Transverse Magnetic Focusing Research Articles

Enhanced e–e interaction in suspended 2DEG evidenced by transverse magnetic focusing

Enhanced e–e interaction in suspended 2DEG evidenced by transverse magnetic focusing

Probing the Anisotropic Fermi Surface in Tetralayer Graphene via Transverse Magnetic Focusing.

Bernal-stacked tetralayer graphene (4LG) exhibits intriguing low-energy properties, featuring two massive sub-bands and showcasing diverse features of topologically distinct, anisotropic Fermi surfaces, including Lifshitz transitions and trigonal warping. Here, we study the influence of the band structure on electron dynamics within 4LG using transverse magnetic focusing. Our analysis reveals two distinct focusing peaks corresponding to the two sub-bands. Furthermore, we uncover a pronounced dependence of the focusing spectra on crystal orientations, indicative of an anisotropic Fermi surface. Utilizing the semiclassical model, we attribute this orientation-dependent behavior to the trigonal warping of the band structure. This phenomenon leads to variations in electron trajectories based on crystal orientation. Our findings not only enhance our understanding of the dynamics of electrons in 4LG but also offer a promising method for probing anisotropic Fermi surfaces in other materials.

Andreev reflection imaging and longitudinal shift of pseudospin-1 fermions in two-dimensional materials

In this work, the transverse magnetic focusing of pseudospin-1 fermions in two-dimensional materials with a superconducting contact is studied. The transmission coefficient and Andreev reflection coefficient from injector lead to collector lead are calculated based on the Landauer–Büttiker formula combined with the non-equilibrium Green's function method. The local state density diagrams of the electron and hole transmission in the center region are given, which provide a complete image of the ballistic electron–hole cyclotron orbits. Using these images, the positive longitudinal shift in Andreev reflection of the pseudospin-1 fermion can be obviously seen. Furthermore, it is found that the total offset is proportional to the number of Andreev reflection occurrence. This setup provides the possibility of detecting this shift experimentally.

Ballistic transport spectroscopy of spin-orbit-coupled bands in monolayer graphene on WSe2

Van der Waals interactions with transition metal dichalcogenides were shown to induce strong spin-orbit coupling (SOC) in graphene, offering great promises to combine large experimental flexibility of graphene with unique tuning capabilities of the SOC. Here, we probe SOC-driven band splitting and electron dynamics in graphene on WSe2 by measuring ballistic transverse magnetic focusing. We found a clear splitting in the first focusing peak whose evolution in charge density and magnetic field is well reproduced by calculations using the SOC strength of ~ 13 meV, and no splitting in the second peak that indicates stronger Rashba SOC. Possible suppression of electron-electron scatterings was found in temperature dependence measurement. Further, we found that Shubnikov-de Haas oscillations exhibit a weaker band splitting, suggesting that it probes different electron dynamics, calling for a new theory. Our study demonstrates an interesting possibility to exploit ballistic electron motion pronounced in graphene for emerging spin-orbitronics.

Probing miniband structure and Hofstadter butterfly in gated graphene superlattices via magnetotransport

The presence of periodic modulation in graphene leads to a reconstruction of the band structure and formation of minibands. In an external uniform magnetic field, a fractal energy spectrum called Hofstadter butterfly is formed. Particularly interesting in this regard are superlattices with tunable modulation strength, such as electrostatically induced ones in graphene. We perform quantum transport modeling in gate-induced square two-dimensional superlattice in graphene and investigate the relation to the details of the band structure. At low magnetic field the dynamics of carriers reflects the semi-classical orbits which depend on the mini band structure. We theoretically model transverse magnetic focusing, a ballistic transport technique by means of which we investigate the minibands, their extent and carrier type. We find a good agreement between the focusing spectra and the mini band structures obtained from the continuum model, proving usefulness of this technique. At high magnetic field the calculated four-probe resistance fit the Hofstadter butterfly spectrum obtained for our superlattice. Our quantum transport modeling provides an insight into the mini band structures, and can be applied to other superlattice geometries.

Spin polarization and spin-dependent scattering of holes observed in transverse magnetic focusing

In two-dimensional hole systems, the Rashba spin-orbit interaction leads to a spin dependent momentum. A perpendicular magnetic field spatially separates holes with different spins, creating a mass spectrometer for spin. Spin-resolved magnetic focussing has been used to measure spin polarization from the amplitude of magnetic focussing peaks. In this work, the authors show that the ${k}^{3}$ form of the Rashba spin-orbit interaction term for semiconductor holes changes the scattering rate, which has an exponential effect on the focussing peak amplitude. This result further demonstrates the impact of the Rashba spin-orbit interaction in the field of semiconductor hole spin physics.

Influence of Device Geometry and Imperfections on the Interpretation of Transverse Magnetic Focusing Experiments

Spatially separating electrons of different spins and efficiently generating spin currents are crucial steps towards building practical spintronics devices. Transverse magnetic focusing is a potential technique to accomplish both those tasks. In a material where there is significant Rashba spin–orbit interaction, electrons of different spins will traverse different paths in the presence of an external magnetic field. Experiments have demonstrated the viability of this technique by measuring conductance spectra that indicate the separation of spin-up and spin-down electrons. However the effect that the geometry of the leads has on these measurements is not well understood. By simulating an InGaAs-based transverse magnetic focusing device, we show that the resolution of features in the conductance spectra is affected by the shape, separation and width of the leads. Furthermore, the number of subbands occupied by the electrons in the leads affects the ratio between the amplitudes of the spin-split peaks in the spectra. We simulated devices with random onsite potentials and observed that transverse magnetic focusing devices are sensitive to disorder. Ultimately we show that careful choice and characterisation of device geometry are crucial for correctly interpreting the results of transverse magnetic focusing experiments.

Electron beam splitting at topological insulator surface states and a proposal for electronic Goos-Hänchen shift measurement

The hexagonal warping effect on transport properties and Goos-H\"anchen (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 derive expressions for calculating lateral shifts in the presence of warping effect. We study the dependence of transmission probabilities and GH shifts of transmitted beams on system parameters in detail by giving an explanation for the appearance of large peaks in the lateral shifts corresponding to their transmission peaks. It is shown that the separation between two transmitted beams through their different GH shifts can be as large as a few micrometers, which is large enough to be observed experimentally. Finally, we propose a method to measure the GH shift of electron beams based on the transverse magnetic focusing technique in which, by tuning an applied magnetic field, a detectable resonant path for electrons can be induced.

Imaging chiral Andreev reflection in the presence of Rashba spin-orbit coupling

In this work, we theoretically study transverse magnetic focusing in a two-dimensional electron gas with strong Rashba spin-orbit interaction when proximitized along its edge with a superconducting contact. The presence of superconducting correlations leads to the emergence of chiral Andreev edge states which -- within this weak magnetic field regime -- may be pictured as states following semiclassical skipping orbits with alternating electron-hole nature. The spin-orbit induced splitting of the Fermi surface causes these carriers to move along cyclotron orbits with different radii, allowing for their spatial spin separation. When Andreev reflection takes place at the superconducting lead, scattered carriers flip both their charge and spin, generating distinguishable features in the transport properties of the device. In particular, we report a notable enhancement of the separation between the spin-split focal points, which scales linearly with the number of Andreev scattering events at the anomalous terminal. We support our results by calculating conductance maps to arbitrary points in the sample that provide a complete image of the ballistic electron-hole cyclotron paths.

Precision measurement of electron-electron scattering in GaAs/AlGaAs using transverse magnetic focusing

Electron-electron (e-e) interactions assume a cardinal role in solid-state physics. Quantifying the e-e scattering length is hence critical. In this paper we show that the mesoscopic phenomenon of transverse magnetic focusing (TMF) in two-dimensional electron systems forms a precise and sensitive technique to measure this length scale. Conversely we quantitatively demonstrate that e-e scattering is the predominant effect limiting TMF amplitudes in high-mobility materials. Using high-resolution kinetic simulations, we show that the TMF amplitude at a maximum decays exponentially as a function of the e-e scattering length, which leads to a ready approach to extract this length from the measured TMF amplitudes. The approach is applied to measure the temperature-dependent e-e scattering length in high-mobility GaAs/AlGaAs heterostructures. The simulations further reveal current vortices that accompany the cyclotron orbits - a collective phenomenon counterintuitive to the ballistic transport underlying a TMF setting.

Significant energy relaxation of quantum dot emitted hot electrons

Mesoscopic quantum dots (QDs) are ubiquitous in quantum devices as reliable sources of hot electrons. However, we have observed an unexpectedly significant energy relaxation of QD-emitted hot electrons up to $\ensuremath{\approx}55%$ of its excitation $\ensuremath{\le}1.5\phantom{\rule{4pt}{0ex}}\mathrm{meV}$ from the Fermi level. The energetics of hot electrons were obtained through transverse magnetic focusing over a few microns using both QD and quantum point contact (QPC) emitters. Unlike the QPC counterparts, QD emissions deviated substantially from Fermi gas predictions---the focusing peak appeared at lower magnetic fields, and excessive broadening was observed. The phenomenon was modeled by a capacitive interaction transferring energy from the hot electron to the QD. Model simulations reproduced the key experimental features, implying the presence of a strong yet overlooked relaxation mechanism that is intrinsic to QD emissions. Our observation calls for the prudent use of QDs as single electron sources.

Cloning of zero modes in one-dimensional graphene superlattices

One-dimensional (1D) graphene superlattices have been predicted to exhibit zero-energy modes a decade ago, but an experimental proof has remained missing. Motivated by a recent experiment that could possibly shed light on this, here we perform quantum transport simulations for 1D graphene superlattices, considering electrostatically simulated potential profiles as realistic as possible. Combined with the analysis on the corresponding miniband structures, we find that the zero modes generated by the 1D superlattice potential can be further cloned to higher energies, which are also accessible by tuning the average density. Our multiterminal transverse magnetic focusing simulations further reveal the modulation-controllable ballistic miniband transport for 1D graphene superlattices. A simple idea for creating a perfectly symmetric periodic potential with strong modulation is proposed at the end of this work, generating well aligned zero modes up to 6 within a reasonable gate strength.

Current Limitations for the Axially Symmetric Hollow Electron Beam in the Drift Tubes of Millimeter-Wave Vacuum Electronic Devices

The conditions for ensuring the equilibrium steady-state of a high-density hollow electron beam transported in cylindrical drift tubes and focused by a uniform magnetic field are investigated. The results of numerical calculations of the currents limited due to the longitudinal deceleration of electrons by the space charge and the violation of equilibrium transverse magnetic focusing conditions when transporting beams of different configurations in klystron-type devices are presented. The limitations of the working length of the device associated with the excitation of diocotron instability are estimated. The calculation results were compared with the data obtained by the quasi-3D Arsenal—MSU computer code and by the approximate analytical formulas of other authors.

Limitations on Currents in Cylindrical Drift Tubes of Millimeter-Wave Vacuum Electronic Devices

The conditions providing the equilibrium stationary state of high-density solid and annular electron beams, transported in cylindrical drift tubes and focused by a homogeneous magnetic field, have been investigated. An efficient numerical model is proposed to determine the distributions of the electrostatic potential, charge density, and electron velocities in the stationary state of these beams, with allowance for the space-charge forces. The results of the numerical calculations for the limiting currents, determined by the longitudinal deceleration (caused by space-charge forces) and conditions for equilibrium transverse magnetic focusing for transported beams of different configurations on the length of a klystrontype vacuum electronic device, are presented. The limitations on the device working length, related to the excitation of diocotron instability, are estimated. The calculation results are compared with the data obtained using the quasi-three-dimensional program “Arsenal-MSU” and analytical formulas suggested by other researchers.

Interfacial Landau levels in bent graphene racetracks

One-dimensional electronic states are critical to many fascinating phenomena in low dimensions. We show that a series of interfacial states can be induced by applying an in-plane bending to a graphene racetrack geometry, an effective superlattice consisting of multiple graphene and hexagonal boron nitride ($h\mathrm{BN}$) domains. These local states, robust against the detailed atomic configuration along interfaces, display a unique coupling of their energies and their spatial locations. This way, it is possible to identify such a state with certain energy by merely referring to its location, mimicking well the transverse magnetic focusing. Further, these local states are ballistically conducting with different group velocities, indicating that each interfacial transport channel has a unique electric conduction. These findings may have implications for the design of nanoelectronics with heterostructures.

Measuring hole g -factor anisotropies using transverse magnetic focusing

Recent theoretical and experimental results from quasi-one dimensional heavy hole systems have suggested that heavy hole gases have a strongly anisotropic g factor. In this theoretical paper, we propose a method for measuring this anisotropy using transverse magnetic focusing (TMF). We demonstrate that for experimentally accessible fields, the g factor anisotropy leads to a relative variation in the characteristic of spin-splitting of the TMF spectrum which allows for the measurement of the anisotropy of the g factor. We show that this variation is insensitive to additional spin-orbit interactions, and is resolvable with current devices.

Interference in spin-orbit coupled transverse magnetic focusing: Emergent phase due to in-plane magnetic fields

Spin-orbit (SO) interactions in two dimensional systems split the Fermi surface, and allow for the spatial separation of spin-states via transverse magnetic focusing (TMF). In this work, we consider the case of combined Rashba and Zeeman interactions, which leads to a Fermi surface without cylindrical symmetry. While the classical trajectories are effectively unchanged, we predict an additional contribution to the phase, linear in the applied in-plane magnetic field. We show that this term is unique to TMF, and vanishes for magnetic (Shubnikov de Haas) oscillations. Finally we propose some experimental signatures of this phase.

Interference effects and Huygens principle in transverse magnetic focusing of electrons and holes

Interference effects form a fundamental pillar of quantum mechanics. In this paper, we examine the interference in spin-orbit coupled transverse magnetic focusing, where a weak magnetic field is used to focus charge carries over mesoscopic scales. We determine a semi-classical form for the Green's function in a weak magnetic field, for the case of both spin-less and spin-orbit coupled charge carriers. The obtained forms for the Greens' function are independent of particle dispersion and are thus applicable to a wide variety of systems.

Controlled spatial separation of spins and coherent dynamics in spin-orbit-coupled nanostructures

The spatial separation of electron spins followed by the control of their individual spin dynamics has recently emerged as an essential ingredient in many proposals for spin-based technologies because it would enable both of the two spin species to be simultaneously utilized, distinct from most of the current spintronic studies and technologies wherein only one spin species could be handled at a time. Here we demonstrate that the spatial spin splitting of a coherent beam of electrons can be achieved and controlled using the interplay between an external magnetic field and Rashba spin–orbit interaction in semiconductor nanostructures. The technique of transverse magnetic focusing is used to detect this spin separation. More notably, our ability to engineer the spin–orbit interactions enables us to simultaneously manipulate and probe the coherent spin dynamics of both spin species and hence their correlation, which could open a route towards spintronics and spin-based quantum information processing.

Understanding magnetic focusing in graphene p−n junctions through quantum modeling

We present a quantum model which provides enhanced understanding of recent transverse magnetic focusing experiments on graphene $p$-$n$ junctions. Spatially resolved flow maps of local particle current density show quantum interference and $p$-$n$ junction filtering effects which are crucial to explaining the device operation. The Landauer-B\"{u}ttiker formula is used alongside dephasing edge contacts to give exceptional agreement between simulated non-local resistance and the recent experiment by Chen $\textit{et al}$ ($\textit{Science}$, 2016). The origin of positive and negative focusing resonances and off resonance characteristics are explained in terms of quantum transmission functions. Our model also captures subtle features from experiment, such as the previously unexplained $p$-$p^-$ to $p$-$p^+$ transition and the second $p$-$n$ focusing resonance.