For the solution of the semiconductor equations a closed domain is required. Boundary conditions for the unknowns in a certain segment must be specified at the boundaries of that segment. Depending on the number of segments (one or two) and the type of the quantity of the unknown (distributed or non-distributed) several model types can be separated . However, in order to illustrate the basic boundary conditions such a differentiation is not needed and therefore not provided here.
The simulation domain usually includes only a single device. In order to separate it from the neighboring devices artificial boundaries, which have no physical origin, are introduced. Through the Neumann boundary conditions a self-contained domain is guaranteed. The fluxes across the boundary are set to zero:
Ohmic contacts are defined by Dirichlet boundary conditions: the contact potential , the carrier contact concentration and , and in the case of a HD simulation the carrier contact temperatures and are fixed. The metal quasi-Fermi level (which is specified by the contact potential ) is equal to the semiconductor quasi-Fermi level. The model assumes charge-neutrality on the boundary. The contact potential at the semiconductor boundary is:
The carrier concentrations in the semiconductor are pinned to the
equilibrium carrier concentrations at the contact:
The carrier temperatures and are set equal to the lattice temperature :
If a contact temperature is specified, the lattice temperature is calculated using and a thermal resistance . The thermal heat flow density at the contact boundary reads:
It is possible to include an electric line resistance of the contact using:
At the Schottky contact mixed boundary conditions apply. The contact potential , the carrier contact concentration and , and in the case of a HD simulation, the contact carrier temperatures and are fixed. The semiconductor contact potential is the difference between the metal quasi-Fermi level and the metal work function difference :
For the equilibrium situation, the quasi-equilibrium concentrations and can be rewritten:
Typical values for the Schottky barrier heights of common semiconductor barriers are listed in Table 4.1. Values measured by different methods (I-V and C-V) for n-GaN/metal differ significantly. This is caused by defects in the surface region, which enhance the tunneling effect and therefore have an impact on the Richardson constant [269,270]. The exact value of the barrier height depends also on orientation, stress, and polarity of the GaN layer . No data is provided for contacts on InN as most metals show ohmic behavior . Schottky barrier height varies with annealing temperature: generally it is reduced after annealing.
|n-GaN/Ni||0.95-1.13||||In Al N/Ni||0.75|||
|p-GaN/Ni||2.68-2.87||||p-In Ga N/Ni||0.39|||
|Al Ga N/Ni||0.94-1.24||||In Ga N/Pt||0.62|||
|Al Ga N/Ni||1.26||||In Ga N/Ni||1.39|||
|Al Ga N/Ni||1.02-1.30||||Al In GaN/Ni||0.98-0.93|||
A model similar to the Schottky contact model is used to calculate the insulator contact potential. The semiconductor contact potential is the difference of the metal quasi-Fermi level and the metal work function difference potential similar to (4.41) and (4.42). The lattice temperature is set equal to the contact temperature (4.38).
In the absence of surface charges the normal component of the dielectric displacement and the potential are continuous:
At the semiconductor/insulator interface the current densities and heat fluxes normal to the interface vanish.
The calculation of the electrostatic potential at the interfaces between two semiconductor segments is similar to that for semiconductor/insulator interfaces:
To calculate the carrier concentrations and the carrier temperatures at the interface, three different approaches are considered:
Each model can be specified for electrons and holes for each semiconductor/semiconductor interface.
In the following denotes the normal to the interface component of the current density , the energy flux density component, and the difference in the conduction band edges . The effective electron mass is denoted by . The subscripts denote the semiconductor segment . Only the equations for electrons are given, those for holes can be deduced accordingly.
Dirichlet boundary conditions are applied, when the CQFL is used. The
carrier concentrations are determined so that the quasi-Fermi levels
remains continuous across the interface .
To consider the band gap alignment as typical in a heterostructure, the
thermionic emission or thermionic field emission interface models must