Wigner Monte Carlo Simulation:Particle Annihilation and Device Applications

Hans Kosina, Viktor Sverdlov, and Tibor GrasserInstitute for Microelectronics, TU Wien, GuBhausstraBe 27-29/E360, A-1040 Wien, AustriaPhone: +43-1-58801-36013, Fax: +43-1-58801-36099, E-mail: hans.kosina@iue.tuwien.ac.at

Abstract- Coherent transport in mesoscopic devices is welldescribed by the Schrodinger equation supplemented by openboundary conditions. When electronic devices are operated atroom temperature, however, a realistic device model needs toinclude carrier scattering. In this work the kinetic equationfor the Wigner function is employed as a model for quantumtransport. Carrier scattering is treated in an approximate mannerthrough a Boltzmann collision operator. A Monte Carlo techniquefor the solution of this kinetic equation has been developed,based on an interpretation of the Wigner potential operator as ageneration term for numerical particles. Details on the algorithmfor particle generation and subsequent particle annihilation arepresented. Including a multi-valley semiconductor model anda self-consistent iteration scheme, the described Monte Carlosimulator can be used for routine device simulations. Applicationsto single barrier and double barrier structures are presented.

I. INTRODUCTION

For FETs with gate lengths below 10 nm quantum effectssuch as direct source-to-drain tunneling become important andstart affecting the device characteristics [1]. Recent studiesshow that scattering will still affect the current [2] and thatthe transition to ballistic transport appears at much shorter gatelengths than previously anticipated [3]. An accurate theory ofMOSFETs near the scaling limit must therefore account forthe interplay between coherent quantum effects and dissipativescattering effects. This mixed transport regime can suitablybe treated by the Wigner equation. Early numerical solutionsof the Wigner equation were obtained using finite differencemethods, assuming simplified scattering models based onthe relaxation time approximation [4]. However, for realisticdevice simulation more comprehensive scattering models arerequired. With the advent of Monte Carlo (MC) methods forthe Wigner equation [5], [6] it became feasible to includethe full Boltzmann collision operator. The development ofMC methods for the Wigner equation, however, is hamperedby the fact that, as opposed to the semi-classical case, theintegral kernel is no longer positive. This so-called negativesign problem will lead to exponentially growing variances ofthe Markov Chain MC method. The Wigner potential operatorcan also be viewed as a generation term of positive andnegative numerical particles. In this picture the sign problemshows up in the avalanche of numerical particles generated.A stable MC method can only be achieved by means of asuitable particle annihilation algorithm.

II. THE PHYSICAL MODEL

Quantum transport is modeled by a time-independent, one-electron Wigner equation for a multi-valley semiconductor.The set of Wigner equations is coupled through the inter-valleyphonon scattering terms.

i (VkiE (k) Vr + F(r) Vk) f, (k, r) =

E([1-v/

(E /[I

+ Vw(k -k', r)fv(k', r) d3k'Silicon: v, v'= [100], [010], [001]

This equation determines the Wigner function ft for valleyv. A valley's energy dispersion relation cv(k) is assumed tobe anisotropic and parabolic. Note that a non-parabolic e(k)relation in the single-electron Hamiltonian would give a non-local diffusion term of the form f c(k, r-r')fv(k, r')d3r'.Although it is straightforward to use a non-parabolic relationin (1), one should be aware that this would approximate thenon-local diffusion term by a local one of the form VkEv Vrf.A spectral decomposition of the potential profile V(r) is

applied [7]. The slowly varying component gives the classicalforce F, whereas the rapidly varying component is taken intoaccount through the Wigner potential V,.

V(r) = Vcl(r) + Vqm(r),

V, (q, r) =

th(2I7)3IL/[ m(r 2)

F(r) = -)Vcl(r)

Vqm r - ) e-i rd3

In (1) scattering is treated semi-classically through a Boltz-mann collision operator, where the transition rate Sv,, (k, k')from initial state (v', k') to final state (v, k) is given by Fermi'sgolden rule. It should be noted that usage of the Boltzmanncollision operator in the Wigner equation represents some adhoc assumption. A rigorous treatment of electron-phonon scat-tering would require a frequency-dependent Wigner function,f (k, r, w). It is related to the non-equilibrium Green's functionG< by G< (r, k, w) = if (k, r, w) and can reasonably be

1-4244-0404-5/06/$20.00 © 2006 IEEE

0 3'-(k.r)]Svv/(k.fv k/) fvl (k/ , r) d kl

-fvo/(kj)]Sv/v(k/.k)d3kl fv (k. r)

(1)

SISPAD 2006 357

approximated as f (k, r, w) = f, (k, r)A(r, k, w) [8]. To arriveat Fermi's golden rule the spectral function A is reduced tothe Dirac 6-function.

Furthermore, in (1) the Pauli blocking factor the equilibriumFermi function f,° is used. The assumption of a Boltzmanncollsion operator in (1) ensures that in the semiclassicalregions, such as the highly doped contact regions, the con-ductivity is finite and that the mean energy increase due todegeneracy is taken into account.

III. NUMERICAL METHODSA stationary MC method for solving (1) has been reported in

[5]. The potential operator e [f1] = f V,(k -k')f(k', r)d3k'is interpreted as a generation term of numerical particles.The strict mass conservation property of this operator canbe satisfied by the numerical particle model exactly if onegenerates the numerical particles only pair-wise, for instance,with statistical weights +1 and -1. As pointed out in [5], asuitable annihilation algorithm for numerical particles needsto be introduced in order to achieve a stable MC method.Since one can devise various algorithms for particle generationand, in particular, for particle annihilation, in the following thelatest developments are described.

A. Particle GenerationA direct numerical representation of the Wigner potential

V, (q, r) would require the discretization of both momentumand space coordinates. The problem can be simplified byexpressing the Wigner potential in terms of V(q), the Fouriertransform of the potential Vq/m(r). The potential operator canbe rewritten as follows.

Ow [fw] (k, r) = (27)3h J V(q) sin [f(q) + q r]

x (fw (k -qr t) f- (k+ q,r,t)) d3q (2)

An advantage of this formulation is that no discretization of thespatial variable r is needed. The expression can be evaluated atthe actual position r of a particle. Only the momentum variableq needs to be discretized in order to numerically represent IV,the modulus, and o, the phase of V.

The structure of (2) suggests the usage of a rejectiontechnique. As a normalization quantity one obtains an upperlimit for the pair generation rate.

'Ymax (2w)3h J V(q) dq (3)

At a rate of 'Tmax the free flight of a particle is interruptedto check for particle pair-generation. From the distributionIV(q)l one generates randomly the momentum transfer q.Then the sine function is evaluated at the actual particleposition r as s = sin [p(q) + q r]. With probability s thepair-generation event is accepted, otherwise a self-scatteringevent is performed. In the former case, two particle states aregenerated with momenta k, k -q/2 and k2 = k + q/2and statistical weights w = wosign(s) and w2 = -WI,respectively, where w0 is the statistical weight of the initial

particle. Since (2) is local in real space, the particle pair isgenerated at the position r of the initial particle.

B. Particle AnnihilationDifferent variants of the single-particle MC method out-

lined in [5] can be devised. The variant discussed below isconstructed such that current is conserved exactly. The onlyinput parameter required is the ratio of negative and positivetrajectories, which makes the algorithm easy to control. Theidea is that from the trajectory tree generated by a particleinjected at the contact only one branch is actually traced.

For steady state problems considered here a phase spacemesh can be utilized, on which numerical particles are tem-porarily stored. After each generation event one has to dealwith three particle states, namely the initial state k and the twogenerated states, k, and k2. In a first step all three particlesare stored on the annihilation mesh, that is, the statisticalweight of each particle is added to a counter associatedwith the mesh element. Then one has to decide which ofthe three states is used to continue the trajectory. One maychoose the weight of the particle continuing the trajectoryto be of the same sign as the incoming one (Fig. 1). Inthis way the statistical weight along one trajectory does notchange, which results in exact current conservation. If theinitial state has a positive statistical weight, out of the threemesh elements one selects that with the largest stored weight.Continuing from that element will reduce the weight of theelement. Conversely, a negative trajectory is to be continuedfrom the element with the smallest stored weight. A certainfraction of negative trajectories needs to be constructed inorder to resolve the negative parts of the Wigner function.This rule for selecting the continuing particle is an attempt tominimize the weights stored in the three elements after eachpair-generation event. The repeated execution of this rule inthe MC main loop results in a minimization of the storedweight on the whole annihilation mesh. Particle annihilationtakes place when positive and negative particles are alternatelystored in the same mesh element. Note that because of themass conservation property of the transport equation and ofthe associated particle model, no net-charge can build up onthe annihilation mesh. The weights stored on the mesh sum up

Fig. 1. The particle annihilation strategy attempts to minimize the weightsstored in the mesh elements. The weights of the initial and continuing particlehave the same sign to ensure current continuity. Particles and mesh elementscarrying a positive weight are in black, the ones carrying a negative weightare in grey.

1-4244-0404-5/06/$20.00 © 2006 IEEE

-r il-< 21----

-0 ---w-k'

SISPAD 2006 358

I* P-A,.I. I

lw 1-

II I .>

to zero. The local weights on the mesh have to be kept small,as they are a measure for the numerical error of the method.This can be controlled by the fraction of negative trajectories,which has to be specified by the user.

Considering Fig. 1, one can develop also the followingnotion of the algorithm. Positive and negative particles are sentthrough the device and interact with the annihilation mesh.A positive particle is likely to recombine in a phase-spaceregion where the weight stored on the mesh is negative. Thismeans that a positive particle is likely to propagate in thoseregions where the stored weight is positive. It is unlikely torecombine there, because this would result in an increase inlocal weight, which would contradict the local minimizationprinciple. For the same reason, a negative particle will beattracted by regions with negative stored weight. Although thetransition probabilities used to propagate the particles are thesame for positive and negative particles, the interaction withthe annihilation mesh causes the trajectories for positive andnegative particles to be systematically different.An annihilation mesh is introduced for each valley-type. For

the three pairs of X-valleys of Si three meshes are required.The meshes are defined in the three-dimensional phase-space,spanned by one spatial and two momentum coordinates.

C. Coupling to Poisson Equation

A self-consistent iteration scheme between Wigner MC andthe Poisson equation is implemented. The adopted scheme,which is similar to the Gummel iteration scheme for the basicsemiconductor equations [9], is commonly used in classicalone-particle MC simulations [10]. Fig. 2 shows the iterationhistory of the current through a Si n-i-n diode for differentwidths of the intrinsic region. Currents computed using Wignerand classical MC show similar convergence behavior.

IV. RESULTS AND DISCUSSION

The described MC method can be used for routine devicesimulations. For the purpose of verification, the first exampleassumes a frozen potential profile from a 10nm gate lengthdouble-gate MOSFET. Fig. 3 compares the quantum ballisticcurrents as obtained from a collision-less Wigner MC simula-tion and from a numerical Schrodinger solver. Good agreementis observed. The quantum ballistic current is higher than theclassical ballistic current due to an additional contributionfrom carriers tunneling through the potential barrier.

To study the effects of scattering and tunneling on the devicecharacteristics we consider Si n-i-n diodes with the length Wof the intrinsic region ranging from 20 nm down to 2.5 nm. Thedoping profile is assumed to increase gradually from the intrin-sic region to the highly doped contact region over the same

distance W. Three transport models are compared: Wignerequation and Boltzmann equation with electron-phonon andionized-impurity scattering included, yielding currents IWIGand IBTE, respectively. The Wigner equation without scatteringinside the intrinsic and transition regions gives the currentICOH (coherent). Fig. 4 shows that the effect of scattering

~- 6XO

c)

t_) 5x10u

2 4 6Number of Iterations

.p

8 10

Fig. 2. Cuffent through Si n-i-n diodes as a function of the number ofself-consistent iterations with the width of the intrinsic region as a parameter.

Schrodinger t1.5 WignerMC

-Classical

C'0.51-- 2 0 30~05

0.1 0.2 0.3 0.4Gate Voltage [V]

Fig. 3. Normalized ballistic currents calculated classically and quantummechanically. Results from Wigner MC and the Schrodinger solver are ingood agreement. The potential profile is obtained from a device simulationof a 10nm gate length DG MOSFET.

reflected in the difference ICOH -IWIG decreases with decreas-ing device length. However, even for W = 2.5 nm the relativedifference in the currents is still of the order of 25%, indicatingthat scattering cannot be neglected. Also shown is the currentdifference due to tunneling, IWIG -IBTE. Clearly, this currentcomponent rises with reduced barrier width.

The next example shows results of self-consistent Wigner-Poisson simulations of a double-barrier tunneling structure. AGaAs/AlGaAs resonant tunneling diode (RTD) is investigated,assuming a barrier height of Eb = 0.3 eV, a barrier width of3 nm, and a well width of 5 nm [5], [11]. Polar optical phonon,acoustic deformation potential and ionized impurity scatteringare included. Fig. 5 shows the effect of degeneracy, whichis introduced in the simulation by the approximated Pauliblocking factors in (1) and through the boundary distributionat the contacts, on the current-voltage characteristics.

1-4244-0404-5/06/$20.00 2006 IEEE

2.5 nm

* . Wigner

D Boltzmann

5.0 nm

..10nm,....o....0....0..?.....I?....I?....I?.... I?....I?.... I(

0

SISPAD 2006 359

3 5 10Length of intrinsic region [nm]

Fig. 4. Relative difference between currents of an n-i-n diode calculatedusing different transport models: Wigner MC with and without scattering inthe intrinsic region (squares); Wigner MC and classical MC (diamonds).

2.0 OK BoltzmannL--l Fermi-Dirac

~1.5

0.5 --

07 100 200 300 400Voltage [mV]

Fig. 5. I-V characteristics of a GaAs resonant tunneling diode at 300 Kwith scattering from PO phonons, acoustic phonons and ionized impuritiesincluded.

Fig. 6 shows the kinetic energy density of electrons in theRTD. In terms of wave functions fij, the eigen energies Ei,and the probabilities pi, the kinetic energy density is defined

as follows.

w(r) = Zpi(Ei- V(r)) 4Fj(r) 2 (4)

This density can become negative in tunneling regions where

the energy of one or more states is below the band edge, Ei <V(r). Transformation of (4) into the Wigner representationgives [12]

w(r) = (2w)3]/ 2m*knY Vr2) fw(k: r, t) d3k . (5)4 r

Fig. 6. Mean kinetic energy of electrons in a resonant tunneling diodecalculated from (5). In the tunneling barriers the mean kinetic energy isnegative.

tional to the second derivative of the electron concentration isrequired.

V. CONCLUSION

A Monte Carlo simulator performing a self-consistent nu-

merical solution of the Wigner equation has been presented.Details of the algorithms for generation and annihilation ofnumerical particles have been described. The quantum MCmethod turns gradually into the classical MC method when thepotential profile becomes smoother. Therefore, the simulationmethod can be used, for instance, to study the gradual emer-

gence of quantum effects when a device structure is scaleddown.

ACKNOWLEDGMENT

This work has been partly supported by the Austrian ScienceFund, project 179-N16.

REFERENCES

[1]

[2]

[3][4][5]

[6][7]

[8]

[9]

[10]

[1 1][12]

Note that the kinetic energy density is not simply given by thesecond moment of the Wigner function. A correction propor-

J. Wang and M. Lundstrom, in Int.Electron Devices Meeting (IEEE,2002), pp. 29.2.1-29.2.4.P. Palestri, D. Esseni, S. Eminente, C. Fiegna, E. Sangiorgi, and L.Selmi, IEEE Trans.Electron Devices 52, 2727 (2005).M. Gilbert, R. Akis, and D. Ferry, J.Appl.Phys. 98, 094303 (2005).W. Frensley, Rev.Mod.Phys. 62, 745 (1990).H. Kosina, M. Nedjalkov, and S. Selberherr, Journal of CompuationalElectronics 2, 147 (2003).L. Shifren, C. Ringhofer, and D. Ferry, Phys.Lett.A 306, 332 (2003).A. Gehring and H. Kosina, Journal of Compuational Electronics 4, 67(2005).W. Hansch, The Drift-Difusion Equation and Its Applications in MOS-FETModeling, Computational Microelectronics (Springer Verlag, 1991).S. Selberheff, Analysis and Simulation of Semiconductor Devices(Springer, 1984).F. Venturi, R. Smith, E. Sangiorgi, M. Pinto, and B. Ricco, IEEETrans.Computer-Aided Design 8, 360 (1989).L. Shifren and D. Ferry, IEEE Trans.Electron Devices 50, 769 (2003).H. Kosina and M. Nedjalkov, in Handbook of Theoretical and Com-putational Nanotechnology, edited by M. Rieth and W. Schommers(American Scientific Publishers, 2006), Vol. 10, Chap. Wigner FunctionBased Device Modeling, pp. 731-763, (in print).

1-4244-0404-5/06/$20.00 2006 IEEE

50

- 40

, 30

.> 20ct

a)

I(I]L

(ICOH - IWIG) / ICOH

(IWIG - IBTE) / 'WIG -

C1)

21

i i

o -o

SISPAD 2006 360