International Journal of Multidisciplinary Research and Publications ISSN (Online): 2581-6187 88 Asmaa A. Salama,Mohamed S. Abdel-wahed, D. A. Hammad, Mourad S. Semary, “Review of numerical solutions of differential equations applied to semiconductor materials,” International Journal of Multidisciplinary Research and Publications (IJMRAP), Volume 5, Issue 12, pp. 88-96, 2023. Review of Numerical Solutions of Differential Equations Applied to Semiconductor Materials Asmaa A. Salama 1 , Mohamed S. Abdel-wahed 1,2 , D. A. Hammad 1 , Mourad S. Semary 1 1 Department of Basic Engineering Sciences, Faculty of Engineering at Benha, Benha University, Egypt. 2 Department of Basic Engineering Sciences, Faculty of Engineering, BADR University in Cairo BUC, Egypt Email address: asmaa.salama@bhit.bu.edu.eg, Mohamed.sayed@bhit.bu.edu.eg, doaa.hammad@bhit.bu.edu.eg, mourad.semary@yahoo.com Abstract— Many differential equations can’t be solved analytically, and for many applications, such as in electronic engineering – numerical solution is often more sufficient. Schrodinger equation, Poisson equation and Continuity equations are the most popular differential equations used in semiconductor materials. In this paper, the numerical solution of these equations is reviewed while focusing on some specific methods which are compared and evaluated. Schrodinger equation is solved using both Numerov method and Finite Difference method. Poisson equation is solved using Finite Difference method at various types of boundary conditions. Continuity equation is solved using Scharfetter-Gummel method. Numerical solutions are compared with the exact results and the dependence of error on the mesh size is shown. Keywords—Numerov method; Finite Difference Method; Scharfetter- Gummel Method. I. INTRODUCTION Semiconductor is a class of solids which have intermediate electrical conductivity between insulators and conductors. Semiconductors are used in the manufacture of different kinds of electronic devices, including transistors, diodes, and integrated circuits. Such devices have been used in a lot of application because of their reliability, compactness, power efficiency, and low cost. Hereinafter, we will discuss the most popular differential equations used in modelling semiconductor materials. We will focus on three main equations: Schrodinger equation Eq. (1.1), Poisson equation Eq. (1.2) and Continuity equation for electrons Eq. (1.3) and for holes Eq. (1.4). First, The Schrödinger equation is a differential equation that governs the wave function of a quantum-mechanical system. It is the basis of in quantum mechanics, and its discovery was a significant landmark in the development of the subject. The form of Schrodinger equation is ( ) ( ) ( ) ( ) 2 2 2 2 d x V x x E x m dx    − + = ħ (1.1) (Where m is the particle mass, ħ is the reduced Planked constant, ( ) x  is the wave function, ( ) V x is the potential, E is the particle energy).In these paper, The 1-D time- independent Schrodinger equation [1, 2] is solved using two methods: Numerov method [3, 4] and Finite Difference method. Numerov method is used to solve ordinary differential equations [5] of second order in which the first- order term does not exist. To be able to derive Numerov Method we start with the expansion of the solution in a Taylor series. Numerical outcomes from Numerov method and Finite Difference method are compared with exact solution. From comparison we know that Numerov method is more accurate than Finite Difference method. Second, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with the potential field known, one can then calculate electrostatic or gravitational (force) field. Also, Poisson equation is a powerful tool for modelling electrostatic systems behaviour. Poisson equation form is ( ) ( ) u x gx  = (1.2) (Where ( ) u x  is second derivative of a given function and, ( ) gx is a given function). In this paper, Poisson equation is solved using Finite Difference method [6, 7] with different boundary conditions (Dirichlet-Dirichlet boundary condition, Neumann-Dirichlet and Dirichlet-Neumann boundary condition). Numerical outcomes are compared with exact solution in all cases and absolute error between numerical and exact outcomes is shown. Third, the continuity equation can describe the transfer of various quantities, such as gas or fluid. The continuity equation can be written in integral form, which is applied at finite region, or in differential form, which is applied at a point. Continuity equation takes two forms as in Eq. (1.3) which is for electrons and Eq. (1.4) which is for holes. ext ( ) 1 p p J p p G t q x       =− + − (1.3) ext ( ) 1 n n J n n G t q x       =+ + − (1.4) Where p p p dp J q p qD dx  = − E (1.5) n n n dn J q p qD dx  = + E (1.6) (where n J is current density for electrons, p J is current density for holes, n is the electrons concentration, p is the