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:

$${{\mathbf{{q}}}_{\mathrm{th}}}{=}- \underbrace{\tilde{\lambda} \c... ...^4 - \varepsilon_{\mathrm{2}}T_2^4).}_{\mathrm{{\textsf{\sc {Lambert}}}'s law}}$$ (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 $\tilde{\lambda}$ and $T$ is the local temperature in Kelvin. The second term describes LAMBERT's law for grey radiators, where ${\sigma_{\mathrm{SB}}}$ denotes the STEFAN-BOLTZMANN constant and $T_1$ and $T_2$ stand for the ambient and the local surface temperature, respectively. The coefficients $\varepsilon_1$ and $\varepsilon_2$ reflect the efficiency of the absorption and the radiation of the considered surfaces. The ``black body'' has the absorption and radiation efficiency $\varepsilon_{\mathrm{BlackBody}} {=}1$ . 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 $C_{\mathrm{p}}$ and the second one describes the heat capacitance at constant volume $C_{\mathrm{V}}$ :

$$C_{\mathrm{p}} {=}T \displaystyle\left( {\frac{\partial{S}}{\part... ...eft( {\frac{\partial{U}}{\partial{T}}} \right)_{\mathrm{N,{p^{\mathrm{mech}}}}}$$ (2.43)

$$C_{\mathrm{V}} {=}T \displaystyle\left( {\frac{\partial{S}}{\part... ...} \displaystyle\left( {\frac{\partial{U}}{\partial{T}}} \right)_{\mathrm{V,N}}.$$ (2.44)

Here, the heat capacitances $C_{\mathrm{p}}$ and $C_{\mathrm{V}}$ determine the change of the internal energy $U$ with regard to the temperature change where different constraints are applied: constant pressure and constant volume. To obtain the specific heat capacitances $c_{\mathrm{i}}$ , the heat capacitances $C_{\mathrm{i}}$ are normalized to their involved mass $m$ :

$$c_{\mathrm{i}} {=}\frac{1}{m} C_{\mathrm{i}}.$$ (2.45)

The unit of the specific heat capacitance is either $\mathrm{J kg^{-1}K^{-1}}$ or $\mathrm{J mol^{-1} K^{-1}}$ 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

$$C_{\mathrm{p}} - C_{\mathrm{V}} = T V\frac{({\alpha^{\mathrm{mech}}})^2}{{{\kappa^{\mathrm{mech}}_{\mathrm{T}}}}},$$ (2.46)

where ${\alpha^{\mathrm{mech}}}$ denotes the thermal expansion coefficient at constant pressure, and ${{\kappa^{\mathrm{mech}}_{\mathrm{T}}}}$ represents the isothermal compressibility coefficient of the material [76]. The thermal volume expansion coefficient ${\alpha^{\mathrm{mech}}}$ and the compressibilities ${{\kappa^{\mathrm{mech}}_{\mathrm{T}}}}$ and ${{\kappa^{\mathrm{mech}}_{\mathrm{S}}}}$ are defined as follows:

$${\alpha^{\mathrm{mech}}}{=}{\frac{1}{V}} \left({\frac{\mathrm{d}{V}}{\mathrm{d}{T}}}\right)_{N,{p^{\mathrm{mech}}}},$$ (2.47)

$${{\kappa^{\mathrm{mech}}_{\mathrm{T}}}}{=}- {\frac{1}{V}} \left({\frac{\partial{V}}{\partial{{p^{\mathrm{mech}}}}}}\right)_{T,N},$$ (2.48)

$${{\kappa^{\mathrm{mech}}_{\mathrm{S}}}}{=}- {\frac{1}{V}} \left({\frac{\partial{V}}{\partial{{p^{\mathrm{mech}}}}}}\right)_{S,N}.$$ (2.49)

Here, the ${\alpha^{\mathrm{mech}}}$ determines the relative volume change with regard to temperature changes, ${{\kappa^{\mathrm{mech}}_{\mathrm{T}}}}$ shows the relative isothermal volume change and ${{\kappa^{\mathrm{mech}}_{\mathrm{S}}}}$ 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.

$$\frac{\partial({p^{\mathrm{mech}}},S,N)}{\partial(V,T,N)} {=} \fr... ...\mathrm{mech}}},T,N)} \frac{\partial({p^{\mathrm{mech}}},T,N)}{\partial(V,T,N)}$$ (2.50)

$${=} \left(\frac{\partial {p^{\mathrm{mech}}}}{\partial V}\right)_... ...hrm{mech}}}} \left(\frac{\partial {p^{\mathrm{mech}}}}{\partial V}\right)_{T,N}$$ (2.51)

$$=\! \Longrightarrow\quad\quad \frac{{{\kappa^{\mathrm{mech}}_{\ma... ...\frac{C_{\mathrm{p}}}{C_{\mathrm{V}}} {=}\frac{c_{\mathrm{p}}}{c_{\mathrm{V}}}.$$ (2.52)

The equations (2.50) and (2.51) show the equality of the different equivalent methods for differentiation according to the chain rules from LEIBNIZ^2.15 and with the previous definitions of the compressibilities (2.48) and (2.49). Since the isobar and isochor heat capacitances describe the same region of matter, their ratio in (2.51) is the same as for the corresponding specific heat capacitances.

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

$$\lambda \Delta{T} {=} c_{\mathrm{p}} \rho_{\mathrm{m}} {{\partial}_{t}}{T} - {H_{\mathrm{th}}},$$ (2.53)

where the maximum of the thermal conductivity has been published for carbo-nano-tubes (CNTs) and nano wires as $4.0-4.6{\times}10^{4} \mathrm{W/K}$ in [77,78]. In comparison to that, the thermal conductivity of diamond is typically in the range of $1.0-2.5{\times}10^{3} \mathrm{W/K}$  [79,80].

To determine the proper heat generation term ${H_{\mathrm{th}}}$ 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 ${\mathbf{{E}}}$ and the resulting local current density ${\mathbf{{J}}}$  [81,82] by

$${H_{\mathrm{th}}}{=}{\mathbf{{E}}} \cdot {\mathbf{{J}}},$$ (2.54)

where ${\mathbf{{E}}}$ and ${\mathbf{{J}}}$ 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 ${\mathbf{{J}}}_{n}$ and ${\mathbf{{J}}}_{p}$ appropriately, the SEEBECK^2.17 effect has to be considered as well, where the phenomenological semiconductor current equations [62,66] can be enhanced by

$${\mathbf{{J}}}_{n} {=}- q n {{{\tilde{\mu}}}^{\mathrm{mob}}}_{n} ... ... \left( {{\boldmath {\nabla}}}\Phi_{n} + P_{n} {{\boldmath {\nabla}}}T \right),$$ (2.55)

$${\mathbf{{J}}}_{p} {=}\quad\! \! q p {{{\tilde{\mu}}}^{\mathrm{mo... ... \left( {{\boldmath {\nabla}}}\Phi_{p} + P_{p} {{\boldmath {\nabla}}}T \right),$$ (2.56)

where $n$ and $p$ represent the carrier concentrations for electrons and holes, ${{{\tilde{\mu}}}^{\mathrm{mob}}}_{n}$ and ${{{\tilde{\mu}}}^{\mathrm{mob}}}_{p}$ denote the mobility tensors for electrons and holes, and $P_{n}$ and $P_{p}$ are the SEEBECK coefficients for electrons and holes. The quantities $\Phi_{n}$ and $\Phi_{p}$ represent the quasi-FERMI^2.18 potentials for electrons and holes in semiconductor materials:

$$% \Phi_{n} {=}\varphi - V_{\mathrm{T}} \ln{\left(\frac{n}{n_{\mathrm{i}}}\right)},$$ (2.57)

$$\Phi_{p} {=}\varphi + V_{\mathrm{T}} \ln{\left(\frac{p}{n_{\mathrm{i}}}\right)},$$ (2.58)

where $\varphi$ denotes the local potential, $V_{\mathrm{T}}$ is the thermal voltage according to $V_{\mathrm{T}}{=}{k_{\mathrm{B}}}T / q$ , and $n_{\mathrm{i}}$ denotes the intrinsic carrier concentration of the semiconductor material.

For semiconductor devices, the temperature $T$ 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

$${{{\mathbf{{q}}}_{\mathrm{th}}}}_{,n} {=}- \tilde{\lambda}_{n} \cdot {{\boldmath {\nabla}}}{T} + P_{n} T {\mathbf{{J}}}_{n},$$ (2.59)

$${{{\mathbf{{q}}}_{\mathrm{th}}}}_{,p} {=}- \tilde{\lambda}_{p} \cdot {{\boldmath {\nabla}}}{T} + P_{p} T {\mathbf{{J}}}_{p},$$ (2.60)

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