2.4.3 General Transport Equations

Following Onsager's general formalism, the electron and hole current densities, $\mathbf{J_n}$ and $\mathbf{J_p}$, and the heat fluxes of the individual subsystems, $\mathbf{Q_n}$, $\mathbf{Q_p}$, and $\mathbf{Q_L}$, are related to the gradients of the state variables by a 15 $\times$ 15 matrix:

$$\begin{pmatrix}\mathbf{J_n} \\ [2ex] \mathbf{Q_n} \\ [2ex] \mathb... ...\\ [2ex] \frac{1}{T_p}\nabla T_p \\ [2ex] \frac{1}{T_L}\nabla T_L \end{pmatrix}$$ (2.4)

The 3 $\times$ 3 sub matrices $L_{\alpha,\beta}$ depend on the quasi-static magnetic field $\mathbf{B}$. Those kinetic coefficients, better known as transport matrices $L_{\alpha,\beta}$, are computed taking into account the reciprocity fact of Onsager's theorem and assuming cubic or higher spatial symmetry of the semiconductor medium. A general structure is as follows:

$$L_{\alpha,\beta}(\mathbf{B}) = a_{\alpha,\beta}(B^2)I + b_{\alpha... ...a}(B^2)(\mathbf{B}\times\cdot) + c_{\alpha,\beta}(B^2)(\mathbf{B}\times\cdot)^2$$ (2.5)

where $a_{\alpha,\beta}=a_{\beta,\alpha}$, $b_{\alpha,\beta}=b_{\beta,\alpha}$, and $c_{\alpha,\beta}=c_{\beta,\alpha}$ are scalar functions of $\vert B^2\vert$, and the symbol $(\mathbf{B}\times\cdot)$ denotes the linear operation of taking the vector product of $\mathbf{B}$ and an arbitrary vector. In matrix representation it reads

$$(\mathbf{B}\times\mathbf{k}) = \begin{pmatrix}0 & -B_z & B_y \\ B... ... \\ -B_y & B_x & 0 \end{pmatrix} \begin{pmatrix}k_x \\ k_y \\ k_z \end{pmatrix}$$ (2.6)

The coefficients $a_{\alpha,\beta}$, $b_{\alpha,\beta}$, and $c_{\alpha,\beta}$ can be given as a parametrization of physical quantities, accessible by measurement and commonly used for the characterization of material properties [7,8]. These are the electric conductivities $\sigma_n$ and $\sigma_p$, the thermal conductivities $\kappa_n$, $\kappa_p$, and $\kappa_L$, and the coefficients describing the thermoelectric, galvanomagnetic, and thermomagnetic effects, namely the thermopowers $P_n$ and $P_p$, the Hall coefficients $R_n$ and $R_p$, the Nernst coefficients $\eta_n$ and $\eta_p$, and the Righi-Leduc coefficients $\pounds_n$ and $\pounds_p$ [22,37,44].