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

The thermal conductivity along ($K_{\mathrm{II}}$) and across ($K_{\perp }$) 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°K. In both directions the conductivity increases with deposition temperature; the $\frac{K_{\mathrm{II}}}{K_{\perp }}$ ratio, however, remains close to 3 in the liquid-helium region and rises rapidly to about 125 at 300°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°K the thermal conductivity measured in this direction is almost temperature-independent ($K_{\perp }\approx 2\times {10}^{-2\mathrm{}}$ W ${\mathrm{cm}}^{-1\mathrm{}}$°${\mathrm{K}}^{-1\mathrm{}}$), while below 20°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 $c_{44}=0.702\mathrm{}\times {10}^{10}$ dyn/${\mathrm{cm}}^2$ as suggested by Komatsu for lampblack-base graphite, a theoretical analysis confirms that below 4°K major contributors to $K_{\perp }$ are out-of-plane modes, and, therefore, $K_{\perp }$ should exhibit the same temperature dependence as $C_L$. By contrast, along layer planes the thermal conductance has an anomalous temperature dependence ($\propto T^{2.5}$ from about 10 to 80°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 $n\approx 2.6$ 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 $T^{n-1\mathrm{}}$ for $T\lesssim 4°$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 Å, 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°C. In conjunction with measurements of the electrical resistivity parallel to the deposition surface, the intercepts point to Lorenz numbers $\frac{K_E{\rho }_{\mathrm{II}}}{T}$ of the order of 3 to 4×${10}^{-8\mathrm{}}$ ${(\mathrm{V}/\mathrm{°}\mathrm{K})}^2$. 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.