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AIP conference proceedings

Magnetic ordering and critical behavior near surfaces

Pierre C. Hohenberg, Kurt Binder

Published Jan 1, 1975• 11 citations
Publisher

Abstract

Films and semi‐infinite systems are studied for Ising and Heisenberg models, using mean‐field theory, scaling, series expansions, and Monte Carlo simulations. Predictions are made for the magnetization as a function of temperature and distance from the surface, and for the critical exponents. Possible experimental consequences of the theory are briefly reviewed.

DOI
10.1063/1.30099

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Surface effects on magnetic phase transitions

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The surface scaling theory previously presented by the authors is developed further, and derived heuristically from a cluster model. Monte Carlo calculations are carried out to obtain the spatial and temperature dependence of the magnetization in Ising and Heisenberg systems with free surfaces. The exponent ${\ensuremath{\beta}}_{1}$ of the (surface) layer magnetization is shown to agree with the scaling value (${\ensuremath{\beta}}_{1}\ensuremath{\approx}\frac{2}{3}$) previously derived. In the Heisenberg system, the results at low temperature agree with a spin-wave calculation by Mills and Maradudin. Ising models with modified exchange ${J}_{s}=J(1+\ensuremath{\Delta})\ensuremath{\ne}J$ on the surface are considered, both in mean-field theory and by means of high-temperature-series expansions. The critical value ${\ensuremath{\Delta}}_{c}$ for surface ordering is found from the series to be 0.6, compared to the mean-field value of 0.25. For $\ensuremath{\Delta}>{\ensuremath{\Delta}}_{c}$ there is a temperature region in which the surface behaves like a bulk two-dimensional Ising model near its phase transition. The critical exponents experience a crossover at $\ensuremath{\Delta}={\ensuremath{\Delta}}_{c}$, which is reflected in poorly behaved series, and effective exponents differing from the true ones for $\ensuremath{\Delta}\ensuremath{\lesssim}{\ensuremath{\Delta}}_{c}$. In the case of weakened surface exchange ($0<{J}_{s}<J$), the layer magnetization is shown to fit a linear temperature dependence over a large temperature range below ${T}_{c}$, thus providing a possible explanation for previous experiments. For sufficiently strong negative ${J}_{s}$, mean-field theory predicts that the surface will order antiferromagnetically while the bulk is ferromagnetic.

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Phase transitions in Ising models with tree surfaces are studied from various points of view, including a phenomenological Landau theory, high-temperature series expansions, and a scaling theory for thermodynamic quantities and correlation functions. In the presence of a surface a number of new critical exponents must be defined. These arise because of the existence of "surface" terms in the thermodynamic functions, and because of the anisotropy of space and lack of translational symmetry introduced by the surface. The need for these new critical exponents already appears in the phenomenological theory, which is discussed in detail and related to the microscopic mean-field approximation. The essential new parameter appearing in this theory is an extrapolation length $\ensuremath{\lambda}$ which enters the boundary condition on the magnetization at the surface. For magnetic systems this length is of the order of the interaction range, in contrast to superconductors, where it is usually much larger. In order to go beyond the mean-field theory, high-temperature series expansions are carried out for the Ising half-space, to tenth order in two dimensions and to eighth order in three dimensions. A scaling theory is developed both for thermodynamic functions and for spin correlations near the surface, and relations are found among the exponents of the half-space. Both the scaling theory and the numerical calculations are compared with the exact solution of the Ising half-plane (two dimensions) by McCoy and Wu, and agreement is found wherever the theory is applicable. In analogy to the bulk situation, the scaling theory is found to agree with mean-field theory in four dimensions. The prediction of the present work which is most easily accessible to experiment is the temperature dependence of the magnetization at the surface, with critical exponent estimated to be ${\ensuremath{\beta}}_{1}=\frac{2}{3}$. The mean-field result, ${\ensuremath{\beta}}_{1}=1$, seems to agree more closely with presently available experiment, and more work is needed to clarify the situation.

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Magnetization Profile in the D= 2 Semi-Infinite Ising Model and Crossover Between Ordinary and Normal Transition

Peter Czerner, Uwe Ritschel

We study the two-dimensional semi-infinite Ising model with a free surface at or near bulk criticality. Special attention is paid to the influence of a boundary magnetic field h1 on the surface-near regime and the crossover between the fixed points at h1=0 and h1=∞. Near the surface, a smallh1 causes a steeply increasing magnetization m(z)~z3/8 log z as the distance z increases away from the surface. By means of a phenomenological scaling analysis, this phenomenon can be related to the well-known logarithmic dependence of the surface magnetization m1 on h1. Our analysis provides a deeper understanding of the existing exact results on m(z) and relates the short-distance phenomena in d=2 to those in higher dimensions. Both the results of the scaling analysis and the exact analytic profiles are corroborated by Monte Carlo simulations.

arXiv:cond-mat/9609120•International Journal of Modern Physics B•Jul 10, 1997•16 citations
Films and semi‐infinite systems are studied for Ising and Heisenberg models, using mean‐field theory, scaling, series expansions, and Monte Carlo simulations. Predictions are made for the magnetization as a function of temperature and distance from the surface, and for the critical exponents. Possible experimental consequences of the theory are briefly reviewed.
The surface scaling theory previously presented by the authors is developed further, and derived heuristically from a cluster model. Monte Carlo calculations are carried out to obtain the spatial and temperature dependence of the magnetization in Ising and Heisenberg systems with free surfaces. The exponent ${\ensuremath{\beta}}_{1}$ of the (surface) layer magnetization is shown to agree with the scaling value (${\ensuremath{\beta}}_{1}\ensuremath{\approx}\frac{2}{3}$) previously derived. In the Heisenberg system, the results at low temperature agree with a spin-wave calculation by Mills and Maradudin. Ising models with modified exchange ${J}_{s}=J(1+\ensuremath{\Delta})\ensuremath{\ne}J$ on the surface are considered, both in mean-field theory and by means of high-temperature-series expansions. The critical value ${\ensuremath{\Delta}}_{c}$ for surface ordering is found from the series to be 0.6, compared to the mean-field value of 0.25. For $\ensuremath{\Delta}>{\ensuremath{\Delta}}_{c}$ there is a temperature region in which the surface behaves like a bulk two-dimensional Ising model near its phase transition. The critical exponents experience a crossover at $\ensuremath{\Delta}={\ensuremath{\Delta}}_{c}$, which is reflected in poorly behaved series, and effective exponents differing from the true ones for $\ensuremath{\Delta}\ensuremath{\lesssim}{\ensuremath{\Delta}}_{c}$. In the case of weakened surface exchange (0<Js<J0<{J}_{s}<J0<Js​<J), the layer magnetization is shown to fit a linear temperature dependence over a large temperature range below Tc{T}_{c}Tc​, thus providing a possible explanation for previous experiments. For sufficiently strong negative Js{J}_{s}Js​, mean-field theory predicts that the surface will order antiferromagnetically while the bulk is ferromagnetic.
Phase transitions in Ising models with tree surfaces are studied from various points of view, including a phenomenological Landau theory, high-temperature series expansions, and a scaling theory for thermodynamic quantities and correlation functions. In the presence of a surface a number of new critical exponents must be defined. These arise because of the existence of "surface" terms in the thermodynamic functions, and because of the anisotropy of space and lack of translational symmetry introduced by the surface. The need for these new critical exponents already appears in the phenomenological theory, which is discussed in detail and related to the microscopic mean-field approximation. The essential new parameter appearing in this theory is an extrapolation length $\ensuremath{\lambda}$ which enters the boundary condition on the magnetization at the surface. For magnetic systems this length is of the order of the interaction range, in contrast to superconductors, where it is usually much larger. In order to go beyond the mean-field theory, high-temperature series expansions are carried out for the Ising half-space, to tenth order in two dimensions and to eighth order in three dimensions. A scaling theory is developed both for thermodynamic functions and for spin correlations near the surface, and relations are found among the exponents of the half-space. Both the scaling theory and the numerical calculations are compared with the exact solution of the Ising half-plane (two dimensions) by McCoy and Wu, and agreement is found wherever the theory is applicable. In analogy to the bulk situation, the scaling theory is found to agree with mean-field theory in four dimensions. The prediction of the present work which is most easily accessible to experiment is the temperature dependence of the magnetization at the surface, with critical exponent estimated to be ${\ensuremath{\beta}}_{1}=\frac{2}{3}$. The mean-field result, ${\ensuremath{\beta}}_{1}=1$, seems to agree more closely with presently available experiment, and more work is needed to clarify the situation.
Motivated by recent experimental activities on surface critical phenomena, we present a detailed theoretical study of the near-surface behavior of the local order parameter m(z) in Ising-like spin systems. Special attention is paid to the crossover regime between “ordinary” and “normal” transition in the three-dimensional semi-infinite Ising model, where a finite magnetic field H1 is imposed on the surface which itself exhibits a reduced tendency to order spontaneously. As the theoretical foundation, the spatial behavior of m(z) is discussed by means of phenomenological scaling arguments, and a finite-size scaling analysis is performed. Then we present Monte Carlo results for m(z) obtained with the Swendsen-Wang algorithm. In particular the sharp power-law increase of the magnetization, m(z) ≈ H1z1−η∼ord, predicted for a smallH1 by previous work of the authors, is corroborated by the numerical results. The relevance of these findings for experiments on critical adsorption in systems where a small effective surface field occurs is pointed out.
We study the surface critical behavior of semi-infinite systems belonging to the bulk universality class of the Ising model. Special attention is paid to the local behavior of experimentally relevant quantities such as the order parameter and the correlation function in the crossover regimes between different surface universality classes, where the surface field h1 and the temperature enhancement c can induce additional macroscopic length scales. Starting from the field-theoretical φ4 model and employing renormalization-group improved perturbation theory (ϵ-expansion), explicit results for the local behavior of the correlation and structure function (zero-loop) and the order parameter (one-loop) are derived. Supplementing earlier studies that focused on the special transition, we here pay particular attention to the situation where a large c suppresses the tendency to order in the surface (ordinary transition) but a surface field h1 generates a small surface magnetization m1. Our results are in good agreement with recent phenomenological considerations and Monte Carlo studies devoted to similar questions and, combined with the latter, provide a much more detailed understanding of the local properties of systems with weak surface fields.
We study the two-dimensional semi-infinite Ising model with a free surface at or near bulk criticality. Special attention is paid to the influence of a boundary magnetic field h1 on the surface-near regime and the crossover between the fixed points at h1=0 and h1=∞. Near the surface, a smallh1 causes a steeply increasing magnetization m(z)~z3/8 log z as the distance z increases away from the surface. By means of a phenomenological scaling analysis, this phenomenon can be related to the well-known logarithmic dependence of the surface magnetization m1 on h1. Our analysis provides a deeper understanding of the existing exact results on m(z) and relates the short-distance phenomena in d=2 to those in higher dimensions. Both the results of the scaling analysis and the exact analytic profiles are corroborated by Monte Carlo simulations.