- 3.1.6.1 Artificial Boundaries
- 3.1.6.2 Semiconductor-Metal Boundaries
- 3.1.6.3 Insulator-Metal Boundaries
- 3.1.6.4 Semiconductor-Insulator Interface
- 3.1.6.5 Insulator-Insulator Interface
- 3.1.6.6 Semiconductor-Semiconductor Interface

3.1.6 Boundary Conditions

Here, denotes an outward oriented vector normal to the boundary. (3.20) and (3.21) give the boundary conditions at the artificial boundaries for semiconductor and insulator segments, respectively.

At Ohmic contacts simple

(3.22) |

(3.23) |

Here, is the net concentration of dopants and other charged defects at the contact boundary. The auxiliary variables and are defined by

(3.24) |

The carrier concentrations in the semiconductor are pinned to the equilibrium carrier concentrations at the contact. They are expressed as

(3.25) | |||

(3.26) |

The carrier temperatures and are set equal to the lattice temperature .

In the case of a thermal contact the lattice temperature is calculated using a specified contact temperature and thermal resistance . The thermal heat flow density at the contact boundary reads:

In case no thermal resistance is specified an isothermal boundary condition is assumed and the lattice temperature is set equal to the contact temperature .

In the case of DD simulation with self-heating an additional thermal energy is accounted for. This thermal energy is produced when the carriers have to surmount the potential difference between the conduction or valence band and the metal quasi-

(3.30) |

At the Schottky contact mixed boundary conditions apply. The contact potential , the carrier contact concentrations and , and in the HD simulation case, the contact carrier temperatures and are fixed. The semiconductor contact potential is the difference of the metal quasi-

The difference between the conduction band energy and the metal workfunction energy gives the workfunction difference energy which is the so-called barrier height of the Schottky contact. The applied boundary conditions are

(3.32) |

(3.33) |

The Schottky contact boundary conditions for the carrier temperatures and and the lattice temperature are similar to the ones which apply for the Ohmic contact, i.e. (3.27) and (3.28), or respectively (3.29).

3.1.6.2.3 Polysilicon Contact

In

(3.34) |

(3.35) |

(3.36) |

Again, is the workfunction difference energy. The lattice temperature is set equal to the contact temperature (3.29).

In the presence of surface charges along the interface the dielectric displacement obeys the law of

At the semiconductor-insulator interface the carrier current densities (or driving forces) and the carrier heat fluxes normal to the interface vanish.

(3.40) | |||

(3.41) |

The lattice temperature at the interface is continuous.

3.1.6.6 Semiconductor-Semiconductor Interface

Here is the interface charge density which can be zero or non-zero. The subscripts are used to distinguish between the two semiconductor segments on both sides of the interface.

To calculate the carrier concentrations and the carrier temperatures at the
interface of two semiconductor segments three different models are considered
These are a model with continuous quasi-*Fermi* level across the
interface (CQFL), a thermionic emission model (TE), and a
thermionic field emission model (TFE). The derivation of these
models is given in [78]. Each model can be specified separately for
electrons and holes for each semiconductor-semiconductor interface.

In the following denotes the current density, the energy flux density, and the difference in the conduction or valence band edges, respectively. The carrier concentration is denoted by . The subscripts denote the semiconductor segment and the carrier type.

(3.44) | |||

(3.45) |

with the thermionic emission velocity (3.50) and the barrier height lowering (3.51).

The barrier height lowering depends on the electric field orthogonal to the interface and the effective tunneling length . For the TFE model reduces to the TE model.

By using the CQFL model a *Dirichlet* interface condition is applied.
The carrier concentrations are directly determined in a way that the
quasi-*Fermi* level across the interface remains continuous. The model
is suitable for use at homojunctions. However, it is erroneous to assume
continuous quasi-*Fermi* levels at abrupt heterojunctions. Also the
bandgap alignment of the adjustent semiconductors is ignored when such
continuous condition is enforced. Therefore, models using a
*Neumann* interface condition, like the TFE model or the TE model,
which determine the current flux across the interface, must be used.
Modeling the electron and hole current as well as the energy flux across
heterointerfaces is a complex task. Several models for different types of
interfaces have been proposed [79,80,81,82]. The TE model is
commonly used to model the current across heterojunctions of compound
semiconductors. The TFE model extends the TE model by accounting for tunneling
effects through the heterojunction barrier by introducing a field dependent
barrier height lowering. In [83] a method for unified treatment of
interface models was presented. It allows a change of the interface condition
from *Neumann* to *Dirichlet* type in the limit case of very
strong barrier reduction due to tunneling.

The lattice temperature is assumed to be continuous across semiconductor-semiconductor interfaces. In the case of DD simulation with self-heating an additional thermal energy is accounted for at heterojunction interfaces. This thermal energy is produced when the carriers have to surmount the energy difference in the conduction and valence bands, and , respectively. The energy equation reads:

(3.52) |

2001-02-28