Orbital Contributions to the Electron $g$ Factor in Semiconductor Nanowires

Recent experiments on Majorana fermions in semiconductor nanowires [S. M. Albrecht, A. P. Higginbotham, M. Madsen, F. Kuemmeth, T. S. Jespersen, J. Nygård, P. Krogstrup, and C. M. Marcus, Nature (London) 531, 206 (2016)] revealed a surprisingly large electronic Landé $g$ factor, several times larger than the bulk value—contrary to the expectation that confinement reduces the $g$ factor. Here we assess the role of orbital contributions to the electron $g$ factor in nanowires and quantum dots. We show that an $\mathbf{L}·\mathbf{S}$ coupling in higher subbands leads to an enhancement of the $g$ factor of an order of magnitude or more for small effective mass semiconductors. We validate our theoretical finding with simulations of InAs and InSb, showing that the effect persists even if cylindrical symmetry is broken. A huge anisotropy of the enhanced $g$ factors under magnetic field rotation allows for a straightforward experimental test of this theory.

Figure 1

(a) Evolution of the energy levels at $k_z=0$ in cylindrical symmetry when SOC is turned on. (b) Energy levels of a cylindrical InSb wire with 40 nm diameter in an axial magnetic field. (c) Zoom in on the $|l|=1$ states marked by the gray rectangle in (b). Dashed lines are $l=+1$, and solid lines $l=-1$, states. The spin alignments are marked by the small arrows and the vertical dashed red line marks $B_{\mathrm{crit}}$.

Figure 2

(a) Diameter dependence of the SOC splitting $\mathrm{\Delta }{E}_1$ and $\mathrm{\Delta }{E}_2$ for InSb and InAs wires. (b) Diameter dependence of the critical magnetic field $B_{\mathrm{crit}}$ defined in Fig. 1. (c) [(d)] Effective $g$ factors at infinitesimal magnetic field of the first five subbands of an InSb (InAs) wire: $l=0,|f|=\frac{1}{2}$ (blue), $|l|=1,|f|=\frac{1}{2}$ (green), $|l|=1,|f|=\frac{3}{2}$ (red), $|l|=2,|f|=\frac{3}{2}$ (cyan), and $|l|=2,|f|=\frac{5}{2}$ (magenta). The dashed lines in the corresponding colors are the prediction of Eq. (2) where we substituted bulk values.

Figure 3

(a) Energy levels of the lowest $|l|=1$ states as a function of $B$ in a tight-binding simulation of a hexagonal InSb wire of 20.1 nm diameter, grown in the 111 direction. (b) SOC splitting as a function of diameter in a cylindrical wurtzite InAs wire.

Figure 4

(a) Magnetic and electric field directions in the hexagonal 111 wire. (b) The $g$ factors measured at 0.2 Tesla ($\alpha =0$) of a hexagonal InSb wire with 40 nm diameter as a function of a perpendicular electric field. (c) [(d)] Energy levels of the $|l|=1$ states as a function of $B$ at an electric field of $\mathcal{E}=0\mathrm{meV}/\mathrm{nm}$ ($\mathcal{E}=3\mathrm{meV}/\mathrm{nm}$). (e) The $g$ factors as a function of $\alpha$ measured at 0.2 Tesla in a hexagonal InSb wire with 40 nm diameter. In (b) and (e) the color code is the same as in Figs. 2 and 2.

Figure 5

(a) [(b)] shows the local density of states (LDOS) at the end of an InAs wire with 40 nm diameter and 2172 nm length in an electric field of $\mathcal{E}=1.2\mathrm{meV}/\mathrm{nm}$ and proximity effect induced superconducting pairing $\mathrm{\Delta }=0.2\mathrm{meV}$. The chemical potential $\mu =39.6\mathrm{meV}$ ($\mu =68.5\mathrm{meV}$) is tuned to the $|l|=1$ ($|l|=2$) subbands. The slope of the whites lines amounts to a $g$ factor of 23 (43).