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Heisenberg Thin Films with Frustrated Surfaces

In this chapter, we show the results obtained by extensive Monte Carlo (MC) simulations and analytical Green’s function (GF) method for a thin film made of stacked triangular layers of atoms using the Heisenberg spin model. We suppose that the in-plane surface interaction Js can be antiferromagnetic or ferromagnetic while all other interactions are ferromagnetic. We suppose the film in addition an Ising-like interaction anisotropy. We show that the ground-state spin configuration is non-linear when Js is lower than a critical value //. The film surfaces are then frustrated. In the frustrated case, there are two phase transitions related to the disordering of the surface and the interior layers. There is a good agreement between MC and GF results. In addition, we show from MC histogram calculation that the value of the ratio of critical exponents y/v of the observed transitions is deviated from the values of two- and three-dimensional Ising universality classes. The origin of this deviation is discussed using general physical arguments.

A part of the results shown here has been published in Ref. [248].

Physics of Magnetic Thin Films: Theory and Simulation Hung T. Diep

Copyright © 2021 Jenny Stanford Publishing Pte. Ltd.

ISBN 978-981-4877-42-8 (Hardcover), 978-1-003-12110-7 (eBook)


The frustration is known to cause a great number of striking effects in various bulk spin systems. Its effects have been extensively studied during the last three decades theoretically, experimentally and numerically. Frustrated models serve not only as testing grounds for theories and approximations, but also to interpret experiments [85].

In the same period, surface physics and systems of nanoscales have been also enormously studied. This is due in particular to applications in magnetic recording, magnetic sensors, spin transport, ... let alone fundamental theoretical interests. Much is understood theoretically and experimentally in thin films where surfaces are 'clean,' i.e., no impurities, no steps, no islands, no defects of any kind [35, 75, 78, 79, 252]. Less is known at least theoretically on complicated thin films with special surface conditions such as defects [63, 320], arrays of dots and magnetization reversal phenomenon [132,164,189, 232, 298, 306, 307].

Section 10.2 is devoted to the description of our model. The ground state in the case of classical spins is determined as a function of the surface interaction. In Section 10.3, we consider the case of quantum spins and we apply the GF technique to determine the layer magnetizations and the transition temperature as a function of the surface interaction. The classical ground state determined in Section 10.2 is used here as starting (input)configuration for quantum spins. We are interested here in the effect of magnetic frustration on magnetic properties of thin films. A phase diagram is established showing interesting surface behaviors. Results from MC simulations for classical spins are shown in Section 10.4 and compared to those obtained by the GF method. We also calculate by the MC histogram technique the critical behavior of the phase transition observed here.


It is known that many well-established theories failed to deal with frustrated spin systems [85]. Among the striking effects due to frustration, let us mention the high ground-state (GS) degeneracy associated often with new symmetries which give rise sometimes to new kinds of phase transition. One of the systems which are most studied is the antiferromagnetic triangular lattice. Due to its geometry, the spins are frustrated under nearest-neighbor (NN) antiferromagnetic interaction. In the case of Heisenberg or XY models, the frustration results in a 120° GS structure: The NN spins form a 1203 angle alternately in the clockwise and counterclockwise senses which are called left and right chiralities (see Fig. 18.6 in the solution to Problem 6 of Chapter 3). Another popular model is Villain's model where the frustration is caused by the competition between ferro- and antiferromagnetic interactions as seen in Problem 7 of Chapter 3 and its solution in Section 18.3).


In this chapter, we consider a thin film made up by stacking Nz planes of triangular lattice of L x L lattice sites.

The Hamiltonian is given by

where S, is the Heisenberg spin at the lattice site i, indicates the sum over the NN spin pairs S, and S;. The last term, which will be taken to be very small, is needed to make the film with a finite thickness to have a phase transition at a finite temperature in the case where all exchange interactions Ду are ferromagnetic. This guarantees the existence of a phase transition at finite temperature, since it is known that a strictly two-dimensional system with an isotropic non-Ising spin model (XY or Heisenberg model) does not have long-range ordering at finite temperature [231].

Interaction between two NN surface spins is equal to Js. Interaction between NN on adjacent layers and interaction between NN on an interior layer are supposed to be ferromagnetic and all equal to ] = 1 for simplicity. The two surfaces of the film are frustrated if Js is antiferromagnetic (Д < 0).

Ground State

In this paragraph, we suppose that the spins are classical. The classical GS can be easily determined as shown below. Note that for antiferromagnetic systems, even for bulk materials, the quantum GS cannot be exactly determined (see Chapter 3). The classical GS is often used as starting configuration for quantum spins. We will follow the same line hereafter.

For Js > 0 (ferromagnetic interaction), the GS is ferromagnetic. However, when )s is negative the surface spins are frustrated. Therefore, there is a competition between the non-collinear surface ordering and the ferromagnetic ordering due to the ferromagnetic interaction from the spins of the beneath layer.

We first determine the GS configuration for / = /* = 0.1 by using the steepest descent method: Starting from a random spin configuration, we calculate the magnetic local field at each site and align the spin of the site in its local field. In doing so for all spins and repeat until the convergence is reached, we obtain in general the GS configuration, without metastable states in the present model. The result shows that when Js is smaller than a critical value // the magnetic GS is obtained from the planar 120° spin structure, supposed to be in the XY plane, by pulling them out of the spin XY plane by an angle p. The three spins on a triangle on the surface form thus an 'umbrella' with an angle a between them and an angle p between a surface spin and its beneath neighbor (see Fig. 10.1). This non-planar structure is due to the interaction of the spins on the beneath layer, just like an external applied field in the z direction. Of course, when Js is larger than // one has the collinear ferromagnetic GS as expected: The frustration is not strong enough to resist the ferromagnetic interaction from the beneath layer.

We show in Fig. 10.2 cos(a) and cos(/J) as functions of Js. The critical value // is found between —0.18 and -0.19. This value can be calculated analytically by assuming the 'umbrella structure'. For GS analysis, it suffices to consider just a cell shown in Fig. 10.1. This is justified by the numerical determination discussed above. Furthermore, we consider as a single solution all configurations obtained from each other by any global spin rotation.

Non collinear surface spin configuration. Angles between spins on layer 1 are all equal (noted a), while angles between vertical spins are fi

Figure 10.1 Non collinear surface spin configuration. Angles between spins on layer 1 are all equal (noted a), while angles between vertical spins are fi.

cos(a) (diamonds) and cos(/J) (crosses) as functions of ]. Critical value of // is shown by the arrow

Figure 10.2 cos(a) (diamonds) and cos(/J) (crosses) as functions of ]s. Critical value of // is shown by the arrow.

Let us consider the full Hamiltonian (10.1). For simplicity, the interaction inside the surface layer is set equal /s (— 1 < ]s < 1) and all others are set equal to / >0. Also, we suppose that = ls for spins on the surfaces with the same sign as )s and all other /,,y are equal to / > 0 for the inside spins including interaction between a surface spin and the spin on the beneath layer.

The spins are numbered as in Fig. 10.1: S 1( S2 and S3 are the spins in the surface layer (first layer), S[, S'2 and S'3 are the spins in the internal layer (second layer). The Hamiltonian for the cell is written


Let us decompose each spin into two components: an xy component, which is a vector, and a z component S, = (S®, Sf). Only surface spins have xy vector components. The angle between these xy components of NN surface spins is which is chosen by (y,-,y is in fact the projection of a defined above on the xy plane)

The angles ft, and p[ of the spin S, and S,' with the z axis are by symmetry

The total energy of the cell (10.2), with S, = S- = can be rewritten as

By a variational method, the minimum of the cell energy corresponds to

We have

For given values of Is and /, we see that the solution (10.6) exists for Js < // where the critical value // is determined by — 1 < cos/3 < 1. For / = —Is = 0.1,7/ ~ -0.1889/ in excellent agreement with the numerical result.

The classical GS determined here will be used as input GS configuration for quantum spins considered in the next section.

Green’s Function Method

Let us consider the quantum spin case. For a given value of ]s, we shall use the GF method to calculate the layer magnetizations as functions of temperature. The details of the method in the case of non-collinear spin configuration have been given in Ref. [285]. We briefly recall it here and show the application to the present model.

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