A numerical Schrödinger–Poisson solver for radially symmetric nanowire core–shell structures
Introduction
The rapid advancement in semiconductor nanowire growth technology has motivated numerous research efforts aimed at different applications of nanowires in electronics, optics and biology [1], [2], [3], [4]. Among various structures under investigation, the nanowire core–shell structure (with a radial variation in material characteristics, such as semiconductor composition) has been very popular, since it provides great versatility for use in devices such as field-effect transistors [5], [6], photoemitters and photodetectors. To exploit the unique traits stemming from the 1-D quantum structure of nanowires, systematic understanding of the electrical and optical properties are important. In general, the determination of the electronic energy levels and potential distribution requires the self-consistent solution of the Schrödinger equation and the Poisson equation under cylindrical coordinates. Previous work has shown that a general-purpose self-consistent Schrödinger–Poisson solver can be formulated for Cartesian coordinates (as needed for representative planar epitaxial structures) [7]. This paper reports on a numerical solver and its applications for nanowires, exploiting cylindrical symmetry.
The numerical Schrödinger Poisson self-consistent solver accounts for quantum confinement as well as the 1-D nature of the density of states in nanowires. The structure considered and its associated coordinate system are illustrated in Fig. 1. The solver is generally applicable to nanowires structures with arbitrary material and doping dependence in the radial direction (the core–shell structure pictured in Fig. 1 has one change in material composition along the radius). The solver uses the conventional finite-difference method to solve the Schrödinger equation. This implies tradeoffs of solution accuracy and computation complexity; values chosen in this work provide reasonable computational times for nanowires of physical interest (R ∼ 5–500 nm).
From the self-consistent solution, the potential profile, wave functions, electron density and further information can be derived. As an example, the paper illustrates the analysis of capacitance–voltage characteristics for a nanowire core–shell structure configured as an FET, with a radial-deposited metallic gate. For the example case shown in the paper, the gate capacitance is reduced to ∼1/3 of the geometrical barrier limited value, and is only ∼70% of the classically predicted value (for which only the Poisson equation is solved).
The paper is organized in the following manner. Governing equations are discussed in Section 2, with emphasis on comparing solutions with cylindrical coordinates and Cartesian coordinates. In Section 3, the solver is tested for a cylindrical constant potential well (where the exact wave functions are known in analytical form) and a core–shell structure with large dimensions (which asymptotically approaches the slab quantum-well solution). The latter result was compared with the solution given by a well-established one-dimensional solver [7] for validation. Section 4 gives an example application of the solver in which a core–shell nanowire structure with metallic gate is characterized in terms of electron distribution and quasi-static C–V characteristics; and mechanisms that cause the gate capacitance reduction are analyzed and de-embedded. Model limitations are briefly discussed in Section 5, while a summary of the paper is provided in Section 6.
Section snippets
Schrödinger solver
In the presence of varying material composition, the corresponding spatially-varying effective masses have to be taken into account in the Schrödinger equation. The following form of the Schrödinger equation results:The effective masses are taken to be energy-independent (thus non-parabolicity effects are neglected). We apply the separation of variables method by assuming the overall wave function can be written asThe angular part of the wave function has
Infinite cylindrical square potential well
The Schrödinger solver was tested under a specific model potential where exact solutions were known as J-type Bessel functions of various orders. The potential was set to be zero inside the core, and infinite in the shell region, which is referred to here as the infinite cylindrical constant potential well. Contributions to the potential from the electron concentration are ignored. The corresponding eigenwave vector values are determined analytically viawhere zn represents the nth root
Model application
Core–shell nanowire structures are of substantial interest for electrical and optoelectronic applications. Core–shell configurations can utilize a great diversity of semiconductor combinations, since the nanowires are relatively tolerant to lattice mismatch between materials, and the associated stress and strain fields. Quantitative understanding of electron concentrations is important for further work, particularly for the two-dimensional electron gas established at the interface between
Limitations of the model
Various simplifications have been introduced into the solver which can limit the accuracy in a number of situations. As noted above, the effective mass is assumed to have a constant value. Thus non-parabolicity effects are not taken into account. The implied isotropy of the effective mass omits a number of details present in the conduction band of Si and other multi-valley semiconductors. Our present treatment of holes is also associated with a simple effective mass; the full complexity of
Summary
A general-purpose Schrödinger–Poisson solver was developed for symmetric structures in cylindrical coordinates. This solver is useful for a wide range of semiconductor nanowire problems. The solver allows evaluation of discrete eigenenergies, and wave functions for a given geometry, based on which the electro-static potential and the electron concentration can be calculated, all in a self-consistent manner. Self-consistent C–V characteristics of a given structure can be readily deduced from the
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