Level structure and spin-orbit effects in quasi-one-dimensional semiconductor nanostructures

We investigate theoretically how the spin-orbit Dresselhaus and Rashba effects influence the electronic structure of quasi-one-dimensional semiconductor quantum dots, similar to those that can be formed inside semiconductor nanorods. We calculate electronic energy levels, eigenfunctions, and effective $g$-factors for coupled, double dots made out of different materials, especially GaAs and InSb. We show that by choosing the form of the lateral confinement, the contributions of the Dresselhaus and Rashba terms can be tuned and suppressed, and we consider several possible cases of interest. We also study how, by varying the parameters of the double-well confinement in the longitudinal direction, the effective $g$-factor can be controlled to a large extent.

Figure 1

Potential-energy profile and schematic drawing of two Al/InSb coupled nanorod quantum dots. For InSb-based systems we take a well height of 100 meV, and for Al/GaAs, 220 meV. In this example, the QD width is 300 Å and barrier width 30 Å, with smoothly changing barriers over a width of a few angstroms. The drawings (a)–(c) illustrate the lateral confinement geometries described in the text.

Figure 2

Ground-state and first-excited-state energy levels of the InSb nanorod QDs shown in Fig. 1. We compare the eigenenergies of (1,4) $H_0=P_z^2∕2m^{*}+V_z\left(z\right)$ to those of (2) $H_0+H_{1dR}$, (5) $H_0+H_{1dD}$, (3) $H_0+H_{1dR}+H_Z$, and (6) $H_0+H_{1dD}+H_Z$. $B=0.2\mathrm{T}$.

Figure 3

Contribution of the Rashba term to the energy levels of InSb (a) and GaAs (b) QDs as a function of $⟨\partial V∕\partial x⟩$. GS: Ground state; 1 and 2: first and second excited states, respectively. Notice the effect is much smaller in GaAs (energy given in $\mu \mathrm{eV}$), as anticipated.

Figure 4

Contribution of the Dresselhaus term to the energy levels of InSb as a function of ${\mathit{\ell }}_x$ for the ground state (GS) and the first excited state (1) for ${\mathit{\ell }}_y=50\mathrm{\AA }$. Level splitting in GaAs is barely visible on the same scale as in InSb.

Figure 5

Effect of the Rashba Hamiltonian on the effective $g$ factor. $g^{*}∕g_0$ for the ground state for different semiconductors as a function of $⟨\partial V∕\partial x⟩$.

Figure 6

Normalized effective $g$ factor for the ground state of InSb structures with Rashba Hamiltonian. (a) For different barrier widths $w=30$, 130, and 330 Å. (b) For different sizes of the QDs. Asymmetric case: $L_{QD1}=100\mathrm{\AA }$ and $L_{QD2}=500\mathrm{\AA }$; symmetric case $L_{QD1}=L_{QD2}=300\mathrm{\AA }$.

Figure 7

Mean value of $S_z$ and effective $g$ factor for InSb systems with symmetric $V_z\left(z\right)$ (two equal dots with $L_{QD1}=L_{QD2}=300\mathrm{\AA }$). (a) $⟨S_z⟩$ as a function of $⟨\partial V∕\partial x⟩$ for the four lowest-energy doublets (pairs of Zeeman-split states). (b) $g^{*}∕g_0$ for the same states.

Figure 8

Same as Fig. 7 for asymmetric $V_z\left(z\right)$ with $L_{QD1}=100\mathrm{\AA }$ and $L_{QD2}=500\mathrm{\AA }$.