Tensor in Low-Symmetry Two-Dimensional Hole Systems*

Zeeman coupling characterized by the g factor is a key ingredient to developing novel spin-based technologies such as quantum information protocols. In low-symmetry systems, the g factor becomes a second-rank tensor (a 3×3 matrix) that couples the spin to the magnetic field B. It has long been believed that this tensor g only affects the energy splitting in a magnetic field. We demonstrate [1] that it also encodes the direction of the axis about which the spins precess in the external field B. In general, this axis is not aligned with B. Using time-resolved Kerr rotation measurements performed on a sequence of low-symmetry two-dimensional hole systems in GaAs/AlAs quantum wells, we show that this feature of the tensor g manifest itself in unusual precessional motion as well as distinct dependencies of hole spin dynamics on the direction of the magnetic field B. A detailed theoretical analysis of these experiments allows us, for the first time, to determine the individual components of the full tensor g for [113]-, [111]- and [110]-grown samples. We also derive transparent analytical expressions for the components of the tensor g, complemented with accurate numerical calculations yielding very good agreement between experiment and theory.

Second-rank tensors characterizing materials properties such as electrical conductivity and dielectric constant are usually symmetric. In contrast, our study demonstrates that the tensor g is generally neither symmetric nor antisymmetric. Opposite off-diagonal components can differ in size by up to an order of magnitude. Consequently, the coupling of spins to the magnetic field varies drastically upon interchanging the direction of magnetic field and spin. This work extends the general concept of optical orientation to the regime of nontrivial Zeeman coupling.

[1] C. Gradl, R. Winkler et al., Phys. Rev. X 8, 021068 (2018)