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Journal of Physics C Solid State Physics

The spin correlation functions of a semi-infinite Heisenberg ferromagnet

Michael G Cottam

Published Jun 14, 1976• 41 citations
Publisher

Abstract

Results are derived for the dynamic correlation functions between a pair of spin operators at any two sites within a semi-infinite Heisenberg ferromagnet at low temperatures, T<<Tc. The correlation functions are found to contain terms describing the effects of the bulk spin-wave modes and the surface spin-wave modes, together with appropriate weighting factors. The results are employed to evaluate the spin deviation in the ferromagnet as a function of temperature and distance from the surface.

DOI
10.1088/0022-3719/9/11/019

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Journal of Physics C Solid State Physics•Feb 14, 1978•17 citations
Results are derived for the dynamic correlation functions between a pair of spin operators at any two sites within a semi-infinite Heisenberg ferromagnet at low temperatures, T<<Tc. The correlation functions are found to contain terms describing the effects of the bulk spin-wave modes and the surface spin-wave modes, together with appropriate weighting factors. The results are employed to evaluate the spin deviation in the ferromagnet as a function of temperature and distance from the surface.
The two-spin and four-spin correlation functions involving one and two pairs of nearest neighbours respectively are evaluated near the free surface of an Ising ferromagnet and the departure from the bulk value is found. These correlation functions could be termed the energy density and the energy density-energy density correlation functions. Near the critical temperature T c the energy density in the surface layer is found to behave as t 2 ln t with t= mod T-T c mod . This contrasts with the bulk energy density which has a t ln t singularity. Correlations between a surface and a bulk spin are evaluated above and below T c . Correlations between surface spins when the coupling between the surface spins is different from those in the bulk are examined.
Using the spin-wave approximation elementary excitations of a semi-infinite two-dimensional S = ½ Heisenberg antiferromagnet are considered. The spectrum consists of bulk modes — standing spin waves and a quasi-one-dimensional mode of boundary spin waves. These latter excitations eject bulk modes from two boundary rows of sites, thereby dividing the antiferromagnet into two regions with different dominant excitations. As a result, absolute values of nearest-neighbor spin correlations on the edge exceed the bulk value.
The dynamical correlation function Gr{\mathit{G}}_{\mathbf{r}}Gr​(t)==〈Sr{\mathbf{S}}_{\mathbf{r}}Sr​(t)\ensuremath{\cdot}S0{\mathbf{S}}_{0}S0​(0)〉 and dynamical structure factor Sq{\mathit{S}}_{\mathbf{q}}Sq​(\ensuremath{\omega}) are calculated by the modified spin-wave theory for the low-dimensional quantum Heisenberg ferromagnets at low temperature. We use the Dyson-Maleev transformation, ideal spin-wave states, and the rotational averaging. Sq{\mathit{S}}_{\mathbf{q}}Sq​(\ensuremath{\omega}) satisfies the dynamic scaling relation. The explicit form of the scaling function is obtained. The classical limit of our results is compared with a molecular-dynamics calculation. The agreement is surprisingly good.
Using the spin-wave approximation elementary excitations of a semi-infinite two-dimensional S=12S=\frac12S=21​ Heisenberg antiferromagnet are considered. The spectrum consists of bulk modes -- standing spin waves and a quasi-one-dimensional mode of boundary spin waves. These latter excitations eject bulk modes from two boundary rows of sites, thereby dividing the antiferromagnet into two regions with different dominant excitations. As a result absolute values of nearest-neighbor spin correlations on the edge exceed the bulk value.
Dynamical spin correlation functions in (q, omega )-space are calculated numerically for linear quantum-mechanical Heisenberg systems containing up to ten spins with s= 1 / 2 and s=1. The authors consider ferro- and antiferromagnets including single-site and exchange anisotropies. They compare their results with various other theoretical treatments and draw conclusions for the dynamics of infinite chains. In the case of the s=1 planar Heisenberg ferromagnet direct comparison is made with inelastic neutron scattering cross sections of CsNiF 3 . The theoretical and experimental results are in good agreement.