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Magnetic properties of crystalline nanoparticles with different sizes and shapes
Author’s Accepted Manuscript Magnetic properties of crystalline nanoparticles with different sizes and shapes Ana T.A. Lima, Ana L. Dantas, N.S. Almeida
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S0304-8853(16)32753-6 http://dx.doi.org/10.1016/j.jmmm.2016.10.124 MAGMA62037
To appear in: Journal of Magnetism and Magnetic Materials Received date: 20 April 2016 Revised date: 15 August 2016 Accepted date: 25 October 2016 Cite this article as: Ana T.A. Lima, Ana L. Dantas and N.S. Almeida, Magnetic properties of crystalline nanoparticles with different sizes and shapes, Journal of Magnetism and Magnetic Materials, http://dx.doi.org/10.1016/j.jmmm.2016.10.124 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Magnetic properties of crystalline nanoparticles with different sizes and shapes. Ana T.A. Lima,1 Ana L. Dantas,2 and N. S. Almeida2 1)
Departamento de F´ısica, Universidade Federal do Cear´ a, Caixa Postal 6030, Campus do Pici, 60455-760 Fortaleza, ´ Cear´ a, Brazil, and Universidade Federal Rural do Semi-Arido, Campus de Caraubas, RN 333, Rio Grande do Norte, Brazil 2) Departamento de F´ısica, Universidade do Estado do Rio Grande do Norte, 59610-210 Mossor´ o, Rio Grande do Norte, Brazil (Dated: 26 October 2016)
The effects of shape and finite size on the physical behavior of nanostructured antiferromagnetic particles are investigated. They were modeled as ellipsoidal systems which preserve the crystalline structure of the correspondent bulk material. In our analysis we consider nanoparticles composed by magnetic ions which are themselves insensitive to the presence of surfaces and/or interfaces. Results are shown for structures similar to M nF2 and N iO crystals. Special attention is given to these last once their singular magnetic arrangement, as well as, their use at different technological and/or biomedical applications, has motivated intense experimental studies at different laboratories. We use the parameters that describe the correspondent bulk material to discuss the magnetic behavior of these particles for different volumes and shapes. PACS numbers: 75.75.-c, 75.50.Tt, 75.60.Ej, 75.75.Jn Keywords: magnetic particles; nanostructured magnetic particles; hysteresis loops; dynamical behavior I.
INTRODUCTION
The use of particles with size in the nanometric scale in the everyday life has increased continuously and, in particular, magnetic nanoparticles have been used for applications that go from the use in the development of reliable large data storage devices1,2 to the use in medical procedures that require non-invasive techniques for imaging and/or localized therapy of serious diseases3–5 . Despite the motivation and the large number of papers dealing with these systems, there are several properties that are not fully understood (as the exchange bias effect6–9 ) and are still challenges for those working in this field. The demand for a better understanding of these systems comes from the fact that in recent years the development of growth techniques has allowed the preparation of this kind of particle with well controlled geometric and structural characteristics. It is well known that physical properties of small systems are modified by the breaking of the lattice symmetry, as well as, broken bonds due to the presence of surface. This requires a major effort in the search for better comprehension of the consequences of the presence of surfaces in systems with size in this domain. Moreover, since several properties depend on these geometric characteristics, the rapid development of well controlled growth techniques also allows one to prescribes that very soon will be possible the obtaining of particles to fit a large range of needs. Most of low dimension magnetic systems studied in the last decades were very thin films. To them, the border effects were not relevant and the main modifications of the physical properties came from the break of symmetry in one direction. In some sense, magnetic nanoparti-
cles might be seen as a thin magnetic film with relevant border effects. This fact gives to these systems a much richer variety of externally controlled properties that can be used to adapt them to different applications. Useful approaches to study these nanoparticles are reported by several authors (see e.g., Zysleka et al.10 and Liu ZhaoSen et al.11 ). A good review on the procedures and approaches used to study small magnetic particles can be found in the papers of R.H. Kodama and co-authors12–14 . The main goal of this work is to discuss the changes imposed by surfaces on the magnetic properties of small nanostructured antiferromagnetic particles. We start modeling the particles as a collection of ions with well localized positions in the crystalline structure. Moreover, we assume that the ions have well defined magnetic moments, which are not sensitive to their location. However, it is assumed that their orientation is the result of the competition of all magnetic interactions present in the system. We start Section II calculating the magnetization of nanoparticles resulting from nano portions of M nF2 crystals. This material was chosen not only for being one of the most studied, but also because it can be seen as a prototype of the antiferromagnetic materials. The results pave the way for the understanding of more complex systems. This calculation is followed by a more thorough study of N iO nanoparticles. In N iO crystals, each magnetic moment of its f cc type II crystalline structure has twelve nearest neighbors and six next nearest neighbors and, in the bulk configuration can be seen as a collection of ferromagnetic planes coupled antiferromagnetically. We investigate the influence of the break of symmetry introduced by the surface of the particle on the hysteresis loop. The magnetic resonances are also in-
vestigate for particles with different forms. The particles studied in this paper have an ellipsoidal format with the shaped described by the parameter e = a/b, where a and b are the semi-axes parallel and perpendicular to the z direction, respectively. Its volume is specified by re , the radius of a spherical particle of the same volume. Among other questions, we are looking for answers that allow us to understand how the breaking of the symmetry due to the presence of the surface, associated to their shape, modify the susceptibility, coercivity, remanent magnetization and resonances of these nanometric systems. The influence of the dipolar interaction is also investigated. For large systems with antiferromagnetic ordering, we found no report or even some evidence of the influence of the shape on the magnetic properties of these materials. We begin our discussion with a bcc antiferromagnetic material. As expected, a behavior similar to that of the equivalent bulk material is found when the surface/volume ratio is close to zero. However, in a small portion of this material, the number of magnetic moments at the surface (non paired) becomes relevant and they give to the particle a resulting magnetic moment which will depend on the crystalline arrangement, shape and volume of the system. In other words, the geometry is determinant for its primary magnetic behavior. For particles with diameter of few lattice parameters, small changes in the form might result in considerable modifications in their magnetic properties. Despite there be no simple mathematical relationship between the crystalline arrangement of the magnetic ions and the physical properties of the system, the numerical results obtained show that small particles are quite sensitivity to both, volume and shape, as well as the crystalline structure.
II. SPATIAL DESCRIPTION OF THE MONO DOMAIN PARTICLES
We will model each particle as a limited portion of the crystalline material. We begin with an unlimited lattice with each site occupied by the magnetic moment of the ion and the particle is obtained excluding the sites not belonging the region limited by its surface. The theoretical description is completed assuming that the interactions between the magnetic moments in a small portion of the material are the same observed in a very large one. As an example, each ion of an antiferromagnetic (AFM) mono domain particles of well known AFM materials as M nF2 or F eF2 , feels the presence of its neighbors through the exchange and dipolar interactions, an uniaxial anisotropy and the Zeeman interaction. The additional dipolar interaction included here, has no action in extensive materials, but small structures might have the low field behavior modified by the presence of this interaction. In this paper we consider that all interactions/anisotropy are the same everywhere in the
FIG. 1. Formats of three particles with a same volume and different values of e: 0.1 (disc shape), 1 (sphere) and 10 (toothpick shape).
system. However, we should mention that, due to symmetry breaking, the ions located on the surface are in a different environment from those within the system. Dangling bonds, changes in atomic coordination and disorder of the crystalline lattice, among others, may be responsible for changes in the anisotropy and exchange interaction. However, if the ratio NS /N is small (NS is the number of ions in the surface and N is the total number of ions), the results shown here should describe the overall behavior of the system and the imperfections in the surface should produce only minor corrections. The equilibrium configuration, first step to have the magnetic properties studied here, is obtained by searching the configuration that has the minimum value for the energy, which is equivalent to find the configuration where the magnetic moment of each site is parallel to the effective magnetic field felt by it (zero torque approach)15–17 . We use this second method to have numerical data for equilibrium configuration of the studied systems. As mentioned before, the geometric description of the particles is given by re , the radius of a sphere that has the same volume of the ellipsoidal particle, and e, the ratio between the length of the axes parallel and perpendicular to the z-direction. Fig.1 illustrates how the shape of ellipsoidal particles depends on e. It is displayed the format of three particles with the same volume and e equal to 1 (sphere), 0.1 (disc) and 10 (long ellipsoid) respectively.
A.
Nanoparticles of uniaxial AFM materials
M nF2 , CoF2 and F eF2 are materials well described by simple model and we use them to illustrate the calculations depicted in this paper. These materials have their magnetic ions at the sites of a body centered tetragonal lattice and they are well described by the two uniaxial
sublattice model18 . If we define the z axis of the coordinate system parallel to the c axis of the crystal, large portions of these materials have, at the equilibrium configuration and in the absence of external applied magnetic field, their magnetic moments pointing along the +/− z axis. In these materials, the exchange interaction between the magnetic moments on different sublattices keep them antiparallel and might be represented by an exchange magnetic field He . Also the magnetic moments are pinned to the z axis by the anisotropy, which here is described by the anisotropy field Ha . Then, one might write the magnetic energy E of these AFM particles as: N N N E Ha 0. = −He μi . μj − [ μi .ˆ z ]2 − γ H μ i MS 2 i i,j i
+D
N μi . μj − 3 ( μi .ˆ r i,j ) (μj .ˆ r i,j ) (1) 3 | r − r | i j i,j
where N is the number of the ions inside of the surface that limit the volume of the particle, and He , Ha and H0 are, respectively, the exchange, anisotropy and the externally applied fields. Also, MS is the saturation magnetization of the material, μi is the magnetization at the site i, in units of MS , D = MS /n, (n is the number of ions per unit cell), ri the vector that locate the site i, in units of the lattice parameter, and rˆi,j is the unit r − r vector given by |rii −rjj | . The equilibrium configuration is obtained when the magnetic moment at each site is parallel to the effective magnetic field at that site. For a site k located by rk , in the ground state, μ k is parallel to H ef f given by k ef 0 H μj + Ha [ μk .ˆ z ]ˆ z+H f = He j=k
+D
3( μj .ˆ rkj )ˆ rkj − μj 3 |rk − rj | j
(2)
The behavior of the averaged magnetic moments per lattice site for different external magnetic fields is showed in Fig. 2 for an uniaxial antiferromagnetic particle. The results were obtained for He = 540kG, Ha = 3.8kG and Ms = 0.6kG which are the parameters associated to M nF2 19 . The diameter of the spherical particle (e = 1) is 8 lattice parameters and the number of localized magnetic moments inside is 537. In order to see the shape effect, this figure also shows the hysteresis loop for a particle with an equal volume and e = 8 (a long ellipsoid). As can be seen, while the spherical particle exhibits a simple hysteresis loop, the ellipsoidal particle has different magnetic configurations at the low field region. The two different plateaus observed at low field region are due to changes in the ordering of magnetic moments with smaller number of neighbors. It should also be mentioned that, our theoretical calculations indicate that this is the field region which the dipolar interaction contributes significantly to stabilize the
FIG. 2. Hysteresis loops for particles with re = 4 lattice parameters (volume equal to 1.57×10−22 cm3 ) and e = 1 (spherical particle) and e = 8 (particle stretched in the z-direction). The inserts show the arrangement of the magnetic moments in the plane (011) that contain the z-axis, corresponding to the points a, b (ellipsoid) and c (sphere) of the hysteresis loops.
magnetic phases of this system. Outside this region, the difference between the results obtained with and without the dipolar interaction is negligible. The inserts in Fig. 2 illustrate the arrangement of the magnetic moments located in the plane (011) that contain the z-axis, at the antiferromagnetic phase corresponding to the points ”a” and ”b” in the hysteresis curve. As can be seen in these inserts, the plateaus correspond to flips of the localized magnetic moments in the furthermost points the z-axis of the ellipsoid. The equilibrium configuration of the magnetic moments in the spherical particle is illustrated by their arrangement in the same (011) plane depicted in the insert ”c”.
B.
NiO nanoparticles
The transition metal oxides have applications ranging from pure science20 to daily use devices21 . Similarly to M nO, CoO and F eO, N iO is antiferromagnetically ordered in a type II face-centered cubic sodium chloride structure with a lattice parameter a = 4.18 ˚ A. The magnetic order occurs below TN = 523 K and consists of the ferromagnetic alignment of the magnetic moments of the Ni ions in the set of (111) planes. The magnetic moments of adjacent (111) planes are antiparallel, characterizing the antiferromagnetic ordering. It is also known that the N iO antiferromagnetic ordering is accompanied a slight rhombohedral distortion that compresses the lattice along the [111] direction. The interaction between the next nearest magnetic moments of the N iO is al-
E 2 =− He (i, j) μi . μj + D 1 [ μi .ηˆ1 ] MS i,j i N
N
+D2
N i
2 0. [ μi .ηˆ2 ] − γ H
N
μi
N
He (k, j) μj − D1 [ μk .ηˆ1 ] ηˆ1
j
0 −D2 [ μk .ηˆ2 ] ηˆ2 + H
0.02
Ho // [11-2] Ho // [111] Ho // [1-10]
0
-0.02
-0.04 -40
-20
0
H0 (T)
20
40
FIG. 3. Hysteresis loop for a spherical particle of N iO with diameter of 4.18 nm for external magnetic fields parallel to [111], [¯ 110] and [11¯ 2] (as indicated in the plot). The saturation magnetization (MS ) of this material is 0.55kG
(3)
i
where He (i, j) is the exchange interaction between the magnetic moments located at ri and rj , which is J1 if the lattice points are nearest neighbor on the same (111) plane and, J2 if they are nearest neighbor located at different (111) planes. He (i, j) is equal to J3 if ri and√rj locate next nearest x + yˆ + zˆ]/ 3, √ neighbors. Also ηˆ1 = [ˆ x + yˆ]/ 2. D1 and D2 are the anisotropy fields ηˆ2 = [−ˆ in the [111] (hard axis) and [¯ 110] directions, respectively. The dipolar interaction was not included in this calculation because in all numerical data obtained for particles with different sizes and shapes, this interaction did not produce any significant modification in the results. This happens because the contribution of the dipolar interaction to the effective field felt by the N iO ions is very small compared with the contribution of the others interactions. As the previous case, the equilibrium configurations were obtained searching the configuration that all magnetic moments are aligned with the effective field at its position, i.e., the magnetic moment μk is parallel to k = H ef f
0.04
M/MS
most 14 times stronger than the interaction between the nearest22 . This suggests that small particles of this material must have a unique behavior due to lack of coordination of the two outermost layers of ions. Therefore, understanding the behavior of these magnetic nanoparticles becomes even more challenging because, in addition to geometrical effects of the system, there are special characteristics of the material itself, which become more relevant if the ratio surface/volume is not small. A good description of the structure and physical properties of the N iO can be found everywhere22–25 and here we have mentioned only the parameters and details essential to development of the calculations presented in this work. The magnetic energy of a N iO particle can be written as22
(4)
Fig. 3 depicts the central part of the hysteresis loop of a N iO spherical particle with diameter of 4.18 nm. The results were obtained considering22 J1 = 15.7 K (≈ 103.8kG, J2 = 16.1K (≈ 106.5kG), J3 = −221K (≈ −1460.0kG), D1 = 1.13K (≈ 7.5kG) and D2 = 0.06K (≈ 0.4kG) and external magnetic fields applied parallel to [11¯ 2], [111] and [1¯ 10] directions, respectively. These results show that if the external field is applied parallel to the easy direction, the particle remain in the antiferromagnetic phase, even for moderate magnetic fields. On
the other hand, if the external field is applied parallel to [111] direction, the particle responds as a antiferromagnetic material in a canted phase. The magnetic response has a very different characteristic if the external field is applied in the [1¯10] direction. As can be seen, for external magnetic fields in this direction, the hysteresis loop around H = 0 is similar to those depicted by ferromagnetic materials with non zero coercive field. In summary, the response to an externally applied dc magnetic field is strongly dependent on the direction of the field relative to the crystallographic axes. Fig. 4 shows the hysteresis loops of particles with the same volume of the particle of the Fig. 3, but different forms; e = 0.1, particle with format like a disk, and e = 10.0, particles with format of a long ellipsoid. These results illustrate how the static magnetic properties of these particles are sensitive to their shape. Unpaired magnetic moments in the outermost layers of these particles produce a nonzero magnetization and any accurate experimental analysis of the static properties of one small portion of a agglomerate with a moderate density of this material must take into account the magnetic field due the remainder of the material. However, the dynamic analysis of these systems must naturally separate the responses due to intra and inter particles interactions. While the response due the internal structure is associated with exchange interaction and anisotropy, the response to inter particles coupling must be associated with dipole interaction, which should be at much lower frequencies. This is the main motivation to investigate the magnetic resonances of these particles. To study the dynamic response of these systems, we begin writing the equation of motion for each magnetic k (t) k (t). Then we write the = γμk (t) × H moment dμdt ef f time dependent solution (μk (t)) as the sum of the equilibrium solution μk with the time dependent fluctuation
0.04
0.12
e = 0.1 e = 10.0
e = 1.0
0.09
D(ω)
M/MS
0.02
0
-0.02
0.06
0.03
-0.04 -40
-20
0
H0 (T)
20
0
40
FIG. 4. Hysteresis loop for a disk and stick-like particles of N iO with volume equal to a spherical particle with diameter of 4.18 nm for external magnetic fields parallel [11¯ 2] .
0
200
400
600
Frequency (cm-1)
800
1000
FIG. 5. Density of modes for a N iO spherical particle with diameter equal to 8.35 nm.
0.12
α k (t), i.e., μk (t) = μk + α k (t). By following this approach, the resulting effective field can be written as k (t) = H 0k + hk (t), where H ef f N
He (k, j) μj − D1 [ μk .ηˆ1 ] ηˆ1
D(ω)
0k = H
j(k)
0 −D2 [ μk .ηˆ2 ] ηˆ2 + H
0.06
(5) 0.03
and hk (t) =
N
He (k, j) αj (t) − D1 [ αk (t).ηˆ1 ] ηˆ1
0
j(k)
where
e = 5.0
0.09
−D2 [ αk (t).ηˆ2 ] ηˆ2
(6)
means the sum over the nearest and next
j(k)
nearest neighbors of the site k. By neglecting nonlinear terms and taking into account that the equilibrium k configuration is obtained searching for μk parallel to H 0 k ( μk × H0 = 0), we have the equation of motion given by d αk (t) k = γ μk × hk (t) + γ αk (t) × H 0 dt
(7)
Then the frequencies of the normal modes are given by the imaginary part of the eigenvalues of the matrix formed by the coefficients of the linear system that relates the components of α k (t). Figs. 5 - 8 show the density of modes for N iO particles with volume equal to 4πre 3 /3, with re = 2.08 nm and different shapes. As could be seen, there is a large dispersion of the modes at the low frequency region. The reason for this is that, besides the absence of translational invariance, there is a distribution of effective inter ions interaction inside the particle and this distribution depends on the particle size and shape. However, at high frequencies region it is observed a quite regular behavior. Particles with different volumes and
0
200
400
600
Frequency(cm-1)
800
1000
FIG. 6. Density of modes for a N iO particle with volume equal to 4πre 3 /3 where re = 2.08 nm and e = 5.0 (particle in the format of a long ellipsoid).
format show the frequency distribution of modes fitted for Gaussian centered at the frequencies w = 900 cm−1 and w = 720 cm−1 . It was observed that the width of these curves depend on the shape of the particle but its maxima value was always at the same frequency.
III.
COMMENTS ON THE RESULTS
The results presented in figures 2, 3 and 4 show the magnetic response of particles to an externally applied dc magnetic field. A variety of phases can be observed for both M nF2 and for N iO particles. The main conclusion is that their static characteristics are strongly dependent on their volume and shape. As far as we know, any experiment to analyze the static magnetic behavior
0.12
e = 2.0
0.09
D(ω)
metric characteristics (shape and volume) only influence the form of the distribution. On the other hand, the low frequencies region exhibits the signature of the geometric characteristics of the particle. Therefore, it must be expected that the response in the low frequency region containing information on the geometrical characteristics of the particles, whereas high frequency peaks provide information on the interactions of the magnetic moments within the particle.
0.06
0.03
IV. 0
0
200
400
600
Frequency(cm-1)
800
1000
FIG. 7. Density of modes for a N iO particle with volume equal to 4πre 3 /3 where re = 2.08 nm and e = 2.0. 0.12
e = 0.5
D(ω)
0.09
0.06
0.03
0
0
200
400
600
Frequency(cm-1)
800
1000
FIG. 8. Density of modes for a N iO particle with volume equal to 4πre 3 /3 where re = 2.08 nm and e = 0.5 (particle in the format close to a disk).
of these systems is done with a large number of particles. Therefore, it would be necessary a precise control on the crystallographic orientation, volume and shape of the particles to obtain accurate results that could be associated with the behavior of one particle. It should be expected that any experimental results for static magnetic properties of these systems should be some kind of average on the volume and shape of the particles that compose it. Our next step was to analyze the dynamical response of these particles. For the smaller one, each magnetic moment is subjected to different forces and, as could be expected, there is a large dispersion of the frequencies of normal modes. However, in all calculations done it was found that the maxima at frequencies w = 900 cm−1 and w = 720 cm−1 were present in the distributions, no matter the volume or format of the particles. The geo-
SUMMARY AND FINAL COMMENTS
We have investigated the effects of finite size on the physical properties of antiferromagnetic particles. We began with the study of the equilibrium configuration of M nF2 particles. This material has the simplest description for a real antiferromagnetic sample and some consequences of the breaking of the symmetry and uncompensated magnetic moments associated with the particle shape were more easily understood. The equilibrium configuration of N iO nanoparticles was also investigated. The crystal structure of this material, as well as the interactions between the ions that make up the particles are quite different of the particles of M nF2 and our calculations have shown a variety of results associated to the relative orientation of the external field with respect to the crystallographic axes. Both systems were modeled as a small portion of a massive material. Certainly a real particle has surface effects not included in this work. Therefore, extensions to this paper including surface effects as local anisotropy and lattice disorder would be appropriated to have a more realistic description for these particles. The calculations of equilibrium configurations allow the understanding of the static properties of these systems, but it is not an easy task connects them with experimental results. The experimental data are usually obtained from measurements of properties of a group of particles, and is not a simple task to separate the collective effects of those resulting from a single particle. Some authors26–30 have pointed out ways to improve the comprehension on this subject, but at the moment it should be said that the knowledge of physical properties of these particles is not complete. An alternative to improve the knowledge of these systems is the study of their dynamical properties. In particular, the spectrum of the vibrational normal modes of an assemble of these particles should contain the results of both inter and intra particles interactions. Since the contribution of the inter particles interaction for the spectrum must be at frequencies much lower than those that come from vibrations of ions inside of the particle, the contribution of the collective behavior can be separated from that of a single particle. In addition, ions located on the surface should contribute to the spectrum of vibration modes in a frequency region different from that which contains the contribution of the internal ions.
Therefore, the frequency distribution can be useful not only to separate the collective from the individual behavior, but also to have information on the characteristics of the surface of the particles. We calculate the spectrum of vibrational modes of NiO particles. We chose this system because, unlike M nF2 , particles of this material are available elsewhere. The results presented here can serve as a basis for understanding the results obtained from models that include surface defects that are certainly present in real systems. V.
ACKNOWLEDGMENTS
Ana T.A. Lima was partially supported by Coordena¸c˜ ao de Aperfei¸coamento de Pessoal de N´ıvel Superior (CAPES); Ana L. Dantas is grateful to the Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ ogico (CNPq). 1 R.
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PII: DOI: Reference:
S0304-8853(16)32753-6 http://dx.doi.org/10.1016/j.jmmm.2016.10.124 MAGMA62037
To appear in: Journal of Magnetism and Magnetic Materials Received date: 20 April 2016 Revised date: 15 August 2016 Accepted date: 25 October 2016 Cite this article as: Ana T.A. Lima, Ana L. Dantas and N.S. Almeida, Magnetic properties of crystalline nanoparticles with different sizes and shapes, Journal of Magnetism and Magnetic Materials, http://dx.doi.org/10.1016/j.jmmm.2016.10.124 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Magnetic properties of crystalline nanoparticles with different sizes and shapes. Ana T.A. Lima,1 Ana L. Dantas,2 and N. S. Almeida2 1)
Departamento de F´ısica, Universidade Federal do Cear´ a, Caixa Postal 6030, Campus do Pici, 60455-760 Fortaleza, ´ Cear´ a, Brazil, and Universidade Federal Rural do Semi-Arido, Campus de Caraubas, RN 333, Rio Grande do Norte, Brazil 2) Departamento de F´ısica, Universidade do Estado do Rio Grande do Norte, 59610-210 Mossor´ o, Rio Grande do Norte, Brazil (Dated: 26 October 2016)
The effects of shape and finite size on the physical behavior of nanostructured antiferromagnetic particles are investigated. They were modeled as ellipsoidal systems which preserve the crystalline structure of the correspondent bulk material. In our analysis we consider nanoparticles composed by magnetic ions which are themselves insensitive to the presence of surfaces and/or interfaces. Results are shown for structures similar to M nF2 and N iO crystals. Special attention is given to these last once their singular magnetic arrangement, as well as, their use at different technological and/or biomedical applications, has motivated intense experimental studies at different laboratories. We use the parameters that describe the correspondent bulk material to discuss the magnetic behavior of these particles for different volumes and shapes. PACS numbers: 75.75.-c, 75.50.Tt, 75.60.Ej, 75.75.Jn Keywords: magnetic particles; nanostructured magnetic particles; hysteresis loops; dynamical behavior I.
INTRODUCTION
The use of particles with size in the nanometric scale in the everyday life has increased continuously and, in particular, magnetic nanoparticles have been used for applications that go from the use in the development of reliable large data storage devices1,2 to the use in medical procedures that require non-invasive techniques for imaging and/or localized therapy of serious diseases3–5 . Despite the motivation and the large number of papers dealing with these systems, there are several properties that are not fully understood (as the exchange bias effect6–9 ) and are still challenges for those working in this field. The demand for a better understanding of these systems comes from the fact that in recent years the development of growth techniques has allowed the preparation of this kind of particle with well controlled geometric and structural characteristics. It is well known that physical properties of small systems are modified by the breaking of the lattice symmetry, as well as, broken bonds due to the presence of surface. This requires a major effort in the search for better comprehension of the consequences of the presence of surfaces in systems with size in this domain. Moreover, since several properties depend on these geometric characteristics, the rapid development of well controlled growth techniques also allows one to prescribes that very soon will be possible the obtaining of particles to fit a large range of needs. Most of low dimension magnetic systems studied in the last decades were very thin films. To them, the border effects were not relevant and the main modifications of the physical properties came from the break of symmetry in one direction. In some sense, magnetic nanoparti-
cles might be seen as a thin magnetic film with relevant border effects. This fact gives to these systems a much richer variety of externally controlled properties that can be used to adapt them to different applications. Useful approaches to study these nanoparticles are reported by several authors (see e.g., Zysleka et al.10 and Liu ZhaoSen et al.11 ). A good review on the procedures and approaches used to study small magnetic particles can be found in the papers of R.H. Kodama and co-authors12–14 . The main goal of this work is to discuss the changes imposed by surfaces on the magnetic properties of small nanostructured antiferromagnetic particles. We start modeling the particles as a collection of ions with well localized positions in the crystalline structure. Moreover, we assume that the ions have well defined magnetic moments, which are not sensitive to their location. However, it is assumed that their orientation is the result of the competition of all magnetic interactions present in the system. We start Section II calculating the magnetization of nanoparticles resulting from nano portions of M nF2 crystals. This material was chosen not only for being one of the most studied, but also because it can be seen as a prototype of the antiferromagnetic materials. The results pave the way for the understanding of more complex systems. This calculation is followed by a more thorough study of N iO nanoparticles. In N iO crystals, each magnetic moment of its f cc type II crystalline structure has twelve nearest neighbors and six next nearest neighbors and, in the bulk configuration can be seen as a collection of ferromagnetic planes coupled antiferromagnetically. We investigate the influence of the break of symmetry introduced by the surface of the particle on the hysteresis loop. The magnetic resonances are also in-
vestigate for particles with different forms. The particles studied in this paper have an ellipsoidal format with the shaped described by the parameter e = a/b, where a and b are the semi-axes parallel and perpendicular to the z direction, respectively. Its volume is specified by re , the radius of a spherical particle of the same volume. Among other questions, we are looking for answers that allow us to understand how the breaking of the symmetry due to the presence of the surface, associated to their shape, modify the susceptibility, coercivity, remanent magnetization and resonances of these nanometric systems. The influence of the dipolar interaction is also investigated. For large systems with antiferromagnetic ordering, we found no report or even some evidence of the influence of the shape on the magnetic properties of these materials. We begin our discussion with a bcc antiferromagnetic material. As expected, a behavior similar to that of the equivalent bulk material is found when the surface/volume ratio is close to zero. However, in a small portion of this material, the number of magnetic moments at the surface (non paired) becomes relevant and they give to the particle a resulting magnetic moment which will depend on the crystalline arrangement, shape and volume of the system. In other words, the geometry is determinant for its primary magnetic behavior. For particles with diameter of few lattice parameters, small changes in the form might result in considerable modifications in their magnetic properties. Despite there be no simple mathematical relationship between the crystalline arrangement of the magnetic ions and the physical properties of the system, the numerical results obtained show that small particles are quite sensitivity to both, volume and shape, as well as the crystalline structure.
II. SPATIAL DESCRIPTION OF THE MONO DOMAIN PARTICLES
We will model each particle as a limited portion of the crystalline material. We begin with an unlimited lattice with each site occupied by the magnetic moment of the ion and the particle is obtained excluding the sites not belonging the region limited by its surface. The theoretical description is completed assuming that the interactions between the magnetic moments in a small portion of the material are the same observed in a very large one. As an example, each ion of an antiferromagnetic (AFM) mono domain particles of well known AFM materials as M nF2 or F eF2 , feels the presence of its neighbors through the exchange and dipolar interactions, an uniaxial anisotropy and the Zeeman interaction. The additional dipolar interaction included here, has no action in extensive materials, but small structures might have the low field behavior modified by the presence of this interaction. In this paper we consider that all interactions/anisotropy are the same everywhere in the
FIG. 1. Formats of three particles with a same volume and different values of e: 0.1 (disc shape), 1 (sphere) and 10 (toothpick shape).
system. However, we should mention that, due to symmetry breaking, the ions located on the surface are in a different environment from those within the system. Dangling bonds, changes in atomic coordination and disorder of the crystalline lattice, among others, may be responsible for changes in the anisotropy and exchange interaction. However, if the ratio NS /N is small (NS is the number of ions in the surface and N is the total number of ions), the results shown here should describe the overall behavior of the system and the imperfections in the surface should produce only minor corrections. The equilibrium configuration, first step to have the magnetic properties studied here, is obtained by searching the configuration that has the minimum value for the energy, which is equivalent to find the configuration where the magnetic moment of each site is parallel to the effective magnetic field felt by it (zero torque approach)15–17 . We use this second method to have numerical data for equilibrium configuration of the studied systems. As mentioned before, the geometric description of the particles is given by re , the radius of a sphere that has the same volume of the ellipsoidal particle, and e, the ratio between the length of the axes parallel and perpendicular to the z-direction. Fig.1 illustrates how the shape of ellipsoidal particles depends on e. It is displayed the format of three particles with the same volume and e equal to 1 (sphere), 0.1 (disc) and 10 (long ellipsoid) respectively.
A.
Nanoparticles of uniaxial AFM materials
M nF2 , CoF2 and F eF2 are materials well described by simple model and we use them to illustrate the calculations depicted in this paper. These materials have their magnetic ions at the sites of a body centered tetragonal lattice and they are well described by the two uniaxial
sublattice model18 . If we define the z axis of the coordinate system parallel to the c axis of the crystal, large portions of these materials have, at the equilibrium configuration and in the absence of external applied magnetic field, their magnetic moments pointing along the +/− z axis. In these materials, the exchange interaction between the magnetic moments on different sublattices keep them antiparallel and might be represented by an exchange magnetic field He . Also the magnetic moments are pinned to the z axis by the anisotropy, which here is described by the anisotropy field Ha . Then, one might write the magnetic energy E of these AFM particles as: N N N E Ha 0. = −He μi . μj − [ μi .ˆ z ]2 − γ H μ i MS 2 i i,j i
+D
N μi . μj − 3 ( μi .ˆ r i,j ) (μj .ˆ r i,j ) (1) 3 | r − r | i j i,j
where N is the number of the ions inside of the surface that limit the volume of the particle, and He , Ha and H0 are, respectively, the exchange, anisotropy and the externally applied fields. Also, MS is the saturation magnetization of the material, μi is the magnetization at the site i, in units of MS , D = MS /n, (n is the number of ions per unit cell), ri the vector that locate the site i, in units of the lattice parameter, and rˆi,j is the unit r − r vector given by |rii −rjj | . The equilibrium configuration is obtained when the magnetic moment at each site is parallel to the effective magnetic field at that site. For a site k located by rk , in the ground state, μ k is parallel to H ef f given by k ef 0 H μj + Ha [ μk .ˆ z ]ˆ z+H f = He j=k
+D
3( μj .ˆ rkj )ˆ rkj − μj 3 |rk − rj | j
(2)
The behavior of the averaged magnetic moments per lattice site for different external magnetic fields is showed in Fig. 2 for an uniaxial antiferromagnetic particle. The results were obtained for He = 540kG, Ha = 3.8kG and Ms = 0.6kG which are the parameters associated to M nF2 19 . The diameter of the spherical particle (e = 1) is 8 lattice parameters and the number of localized magnetic moments inside is 537. In order to see the shape effect, this figure also shows the hysteresis loop for a particle with an equal volume and e = 8 (a long ellipsoid). As can be seen, while the spherical particle exhibits a simple hysteresis loop, the ellipsoidal particle has different magnetic configurations at the low field region. The two different plateaus observed at low field region are due to changes in the ordering of magnetic moments with smaller number of neighbors. It should also be mentioned that, our theoretical calculations indicate that this is the field region which the dipolar interaction contributes significantly to stabilize the
FIG. 2. Hysteresis loops for particles with re = 4 lattice parameters (volume equal to 1.57×10−22 cm3 ) and e = 1 (spherical particle) and e = 8 (particle stretched in the z-direction). The inserts show the arrangement of the magnetic moments in the plane (011) that contain the z-axis, corresponding to the points a, b (ellipsoid) and c (sphere) of the hysteresis loops.
magnetic phases of this system. Outside this region, the difference between the results obtained with and without the dipolar interaction is negligible. The inserts in Fig. 2 illustrate the arrangement of the magnetic moments located in the plane (011) that contain the z-axis, at the antiferromagnetic phase corresponding to the points ”a” and ”b” in the hysteresis curve. As can be seen in these inserts, the plateaus correspond to flips of the localized magnetic moments in the furthermost points the z-axis of the ellipsoid. The equilibrium configuration of the magnetic moments in the spherical particle is illustrated by their arrangement in the same (011) plane depicted in the insert ”c”.
B.
NiO nanoparticles
The transition metal oxides have applications ranging from pure science20 to daily use devices21 . Similarly to M nO, CoO and F eO, N iO is antiferromagnetically ordered in a type II face-centered cubic sodium chloride structure with a lattice parameter a = 4.18 ˚ A. The magnetic order occurs below TN = 523 K and consists of the ferromagnetic alignment of the magnetic moments of the Ni ions in the set of (111) planes. The magnetic moments of adjacent (111) planes are antiparallel, characterizing the antiferromagnetic ordering. It is also known that the N iO antiferromagnetic ordering is accompanied a slight rhombohedral distortion that compresses the lattice along the [111] direction. The interaction between the next nearest magnetic moments of the N iO is al-
E 2 =− He (i, j) μi . μj + D 1 [ μi .ηˆ1 ] MS i,j i N
N
+D2
N i
2 0. [ μi .ηˆ2 ] − γ H
N
μi
N
He (k, j) μj − D1 [ μk .ηˆ1 ] ηˆ1
j
0 −D2 [ μk .ηˆ2 ] ηˆ2 + H
0.02
Ho // [11-2] Ho // [111] Ho // [1-10]
0
-0.02
-0.04 -40
-20
0
H0 (T)
20
40
FIG. 3. Hysteresis loop for a spherical particle of N iO with diameter of 4.18 nm for external magnetic fields parallel to [111], [¯ 110] and [11¯ 2] (as indicated in the plot). The saturation magnetization (MS ) of this material is 0.55kG
(3)
i
where He (i, j) is the exchange interaction between the magnetic moments located at ri and rj , which is J1 if the lattice points are nearest neighbor on the same (111) plane and, J2 if they are nearest neighbor located at different (111) planes. He (i, j) is equal to J3 if ri and√rj locate next nearest x + yˆ + zˆ]/ 3, √ neighbors. Also ηˆ1 = [ˆ x + yˆ]/ 2. D1 and D2 are the anisotropy fields ηˆ2 = [−ˆ in the [111] (hard axis) and [¯ 110] directions, respectively. The dipolar interaction was not included in this calculation because in all numerical data obtained for particles with different sizes and shapes, this interaction did not produce any significant modification in the results. This happens because the contribution of the dipolar interaction to the effective field felt by the N iO ions is very small compared with the contribution of the others interactions. As the previous case, the equilibrium configurations were obtained searching the configuration that all magnetic moments are aligned with the effective field at its position, i.e., the magnetic moment μk is parallel to k = H ef f
0.04
M/MS
most 14 times stronger than the interaction between the nearest22 . This suggests that small particles of this material must have a unique behavior due to lack of coordination of the two outermost layers of ions. Therefore, understanding the behavior of these magnetic nanoparticles becomes even more challenging because, in addition to geometrical effects of the system, there are special characteristics of the material itself, which become more relevant if the ratio surface/volume is not small. A good description of the structure and physical properties of the N iO can be found everywhere22–25 and here we have mentioned only the parameters and details essential to development of the calculations presented in this work. The magnetic energy of a N iO particle can be written as22
(4)
Fig. 3 depicts the central part of the hysteresis loop of a N iO spherical particle with diameter of 4.18 nm. The results were obtained considering22 J1 = 15.7 K (≈ 103.8kG, J2 = 16.1K (≈ 106.5kG), J3 = −221K (≈ −1460.0kG), D1 = 1.13K (≈ 7.5kG) and D2 = 0.06K (≈ 0.4kG) and external magnetic fields applied parallel to [11¯ 2], [111] and [1¯ 10] directions, respectively. These results show that if the external field is applied parallel to the easy direction, the particle remain in the antiferromagnetic phase, even for moderate magnetic fields. On
the other hand, if the external field is applied parallel to [111] direction, the particle responds as a antiferromagnetic material in a canted phase. The magnetic response has a very different characteristic if the external field is applied in the [1¯10] direction. As can be seen, for external magnetic fields in this direction, the hysteresis loop around H = 0 is similar to those depicted by ferromagnetic materials with non zero coercive field. In summary, the response to an externally applied dc magnetic field is strongly dependent on the direction of the field relative to the crystallographic axes. Fig. 4 shows the hysteresis loops of particles with the same volume of the particle of the Fig. 3, but different forms; e = 0.1, particle with format like a disk, and e = 10.0, particles with format of a long ellipsoid. These results illustrate how the static magnetic properties of these particles are sensitive to their shape. Unpaired magnetic moments in the outermost layers of these particles produce a nonzero magnetization and any accurate experimental analysis of the static properties of one small portion of a agglomerate with a moderate density of this material must take into account the magnetic field due the remainder of the material. However, the dynamic analysis of these systems must naturally separate the responses due to intra and inter particles interactions. While the response due the internal structure is associated with exchange interaction and anisotropy, the response to inter particles coupling must be associated with dipole interaction, which should be at much lower frequencies. This is the main motivation to investigate the magnetic resonances of these particles. To study the dynamic response of these systems, we begin writing the equation of motion for each magnetic k (t) k (t). Then we write the = γμk (t) × H moment dμdt ef f time dependent solution (μk (t)) as the sum of the equilibrium solution μk with the time dependent fluctuation
0.04
0.12
e = 0.1 e = 10.0
e = 1.0
0.09
D(ω)
M/MS
0.02
0
-0.02
0.06
0.03
-0.04 -40
-20
0
H0 (T)
20
0
40
FIG. 4. Hysteresis loop for a disk and stick-like particles of N iO with volume equal to a spherical particle with diameter of 4.18 nm for external magnetic fields parallel [11¯ 2] .
0
200
400
600
Frequency (cm-1)
800
1000
FIG. 5. Density of modes for a N iO spherical particle with diameter equal to 8.35 nm.
0.12
α k (t), i.e., μk (t) = μk + α k (t). By following this approach, the resulting effective field can be written as k (t) = H 0k + hk (t), where H ef f N
He (k, j) μj − D1 [ μk .ηˆ1 ] ηˆ1
D(ω)
0k = H
j(k)
0 −D2 [ μk .ηˆ2 ] ηˆ2 + H
0.06
(5) 0.03
and hk (t) =
N
He (k, j) αj (t) − D1 [ αk (t).ηˆ1 ] ηˆ1
0
j(k)
where
e = 5.0
0.09
−D2 [ αk (t).ηˆ2 ] ηˆ2
(6)
means the sum over the nearest and next
j(k)
nearest neighbors of the site k. By neglecting nonlinear terms and taking into account that the equilibrium k configuration is obtained searching for μk parallel to H 0 k ( μk × H0 = 0), we have the equation of motion given by d αk (t) k = γ μk × hk (t) + γ αk (t) × H 0 dt
(7)
Then the frequencies of the normal modes are given by the imaginary part of the eigenvalues of the matrix formed by the coefficients of the linear system that relates the components of α k (t). Figs. 5 - 8 show the density of modes for N iO particles with volume equal to 4πre 3 /3, with re = 2.08 nm and different shapes. As could be seen, there is a large dispersion of the modes at the low frequency region. The reason for this is that, besides the absence of translational invariance, there is a distribution of effective inter ions interaction inside the particle and this distribution depends on the particle size and shape. However, at high frequencies region it is observed a quite regular behavior. Particles with different volumes and
0
200
400
600
Frequency(cm-1)
800
1000
FIG. 6. Density of modes for a N iO particle with volume equal to 4πre 3 /3 where re = 2.08 nm and e = 5.0 (particle in the format of a long ellipsoid).
format show the frequency distribution of modes fitted for Gaussian centered at the frequencies w = 900 cm−1 and w = 720 cm−1 . It was observed that the width of these curves depend on the shape of the particle but its maxima value was always at the same frequency.
III.
COMMENTS ON THE RESULTS
The results presented in figures 2, 3 and 4 show the magnetic response of particles to an externally applied dc magnetic field. A variety of phases can be observed for both M nF2 and for N iO particles. The main conclusion is that their static characteristics are strongly dependent on their volume and shape. As far as we know, any experiment to analyze the static magnetic behavior
0.12
e = 2.0
0.09
D(ω)
metric characteristics (shape and volume) only influence the form of the distribution. On the other hand, the low frequencies region exhibits the signature of the geometric characteristics of the particle. Therefore, it must be expected that the response in the low frequency region containing information on the geometrical characteristics of the particles, whereas high frequency peaks provide information on the interactions of the magnetic moments within the particle.
0.06
0.03
IV. 0
0
200
400
600
Frequency(cm-1)
800
1000
FIG. 7. Density of modes for a N iO particle with volume equal to 4πre 3 /3 where re = 2.08 nm and e = 2.0. 0.12
e = 0.5
D(ω)
0.09
0.06
0.03
0
0
200
400
600
Frequency(cm-1)
800
1000
FIG. 8. Density of modes for a N iO particle with volume equal to 4πre 3 /3 where re = 2.08 nm and e = 0.5 (particle in the format close to a disk).
of these systems is done with a large number of particles. Therefore, it would be necessary a precise control on the crystallographic orientation, volume and shape of the particles to obtain accurate results that could be associated with the behavior of one particle. It should be expected that any experimental results for static magnetic properties of these systems should be some kind of average on the volume and shape of the particles that compose it. Our next step was to analyze the dynamical response of these particles. For the smaller one, each magnetic moment is subjected to different forces and, as could be expected, there is a large dispersion of the frequencies of normal modes. However, in all calculations done it was found that the maxima at frequencies w = 900 cm−1 and w = 720 cm−1 were present in the distributions, no matter the volume or format of the particles. The geo-
SUMMARY AND FINAL COMMENTS
We have investigated the effects of finite size on the physical properties of antiferromagnetic particles. We began with the study of the equilibrium configuration of M nF2 particles. This material has the simplest description for a real antiferromagnetic sample and some consequences of the breaking of the symmetry and uncompensated magnetic moments associated with the particle shape were more easily understood. The equilibrium configuration of N iO nanoparticles was also investigated. The crystal structure of this material, as well as the interactions between the ions that make up the particles are quite different of the particles of M nF2 and our calculations have shown a variety of results associated to the relative orientation of the external field with respect to the crystallographic axes. Both systems were modeled as a small portion of a massive material. Certainly a real particle has surface effects not included in this work. Therefore, extensions to this paper including surface effects as local anisotropy and lattice disorder would be appropriated to have a more realistic description for these particles. The calculations of equilibrium configurations allow the understanding of the static properties of these systems, but it is not an easy task connects them with experimental results. The experimental data are usually obtained from measurements of properties of a group of particles, and is not a simple task to separate the collective effects of those resulting from a single particle. Some authors26–30 have pointed out ways to improve the comprehension on this subject, but at the moment it should be said that the knowledge of physical properties of these particles is not complete. An alternative to improve the knowledge of these systems is the study of their dynamical properties. In particular, the spectrum of the vibrational normal modes of an assemble of these particles should contain the results of both inter and intra particles interactions. Since the contribution of the inter particles interaction for the spectrum must be at frequencies much lower than those that come from vibrations of ions inside of the particle, the contribution of the collective behavior can be separated from that of a single particle. In addition, ions located on the surface should contribute to the spectrum of vibration modes in a frequency region different from that which contains the contribution of the internal ions.
Therefore, the frequency distribution can be useful not only to separate the collective from the individual behavior, but also to have information on the characteristics of the surface of the particles. We calculate the spectrum of vibrational modes of NiO particles. We chose this system because, unlike M nF2 , particles of this material are available elsewhere. The results presented here can serve as a basis for understanding the results obtained from models that include surface defects that are certainly present in real systems. V.
ACKNOWLEDGMENTS
Ana T.A. Lima was partially supported by Coordena¸c˜ ao de Aperfei¸coamento de Pessoal de N´ıvel Superior (CAPES); Ana L. Dantas is grateful to the Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ ogico (CNPq). 1 R.
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