C. W. J. Beenakker, R. A. Sepkhanov, A. R. Akhmerov, J. Tworzydlo
备注：
5 pages, 6 figures; expanded version of the published paper (one extra page, one extra figure), including also a reference to J.M. Pereira et al, PRB 74, 045424 (2006)
The Goos-Hanchen (GH) effect is an interference effect on total internal reflection at an interface, resulting in a shift sigma of the reflected beam along the interface. We show that the GH effect at a p-n interface in graphene depends on the pseudospin (sublattice) degree of freedom of the massless Dirac fermions, and find a sign change of sigma at angle of incidence alpha*=arcsin[sin alpha_c]^1/2 determined by the critical angle alpha_c for total reflection. In an n-doped channel with p-doped boundaries the GH effect doubles the degeneracy of the lowest propagating mode, introducing a two-fold degeneracy on top of the usual spin and valley degeneracies. This can be observed as a stepwise increase by 8e^2/h of the conductance with increasing channel width.
We present a theoretical investigation of the Goos-Hänchen effect, i.e., the lateral shift of the light beam transmitted through one-dimensional biperiodic multilayered photonic systems consisting of equidistantmagnetic layers separated by finite size dielectric photonic crystals. We show that the increase of the number of periods in the photonic-magnonic structure leads to increase of the Goos-Hänchen shift in the vicinity of the frequencies of defect modes located inside the photonic band gaps. Presence of the linear magnetoelectric coupling in the magnetic layers can result in a vanishing o…
We investigate the strain effect on the Goos-Hänchen (GH) shifts and group delay time for transmitted Dirac fermions in gapped graphene through a single barrier potential. The solutions of energy spectrum are used to compute the transmission probabilities together with the GH shifts and group delay time. Our results show that the two last quantities are strongly depending to weather the strain is applied along armchair or zigzag directions. In particular it found that both of quantities can be enhanced with the applied strain.
Strong Goos-Hänchen (GH) effect at a prism-graphene interface in the quantum Hall effect (QHE) condition is reported. Based on the full quantum description of the temperature-dependent surface conductivity of graphene present in the unconventional quantum Hall regime, magnetically strong tunable QHE GH shifts emerge. Our approach is based on deriving the generalized Fresnel coefficients with antisymmetric conductivity tensor for the Kerr phase of the incident linearly polarized light. Moreover, it is demonstrated that at low temperatures, GH shifts map plateaus as the intensity of the magnetic…
Spin waves are promising information carriers which can be used in modern magnonic devices, characterized by higher performance and lower energy consumption than presently used electronic circuits. However, before practical application of spin waves, the efficient control over spin wave amplitude and phase needs to be developed. We analyze analytically reflection and refraction of the spin waves at the interface between two ferromagnetic materials. In the model we consider the system consisting of two semi-infinite ferromagnetic media, separated by the ultra-narrow interface region with the ma…
At a normal-metal/superconductor interface, an incident electron from the normal-metal (N) side can be normally reflected as an electron or Andreev reflected as a hole. We show that pronounced lateral shifts along the interface between the incident and the reflected quasiparticles can happen in both reflection processes, which are analogous to the Goos-Hänchen effect in optics. Two concrete model systems are considered. For the simplest model in which the N side is of the two-dimensional electron gas, we find that while the shift in Andreev reflection stays positive, the shift in normal reflec…
Based on a Dirac-like Hamiltonian and coherent scattering formalism, we study spin-valley transport and Goos-Hänchen like (GHL) effect of transmitted and reflected electrons in a gated monolayer WS$_2$. Our results show that the lateral shift of spin-polarized electrons is strongly dependent on the width of the gated region and can be positive or negative in both Klein tunneling and classical motion regimes. The absolute values of the lateral displacements at resonance positions can be considerably enhanced when the incident angle of electrons is close to the critical angle. In contrast to the…
We study Dirac fermions in gapped graphene that are subjected to a magnetic field and a potential barrier harmonically oscillating in time. The tunneling modes inside the gap and the associated Goos-Hänchen (GH) shifts are analytically investigated. We show that the GH shifts in transmission for the central band and the first two sidebands change sign at the Dirac points $ε+l\hbar\tildeω$ $(l=0,\pm 1)$. We also find that the GH shifts can be either negative or positive and becomes zero at transmission resonances.
The main object of investigation in magnonics, spin waves (SWs) are promising information carriers. Presently the most commonly studied are plane wave-like SWs and SWs propagating in confined structures, such as waveguides. Here we consider a Gaussian SW beam obliquely incident on an ultra-narrow interface between two identical ferromagnetic materials. We use an analytical model and micromagnetic simulations for an in-depth analysis of the influence of the interface properties, in particular the magnetic anisotropy, on the transmission of the SW beam. We derive analytical formulas for the refl…
We study the Goos-Hänchen shifts for Dirac fermions in graphene scattered by a triangular double barrier potential. The massless Dirac-like equation was used to describe the scattered fermions by such potential configuration. Our results show that the GHL shifts is affected by the geometrical structure of the double barrier. In particular the GHL shifts change sign at the transmission zero energies and exhibit enhanced peaks at each bound state associated with the double barrier when the incident angle is less than the critical angle associated with the total reflection.
Using the energy spectrum of a system made of graphene subjected to a linear barrier potential, we study the Goos-Hänshen shifts. The calculation is done by first determining the corresponding phase shifts via the transmission and reflection probabilities. Numerical analysis shows that the Goos-Hänshen shifts depend strongly on the incident energy, barrier height and width, and vary positively or negatively under suitable conditions.
Using the energy spectrum of graphene with spatially modulated potential, we study the Goos-Hänshen shifts near extra Dirac points located at finite energy $\varepsilon=mπ$, with $m$ integer. On both sides of such points, we show that the Goos-Hänshen shifts can be negative as well as positive under various conditions. It is found that the shifts are strongly depending on the effects of the incident energy, potential height, incident angle and width of central region of unit cell. Such effects tell us that such width can be used to tune the shifts at the extra Dirac points.
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 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 …
We study the Goos-Hänchen (GH) shifts for transmitted Dirac fermions in gapped graphene through a single barrier structure having a time periodic oscillating component. Our analysis shows that the GH shifts in transmission for central band $l=0$ and two first sidebands $l=\pm1$ change sign at the Dirac points $E=V+l\hbarω$. In particular the GH shifts inbtransmission exhibit enhanced peaks at each bound state associated with the single barrier when the incident angle is less than the critical angle associated with total reflection. Klein tunneling, reflected by perfect transmission at normal i…
We theoretically investigate the Goos-Hänchen (GH) effect of spin-wave beams reflected from the interface between two ferromagnetic films with different Dzyaloshinskii-Moriya interactions (DMIs). The formula of the GH shift as functions of the incident angle and material parameters is derived analytically. We show that the GH effect occurs only when spin waves are totally reflected at the interface and vanishes otherwise. We further explore the GH shift of spin waves by narrow DMI strips of different widths. It is found that the induced shift is independent of the strip width down to $10$ nm, …
We investigate the Goos-Hänchen shift for ballistic electrons (i) reflected from a step-like inhomogeneity of the potential energy and (or) effective mass, and (ii) transmitted through a ferromagnetic barrier region in monolayer silicene or gapped graphene. For the electrons reflected from a single interface we found that the Goos-Hänchen shift is valley-polarized for gapped graphene structure, and valley- and spin-polarized for silicene due to the spin-valley coupling. Incontrast, for example, to gapless graphene the lateral beam shift in gapped structures occurs not only in the case of total…
The hexagonal warping effect on transport properties and Goos-Hänchen (GH) lateral shift of electrons on the surface of a topological insulator with a potential barrier is investigated theoretically. Due to the warped Fermi surface for incident electron beams, we can expect two propagating transmitted beams corresponding to the occurrence of double refraction. The transmitted beams have spin orientations locked to their momenta so one of the spin directions rotates compared to the incident spin direction. Based on a low-energy Hamiltonian near the Dirac point and considering Gaussian beams, we…
We predict quantized Imbert-Fedorov, Goos-Hänchen, and photonic spin Hall shifts for light beams impinging on a graphene-on-substrate system in an external magnetic field. In the quantum Hall regime the Imbert-Fedorov and photonic spin Hall shifts are quantized in integer multiples of the fine structure constant $α$, while the Goos- Hänchen ones in multiples of $α^2$. We investigate the influence on these shifts of magnetic field, temperature, and material dispersion and dissipation. An experimental demonstration of quantized beam shifts could be achieved at terahertz frequencies for moderate …
Recently, the electronic analogy of the anomalous spatial shift, including Goos-Hänchen and Imbert-Fedorov effects, has been attracting widespread interest. The current research on the anomalous spatial shift in interface electronic reflection is based on the paradigm of linear approximation, under which the center position of the incident and reflected beams are obtained by expanding the phases of relevant basis states and scattering amplitudes to the first order of incident momentum. However, in a class of normal cases, the linear approximation can lead to a divergent spatial shift in reflec…
We solve the Dirac equation in three regions of graphene to get the solutions of the energy spectrum in connection to the strain, energy gap, and magnetic field. The Goos-Hänchen shifts and group delay time will be obtained by applying the stationary phase approximation after the wave functions at the interfaces have been matched. Our results suggest that the group delay time is influenced by the presence of strain along the armchair and zigzag directions. We show that the gate voltage and strain have the ability to change the group delay from subluminality to superluminality. This may have si…
The intrinsic spin-orbit coupling in the 2D staggered monolayer semiconductors is very large as compared to graphene. The large spin orbit interaction in these materials leads to the opening of a gap in the energy spectrum and spin-splitting of the bands in each valley. In this paper, we theoretically investigate the mechanical steering of beams from these spin-orbit rich, staggered 2D materials. Mechanical steering results in noticeable deviations of the reflected and transmitted ray profiles as predicted from classical laws of optics. These effects are generally called the Goos-Hanchen (GH) …
The fourfold spin-valley degenerate degrees of freedom in bulk graphene can support rich physics and novel applications associated with multicomponent quantum Hall effects and linear conductance filtering. In this work, we study how to break the spin-valley degeneracy of electron beams spatially. We propose a spin-valley beam splitter in a gated ferromagnetic/pristine/strained graphene structure. We demonstrate that, in a full resonant tunneling regime for all spin-valley beam components, the formation of quasi-standing waves can lead four giant lateral Goos-Hänchen shifts as large as the tran…
Physically, one tends to think of non-Hermitian systems in terms of gain and loss: the decay or amplification of a mode is given by the imaginary part of its energy. Here, we introduce an alternative avenue to the realm of non-Hermitian physics, which involves neither gain nor loss. Instead, complex eigenvalues emerge from the amplitudes and phase-differences of waves backscattered from the boundary of insulators. We show that for any strong topological insulator in a Wigner-Dyson class, the reflected waves are characterized by a reflection matrix exhibiting the non-Hermitian skin effect. This…
The formation of a superstructure - with a related Moiré pattern - plays a crucial role in the extraordinary optical and electronic properties of twisted bilayer graphene, including the recently observed unconventional superconductivity. Here we put forward a novel, interdisciplinary approach to determine the Moiré angle in twisted bilayer graphene based on the photonic spin Hall effect. We show that the photonic spin Hall effect exhibits clear fingerprints of the underlying Moiré pattern, and the associated light beam shifts are well beyond current experimental sensitivities in the near-infra…
