Ordering in magnetic films with surface anisotropy

TL;DR

This paper investigates how surface anisotropy influences the magnetic ordering temperature in ferromagnetic films, revealing specific scaling behaviors and phase transition phenomena through an exactly solvable classical spin model.

Contribution

It provides an exact analysis of surface anisotropy effects on ordering in ferromagnetic films, highlighting unique scaling laws and phase transition conditions.

Findings

For small surface anisotropy, the shift in critical temperature scales as (1/N)ln(1/η'_s).

Finite-size scaling behavior differs depending on bulk versus surface anisotropy.

Surface phase transition occurs at T_c^{bulk} for any small surface anisotropy in three dimensions.

Abstract

Effects of the surface exchange anisotropy on ordering of ferromagnetic films are studied for the exactly solvable classical spin-vector model with D \to \infty components. For small surface anisotropy \eta'_s << 1 (defined relative to the exchange interaction), the shift of T_c in a film consisting of N >> 1 layers behaves as T_c^{\rm bulk} - T_c(N) ~ (1/N)\ln(1/\eta'_s) in three dimensions. The finite-size-scaling limit T_c^{\rm bulk} - T_c(N) \propto 1/(\eta'^{1/2}N^2), which is realized for the model with a bulk anisotropy \eta' << 1 in the range N\eta'^{1/2} >~ 1, never appears for the model with the pure surface anisotropy. Here for N\exp(-1/\eta'_s) >~ 1 in three dimensions, film orders at a temperature above T_c^{\rm bulk} (the surface phase transition). In the semi-infinite geometry, the surface phase transition occurs for whatever small values of \eta'_s (i.e., the special phase transition corresponds to T_c^{\rm bulk}) in dimensions three and lower.