2.4.5 Transport Coefficients with a Magnetic Field

Even if the medium is isotropic, the transport coefficients in presence of a magnetic field show an asymmetry. The way in which driving force and magnetic field are oriented (perpendicular or parallel with respect to each other) implies a non-symmetric tensor representation of the transport coefficients that can be written as

$$L = L_\bot(1-P_B)+L_\Vert P_B$$ (2.10)

where $P_B$ denotes the operation of projecting a vector along the direction of $B$.

The transport coefficients with magnetic field are the following:

2.4.5.1 Hall Coefficients

The isothermal Hall coefficients $R_n$ ($R_p$) characterize the transverse quasi Fermi level gradient caused by the magnetic field acting on the electron (hole) current:

$$R_{n,p} = - \frac{\nabla\phi_{n,p}\cdot(\mathbf{B}\times \mathbf{J_{n,p}})}{(\mathbf{B}\times \mathbf{J_{n,p}})^2} \quad \nabla T = 0$$ (2.11)

2.4.5.2 Nernst Coefficients

The isothermal Nernst coefficients $\eta_n$ ($\eta_p$) characterize the transverse gradient of the quasi Fermi potential caused by the deflection of an electron (hole) heat current flowing 'down a temperature gradient':

$$\eta_{n,p} = - \frac{\nabla\phi_{n,p}\cdot (\mathbf{B}\times\nabla T_{n,p})} {(\mathbf{B}\times\nabla T_{n,p})^2} \quad \mathbf{J_{n,p}}=0$$ (2.12)

2.4.5.3 Righi-Leduc Coefficients

The Righi-Leduc coefficients $\pounds_n$ ($\pounds_p$) characterize the transverse temperature gradient caused by the deflection of an electron (hole) heat current flowing 'down a temperature gradient':

$$\pounds_{n,p} = \frac{\nabla T\cdot (\mathbf{B}\times \mathbf{Q_{n,p}})} {(\mathbf{B}\times\nabla T)\cdot (\mathbf{B}\times \mathbf{Q_{n,p}})} \quad \mathbf{J_{n,p}}=0$$ (2.13)