Small polaron hopping in spinel manganates

Abstract

The temperature dependence of small polaron hopping conduction in ceramic spinel $\mathrm{Ni}{\mathrm{Mn}}_{2}{\mathrm{O}}_{4+\ensuremath{\delta}}$ thermistor material has been investigated. We used a theoretical framework based on a random resistor network model to describe small polaron nearest-neighbor hopping (NNH) and variable range hopping (VRH), following the principles of Shklovskii and Efros. We find that in printed thick films and in pressed pellets resistivity is described best by a VRH model of the form $\ensuremath{\rho}\ensuremath{\sim}{T}^{2p}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}{({T}_{0}∕T)}^{p}$, whereas in thin $\mathrm{Ni}{\mathrm{Mn}}_{2}{\mathrm{O}}_{4+\ensuremath{\delta}}$ films resistivity is better described by NNH, $\ensuremath{\rho}\ensuremath{\sim}T\phantom{\rule{0.2em}{0ex}}\mathrm{exp}({T}_{0}∕T)$. Steady state dc resistance vs temperature measurements for $\mathrm{Ni}{\mathrm{Mn}}_{2}{\mathrm{O}}_{4+\ensuremath{\delta}}$ thick films, thin films, and pellets have been carried out and the parameters $p$ and ${T}_{0}$ determined. For thick films $p$ was found to be $\ensuremath{\sim}0.5$, indicating VRH with an approximately parabolic distribution of the density of states (DOS) around Fermi level. For pellets $p$ was $\ensuremath{\sim}0.65$, and in thin films $\ensuremath{\sim}1$ indicating NNH. The increase of $p$ was interpreted as an increase of disorder in the system, leading to strong electron localization effects and narrowing ${\mathrm{Mn}}^{3+}∕{\mathrm{Mn}}^{4+}$ bandwidth. In thick films and pellets the DOS was determined by a parametrization related to the $p$ value, giving ${10}^{20}--{10}^{21}\phantom{\rule{0.3em}{0ex}}{\mathrm{eV}}^{\ensuremath{-}1}\phantom{\rule{0.2em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}3}$. The characteristic temperature ${T}_{0}$ was in the range of $2\ifmmode\times\else\texttimes\fi{}{10}^{5}\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ in thick films, $3\ifmmode\times\else\texttimes\fi{}{10}^{4}\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ in pellets, and $5\ifmmode\times\else\texttimes\fi{}{10}^{3}\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ in thin films.