Exact Coulomb cutoff technique for supercell calculations in two dimensions

We present a reciprocal space technique for the calculation of the Coulomb integral in two dimensions in systems with reduced periodicity, i.e., finite systems, or systems that are periodic only in one dimension. The technique consists of cutting off the long-range part of the interaction by modifying the expression for the Coulomb operator in reciprocal space. The physical result amounts in an effective screening of the spurious interactions originated by the presence of ghost periodic replicas of the system. This work extends a previous report [C. A. Rozzi et al., Phys. Rev. B 73, 205119 (2006)], where three-dimensional systems were considered. We show that the use of the cutoffs dramatically enhances the accuracy of the calculations, and it allows description of two-dimensional systems of reduced periodicity with substantially less computational effort. In particular, we consider quantum-dot arrays having potential applications in quantum information technology.

Figure 1

(Color online) Fermi energy of a quantum-dot chain with three electrons per dot as a function of the supercell size perpendicular to the chain.

Figure 2

(Color online) Band structure at lattice constants $a_x=5.06$ (left) and $a_x=8.55$ (right) for a quantum-dot chain with three electrons in each dot. The dotted lines in the left panel correspond to the result with multiple parallel chains at $y=n{a}_x=n{a}_y$, $n=\pm 1,\pm 2,\dots$. The straight lines correspond to the Fermi energies.

Figure 3

(Color online) Band structure at a lattice constant $a_x=5$ for quantum-dot chains with two electrons in each dot. The left and right panels show the results for single and multiple chains, leading to metallic and insulating band structures, respectively. In the latter case, the parallel chains are located at $y=n{a}_x=n{a}_y$, $n=\pm 1,\pm 2,\dots$.