Magnetic electron collimation in three-dimensional semi-metals

Controlling electron beams using electric and magnetic fields is a cornerstone of technologies like cathode-ray tubes and electron microscopes. Translating this to solid-state devices is a significant challenge. While long mean-free paths are needed for unperturbed electron motion, high conductivity materials effectively screen external electric fields. Geometric shaping of constrictions in ballistic devices offers a solution, successfully explored in two-dimensional electron gases (2DEGs). However, 2DEGs limit magnetic field control to a scalar quantity, fixing the beam shape during fabrication. This work aims to demonstrate electron beam control in three-dimensional (3D) semi-metals, leveraging additional degrees of freedom for electron motion. Unlike 2DEGs, in-plane magnetic fields in 3D materials induce an out-of-plane Hall component, significantly reshaping current flow. This is visualized by helical electron trajectories entering a bulk metal isotropically, leading to anisotropic conductivity even in isotropic zero-field conductors. This anisotropy, captured by a conductivity tensor, is particularly pronounced in low carrier density materials with high Hall conductivity, enabling long-range current beams. The Dirac semi-metal Cd3As2, with its high mobility and low carrier density, serves as an ideal experimental platform to investigate these phenomena. Current jetting, caused by imperfections in current injection contacts, has previously hampered research, but here it is intentionally exploited for generating and controlling electron beams.

Previous research in electron beam generation and control has largely focused on two-dimensional electron gases (2DEGs). Studies have demonstrated methods for shaping electron flow using electrostatic lenses and geometric constrictions in ballistic devices [2–11]. However, these approaches often lack the flexibility to dynamically steer or reshape the electron beams. In contrast, three-dimensional semi-metals offer a potential pathway for greater control, due to the added complexity of the electron trajectories in the presence of magnetic fields [14]. The phenomenon of current jetting, characterized by the formation of long-range current filaments along the magnetic field lines near point contacts, has been previously observed in semimetals [17, 18] and recently studied in topological semimetals [24, 25] where it can confound the interpretation of magnetotransport experiments [19, 20]. Longitudinal electron focusing (LEF) experiments using lasers to excite carriers provide a related approach for mapping quasi-particle propagation [27], but lack the precise control of the present study. The use of focused ion beam (FIB) machining to create micro-structures within crystals to control and induce current jetting has not been explored in detail.

The study uses micro-devices fabricated from flux-grown single crystals of Cd3As2. Focused ion beam (FIB) machining is employed to create artificial constrictions on the micrometer scale, shaping the current flow. The devices consist of a main cuboid body connected by four narrow bridges (~1 μm wide) serving as artificial constrictions (Figure 2a). The thickness of the samples (1.4 μm) significantly exceeds the Fermi wavelength, ensuring genuine 3D electron behavior. Electrical contacts are formed via gold deposition. Current is applied to one pair of contacts, and the voltage difference between another pair is measured. A magnetic field is applied in the plane of the device, and its angle is varied to control the direction of the electron beams. Finite element simulations, using the Drude model to approximate conductivity, are used to model the electric potential distribution. The simulations incorporate the experimentally determined parameters, such as zero-field conductivity (σ0) and the cyclotron frequency (μB). The temperature and magnetic field dependence of the electron beam collimation is also investigated. The conductivity tensor σij is used to model the anisotropic conductivity in the presence of a magnetic field, and the Drude model approximates the components of this tensor [14]. The effect of the Hall conductivity, σxy, is explicitly considered in the simulations. Measurements are performed in a Physical Property Measurement System (PPMS) with a 14T superconducting magnet, using AC excitation current at 177.77 Hz. Angular-dependent measurements are obtained using a sample rotator. A multi-contact device (Figure 5a) is also studied to demonstrate more complex architectures and functionalities. This device consists of two rows of four contacts separated by a channel, allowing the investigation of signal propagation and selectivity.

The experiments demonstrate the controlled generation and steering of tightly focused electron beams in Cd3As2 using magnetic fields. The characteristic 'W-shape' voltage response as a function of magnetic field angle (Figure 2b) is observed, exhibiting regions of positive and negative voltages. The position of the voltage minimum quantitatively matches the geometric condition of the beam hitting a diagonal electrode. Finite element simulations, using experimentally determined parameters and the Drude model, accurately capture the angle-dependent voltage signal, strongly supporting the picture of field-guided current beams. The temperature dependence of the collimation strength (Figure 3a, c) shows that increasing the temperature reduces the effect, but the characteristic W-shape remains visible even above room temperature. The effect also decreases with decreasing magnetic field (Figure 3b, c). Simulations accurately reproduce this behavior and show that the field-induced anisotropy is tuned by the parameter μ(T)B, where μ(T) is the temperature-dependent mobility. The study further demonstrates that the 3D nature of the sample and the out-of-plane Hall conductivity are essential for establishing the current beams (Figure 4). By suppressing the Hall effect in the simulations, the characteristic features of the voltage response are lost, showing that the helical trajectory is critical for beam collimation. The multi-contact device (Figure 5) demonstrates selective signal propagation to different contacts by varying the magnetic field angle. The behavior is well-captured by the simulations, allowing a level of control suggesting new solid-state device applications.

The results demonstrate the ability to control current flow in a highly non-local fashion using magnetic fields in 3D semi-metals. This contrasts with 2D systems where this level of dynamic control is not possible. The quantitative agreement between experimental observations and simulations supports the understanding of current jetting as a mechanism to create and manipulate electron beams in solids. The temperature and magnetic field dependence of the collimation strength shows a good correspondence to the simulations, validating the use of the Drude model in this context, although some deviations remain likely due to finite size effects, and complex magneto-resistance of the Cd3As2 material. The study suggests potential applications in fields such as neuromorphic computing, leveraging the non-local signal propagation to simulate neuronal behavior. The findings call for a complete theoretical framework that unifies the semi-classical and topological aspects of non-local transport in topological semi-metals.

This study successfully demonstrates the controlled generation and manipulation of narrow electron beams in Cd3As2 using magnetic fields. The results highlight the importance of three-dimensionality and the Hall effect in achieving this control. While the high magnetic fields required for room-temperature operation currently limit practical applications, improvements in material properties or the use of magnetic thin films could pave the way for future applications in areas such as neuromorphic computing and sensing.

The current study is limited to low temperatures and high magnetic fields. Room temperature operation would require materials with higher mobility. The Drude model, while providing a good approximation, is a simplification and does not capture all aspects of the complex electronic behavior in Cd3As2. The devices explored represent a preliminary step in device design; more sophisticated architectures are necessary for widespread applications.