semiconductor devicemodeling

7/27/2019 Semiconductor DeviceModeling

1/33

1

Semiconductor Device Modeling

D. Vasileska, D. Mamaluy, H. R. Khan, K. Raleva and S. M. Goodnick

Department of Electrical Engineering

Arizona State University, Tempe, AZ 85287-5706

Corresponding author: [email protected]

In this review paper we describe a hierarchy of simulation models for modelingstate of the art devices. Within the semiclassical simulation arena, emphasis isplaced on particle-based device simulations that can model devices operatingfrom diffusive down to ballistic regime. In here, we also describe in detail theproper inclusion of the short-range Coulomb interactions using real-space ap-proach that eliminates double-counting of the Coulomb interaction (due to its par-

tial inclusion via the solution of the Poisson equation). Regarding the quantumtransport approaches, emphasis is placed on the description of the CBR methodthat is implemented in ASUs 2D and 3D NEGF device simulator (that is used formodeling 10 nm gate length FinFETs, which are likely to be the next generationof devices that the Industry will be mass-producing in year 2015). Comparisonwith existing experimental data is presented to verify the accuracy and speed ofthe quantum transport simulator. We conclude this review paper by emphasizingwhat kind of semiconductor tools will be needed to model next generation devic-es.

Keywords: Semiclassical and Quantum Transport, particle-based device simu-lations, Boltzmann transport equation, electron-electron and elec-tron-ion interactions, quantum transport, Landauers approach,Greens functions, Contact Block Reduction method, FinFET devic-es


2/33

2

CONTENTS

1. Computational Electronics2. Semiclassical Transport Approaches

2.1. Drift-Diffusion Model2.2. Hydrodynamic Model

2.2.1. Extension of the Drift-Diffusion Model2.2.2. Stratons Approach2.2.3. Balance Equation Model

2.3 Particle-Based Device Simulations2.3.1. Free-Flight Generation2.3.2. Final State After Scattering2.3.3. Ensemble Monte Carlo Simulation2.3.4. Device Simulation Using Particles2.3.5. Simulation Example2.3.6. Direct Treatment of the Inter-Particle Interactions

3. Quantum Transport3.1. Open System3.2. Evaluation of the Current Density3.3. Landauer-Buttiker Formalism and Related Methods3.4. Contact Block Reduction Method

3.4.1. Bound States Treatment3.4.2. Energy Discretization3.4.3. Self-Consistent Solution3.4.4. Device Hamiltonian, Algorithm and SomeNumerical Details3.4.5. Simulation Example

4. ConclusionsReferences

1. COMPUTATIONAL ELECTRONICS

As semiconductor feature sizes shrink into the nanome-ter scale regime, even conventional device behaviorbecomes increasingly complicated as new physicalphenomena at short dimensions occur, and limitationsin material properties are reached 1. In addition to theproblems related to the understanding of actual opera-

tion of ultra-small devices, the reduced feature sizesrequire more complicated and time-consuming manu-facturing processes. This fact signifies that a pure trial-and-error approach to device optimization will becomeimpossible since it is both too time consuming and tooexpensive. Since computers are considerably cheaperresources, simulation is becoming an indispensabletoolfor the device engineer. Besides offering the possibilityto test hypothetical devices which have not (or couldnot) yet been manufactured, simulation offers uniqueinsight into device behavior by allowing the observa-tion of phenomena that can not be measured on realdevices. Computational Electronics 2,3,4in this contextrefers to the physical simulation of semiconductor de-vices in terms of charge transport and the correspondingelectrical behavior. It is related to, but usually separatefrom process simulation, which deals with variousphysical processes such as material growth, oxidation,impurity diffusion, etching, and metal deposition inher-ent in device fabrication 5 leading to integrated circuits.Device simulation can be thought of as one componentof technology for computer-aided design (TCAD),which provides a basis for device modeling, which

deals with compact behavioral models for devices andsub-circuits relevant for circuit simulation in commer-cial packages such as SPICE6. The relationship betweenvarious simulation design steps that have to be followedto achieve certain customer need is illustrated in Figure1.

The goal of Computational Electronics is to pro-vide simulation tools with the necessary level of sophis-tication to capture the essential physics while at thesame time minimizing the computational burden so thatresults may be obtained within a reasonable time frame.Figure 2 illustrates the main components of semicon-ductor device simulation at any level. There are twomain kernels, which must be solved self-consistentlywith one another, the transport equations governingcharge flow, and the fields driving charge flow. Bothare coupled strongly to one another, and hence must besolved simultaneously. The fields arise from externalsources, as well as the charge and current densitieswhich act as sources for the time varying electric and

magnetic fields obtained from the solution of Max-wells equations. Under appropriate conditions, onlythe quasi-static electric fields arising from the solutionof Poissons equation are necessary.

Customer Need

Process Simulation

Device Simulation

Parameter Extraction

Circuit Level Simulation

yes

Computational

Electronics

no

Fig. 1. Design sequence to achieve desired customer need.

The fields, in turn, are driving forces for chargetransport as illustrated in Figure 3 for the various levels

of approximation within a hierarchical structure rangingfrom compact modeling at the top to an exact quantummechanical description at the bottom. At the very be-ginnings of semiconductor technology, the electricaldevice characteristics could be estimated using simpleanalytical models (gradual channel approximation forMOSFETs) relying on the drift-diffusion (DD) formal-ism. Various approximations had to be made to obtainclosed-form solutions, but the resulting models cap-tured the basic features of the devices7. These approxi-


3/33

3

mations include simplified doping profiles and devicegeometries. With the ongoing refinements and im-provements in technology, these approximations losttheir basis and a more accurate description was re-quired. This goal could be achieved by solving the DDequations numerically. Numerical simulation of carriertransport in semiconductor devices dates back to thefamous work of Scharfetter and Gummel8, who pro-posed a robust discretization of the DD equations whichis still in use today.

Fig. 2 Schematic description of the device simulation se-quence.

Model Improvements

Compact models Appropriate for Circuit

Design

Drift-Diffusion

equations

Good for devices down to

0.5 m, include (E)

HydrodynamicEquations

Velocity overshoot effect can

be treated properly

Boltzmann Transport

EquationM o n t e C a r l o / C A m e t h o d s

Accurate up to the classicallimits

QuantumHydrodynamics

Keep all classical

hydrodynamic features +quantum corrections

Semi-classicalapproaches

Quantumapproaches

Q u a n t u m - K i n e t i c E q u a t i o n

( L i o u v i l l e , W i g n e r - B o l t z m a n n )

Accurate up to single particledescription

Green's Functions methodIncludes correlations in both

space and time domain

QuantumM o n t e C a r l o / C A m e t h o d s

Keep all classicalfeatures + quantum corrections

Direct solution of the n-bodySchrdinger equation

Can be solved only for smallnumber of particles

Model Improvements

Compact models Appropriate for Circuit

Design

Drift-Diffusion

equations

Good for devices down to

0.5 m, include (E)

HydrodynamicEquations

Velocity overshoot effect can

be treated properly

Boltzmann Transport

EquationM o n t e C a r l o / C A m e t h o d s

Accurate up to the classicallimits

QuantumHydrodynamics

Keep all classical

hydrodynamic features +quantum corrections

Semi-classicalapproaches

Quantumapproaches

Q u a n t u m - K i n e t i c E q u a t i o n

( L i o u v i l l e , W i g n e r - B o l t z m a n n )

Accurate up to single particledescription

Green's Functions methodIncludes correlations in both

space and time domain

QuantumM o n t e C a r l o / C A m e t h o d s

Keep all classicalfeatures + quantum corrections

Direct solution of the n-bodySchrdinger equation

Can be solved only for smallnumber of particles

Fig. 3. Illustration of the hierarchy of transport models.

However, as semiconductor devices were scaled in-to the submicrometer regime, the assumptions underly-ing the DD model lost their validity. Therefore, thetransport models have been continuously refined andextended to more accurately capture transport phenom-ena occurring in these devices. The need for refinementand extension is primarily caused by the ongoing fea-ture size reduction in state-of-the-art technology. As the

supply voltages can not be scaled accordingly withoutjeopardizing the circuit performance, the electric fieldinside the devices has increased. A large electric field,which rapidly changes over small length scales, givesrise to non-local and hot-carrier effects which begin todominate device performance. An accurate descriptionof these phenomena is required and is becoming a pri-mary concern for industrial applications.

To overcome some of the limitations of the DDmodel, extensions have been proposed which basicallyadd an additional balance equation for the average car-rier energy9. Furthermore, an additional driving term isadded to the current expression which is proportional tothe gradient of the carrier temperature. However, a vastnumber of these models exist, and there is a considera-ble amount of confusion as to their relation to each oth-er. It is now a common practice in industry to usestandard hydrodynamic models in trying to understandthe operation of as-fabricated devices, by adjusting anynumber of phenomenological parameters (e.g. mobility,

impact ionization coefficient, etc.). However, suchtools do not have predictive capability for ultra-smallstructures, for which it is necessary to relax some of theapproximations in the Boltzmann transport equation10.Therefore, one needs to move downward to the quan-tum transport area in the hierarchical map of transportmodels shown in Figure 3, where, at the very bottomwe have the Green's function approach11,12,13. The latteris the most exact, but at the same time the most difficultof all. In contrast to, for example, the Wigner functionapproach (which is Markovian in time), the Green'sfunctions method allows one to consider simultaneouslycorrelations in space and time, both of which are ex-

pected to be important in nano-scale devices. However,the difficulties in understanding the various terms in theresultant equations and the enormous computationalburden needed for its actual implementation make theusefulness in understanding quantum effects in actualdevices of limited values. For example, the only suc-cessful utilization of the Green's function approachcommercially is the NEMO (Nano-Electronics Model-ing) simulator14, which is effectively 1D and is primari-ly applicable to resonant tunneling diodes.

From the discussion above it follows that, contraryto the recent technological advances, the present stateof the art in device simulation is currently lacking in the

ability to treat these new challenges in scaling of devicedimensions from conventional down to quantum scaledevices. For silicon devices with active regions below0.2 microns in diameter, macroscopic transport descrip-tions based on drift-diffusion models are clearly inade-quate. As already noted, even standard hydrodynamicmodels do not usually provide a sufficiently accuratedescription since they neglect significant contributionsfrom the tail of the phase space distribution function inthe channel regions15,16. Within the requirement of self-


4/33

4

consistently solving the coupled transport-field problemin this emerging domain of device physics, there areseveral computational challenges, which limit this abil-ity. One is the necessity to solve both the transport andthe Poisson's equations over the full 3D domain of thedevice (and beyond if one includes radiation effects).As a result, highly efficient algorithms targeted to high-end computational platforms (most likely in a multi-processor environment) are required to fully solve eventhe appropriate field problems. The appropriate level ofapproximation necessary to capture the proper non-equilibrium transport physics, relevant to a future de-vice model, is an even more challenging problem bothcomputationally and from a fundamental physicsframework.

2. SEMICLASSICAL TRANSPORTAPPROACHES

2.1 Drift-Diffusion Model

In Section 1, we discussed the various levels of approx-imations that are employed in the modeling of semi-conductor devices. The direct solution of the full BTEis challenging computationally, particularly when com-bined with field solvers for device simulation. There-fore, for traditional semiconductor device modeling, thepredominant model corresponds to solutions of the so-called drift-diffusion equations, which are local interms of the driving forces (electric fields and spatialgradients in the carrier density), i.e. the current at a par-ticular point in space only depends on the instantaneouselectric fields and concentration gradient at that point.

The complete drift-diffusion model is based on the fol-lowing set of equations:

1. Current equations

( ) ( )

( ) ( )

n n n

p p p

dnJ qn x E x qD

dx

dnJ qp x E x qD

dx

= +

= (1)

2. Continuity equations1

1

n n

p p

nU

t q

pU

t q

= +

= +

J

J

(2)

3. Poisson's equation( ) ( )D AV p n N N

+ = + , (3)where Unand Up are the net generation-recombinationrates. The continuity equations are the conservationlaws for the carriers. A numerical scheme which solvesthe continuity equations should

1. Conserve the total number of particles insidethe device being simulated.

2. Respect local positive definite nature of carrierdensity. Negative density is unphysical.

3. Respect monotonicity of the solution (i.e. itshould not introduce spurious space oscilla-tions).

Conservative schemes are usually achieved by subdivi-sion of the computational domain into patches (boxes)surrounding the mesh points. The currents are then de-fined on the boundaries of these elements, thus enforc-ing conservation (the current exiting one element side isexactly equal to the current entering the neighboringelement through the side in common). In the absence ofgeneration-recombination terms, the only contributionsto the overall device current arise from the contacts.Remember that, since electrons have negative charge,the particle flux is opposite to the current flux. Whenthe equations are discretized, using finite differences forinstance, there are limitations on the choice of meshsize and time step17:

1. The mesh size x is limited by the Debyelength.

2. The time step is limited by the dielectric relax-ation time.

A mesh size must be smaller than the Debye lengthwhere one has to resolve charge variations in space. Asimple example is the carrier redistribution at an inter-face between two regions with different doping levels.Carriers diffuse into the lower doped region creatingexcess carrier distribution which at equilibrium decaysin space down to the bulk concentration with approxi-mately exponential behavior. The spatial decay constantis the Debye length

2

B

D

k T

L q N

= (4)whereNis the doping density. In GaAs and Si, at roomtemperature the Debye length is approximately 400 when 16 310N cm and decreases to about only 50 when 18 310N cm .

The dielectric relaxation time, on the other hand, isthe characteristic time for charge fluctuations to decayunder the influence of the field that they produce. Thedielectric relaxation time may be estimated using

drtqN

= (5)

The drift-diffusion semiconductor equations consti-

tute a coupled nonlinear set. It is not possible, in gen-eral, to obtain a solution directly in one step, but a non-linear iteration method is required. The two most popu-lar methods for solving the discretized equations are theGummel's iteration method18 and the Newton's meth-od19. It is very difficult to determine an optimum strate-gy for the solution, since this will depend on a numberof details related to the particular device under study.


5/33

5

Finally, the discretization of the continuity equa-tions in conservation form requires the determination ofthe currents on the mid-points of mesh lines connectingneighboring grid nodes. Since the solutions are accessi-ble only on the grid nodes, interpolation schemes areneeded to determine the currents. The approach byScharfetter and Gummel8 has provided an optimal solu-tion to this problem, although the mathematical proper-ties of the proposed scheme have been fully recognizedmuch later.

.2.2. Hydrodynamic Model

The current drive capability of deeply scaled MOSFETsand, in particular, n-MOSFETs has been the subject ofinvestigation since the late 1970s. First it was hypothe-sized that the effective carrier injection velocity fromthe source into the channel would reach the limit of thesaturation velocity and remain there as longitudinalelectric fields increased beyond the onset value for ve-locity saturation. However, theoretical work indicatedthat velocity overshoot can occur even in silicon20, andindeed it is routinely seen in the high-field region nearthe drain in simulated devices using energy balancemodels or Monte Carlo. While it was understood thatvelocity overshoot near the drain would not help currentdrive, experimental work21,22 claimed to observe veloci-ty overshoot near the source, which of course would bebeneficial and would make the drift-diffusion modelinvalid.

In the computational electronics community, thenecessity for the hydrodynamic (HD) transport model isnormally checked by comparison of simulation results

for HD and DD simulations. Despite the obvious factthat, depending on the equation set, different principalphysical effects are taken into account, the influence onthe models for the physical parameters is more subtle.The main reason for this is that in the case of the HDmodel, information about average carrier energy isavailable in form of carrier temperature. Many parame-ters depend on this average carrier energy, e.g., the mo-bilities and the energy relaxation times. In the case ofthe DD model, the carrier temperatures are assumed tobe in equilibrium with the lattice temperature, that is

C LT T= , hence, all energy dependent parameters haveto be modeled in a different way.

2.2.1 Extensions of the Drift-Diffusion model

In the DD approach, the electron gas is assumed to bein thermal equilibrium with the lattice temperature( )n LT T= . However, in the presence of a strong elec-tric field, electrons gain energy from the field and thetemperature nT of the electron gas is elevated. Since

the pressure of the electron gas is proportional to

B nnk T , the driving force now becomes the pressuregradient rather then merely the density gradient. Thisintroduces an additional driving force, namely, the tem-perature gradient besides the electric field and the den-sity gradient. Phenomenologically, one can write

( )J En n T n q n D n nD T = + + (6)

where TD is the thermal diffusivity.

2.2.2. Strattons Approach

One of the first derivations of extended transport equa-tions was performed by Stratton23. First the distributionfunction is split into the even and odd parts

0 1(k, r) (k, r) (k, r)f f f= + . (7)

From 1 1( k, r) (k, r)f f = , it follows that 1 0f = .Assuming that the collision operator C is linear andinvoking the microscopic relaxation time approxima-tion for the collision operator

[ ]( , r)

eqf fC f

= (8)

the BTE can be split into two coupled equations. Inparticularf1 is related tof0 via

( )1 r 0 k 0, r v Eq

f f f =

. (9)

The microscopic relaxation time is then expressed by apower law

0( )

p

B Lk T

= . (10)

When f0 is assumed to be heated Maxwellian distribu-

tion, the following equation system is obtainedJ

nq

t

=

(11)

( )J E B nqn k n T = +

( ) ( )3 3

S E J2 2

n LB n B

T Tn k nT k n

= +

( )25

S E+2

BB n n

kn p nk T n T

q

=

Equation for the current density can be rewritten as:

( )1B Bn n nk k

J q nE T n n T

q q

= + + +

(12a)

withln

lnn

n

n n

T

T T

= =

(12b)

which is commonly used as a fit parameter with valuesin the range [-0.5,-1.0]. For n =-1.0, the thermal distri-bution term disappears. The problem with Eq. (10) for is that p must be approximated by an average value to


6/33

6

cover the relevant processes. In the particular case ofimpurity scattering, p can be in the range [-1.5,0.5],depending on charge screening. Therefore, this averagedepends on the doping profile and the applied field;thus, no unique value forp can be given. Note also thatthe temperature Tn is a parameter of the heated Max-wellian distribution, which has been assumed in thederivation. Only for parabolic bands and a Maxwelliandistribution, this parameter is equivalent to the normal-ized second-order moment.

2.2.3. Balance Equations Model

The first three balance equations, derived by takingmoments of Boltzmann Transport Equation (BTE), takethe form

( )

2

0

1J

2 1

* *

1F E J

n n

z izz z

i i m

W

E

nS

t e

J We neE J

t m x m

WW W

t

= +

= +

= +

(13)

The balance equation for the carrier density introducesthe carrier current density, which balance equation in-troduces the kinetic energy density. The balance equa-tion for the kinetic energy density, on the other hand,introduces the energy flux. Therefore, a new variableappears in the hierarchy of balance equations and theset of infinite balance equations is actually the solutionof the BTE. The momentum and energy relaxation rates,that appear in Eq. (13) are ensemble averaged quantities.

For simple scattering mechanisms one can utilize thedrifted-Maxwellian form of the distribution function,but for cases where several scattering mechanisms areimportant, one must use bulk Monte Carlo simulationsto calculate these quantities.

One can express the energy flux that appears in Eq.(13) in terms of the temperature tensor. The energy flux,is calculated using

1( ) ( , , )

p

F v p r pW E f tV

= , (14)

which means that the i-th component of this vectorequals to

Wi di B ij dj i

j

F v W nk T v Q= + + (15)

wherei

Q is the component of the heat flux vectorwhich describes loss of energy due to flow of heat outof the volume. To summarize, the kinetic energy fluxequals the sum of the kinetic energy density times ve-locity plus the velocity times the pressure, which actu-ally represents the work to push the volume plus theloss of energy due to flow of heat out. In mathematicalterms this is expressed as

F v v QW BW nk T = + +

. (16)

With the above considerations, the momentum and theenergy balance equations reduce to

( ) ( )

2

0

2 1 1* 2 *

1v Q v E J

ziz B iz z z

i i m

B n

E

J e neK nk T E J

t m x m

W W nk T W W t

= + +

= + + +

(17)

For displaced-Maxwellian approximation for the distri-bution function, the heat flux Q = 0. However,Blotekjaer24 has pointed out that this term must be sig-nificant for non-Maxwellian distributions, so that aphenomenological description for the heat flux, of theform described by Franz-Wiedermann law, which statesthat

cT= Q (18)is used, where is the thermal or heat conductivity. Insilicon, the experimental value of is 142.3 W/mK.The above description for Q actually leads to a closedset of equations in which the energy balance equation isof the form

( )

( )01

c B c n

E

WW T nk T

t

W W

= + +

v v E J

(19)

It has been recognized in recent years that this approachis not correct for semiconductors in the junction regions,where high and unphysical velocity peaks are estab-lished by the Franz-Wiedemann law. To avoid thisproblem, Stettler, Alam and Lundstrom25 have suggest-ed a new form of closure

( )5

12

B L

c

k TT r

e= + Q J (20)

where J is the current density and ris a tunable parame-ter less than unity. Now using

( ) ( ) ( )2 * *

*

iz di dz di dz

i

di dz

dz dz

i z

K nm v v nm v vx x x

v vnm v v

x x

= =

= +

(21)

and assuming that the spatial variations are confinedalong the z-direction, we have

( ) ( )2

2 *iz dzz zK nm vx x

= . (22)To summarize, the balance equations for the drifted-Maxwellian distribution function simplify to


7/33

7

( )

( )

( )

2

2

0

1

**

1

*

1

n n

z

dz B c

z

z z

m

cB c dz

z z

z z

E

nJ S

t e

J enm v nk T

t m x

neE J

m

TWW nk T v

t x x

J E W W

= +

= +

+

= +

+

(23)

where

2

*1 3

*

2 2

z dz z

dz B c

eJ env P

m

W nm v nk T

= =

= +

(24)

2.3. Particle Based Device Simulation Methods

In the previous sections we have considered continuummethods of describing transport in semiconductors,specifically the drift-diffusion and hydrodynamic mod-els, which are derived from moments of the semi-classical Boltzmann Transport Equation (BTE). Asapproximations to the BTE, it is expected that at somelimit, such approaches become inaccurate, or fail com-pletely. Indeed, one can envision that, as physical di-mensions are reduced, at some level a continuum de-scription of current breaks down, and the granular na-ture of the individual charge particles constituting thecharge density in the active device region becomes im-portant.

The microscopic simulation of the motion of indi-vidual particles in the presence of the forces acting onthem due to external fields as well as the internal fieldsof the crystal lattice and other charges in the system haslong been popular in the chemistry community, wheremolecular dynamics simulation of atoms and moleculeshave long been used to investigate the thermodynamicproperties of liquids and gases. In solids, such as semi-conductors and metals, transport is known to be domi-nated by random scattering events due to impurities,

lattice vibrations, etc., which randomize the momentumand energy of charge particles in time. Hence, stochas-tic techniques to model these random scattering eventsare particularly useful in describing transport in semi-conductors, in particular theMonte Carlo method.

The Ensemble Monte Carlo techniques have beenused for well over 30 years as a numerical method tosimulate nonequilibrium transport in semiconductormaterials and devices and has been the subject of nu-

merous books and reviews 26,27,28 . In application totransport problems, a random walk is generated usingthe random number generating algorithms common tomodern computers, to simulate the stochastic motion ofparticles subject to collision processes. This process ofrandom walk generation is part of a very general tech-nique used to evaluate integral equations and is con-nected to the general random sampling technique usedin the evaluation of multi-dimensional integrals29.

The basic technique as applied to transport prob-lems is to simulate the free particle motion (referred toas the free flight) terminated by instantaneous randomscattering events. The Monte Carlo algorithm consistsof generating random free flight times for each particle,choosing the type of scattering occurring at the end ofthe free flight, changing the final energy and momen-tum of the particle after scattering, and then repeatingthe procedure for the next free flight. Sampling theparticle motion at various times throughout the simula-tion allows for the statistical estimation of physically

interesting quantities such as the single particle distri-bution function, the average drift velocity in the pres-ence of an applied electric field, the average energy ofthe particles, etc. By simulating an ensemble of parti-cles, representative of the physical system of interest,the non-stationary time-dependent evolution of the elec-tron and hole distributions under the influence of atime-dependent driving force may be simulated.

This particle-based picture, in which the particlemotion is decomposed into free flights terminated byinstantaneous collisions, is basically the same approxi-mate picture underlying the derivation of the semi-classical Boltzmann Transport Equation (BTE). In fact,

it may be shown that the one-particle distribution func-tion obtained from the random walk Monte Carlo tech-nique satisfies the BTE for a homogeneous system inthe long-time limit30. This semi-classical picture breaksdown when quantum mechanical effects become pro-nounced, and one cannot unambiguously describe theinstantaneous position and momentum of a particle, asubject which we will comment on later. In the follow-ing, we develop the standard Monte Carlo algorithmused to simulate charge transport in semiconductors.We then discuss how this basic model for chargetransport within the BTE is self-consistently solvedwith the appropriate field equations to perform particle

based device simulation.2.3.1 Free Flight Generation

In the Monte Carlo method, particle motion is assumedto consist of free flights terminated by instantaneousscattering events, which change the momentum andenergy of the particle after scattering. So the first taskis to generate free flights of random time duration foreach particle. To simulate this process, the probability


8/33

8

density, P(t), is required, in which P(t)dt is the jointprobability that a particle will arrive at time t withoutscattering after a previous collision occurring at time t=0, and then suffer a collision in a time interval dtaroundtime t. The probability of scattering in the time intervaldtaround tmay be written as [k(t)]dt, where [k(t)] is

the scattering rate of an electron or hole of wavevectork. The scattering rate, [k(t)], represents the sum ofthe contributions from each individual scattering mech-anism, which are usually calculated quantum mechani-cally using perturbation theory, as described later. Theimplicit dependence of [k(t)] on time reflects thechange in k due to acceleration by internal and externalfields. For electrons subject to time independent electricand magnetic fields, the time evolution of k betweencollisions is represented as

( ) ( )( )

0e t

t+

= E v B

k k

, (25)

where E is the electric field, v is the electron velocity

and B is the magnetic flux density. In terms of the scat-tering rate, [k(t)], the probability that a particle hasnot suffered a collision after a time t is given by

( )0

expt

t dt

k . Thus, the probability of scatter-

ing in the time interval dt after a free flight of time tmay be written as the joint probability

( ) ( )[ ] ( )0

expt

P t dt t t dt dt =

k k . (26)

Random flight times may be generated according to theprobability density P(t) above using, for example, the

pseudo-random number generator implicit on mostmodern computers, which generate uniformly distribut-ed random numbers in the range [0,1]. Using a directmethod (see, for example Ref. [26]), random flighttimes sampled from P(t) may be generated according to

( )0

rt

r P t dt = , (27)

where r is a uniformly distributed random number andtr is the desired free flight time. Integrating Eq, (27)with P(t) given by Eq. (26) above yields

( )0

1 exprt

r t dt =

k . (28)

Since 1-ris statistically the same as r, Eq. (28) may besimplified to

( )0

lnrt

r t dt = k . (29)

Eq. (29) is the fundamental equation used to generatethe random free flight time after each scattering event,resulting in a random walk process related to the under-lying particle distribution function. If there is no exter-

nal driving field leading to a change ofk between scat-tering events (for example in ultrafast photoexcitationexperiments with no applied bias), the time dependencevanishes, and the integral is trivially evaluated. In thegeneral case where this simplification is not possible, itis expedient to introduce the so called self-scatteringmethod31, in which we introduce a fictitious scatteringmechanism whose scattering rate always adjusts itselfin such a way that the total (self-scattering plus realscattering) rate is a constant in time

( ) ( )selft t = + k k , (30)

where self[k(t )] is the self-scattering rate. The self-scattering mechanism itself is defined such that the finalstate before and after scattering is identical. Hence, ithas no effect on the free flight trajectory of a particlewhen selected as the terminating scattering mechanism,yet results in the simplification of Eq. (29) such that thefree flight is given by

1

lnrt r

= . (31)

The constant total rate (including self-scattering) ,must be chosen at the start of the simulation interval(there may be multiple such intervals throughout anentire simulation) so that it is larger than the maximumscattering encountered during the same time interval.In the simplest case, a single value is chosen at the be-ginning of the entire simulation (constant gamma meth-od), checking to ensure that the real rate never exceedsthis value during the simulation. Other schemes may bechosen that are more computationally efficient, andwhich modify the choice of at fixed time incre-ments32.

2.3.2 Final State After Scattering

The algorithm described above determines the randomfree flight times during which the particle dynamics istreated semi-classically. For the scattering process it-self, we need the type of scattering (i.e. impurity,acoustic phonon, photon emission, etc.) which termi-nates the free flight, and the final energy and momen-tum of the particle(s) after scattering. The type of scat-tering which terminates the free flight is chosen using auniform random number between 0 and , and usingthis pointer to select among the relative total scattering

rates of all processes including self-scattering at thefinal energy and momentum of the particle[ ] [ ] [ ] [ ]1 2, , , ,self N n n n n = + + + k k k k , (32)

with n the band index of the particle (or subband in thecase of reduced-dimensionality systems), and k thewavevector at the end of the free-flight. This process isillustrated schematically in Figure 4.


9/33

9

( )( )rtE121 +

321 ++

4321 +++

Self

54321 ++++

1

3

2

4

5

r

( )( )rtE121 +

321 ++

4321 +++

Self

54321 ++++

1

3

2

4

5

r

Fig. 4. Selection of the type of scattering terminating a freeflight in the Monte Carlo algorithm.

Once the type of scattering terminating the freeflight is selected, the final energy and momentum (aswell as band or subband) of the particle due to this typeof scattering must be selected. For elastic scatteringprocesses such as ionized impurity scattering, the ener-

gy before and after scattering is the same. For the in-teraction between electrons and the vibrational modesof the lattice described as quasi-particles known asphonons, electrons exchange finite amounts of energywith the lattice in terms of emission and absorption ofphonons. For determining the final momentum afterscattering, the scattering rate, j[n,k;m,k] of the jthscattering mechanism is needed, where n and m are theinitial and final band indices, and k and k are the parti-cle wavevectors before and after scattering. Defining aspherical coordinate system as shown in Figure 5around the initial wavevector k, the final wavevector kis specified by |k| (which depends on conservation of

energy) as well as the azimuthal and polar angles, andaround k. Typically, the scattering rate, j[n,k;m,k],only depends on the angle between k and k. There-fore, may be chosenusing a uniform random numberbetween 0 and 2 (i.e. 2r), while is chosen accord-ing to the angular dependence for scattering arisingfrom j[n,k;m,k]. If the probability for scattering intoa certain angle P()dis integrable, then random anglessatisfying this probability density may be generatedfrom a uniform distribution between 0 and 1 throughinversion of Eq. (27). Otherwise, a rejection technique(see, for example, [26,27]) may be used to select ran-dom angles according to P().

The rejection technique for sampling a randomvariable over some interval corresponds to choosing amaximum probability density (referred to here as amaximizing function) that is integrable in terms of Eq.(27) (for example a uniform or constant probability),and is always greater than or equal to the actual proba-bility density of interest. A sample value of the randomvariable is then selected using a uniform number be-tween 0 and 1, and then applying Eq. (27) to the max-imizing function to select a value of the random varia-

ble analytically according to the probability density ofthe maximizing function. To now sample according tothe desired probability density, a second random num-ber is picked randomly between 0 and the value of themaximizing function at the value of the random varia-ble chosen. If the value of this random number is lessthan the true value of the probability density (i.e. liesbelow it) at that point, the sampled value of the randomvariable is selected. If it lies above, it is rejected,and the process repeated until one satisfying the condi-tion of selected is generated. In choosing random sam-ples via this technique, one then samples according tothe desired probability density.

zk

xk

yk

kk

zk

xk

yk

kk

Fig. 5. Coordinate system for determining the final state afterscattering.

2.3.3 Ensemble Monte Carlo Simulation

The basic Monte Carlo algorithm described in the pre-

vious sections may be used to track a single particleover many scattering events in order to simulate thesteady-state behavior of a system. However, for im-proved statistics over shorter simulation times, and fortransient simulation, the preferred technique is the useof a synchronous ensemble of particles, in which thebasic Monte Carlo algorithm is repeated for each parti-cle in an ensemble representing the (usually larger)system of interest until the simulation is completed.Since there is rarely an identical correspondence be-tween the number of simulated charges, and the numberof actual particles in a system, each particle is really asuper-particle, representing a finite number of real par-

ticles. The corresponding charge of the particle isweighted by this super-particle number. Figure 6 illus-trates an ensemble Monte Carlo simulation in which afixed time step, t, is introduced over which the motionof all the carriers in the system is synchronized. Thesquares illustrate random, instantaneous, scatteringevents, which may or may not occur during a giventime-step. Basically, each carrier is simulated only upto the end of the time-step, and then the next particle inthe ensemble is treated. Over each time step, the mo-


10/33

10

tion of each particle in the ensemble is simulated inde-pendent of the other particles. Nonlinear effects such ascarrier-carrier interactions or the Pauli exclusion princi-ple are then updated at each times step, as discussed inmore detail below.

1=n23456

N

0 t t2 t3 t4 st

1=n23456

N

0 t t2 t3 t4 st

Fig. 6. Ensemble Monte Carlo simulation in which a time step,t, is introduced over which the motion of particles is syn-

chronized. The squares represent random scattering events.

The non-stationary one-particle distribution func-tion and related quantities such as drift velocity, valleyor subband population, etc., are then taken as averagesover the ensemble at fixed time steps throughout thesimulation. For example, the drift velocity in the pres-ence of the field is given by the ensemble average ofthe component of the velocity at the nth time step as

( ) ( )1

1 N jz z

j

v n t v n t N =

, (33)where N is the number of simulated particles and jlabels the particles in the ensemble. This equation

represents an estimator of the true velocity, which has astandard error given by

sN

= , (34)

where 2 is the variance which may be estimated from29

( )22 2

1

1

1

Nj

z z

j

Nv v

N N

=

. (35)

Similarly, the distribution functions for electronsand holes may be tabulated by counting the number ofelectrons in cells ofk-space. From Eq. (35), we see thatthe error in estimated average quantities decreases as

the square root of the number of particles in the ensem-ble, which necessitates the simulation of many parti-cles. Typical ensemble sizes for good statistics are inthe range of 104 105 particles. Variance reductiontechniques to decrease the standard error given by Eq.(35) may be applied to enhance statistically rare eventssuch as impact ionization or electron-hole recombina-tion [27].

An overall flowchart of a typical Ensemble MonteCarlo (EMC) simulation is illustrated in Figure 7. Afterinitialization of run parameters, there are two mainloops, and outer one which advances the time step byincrements ofT until the maximum time of the simu-lation is reached, and an inner loop over all the particles

in the ensemble (N

), where the Monte Carlo algorithmis applied to each particle individually over a giventime step.

Fig. 7. Flow chart of an Ensemble Monte Carlo (EMC) simu-lation.

2.3.4. Device Simulation Using Particles

Within an inhomogeneous device structure, both thetransport dynamics and an appropriate field solver arecoupled to each other. For quasi-static situations, thespatially varying fields associated with the potentialarising from the numerical solution of Poisson's equa-tion are the driving force accelerating particles in theMonte Carlo phase. Likewise, the distribution of mo-

bile (both electrons and holes) and fixed charges (e.g.donors and acceptors) provides the source of the elec-tric field in Poisson's equation. By decoupling thetransport portion from the field portion over a smalltime interval (discussed in more detail below), a con-vergent scheme is realized in which the Monte Carlotransport phase is self-consistently coupled to Poissonsequation, similar to Gummels algorithm. In the fol-lowing section, a description of Monte Carlo particle-based device simulators is given, with emphasis on the

INITIALIZATION: Input run parameters, read materialparameters, tabulate scattering rates, choose maximum gamma,initialize distribution, choose initial flight times for each electron

Main Loop

T=T+T

N=N+1

Check time left, TL; TL


11/33

11

particle-mesh coupling and the inclusion of the short-range Coulomb interactions.

As mentioned above, for device simulation basedon particles, Poisson's equation is decoupled from theparticle motion (described e.g. by the EMC algorithm)over a suitably small time step, typically less than theinverse plasma frequency corresponding to the highestcarrier density in the device. Over this time interval,carriers accelerate according to the frozen field profilefrom the previous time-step solution of Poisson's equa-tion, and then Poisson's equation is solved at the end ofthe time interval with the frozen configuration of charg-es arising from the Monte Carlo Phase. It is importantto note that Poisson's equation is solved on a discretemesh, whereas the solution of charge motion usingEMC occurs over a continuous range of coordinatespace in terms of the particle position. An illustrationof a typical device geometry and the particle meshscheme is shown in Figure 8. Therefore, a particle-mesh (PM) coupling is needed for both the charge as-

signment and the force interpolation. The size of themesh and the characteristic time scales of transport setconstraints on both the time-step and the mesh size.We must consider how particles are treated in terms ofthe boundaries, and how they are injected. Finally, thedetermination of the charge motion and correspondingterminal currents from averages over the simulationresults are necessary in order to calculate the I-V char-acteristics of a device. These issues are discussed brief-ly below, along with some typical simulation results.

Source DrainGate

Substrate

V(x,y)Boundary

Source DrainGate

Substrate

V(x,y)Boundary

Fig. 8. Schematic diagram of a prototypical three-terminaldevice where charge flow is described by particles, while thefields are solved on a finite mesh.

As in the case of any time domain simulation, forstable Monte Carlo device simulation, one has tochoose the appropriate time step, t, and the spatialmesh size (x, y, and/or z). The time step and themesh size may correlate to each other in connectionwith the numerical stability. For example, the time stept must be related to the plasma frequency. From theviewpoint of numerical stability, t must be muchsmaller than the inverse plasma frequency above. Sincethe inverse plasma frequency goes as 1/ n , the high-est carrier density occurring in the modeled devicestructure corresponds to the smallest time used to esti-mate t. If the material is a multi-valley semiconductor,

the smallest effective mass encountered by the carriersmust be used as well.

The mesh size for the spatial resolution of the po-tential is dictated by the spatial variation of charge vari-ations. Hence, one has to choose the mesh size to besmaller than the smallest wavelength of the charge vari-ations. The smallest wavelength is approximately equalto the Debye length (for degenerate semiconductors therelevant length is the Thomas-Fermi wavelength).

Based on the discussion above, the time step (t),and the mesh size (x, y, an/or z) are chosen inde-pendently based on the physical arguments givenabove. However, there are numerical constraints cou-pling both as well. More specifically, the relation of tto the grid size must also be checked by calculating thedistance lmax, defined as

max maxvl t= , (36)where vmax is the maximum carrier velocity, that can beapproximated by the maximum group velocity of the

electrons in the semiconductor (on the order of 108

cm/s). The distance lmax is the maximum distance thecarriers can propagate during t. The time step istherefore chosen to be small enough so that lmax issmaller than the spatial mesh size chosen.. This con-straint arises because for too large of a time step, t,there may be substantial change in the charge distribu-tion, while the field distribution in the simulation isonly updated every t, leading to unacceptable errors inthe carrier force.

H/D

10

1

0.1

0.1 1 10

H/D

10

1

0.1

0.1 1 10

Fig. 9. Illustration of the region of stability (unshaded re-gions) of the time step, t, and the minimum grid size,H. peis the plasma frequency corresponding to the maximum carri-er density.

To illustrate these various constraints, Figure 9 il-lustrates the range of stability for the time step and min-imum grid size. The unshaded region corresponds tostable selections of both quantities. The right region isunstable due to the time step being larger than the in-verse plasma frequency, whereas the upper region isunstable due to the grid spacing being larger than theDebye length. The velocity constraint bounds the lowerside with its linear dependence on time-step.


12/33

12

An issue of importance in particle-based simulationis the real space boundary conditions for the particlepart of the simulation. Reflecting or periodic boundaryconditions are usually imposed at the artificial bounda-ries. For Ohmic contacts, they require more carefulconsideration because electrons (or holes) crossing thesource and drain contact regions contribute to the corre-sponding terminal currents. In order to conserve chargein the device, the electrons exiting the contact regionsmust be re-injected. Commonly employed models forthe contacts include 33: Electrons are injected at the opposite contact with

the same energy and wavevector k. If the sourceand drain contacts are in the same plane, as in thecase of MOSFET simulations, the sign of k, nor-mal to the contact will change. This is an unphysi-cal model, however34.

Electrons are injected at the opposite contact with awavevector randomly selected based upon a ther-mal distribution. This is also an unphysical model.

Contact regions are considered to be in thermalequilibrium. The total number of electrons in asmall region near the contact are kept constant,with the number of electrons equal to the numberof dopant ions in the region. This approximation ismost commonly employed in actual particle baseddevice simulation.

Another method uses reservoirs of electrons adja-cent to the contacts. Electrons naturally diffuse in-to the contacts from the reservoirs, which are nottreated as part of the device during the solution ofPoissons equation. This approach gives resultssimilar to the velocity-weighted Maxwellian, but at

the expense of increased computational time due tothe extra electrons simulated. It is an excellentmodel employed in some of the most sophisticatedparticle-based simulators. There are also severalpossibilities for the choice of the distribution func-tionMaxwellian, displaced Maxwellian, and ve-locity-weighted Maxwellian33.

The particle-mesh (PM) coupling is broken intofour steps: (1) assignment of particle charge to themesh; (2) solution of Poissons equation on the mesh;(3) calculation of the mesh-defined forces; and (4) in-terpolation to find the forces acting on the particle. The

charge assignment and force interpolation schemes usu-ally employed in self-consistent Monte Carlo devicesimulations are the nearest-grid-point (NGP) and thecloud-in-cell (CIC) schemes 35 . Figure 10 illustratesboth methods. In the NGP scheme, the particle positionis mapped into the charge density at the closest gridpoint to a given particle. This has the advantage ofsimplicity, but leads to a noisy charge distribution,which may exacerbate numerical instability. Alternate-ly, within the CIC scheme, a finite volume is associated

with each particle spanning several cells in the mesh,and a fractional portion of the charge per particle isassigned to grid points according to the relative volumeof the cloud occupying the cell corresponding to thegrid point. This method has the advantage of smooth-ing the charge distribution due to the discrete charges ofthe particle based method, but may result in an artificialself-force acting on the particle, particularly if an in-homogeneous mesh is used.

NGP CIC

Fig. 10. Illustration of the charge assignment based on thenearest grid point method (NGP) and the cloud in cell method(CIC).

The requirements for constant permittivity (P) andconstant mesh (M) severely limit the scope of devicesthat may be considered in device simulations using theNGP and the CIC schemes. Laux36 proposed a newparticle-mesh coupling scheme, namely, the nearest-element-center (NEC) scheme, which relaxes the re-strictions (P) and (M). The NEC charge assign-ment/force interpolation scheme attempts to reduce the

self-forces and increase the spatial accuracy in the pres-ence of nonuniformly spaced tensor-product meshesand/or spatially-dependent permittivity. In addition, theNEC scheme can be utilized in one axis direction(where local mesh spacing is nonuniform) and the CICscheme can be utilized in the other (where local meshspacing is uniform). Such hybrid schemes offersmoother assignment/ interpolation on the mesh com-pared to the pure NEC. The NEC designation derivesfrom the appearance, of moving the charge to the centerof its element and applying a CIC-like assignmentscheme. The NEC scheme involves only one meshelement and its four nodal values of potential. Thislocality makes the method well-suited to non-uniformmesh spacing and spatially-varying permittivity. Theinterpolation and error properties of the NEC schemeare similar to the NGP scheme.

The motion in real space of particles under the in-fluence of electric fields is somewhat more complicateddue to the band structure. The velocity of a particle inreal space is related to theE-k dispersion relation defin-ing the bandstructure as


13/33

13

( ) ( )( )

( )

1dt E t

dt

d q

dt

= =

=

k

rv k

k E r

(37)

where the rate of change of the crystal momentum isrelated to the local electric field acting on the particle

through the acceleration theorem expressed by the se-cond equation. In turn, the change in crystal momen-tum, k(t), is related to the velocity through the gradientofEwith respect to k. If one has to use the full band-structure of the semiconductor, then integration of theseequations to find r(t) is only possible numerically, us-ing for example a Runge-Kutta algorithm. If a threevalley model with parabolic bands is used, then theexpression is integrable

( );

*

d d q

dt m dt = = =

r k k E rv

(38) (0.3)

Therefore, for a constant electric field in thex direction,the change in distance along thex direction is found by

integrating twice( ) ( ) ( )

0 2

0 02 *

xx

qE tx t x v t

m= + + (39)

To simulate the steady-state behavior of a device,the system must be initialized in some initial condition,with the desired potentials applied to the contacts, andthen the simulation proceeds in a time stepping manneruntil steady-state is reached. This process may takeseveral picoseconds of simulation time, and conse-quently several thousand time-steps based on the usualtime increments required for stability. Clearly, thecloser the initial state of the system is to the steady statesolution, the quicker the convergence. If one is, for

example, simulating the first bias point for a transistorsimulation, and has no a priori knowledge of the solu-tion, a common starting point for the initial guess is tostart out with charge neutrality, i.e. to assign particlesrandomly according to the doping profile in the deviceand based on the super-particle charge assignment ofthe particles, so that initially the system is charge neu-tral on the average. For two-dimensional device simu-lation, one should keep in mind that each particle actu-ally represents a rod of charge into the third dimension.Subsequent simulations at the same device at differentbias conditions can use the steady state solution at theprevious bias point as a good initial guess. After as-

signing charges randomly in the device structure,charge is then assigned to each mesh point using theNGP or CIC or NEC particle-mesh methods, and Pois-sons equation solved. The forces are then interpolatedon the grid, and particles are accelerated over the nexttime step. A flow-chart of a typical Monte Carlo devicesimulation is shown in Figure 11.

Simulation timeend?

Initialize Data

Compute Charge

Solve Poisson Equation

Carrier Dynamicsusing Monte Carlo

Transport kernel

Collect Datayes

no

STARTSTART

STOP

Fig. 11. Flow-chart of a typical particle based device simula-tion.

As the simulation evolves, charge will flow in andout of the contacts, and depletion regions internal to thedevice will form until steady state is reached. Thecharge passing through the contacts at each time step

can be tabulated, and a plot of the cumulative charge asa function of time gives the steady-state current. Figure12 shows the particle distribution in 3D of a MESFET,where the dots indicate the individual simulated parti-cles for two different gate biases. Here, the heavilydoped MESFET region (shown by the inner box) issurrounded by semi-insulating GaAs forming the rest ofthe simulation domain. The upper curve corresponds tono net gate bias (i.e. the gate is positively biased toovercome the built-in potential of the Schottky contact),while the lower curve corresponds to a net negative biasapplied to the gate, such that the channel is close topinch-off. One can see the evident depletion of carriersunder the gate under the latter conditions.


14/33

14

Fig. 12. Example of the particle distribution in a MESFETstructure simulated in 3D using an EMC approach. The upperplot is the device with zero gate voltage applied, while thelower is with a negative gate voltage applied, close to pinch-off.

After sufficient time has elapsed, so that the systemis driven into a steady-state regime, one can calculatethe steady-state current through a specified terminal.The device current can be determined via two different,but consistent methods. First, by keeping track of thecharges entering and exiting each terminal, the net

number of charges over a period of the simulation canbe used to calculate the terminal current. This method,however, is relatively noisy due to the discrete nature ofthe carriers, and the fact that one is only counting thecurrents crossing a 2D boundary in the device, whichlimits the statistics. A second method uses the sum ofthe carrier velocities in a portion of the device are usedto calculate the current. For this purpose, the device isdivided into several sections along, for example, the x-axis (from source to drain for the case of a MOSFET orMESFET simulation). The number of carriers and theircorresponding velocity is added for each section aftereach free-flight time step. The totalx-velocity in eachsection is then averaged over several time steps to de-termine the current for that section. The total devicecurrent can be determined from the average of severalsections, which gives a much smoother result comparedto counting the terminal charges. By breaking the de-vice into sections, individual section currents can becompared to verify that the currents are uniform. Inaddition, sections near the source and drain regions of aMOSFET or a MESFET may have a high y-component

in their velocity and should be excluded from the cur-rent calculations.

2.3.5 Simulation Example

The fully depleted (FD) Silicon-On-Insulator (SOI)MOSFET (Figure 13) was of much interest a decadeago, because of its projected superiority over the par-tially depleted (PD) and the bulk-silicon counterparts37.Its advantages, due mainly to the gate-substrate chargecoupling enabled by the thin FD Si-film body on a thickburied oxide (BOX) 38 ,39 included higher drive cur-rent/transconductance, near-ideal subthreshold slope,low-junction capacitance and suppression of the float-ing-body effects. However, in the deep sub-micrometerregime, because of velocity saturation, two-dimensionaleffects in the BOX and the technological limits of scal-ing the SOI-body thickness, these advantages dimin-ished 40, the interest subsided and classical (i.e., bulk-Siand PD SOI) CMOS prevailed.Now, as the scaling of classical CMOS approaches its

limit, interest in non-classical FD devices - particularlydouble-gate (DG) and ultra-thin-body (UTB)MOSFETs - is rapidly growing. These devices deliverfundamental improvements over the performance of thebulk Si MOSFET devices41. And, while DG FinFETs42seem most promising, their complex and immature pro-cess technology has led to a renewed interest in single-gate FD SOI UTB MOSFETs43.

Silicon substrate

Drain

VS

VG

VB

Source

BOX (Buried Oxide)

VD

Fig. 13. FD SOI MOSFET device structure.

The dimensions of the n-channel FD SOI MOSFETbeing investigated using the particle-based device simu-lator discussed in the previous section are: the channellength is 25nm, the silicon film width, which is equal tothe source/drain junction depth is 10nm, the gate oxidewidth is 2nm, the BOX width is 50nm, the source/draindoping is 11019 cm-3 and the channel doping is 11018cm-3.

To simulate the steady-state state behavior of a de-vice, the system is started in some initial condition,with the desired potential applied to the contacts, andthen the simulation proceeds in a time stepping manner


15/33

15

until steady-state is reached. This takes several picose-conds of simulation time and consequently severalthousand time steps based on the usual time incrementsrequired for stability. A common starting point for theinitial guess is to start out with charge neutrality, i.e., toassign particles randomly according to the doping pro-file in the device, so that initially the system is chargeneutral on the average. After assigning charges random-ly in the device structure, charge is then assigned toeach mesh point using an adequate PM coupling meth-od, and Poissons equation is solved. The forces arethen interpolated on the grid, and particles are acceler-ated over the next time step. At this stage, typicalsimulation result that is shown is the variation of thescattering rates of the various scattering mechanismsincluded in the model. In our case, we have includedacoustic phonon scattering, g- and f- intervalley phononscattering (see Figure 14). Afterwards, bias is appliedand the carriers undergo the free-flight scattering se-quence until steady-state is achieved. At this point, it is

important to present the simulation results for the aver-age drift velocity and the average carrier energy in thechannel region of the device. These results are shown inFigures 15a and 15b, respectively. They demonstratethe need for performing Monte Carlo device simula-tions that are more time consuming then solving eitherthe drift-diffusion or the hydrodynamic model dis-cussed previously.

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

13

energy (eV)

scatteringrate(1/s)

f-process

absorption

f-process

emission

g-process

absorptiong-process

emission

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

14x 10

12

energy (eV)

scatteringrat

e(1/s)

acoustic

Fig. 14. Scattering rate variation versus energy for the variousscattering mechanisms included in the model.

0 1 2 3 4 5 6 7

x 10-8

-1

0

1

2

3x 10

5

x (m)

velocity(m/s)

4

(a)

0 1 2 3 4 5 6 7

x 10-8

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

x (m)

carrierenergy(eV)

source channel

drain

(b)Fig. 15. (a) Average carrier drift velocity along the channel.(b) Average carrier energy. Different currents correspond tothe different gate biases denoted on Figure 16.

The choice of the Monte Carlo device simulation isjustified with the fact that in the devices simulated, weobserve significant velocity overshoot near the drainend of the channel. Namely, the saturation velocity ofthe electrons in Si is 1.1105 m/s and from the resultsshown in Figure 15a it is evident that the electrons arein the overshoot regime near the drain end of the chan-nel and their average drift velocity exceeds 2105m/s.

Proper modeling of the velocity overshoot effect, whichleads to larger current drive, is only possible via a Mon-te Carlo device simulation scheme. Another issue that isworth mentioning is the fact that the average carrierenergy in the channel region of the device is less than0.5 eV which justifies the use of the non-parabolicmodel that is adopted in this work..

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10-12

-4

-3

-2

-1

0

1

2

3

4x 10

4

time (s)time (s)time (s)time (s)

charge(numberofcarriersperum)

cumulative drain charge

cumulative source charge

Vg=0.6V

Vg=0.4V

Vg=0.2V

Vg=0.05V

(a)


16/33

16

0 1 2 3 4 5 6 7

x 10-8

-2

0

2

4

6

8

10

12x 10

-4

x (m)

currentdensity(A/um)

Vg=0.6V

Vg=0.4V

Vg=0.2V

Vg=0.05V

source channel drain

(b)Fig. 16. (a) Cumulative charge versus time for drain bias VD=0.6V and different gate biases. The slope of the curve givesthe source and drain currents. (b) Current density calculatedby using the average drift velocity of the carriers in the x-direction. Both methods give the same value of the current

through the device which suggests that conservation of parti-cles in the system is being preserved.

After sufficient time has elapsed, so that the system isdriven into a steady-state regime, one can calculate thesteady-state current through a specified terminal. Asalready discussed, the device current can be determinedvia two different, but consistent methods. First, bykeeping track of the charges entering and exiting eachterminal, the net number of charges over a period of thesimulation can be used to calculate the current (Figure16a). The method is quite noisy due to the discrete na-ture of the carriers. In a second method, the sum of the

carrier velocities in a portion of the device are used tocalculate the current (Figure 16b). For this purpose, thedevice is divided into several sections along, for exam-ple, the x-axis (from source to drain for the case of aMOSFET simulation).

-0.4 -0.2 0 0.2 0.4 0.60

200

400

600

800

1000

1200

1400

Vg (V)

Id(uA/um)

Vd=0.1V

Vd=0.6V

Vd=1.0V

Fig. 17. Transfer Characteristics of the device shown in Fig. 1.Notice the increase in threshold voltage with increasing drainbias.

0 1 2 3 4 5 6 7 8

x 10-8

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

x (m)

conductionband(eV)

Drain-induced barrier lowering effect

Vd=0.1V

Vd=0.6V

Vd=1.0V

source injection

barrier (y=0)

Vg = -0.3V

Fig. 18. Demonstration of Drain induced barrier lowering forgate bias VG = -0.3V and different drain bias.

The number of carriers and their corresponding velocityis added for each section after each free flight time step.The total x-velocity in each section is then averagedover several time steps to determine the current for thatsection. The total device current can be determinedfrom the average of several sections, which gives amuch smoother result compared to counting the termi-nal charges. By breaking the device into sections, indi-vidual section currents can be compared to verify thatthe currents are uniform. In addition, sections near thesource and the drain regions of a MOSFET may have ahigh y-velocity and should be excluded from the currentcalculations. Finally, by using several sections in thechannel, the average energy and velocity of electronsalong the channel is checked to ensure proper physicalcharacteristics. The two ways of determining currentthrough the device are demonstrated in Figures 16a and

16b.In Figure 17 we show the device transfer characteristicsfor different drain biases. It is obvious from the resultspresented that the threshold voltage shifts due to theDrain Induced Barrier Lowering (DIBL) of the sourcebarrier (see Figure 18). This observation also demon-strates the need of using computer simulations for mod-eling semiconductor devices as fields and potentials aretwo-dimensional quantities and one-dimensional mod-els can not properly capture effects such as DIBL.Finally a snapshot of the electron density in the channel,when the transistor is turned on, is shown in Figure 19a.We see the existence of electrons in the channel region

of the device. The corresponding conduction band pro-file, for the same biasing conditions, that is smoothedover time, is shown in Figure 19b and demonstrates thetwo-dimensional character of the potential and the elec-tric field profiles in the active portion of the device.


17/33

17

0

20

40

60

0

20

40

60

800

5

10

15

x 1024

x (nm)y (nm)

(m

-3)

source

drain

BOX

(a)

0

20

40

60

0

20

40

60

80-0.8

-0.6

-0.4

-0.2

0

0.2

x (nm)y (nm)

(eV)

S

D

(b)

Fig. 19. (a) Snapshot of the Electron density in the device. Weuse Vg = 0.6V and Vd=0.6V in these simulations. (b) Varia-

tion of the conduction band edge for the same bias conditions.

2.3.6 Direct Treatment of Inter-Particle Interac-

tion

In modern deep-submicrometer devices, for achievingoptimum device performance and eliminating the so-called punch-through effect, the doping densities mustbe quite high. This necessitates a careful treatment ofthe electron-electron (e-e) and electron-impurity (e-i)interactions, an issue that has been a major problem forquite some time. Many of the approaches used in thepast have included the short-range portions of the e-e

and e-i interactions in the k-space portion of the MonteCarlo transport kernel, thus neglecting many of the im-portant inelastic properties of these two interactionterms 44, 45]. An additional problem with this screenedscattering approach in devices is that, unlike the otherscattering processes, e-e and e-i scattering rates need tobe re-evaluated frequently during the simulation pro-cess to take into account the changes in the distributionfunction in time and spatially. The calculation and tab-ulation of a spatially inhomogeneous distribution func-

tion may be highly CPU and memory intensive. Fur-thermore, ionized impurity scattering is usually treatedas a simple two-body event, thus ignoring the multi-ioncontributions to the overall scattering potential. A sim-ple screening model is usually used that ignores thedynamical perturbations to the Coulomb fields causedby the movement of the free carriers. To overcome theabove difficulties, several authors have advocated cou-pling of the semi-classical molecular dynamics ap-proach to the ensemble Monte-Carlo approach 46,47,48.Simulation of the low field mobility using such a cou-pled approach results in excellent agreement with theexperimental data for high substrate doping levels 48.However, it is proven to be quite difficult to incorporatethis coupled ensemble Monte-Carlo-molecular dynam-ics approach when inhomogeneous charge densities,characteristic of semiconductor devices, are encoun-tered 45,49 . An additional problem with this approach ina typical particle-based device simulation arises fromthe fact that both the e-e and e-i interactions are already

included, at least within the Hartree approximation(long-range carrier-carrier interaction), through the self-consistent solution of the Poisson equation via the PMcoupling discussed in the previous section. The magni-tude of the resulting mesh force that arises from theforce interpolation scheme, depends upon the volume ofthe cell, and, for commonly employed mesh sizes indevice simulations, usually leads to double-counting ofthe force.

To overcome the above-described difficulties ofincorporation of the short-range e-e and e-i force intothe problem, one can follow two different paths. Oneway is to use the P3M scheme introduced by Hockney

and Eastwood35

. An alternative to this scheme is to usethe corrected-Coulomb approach due to Gross etal.50,51,52,53.

The P3M Method

The particle-particle-particle-mesh (P3M) algorithmsare a class of hybrid algorithms developed by Hockneyand Eastwood35. These algorithms enable correlatedsystems with long-range forces to be simulated for alarge ensemble of particles. The essence of the methodis to express the interparticle forces as a sum of twocomponent parts; the short range part Fsr, which is non-

zero only for particle separations less than some cutoffradius re, and the smoothly varying part F, which has atransform that is approximately band-limited. The totalshort-range force on a particle Fsr is computed by directparticle-particle (PP) pair force summation, and thesmoothly varying part is approximated by the particle-mesh (PM) force calculation.


18/33

18

The Corrected Coulomb Approach

This second approach is a purely numerical scheme thatgenerates a corrected Coulomb force look-up table forthe individual e-e and e-i interaction terms. To calculatethe proper short-range force, one has to define a 3D boxwith uniform mesh spacing in each direction. A single(fixed) electron is then placed at a known position with-in a 3D domain, while a second (target) electron isswept along the device in, for example, 0.2 nm incre-ments so that it passes through the fixed electron. The3D box is usually made sufficiently large so that theboundary conditions do not influence the potential solu-tion. The electron charges are assigned to the nodesusing one of the charge-assignment schemes discussedpreviously 36 . A 3D Poisson equation solver is thenused to solve for the node or mesh potentials. At self-consistency, the force on the swept electron F= Fmesh isinterpolated from the mesh or node potential. In a sepa-rate experiment, the Coulomb force Ftot= F

coul

is calcu-lated using standard Coulomb law. For each electronseparation, one then tabulates Fmesh , Fcoul and the dif-

ference between the two F' F Fcoul mesh = =Fsr, whichis called the corrected Coulomb force or a short-rangeforce. The later is stored in a separate look-up table.

104

105

106

107

-60 -40 -20 0 20 40 60

|Emesh

|

|Ecoul

|

|E'|

Electricfield[V/m]

Range [nm]

targetelectron

fixedelectron

Range

Fig. 20. Mesh, Coulomb and corrected Coulomb field versusthe distance between the two electrons. Note: F=-eE.

As an example, the corresponding fields to these threeforces for a simulation experiment with mesh spacingof 10 nm in each direction are shown in Figure 20. It isclear that the mesh force and the Coulomb force are

identical when the two electrons are separated severalmesh points (30-50 nm apart). Therefore, adding thetwo forces in this region would result in double-counting of the force. Within 3-5 mesh points, Fmesh

starts to deviate from Fcoul . When the electrons arewithin the same mesh cell, the mesh force approacheszero, due to the smoothing of the electron charge whendivided amongst the nearest node points. The generatedlook-up table for F' also provides important infor-

mation concerning the determination of the minimumcutoff range based upon the point where Fcoul and Fmesh begin to intersect, i.e. F' goes to zero.

Figure 21 shows the simulated doping dependenceof the low-field mobility, derived from 3D resistor sim-ulations, which is a clear example demonstrating the

importance of the proper inclusion of the short-rangeelectron-ion interactions. For comparison, also shownin this figure are the simulated mobility results reportedin17 , calculated with a bulk EMC technique using theBrooks-Herring approach 54 for the e-i interaction, andfinally the measured data 55 for the case when the ap-plied electric field is parallel to the (100) crystallo-graphic direction. From the results shown, it is obviousthat adding the corrected Coulomb force to the meshforce leads to mobility values that are in very goodagreement with the experimental data. It is also im-portant to note that, if only the mesh force is used in thefree-flight portion of the simulator, the simulation mo-

bility data points are significantly higher than the exper-imental ones due to the omission of the short-rangeportion of the force.

The short-range e-e and e-i interactions also playsignificant role in the operation of semiconductor de-vices. For example, carrier thermalization at the drainend of the MOSFET channel is significantly affected bythe short-range e-e and e-i interactions. This is illus-trated in Figure 22 on the example of a 80 nm channel-length n-MOSFET. Carrier thermalization occurs overdistances that are on the order of few nm when the e-eand e-i interactions are included in the problem. Usingthe mesh force alone does not lead to complete thermal-ization of the carriers along the whole length of thedrain extension, and this can lead to inaccuracies whenestimating the device on-state current.

0

500

1000

1500

2000

2500

1017

1018

Experimental data from Refs. [12,13]

Monte Carlo results from Ref. [10]

Resistor simulations

Resistor simulations - mesh force only

M

obility

[cm2/V-s]

Doping [cm-3

] Fig. 21. Low-field electron mobility derived from 3D resistorsimulations versus doping. Also shown on this figure are theEnsemble Monte Carlo results and the appropriate experi-mental data.


19/33

19

0

100

200

300

400

100 110 120 130 140 150 160 170 180

with e-e and e-imesh force only

Electrone

nergy

[meV]

Length [nm]

(b)

VD=1 V, V

G=1 V

channel drain

0

100

200

300

400

100 110 120 130 140 150 160 170 180

with e-e and e-imesh force only

Electrone

nergy

[meV]

Length [nm]

(b)

VD=1 V, V

G=1 V

channel drain

Fig. 22. Average energy of the electrons coming to the drainfrom the channel. Filled (open) circles correspond to the casewhen the short-range e-e and e-i interactions are included(omitted).

3. QUANTUM TRANSPORT

Semiconductor transport in the nanoscale region hasapproached the regime of quantum transport. This issuggested by two trends: (1) within the effective-massapproximation, the thermal de Broglie wavelength forelectrons in semiconductors is on the order of the gatelength of nano-scale MOSFETs, thereby encroachingon the physical optics limit of wave mechanics; (2) thetime of flight for electrons traversing the channel withvelocity well in excess of 107 cm/sec is in the 10-15 to10-12 sec regiona time scale which equals, if not be-ing less than the momentum and energy relaxationtimes in semiconductors which precludes the validity of

the Fermis golden rule.The static quantum effects, such as tunneling

through the gate oxide and the energy quantization inthe inversion layer of a MOSFET are also significant innanoscale devices. The current generation of MOS de-vices has oxide thicknesses of roughly 15-20 and isexpected that, with device scaling deeper into the na-noscale regime, oxides with 8-10 thickness will beneeded. The most obvious quantum mechanical effect,seen in the very thinnest oxides, is gate leakage via di-rect tunneling through the oxide. The exponential turn-on of this effect sets the minimum practical oxidethickness (~10). A second effect due to spatial/size-

quantization in the device channel region is also ex-pected to play significant role in the operation of na-noscale devices. To understand this issue, one has toconsider the operation of a MOSFET device based ontwo fundamental aspects: (1) the channel charge in-duced by the gate at the surface of the substrate, and (2)the carrier transport from source to drain along thechannel. Quantum effects in the surface potential willhave a profound impact on both, the amount of chargewhich can be induced by the gate electrode through the

gate oxide, and the profile of the channel charge in thedirection perpendicular to the surface (the transversedirection). The critical parameter in this direction is thegate-oxide thickness, which for a nanoscale MOSFETdevice is, as noted earlier, on the order of 1 nm. Anoth-er aspect, which determines device characteristics, isthe carrier transport along the channel (lateral direction).Because of the two-dimensional (2D), and/or one-dimensional (1D) in the case of narrow-width devices,confinement of carriers in the channel, the mobility (ormicroscopically speaking, the carrier scattering) will bedifferent from the three-dimensional (3D) case. Theo-retically speaking, the 2D/1D mobility should be largerthan its 3D counterpart due to reduced density of statesfunction, i.e. reduced number of final states the carrierscan scatter into, which can lead to device performanceenhancement. A well known approach that takes thiseffect into consideration is based on the self-consistentsolution of the 2D Poisson1D Schrdinger2D MonteCarlo, and requires enormous computational resources

as it requires storage of position dependent scatteringtables that describe carrier transition between varioussubbands56. More importantly, these scattering tableshave to be re-evaluated at each iteration step as the Har-tree potential (the confinement) is a dynamical functionand slowly adjusts to its steady-state value. It is im-portant to note, however, that in the smallest size devic-es ( 10 nm feature size), carriers experience very littleor no scattering at all (ballistic limit), which makes thissecond issue less critical when modeling these na-noscale devices (e.g. [57,58,59]).

On the other hand, the dynamical quantum effectsin nanoscale MOSFETs, associated with energy dissi-

pating scattering in electron transport can be physicallymuch more involved60. There are several other funda-mental problems one must overcome in this regard. Forexample, since ultrasmall devices, in which quantumeffects are expected to be significant, are inherentlythree-dimensional (3D), one must solve the 3D open-Schrdinger equation.

Table 1. Quantum Effects.

1. Static Quantum Effects Periodic crystal potential and band struc-

ture effects

Scattering from defects, phonons Strong electric and magnetic field Inhomogeneous electric field Tunnelinggate oxide tunneling and

source-to-drain tunneling Quantum wells and band-engineered bar-

riers

2. Dynamical Quantum Effects


20/33

20

Collisional broadening Intra-collisional field effects Temperature dependence Electron-electron scattering Dynamical screening Many-body effects

Pauli exclusion principle

Another question that becomes important in nanoscaledevices is the treatment of scattering processes. Withinthe Born approximation, the scattering processes aretreated as independent and instantaneous events. It is,however, a nontrivial question to ask whether such anapproximation is actually satisfactory under high tem-perature, in which the electron strongly couples withthe environment (such as phonons and other carriers).In fact, many dynamical quantum effects, such as thecollisional broadening of the states or the intra-collisional field effect, are a direct consequence of the

approximation employed for the scattering kernel in thequantum kinetic equation. Depending on the orders ofthe perturbation series in the scattering kernel, the mag-nitude of the quantum effects could be largely changed.Many of these issues relevant to quantum transport insemiconductors are highlighted inTable 1. Note that at present there is no consensus as towhat can be the best approach to model quantumtransport in semiconductors. Density matrices, and theassociated Wigner function approach, Greens functions,and Feynman path integrals all have their applicationstrengths and weaknesses.

3.1 Open Systems

A general feature of electron devices is that they are ofuse only when connected to a circuit, and to be so con-nected any device must possess at least two terminals,contacts, or leads. As a consequence, every device is anopen system with respect to carrier flow61. This is theoverriding fact that determines which theoretical mod-els and techniques may be appropriately applied to thestudy of quantum devices. For example, the quantummechanics of pure, normalizable states, such as thoseemployed in atomic physics, does not contribute signif-icantly to an understanding of devices, because suchstates describe closed systems.

To understand devices, one must consider the un-normalizable scattering states, and/or describe the stateof the device in terms of statistically mixed states,which casts the problem in terms of quantum kinetictheory. As a practical matter of fact, a device is of useonly when its state is driven far from thermodynamicequilibrium by the action of the external circuit. Thenon-equilibrium state is characterized by the conductionof significant current through the device and/or the ap-

pearance of a non-negligible voltage drop across thedevice.

In classical transport theory, the openness of thedevice is addressed by the definition of appropriateboundary conditions for the differential (or integro-differential) transport equations. Such boundary condi-tions are formulated so as to approximate the behaviorof the physical contacts to the device, typically Ohmicor Schottky contacts62. In the traditional treatments ofquantum transport theories, the role of boundary condi-tions is often taken for granted, as the models are con-structed upon an unbounded spatial domain. The properformulation and interpretation of the boundary condi-tions remains an issue, however. It should be under-stood that, unless otherwise specified, all models to beconsidered here are based upon a single-band, effective-mass open-system Schrodinger equation.

3.2 Evaluation of the Current Density

To investigate the transport properties of a quantumsystem one must generally evaluate the current flowthrough the system, and this requires that one examinesystems that are out of thermal equilibrium. A commonsituation, in both experimental apparatus and techno-logical systems, is that one has two (or more) physicallylarge regions densely populated with electrons in whichthe current density is low, coupled by a smaller regionthrough which the current density is much larger. It isconvenient to regard the large regions as electron res-ervoirs within which the electrons are all in equilibri-um with a constant temperature and Fermi level, andwhich are so large that the current flow into or out

semiconductor devicemodeling

Documents