{"id":31178,"date":"2004-02-01T10:00:23","date_gmt":"2004-02-01T10:00:23","guid":{"rendered":"https:\/\/silvaco-stage.betagentechnologies.com\/%eb%b6%84%eb%a5%98%eb%90%98%ec%a7%80-%ec%95%8a%ec%9d%8c\/schrodinger-approach-and-density-gradient-model-for-quantum-effects-modeling\/"},"modified":"2021-07-16T21:53:45","modified_gmt":"2021-07-17T04:53:45","slug":"schrodinger-approach-and-density-gradient-model-for-quantum-effects-modeling","status":"publish","type":"post","link":"https:\/\/silvaco-stage.betagentechnologies.com\/ko\/schrodinger-approach-and-density-gradient-model-for-quantum-effects-modeling\/","title":{"rendered":"Schr\u00f6dinger Approach and Density Gradient Model for Quantum Effects Modeling"},"content":{"rendered":"<h1>Schr\u00f6dinger Approach and Density Gradient Model for Quantum Effects Modeling<\/h1>\n<p class=\"regular\" align=\"center\">A.Ferron1, B.Cottle2, G.Curatola3, G.Fiori3, E.Guichard1<br \/>\n1 Silvaco Data Systems, 55 rue Blaise Pascal, 38330 Montbonnot Saint-Martin, France<br \/>\n2 Silvaco International, 2811 Mission Blvd. 6th Floor, Santa Clara, CA 95054, USA<br \/>\n3 University of Pisa, Via Diotisalvi 2, I-56122, Pisa, Italy<\/p>\n<p class=\"feature\">Abstract<\/p>\n<p>We describe here two approaches to model the quantum effects that can no more be neglected in actual and future devices. These models are the Schr\u00f6dinger-Poisson and Density-Gradient methods fully integrated in the device simulator\u00a0<strong><em>ATLAS<\/em><\/strong>. Simulations based on such methods are compared to each other on electron concentration and C-V curves in a MOS-capacitor.<\/p>\n<p class=\"feature\">Introduction<\/p>\n<p>Advanced silicon technology tends towards ever thinner and shorter gate oxide resulting in significant quantum effects. The most relevant effect is the confinement of the carriers. For instance, in a Metal-Oxide-Semiconductor capacitor C-V characteristic, the threshold voltage is shifted and the apparent oxide thickness is increased compared to the C-V characteristic expected with a semi-classical approach. To model this confinement accurately in a device simulator based on a drift-diffusion approach, two methods are treated in this paper. The first one, and the most accurate, is to include the Schr\u00f6dinger equation into a self-consistent computation with the Poisson equation. Unfortunately this solution, due to its non-locality, has a significant numerical cost and cannot be efficiently coupled with the continuity equations giving the current flow in practical applications. All the same this method is used in 1D as a reference: the C-V characteristic and the carrier density profiles are useful to validate simpler methods. Different simpler methods compatible with the drift-diffusion approach have been developed [1, 2]. In this paper we describe a density gradient model which introduces a quantum potential correction in the continuity equations. In the following, we present first the Schr\u00f6dinger-Poisson model, then the density gradient model and the comparison to each other.<\/p>\n<p><img loading=\"lazy\" width=\"600\" height=\"800\" src=\"https:\/\/silvaco-stage.betagentechnologies.com\/wp-content\/uploads\/simulationstandard\/simstd_feb_2004_a1-e1611194549782.jpg\" class=\"cleaned-enfold-image\" alt=\"\" decoding=\"async\" loading=\"lazy\" \/><\/p>\n","protected":false},"excerpt":{"rendered":"<p>We describe here two approaches to model the quantum effects that can no more be neglected in actual and future devices. These models are the Schr\u00f6dinger-Poisson and Density-Gradient methods fully integrated in the device simulator\u00a0ATLAS. Simulations based on such methods are compared to each other on electron concentration and C-V curves in a MOS-capacitor.<\/p>\n","protected":false},"author":3,"featured_media":22707,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[7486],"tags":[],"class_list":["post-31178","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-simulation-standard-ko","entry","has-media"],"_links":{"self":[{"href":"https:\/\/silvaco-stage.betagentechnologies.com\/ko\/wp-json\/wp\/v2\/posts\/31178","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/silvaco-stage.betagentechnologies.com\/ko\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/silvaco-stage.betagentechnologies.com\/ko\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/silvaco-stage.betagentechnologies.com\/ko\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/silvaco-stage.betagentechnologies.com\/ko\/wp-json\/wp\/v2\/comments?post=31178"}],"version-history":[{"count":1,"href":"https:\/\/silvaco-stage.betagentechnologies.com\/ko\/wp-json\/wp\/v2\/posts\/31178\/revisions"}],"predecessor-version":[{"id":31183,"href":"https:\/\/silvaco-stage.betagentechnologies.com\/ko\/wp-json\/wp\/v2\/posts\/31178\/revisions\/31183"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/silvaco-stage.betagentechnologies.com\/ko\/wp-json\/wp\/v2\/media\/22707"}],"wp:attachment":[{"href":"https:\/\/silvaco-stage.betagentechnologies.com\/ko\/wp-json\/wp\/v2\/media?parent=31178"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/silvaco-stage.betagentechnologies.com\/ko\/wp-json\/wp\/v2\/categories?post=31178"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/silvaco-stage.betagentechnologies.com\/ko\/wp-json\/wp\/v2\/tags?post=31178"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}