A numerical Schro ¨ dinger–Poisson solver for radially symmetric nanowire core–shell structures Lingquan Wang * , Deli Wang, Peter M. Asbeck Department of Electrical and Computer Engineering, University of California, San Diego, 9500 Gilman Drive CallT2 Building, La Jolla, CA 92093, United States Received 4 April 2006; accepted 21 September 2006 Available online 9 November 2006 The review of this paper was arranged by Prof. Y. Arakawa Abstract We present here a general purpose numerical Schro ¨ dinger–Poisson solver for radially symmetric nanowire core–shell structures for electronic and optoelectronic applications. The solver provides self-consistent solutions of the Schro ¨ dinger equation and the Poisson equation in cylindrical coordinates, for nanowire core–shell structures with radial compositional variation. Quantized energy levels as well as their associated electron wavefunctions and populations can be obtained from the solutions. Individual equation solvers were verified by comparison with scenarios where analytical results exist; verification of the self-consistent solution process was done by com- paring results in the large radius limit with numerical solutions for a rectangular slab structure. We apply this solver to compute the charge/capacitance–voltage characteristics for a nanowire field effect device with wrap-around gate. It is shown that quantum confine- ment and the low dimensionality can give rise to, for representative nanowire FETs considered, 30% reduction in gate capacitance com- pared to the classically predicted value, and is 1/3 of the geometrical barrier limited capacitance. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Nanowire; Core–shell structure; Self-consistent calculation; Quantum capacitance 1. Introduction The rapid advancement in semiconductor nanowire growth technology has motivated numerous research efforts aimed at different applications of nanowires in elec- tronics, optics and biology [1–4]. Among various structures under investigation, the nanowire core–shell structure (with a radial variation in material characteristics, such as semi- conductor 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 quan- tum structure of nanowires, systematic understanding of the electrical and optical properties are important. In gen- eral, the determination of the electronic energy levels and potential distribution requires the self-consistent solution of the Schro ¨dinger equation and the Poisson equation under cylindrical coordinates. Previous work has shown that a general-purpose self-consistent Schro ¨ dinger–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 applica- tions for nanowires, exploiting cylindrical symmetry. The numerical Schro ¨dinger Poisson self-consistent sol- ver 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 illus- trated in Fig. 1. The solver is generally applicable to nano- wires structures with arbitrary material and doping dependence in the radial direction (the core–shell structure 0038-1101/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.sse.2006.09.013 * Corresponding author. Tel.: +1 858 822 6943; fax: +1 858 534 2486. E-mail address: liw001@ucsd.edu (L. Wang). www.elsevier.com/locate/sse Solid-State Electronics 50 (2006) 1732–1739