comput. methods appl. mech. engrg.nite element methods originally devised to solve hyperbolic...

Comput. Methods Appl. Mech. Engrg. 198 (2009) 3130–3150

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Comput. Methods Appl. Mech. Engrg.

journal homepage: www.elsevier .com/locate /cma

A discontinuous Galerkin solver for Boltzmann–Poisson systems in nano devices q

Yingda Cheng a, Irene M. Gamba a,*,1, Armando Majorana b,2, Chi-Wang Shu c,3

a Department of Mathematics and ICES, University of Texas, Austin, TX 78712, United Statesb Dipartimento di Matematica e Informatica, Università di Catania, Catania, Italyc Division of Applied Mathematics, Brown University, Providence, RI 02912, United States

a r t i c l e i n f o a b s t r a c t

Article history:Received 9 February 2009Received in revised form 18 May 2009Accepted 21 May 2009Available online 3 June 2009

Keywords:Deterministic numerical methodsDiscontinuous Galerkin schemesBoltzmann–Poisson systemsStatistical hot electron transportSemiconductor nano-scale devices

0045-7825/$ - see front matter � 2009 Elsevier B.V. Adoi:10.1016/j.cma.2009.05.015

q Support from the Institute of Computational EngiUniversity of Texas Austin is gratefully acknowledged

* Corresponding author.E-mail addresses: [email protected] (Y. Che

(I.M. Gamba), [email protected] (A. Majorana),Shu).

1 Research supported by NSF-0807712 and NSF-FRG2 Research supported by Italian PRIN 2006: Kineti

particle transport in gases and semiconductors: analytic3 Research supported by NSF Grant DMS-080908

08ER25863.

In this paper, we present results of a discontinuous Galerkin (DG) scheme applied to deterministic com-putations of the transients for the Boltzmann–Poisson system describing electron transport in semicon-ductor devices. The collisional term models optical-phonon interactions which become dominant understrong energetic conditions corresponding to nano-scale active regions under applied bias. The proposednumerical technique is a finite element method using discontinuous piecewise polynomials as basis func-tions on unstructured meshes. It is applied to simulate hot electron transport in bulk silicon, in a siliconnþ—n—nþ diode and in a double gated 12 nm MOSFET. Additionally, the obtained results are compared tothose of a high order WENO scheme simulation and DSMC (Discrete Simulation Monte Carlo) solvers.

� 2009 Elsevier B.V. All rights reserved.

1. Introduction

The evolution of the electron distribution function f ðt;x;kÞ insemiconductors in dependence of time t, position x and electronwave vector k is governed by the Boltzmann transport equation(BTE) [24,28,21]

@f@tþ 1

�hrke � rxf � q

�hE � rkf ¼ Qðf Þ; ð1:1Þ

where �h is the reduced Planck constant, and q denotes the positiveelementary charge. The function eðkÞ is the energy of the consideredcrystal conduction band measured from the band minimum;according to the Kane dispersion relation, e is the positive root of

eð1þ aeÞ ¼ �h2k2

2m�; ð1:2Þ

where a is the non-parabolicity factor and m� the effective electronmass. The electric field E is related to the doping density ND and the

ll rights reserved.

neering and Sciences and the.

ng), [email protected]@dam.brown.edu (C.-W.

-0757450.c and continuum models foral and computational aspects.

6 and DOE Grant DE-FG02-

electron density n, which equals the zero-order moment of the elec-tron distribution function f, by the Poisson equation

rx½�rðxÞrxV � ¼ q�0½nðt;xÞ � NDðxÞ�; E ¼ �rxV ; ð1:3Þ

where �0 is the dielectric constant of the vacuum, �rðxÞ labels therelative dielectric function depending on the material and V theelectrostatic potential. The collision operator Qðf Þ takes into ac-count acoustic deformation potential and optical intervalley scat-tering [31,29]. For low electron densities, it reads

Qðf Þðt;x;kÞ ¼Z

R3½Sðk0;kÞf ðt;x;k0Þ � Sðk;k0Þf ðt;x;kÞ�dk0 ð1:4Þ

with the scattering kernel

Sðk;k0Þ ¼ ðnqþ1ÞKdðeðk0Þ�eðkÞþ�hxpÞþnqKdðeðk0Þ�eðkÞ��hxpÞþK0dðeðk0Þ�eðkÞÞ ð1:5Þ

and K and K0 being constant for silicon. The symbol d indicates theusual Dirac distribution and xp is the constant phonon frequency.Moreover,

nq ¼ exp�hxp

kBTL

� �� 1

� ��1

is the occupation number of phonons, kB is the Boltzmann constantand TL is the constant lattice temperature.

Semiclassical description of electron flow in semiconductors isan equation in six dimensions (plus time if the device is not insteady state) for a truly 3-D device, and four dimensions for a

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Y. Cheng et al. / Comput. Methods Appl. Mech. Engrg. 198 (2009) 3130–3150 3131

1-D device. This heavy computational cost explains why the BPsystem is traditionally simulated by the Direct Simulation MonteCarlo (DSMC) methods [23]. In recent years, deterministic solversto the BP system were proposed [20,27,3,2,4–6,22]. These methodsprovide accurate results which, in general, agree well with thoseobtained from Monte Carlo (DSMC) simulations, often at a frac-tional computational time. Moreover, they can resolve transientdetails for the pdf, which are difficult to compute with DSMC sim-ulators. The methods proposed in [4,6] used weighted essentiallynon-oscillatory (WENO) finite difference schemes to solve theBoltzmann–Poisson system. The advantage of the WENO schemeis that it is relatively simple to code and very stable even on coarsemeshes for solutions containing sharp gradient regions. A disad-vantage of the WENO finite difference method is that it requiressmooth meshes to achieve high order accuracy, hence it is not veryflexible for adaptive meshes.

On the other hand, motivated by the easy hp-adaptivity andsimple communication pattern of the discontinuous Galerkin(DG) methods for macroscopic (fluid level) models [7,8,25,26],we proposed in [11,10] to implement a DG solver to the full Boltz-mann equation, that is capable of capturing transients of the prob-ability density function (pdf).

The type of DG method that we will discuss here is a class of fi-nite element methods originally devised to solve hyperbolic con-servation laws containing only first order spatial derivatives, e.g.[16,15,14,13,17]. Using completely discontinuous polynomialspace for both the test and trial functions in the spatial variablesand coupled with explicit and nonlinearly stable high order Run-ge–Kutta time discretization, the method has the advantage offlexibility for arbitrarily unstructured meshes, with a compactstencil, and with the ability to easily accommodate arbitrary hp-adaptivity. For more details about DG scheme for convection dom-inated problems, we refer to the review paper [19]. The DG methodwas later generalized to the local DG (LDG) method to solve theconvection diffusion equation [18] and elliptic equations [1]. It isL2 stable and locally conservative, which makes it particularly suit-able to treat the Poisson equation.

In our previous preliminary work [11,10], we proposed the firstDG solver for (1.1) and showed some numerical calculations forone- and two-dimensional devices. In this paper, we will carefullyformulate the DG-LDG scheme for the Boltzmann–Poisson systemand perform extensive numerical studies to validate our calcula-tion. In addition, using this new scheme, we are able to implementa solver to full energy bands [9] that no other deterministic solverhas been able to implement so far. It would be more difficult oreven unpractical to produce the full band computation with othertransport scheme.

This paper is organized as follows: in Section 2, we review thechange of variables in [27,5]. In Section 3, we study the DG-BTE sol-ver for 1D diodes. Section 4 is devoted to the discussion of the 2Ddouble gate MOSFET DG solver. Conclusions and final remarks arepresented in Section 5. Some technical implementations of the DGsolver are collected in Appendix A.

2. Change of variables

In this section, we review the change of variables proposed in[27,4].

For the numerical treatment of the system (1.1) and (1.3), it isconvenient to introduce suitable dimensionless quantities andvariables. We assume TL ¼ 300 K. Typical values for length, timeand voltage are ‘� ¼ 10�6 m;t� ¼ 10�12 s and V� ¼ 1 V, respectively.Thus, we define the dimensionless variables

ðx; yÞ ¼ x‘�; t ¼ t

t�; W ¼ V

V�; ðEx; Ey;0Þ ¼

EE�

with E� ¼ 0:1V�‘�1� and

Ex ¼ �cv@W@x

; Ey ¼ �cv@W@y

; cv ¼V�‘�E�

:

In correspondence to [27,5], we perform a coordinate transforma-tion for k according to

k ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2m�kBTL

p�h

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiwð1þ aK wÞ

pl;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� l2

qcos u;


qsin u

� �;

ð2:6Þ

where the new independent variables are the dimensionless en-ergy w ¼ e

kBTL, the cosine of the polar angle l and the azimuth an-

gle u with aK ¼ kBTLa. The main advantage of the generalizedspherical coordinates (2.6) is the easy treatment of the Dirac dis-tribution in the kernel (1.5) of the collision term. In fact, this pro-cedure enables us to transform the integral operator (1.4) with thenonregular kernel S into an integral-difference operator, as shownin the following.

We are interested in studying problems that are two-dimen-sional in real space and three-dimensional in k-space. It wouldbe useful to consider the new unknown function U related to theelectron distribution function via

Uðt; x; y;w;l;uÞ ¼ sðwÞf ðt;x;kÞ;

where

sðwÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiwð1þ aK wÞ

pð1þ 2aK wÞ ð2:7Þ

is proportional to the Jacobian of the change of variables (2.6) and,apart from a dimensional constant factor, to the density of states.This allows us to write the free streaming operator of the dimen-sionless Boltzmann equation in a conservative form, which isappropriate for applying standard numerical schemes used forhyperbolic partial differential equations. Due to the symmetry ofthe problem and of the collision operator, we have

Uðt; x; y;w;l;2p�uÞ ¼ Uðt; x; y;w;l;uÞ: ð2:8Þ

Straightforward but cumbersome calculations end in the followingtransport equation for U:

@U@tþ @

@xðg1UÞþ

@

@yðg2UÞþ

@

@wðg3UÞþ

@

@lðg4UÞþ@

@uðg5UÞ¼CðUÞ:

ð2:9Þ

The functions gi ði ¼ 1;2; . . . ;5Þ in the advection terms depend onthe independent variables w;l;u as well as on time and positionvia the electric field. They are given by

g1ð�Þ ¼ cxl


p1þ 2aK w

;

g2ð�Þ ¼ cx


p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiwð1þ aK wÞ

pcos u

1þ 2aK w;

g3ð�Þ ¼ �2ck


p1þ 2aK w

lExðt; x; yÞ þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� l2

qcos uEyðt; x; yÞ

� �;

g4ð�Þ ¼ �ck


pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiwð1þ aK wÞ

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� l2

qExðt; x; yÞ � l cos uEyðt; x; yÞ

� �;

g5ð�Þ ¼ cksin uffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

wð1þ aK wÞp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� l2p Eyðt; x; yÞ

with

cx ¼t�‘�

ffiffiffiffiffiffiffiffiffiffiffiffiffi2kBTL

m�

rand ck ¼

t�qE�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2m�kBTL

p :

The right hand side of (2.9) is the integral-difference operator

3132 Y. Cheng et al. / Comput. Methods Appl. Mech. Engrg. 198 (2009) 3130–3150

CðUÞðt; x; y;w;l;uÞ ¼ sðwÞ c0

Z p

0du0

Z 1

�1dl0Uðt; x; y;w;l0;u0Þ

�þZ p

0du0

Z 1

�1dl0½cþUðt; x; y;wþ c;l0;u0Þ

þc�Uðt; x; y;w� c;l0;u0Þ�� 2p½c0sðwÞ

þ cþsðw� cÞ þ c�sðwþ cÞ�Uðt; x; y;w;l;uÞ;

where

ðc0; cþ; c�Þ ¼2m�t�

�h3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2m�kBTL

pðK0; ðnq þ 1ÞK;nqKÞ; c ¼ �hxp

kBTL

are dimensionless parameters. We remark that the d distributionsin the kernel S have been eliminated which leads to the shiftedarguments of U. The parameter c represents the jump constant cor-responding to the quantum of energy �hxp. We have also taken intoaccount (2.8) in the integration with respect to u0. Since the energyvariable x is not negative, we must consider null U and the functions, if the argument x� c is negative.

In terms of the new variables the electron density becomes

nðt�t; ‘�x; ‘�yÞ ¼Z

R3f ðt�t; ‘�x; ‘�y;kÞdk ¼


p�h

!3

qðt; x; yÞ;

x00.2

0.40.6

0.81t

12

34

5

density

0

2E+17

4E+17

5

x0

0.20.4

0.60.8

1t

1

2

3

4

5

energy

0.1

0.2

5

Fig. 3.1. Time evolution of macroscopic quantities using DG method for 400 nm channelleft: energy in eV; bottom right: momentum in cm�2 s�1.

where

qðt; x; yÞ ¼Z þ1

0dwZ 1

�1dlZ p

0duUðt; x; y;w;l;uÞ: ð2:10Þ

Hence, the dimensionless Poisson equation is

@

@x�r@W@x

� �þ @

@y�r@W@y

� �¼ cp½qðt; x; yÞ �NDðx; yÞ� ð2:11Þ

with

NDðx; yÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2m�kBTL

p�h

!�3

NDð‘�x; ‘�yÞ and

cp ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2m�kBTL

p�h

!3‘2�q�0

:

Choosing the same values of the physical parameters as in [27], weobtain

x0

0.20.4

0.60.8

1t

1

2

3

4

velocity

0

1E+07

x0

0.20.4

0.60.8

1t

12

34

momentum

-2E+23

0

2E+23

at Vbias ¼ 1:0 V. Top left: density in cm�3; top right: mean velocity in cm/s; bottom


Moreover, the dimensional x-component of the velocity is given byRþ10 dw

R 1�1 dl

R p0 dug1ðw;lÞUðt; x; y;w;l;uÞ

qðt; x; yÞ ;

the dimensional density by

1:0115� 1026 � qðt; x; yÞ;

and the energy by

0:025849�Rþ1

0 dwR 1�1 dl

R p0 duwUðt; x; y;w;l;uÞqðt; x; yÞ :

x

density

0.2 0.4 0.6 0.8

2E+17

4E+17

x

energy

0.2 0.4 0.6 0.8

0.05

0.1

0.15

x

potential

0.2 0.4 0.6 0.8

0

0.2

0.4

0.6

0.8

1

Fig. 3.2. Comparison of macroscopic quantities using DG (symbols) and WENO (solid linemean velocity in cm/s; middle left: energy in eV; middle right: electric field in kV/cm; bosteady state.

In a simplified model, we consider our device in the x�directionby assuming that the doping profile, the potential and thus theforce field are only x-dependent. By cylindrical symmetry, theresulting distribution function does not depend on u. In this case,the Boltzmann transport equation is reduced to

@U@tþ @

@xðg1UÞ þ

@

@wðg3UÞ þ

@

@lðg4UÞ ¼ CðUÞ: ð2:12Þ

with

x

velocity

0.2 0.4 0.6 0.80

4E+06

8E+06

x

E

0.2 0.4 0.6 0.8-60

-40

-20

0

20

x

momentum

0.2 0.4 0.6 0.8

1E+23

) for 400 nm channel at t ¼ 5:0 ps, Vbias ¼ 1:0 V. Top left: density in cm�3; top right:ttom left: potential in V; bottom right: momentum in cm�2 s�1. Solution has reached


g1ð�Þ ¼ cxl


p1þ 2aK w

;

g3ð�Þ ¼ �2ck


p1þ 2aK w

lEðt; xÞ;

g4ð�Þ ¼ �ck1� l2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

wð1þ aK wÞp Eðt; xÞ

and

CðUÞðt;x;w;lÞ¼ sðwÞ c0pZ 1

�1dl0Uðt;x;w;l0Þ

�þpZ 1

�1dl0½cþUðt;x;wþc;l0Þþc�Uðt;x;w�c;l0Þ�

��2p½c0sðwÞþcþsðw�cÞþc�sðwþcÞ�Uðt;x;w;lÞ;

In terms of the new variables the electron density becomes

nðt�t; ‘�xÞ ¼Z

R3f ðt�t; ‘�x;kÞdk ¼


p�h

!3

qðt; xÞ;

w0 10 20 30

μ

-1

0

1

Φ

0

0.0001

0.0002

0.0003

w0 10 20 30

μ

-1

0

1

Φ0

4E-07

8E-07

w0 10 20 30

μ

-101

Φ

0

0.0002

0.0004

Fig. 3.3. Comparison of the snapshot for Uðx0;w;lÞ using DG (left) and WENO (right) forbottom: x0 ¼ 0:7 lm. Solution has not yet reached steady state.

where

qðt; xÞ ¼ pZ þ1

0dwZ 1

�1dlUðt; x;w;lÞ: ð2:13Þ

Hence, the dimensionless Poisson equation writes

@

@x�r@W@x

� �¼ cp½qðt; xÞ �NDðxÞ� ð2:14Þ

and

E ¼ �cv@W@x

:

3. DG-BTE solver for 1D diodes simulation

3.1. Algorithm description

We begin with formulating the DG-BTE solver for 1D diodes.These examples have been thoroughly studied and tested byWENO in [4].

Φ

w0 10 20 30

μ

-1

0

1

Φ

0

0.0001

0.0002

0.0003

w0 10 20 30

μ

-1

0

10

4E-07

8E-07

w0 10 20 30

μ

-1

0

1

Φ

0

0.0001

0.0002

0.0003

0.0004

0.0005

400 nm channel at t ¼ 0:5 ps, Vbias ¼ 1:0 V. Top: x0 ¼ 0:3 lm; middle: x0 ¼ 0:5 lm;


The Boltzmann–Poisson system (2.12) and (2.14) will be solvedon the domain

x 2 ½0; L�; w 2 ½0;wmax�; l 2 ½�1;1�;

where L is the dimensionless length of the device and wmax is themaximum value of the energy, which is adjusted in the numericalexperiments such that

Uðt; x;w;lÞ � 0 for w P wmax and every t; x;l:

In (2.12), g1 and g3 are completely smooth in the variable w andl, assuming E is given and smooth. However, g4 is singular for theenergy w ¼ 0, although it is compensated by the sðwÞ factor in thedefinition of U.

The initial value of f is a locally Maxwellian distribution at thetemperature TL,

Uð0; x;w;lÞ ¼ sðwÞNDðxÞe�wM

with the numerical parameter M chosen so that the initial value forthe density is equal to the doping NDðxÞ.

w0 10 20 30

μ

-1

0

1

Φ

0

0.0001

0.0002

0.0003

w0 10 20 30

μ

-1

0

1

Φ

0

2E-06

4E-06

6E-06

8E-06

w0 10 20 30

μ

-1

0

1

Φ

0

5E-05

0.0001

0.00015

0.0002

0.00025

Fig. 3.4. Comparison of the snapshot for Uðx0;w;lÞ using DG (left) and WENO (right) forbottom: x0 ¼ 0:7 lm. Solution has reached steady state.

We choose to perform our calculations on the following rectan-gular grid:

Xikm ¼ xi�12; xiþ1

2

h i� wk�1

2;wkþ1

2

h i� lm�1

2;lmþ1

2

h i; ð3:15Þ

where i ¼ 1; . . . Nx; k ¼ 1; . . . Nw;m ¼ 1; . . . Nl, and

xi�12¼ xi �

Dxi

2; wk�1

2¼ wk �

Dwk

2; lm�1

2¼ lm �

Dlm

2:

It is useful that we pick Nl to be even, so the function g1 will as-sume a constant sign in each cell Xikm.

The approximation space is thus defined as

V ‘h ¼ fv : v jXikm

2 P‘ðXikmÞg; ð3:16Þ

where P‘ðXikmÞ is the set of all polynomials of degree at most ‘ onXikm. The DG formulation for the Boltzmann equation (2.12) wouldbe: to find Uh 2 V ‘

h, such that

w0 10 20 30

μ

-1

0

1

Φ

0

0.0001

0.0002

0.0003

w0 10 20 30

μ

-1

0

1

Φ

0

2E-06

4E-06

6E-06

8E-06

w0 10 20 30

μ

-1

0

1

Φ

0

5E-05

0.0001

0.00015

0.0002

0.00025


ZXikm
ðUhÞtvh dX�Z

Xikm

g1UhðvhÞx dX�Z

Xikm

g3UhðvhÞw dX

�Z

Xikm

g4UhðvhÞl dXþ Fþx � F�x þ Fþw � F�w þ Fþl � F�l

¼Z

Xikm

CðUhÞvh dX ð3:17Þ

for any test function vh 2 V ‘h. In (3.17),

Fþx ¼Z w

kþ12

wk�1

2

Z lmþ1

2

lm�1

2

g1�Uv�h xiþ1

2;w;l

dwdl;

F�x ¼Z w

kþ12

wk�1

2

Z lmþ1

2

lm�1

2

g1�Uvþh xi�1

2;w;l

dwdl;

Fþw ¼Z x

iþ12

xi�1

2

Z lmþ1

2

lm�1

2

g3bUv�h x;wkþ1

2;l

dxdl;

F�w ¼Z x

iþ12

xi�1

2

Z lmþ1

2

lm�1

2

g3bUvþh x;wk�1

2;l

dxdl;

Fþl ¼Z x

iþ12

xi�1

2

Z wkþ1

2

wk�1

2

g4eUv�h x;w;lmþ1

2

dxdw;

F�l ¼Z x

iþ12

xi�1

2

Z wkþ1

2

wk�1

2

g4eUvþh x;w;lm�1

2

dxdw;

where the upwind numerical fluxes �U; bU; eU are chosen according tothe following rules:

x0

0.050.1

0.150.2

0.25t1

2

3

density

2E+18

4E+18

x0

0.050.1

0.150.2

0.25t1

2

3

energy

0

0.2

0.4

0.6

0.8

Fig. 3.5. Time evolution of macroscopic quantities using DG method for 50 nm channelleft: energy in eV; bottom right: momentum in cm�2 s�1.

The sign of g1 only depends on l, if lm > 0; �U ¼ U�; otherwise,�U ¼ Uþ.

The sign of g3 only depends on lEðt; xÞ, if lmEðt; xiÞ < 0; bU ¼ U�;otherwise, bU ¼ Uþ.

The sign of g4 only depends on Eðt; xÞ, if Eðt; xiÞ < 0; eU ¼ U�;otherwise, eU ¼ Uþ.

At the source and drain contacts, we implement the sameboundary condition as proposed in [6] to realize neutral charges.In the ðw;lÞ-space, no boundary condition is necessary, since

At w ¼ 0; g3 ¼ 0. At w ¼ wmax;U is machine zero. At l ¼ �1; g4 ¼ 0,

Fþw; F�w; F

þl ; F

�l are always zero. This saves us the effort of construct-

ing ghost elements in comparison with WENO.The Poisson equation (2.14) is solved by the LDG method on a

consistent grid of (3.15) in the x-direction. It involves rewritingthe equation into the following form:

q ¼ @W@x ;

@@x ð�rqÞ ¼ Rðt; xÞ;

(ð3:18Þ

where Rðt; xÞ ¼ cp½qðt; xÞ �NDðxÞ� is a known function that can becomputed at each time step once U is solved from (3.17), and thecoefficient �r here is a constant. The grid we use is Ii ¼ xi�1

2; xiþ1

2

h i,

with i ¼ 1; . . . ;Nx. The approximation space is

W ‘h ¼ fv : v jIi

2 P‘ðIiÞg;

x00.05

0.10.15

0.20.25t

1

2

3velocity

1E+07

x0

0.050.1

0.150.2

0.25t

1

2

3

momentum

6E+24

9E+24

1.2E+25

at Vbias ¼ 1:0 V. Top left: density in cm�3; top right: mean velocity in cm/s; bottom


with P‘ðIiÞ denoting the set of all polynomials of degree at most ‘ onIi. The LDG scheme for (3.18) is given by: to find qh;Wh 2 V ‘

h, suchthatZ

Ii

qhvhdxþZ

Ii

WhðvhÞx dx� bWhv�h xiþ12

þ bWhvþh xi�1

2

¼ 0;

�Z

Ii

�rqhðphÞx dxþd�rqhp�h xiþ12

�d�rqhpþh xi�1

2

¼Z

Ii

Rðt; xÞph dx

ð3:19Þ

hold true for any vh;ph 2W ‘h. In the above formulation, the flux is

chosen as follows, bWh ¼ W�h ;d�rqh ¼ �rqþh � ½Wh�, where ½Wh� ¼ Wþh�W�h . At x ¼ L we need to flip the flux to bWh ¼ Wþh ;d�rqh ¼�rq�h � ½Wh� to adapt to the Dirichlet boundary conditions. Solving

x

density

0.05 0.1 0.15 0.2

2E+18

4E+18

x

energy

0.05 0.1 0.15 0.2

0.1

0.2

0.3

0.4

x

potential

0.05 0.1 0.15 0.20

0.2

0.4

0.6

0.8

1

Fig. 3.6. Comparison of macroscopic quantities using DG (symbols) and WENO (solid linmean velocity in cm/s; middle left: energy in eV; middle right: electric field in kV/cm; botsteady state.

(3.19), we can obtain the numerical approximation of the electricpotential Wh and electric field Eh ¼ �cvqh on each cell Ii.

We include some of the technical details of the implementa-tions for a 2D device in Appendix A. 1D diodes simulation carriessimilar steps and is simpler. We start with an initial value for Uh

described in Appendix A.5. The DG-LDG algorithm advances fromtn to tnþ1 in the following steps:

Step 1: Compute qhðt; xÞ ¼ pRþ1

0 dwR 1�1 dlUhðt; x;w;lÞ as

described in Appendix A.6.Step 2: Use qhðt; xÞ to solve from (3.19) the electric field, and com-

pute gi; i ¼ 1;3;4 on each computational element.Step 3: Solve (3.17) and get a method of line ODE for Uh, see

Appendices A.2, A.3 and A.4.

x

velocity

0.05 0.1 0.15 0.2

1E+07

x

E

0.05 0.1 0.15 0.2-400

-300

-200

-100

0

x

momentum

0.05 0.1 0.15 0.26E+24

8E+24

e) for 50 nm channel at t ¼ 3:0 ps, Vbias ¼ 1:0 V. Top left: density in cm�3; top right:tom left: potential in V; bottom right: momentum in cm�2 s�1. Solution has reached


Step 4 Evolve this ODE by proper time stepping from tn to tnþ1, ifpartial time step is necessary, then repeat Steps 1–3 asneeded.

Finally, the hydrodynamic moments can be computed at anytime step by the method in Appendix A.6.

We want to remark that, unlike WENO, the DG formulationabove has no restriction on the mesh size. In fact, nonuniformmeshes are more desirable in practice. For small semiconductordevices with highly doped, non-smooth regions, strong relativeelectric fields give high energy to charged particles, so it is ex-pected to create states whose distribution functions f have highdensities in some regions but low densities in others. Only a non-uniform grid can guarantee accurate results without using a largenumber of grid points. In particular the evolving distribution func-tion f ðt; x; kÞ has, at some regions points in the physical domaincorresponding to strong force fields, a shape that is very differentfrom a Maxwellian distribution (the expected equilibrium statisti-cal state in the absence of electric field), see for example Fig. 3.11.Moreover, taking into account the rapid decay of f for large value of

w0 20 40 60

μ

-1

0

1

Φ

0

0.001

0.002

0.003

0.004

0.005

w0 20 40 60

μ

-1

0

1

Φ0

0.0005

0.001

w0 20 40 60

μ

-1

0

1

Φ

0

0.0001

0.0002

0.0003

0.0004

0.0005

Fig. 3.7. Comparison of the snapshot for Uðx0;w;lÞ using DG (left) and WENO (right) forbottom: x0 ¼ 0:15 lm. Solution has not yet reached steady state.

jkj, a nonuniform grid is more computational efficient without sac-rificing accuracy of the calculation.

In the following simulations, we have implemented, withoutself-adaptivity, a nonuniform mesh that refines locally near the re-gions where most of the interesting phenomena happen, that is thejunctions of the channel, addressing one of the main difficulties insolving the Boltzmann–Poisson system with the presence of thedoping, which creates strong gradients of the solution near itsjumps, and the polar angle direction l ¼ 1, corresponding to thealignment with the force electric field.

Consequently, for this first computational implementations ofthe DG solver, using the ansatz that the doping profile functionNDðxÞ in the Poisson equation does not depend on time, we havethe advantage to know where we need a fine grid in space to guar-antee a reasonable accuracy.

Even in the simplest test problem of the bulk case, the electricfield is constant and the solution is spatially homogeneous. If theinitial condition is a Maxwellian distribution function, then, forhigh constant electric field, one observes the overshoot velocityproblem which stabilizes in time. This fact implies that the final

w0 20 40 60

μ

-1

0

1

Φ

0

0.001

0.002

0.003

0.004

0.005

w0 20 40 60

μ

-1

0

1

Φ

0

0.0005

0.001

w0 20 40 60

μ

-1

0

1

Φ

0

0.0001

0.0002

0.0003

0.0004

0.0005



state distribution tails change with respect to the initial one, andthe distribution has also a shift proportional to the potential en-ergy U, therefore a simple choice a fixed mesh in ðw;lÞ worksfor a given constant electric field that depends on a potential biasat contacts. In general, due to the stabilization nature of the Boltz-mann–Poisson system, the choice of a fixed nonuniform meshworks for this preliminary implementation. However, the magni-tude of the gradient variation of the solutions depends on the ap-plied bias, ultimately adaptivity will be the most efficient way tocalculate the solution, and so self adaptive DG schemes will bestudied in our future work. Nevertheless, we anticipate that a selfadaptive scheme has the advantage of a smaller number of cells,but the overhead for the adjustment of the solution to a new gridmight be CPU time consuming.

3.2. Discussions on benchmark test problems

We consider two benchmark test examples: Si nþ—n—nþ diodesof a total length of 1 and 0.25 lm, with 400 and 50 nm channelslocated in the middle of the device, respectively. For the 400 nm

w0 20 40 60

μ

-1

0

1

Φ

0

0.001

0.002

0.003

0.004

0.005

Φ

w0 20 40 60

μ

-1

0

1

Φ

0

0.0005

0.001

0.0015

w0 20 40 60

μ

-1

0

1

Φ

0

0.0002

0.0004

0.0006

Fig. 3.8. Comparison of the snapshot for Uðx0;w;lÞ using DG (left) and WENO (rightx0 ¼ 0:125 lm; bottom: x0 ¼ 0:15 lm. Solution has reached steady state.

channel device, the dimensional doping is given by ND ¼ 5�1017 cm�3 in the nþ region and ND ¼ 2� 1015 cm�3 in the n� region.For the 50 nm channel device, the dimensional doping is given byND ¼ 5� 1018 cm�3 in the nþ region and ND ¼ 1� 1015 cm�3 in then� region. Both examples were computed by WENO in [5].

In our simulation, we use piecewise linear polynomials, i.e.‘ ¼ 1, and second-order Runge–Kutta time discretization. The dop-ing ND is smoothed in the following way near the channel junctionsto obtain non-oscillatory solutions. Suppose ND ¼ NþD in the nþ re-gion, ND ¼ N�D in the n� region and the length of the transition re-gion is 2 cells, then the smoothed function is ðNþD � N�DÞð1� y3Þ3

þN�D , where y ¼ ðx� x0 þ DxÞ=ð2Dxþ 10�20Þ is the coordinatetransformation that makes the transition region ðx0 � Dx; x0 þ DxÞvaries from 0 to 1 in y.

The nonuniform mesh we use for 400 nm channels is defined asfollows. In the x-direction, if x < 0:2 lm or x > 0:4 lm,Dx ¼ 0:01 lm. In the region 0:2 lm < x < 0:4 lm, Dx ¼ 0:005 lm.Thus, the total number of cells in x direction is 120. In the w-direc-tion, we use 60 uniform cells. In the l-direction, we use 24 cells, 12in the region l < 0:7, 12 in l > 0:7. Thus, the grid consists of

w0 20 40 60

μ

-1

0

10

0.001

0.002

0.003

0.004

0.005

w0 20 40 60

μ

-1

0

1

Φ

0

0.0005

0.001

0.0015

w0 20 40 60

μ

-1

0

1

Φ

0

0.0002

0.0004

0.0006

) solution for 50 nm channel at t ¼ 3:0 ps, Vbias ¼ 1:0. Top: x0 ¼ 0:1 lm; middle:

V1

-10-5

05

10

V2-10

-50

510

pdf

0

0.0002

0.0004

Fig. 3.10. PDF for 50 nm channel at t ¼ 3:0 ps, x ¼ 0:125 lm. Center of the channel.


120 � 60 � 24 cells, compared to the WENO grid of 180 � 60 � 24uniform cells. We should remark that there are more degrees offreedom per cell for DG than for WENO, hence a more adequatecomparison is the CPU cost to obtain the same level of errors, seefor example [30] in which such a comparison is made for somesimple examples indicating the advantage of DG in certainsituations.

We plot the evolution of density, mean velocity, energy andmomentum in Fig. 3.1. Due to the conservation of mass, whichthe DG scheme satisfies for our boundary condition, one can seethat when the momentum becomes constant, the solution is sta-ble. The solution has already stabilized at t = 5.0 ps from themomentum plots. The macroscopic quantities at steady stateare plotted in Fig. 3.2. The results are compared with the WENOcalculation. They agree with each other in general, with DG offer-ing more resolution and a higher peak in energy near the junc-tions. This is crucial to obtain a more agreeable kinetic energywith experiments and DSMC simulations. Figs. 3.3 and 3.4 showcomparisons for the pdf at transient and steady state. We plotat different positions of the device, namely, the left, center andright of the channel. We notice a larger value of pdf especiallyat the center of the channel, where the pdf is no longer Maxwell-ian. Moreover, at t = 0.5 ps, x0 ¼ 0:5 lm, the pdf shows a doublehump structure, which is not captured by the WENO solver. Allof these advantages come from the fact that we are refining morenear l ¼ 1. To have a better idea of the shape of the pdf, we plotUðt ¼ 5:0 ps;x ¼ 0:5 lmÞ in the cartesian coordinates in Fig. 3.9.The coordinate V1 in the plot is the momentum parallel to theforce field k1;V2 is the modulus of the orthogonal component.The peak is captured very sharply compared to WENO, see Figs.3 and 4 in [5].

The nonuniform mesh we use for 50 nm channels is defined asfollows. In the x-direction, near the junctions, in 0:09 lm <

x < 0:11 lm and 0:14 lm < x < 0:16 lm;Dx ¼ 0:001 lm; in cen-ter of the channel 0:11 lm < x < 0:14 lm;Dx ¼ 0:005 lm; ateverywhere else, Dx ¼ 0:01 lm. Thus, the total number of cells inx direction is 64. In the w-direction, we use 60 uniform cells. Inthe l-direction, we use 20 cells, 10 in the region l < 0:7, 10 inl > 0:7. Thus, the grid consists of 64 � 60 � 20 cells, comparedto the WENO calculation of 150 � 120 � 16 uniform cells. The evo-lution and steady state plots are listed in Figs. 3.5–3.7 and 3.8. Theconclusions are similar with 400 nm, that we obtain better resolu-tions near the channel junctions and the peak for pdf is much high-er. Fig. 3.10 plots Uðt ¼ 5:0 ps;x ¼ 0:125 lmÞ in the cartesian

V1-5

0

5

V2-5

0

5

pdf

0

2E-06

4E-06

6E-06

8E-06

1E-05

Fig. 3.9. PDF for 400 nm channel at t ¼ 5:0 ps, x ¼ 0:5 lm. Center of the channel.

coordinates. The peak is twice the height of WENO and is verysharp. Fig. 3.11 plots the pdf near x ¼ 0:15 lm, the drain junction.We obtain distributions far away from statistical equilibrium, thatreflects the lack of suitability of the classical hydrodynamical mod-els for the drain region of a small gated device under even moder-ate voltage bias.

3.3. Comparisons with DSMC simulations

We compare the results from DG-BTE solver with those ob-tained from DSMC simulations (see Figs. 3.12 and 3.13). Both sim-ulations show good agreement except for the kinetic energy nearthe boundaries.

This mismatch is due to the modeling of the contact boundaryconditions in DSMC simulations, which require the knowledge ofthe distribution function f (or the corresponding energy-bandtransform U) for entering particles. A classical way to solve thisproblem for DSMC simulations is done by the inclusion of a trans-port kinetic equation for the dynamic of the electron at the metaljunctions. However, this is not realistic due to the complexity ofthis new kinetic equation, where the importance of electron–elec-tron interaction requires a nonlinear collisional operator, similar tothe classical one of the Boltzmann equation for a rarefied perfectgas.

Nevertheless, traditionally, boundary conditions associatedwith these DSMC models for charge electron transport have beenimplemented by the following two simplistic rules consisting onthe assumption that the distribution function f, near the contactbut outside the computational domain is, either proportional to aMaxwellian equilibrium distribution function for incoming veloci-ties, or related to f evaluated near the contact and inside the com-putational domain. (For strong electric fields near the boundaries,these choices are not correct, as they may required a shift Max-wellian or a condition that generates momentum in the directionof the field.)

The first option is well accepted when a high doping concentra-tion near the boundaries yields a very small electric field: so, theelectron gas, in these zones, is almost in the thermal equilibriumand has a constant velocity approximately. That is, we could usethe ansatz that the distribution function U, in the outer side ofthe boundary is taken to be proportional to a energy-band spheri-cally transformed Maxwellian equilibrium distribution function

V1

-10-5

05

10

V2-10

-50

510

pdf

0

2E-05

4E-05

6E-05

8E-05

0.0001

V1

-10-5

05

10

V2-10

-50

510

pdf

0

4E-05

8E-05

0.00012

V1

-10-5

05

10

V2-10

-50

510

pdf

0

0.0002

0.0004

V1

-10-5

05

10

V2-10

-50

510

pdf

0

0.003

0.006

0.009

Fig. 3.11. Non-equilibrium PDF states near the drain junction for 50 nm channel at t ¼ 3:0 ps. They exhibit the large deviation from classical statistical states even at placescorresponding to highly doped region. This is a signature of hot electron transport. Top left: x ¼ 0:149 lm. Top right: x ¼ 0:15 lm. Bottom left: x ¼ 0:152 lm. Bottom right:x ¼ 0:16 lm.


centered at zero energy, whose average matches the doping profilefunction NDðxÞ at the boundary. However this approach will createboundary layers of the mean free path order and so the conditiondo not secure an absolute charge neutrality.

Hence, we rather use the second option which imposes thatf ¼ fout , near the contact and outside the computational domainðk � gxb

< 0Þ where xb is a boundary point and gxbthe outer normal,

is related to f ¼ fin evaluated near the contact and inside the com-putational domain ðk � gxb

> 0Þ. This approach allows us to get anexact neutral charge at the source and drain contact boundaries,which imposes that the probability and doping density must coin-cide, as well as continuity in velocity space. Then the simpleformula,

Uoutðt; xb;w;lÞ ¼NDðxbÞqðt; xbÞ

Uinðt; xb;w;lÞ ð3:20Þ

is assumed in our simulations, where Uout is the band-energy spher-ically coordinated transformed fout , in the corresponding transformdomain.

This very same boundary type condition (3.20) at the gate anddrain contacts has been used in the deterministic simulations ofBoltzmann–Poisson systems for semiconductor devices by WENOschemes (see [6]) and are use here for the DG schemes to approx-imate the same boundary value problem. So the agreement of low-er moments is very good, even though at the computational level,(3.20) is approximated in different ways depending on the numer-ical scheme.

For example, in the case of DSMC simulations, particles must becreated or destroyed. However in the case of finite differenceWENO schemes, we introduced suitable ghost points [4,6]; andin the case of DG schemes it is enough to give values of U at bound-aries according to condition (3.20).

Hence, we can not have a unique exact boundary conditions inthe computational experiments, which makes it clear that thesedifferences in the numerical boundary treatment have an influencefor the solutions at the stationary regime, which explains the dis-crepancies of the numerical comparisons of moments near theboundaries. Nevertheless, we stress that, as Figs. 3.5 and 3.6 show

x

density

0.2 0.4 0.6 0.80

2E+17

4E+17

x

velocity

0 0.2 0.4 0.6 0.8 10

4E+06

8E+06

x

energy

0 0.2 0.4 0.6 0.8 1

0.05

0.1

0.15

x

E

0 0.2 0.4 0.6 0.8 1

-40

-20

0

20

x

potential

0 0.2 0.4 0.6 0.8 1

0

0.4

0.8

x

momentum

0.2 0.4 0.6 0.8 1

1E+23

Fig. 3.12. Comparison of macroscopic quantities using DG (dashed line) and DSMC (solid line) for 400 nm channel at t ¼ 5:0 ps, V bias ¼ 1:0 V. Top left: density in cm�3; topright: mean velocity in cm/s; middle left: energy in eV; middle right: electric field in kV/cm; bottom left: potential in V; bottom right: momentum in cm�2 s�1. Solution hasreached steady state.


in all cases, there is a very good agreement of the comparisons in-side the device.

3.4. Summary

We have benchmarked the DG algorithm on two test cases: Sinþ—n—nþ diodes with 400 and 50 nm channels. The DG schemehas the ability to quickly stabilize to the steady state and accu-rately resolve transient details. In particular, the pdf along thedirection of the electric field is captured sharply due to the nonuni-form mesh we use. For the more energetic device with 50 nm chan-nel, DG computes a much higher peak and a ‘‘donut” shaped tail inthe pdf near the drain junction, thus resulting in an accuratedescription of higher order moments in those regions.

4. DG-BTE solver for 2D double gate MOSFET simulation

In this section, we turn to a 2D double gate MOSFET device. Thealgorithm is similar to those described in the previous section. Forthe benefit of the reader, we repeat some of the procedures in Sec-tion 3. We will use a simple rectangular grid and let

Xijkmn ¼ xi�12; xiþ1

2

h i� yj�1

2; yjþ1

2

h i� wk�1

2;wkþ1

2

h i� lm�1

2;lmþ1

2

h i� un�1

2;unþ1

2

h i;

wherei ¼ 1; . . . Nx; j ¼ 1; . . . Ny; k ¼ 1; . . . Nw; m ¼ 1; . . . Nl; n ¼ 1; . . . Nu,and

x

density

0 0.05 0.1 0.15 0.2 0.25

1E+18

2E+18

3E+18

4E+18

5E+18

x

velocity

0 0.05 0.1 0.15 0.2 0.25

6E+06

1.2E+07

1.8E+07

x

energy

0 0.05 0.1 0.15 0.2 0.25

0.1

0.2

0.3

0.4

x

E

0 0.05 0.1 0.15 0.2 0.25-400

-300

-200

-100

0

100

x

potential

0 0.05 0.1 0.15 0.2 0.25

0

0.2

0.4

0.6

0.8

1

x

momentum

0 0.05 0.1 0.15 0.2 0.25

6E+24

7.5E+24

9E+24

Fig. 3.13. Comparison of macroscopic quantities using DG (dashed line) and DSMC (solid line) for 50 nm channel at t ¼ 3:0 ps, V bias ¼ 1:0 V . Top left: density in cm�3; topright: mean velocity in cm/s; middle left: energy in eV; middle right: electric field in kV/cm; bottom left: potential in V; bottom right: momentum in cm�2 s�1. Solution hasreached steady state.

xdrai

n

source

top gate

bottom gate

y

Fig. 4.14. Schematic representation of a 2D double gate MOSFET device.


xi�12¼ xi �

Dxi

2; yj�1

2¼ yj �

Dyj

2; wk�1

2¼ wk �

Dwk

2;

lm�12¼ lm �

Dlm

2; un�1

2¼ un �

Dun

2:

The approximation space is defined as

V ‘h ¼ fv : v jXijkmn

2 P‘ðXijkmnÞg: ð4:21Þ

Here, P‘ðXijkmnÞ is the set of all polynomials of degree at most ‘ onXijkmn. The DG formulation for the Boltzmann equation (2.9) wouldbe: to find Uh 2 V ‘

h, such that

0.00 0.03 0.06 0.09 0.12 0.15

0.0030.006

0.0090.012

0 2e+24 4e+24 6e+24 8e+24 1e+25

1.2e+25

xy 0.00 0.03 0.06 0.09 0.12 0.15

0.0030.006

0.0090.012

0.1 0.15 0.2

0.25 0.3

0.35 0.4

xy

0.00 0.03 0.06 0.09 0.12 0.15

0.0030.006

0.0090.012

0 2e+06 4e+06 6e+06 8e+06 1e+07

1.2e+071.4e+071.6e+071.8e+07

x

y 0.00 0.03 0.06 0.09 0.12 0.15

0.0030.006

0.0090.012

-1e+06

-500000

0

500000

1e+06

1.5e+06

xy

Fig. 4.15. Macroscopic quantities of double gate MOSFET device at t ¼ 0:5 ps. Top left: density in cm�3; top right: energy in eV; bottom left: x-component of velocity in cm/s;bottom right: y-component of velocity in cm/s. Solution reached steady state.

0.00 0.03 0.06 0.09 0.12 0.15

0.0030.006

0.0090.012-150

-100

-50

0

xy 0.00 0.03 0.06 0.09 0.12 0.15

0.0030.006

0.0090.012-800

-600-400-200

0

xy

0.00 0.03 0.06 0.09 0.12 0.15

0.0030.006

0.0090.012

0 0.2 0.4 0.6 0.8

1 1.2 1.4 1.6

xy

Fig. 4.16. Macroscopic quantities of double gate MOSFET device at t ¼ 0:5 ps. Top left: x-component of electric field in kV/cm; top right: y-component of electric field in kV/cm; bottom: electric potential in V. Solution has reached steady state.


ZXijkmn

ðUhÞtvh dX�Z

Xijkmn

g1UhðvhÞx dX�Z

Xijkmn

g2UhðvhÞy dX

�Z

Xijkmn

g3UhðvhÞw dX�Z

Xijkmn

g4UhðvhÞl dX

�Z

Xijkmn

g5UhðvhÞu dXþ Fþx � F�x þ Fþy � F�y þ Fþw � F�w

þ Fþl � F�l þ Fþu � F�u ¼Z

Xijkmn

CðUhÞvh dX ð4:22Þ

for any test function vh 2 V ‘h. In (4.22),

F�x ¼Z y

jþ12

yj�1

2

Z wkþ1

2

wk�1

2

Z lmþ1

2

lm�1

2

Z unþ1

2

un�1

2

g1�Uvh xi�1

2; y;w;l;u

dydwdldu;

F�y ¼Z x

iþ12

xi�1

2

Z wkþ1

2

wk�1

2

Z lmþ1

2

lm�1

2

Z unþ1

2

un�1

2

g2Uvh x; yj�12;w;l;u

dxdwdldu;

F�w ¼Z x

iþ12

xi�1

2

Z yjþ1

2

yj�1

2

Z lmþ1

2

lm�1

2

Z unþ1

2

un�1

2

dg3Uvh x; y;wk�12;l;u

dxdydldu;


F�l ¼Z x

iþ12

xi�1

2

Z yjþ1

2

yj�1

2

Z wkþ1

2

wk�1

2

Z unþ1

2

un�1

2

gg4Uvh x; y;w;lm�12;u

dxdydwdu;

F�u ¼Z x

iþ12

xi�1

2

Z yjþ1

2

yj�1

2

Z wkþ1

2

wk�1

2

Z lmþ1

2

lm�1

2

g5_Uvh x; y;w;l;un�1

2

dxdydwdl;

where the upwind numerical fluxes �U;U; dg3U; gg4U; _U are defined inthe following way:

The sign of g1 only depends on l, if lm > 0, then �U ¼ U�; other-wise, �U ¼ Uþ.

The sign of g2 only depends on cos u, if cos un > 0, then U ¼ U�;otherwise, U ¼ Uþ. Note that in our simulation, Nu is alwayseven.

w

020

4060

μ-0.500.5

00.00020.00040.00060.00080.0010.00120.00140.0016

w

020

4060

μ-0.500.5

0

0.0002

0.0004

0.0006

0.0008

0.001

w

020

4060

μ-0.500.5

0

0.0005

0.001

0.0015

Fig. 4.17. PDF of double gate MOSFET device at t ¼ 0:5 ps. Top left: at ð0:025 lm;0:012 lright: at ð0:1375 lm;0:006 lmÞ; bottom left: at ð0:09375 lm; 0:010 lmÞ; bottom right:

For dg3U, we let

dg3U¼�2ck

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiwð1þaK wÞ

p1þ2aK w

lExðt;x;yÞbUþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�l2

qcosuEyðt;x;yÞeU� �

:

If lmExðt; xi; yjÞ < 0, then bU ¼ U�; otherwise, bU ¼ Uþ.Ifðcos unÞEyðt; xi; yjÞ < 0, then ~U ¼ U�; otherwise, ~U ¼ Uþ:

For gg4U, we let

gg4U¼�ck

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�l2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiwð1þaK wÞ

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�l2

qExðt;x;yÞ bU�lcosuEyðt;x;yÞeU� �

;

If Exðt; xi; yjÞ < 0, then bU ¼ U�; otherwise, bU ¼ Uþ.If lm cosðunÞEyðt; xi; yjÞ > 0, then eU ¼ U�; otherwise, eU ¼ Uþ.

w

020

4060

μ-0.500.5

0

0.0005

0.001

0.0015

0.002

w

020

4060

μ-0.500.5

0

0.0005

0.001

0.0015

0.002

w

020

4060

μ-0.500.5

0

0.0002

0.0004

0.0006

0.0008

0.001

mÞ; top right: at ð0:075 lm;0:012 lmÞ; middle left: at ð0:125 lm;0:012 lmÞ; middleat ð0:09375 lm;0 lmÞ. Solution reached steady state.

V1-10

-50

510

V2-10

-50

510

pdf

0

0.0001

0.0002

0.0003

Fig. 4.18. PDF for 2D double gate MOSFET at t ¼ 0:5 ps,


The sign of g5 only depends on Eyðt; x; yÞ, if Eyðt; xi; yjÞ > 0, then_U ¼ U�; otherwise, _U ¼ Uþ.

The schematic plot of the double gate MOSFET device is given inFig. 4.14. The shadowed region denotes the oxide-silicon region,whereas the rest is the silicon region. Since the problem is sym-metric about the x-axis, we will only need to compute for y > 0.At the source and drain contacts, we implement the same bound-ary condition as proposed in [6] to realize neutral charges. A bufferlayer of ghost points of i ¼ 0 and i ¼ Nx þ 1 is used to make

Uði ¼ 0Þ ¼ Uði ¼ 1ÞNDði ¼ 1Þqði ¼ 1Þ ;

and

Uði ¼ Nx þ 1Þ ¼ Uði ¼ NxÞNDði ¼ NxÞqði ¼ NxÞ

:

At the top and bottom of the computational domain (the siliconregion), we impose the classical elastic specular boundaryreflection.

In the ðw;l;uÞ-space, no boundary condition is necessary be-cause of similar reason as in the 1D case,

at w ¼ 0; g3 ¼ 0. At w ¼ wmax;U is machine zero; at l ¼ �1; g4 ¼ 0; at u ¼ 0;p; g5 ¼ 0,

so at those boundaries, the numerical flux always vanishes, henceno ghost point is necessary.

For the boundary condition of the Poisson equation, W ¼0:52354 at source, W ¼ 1:5235 at drain and W ¼ 1:06 at gate. Forthe rest of boundaries, we impose homogeneous Neumann bound-ary condition, i.e., @W

@n ¼ 0. The relative dielectric constant in theoxide-silicon region is �r ¼ 3:9, in the silicon region is �r ¼ 11:7.

The Poisson equation (2.11) is solved by the LDG method. It in-volves rewriting the equation into the following form:

q ¼ @W@x ; s ¼ @W

@y ;

@@x ð�rqÞ þ @

@y ð�rsÞ ¼ Rðt; x; yÞ;

(ð4:23Þ

where Rðt; x; yÞ ¼ cp½qðt; x; yÞ �NDðx; yÞ� is a known function thatcan be computed at each time step once U is solved from (4.22),and the coefficient �r depends on x; y. The Poisson system is only

on the ðx; yÞ domain. Hence, we use the grid Iij ¼ xi�12; xiþ1

2

h i�

yj�12; yjþ1

2

h i, with i ¼ 1; . . . ;Nx; j ¼ 1; . . . ;Ny þMy, where j ¼ Nyþ

1; . . . ;Ny þMy denotes the oxide-silicon region, and the grid inj ¼ 1; . . . ;Ny is consistent with the five-dimensional rectangular gridfor the Boltzmann equation in the silicon region. The approximationspace is defined as

W ‘h ¼ fv : v jIij

2 P‘ðIijÞg: ð4:24Þ

Here P‘ðIijÞ denotes the set of all polynomials of degree at most ‘ onIij. The LDG scheme for (4.23) is: to find qh; sh;Wh 2 V ‘

h, such thatZIi;j

qhvh dxdyþZ

Ii;j

WhðvhÞx dxdy�Z y

jþ12

yj�1

2

bWhv�h xiþ12; y

dy

þZ y

jþ12

yj�1

2

bWhvþh xi�12; y

dy ¼ 0;

ZIi;j

shwh dxdyþZ

Ii;j

WhðwhÞy dxdy�Z x

iþ12

xi�1

2

eWhw�h x; yjþ12

dx

þZ x

iþ12

xi�1

2

eWhwþh x; yj�12

dx ¼ 0;

�Z

Ii;j

�rqhðphÞx dxdyþZ y

jþ12

yj�1

2

d�rqhp�h xiþ12; y

dy

�Z y

jþ12

yj�1

2

d�rqhpþh xi�12; y

dy�

ZIi;j

�rshðphÞy dxdy

þZ x

iþ12

xi�1

2

f�rshp�h x; yjþ12

dx�

Z xiþ1

2

xi�1

2

f�rshpþh x; yj�12

dx

¼Z

Ii;j

Rðt; x; yÞph dxdy

hold true for any vh;wh;ph 2W ‘h. In the above formulation, we

choose the flux as follows, in the x-direction, we usebWh ¼ W�h ;d�rqh ¼ �rqþh � ½Wh�. In the y-direction, we useeWh ¼ W�h ; f�rsh ¼ �rsþh � ½Wh�. On some part of the domain boundary,the above flux needs to be changed to accommodate various bound-ary conditions [12]. Near the drain, we are given Dirichlet boundarycondition, so we need to flip the flux in x-direction: letbWh xiþ1

2; y

¼ Wþh xiþ1

2; y

and d�rqh xiþ1

2; y

¼ �rq�h xiþ1

2; y

�

½Wh� xiþ12; y

, if the point xiþ1

2; y

is at the drain. For the gate, we

need to flip the flux in y-direction: let eWh x; yjþ12

¼ Wþh x; yjþ1

2

and f�rsh x; yjþ1

2

¼ �rs�h x; yjþ1

2

� ½Wh� x; yjþ1

2

, if the point x; yjþ1

2

is at the gate. For the bottom, we need to use the Neumann condi-

tion, and flip the flux in y-direction, i.e., eWh ¼ Wþh ; f�rsh ¼ �rs�h . Thisscheme described above will enforce the continuity of W and �r

@W@n

across the interface of silicon and oxide-silicon interface. The solu-tion of (4.25) gives us approximations to both the potential Wh andthe electric field ðExÞh ¼ �cvqh, ðEyÞh ¼ �cvsh.

To summarize, start with an initial condition for Uh (AppendixA.5), the DG-LDG algorithm for the 2D double gate MOSFET ad-vances from tn to tnþ1 in the following steps similar to the 1Ddiodes simulation:

Step 1: Compute qhðt; x; yÞ ¼Rþ1

0 dwR 1�1 dl

R p0 duUhðt; x; y;w;

l;uÞ as described in Appendix A.6.Step 2: Use qhðt; x; yÞ to solve from (4.25) the electric field ðExÞh

and ðEyÞh, and compute gi; i ¼ 1; . . . ;5.Step 3: Solve (4.22) and get a method of line ODE for Uh, see

Appendices A.2, A.3 and A.4.

ðx; yÞ ¼ ð0:9375 lm;0:10 lmÞ.


Step 4: Evolve this ODE by proper time stepping from tn to tnþ1, ifpartial time step is necessary, then repeat Steps 1–3 asneeded.

Finally, the hydrodynamic moments can be computed at any timestep by the method in Appendix A.6.

All numerical results are obtained with a piecewise linearapproximation space and first order Euler time stepping. The dissi-pative nature of the collision term makes the Euler forward timestepping stable. We use a 24 � 14 grid in space, 120 points in w,8 points in l and 6 points in u. In Figs. 4.15 and 4.16, we showthe results of the macroscopic quantities. We also show the pdfat six different locations in the device in Fig. 4.17. These pdf’s havebeen computed by averaging the values of Uh over u. In Fig. 4.18,we present the cartesian plot for pdf at ðx; yÞ ¼ ð0:125 lm;

0:012 lmÞ, where a very non-equilibrium pdf is observed.

5. Conclusions and final remarks

We have developed a DG scheme for BTEs of type (1.1),which takes into account optical-phonon interactions that be-come dominant under strong energetic conditions. We used thecoordinate transformation proposed in [27,4] and changed thecollision into an integral-difference operator by using energyband as one of the variables. The Poisson equation is treatedby LDG on a mesh that is consistent with the mesh of the DG-BTE scheme. The results are compared to those obtained froma high order WENO scheme simulation. By a local refinementin mesh, we were able to capture the subtle kinetic effectsincluding very non-equilibrium distributions without a great in-crease of memory allocation and CPU time. The advantage of theDG scheme lies in its potential for implementation on unstruc-tured meshes and for full hp-adaptivity. The simple communica-tion pattern of the DG method also makes it a good candidatefor the domain decomposition method for the coupled kineticand macroscopic models.

Appendix A

In this appendix, we collect some technical details for theimplementation of the 2D DG-BTE solver. The discussion for 1Dsolver is similar and omitted here.

A.1. The basis of the finite dimensional function space

In every cell Xijkmn, we use piecewise linear polynomials andassume

Uhðt; x; y;w;l;uÞ ¼ TijkmnðtÞ þ XijkmnðtÞ2ðx� xiÞ

Dxiþ YijkmnðtÞ

2ðy� yjÞDyj

þWijkmnðtÞ2ðw�wkÞ

DwkþMijkmnðtÞ

2ðl� lmÞDlm

þ PijkmnðtÞ2ðu�unÞ

Dun: ðA:1Þ

It will be useful to note that

Uhðt;x;y;w;l;uÞ

¼XNw

k¼1

TijkmnðtÞþXijkmnðtÞ2ðx�xiÞ

DxiþYijkmnðtÞ

2ðy�yjÞDyj

"

þWijkmnðtÞ2ðw�wkÞ

DwkþMijkmnðtÞ

2ðl�lmÞDlm

þPijkmnðtÞ2ðu�unÞ

Dun

�vkðwÞ

for every ðx; y;w;l;uÞ 2SNw

k¼1Xijkmn. Here, vkðwÞ is the characteristic

function in the interval wk�12;wkþ1

2

h i.

A.2. Treatment of the collision operator

The gain term of the collisional operator is

GðUhÞðt; x; y;wÞ ¼ sðwÞ c0

Z p

0du0

Z 1

�1dl0Uhðt; x; y;w;l0;u0Þ

�þZ p

0du0

Z 1

�1dl0½cþUhðt; x; y;wþ c;l0;u0Þ

þc�Uhðt; x; y;w� c;l0;u0Þ��: ðA:2Þ

Now, we define

ðvhÞmnðx; y;wÞ :¼Z u

nþ12

un�1

2

duZ l

mþ12

lm�1

2

dlvhðx; y;w;l;uÞ;

and, for r ¼ �c, 0, c, we haveZXijkmn

vhðx; y;w;l;uÞ sðwÞZ p

0du0

Z 1

�1dl0Uðt; x; y;wþ r;l0;u0Þ

� �� dxdydwdldu

¼Z p

0du0

Z 1

�1dl0

Z xiþ1

2

xi�1

2

dxZ y

jþ12

yj�1

2

dyZ w

kþ12

wk�1

2

dw sðwÞ

�Uðt; x; y;wþ r;l0;u0ÞðvhÞmnðx; y;wÞ

¼XNl

m0¼1

XNu

n0¼1

ZXijkm0n0

sðwÞUhðt; x; y;wþ r;l0;u0Þ

� ðvhÞmnðx; y;wÞdxdydwdl0 du0:

Now we discuss the following integral for different test functionvh,

I¼Z

Xijkmn

vhðx;y;w;l;uÞ sðwÞZ p

0du0

Z 1

�1dl0Uhðt;x;y;wþr;l0;u0Þ

� ��dxdydwdldu:

For vhðx; y;w;l;uÞ ¼ 1,

I¼XNl

m0¼1

XNu

n0¼1

XNw

k0¼1

Dlm0Dun0 Tijk0m0n0 ðtÞZ w

kþ12

wk�1

2

sðwÞvk0 ðwþrÞdw

24þWijk0m0n0 ðtÞ

Z wkþ1

2

wk�1

2

sðwÞ2ðwþr�wk0 ÞDwk0

vk0 ðwþrÞdw

35Dxi Dyj Dlm Dun:

For vhðx; y;w;l;uÞ ¼ 2ðx�xiÞDxi

,

I ¼ 13

Dxi Dyj Dlm Dun

XNl

m0¼1

XNu

n0¼1

XNw

k0¼1

Dlm0 Dun0Xijk0m0n0 ðtÞ

�Z w

kþ12

wk�1

2

sðwÞvk0 ðwþ rÞdw:

For vhðx; y;w;l;uÞ ¼2ðy�yjÞ

Dyj,

I ¼ 13

Dxi Dyj Dlm Dun

XNl

m0¼1

XNu

n0¼1

XNw

k0¼1

Dlm0 Dun0Yijk0m0n0 ðtÞ

�Z w

kþ12

wk�1

2

sðwÞvk0 ðwþ rÞdw:

For vhðx; y;w;l;uÞ ¼ 2ðw�wkÞDwk

,

I¼XNl

m0¼1

XNu

n0¼1

XNw

k0¼1

Dlm0 Dun0 Tijk0m0n0 ðtÞZ w

kþ12

wk�1

2

sðwÞ2ðw�wkÞDwk

vk0 ðwþrÞdw

24þWijk0m0n0 ðtÞ

Z wkþ1

2

wk�1

2

sðwÞ4ðwþr�wkÞðw�wkÞDwk0 Dwk

vk0 ðwþrÞdw

35Dxi Dyj Dlm Dun:


For vhðx; y;w;l;uÞ ¼ 2ðl�lmÞDlm

; I ¼ 0.

For vhðx; y;w;l;uÞ ¼ 2ðu�unÞDun

; I ¼ 0.The lost term in the collision operator is

2p½c0sðwÞ þ cþsðw� cÞ þ c�sðwþ cÞ�Uðt; x; y;w;l;uÞ: ðA:3Þ

LetmðwÞ ¼ 2p½c0sðwÞ þ cþsðw� cÞ þ c�sðwþ cÞ�;then we need to evaluate numerically,

I0 ¼Z

Xijkmn

mðwÞUhðt; x; y;w;l;uÞvhðx; y;w;l;uÞdxdydwdldu:

For vhðx; y;w;l;uÞ ¼ 1,

I0 ¼Dxi Dyj Dlm Dun

� TijkmnðtÞZ w

kþ12

wk�1

2

mðwÞdwþWijkmnðtÞZ w

kþ12

wk�1

2

mðwÞ2ðw�wkÞDwk

dw

24 35: For vhðx; y;w;l;uÞ ¼ 2ðx�xiÞ

Dxi,

I0 ¼ 13

Dxi Dyj Dlm DunXijkmnðtÞZ w

kþ12

wk�1

2

mðwÞdw:

For vhðx; y;w;l;uÞ ¼2ðy�yjÞ

Dyj,

I0 ¼ 13

Dxi Dyj Dlm DunYijkmnðtÞZ w

kþ12

wk�1

2

mðwÞdw:

For vhðx; y;w;l;uÞ ¼ 2ðw�wkÞDwk

,

I0 ¼ Dxi Dyj Dlm Dun TijkmnðtÞZ w

kþ12

wk�1

2


dw

24þWijkmnðtÞ

Z wkþ1

2

wk�1

2

mðwÞ4ðw�wkÞ2

ðDwkÞ2dw

35: For vhðx; y;w;l;uÞ ¼ 2ðl�lmÞ

Dlm,

I0 ¼ 13

Dxi Dyj Dlm DunMijkmnðtÞZ w

kþ12

wk�1

2

mðwÞdw:

For vhðx; y;w;l;uÞ ¼ 2ðu�unÞDun

,

I0 ¼ 13

Dxi Dyj Dlm DunPijkmnðtÞZ w

kþ12

wk�1

2

mðwÞdw:

A.3. Integrals related to the collisional operator

We need to evaluate (some numerically) the following integralsZ wkþ1

2

wk�1

2

sðwÞvk0 ðwþ rÞdw;Z wkþ1

2

wk�1

2

sðwÞ2ðwþ r�wk0 ÞDwk0

vk0 ðwþ rÞdw;Z wkþ1

2

wk�1

2

sðwÞ2ðw�wkÞDwk


2

wk�1

2

sðwÞ4ðwþ r�wk0 Þðw�wkÞDwk0 Dwk


2

wk�1

2

mðwÞdw;Z wkþ1

2

wk�1

2


dw;

Z wkþ1

2

wk�1

2

mðwÞ 2ðw�wkÞDwk

� �2

dw:

If we evaluate these integrals by means of numerical quadratureformulas, then it is appropriate to eliminate the singularity of thefunction sðwÞ at w ¼ 0 by change of variables.Z b

asðwÞdw¼

Z b

a

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiwð1þaK wÞ

pð1þ2aK wÞdw

¼Z ffiffi

bp

ffiffiap

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þaK r2

p1þ2aK r2� �

2r2 dr; ðw¼ r2Þ;Z b

asðw� cÞdw¼

Z b�c

a�csðwÞdw

¼Z ffiffiffiffiffiffi

b�cp

ffiffiffiffiffiffia�cp


p1þ2aK r2� �

2r2 dr; ðw¼ r2 þ cÞ;Z b

asðwþ cÞdw¼

Z bþc

aþcsð �wÞd �w

¼Z ffiffiffiffiffiffi

bþcp

ffiffiffiffiffiffiaþcp


p1þ2aK r2� �

2r2 dr; ðw¼ r2 � cÞ:

A.4. Integrals related to the free streaming operator

We recall that

g1ð�Þ ¼ cxl


p1þ 2aK w

;

g2ð�Þ ¼ cx


p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiwð1þ aK wÞ

pcos u

1þ 2aK w;

g3ð�Þ ¼ �2ck


p1þ 2aK w

lExðt; x; yÞ þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� l2

qcos uEyðt; x; yÞ

� �;

g4ð�Þ ¼ �ck


pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiwð1þ aK wÞ

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� l2

qExðt; x; yÞ � l cos uEyðt; x; yÞ

� �;

g5ð�Þ ¼ cksin uffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

wð1þ aK wÞp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� l2p Eyðt; x; yÞ:

Now, we define:

s1ðwÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiwð1þ aK wÞ

p1þ 2aK w

; s2ðwÞ ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

wð1þ aK wÞp :

We need to evaluate the integrals.Z b

as1ðwÞdw ¼

Z b

a


p1þ 2aK w

dw ¼Z ffiffi

bp

ffiffiap

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ aK r2

p1þ 2aK r2 2r2 dr;

Z b

as2ðwÞdw ¼

Z b

a

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiwð1þ aK wÞ

p dw ¼Z ffiffi

bp

ffiffiap

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ aK r2

p dr;

Z b

as1ðwÞ

2ðw�wkÞDwk

dw ¼Z b

a


p1þ 2aK w

2ðw�wkÞDwk

dw

¼Z ffiffi

bp

ffiffiap


p1þ 2aK r2

2ðr2 �wkÞDwk

2r2 dr;

Z b

as2ðwÞ

2ðw�wkÞDwk

dw ¼Z b

a


p 2ðw�wkÞDwk

dw

¼Z ffiffi

bp

ffiffiap


p 2ðr2 �wkÞDwk

dr;

Z b

as1ðwÞ

2ðw�wkÞDwk

� �2

dw ¼Z b

a


p1þ 2aK w

2ðw�wkÞDwk

� �2

dw

¼Z ffiffi

bp

ffiffiap


p1þ 2aK r2

2ðr2 �wkÞDwk

� �2

2r2 dr;


Z b

as2ðwÞ

2ðw�wkÞDwk

� �2

dw ¼Z b

a


p 2ðw�wkÞDwk

� �2

dw

¼Z ffiffi

bp

ffiffiap


p 2ðr2 �wkÞDwk

� �2

dr:

A.5. Initial condition

Since measureðXijkmnÞ ¼ Dxi Dyj Dwk Dlm Dun, we haveZXijkmn

Uhð0; x; y;w;l;uÞdxdydwdldu ¼ measureðXijkmnÞTijkmnð0Þ;

ZXijkmn

Uhð0; x; y;w;l;uÞ2ðx� xiÞ

Dxidxdydwdldu

¼ 13

measureðXijkmnÞXijkmnð0Þ;

ZXijkmn

Uhð0; x; y;w;l;uÞ2ðy� yjÞ

Dyjdxdydwdldu

¼ 13

measureðXijkmnÞYijkmnð0Þ;

ZXijkmn

Uhð0; x; y;w;l;uÞ2ðw�wkÞ

Dwkdxdydwdldu

¼ 13

measureðXijkmnÞWijkmnð0Þ;

ZXijkmn

Uhð0; x; y;w;l;uÞ2ðl� lmÞ

Dlmdxdydwdldu

¼ 13

measureðXijkmnÞMijkmnð0Þ;

ZXijkmn

Uhð0; x; y;w;l;uÞ2ðu�unÞ

Dundxdydwdldu

¼ 13

measureðXijkmnÞPijkmnð0Þ:

If, for each ðx; yÞ and ðl;uÞ,

Uhð0; x; y;w;l;uÞ ¼Fðx; yÞsðwÞe�w for w < wNwþ1

2;

0 otherwise

(and

Fðx; yÞ ¼ FijðconstantÞ; 8ðx; yÞ 2 xi�12; xiþ1

2

h i� yj�1

2; yjþ1

2

h i;

then it is reasonable to assume

Tijkmnð0Þ ¼Fij

Dwk

Z wkþ1

2

wk�1

2

sðwÞe�w dw; ðA:4Þ

Xijkmnð0Þ ¼ Yijkmnð0Þ ¼ 0; ðA:5Þ

Wijkmnð0Þ ¼ 3Fij

Dwk

Z wkþ1

2

wk�1

2

sðwÞe�w 2ðw�wkÞDwk

dw; ðA:6Þ

Mijkmnð0Þ ¼ Pijkmnð0Þ ¼ 0: ðA:7Þ

Recalling the definition (2.10) of the dimensionless charge density,we have

qð0; x; yÞ ¼ 2pFðx; yÞXNw

k¼1

Z wkþ1

2

wk�1

2

sðwÞe�w dw;

which gives a relationship between Fðx; yÞ and the initial chargedensity q.

A.6. Hydrodynamical variables

If ðx; yÞ 2 xi�12; xiþ1

2

h i� yj�1

2; yjþ1

2

h i, then

qhðt; x; yÞ ¼XNw

k¼1

XNl

m¼1

XNu

n¼1

TijkmnðtÞ þ XijkmnðtÞ2ðx� xiÞ

Dxi

�

þYijkmnðtÞ2ðy� yjÞ

Dyj

#Dwk Dlm Dun: ðA:8Þ

We define

bT ijðtÞ ¼XNw

k¼1

XNl

m¼1

XNu

n¼1

TijkmnðtÞDwk Dlm Dun;

bXijðtÞ ¼XNw

k¼1

XNl

m¼1

XNu

n¼1

XijkmnðtÞDwk Dlm Dun;

bY ijðtÞ ¼XNw

k¼1

XNl

m¼1

XNu

n¼1

YijkmnðtÞDwk Dlm Dun:

Therefore, for every ðx; yÞ 2 xi�12; xiþ1

2

h i� yj�1

2; yjþ1

2

h i,

qhðt; x; yÞ ¼ bT ijðtÞ þ bX ijðtÞ2ðx� xiÞ

Dxiþ bY ijðtÞ

2ðy� yjÞDyj

: ðA:9Þ

For every ðx; yÞ 2 xi�12; xiþ1

2

h i� yj�1

2; yjþ1

2

h i, the approximate momen-

tum in x-direction is

XNw

k¼1

XNl

m¼1

XNu

n¼1

Dun½g1;kmTijkmnðtÞ þ g1w;kmWijkmnðtÞ

þ g1l;kmMijkmnðtÞ�

þXNw

k¼1

XNl

m¼1

XNu

n¼1

Dung1;kmXijkmnðtÞ" #

2ðx� xiÞDxi

þXNw

k¼1

XNl

m¼1

XNu

n¼1

Dung1;kmYijkmnðtÞ" #

2ðy� yjÞDyj

; ðA:10Þ

and the approximate momentum in y-direction is

XNw

k¼1

XNl

m¼1

XNu

n¼1

½g2;kmnTijkmnðtÞ þ g2w;kmnWijkmnðtÞ

þ g2l;kmnMijkmnðtÞ þ g2u;kmnPijkmnðtÞ�

þXNw

k¼1

XNl

m¼1

XNu

n¼1

g2;kmnXijkmnðtÞ" #

2ðx� xiÞDxi

þXNw

k¼1

XNl

m¼1

XNu

n¼1

g2;kmnYijkmnðtÞ" #

2ðy� yjÞDyj

; ðA:11Þ

where

g1;km ¼Z w

kþ12

wk�1

2

Z lmþ1

2

lm�1

2

g1ðw;lÞdwdl;

g1w;km ¼Z w

kþ12

wk�1

2

Z lmþ1

2

lm�1

2

g1ðw;lÞ2ðw�wkÞ

Dwkdwdl;

g1l;km ¼Z w

kþ12

wk�1

2

Z lmþ1

2

lm�1

2

g1ðw;lÞ2ðl� lmÞ

Dlmdwdl;

g2;kmn ¼Z w

kþ12

wk�1

2

Z lmþ1

2

lm�1

2

Z unþ1

2

un�1

2

g2ðw;l;uÞdwdldu;


g2w;kmn ¼Z w

kþ12

wk�1

2

Z lmþ1

2

lm�1

2

Z unþ1

2

un�1

2

g2ðw;l;uÞ2ðw�wkÞ

Dwkdwdldu;

g2l;kmn ¼Z w

kþ12

wk�1

2

Z lmþ1

2

lm�1

2

Z unþ1

2

un�1

2

g2ðw;l;uÞ2ðl� lmÞ

Dlmdwdldu;

g2u;kmn ¼Z w

kþ12

wk�1

2

Z lmþ1

2

lm�1

2

Z unþ1

2

un�1

2

g2ðw;l;uÞ2ðu�unÞ

Dundwdldu:

Analogously, the energy multiplied by the charge density qhðt; x; yÞis

XNw

k¼1

XNl

m¼1

XNu

n¼1

DunDlm wkDwkTijkmnðtÞ þðDwkÞ2

6WijkmnðtÞ

" #

þXNw

k¼1

XNl

m¼1

XNu

n¼1

DunDlmwkDwkXijkmnðtÞ" #

2ðx� xiÞDxi

þXNw

k¼1

XNl

m¼1

XNu

n¼1

DunDlmwkDwkYijkmnðtÞ" #

2ðy� yjÞDyj

ðA:12Þ

for every ðx; yÞ 2 xi�12; xiþ1

2

h i� yj�1

2; yjþ1

2

h i.

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