Mapping the anisotropic Lande g-factor tensor of 1D GaAs holes in all 3 spatial directions

We have studied the Zeeman splitting of 1D holes formed on a (100) GaAs/AlGaAs heterostructure on a single cooldown. The strong spin orbit coupling and 1D confinment give rise to a highly anisotropic spin splitting. By use of the high-symmetry (100) crystal, we eliminate the effects of crystal anisotropy on our measurements. In measuring the spin splitting as a function of angle between the wire and the applied magnetic field, we are able to identify the principle axes of the $g$-tensor. We show that the principle axes are defined by the potential confining the 1D holes, and are not affected by the crystal axes. We find that $g_{\parallel}^{\perp} < g_{\parallel}^{\parallel} < g_{\perp}$, where $g_{\parallel}$ refers to the in-plane $g$-factors parallel and perpendicular to the wire, and $g_{\perp}$ refers to the $g$-factor perpendicular to the 2D well.