Point defects and impurities strongly affect the physical properties of materials and have a decisive impact on their performance in applications. First-principles calculations have emerged as a powerful approach that complements experiments and can serve as a predictive tool in the identification and characterization of defects. The theoretical modeling of point defects in crystalline materials by means of electronic-structure calculations, with an emphasis on approaches based on density functional theory (DFT), is reviewed. A general thermodynamic formalism is laid down to investigate the physical properties of point defects independent of the materials class (semiconductors, insulators, and metals), indicating how the relevant thermodynamic quantities, such as formation energy, entropy, and excess volume, can be obtained from electronic structure calculations. Practical aspects such as the supercell approach and efficient strategies to extrapolate to the isolated-defect or dilute limit are discussed. Recent advances in tractable approximations to the exchange-correlation functional ($\mathrm{DFT}+U$, hybrid functionals) and approaches beyond DFT are highlighted. These advances have largely removed the long-standing uncertainty of defect formation energies in semiconductors and insulators due to the failure of standard DFT to reproduce band gaps. Two case studies illustrate how such calculations provide new insight into the physics and role of point defects in real materials.

First-principles calculations for point defects in solids

Diffusion in silicon of elements from columns III and V of the Periodic Table is reviewed in theory and experiment. The emphasis is on the interactions of these substitutional dopants with point defects (vacancies and interstitials) as part of their diffusion mechanisms. The goal of this paper is to unify available experimental observations within the framework of a set of physical models that can be utilized in computer simulations to predict diffusion processes in silicon. The authors assess the present state of experimental data for basic parameters such as point-defect diffusivities and equilibrium concentrations and address a number of questions regarding the mechanisms of dopant diffusion. They offer illustrative examples of ways that diffusion may be modeled in one and two dimensions by solving continuity equations for point defects and dopants. Outstanding questions and inadequacies in existing formulations are identified by comparing computer simulations with experimental results. A summary of the progress made in this field in recent years and of directions future research may take is presented.

Point defects and dopant diffusion in silicon

In this thesis, the existence and uniqueness of gradient trajectories near an A2singularity are analysed. The A2-singularity is called a birth-death critical point in the real case. The birth-death critical point appears in a one-parameter family of functions. Such a family of functions has precisely two Morse critical points of index difference one, on the birth side. The result of the real case states that these two critical points are joined by a unique gradient trajectory up to time-shift. Here the gradient flow is defined with respect to any family of Riemannian metrics. This can be viewed as a converse to Smale’s cancellation theorem. We also look at the complex analogue of the result in Picard–Lefschetz theory. This analogue considers a holomorphic one-parameter family with an A2-singularity. Such a family has two critical Morse critical points near the singularity for every small non-zero parameter. We prove that the two Lagrangian vanishing cycles associated to these critical points intersect transversally in exactly one point in all regular fibres along a straight line. The result is obtained by analysing the gradient trajectories of the real part of these functions. Both proofs start with a normal form in local coordinates for such families of functions. The gradient equations in these coordinates can be rescaled into a fast-slow system of non-linear differential equation. Existence will rely on an adiabatic limit analysis whereas uniqueness follows from a Conley index pair construction. The latter construction will also show that connecting gradient trajectories cannot leave the local charts. Even though the proof of these two results follow from similar lines of argument, the real case cannot be reduced to the complex case and vice versa.

Doctoral Thesis by

This paper summarizes some of the essential aspects of silicon-nanowire growth and of their electrical properties. In the first part, a brief description of the different growth techniques is given, though the general focus of this work is on chemical vapor deposition of silicon nanowires. The advantages and disadvantages of the different catalyst materials for silicon-wire growth are discussed at length. Thereafter, in the second part, three thermodynamic aspects of silicon-wire growth via the vapor–liquid–solid mechanism are presented and discussed. These are the expansion of the base of epitaxially grown Si wires, a stability criterion regarding the surface tension of the catalyst droplet, and the consequences of the Gibbs–Thomson effect for the silicon wire growth velocity. The third part is dedicated to the electrical properties of silicon nanowires. First, different silicon nanowire doping techniques are discussed. Attention is then focused on the diameter dependence of dopant ionization and the influence of interface trap states on the charge carrier density in silicon nanowires. It is concluded by a section on charge carrier mobility and mobility measurements.

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Silicon Nanowires: A Review on Aspects of their Growth and their Electrical Properties

Implanted B and P dopants in Si exhibit transient enhanced diffusion (TED) during annealing which arises from the excess interstitials generated by the implant. In order to study the mechanisms of TED, transmission electron microscopy measurements of implantation damage were combined with B diffusion experiments using doping marker structures grown by molecular-beam epitaxy (MBE). Damage from nonamorphizing Si implants at doses ranging from 5×1012 to 1×1014/cm2 evolves into a distribution of {311} interstitial agglomerates during the initial annealing stages at 670–815 °C. The excess interstitial concentration contained in these defects roughly equals the implanted ion dose, an observation that is corroborated by atomistic Monte Carlo simulations of implantation and annealing processes. The injection of interstitials from the damage region involves the dissolution of {311} defects during Ostwald ripening with an activation energy of 3.8±0.2 eV. The excess interstitials drive substitutional B into electric...

/pdf/physical-mechanisms-of-transient-enhanced-dopant-diffusion-zkysm82pgr.pdf

Physical mechanisms of transient enhanced dopant diffusion in ion-implanted silicon

A conceptually simple experiment to measure silicon interstitial diffusivity is described. The structure uses a silicon wafer with buried layers deep in the bulk. Oxidation of the wafer surface generates interstitials that diffuse into the wafer and enhance the buried layer diffusion. The results show that the interstitial profiles are flat out to 40 μm for the times and temperatures investigated. We can thus conclude that the interstitials move significantly faster than previously thought.

P. M. Fahey

Papers

Point defects and dopant diffusion in silicon

Measurement of silicon interstitial diffusivity