Size, shape, and strain dependence of the $g$ factor in self-assembled In(Ga)As quantum dots

We have investigated Zeeman splitting in single self-assembled InAs and InGaAs quantum dots experimentally and theoretically. By measuring photoluminescence from single dots, in a wide spectral region, we have obtained the exciton $g$ factors of quantum dots with various photoluminescence energies. We find that the absolute value of the exciton $g$ factors of InAs dots are smaller than those of the InGaAs dots, which differs from the composition dependence expected from that of the bulk ones. The experimentally obtained $g$ factors are compared with calculated ones based on the eight-band $\mathbf{k}\bullet \mathbf{p}$ model where the influence of strain and the Zeeman effect are included. We find a good agreement between the calculation and the experiment qualitatively and quantitatively. The calculation reproduces the nontrivial composition dependence of the $g$ factor of the quantum dots. In addition, the eight-band model predicts a size and shape dependence of the electron and hole $g$ factors of the pyramidal quantum dots.

Figure 1

(a) PL spectra of the InGaAs dots as a function of magnetic field from $0\text{to}5\mathrm{T}$. All the spectra are obtained without polarization selectivity except the upper two traces that are recorded for ${\sigma }^{+}$ and ${\sigma }^{-}$ circular polarizations, respectively. (b) The center position of the doublet in (a), or the diamagnetic shift, plotted as a function of magnetic field. The solid curve is a quadratic fit. (c) Peak positions after the subtraction of the diamagnetic shift, plotted as a function of magnetic field. Open (closed ) circles represent the energy of ${\sigma }^{-}$ $\left({\sigma }^{+}\right)$ polarized emissions. The solid lines are linear fits to the data.

Figure 2

Exciton $g$ factors plotted as a function of the ground state emission energy. Open (closed) circles represent the experimental $g$ factor of InGaAs (InAs) dots. Solid (dashed) curve gives the calculation made for the dots with a base length of $15\mathrm{nm}$ and a height of $7.5\mathrm{nm}$ $\left(4.3\mathrm{nm}\right)$.

Figure 3

Magnetic field dependence of the energy of (a) the lowest CB state and (b) the highest VB state. The zero of the energy scale is taken at the VB edge of unstrained GaAs. (c) Magnetic field dependence of the calculated energy of the center of the spin-splitted transition energies. The solid curve shows a quadratic fit to the calculated energies. (d) Magnetic field dependence of the calculated transition energies of ${\sigma }^{+}$ polarized emission (closed symbols) and ${\sigma }^{-}$ polarized emission (open symbols). The solid lines are linear fits to the data.

Figure 4

Calculated $g$ factors of the lowest CB state $\left(g_c\right)$, the highest VB state $\left(g_v\right)$, and the electron-hole pair $\left(g_{\mathrm{ex}}\right)$, as a function of the emission energy of the quantum dots with a base length of $15\mathrm{nm}$ and a height of $7.5\mathrm{nm}$. The lines are to guide the eye. The composition of the quantum dots, ranging from pure InAs to ${\mathrm{In}}_{0.5}{\mathrm{Ga}}_{0.5}\mathrm{As}$, are represented by the emission energy.

Figure 5

Calculated $g$ factors of (a) the lowest CB state and (b) the highest VB state for quantum dots with various size and shape, plotted as a function of the emission energy. The dot size and shape are labeled by the base length $\left(b\mathrm{nm}\right)$ and the height $\left(h\mathrm{nm}\right)$. The composition of the quantum dots, ranging from pure InAs to maximum allowable Ga-rich InGaAs for carriers to be confined, are represented by the emission energy.

Figure 6

Size dependence of the $g$ factors in the pyramidal InAs dots with a base to height ratio of 2.