Micromagnetic study of equilibrium states in nano hemispheroidal shells

Accepted Manuscript Micromagnetic study of equilibrium states in nano hemispheroidal shells Keren Schultz, Moty Schultz PII: DOI: Reference:

S0304-8853(16)33264-4 http://dx.doi.org/10.1016/j.jmmm.2017.07.012 MAGMA 62943

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Journal of Magnetism and Magnetic Materials

Please cite this article as: K. Schultz, M. Schultz, Micromagnetic study of equilibrium states in nano hemispheroidal shells, Journal of Magnetism and Magnetic Materials (2017), doi: http://dx.doi.org/10.1016/j.jmmm.2017.07.012

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Micromagnetic study of equilibrium states in nano hemispheroidal shells Keren Schultz and Moty Schultz Department of Physics, Nano-magnetism Research Center, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat Gan 5290002, Israel

Abstract We present results of micromagnetic simulations of thin ferromagnetic nano hemispheroidal shells with sizes ranging from 5 to 50 nm (inside dimensions). Depending on the geometrical and magnetic parameters of the hemispheroidal shell, there exist three different magnetic phases: easy axis, onion and vortex. The profile for the vortex magnetization distribution is analyzed and the limitations and applicability of different vortex ansatzes are discussed. In addition, we investigate the total energy density for each of the magnetic distributions as a function of the hemispheroidal shell dimensions. Keywords: Micromagnetic simulations; Janus particles; Magnetic phases; Vortex PACS: 75.78.Cd, 75.10.Hk, 75.40.Mg

1. Introduction Synthesis and application of magnetic nanoparticles are a fast burgeoning field which has potential to bring significant advance in many fields, ranging from magnetic nanoparticles based cancer therapy to sensors [1]. The magnetic properties of a nanoparticle are important since they enable the manipulation of particle’s behaviour remotely and therefore provide the means to direct a particles’ orientation and translation. Janus particles (JPs) are particles with two dissimilar faces having unique material properties [2, 3]. JPs provide us with the ability to complete challenging tasks that would be impossible with isotropic particles. Magnetic JPs combine magnetic properties with anisotropy and thus are potential building blocks for complex structures that can be assembled from a particle suspension and can be directed easily by

Preprint submitted to Journal of LATEX Templates

July 4, 2017

external magnetic fields. JPs with ferromagnetic or paramagnetic properties of various anisotropic shapes have been proposed to have potential application in probing the rheology of complex fluid interfaces [4]. Magnetic JPs have interesting properties for their application in microfluidics and as nanodelivery systems. In order to control magnetic JPs, it is very important to understand their magnetization behaviour. Masking is the simplest technique and was one of the first developed for the synthesis of magnetic JPs. In a first step, particles were deposited onto a substrate and then the above-lying surface was coated with a thin shell of a magnetic metal. During the last years, there was an increasing interest in studying the magnetic properties of spherical and hemispherical soft magnetic shells. In all these geometries which were investigated so far, experiment measurements, analytical calculations and micromagnetic simulations, suggest the existence of two magnetic ground states: vortex state and onion state [5, 6, 7]. Considerably less attention has been devoted to non spherical shells, such as spheroids. In addition to the inhomogeneous magnetization states, micromagnetic simulations of extruded hemispherical shells show the existence of a uniform magnetic ground state [8]. Moreover, out of plane magnetized caps were made by using multilayers of [Co/Pt]N and [Co/Pd]N [9, 10]. Stacking the out of plane magnetized caps and soft magnetic caps with an interlayer exchange coupling allows for imprinting non-collinear spin textures [11]. In this work, we present micromagnetic simulations of nano hemispheroidal soft magnetic shells, by using Nmag simulator [12]. The simulations are made for a variety of shells’ thicknesses (5 to 30 nm) and sizes ranging from 5 to 50 nm (inside dimensions). We find the phase diagram of the magnetic ground states in these shells. In contrast to the hemispherical soft magnetic shells which have only two magnetic ground state phases (onion and vortex) [6], the hemispheroidal soft magnetic shells have three magnetic ground state configurations: one phase with homogeneous magnetization (easy axis) and two other phases with inhomogeneous magnetization (onion and vortex). The easy axis structure is encountered only in the small and elongated hemispheroidal magnetic shells. Hereafter we consider the vortex magnetization profiles; the limitation and applicability of different vortex ansatzes are discussed. Moreover, we investigate the total energy density for each of the magnetic distributions as a 2

function of the hemispheroidal shell dimensions.

2. method The starting point of magnetization dynamics at zero temperature is from the LandauLifshitz-Gilbert (LLG) equation, ∂M α ∂M = −γM × Hef f + M× , ∂t |M| ∂t

(1)

where γ is the gyromagnetic ratio, Hef f is the effective magnetic field and α is the damping constant. There are some publicly available packages for solving LLG equation, like OOMMF, mumax3 and Nmag.[12, 13, 14] OOMMF and mumax3 both are based on discretizing space into small cuboid cells. One advantage of this method (often called finite differences) is that the demag field can be computed very efficiently. On the other hand, this method works less well if the geometry shape does not align with a Cartesian grid since the boundary is represented as a staircase pattern. Nmag’s finite elements method discretizes space into many small tetrahedra. The corresponding approach is based on the Fredkin Koehler hybrid Finite Element/Boundary Element method (FEM/BEM) [15]. The advantage of this method (over mumax3 and OOMMF’s approach) is that curved and spherical geometries can be spatially resolved much more accurately. However, this method of calculating the demagnetization field is less efficient than OOMMF’s approach for thin films. In particular, memory requirements for the boundary element method grow as the square of the number of surface points. Note that for simulation of thin films, the hybrid FEM/BEM is likely to require a lot of memory. In order to improve the efficiency of the hybrid FEM/BEM, one can employ techniques which involve some kind of approximation, for example using hierarchical matrices [16]. The library Hlib [http://www.hlib.org] contains implementations of this hierarchical matrix methodology. For our purpose, Nmag is a better choice than OOMMF and mumax3 due to its superior calculation of curved geometries. Using Hlib library for the big samples simulations, we managed to overcome the memory requirements. 3

The cell size of micromagnetic simulations should be equal or smaller than the half size of the smallest feature of interest, so sampling at that length scale is sufficient to capture all relevant details. In our micromagnetic simulations, it is the domain wall. For soft materials, the domain wall profile is determined by competition between the exchange and demagnetization energies. Therefore, it is common to define magnetop static exchange length as lex = 2A/(µ0 Ms2 ) , where A is the exchange constant in J/m and Ms is the saturation magnetization in A/m. In order to get reliable results, in all our simulations the cell size was kept smaller than half of the exchange length. A good criterion for testing the reliability of the simulations is the maximum angle between two neighboring cells. Because this is such important issue, Nmag provides this data. A spheroid, or ellipsoid of revolution, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semidiameters. If the ellipse is rotated about its major axis, the result is a prolate (elongated) spheroid. While rotating about its minor axis, an oblate (flattened) spheroid is obtained. The systems studied are hemispheroidal shells with Permalloy parameters: exchange coupling A = 13 × 10−12 J/m, saturation magnetization Ms = 0.86 × 106 A/m and magnetocrystalline anisotropy is neglected. Therefore, the exchange length is about 5 nm. We create the shells by rotating a half elliptical ring around the z axis. The geometry under study and accompanying relevant parameters are represented in Fig. 1. The internal dimensions of the shells are of a half spheroid with semi-principal axes of length Ain , Bin and Cin , where Bin = Ain . The outside dimensions of the shells are of a half spheroid with semi-principal axes of length Aout , Bout and Cout , where Bout = Aout so thickness = Aout − Ain = Bout − Bin = Cout − Cin .

3. Results and discussion 3.1. Phase diagram We simulate the remanent states starting with four different configurations of initial magnetization: single domain configuration in the xy plane (the cutting plane of the hemispheroidal shell), single domain along the z direction (perpendicular to the cutting

4

Figure 1: Sketch of the hemispheroidal shell.

plane), rough vortex state and rough onion state. Depending on the dimensions of the hemispheroidal shell, there are one or two remanent states. The ground state is the one with the minimum energy. Following such a scheme, we construct a phase diagram of the magnetic ground states in the hemispheroidal shell. Since we have three parameters, which define the hemispheroidal shell, Ain , Cin and the thickness of the shell, our phase diagram is three dimensional. We choose to show the phase diagram in two dimensional plots for clarity (see Figure 2). The general properties of the phase diagram are as follows. The ground state of the small hemispheroidal shells is the onion state, while the ground state of the small and elongated hemispheroidal shells is the homogeneous easy axis state (z axis). As Ain and Cin increase, the ground state switches to the vortex state. The vortex state becomes much more preferable while moving to larger thicknesses. Therefore, all the hemispheroidal shells with thickness ≥ 25 nm have the vortex state as the ground state. Our phase diagram is in a good agreement with the phase diagram of hemispherical shells [6]. This comparison is made by analyzing the ground state along a line Ain = Cin . The total energy densities were obtained for different hemispheroidal shells’ sizes

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Figure 2: Phase diagrams of simulated magnetic ground states in Permalloy hemispheroidal shells: blue triangles–single domain z axis, red squares–onion, green disks–vortex. Shell thickness: (a) 5 nm; (b) 10 nm and (c) 15 nm.

6

and magnetization distributions, the results of these simulations are presented in Figures 3, 4 and 5. Figure 3(a) depicts the total energy density of wide hemispheroidal shells (Ain = 40 nm) with a thickness of 5 nm. Due to the magnetostatic energy, the total energy density of the single domain xy plane increases with Cin . While Cin increases, the total energy density of the onion state increases when Cin < 30 nm and decreases when Cin > 30 nm; the decrease of the total energy density when Cin > 30 nm is very small and therefore unnoticeable in the figure. While Cin decreases, the total energy density of the vortex state and the single domain z axis increase. The total energy density of the single domain z axis is very high for all values of Cin and decreases with increasing length of Cin . Figure 3(b) shows the total energy densities of narrow hemispheroidal shells (Ain = 10 nm) with a thickness of 5 nm. As can be seen the energies for the single domain easy axis and the single domain easy plane behave qualitative the same as shown in Figure 3(a). There are two main differences between Figure 3(a) and Figure 3(b): for Ain = 10 nm the vortex sate is no more a reachable metastable state and the single domain z axis is the ground state for Cin > 22 nm. The total energy density of narrow hemispheroidal shells (Ain = 10 nm) with a thickness of 10 nm is shown in Figure 4. While the vortex state is not a reachable state for a thickness of 5 nm and Ain = 10 nm, in Figure 4 the vortex state is a reachable state for all Cin and the ground state for Cin ≥ 22 nm. Furthermore, even for the most elongated hemispheroidal shells the single domain z axis is not the ground state. The energy densities of narrow hemispheroidal shells (Ain = 10 nm) with a thickness of 15 nm are plotted in Figure 5. One can see the same trend as in Figure 4, the vortex state becomes much more preferable while moving to larger thicknesses. In the absence of intrinsic anisotropy energy, the reason for all these various ground states is the competition between two energies: the exchange energy and stray field energy. A general solution of the stray field is given by potential theory. The volume magnetostatic charge density λ and the surface magnetostatic charge density σ are defined in terms of the normalized magnetization: λ = −∇ · m, σ = m · n, where n is the outward surface normal. For the vortex state, the surface and the volume magnetostatic charges are almost absent. In the vortex distribution, there is a stray field only inside the vortex core. For the homogeneous magnetization distribution the total 7

3

Total energy density (J/cm )

0.4 Single domain xy plane Singe domain z axis Onion Vortex

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Figure 3: Total energy density of different magnetization distributions as a function of the semi-principal axis Cin for a thickness of 5 nm and different semi-principal axis (a) Ain = 40 nm (b) Ain = 10 nm . The dashed lines are guides for eyes.

energy contains only the contribution of surface magnetostatic charges. These surface magnetostatic charges are the source of demagnetization field which is opposite to the sample magnetization. In general, the demagnetization field along a short axis is stronger than that along a long axis. Due to demagnetization field, the single domain z axis becomes energetically favorable with the particle size decreasing and the aspect ratio Cin /Ain increasing. For nano particles the single domain configuration becomes the lowest energy state when the particle diameter becomes comparable or smaller than the exchange length. Therefore, if we extrapolate the total energy density for Cin < 5 nm, well below 5 nm the energy of a single domain configuration is the smallest one for all hemispheroidal shells.

8

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Figure 4: Total energy density of different magnetization distributions as a function of the semi-principal axis Cin for a thickness of 10 nm and Ain = 10 nm. The dashed lines are guides for eyes.

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Figure 5: Total energy density of different magnetization distributions as a function of the semi-principal axis Cin for a thickness of 15 nm and Ain = 10 nm. The dashed lines are guides for eyes.

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Figure 6: The magnetization distribution in vortex state of an oblate hemispheroidal shell with Ain = 40 nm, Cin = 20 nm and a thickness of 5 nm. The color code, represents the normalized z component of the magnetization, emphasizes that the core is only in the center of the vortex.

3.2. Vortex state The typical vortex magnetization configuration for an oblate hemispheroidal shell and a prolate hemispheroidal shell are depicted in Figures 6 and 7, respectively. While for the oblate hemispheroidal shell we can see clearly a core in the middle of the vortex, in the prolate hemispheroidal shell the core is spread all over the vortex. Using the angular parametrization for the normalized magnetization m=

M = (sin θ cos φ, sin θ sin φ, cos θ), M

(2)

one can describe the vortex solution as follows:

cos θ = pf (r),

φ=ǫ

π + χ. 2

(3)

Here, (r, χ, z) are the cylinder coordinates, p = ±1 is the vortex polarity which describes the vortex core magnetization (up or down) and ǫ = ±1 is the vortex chirality (clockwise or counterclockwise). The function f (r) describes the out of surface structure of the vortex. In order to find a good fit to the oblate hemispheroidal shell, it is

10

Figure 7: The magnetization distribution in vortex state of a prolate hemispheroidal shell with Ain = 15 nm, Cin = 20 nm and a thickness of 5 nm. The color code represents the normalized z component of the magnetization, shows that the core is not only in the center of the vortex.

instructive to use an analogy with a vortex profile of the planar disk. There are known several models for describing the vortex in a disk shaped particles. For example the ansatz by Usov and Peschany [17, 18] and the Gaussian ansatz [19] which describes the vortex structure in disks and rings: f (r) = e

−(

r 2 ) ξ ,

(4)

where the parameter ξ determines the radius of the vortex core. For the hemispherical shells Sheka et al. used a modified version of Usov and Peschany’s model [6]:  2 r − r2   c if r < rc 2 2 . f (r) = rc + r  0 if r ≥ rc

(5)

Figure 8(a) shows the profile of the normalized z component of the magnetization as a function of r, for Ain = 40 nm and a shell thicknesses of 5 nm. Gaussian ansatz (Eq. 4) can be a good approximation only for Cin /Ain ≤ 0.75. The deviation from the Gaussian ansatz increases with increasing the aspect ratio (Cin /Ain ). The magnetization distributions for 1 <

Cin Ain

< 3 do not fit at all to the Gaussian ansatz and do not 11

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Figure 8: The normalized z component of the magnetization as a function of r for hemispheroidal shells (Thickness = 5 nm). (a) Symbols correspond to the micromagnetic simulations for semi-principal axis Ain = 40 nm and various Cin . (b) Symbols indicate the micromagnetic simulations results for semi-principal axis Ain = 15 nm and three different Cin values. Solid lines are the best ansatzes which fit to the simulations’ results.

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Figure 9: The normalized z component of the magnetization as a function of r for hemispheroidal shells (Thickness = 10 nm). (a) Symbols correspond to the micromagnetic simulations for semi-principal axis Ain = 40 nm and various Cin . (b) Symbols indicate the micromagnetic simulations results for semiprincipal axis Ain = 15 nm and three different Cin values. Solid lines are the best ansatzes which fit to the simulations’ results.

13

correspond to Usov and Peschany’s ansatz given by Eq. 5. Therefore, to describe mz vs r in this range we propose the one parameter ansatz:

f (r) =

λ2 . λ2 + r 2

(6)

The profile of the normalized z component of the magnetization as a function of r, for Ain = 15 nm and shell thickness of 5 nm is plotted in Figure 8(b). Equation 6 can be a good approximation only for Cin /Ain ≤ 1.33. The magnetization distributions for Cin Ain

> 2 do not fit at all to the Gaussian ansatz or to Eq. 6 and do not correspond to

Usov and Peschany’s ansatz given by Eq. 5 as well. Therefore, to describe mz vs r for Cin Ain

> 2 we suggest the two parameters ansatz: f (r) = 1 − (r/Rc )α .

(7)

Although the fit is not good enough but it is the best simple ansatz. The profile of the normalized z component of the magnetization as a function of r, for Ain = 40 nm and a shell thickness of 10 nm is presented in Figure 9(a). The Gaussian ansatz is the best fit throughout the range (Cin /Ain ≤1.25). Figure 9(b) depicts the profile of the normalized z component of the magnetization as a function of r, for Ain = 15 nm and a shell thickness of 10 nm. The best fitting for Cin Ain

≥ 2.66 follows equation 7, whereas in the range 0.33 <

Cin Ain

< 0.66 follows eq. 6.

None of the above ansatzes was good enough to describe mz vs r for the middle size aspect ratios.

4. Conclusions We present a detailed study of the ground state of soft magnetic nano hemispheroidal shells. In addition to the two inhomogeneous magnetic ground states which exist in hemispherical magnetic shells we find another ground state, the single domain easy axis. Similar uniform magnetic ground state was found in micromagnetic simulations of extruded hemispherical shells (caps)[8]. But there is a big difference, in our research this additional magnetic structure is the ground state only for thin, small and elongated hemispheroidal shells while for the henispherical caps it is the ground state for thick 14

samples with small radius. The disparity between them stems from the thickness profile, while in our work the thickness is kept constant in the previous work there is a thickness gradient with a nominal thickness at the very center of the cap and a persistently decreasing thickness towards the equator. Like hemispherical shells [6], as the dimensions increase there is more preference for vortex state than the onion state. The vortex profile cannot be described by only one ansatz. We need to choose the right ansatz for each hemispheroidal shell. Start with Gaussian ansatz for the wide hemispheroidal shells with low aspect ratio (Cin /Ain ), through Eq. 6 and end with the ansatz which is given by Eq. 7 for the narrow hemispheroidal shells with large aspect ratio (Cin /Ain ). Magnetic nano-particles are the building blocks of many experimental uses, from magnetic manipulation to magnetic imaging. The above simulations provide a primary tool to design the proper nano-particles with specific properties.

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arXiv:0604228,

• • • •

Results of micromagnetic simulations of ferromagnetic nano hemispheroidal shells. There exist three different magnetic phases: easy axis, onion and vortex. The profile for the vortex magnetization distribution is analyzed. We investigate the total energy density for each of the magnetic distributions.