Spin-entropy contribution to thermopower in the [Ca$_2$CoO$_{3-t}$]$_{0.62}$(CoO$_2$) misfits

Two samples of the [Ca$_2$CoO$_{3-t}$]$_{0.62}$(CoO$_2$) misfit cobaltate, often denoted as the Ca$_{3}$Co$_{3.93}$O$_{9}$ phase, were prepared from the same ceramic material by the oxygen and argon annealing, resulting in different carrier concentrations in the conducting CoO$_{2}$ layers, n=0.31 and 0.19 hole/Co, respectively. Electrical and thermal transport properties were studied in dependence of magnetic field up to 140 kOe. The magnetothermopower data reveal an extra spin-entropy contribution to Seebeck coefficient that is not expected for carriers of Fermi liquid character. Its magnitude is unprecedentedly large and makes at zero field up to 50$\%$ of the theoretical limit k$_B$/$e$ ln2$ = 59 \mu VK^{-1}$. This spin-entropy contribution is gradually suppressed with increasing magnetic field, and the saturation is even observed when temperatures are low enough. To understand the results, the thermopower is treated in terms of purely thermodynamic Kelvin formula, and so-called Spin liquid model is evoked, providing a reason for the spin-entropy manifestation in the [Ca$_2$CoO$_{3-t}$]$_{0.62}$(CoO$_2$) misfits.