Transverse magnetic focusing in two-dimensional hole gases

TL;DR

This work addresses the complex spin dynamics of two-dimensional hole gases in transverse magnetic focusing by developing a 2D transport model from the 4×4 Luttinger Hamiltonian, including Zeeman and Peierls effects and spin projections. Band-structure analysis reveals strong heavy-hole/light-hole mixing at finite $k$, causing HH subbands to be non-polarised even with Rashba splitting, which challenges conventional spin-filter interpretations. The TMF response is studied in idealised devices and with quantum point contacts, showing that extra peaks and fringe-like features arise from interface mismatch and Rashba-related trajectories, not spin-polarised transport. The results emphasize the need for detailed, spin-resolved modeling of 2DHG TMF experiments and suggest that confinement engineering can tune (or restore) spin-polarised behavior in narrow wells, with implications for spintronic device design.

Abstract

Two-dimensional hole gases (2DHGs) have strong intrinsic spin-orbit coupling and could be used to build spin filters by utilising transverse magnetic focusing (TMF). However, with an increase in the spin degree of freedom, holes demonstrate significantly different behaviour to electrons in TMF experiments, making it difficult to interpret the results of these experiments. In this paper, we numerically model TMF in a 2DHG within a GaAs/Al$_{\mathrm{x}}$Ga$_{\mathrm{1-x}}$As heterostructure. Our band structure calculations show that the heavy $(\langle J_{z} \rangle = \pm\frac{3}{2})$ and light $(\langle J_{z} \rangle = \pm\frac{1}{2})$ hole states in the valence band mix at finite $k$, and the heavy hole subbands which are spin-split due to the Rashba effect are not spin-polarised. This lack of spin polarisation casts doubt on the viability of spin filtering using TMF in 2DHGs within conventional GaAs/Al$_{\mathrm{x}}$Ga$_{\mathrm{1-x}}$As heterostructures. We then calculate transport properties of the 2DHG with spin projection and offer a new perspective on interpreting and designing TMF experiments in 2DHGs.

Figures (15)

• Figure 1: Schematic of a typical transverse magnetic focusing experimental setup. Carriers are injected through an injector lead ($\mathrm{L}_1$) into a scattering area where a perpendicular magnetic field $B_z$ is applied. Tuning the magnetic field allows the carriers to be focused onto the collector lead ($\mathrm{L}_2$) when the diameter of the cyclotron orbit coincides with the distance between the injector and collector leads ($D$).
• Figure 2: Diagram of the confinement potential at the interface of a GaAs/AlxGa1-xAs heterostructure (black solid line). The red dashed line shows a triangular potential well used in our calculations to approximate the confinement potential.
• Figure 3: (a) Diagram of heavy hole subbands for an idealised 2DHG with Rashba spin-orbit coupling. (b-f) Band structures calculated for a 2DHG in GaAs/AlxGa1-xAs. (b) and (c) are plotted along the $k^{}_{y}$ axis and the dashed lines here mark $1.6$ meV below the valence band edge, the energy at which (e) and (f) were calculated. (d) is plotted along the along the $k^{}_{x}=k^{}_{y}$ axis, and the dotted line marks $1.6$ meV below the valence band edge. (e) and (f) show the Fermi surface sliced at $1.6$ meV below the valence band edge and the dashed and dotted lines here indicate the $k^{}_{y}$ and $k^{}_{x}=k^{}_{y}$ axes respectively. The colour bars in (b), (d) and (e) indicate the expectation value of $\left| J^{}_{z} \right|$, whereas the colour bars in (c) and (f) indicate the expectation value of $J^{}_{x}$.
• Figure 4: Example diagram of a discretised grid used to model a TMF device with 50 nm wide injector and collector leads separated by a distance of 150 nm. The Hamiltonian is discretised in real space, giving onsite terms shown as blue dots, and hopping terms shown as solid black lines that connect neighbouring sites. Semi-infinite leads are indicated by the red dots. Here the grid spacing is set to be 10 nm for simplicity. In our calculations, the grid spacing is set to 2 nm and the separation between the leads is much larger as seen in Fig. \ref{['fig:LDOS']}.
• Figure 5: Conductance spectra calculated for TMF devices with 50 nm wide injector and collector leads separated by distances ranging from 850 nm to 2350 nm. The Fermi energy was set to $1.6$ meV below the valence band edge in each calculation. The dotted lines indicate the first and second focusing magnetic field strengths corresponding to $k^{}_{y}$ = 0.07 nm$^{-1}$ and $k^{}_{y}$ = 0.13 nm$^{-1}$ respectively, calculated from Eq. \ref{['Eq:larmor']}. The perpendicular magnetic field strengths are normalised to the second focusing field strengths for each $D$, such that $B_{z} = \widetilde{B} \times D^{-1} \times (-175~\mathrm{T}\cdot\mathrm{nm})$.