Quantum Information in Silicon Devices Based on Individual Dopants

Abstract

Several solid-state quantum information processor proposals arebased on silicon [16-18, 20, 52], including the original architectureconceived by Kane in 1998 [22], in which the quantum informationwas stored in the spin states of 31P donors. The interest raisedby silicon in the field of quantum information is twofold. On theone hand, it represents the main semiconductor used for large-scale fabrication. On the other hand, it owns interesting physicalproperties suitable for preserving andmanipulating quantum states,such as long coherence times and relatively low disturbance ofnuclear spins, which are further enhanced by employing purified 28Si. The employment of individual donors in silicon covers allthe aspects of quantum information encoding and processing, likequantumbit (qubit) storage, implementation of quantum logic gates,coherent transfer of quantum states, and readout schemes. Thischapter is devoted to the physics of few dopants in nanometricsilicon devices and their applications to quantum informationprocessing purposes. In the first part, an overview of the physics ofthemost common donors and acceptors in silicon is presented. Next,the quantum information key concepts and their implementationin donor-based architectures are discussed. Finally, decoherenceeffects and the concept of measurement of quantum states arepresented 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 dopingof bulk silicon have been investigated and explained. A siliconcrystal consists of a diamond lattice constituted by two interleavedcubic lattices (face-centered cubic, FCC), the second of which hasthe 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 fourvalence electrons, which create a covalent bond with the sharedelectrons of four other Si atoms. The substitution of silicon atomswith 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 providesnew energy levels in the band gap, close to the conduction bandedge. Similarly, substituting silicon atoms with atoms of a groupIII 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, closeto the valence band. The diffusion of impurities in semiconductorsalters the conduction properties from an insulating to a metallicregime. Depending on the different doping concentration, thewavefunction of the electron (hole) states introduced by eachdonor (acceptor) may overlap negligibly or substantially with thoseof neighboring sites. Such a transition from low doping to highdoping as a function of the average distance between neighboringsites is described in terms of an Anderson-Mott transition andproduces additional impurity bands (Hubbard bands) below theconduction band edge at sufficiently high concentration [1, 34]. Thetransport is governed by mechanisms based on localized states atlow 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. Above2 · 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 Hubbardband 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 donorconcentration and the Anderson-Mott transition are described. TheAnderson-Mott transition has been observed down to microscopicscale by employing arrays of few deterministically implanted Asions (see chapter 5) in Si transistors [39]. Analogous arguments andtreatment can be given to acceptor concentration regimes.