Electrostatic Formation of Coupled Si / SiOz Quantum Dot Systems

We present three-dimensional numerical modeling results for gated Si/SiO2 quantum dot systems in the few-electron regime. In our simulations, the electrostatic confining potential results from the Poisson equation assuming a self-consistent Thomas-Fermi charge model. We find that a very thin SiO2 top insulating layer allows an effective control with single-electron confinement in quantum dots with radius less than 10nm and investigate the detailed potential and resulting charge densities. Our threedimensional finite-element modeling tool allows future investigations of the charge coupling in gated few-electron quantum-dot cellular automata.

We present numerical simulations for electrostatically confined few-electron quantum dot systems in the technologically important Si/SiO2 material system. Our emphasis is modeling a possible socalled Quantum-dot cellular automata (QCA) structure [1] in which a bi-stable occupation by two excess electrons in a small and strongly chargecoupled quantum-dot system defines logic 0/1.
The bottom panel of Figure shows a schematic of the Si/SiO2 material system: a thin silicondioxide layer serves as excellent insulation of the bottom silicon slab from the set of top gates. Applying finite biases at these gates allows the formation of electrostatically confined quantum dots just below the heterostructure interface (at z=z0). Mesoscopic transport investigations in gate-induced quantum-dot arrays [2] documents the feasibility of fabricating few-electron quantum-dot systems in the Si/SiOz material system. This development has in turn resulted in a proposal for room-temperature single-electron Si/ SiO2 memory cells [3]. Previous (two-dimensional) modeling results [4,5] of few-electron Si/SiO. quantum dot systems exploited an axial symmetry to investigate the electrostatic confinement within an individual dot. Encouraged by recent three-dimensional modeling of larger quantum-dot systems [6] we investigate *Corresponding author. In our T 100K simulations, the confining potential is obtained from the Poisson equation within a self-consistent Thomas-Fermi charge model. The silicon is assumed to have a small unintended but fully ionized doping, p= 1015 cm -3 and to ensure convergence we investigate a 1.5 nm thick bottom slab with in-plane extension of approximately 300nm by 300nm. The top metal depletion and attractor gates are described by Dirichlet boundary conditions. For the exposed SiO2 surface we assume for simplicity a potential fixed at the mid-gap SiO2 value, that is, again a Dirichlet boundary condition.
Our finite-element calculation uses a 129 by 129 nonuniform grid to allow a nm resolution from the surface and well below the Si/SiO2 interface, that is, around the quantum dot system. The top panel of Figure 2 shows how most nodal layers (at constant z) are connected in a mesh with alternating tetrahedron orientation to eliminate a geometrical bias. The bottom panel of Figure 2 illustrates the repeated thinning of our finite-element mesh undertaken deep below the interfaces where a high in-plane resolution is no longer needed. However, our numerical simulation still involves 6105 nodal points for which we determine the electrostatic potential within the self-consistent Thomas Fermi screening model. Using the Newton-Ralphson procedure we solve in each iteration the resulting huge linear system using a quasi-minimal residual implementation [7].  Figure 4 shows our modeling results for the simpler double-quantum-dot system in which we are preparing to investigate the mutual charge coupling between the quantum dots. We assume again r-10nm attractor gates with a mutual separation of 30 nm and adjust the positive bias to achieve a single-electron equilibrium occupation of each of the quantum dots. The upper panel shows the variation of the confinement potential both along the axes (x) connecting the two quantum dots and in the growth direction (z). The heterostructure-cut panel illustrates the excellent topgate control of the electrostatic potential into the Si/SiO2 slab well below the heterostructure interface, z z0.
The lower panel of Figure 4 shows the corresponding equilibrium charge distribution, ne. Potential is calculated for an experimentally accessible gate geometry: four attractor gate of radius 10 nm with mutual 30nm separation. The upper panel shows potential dips (measured relative to Fermi energy) with corresponding contour plot and illustrates the crisp gate control allowed by the thin SiO2 top layer. The electron potential confines in equilibrium exactly four electrons with very well-defined charge distribution nel (lower panel).
Note that this electron distribution, i.e., the equilibrium quantum dot, is confined within nm of the interface and about 5 nm of the attractorgate center. Future modeling will investigate the charge coupling of such quantum-dot disks in the presence of the attractor and depletion gates.
In summary, we have presented three-dimensional finite-element calculations for gate-confined few-electron Si/SiO2 quantum-dot systems. We have documented the feasibility of crisp electrostatic gate control for also few-electron quantumdot systems and have investigated the detailed charge distribution and confinement potentials. Our modeling tool allows future investigations of charge coupling in few-electron quantum-dot cellular automata structures.  This system is defined by two 10 nm attractor dots also with a mutual separation of 30 nm. All z-positions are given relative to the interface Z=Zo. Note that the equilibrium electron distribution nel is confined within nm of the interface and about 5 nm of the attractor-gate center.