Mechanisms for Strong Anisotropy of In-Plane $g$-Factors in Hole Based Quantum Point Contacts

In-plane hole $g$ factors measured in quantum point contacts based on $p$-type heterostructures strongly depend on the orientation of the magnetic field with respect to the electric current. This effect, first reported a decade ago and confirmed in a number of publications, has remained an open problem. In this work, we present systematic experimental studies to disentangle different mechanisms contributing to the effect and develop the theory which describes it successfully. We show that there is a new mechanism for the anisotropy related to the existence of an additional $B_{+}{k}_{-}^4{\sigma }_{+}$ effective Zeeman interaction for holes, which is kinematically different from the standard single Zeeman term $B_{-}{k}_{-}^2{\sigma }_{+}$ considered until now.

Figure 1

Panels (a),(b): Grey-scale plots of the transconductance showing Zeeman spin splitting of 1D hole subbands in a magnetic field applied parallel and perpendicular the QPC. Panel a (Panel b) corresponds to the electric field along the $z$ axis $E_z=1.2\times {10}^6\mathrm{V}/\mathrm{m}$ ($E_z=2.5\times {10}^6\mathrm{V}/\mathrm{m}$). Panel (c): Absolute values of the subband $g$ factors extracted from data in panels (a) and (b). The circles (squares) correspond to the direction of magnetic field along (perpendicular) the QPC. The red (blue) symbols correspond to the out-of-plane electric field $E_z=1.2\times {10}^6\mathrm{V}/\mathrm{m}$ ($E_z=2.5\times {10}^6\mathrm{V}/\mathrm{m}$).

Figure 2

Panel (a): Functions $g_1$ (solid lines) and $g_2$ (dashed lines) for the 2D system versus momentum, see Eqs. (3) and (8). Panel (b): Functions $g_{||}^{(0)}$ (solid lines) and $g_{\perp }^{(0)}$ (dashed lines) for the 1D system versus momentum, see Eqs. (4). Both panels are calculated for a rectangular GaAs quantum well of width $d=15\mathrm{nm}$ with a superimposed electric field $E_z$. We present plots for $E_z=0$, 1, 2, $3\mathrm{MV}/\mathrm{m}$, with black, red, green, and blue lines, respectively. The value of the electric field in MV/m is pointed out near each line.

Figure 3

(a) Self-consistent QPC transverse potential for 1D channels with $n=1$, 3, 5, 8 subbands occupied. (b) QPC $g$ factors $g_{\parallel }$ and $g_{\perp }$ for $n\ge 3$ versus electric field. The heterostructure is modeled as a 15 nm wide infinite rectangular quantum well with a superimposed electric field $E_z$. The hole density in the 2D leads is $1.1\times {10}^{11}{\mathrm{cm}}^{-2}$. The dashed lines account only for the leading spherical contribution. The solid ([110] QPC orientation) and dotted ([100] QPC orientation) lines account for the leading contribution and for the first subleading one proportional to ${\gamma }_3-{\gamma }_2$.