An Immersed Interface Method for Modeling Semiconductor Devices
Students Daniel J. Costinett
Dates of Involvement 2008
Faculty Advisor Theodoros P. Horikis
Semiconductors are present in nearly all modern electronic devices and are a crucial component of integrated circuits. Because of their widespread use, it has become desirable to be able to create semiconductor material with highly specific conduction properties. For this purpose, many different materials are used in the fabrication of semiconductors. Additionally the conduction properties of a material can be altered through a process known as 'doping' in which impurities are introduced into a purified host material in a specific pattern (Fig. 1a). Doping can occur uniformly throughout an entire semiconductor to alter its overall properties, or in specific areas to define circuit elements within the material. The impurities may be as dispersed as sparsely as 1 atom per 100,000,000 atoms of semiconductor, requiring sophisticated instruments to accurately dope the material. Because of this, and the relatively high rate of manufacturing defects, it is crucial to have a method for testing new semiconductor formations without requiring a sample to be fabricated. The behavior of a doped semiconductor can be examined mathematically using the Schrödinger equation under the effective mass approximation (Eq. 1). In this equation, both the mass of the material in the semiconductor and the shape of the doping are modeled by the functions m and u(x), respectively. E is the total energy.
(1) 
Figure 1: Semiconductor Modeling Using the Immersed Interface Method
A typical silicon semiconductor with copper doping is shown in (a); (b) shows the construction of functions for the mass (orange) and the shape of the potential (red). Finally, (c) shows the ground state and first two excited states solved using this method.
Solving Eq. (1) analytically is very difficult and often even impossible making the use of numerical techniques unavoidable. The shape of the semiconductor and the mass of the two materials are modeled as piecewise functions (Figs. 1b, 1c). The problem arises, since the piecewise functions result in discontinuities in the function and/or its derivatives. When this is the case, it becomes necessary to solve the equation using a modified difference scheme such as the Immersed Interface Method. For all points not affected by the interface, a common central differences scheme can be used; for example Eq. (2) which uses a second order scheme. At points where the expansion crosses the interface, however, a modified expansion must be constructed (Fig. 2). Hence, at the points ψ(xj) and ψ(xj+1), a modified expansion is necessary because points on
both sides of the interface are used.
(2) γ1 · ψn− 1 + γ2 · ψn + γ3 · ψn+1 = Eψn
Figure 2: Discrete Interface in Semiconductor Potential
While normal central differences can be used elsewhere, a modified expansion must be used at any point where the difference terms cross the interface. The number of terms replaced and the order of the expansion at each term determine the order of the solution produced.
In order to use this method a jump condition needs to be defined for the function, say, ψ(xj) = τψ(xj+1) and its derivatives (this is done by integrating Eq. (1) around x*). With the jump conditions it is then possible to find a modified central differences expansion for the function near the interfaces using the Taylor expansion of the surrounding points. With the modifications, the eigenvalue equation can be solved completely using sparse matrices shown in Eq. (3).
The resulting equation, Eq. (3), is an algebraic equation with eigenvalue the energy. An eigenvalue problem involving a differential equation is now a typical algebraic eigenvalue problem that can be solved with any known numerical routine. The only modification is that a few entries in the matrix have been replaced accordingly to take into account the jump conditions of the problem. In addition, the matrix remains sparse, i.e. the zero entries are not affected making the diagonalization even faster. The total accuracy of the method is also maintained (here we used a second order expansion).
Future plans for research include the solution of the Schrödinger equation to fourth order and applying the Immersed Interface Method or its variants to nonlinear forms of the Schrödinger equation thus taking into account the so-called nonparabolicity effect.
Figure 3: Silicon Wafer
Many modern processors are etched onto silicon wafers, where doping inserts alter the conductivity of the material as a whole, or only in certain areas.
