Anisotropic $g$ factor in InAs self-assembled quantum dots

We investigate the wave functions, spectrum, and $g$-factor anisotropy of low-energy electrons confined to self-assembled, pyramidal InAs quantum dots (QDs) subject to external magnetic and electric fields. We present the construction of trial wave functions for a pyramidal geometry with hard-wall confinement. We explicitly find the ground and first excited states and show the associated probability distributions and energies. Subsequently, we use these wave functions and 8-band $\mathit{k}·\mathbf{p}$ theory to derive a Hamiltonian describing the QD states close to the valence band edge. Using a perturbative approach, we find an effective conduction band Hamiltonian describing low-energy electronic states in the QD. From this, we further extract the magnetic field dependent eigenenergies and associated $g$ factors. We examine the $g$ factors regarding anisotropy and behavior under small electric fields. In particular, we find strong anisotropies, with the specific shape depending strongly on the considered QD level. Our results are in good agreement with recent measurements [Takahashi et al., Phys. Rev. B 87, 161302 (2013)] and support the possibility to control a spin qubit by means of $g$-tensor modulation.

Figure 1

Sketch of the QD geometry and the coordinate system used in this work with $x$, $y$, and $z$ axes pointing along the growth directions [100], [010], and [001], respectively. The externally applied fields under consideration are $\mathit{B}=(B_x,B_y,B_z)$ and $\mathit{E}=(E_x,0,0)$.

Figure 2

Probability distributions of the smallest nontrivial set of trial wave functions ${\psi }_{\mathit{m}}\left(\mathit{r}\right)$, i.e., $max\left(m_i\right)=3$, satisfying the hard-wall boundary conditions for the geometry given in Fig. 1. We show contour plots of $|{\psi }_{\mathit{m}}{\left(\mathit{r}\right)|}^2=0.1$ inside the pyramidal geometry assumed for the QD; see Fig. 1. Note the degenerate pairs: ${\psi }_{321}$ and ${\psi }_{231}$, ${\psi }_{312}$ and ${\psi }_{132}$, ${\psi }_{213}$ and ${\psi }_{123}$.

Figure 3

Left: Spectrum of the lowest six QD levels $E_n$ of ${\tilde{H}}_d^{\mathrm{CB}}$ as a function of the magnetic field $\mathit{B}=(0,0,B_z)$, where we increase $\left|\mathit{B}\right|=0\mathrm{T}\mathrm{to}1\mathrm{T}$. We assume $\mathit{E}=0$. Right: Enlarged plots of the $\mathit{B}$-dependent splitting of the single-QD levels. For most QD levels, except for $n=5$, we observe a nonlinear dependence of $E_n^{\pm }$ on $\mathit{B}$.

Figure 4

Ground state $g$ factor $|g_1|$ as a function of the magnetic field direction for $\left|\mathit{B}\right|=1\mathrm{T}$ shown in (a) 3D plot, and cuts along the planes (b) $xy$, and (c) $(x-y)z$ with electric field $\mathit{E}=(E_x,0,0)$.

Figure 5

$|g_2|$ and $|g_3|$ as functions of the magnetic field direction for $\left|\mathit{B}\right|=1\mathrm{T}$ shown in 3D plots for (a) $n=2$, (b) $n=3$, and cuts along the $xz$ plane for (c) $n=2$ and (d) $n=3$ with electric field $\mathit{E}=(E_x,0,0)$.

Figure 6

$|g_4|$ as a function of the magnetic field direction for $\left|\mathit{B}\right|=1\mathrm{T}$ shown in (a) 3D plot, and cuts along the planes (b) $xy$, and (c) $(x-y)z$ with electric field $\mathit{E}=(E_x,0,0)$.

Figure 7

$|g_5|$ as a function of the magnetic field direction shown for $\left|\mathit{B}\right|=1\mathrm{T}$ in cuts along the planes (a) $xy$, and (b) $xz$ with electric field $\mathit{E}=(E_x,0,0)$. Here we omit the 3D plot since the $g$ factor shows a spherical distribution, where such a plot does not yield further insight.

Figure 8

$|g_6|$ as a function of the magnetic field direction for $\left|\mathit{B}\right|=1\mathrm{T}$ shown in (a) 3D plot, and cuts along the planes (b) $xy$ and (c) $(x+y)z$ with electric field $\mathit{E}=(E_x,0,0)$. Note that the color scale changed because $|g_6|$ reaches larger values than the $|g_n|$ of the QD levels with $n<6$.

Figure 9

We span the pyramid volume by multiplying two upright isosceles triangles (red and blue). Note that $\tilde{\mathit{r}}$ and $\tilde{a}$ correspond to $\mathit{r}$ and $a$ in the main text, respectively.