Magnetic field induced valence band mixing in [111] grown semiconductor quantum dots

We present a microscopic theory of the magnetic field induced mixing of heavy-hole states $\pm 3/2$ in GaAs droplet dots grown on (111)A Ga-rich surfaces. The proposed theoretical model takes into account the striking dot shape with trigonal symmetry revealed in atomic force microscopy. Our calculations of the hole states are carried out within the Luttinger Hamiltonian formalism, supplemented with allowance for the triangularity of the confining potential. They are in quantitative agreement with the experimentally observed polarization selection rules, emission line intensities and energy splittings in both longitudinal and transverse magnetic fields for neutral and charged excitons in all measured single dots.

Figure 1

AFM images of GaAs quantum dots grown on (111)A substrates reveal the triagonal symmetry of the dots, with shapes varying from irregular hexagons to equilateral triangles depending on crystallisation temperature.

Figure 2

(a) (QD1 left) Measured transition energies as a function of the applied magnetic field $\mathit{B}\parallel \left[111\right]$. (Right) Measurements for $\mathit{B}\perp \left[111\right]$. (b) Same as (a), but for QD5. (c) Polarization selection rules for X${}^{+}$ recombination in the Faraday geometry ($B\parallel \left[111\right]$) with mixed hole states, see Eq. (2), where $C_{1,2}$ are coefficients determining hole eigenstates. (d) Polarization selection rules for X${}^{+}$ recombination in the Voigt geometry ($B\perp \left[111\right]$).

Figure 3

QD6 (a) Transition energy of the nominally dark (squares) and bright (circles) neutral exciton X${}^0$ state as a function of the applied magnetic field (Faraday geometry). Left- and right-hand sides represent the states with $s_z=1/2$ and $s_z=-1/2$, respectively. Fit (green curve) of the data using Eq. (3) allows to extract ${\delta }_0=422$ $\mu$eV, $g_h=1.16$, $g_e=0.49$, $|g_{h1}|=1.05$ and $g_{h2}=0.42$. The fit included the diamagnetic shift for both curves as $E=E_0+\alpha B_z^2$ with $E_0=1.819676$ eV and $\alpha =4.15$ $\mu$eV/T${}^2$. Blue curve: identical parameters apart from $g_{h2}=0$. (b) X${}^0$ exciton bright splitting as a function of the magnetic field. Solid line is the fit obtained by using Eq. (3).

Figure 4

Equipotential surface of the dot confining potential for the two values of triangularity parameter $\beta =0$ (a) and $\beta =-0.35$ (b).

Figure 5

The off-diagonal $g$ factor $g_{h2}$ as a function of $L/a$ depicted for three values of the trigonal parameter $\beta$: $-0.1$, $-0.15,$ and $-0.2$. The inset shows the behavior of $g_{h1}$.

Figure 6

Correlation between the values of $g_{h1}$ and $|g_{h2}|$. Symbols represent experimental data for different GaAs dots measured on the neutral as well as on charged complexes, dashed curve stands for theoretical fit with parameters $\beta =-0.15$, $g_{h1}^0=-1.5$, $c_1=6.7$, $c_2=10.2$. Thin lines are calculated for the same set of parameters, except for $\beta =-0.13$ and $-0.17$. The corresponding values of $L/a$ (for $\beta =-0.15$) are presented at the right-hand axis.