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Measurements of the Fermi surface are a fundamental technique for determining the electrical and magnetic properties of solids. In 2D systems, the area and diameter of the Fermi surface is typically measured using Shubnikov-de Haas oscillations and commensurability oscillations respectively. However, these techniques are unable to detect changes in the parity of the Fermi surface (i.e. when +k $\neq$ -k). Here, we show that transverse magnetic focussing can be used to detect such changes, because focussing only measures a well defined section of the Fermi surface and does not average over +k and -k. Furthermore, our results show that focussing is an order of magnitude more sensitive to changes in the Fermi surface than other 2D techniques, and could be used to investigate similar Fermi surface changes in other 2D systems.

In 2D systems with a spin-orbit interaction, magnetic focussing can be used to create a spatial separation of particles with different spin. Here we measure hole magnetic focussing for two different magnitudes of the Rashba spin-orbit interaction. We find that when the Rashba spin-orbit magnitude is large there is significant attenuation of one of the focussing peaks, which is conventionally associated with a change in the spin polarisation. We instead show that in hole systems with a $k^3$ spin-orbit interaction, this peak suppression is due to a change in the scattering of one spin state, not a change in spin polarisation. We also show that the change in scattering length extracted from magnetic focussing is consistent with results obtained from measurements of Shubnikov-de Haas oscillations. This result suggests that scattering must be considered when relating focussing peak amplitude to spin polarisation in hole systems

Magnetic focussing of charge carriers in two-dimensional systems provides a solid state version of a mass spectrometer. In the presence of a spin-orbit interaction, the first focussing peak splits into two spin dependent peaks, allowing focussing to be used to measure spin polarisation and the strength of the spin-orbit interaction. In hole systems, the k^3 dependence of the Rashba spin-orbit term allows the spatial separation of spins to be changed in-situ using a voltage applied to an overall top gate. Here we demonstrate that this can be used to control the splitting of the magnetic focussing peaks. Additionally, we compare the focussing peak splitting to that predicted by Shubnikov-de Haas oscillations and k.p bandstructure calculations. We find that the focussing peak splitting is consistently larger than expected, suggesting further work is needed on understanding spin dependent magnetic focussing.

Single holes confined in semiconductor quantum dots are a promising platform for spin qubit technology, due to the electrical tunability of the $g$-factor of holes. However, the underlying mechanisms that enable electric spin control remain unclear due to the complexity of hole spin states. Here, we study the underlying hole spin physics of the first hole in a silicon planar MOS quantum dot. We show that non-uniform electrode-induced strain produces nanometre-scale variations in the HH-LH splitting. Importantly, we find that this \RR{non-uniform strain causes} the HH-LH splitting to vary by up to 50\% across the active region of the quantum dot. We show that local electric fields can be used to displace the hole relative to the non-uniform strain profile, allowing a new mechanism for electric modulation of the hole g-tensor. Using this mechanism we demonstrate tuning of the hole $g$-factor by up to 500\%. In addition, we observe a \RR{potential} sweet spot where d$g_{(1\overline{1}0)}$/d$V$ = 0, offering a configuration to suppress spin decoherence caused by electrical noise. These results open a path towards a previously unexplored technology: engineering of \RR{non-uniform} strains to optimise spin-based devices., Comment: 12 pages, 4 figures