Critical behavior of the resistivity in magnetic systems. II. BelowTcand in the presence of a magnetic field

The effect of critical fluctuations on the electrical resistivity of magnetic materials is discussed in detail. The temperature and magnetic field dependences of this critical resistivity $\rho (T, H)$ are given for both cases: when the critical temperature $T_c$ is approached from above and when it is approached from below. The results obtained apply to the two basic classes of magnetic materials: ferromagnets and antiferromagnets, as well as to the two basic electronic systems: metals and semiconductors. The present work is founded on the conclusion that close enough to $T_c$ the critical resistivity of all systems has a magnetic-energy-like behavior. This behavior extends father away from $T_c$ for all systems except for ferromagnetic semiconductors. It is shown that the critical magnetoresistance, $\Delta \rho =\rho (T, H)-\rho (T, 0)$ is negative except for antiferromagnetic semiconductors. $\Delta \rho$ is found to be peaked at $T_c$ for all systems but it never diverges as a function of temperature or field. The results are expressed in terms of power-law dependences of $\Delta \rho$ on $T-T_c$ and $H$ for all the different material classes and for all interesting temperature regions. The corresponding powers, under various conditions, are combinations of the well-known critical exponents, $\alpha$, $\beta$, $\gamma$, $\delta$, and $\nu$. In the mean-field regime the powers are predicted to be those of the critical regime except that the classical values of these exponents have to be used. The present results predict more details of the critical resistivity than can be deduced from the existing experimental data. However, the experimental data available are in accord with the results. In the case of antiferromagnetic metals, the present work explains the features of the magnetoresistance in the rare-earth metals, features that have not been understood before. It is suggested that by proper fit of critical resistance and magnetoresistance data to the power-law behaviors predicted here, critical exponents can be deduced. Recent demonstrations of such fits show that this is indeed feasible.