As thermal circuit simulation is an equivalent problem to electrical circuit simulation
*MINIMOS-NT* makes use of similar formulations. The thermal heat flow over the
contact replaces the electrical current and the contact temperature the contact
voltage, hence the contact condition reads

with

In *MINIMOS-NT* two different thermal contact models are implemented. The first
model implements an isothermal contact by simply setting the lattice
temperature at the interface points equal to the contact temperature

T_{L} = T_{C} . |
(5.30) |

The second model considers a thermal contact resistance and the thermal heat flow density

S_{L} = n^{ . } |
(5.31) |

with being the thermal contact resistivity. The contact conductance

The choice of the contact model has a fundamental influence on both the
electrical and the thermal behavior of the device. Both models result in
approximately the same temperature difference inside the device relative to the
boundary values. However, the absolute temperature values can be completely
different. Simulated temperature distributions inside the device for the HBT
from Section 7 are shown in Fig. 5.4 and
Fig. 5.5 for the isothermal and the resistance contact model,
respectively, with
*V*_{CE} = 3.5 *V*. Fig. 5.4 shows
the temperature distribution for different base-emitter voltages
*V*_{BE}
whereas for Fig. 5.5
*V*_{BE} = 1.0 *V* was used and
*G*_{th} was varied. For
*V*_{BE} = 1.0 *V* both contact
models generate the same relative temperature distribution but in the case of
the resistance contact model the temperature is shifted by an offset which
exponentially depends on
*G*_{th}. For
*G*_{th} as small as
10 *mW*/*K* no solution could be found at all as the lattice temperature
would exceed 600 *K* which inhibits a successful simulation. Although
*V*_{BE} = 1.0 *V* is quite high it must be pointed out that even for
lower bias conditions the same situation occurs for improper choice of
*G*_{th}.

These investigations show that the isothermal model can only be used when the exact contact temperatures are known. Simply assuming ambient temperature delivers completely wrong results as the simulated region of the device is normally reduced to the electrically active region which is only a small portion of the whole geometry. Hence, especially for mixed-mode simulations use of thermal contact resistances is mandatory.

In Fig. 5.6 the heat generation inside the device is shown for
different values of
*G*_{th}. As the current density remains
approximately constant within this cross-section, the maximum of the heat
generation is located at the base-collector space charge region where the
electric field is maximal. As
*V*_{CE} = 3.5 *V* was assumed
which is quite moderate, even higher heat generation rates can be expected for
the circuits simulated in Section 7. Although the final values may
give reasonable temperature distributions, during iteration the bias voltages
of the device may vary considerably and can easily exceed
*V*_{BE} = 1.5 *V* and
*V*_{CE} = 20 *V*. This situation can occur
during mixed-mode simulation of circuits with large supply voltages. Under
these bias conditions, the thermal problem cannot be solved as the melting
point of the device would be exceeded considerably. This causes excessive
problems when simulating fully-coupled electro-thermal systems especially
because measured values for
*G*_{th} are in the range
1-10 *mW*/*K*.

In Fig. 5.7 the temperature distribution for different
base-emitter voltages is shown generated with a quite large value of
100 *mW*/*K* for
*G*_{th}. All these figures indicate, that
the temperature difference *inside* the device is normally much smaller
than the temperature difference induced by the contact model. For some
simulations it might therefore be beneficial to substitute the heat flow
equation by a spatially constant lattice temperature which is determined by
the dissipated power of the device and thermal resistances at the contacts.
For example, when the device is known to operate at 400 *K*, the
local generated heat may be neglected.

1999-05-31