Nature of phase transition in magnetic thin films

TL;DR

This study investigates the critical behavior of magnetic thin films using Monte Carlo simulations, revealing a gradual deviation from 2D to 3D universality and emphasizing the importance of finite size scaling in the film thickness direction.

Contribution

It demonstrates that magnetic thin films exhibit a continuous crossover from 2D to 3D critical behavior, highlighting the role of finite size scaling in the thickness direction.

Findings

Critical exponents deviate systematically from 2D values with increasing thickness.

The effective dimension of thin films lies between 2 and 3.

Finite size scaling in the thickness direction is essential for 3D universality.

Abstract

We study the critical behavior of magnetic thin films as a function of the film thickness. We use the ferromagnetic Ising model with the high-resolution multiple histogram Monte Carlo (MC) simulation. We show that though the 2D behavior remains dominant at small thicknesses, there is a systematic continuous deviation of the critical exponents from their 2D values. We observe that in the same range of varying thickness the deviation of the exponent $\nu$ is very small from its 2D value, while exponent $\beta$ suffers a larger deviation. Moreover, as long as the film thickness is fixed, i. e. no finite size scaling is done in the $z$ direction perpendicular to the film, the 3D values of the critical exponents cannot be attained even with very large (but fixed) thickness. The crossover to 3D universality class cannot therefore take place without finite size scaling applied in the $z$ direction, in the limit of numerically accessible thicknesses. From values of exponent $\alpha$ obtained by MC, we estimate the effective dimension of the system. We conclude that with regard to the critical behavior, thin films behave as systems with effective dimension between 2 and 3.