Surface spin slips in thin holmium films

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Surface spin slips in thin holmium films

F. H. S. Sales, A. L. Dantas, and A. S. Carriço

Citation: AIP Advances 2, 032158 (2012); View online: https://doi.org/10.1063/1.4750030

View Table of Contents: http://aip.scitation.org/toc/adv/2/3

Published by the American Institute of Physics

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Surface spin slips in thin holmium films

F. H. S. Sales,1,a_{A. L. Dantas,}2_{and A. S. Carric¸o}3

1_{Department of Physics, IFMA, S˜ao Lu´ıs, MA, Brazil 65030-005, Brazil} 2_{Department of Physics, UERN, Mossor´o, RN, Brazil 59610-210, Brazil} 3_{Department of Physics, UFRN, Natal, RN, Brazil 59072-970, Brazil}

(Received 5 June 2012; accepted 21 August 2012; published online 29 August 2012)

We report a theoretical investigation of new spin slips phases of thin holmium (Ho) films. The new phases originate from the loss of coordination of atoms in the near surface region, which affects the balance between exchange and anisotropy energies, favoring the alignment of near surface spins along the basal plane easy axis directions.

There is currently great interest in confinement effects in artificial magnetic systems. Magnetic systems with bulk periodic patterns deserve special attention. Rare-earth (RE) elements have a rich variety of periodic magnetic phases, induced either by external magnetic fields or by temperature. The helix period is an intrinsic length scale of RE materials bulk equilibrium phases. As a result confinement effects are likely to be stronger for thin RE films thicknesses of the order of the helix period.1–9

There is a good deal of theoretical work on the dramatic changes in the magnetic phases of thin Ho films for thicknesses close to and smaller than the helix period.7–9

These studies focused on the impact of confinement on the helical and fan phase, which are genuine manifestations of the RE competing exchange energy interactions.

As appropriate to temperatures close to the Neel temperature, these studies have used a Heisen-berg hamiltonian stressing the prevalence of exchange energies over the basal plane hexagonal anisotropy energy.

In this paper we focus on anisotropy energy effects on Ho thin films.

The balance between the exchange and basal anisotropy energies in Ho is a key feature of the Ho magnetic phases at low temperatures.

There are prominent anisotropy effects for low temperature helimagnetic RE metals. New states emerge from the prevalence of the anisotropy energy over the exchange energy.

For holmium the basal-plane moments bunch strongly around the easy b directions, leading to a magnetic pattern commensurable with the lattice. The modifications in this basic figure as the temperature raises, lead to various spin slip patterns at certain temperature intervals.10–13,15

The existence of lattice commensurate phases in Holmium bulk samples, fulfills the requirement imposed by the hexagonal anisotropy. The exchange energy alone would lead to a basal plane regularly spiraling helix.

The stability of the commensurate phases depends on how well the gain in anisotropy energy, with spins locked to near easy axis directions in the basal plane, counter-balances the increase in the exchange energy.

In thin Ho films confinement effects and the lack of surface neighbors lead to new features. The exchange energy balance is changed in favor of parallel orientation of near surface spins.2,4–6 We show presently that this may affect the commensurate phases of Ho.

In this paper, we report a theoretical study of surface and confinement effects on the commen-surate phases of Ho at low temperatures. Our results indicate that near the surfaces the spins from a few atomic layers bunch together leading to modified spin slip patterns.

a_{Author to whom correspondence Should be addressed. Electronic mail:}_{[email protected]}_.

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032158-2 Sales, Dantas, and Carric¸o AIP Advances 2, 032158 (2012)

2(221)

( a )

( b )

( c )

( d )

6(2)

2(222221)

2(212121)

FIG. 1. Schematic representation of spins per plane of bulk commensurate structures in phases with magnetic unit cells containing (a) 12, (b) 11, (c) 9 and (d) 10 atomic layers. The temperature is 20 K and the external field is zero. The red arrows indicate spin slip planes.

The equilibrium states are found using a theoretical model based on a self-consistent molecular field algorithm,2–6_{including simultaneously the effects of external magnetic field, temperature and}

reduced coordination in the near surface region.

We investigated thin Ho films, consisting of a stack of atomic layers with equivalent spins, infinitely extended in the x-y direction. The x-axis is along one of the easy directions of the hexagonal anisotropy energy. The spins in each monolayer are coupled by exchange interaction with spins in the nearest and next-nearest neighbor monolayers. The anisotropy is uniform throughout the film, and in the surface region the spin exchange energy is reduced. The magnetic Hamiltonian is given by E = J1(g − 1)2 N −1 n=1 J(n) · J(n + 1) + J2(g − 1)2 N −2 n=1 J(n) · J(n + 2) + N n=1 K₆6cos 6ϕn (1)

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( a )

( b )

( c )

( d )

(212)

(221221)

FIG. 2. Schematic configuration of spins per plane for thin Ho films, for H = 0 and T = 20 K, with (a) 5, (b) 7 (c) 9 and (d) 13 atomic layers. As shown in the panel (d) the 13 layers film display a surface modified (221221) spin slip phase.

where J1and J2describe the exchange energy interactions of a given spin with spins in the first and second neighbor atomic layers. J(n) represents the total angular momentum per atom in the n-th atomic layer. K6

6 describes the hexagonal anisotropy energy and ϕnis the angle with x-axis for spins

in the n-th atomic layer.

We use bulk Ho energy parameters, with J = 8, J1= 47kB5, J2= −J1/4cos φ(T), where φ(T) is the value of the temperature dependent turn angle. Although we are not presently focusing in surface modifications of the helical state, it is necessary to have a good estimate of the relation between the exchange energy parameters. By using the measured bulk values14_{of the turn angle φ(T), for each}

temperature T, we are neglecting variations of the turn angle near the surfaces. g = 5/4 is the Land´e factor, which corresponds to a saturation magnetic moment per atom of 10µB, and K66(T ) describes the temperature dependence of the hexagonal anisotropy energy.16,17

Before considering surface effects, we show in Fig.1typical bulk spin slips patterns.18This is in a certain way a means of validating the theoretical model and the efficiency of the self-consistent algorithm used in this paper. Furthermore, some of the spin slip patterns shown in Fig.1will help to identify the surface features in the corresponding spin slip phases of thin films, as discussed below. In order to reproduce the bulk commensurate phases shown in Fig.1we use a system consisting of a thin film with the number of layers equal to the number of layers in the magnetic unit cell of each particular spin slip phase, and impose cyclic boundary conditions.

In Fig.1we show typical bulk Ho spin slips patterns at 20 K in the absence of external field. The chosen bulk spin slip phases shown in Fig.1(a)–1(d)have magnetic unit cells with 12, 11, 9 and 10 atomic layers, and correspond to known spin slip phases in the absence of external field.10,18

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( a ) 20 K

( b ) 80K

13 Layers

( c ) 20 K

( d ) 80K

15 Layers

(221221)

(212121)

FIG. 3. Schematic representation of spins per plane for 13 and 15 atomic layers thin Ho films, at temperatures of 20 K and 80 K, in the absence of external field. The shaded region (in orange) show spins in the near surface region.

The 12 atomic layers structure 6(2) (shown in panel (a)) consists of six spin pairs arranged at small angles with easy axis of the anisotropy energy. The 11 atomic layers structure 2(222221) (shown in panel (b)) contains a spin slip plane (shown in red color), with the spins oriented along the anisotropy easy axis, for every set of five spin pairs oriented at small angles with easy axis of the anisotropy energy. Notice that in the 11-layer one-spin-slip structure, the bunched pairs of moments in the vicinity of the spin slip are disposed unsymmetrically with respect to the easy axis.

The 9 atomic layers structure 2(212121) (shown in panel (c)) contains a spin slip for each pair of spins, oriented at small angles with easy axis of the anisotropy energy. The 10 atomic layers structure 2(221) (shown in panel (d)), contains a spin slip plane for every set of two pairs of spins oriented at small angles with easy axis of the anisotropy energy. Symbolically, each spin slip plane (singlet) is represented by the number “1” and each set of two atomic layers with spins arranged symmetrically at small angles spin slip is represented by the number “2”.

In Fig.2we show the low temperature (T = 20 K) equilibrium patterns for ultrathin films with thicknesses ranging from 5 to 13 atomic layers, in the absence of external field.

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20 K

70 K

75 K

80 K

( a )

_{( b )}

( c )

(212121)

_{( d )}

(212121)

Helix

FIG. 4. Schematic representation of spins per plane for the (212121) spin slip phase of bulk Ho, at temperatures of (a) 20 K, (b) 70 K, (c) 75 K, and (d) 80 K, in the absence of external field.

The 5 layers film is in the ferromagnetic state, with all spins pointing along the easy axis of the anisotropy energy. The 7 layers film has the surface spins and the spins in the middle plane along an easy axis direction.

The 9 layers film has spins along the easy axis, like a spin slip, and also pair of spins oriented at small angles with the easy axis direction. However the film is not thick enough so as to fit the unit cell of a spin slip phase.

The 13 layers film displays a modified (221221) pattern, with surface spins (of atomic planes 1, 2, 3 and 13) pointing nearly along the easy axis.

In Fig.3we show that low temperature surface modified (221221) phase evolves, by heating to 80 K, to a surface modified (212121) phase, with an increase (from 4 to 6) of the number of surface spins pointing nearly parallel along an easy axis direction.

The 15 layers film has a more stable surface phase. At T = 20 k it displays a surface modified (212121) spin slip phase, which remains almost unchanged up to the temperature of 80 K, beyond which the surface spin slip turns into a helical phase. In this temperature range the surface modified (212121) spin slip phase suffer minor changes in the angles between spins in the doublets (31 degrees at T = 20 K and 39 degrees at T = 80 K), keeping spins locked in the easy axis directions as shown in the T = 20 K and T = 80 K diagrams in Fig.3.

In Fig.4we show the temperature effects in the bulk (212121) spin slip phase. It remains almost unchanged from T = 20 K up to T = 70 K, only the relative orientation of spins in the doublets is

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changed from 32 degrees at T = 20 K to 40 degrees at T = 75 k. At T = 80 K it is no longer stable. We have found that at T = 78 K the (212121) spin slip phase turns into a helical phase, with a turn angle of 40 degrees.

The extra stability of the surface modified (212121) spin slip phase of the 15 layers thin Ho film, originates, as shown in Fig.3(d), in the fact that there are two spin slips in the near surface region (one near each surface) accommodating spins nearly aligned along the easy axis, and a surface doublet consisting of the first and second layer planes from each surface.

In summary we have shown that the reduction in the coordination number of surface atoms modifies the balance between exchange and hexagonal anisotropy energies, favoring the formation of surface modified spin slips phases. Our theoretical model includes the temperature effects on the basal plane hexagonal anisotropy as well as on the exchange energies, allowing the discussion of both the bulk Ho commensurate spin slips phases, by using suitable cyclic boundary conditions on a cell consisting of the basic unit of the bulk commensurate phases, as well as the modifications of the bulk commensurate phases produced by confinement and surface effects in thin Ho films.

The authors acknowledge support from CNPq, CAPES, FAPERN, and FAPEMA. The work of A. S. Carric¸o was supported by CNPq Grant 350773. The work of A. L. Dantas was supported by CNPq Grant 309676. The work of F. H. S. Sales was supported by FAPEMA Grant 00415/11.

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