Ge as an orbitronic platform: giant orbital Hall effect

TL;DR

The paper argues that bulk Ge holes form an exceptional platform for orbitronics by exhibiting a giant intrinsic orbital Hall effect (OHE) that far surpasses the spin Hall effect. Using the spherical Luttinger-Kohn framework and the modern theory of orbital magnetisation, it demonstrates that quantum corrections to the OHE dominate and that the OHE persists in inversion-symmetric systems where spin- and orbital-Edelstein effects are symmetry-forbidden. The authors show that the OHE in Ge is SOC-driven and remains robust against realistic parameters, outperforming Bi2Se3 and offering practical advantages for Ge-based orbitronic devices, including high mobility and compatibility with silicon technology. They also discuss implications for orbital torques in Ge/ferromagnet interfaces and underscore the need for experimental verification. Overall, the work provides a theoretical blueprint for leveraging Ge to realize strong orbital torques in magnetic devices and expands the application of the modern theory to inversion-symmetric, spin-orbit coupled systems.

Abstract

State-of-the-art developments in magnetic devices rely on manufacturing faster, more efficient memory elements. A significant development in this direction has been the discovery of orbital torques, which employ the orbital angular momentum of Bloch electrons to switch the magnetisation of an adjacent ferromagnet, and has motivated the search for orbitronic materials displaying strong orbital dynamics exemplified, by the orbital Hall effect (OHE). In this work we propose Ge, as an optimal orbitronic platform. We demonstrate that holes in bulk Ge exhibit a giant OHE, exceeding that of the bulk states of topological insulators, and exceeding the spin-Hall effect by four orders of magnitude. The calculation is performed within the framework of the Luttinger model and the modern theory of orbital magnetisation, while incorporating recently-discovered quantum corrections to the OHE. Our study constitutes a fundamental milestone in applying the modern theory to a system with inversion symmetry. Moreover, we argue that bulk Ge serves as an ideal testbed for the orbital torque resulting from a charge current, since the spin- and orbital-Edelstein effects in Ge are forbidden by symmetry. Our results provide a blueprint for producing strong orbital torques in magnetic devices with Ge, guiding future experimental work in this direction.

Figures (5)

• Figure 1: Dispersion for holes in Ge. In this figure the bands have been inverted so the energy is positive. Here we have chosen a Fermi energy of $5$ meV and have indicated the two Fermi wave vectors for the heavy/light hole bands.
• Figure 2: The orbital Hall conductivity vs the Fermi energy in Ge in the spherical approximation. We have also plotted the conventional part of the orbital Hall conductivity $\sigma_{\text{conv}}$ along with the quantum correction $\Delta\sigma$. Here we use the Luttinger parameters $\gamma_1=13.38$ and $\bar{\gamma}=4.97$.
• Figure 3: The proper spin Hall conductivity vs the Fermi energy in Ge in the spherical approximation. Here we use the Luttinger parameters $\gamma_1=13.38$ and $\bar{\gamma}=4.97$. Note the difference of three orders of magnitude in the scale of the $y$-axis as compared to the OHE.
• Figure 4: Diagram of the orbital Hall torque in a Ge/Co heterostructure. Here the applied electric field generates transverse orbital currents via the intrinsic orbital Hall effect. The orbital current will generate an orbital accumulation. The orbital angular momentum is then converted into spin which generates a torque on the magnetisation in the Co layer. A barrier layer may be used to assist with orbital-to-spin conversion.
• Figure 5: The orbital Hall conductivity vs Fermi energy for various hole-doped semiconductors described by the Luttinger Hamiltonian in the spherical approximation. Si has been included for comparison yet as noted in the text the spherical approximation is of very limited applicability for Si.