ALmost EXact boundary conditions for transientSchrodinger-Poisson system

Lei Biana, Gang Pangb, Shaoqiang Tanga,∗, Anton Arnoldc

aHEDPS, CAPT, College of Engineering, Peking University, Beijing 100871, ChinabInstitute of Applied Physics and Computational Mathematics, Beijing 100088, China

cInstitut fur Analysis und Scientific Computing, TU-Wien, Wiedner Hauptstr. 8, 1040 Wien, Austria

Abstract

For the Schrodinger-Poisson system, we propose an ALmost EXact(ALEX) boundary conditionto treat accurately the numerical boundaries. Being local in both space and time, the ALEXboundary conditions are demonstrated to be effective in suppressing spurious numerical reflec-tions. Together with the Crank-Nicolson scheme, we simulate a resonant tunneling diode. Thealgorithm produces numerical results in excellent agreement with those in Mennemann et al. [1],yet at a much reduced complexity. Primary peaks in wave function profile appear as a conse-quence of quantum resonance, and should be considered in selecting the cut-offwave number fornumerical simulations.

Keywords: Schrodinger-Poisson simulation, ALEX boundary condition, resonant tunnelingdiode, primary peak

1. Introduction

The Schrodinger-Poisson equations govern many importantphysical systems, such as thequantum many body system, time-dependent density functional theory, and quantum chargetransport in semiconductor devices [1][2][3].

One major challenge in numerical simulations of these equations is a broad band of wave-numbers that require well resolved to reproduce the correctphysics. This leads to a huge systemcoupled through the electron density and self-consistent potential. When resolving each discretewave number, we need to treat the numerical boundaries in an effective way to suppress non-physical reflections. Otherwise, wrong physics and even numerical instability will destroy thewhole simulation.

In the literature, there are various treatments developed over the past two decades. An accu-rate approach is of DtN or NtD type[4][5]. As convolutions are involved, the boundary treatmentis rather expensive numerically. Also, it does not apply to nonlinear problems in general. On theother hand, the perfectly match layer (PML) approach applies to quite general systems includ-ing the Schrodinger-Poisson system with a rather empirical choice of parameters[6][7]. Moredelicate approaches were developed as well [8].

∗Corresponding authorEmail addresses:bianlei@pku.edu.cn (Lei Bian),1517191281@qq.com (Gang Pang),maotang@pku.edu.cn

(Shaoqiang Tang),anton.arnold@tuwien.ac.at (Anton Arnold)

Preprint submitted to Journal of Computational Physics December 25, 2015

In the study of accurate numerical boundary treatments for crystalline solids, we have discov-ered that the Bessel functions of the first kind satisfy approximate linear relations to any requiredaccuracy [9]. This allows us to design an ALmost EXact(ALEX)boundary condition that worksextremely well in that application. The time history kernelfunctions of the Schrodinger equa-tion were figured out to be also based on the Bessel functions.Therefore, we constructed anALEX boundary condition for the Schrodinger equation, anddemonstrated its high efficiency inboth linear and nonlinear systems [10]. It is a boundary condition local in both space and time,with precomputed coefficients. In this work, we extend the ALEX boundary conditionsto theSchrodinger-Possion system, and demonstrate its effectiveness by applications to the resonanttunneling diode simulations. From the numerical simulations, we observe quantum resonance inthe wave function profile. Two primary peaks appear inside the potential well. Their role in nu-merical simulation is examined. Accordingly we propose to account for the quantum resonancein selecting the cut-off wave number.

The rest of this paper is organised as follows. In Section 2, we describe the governing equa-tions for both the stationary and the transient problem. Theformer is adopted for preparing theinitial data to transient simulation at zero applied voltage, in order to avoid initial layer. Then inSection 3, we describe the numerical algorithms for both simulations. In particular, we providethe ALEX boundary conditions and the two-way boundary conditions. Numerical tests for aresonant tunneling diode are presented in Section 4. We explore the primary peaks in Section 5.Finally, we draw some concluding remarks in Section 6.

2. Governing equations

In this section, we first describe the transient system whichgoverns the dynamics of chargetransport in a resonant tunneling diode. The initial data for this transient system are prescribedby the solution to the corresponding stationary system. Theboundary conditions are also given.

2.1. Transient Schrodinger-Poisson system

In a resonant tunneling diode of lengthL, the electron wave functionψ(x, k, t) at a wavenumberk is described by the Schrodinger equation

i~ψt = −~

2

2mψxx + V(x, t)ψ, x ∈ R, k ∈ R, t > 0. (1)

Because the wave number does not appear explicitly in the equation, we may solve for eachk separately. Here~ is the reduced Planck constant, andm is the electron mass. The potential en-ergyV(x, t) consists of two parts, a self-consistent potential energyVs(x, t) and a built-in barrierpotential energyVB(x) for a resonant tunneling diode. The latter is compactly supported withinthe device. In particular, the potential outside of the device is assumed to be

V(x, t) =

0, x ≤ 0;

−eU(t), x ≥ L,(2)

whereU(t) is a given applied voltage, ande represents the elementary charge. To avoid confu-sion, throughout this paper we use exp to denote the exponential function.

2

This potential gives rise to plane waves for the Schrodinger equation away from the spaceinterval [0, L]. In particular, fork > 0, a plane wave takes the form of

ψ(x, k, t) =

exp

(

−iE(k)t~+ ikx

)

+ R(k) exp

(

−iE(k)t~− ikx

)

, x ≤ 0;

T(k) exp

(

−iE(k)t~+ ikx

)

, x ≥ L.(3)

HereE(k) =~

2k2

2m. The reflection coefficientR(k) and transmission coefficientT(k) are deter-

mined by the potential energy inside the device.Similarly, for k < 0, a plane wave takes the form of

ψ(x, k, t) =

T(k) exp

(

−iE(k)t~+ ikx

)

, x ≤ 0;

exp

(

−iE(k)t~− ikx

)

+ R(k) exp

(

−iE(k)t~+ ikx

)

, x ≥ L.(4)

HereE(k) =~

2k2

2m− eU(t). We remark that these plane waves are bounded inL∞(R), not in

L2(R).The electron density is calculated from

n(x, t) =∫

R

g(k) |ψ(x, k, t)|2 dk, x ∈ R, (5)

with an injection profile given by the Fermi-Dirac statistics

g(k) =mkBT0

2π2~2ln

[

1+ exp

(

EF − ~2k2/(2m)kBT0

)]

. (6)

HerekB is the Boltzmann constant,T0 is the electron temperature, andEF is the Fermi energy.We remark that the plane waves may be regarded as injected from ±∞, henceg(k) is called theinjection profile.

The self-consistent potential energyVs(x, t) is coupled with the electron density by the Pois-son equation in the device.

−ε∂2Vs

∂x2= e2(n(x, t) − nD(x)), x ∈ [0, L]. (7)

The permittivityε and doping profilenD(x) are given constant and function, respectively. Wenotice thatVs(x, t) = V(x, t) outside of the device, given by (2). The boundary conditions aretherefore furnished with the continuity requirements atx = 0, L.

Instead of simulating the whole spatial domainR, we perform numerical studies only overthe device forx ∈ [0, L], k ∈ [−kM, kM] with a suitable cut-off momentumkM, and t > 0.Accordingly, we need to impose suitable initial and boundary conditions for the Schrodingerequation (1).

3

2.2. Initial data via stationary Schrodinger-Poisson system

It has been known that quantum device simulations require well-prepared initial data to avoidstrong initial layer [11][12]. Following [1], we adopt the stationary solutionϕ(x, k) as the initialdataψ(x, k, 0) for x ∈ [0, L]. Furthermore, in this study we shall takeU(0) = 0 for the transientsimulation.

With an energyE(k) =~

2k2

2m, the wave functionϕ(x, k) and the stationary self-consistent

potentialVs solve the following equations,

E(k)ϕ = − ~2

2mϕxx+ [VB(x) + Vs(x)]ϕ, (8)

−εd2Vs

dx2= e2(n(x) − nD(x)). (9)

Heren(x) =∫

Rg(k) |ϕ(x, k)|2 dk is the electron density. The equations are solved in [0, L]. The

boundary conditions for the Poisson equation are

Vs(0) = Vs(L) = 0. (10)

Following [6][13], the boundary conditions for the Schrodinger equation (8) are as follows.

ϕx(0, k) + ikϕ(0, k) = 2ik, ϕx(L, k) =i~

√

2mE(k)ϕ(L, k), k > 0, (11)

ϕx(0, k) = −i~

√

2mE(k)ϕ(0, k), ϕx(L, k) + ikϕ(L, k) = 2ikeikL, k < 0. (12)

2.3. Boundary conditions

On the boundaries of the device, incoming waves enter fromx = 0 for k > 0, and fromx = Lfor k < 0.

Fork > 0, we notice that exp(−i E(k)~

t+ ikx) for x ≤ 0 represents an incoming wave for the leftboundaryx = 0. Therefore,ψ(x, k, t)−exp(−i E(k)

~t+ikx) = R(k) exp(−i E(k)

~t−ikx) satisfies a trans-

parent boundary condition for outgoing waves atx = 0, andψ(x, k, t) = T(k) exp(−i E(k)~

t + ikx)satisfies a transparent boundary condition for outgoing waves atx = L. We remark that depend-ing on the formulation of transparent boundary condition, one should be careful of consistencyat t = 0 [14].

In the same way, fork < 0,ψ(x, k, t) at x = 0 andψ(x, k, t)−exp(−i E(k)~

t+ ikx) at x = L satisfytransparent boundary conditions for outgoing waves, respectively.

We remark that in the literature, there are only several numerically effective ways in imposingsuch transparent boundary conditions, in spite of considerable efforts put in this important yetdifficult topic [7][15][16][17][18].

3. Numerical methods

In this section, we first briefly describe the Gummel iteration method for solving the station-ary Schrodinger-Poisson system [1]. Then we present the discretization and boundary treatmentfor the transient problem. A flow chart is also given.

4

3.1. Stationary Schrodinger-Poisson system

We discretize the spacex and the momentumk with equidistant grids, respectively. Letx j = j∆x, for j = 0, . . . , J with xJ = L; and kl = l∆k, for l = −M, . . . ,−1, 1, . . .M, withkM =

√2m(EF + 7kBT0)/~.

The equations (8) and (9) are iteratively solved for eachk = kl as follows, withp countingthe iteration step,

Ed(k)ϕ(p)j,l = −

~2

2m(∆x)2(ϕ(p)

j+1,l − 2ϕ(p)j,l + ϕ

(p)j−1,l) + [VB(x j) + Vp

j ]ϕ(p)j,l , (13)

− ε

(∆x)2(V(p+1)

j−1 − 2V(p+1)j + V(p+1)

j+1 ) = e2

n(p)j exp

V(p)j − V(p+1)

j

Vre f

− nD(x j)

. (14)

HereEd(k) = ~2

2m(∆x)2 (1 − cos (k∆x)) is the discrete energy level,Vre f = 0.03eV is a relaxationparameter, and the electron density is calculated from

n(p)j = ∆k

M∑

|l|=1

g(kl)∣

∣

∣

∣ϕ

(p)j,l

∣

∣

∣

∣

2. (15)

Following [1][14], we take boundary conditions for (13) as follows,

−1α+ϕ

(p)−1,l + ϕ

(p)0,l = 1−

1(α+)2

, ϕ(p)J,l −

1α+ϕ

(p)J+1,l = 0, k > 0, (16)

ϕ(p)0,l − α

−ϕ(p)−1,l = 0, ϕ

(p)J,l −

1α−ϕ

(p)J+1,l =

(

1α−

)J

−(

1α−

)J+2

, k < 0, (17)

with

α± = 1− mEd(k)(∆x)2

~2± i

√

2mEd(k)(∆x)2

~2− m2(Ed(k))2(∆x)4

~4, (18)

The boundary conditions for (14) are homogeneousV(p+1)0 = V(p+1)

J = 0.

3.2. Transient Schrodinger-Poisson system

For the transient problem, the discretized Poisson equation is

− ε

(∆x)2(Vs

j−1 − 2Vsj + Vs

j+1) = e2(n j(t) − nD(x j)), j = 1, . . . , J − 1, (19)

with Vs0 = 0 andVs

J = −eU(t). Here the electron density is calculated from

n j(t) = ∆kM∑

|l|=1

g(kl)∣

∣

∣ψ j,l(t)∣

∣

∣

2. (20)

The Schrodinger equation is semi-discretized as

i~ddtψ j,l = −

~2

2m(∆x)2(ψ j+1,l − 2ψ j,l + ψ j−1,l) + V jψ j,l , (21)

5

with V j = Vsj + VB(x j).

In the previous work [10], we have discovered that the semi-discretized system (21) admitstime history kernel function (lattice Green’s function) inthe form of Bessel functions. That is, asV j = 0 for j ≤ 0 in (21), we have

ψ−p,l(t) =ptip exp(−2it)Jp

(

~tm(∆x)2

)

∗ ψ0,l(t). (22)

Meanwhile, through asymptotic expansion it was discoveredthat the first kind Bessel func-tions are approximately linearly dependent to any requiredaccuracy, provided the orders of themare big enough [9]. More precisely, forMp ≥ 5 and any givenε > 0, if Q is big enough, thereexist a set of real numbersap (1 ≤ p ≤ Mp), such that

∣

∣

∣

∣

∣

∣

∣

∣

Mp∑

p=1

apJQ−p(t) − JQ(t)

∣

∣

∣

∣

∣

∣

∣

∣

<ε

3√

Q. (23)

This naturally yields a transparent boundary condition

ψ−Q,l(t) ≈Mp∑

p=1

apψ−Q+p,l(t). (24)

Using the recursive relation for the derivative of the Bessel functions, we may further enhancethe efficiency for numerical boundary treatments. After some calculations, the approximate lineardependency leads to an ALmost EXact (ALEX) boundary condition as follows.

i~ddtψ−Q,l =

Mp∑

p=0

bpψ−Q+p,l . (25)

In this study, we take the same formulation as [10] withMp = 15 andb0 to b15 as follows.

−~2/(

2m(∆x)2)

×−2+ 12.052110193235i, 71.177637000065,

262.693855125265i, 708.598614518240,1461.844046849774i, 2387.382683759241,3149.979100974855i, 3395.091578678526,3000.639882589720i, 2168.648792241722,1268.957016689205i, 589.685213345064,210.491232766998i, 54.442223074977,

9.111658309311i, 0.742160906401.

Now we present the boundary conditions based on ALEX [19].First, we consider the left boundaryx = 0. Fork < 0, we directly take the boundary condition

i~ddtψ0,l =

Mp∑

p=0

bpψp,l . (26)

6

symbol physical meaning valuee elementary charge 1.602s10−19C~ reduced Planck constant 1.0546× 10−34J · sm effective mass 6.1033× 10−32kgε permittivity 1.012319× 10−10F/mkB Boltzmann constant 1.3806505× 10−23J/KEF Fermi energy 6.7097× 10−21JT0 semiconductor temperature 300K

Table 1: Physical constants and parameters in this simulation.

For k > 0, we adopt the ALEX boundary condition for the reflected componentψ j,l −exp(i(k j∆x− Ed(kl )

~)). This gives

i~ddtψ0,l =

Mp∑

p=0

bpψp,l + exp

(

−iEd(kl)~

t

)

Ed(kl) + exp

(

−iEd(kl)~

t

) Mp∑

p=0

bp exp(ikp∆). (27)

Next, we consider the right boundaryx = L. Using a transformψ j,l = ψ j,l exp (−ie∫ t

0U(s)ds),

one may show thatψ j,l is governed by the free discrete Schrodinger equation, namely, withoutthe electric potential term. Fork < 0, we adopt the ALEX boundary condition for the reflectedpartψ j,l − exp(i(k j∆x− Ed(k)

~)). Going back toψ j,l = ψ j,l exp (ie

∫ t

0U(s)ds), we get the boundary

condition

i~ddtψJ,l =

Mp∑

p=0

bpψJ−p,l+VJψJ,l+exp

(

ie~

∫ t

0U(s)ds− i

Ed(kl)~

t

)

Ed(kl) exp(ikL) −Mp∑

p=0

bpeik(L−p∆x)

.

(28)Fork > 0, the ALEX boundary condition forψJ,l leads to

i~ddtψJ,l =

Mp∑

p=0

bpψJ−p,l + VJψJ,l . (29)

3.3. Flow-chart for transient computation

We summarize one-step advancing for the transient computation as follows.Let the wave function at a steptr = r∆t beψr

j,l for j = 0, . . . , J and|l| = 1, . . . ,M.

Step 1 Evaluate the electron densitynrj according to (20) forj = 1, . . . , J − 1;

Step 2 Solve the Poisson equation (19) by standard solver to obtain the self-consistent electronpotential energy and hence the total potential energyVr

j for j = 1, . . . , J − 1;Step 3 Under potential energyVr

j , use the Crank-Nicolson scheme to solve the ordinary differ-ential equation (21) atj = 1, . . . , J − 1 with the ALEX boundary conditions (26)-(29).

3.4. Implementation details

The physical constants used in this study are listed in Table1.

7

The electron doping profile is

nD(x) =

1× 1024m−3, x ∈ [0, 50nm]∪ [85nm, 135nm],

5× 1021m−3, elsewhere.(30)

The barrier potential is

Vbarr(x) =

0.3eV, x ∈ [60nm, 65nm]∪ [70nm, 75nm],

0, elsewhere.(31)

They are schematically shown in Fig.1.

0 50 60657075 85 135

1022

1023

1024

n D in

m−

3

x in nm

0 50 60657075 85 135

0.1

0.2

0.3

VB in

eV

nD

VB

Figure 1: The doping profile and the given barrier potential energy.

We takeJ = 1350 andM = 1500. We note thatkM = 6.26× 108m−1, according to the choiceof physical constants and parameters.

4. Numerical test

We first present the stationary simulation results at zero applied voltage. Then we show thetransient simulation results with initial data given by thestationary solution.

4.1. Numerical solution of stationary Schrodinger-Poisson system

For the stationary solution to the Schrodinger-Poisson system at zero applied voltage, Fig.2shows|ϕ(x, k)| in the (x, k) plane. A pair of peaks appear. They are located aroundk = ±1174∆k,and within the barrier region [60nm, 75nm]. See Fig.3. The peak amplitude is 25.5385, about 10times higher than elsewhere. They contribute to electrons trapped in the barrier. See Fig.4. Wecall these peaks the primary peaks.

The conduction current density

Jcond(x) =e~m

∫ kM

−kM

g(k)Im(ϕ∗(x, k)∂ϕ(x, k)∂x

)dk, (32)

is depicted in Fig.5. Noticing that the current density is onthe order of 109A/m2 when there is anapplied voltage, this current density is practically regarded as 0, as is expected for zero appliedvoltage.

The self-consistent potential energyVs is shown in Fig.6. There is an effect of the doping pro-file, which is low for [50nm, 85nm], as well as a minor effect from the electron density variationdue to the double barrier.

8

−5

0

5

x 108

0

50

100

0

10

20

30

kx

Figure 2: Wave function|ϕ(x, k)| under (x, k) coordinates.

4.2. Numerical solution of transient Schrodinger-Poisson system

We solve the transient system with the ALEX boundary condition and the Crank-Nicolsonscheme. The time step is chosen as∆t = 0.5 f s. The applied voltage is

U(t) =

U0

2 (1− cos2πtT ), 0 ≤ t ≤ 1ps;

U0, t > 1ps,(33)

whereU0 = 0.25V andT = 2ps.First, we demonstrate that the proposed boundary treatments effectively suppress spurious

reflections. In Fig.7, we show the wave function modulus|ψ(x, k, 0.2ps)|. As mentioned before,at the left boundary of the device (x = 0) there is no incoming wave fork < 0. The numericalsolution does not show any spurious reflection. Similarly, no reflection is observed at the rightboundary (x = 135nm) fork > 0. Moreover, the primary peaks remain unmoved, yet theiramplitudes decrease slightly to 25.3392. The change in wave function modulus is shown inFig.8. The biggest change occurs along±1174∆k.

To check the effectiveness in more detail, we focus on a particular wave number k = ∆k,and plot the evolution of the wave function modulus|ψ(x, k, t)| for t ∈ [0, 0.2ps] in the upper-left subplot of Fig.9. In the upper-right plot, we zoom in theright-boundary vicinityx ∈[130nm, 135nm]. The directions of contour lines clearly demonstrate that the wave propagatesfrom left to right as time elapses, without observable reflections. For the left boundary vicini-ty x ∈ [0nm, 5nm] in the lower-left subplot, the wave function consists both an incoming part,

9

0 50 85 135

10

20

|φ|

x in nm

0.1

0.2

0.3

VB in

eV

Figure 3: Wave function|ϕ(x, k)|: a side view (maxk |ϕ(x, k)|).

0 50 85 13510

21

1022

1023

1024

n D in

m−

3

x in nm0 50 85 135

0.1

0.2

0.3

VB in

eV

Figure 4: The electron density.

and the rest part that comes from wave propagation and complex dynamics. In the lower-rightsubplot, we show that the latter is indeed an outgoing wave that propagates from right to left.

With the numerical boundary treatment validated, we now briefly describe the behavior ofthe device, which agrees with those in the literature [10][11][12]. As seen from Fig.7, the peakspersist over the whole simulation, yet the amplitude varies. A close check shows that the profilesat other wave numbers and positions also change along with time. The evolution in the wavefunction leads to change in the electron density, current and self-consistent potential.

From Fig.10, we observe that when the applied voltage increases from 0 toU02 , the electron

density increases in the left barrier and the intermediate region between the two barriers, whichclearly reflects the resonant tunnelling mechanism. Electron density inside the right barrier doesnot change much, and to the right of it decreases considerably. Accordingly, the conduction cur-rent increases in this period. Afterwards, when the appliedvoltage further increases up toU0,the electron density moves up in the left barrier and the intermediate region. The conductioncurrent density also moves up accordingly. However, at the left doping gradient boundary, theredevelops a minor kink at aboutx = 52nm. This may mean that there are not enough of electronsfrom the right side, yet the increment in applied voltage attracts more electrons to the left. Theconduction current density moves downward away from the barriers, leading to a smaller totalcurrent. Afterwards, when the applied voltage remains atU0, there are more electrons accumu-

10

0 50 85 135−2

0

2x 10

−5

J cond

in A

/m2

x in nm

Figure 5: Current density in stationary Schrodinger-Poisson system.

0 50 85 135

0.02

0.04

0.06

Vs in

eV

x in nm0 50 85 135

0.1

0.2

0.3

VB in

eV

Figure 6: Self-consistent potential energy in stationary Schrodinger-Poisson system.

lated in the intermediate region, and the conduction current density profile becomes flatter in thedevice.

In Fig.11, we plot the total current density averaged over [0, L] during the whole time period,which is defined as

Jtotal = Jcond−ε

e∂

∂t∇Vs. (34)

This agrees with [1]. If we take the discrete transparent boundary conditions with the exactconvolutions in [1] as the reference solution, it may be observed that the numerical error in oursolution is negligible. It is shown in Fig.12.Λ is the parameter which appeared in the fastevaluation for the discrete convolution terms. Quantitatively, we may define the relativeL2-erroras

εr (t) =

√

∫ L

0(Jcond(x, t) − Jre f

cond(x, t))2dx

√

∫ L

0(Jre f

cond(x, t))2dx

. (35)

Thenεr (1.5) = 6.45× 10−10. This is much smaller than that forΛ = 100 in the fast evaluation[1]. This demonstrates the effectiveness of our boundary treatment.

11

−5

0

5

x 108

0

50

100

0

10

20

30

kx

Figure 7: Wave function|ψ(x, k, 0.2ps)|.

5. Primary peaks

The primary peaks are prominent in the wave function profile of resonant tunneling diode.They are induced by the double barrier structure, and influence the physics in the device.

To better understand these peaks, we first double the computational domain in wave numberspace, namely,k ∈ [−3000∆k, 3000∆k]. As shown in Fig.13, a pair of secondary peaks appearat k = ±1866∆k, with a much smaller amplitude. Because of the fast decay in the Fermi-Diracdistribution (6), the extended computational domain, and in particular the secondary peaks donot lead to significant change in the electron densityn or the potential energyV. Moreover, it isobserved that the secondary peaks are at an energy level equal to the barrier potential, namely,~

2k2

2m= e(VB + Vs) = 5.5269× 10−20J.

To further explore the choice of the wave number domain, we calculate the electron den-sity n0, j with different cut-off wave number. That is, we depict the partial sums in (15) alongwith different number of terms. In Fig.15, we observe that the series converge if we sum over[−800∆k, 800∆k] for most positionx. Yet atx = 62.5nm, there is a meaningful contribution whenwe expand the domain from [−1000∆k, 1000∆k] to [−1200∆k, 1200∆k]. More careful inspectionshows that actually corresponds to the primary peaks around±1174∆k. This explains that thechoice ofkM =

√2m(EF + 7kBT0)/~ is sufficient in (6) for 0 applied voltage. In Fig.4, we plot

the electron density for summation over the whole computational domain of wave number. Thehump inside the barrier is caused by the two peaks, as explained above.

The primary peaks may be explained as a resonance phenomenonin quantum device. In a

12

Figure 8: Change in wave function modulus|ψ(x, k, 0.2ps)| − |ψ(x, k, 0)|.

much simplified manner, we may consider the following stationary Schrodinger equation

E(k)φ = − ~2

2mφxx+ V(x)φ, x ∈ R, (36)

under potential

V(x) =

0, x < 60nm;0.345eV, 60nm< x < 65nm;0.045eV, 65nm< x < 70nm;0.345eV, 70nm< x < 75nm;0, x > 75nm.

(37)

We rescaleV(x) by 0.345eV, shiftx by 67.5nm and rescale by~/√

2m∗ 0.345eV, and rescalek by

√2m∗ 0.345eV/~. Letw = 2.5nm/[~/

√2m∗ 0.345eV]= 1.9471096,q = 0.045eV/0.345eV=

0.1304348. The rescaled equation and potential are as follows.

E(k)φ = −φxx+ V(x)φ, x ∈ R, (38)

V(x) =

0, x < −3w;1, −3w < x < −w;q, −w < x < w;1, w < x < 3w;0, x > 3w.

(39)

13

5e−095e−09

5e−085e−08

5e−

085e−07

5e−07

5e−0

7

5e−075e−075e−07 5e−07

5e−075e−075e−07 5e−07

5e−075e−075e−07

5e−07

5e−07

5e−

06

5e−0

6

5e−06

5e−06

5e−

055e

−05

5e−05

5e−05

0.00

05

0.00

05

0.0005

0.00

18

0.00

18

0.0018

0.00

48

0.00

48

0.0048

0.00

96

0.00

96

0.0096

0.01

44

0.01

44

0.0144

0.01

93

0.01

93

0.0193

0.02

41

0.02

41

0.0241

0.02

89

0.02

89

0.0289

0.03

3

0.03

3

0.0330.033

0.0380.038

x in nm

t in

ps

0 50 85 135

0.05

0.1

0.15

0.2

t in

ps

0.025

0.05

0.075

0.1

0.125

0.15

0.175

0.2

0.032533

0.03

2533

0.0338960.033896

0.033896

0.035260.03526 0.03526 0.035260.036623 0.036623 0.036623 0.0366230.037986 0.037986 0.037986 0.037986

0.037986

0.039349 0.039349 0.0393490.039349

0.039349

0.039349

0.040712 0.0407120.040712

0.040712

0.040712

0.040712

0.042075

0.042075

0.042075

x in nm

t in

ps

0 2.5 5

0.05

0.1

0.15

0.2

x in nm

t in

ps

0 2.5 5

0.05

0.1

0.15

0.2

Figure 9:|ψ(x,∆k, t)| for t ∈ [0, 0.2ps]. Upper-left: wave function in the whole device; upper-right: blow-up view near theright boundary forx ∈ [130nm, 135nm], timet ∈ [0, 0.2ps] in four equal length intervals; lower-left: blow-up view near

the left boundary forx ∈ [0, 5nm]; lower-right:∣

∣

∣

∣

ψk − exp(

−iEd(k)/~t + ikx)

∣

∣

∣

∣

near the left boundary forx ∈ [0, 5nm].

Further letα =√

1− k2 andβ =√

k2 − q. The analytical solution with an incident waveexp(ikx) takes the following form

φ(x, k) =

exp(ikx) + R(k) exp(−ikx), x < −3w;A− exp(αx) + B− exp(−αx), −3w < x < −w;A0 exp(iβx) + B0 exp(−iβx), −w < x < w;A+ exp(αx) + B+ exp(−αx), w < x < 3w;T(k) exp(ikx), x > 3w.

(40)

Here the coefficientsA−,A0,A+, B−, B0, B+ are to be determined,R(k) andT(k) are reflection andtransmission coefficients, respectively. Making use ofφ(x) ∈ C1, we may form a linear algebraicsystem for the above coefficients. After some calculations, we end up with

[

T(k) exp(3ikw)0

]

= − 116α2βk

[

α + ik −α + ik−α + ik α + ik

]

Λ(2α)

[

α + iβ α − iβα − iβ α + iβ

]

Λ(2iβ)[

α + iβ −α + iβ−α + iβ α + iβ

]

Λ(2α)

[

α + ik α − ikα − ik α + ik

]

Λ(−3ik)

[

1R(k)

]

,

(41)with

Λ(s) =

[

exp(sw)exp(−sw)

]

.

14

Solving this linear system, we find that resonance occurs at about k∗ = 0.62458. Thisgives, in physical units, a wave number of 0.62458×

√2m∗ 0.345eV/~ = 4.8645× 108m−1.

In our stationary solution, the primary peaks appear at 1174∆k = 4.8995× 109m−1. Therelative error is less than 1%. Numerically, the transmission and reflection coefficients are|T(k∗)| = 0.999972, |R(k∗)| = 0.007418. Moreover, the maximal amplitude of|φ(x, k∗)| = 24.5175at x = 0. Again it agrees well with the full stationary Schrodinger-Poisson simulation, whichgives 25.5385.

With the above discussions, we propose to prescribe the cut-off wave number in a new way,namely,kM = max(

√2m(EF + 7kBT0)/~,

√2m∗ VB/~). Under the settings described before,

this amounts to 7.26× 108m−1.

6. Conclusion

In this paper, we propose the ALEX boundary treatment to effectively suppress spurious re-flections at the ends of the resonant-tunneling diode device. It reaches high accuracy, comparableto the theoretically exact boundary treatments developed in [15] and applied in [1]. The numer-ical cost and programming complexity are reduced considerably. Local in both space and time,it bears nice features of being simple, clear, and without empirical parameters. When the meshsize or time step sizes changes, we do not need to restart the simulation from the beginning, incontract to the time history dependent boundary treatmentsin [1]. The numerical results with theproposed boundary treatments demonstrate the effectiveness.

In the wave function profile, we observe two primary peaks around±1174∆k, inside thebarrier potential well in space. With the choice of cut-off wave number proposed in previousworks, these peaks are included in the computational domain. The important role of the primarypeaks is revealed by a careful study of terms for summation incalculating the electron density, aswell as the prominent wave function profile changes at the wave numbers±1174∆k along with avoltage applied. Accordingly we propose to take the primarypeaks into account for determiningcut-off wave number in numerical simulations. Moreover, the position and amplitude of thepeaks are accurately modeled by a simplified model at zero applied voltage.

Acknowledgement

This study is partially supported by National Natural Science Foundation of China undercontract numbers 11272009 and 11521202. Anton Arnold was supported by the FWF-DoctoralSchool W1245. The authors would like to thank Prof. Jungel for providing the electronic file ofcurrent density att = 1.5ps, and the anonymous referees for stimulating discussions.

References

[1] J.-F. Mennemann, A. Jungel, H. Kosina, Transient Schr¨odinger–Poisson simulations of a high-frequency resonanttunneling diode oscillator, Journal of Computational Physics 239 (2013) 187–205.

[2] C. Bardos, L. Erdos, F. Golse, N. Mauser, H.-T. Yau, Derivation of the Schrodinger–poisson equation from thequantum N-body problem, Comptes Rendus Mathematique 334 (2002) 515–520.

[3] M. A. Marques, E. Gross, Time-dependent density functional theory, Annual Review of Physical Chemistry 55(2004) 427–455.

[4] B. Mayfield, Non–local boundary conditions for the Schr¨odinger equation, Ph.D. thesis, University of Rhode Island,Providence, RI, 1989.

15

[5] V. Baskakov, A. Popov, Implementation of transparent boundaries for numerical solution of the Schrodinger equa-tion, Wave Motion 14 (1991) 123–128.

[6] N. Ben Abdallah, S. Tang, On hybrid quantum–classical transport models, Mathematical Methods in the AppliedSciences 27 (2004) 643–667.

[7] J.-F. Mennemann, A. Jungel, Perfectly matched layers versus discrete transparent boundary conditions in quantumdevice simulations, Journal of Computational Physics 275 (2014) 1–24.

[8] X. Antoine, A. Arnold, C. Besse, M. Ehrhardt, A. Schadle, A review of transparent and artificial boundary con-ditions techniques for linear and nonlinear Schrodinger equations, Communications in Computational Physics 4(2008) 729–796.

[9] G. Pang, S. Tang, Approximate linear relations for bessel functions, preprint (2015).[10] G. Pang, L. Bian, S. Tang, Almost exact boundary condition for one-dimensional Schrodinger equations, Physical

Review E 86 (2012) 066709.[11] A. Jungel, S. Tang, Numerical approximation of the viscous quantum hydrodynamic model for semiconductors,

Applied Numerical Mathematics 56 (2006) 899–915.[12] X. Hu, S. Tang, Transient and stationary simulations for a quantum hydrodynamic model, Chinese Physics Letters

24 (2007) 1437.[13] N. B. Abdallah, P. Degond, P. A. Markowich, On a one-dimensional Schrodinger-Poisson scattering model,

Zeitschrift fur angewandte Mathematik und Physik ZAMP 48 (1997) 135–155.[14] A. Arnold, Mathematical concepts of open quantum boundary conditions, Transport Theory and Statistical Physics

30 (2001) 561–584.[15] A. Arnold, Numerically absorbing boundary conditionsfor quantum evolution equations, VLSI Design 6 (1998)

313–319.[16] A. Arnold, M. Ehrhardt, I. Sofronov, et al., Discrete transparent boundary conditions for the Schrodinger equation:

Fast calculation, approximation, and stability, Communications in Mathematical Sciences 1 (2003) 501–556.[17] X. Antoine, C. Besse, Unconditionally stable discretization schemes of non-reflecting boundary conditions for the

one-dimensional Schrodinger equation, Journal of Computational Physics 188 (2003) 157–175.[18] C. Zheng, A perfectly matched layer approach to the nonlinear Schrodinger wave equations, Journal of Computa-

tional Physics 227 (2007) 537–556.[19] S. Tang, A two-way interfacial condition for lattice simulations, Advances in Applied Mathematics and Mechanics

2 (2010) 45–55.

16

0 50 85 135

1022

1023

1024

n

x0 50 85 135

107

108

109

J cond

x0 50 85 135

−0.2

−0.1

0

0.1

Vs

x

0 50 85 135

1022

1023

1024

n

x0 50 85 135

107

108

109

J cond

x0 50 85 135

−0.2

−0.1

0

0.1

Vs

x

0 50 85 135

1022

1023

1024

n

x0 50 85 135

107

108

109

J cond

x0 50 85 135

−0.2

−0.1

0

0.1

Vs

x

0 50 85 135

1022

1023

1024

n

x0 50 85 135

107

108

109

J cond

x

0 50 85 135

−0.2

−0.1

0

0.1

Vs

x

0 50 85 135

1022

1023

1024

n

x0 50 85 135

107

108

109

J cond

x0 50 85 135

−0.2

−0.1

0

0.1

Vs

x

0 50 85 135

1022

1023

1024

n

x0 50 85 135

107

108

109

J cond

x0 50 85 135

−0.2

−0.1

0

0.1

Vs

x

Figure 10: Left column: electron density in m−3; middle column: current density in A/m2; right column: self-consistentpotential energy in eV; different rows show different timet =0.3ps, 0.5ps, 0.8ps, 1.0ps, 1.5ps and 2.0ps, respectively.

17

0 0.5 1 1.5 20

5

10

x 108

t in ps

J tota

l in A

/m2

0.1

0.2

U in

V

Jtotal

U

Figure 11: Total current density.

Figure 12: Conduction current density att = 1.5ps with comparison to [1].

18

−1−0.5

00.5

1

x 109

0

50

100

0

10

20

30

kx

Figure 13: Wave function|ϕ(x, k)| computed with cut-off wave numberk′M = 2kM .

0 1000 2000 3000

3

6

9

x 1023

M

n D in

m−

3

5nm40nm60nm

0 1000 2000 30000

1

2

3

4

x 1022

M

n D in

m−

3

62.5nm65nm67.5nm

Figure 14: Electron density computed with different number of termsM.

19

0.3 0.4 0.5 0.6 0.7 0.8

0

0.2

0.4

0.6

0.8

1

k0.3 0.4 0.5 0.6 0.7 0.80

4

8

12

k

Figure 15: Resonance computed with simplified model. Left: reflection coefficient |R(k)| (solid line) and transmissioncoefficient |T(k)| (dashed line). Right: coefficients|A0| (solid line) and|B0| (dashed line), which coincide with each other.The horizontal axis stands for undimensionalized wave number k.

20