Quasi-two-dimensional behavior of the surface magnetization in a ferromagnet with softened surface exchange

The effect of softening of the surface exchange on the temperature dependence of the surface magnetization $M_s\mathrm{}\left(T\right)\mathrm{}$ in the spin-wave regime is investigated. A recursion method for calculating the surface density of spin-wave states is developed. It is used to prove that neither softening of exchange nor enhancement of the magnetization in an arbitrary (finite) number of atomic layers parallel to the surface has any effect on the initial $T^{\frac{3}{2}}$ law for $M_s\mathrm{}\left(T\right)\mathrm{}$. This result holds, however, only at temperatures of the order of 1% of the Curie temperature $T_c$. It is shown that a softening of the exchange perpendicular to the surface causes a gradual transition to a quasi-two-dimensional spin-wave density of states, which results in a crossover to a second extended region of a $T^{\frac{3}{2}}$ law with a prefactor determined by the surface exchange. This second region of the $T^{\frac{3}{2}}$ law, which extends up to temperatures $T\approx 0.4T_C$, provides a complete explanation for the observed $M_s\mathrm{}\left(T\right)\mathrm{}$ which was determined recently by spin-polarized low-energy electron diffraction for ${\mathrm{Ni}}_{40}$ ${\mathrm{Fe}}_{40}$ ${\mathrm{B}}_{20}$ and by Mössbauer spectroscopy for an Fe(110) surface.