Theory of the Hall coefficient of polycrystals: Application to a simple model for ${\mathrm{La}}_{2-x}{M}_x\mathrm{Cu}{\mathrm{O}}_4$ ($M=\mathrm{Sr},\mathrm{Ba}$)

We calculate the low-field Hall coefficient $R_H$ of a polycrystalline sample of an anisotropic metal, using the effective-medium approximation. For a quasiplanar metal, with a single band of carriers, we find $R_H=\frac{4}{5}{R}_H^0$, where $R_H^0=\frac{1}{\mathrm{}\left(\mathrm{nqc}\right)\mathrm{}}$ is the free-electron Hall coefficient for a density of $n$ carriers of charge $q$. With both electron and hole carriers, we obtain to first order in the ratios $\frac{m_{\mathrm{hx}}}{m_{\mathrm{hz}}}$ and $\frac{m_{\mathrm{ex}}}{m_{\mathrm{ez}}}$, $R_H=\frac{4}{5}\frac{1}{n_h\mathrm{ec}}\frac{[1-(\frac{n_e}{n_h}\left)Q_x^2\right]}{{[1+(\frac{n_e}{h_n}\left)Q_x\right]}^2} \left[1+4\frac{m_{\mathrm{hx}}}{m_{\mathrm{hz}}}\frac{[1-(\frac{n_e}{n_h}\left)Q_x{Q}_z\right]}{[1-(\frac{n_e}{n_h}\left)Q_x^2\right]}\right],$ where $n_h$ and $n_e$ are the number densities of holes and electrons, $\frac{m_{\mathrm{hx}}}{m_{\mathrm{hz}}}$ the ratio of in-plane to out-of-plane hole masses, and $Q_i$ is the ratio of electron-to-hole mobilities in the ith direction. The model agrees well with experimental results of Penney, Shafer, Olson, and Plaskett for ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x\mathrm{Cu}{\mathrm{O}}_4$ as a function of $x$ at $T=50\mathrm{}$ K.