Electron states in a double quantum dot with broken axial symmetry

We study theoretically the electron states in a system of two vertically stacked quantum dots. We investigate the influence of the geometrical symmetry breaking (caused by the displacement as well as the ellipticity of the dots) on the electron states. Our modeling is based on the eight-band $\mathit{k}·\mathit{p}$ method. We study the coupling of the $s$ state from one dot with the $p$ and $d$ states from the other induced by the absence of axial symmetry. Our findings indicate that this coupling can produce a strong energy splitting at resonance (on the order of several meV) in the case of closely spaced quantum dots. Furthermore, we show that in the presence of a piezoelectric field, the direction of the displacement plays an important role in the character of the coupling.

Figure 1

The in-plane probability density of the six lowest electron states. The first column corresponds to a circular lens-shaped QD in an ideal case (without a PZ field). The second one shows the same QD but in the presence of the PZ field. The third column presents the results for an elliptical QD without a PZ field. The last column contains results for an elliptical QD with the PZ field.

Figure 2

Schematic electron energy structure in the investigated DQD without (a) and with different values of an electric field (b)–(d) indicating resonances between energy levels in the two dots.

Figure 3

(a) Lowest electron energy branches as a function of the electric field without a PZ field. (b) The same as (a) but with the piezoelectric field included. (c) Enlarged region of (a) with $s$-$d$ resonances (indicated by the blue rectangle). (d) Enlarged region of (b) with $s$-$d$ resonances.

Figure 4

Energy branches of $s$ and $p$ states (a) without piezoelectric field for a shift of $x_{\mathrm{s}}=1.8$ nm, (b) with included piezoelectric field and a shift of $x_{\mathrm{s}}=1.8$ nm, (c) with included piezoelectric field and $x_{\mathrm{s}}=y_{\text{s}}=1.8$ nm, and (d) with included piezoelectric field and $x_{\mathrm{s}}=1.8$ nm, $y_{\mathrm{s}}=-1.8$ nm.

Figure 5

(a) The values of the energy splitting at the resonances between the $s$ and lower $p$ state (red circles) and higher $p$ state (blue boxes) for $D=10.2$ nm as a function of the value of the relative displacement of the lower dot $r_{\mathrm{s}}$. (b) The values of the energy splitting at the resonances between the $s$ state and lower $p$ state for $x_{\mathrm{s}}=y_{\mathrm{s}}=1.8$ nm, i.e., $r_{\mathrm{s}}=14$% as a function of $D$. The red points represents the simulation results and the blue dashed line is an exponential fitting.

Figure 6

(a) Energy branches as a function of the electric field in the region of the $s$-$d$ resonances at $x_{\mathrm{s}}=1.8$ nm with $y_{\mathrm{s}}=0$. (b) Energy branches as a function of electric field in the region of the $s$-$d$ resonances in the case of elliptical dots.