1 Introduction

Effect of Magnetic Field on Goos-Hänchen Shifts

in Gaped Graphene Triangular Barrier

Miloud Mekkaoui, Ahmed Jellal***a.jellal@ucd.ac.ma and Hocine Bahlouli

Laboratory of Theoretical Physics, Faculty of Sciences, Chouaïb Doukkali University,

PO Box 20, 24000 El Jadida, Morocco

Saudi Center for Theoretical Physics, Dhahran, Saudi Arabia

Physics Department, King Fahd University of Petroleum Minerals,

Dhahran 31261, Saudi Arabia

We study the effect of a magnetic field on Goos-Hänchen shifts in gaped graphene subjected to a double triangular barrier. Solving the wave equation separately in each region composing our system and using the required boundary conditions, we then compute explicitly the transmission probability for scattered fermions. These wavefunctions are then used to derive with the Goos-Hänchen shifts in terms of different physical parameters such as energy, electrostatic potential strength and magnetic field. Our numerical results show that the Goos-Hänchen shifts are affected by the presence of the magnetic field and depend on the geometrical structure of the triangular barrier.

PACS numbers: 72.80.Vp, 73.21.-b, 71.10.Pm, 03.65.Pm

Keywords: graphene, triangular double barrier, scattering, Goos-Hänchen shifts.

## 1 Introduction

Quantum and classical analogies between phenomena occurring in two different physical systems can be at the origin of discovering new effects that are relevant in device applications, and often help to understand both systems better [1]. It is well known that a light beam totally reflected from an interface between two dielectric media undergoes lateral shift from the position predicted by geometrical optics [2]. The close relationship between optics and electronic results from the fact that the electrons behave as de Broglie wave due to the ballistic transport properties of a highly mobility two-dimensional electron gas created in semiconductor heterostructures [3]. The recent discovery of graphene [4, 5] added a new twist to this well established optical analogy of ballistic electron transport.

Graphene remains among the most fascinating and attractive subject in condensed matter physics. This is because of its exotic physical properties and the apparent similarity of its mathematical model to the one describing relativistic fermions in two-dimensions. As a consequence of this relativistic-like behavior particle could tunnel through very high barriers in contrast to the conventional tunneling of non-relativistic particles, an effect known in relativistic field theory as Klein Tunneling. This tunneling effect has already been observed experimentally [6] in graphene systems. There are various ways for creating barrier structures in graphene [7, 8], for instance it can be done by applying a gate voltage, cutting the graphene sheet into finite width to create nanoribbons, using doping or through the creation of a magnetic barrier. In the case of graphene, computation of the transmission coefficient and the tunneling conductance were already reported for electrostatic barriers [8, 9, 10, 11], magnetic barriers [10, 12, 13], potential barrier [14] and triangular barrier [16].

During the past few years there was a progress in studying the optical properties in graphene systems such as the quantum version of the Goos-Hänchen (GH) effect originating from the reflection of particles from interfaces. The GH effect was discovered by Hermann Fritz Gustav Goos and Hilda Hänchen [2] and theoretically explained by Artman [17] in the late of 1940s. Many works in various graphene-based nanostructures, including single [18], double barrier [19], and superlattices [20], showed that the GH shifts can be enhanced by the transmission resonances and controlled by varying the electrostatic potential and induced gap [18]. Similar to observations of GH shifts in semiconductors, the GH shifts in graphene can also be modulated by electric and magnetic barriers [21], an analogous GH like shift can also be observed in atomic optics [22]. It has been reported that the GH shift plays an important role in the group velocity of quasiparticles along interfaces of graphene p-n junctions [23, 24]. Very recently, we have studied the GH shift exhibited by Dirac fermions in graphene scattered by triangular double barrier [25].

We extend our former work [25] to include the effect of an external magnetic field on the Goos-Hänchen shifts in a gaped graphene triangular barrier. By separating our system into five regions, we determine the solutions of the energy spectrum in terms of different physical quantities in each region. After matching the wavefunctions at different interfaces of potential width, we determine the transmission probability and subsequently the GH shifts. To acquire a better understanding of our results, we plot the GH shifts for different values of the physical parameters characterizing our system. We also show that the GH shifts can be influenced by the applied magnetic field and can be positive or negative according to the values taken by the physical parameters. In addition, interesting discussions and comments will be reported in different occasions.

The present paper is organized as follows. In section 2, we formulate our model by setting the Hamiltonian system describing particles scattered by a triangular double barrier whose intermediate zone is subject to a mass term. We also obtain the spinor solution corresponding to each regions composing our system in terms of different physical parameters. Using the transfer matrix resulting from the boundary conditions, we determine the corresponding transmission probability in section 3. In section 4, we derive the analytical form of the GH shifts and in section 5 we present the main numerical results for the GH shifts and transmission probability of the particle beam transmitted through graphene triangular double barrier. Finally, in section 6 we conclude our work and summarize our main findings.

## 2 Model of the system

We consider massless Dirac particles with energy and incident angle with respect to the incident -direction of a gaped graphene triangular double barrier. This system is a flat sheet of graphene subject to a triangular potential barrier along the -direction while particles are free in the -direction. To ease our task let us first describe the geometry of our system which is made of five regions denoted by . Each region is characterized by its potential and interaction with external sources. The barrier regions are formally described by the Dirac-like Hamiltonian

 H=vFσ⋅(p+ecA(x,y))+V(x)I2+Δσz (1)

where is the Fermi velocity, are the Pauli matrices, , the unit matrix. The vector potential will be chosen in the Landau gauge with the magnetic field . The parameter is the energy gap owing to the sublattice symmetry breaking or can be seen as originating from spin-orbit interaction , and is confined to the region

 Δ=t′Θ(d21−x2) (2)

where is the Heaviside step function. The double triangular barrier is described by the following potential configuration

 V(x)=Vj=⎧⎪⎨⎪⎩(γx+d2)F,d1≤|x|≤d2V2,|x|≤d10,otherwise (3)

where for , for and , it is presented schematically in Figure 1.

We introduce a uniform perpendicular magnetic field, along the -direction, constrained to the well region between the two barriers such that

 B(x)=BΘ(d21−x2) (4)

where is the strength of the magnetic field within the strip located in the region and otherwise. Choosing the Landau gauge and imposing continuity of the vector potential at the boundaries to avoid nonphysical effects, we end up with the following vector potential

 Ay(x)=Aj=cel2B×⎧⎪⎨⎪⎩−d1,x<−d2x,∣x∣

where the magnetic length is in the unit system . Recall that our system contains five regions denoted . The left region () describes the incident electron beam with energy and incident angle where is the Fermi velocity. The right region () describes the transmitted electron beam with angle .

The time-independent Dirac equation for the spinor at energy reads as

 [σ⋅(p+ecA(x,y))+vjI2+μΘ(d21−x2)σz]ψj(x,y)=ϵψj(x,y) (6)

where we have defined and and . To proceed further, we need to find the solutions of the corresponding Dirac equation according to each region . Indeed, for (region 1):

 ϵ=⎡⎣p2x1+(py−1l2Bd1)2⎤⎦12 (7) ψ1(x,y)=1√2(1z1)e\emph{i}(px1x+pyy)+r1√2(1−z∗1)e\emph{i}(−px1x+pyy) (8) z1=s1px1+i(py−1l2Bd1)√p2x1+(py−1l2Bd1)2 (9)

For (region 5):

 ϵ=⎡⎣p2x5+(py+1l2Bd1)2⎤⎦12 (10) ψ5(x,y)=1√2t(1z5)e\emph{i}(px5x+pyy),z5=s5px5+i(py+1l2Bd1)√p2x1+(py+1l2Bd1)2 (11)

As far as (region 3) is concerned, we first write down the corresponding Hamiltonian in terms of annihilation and creation operators. This can be obtained from (1)

 H=vF⎛⎜ ⎜⎝m+−i√2lBa−i√2lBa+m−⎞⎟ ⎟⎠ (12)

where we have introduced the shell operators

 a±=lB√2(∓∂x+ky+xl2B) (13)

and the parameters . We show that the involved operators obey the canonical commutation relation . Note that, the energy gap behaves like a mass term in Dirac equation, which this will affect the above results and leads to interesting consequences on the transport properties of our system. We determine the eigenvalues and eigenspinors of the Hamiltonian by considering the time independent equation for the spinor using the fact that the transverse momentum is conserved to write with . Then the eigenvalue equation

 H(φ+3φ−3)=ϵ(φ+3φ−3) (14)

gives the coupled equations

 m+φ+3−i√2lBa−φ−3=ϵφ+3 (15) i√2lBa+φ+3+m−φ−3=ϵφ−3. (16)

Injecting (16) into (15) we end up with a differential equation of second order for

 (ϵ−m+)(ϵ−m−)φ+3=2l2Ba−a+φ+3. (17)

This is in fact an equation of the harmonic oscillator and therefore we identify with its eigenstates corresponding to the eigenvalues

 ϵ−v2=s3kη=s31lB√(μlB)2+2n (18)

where we have set , correspond to positive and negative energy solutions. The second spinor component can be derived from (16) to obtain

 φ−3=s3i√kηlB−s3μlBkηlB+s3μlB∣n⟩. (19)

Introducing the parabolic cylinder functions

 Dn(x)=2−n2e−x24Hn(x√2) (20)

to express the solution in region 3 as

 ψ3(x,y)=b1ψ+3(x,y)+b2ψ−3(x,y) (21)

with the two components

 ψ±3(x,y)=1√2⎛⎜ ⎜ ⎜ ⎜⎝√kηlB+s3μlBkηlBD((kηlB)2−(μlB)2)/2−1(±√2(xlB+kylB))±is3√2√kηlB(kηlB+s3μlB)D((kηlB)2−(μlB)2)/2(±√2(xlB+kylB))⎞⎟ ⎟ ⎟ ⎟⎠eikyy (22)

In (regions 2 () and 4 ()): the general solution can be written in terms of the parabolic cylinder function [26, 27, 16] as

 χ+γ=c1Dνγ−1(Qγ)+c2D−νγ(−Q∗γ) (23)

where we have set the parameters , and made the change of variable , and are two constants. The second component is given by

 χ−γ = −c21ky−γd1l2B[2(ϵ0+γϱx)D−νγ(−Q∗γ)+√2ϱeiπ/4D−νγ+1(−Q∗γ)] (24) −c1ky−γd1l2B√2ϱe−iπ/4Dνγ−1(Qγ)

The components of the spinor solution of the Dirac equation (6) in regions m=2 () and m=4 () can be obtained from (23) and (24) by setting

 φ+γ(x)=χ+γ+iχ−γ,φ−γ(x)=χ+γ−iχ−γ (25)

and therefore obtain the eigenspinor

 ψm(x,y) = am−1(u+γ(x)u−γ(x))eikyy+am(v+γ(x)v−γ(x))eikyy (26)

where the functions and are given by

 u±γ(x) = Dνγ−1(Qγ)∓1ky−γd1l2B√2ϱeiπ/4Dνγ(Qγ) (27) v±γ(x) = ±1ky−γd1l2B√2ϱe−iπ/4D−νγ+1(−Q∗γ) (28) ±1ky−γd1l2B(−2iϵ0±(ky−γd1l2B)−γ2iϱx)D−νγ(−Q∗γ).

with and are four constants. The above established results will be used in the next section to derive the transmission and refection amplitudes.

## 3 Transmission amplitude

The coefficients can be determined using the boundary conditions, continuity of the eigenspinors at each interface. Based on different considerations, we study the interesting properties of our system in terms of the corresponding transmission probability. Before doing so, let us simplify our writing using the following shorthand notation

 ϑ±τ1=D[(kηlB)2−(μlB)2]/2−1[±√2(τd1lB+kylB)] (29) ζ±τ1=D[(kηlB)2−(μlB)2]/2[±√2(τd1lB+kylB)] (30) f±1=√kη±μkη,f±2=√2/l2B√kη(kη±μ) (31) u±γ(τd1)=u±γ,τ1,u±γ(τd2)=u±γ,τ2 (32) v±γ(τd1)=v±γ,τ1,v±γ(τd2)=v±γ,τ2 (33)

where . Now, requiring the continuity of the spinor wavefunctions at each junction interface gives rise to a set of equations. We prefer to express these relationships in terms of transfer matrices between different regions, . Then the full transfer matrix over the whole triangular double barrier can be written as

 (34)

which is the product of four transfer matrices that couple the wave function in the -th region to the wave function in the -th region

 M=M12⋅M23⋅M34⋅M45 (35)

and are given by

 M=(~m11~m12~m21~m22) (36) M12=(e−\emph{i}px1d2e\emph{i}px1d2z1e−\emph{i}px1d2−z∗1e\emph{i}px1d2)−1(u+1,−2v+1,−2u−1,−2v−1,−2) (37) M23=(u+1,−1v+1,−1u−1,−1v−1,−1)−1(ϑ+1ϑ−1ζ+1ζ−1) (38) M34=(ϑ+−1ϑ−−1ζ+−1ζ−−1)−1(u+−1,1v+−1,1u−−1,1v−−1,1) (39) M45=(u+−1,2v+−1,2u−−1,2v−−1,2)−1(e\emph{i}px5d2e−\emph{i}px5d2z5e\emph{i}px5d2−z∗5e−\emph{i}px5d2). (40)

Using these we obtain the transmission and reflection amplitudes

 t=1~m11,r=~m21~m11 (41)

and explicitly, takes the form

 t=eid2(px1+px5)(1+z25)(ϑ−1ζ+1+ϑ+1ζ−1)f+2(f−1L1+if−2L2)+f+1(f−2L3+if−1L4)D (42)

where we have set

 D = (u−−1,1v+−1,1−u+−1,1v−−1,1)(u+1,−2v−1,−2−u−1,−2v+1,−2) (43) L1 = ϑ−−1ζ+1FG−ϑ−1ζ+−1KJ (44) L2 = (ζ+1ζ−−1−ζ−1ζ+−1)FJ (45) L3 = ϑ+−1ζ−1FG−ϑ+1ζ−−1KJ (46) L4 = =(ϑ+1ϑ−−1−ϑ−1ϑ+−1)KG (47)

with the quantities

 F = [u+1,−1v−1,−2−u−1,−2v+1,−1−z1(u+1,−1v+1,−2−u+1,−2v+1,−1)] (48) G = [u−−1,1v+−1,2−u+−1,2v−−1,1+z5(u−−1,1v−−1,2−u−−1,2v−−1,1)] (49) K = [u−1,−1v−1,−2−u−1,−2v−1,−1−z1(u−1,−1v+1,−2−u+1,−2v−1,−1)] (50) J = [u+−1,1v+−1,2−u+−1,2v+−1,1+z5(u+−1,1v−−1,2−u−−1,2v+−1,1)] (51)

We can also write (41) in complex notation as

 t=ρteiφt,r=ρreiφr (52)

where the phase of the transmitted and reflected amplitudes are given by

 φt=arctan(it∗−tt∗+t),φr=arctan(ir∗−rr∗+r). (53)

Actually what we exactly need are the transmission and reflection probabilities, which can be obtained by introducing the electric current density corresponding to our system. Then from the previous Hamiltonian, we derive the incident, reflected and transmitted current

 Jinc=eυF(ψ+1)†σxψ+1 (54) Jref=eυF(ψ−1)†σxψ−1 (55) Jtra=eυFψ†5σxψ5. (56)

These can be used to write the transmission and reflection probabilities as

 T=px5px1(Im[t]2+Re[t]2),R=Im[r]2+Re[r]2. (57)

These will be numerically computed by choosing different values of the parameters characterizing the present system. In fact, we will present different plots to underline and understand the basic properties of our system.

## 4 The Goos-Hänchen shifts

The Goos-Hänchen shifts in graphene can be analyzed by considering an incident, reflected and transmitted beams around some transverse wave vector and the angle of incidence , denoted by the subscript . These can be expressed in integral forms as

 Ψi(x,y) = ∫+∞−∞dky f(ky−ky0) ei(kx1(ky)x+kyy)(1eiϕ1(ky)) (58) Ψr(x,y) = ∫+∞−∞dky r(ky) f(ky−ky0) ei(−kx1(ky)x+kyy)(1−e−iϕ1(ky)) (59)

and the reflection coefficient is . This fact is represented by writing the -component of wave vector, as well as in terms of , where each spinor plane wave is a solution of (6) and is the angular spectral distribution. We can approximate the -dependent terms by a Taylor expansion around , retaining only the first order term to get

 ϕ1(ky)≈ϕ1(ky0)+∂ϕ1∂ky|ky0(ky−ky0) (60) kx1(ky)≈kx1(ky0)+∂kx1∂ky|ky0(ky−ky0). (61)

Finally, the transmitted beams are

 Ψt(x,y) = ∫+∞−∞dky t(ky) f(ky−ky0) ei(kx1(ky)x+kyy)(1eiϕ1(ky)) (62)

and the transmission coefficient is .

The stationary-phase approximation indicates that the GH shifts are equal to the negative gradient of transmission phase with respect to . To calculate the GH shifts of the transmitted beam through our system, according to the stationary phase method [28], we adopt the definition [18, 29, 30]

 St=−∂φt∂ky0,Sr=−∂φr∂ky0. (63)

Assuming a finite-width beam with the Gaussian shape, , around , where , and is the half beam width at waist, we can evaluate the Gaussian integral to obtain the spatial profile of the incident beam, by expanding and to first order around when satisfying the condition

 δϕ1=λF/(πw)≪1 (64)

where is Fermi wavelength. Comparison of the incident and transmitted beams suggests that the displacements of up and down spinor components are both equal to and the average displacement is

 St=12(σ++σ−)=−∂φt∂ky0. (65)

It should be noted that when the above-mentioned condition is satisfied, that is, the stationary phase method is valid [18], the definition (63) can be applicable to any finite-width beam, not necessarily a Gaussian-shaped beam.

## 5 Discussion of numerical simulations

The numerical results for the GH shifts of Dirac electrons in graphene scattered by a triangular double barrier potential under a uniform vertical magnetic field are now presented. In fact, we numerically evaluate the GH shifts in transmission and in reflection regions, respectively, as a function of structural parameters of our system, including the energy , the -direction wave vector , the energy gap , the barriers widths and , the strength of potential barriers and . First we present in Figures 2 and 3 the GH shifts in the transmission region a) and the corresponding transmission probabilities b) as function of the incident energy . We have chosen the parameters (, ) in Figure 2 and (, ) in Figure 3, with and three different values of the barrier width , , corresponding to red, green and blue colors, respectively.

It is shown that the GH shifts are closely related to the transmission probabilities. For the sake of simplicity, we will find the explicit expressions at zero-gap . One can notice that, at the Dirac points , the GH shifts change their sign and behave differently. This change in sign of the GH shifts show clearly that they are strongly dependent on the barrier heights. We also observe that the GH shifts are negative and positive in Figures 2 and 3. Recall that, the Dirac points represent the zero modes for Dirac operator [21] and lead to the emergence of new Dirac points, which have been discussed in different works [31, 32]. Such points separate the two regions of positive and negative refraction. In the cases of and , the shifts are, respectively, in the forward and backward directions, due to the fact that the signs of group velocity are opposite. It is clearly seen that are oscillating between negative and positive values around the critical point , in the interval when the usual high energy barrier oscillations appear either in Figure 2 or Figure 3.

We further explore the effects of the triangular double barrier widths and on the GH shifts in Figure 4 for both cases and . Figure 4a) shows an interesting behavior of the GH shifts in terms of the barrier width where oscillations with different amplitudes appeared for the configuration . However such behavior completely changes when we reverse the choice of parameters, i.e. then crosses from positive to negative behaviors. As far as the second barrier width is concerned, from Figure 4b) we observe different oscillations by considering some values of the couple .

At this level, we turn to the discuss the influence of the induced gap in our system in the presence of a triangular double barrier and a magnetic field. Note that, the gap is introduced as shown in Figure 1 and therefore it affects the system energy according to the solution of the energy spectrum obtained in region .

Figure 5 shows that the GH shifts in the propagating case can be enhanced by a gap opening at the Dirac point. This has been performed by fixing the parameters , , , and making different choices for the energy and potential . Figure 5a) presents the GH shift in transmission and transmission probability as a function of energy gap . For the configuration it is clear that one can still have positive shifts (blue line) and for the configuration the GH shifts are negative (red line). Note that for certain energy gap , there is no transmission possible and therefore the GH shifts in transmission vanish. As shown in Figure 5b), we plot the GH shifts in reflection and the reflection probability as a function of energy gap and found that the GH shifts display sharp peaks inside the transmission gap. It is clearly seen that the GH shifts can be enhanced by a certain gap opening. Indeed, by increasing the gap we observe that the gap of transmission becomes broader, changing the transmission resonances and the modulation of the GH shifts. Note that for certain energy gap , there is total reflection and therefore the GH shifts in reflection does not vanish. In fact, under the condition every incoming wave is reflected.