Measuring hole $g$-factor anisotropies using transverse magnetic focusing

Recent theoretical and experimental results from quasi-one-dimensional heavy-hole systems have suggested that heavy-hole gases have a strongly anisotropic $g$ factor. In this theoretical work, we propose a method for measuring this anisotropy using transverse magnetic focusing (TMF). We demonstrate that for experimentally accessible fields, the $g$-factor anisotropy leads to a relative variation in the characteristic of spin splitting of the TMF spectrum which allows for the measurement of the anisotropy of the $g$ factor. We show that this variation is insensitive to additional spin-orbit interactions, and is resolvable with current devices.

Figure 1

Magnetic focusing setup, with focusing length $l$. The source and detector in hole systems are typically quantum point contacts (QPCs).

Figure 2

Cyclotron orbits for spin down (red) and up (blue) with the spin ($〈\mathbf{\sigma }〉$) orientation in the adiabatic limit. The left panel has ${\tilde{g}}_2/{\tilde{g}}_1=0.5$, while the right panel has ${\tilde{g}}_1/{\tilde{g}}_2=0.5$. We note that while the Fermi surfaces have a nearly identical shape, due to the different momentum dependence of the two interactions, the spin dynamics are qualitatively different.

Figure 3

Cartoon of the variation in the focusing splitting. The red dashed curve is the TMF spectrum without any $g$-factor asymmetry, with splitting magnitude ${\mathrm{\Delta }}_0$, with ${\mathrm{\Delta }}_0={\tilde{g}}_1{\mu }_B{B}_{\left|\right|}/{\varepsilon }_F$. The blue curve is the TMF spectrum, including the $g$-factor anisotropy, with $\phi =0$ for the left panel, and $\phi =\pi /2$ for the right panel. The focusing field without spin splitting, $B_z=2k_F/el$, is marked with a dashed black line.

Figure 4

The field angle dependence of the first focusing peak splitting, with increasing in-plane magnetic field, from 1 to 3.5 T, with ${\tilde{g}}_2=1$ and ${\tilde{g}}_1=-2.5$. The deviation from Eq. (14) is the result of the nonparabolic terms in the dispersion.

Figure 5

Relative focusing field splitting as a function of the in-plane magnetic field angle $\phi$ with $B_{\left|\right|}=2.5\mathrm{T}$, ${\tilde{g}}_1=-2.5$, ${\tilde{g}}_2=1$. Here, we vary the strength of the Rashba interaction ${\mathrm{\Delta }}_R$ in the range $0<{\mathrm{\Delta }}_R<0.1{\epsilon }_F$. The maximum value is ${\mathrm{\Delta }}_R\sim 0.2\mathrm{meV}$.