Surface Effects in Magnetic Crystals near the Ordering Temperature

A form of the Landau-Ginsberg equations applicable to semi-infinite magnetic crystals is derived from the molecular-field theory. We consider both the Heisenberg antiferromagnet and the Heisenberg ferromagnet. We find simple analytic expressions that describe the temperature dependence of the order parameter in the surface region near the ordering temperature. In both cases we find that the order parameter in the surface vanishes linearly with temperature as the ordering temperature is approached from below. This result is in good agreement with the temperature variation of the sublattice magnetization in the surface of antiferromagnetic NiO inferred from the low-energy-electron-diffraction (LEED) data of Palmberg and co-workers, for the entire range of temperatures studied ($0.8T_N<T<\mathrm{}{T}_N$). We find that one cannot use the existing LEED data to determine the value of the exchange constants in the surface layer without a measurement of the absolute value of the sublattice magnetization or measurements over a wider range of temperatures. This conclusion differs from that reached in an earlier study based on a numerical solution of the molecular-field equations. We also examine the behavior of the static spin correlation function $〈S_z\left(\overrightarrow{\mathrm{l}}\right)S_z\left({\overrightarrow{\mathrm{l}}}^{\prime }\right)〉$ in the paramagnetic state, when the sites $\overrightarrow{\mathrm{l}}$ and/or ${\overrightarrow{\mathrm{l}}}^{\prime }$ lie near the surface. We find that there should be no magnetic critical scattering of low-energy electrons from the surface, as the ordering temperature is approached from above. The correlation length for spins in or near the surface remains the order of a lattice constant, even at the ordering temperature for both the ferromagnet and the antiferromagnet.