EECS 105 Fall 2003, Lecture 13
Lecture 13:
Part I: MOS Small-Signal Models
Prof. Niknejad
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 13
Prof. A. Niknejad
Lecture Outline


Department of EECS
MOS Small-Signal Model (4.6)
Diode Currents in forward and
reverse bias (6.1-6.3)
University of California, Berkeley
EECS 105 Fall 2003, Lecture 13
Prof. A. Niknejad
Total Small Signal Current
iDS (t )  I DS  ids
iDS
iDS
ids 
vgs 
vds
vgs
vds
1
ids  g m vgs  vds
ro
Transconductance
Department of EECS
Conductance
University of California, Berkeley
EECS 105 Fall 2003, Lecture 13
Prof. A. Niknejad
Role of the Substrate Potential
Need not be the source potential, but VB < VS
Effect: changes threshold voltage, which
changes the drain current … substrate acts
like a “backgate”
g mb
iD

v BS
Q
iD

v BS
Q
Q = (VGS, VDS, VBS)
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 13
Prof. A. Niknejad
Backgate Transconductance
VT  VT 0  
Result:
Department of EECS
g mb 
iD
vBS

Q

iD
VTn
Q
VSB  2 p  2 p
VTn
vBS

Q

 gm
2 VBS  2 p
University of California, Berkeley
EECS 105 Fall 2003, Lecture 13
Prof. A. Niknejad
Four-Terminal Small-Signal Model
1
ids  g m vgs  g mb vbs  vds
ro
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 13
Prof. A. Niknejad
MOSFET Capacitances in Saturation
Gate-source capacitance: channel charge is not
controlled by drain in saturation.
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 13
Prof. A. Niknejad
Gate-Source Capacitance Cgs
Wedge-shaped charge in saturation  effective area is (2/3)WL
(see H&S 4.5.4 for details)
C gs  (2 / 3)WLC ox  Cov
Overlap capacitance along source edge of gate 
Cov  LDWCox
(Underestimate due to fringing fields)
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 13
Prof. A. Niknejad
Gate-Drain Capacitance Cgd
Not due to change in inversion charge in channel
Overlap capacitance Cov between drain and source
is Cgd
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 13
Junction Capacitances
Prof. A. Niknejad
Drain and source diffusions have (different) junction
capacitances since VSB and VDB = VSB + VDS aren’t
the same
Complete model (without interconnects)
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 13
Prof. A. Niknejad
P-Channel MOSFET
Measurement of –IDp versus VSD, with VSG as a parameter:
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 13
Prof. A. Niknejad
Square-Law PMOS Characteristics
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 13
Prof. A. Niknejad
Small-Signal PMOS Model
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 13
Prof. A. Niknejad
MOSFET SPICE Model
Many “levels” … we will use the square-law
“Level 1” model
See H&S 4.6 + Spice refs. on reserve for details.
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 13
Part II: Currents in PN Junctions
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 13
Prof. A. Niknejad
Diode under Thermal Equilibrium
Minority Carrier Close to Junction
Thermal
+
p-type
Generation
−
- +
+
- +
+
−
- +
+
J p ,drift - - ++
+
J n ,drift - - ++
ND
- +
+
n-type
J n ,diff
J p , diff
−
NA
E0
Recombination
+
qbi
−




Carrier with energy
below barrier height
Diffusion small since few carriers have enough energy to penetrate barrier
Drift current is small since minority carriers are few and far between: Only
minority carriers generated within a diffusion length can contribute current
Important Point: Minority drift current independent of barrier!
Diffusion current strong (exponential) function of barrier
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 13
Prof. A. Niknejad
Reverse Bias


Reverse Bias causes an increases barrier to
diffusion
Diffusion current is reduced exponentially
p-type
-
ND
-
-
+
+
+
+
+
+
+
n-type
NA
+
−
q(bi  VR )


Drift current does not change
Net result: Small reverse current
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 13
Prof. A. Niknejad
Forward Bias


Forward bias causes an exponential increase in
the number of carriers with sufficient energy to
penetrate barrier
Diffusion current increases exponentially
p-type
-
ND
-
-
+
+
+
+
+
+
+
n-type
NA
+
−
q(bi  VR )


Drift current does not change
Net result: Large forward current
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 13
Prof. A. Niknejad
Diode I-V Curve
Id
Is
d
 qV

kT
I d  I S  e  1


I d (Vd  )   I S
1
qVd
kT

Diode IV relation is an exponential function

This exponential is due to the Boltzmann distribution of carriers versus
energy

For reverse bias the current saturations to the drift current due to minority
carriers
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 13
Prof. A. Niknejad
Minority Carriers at Junction Edges
Minority carrier concentration at boundaries of
depletion region increase as barrier lowers …
the function is
(minority) hole conc. on n-side of barrier
p n ( x  xn )

p p ( x   x p ) (majority) hole conc. on p-side of barrier
 e ( Barrier Energy ) / kT
p n ( x  xn )
 e q(B VD ) / kT
NA
Department of EECS
(Boltzmann’s Law)
University of California, Berkeley
EECS 105 Fall 2003, Lecture 13
Prof. A. Niknejad
“Law of the Junction”
Minority carrier concentrations at the edges of the
depletion region are given by:
pn ( x  xn )  N Ae q ( B VD ) / kT
n p ( x   x p )  N De
q (B VD ) / kT
Note 1: NA and ND are the majority carrier concentrations on
the other side of the junction
Note 2: we can reduce these equations further by substituting
VD = 0 V (thermal equilibrium)
Note 3: assumption that pn << ND and np << NA
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 13
Prof. A. Niknejad
Minority Carrier Concentration
pn 0 e
p side
qVA
kT
n side
n p 0e
qVA
kT
x
 qVkTA
  Lp
pn ( x)  pn 0  pn 0  e  1 e


pn 0
np0
-Wp
-xp
xn
Minority Carrier
Diffusion Length
Wn
The minority carrier concentration in the bulk region for
forward bias is a decaying exponential due to recombination
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 13
Prof. A. Niknejad
Steady-State Concentrations
Assume that none of the diffusing holes and
electrons recombine  get straight lines …
pn 0 e
p side
qVA
kT
n side
n p 0e
qVA
kT
pn 0
np0
-Wp
-xp
xn
Wn
This also happens if the minority carrier
Ln , p  Wn , p
diffusion lengths are much larger than Wn,p
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 13
Prof. A. Niknejad
Diode Current Densities
pn 0 e
p side
n p 0e
qVA
kT
dn p
n side
qVA
kT
dx
( x) 
n p 0e
qVA
kT
 np0
 x p  (Wp )
pn 0
np0
np0
-Wp
-xp
xn
Wn
ni2

Na
qVA


D
diff
n
kT
J n  qDn
q
n p 0  e  1
dx x  x
Wp


p
qVA
D


dp
p
diff
n
kT
J p  qDp
 q
pn 0 1  e 
dx x  xn
Wn


dn p
J
Department of EECS
diff
 Dp
Dn
 qn 

 N dWn N aW p

2
i
  qVA

kT
  e  1


University of California, Berkeley
EECS 105 Fall 2003, Lecture 13
Prof. A. Niknejad
Fabrication of IC Diodes
p+
cathode
annode
p
n+
n-well
p-type
p-type





Start with p-type substrate
Create n-well to house diode
p and n+ diffusion regions are the cathode and annode
N-well must be reverse biased from substrate
Parasitic resistance due to well resistance
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 13
Prof. A. Niknejad
Diode Small Signal Model

The I-V relation of a diode can be linearized
qVd qvd
d  vd )
 q (VkT

I D  iD  I S  e
 1  I S e kT e kT


2
3
x
x
ex  1  x   
2! 3!
 q (Vd  vd )
I D  iD  I D 1 

kT




qvd
iD 
 g d vd
kT
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 13
Prof. A. Niknejad
Diode Capacitance

We have already seen that a reverse biased diode
acts like a capacitor since the depletion region
grows and shrinks in response to the applied field.
the capacitance in forward bias is given by
Cj  A



S
X dep
 1.4C j 0
But another charge storage mechanism comes into
play in forward bias
Minority carriers injected into p and n regions
“stay” in each region for a while
On average additional charge is stored in diode
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 13
Prof. A. Niknejad
Charge Storage
pn 0 e
p side
n p 0e
q (Vd  vd )
kT
n side
q (Vd  vd )
kT
pn 0
np0
-Wp



-xp
xn
Wn
Increasing forward bias increases minority charge density
By charge neutrality, the source voltage must supply equal
and opposite charge
1 qI d

A detailed analysis yields: Cd 
2 kT
Time to cross junction
(or minority carrier lifetime)
Department of EECS
University of California, Berkeley
EECS 105 Fall 2003, Lecture 13
Prof. A. Niknejad
Diode Circuits





Rectifier (AC to DC conversion)
Average value circuit
Peak detector (AM demodulator)
DC restorer
Voltage doubler / quadrupler /…
Department of EECS
University of California, Berkeley