mosfet and set co-simulation -...

MOSFET and SET co-simulation

Chaitanya Sathe

II MEDepartment of Electrical communication Engineering

Indian Institute of ScienceBangalore

India

December 7, 2006

Outline

I Brief introduction of Monte Carlo Method

I Monte Carlo Simulator for Single-Electron Tunnel Devices (SIMON)

I Master Equation

I Macro-modeling of SET

I CAD framework for SET-MOS Co-simulation

I Case study

Chaitanya, IISc MOSFET and SET co-simulation 2/28

What are Monte Carlo Methods ?

I Can be viewed as a branch of experimental mathematics in whichone uses random numbers to conduct experiments.

I Typically the experiments are done on a computer using anywherefrom hundreds to billions of random numbers

I Very roughly speaking, we can categorize Monte Carlo experimentsinto the following two broad classes

I Direct simulation of a naturally random systemI Addition of artificial randomness to a system,followed by simulation

of the new system


Estimating π using Monte Carlo Method

+1

+1

−1

− 1

π =40

13= 3.079

If we use 10000 points we get π = 3.1728


Monte Carlo Method

I Tunnel events can be modeled as discrete events as long as theelectrons are confined on quantum dots

RT > Rq = he2 = 25.813Ω

I The tunnel events are assumed to be independent and exponentiallydistributed.


Flow Chart of Monte Carlo Method

PSfrag replacements

Parse circuit file

Build capacitance matrix and prepare

for computation of node potential and free energy

for each possible tunnel event:

compute free energy change

compute tunnel rate

compute duration to next tunnel event

Choose event with smallest duration and

update charges accordingly

time limit ?

accuracy limit ?

event limit?

yes

no

stop


Monte Carlo Approach

I Extract the capacitance matrix

(

q

Q

)

=

(

Ca Cb

Cb Cd

) (

V

v

)

(1)

I The tunnel rate for one tunnel junction is given by

Γ =∆F

e2RT (1 − exp(−∆FkT

))(2)

I Helmholtz free energy is given by

F = U − W =1

2(q, v).

(

V

Q

)

− W (3)

with

W =∑

n

∫

Vn(t)in(t)dt (4)


Monte Carlo ApproachI Once all tunnel rates are known, the actually occurring event is

determined with a Monte Carlo MethodI The probability that a tunnel event out of a state happens at τ and

not earlier isP0(τ) = e−Γτ (5)

I The expression for the duration to the next tunnel event is therefore

τ = −log r

Γ(6)

where r is an evenly distributes random number from the interval[0,1].

I The event among all possible ones, with the shortest duration istaken.

I After a tunnel event, node charges and node voltages generallychange hence the free energy, and one has to compute all possiblerates again.

I The loop is performed many times to simulate the transport ofelectrons through the network


Master Equation method

I If we assume that electrons cannot probe the past, that is, theypossess no memory, and thus their tunnel rate depends only on themomentary state of the system then one can describe such a systemusing master equation

∂p(S , t)

∂t=

∫

dS′

[Γ(S |S′

)p(S′

, t) − Γ(S′

|S)p(S , t)] (7)

I If the states are discrete, the master equation becomes

∂p(S , t)

∂t=

∑

j 6=i

[ΓijPj (t) − ΓjiPi (t)] (8)


Master Equation methodI The master equation is a description of the underlying Markov

Process of electrons tunneling from island to island.

I A state is a specific charge distribution , each node or quantum dotis occupied by a certain number of electrons.

I The master equation method for the simulation of single-electroncircuits tries to solve these equations.

PSfrag replacements1

2

3

4

5

Figure: State transition diagram


Master Equation method

I Although the Master equation method gives theoretically accurateresults, it has many other impracticability’s that limit its accuracyand usability.

I The starting point of the ME is the set of all relevant states a circuitwill occupy during operation.

I There is no way to filter out these states a priori from an arbitrarycircuit.

I One way is to include many more states than would be relevant,which results in extremely long simulation times and sometimes badnumerical stability.

I Another way out is to use an adaptive algorithm which alleviates thisproblem.


Advantages AND Disadvantages of A Monte Carlo

Approach

I It gives better transient and dynamic characteristics of SET circuitsbecause it models the underlying microscopic physics in a very directmanner.

I It is not required to find the relevant states before one can start withthe actual simulation as in the case of a master equation.

I It is easy to trade accuracy with simulation time, and therefore canquickly achieve approximate results of very large circuits.

I One major disadvantage of the Monte Carlo method is simulatingco-tunneling events


Problems When Simulating Co-tunneling

PSfrag replacements

jail topocean

elasticinelastic

Figure: Co-tunneling via a virtual intermediate state

I A co-tunnel event has a very rare occurrence compared to a normaltunnel event.

I Standard variance reducing techniques do not work because in atypical MC simulation run, rare states are very likely not even visitedonce.

I This problem is addressed using an MC-ME approach.


Lets Summarize ...

I Monte Carlo (MC) Simulation Technique:I MC approach starts with all possible tunneling events, calculates

their probabilities, and choose one of the possible events randomly

and weighted according to their probabilities.I Examples : SIMON, MOSES, KOSEC, and SENECAI Probably most accurate method, but time consuming for large circuit

simulation.I None of these reported simulators offer any co-simulation

environment with MOSFET devices

I Master Equation (ME) Simulation MethodI The ME , is a description for the underlying Markov process of

electron tunneling from island to island.I Need to know all possible states of the circuitI Example : SETTRANSI Co-simulation ?


Motivation For CMOS-SET Hybridization

SET CMOSN Nano-scale feature size N High gain and current driveN Unique Coulomb blockade N High speed

Adv oscillation characteristicsN Ultra-low power dissipation N Very matured fabrication

technologyH Low current drive H Sub-10nm Physical limits

Limi H Lack of room temperature H Power densityoperable technology

H Back-ground charge effect


Possible CAD framework for CMOS-SET Co-simulation

I Macro Modeling of SET device using any SPICE Simulator

I CAD framework using AHDL (Analog Hardware DescriptionLanguage ) for implementing compact models like MIB


Macro Modeling of SET deviceI The following assumptions are made in compact simulators like

SPICE to simulate the characteristics of any given circuit topology.I Once the model parameters of a device are determined from the

device simulator or other modeling tools, it can be used in the whole

circuit.I The I-V characteristics of the device are affected by neighboring

transistors only through the changes of the terminal voltages of

those transistors, the interaction between adjacent devices is usually

neglected.

I The second assumption in the case of SET circuits may not ingeneral be valid.

I When the size of interconnection between the SET devices is largeenough, the interconnection acts like a reservoir rather than aCoulomb island.

I In that case, the Coulomb islands of SET’s become isolated by theinterconnection and the interaction among neighboring SET’s is notsignificant.

I CL ≥ 6.25Cj for the compact model to be valid


Macro Modeling of SET device

−

+

+

−

PSfrag replacements

gate

drain

source

gate drain

source

RG

R1R2 R3

D1 D2

vpvp

Figure: Equivalent circuit of a single-electron transistor



I Symmetric features of the drain-source current-voltage (Ids − Vds)characteristics are incorporated with two branches consisting of thecombinations of resistors, diodes and voltagesources(R2/D1/VPandR3/D2/VP).

I The directions of D1 and VP are opposite with those of D3 and V3

to have adequate current flow in both positive and negativedrain-source bias.

I The charging energy, periodically changing as a function of the gatebias, is included in R1, R2, and R3 where the cosine of the gate biasis used

R1(VG ) = CR1 + CR2cos(CF1VG ) (9)

R2(VG ) = R3(VG ) =CVp

CI2 −2CVp

R1(VG )

(10)



I The parameters, CF1, CVp , CI2, CR1, andCR2 are used to fit the I-Vcharacteristics at various gate biases.

I Since the SET characteristics of SET’s strongly depend on T, themacromodel parameters are a function of T.

I SPICE macromodeling of single-electron transistors can be used forefficient circuit simulations, these macromodels produce simulationresults with reasonable accuracy and are faster than Monte Carlosimulations.

I This technique is non-physical (empirical) in nature.

I The technique is also not scalable


CMOS-SET CO-SIMULATION using VERLOG-A

I Verilog-A Hardware Description Language (HDL) is a behaviorallanguage for analog and mixed signal systems.

I Verlog-A enables to compactly model the device behaviour

PSfrag replacements Verilog-A

SET model

VerilogA

Compiler

C file

C-compiler

.so fileSMARTSPICE

RUN

SOURCE

SPICE

NETLIST


CMOS-SET CO-SIMULATION using VERLOG-A

VerilogA SET Module Standard spice Netlistmodule set (drain,gate1,gate2,source); .verilog ”set.va” // includes the SET module

inout drain,gate1,gate2,source;

electrical drain,gate1,gate2,source; M1 2 2 4 4 MOD1 L1=0.5U W=0.8U

M2 3 2 4 4 MOD1 L=0.5U W=0.8U

//Default value of the model parameters .....................

parameter real CTS = 1e-18,CTD = 1e-18 .....................

parameter real CG = 2e-18,CG2 =0

parameter real RTD = 1e6,RTS = 1e6 VDD 1 0 5

parameter real XI=0 VSS 4 0 5

.....................

.....................

analog YVLGmyset 1 3 6 7 0 set CG1=2e-18

begin // Instantiate SET

//MIB subroutine

I(drain,source) =.... .dc VIN 0.0 0.04 0.008

end .end

endmodule


Case Study: Modeling Analysis of Noise Margin in SET

Logic

VOH

VOL

VIH

VIL

"0"

"1"

NMH

NML

GateOutput

GateInput

Figure: Definition of Noise Margin



Logic

−20 −15 −10 −5 0 5 10 15 20−20

−15

−10

−5

0

5

10

15

20

VOUT

VIN

VOH

VOL

VIL

VIH

A

B

C

D

VMAX

VMIN

region 1

region 2

region 3

VSS

VDD

Figure: Definition of VOH , VOL, VIL, VIH

NMH = NML =(

1 −CT

CG

) VDD

CG

CT+ 2CT

CG+ 1

(11)



Logic

10−2

10−1

100

101

102

−0.5

0

0.5

1

1.5

2

2.5

3

Temperature in K

NM

H (

NM

L )

in m

V

CG

:CT=3C

G:C

T=5

CG

:CT=7

β>40

T=11.5K

Figure: Effect of Temperature on Noise Margin



Logic

−0.2 −0.1 0 0.1 0.2−20

−15

−10

−5

0

5

10

15

20

25

Background charge in e

NM

in m

V

NMH

NML

Figure: Effect of Background on Noise Margin



Logic

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−20

−10

0

10

20V

1 ,V

2,V

10 (

mV

)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.4

−0.2

0

0.2

0.4

Time (µs)

VIN

(m

V)

VIN

V1

V2

V10

Figure: Simulation result of cascaded chain of inverters


References

I Lectures on Monte Carlo Methods. Neal Madras, AmericanMathematical Society

I SIMON - A Simulator for Single-Electron Tunnel Devices andCircuits. Christoph Wasshuber et al, IEEE Transactions ofComputer-Aided Design of Integrated Circuits and Systems VOL.16,NO. 9,SEP 1997

I Computational Single - Electronics. Christoph Wasshuber, Springer

I Macromodeling of single electron transistors for efficient circuitsimulation. Y.S.Yu et al ,IEEE Trans.Elec.Dev vol46 no8

I Hybrid CMOS Single Electron Transistor Device and Circuit Design.Santanu Mahapatra et al , Artech House Publication 2006

I Modeling and analysis of noise Margin in SET Logic. Chaitanya etal, International Conference on VLSI Design 2007


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