Semiconductor device equations

P. A. Markovich, The Stationary Semiconductor Device Equations, Springer-Verlag, Vienna, New York, 1986. 

F. Brezzi, A. Capelo, and L. Gastaldi, A Singular Perturbation for Semiconductor Device Equations, to appear. 

H. Steinriick, A bifurcation analysis of the steady state semiconductor device equation, SIAM J. Appl. Math., 49 (1989), pp. 1102-1121. 

C. Jacoboni and P. Lugli, The Monte Carlo Method for Semiconductor Device Equations,... 

In terms of electric potential and Quasi-Fermi potentials, the three semiconductor device equations then become ... 

From this discussion it can be concluded that the route to obtaining converged solutions to the set of semiconductor device equations in this BV problem is not to be found in searching for some improved numerical techniques that can handle extremely large solution derivatives at the boundaries, but in modifying the boundary conditions so that known semiconductor physical limits are not exceeded. A more realistic set of boundary conditions on the minority carriers at the boundaries of this problem is needed. This means the boundary condition for on the left boundary and on the right boundary. Without much discussion an acceptable set of boundary conditions can be formulated as ... 

For the manufacture of silicon semiconductor devices, oxide thicknesses of from <10 to >1000 nm are required on sHces of single-crystal silicon. These oxide layers are formed at elevated temperatures, generally at about 1000°C, in an atmosphere of either oxygen or steam. Usually the oxidation is at atmospheric pressure, but sometimes, to speed the oxidation rate, pressures of several atmospheres are used. Oxidation consumes a silicon thickness equal to about 0.4 the thickness of the oxide produced (grown). The thickness of the oxide, V (4) is approximately given by equation 1 ... 

This book treats a selection of topics in electro-diffusion—a nonlinear transport process whose essence is diffusion of charged particles, combined with their migration in a self-consistent electric field. Basic equations of electro-diffusion were formulated about 100 years ago by Nernst and Planck in the ionic context [1]—[3]. Sixty years later Van Roosbroeck applied these equations to treat the transport of holes and electrons in semiconductors [4]. Correspondingly, major applications of the theory of electro-diffusion still lie in the realms of chemical and electrical engineering, related to ion separation and semiconductor device technology. Some aspects of electrodiffusion are relevant for electrophysiology. 

Describe the preparation of silicon from silica sand, and tell how silicon is purified for use in semiconductor devices. Write balanced equations for all reactions. 

The natural abundance of °Si is 3.1%. Upon irradiation with neutrons, this isotope is converted to Si, which decays to the stable isotope P. This provides a way of introducing trace amounts of phosphorus into silicon in a much more uniform fashion than is possible by ordinary mixing of silicon and phosphorus and gives semiconductor devices the capability of handling much higher levels of power. Write balanced nuclear equations for the two steps in the preparation of P from °Si. 

For the fabrication of semiconductor devices, tin chalcogenides are deposited by the thermal decomposition (MOCVD) of organotin chalcogenides R3SnXR ( X = S, Se, or Te), which must be free of any contaminants which would detract from the performance of the semiconductors. These pure precursors can be prepared by the light-induced com-proportionation of R3SnSnR3 and R XXR (equation 17-12).13 The obvious mechanism... 

As with all optoelectronic semiconductor devices, the current-voltage characteristics of an organic solar cell can be modelled using the continuity equations for each charge type. For electrons in the steady state we have... 

The above set of equations applies as shown to capacitively coupled systems. An identical set applies to the modeling of semiconductor devices [149]. Also, the similarity with the equations governing electrochemical systems (Section 8) is striking. The main difference is that continuity equations for electrons are not necessary, since there are no free electrons in the solution. 

M. S. Lundstrom, R. J. Schwartz, and J. L. Gray, Transport equations for the analysis of heavily doped semiconductor devices, Solid State Electron. 24 (1981) 195-202. 

In order to model electron and hole transport in a semiconductor device, such as a transistor, we need solutions to a set of equations for the concentrations n and p, the electric current densities jn and jp, and for the electrostatic... 

In an intrinsic semiconductor, the number of holes win equal the number of electrons. However, the equilibrium equation also applies to doped semiconductors, and the equilibrium constant derived for a pure intrinsic semiconductor is valid for a doped sample of the same semiconductor. This is an extremely useful finding because it means that as the concentration of electrons in a semiconductor is increased by doping the concentration of holes decreases, and vice versa. Thus an n-type semiconductor can be changed to a p-type semiconductor simply by increasing the number of holes present, by appropriate doping, and vice versa. This possibility underlies the fabrication of semiconductor devices. This information is used in Sections 13.2.2 and 13.2.4. 

The type (1) interface is an ideal Schottky barrier contact in which the barrier height varies directly with the metal work function in accordance with Equation [3.18], The type (2) interface approximates to a Bardeen barrier, provided that the surface states are assumed to be spaced inside the semiconductor so as to allow a potential drop across this region. In the clean contacts of this type, one would expect the barrier height to show a weak dependence on 0 ,.The type (3) interface represents a case of strong chemical bonding between the metal and the semiconductor and, hence, we would expect the barrier height to depend on some quantity related to chemical or metallurgical reactions at the interface. The type (4) contact is the one most frequently encountered in actual metal-semiconductor devices. 

The semianalytical model by Roichman et al. [42] is based on the mean medium approximation (MMA) and is probably the only one that shares the assumptions found in standard semiconductor models (for better or worse). A percolation-type model, which predicts that there are bottlenecks or small regions in which the current density is much larger than average, do not agree with the assumptions behind the semiconductor device model equations. The assumptions behind the MMA model include... 

In this paper our task is to determine the average lifetime t of the semiconductor devices at functioning temperature. There is applied the Arrhenius model while considering that the lifetime to failure at a temperature is proportional with the rate of the chemical degradation reaction, which takes place at that temperature. The equation of Arrhenius for the lifetime may be written as follows ... 

NEC corporation [20] has developed an interstitial concentration simulation method. Here, a mesh is set in a simulation region of a semiconductor device. Under the condition that an area outside of the simulation region is infinite, a provisional interstitial diffusion flux at the boundary of the simulation region is calculated. Then, an interstitial diffusion rate at the boundary of the simulation is calculated by a ratio of the provisional interstitial diffusion flux to the provisional interstitial concentration. Finally, an interstitial diffusion equation is solved for each element of the mesh using the interstitial diffusion rate at the boundary. 

In any semiconductor device the charge densities are governed by three differential equations, Poisson s equation which expresses the minimization of electrostatic potential energy and continuity equations that express the conservation of charges. Typically, multilayer stacks of thin-film solar cells are simulated in the dimension parallel to the surface normal of the stack. For such a one-dimensional device model, these equations become... 

The preceding semiconductor devices require large quantities of silicon that must be pure to 1 part per billion. To achieve such a purity, the crude silicon produced from the reduction of silica with carbon [Equation (15.1)] is converted to the liquid silicon tetrachloride, as shown in Equation (15.17) ... 

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