Interactions of metallic quantum dots on a semiconductor substrate

Experiments have shown that uniform metallic quantum dots may self-assemble on a semiconductor substrate. The observation calls for a repulsive force when the dots are close. In a traditional quantum dot system, such as Ge dots on a Si substrate, such an action is achieved by elastic interaction. This paper proposes a mechanism for metallic dots without coherent lattice or lattice mismatch, so that elastic effect may not account for the phenomena. We show that electric double layers due to contact potential can lead to size-dependent repulsion, which counterbalances van der Waal attraction and determines feature sizes.

Figure 1

Contour of $4.5\%{\psi }_s$ for disks with radius (a) $0.232l_D$ $\left(30\mathrm{nm}\right)$, (b) $0.773l_D$ $\left(100\mathrm{nm}\right)$, (c) $2.32l_D$ $\left(300\mathrm{nm}\right)$, (d) $7.73l_D$ $\left(1\mu \mathrm{m}\right)$, (e) $12.36l_D$ $\left(1.6\mu \mathrm{m}\right)$, (f) infinite contact.

Figure 2

Large disks show a linear relationship between electrostatic free energy and disk area. The relation is nonlinear for small disks.

Figure 3

Electrostatic interaction as a function of disk separation.

Figure 4

The electrostatic and van der Waals energies for disks with (a) various radii (b) radius of $0.08l_D$ for various contact potentials.