# Preamble: Transport Coefficients Definition

Chapter
Part of the Springer Theses book series (Springer Theses)

## Abstract

In this manuscript we will analyse the thermo-electric transport properties of two-dimensional condensed matter systems.

In this manuscript we will analyse the thermo-electric transport properties of two-dimensional condensed matter systems. Then, in order to fix the notations in this brief preamble we will define the transport coefficients for the case at hand.

We will consider a two-dimensional system living in the plane $$x-y$$ and, in some cases, we will analyse also the effects due to an external magnetic field B applied in the direction perpendicular to the $$x-y$$ plane, z (see Fig. 1.1).

We are interested in the response of the electrical current $$\vec {J}$$ and the heat current $$\vec {Q}$$ to an applied electric field $$\vec {E}$$ and a temperature gradient $$\vec {\nabla }T$$. By definition, the transport coefficients relate the previous quantities in the following way:
\begin{aligned} \begin{pmatrix} \vec {J} \\ \vec {Q} \end{pmatrix} =\begin{pmatrix} \hat{\sigma } &{} \hat{\alpha } \\ T \hat{\alpha } &{} \hat{\bar{\kappa }} \end{pmatrix} \begin{pmatrix} \vec {E} \\ -\vec {\nabla } T \end{pmatrix} \ . \end{aligned}
(1.1)
In the presence of an external magnetic field B in the z-direction (see Fig. 1.1) the transport coefficients $$\hat{\sigma }, \; \hat{\alpha }$$ and $$\hat{\bar{\kappa }}$$ are matrices, which, due to Onsager reciprocity, assume the following form:
\begin{aligned} \hat{\sigma }=\sigma _{xx} \hat{1}+ \sigma _{xy} \hat{\varepsilon } \ , \end{aligned}
(1.2)
where $$\hat{1}$$ is the identity, and $$\hat{\varepsilon }$$ is the antisymmetric tensor $$\varepsilon _{ij}=-\varepsilon _{ji}$$. $$\sigma _{xx}$$ and $$\sigma _{xy}$$ describe the longitudinal and Hall conductivity, respectively. The resistivity $$\hat{\rho }$$ is defined as the inverse of the conductivity matrix, namely $$\hat{\rho }=\hat{\sigma }^{-1}$$. Similarly, the thermo-electric conductivity $$\hat{\alpha }$$ has a form analogous to (1.2), and determines the Seebeck coefficient S via the relation:
\begin{aligned} S=\frac{\alpha _{xx}}{\sigma _{xx}} \ , \end{aligned}
(1.3)
Finally $$\hat{\bar{\kappa }}$$, which governs thermal transport in the absence of electric fields, assumes a similar structure to that described before for $$\hat{\sigma }$$ and $$\hat{\alpha }$$. In contrast to $$\hat{\bar{\kappa }}$$, the thermal conductivity, $$\hat{\kappa }$$, is defined as the heat current response to $$-\vec {\nabla }T$$ in the absence of an electric current, namely, in the presence of electrically isolated boundaries. It is given by
\begin{aligned} \hat{\kappa }=\hat{\bar{\kappa }}-T \hat{\alpha } \cdot \hat{\sigma }^{-1} \cdot \hat{\alpha } \ . \end{aligned}
(1.4)
Eventually, the Nernst coefficient is defined as the electric field induced by a thermal gradient in the absence of an electric current. It is defined by the linear response relation $$\vec {E} = -\hat{\theta } \vec {\nabla } T$$, with
\begin{aligned} \hat{\theta }=-\hat{\sigma }^{-1} \cdot \hat{\alpha } \ . \end{aligned}
(1.5)

With these definitions at hand, we are now ready to start the analysis of the exotic and exciting properties of the cuprates superconductors, starting by understand how they differ from the Fermi Liquid theory.