Use of WKB approximation for analytical boundary conditions in numerical solution of Schrödinger equation: application to semiconductor–high‐k dielectric interfaces. (7th July 2015)

• Record Type: Journal Article
• Authors: Bayoumi, Amr M.
• Is Part Of: International journal of numerical modelling. Volume 29:Number 3(2016:May/Jun.)
• Page Start: 392
• Page End: 406
• Publication Date: 2015-07-07
• Subjects: WKB approximation -- Schrödinger's equation boundary conditions -- Numerov's/shooting method -- high‐k dielectrics -- tunneling -- MOS device simulation; Electric networks -- Mathematical models -- Periodicals; Electronics -- Mathematical models -- Periodicals; 621.3011
• DOI: 10.1002/jnm.2084
• Languages: English
• ISSNs: 0894-3370

Abstract

Summary: Numerical methods used to solve 1D Schrödinger's equation in quantum structures, such as Numerov's integration of wavefunction or the shooting method iterative solution of energy levels, require knowledge of two‐point boundary conditions at interfaces. This is especially true when the interfaces are not symmetrical or where exponential decay of wavefunction at asymptotically large distances does not hold. A closed‐form expression for boundary conditions, which is not sensitive to intermediate solutions at interfaces, can minimize possible divergence during iterations and relax simulation grid size and simulation time. In this work, the Wentzel‐Kramers‐Brillouin (WKB) approximation within potential barriers is proposed to analytically calculate the boundary conditions for abrupt interfaces, such as dielectric–semiconductor interface. An analytical expression for the slope at the interface is derived, and the errors are estimated with respect to numerical methods. An application is shown for self‐consistent solution of coupled Poisson–Schrödinger's equations at multi‐layer HfO2 ‐SiO2 dielectric gate stack corresponding to International Technology Roadmap for Semiconductors‐projected 10 nm bulk single‐gate Complementary Metal‐Oxide‐Semiconductor (CMOS) technology node, where wavefunction penetration into the dielectric is of critical importance. Application to dual gate structures with 5 nm fin width and high‐k dielectric with 0.5 nm equivalent oxide thickness is also shown. A quantum mechanical simulator 'hksim' based on this principle is posted for public domain usage. Copyright © 2015 John Wiley & Sons, Ltd.