Thermal Conductivity of Pyrolytic Graphite at Low Temperatures. I. Turbostratic Structures

The thermal conductivity along (${K}_{\mathrm{II}}$) and across (${K}_{\ensuremath{\perp}}$) the layer planes of 5 specimens of pyrolytic graphite (PG) in their as-deposited condition has been investigated at temperatures ranging from 1.7 to 300\ifmmode^\circ\else\textdegree\fi{}K. In both directions the conductivity increases with deposition temperature; the $\frac{{K}_{\mathrm{II}}}{{K}_{\ensuremath{\perp}}}$ ratio, however, remains close to 3 in the liquid-helium region and rises rapidly to about 125 at 300\ifmmode^\circ\else\textdegree\fi{}K. In part, these anisotropies are presumed to reflect a situation in which only a fraction of the deposit permits regular heat flow perpendicular to the deposition surface. Above 100\ifmmode^\circ\else\textdegree\fi{}K the thermal conductivity measured in this direction is almost temperature-independent (${K}_{\ensuremath{\perp}}\ensuremath{\approx}2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}$ W ${\mathrm{cm}}^{\ensuremath{-}1}$\ifmmode^\circ\else\textdegree\fi{}${\mathrm{K}}^{\ensuremath{-}1}$), while below 20\ifmmode^\circ\else\textdegree\fi{}K it is nearly proportional to ${T}^{2.3}$, in accordance with the lattice specific heat ${C}_{L}$. At these temperatures the thermal resistance arises from boundary scattering, and judging from x-ray evidence the phonon mean free path may be taken as isotropic. Turbostratic stacking affects the shearing elastic constant quite drastically; with ${c}_{44}=0.702\ifmmode\times\else\texttimes\fi{}{10}^{10}$ dyn/${\mathrm{cm}}^{2}$ as suggested by Komatsu for lampblack-base graphite, a theoretical analysis confirms that below 4\ifmmode^\circ\else\textdegree\fi{}K major contributors to ${K}_{\ensuremath{\perp}}$ are out-of-plane modes, and, therefore, ${K}_{\ensuremath{\perp}}$ should exhibit the same temperature dependence as ${C}_{L}$. By contrast, along layer planes the thermal conductance has an anomalous temperature dependence ($\ensuremath{\propto}{T}^{2.5}$ from about 10 to 80\ifmmode^\circ\else\textdegree\fi{}K), which reflects mainly the dispersive nature of bond-bending vibrations in graphite. At still lower temperatures ${K}_{\mathrm{II}}$ involves electronic contributions ${K}_{E}$ and can be expressed as a sum of two terms $AT+B{T}^{n}$, where $n\ensuremath{\approx}2.6$ as predicted from a long-wave-length treatment of the "effective" phonon velocity, which shows that in-plane modes are seriously enhanced relative to the part they play in ${C}_{L}$. Plots of $\frac{{K}_{\mathrm{II}}}{T}$ versus ${T}^{n\ensuremath{-}1}$ for $T\ensuremath{\lesssim}4\ifmmode^\circ\else\textdegree\fi{}$K yield straight lines with intercepts A and slopes B that correlate closely with the deposition temperature. The slopes are direct measures of the phonon mean free paths (600 to 1200 \AA{}, which corresponds to the polygonal zones of the wrinkled-sheet structure), and illustrate the gradual improvement in preferred orientation as the deposition temperature covers the range 1700 to 2300\ifmmode^\circ\else\textdegree\fi{}C. In conjunction with measurements of the electrical resistivity parallel to the deposition surface, the intercepts point to Lorenz numbers $\frac{{K}_{E}{\ensuremath{\rho}}_{\mathrm{II}}}{T}$ of the order of 3 to 4\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}8}$ ${(\mathrm{V}/\mathrm{\ifmmode^\circ\else\textdegree\fi{}}\mathrm{K})}^{2}$. Within the framework of a simple two-band-model description of graphite, there is strong evidence for a bipolar term in the electronic thermal conductivity of turbostratic PG, which indicates that these Lorenz numbers may not be inconsistent.