Excitation spectra of harmonic quantum dot lattices with Coulomb interaction between the dots and the broken generalized Kohn theorem

Lattices of parabolic quantum dots with different dot species per unit cell and Coulomb interaction between the dots are investigated. As examples, we solve the Schrödinger equation for square lattices with two different dots per unit cell: (i) two different circular dots, and (ii) two elliptical dots, which are rotated by $90°$ relative to each other. The interaction between the dots is considered in a dipole approximation, and excitation spectra are calculated. For vanishing momentum transfer $(\mathbf{q}=0),$ the energy spectrum of the first case can be expressed as a superposition of two noninteracting dots with an effective confinement frequency, which includes the effect of dot interaction. Only in the second case is there a splitting of degenerate absorption lines, and an anticrossing occurs, which is a qualitative indication of interdot interaction. If the interaction becomes very strong and if all lattice sites (not necessarily confinement potentials) are equivalent, then the contribution of the dot interaction outweighs possible differences in the confinement potentials and the generalized Kohn theorem gradually reenters, in the sense that one pair of excitation modes (pseudo-Kohn modes) becomes independent of the interaction strength. For finite momentum transfer $\left(\mathbf{q}\ne 0\right),$ we investigated mode softening and the influence of changing the interaction strength between dots of different sublattices. The latter effect may be implemented by putting different electron numbers in different dot species. It is shown that strengthening the next-nearest-neighbor interaction versus the nearest-neighbor interaction stabilizes the square lattice.