Asymmetry of single-particle hole states in a strained Ge/Si double quantum dot

A six-band $\mathbf{k}\cdot \mathbf{p}$ formalism was used to study single-particle hole states of two vertically aligned pyramidal Ge quantum dots embedded in Si and separated by a distance $t_{\text{Si}}$. The elastic strain due to the lattice mismatch between Ge and Si was included into the problem via Bir-Pikus Hamiltonian. The three-dimensional spatial strain distribution was obtained by finite element method. We found that at small interdot separation $\left(t_{\text{Si}}<3.5\text{nm}\right)$, when the quantum-mechanical coupling between the dots is significant, the molecule-type hole orbitals delocalized fairly over the two dots are formed. The ground (excited) states correspond to symmetric ${\sigma }_S$ (antisymmetric ${\sigma }_{\text{AS}}$) linear combination of single-dot states. However the splitting of ${\sigma }_S$ from ${\sigma }_{\text{AS}}$ is not symmetric, the average hole binding energy decreases with decreasing interdot separation. Strain effects start to play the dominant role at larger $t_{\text{Si}}$. In this region hole wave functions are localized on different dots, showing symmetry breaking. The most interesting property of energy spectrum is the crossing of levels with different symmetry which occurs with changing $t_{\text{Si}}$. At $t_{\text{Si}}>3.5\text{nm}$, ${\sigma }_{\text{AS}}$ becomes the ground state of the system, replacing ${\sigma }_S$.

Figure 1

Schematic picture of a Ge/Si double QD used for simulation of hole states.

Figure 2

(a) Evolution of hole binding energy in symmetric $\left({\sigma }_S\right)$ and antisymmetric $\left({\sigma }_{\text{AS}}\right)$ states as a function of the distance between QDs. As reference, also the ground-state energy of isolated single dot $E_{\text{dot}}$ is shown. All energies are counted with respect to the valence-band edge in bulk Si. The dashed line is the average of $E_{{\sigma }_S}$ and $E_{{\sigma }_{\text{AS}}}$. The ${\sigma }_S$ and ${\sigma }_{\text{AS}}$ cross at about 3.9 nm, as indicated by the arrow. (b) The energy difference between symmetrical and antisymmetrical states ${\Delta }_{\text{SAS}}=\left|E_{{\sigma }_S}-E_{{\sigma }_{\text{AS}}}\right|$ (filled squares) and between the isolated levels $\delta$ (empty squares). The dashed line corresponds to ${\Delta }_{\text{SAS}}=3.7\text{exp}\left(-t_{\text{Si}}/0.5\right)\text{eV}$.

Figure 3

Spatial distributions of the ground-state (left panel) and excited-state (right panel) hole wave functions in the $\left(yz\right)$ plane, where the $z$ axis goes though the vertical symmetry axis of the pyramids. Here we show only real part of the $|y⟩$ component calculated in a basis of Bloch functions.

Figure 4

Biaxial component of the strain in single dot (solid line) and double-dot structures for two selected interdot distances of 2 nm (dashed-dotted line) and 3 nm (dashed line) as a function of the position from the bottom boundary of the bottom wetting layer along the $z$ axis through the pyramid apex.

Figure 5

Contribution of the HH, LH, and SO states for ${\sigma }_S$ and ${\sigma }_{\text{AS}}$ hole orbitals.

Figure 6

Calculated spatial distribution of the relative contribution of HH, LH, and SO components in the wave function of the hole ground state for a single Ge/Si dot. The profiles are plotted along the vertical symmetry axis $z$ of a pyramid. The shaded region corresponds to Ge pyramid and wetting layer.