Edge modes and Fabry-Perot Plasmonic Resonances in anomalous-Hall Thin Films
EEdge modes and Fabry-Perot Plasmonic Resonances in anomalous-Hall Thin Films
Thomas Benjamin Smith, Iacopo Torre, and Alessandro Principi School of Physics and Astronomy, University of Manchester, Manchester, M13 9PY, United Kingdom ICFO-Institut de Ci`encies Fot`oniques, The Barcelona Institute of Science and Technology,Av. Carl Friedrich Gauss 3, 08860 Castelldefels (Barcelona), Spain
We study plasmon propagation on a metallic two-dimensional surface partially coated with a thinfilm of anomalous-Hall material. The resulting three regions, separated by two sharp interfaces, arecharacterised by different Hall conductivities but identical normal conductivities. A single boundmode is found, which can localise to either interface and has an asymmetric potential profile acrossthe region. For propagating modes, we calculate the reflection and transmission coefficients throughthe magnetic region. We find Airy transmission patterns with sharp maxima and minima as afunction of the plasmon incidence angle. The system therefore behaves as a high-quality filter.
Keywords: Plasmonics, topological insulator, zero group velocity, Fabry-P´erot resonator
I. INTRODUCTION
Much theoretical and experimental work of re-cent years has highlighted the potential of graphene’stwo-dimensional (2D) surface plasmon polaritons in next-generation transistors, emitters and detectors. Their ex-traordinary properties include, but are not limited to,small confinement scales at high-field, long lifetimesand low losses, and gate-tunability of the propaga-tion wavelength.
Plasmons are high-frequency, electronic density wavesthat occur at frequencies at which the metal dielectricfunction vanishes. The long lifetimes of plasmons atsmall momenta stem from their inability, without the aidof impurities and phonons, to excite single-electron-hole pairs. On the other hand, their small confine-ment scales are due to their weak self-interaction, whichsuppresses any incoherence-causing diffraction. Surfaceplasmons are exponentially localised to interfaces be-tween a metal and a dielectric (or the vacuum).Provided that the surface in question is capable of host-ing metallic conducting electronic states, plasmonic os-cillations may be supported. Such systems include the2D surface of a general 3D topological insulator.
Insuch cases, the low-energy electronic states possess lin-ear dispersions and behave as massless Dirac fermions.When time reversal symmetry is broken by, e.g. , a lo-cal magnetisation or a magnetic field then gapsopen in the surface band structure. As a result, when theFermi energy is tuned to reside in such a magnetisationgap, electrons are characterised by a finite, frequency in-dependent Hall conductance, in units of e /h , where e and h are the electronic charge and Planck’s constantrespectively. The Hall conductivity decreases, and even-tually vanishes, when the Fermi energy is pushed faraway from the middle of the magnetisation gap, in ei-ther the conduction or valence band. Regardless, all ofthese situations have the effect of causing the emergenceof a frequency independent Hall conductance of the or-der of e /h . A similar phenomenon occurs in spin-orbitcoupled metallic thin films in the presence of a finite mag-netisation or magnetic disorder. ˆ z ˆ y ˆ x dσ H ,A σ H ,B σ H ,A x L = 0 x R = d FIG. 1. (Colour on-line) A diagram of the system under con-sideration. The central region B of width d is the thin 2Dmagnetic film. It is bounded on both sides by sharp interfacesand is characterised by a Hall conductance of σ H ,B , whilst inthe other two regions the Hall conductance is σ H ,A . The ˆ y direction is assumed to extend uniformly to ±∞ and the ˆ x direction to −∞ for x < ∞ for x > d with the thin-film located at 0 ≤ x ≤ d . (Note that, due to the mirrorsymmetry of the problem in x , this configuration is identicalto any other so long as x R − x L = d .) Finally, the dielectricenvironment is assumed uniform and equivalent to air suchthat (cid:15) = (cid:15) = 1. In this paper, we investigate a “2D thin-film geom-etry”, whereby a narrow region of a 2D metallic sur-face exhibits a finite Hall conductivity (due to, e.g. alocal nonvanishing magnetisation)–see Fig. 1. The in-terfaces separating the three regions are assumed to besharp relative to the plasmon wavelength so that bound-ary effects may be ignored. Furthermore, the conduc-tivity is assumed to be local so that it does not dependon the plasmon wavevector. Finally, it is assumed to beisotropic so that the conductivity tensor may be decom-posed into a normal diagonal part and an antisymmetric,off-diagonal Hall part. By allowing for these approxima-tions, we are implicitly assuming that the inverse of theplasmon wavevector is much larger than both the mag-netic thin-film size and the domain-wall length. (In pass-ing, we note that the impact of sharp variations of the(valley-)Hall conductivity, at domain walls between ABand BA regions, on the plasmons of bilayer graphene hasbeen studied in Ref. 41.)The frequency-independent off-diagonal Hall conduc-tivity therefore varies step-wise between the three re-gions. Conversely, we assume that the normal frequency- a r X i v : . [ c ond - m a t . m e s - h a ll ] M a r dependent conductivity is spatially independent, i.e. itassumes the same value across the magnetised thin film.This approximation is justified by the fact that the nor-mal conductivity is much less sensitive to variations ofthe magnetisation whereas the Hall conductivity, underthe same conditions, jumps from zero to a finite value.Although simplified, this model captures the fundamen-tal physics of the problem, and lends well to experimentaltesting. There, the typical plasmon wavelength is of theorder of 2 π/q p ∼
100 nm. The paper is organised into four parts. Firstly, a semi-classical model for plasmon propagation is derived.Secondly, the dispersions, lifetimes and potential pro-files of bound interface states, which exist within thethin-film region and are exponentially localised at theinterfaces, are found. Plots of these quantities with vary-ing interface separation and wavevector parallel to theinterfaces are then shown and discussed.Thirdly, de-localised states propagating through thethin film are investigated. Their frequency is given bythe classical 2D plasmon frequency ω p ( q p ) and are as-sumed to be undamped. The reflection and transmissioncoefficients that characterise the region are found andare used to plot transmittance spectra for varying regionthickness and Hall conductance of the thin-film.Finally, typical experimental conditions along with anypotential applications and possible extensions are dis-cussed. II. THE SEMI-CLASSICAL MODEL
To highlight the fundamental physics at play, and with-out the pretense of describing a particular experimentalrealisation of the setup, we consider the simplest possi-ble model: a conducting 2D surface of helical masslessDirac fermions. In addition, the dielectric environmentis assumed to be air. In the magnetised strip, 0 < x < d ,Dirac fermions acquire a finite mass and the Hall conduc-tivity becomes non-zero. The system is assumed isotropicin the ˆ y -direction. Such a description applies, e.g., toelectrons at the surface of a thick 3D topological insula-tor (such that its surfaces are electrostatically decou-pled), once the correct dielectric environment is takeninto account. It can also qualitatively describe plasmonsin spin-orbit coupled metallic thin films. Surface-state electrons are described with a continuummassless-Dirac-fermion model. We assume the systemto be n -doped with a surface carrier density n , whichdefines a Fermi wavevector and energy k F = (cid:112) πn/N f and ε F = (cid:126) v F k F , respectively. Here N f is the number offermion flavors and v F is the density-independent Fermivelocity.We employ a continuum semi-classical description,whereby the electronic flow is modelled by collectiveproperties, i.e the deviation of the charge density from itsequilibrium value ρ ( r , t ) and the charge current j ( r , t ). The two are connected by the continuity equation: − i ˜ ωρ ( r , ˜ ω ) + ∇ · j ( r , ˜ ω ) = 0 , (1)whereas the response of the charge current to the (self-induced) electric field obeys the linear-response Ohm’slaw: j i ( r , ˜ ω ) = σ ij ( r , ˜ ω ) E j ( r , z = 0 , ˜ ω ) . (2)Here Einstein’s summation convention for Roman indices(standing for in-plane Cartesian components) is under-stood, and the local conductivity σ ij ( r , ˜ ω ) is determinedmicroscopically. In Eqs. (1) and (2), ρ ( r , ˜ ω ), j i ( r , ˜ ω ), σ ij ( r , ˜ ω ) and E j ( r , z = 0 , ˜ ω ) are the Fourier componentsin complex frequency space, ˜ ω = ω + i/τ with τ = 1 / Γ,of the charge density, current, conductivity and electricfield, respectively, and r = x ˆ x + y ˆ y .We assume the system to be locally isotropic. There-fore: σ ij ( r , ˜ ω ) ≡ σ (˜ ω ) δ ij + σ H ( x, ˜ ω ) ε ij . (3)where δ ij and ε ij are the Kronecker-delta and 2D an-tisymmetric Levi-Civita symbols, respectively. To sim-plify the further analysis, we assume σ ( x ) to be spatiallyindependent, whereas the Hall component σ H ( x ) variesstepwise across the interfaces: σ H ( x, ˜ ω ) = σ H ,A [ θ ( x − d )+ θ ( − x )]+ σ H ,B θ ( d − x ) θ ( x ) . (4)Hereafter we assume σ H ,A and σ H ,B to be frequency in-dependent and the interfaces to be infinitely sharp. Thisapproximation is valid as long as the typical wavelengthsof the problem are much longer than the length scales ofthe interface. For the sake of definitiveness, we write: σ (˜ ω ) = i D ˜ ω + iγ , (5)where D is the Drude weight and γ = 1 /τ sc is the scat-tering rate of the underlying electronic carriers.The problem as defined by the constitutive rela-tions (1)-(4) is solved together with the self-induced 3DPoisson’s equation: ∇ φ ( r , z, ˜ ω ) = − πρ ( r , ˜ ω ) δ ( z ) . (6)Note that, whilst electrons are bounded to the 2D sur-face, the electric potential extends to the whole 3D space.To determine the plasmons of the heterostructure, we as-sume that no external electric field is applied, and that E ( r , z, t ) = − ∇ φ ( r , z, t ). Since the system is assumed tobe translationally invariant in the ˆ y -direction, all quan-tities may be expanded in Fourier components along ˆ y ,e.g.: ρ ( r , ˜ ω ) = ρ ( x, ˜ ω ) e iq y y . Thus the problem reduces tothat of a 1D well/barrier.Solving the Poisson’s equation given by Eq.(6) and tak-ing the Fourier transform of the solution we achieve: ˆ φ ( q x ) = L ( q )ˆ ρ ( q x ) , (7)where q = [ q x + q y ] / and: L ( q ) = 4 π ( (cid:15) + (cid:15) ) q = 2 πq , (8)since the dielectric environment is assumed to be air sothat (cid:15) = (cid:15) = 1. It must be noted here, however, thatif the system were a Bismuth based topological insulatorthis assumption would not hold: the dielectric constantscan be orders of magnitude greater than one. Fourier-transforming Eq. (1), and using Eqs. (2), (3)and (4), we achieve:ˆ ρ ( q x ) = − iσ (˜ ω )˜ ω q ˆ φ ( q x ) − δσ H q y ˜ ω [ φ (0) − φ ( d ) e − iq x d ] , (9)where: δσ H = σ HB − σ HA = ν e π (cid:126) , (10)with ν as a parameter that characterises the differencebetween the Hall conductivities of each region. Note that ν can take either positive or negative values dependingon whether the Hall conductivity varies as a well ( ν < ν > (cid:15) ( q, ˜ ω ) ˆ φ ( q x ) = − δσ H q y ˜ ω L ( q )[ φ (0) − φ ( d ) e − iq x d ] , (11)where: (cid:15) ( q, ˜ ω ) = 1 + iσ (˜ ω )˜ ω q L ( q ) = 1 − qQ (˜ ω ) , (12)is the dielectric function of the homogeneous surface inthe absence of the magnetised strip. (cid:15) ( q, ˜ ω ) clearly has azero at: q = Q (˜ ω ) ≡ ˜ ω π D (˜ ω + iγ ) = ω π D [ ω + i ( γ + 2Γ)] , (13)where ˜ ω = ω + i Γ has been used and any quadratic decayterms are assumed negligible since ω (cid:29) γ, Γ. We denote,from this point on, the real part of Q ( ω ) with q p ( ω ) = ω / (2 π D ), which is the bulk plasmon wavevector. Boundinterface plasmons exist for frequencies such that q y >q p ( ω ) whilst propagating solutions for q y ≤ q p ( ω ), whichwe hereafter name the “continuum region”. III. BOUNDED INTERFACE STATES
We start by determining the dispersion and field dis-tribution of interface-localised plasmons, which occur forwavevectors q y > q p ( ω ), i.e. to the right and belowthe bulk plasmon continuum. These modes are exponen-tially localised around the interfaces in a region of size ξ − = [ q y − q ( ω )] − / . At these frequencies, (cid:15) ( q, ˜ ω ) (cid:54) = 0. We are thus free to divide Eq. (11) by the dielectric func-tion and inverse Fourier transform it back into coordinatespace. We therefore obtain: φ ( x ) = − δσ H q y ˜ ω [ I ( x, q y , ˜ ω ) φ (0) − I ( x − d, q y , ˜ ω ) φ ( d )] , (14)where: I ( x, q y , ˜ ω ) = P (cid:90) + ∞−∞ dq x π L ( q ) (cid:15) ( q, ˜ ω ) e iq x x . (15)To determine the bound states we impose that thepotential is continuous at the interfaces. By evaluatingEq. (14) at x = 0 and x = d we thus arrive at the follow-ing matrix equation: (cid:18) KI (0 , q y , ˜ ω ) − KI ( d, q y , ˜ ω ) KI ( d, q y , ˜ ω ) 1 − KI (0 , q y , ˜ ω ) (cid:19) (cid:18) φ (0) φ ( d ) (cid:19) = 0 , (16)where: K = δσ H q y ˜ ω . (17)Note that we have used I ( d, q y , ˜ ω ) = I ( − d, q y , ˜ ω ); a rela-tion to be shown subsequently.Non-trivial solutions of the matrix equation (16) arefound whenever its determinant is zero. This yields thefollowing transcendental equation:˜ ω − δσ q y [ I (0 , q y , ˜ ω ) − I ( d, q y , ˜ ω )] = 0 , (18)which, ultimately, may only be solved numerically in or-der to determine the plasmon dispersion relation.Due to the complex-valued nature of both the fre-quency and the conductivity, the integral in Eq. (15) maybe decomposed into real and imaginary parts, based onthe assumption that any term quadratic in the scatteringrates γ and Γ vanishes since ω (cid:29) γ, Γ, as: I ( x, q y , ω, Γ) = I ( x, q y , ω ) − iω ( γ + 2Γ) J ( x, q y , ω ) . (19)Then, using this decomposition with ˜ ω = ω + i Γ withinEq. (18), two simultaneous equations for ω and Γ may befound. The numerical solution of the first yields ω withwhich the second may be solved (also numerically) for Γ,as shall be seen.Furthermore, the I and J integrals may be evalu-ated using contour integration. Due to the the oscilla-tory Fourier exponential in the integrals, the contoursare closed in the upper (lower) half of the complex planewhen x > x < q y < q p ( ω ) there exist poles at q x = ± iq ,where q ( ω ) = q y − q ( ω ) ≡ ξ − , and branch cuts alongthe imaginary axis from ± iq y to ± i ∞ . On the otherhand, when q y > q p ( ω ) the poles exist upon the real axisat q x = ± k , where k ( ω ) = − q ( ω ) = q ( ω ) − q y , whilstthe branch cuts remain unchanged.Once this is all taken into account we find: ω − δσ q y (cid:2) I (0 , q y , ω ) − I ( d, q y , ω ) (cid:3) = 0 , (20)Γ + δσ q y ( γ + 2Γ)[ I (0 , q y , ω ) J (0 , q y , ω ) − I ( d, q y , ω ) J ( d, q y , ω )] = 0 , (21)where: {I / J } ( x, q y , ω ) = {I / J } P ( x, q y , ω ) + {I / J } B ( x, q y , ω ) , (22)with the first terms as the contributions from the poles,which are given by: I P ( x, q y , ω ) = − π q p ( ω ) q ( ω ) e − q ( ω ) | x | , for | q y | > q p ( ω ) , π q p ( ω ) k ( ω ) sin[ k ( ω ) | x | ] , for | q y | ≤ q p ( ω ) , (23) J P ( x, q y , ω ) = 2 π (cid:32) q y q ( ω ) + q ( ω ) | x | q ( ω ) (cid:33) e − q ( ω ) | x | , (24)and the second terms as the contributions from thebranch cuts, which are given by: I B ( x, q y , ω ) = 2 q ( ω ) (cid:90) ∞ dη e −| q y x | cosh( η ) q ( ω ) + q y sinh ( η ) , (25) J B ( x, q y , ω ) = 4 q p ( ω ) q y (cid:90) ∞ dη e −| q y x | cosh( η ) sinh ( η )[ q ( ω ) + q y sinh ( η )] . (26)Note that Eqs. (22)-(25) depend on the absolute valueof x . This is due to the fact that, for x <
0, one closes thecontour in the lower half of the complex plane and findsan identical result. This therefore proves the relation I ( x, q y , ˜ ω ) = I ( − x, q y , ˜ ω ), which was used earlier.From Eq. (22) it can be seen that the contribution ofthe poles is exponentially localised around x with charac-teristic length ξ . On the other hand, the branch cut con-tributions are superpositions of evanescent waves causedby the non-locality of the Coulomb interaction that actsas a transient decay around the interfaces. As such, thebehaviour of the plasmonic field is extremely complicatedin the immediate vicinity of the interface.For q y → ∞ equation (18) simplifies and can be solvedanalytically yielding: ω ∞ = D| δσ H | , (27)whilst in the limit of small q y , the plasmon tends to thecontinuum region and so q y → q p .The normalised eigenvectors of Eq. (16), correspond-ing to the positive frequencies (negative frequencies aresimply plasmons moving backwards in time) as obtainedfrom Eq. (18), are: (cid:18) φ (0) φ ( d ) (cid:19) = N c (cid:18) − KI (0 , q y , ω ) − KI ( d, q y , ω ) (cid:19) , (28) where N c = (cid:2) (1 − KI (0 , q y , ω, Γ)) + K I ( d, q y , ω, Γ) (cid:3) − .Using Eq. (28) in Eq. (14) allows for solution of thespatially dependent potential relation as: φ ( x ) = [ KI (0) − I ( x ) − KI ( d ) I ( x − d ) (cid:113) [ I (0) − K − ] + I ( d ) , (29)where the q y , ω and Γ dependencies of I ( x, q y , ω, Γ) havebeen dropped for brevity. Note that K = δσ H q y / ( ω + i Γ).Although not immediately obvious, this shows that,depending on the combined sign of δσ H q y , the plasmonwill not only prefer to localise to a specific interface butalso have . It is in fact the K terms in the numeratorthat cause this behaviour since I ( x ) depends on | q y | and q y only.In the figures and results to come, we adimension-alise the variables using the linear Dirac dispersion ofthe electronic system ε F = (cid:126) v F k F where k F = √ πn and mv F = (cid:126) k F . Note that we set the number of fermion fla-vors N F = 1, which is equivalent to measuring k F inunits of √ N F . In addition, α ∗ = e / ( (cid:126) v F ) (in CGSunits) is the fine-structure constant of said Dirac sys-tem. In these units the factor 2 π D may be expressed as2 π D = α ∗ ε / (2 (cid:126) k F ) and so: Q (˜ ω ) = Q ( ω + i Γ) k F = 2 (cid:126) α ∗ ε [ ω + iω ( γ + 2Γ)] , (30)Furthermore, the change in Hall conductivity may beexpressed as: δσ H = να ∗ ε F / (2 π (cid:126) k F ), in these units. Wetake α ∗ ≈ On the other hand, α ∗ ≈ Fig. 2(a) shows the energy dispersion (cid:126) ω in units of ε F ,calculated by numerically solving Eq. (18), as a functionof the momentum q y along the interface (in units of k F ).Note that Eq. (18) has only one undamped solution thatexists outside the continuum. Its key feature, comparedto the single-interface case, is the appearance of a peakwhose position in q y k − and height in (cid:126) ωε − depends onboth d and ν .In Fig. 2(a) we plot two curves, for dk F = 1 and dk F =2, which show that the position in q y and the magnitudeof the peak increase with decreasing interface separation.This is due to the fact that, at long wavelengths, a smallregion, with respect to the plasmon wavelength, will haveno effect on it. In this case, the bound mode tracks thecontinuum region closely and resembles a propagatingstate due to its poor localisation.It must be noted, however, that for | ν | < dk F > q y ≈ q cy . Similar phenomena have been observed in standardmetallic thin films. . . . . . . q y k − . . . . . . ¯ h ω ε − F Single interfaceContinuum regionBound state, dk F = 1 Bound state, dk F = 2 (a) . . . . . . q y k − . . . . . . . τ γ dk F = 1 dk F = 10 (b) − −
10 0 10 20 xk F . . . . . . . R e [ φ ( x k F ) ] q y = +0 . k F q y = +0 . k F q y = − . k F (c)FIG. 2. (Colour on-line) Panel (a): The peak of the energydispersion of bound states decreases with increasing the in-terface separation from dk F = 1 (solid line) to dk F = 2 (long-dashed line). Here | ν | = 2, whereas the short-dashed anddotted lines stand for the single-interface result and the con-tinuum, respectively. Panel (b): Plotted are the plasmon life-times as a function of q y for dk F = 1 (solid line) and dk F = 10(dashed line). The lifetime of the bound state is seen to begreatest at q y = 0, have a turning point in q y that recedes in q y as d increases and to be roughly constant for all q y around τ γ ∼ .
8. Panel (c): The potential of the bound state local-izes at one of the two interfaces, depending on the combinedsign of νq y , with a localization length that is inversely propor-tional to q y . The solid, long dashed, and small dashed linescorrespond to q y = +0 . k F , +0 . k F , − . k F respectively. Such a plasmon wavepacket with negative group ve-locity would then propagate backwards against its ini-tial momentum direction. However, due to its shorterwavelength and thus closer proximity to the electron-holecontinuum, it would likely decay at a quicker rate thanwavepackets of longer wavelengths.Furthermore, if one were to generate a plasmonwavepacket of a single frequency with constituent wavesof momenta (say) q and q (before and after the peak, respectively) then it would exhibit a beating effect dueto the constructive and destructive interference of theseconstituent waves within the wavepacket.In Fig. 2(c) we plot the real part of the potential profile(assuming that quadratic decay terms may be ignored)as a function of the dimensionless x coordinate for aninterface separation dk F = 1. We show curves for fourdifferent values of q y , namely q y = ± . k F and q y = ± . k F . The former (latter) occurring before (after) theturning point of the dispersion curve of Fig. 2(a).Interestingly, the potential may be seen to decay acrossthe region in a non-exponential manner reflecting the na-ture of the plasmon to localise within the region to theinterfaces and to then decay outside.The effect of the interfaces can be seen: the modetransitions from being confined within the whole region0 < x < d at small wavevectors, to being completelylocalised at large q y to only one of the two interfaces,depending on the combined sign of νq y . In the lattercase the mode reproduces the short-wavelength limit ofthe single-interface result, as shown in Fig. 2(a). Thisexplains the origin of the turning point in the energy dis-persion, which develops because of the transition betweenthese two extremes.At all wavelengths the energy of the double-interfacemode exceeds that of the single-interface one. This isdue to the fact that the two interfaces, due to the oppo-site jumps in δσ H , have opposite chiralities. As a con-sequence, if they would be infinitely separated, each ofthem would host low-energy plasmons propagating in onepreferred direction. The latter, being determined by thecombined sign of νq y , is opposite for the two interfaces.Since the plasmon mode is shared by both of them, oneof which has the “wrong” chirality, a higher energy isrequired for it to exist.This “wrong chirality” effect may be seen most appar-ently in the potential plot of Fig. 2(c). When νq y > x = 0. On theother hand, when νq y < ν and q y areboth negative or both positive.In Fig. 2(b) we plot the dimensionless lifetime of theplasmon mode as a function of the wavevector q y . Asmay be seen, the lifetime possesses a turning point thatrecedes in q y as d is increased. However, the lifetimeremains roughly constant for all wavevectors and is oforder τ γ ∼ .
8. This minimum corresponds with thepoint at which the plasmon is shared equally between thetwo interfaces of opposing chiralities. Hence, its lifetimeis negatively affected (albeit minimally) as a result of thisenergetically unfavourable sharing mechanism.
IV. PROPAGATING STATES
Propagating states cannot be found by using the abovemethod. The latter is in fact only applicable for boundstates, whose wavevectors satisfy q y > q p ( ω ), and forwhich the bulk dielectric function (cid:15) ( q, ω ) [Eq. (12)] isnon-zero. In the present case, as we will show momen-tarily, the plasmon wavevector is q = q p ( ω ). The bulkdielectric function therefore vanishes, and thus care mustbe taken in performing the inverse Fourier transform ofEq. (11). Furthermore, since the plasmonic energies con-sidered here are smaller yet than the bound state ener-gies, the approximation that ω (cid:29) Γ will not hold. Thus,a more in depth analysis will be required to include thedecay within this section. As such, we now set γ = Γ = 0such that J ( x, q y , ω ) = 0 and so I ( x, q y , ω ) = I ( x, q y , ω ).When Eq. (11) by (cid:15) ( q, ω ), since the latter is zero at q = q p ( ω ), we have to introduce terms proportional to theDirac delta functions δ ( q ± q p ) on its right-hand side. Thelatter indeed vanish when multiplied by (cid:15) ( q, ω ), returningEq. (11).As a result, after this treatment and an inverse Fouriertransform, Eq. (11) becomes: φ ( x ) = − δσ H q y ω [ I ( x, q y , ω ) φ (0) − I ( x − d, q y , ω ) φ ( d )]+ C + π e ik x + C − π e − ik x . (31) The last two terms in this equation appear as a resultof the added Dirac delta functions. Here ± k are thepositions of the poles on the real axis, with k ( ω ) = q ( ω ) − q y , and I ( x, q y , ω ) is the same integral as de-fined in Eq. (15). The poles of its integrand are now onthe real axis at ± k , rather than at ± iq as previously.In order to determine the scattering coefficients r and t that characterise the magnetic region, we impose thatthere are no left-moving waves in the region x > d , i.e. terms proportional to e − ik x sum to zero. Thus, the po-tentials in x < x > d , far from the interfaces, are: φ ( x ) = (cid:40) e ik x + re − ik x , x → −∞ ,te ik x , x → + ∞ , (32)where the calculation of r and t is laid out in the ap-pendix. The reflectance and transmittance of the regionare then given by R = | r | and T = | t | , respectively,where R and T must satisfy R + T = 1 by construction.We express the coefficients r and t in terms of the angleof incidence at x = 0 using q y = q p sin( θ ) and k = q p cos( θ ). They thus read: r = 4 πi (cid:20) cos( θ ) (cid:104) I B ( d, θ ) − I B (0 , θ ) cos (cid:16) ˜ d cos( θ ) (cid:17)(cid:105) + (cid:16) π − i ˜ K − cot( θ ) (cid:17) sin (cid:16) ˜ d cos( θ ) (cid:17) (cid:21) e i ˜ d cos( θ ) cos ( θ ) (cid:104) [ ˜ K sin( θ )] − + I B ( d, θ ) − I B (0 , θ ) (cid:105) + 4 iπ cos( θ ) (cid:104) I B (0 , θ ) − I B ( d, θ ) e i ˜ d cos( θ ) (cid:105) + 4 π (cid:104) − e i ˜ d cos( θ ) (cid:105) , (33) t = cos ( θ ) (cid:104) [ ˜ K sin( θ )] − + I B ( d, θ ) − I B (0 , θ ) (cid:105) + 4 π cos( θ ) I B ( d, θ ) sin[ ˜ d cos( θ )]cos ( θ ) (cid:104) [ ˜ K sin( θ )] − + I B ( d, θ ) − I B (0 , θ ) (cid:105) + 4 iπ cos( θ ) (cid:104) I B (0 , θ ) − I B ( d, θ ) e i ˜ d cos( θ ) (cid:105) + 4 π (cid:104) − e i ˜ d cos( θ ) (cid:105) , (34)where the ω dependencies are still dropped for brevity.Here we have also introduced ˜ K = δσ H q p ( ω ) /ω , ˜ d = q p ( ω ) d and I B ( x, θ, ω ) ≡ I B ( x, q y = q p ( ω ) sin( θ ) , ω ).Interestingly, the sign of δσ H has no effect on the re-flectance, R = | r | , and transmittance, T = | t | , of theregion. This is as a result of its appearance in r and t aspart of either a squared term or an imaginary term.As a final note, we have presented r ≡ r L and t ≡ t L , i.e. the scattering coefficients from left to right. How-ever, due to the mirror symmetry of the magnetic region,it follows that r R = r and t R = t .In Fig. 3 we plot the transmittance of propagatingmodes as a function of the plasmon angle of incidence θ from the normal to the interface. In panel (a) we showtwo curves for two distinct values of the dimensionlessinterface separation dk F = 40 and dk F = 400 with thesame ν parameter | ν | = 1.The interface separation has a dramatic effect: small-width regions exhibit selective angle-dependent trans-mission, however with rather poor quality factor. Onthe other hand, for large-width regions, many sharp sidetransmission peaks are seen to appear. The central peakremains broad in both cases. Furthermore, we find that if dk F <
40, the peaks disappear. This is because the plas-mon will again not see the region and will instead propa-gate through it unaffected. Note that the spectrum shows − π/ − π/ π/ π/ θ . . . . . . T ( θ , ω ) dk F = 400 dk F = 40 (a) − π/ − π/ π/ π/ θ . . . . . . T ( θ , ω ) | ν | = 2 | ν | = 1 / (b)FIG. 3. (Colour on-line) Panel (a): The number of trans-mittance peaks at fixed plasmon frequency increases with de-creasing dimensionless interface separation dk F . In this plot,the plasmon frequency is (cid:126) ω = ε F / | ν | = 1. Impor-tantly, the peak strengths do not decay heavily as the angleis increased for small ν , this is shown further in the | ν | = 1 / | ν | . Here, (cid:126) ω = ε F / dk F = 200.This strongly improves the quality factors of all transmissionpeaks but suppresses the intensity of any side peaks, i.e. alarge ν causes the film to become a strong mirror except ina narrow region around θ ≈ ν causes thefilm to become transparent as the quality factors of side peaksdecreases. the typical “Airy-disk” characteristic of Fabry-P´erot res-onance, wherein the linewidth is directly related to theregion width. In panel (b) we plot instead the transmittance as afunction of the incident angle but for a fixed dk F = 200and two values of the parameter | ν | : | ν | = 1 / | ν | = 2. In this case it can be seen that the qualityfactor (sharpness) of all peaks is increased. However, thetransmittance of the side peaks is suppressed as a resultof the increasing ν parameter.Thus there is a trade-off. To have transmission peakswith high quality factor, the ν parameter must be largeyet this increase diminishes the strength of said peaks.The same goes by changing the width d . The qualityfactors of the peaks increase as d increases. However, thepeaks become closer to each other, and hence more andmore difficult to resolve. V. SUMMARY AND CONCLUSIONS
To summarise, we developed a semi-classical descrip-tion of plasmonic excitations in the presence of a fre-quency independent step-wise-varying off-diagonal Hallconductivity.We found that a plasmon can propagate confined be-tween the interfaces. For a given energy, said plasmon hasa larger wavevector than the bulk ones, and therefore canbe excited separately. Its energy dispersion shows a turn-ing point at which the plasmon has zero group velocity.The mode is bound to one of the interfaces dependingon the combined sign of the momentum along the in-terface and the “filling factor” ν that parametrizes thedifferences in Hall conductivities between the regions, inunits of e /h . The bound plasmon also shows a typi-cal localisation length which is inversely proportional tothis momentum. By studying the scattering process foran incident plasmon through the region, we calculate thereflection and transmission coefficients. The number ofside transmission peaks depends heavily on the interfaceseparation, whilst their intensity and sharpness decreasewith the parameter ν .As can be seen in figure 2, the interface state localisesvery strongly to the region as a whole with preferencefor either of the interfaces, depending on the sign of thejump is Hall conductance, as the wavevector increases.As such, the thin film geometry could find application asa plasmonic waveguide.The fact that the bound state dispersion curve exhibitsa maximum, at which point the group velocity vanishes,may be exploited to confine interface plasmons withina finite region without the need of a solid barrier. Werecall indeed that the wavevector at which the plasmondispersion peaks, as well as the peak energy, depend onthe geometrical parameters of the structure. In partic-ular, the peak lowers in frequency with increasing inter-face separation. Therefore, one could imagine to shapethe region in such a way that a wavepacket with a fixedfrequency will eventually stop propagating and bounceback when the group velocity vanishes. This is achievedby adiabatically increasing the interface separation awayfrom the point where the wavepacket is created in sucha way that its dispersion evolves adiabatically while itpropagates. If the thin film widens in both directions,then the wavepacket would be confined within the regionand would thus become a confined standing wave. Sucha plasmon may be seen in Fig. 4.For propagating modes, the fact that the side trans-mission peaks may be modulated in number, intensityand quality factor through the variation of d , ν and ω could be used to generate monochromatic plasmons. Byconstructing a resonator with a given width d and ν , aplasmon with a certain frequency may be made to passthrough alone by sending it at a specific angle θ . Thus,incident plasmons of specific frequencies may be selectedfor by detecting them at an angle after the magnetic re-gion. Furthermore, the opacity of the region to plasmons FIG. 4. A contour plot of the potential profile of a boundplasmon within an adiabatically widening thin-film. Here, forconsistency, the parameters are identical to those of Fig. 2: ν = 2, ω = 0 . ε F with q y as the smaller positive solutionto Eq. (18) with this chosen frequency. The width of theregion is varied from dk F = 1 to dk F = 2 over a suitablylarge range of y : 0 ≤ yk F ≤ y -directional propagation whereat the plasmon ceases to existand so ‘reflects’ back. of certain energies depending on d and ν could be usedto confine plasmons of such energies between two regions.However, if a plasmon were to lose energy during its prop-agation, i.e. through any form of decay, then the regionswould appear transparent to the plasmon at certain en-ergies thus hampering the confinement quality.Finally, we wish to comment on the feasibility of ourset-up. Candidates for the realisation of these phenom-ena are metallic Dirac-like 2D surface states (e.g., thoseat the surface of a 3D TI, accounting for the properdielectric environment) as mentioned in the introduc-tion. Such systems exhibit typical surface electronicnumber densities of n ∼ cm − and a Fermi en-ergy of ε F = (cid:126) v F k F . Thus, the Fermi momentum (inunits of √ N F ) for the system is given by k F = √ πn ∼ . × cm − = 0 .
35 nm − . Moreover, their typicalFermi velocity is v F ∼ cms − and so the Fermi en-ergy is ε F ∼ .
23 eV. (Note that: taking N F = 4, as ingraphene, would halve both the Fermi momentum andthe Fermi energy.) Finally, taking a typical experimentalelectron scattering time of τ sc = 50 fs, we may see thatthe lifetime of the bound plasmon is τ ∼ . τ sc and so τ ≈
100 fs = 0 . d (cid:46) | ν | (cid:38)
2. For dk F (cid:29) d < q y ∼ k F . Such a large step-wise change in the Hallconductivity is also unlikely to be able to implementedexperimentally.The use of the studied heterostructure as a plasmonicwaveguide has much better chances. We find that plas-mons can be bound within magnetic strips satisfying d (cid:38) k F ∼ | ν | <
1. In this case, the boundplasmon would localise to either of the interfaces, de-pending on the sign of νq y , rather than inside the region,as explained above. For, say positive ν , a plasmon with q y > x = 0 whilst a plasmon with q y < x = d . Yet, therather small lifetime might render it difficult to utilise.Nevertheless observation ought not to be impossible.Peaked transmission spectra require rather large sepa-rations in the range 40 nm < d (cid:46)
400 nm, well within thestudied semi-classical regime. Heterostructures workingas plasmon filters could therefore be well realised exper-imentally, and their theoretical description does not re-quire the consideration of quantum effects. The upperlimit of 400 nm is not a strong one since a larger regionwould simply see an increase in the number of peaks inthe transmission spectrum. Admittedly, when the num-ber of peaks becomes too large they blur together andcease to be resolvable, thus making the region practi-cally transparent for all angles. Conversely, the lowerlimit of 40 nm is a stringent one: below that, peaks donot occur. Frequency selection of plasmons could wellbe seen within experimental conditions. In fact, typi-cal plasmonic energies of metallic surfaces are of order (cid:126) ω ∼ ε F / ∼ . and therefore observable undertypical experimental conditions. VI. ACKNOWLEDGEMENTS
We would like to thank the referees for their pertinentcriticisms and valuable suggestions without which thiswork would be of a considerably lower standard.T.B.S. acknowledges the support of the EPSRCthrough a PhD studentship grant. A.P. and T.B.S. ac-knowledge support from the Royal Society InternationalExchange grant IES \ R3 \ REFERENCES S. Huang, C. Song, G. Zhang, and H. Yan, Nanophotonics , 1191 (2017). B. Wunsch, T. Stauber, F. Sols, and F. Guinea, New Jour-nal of Physics , 318 (2006). M. Polini, R. Asgari, G. Borghi, Y. Barlas, T. Pereg-Barnea, and A. H. MacDonald, Phys. Rev. B ,081411(R) (2008). A. N. Grigorenko, M. Polini, and K. S. Novoselov, NaturePhotonics , 749 (2012). H. Yan, T. Low, W. Zhu, Y. Wu, M. Freitag, X. Li,F. Guinea, P. Avouris, and F. Xia, Nature Photonics ,394 (2013). A. Principi, G. Vignale, M. Carrega, and M. Polini, Phys.Rev. B , 195405 (2013). A. Principi, G. Vignale, M. Carrega, and M. Polini, Phys.Rev. B , 121405(R) (2013). A. Principi, M. Carrega, M. B. Lundeberg, A. Woessner,F. H. L. Koppens, G. Vignale, and M. Polini, Phys. Rev.B , 165408(R) (2014). Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S.McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens,G. Dominguez, M. M. Fogler, A. H. C. Neto, C. N. Lau,F. Keilmann, and D. N. Basov, Nature , 82 (2012). J. Chen, M. Badioli, P. Alonso-Gonz´alez, S. Thongrat-tanasiri, F. Huth, J. Osmond, M. Spasenovi´c, A. Centeno,A. Pesquera, P. Godignon, A. Zurutuza Elorza, N. Ca-mara, F. J. G. de Abajo, R. Hillenbrand, and F. H. L.Koppens, Nature , 77 (2012). A. Woessner, M. B. Lundeberg, Y. Gao, A. Principi,Alonso-Gonz´alez, M. Carrega, K. Watanabe, T. Taniguchi,G. Vignale, M. Polini, J. Hone, R. Hillenbrand, andF. H. L. Koppens, Nature Materials , 421 (2014). P. Alonso-Gonz´alez, A. Y. Nikitin, Y. Gao, A. Woess-ner, M. B. Lundeberg, A. Principi, N. Forcellini, W. Yan,S. V´elez, A. J. Huber, K. Watanabe, T. Taniguchi,F. Casanova, L. E. Hueso, M. Polini, J. Hone, F. H. L.Koppens, and R. Hillenbrand, Nature Nanotechnology ,31 (2016). G. X. Ni, L. Wang, M. D. Goldflam, M. Wagner, Z. Fei,A. S. McLeod, M. K. Liu, F. Keilmann, B. ¨Ozyilmaz, A. H.Castro Neto, J. Hone, M. M. Fogler, and D. N. Basov,Nature Photonics , 244 (2016). M. B. Lundeberg, Y. Gao, A. Woessner, C. Tan, P. Alonso-Gonz´alez, K. Watanabe, T. Taniguchi, J. Hone, R. Hillen-brand, and F. H. L. Koppens, Nature Materials , 204(2016). F. J. Bezares, A. D. Sanctis, J. R. M. Saavedra, A. Woess-ner, P. Alonso-Gonz´alez, I. Amenabar, J. Chen, T. H.Bointon, S. Dai, M. M. Fogler, D. N. Basov, R. Hillen-brand, M. F. Craciun, F. J. Garc´ıa de Abajo, S. Russo,and F. H. L. Koppens, Nano Letters , 5908 (2017). T. Low, P.-Y. Chen, and D. N. Basov, Phys. Rev. B ,041403(R) (2018). E. J. C. Dias, D. A. Iranzo, P. A. D. Gon¸calves, Y. Hajati,Y. V. Bludov, A.-P. Jauho, N. A. Mortensen, F. H. L.Koppens, and N. M. R. Peres, Phys. Rev. B , 245405(2018). G. X. Ni, H. Wang, J. S. Wu, Z. Fei, M. D. Goldflam,F. Keilmann, B. ¨Ozyilmaz, A. H. Castro Neto, X. M. Xie,M. M. Fogler, and D. N. Basov, Nature Materials , 1217(2015). G. X. Ni, A. S. McLeod, Z. Sun, L. Wang, L. Xiong, K. W.Post, S. S. Sunku, B. Y. Jiang, J. Hone, C. R. Dean, M. M.Fogler, and D. N. Basov, Nature , 530 (2018). D. Alcaraz Iranzo, S. Nanot, E. J. C. Dias, I. Epstein,C. Peng, D. K. Efetov, M. B. Lundeberg, R. Parret, J. Os-mond, J.-Y. Hong, J. Kong, D. R. Englund, N. M. R. Peres,and F. H. L. Koppens, Science , 291 (2018). D. N. Basov, M. M. Fogler, and F. J. Garc´ıa de Abajo,Science , aag1992 (2016). F. H. Koppens, D. E. Chang, and F. J. G. de Abajo, NanoLetters , 3370 (2011). G. Giuliani and G. Vignale,
Quantum Theory of the Elec-tron Liquid , 1st ed. (Cambridge University Press, 2005). R. F. Oulton, G. Bartal, D. F. P. Pile, and X. Zhang, NewJournal of Physics , 105018 (2008). X.-L. Qi and S.-C. Zhang, Review of Modern Physics ,1057 (2010). M. Hasan and C. Kane, Review of Modern Physics ,3045 (2010). M. Salehi, M. Brahlek, N. Koirala, J. Moon, L. Wu, N. P.Armitage, and S. Oh, APL Materials , 91101 (2015). R. Yu, W. Zhang, H.-J. Zhang, S.-C. Zhang, X. Dai, andZ. Fang, Science , 61 (2010). J. Wang, B. Lian, and S.-C. Zhang, Physica Scripta ,014003 (2015). S.-J. Zhang, W.-X. Ji, C.-W. Zhang, P. Li, and P. ji Wang,Scientific Reports , 45923 (2004). K. Dybko, M. Szot, A. Szczerbakow, M. U. Gutowska,T. Zajarniuk, J. Z. Domagala, A. Szewczyk, T. Story, andW. Zawadzki, Phys. Rev. B , 205129 (2017). M. Z. Hasan, S.-Y. Xu, D. Hseih, L. A. Wray, and Y. Xia,Contemporary Concepts of Condensed Matter Science ,143 (2013). H. Yi, Z. Wang, C. Chen, Y. Shi, Y. Feng, A. Liang, Z. Xie,S. He, J. He, Y. Peng, X. Liu, Y. Liu, L. Zhao, G. Liu,X. Dong, J. Zhang, M. Nakatake, M. Arita, K. Shimada,H. Namatame, M. Taniguchi, Z. Xu, C. Chen, X. Dai,Z. Fang, and J. Zhou, Scientific Reports , 6106 (2014). X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. , 1057(2011). C.-Z. Chang, J. Zhang, M. Liu, Z. Zhang, X. Feng, K. Li,L.-L. Wang, X. Chen, X. Dai, Z. Fang, X.-L. Qi, S.-C.Zhang, Y. Wang, K. He, X.-C. Ma, and Q.-K. Xue, Ad-vanced Materials , 1065 (2013). P. Cheng, C. Song, T. Zhang, Y. Zhang, Y. Wang, J.-F.Jia, J. Wang, Y. Wang, B.-F. Zhu, X. Chen, X. Ma, K. He,L. Wang, X. Dai, Z. Fang, X. Xie, X.-L. Qi, C.-X. Liu, S.-C. Zhang, and Q.-K. Xue, Phys. Rev. Lett. , 076801(2010). Z. Qiao, W. Ren, H. Chen, L. Bellaiche, Z. Zhang, A. H.MacDonald, and Q. Niu, Phys. Rev. Lett. , 116404(2014), arXiv:1501.04828 [cond-mat.mes-hall]. S.-B. Zhang, H.-Z. Lu, and S.-Q. Shen, Scientific Reports , 13277 (2015). X. Liu, H.-C. Hsu, and C.-X. Liu, Phys. Rev. Lett. ,086802 (2013). N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, andN. P. Ong, Rev. Mod. Phys. , 1539 (2010). E. H. Hasdeo and J. C. W. Song, Nano Letters , 7252(2017), pMID: 29164888. A. L. Fetter, Physical Review B , 7676 (1985). O. Madelungu, R¨ossler, and M. Schulz,
Semiconductors:Non-Tetrahedrally Bonded Elements and Binary Com-pounds I , 1st ed., Vol. 41C (Springer, Berlin, Heidelberg, H. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang, and S.-C.Zhang, Nature Physics , 438 (2009). F. Zhang, C. L. Kane, and E. J. Mele, Phys. Rev. B ,081303(R) (2012). A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S.Novoselov, and A. K. Geim, Rev. Mod. Phys. , 109(2009). D. Y. Fedyanin, A. V. Arsenin, V. G. Leiman, and A. D.Gladun, Quantum Electronics , 745 (2009). X. Luo, T. Qiu, W. Lu, and Z. Ni, Materials Science andEngineering R , 351 (2013). L. L. S´anchez-Soto, J. J. Monz´on, and G. Leuchs, Euro-pean Journal of Physics , 064001 (2016). A. J. Cox and D. C. Dibble, J. Opt. Soc. Am. A , 282(1992). J. Yin, H. N. Krishnamoorthy, G. Adamo, A. M.Dubrovkin, Y. Chong, N. I. Zheludev, and C. Soci, NPGAsia Materials (2017), https://doi:10.1038/am.2017.149.
Appendix A: Calculation of the scattering coefficients r and t The boundary conditions that specify C + and the ratio of φ (0) and φ ( d ) are generated by the imposition ofcontinuity of the potential at the interfaces, whilst C − is specified by the imposition that there are only right-movingwaves in x > d . By evaluating Eq. (31) at x = 0 and x = d we find:12 π (cid:0) C + + C − (cid:1) = [1 + KI B (0)] φ (0) − K (cid:20) π q p k sin( k d ) + I B ( d ) (cid:21) φ ( d ) , (A1)12 π (cid:0) C + e ik d + C − e − ik d (cid:1) = [1 − KI B (0)] φ ( d ) + K (cid:20) π q p k sin( k d ) + I B ( d ) (cid:21) φ (0) , (A2)where any dependence on q y and ω in q p ( ω ), k ( ω ) and I B ( x, q y , ω ) has been dropped for brevity and K = δσ H q y /ω .To find the reflectance and transmittance through the central region from x < x > d , we impose that there areno left moving waves ( ∝ e − ik x ) in the region x > d , i.e. (from Eq. (31)): C − π = iπK q p k (cid:2) φ (0) − φ ( d ) e ik d (cid:3) . (A3)Then, far from the interfaces, the potentials have the following form: φ ( x ) = (cid:40) Ae ik x + Be − ik x , x → −∞ ,Ce ik x , x → + ∞ , (A4)such that the reflection and transmission coefficients are simply r = B/A and t = C/A . A , B and C may then befound by solving Eqs. (A1) and (A2) for C + and φ (0) /φ ( d ), with C − given by Eq. (A3), and then plugging the resultsback into Eq. (31).Firstly, solving for C + by substituting Eq. (A3) into Eq. (A1) yields: C + π = (cid:20) KI B (0) − iπK q p k (cid:21) φ (0) + K (cid:20) iπ q p k (cid:0) e ik d − e − ik d (cid:1) − I B ( d ) (cid:21) φ ( d ) . (A5)Then, secondly, the ratio φ (0) /φ ( d ) may be determined by using Eqs. (A2,A3,A5) together as: φ (0) φ ( d ) = k (cid:0) K (cid:2) I B ( d ) e ik d − I B (0) (cid:3)(cid:1) − iπKq p (cid:0) e ik d − (cid:1) k ([1 + KI B (0)] e ik d − KI B ( d )) . (A6)Thus, A , B and C may be found now as the appropriate coefficients of e ± ik x as in Eq. (31). Explicitly, we have: A = C + π − iπK q p k (cid:2) φ (0) − φ ( d ) e − ik d (cid:3) , (A7) B = C − π + iπK q p k (cid:2) φ (0) − φ ( d ) e ik d (cid:3) , (A8) C = C + π + iπK q p k (cid:2) φ (0) − φ ( d ) e − ik d (cid:3) , (A9)1and thus, through the use of Eqs. (A3,A5,A6) and after some lengthy algebra, we arrive at: A = (cid:34) k (cid:0) K (cid:2) I B ( d ) − I B (0) (cid:3)(cid:1) + 4 iπK q p k (cid:2) I B (0) − e ik d I B ( d ) (cid:3) + 4 π K q (cid:0) − e ik d (cid:1) k ( e ik d + K [ I B (0) e ik d − I B ( d )]) (cid:35) φ ( d ) , (A10) B = (cid:20) iπKq p [ Kk [ I B ( d ) − I B (0) cos( k d )] + (2 πKq p − ik ) sin( k d )] e ik d k ( e ik d + K [ I B (0) e ik d − I B ( d )]) (cid:21) φ ( d ) , (A11) C = (cid:34) k (cid:0) K (cid:2) I B ( d ) − I B (0) (cid:3) + 4 πK q p I B ( d ) sin( k d ) (cid:1) k ( e ik d + K [ I B (0) e ik d − I B ( d )]) (cid:35) φ ( d ) , (A12)from which we find the scattering coefficients as r = B/A and t = C/A . Finally, taking q y = q p ( ω ) sin( θ ) and k = q p ( ω ) cos( θ ) along with simple rearrangement yields the forms of r and t as quoted in Eqs. (33,34).The same analysis may be applied to the reverse case where the scattering occurs from right to left. The resultmay be seen to be identical in such a case: r L = r R = r and t L = t R = t , due to the mirror symmetry of the regionin the line x = d/d/