Spontaneous Magnetism of Quantum Dot Lattices

The magnetism of square lattices of quantum dots with up to 12 electrons per dot is studied using the spin-density functional formalism. At small values of the lattice constant, all lattices are nonmagnetic and gapless. When the lattice constant is increased, the shell structure of the single dots governs the magnetism of the lattice. At closed shells, the lattices are nonmagnetic and have a gap at the Fermi level. At the beginning and at the end of a shell, they become ferromagnetic and stay gapless up to large values of the lattice constant. Antiferromagnetism was observed only at midshell after a band gap was opened.

Figure 1

Schematic picture of the square lattice of quantum dots with the unit cell (green) containing two dots, allowing antiferromagnetic (AF) and ferromagnetic (F) order, as shown at the right-hand side.

Figure 2

Magnetism of square lattices of quantum dots with $N\le 12$ electrons per dot, as a function of the lattice constant $a$ (in $a_B^{*}$). ($r_s=2a_B^{*}$, see text.) The numbers on the right give the ground-state spin of a single quantum dot. The gapless phase is shown as blue and the phase with a Fermi gap as yellow. The arrows indicate antiferromagnetic or ferromagnetic order of the quantum dot spins. The dashed line is a parabola fitted to the transition points of the dots with half-filled shells.

Figure 3

$2s$- and $1d$-electron bands of the antiferromagnetic dot lattice with eight electrons per dot ($a=19a_B^{*}$, $r_s=2a_B^{*}$). The Brillouin zone is shown in the inset. The right-hand side shows the symmetry and spin of the single dot state responsible for the band (assuming that we look at the states mainly localized at sites where the majority spin is up). The Fermi level is denoted by a dashed line. (The energy is given in effective atomic units.)

Figure 4

Total electron density for $N=9$ ($a=19a_B^{*}$) and spin density ($n_{\uparrow }-n_{\downarrow })$ for the antiferromagnetic $N=9$ and $N=8$ ($a=19a_B^{*}$) and ferromagnetic $N=7$ ($a=18a_B^{*}$) quantum dot lattices. Note that four unit cells are shown. (The maximum electron density for $N=9$ is about $1/\pi r_s^2$, with $r_s=2a_B^{*}$. The spin densities are scaled by a factor of 2.5.)

Figure 5

Fermi gap for quantum dot lattices with $N=4$, 6, 9, and 12 electrons per dot as a function of the overlap parameter $\Delta$. Lower panel: Spin per dot in the magnetic phases for $N=4$, 5, 9, and 10, as calculated over a single-dot primitive cell and shown as absolute values in the antiferromagnetic cases. The points show the calculated values; the lines are guides for the eye.