Quantum Information in Silicon Devices Based on Individual Dopants

Abstract

Several solid-state quantum information processor proposals are based on silicon [16-18, 20, 52], including the original architecture conceived by Kane in 1998 [22], in which the quantum information was stored in the spin states of 31P donors. The interest raised by silicon in the field of quantum information is twofold. On the one hand, it represents the main semiconductor used for large- scale fabrication. On the other hand, it owns interesting physical properties suitable for preserving andmanipulating quantum states, such as long coherence times and relatively low disturbance of nuclear spins, which are further enhanced by employing purified 28Si. The employment of individual donors in silicon covers all the aspects of quantum information encoding and processing, like quantumbit (qubit) storage, implementation of quantum logic gates, coherent transfer of quantum states, and readout schemes. This chapter is devoted to the physics of few dopants in nanometric silicon devices and their applications to quantum information processing purposes. In the first part, an overview of the physics of themost common donors and acceptors in silicon is presented. Next, the quantum information key concepts and their implementation in donor-based architectures are discussed. Finally, decoherence effects and the concept of measurement of quantum states are presented for the case of electron states bound to donors in silicon. Between the sixties and the early eighties of the twentieth century, all the theoretical and experimental aspects related to the doping of bulk silicon have been investigated and explained. A silicon crystal consists of a diamond lattice constituted by two interleaved cubic lattices (face-centered cubic, FCC), the second of which has the origin in the center of the tetrahedron given by the origin (0,0,0) and the centers of the faces 100. Each Si atom has four valence electrons, which create a covalent bond with the shared electrons of four other Si atoms. The substitution of silicon atoms with atoms of a group V element (indicated as donors, typically P, As, and Sb in silicon) generates an n-doped silicon crystal (n-Si). Each donor introduces an eccess electron in the crystal, and it provides new energy levels in the band gap, close to the conduction band edge. Similarly, substituting silicon atoms with atoms of a group III element (an acceptor, typically B in silicon) generates a p-doped silicon crystal (p-Si). Each acceptor introduces an electron hole in the crystal, and it provides new energy levels in the band gap, close to the valence band. The diffusion of impurities in semiconductors alters the conduction properties from an insulating to a metallic regime. Depending on the different doping concentration, the wavefunction of the electron (hole) states introduced by each donor (acceptor) may overlap negligibly or substantially with those of neighboring sites. Such a transition from low doping to high doping as a function of the average distance between neighboring sites is described in terms of an Anderson-Mott transition and produces additional impurity bands (Hubbard bands) below the conduction band edge at sufficiently high concentration [1, 34]. The transport is governed by mechanisms based on localized states at low density, while it is based on delocalized states at high density. Four conventional regimes of impurity concentration are defined [50]. The dilute concentration of impurities holds for n < 1 · 1016 cm−3. There, the problem of the donor is that of a hydrogen atom with a scaled Rydberg and radius. Between the densities n of 1 ·1016 cm−3 and 2 · 1017 cm−3 (rNN = 11.9 nm), respectively, the regime is called semidilute, and it is characterized by formation of pairs. Above 2 · 1017 cm−3 and below the metallic behavior, which occurs at 3.7 · 1018 cm−3, the regime is called intermediate, and the formation of random clusters leads to effects generally accounted by the Hubbard band formation. Above n > 3.7 · 1018 cm−3, silicon is treated as a metal. In this section the physics of the first three regimes of donor concentration and the Anderson-Mott transition are described. The Anderson-Mott transition has been observed down to microscopic scale by employing arrays of few deterministically implanted As ions (see chapter 5) in Si transistors [39]. Analogous arguments and treatment can be given to acceptor concentration regimes.