2.2.1 Heat Flux

It can be observed from various experiments that heat flows from the hotter to the colder side. Since the matter that is involved shows a statistical behavior at micro and nano-scale level in terms of the BROWNian molecular motion, the previous statement can be formulated as: The most probable consequence is that heat flows spontaneously from the hotter to the colder side by diffusion and relaxation mechanisms. This is exactly the definition of the second law of thermodynamics found in [73].

The time derivative of the heat can be expressed by FOURIER's and
LAMBERT's^{2.12} law, which is equivalent to the
STEFAN^{2.13}-BOLTZMANN law for grey radiators.
These laws consider the spatial temperature gradient
plus the heat flux density due to the surface radiation to the ambient, respectively:

(2.42) |

Here, the first term on the right hand side is determined by FOURIER's law, where the thermal conductivity tensor is denoted by and is the local temperature in Kelvin. The second term describes LAMBERT's law for grey radiators, where denotes the STEFAN-BOLTZMANN constant and and stand for the ambient and the local surface temperature, respectively. The coefficients and reflect the efficiency of the absorption and the radiation of the considered surfaces. The ``black body'' has the absorption and radiation efficiency . In most TCAD applications, the radiation can be neglected, except for areas at the surface of semiconductor devices, for instance, in passivation layers and heat sinks.

As the electric and magnetic fields store energy, also the matter
stores heat energy. If heated bodies are put into a colder environment,
they show a certain thermal relaxation behavior.
A possible way to describe this relaxation behavior is to assign a quantity to
each material, where the value of the quantity determines how much energy can be
stored per mass or per mole. This quantity is called specific heat
capacitance^{2.14}.
Historically, the heat capacitance is distinguished by two types. The first
one determines the heat capacitance at constant pressure
and
the second one describes the heat capacitance at constant volume
:

Here, the heat capacitances and determine the change of the internal energy with regard to the temperature change where different constraints are applied: constant pressure and constant volume. To obtain the specific heat capacitances , the heat capacitances are normalized to their involved mass :

The unit of the specific heat capacitance is either or according to the type of the mass used in (2.45) (mass or molar mass). The different values for the specific heat capacitances of a particular material can be easily transformed into each other.

Both heat capacitances of a given material, the one at constant pressure (cf. (2.43)) and the one at constant volume (cf. (2.44)), differ from each other by the identity

where denotes the thermal expansion coefficient at constant pressure, and represents the isothermal compressibility coefficient of the material [76]. The thermal volume expansion coefficient and the compressibilities and are defined as follows:

Here, the determines the relative volume change with regard to temperature changes, shows the relative isothermal volume change and the relative isentropic volume change with regard to changes of the local pressure.

Together with the thermodynamic potentials (2.31) and (2.32) and the
thermodynamic identities (2.36) and (2.37), another correlation between
heat capacitances and compressibilities can be derived by using the chain rule
for differentiation.

The equations (2.50) and (2.51) show the equality of the different equivalent methods for differentiation according to the chain rules from LEIBNIZ

For
isotropic and temperature-independent materials, the left hand side
of (2.9) becomes
times the LAPLACEian^{2.16}operator and (2.9) can be written as

where the maximum of the thermal conductivity has been published for carbo-nano-tubes (CNTs) and nano wires as in [77,78]. In comparison to that, the thermal conductivity of diamond is typically in the range of [79,80].

To determine the proper heat generation term for a particular problem, several proposals have been made for semiconductor and interconnect models. The simplest model is to calculate the power loss with the local electrical field and the resulting local current density [81,82] by

where and can be calculated using the appropriate models to describe the observed behavior of the electrical field and the electrical current density. In order to account for the current densities and appropriately, the SEEBECK

where and represent the carrier concentrations for electrons and holes, and denote the mobility tensors for electrons and holes, and and are the SEEBECK coefficients for electrons and holes. The quantities and represent the quasi-FERMI

(2.57) | |||

(2.58) |

where denotes the local potential, is the thermal voltage according to , and denotes the intrinsic carrier concentration of the semiconductor material.

For semiconductor devices, the
temperature
is often assumed to be the lattice temperature of the semiconductor crystal
since the carriers and the lattice can be considered as two systems in thermal
quasi equilibrium [66].
For a rigorous treatment of
the SEEBECK effect, also the FOURIER law for the heat conduction
equation (2.8) has to be adapted to

(2.59) | |||

(2.60) |

where the first part is due to FOURIER's law and the second part due to SEEBECK's effect.

Stefan Holzer 2007-11-19