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CMOS Device Model
• Objective
– Hand calculations for analog design
– Efficiently and accurately simulation
• CMOS transistor models
– Large signal model
– Small signal model
– Simulation model
– Noise model
Large Signal Model
• Nonlinear equations for solving dc values of
device currents given voltages
• Level 1: Shichman-Hodges (VT, K', g, l, f, and
NSUB)
• Level 2: with second-order effects (varying
channel charge, short-channel, weak inversion,
varying surface mobility, etc.)
• Level 3: Semi-empirical short-channel model
• Level 4: BSIM models. Based on automatically
generated parameters from a process
characterization. Good weak-strong inversion
transition.
Transconductance when VDS is small
Transconductance when VDS is small
Transconductance when VDS is small
Effect of changing VDS for a large VGS
Effect of changing VDS for a given VGS
Effect of changing VDS for a given VGS
Effect of changing VDS for various VGS
VGS<=VT
Effect of changing VDS for various VGS
Effect of changing VDS for various VGS
MOST Regions of Operation
• Cut-off, or non-conducting: VGS <VT
– ID=0
• Conducting: VGS >=VT
– Saturation: VDS > VGS – VT
iD
μCoxW

(vGS - VT )2
2L
– Triode or linear or ohmic or non-saturation: VDS <=
VGS – VT
iD
2
μCoxW
VDS

((vGS - VT )VDS - 2 )
L
With channel length modulation
iD
μCoxW
2

(vGS - VT ) ( 1  λVDS )
2L
VT  VT 0  g ( 2|φ f |  |v BS| μCoxW
W
 
 K'
L
L
2|φ f | )
Capacitors Of The Mosfet
CBD and CBS include both the diffusion-bulk
junction capacitance as well as the side wall
junction capacitance. They are highly nonlinear
in bias voltages.
C4 is the capacitance between the channel and
the bulk. It is highly nonlinear and depends on
the operation of the device. C4 is not
measurable from terminals.
/2
Gate related capacitances
Small signal
model
Typically: VDB, VSB are in such a way that there is
a reversely biased pn junction.
Therefore:
gbd ≈ gbs ≈ 0
In saturation:
But
In non-saturation region
High Frequency Figures of Merit wT
•
•
•
•
•
•
AC current source input to G
AC short S, D, B to gnd
Measure AC drain current output
Calculate current gain
Find frequency at which current gain = 1.
Ignore rs and rd,  Cbs, Cbd, gds, gbs, gbd all have
zero voltage drop and hence zero current
• Vgs = Iin /jw(Cgs+Cgb+Cgd) ≈ Iin /jwCgs
• Io = − (gm − jw Cgd)Vgs ≈ − gmVgs
• |Io/Iin| ≈ gm/wCgs
• At wT, current gain =1
• wT ≈ gm/(Cgs+Cgd)≈ gm/Cgs
• or
High Frequency Figures of Merit wmax
•
•
•
•
•
•
•
AC current source input to G
AC short S, D, B to gnd
Measure AC power into the gate
Assume complex conjugate load
Compute max power delivered by the transistor
Find maximum power gain
Find frequency at which power gain = 1.
• wmax: frequency at which power gain
becomes 1
PL=
BSIM models
• Non-uniform charge density
• Band bending due to non-uniform gate voltage
• Non-uniform threshold voltage
– Non-uniform channel doping, x, y, z
– Short channel effects
• Charge sharing
• Drain-induced barrier lowering (DIBL)
– Narrow channel effects
– Temperature dependence
• Mobility change due to temp, field (x, y)
• Source drain, gate, bulk resistances
“Short Channel” Effects
• VTH decreases for small L
– Large offset for diff pairs with small L
• Mobility reduction:
– Velocity saturation
– Vertical field (small tox=6.5nm)
– Reduced gm: increases slower than root-ID
Threshold Voltage VTH
• Strong function of L
– Use long channel for VTH matching
– But this increases cap and decreases speed
• Process variations
– Run-to-run
– How to characterize?
– Slow/nominal/fast
– Both worst-case & optimistic
Effect of Velocity Saturation
• Velocity ≈ mobility * field
• Field reaches maximum Emax
– (Vgs-Vt)/L reaches ESAT
• gm become saturated:
– gm ≈ ½mnCoxW*ESAT
• But Cgs still 2/3 WL Cox
• wT ≈ gm/Cgs = ¾ mnESAT /L
• No longer ~ 1/L^2
Threshold Reduction
• When channel is short, effect of Vd extends to S
• Cause barrier to drop, i.e. Vth to drop
• Greatly affects sub-threshold current: 26 mV Vth
drop  current * e
• 100~200 mV Vth drop due to Vd not uncommon
 100s or 1000 times current increase
• Use lower density active near gate but higher
density for contacts
Other effects
• Temperature variation
• Normal-Field Mobility Degradation
• Substrate current
– Very nonlinear in Vd
• Drain to source leakage current at Vgs=0
– Big concern for static power
• Gate leakage currents
– Hot electron
– Tunneling
– Very nonlineary
• Transit Time Effects
Consequences for Design
• SPICE (HSPICE or Spectre)
– BSIM3, BSIM4 models
– Accurate but inappropriate for hand analysis
– Verification (& optimization)
• Design:
– Small signal parameter design space:
• gm, CL
• gm/ID, ID
• Av0= gmro
(speed, noise)
(power, output range, speed)
(gain)
– Device geometries from SPICE (table, graph);
– may require iteration (e.g. CGS)
Intrinsic voltage gain of MOSFET
Sweep V1
Measure vgs
Intrinsic voltage gain = gm/go = Dvds/Dvgs for constant Id
Electronic Noise
• Noise phenomena
• Device noise models
• Representation of noise (2-ports):
–
–
–
–
–
Motivation
Output spectral density
Input equivalent spectral density
Noise figure
Sampling noise (“kT/C noise”)
• SNR versus Bits
• Noise versus Power Dissipation
– Dynamic range
– Minimum detectable signal
Noise in Devices and Circuits
•Noise is any unwanted excitation of a circuit, any
input that is not an information-bearing signal.
• External noise: Unintended coupling with other
parts of the physical world; in principle, can be
virtually eliminated by careful design.
• Intrinsic noise: Unpredictable microscopic events
inherent in the device/circuit; can be reduced, but
never eliminated.
•Noise is especially important to consider when
designing low-power systems because the signal
levels (typically voltages or currents) are small.
Noise vs random process
variations
• random process variations
– Variations from one device to another
– For any device, it is fixed after fabrication
• Noise
– Unpredictable variations during operation
– Unknown after fabrication
– Remains unknown after measurement during
operation
– May change with environment
Time domain description of noise
What is signal and what
is noise?
Signal and noise power:
x(t )  s(t )  n(t )
1 T 2
Ps   s (t ) dt , S (rms)  Srms  Ps
T 0
1 T 2
Pn   n (t ) dt , N (rms)  N rms  Pn
T 0
Physical interpretation
If we apply a signal (or noise) as a voltage
source across a one Ohm resistor, the power
delivered by the source is equal to the signal
power.
Signal power can be viewer as a measure of
normalized power.
power
Signal to noise ratio
Ps
S rms
SNR  10 log 10 ( )  20 log 10 (
)
Pn
N rms
SNR = 0 dB when signal power = noise power
Absolute noise level in dB:
w.r.t. 1 mW of signal power
Pn
Pn in dB m  10 log
1mW
 30 dB  10 log( Pn )
SNR in bits
• A sine wave with magnitude 1 has power
= 1/2.
• Quantize it into N=2n equal levels between
-1 and 1 (with step size = 2/2n)
• Quantization error uniformly distributed
between +–1/2n
• Noise (quantization error) power
=1/3 (1/2n)2
• Signal to noise ratio
= 1/2 ÷ 1/3 (1/2n)2 =1.5(1/2n)2
= 1.76 + 6.02n dB or n bits
-1<=C<=+1
C=0: n1 and n2 uncorrelated
C=1: perfectly correlated
Adding
uncorrelated
noises
Adding
correlated
noises
For independent noises
Frequency domain description of
noise
Given n(t) stationary, its autocorrelation is:
1 T
Rn ( )  lim
n(t )n(t   ) dt

T  2T T
The power spectral density of n(t) is:
PSDn ( f )  Sn ( f )  F ( Rn ( ))
Pn  


PSDn ( f ) df
For real signals, PSD is even.  can use single sided
spectrum: 2x positive side
Pn  

0
PSDn ( f ) df
↑ single sided PSD
Parseval’s Theorem:
x(t )  X ( f )
If

 
2

x(t ) dt  


2
X ( f ) df
If x(t) stationary,
Rx ( )  PSDx ( f )

lim
T 

T
T
2
x(t ) dt  Rx (0)  


PSDx ( f ) df
Interpretation of PSD
Pxf1 = PSDx(f1)
PSDx(f)
Types of “Noise”
• “man made”
– Interference
– Supply noise
–…
– Use shielding, careful layout, isolation, …
• “intrinsic” noise
– Associated with current conduction
– “fundamental” –thermal noise
– “manufacturing process related”
– flicker noise
Thermal Noise
• Due to thermal excitation of charge carriers in a
conductor. It has a white spectral density and is
proportional to absolute temperature, not
dependent on bias current.
• Random fluctuations of v(t) or i(t)
• Independent of current flow
• Characterization:
– Zero mean, Gaussian pdf
– Power spectral density constant or “white” up to about
80THz
Thermal noise dominant in
resisters
Example:
R = 1kΩ, B = 1MHz, 4µV rms or 4nA rms
HW
Equivalently, we can model a real resistor with an
ideal resistor in parallel with a current noise source.
What rms value should the current source have?
Show that when two resistors are connected in
series, we can model them as ideal series resistors
in series with a single noise voltage source. What’s
the rms value of the voltage source?
Show that two parallel resistors can be modeled as
two ideal parallel resistors in parallel with a single
noise current source. What’s the rms value of the
current source?
Noise in Diodes
Shot noise dominant
– DC current is not continuous and smooth but
instead is a result of pulses of current caused by
the individual flow of carriers.
It depends on bias, can be modeled as a
white noise source and typically larger than
thermal noise.
− Zero mean
– Gaussian pdf
– Power spectral density flat
– Proportional to current
– Dependent on temperature
Example:
ID= 1mA, B = 1MHz, 17nA rms
MOS Noise Model
Flicker noise
–Kf,NMOS 6 times larger than Kf,PMOS
–Strongly process dependent
−when referred to as drain current noise, it
is inversely proportional to L2
BJT Noise
Sampling Noise
• Commonly called “kT/C” noise
• Applications: ADC, SC circuits, …
R
von
C
Used:
Filtering of noise
x(t)
y(t)
H(s)
|H(f )|2 = H(s)|s=j2pf H(s)|s=-j2pf
Noise Calculations
1) Get small-signal model
2) Set all inputs = 0 (linear superposition)
3) Pick output vo or io
4) For each noise source vx, or ix
Calculate Hx(s) = vo(s) / vx(s) (or … io, ix)
5) Total noise at output is
6) Input Referred Noise: Fictitious noise source at
input:
2
in,eff
v
v
2
on,T
/ A( s )
2
Example: CS Amplifier
Von=(inRL +inMOS)/goT
VDD
goT = 1/RL + sCL
RL
M1
2
nRL
i
CL
2
nMOS
i
1
 4 k BT
RL
2
 4 k BT g m
3
wo=1/RLCL
Some integrals
HW
In the previous example, if the transistor is
in triode, how would the solution change?
HW
If we include the flicker noise source, how
would that affect the computation? What do
you suggest we should modify?
HW
In the example, if RL is replaced by a PMOS
transistor in saturation, how would the
solution change? Assume appropriate bias
levels.