Weak anti-localization of two-dimensional holes in germanium beyond the diffusive regime

Abstract

Gate-controllable spin-orbit coupling is often one requisite for spintronic devices. For practical spin field-effect transistors, another essential requirement is ballistic spin transport, where the spin precession length is shorter than the mean free path such that the gate-controlled spin precession is not randomized by disorder. In this letter, we report the observation of a gate-induced crossover from weak localization to weak anti-localization in the magneto-resistance of a high-mobility two-dimensional hole gas in a strained germanium quantum well. From the magneto-resistance, we extract the phase-coherence time, spin-orbit precession time, spin-orbit energy splitting, and cubic Rashba coefficient over a wide density range. The mobility and the mean free path increase with increasing hole density, while the spin precession length decreases due to increasingly stronger spin-orbit coupling. As the density becomes larger than ∼6 × 1011 cm-2, the spin precession length becomes shorter than the mean free path, and the system enters the ballistic spin transport regime. We also report here the numerical methods and code developed for calculating the magneto-resistance in the ballistic regime, where the commonly used HLN and ILP models for analyzing weak localization and anti-localization are not valid. These results pave the way toward silicon-compatible spintronic devices.

The germanium quantum information route

“…However, it is known from studying conduction-band electrons in lateral quantum dots [76][77][78] that many spin-orbit interaction terms exist which can be of high relevance. For holes in quasi-2D systems, great efforts in both theory and experiment have led to detailed insights and knowledge about the spin-orbit interaction 10,29,37,40,42,[79][80][81][82][83][84][85][86][87][88][89][90] . Calculations for quantum dots in planar Ge/SiGe heterostructures 42 predicted that the spin-orbit interaction of hole-spin qubits therein can be harnessed to perform rapid (few nanoseconds) qubit rotations by means of EDSR, as measured in experiments 15,19 .…”

“…The superior quality achieved in reverse-graded Ge/SiGe heterostructures enabled a plethora of quantum transport studies in modulation-doped etched Hall-bar devices 89,141,[155][156][157][158][159][160][161][162][163][164][165][166][167][168] and, more recently, in undoped H-FETs 45,90,140,154,[169][170][171] . Initial expectations from bandstructure considerations were confirmed and the knowledge-base of confined holes in planar Ge was advanced.…”

Recent progress in undoped group-IV heterostructures for quantum technologies

“…Using an undoped structure, a lot of quantum physics can be probed by modulating the carrier density via top gating. For example, the SOC strength in the Ge layer was modulated and a ballistic transport of holes was demonstrated [26]. For a lower hole density, only weak localization effect was observed while inducing more holes into the Ge channel strongly enhances the SOC effect.…”

“…In 2017, high hole mobility (∼200 000 cm 2 (V s) −1 ) in an undoped Ge/GeSi heterostructure was demonstrated [23], enabling the Ge-based qubit development utilizing its strong SOC effects [24][25][26]. The SOC effects in Ge originates from the relativistic interaction of holes' spin with their motion in electric potentials created by the nuclei.…”

“…A fast two-qubit gate reached a Rabi frequency exceeding 100 MHz in 2020 [15]. In 2021, Tai et al demonstrated even larger SOC strengths in GeSn [30] than in Ge [26]. With its direct-bandgap characteristics [31] and high carrier mobility [32,33], GeSn is a promising enabler to accomplish a full-functional chip including spintronic, photonic, and electronic devices compatible with the Si CMOS platform.…”