2.5 Compact Vector Formulation

The inversion of (2.4) will give us a better representation of the driving forces with respect to the fluxes. Making use of all the definitions given in the previous chapter, the current relations can be rewritten in a very compact formulation.

$$\nabla\phi_n = -\sigma_n^{-1}\mathbf{J_n} - P_n\nabla T_n - \mathbf{B} \times (R_n\mathbf{J_n} + \eta_n\nabla T_n)$$ (2.14)

$$\mathbf{Q_n} = P_n T_n \mathbf{J_n} - \kappa_n\nabla T_n + \mathbf{B} \times (\eta_n T_n \mathbf{J_n} + \kappa_n\pounds_n\nabla T_n)$$ (2.15)

An expansion of $\nabla\phi_n$ and $\nabla\phi_p$ in terms of electric potential $\psi$, carrier concentrations, and temperature reveals that actually drift, chemical diffusion, and thermal diffusion appear as driving forces. The deflection that the magnetic field causes on electric and thermal currents is reflected by vector products with $\mathbf{B}$. Solving (2.14) for the current density $\mathbf{J_n}$ yields

$$\mathbf{J_n} = -\sigma_n(\nabla\phi_n + P_n\nabla T) -\sigma_n\fr... ...\phi_n) + (\eta_n + \mu^*_nP_n)\mathbf{B} \times (\mathbf{B} \times \nabla T)\}$$ (2.16)

with $\mu^*_n = r_n \cdot \mu_n$ denoting the Hall mobility of electrons [37].

The current density $\mathbf{J_p}$ reads

$$\mathbf{J_p} = -\sigma_p(\nabla\phi_p + P_p\nabla T) -\sigma_p\fr... ...\phi_p) + (\eta_p + \mu^*_pP_p)\mathbf{B} \times (\mathbf{B} \times \nabla T)\}$$ (2.17)

with $\mu^*_p = r_p \cdot \mu_p$ denoting the Hall mobility of holes [37].