Landé $g$ Tensor in Semiconductor Nanostructures

Understanding the electronic structure of semiconductor nanostructures is not complete without a detailed description of their corresponding spin-related properties. Here we explore the response of the shell structure of InAs self-assembled quantum dots to magnetic fields oriented in several directions, allowing mapping of the $g$-tensor modulus for the $s$ and $p$ shells. We find that the $g$ tensors for the $s$ and $p$ shells exhibit a very different behavior. The $s$ state, being more localized, probes the confinement potential details by sweeping the magnetic-field orientation from the growth direction towards the in-plane direction. For the $p$ state, the $g$-tensor modulus is closer to that of the surrounding GaAs, consistent with a larger delocalization. In addition to the assessment of the $g$ tensor, these results reveal further details of the confining potentials of self-assembled quantum dots that have not yet been probed.

Figure 1

(a) Measured data (derivative) for the polar scan, i.e., the magnetic field sweeping from parallel to the [001] direction towards the [110] direction. The gray scale is keyed to the second derivative of the capacitance spectra, with light colors indicating a higher density of states. Each of the 13 frames corresponds to $B$ sweeps from 0 to 15 T, taken at 15° intervals with $\theta$ the angle between $B$ and [001]. The $s$, $p$, and $d$ states, as well as the wetting layer 2D levels are indicated also. (b) Different capacitance spectra showing the filling of the $s$ shell with 1 and 2 electrons, dots correspond to $\mathbf{B}=0\mathrm{T}$ and triangles to $|\mathbf{B}|=15\mathrm{T}$, and $\theta =\pi /12$. The solid lines correspond to two Gaussian fits. (c) Fits to Eq. (1) for $s$ and $p$ shell electron addition energies for $\theta =\pi /2$.

Figure 2

(Top) Wave function for the $s$ and $p$ shells for three directions on the applied magnetic field $\theta =0$, $\theta =\pi /3$, and $\theta =\pi /2$. $x$, $y$, and $z$ correspond to the [110], $[1\overline{1}0]$, and [001] directions, respectively; a QD is schematically represented in the same plot. (Bottom) $g$ tensor for the $s$ and $p$ shells for the polar scan.

Figure 3

(Top) Wave function for the $s$ and $p$ shells for three directions on the applied magnetic field $\varphi =0$, $\varphi =\pi /4$, and $\varphi =\pi /2$. $x$, $y$, and $z$ correspond to the [110], $[1\overline{1}0]$, and [001] directions, respectively; a QD is schematically represented in the same plot. (Bottom) $g$ tensor for the $s$ and $p$ shells for the azimuthal scan.

Figure 4

(Left) Calculated and measured addition energies for the polar scan; (right) same for the azimuthal scan.