c-axis transport in highly anisotropic metals: role of small polarons

We show in a simple model of interlayer hopping of single electrons, that transport along the weakly coupled c-axis of quasi-two-dimensional metals does not always probe only the in-plane electron properties. In our model where there is a strong coupling between electrons and a bosonic mode that propagates in the $c$ direction only, we find a broad maximum in the c-axis resistivity at a temperature near the characteristic energy of the bosonic mode, while no corresponding feature appears in the $ab$ plane transport. At temperatures far from this bosonic energy scale, the c-axis resistivity does track the in-plane electron scattering rate. We demonstrate a reasonable fit of our theory to the apparent metallic to non-metallic crossover in the c-axis resistivity of the layered ruthenate $\rm Sr_2 Ru O_4$.