rasit onur topaloglu [email protected] university of california at san diego computer science...
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Rasit Onur Topaloglu [email protected] University of California at San Diego Computer Science and Engineering Department
La Jolla, CA, 92093, USA
Mismatch Modeling of MOS Transistors for Deep Sub-micron
Technologies
Outline
-Mismatch at transistor, circuit and VLSI levels-Modeling Challenges and Requirements in Deep Sub-Micron -Optimizations to Avoid Mismatch-Past and Present Mismatch Modeling Approaches:
-Electrical & Empirical Models-Layout-based Models-BSIM-based Models-Physics-based Models
-Insights for Future Models for Mismatch-Summary & Conclusions
silicon
silicon dioxidesilicon •Wafer is oxidized
silicon dioxidesilicon
photoresist
•Wafer is covered with photoresist
silicon dioxidesilicon
photoresistphotomask
•A photomask is placed on wafer
Semiconductor Manufacturing Steps
•Wafer is created
Semiconductor Manufacturing Steps
silicon dioxidesilicon
photoresistphotomask
UV radiation
•Wafer is exposed to ultra-violet (UV) radiation
silicon dioxidesilicon
photoresist •Unexposed regions dissolved
silicon dioxidesilicon
photoresist•Unprotected oxide etched
silicon dioxidesilicon
•Photoresist removed. Wafer is ready for doping
Possible Causes of Mismatch
silicon dioxidesilicon
doping
•Wafer is doped
silicon dioxidedoped regions •Doping creates n or p-type wells
•These variations can depend on the process (PVE), or they can be random effects (RE) •Mismatch negatively influences the design, which assumes accurate matching of electrical parameters between matched transistors
•Mismatch is caused by variations in the processing stagesex. photomask misalignment, difference in doping gradients, etc.
Transistor Operation and Mismatch
VGS
VG
VDD
ID
VDS
GD
S
GND
VG
Increasing VGS
ID
VDS
Vth=threshold voltage
D=drain
G=gate
S=source
V=voltage
I=current
L=channel length
W=channel width
n=channel mobility
Cox=oxide capacitance
Mismatch = variation in drain current of matched transistors
21
2D n ox GS th
WI C V V
L
Drain current formula:
Schematic view of transistor
G
DS
VG VDDGND
ID
L
Physical view
Vth
ID
VGS
quadraticincrease
Impact of Mismatch on a Circuit
Vi1
•Proper circuit operation requires precision matching of certain transistors
M1 M2
Vi2
A current mirror operational amplifier realizing the function Vout=G(Vi1-Vi2)
Mismatch = variation in drain current of matched transistors
M=transistori=inputG=gain
•Mismatch most important in analog circuits•Analog circuits implement a linear function within a local input space by strictly optimizing circuit parameters
VOUT
Impact of Mismatch on a Circuit
W1=300mW2=300m
W=width
•Proper circuit operation requires precision matching of certain transistors A current mirror operational amplifier realizing the function Vout=G(Vi1-Vi2)
Mismatch = variation in drain current of matched transistors•Mismatch most important in analog circuits•Analog circuits implement a linear function within a local input space by strictly optimizing circuit parameters
Vi1
Vi2
VOUT
Impact of Mismatch on a Circuit
Vth1=0.7V
•Proper circuit operation requires precision matching for certain transistors
Vth2=0.7V
A current mirror operational amplifier realizing the function Vout=G(Vi1-Vi2)
Vth=threshold
voltageG=gain
Mismatch = variation in drain current of matched transistors•Mismatch most important in analog circuits•Analog circuits implement a linear function within a local input space by strictly optimizing circuit parameters
Vi1
Vi2
VOUT
Impact of Mismatch on a Circuit
Id1=1mAId2=1mA
Id=drain current
•Proper circuit operation requires precision matching for certain transistors A current mirror operational amplifier realizing the function Vout=G(Vi1-Vi2)
Mismatch = variation in drain current of matched transistors•Mismatch most important in analog circuits•Analog circuits implement a linear function within a local input space by strictly optimizing circuit parameters
Vi1
Vi2
VOUT
Process Variations and Mismatch
Iref
300 300
300+2+1300+2+-2
300 303
W1 W2 GNominal :300 300 100PVE+RE causing mismatch:303 303 102297 297 98PVE+RE not causing mismatch:303 300 99297 300 90
PVE RE PVE RE
W1when W2=300
G G
W1when W2=W1
•PVE and RE’s add up; they may or may not create a mismatch that effects circuit operation according to specifications
•Mismatch deteriorates circuit performance, in the limit, causes the circuit to escape optimal operating region
PVE=process variation effectsRE=random effects
Vi1
Vi2
VOUT
Impact of Mismatch on VLSI Design
Mismatch causes soft errors (reduction in gain, higher output R)
-Yield loss
-Increased time to market
•We need to design for mismatch
•We need to estimate effects of mismatch : both require models
Critical mismatch necessitates re-design
-Optimization of circuit without accurate consideration of mismatch is barely lost time
Mismatch effects are not easily predictable
Why Mismatch Models Necessary for Analog Design Flow?
Manual design
SPICE simulations
Select architecture and technology
Optimizations
Test & diagnosis after production
Behavioral level design and simulations
Costly redesign needed due to late observation of mismatch effects
SPICE simulations
Optimizations
Manual design
SPICE simulations
Select architecture and technology
Optimizations
Test & diagnosis after production
Behavioral level design and simulations
Essential to help model mismatch as soon as possible so as to accomplish heavy reduction in number of iterations
Costly redesign needed due to late observation of mismatch effects
Manual design
Why Mismatch Models Necessary for Analog Design Flow?
Outline
-Mismatch at transistor, circuit and VLSI levels-Modeling Challenges and Requirements in Deep Sub-Micron -Optimizations to Avoid Mismatch-Past and Present Mismatch Modeling Approaches:
-Electrical & Empirical Models-Layout-based Models-BSIM-based Models-Physics-based Models
-Insights for Future Models for Mismatch-Summary & Conclusions
Model Requirements for each Step of the Design Flow
Manual design
Select architecture and technology
SPICE simulations
Behavioral level design and simulations
Optimizations
Test & diagnosis after production
Manually applicable models
Accurate models for simulation
Mismatch Predictive models
Models for Test and diagnosis
Worst-Case Estimations Fail in Deep Sub-Micron
Iref
Current mirror
•Worst-case propagation between blocks results in overestimation of errors
•Designing while considering such large variations impossible in DSM
•Correlations between parameters accentuate this error
Bandgap reference circuit
Iref
Vb1
Vb2
Vb3
OpAmp
99 % falls in this range
actual pdf
New models should avoid worst-case estimations
Worst-case limits for an output parameter
RE
Fabrication accuracy cannot keep up with feature size shrinkage rate, as: •Errors occurring from diffusion cause PV distributions that have similar across different technologies
Wafer radius
tox (x40nm)
Wafer radius
tox (x4nm) newer technology
The Increasing Importance of Random Effects
•Mismatch groups closer than ever before, therefore: REs assume increased importance compared to previous technology
PVE
New models should be able to consider random effects
Outline
-Mismatch at transistor, circuit and VLSI levels-Modeling Challenges and Requirements in Deep Sub-Micron -Optimizations to Avoid Mismatch-Past and Present Mismatch Modeling Approaches:
-Electrical & Empirical Models-Layout-based Models-BSIM-based Models-Physics-based Models
-Insights for Future Models for Mismatch-Summary & Conclusions
Layout Optimizations
A B
standard layout style
•Layout optimizations try to disperse the effects of on-chip gradients between matched transistors A and B equally
•Statistical average of a parameter would differ a lot on matched transistors without optimizations causing significant mismatch
Common centroid layout style
A B B A
iso-parameter contours for a physical parameter such as tox
tox=4.0nm
tox=4.1nm
tox=4.2nm
tox=4.0nmtox=4.1nm
tox=4.2nm
tox=oxide thickness
-Connecting gates, or avoiding from deteriorating effects of other blocks on layout may be problematic
+Sufficient for minimal to moderate matching-Suffers linear optimization limitations as transistors rectangular, yet physical parameter distributions have curves
Common centroid layout style
Criticism of Layout Optimization
A B B A
-Not quite suitable for matching ratios other than unity
iso-parameter contours for a physical parameter such as tox
tox=4.0nmtox=4.1nm
tox=4.2nm
Circuit Optimizations
The
rmom
eter
Enc
oder
I. Galton and P. Carbone, “A Rigorous Error Analysis of D/A Conversion with Dynamic Element Matching,” IEEE TCAS-II, 1995
.
.x[n]
.
. y[n]
x1[n]
b
x2b[n]
y1[n]
y2[n]
y2b [n]
1-BitDAC1-BitDAC
1-BitDAC
A Digital-to-Analog Converter (DAC)
x[n]
y[n]
frequency
PSD
PSD=power spectral densityDAC=digital to analog converter
•The circuit implements an ideal staircase transfer function between input and output
•Mismatch within DAC’s cause a related performance parameter, PSD, to fail specifications
Circuit Optimizations
The
rmom
eter
Enc
oder
I. Galton and P. Carbone, “A Rigorous Error Analysis of D/A Conversion with Dynamic Element Matching,” IEEE TCAS-II, 1995
Scra
mbl
er..
x[n]..
.
. y[n]
x1[n]
b
x2b[n]
y1[n]
y2[n]
y2b [n]
1-BitDAC1-BitDAC
1-BitDAC
•Scrambler randomly selects 1-Bit DACs to be used in computation
A Low Harmonic DAC
frequency
PSD
Circuit Optimizations
The
rmom
eter
Enc
oder
I. Galton and P. Carbone, “A Rigorous Error Analysis of D/A Conversion with Dynamic Element Matching,” IEEE TCAS-II, 1995
Scra
mbl
er
.
.x[n]..
.
. y[n]
x1[n]
b
x2b[n]
y1[n]
y2[n]
y2b [n]
1-BitDAC1-BitDAC
1-BitDAC
•Scrambler randomly selects 1-Bit DACs to be used in computation
•Mismatch in 1-Bit DAC blocks averaged, compensating deteriorating effects of mismatch on distortion
A Low Harmonic DAC
frequency
PSD
-Architecture specific, hence requires design time
+May be the best way to optimize for a single parameter
-It is usually necessary to optimize a circuit for more than one parameter
Criticism of Circuit Optimizations
The
rmom
eter
Enc
oder
Scra
mbl
er
.
.x[n]..
.
. y[n]
x1[n]
b
x2b[n]
y1[n]
y2[n]
y2b [n]
1-BitDAC1-BitDAC
1-BitDAC
A Low Harmonic DAC
Outline
-Mismatch at transistor, circuit and VLSI levels-Modeling Challenges and Requirements in Deep Sub-Micron -Optimizations to Avoid Mismatch-Past and Present Mismatch Modeling Approaches:
-Electrical & Empirical Models-Layout-based Models-BSIM-based Models-Physics-based Models
-Insights for Future Models for Mismatch-Summary & Conclusions
Electrical / Empirical Parameter-based Mismatch Modeling
•Classical ad-hoc approach by designers
•Threshold voltage mismatch is the most common model
•Worst-case conditions algebraically calculated for parameters
2211 RIRI DD
Derivation of Vos starts with equating drain voltages:
21
2D n ox GS th
WI C V V
L
Drain current formula:
B. Razavi, “Analog CMOS Integrated Circuits,” McGraw-Hill, 2000
Ex:Mismatch modeled at input as common-mode offset voltage for a differential pair
Vos
Iref
VGS1 VGS2
ID1 ID2
+
-
+
-
R1 R2Vos=offset voltage
R=resistance
Differential stage
Derivation of Optimization Equations
21 GSGSos VVV
Vth
LW
LW
R
R
LW
C
IV
D
D
oxn
Dos
/
/2
2
1
•Terms with come from first order Taylor series expansion
Derive Vos to compensate for mismatch:
+
Iref
Vos
VGS1 VGS2
ID1 ID2
+
- -
R1 R2
•Assumption: Independence between parameters
•By adding Vos, mismatch caused degeneration of some parameters like CMRR avoided
CMRR=common mode reject ratio
C. J. Abel, C. Michael, M. Ismail, C.S. Teng and R. Lahri, “Characterization of Transistor Mismatch for Statistical CAD of Submicron CMOS Analog Circuits,” ISCAS, 1993
Consideration of Correlations Between Parameters
•First order Taylor series taken around nominal bias point
,..),( GSth VVfI
Method of Moments formula is applied this formula:
n
ij
n
iji
jiij
n
ii
in PP
P
f
P
fP
P
fPPf
1 11
221
2 )()())((2)()()),..,((
Pi=parameters to be matched
•Correlations are considered through second term in the sum
•Effect of each parameter on the variance of a function of these parameters is individually added by first term in the sum
Critical Analysis of Electrical-Empirical Parameter-based Mismatch Modeling
-Usually used to acquire a worst-case estimation
+Suitable for back-of-the-envelope calculations
-Real results are seldom worst-case, but occur according to a non-uniform probability distribution
-No layout consideration
+Used when starting a design
Outline
-Mismatch at transistor, circuit and VLSI levels-Modeling Challenges and Requirements in Deep Sub-Micron -Optimizations to Avoid Mismatch-Past and Present Mismatch Modeling Approaches:
-Electrical & Empirical Models-Layout-based Models-BSIM-based Models-Physics-based Models
-Insights for Future Models for Mismatch-Summary & Conclusions
Layout Dependent Mismatch Modeling
WL
AP p
22 )(
K. R. Lakshmikumar, R. A. Hadaway, M. A. Copeland, “Characterization and Modeling of Mismatch in MOS Transistors for Precision Analog Design,” JSSC, 1986
Ap=fitting constant for area
•Variance of deviation in a parameter is inversely proportional to the area of the transistors to be matched
J. M. Pelgrom, C. J. Duinmaijer and P. G. Welbers, “Matching Properties of MOS Transistors,” JSSC, 1989
Layout Dependent Mismatch Modeling
22 2 2( ) p
p
AP S D
WL
•Variance of deviation in a parameter is inversely proportional to the area of the transistors to be matched
•Variance of deviation in a parameter is directly proportional to the squared distance between the transistors to be matched
K. R. Lakshmikumar, R. A. Hadaway, M. A. Copeland, “Characterization and Modeling of Mismatch in MOS Transistors for Precision Analog Design,” JSSC, 1986
Ap=fitting constant for area, WL
Sp=fitting constant for distance D
between matched transistors
•A consequent design strategy is laying out matched pairs closer and selecting their areas as large as possible
Extending Distance Parameter in the Fitting Model
G. Tulunay, G. Dundar and A. Ataman “A New Approach to Modeling Statistical Variations in MOS Transistors,” ISCAS, 2002
PyxfP ),()(2
•Distance parameter in Pelgrom’s model extended to a polynomial model by including first order terms => higher accuracy
x,y is location on wafera,b,c,d fitting constants
dcybxyxayxf )(),( 22
•Change in parameter P is modeled as a function of systematic and local variations
•Function f is obtained by fitting regression curves on factory provided manufacturing data
Improved Stochastic Estimation
M. Conti, P. Crippa, S. Orcioni and C. Turchetti, “Layout-Based Statistical Modeling for the Prediction of the Matching Properties of MOS Transistors,” IEEE TCAS-I, 2002
•Extensions to include inter-digitated and cross-coupled geometries through usage of stochastic theory
x
y
M1 and M2 are matched transistors in common-centroid layout style
M1 M2
•Parameters (x,y) formulated using integration over pairs’ areas
•A covariance matrix for (x,y) is formulated using Gaussians as an autocorrelation function•Levenberg-Marquardt least squared method used to fit parameters in the formulations to on-chip measurement data
Incorporation of Effective Lengths
S. J. Lovett, M. Welten and B. Mason “Optimizing MOS Transistor Mismatch,” JSSC, 1998
•Suggestion of usage of effective width and lengths instead, as dotted effective area important for matching
equal initial areas
for MOS channels
(solid lines)
M2M1
transistor channels
•Pelgrom’s equation used with effective lengths of transistors
•Algebraic estimation of L is possible using SPICE models
•Equal nominal mask areas for differing transistor shapes may result in mismatch when lengths in real chip are considered
•If L is nominal, L-L is the effective length caused by penetration of doping under channel region L
Leffective
Higher Regression Order for Area Term
MN
mnm
mn
n
nm
LLWW
CP
,
,
2
)()()(
T. Serrano-Gotarredona and B. Linares-Barranco, “Systematic Width and Length Dependent CMOS Transistor Mismatch Characterization and Simulation,” Analog IC and Signal Processing, 1999
•Higher order regression using Cnm terms as fitting constants
Cnm : fitting constants
•The choice of maximum regression order is questionable
•Builds on previous work and uses effective lengths in denominator
Criticism of Layout Dependent Mismatch Modeling
222
2 )( xpp DS
WL
AP
+Practical for an initial estimate
-More accurate ones requires a costly extraction procedure of fitting constants from the process
-Pelgrom’s model loses accuracy for longer distances
All models based on improving Pelgrom’s Equation:
Outline
-Mismatch at transistor, circuit and VLSI levels-Modeling Challenges and Requirements in Deep Sub-Micron -Optimizations to Avoid Mismatch-Past and Present Mismatch Modeling Approaches:
-Electrical & Empirical Models-Layout-based Models-BSIM-based Models-Physics-based Models
-Insights for Future Models for Mismatch-Summary & Conclusions
C. Michael and M. Ismail, “Statistical Modeling of Device Mismatch for Analog MOS Integrated Circuits,” JSSC, 1992
Statistical Mismatch Modeling
2
M
M MI II
P A R
•Model for space dependent mismatch:
ij : standard deviation between Pi and Pj
Pi : a parameter of i’th transistor
Ri : independent N(0,1) random numbers
AMI : coefficients of the random numbers
R2
-space for parameter P :
|13||23|
|12|P1
P2
P3
MNPMN Ds
•Distances represent variances in -space analysis :
DMN : distance between transistors
sp : fitting constant
R3
A22 A32
A33
-space analysis relates variances in parameters to distances between transistors thus preserving space correlations
•Once P1 is fixed to origin, location of other transistors are found using geometry, then Aij values can be extracted on the axes
C. Michael and M. Ismail, “Statistical Modeling of Device Mismatch for Analog MOS Integrated Circuits,” JSSC, 1992
Principal Component Analysis (PCA) for Preserving Correlations
p
pPP
` •Normalize each parameter
n
i iPQ QPn
ρ1
``1
•Find correlations
•PCA helps preserve parameter correlations by writing each parameter as a function of independent principal components
P,Q : parametersP` : normalized parameter
`12/1 PUC
CUP 2/1` C : principal component vector : diagonal eigenvalue matrixU : eigenvector matrixP` : normalized parameter matrix
•Apply PCA by finding eigenvalues and eigenvectors of C first
C. Michael and M. Ismail, “Statistical Modeling of Device Mismatch for Analog MOS Integrated Circuits,” JSSC, 1992
An Example of PCA Application
654321 19.026.009.046.041.069.0` CCCCCCVFB
654321 12.049.033.009.077.005.0` CCCCCCMUZ
•P` values are unit normal, they are used in -space analysis as random numbers so that parameters correlations are preserved
•Area relationship is obtained through using a fitting constant
VFB` : normalized flat-band voltageMUZ` : normalized zero bulk bias mobilityCi : principal components, chosen unit normal
•PCA helps formulate normalized parameters in terms of independent principal components:
CUP 2/1`
•Remember that -space preserved distance based correlations
Q. Zhang, J. J. Liou, J. R. McMacken, J. Thomson and P. Layman, “SPICE Modeling and Quick Estimation of MOSFET Mismatch Based on BSIM3 Model and Parametric Tests,” JSSC, 2001
An Approach for SPICE Implementation
1xdelvtthV
•Unit normal random numbers, x1 and x2, generated
•SPICE parameters, delvt and delu0, are perturbed using different x1 and x2 each time to simulate a new process
)1(0 22
1/ xrxrdeluVthVth
x1, x2 : unit normal random numbers
r : correlation constantdelvt=deviation in threshold voltagedelu0=deviation in mobility
•SPICE parameters are perturbed directly:
Criticism of BSIM-based Mismatch Modeling
+Considers correlations
-Correlation constants are somewhat inaccurate themselves
-Requires fitting and process related constants
-Does not provide an intuitive understanding of mismatch
+Considers layout
-Parameter inaccuracies due to extraction from wafer may be magnified through PCA
Outline
-Mismatch at transistor, circuit and VLSI levels-Modeling Challenges and Requirements in Deep Sub-Micron -Optimizations to Avoid Mismatch-Past and Present Mismatch Modeling Approaches:
-Electrical & Empirical Models-Layout-based Models-BSIM-based Models-Physics-based Models
-Insights for Future Models for Mismatch-Summary & Conclusions
P. G. Drennan and C. C. McAndrew, “Understanding MOSFET Mismatch for Analog Design,” JSSC, 2003
Physics-based Mismatch Modeling
2
2 2
ie pi i
e
p
•Basic idea : Mismatch is a physical phenomenon and physical parameters are independent
•If enumeration factors are such that |e| > |i|, estimation of physical parameter variances from electrical measurements is also possible
•pi’s can be Vfb, Tox, W, L, 0, Nsub, etc.
e : electrical parameterp : physical parameter
• Variance in electrical parameters written in terms of geometry dependent variances of physical parameters
pi’s are dependant on size and distance of transistors
•Due to complex formulas, CAD tools required
gm
Independent Normal
Correlated?
pdf?Level1
Level4
Level0
•Random effects mimicked through assigning pdf’s to Level0 parametersAn Tractable Physics-based Mismatch Model
nVFBNSUBLW
Vth Cox
tox
ID
k Level2
Level3
Correlated?
pdf?
Correlated?
pdf?
Correlated?
pdf?
R. O. Topaloglu and A. Orailoglu, “Mismatch in the Deep Sub-Micron Era : From Physics to Circuits,” ASP-DAC, 2004
Graph structured using SPICE formula hierarchy
Each node is a parameter
Connectivity Based Traversal•Proposed approach provides a manually tractable estimation
Vth
VT0
NSUB
PHI
Vth=f3(PHI,VT0)
VT0=f2(PHI,NSUB)
VT0=f1(NSUB) L1
L0
L2
L3
SVth = SVth * SPHI + SVth * ( SVT0 + SVT0 * SPHI )NSUB PHI NSUB VT0 NSUB PHI NSUB
R. O. Topaloglu and A. Orailoglu, “Mismatch in the Deep Sub-Micron Era : From Physics to Circuits,” ASP-DAC, 2004
•Chain rule used to relate a high level parameter to physical onesL0 : level 0
x
y
y
xS x
y
Connectivity Based Traversal
Vth
VT0
NSUB
PHI
Vth=f3(PHI,VT0)
VT0=f2(PHI,NSUB)
VT0=f1(NSUB) L1
L0
L2
L3
SVth = SVth * SPHI + SVth * ( SVT0 + SVT0 * SPHI )NSUB PHI NSUB VT0 NSUB PHI NSUB
R. O. Topaloglu and A. Orailoglu, “Mismatch in the Deep Sub-Micron Era : From Physics to Circuits,” ASP-DAC, 2004
•Proposed approach provides a manually tractable solution•Chain rule used to relate a high level parameter to physical ones
Connectivity Based Traversal
Vth
VT0
NSUB
PHI
Vth=f3(PHI,VT0)
VT0=f2(PHI,NSUB)
VT0=f1(NSUB) L1
L0
L2
L3
SVth = SVth * SPHI + SVth * ( SVT0 + SVT0 * SPHI )NSUB PHI NSUB VT0 NSUB PHI NSUB
R. O. Topaloglu and A. Orailoglu, “Mismatch in the Deep Sub-Micron Era : From Physics to Circuits,” ASP-DAC, 2004
•Proposed approach provides a manually tractable solution•Chain rule used to relate a high level parameter to physical ones
Bridging Physical Aspects to Circuit Parameters
circuit parameters
design parameters
SPICE parameters
L0
L1
L2
L3
L4
L5
L6
L7
L8
gm
CMRRrout
W LNSS T NSUB TOX
COX
GAMMA
PHI
egap
Vth
Id
PHIms
VFB
VT0
+Obviates the need for the use of correlations
+Provides an intuitive understanding for mismatch
Critical Analysis of Physics-based Mismatch Modeling
-Are all physical reasons accounted for?
+Suitable for diagnosis
-May physical reasons be correlated to chemical or even quantum-based reasons?
Outline
-Mismatch at transistor, circuit and VLSI levels-Modeling Challenges and Requirements in Deep Sub-Micron -Optimizations to Avoid Mismatch-Past and Present Mismatch Modeling Approaches:
-Electrical & Empirical Models-Layout-based Models-BSIM-based Models-Physics-based Models
-Insights for Future Models for Mismatch-Summary & Conclusions
Sense Amplifier
CLK M2
M3 M4
M5 M1
Dependence of gain to variations in M4 when M3 nominal @800MHz
Gain vs NCH Gain vs TOXGain vs W
Insights for Future Models
Techniques for estimation of non-Gaussian distributions necessary
•Higher level circuit parameters for large variations indicate highly non-linear relationships; as opposed to linear ones observed for lower level parameters•Signifies that a Gaussian assumption is not accurate since linear sums of Gaussians are Gaussians, which can be completely described by (,)
Differential stage biased with current mirror
Iref
A simulation-based Proof by Contradiction
•Matched transistor group G1 introduced random mismatch to predict probability distribution function (pdf) of circuit gain
G1
pdf of gain
~Gaussian
Differential stage biased with current mirror
Iref
G2
G3
pdf of gain
•Matched transistor groups G2 and G3 introduced random mismatch to predict probability distribution function (pdf) of circuit gain
non-Gaussian
A simulation-based Proof by Contradiction
Differential stage biased with current mirror
Iref
G2
G3
G1
pdf of gain
Closer-to-real probability distribution should be estimated through models
non-Gaussian
•High level circuit parameters may not exhibit a Gaussian-like pdf when physical input parameters are assigned independent Gaussian distributions.
A simulation-based Proof by Contradiction
Summary:
•Circuit optimizations : remedy for a single circuit parameter
•Layout optimizations : used whenever possible yet insufficient
•Electrical / Empirical models : used when starting a design
•Layout-based models : Pelgrom’s Equation used to incorporate layout information
•BSIM-based models : used for direct SPICE implementation
•Physics-based models : used for better intuition and accuracy
Conclusions
•Deep sub-micron modeling needs has been identified as avoiding worst-case limits, consideration of random effects and an early estimation of mismatch
•Future models should target to obtain closer-to-real probability distribution functions of performance parameters
•A spectrum of mismatch models has been presented