Accepted Manuscript
Characterization of the Magnetic Phases of Holmium Nanofilms via Magnetic
Neutron Scattering
V.D. Mello, F.A.L. Santiago, D.H.A.L. Anselmo, M.S. Vasconcelos, N.S.
Almeida
PII:
S0304-8853(18)31339-8
DOI:
https://doi.org/10.1016/j.jmmm.2018.12.006
Reference:
MAGMA 64697
To appear in:
Journal of Magnetism and Magnetic Materials
Please cite this article as: V.D. Mello, F.A.L. Santiago, D.H.A.L. Anselmo, M.S. Vasconcelos, N.S. Almeida,
Characterization of the Magnetic Phases of Holmium Nanofilms via Magnetic Neutron Scattering, Journal of
Magnetism and Magnetic Materials (2018), doi:
https://doi.org/10.1016/j.jmmm.2018.12.006
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Characterization of the Magnetic Phases of Holmium
Nanofilms via Magnetic Neutron Scattering
V.D. Melloa,∗, F.A.L. Santiagob,a, D.H.A.L. Anselmob, M.S. Vasconcelosc,
N.S. Almeidaa
aDepartamento de F´ısica, Universidade do Estado do Rio Grande do Norte, Mossor´o - RN 59625-620, Brazil
bDepartamento de F´ısica Te´orica e Experimental, Universidade Federal do Rio Grande do Norte, Natal - RN 59600-900, Brazil
cEscola de Ciˆencia e Tecnologia, Universidade Federal do Rio Grande do Norte, 59072-970, Natal- RN, Brazil
Abstract
Magnetic phases of holmium films are associated with the intensity of the mag-netic neutron scattering. It is shown that each magmag-netic phase of the system (fan, helifan, spin-slip, helix and ferromagnetic) exhibits a characteristic profile of scattering which can be used as a fingerprint to identify it. In this paper, we present theoretical results obtained for holmium films 24 monolayers thick at a fixed temperature and in the presence of a dc magnetic field applied along the basal plane. A self-consistent local field algorithm was used to obtain the equilibrium configurations of the magnetic moments of the film and, with these results, the spin-spin correlation functions which determine the shape of the neutron scattering intensity were calculated.
Keywords: A. Rare-Earth, B. Magnetic Thin Films Phases, C. Magnetic
Phases, D. Magnetic Neutron Scattering PACS: 75.50.Cc, 73.22.-f, 75.40.-s
∗Corresponding author
1. Introduction
It is well known that neutron scattering is a useful experimental technique
to identify magnetic structures of solids [1, 2]. Besides the structures, this
technique also allows to one obtain information on the magnetic coupling be-tween layers of layered systems [3, 4], magnetic excitations [5], phase transitions 5
[6], superfluidity and superconductivity [7], and characterization of magnetic nanoparticles [8]. These facts make this technique one of the most powerful tools to acquire information on the physical properties of small size magnetic systems.
In the early 1960s, several investigations using neutron diffraction measure-10
ments were performed to identify the rare earth magnetic configurations. A brief review of the theoretical and experimental aspects of neutron diffraction, as well as their application in magnetic systems, was discussed at the time by M.K.
Wilkinson [9]. Neutron diffraction measurements in Dysprosium
monocrys-tals [10] showed good agreement with the experimental results investigated by 15
Behrendt [11], where it was observed that the dysprosium had a ferromag-netic arrangement below 85K, antiferromagferromag-netic between 85K and 179K, and paramagnetic above 179 K. Measurements of neutron diffraction performed on Erbium [12] indicated that it is antiferromagnetic between 20K and 80K and fer-romagnetic below 20K. Preliminary investigations of neutron diffraction applied 20
to helical magnetic structures of Holmium were carried out by W.C. Koehler [13, 14], which showed agreement with the experimental results performed by B.L. Rhodes [15]. Later, by using neutron diffraction, W.C. Koehler and col-leagues [16, 17] carried out a detailed study of the field-dependent Holmium structures. In their experiment, the magnetic field and temperature were used 25
to have the holmium sample at a specific configuration, and it was shown that the intensity of the neutron scattering has a signature of the phases. With this technique they were able to identify the basal-plane helix structure of this rare earth at temperatures between 20 K and 132 K, and the cone phase at tempera-tures below 20 K. Few years after, Elliot [18], and more recently Michels and col-30
laborators [19], pointed out that neutron scattering experiments might be used to identify the magnetic phases (helix, cone, longitudinal wave, ferromagnetic) present in a bulk sample of terbium, dysprosium, and holmium. The influence of a dc magnetic field on the magnetic structures of the holmium/yttrium su-perlattices at 80 K was also investigated by Fuente et al. [20] with the help of 35
this technique. From the data obtained from their experiments, these authors identified four different magnetic phases (helix, helifan, fan, and ferromagnetic phases) when a dc magnetic field with intensities between zero and six tesla was applied in the basal plane of this layered system. Their results confirmed those obtained by Jehan and colleagues [21] who found that the cone phase is 40
not present in these rare earth superlattices. These are examples of the use of neutron scattering to study layered magnetic systems. An excellent review on applications of this experimental technique can be found in the publication of Thomas and Frank [22]. Also, current perspectives on neutron scattering were reported by Velthuis and Pappas [23].
45
The interaction between neutron and matter reveals the character of the system under investigation, and this is materialized through the analysis of the scattering cross-section. This quantity describes the behavior of a neutron beam after being modified by some mechanism of obscuration and/or scattering present in the analyzed system, and depends on the structure of the sample as 50
well as on the energy of the incident neutrons. A detailed review on the cross section for the thermal neutron scattering can be found in the books of Squires [24], Chaterji [25], and Stephen [26].
In this paper we investigate theoretically the magnetic phases in Holmium films, at a fixed temperature and in the presence of a dc magnetic field applied 55
in the basal-plane, via the analysis of the magnetic neutron scattering. The results obtained show that each magnetic phase is unequivocally associated with a specific configuration of peaks of the neutron scattering, which have shapes and intensities well defined. We organize this report as follows: in section 2 we show the main steps to obtain the shape of the scattering cross-section, in 60
monolayers thick at T = 120 K, and in the last section we offer some comments on the results presented.
2. Theoretical Model
Besides the neutron-magnetic moment interaction, some others contributions 65
(like neutron-nucleon interaction) are present in the neutron scattering exper-iments. It is also known that the magnetic interaction is very weak compared with some other forces present in the system. However, in general the peaks generated by these stronger interactions appear in a narrow range of the wave-length and can easily be identified and separated from the results that arise from 70
the magnetic structures. In this work, we will deal only with neutron-magnetic moment scattering.
Our model describes the scattering of high-angle elastic neutrons, usually called neutron diffraction, which has a high sensitivity to magnetic structures, through the occurrence of half-order Bragg diffraction reflections. The observed 75
intensities are directly related to the magnitude of the atomic magnetic
mo-mentum, which can be obtained up to close to the N`eel temperature. The
neutron diffraction technique is particularly useful for the study of antiferromag-netic structures and more complex systems, but its application to thin films is harmed by the small amount of magnetic material. However, this technique has 80
been used quite successfully to determine the magnetic structures of thin films [27, 28, 29], multilayers [30] and complex systems such as multiferroic materials [31, 32]. Since the absorption cross section of neutron scattering is strongly determined by the scattering length [33] and, since the rare earths possess the largest magnetic moments of known elements (and therefore the largest possible 85
cross-section for magnetic neutron scattering) [34], then the absorption lengths for, e.g., Holmium, are long enough, such that absorption is not an issue for nanofilms.
The intensity of the neutron magnetic scattering due to a magnetic moment ~
Jl=PαJ
α
leˆα at the position ~Rl may be written as [35]
Imag( ~Q) ∝ |F ( ~Q)|2
X
α,β
(δαβ− qαqβ)Sαβ( ~Q), (1)
where ~Q is the scattering vector (~ ~Q is the momentum transfer), qα =
~ Q·ˆeα | ~Q| , qβ = ~ Q·ˆeβ
| ~Q| , where {α, β} = {x, y, z} are the scattering coordinates. F ( ~Q) is the
magnetic form factor, and Sαβ( ~Q) are the static spin-spin correlation functions
given by Sαβ( ~Q) = 1 N X ll0 hJα l ihJ β l0iei( ~Rl− ~Rl0)· ~Q, (2)
where Jlα, Jlβ0 are the components of the magnetic moments located at layers
95
l, l0. In our calculations we consider the wave vector to be parallel to the z-axis,
which in turn we chose parallel to the c-axis of the sample ( ~Q = Q ˆez), to rewrite
Imag( ~Q) as
Imag( ~Q) ∝ |F ( ~Q)|2[Sxx( ~Q) + Syy( ~Q)] (3)
we write the distance between the layers l and l0, ( ~Rl− ~Rl0) as (l − l0)c
2eˆz to write Sii (i = x or y) as: 100 Sii( ~Q) = 1 N X l,l0 hJliihJ i l0ie−iQ(l−l 0)c 2 (4)
We name φl= ql2c the angle between the magnetic moments at the lthlayer
and the x direction to have hJx
li and hJ y li given by hJlxi = 1 2Jo(e iqlc2 + e−iqlc2) and hJlyi = 1 2iJo(e
iql2c − e−iql2c). Then, after a bit of algebra, the intensity of
the scattering might be written as:
Imag( ~Q) ∝ |F ( ~Q)|2(Jo)2( fN+(Q) 2 + fN−(Q) 2 ), (5) where 105
- 0 . 7 8 5 0 . 0 0 0 0 . 7 8 5 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 [S x x (Q )+ S y y (Q )] (a .u ) Q ( 2π/ c )
Figure 1: Intensity of magnetic neutron scattering by bulk holmium.
fN±(Q) = 1 N1/2 N X l=0 ei(Q±ql)l2c. (6)
It is well known in the literature that the bulk Holmium has (ql+1−ql)c2= π4
[36] for all l. We show in Fig. 1 the behavior of Imag(Q) × Q for this case, where
one can see clearly the well defined peaks of the scattering for Qc2 = ±π4.
In the general case it is necessary to know the equilibrium configuration of
the magnetic moments present in the system, which means to know φl = ql2c
110
associated to each layer. Recently, Rodrigues et al. [37] studied the magnetic behavior of a holmium film with 24 layers in presence of dc magnetic fields at different temperatures. These authors describe the magnetic system as a stack-ing of infinitely extended atomic layers, parallel to x − y plane, each one with equivalent magnetic moment. These magnetic moments have their directions 115
E = J1 N −1 X n=1 ~ J (n) · ~J (n + 1) + J2 N −2 X n=1 ~ J (n) · ~J (n + 2) + N X n=1 [K66cos (6φn) − gµBJ (n) · ~~ H0], (7)
where J1 (J2) describes the exchange interaction between the nearest (next
nearest) magnetic layers, ~J (n) is the total angular momentum per atom in n-th
monolayer, while the term proportional to K6
6 depends on the temperature and
represents the hexagonal anisotropy. The last term corresponds to the Zeeman 120
energy due to the presence of an external dc magnetic field ~H0which is applied
parallel to one of the easy direction in the hexagonal plane which makes an
angle of π6 with the x-axis. We use the Ho bulk energy parameters [38], where
J = 8, J1 = 47kB, J2 = −J1/4 cos φ(T ), is the temperature dependent helix
turn angle [39] and K66is adjusted to reproduce the temperature dependence of
125
the hexagonal anisotropy energy [40]. The physical properties of this holmium film were investigated for temperatures between 20 K and 140 K and magnetic fields smaller than 20 kOe. The results presented by these authors show that, at
T = 120 K, this layered system should be found in the helix phase (for H0< 5.8
kOe), in the helifan phase for H0 between 5.8 and 8.0 kOe, in the fan phase
130
for H0between 8.0 and 16.8 kOe and in the ordered paramagnetic phase for H0
greater than 16.8 kOe. We chose the parameters that describe the system at this temperature to investigate the neutron scattering for different values of the magnetic field with the purpose to associate the shape of the cross-section and the phase of the system. Below we will present and discuss these results. 135
3. Results and Discussion
Before to present the evolution of the intensity of the neutron scattering with the increase of the magnetic field, applied parallel to the x-axis (direction of one of the ease axis of the holmium film), we use the results obtained by Rodrigues et al. [37] to show the profile of the intensity of the scattering with the momentum 140
- 1 . 5 - 1 . 2 - 0 . 9 - 0 . 6 - 0 . 3 0 . 0 0 . 3 0 . 6 0 . 9 1 . 2 1 . 5 0 6 0 0 1 2 0 0 1 8 0 0 n S H [ S x x(Q )+ S y y(Q )] (a .u ) H e l i x P h a s e Q ( 2π/ c ) ( a ) - 1 . 5 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0 1 . 5 0 6 0 0 1 2 0 0 1 8 0 0 ( b ) Sn H [S x x(Q )+ S y y(Q )] (a .u ) Q ( 2π/ c ) S p i n - S l i p P h a s e - 1 . 5 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0 1 . 5 0 4 0 0 8 0 0 ( c ) [S x x(Q )+ S y y(Q )] (a .u ) n S H H e l i f a n P h a s e Q ( 2π/ c ) - 0 . 5 0 . 0 0 . 5 0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 F a n P h a s e F e r r o m a g n e t i c P h a s e ( d ) [S x x(Q )+ S y y(Q )] (a .u ) n S H n S H Q ( 2π/ c )
Figure 2: Intensity of magnetic neutron scattering obtained for a holmium film 24 monolayer thick in the helix (a), Spin-Slip (b), Helifan (c) and Fan and Ferromagnetic (d) phases. The inserts show the projections of the magnetic moments on the plane perpendicular to the c-axis for each phase.
transfer (Imag(Q)) at different magnetic phases. Fig. 2(a) shows in its insert the
projection of the magnetic moments of each layer on the plane perpendicular to the c-axis. This spatial distribution of the moments is associated with the peaks distribution showed in the main curve of this figure. The peaks position and the shoulders characterize this magnetic phase.
145
Fig. 2(b) shows the behavior of Imag(Q) for the spin-slip phase. In this
phase, the magnetic moments are oriented in directions symmetric with respect to the easy axis of holmium film (as illustrated in the insert of this figure)
and Imag(Q) has well-defined peaks at symmetric positions with respect to this
axis. Fig. 2(c) exhibits the profile of Imag(Q) for the helifan phase. The
150
magnetic configuration of this phase is presented in the insert of the figure and corresponds to an intermediate phase that might be seen as a mixture of the
- 1 . 2 - 0 . 6 0 . 0 0 . 6 1 . 2 0 9 0 0 1 8 0 0 2 7 0 0 S p i n - S l i pH e l i x H e l i f a n F a n F M 1 8 k O e 1 5 k O e 1 3 k O e 8 k O e A pp l i ed F i e ld Q ( 2π / c ) [ S x x (Q )+ S y y (Q )] ( a .u ) 0 k O e 1 k O e 3 k O e 5 k O e 6 k O e
Figure 3: Intensity of magnetic neutron scattering for a holmium film 24 monolayer thick in presence of a dc magnetic field applied parallel to the x-axis (ease direction). The intensity of the field is indicated beside each curve.
Helix and Fan phases. In this phase it is observed, besides central and symmetric peaks, the presence of shoulders near the central peak, characterizing this phase univocally. Finally, Fig. 2(d) illustrates the fan and ferromagnetic (ordered 155
paramagnetic) phases. In this configuration Imag(Q) has only one central peak,
and the intensity might be used to distinguish this phase.
Fig. 3 shows a collection of curves Imag(Q) × Q for the Holmium film at
T = 120 K in the presence of dc magnetic fields between 0 and 18 kOe. From this figure, one can see the evolution of the magnetic structure of the layered 160
system through the shape of the intensity of neutron scattering when the system evolutes from the helix to the ferromagnetic phase, with the intermediate phases
registered by the shapes of Imag(Q).
4. Conclusion
In this work, we discussed the theoretical neutron scattering profiles of 165
due the lack of next-nearest neighbors, and film thickness favors the nucleation of modified helical states, which are strongly affected by magnetocrystalline
anisotropy K66, external magnetic field, and temperature, differently from that is
observed in bulk, where the exchange interaction between momenta is complete. 170
As a result of this reduced coordination, the helifan and spin-slip structures are formed. Often, these magnetic structures are not easily detected through the magnetization curves. Thus, the detection of these structures requires methods, such as neutron scattering, that can see details of the magnetic pattern. Also, when one changes the film thickness, due to the redistribution of the positions of 175
the magnetic moments, the scattering function S( ~Q) is modified to incorporate
the new geometrical arrangement of the sample. Moreover, the different profiles associated with each magnetic phase might be used to follow their evolution and stability, i.e., how far the system is from the phase transition. An interesting point is the possibility to follow any magnetic configuration up to phase transi-180
tion of the system by analysis of the cross-section of the neutron scattering. By doing that, it is possible to have information on the dynamical of the process
inside the sample. Indeed, this procedure can not be used for abrupt (1storder)
transitions. However, the analysis of the neutron scattering in both sides of the threshold of the transition might be useful to have a better understanding of 185
the process and on the system itself.
Acknowledgements: The authors acknowledge the financial support from the Brazilian Research Agencies CNPq and FAPERN.
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