Transverse magnetic focusing in two-dimensional hole gases

Two-dimensional hole gases (2DHGs) have strong intrinsic spin-orbit coupling and could be used to build spin filters by utilizing transverse magnetic focusing (TMF). However, with an increase in the spin degree of freedom, holes demonstrate significantly different behavior to electrons in TMF experiments, making it difficult to interpret the results of these experiments. In this paper, we numerically model TMF in a 2DHG within a $\mathrm{GaAs}/{\mathrm{Al}}_{x}{\mathrm{Ga}}_{1\ensuremath{-}x}\mathrm{As}$ heterostructure. Our band structure calculations show that the heavy $(\ensuremath{\langle}{J}_{z}\ensuremath{\rangle}=\ifmmode\pm\else\textpm\fi{}3/2)$ and light $(\ensuremath{\langle}{J}_{z}\ensuremath{\rangle}=\ifmmode\pm\else\textpm\fi{}1/2)$ hole states in the valence band mix at finite $k$, and the heavy-hole subbands which are spin-split due to the Rashba effect are not spin-polarized. This lack of spin polarization casts doubt on the viability of spin filtering using TMF in 2DHGs within conventional $\mathrm{GaAs}/{\mathrm{Al}}_{x}{\mathrm{Ga}}_{1\ensuremath{-}x}\mathrm{As}$ heterostructures. We then calculate the transport properties of the 2DHG with spin projection and offer a different perspective on interpreting and designing TMF experiments in 2DHGs.