Ge as an orbitronic platform: giant in-plane orbital magneto-electric effect in a 2-dimensional hole gas

Increasing demand for computational power has initiated the hunt for energy efficient and stable memory devices. This is the overarching motivation behind the recent rise of \textit{orbitronics}, which looks to harness the orbital angular momentum of charge carriers in computing devices. Orbitronic devices require materials with efficient generation of orbital angular momentum (OAM). In 2D materials, OAM can be electrically generated via the orbital magneto-electric effect (OME). In this paper we report the calculation of the OME in 2 dimensional hole gases (2DHGs). We show that the OME in Ge holes is very large, for an applied electric field of the order $10^4$ V$/$m the OAM density is of the order $10^{12}$ $\hbar/$cm$^{2}$. Furthermore, we find the OME to be an order of magnitude larger than the Rashba-Edelstein effect in 2DHGs. The OME we calculated in 2DHGs generates OAM aligned in the plane and arises due to transitions between heavy and light hole states, which is unique to this system. Our results put Ge, as well as other p-type semiconductors, forward as strong candidates for building future orbitronic devices.

💡 Research Summary

The paper investigates the orbital magneto‑electric effect (OME) in a two‑dimensional hole gas (2DHG) formed in germanium (Ge), aiming to assess its suitability for future orbitronic devices that exploit the orbital angular momentum (OAM) of carriers for low‑power information processing. After a concise introduction that places OME alongside the orbital Hall effect (OHE) as the two primary routes to generate OAM electrically, the authors argue that in 2D systems OME is more advantageous because it creates a bulk orbital polarization rather than merely edge accumulation as in OHE.

Methodologically, the authors employ the Luttinger‑Kohn 4 × 4 Hamiltonian in the spherical approximation (γ₂≈γ₃) to describe the heavy‑hole (HH) and light‑hole (LH) bands (total angular momentum J = 3/2). The quantum well confinement in the growth (z) direction is modeled as an infinite square well of width d, with the ground‑state envelope functions taken from the Bastard solution. An external gate field F z and an in‑plane electric field Eₓ are added to the Hamiltonian. The non‑equilibrium density matrix ρ_E is obtained from the quantum Liouville equation using a simple relaxation‑time approximation (τ) for the scattering term, separating diagonal (extrinsic) and off‑diagonal (intrinsic) contributions.

The OAM operator is defined as L = m(r × v − v × r) and its expectation value is evaluated via Tr(L ρ). Because the HH and LH envelope functions have different average positions ⟨z⟩_HH and ⟨z⟩_LH when the confinement potential is asymmetric (non‑zero F z), a finite Δz = ⟨z⟩_HH − ⟨z⟩_LH appears. The authors show that only the y‑component of OAM, ⟨L_y⟩, survives; it is proportional to Δz, the relaxation time τ, and the applied in‑plane field Eₓ. The intrinsic (band‑off‑diagonal) part vanishes, leaving an entirely extrinsic OME that scales linearly with τ.

Numerical estimates are performed for a 20 nm wide Ge quantum well, τ = 100 ps (consistent with experimentally reported mobilities μ ≈ 10⁶ cm² V⁻¹ s⁻¹), gate field F = 10⁵ V m⁻¹, and carrier Fermi energies ranging from the HH band edge up to the first excited subband. The resulting OAM density per unit field reaches ≈10¹² ℏ cm⁻² for Eₓ ≈ 10⁴ V m⁻¹, i.e., an OME comparable to a Rashba‑Edelstein spin polarization of ~100 % in the same carrier population. The effect grows with increasing well width and gate field, although the authors caution that neglecting higher‑lying subbands may over‑estimate the trend for very wide wells.

A direct comparison with the Rashba‑Edelstein effect (REE) in 2DHGs shows that the OME in Ge exceeds the REE by roughly one order of magnitude, a consequence of the exceptionally long τ in high‑quality Ge. Similar calculations for GaAs yield comparable OME magnitudes, indicating that the phenomenon is generic to p‑type semiconductors with strong spin‑orbit coupling.

In the discussion, the authors emphasize that the OME generates a uniform in‑plane orbital polarization, which can exert an orbital torque on adjacent magnetic layers—an orbital analogue of the spin‑orbit torque widely studied in spintronics. Because the OME does not rely on edge accumulation, it is more amenable to device scaling and integration with existing Si/Ge CMOS technology. The paper also notes that the OME is fundamentally different from previously reported 2D OME cases, as it originates from HH‑LH interband matrix elements that become active only when inversion symmetry is broken along z.

The conclusion positions Ge 2DHGs as a promising platform for orbitronic applications, combining a giant OME, a theoretically predicted large OHE, and compatibility with mature semiconductor processing. Future work is suggested on incorporating higher subband contributions, exploring non‑linear field regimes, and developing experimental probes (e.g., X‑ray magnetic circular dichroism or Kerr rotation) to directly detect the predicted orbital polarization.