Department of Information Technology

Simulation of Klein-tunneling inGraphene

Daniel Salvador, Fredrik Harlin

Project in Computational Science - Report #14

January 2015

PROJECTREPORT

Abstract

We study the dynamics of Klein-tunneling in single-layer graphene. Inparticular we compare the numerical transmission probabilities of time-dependent wave-packets across a potential barrier with the analyticalplane wave time-independent results presented in [3].

A strictly stable high-order accurate finite difference scheme for thesimulation of time-dependent Klein-tunneling in single-layer graphene isconstructed using the Summation-by-Parts–Simultaneous ApproximationTerm (SBP–SAT) method.

Particle transmission probabilities are obtained for angles of incidencebetween 0o and 75o. It is observed that the numerical transmission coef-ficient decays with an increasing angle of incidence, which does not agreewith the analytical stationary results. Experiments with different wave-packet sizes indicate that localization is not responsible for the discrep-ancy; thus it must be caused by time-dependence. This suggests thattime-dependent models are essential even for a qualitative understandingof Klein-tunneling in graphene.

Keywords: Klein-tunneling · Single-layer graphene · Massless Diracfermions · High-order finite difference methods · SBP-SAT.

Contents

1 Introduction 2

2 The equation 32.1 Analytical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Steger-Warming Flux Splitting . . . . . . . . . . . . . . . . . . . 4

3 The 1-D Problem 43.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 Continuous analysis . . . . . . . . . . . . . . . . . . . . . . . . . 63.3 Semi-discrete analysis . . . . . . . . . . . . . . . . . . . . . . . . 7

3.3.1 1-D Interface treatment . . . . . . . . . . . . . . . . . . . 83.4 Convergence study . . . . . . . . . . . . . . . . . . . . . . . . . . 93.5 Simulations of Klein-tunneling . . . . . . . . . . . . . . . . . . . 10

4 The 2-D Problem 124.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2 Continuous analysis . . . . . . . . . . . . . . . . . . . . . . . . . 134.3 Semi-discrete analysis . . . . . . . . . . . . . . . . . . . . . . . . 14

4.3.1 2-D Interface treatment . . . . . . . . . . . . . . . . . . . 154.4 Convergence study . . . . . . . . . . . . . . . . . . . . . . . . . . 184.5 Simulations of Klein-tunneling . . . . . . . . . . . . . . . . . . . 18

5 Conclusion 21

6 Acknowledgments 22

1

1 Introduction

The tunnel effect of a particle traveling through a potential barrier is awell-known phenomenon in quantum mechanics. When modeled with the non-relativistic Schrodinger equation, the probability that a particle crosses the bar-rier decays exponentially with the increase of the width and height of the barrier.However, if relativistic effects are taken into account, and instead one uses themassive Dirac equation to model this phenomenon, then the particle travelsthrough the potential barrier with a certain probability which does not tend tozero regardless of the width and the potential height of the barrier [1]. This ef-fect is called Klein-tunneling, after being theoretically described by Oskar Klein,a Swedish physicist, in 1929. See the illustration in Figure 1 for a conceptualdescription of a possible setting for Klein-tunneling.

Figure 1: Illustration of a particle with energy E and momentum k travelingtowards a potential barrier greater than its own energy, figure from [3].

Recent research has predicted this phenomenon for massless Dirac fermionsin single-layer graphene. The 2-D honeycomb lattice structure of single-layergraphene is a possible physical environment for testing Klein-tunneling experi-mentally and the behavior of this type of fermions in this context can be modeledby the equation

i~ψt = −i~νF σ · ∇ψ + V0ψ, (1.1)

which is further described in the next section.The aim of this work is to study the dynamics of Klein-tunneling in single-

layer graphene by solving a time-dependent problem with an efficient numer-ical method and then to compare the results with the theoretical plane wavetime-independent results presented in [3]. Our report describes one method tonumerically simulate dynamical fermions as wave-packets, by solving (1.1). Ananalysis of the transmission probability of fermions across a potential barrier asa function of angle of incidence is also presented.

The numerical simulations are performed using a Summation-by-Parts–Simultaneous Approximation Term (SBP–SAT) finite difference scheme as thediscretization method for the spatial domain and a fourth order Runge-Kuttamethod for time-integration. There are several reasons to use SBP–SAT. Firstly,the summation-by-parts property of SBP operators, combined with the SATboundary treatment, leads to discrete energy estimates that mimic the continu-ous energy estimates, and strict stability of the scheme can be proven. Moreover,the existence of high-order accuracy SBP operators provides a practical way ofattaining a global high-order convergence rate. This proves to be crucial sincehigh-order of accuracy schemes yield low dispersion errors, which makes themadequate for propagating disturbances over long distances. Additionally, byusing SBP–SAT, important properties of a single-block configuration, such as

2

strict stability and accuracy, are preserved upon extension to multi-block ge-ometries. The method has also been used previously to successfully simulatesimilar effects for massive Dirac fermions, see for example [2].

This report contains five sections. In the second section, the Equation (1.1)describing massless Dirac fermions in single-layer graphene is contextualizedand analytical solutions for the plane wave case are derived. The third sectioncontains the complete problem set-up in a 1-D domain, with an analysis of bothwell-posedness and error convergence rates for SBP operators of several orders ofaccuracy. In addition, numerical simulations of Klein-tunneling are performedand the results are compared to those obtained in [3]. The fourth section ex-pands the definitions and analysis in the third section to two dimensions. Inthe fifth and final section, some conclusions regarding the results are drawn.

2 The equation

Due to graphene’s particular material properties, charge carriers near theFermi energy behave like 2-D massless relativistic particles (Dirac fermions), itsenergy being described by the linear dispersion relation

E = ~||k||νF , (2.1)

where k is the wavevector and νF ≈ 106ms−1 is the Fermi velocity. As a conse-quence, at low energies, the electrons in single-layer graphene can be describedby the Dirac-like equation

i~ψt = −i~νF σ · ∇ψ + V0ψ, (2.2)

where σ = (σx, σy) is the 2-D vector of Pauli matrices

σx =

[0 11 0

], σy =

[0 −ii 0

],

and V0 is the height of the potential barrier.Considering a rectangular subdomain of R2,

Ωe,nw,s =

(x, y) ∈ R2 : w ≤ x ≤ e ; s ≤ y ≤ n,

the equation (2.2) describing massless Dirac fermions takes the form

ψt +Aψx +Bψy = −(i/~)V0ψ, t ≥ 0, (x, y) ∈ Ωe,nw,s, (2.3)

where

A =

[0 νFνF 0

], B =

[0 −iνFiνF 0

].

2.1 Analytical Solution

For the case of a particle free in space, that is, in the absence of a potential,when V0 = 0, solutions to (2.3) can be found if we consider ψ to be a 2-D planewave, whose general form is

ψ (r, t) = ψ0ei(k·r−ωt), (2.4)

3

where r = (x, y), ψ0 =(ψ(1)0 , ψ

(2)0

), k =

(k(1), k(2)

)denotes the wave vector

and ω is the angular frequency. Moreover, for a plane wave with Fermi velocityνF , these two latter physical quantities are related by the dispersion relation

ω2 = ν2F

(k(1)

2+ k(2)

2). (2.5)

Plugging the general 2-D plane wave expression (2.4) into Eq. (2.3) andsolving for ψ, we arrive at a linear system which, bearing in mind the dispersionrelation (2.5), yields two possible solutions

case I: ω = −νF√k(1)

2+ k(2)

2

⇒ ψ (r, t) =

[1

− k(1)+ik(2)√k(1)2+k(2)2

]ei(k·r+νF

√k(1)2+k(2)2t

); (2.6)

case II: ω = +νF√k(1)

2+ k(2)

2

⇒ ψ (r, t) =

[1

k(1)+ik(2)√k(1)2+k(2)2

]ei(k·r−νF

√k(1)2+k(2)2t

). (2.7)

2.2 Steger-Warming Flux Splitting

The matrices A and B in the system (2.3) are Hermitian , i.e. A = A∗ andB = B∗. Thus, they can be diagonalized by the unitary similarity transforma-tions TA and TB , respectively, that is, we can write

A = TAΛAT∗A, B = TBΛBT

∗B ,

where ΛA = diag(−νF ,+νF ) = ΛB , since σ(A) = ±νF = σ(B), and

TA =1√2

[−1 11 1

], TB =

1√2

[−i 1i 1

].

We can use the Steger-Warming flux splitting to split A and B into positiveand negative parts, A = A+ +A− and B = B+ +B− by defining

A± = T ∗A

(ΛA ± |ΛA|

2

)TA, B± = T ∗B

(ΛB ± |ΛB |

2

)TB .

Note that A+ and B+ are Hermitian positive semidefinite (HPSD), while A−and B− are Hermitian negative semidefinite (HNSD). This splitting will proveto be very useful to conveniently define appropriate boundary conditions for ourproblem.

3 The 1-D Problem

In one dimension, the equation describing massless Dirac fermions in singlelayer graphene can be written

ψt +Aψx = −(i/~)V0ψ. (3.1)

4

Considering the domain Ω = [0, 1] and bearing in mind the Steger-Warmingflux splitting defined in Section 2.2, we can impose boundary conditions on theingoing characteristic variables by specifying A+ψ and A−ψ at the left andright boundaries, respectively. We then obtain the 1-D problem

ψt +Aψx = −(i/~)V0ψ, x ∈ Ω, t > 0,A+ψ = A+g0(t), x = 0, t ≥ 0,A−ψ = A−gN (t), x = 1, t ≥ 0,ψ = f , x ∈ Ω, t = 0,

(3.2)

where g0 and gN are, respectively, the data corresponding to the left and rightboundaries.

3.1 Definitions

Before we start analyzing problem (3.2), some definitions are needed.

Let u,v ∈ L2[0, 1], where u =[u(1), u(2)

]T, v =

[v(1), v(2)

]Tare complex-

valued vector functions with 2 components, and let u∗ denote the Hermitian

transpose of u. Let the inner product be defined by (u,v) =∫ 1

0u∗v dx, and let

the corresponding norm be ||u||2 = (u,u). The domain Ω = [0, 1] is discretizedwith an equidistant grid xiNi=0, such that

xi = ih, i = 0, 1, ..., N, h =1

N.

The approximate solution at gridpoint xi is a (2 × 1) vector vi =[v(1)i , v

(2)i

]Tand the discrete solution vector is

v =[v(1)0 , v

(1)1 , ..., v

(1)N , v

(2)0 , ..., v

(2)N

]T2(N+1)×1

.

Mimicking the inner product for continuous functions, we define an inner prod-uct for discrete complex-valued vector functions, ψ, φ ∈ C2×(N+1) by (ψ, φ)H =ψ∗Hφ, where H is Hermitian and positive definite. The corresponding norm is||ψ||2H = ψ∗Hψ.

The following vectors will be frequently used:

e0 = [1, 0, . . . , 0]T , eN = [0, 0, . . . , 1]T ,

v0 =(I2 ⊗ eT0

)v, vN =

(I2 ⊗ eTN

)v.

Definition 3.1. An explicit pth-order accurate finite difference scheme withminimal stencil width of a Cauchy problem is called a pth-order accurate narrow-stencil.

Definition 3.2. A difference operator D1 = H−1Q = H−1(Q − 12e0e

T0 +

12eNe

TN ) approximating ∂/∂x, using a pth-order accurate narrow-stencil, is said

to be a pth-order accurate first-derivative SBP operator if H is symmetric andpositive definite, and Q+ QT = 0.

5

To make the notation for systems of equations more compact we introducethe Kronecker product,

C ⊗D =

c0,0D · · · c0,q−1D...

...cp−1,0D · · · cp−1,q−1D

,where C is a p × q matrix and D is an r × s matrix. The following propertiesof the Kronecker product will be used

• (C ⊗D)(O ⊗ P ) = (CO)⊗ (DP ), whenever CO and DP are defined;

• (C ⊗D)−1 = (C−1 ⊗D−1), whenever C−1 and D−1 exist;

• (C ⊗D)∗ = C∗ ⊗D∗;

• if C and P are HPSD and O is HNSD, then (C⊗P ) is HPSD and (C⊗O)is HNSD.

Throughout, we let In denote the n× n identity matrix.

3.2 Continuous analysis

Consider the case of a free particle in space, that is, when V0 = 0 in (3.2).The problem becomes

ψt +Aψx = 0, x ∈ Ω, t > 0,A+ψ = A+g0(t), x = 0, t ≥ 0,A−ψ = A−gN (t), x = 1, t ≥ 0,ψ = f , x ∈ Ω, t = 0.

. (3.3)

Integrating by parts (ψ,ψt), we get

(ψ,ψt) = (ψ,−Aψx) = − ψTAψ∣∣x=1

x=0+ (ψx, Aψ) . (3.4)

Also, as A is Hermitian, we have

(ψt,ψ) = (−Aψx,ψ) = − (ψx, Aψ) . (3.5)

Adding expressions (3.4) and (3.5), we arrive at

(ψ,ψt) + (ψt,ψ)︸ ︷︷ ︸= ddt‖ψ‖2

= − ψ∗Aψ|x=1x=0 = − (ψ∗A+ψ +ψ∗A−ψ)|x=1

x=0 , (3.6)

where ‖ψ‖2 is the probability of finding the particle anywhere in the domain.This quantity can also be interpreted as the mathematical energy of the system.Thus, the left-hand side of (3.6) corresponds to the rate of change of energy.

Expanding (3.6) according to the Steger-Warming flux splitting and bearingin mind the ingoing characteristic boundary conditions set in (3.3), we obtainthe following energy estimate

d

dt‖ψ‖2 = ψ∗A−ψ|x=0 − ψ

∗A+ψ|x=1︸ ︷︷ ︸≤0

− g∗NA−gN︸ ︷︷ ︸known data

+ g∗0A+g0︸ ︷︷ ︸known data

. (3.7)

6

Integrating (3.7) in time, we finally arrive at

‖ψ(·, t)‖2 ≤ ‖ψ(·, 0)‖2 +

∫ t

0

g∗0A+g0 − g∗NA−gN dτ. (3.8)

Hence the energy grows only in terms of known data so, for the case of afree particle, assuming the existence of a solution and bearing in mind that itsuniqueness is easily proven using (3.8), problem (3.3) is strongly well posed.

3.3 Semi-discrete analysis

Discretizing the system of PDEs in (3.3) in space with the SBP–SAT methodleads to the semi-discrete problem for the particle in 1-D

vt = −AI2 ⊗D1︸ ︷︷ ︸SBP

v + τ0A+ ⊗H−1e0(v0 − g0)︸ ︷︷ ︸SAT at x=0

+ τNA− ⊗H−1eN (vN − gN )︸ ︷︷ ︸SAT at x=1

,

(3.9)where τ0 and τN are penalty parameters yet to be determined.

The undermentioned definition characterizes discretization methods that donot allow nonphysical solution growth in time.

Definition 3.3. A semi-discretization is said to be strictly stable if the solutiongrowth rate of the discrete problem is bounded by the growth rate of the (well-posed) continuous problem.

In the following lemma, we present penalty parameters that yield a strictlystable semi-discretization.

Lemma 3.1. The scheme (3.9) is strictly stable if τ0 = −1 and τN = 1.

Proof. Multiplying (3.9) by v∗(I2⊗H) and then adding the conjugate transposeleads to

d

dt||v||2H = (v,vt)H + (vt,v)H = v∗(I2 ⊗H)vt + v∗t (I2 ⊗H)v

= −v∗(A⊗ (Q+ Q∗)

)︸ ︷︷ ︸

=0

v + v∗0Av0 − v∗NAvN

+ τ0v∗0A+(v0 − g0) + τNv

∗NA−(vN − gN )

+ τ∗0 (v0 − g0)∗A+v0 + τ∗N (vN − gN )∗A−vN .

Choosing the penalty parameters such that τ0 = −1 and τN = 1, we finallyarrive at

d

dt||v||2H = v∗0A−v0 − v∗NA+vN − g∗NA−gN + g∗0A+g0

−(v0 − g0)∗A+(v0 − g0) + (vN − gN )∗A−(vN − gN )︸ ︷︷ ︸damping terms ≤ 0

,

which is analogous to the continuous energy estimate (3.8), with the additionof two damping terms that vanish with grid refinement. Therefore, the discretesolution growth rate is bounded by the continuous growth rate, and hence thescheme (3.9) is strictly stable.

7

Figure 2: Relation between the physical domain with a potential barrier andthe SAT terms.

3.3.1 1-D Interface treatment

In order to introduce a barrier, we now consider our domain to be Ω =[−1, 1] = Ω1 ∪Ω2, with Ω1 = [−1, 0] and Ω2 = [0, 1], as illustrated in Figure 2.

The potential barrier will be introduced in Ω2, while in Ω1 the particle isstill free. Thus, the problem we are considering takes the form

ψ(1)t +Aψ

(1)x = 0, x ∈ Ω1, t > 0,

ψ(2)t +Aψ

(2)x = −(i/~)V0ψ

(2), x ∈ Ω2, t > 0,A−ψ = A−gN (t), x = 1, t ≥ 0,A+ψ = A+g0(t), x = −1, t ≥ 0,ψ = f , x ∈ Ω, t = 0,

(3.10)

where ψ(1) and ψ(2) denote the solutions in Ω1 and Ω2, respectively.Moreover, for our solution to make physical sense, it has to be continuous

across the interface between Ω1 and Ω2, that is, we have to impose

ψ(1) = ψ(2), x = 0. (3.11)

For the sake of notational simplicity, assuming the same number of gridpoints in the spatial discretization of each subdomain, a semi-discrete approxi-mation of (3.10) is given by

v(1)t =−AI2 ⊗D1v

(1)+

+ τ0A+ ⊗H−1e0(v(1)0 − g0

)+ γNA− ⊗H−1eN

(v(1)N − v

(2)0

),

v(2)t =−AI2 ⊗D1v

(2) − (i/~)v0v(2)

+ τNA− ⊗H−1eN(v(2)N − gN

)+ γ0A+ ⊗H−1e0

(v(2)0 − v

(1)N

).

(3.12)

Lemma 3.2. The scheme (3.12) is strictly stable if τ0 = −1 = γ0, τN = 1 = γN .

Proof. We start by setting τ0 = −1, τN = 1, γ0 = −1 and γN = 1 and thenmultiply the first and second equations in (3.12) by v(1)

∗I2⊗H and v(2)

∗I2⊗H,

8

respectively. Then, adding the conjugate transpose, we arrive at the followingdiscrete energy estimate

d

dt

(||v(1)||2H + ||v(2)||2H

)= (v(1),v

(1)t )H + (,v

(1)t ,v(1))H

+ (v(2),v(2)t )H + (,v

(2)t ,v(2))H

= v(1)∗I2 ⊗Hv(1)t + v

(1)t

∗I2 ⊗Hv(1)+

+ v(2)∗I2 ⊗Hv(2)t + v

(2)t

∗I2 ⊗Hv(2)

= −(v(1) + v(2)

)∗(A⊗ (Q+Q∗))︸ ︷︷ ︸

=0

(v(1) + v(2)

)−(i/~)v0v

(2)∗I2 ⊗H + (i/~)v0v(2)∗I2 ⊗H︸ ︷︷ ︸

=0

+ v(1)0

∗A−v

(1)0 −

(v(1)0 − g0

)∗A+

(v(1)0 − g0

)+ g∗0A+g0

− v(2)N∗A+v

(2)N +

(v(2)N − gN

)∗A−

(v(2)N − gN

)− g∗NA+gN

+(v(1)N − v

(2)0

)∗(A− −A+)

(v(1)N − v

(2)0

).

Thus, this case is analogous to the discrete energy estimate for the free parti-

cle case, except for the interface contribution(v(1)N − v

(2)0

)∗(A−−A+)

(v(1)N − v

(2)0

)which is a damping term that vanishes with grid refinement, since (A− − A+)is HNSD. Therefore, the multi-block discretization (3.12) is strictly stable.

3.4 Convergence study

The convergence study is performed in a single-block domain with no po-tential barrier, using the analytical plane wave solution derived in Section 2.1as a reference solution. Since these simulations are performed only in 1-D, k(2)

is set to zero in the analytical solution. The analytical solution is used as initialdata.

For each refinement of the grid, the error is analyzed at t = 5 ·10−14, that is,at the 500th time step, with a fixed time step size of k = 10−16 s, conservativelybelow the requirements of the CFL condition.

We now compute the error convergence rate q, which is defined in [2] as:

q =log10

(||vref−v(N2)||h||vref−v(N1)||h

)log10

(N1

N2

) , (3.13)

where ||v||h =√h∑j v

2j and the reference solution, vref , is the analytical

solution derived in Section 2.1. The magnitude of the error and the resultsof the error convergence analysis are presented in Table 1. It is seen that themethod asymptotically reaches the expected overall error convergence rates,p+ 1 for an SBP operator of order 2p, which was theoretically expected.

9

Table 1: Error and error convergence rates, according to Eq. 3.13, for SBPoperators with different orders of accuracy. Note that the global convergencerate asymptotically approaches p+ 1 for an SBP operator of order 2p. N is thenumber of grid points in the domain, whose length is 500 nm.

N 2nd q(2) 4th q(4) 6th q(6)

101 3.56e-04 3.37e-05 3.94e-05201 9.01e-05 2.00 3.09e-06 3.47 3.05e-06 3.72401 2.26e-05 2.01 3.36e-07 3.21 2.00e-07 3.94801 5.64e-06 2.00 4.01e-08 3.07 1.26e-08 3.99

N 8th q(8) 10th q(10)

101 2.60e-05 3.98e-05201 8.95e-07 4.89 7.44e-07 5.78401 2.66e-08 5.09 1.23e-08 5.94801 8.24e-10 5.02 1.96e-10 5.99

3.5 Simulations of Klein-tunneling

In a physical sense, it is more realistic to let a wave-packet travel towards thepotential barrier than to use a plane wave solution. In order to create the initialconditions of the simulations, the plane wave solution is used with a Gaussianenvelope. The initial conditions for the simulations then become

ψ (x, 0) =

[11

]ei(kx−νF kt) · e− x

2

4σ , (3.14)

where the last term creates a Gaussian envelope, and σ is a measure of itswidth. In this case, the characteristic boundary conditions are set to zero, thatis g0 = gN = 0, in order to create non-reflecting boundary conditions.

Since ψ is a two component complex spinor function, the representation ofthe solution, in Figure 3, is done only for the real part of the first component,on the left column. ψ∗ψ, which is often interpreted as the probability densityfunction, is shown on the right column.

As expected from the theoretical results, the 1-D case is similar to a particletraveling perpendicularly towards the potential barrier, which means that theparticle would travel through the barrier with absolute certainty. This phe-nomenon, which, as mentioned earlier, is called Klein-tunneling, can be seenfrom Figure 3, where the shape of the probability density function remains thesame and, as observed in our numerical experiments, would remain so, regardlessof the height or width of the potential barrier. Regarding the grid resolution,we have used about 40 grid points per wavelength in this case, which is morethan enough to make discretization errors negligible. Another observation thatcan be made from looking at Figure 3 is that the frequency of the componentsincrease within the barrier. Our observations also indicate that the higher thepotential within the barrier, the more the frequency increases, which in extremecases creates a requirement for a finer grid within the barrier.

10

(a) Initial condition

(b) t = 1.95 · 10−13 s

(c) t = 2.85 · 10−13 s

(d) t = 3.75 · 10−13 s

(e) t = 4.95 · 10−13 s

Figure 3: To the right: a non-normalized probability density function simulatinga massless fermion of energy 80 meV traveling in single-layer graphene towardsa barrier of 300 meV. To the left: the corresponding real part of the first compo-nent of the spinor. Note that the frequency of the components increases withinthe barrier, while the shape of the probability function remains the same.

11

4 The 2-D Problem

In two dimensions, the equation describing massless Dirac fermions in single-layer graphene can be written

ψt +Aψx +Bψy = −(i/~)V0ψ. (4.1)

Considering the domain to be the unit square

Ω1,10,0 =

(x, y) ∈ R2 : 0 ≤ x ≤ 1 ; 0 ≤ y ≤ 1

and bearing in mind the Steger-Warming flux splitting defined in Section 2.2,we can impose boundary conditions on the ingoing characteristic variables byspecifying A+ψ and A−ψ at the west (W ) and east (E) boundaries, and B+ψand B−ψ at the south (S) and north (N) boundaries, respectively. We thenobtain the following 2-D problem

ψt +Aψx +Bψy = −(i/~)V0ψ, r ∈ Ω1,10,0 , t > 0,

A+ψ = A+gW (y, t) r ∈W, t ≥ 0,A−ψ = A−gE(y, t) r ∈ E, t ≥ 0,B+ψ = B+gS(x, t) r ∈ S, t ≥ 0,B−ψ = B−gN (x, t) r ∈ N, t ≥ 0,

ψ = f , r ∈ Ω1,10,0 , t = 0,

(4.2)

where gW , gE , gS and gN are, respectively, the data corresponding to the east,west, south and north boundaries.

4.1 Definitions

Before we start solving Equation (4.2) some definitions are needed.Let u,v ∈ L2[Ωw,ne,s ] where u = [u(1), u(2)]T and v = [v(1), v(2)]T are complex-

valued vector functions with 2 components. Let the inner product be definedby (u,v) =

∫ we

∫ nsu∗vdx, and let the corresponding norm be ||u||2 = (u,u).

Let r = (x, y) denote the position vector in two dimensions and considerthe computational domain Ωw,ne,s =

(x, y) ∈ R2 : w ≤ x ≤ e ; s ≤ y ≤ n

. This

domain is discretized using an (Nx+1)×(Ny+1)-point equidistant grid definedas:

xi = w + ihx, i = 0, 1, . . . , Nx, hx =e− wNx

,

yj = s+ jhy, i = 0, 1, . . . , Ny, hy =n− sNy

.

In order to simplify the notation, we take Nx = Ny = N − 1. The approxi-

mate solution at the grid point (xi, yj) is vi,j =

[v(1)i,j

v(2)i,j

]and, for computational

purposes, the entire solution vector should be defined as

v =[v(1)00 , . . . , v

(1)0Ny

, v(1)10 , . . . , v

(1)1Ny

, . . . , v(1)NxNy

, . . . , v(2)00 , . . . , v

(2)0Ny

, . . . , v(2)NxNy

]1×2N2

.

To distinguish whether a 2-D difference operator R is operating in the x- orthe y-direction, we use the notations Rx and Ry, respectively. The following

12

2-D operators will be used:

Dx = I2 ⊗D1 ⊗ IN , Dy = I2 ⊗ IN ⊗D1,Hx = I2 ⊗H ⊗ IN , Hy = I2 ⊗ IN ⊗H,EW = I2 ⊗ e0 ⊗ IN , ES = I2 ⊗ IN ⊗ e0,EE = I2 ⊗ eN ⊗ IN , EN = I2 ⊗ IN ⊗ eN ,H = HxHy = I2 ⊗H ⊗H,

where we refer to the 4 boundaries of the domain as W (West), E (East), S(South) and N (North) boundaries, D1 and H are the 1-D operators introducedearlier, and e0 and eN are the 1-D ”boundary” vectors defined earlier. Notethat the matrix EW is defined so that vW = E∗Wv is a vector that contains onlythe elements of v that correspond to the west boundary. The matrices EE , ESand EN , and respective vectors vE , vS and vN , are defined in a similar way foreach of the other boundaries.

4.2 Continuous analysis

Consider the case of a free particle in 2-D space, that is, when V0 = 0 in(4.2), by solving

ψt +Aψx +Bψy = 0, r ∈ Ω1,10,0 , t > 0,

A+ψ = A+gW (y, t), r ∈W, t ≥ 0,A−ψ = A−gE(y, t), r ∈ E, t ≥ 0,B+ψ = B+gS(x, t), r ∈ S, t ≥ 0,B−ψ = B−gN (x, t), r ∈ N, t ≥ 0,

ψ = f , r ∈ Ω1,10,0 , t = 0,

(4.3)

Multiplying the first equation in problem (4.3) by ψ∗, adding the transposeand integrating over the domain Ω1,1

0,0 results in the following energy estimate

d

dt||ψ||2 =

∫Ω

ψ∗ψt +ψ∗tψdΩ

= −∫Ω

∂

∂x(ψ∗Aψ) dΩ −

∫Ω

∂

∂y(ψ∗Bψ) dΩ

=

∫W

ψ∗Aψdy︸ ︷︷ ︸=BTW

−∫E

ψ∗Aψdy︸ ︷︷ ︸=BTE

+

∫S

ψ∗Bψdx︸ ︷︷ ︸=BTS

∫N

ψ∗Bψdx︸ ︷︷ ︸=BTN

= BTW +BTE +BTS +BTN (4.4)

where the boundary terms are

BTW =

∫W

ψ∗A+ψdy︸ ︷︷ ︸=∫Wg∗WA+gW dy

+

∫W

ψ∗A−ψdy︸ ︷︷ ︸≤0

, BTE = −∫E

ψ∗A+ψdy︸ ︷︷ ︸≤0

−∫E

ψ∗A−ψdy︸ ︷︷ ︸=−

∫Eg∗EA−gEdy

,

BTS =

∫S

ψ∗B+ψdx︸ ︷︷ ︸=∫Sg∗SB+gSdx

+

∫S

ψ∗B−ψdx︸ ︷︷ ︸≤0

, BTN = −∫N

ψ∗B+ψdx︸ ︷︷ ︸≤0

−∫N

ψ∗B−ψdx︸ ︷︷ ︸=−

∫Ng∗NB−gNdx

.

13

Thus

d

dt||ψ(r, t)||2 ≤

∫W

g∗WA+gW dy︸ ︷︷ ︸=IW+

−∫E

g∗EA−gEdy︸ ︷︷ ︸=IE−

+

∫S

g∗SB+gSdx︸ ︷︷ ︸IS+

−∫N

g∗NB−gNdx︸ ︷︷ ︸IN−

.

Integrating the last expression in time, we finally arrive at

||ψ(·, t)||2 ≤ ||ψ(·, 0)||2 +

∫ t

0

(IW+ − IE− + IS+ − IN−) dτ. (4.5)

Hence we obtain an energy growth only in terms of known data so, for thecase of a free particle, assuming the existence of a solution and bearing in mindthat its uniqueness is easily proven using (4.5), problem (4.3) is strongly wellposed.

4.3 Semi-discrete analysis

Discretizing the system of PDEs in (4.3) in space with the SBP–SAT methodleads to the semi-discrete problem for a free particle in 2-D

vt = −ADxv −BDyv + SAT, (4.6)

where SAT = SATW + SATE + SATS + SATN is the sum of the SAT termsfrom each boundary

SATW = −(A+ ⊗H−1e0 ⊗ IN

)(vW − gW ),

SATE =(A− ⊗H−1eN ⊗ IN

)(vE − gE),

SATS = −(B+ ⊗ IN ⊗H−1e0

)(vS − gS),

SATN =(B− ⊗ IN ⊗H−1eN

)(vN − gN ).

In the following lemma, we prove the strict stability of the suggested semi-discretization.

Lemma 4.1. The scheme (4.6) is strictly stable.

Proof. We first multiply (4.6) by v∗H and then add the conjugate transpose,obtaining

d

dt‖v‖2

H= v∗Hvt + v∗t Hv

= −v∗H(ADx +BDy)v + v∗H(SATW ) + v∗H(SATE) + v∗H(SATS) + v∗H(SATN )

− v∗(ADx +BDy)∗Hv + (SATW )∗Hv + (SATE)∗Hv + (SATS)∗Hv + (SATN )∗Hv

= BTW +BTE +BTS +BTN

14

where

BTW = v∗W (A− ⊗H)vW + v∗W (A+ ⊗H) gW − (vW − gW )∗ (A+ ⊗H)vW

= v∗W (A− ⊗H)vW︸ ︷︷ ︸≤0

+ g∗W (A+ ⊗H) gW︸ ︷︷ ︸known data

−(vW − gW )∗ (A+ ⊗H) (vW − gW )︸ ︷︷ ︸≤0; damping term

;

BTE = −v∗E (A+ ⊗H)vE − v∗E (A− ⊗H) gE + (vE − gE)∗ (A− ⊗H)vE

= −v∗E (A+ ⊗H)vE︸ ︷︷ ︸≤0

− g∗E (A− ⊗H) gE︸ ︷︷ ︸known data

+(vE − gE)∗ (A− ⊗H) (vE − gE)︸ ︷︷ ︸≤0; damping term

;

BTS = v∗S (B− ⊗H)vS + v∗S (B+ ⊗H) gS − (vS − gS)∗ (B+ ⊗H)vS

= v∗S (B− ⊗H)vS︸ ︷︷ ︸≤0

+ g∗S (B+ ⊗H) gS︸ ︷︷ ︸known data

−(vS − gS)∗ (B+ ⊗H) (vS − gS)︸ ︷︷ ︸≤0; damping term

;

BTN = −v∗N (B+ ⊗H)vN − v∗N (B− ⊗H) gN + (vN − gN )∗ (B− ⊗H)vN

= −v∗N (B+ ⊗H)vN︸ ︷︷ ︸≤0

− g∗N (B− ⊗H) gN︸ ︷︷ ︸known data

+(vN − gN )∗ (B− ⊗H) (vN − gN )︸ ︷︷ ︸≤0; damping term

.

which is analogous to the continuous energy estimate (4.4),with the addition ofsome damping terms that vanish with grid refinement. Therefore, the discretesolution growth rate is bounded by the continuous growth rate, and hence thescheme (4.6) is strictly stable.

4.3.1 2-D Interface treatment

Figure 4: Illustration of the domain in consideration, composed by two 2-D blocks with different potentials, connected by an interface, and respectiveboundary conditions.

In order to introduce a potential barrier, one has now to consider two differ-ent blocks with different potentials, connected by an interface. See Figure 4 foran illustration of the domain and the corresponding boundary conditions.

15

The problem we are considering is

ψ(1)t +Aψ

(1)x +Bψ

(1)y = 0, r ∈ Ω1, t > 0,

ψ(2)t +Aψ

(2)x +Bψ

(2)y = − i

~V0ψ(2), r ∈ Ω2, t > 0,

A+ψ(1) = A+gW , r ∈W, t > 0,

A−ψ(2) = A+gE , r ∈ E, t > 0,

B+ψ(1,2) = B+g

(1,2)S , r ∈ S(1,2), t > 0,

B−ψ(1,2) = B−g

(1,2)N , r ∈ N (1,2), t > 0,

ψ(1) = ψ(2), r ∈ I, t > 0,ψ = f , r ∈ Ω, t = 0,

(4.7)

Note that, as in the 1-D interface treatment, for the solution to make physicalsense, it has to be continuous across the interface between Ω1 and Ω2, that is,we have to impose

ψ(1) = ψ(2), r ∈ I. (4.8)

The resulting spatial semi-discretization of (4.7) is then

v(1)t = −ADxv

(1) −BDyv(1) + SAT (1)

v(2)t = −ADxv

(2) −BDyv(2) + i

~V0v(2) + SAT (2)

(4.9)

where the SAT terms are defined as SAT (1) = SAT(1)W + SAT

(1)I + SAT

(1)S +

SAT(1)N and SAT (2) = SAT

(2)I + SAT

(2)E + SAT

(2)S + SAT

(2)N , with

SAT(1)W = −(A+ ⊗H−1e0 ⊗ IN )(v

(1)W − gW )

SAT(1)I = (A− ⊗H−1eN ⊗ IN )(v

(1)I − v

(2)I )

SAT(1)S = −(B+ ⊗ IN ⊗H−1e0)(v

(1)S − g

(1)S )

SAT(1)N = (B− ⊗ IN ⊗H−1eN )(v

(1)N − g

(1)N )

SAT(2)I = −(A+ ⊗H−1e0 ⊗ IN )(v

(2)I − v

(1)I )

SAT(2)E = (A− ⊗H−1eN ⊗ IN )(v

(2)E − gE)

SAT(2)S = −(B+ ⊗ IN ⊗H−1e0)(v

(2)S − g

(2)S )

SAT(2)N = (B− ⊗ IN ⊗H−1eN )(v

(2)N − g

(2)N )

The stability analysis is done by the energy method.

Lemma 4.2. The scheme (4.9) is strictly stable.

Proof. Multiplying the equations in (4.9) with v(1)∗H and v(2)

∗H, respectively,

adding the transpose, and proceeding as in the analysis for the case of a freeparticle in 2-D, we end up with the following discrete energy estimate

16

d

dt

(‖v(1)‖2

H+ ‖v(2)‖2

H

)= v(1)

∗Hv

(1)t + (v

(1)t )∗Hv(1) + v(2)

∗Hv

(2)t + (v

(2)t )∗Hv(2)

= −v(1)∗H(ADx +BDy)v(1) + v(1)

∗H(SAT (1))

− v(1)∗(ADx +BDy)∗Hv(1) + (SAT (1))∗Hv(1)

− v(2)∗H(ADx +BDy)v(2) + v(2)

∗H(SAT (2))

− v(2)∗(ADx +BDy)∗Hv(2) + (SAT (2))∗Hv(2)

+v(2)∗H(

i

~V0)v(2) + v(2)

∗(i

~V0)∗Hv(2)︸ ︷︷ ︸

=0

= BTW +BTI +BTE +BTS +BTN

where the boundary terms are as follows

Western boundary

BTW = v(1)W

∗(A+ ⊗H)v

(1)W + v

(1)W

∗(A− ⊗H)v

(1)W︸ ︷︷ ︸

from SBP (1)

−v(1)W∗(A+ ⊗H)(v

(1)W − gW )︸ ︷︷ ︸

from SAT(1)W

−(v(1)W

∗− gW )∗(A+ ⊗H)v

(1)W︸ ︷︷ ︸

from (SAT(1)W )∗

= v(1)W

∗(A− ⊗H)v

(1)W︸ ︷︷ ︸

≤0

−(v(1)W − gW )∗(A+ ⊗H)(v

(1)W − gW )︸ ︷︷ ︸

≤0

+ g∗W (A+ ⊗H)gW︸ ︷︷ ︸known data

Interface

BTI = −v(1)I∗(A+ ⊗H)v

(1)I − v

(1)I

∗(A− ⊗H)v

(1)I︸ ︷︷ ︸

from SBP (1)

+v(1)I

∗(A− ⊗H)(v

(1)I − v

(2)I ) + (v

(1)I − v

(2)I )∗(A− ⊗H)v

(1)I︸ ︷︷ ︸

from SAT(1)I ,(SAT

(1)I )∗

+v(2)I

∗(A+ ⊗H)v

(2)I + v

(2)I

∗(A− ⊗H)v

(2)I︸ ︷︷ ︸

from SBP (2)

−v(2)I∗(A+ ⊗H)(v

(2)I − v

(1)I )− (v

(2)I − v

(1)I )∗(A+ ⊗H)v

(2)I︸ ︷︷ ︸

from SAT(2)I ,(SAT

(2)I )∗

= −(v(1)I − v

(2)I )∗(A+ ⊗H)(v

(1)I − v

(2)I )︸ ︷︷ ︸

≤0

+ (v(1)I − v

(2)I )∗(A− ⊗H)(v

(1)I − v

(2)I )︸ ︷︷ ︸

≤0

Eastern boundary Similarly as for the western boundary and analogous tothe one dimensional case, BTE will be

BTE = −v(2)E∗(A+ ⊗H)v

(2)E︸ ︷︷ ︸

≤0

+(v(2)E − gE)∗(A− ⊗H)(v

(2)E − gE)︸ ︷︷ ︸

≤0

−g∗E(A− ⊗H)gE︸ ︷︷ ︸known data

17

Southern boundary

BTS = v(1)S

∗(B− ⊗H)v

(1)S + (v

(2)S )∗(B− ⊗H)v

(2)S︸ ︷︷ ︸

≤0

−(v(1)S − g

(1)S )∗(B+ ⊗H)(v

(1)S − g

(1)S )− (v

(2)S − g

(2)S )∗(B+ ⊗H)(v

(2)S − g

(2)S )︸ ︷︷ ︸

≤0

+ g(1)S

∗(B+ ⊗H)g

(1)S + g

(2)S

∗(B+ ⊗H)g

(2)S︸ ︷︷ ︸

known data

Northern boundary

BTN = −v(1)N∗(B+ ⊗H)v

(1)N − v

(2)N

∗(B+ ⊗H)v

(2)N︸ ︷︷ ︸

≤0

+(v(1)N − g

(1)N )∗(B− ⊗H)(v

(1)N − g

(1)N ) + (v

(2)N − g

(2)N )∗(B− ⊗H)(v

(2)N − g

(2)N )︸ ︷︷ ︸

≤0

− g(1)N∗(B− ⊗H)g

(1)N − g

(2)N

∗(B− ⊗H)g

(2)N︸ ︷︷ ︸

known data

Thus, the discrete solution growth rate is bounded by the continuous growthrate and hence the semi-discretization (4.9) is strictly stable.

4.4 Convergence study

To verify the performance of the discretization, an error convergence analysisis performed, using the same method and definitions as for the one dimensionalcase, in Section 3.4, except of course for the extra dimension and related impli-cations. Again, the analytic plane wave solution derived in Section 2.1 is used asa reference solution, but we now consider an angle of incidence α = 30. The do-main is composed by a rectangle with dimensions x ∈ [0, 500]nm, y ∈ [0, 250]nm.The error is analyzed at t = 1 · 10−13 s for each numerical solution, correspond-ing to the 257th time step, having each time step a size of k = 3.8911 · 10−16s,conservatively below the CFL condition. The results of the convergence study,which is otherwise carried out in the same way as for the one dimensional prob-lem, are presented in Table 2. The achieved global convergence rate is p+ 1 foran SBP operator of order 2p, which was theoretically expected.

4.5 Simulations of Klein-tunneling

The transmission probability as a function of angle of incidence will lookdifferent for different fermion energies and for different barrier height energies.In our simulations we have used the energies presented in [3] - a fermion energyof 80 meV and a barrier height of 200 meV. An illustration of the simulatedphenomenon for an angle of incidence of 15 is shown in Figure 5.

As in the one-dimensional case, it can be seen that the frequency of the wave-packet components changes within the barrier, increasing with the heighten ofthe barrier’s potential. An additional observation in this two-dimensional caseis that the direction of travel of the fermion changes within the barrier. This

18

Table 2: Error and error convergence rates, according to Eq. 3.13, for SBPoperators with different orders of accuracy. Note that the global convergencerate is p + 1 for an SBP operator of order 2p, which is expected, exactly as inthe one dimensional case. The domain is 500 nm wide and 250 nm high.

Nx Ny 2nd q(2) 4th q(4) 6th q(6)

41 21 9.19e-07 3.66e-07 2.36e-0781 41 3.18e-07 1.53 3.07e-08 3.57 3.21e-08 2.88161 81 8.07e-08 1.98 2.68e-09 3.52 2.61e-09 3.62321 161 2.02e-08 2.00 2.82e-10 3.25 1.74e-10 3.90

Nx Ny 8th q(8) 10th q(10)

41 21 2.66e-07 4.33e-0781 41 2.39e-08 3.48 3.56e-08 3.60161 81 8.78e-10 4.77 7.21e-10 5.63321 161 2.70e-11 5.02 1.21e-11 5.89

suggests that a description similar to Snell’s law for electromagnetic refractioncould be appropriate also for describing Klein-tunneling in single-layer graphene,as discussed in [1].

This type of simulation was carried out with every integer angle of incidencebetween 0 and 75. In this case 40 grid points per wave-length are used, whichis sufficient to make the discretization errors negligible. The probability for thefermion to travel through the barrier is estimated for every angle by taking thetransmitted mathematical energy and dividing it by the initial mathematicalenergy of the fermion. From this, the transmission probability as a function ofangle of incidence can be computed. In Figure 6 the results are presented ina polar plot, which includes the theoretically expected transmission probabilityfor the time-independent plane wave solution derived in [3]. As can be seen, theresults do not agree, except for small angles of incidence, up to 15.

This disagreement is probably due to the different approaches to the prob-lem. Our approach takes into account the time-dependence and the fact thatfermions have a finite size. In the simulations the Gaussian envelope used tocreate an initial wave-packet is about three wavelengths. Even though differentwidths were tested, the largest more than twice as big as the simulations inFigure 5, no significant change in transmission coefficient was observed. Thissuggests that the discrepancy is not caused by the localization but instead thatit is due to the time-dependent approach to the problem.

19

(a) Initial condition

(b) t = 2.4365 · 10−13 s

(c) t = 3.586 · 10−13 s

(d) t = 5.4539 · 10−13 s

Figure 5: To the right: a non-normalized probability density function of asimulated massless fermion of energy 80 meV, traveling in single-layer graphenetowards a barrier of 200 meV, with an angle of incidence of 15. To the left: thereal part of the first component of the spinor. Note that there is a probabilityfor the fermion to reflect back into the left domain when hitting the barrier.

20

Figure 6: Transmission probability as function of the angle of incidence fora massless Dirac fermion of energy 80 meV and a barrier energy of 200 meV.Our numerical results for a tenth order semi-discretization in space and a fourthorder Runge-Kutta as a time integrator (in blue) are compared with the analytictime-independent plane wave results from [3] (in red).

5 Conclusion

A strictly stable high-order accurate finite difference scheme for the simula-tion of time-dependent Klein-tunneling in single-layer graphene was constructedusing the Summation-by-Parts–Simultaneous Approximation Term (SBP–SAT)method. The numerical simulations achieved a global error convergence rate ofsixth order, using a tenth order SBP-operator, in agreement with the theoreticalexpectations.

The dynamics of Klein-tunneling in single-layer graphene were studied. Weobserved that the frequency of the components of the wave-packet changeswithin the barrier, depending on the height of the potential barrier. Moreover,in the two-dimensional case, the direction of travel of the fermion changes withinthe barrier, which suggests that a description similar to Snell’s law for electro-magnetic refraction could be appropriate also for describing Klein-tunneling insingle-layer graphene, as discussed in [1].

We also compared the numerical transmission probabilities of time-dependentwave-packets across a potential barrier with the analytical plane wave time-independent results presented in [3], by computing the particle transmissionprobabilities for angles of incidence between 0o and 75o. The numerical time-dependent simulations do not agree with the theoretical stationary results. Thisdiscrepancy in the transmission probability may be due to the fact that, insteadof plane waves, we model our particles more realistically as time-dependentmoving wave-packets. Experiments with different wave-packet sizes indicatethat localization is not responsible for the discrepancy, thus it must be causedby time-dependence. This suggests that time-dependent models are essentialeven for a qualitative understanding of Klein-tunneling in graphene.

21

6 Acknowledgments

The authors would like to thank their supervisor Martin Almquist for in-troducing them to the subject and for his constant support, Michael Thune forsharing with them his experience in research on Scientific Computing and MayaNeytcheva for both her insightful comments and the nice mugs.

References

[1] P. E. Allain and J. N. Fuchs. Klein tunneling in graphene: optics withmassless electrons. The European Physical Journal B, 83(3):301–317, Octo-ber 2011.

[2] Martin Almquist, Ken Mattsson, and Tomas Edvinsson. High-fidelity nu-merical solution of the time-dependent Dirac equation. Journal of Compu-tational Physics, 262:86–103, April 2014.

[3] M. I. Katsnelson, K. S. Novoselov, and a. K. Geim. Chiral tunnelling andthe Klein paradox in graphene. Nature Physics, 2(9):620–625, August 2006.

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