A Wigner Potential Decompositionin the Signed-Particle Monte Carlo

Approach

Majid Benam(B), Mihail Nedjalkov, and Siegfried Selberherr

Institute for Microelectronics, TU Wien, Vienna, Austriabenam@iue.tuwien.ac.at

Abstract. The description of the electron evolution, provided by theWigner equation, involves a force-less Liouville operator, which is asso-ciated with particles moving over Newtonian trajectories, and a Wignerpotential operator associated with generation of positive and negativeparticles. These concepts can be combined to develop stochastic algo-rithms for solving the Wigner equation, consolidated by the so-calledsigned particle approach. We investigate the option to split the Wignerpotential into two parts and to approximate one of them by a classicalforce term. The purpose is two-fold: First, we search for ways to sim-plify the numerical complexity involved in the simulation of the Wignerequation. Second, such a term offers a way to a self-consistent couplingof the Wigner and the Poisson equations. The particles in the signed-particle approach experience a force through the classical component ofthe potential. A cellular automaton algorithm is used to update the dis-crete momentum of the accelerated particles, which is then utilized alongwith the Wigner-based generation/annihilation processes. The effect ofthe approximation on generic physical quantities such as current anddensity are investigated for different cut-off wavenumbers (wavelengths),and the results are promising for a self-consistent solution of the Wignerand Poisson equations.

Keywords: Wigner function · Potential splittingSigned-particle approach

1 Introduction

The Wigner function is defined with the Fourier transform of the density matrixexpressed in the mean and difference of coordinates in two dimensions:

fw(r,k, t) =1

(2π)2

∫ +∞

−∞dse−ik.sρ

(r +

s2, r − s

2, t).

Furthermore, the finite dimensions of the simulation domain allow the Wignerfunction to be calculated over finite dimensions and discretized k values. Thephysical domain in the simulations analyzed in this paper is a two-dimensionalc© Springer Nature Switzerland AG 2019G. Nikolov et al. (Eds.): NMA 2018, LNCS 11189, pp. 263–272, 2019.https://doi.org/10.1007/978-3-030-10692-8_29

264 M. Benam et al.

area of size (Lx, Ly) = (20 nm, 30 nm). We choose the center of the domain to bethe origin, and therefore, the position and momentum vectors are discretized as:

r ≡(xΔr, yΔr) −M

2≤ x <

M

2, −N

2≤ y <

N

2

s ≡(mΔs, nΔs)

L =(Lx, Ly) ≡ (MΔs, NΔs)

qΔk =(p, q).(π

Lx,

π

Ly) ≡ (p

π

MΔs, q

π

NΔs) −M

2≤ q <

M

2, −N

2≤ p <

N

2

q′Δk =(p′, q′).(π

Lx,

π

Ly) ≡ (p′ π

MΔs, q′ π

NΔs).

Δr and Δs represent the spatial spacing between nodes and are assumed to beequal. s and q′ are used for performing summations over position and momentumvariables. We use the short notation:

f(x, y, p, q) ≡ f(xΔr, yΔr, pπ

MΔs, q

π

NΔs).

The Wigner equation, which follows from the von Neumann equation for thedensity matrix [6], is written in the semi-discrete form as:

(∂

∂t+

�qΔk

m∗ ∇r

)fw(r,q, t) =

∑q

VW (r,q − q′)fw(r,q′, t). (1)

The semi-discrete Wigner potential (WP), which is of central importance in thesigned-particle approach, is defined as [4]:

VW (r,q) ≡ 1i�L

∫ L2

− L2

dse−iqΔk.s[V (r +

s2) − V (r − s

2)]. (2)

2 Wigner Potential Decomposition

Here, we focus on a full discretization, and thus use the fully discretized WPwhich must be computed at each node [1] becomes:

VW (x, y, p, q) =1

i�MN

M2 −1∑

m=− M2

N2 −1∑

n=− N2

e−i(pm πM +qn π

N )

×[V

(x +

m

2, y +

n

2) − V

(x − m

2, y − n

2)]

. (3)

The two-dimensional discrete Fourier transform of a potential V (x, y) in a regionof size M × N is a function V̂ (p, q). The pair is given by the equations [5]:

V̂ (p, q) =

M2 −1∑

x=− M2

N2 −1∑

y=− N2

V (x, y)e−i2π( pxM + qy

N ),

Wigner Potential Decomposition 265

V (x, y) =1

MN

M2 −1∑

p=− M2

N2 −1∑

q=− N2

V̂ (p, q)ei2π( pxM + qy

N ).

V̂ (p, q) can be expressed in polar form, V̂ (p, q) = A(p, q)eiφ(q,p). Since the two-dimensional discrete Fourier transform pair is periodic (k and l being integers),

V̂ (p, q) = V̂ (p + kM, q + lN), V (x, y) = V (x + kM, y + lN),

we also assume the physical potential V (x, y) to be periodic so that a shift equalto a multiple of the physical region lengths in the corresponding argument ofthe potential does not change the value of the potential. It holds:

M2 −1∑

m=− M2

N2 −1∑

n=− N2

e−i(πmpM +πnq

N )

[V

(x +

m

2, y +

n

2)]

= ei2(πxpM +πyq

N )

×x+M

4 −1∑m′=x− M

4

y+N4 −1∑

n′=y− N4

e−i2(πm′pM +πn′q

N )

[V

(m′, n′)] = ei2(πxp

M +πyqN )V̂ (2p, 2q),

M2∑

m=− M2

N2∑

n=− N2

e−i( πmpM

+ πnqN

)

[V

(x − m

2, y − n

2

)]= e−i2( πxp

M+ πyq

N)

×x+ M

4 −1∑

m′=x− M4

y+ N4 −1∑

n′=y− N4

ei2( πm′pM

+ πn′qN

)

[V

(m′, n′)

]=

[ei2( πxp

M+ πyq

N)V̂ (2p, 2q)

]∗.

We have used the properties of the discrete Fourier transform, namely periodicityand scaling. Thus, the Wigner potential given in Eq. 3 becomes:

Vw(x, y, p, q) =1

i�MN

{ei2(πxp

M +πyqN )V̂ (2p, 2q)−[

ei2(πxpM +πyq

N )V̂ (2p, 2q)]∗

}. (4)

Using Euler’s formula, Eq. 4 can be rewritten in a polar form as:

Vw(x, y, p, q) =2

�MNA(2p, 2q)sin

[φ(2p, 2q) + 2

πxp

M+ 2

πyq

N

]. (5)

In the following, we show that treating the summation in the right-hand sideof Eq. 1 in two separate regions results in the spectral decomposition of thepotential profile into a slowly varying classical component and a rapidly vary-ing quantum mechanical component [2]. For each direction we specify a cut-off wavenumber determining the sharpness of the corresponding low-pass filterwhich is discussed later in more details. The cut-off wavenumber is specified by acut-off wavelength, λcx

= 2πqcx Δkx

and λcy= 2π

qcy Δky. We assume qcx

and qcyto be

266 M. Benam et al.

equal and use qc for both directions. Applying the decomposition, the potentialoperator on the right-hand side of Eq. 1 can be rewritten as [3]:

Qfw(x, y, p, q)=∑

p′,q′Vw(x, y, p

′, q

′)fw(x, y, p − p

′, q − q

′) =

∑

|q′|,|p′|≤ qc2

+∑

|q′|,|p′|> qc2

=Qclfw + Qqmfw

Qcl and Qqm represent the classical and potential parts of the potential operator,respectively. We recall that Lagrange’s mean value theorem allows to expressthe increment of a continuous function on an interval through the value of thederivative at an intermediate point of the segment (for small (m − n)Δx)

f(mΔx) − f(nΔx) � f ′(kΔx)(m − n)Δx � Δf(m − n), n ≤ k ≤ m.

Using the notations Δp and Δq, we calculate the classical potential operator as:

Qclfw(x, y, p, q) =∑

|q′|,|p′|≤ qc2

Vw(x, y, p′, q′)fw(x, y, p − p′, q − q′)

≈∑

|q′|,|p′|≤ qc2

Vw(x, y, p′, q′)[fw(x, y, p, q) − πp′

MΔpfw(x, y, p, q) − πq′

NΔqfw(x, y, p, q)

].

In the second line, the summation over fw vanishes as Vw is an odd function inboth p and q. Using the polar form of the Wigner potential (Eq. 5), we obtain:

−∑

|q|,|p|≤ qc2

πp

MVw(x, y, p, q) =

−2

�MN

∑

|q|,|p|≤ qc2

πp

MA(2p, 2q) sin[φ(2p, 2q) + 2

πxp

M+ 2

πyq

N]

=−1

�MN

∑

|q|,|p|≤qc

πp

MA(p, q) sin[φ(p, q) +

πxp

M+

πyq

N]

=Δx1

�MN

∑

|q|,|p|≤qc

A(p, q) cos[φ(p, q) +πxp

M+

πyq

N]

=1

�Δx�

{ 1

MN

∑

|q|,|p|≤qc

A(p, q)eiφ(p,q)ei πxpM ei πyq

N

}

=1

�Δx�

{ 1

MN

∑

|q|,|p|≤qc

V̂ (p, q)ei( πxpM

+ πyqN

)}

=1

�ΔxVcl(x, y).

With a similar approach, we can show that:

−∑

|q|≤ qc2

∑|p|≤ qc

2

πq

NVw(x, y, p, q) =

1�ΔyVcl(x, y).

Here, we have introduced the classical potential component:

Vcl(x, y) =1

MN

∑|q|,|p|≤qc

V̂ (p, q)ei(πxpM +πyq

N ).

Wigner Potential Decomposition 267

This function is real, as can be easily shown by substituting V̂ (p, q):

Vcl(x, y) =1

MN

∑|q|,|p|≤qc

M2 −1∑

m=− M2

N2 −1∑

n=− N2

V (m,n)eiπ(x−m)p

M .eiπ(y−n)q

N

=

M2 −1∑

m=− M2

N2 −1∑

n=− N2

V (m,n).sin[πqc(x−m)

M ]π(x − m)

.sin[πqc(y−n)

N ]π(y − n)

.

In order to calculate Vcl, we have a convolution of real functions, the potentialV (x, y) and the sinc functions in both x and y directions acting as low-pass fil-ters. The convolution involves an infinite summation for an ideal filter. However,we choose our low-pass filter to be bounded to the physical region. Vcl at eachnode in the spatial domain is then calculated as:

Vcl(x, y) =

+∞∑

m,n=−∞Vlp(x − m, y − n).V (m, n) � 1

Ω(x, y)

M2 −1∑

m=− M2

N2 −1∑

n=− N2

ωxymnV (m, n).

The coefficients ωxymn and Ω(x, y) are:

ωxymn =sin[πqc(x−m)

M ]π(x − m)

.sin[πqc(y−n)

N ]π(y − n)

, Ω(x, y) =

M2 −1∑

m=− M2

N2 −1∑

n=− N2

ωxymn.

Therefore, through the introduction of qc, the input potential V (x, y) is splitinto a classical and a quantum-mechanical part.

V (x, y) = Vcl(x, y) + Vqm(x, y). (6)

Vcl(x, y) is the slowly varying part of the potential calculated above by filteringout the high frequency components. Vqm(x, y) contains only the high-frequencycomponents and represents the rapidly varying part of V (x, y). It is easily calcu-lated from Eq. 6 after knowing Vcl(x, y). Using Eqs. 2 and 3, the fully discretizedWP can then be computed at each node using our new approach:

Vqm,W (x, y, p, q) =1

i�MN

M2 −1∑

m=− M2

N2 −1∑

n=− N2

e−i( πmp

M+ πqn

N)[

V(x +

m

2, y +

n

2

) − V (x − m

2, y − n

2)].

The Wigner equation in the light of the spectral decomposition will be:(

∂

∂t+

�qΔk

m∗ Δr − 1

�

[ΔrVcl(r)

]Δq

)fw(r,q, t) =

∑

q

Vqm,W (r,q − q′)fw(r,q′, t).

As shown in the left-hand side of the modified Wigner equation, Vcl gives riseto a local force term which is calculated using the finite difference method.Furthermore, the new Wigner potential (Vqm,W ) on the right-hand side of theequation is calculated from the non-local component of the potential (Vqm).

268 M. Benam et al.

3 Calculating Local Force and Evolving the Particles

The finite difference method suggests to use the two classical potential valuesfrom two adjacent nodes to approximate the force:

Fx(x, y) =Vcl(x + 1, y) − Vcl(x − 1, y)

2Δr, Fy(x, y) =

Vcl(x, y + 1) − Vcl(x, y − 1)

2Δr.

Here, we focus on the force due to the presence of a dopant in the center; asit is exerted on each particle, the value of the momentum in each direction isupdated and lies somewhere between two momentum grid points in the corre-sponding direction. In the following, a probabilistic approach is discussed andthe above-mentioned grid points in the x-direction are named A and B. Whenthe particles change their momenta, they might jump to points on the momen-tum grid which are several Δk farther afield. To explain our algorithm, twovariables are introduced. The first variable, krnd, is a random number between0 and 1. The second one, kjump, is the remaining fractional part of momentumafter it is rounded to the smaller point on the momentum grid (A), and dividedby Δk, therefore, kjump is also always a real number between 0 and 1. Figure 1illustrates these variables in more details.

Fig. 1. Left : Two possible jumps on the momentum grid. Right : A schematic of apossible set of variables (Color figure online)

Without the probabilistic approach, for small momentum changes (comparedto Δk), the new momentum is always rounded back to the same point on themomentum grid. In our approach, however, the decision whether to jump to thenearby point on the grid or not is based on a comparison. krnd is comparedto kjump; if kjump is larger than krnd (the green case in Fig. 1) the momentumof the particle will jump to the nearby grid point (B). Otherwise (the red casein Fig. 1), it remains on the initial grid point A. Higher kjump means higherprobability of jumping to the nearby grid. The same approach is used in they-direction.

4 Results

For comparison purposes, we introduced ( λc

Δx ) as a dimensionless input parame-ter in the ViennaWD simulator [7]. For λc

Δx = 2, Vqm vanishes as all the possiblecontributions of the adjacent nodes are canceled out due to the nature of the

Wigner Potential Decomposition 269

sinc function; however, as we increase λc

Δx , Vcl becomes smoother and Vqm getscloser to V (x, y). Potential values at each node in the physical region contributeto Vcl (and hence Vqm) at all the other nodes through a weighted-average pro-cess. The contribution of each node depends on two factors: (1) How far is thenode from the target node, (2) How big is qc (i.e. how small is λc). The resultsfor the potential decomposition in the case of a single charge in the center of a20 nm × 30 nm region for three different values of λc are shown in Fig. 2:

Fig. 2. The potential input (left) and its classical (middle) and quantum (right) com-ponents for a single charge in the center of a 20 nm × 30 nm region for λc

Δx= 5 (top

row), λcΔx

= 10 (middle row), and λcΔx

= 30 (bottom row).

Increasing λc

Δx results in a smoother Vcl as can be seen in Fig. 2. The right-mostgraphs show the rapidly varying quantum component of the potential (Vqm),which tends to V (x, y) at higher values of λc. Vqm contains only the rapidlyvarying part of the potential and dictates the generation rate of particles (γ)at each node. Since the process of particle generation is exponentially relatedto γ, Ntn+1 = Ntn

e2γ(tn+1−tn), the increase in the number of particles becomesslower which results in fewer annihilation processes and hence improvementsin the simulation time. It also reduces the undesirable effects of approximationsinherent in the annihilation process [6]. In Fig. 3, the results for different λc valuesand the pure quantum case for t = 95fs are shown. The tendency towards purequantum behaviour is noticed as we increase λc.

270 M. Benam et al.

Fig. 3. The density of particles for different values of λcΔx

. From left to right: λcΔx

= 2

(classical case), λcΔx

= 10, λcΔx

= 30, and pure quantum (given for reference)

The density values at each point in the physical region are compared to thedensity values of the pure quantum case by introducing the error ratio:

ErrD(xi, yj) =Dλc

(xi, yj) − Dqm(xi, yj)Dqm(xi, yj)

.

Figure 4 shows the results of the averaged value of this error for different λc

values. Comparing Figs. 3 and 4, it can be noticed that even for low valuesof λc (high values of qc), the error remains close to zero for the points in thephysical region where the density values are significant. Therefore, we can claimthat a spectral decomposition provides a promising step towards coupling thePoisson and Wigner equations. For the regions with lower density values, theimprovement in ErrD is evident as we increase λc. If quantum effects such astunneling are to be analyzed, higher values of λc are preferred and more reliable.

Fig. 4. The error ratio of particle density for different values of λcΔx

. From left to right:λcΔx

= 2 (classical case), λcΔx

= 5, λcΔx

= 20, and λcΔx

= 30.

As can be seen in Fig. 5, the current values come to a saturated value after around50fs, and for different cut-off wavelengths, this saturated value lies within the10% range of the quantum case.It is important to note that the accuracy of the decomposition is not linearlyrelated to the value of λc as depicted in Fig. 5.

Wigner Potential Decomposition 271

Fig. 5. Current curves for different values of λc and the pure quantum case (reference).

5 Discussion and Conclusion

The advantage of utilizing a potential splitting approximation is two-fold. Onone hand, the statistics governing the generation of particles are modified sothat the ensemble of particles experiences fewer annihilation processes. On theother hand, using this approach, the momentum values are updated according toVcl and the local force term. Particles are accelerated in each time step and thevalue of the momentum, while remaining on the momentum grid, is no longerconstant through the simulation time. Averaged physical quantities such as cur-rent and density show a decent similarity to the pure quantum case, especiallyfor λc � Δr. Coupling the quantum character of carrier transport with theclassical evolution of particles seems to be a promising step towards an efficientself-consistent coupling of the Wigner and Poisson equations.

Acknowledgements. This research has been supported by the Austrian Science Fundthrough the project FWF-P29406-N30.

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