Gauge-invariant discretization in multiband envelope function theory and $g$ factors in nanowire dots

We present a gauge-invariant discretization scheme for the multiband envelope function approximation including strain as well as relativistic effects. Our procedure is based on Wilson’s formulation of gauge theories. The magnetic field couples to the envelope functions via phase factors that result from spatial discretization of the gauge covariant derivative. These phase factors contain a discretized curve integral over the vector potential. In addition, the carrier’s spin couples to the magnetic field via a Zeeman term. In the case of infinitesimal grid spacings, our method becomes equivalent to the minimal substitution method. Applying our procedure, we calculate the effective electron and hole $g$ tensor of InAs/InP nanowire dots and obtain excellent agreement with experimental data. We show that the correct momentum operator ordering in the Hamiltonian grossly affects the hole $g$ factors. Furthermore, we investigate the influence of strain on $g$ factors and nonlinear Zeeman splittings in high magnetic fields.

Figure 1

Alternative integration paths in the discretization of mixed second-order derivatives.

Figure 2

Illustration of the integration paths associated with (a) the connection $U\left(\mathbf{m},\mathbf{n},ij\right)$ and (b) its Hermitian conjugate $U\left(\mathbf{n},\mathbf{m},ji\right)$.

Figure 3

Comparison of calculated strained (solid lines), calculated unstrained (dashed lines), and experimental (circles) effective electron $g$ factors for wire dots of length $L$. The $g_{\parallel }$ and $g_{\perp }$ components correspond to the magnetic-field lying parallel and perpendicular to the wire axis, respectively. Experimentally, only the $g_{\perp }$ components have been determined (Ref. 12).

Figure 4

Comparison of calculated strained (solid lines) and unstrained (dashed lines) effective hole $g$ factors for wire dots of length $L$ and magnetic-fields pointing perpendicular $\left(g_{\perp }\right)$ and parallel $\left(g_{\parallel }\right)$ to the wire axis. The dotted line illustrates the importance of the correct operator ordering in Eq. (A3) by showing the unstrained $g_{\parallel }$ for a Hermitian but naively and therefore incorrectly symmetrized Hamiltonian.

Figure 5

(a) Cross section of calculated strain components along the wire axis $z$ for a dot with a length of $L=20\text{nm}$ and a barrier thickness of $w=6\text{nm}$. (b) Calculated energies of conduction band (cb), heavy hole (hh), and light hole (lh) valence-band edges.

Figure 6

(a) Calculated electron (dashed lines) and hole (solid lines) Zeeman splittings and (b) effective $g$ factors for electrons and holes as a function of the magnetic-field lying parallel to the wire axis for a dot of length $L=10\text{nm}$.

Figure 7

Cross section of the density of the electron ground state at the dot center in units of ${10}^{18}{\text{cm}}^{-3}$ for different magnetic fields in tesla. The density of the ground state shows a noticeable deformation if the ratio $\alpha =\hslash {\omega }_c/\Delta E$ is close to one. (a) $\alpha =0$, (b) $\alpha =0.2$, and (c) $\alpha =0.8$.