Physics for Semiconductor Modeling
The physics of semiconductor devices is highly dependent on the size of the device. Over the last 50 years logic devices have been progressively miniaturized, enabling improvements in speed and reductions in power consumption. Nonetheless semiconductors are so ubiquitous that devices of larger sizes such as photodiodes and power transistors are still widely used for many applications.
The Semiconductor interface was initially directed at larger scale devices (with length scales of 100s of nm or more), which can be modeled by a conventional drift–diffusion approach using partial differential equations.
More recently, the density-gradient theory has been added to the Semiconductor interface as a computationally efficient option to include the effect of quantum confinement in the drift–diffusion method, pushing its applicability to nanometer length scales.
The Schrödinger Equation Interface and The Schrödinger–Poisson Equation Multiphysics Interface have also been added to the Semiconductor Module to provide more tools for analyzing quantum systems.
It is useful to note some of the important assumptions implicit to the drift–diffusion type of approaches:
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The relaxation-time approximation is used to describe the scattering process. This is a much simplified form of the scattering probability, which is elastic and isotropic.
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Magnetic fields are not included in the model.
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The carrier temperature is assumed to be equal to the lattice temperature and, consequently, the diffusion of hot carriers is not properly described.
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The energy bands are assumed to be parabolic. In reality the band structure is significantly altered in a complex manner in the vicinity of free surfaces or grain boundaries.
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Velocity overshoot, and other complex time-dependent conductivity phenomena, are not included in the model.
In addition to these intrinsic assumptions, The Semiconductor Interface allows you to make additional assumptions to simplify the solution process:
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For nondegenerate semiconductors it is possible to assume a Maxwell–Boltzmann distribution for the carrier energies at a given temperature, which reduces the nonlinearity of the semiconductor equations. If degenerate semiconductors are present within the model, or at lower temperatures, it is necessary to use Fermi–Dirac statistics.
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In majority carrier devices, it is often only necessary to solve for one of the carrier concentrations (the majority carrier). The minority carrier concentration is usually unimportant for the device operation and can be estimated by assuming the mass action law.
By default Maxwell–Boltzmann statistics are assumed by the Semiconductor interface and the interface explicitly solves for both the electron and hole concentrations.
The Semiconductor interface solves Poisson’s equation in conjunction with continuity equations for the charge carriers. An important length scale to consider when modeling electrostatic fields in the presence of mobile carriers is the Debye length:
where kB is Boltzmann’s constant, T is the lattice temperature, ε0 is the permittivity of free space, εr is the relative permittivity of the semiconductor, N is the total concentration of the electrons and holes, and q is the electron charge. The Debye length is the length scale over which the electric field decays in the presence of mobile carriers; it is important to resolve this length scale with the mesh in semiconductor models.
Theory for the Semiconductor Interface