
−−−−−− V
++++
−+
+−+−+−+−+− +
ND
n-type
University of California, Berkeley
EECS 105 Fall 2003, Lecture 8
Prof. A. Niknejad
PN Junction Currents


Consider the PN junction in thermal equilibrium
Again, the currents have to be zero, so we have
dno
J n  0  qn0  n E0  qDn
dx
dno
qn0  n E0  qDn
dx
E0 
dno
dx   kT 1 dn0
n0  n
q n0 dx
 Dn
dpo
Dp
kT 1 dp0
dx
E0 

n0  p
q p0 dx
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 8
Prof. A. Niknejad
PN Junction Fields
p-type
n-type
NA
ND
p0  N a
p0 ( x)
J diff
E0
 x p0
ni2
n0 
Na
xn 0
ni2
p0 
Nd
n0  N d
J diff
E0
– –++
Transition Region
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 8
Prof. A. Niknejad
Total Charge in Transition Region

To solve for the electric fields, we need to write
down the charge density in the transition region:
0 ( x)  q( p0  n0  N d  N a )

In the p-side of the junction, there are very few
electrons and only acceptors:
0 ( x)  q( p0  N a )

 x p0  x  0
Since the hole concentration is decreasing on the pside, the net charge is negative:
N a  p0
Department of EECS
 0 ( x)  0
University of California, Berkeley
EECS 105 Fall 2003, Lecture 8
Prof. A. Niknejad
Charge on N-Side

Analogous to the p-side, the charge on the n-side is
given by:
0 ( x)  q(n0  N d )

0  x  xn 0
The net charge here is positive since:
 0 ( x)  0
N d  n0
n0  N d
ni2
n0 
Na
J diff
E0
– –++
Department of EECS
Transition Region
University of California, Berkeley