Thermoelectric properties of Na$_{0.68}$CoO$_{2}$ on a 2D triangular lattice

Na$_{x}$CoO$_2$ at $x=0.68$ is a material with important and interesting thermopower. Using a new formalism for computing thermal response functions, via the response to dynamical temperature gradients, in the high frequency limit a nearly frequency independent term is identified for the thermopower $S^*$, the Lorentz ratio $L^*$, and the dimensionless figure of merit $Z^*T$. We calculate, via exact diagonalization for small systems, $S^*$, $L^*$, and $Z^*T$, at all temperatures for the $t$-$J$ model on a 2D triangular lattice exploring the model parameters relevant to the experiments by I. Terasaki {\em{et al}}, (PRB {\bf{56}}, R12685 (1997)) and Y. Wang {\em{et al}}, (Nature {\bf{423}}, 425 (2003)). Our objective is to understand the (large) magnitude of the thermopower, and its remarkable sensitivity to a magnetic field. We also consider the hypothetical and interesting case of the opposite sign of hopping $t\rightarrow -t$.