Eigensolutions of the Wigner-Eisenbud problem for a cylindrical nanowire within finite volume method

We present a node-centered finite volume method for computing a representative range of eigenvalues and eigenvectors of the Schrodinger operator on a three-dimensional cylindrically symmetric bounded domain with mixed boundary conditions. The three-dimensional Schrodinger operator is reduced to a family of two-dimensional Schrodinger operators distinguished by a centrifugal potential. We consider a uniform, boundary conforming Delaunay mesh, which additionally conforms to the material interfaces. We study how the anisotropy of the effective mass tensor acts on the uniform approximation of the first K eigenvalues and eigenvectors and their sequential arrangement. There exists an optimal uniform Delaunay discretization with matching anisotropy with respect to the effective masses of the host material. For a centrifugal potential one retrieves the theoretically established first-order convergence, while second-order convergence is recovered only on uniform grids with an anisotropy correction.

[1]  P. N. Racec,et al.  Fano regime of transport through open quantum dots , 2010, 1004.0472.

[2]  Hans Lüth,et al.  Resonant tunneling in nanocolumns improved by quantum collimation. , 2005, Nano letters.

[3]  Walter A. Harrison.  Applied Quantum Mechanics , 2000 .

[4]  R. Rurali.  Structural, electronic, and transport properties of silicon nanowires , 2009, 0910.2553.

[5]  R. Eymard,et al.  Weierstraß-institut F ¨ Ur Angewandte Analysis Und Stochastik Convergence of a Finite Volume Scheme to the Eigenvalues of a Schrödinger Operator. Convergence of a Finite Volume Scheme to the Eigenvalues of a Schrödinger Operator , 2022 .

[6]  E. Wigner,et al.  Higher Angular Momenta and Long Range Interaction in Resonance Reactions , 1947 .

[7]  Fang Qian,et al.  Nanowire electronic and optoelectronic devices , 2006 .

[8]  K. Gärtner,et al.  Boundary conforming Delaunay mesh generation , 2010 .

[9]  Observation of three-dimensional shell filling in cylindrical silicon nanowire single electron transistors , 2007 .

[10]  L. Segerlind.  Applied Finite Element Analysis , 1976 .

[11]  P. Daly.  IMPROVED CONVERGENCE RATES FOR FINITE‐ELEMENT APPROXIMATIONS IN CYLINDRICAL COORDINATE GEOMETRIES , 1983 .

[12]  Jonathan Richard Shewchuk,et al.  What is a Good Linear Element? Interpolation, Conditioning, and Quality Measures , 2002, IMR.

[13]  P. Daly.  An alternative high-order finite element formulation for cylindrical field problems , 1983 .

[14]  J. Guermond,et al.  Theory and practice of finite elements , 2004 .

[15]  Transport through a quantum ring, dot, and barrier embedded in a nanowire in magnetic field , 2004, cond-mat/0412402.

[16]  I. Babuska.  The Finite Element Method with Penalty , 1973 .

[17]  Chii-Chang Chen,et al.  Interband optical transitions in GaP nanowires encapsulated in GaN nanotubes , 2003 .

[18]  Christophe Besse,et al.  A Review of Transparent and Artificial Boundary Conditions Techniques for Linear and Nonlinear Schrödinger Equations , 2008 .

[19]  Hang Si,et al.  TetGen: A quality tetrahedral mesh generator and a 3D Delaunay triangulator (Version 1.5 --- User's Manual) , 2013 .

[20]  On Eisenbud's and Wigner's R-matrix: A general approach , 2007, math-ph/0701052.

[21]  Jing Wang,et al.  A three-dimensional quantum simulation of silicon nanowire transistors with the effective-mass approximation , 2004, cond-mat/0403739.

[22]  Jens Lang,et al.  A new anisotropic mesh adaptation method based upon hierarchical a posteriori error estimates , 2010, J. Comput. Phys..

[23]  Jonathan Richard Shewchuk,et al.  Triangle: Engineering a 2D Quality Mesh Generator and Delaunay Triangulator , 1996, WACG.

[24]  Thierry Gallouët,et al.  The nite volume method , 2000 .

[25]  T. Schapers,et al.  Spin precession and modulation in ballistic cylindrical nanowires due to the Rashba effect , 2010, 1011.4557.

[26]  Walter Riess,et al.  Nanowire-based one-dimensional electronics , 2006 .

[27]  Weizhang Huang,et al.  Anisotropic Mesh Adaptation for Variational Problems Using Error Estimation Based on Hierarchical Bases , 2010, 1006.0191.

[28]  Olaf Schenk,et al.  Solving unsymmetric sparse systems of linear equations with PARDISO , 2004, Future Gener. Comput. Syst..

[29]  R. Eymard,et al.  Finite Volume Methods , 2019, Computational Methods for Fluid Dynamics.

[30]  Charles M. Lieber,et al.  Ge/Si nanowire heterostructures as high-performance field-effect transistors , 2006, Nature.

[31]  Lars-Erik Wernersson,et al.  Vertical wrap-gated nanowire transistors , 2006 .

[32]  A. Lisyansky,et al.  Dynamics of the radiative recombination in cylindrical nanostructures with type-II band alignment , 2009 .

[33]  Christophe Delerue,et al.  Electronic structure of semiconductor nanowires , 2006 .

[34]  A. García-Cristóbal,et al.  Inhomogeneous free-electron distribution in InN nanowires: Photoluminescence excitation experiments , 2010 .

[35]  Paul C. Abbott,et al.  Efficient computation of Wigner-Eisenbud functions , 2013, Comput. Phys. Commun..

[36]  Efficient quantum three-dimensional modeling of fully depleted ballistic silicon-on-insulator metal-oxide-semiconductor field-effect-transistors , 2004 .

[37]  D. Vasileska,et al.  Semiconductor Device Modeling , 2008 .

[38]  Charles M. Lieber,et al.  High Performance Silicon Nanowire Field Effect Transistors , 2003 .

[39]  Charles M. Lieber,et al.  Growth of nanowire superlattice structures for nanoscale photonics and electronics , 2002, Nature.

[40]  Yong Ding,et al.  Multi-quantum-well nanowire heterostructures for wavelength-controlled lasers. , 2008, Nature materials.

[41]  Evanescent channels and scattering in cylindrical nanowire heterostructures , 2008, 0812.3305.

[42]  Lars Samuelson,et al.  Nanowire resonant tunneling diodes , 2002 .

[43]  Experimental evidence of ballistic transport in cylindrical gate-all-around twin silicon nanowire metal-oxide-semiconductor field-effect transistors , 2008 .

[44]  F. Peeters,et al.  Optical properties of free-standing GaAs semiconductor nanowires and their dependence on the growth direction , 2008 .

[45]  O. Schenk,et al.  ON FAST FACTORIZATION PIVOTING METHODS FOR SPARSE SYMMETRI C INDEFINITE SYSTEMS , 2006 .

[46]  R-matrix and the coherent transport in mesoscopic systems , 1990 .

[47]  T. Bryllert.  Vertical wrap-gated nanowire FETs , 2005 .

[48]  Gerhard Klimeck,et al.  Performance analysis of a Ge/Si core/shell nanowire field-effect transistor. , 2006, Nano letters.