Electrical manipulation of a hole ‘spin’–orbit qubit in nanowire quantum dot: The nontrivial magnetic field effects

Abstract

Strong ‘spin’-orbit coupled one-dimensional hole gas is achievable in a Ge nanowire in the presence of a strong magnetic field. The strong magnetic field lifts the two-fold degeneracy in the hole subband dispersions, such that the effective low-energy subband dispersion exhibits strong ‘spin’-orbit coupling. Here, we study the electrical ‘spin’ manipulation in a Ge nanowire quantum dot for both the lowest and second lowest hole subband dispersions. Using a finite square well to model the quantum dot confining potential, we calculate exactly the level splitting of the ‘spin’-orbit qubit and the Rabi frequency in the electric-dipole ‘spin’ resonance. The ‘spin’-orbit coupling modulated longitudinal g-factor g so is not only non-vanishing but also magnetic field dependent. Moreover, the ‘spin’-orbit couplings of the lowest and second lowest subband dispersions have opposite magnetic dependences, such that the results for these two subband dispersions are totally different.

Two-band description of the strong `spin'-orbit coupled one-dimensional hole gas

“…Also, the magnetic field dependent spin-orbit coupling α(B) and effective g-factor g * h (B) now have explicit expressions. Note that the effective hole mass m * h here does not depend on the magnetic field, this result consists with the previous band fitting result where m * h only has a very small magnetic field dependence [43].…”

“…where σ x,z are Pauli 'spin' matrices. We also emphasize that the effective hole mass m * h ≡ m * h (B), the Rashba [48] type 'spin'-orbit coupling α ≡ α(B), and the effective gfactor g * h ≡ g * h (B) are all magnetic field dependent [40,43]. In the following, we show the correctness of Eq.…”

“…When a strong magnetic field is applied. We can solve the Hamiltonian H = H 0 + H 1 using quasi-degenerate perturbation theory [40,43], where the perturbation H 1 contains all the magnetic terms (∝ B, B 2 ). The strong magnetic field lifts the two-fold degenerate in the hole subband dispersions.…”

“…These subband crossings imply we can label the dispersion lines with a definite spin index, i.e., we can classify the subband dispersions into different spin subsets. Each subset of the subband dispersions can be modeled by [40,43]…”

“…Note that each dispersion curve is two-fold degenerate, i.e., spin degeneracy [38,39]. We simply lift this spin degeneracy by using an strong magnetic field, such that a pure (without hole spin degeneracy) 'spin'-orbit coupled 1D hole gas can be obtained [40,43]. In this paper, basing on the Luttinger-Kohn Hamiltonian in the spherical approximation, we show both numerically and analytically this strong 'spin'-orbit coupled 1D hole gas is accurately described by an effective two-band Hamiltonian…”