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Theoretical investigation on the magnetostatic interaction between two wire-tube nanostructures
Author’s Accepted Manuscript Theoretical investigation on the magnetostatic interaction between two wire-tube nanostructures A. Riveros, D. Salazar-Aravena, J. Escrig
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PII: DOI: Reference:
S0304-8853(16)31985-0 http://dx.doi.org/10.1016/j.jmmm.2016.12.127 MAGMA62335
To appear in: Journal of Magnetism and Magnetic Materials Received date: 30 August 2016 Revised date: 21 November 2016 Accepted date: 23 December 2016 Cite this article as: A. Riveros, D. Salazar-Aravena and J. Escrig, Theoretical investigation on the magnetostatic interaction between two wire-tube nanostructures, Journal of Magnetism and Magnetic Materials, http://dx.doi.org/10.1016/j.jmmm.2016.12.127 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.
Theoretical investigation on the magnetostatic interaction between two wire-tube nanostructures A. Riveros1 , D. Salazar-Aravena2,3 , and J. Escrig1,4
1
Departamento de F´ısica, Universidad de Santiago de Chile (USACH), Av. Ecuador 3493, 9170124 Santiago, Chile 2 Instituto de F´ısica Gleb Wataghin (IFGW), Universidade Estadual de Campinas (Unicamp), Campinas, SP 13083-859, Brazil 3 Departamento de F´ısica, Facultad de Ciencias, Universidad de Tarapac´ a, 1000007 Arica, Chile 4 Center for the Development of Nanoscience and Nanotechnology (CEDENNA), 9170124 Santiago, Chile In this paper we have calculated analytically the magnetostatic interaction between two wire-tube nanostructures as a function of their magnetic and geometric parameters. As expected, the interaction energy increases as the nanostructures approach, but interestingly when the nanostructures are close enough, a non-monotonic behavior with the wire-tube portions is reported. Besides, we investigate the hysteresis loop for two interacting Ni81 Fe19 wire-tube nanostructures by micromagnetic simulations in order to study how the interaction affects the magnetic properties of these nanostructures. This expression allows for the study of magnetostatic interactions between wire-tube nanostructures that have been proposed as an interesting alternative to store information or even perform logic functions, because to their ability to pin a domain wall.
I.
INTRODUCTION
In recent years, the nanometric systems have received considerable attention mainly due to the fact that their sizes are in the range of many physical phenomena. Of course magnetic nanostructures are no exception, and therefore their magnetic properties depend strongly on their size and shape. Thus, it is important to consider that the implementation of these nanostructures in potential applications will require a thorough understanding of how their magnetic properties change as a function of their magnetic and geometric parameters. Among the different forms that can acquire magnetic nanostructures, the pseudo-one-dimensional systems have been widely investigated because their shape anisotropy generates an axis of easy magnetization, producing square hysteresis curves when an external field is applied parallel to axis of the structure. Recently, Shahid et al. [1] have synthesized wire-tube nanostructures consisting of a pair of nanotube and nanowire segments and investigated systematic changes in the angular dependence of the coercivity behavior due to the change in the length of the nanowire segment. Leveraging the advantages of both the nanowires and nanotubes, these complex nanostructures that combine both structures may be useful for potential applications [1–5]. An interesting property of this nanostructure is that because the nanowires reverse their magnetization to a different field to which reverse the nanotubes, it is expected that the hysteresis curve of this complex nanostructure exhibits two clearly defined Barkhausen jumps (corresponding to the reversal of each of the segments), producing a step in the hysteresis curve, where one can expect two head-to-head domains with a vortex domain wall pinned in the center of the nanostructure [2, 3]. Interestingly, it has been shown that the step size changes as a function of angle in which the external magnetic field is applied [4]. Additionally, because these nanostructures may pin or release a domain wall as a function of their magnetic and geometric parameters,
these nanostructures have been proposed as an alternative to perform logic functions and even as information storage systems [5]. To increase information storage density in magnetic recording systems is necessary to decrease the size of the nanostructures that comprise the system, while decrease the distance between them. However, this approach between the nanostructures can produce changes on the magnetic properties of the system due to the effect of magnetostatic interactions between nanostructures [6– 8]. As mentioned above, the shape anisotropy of magnetic nanowires and nanotubes favors that the magnetization aligns with the axis of the structure. Furthermore, the magnetostatic interaction between two parallel nanostructures will favor an antiferromagnetic coupling, causing the reversal of one of the structures. In the case of a structure with complex geometry, such as a wire-tube nanostructure, it is expected that magnetostatic interactions also play a fundamental role changing their magnetic behavior. In a previous work [9] we obtained numerically that the hysteresis loop of a hexagonal cell with seven wire-tube nanostructures evolves from displaying a single plateau when the interaction vanishes, to several plateaus or even no plateaus as the interaction increases. However, in that study we studied a particular case, so that the influence of the interaction as a function of the geometric and magnetic parameters of the nanostructures is not easily identifiable, and then a theoretical study can shine light on this problem. The purpose of this work is to calculate analytically the magnetostatic interaction between two wire-tube nanostructures (see Fig. 1) exploring the possibility of varying the magnetic coupling as a function of their geometric and magnetic parameters. II.
ANALYTICAL MODEL AND RESULTS
The geometry of the wire-tube nanostructures is characterized by their external and internal radii, R and a,
2 After some manipulation, the integral expression for the scalar potential (defined in Eq. (1)) can be written as (details of the calculations can be found in Appendix A) M0 σ1 ∞ dk J0 (kr) [RJ1 (kR)− aJ1 (ka)] e−k|L−z| U1 (r, z) = 2 0 k
z a
d
L h
x
−RJ1 (kR)e−k|z| + aJ1 (ka)[e−k(h−z) Θ(−z + h) +e−k(z−h) Θ(z − h)] , (2)
y
FIG. 1. (Color online) Geometric parameters of wire-tube nanostructures.
respectively, and the total length, L, which includes the magnetic wire, h, and tube, L − h, segments. The separation between the nanostructures is written in terms of the interaxial distance, d. It is convenient to define the ratios γ = R/L, η = h/L, β = a/R and δ = 2R/d. We adopt a simplified description of the system, in which the discrete distribution of magnetic moments is replaced with a continuous one characterized by a slowly vary (r) [10]. The total magnetization ing magnetization M 2 (r), where M 1 (r) 1 (r) + M can be written as M (r) = M and M2 (r) are the magnetization of wire-tube nanostructures 1 and 2, respectively. In this case, the magnetostatic potential U (r) splits up into two components, U1 (r) and U2 (r), associated with the magnetization of each individual nanostructure. Then, the magnetostatic energy of two interacting magnetic wire-tube nanostructures may be written in terms of their magnetizations and the fields generated by each one. The general expression, after using the reciprocity theorem [10, 11] is Ed = (1) (2) (i) i (r)dVi i (r)· ∇U Eself +Eself +Eint . The Eself = (μ0 /2) M terms correspond to the self-energy of the ith structure, 1 (r)dV2 is the interaction en 2 (r) · ∇U and Eint = μ0 M ergy between two wire-tube nanostructures, which is the focus of this paper. To continue, we first need to calculate the magnetostatic potential U (r) of a single nanostructure. For this purpose, it is necessary to specify the functional form of the magnetization for each wire-tube nanostructure. We consider nanostructures with an axial magnetization i (r) = M0 σi zˆ, where M0 is the saturation defined by M magnetization of each nanostructure, zˆ is the unit vector parallel to the axis of the nanostructure, and σi takes the value ±1, allowing the magnetization of structure i to point up (σi = +1) or down (σi = −1) along zˆ. The magnetostatic potential produced by the structure 1 with 1 (r) is given by volume and surface conmagnetization M tributions and can be written as [10] 4πU1 (r) = −
dV V1
M 1 (r ) ∇ + |r − r |
dS S1
1 (r ) n ˆ · M . (1) |r − r |
here Θ(x) is the well-known heaviside step function and Jn (x) is the nth order Bessel function of first kind. The magnetostatic field produced by such structure is given (1) 1 = Hr(1) (r, z)ˆ 1 = −∇U r + Hz (r, z)ˆ z . Figure 2 by H shows this magnetostatic field in the space region (r, z) for a tube (a), a wire-tube nanostructure (b) and a wire (c). In this figure we also show the strength of U1 (r, z) by a density color plot. From this figure we show that the magnetostatic field produced by a wire-tube nanostructure is very different than that produced by a homogeneous nanowire or nanotube. a )
b)
c )
6 0 0
z
R
4 0 0 2 0 0 0
2 0 0 2 0 0
1 0 0
0
1 0 0
r
2 0 02 0 0
1 0 0
0
1 0 0
r
2 0 02 0 0
1 0 0
0
1 0 0
r
2 0 0
FIG. 2. (Color online) Magnetostatic field in the (r, z) region produced by (a) a tube, (b) a wire-tube nanostructure and (c) a wire with L = 500 nm, R = 50 nm, a = 40 nm, and with a magnetization pointing in +z.
Now it is possible to calculate the dipolar self-energy produced by a wire-tube nanostructure using the magnetostatic field experienced by one nanostructure due to itself. For notation we will use a tilde for the energies ˜ = E/μ0 M 2 V . The final rein units of μ0 M02 V , i.e.: E 0 sult reads as (details of the calculations can be found in Appendix B) ∞ dq 1 ˜self = E gγηβ (q). (3) 2 2 (1 − β + β η) 0 q 2 Here V = (1 − β 2 + β 2 η)πR2 L is the volume of a wiretube nanostructure and the function gγηβ (q) is defined by: gγηβ (q) = J1 (qγ) [J1 (qγ) + βJ1 (qβγ)] (1 − e−qη ) + J1 (qγ) [J1 (qγ) − βJ1 (qβγ)] (e−q(1−η) + e−qη − 2e−q ) + [J1 (qγ) − βJ1 (qβγ)] [J1 (qγ) − 2βJ1 (qβγ)] (1 − e
(4)
−q(1−η)
˜self as a function of η = h/L, In Fig. 3 we have plotted E for different β values. It can be seen that if the tube
).
3 segment predominates over the wire η 0.5, then as ˜self becomes larger, where the maximum β increases E occurs at β → 1. As the wire segment increases in length, ˜self decreases until a constant value (independent of β) E for wires (η → 1) as should be.
β = 0.4
β = 0.95
δ
R a h 1
η
η
FIG. 3. (Color online) Dipolar self-energy (in units of μ0 M02 V ) of a wire-tube nanostructure with R = 50 nm and L = 500 nm due to its own magnetostatic field as a function of the tube-wire aspect ratio η, for different β values.
The magnetostatic interaction energy between two identical wire-tube nanostructures using the magnetostatic field experienced by one of the structures due to the other is given by (details of the calculations can be found in Appendix C) ∞ dq σ1 σ 2 ˜ J0 (2γq/δ) gγηβ (q) (5) Eint = 2 2 (1 − β + β η) 0 q 2 By taking the limit η → 0 in Eq. (5), we obtain the interaction energy of two tubes separated by a distance d [12]. By the other hand, putting η → 1, the magnetostatic interaction energy of two nanowires [13, 14] is recovered. To study the behaviour of the magnetostatic interaction we consider wire-tube nanostructures with aligned magnetization (σ1 = σ2 = 1). Since the interaction energy is proportional to σ1 σ2 , the behaviour of this energy for wire-tube nanostructures with antiparallel magnetization is only changed by an overall −1 factor. In Fig. 4 using a density color plot we show the behavior of the interaction (Eq. (5)) in the (η, δ) region, for β = 0.4 and 0.95 (see Figs. 4a and 4b, respectively). We have used γ = 0.1. As can be seen when the tube segment is made by a small hole (Fig. 4a), the change ˜int as the structures approach each other is slow rate of E and quite similar for any tube-wire aspect ratio η. As ˜int the wire segment decreases until disappears (η = 0), E slightly decreases as compared to the corresponding behavior of two wires (η = 1). This monotonic behaviour as the nanostructures approach (with respect to η) completely changes as the hole size grows (see Fig. 4b) in ˜int is greater for wire-tube nanostructures with which E predominating tube segment, and can be disregarded for η = 0. For completeness, we show by a line plot the interaction energy as a function of η for different δ values for β = 0.8 (see Fig. 5a) and as a function of β for different η values, when both nanostructures are joined, δ = 1
η
FIG. 4. (Color online) Magnetostatic interaction energy of two wire-tube nanostructures (with aspect ratio γ = R/L = 0.1) in the (η = h/L, δ = 2R/d) region when (a) β = a/R = 0.4 and (b) β = a/R = 0.95.
(see Fig. 5b). As can be seen in Fig. 5a the interaction energy increases for η values between 0 < η < 0.3, and then remains approximately constant for other values of η. As expected, the interaction energy increases as the nanostructures approach. Furthermore, in Fig. 5b, it can be noted that the interaction energy remains almost constant for small β values (independent of the aspect ratio η). Moreover after this plateau regime, for tubes (η = 0), the interaction energy decreases as β increases. Instead, for wire-tube nanostructures, a change in the curvature depending of the magnitude of β can be observed (which is consistent with the behaviors reported in the previous figures), decreasing for intermediate β values and increasing for large β values. This effect is more notorious when the tube segment predominates over the wire (small η values). b)
a )
~ Eint
~ Eself
L
R a
η
β
˜int (for β = 0.8) as a function FIG. 5. (Color online) (a) E of the tube-wire aspect ratio η = h/L, for different values of ˜int for two joined wire-tube nanostructures δ = 2R/d. (b) E as a function of β for different η values. Here γ = R/L = 0.1.
III.
NUMERICAL SIMULATIONS AND DISCUSSION
In the previous section we have investigated analytically how the interaction energy between two nanostruc-
4 tures changes as a function of their geometric parameters. In order to show how the interaction affects the magnetic properties of these nanostructures, we have performed the hysteresis loop for two interacting Ni81 Fe19 wire-tube nanostructures using a micromagnetic simulator [15]. We use the material parameters M0 = 796 × 103 A m−1 and a stiffness constant A = 13 × 10−12 J m−1 . Since the structures are polycrystalline, we neglected the crystalline anisotropies. Besides, to mimic analytical results we used an outer radii of R = 50 nm, β = 0.8 and a total length of L = 500 nm, and chose cubic cells of 2 × 2 × 5 nm3 . Besides, we have considered the damping constant to be equal to 0.5. Figure 6 shows the hysteresis curves for two wiretube nanostructures for (a) δ → 0, (b) δ = 0.5 and (c) δ = 1 as a function of the parameter η when the magnetic field is applied parallel to the z-axis. From these hysteresis curves we observed that indeed the number of Barkhausen jumps depends on both the η parameter and the distance between the nanostructures. In particular, we note that for both isolated (Fig. 6a) and weakly interacting (Fig. 6b) nanostructures, the number of Barkhausen jumps exhibited by hysteresis curves follows a nonmonotonic behavior with η. That is, for η = 0.25 and 0.5 we obtain three Barkhausen jumps, but for η = 0.75 the hysteresis curve is simplified, exhibiting a single Barkhausen jump. However, the hysteresis curves for strongly interacting nanostructures (Fig. 6c) exhibit a completely different behavior. In this case we have that for η = 0.25 hysteresis curves exhibit two Barkhausen jumps, while for η > 0.25, hysteresis curves exhibit three Barkhausen jumps, which means a significant change from what was observed for weakly interacting systems. This is in perfect accordance with our analytical calculations, because as we saw in Figs. 5a and 5b, the interaction energy increases for η values between 0 < η < 0.3, and then remains approximately constant for other values of η. 1.0
c)
b)
a)
M/M0
0.5 0.0 η = 0.25 η = 0.5 η = 0.75
-0.5 -1.0 -200
-100
0
100
200 -200
Applied field (mT)
η = 0.25 η = 0.5 η = 0.75
η = 0.25 η = 0.5 η = 0.75
-100
0
100
200 -200
Applied field (mT)
-100
0
100
200
Applied field (mT)
FIG. 6. (Color online) Hysteresis curves for two wire-tube nanostructures for (a) δ → 0, (b) δ = 0.5 and (c) δ = 1 as a function of the parameter η when the magnetic field is applied parallel to the z-axis.
IV.
CONCLUSIONS
In conclusion, we have obtained an analytical expression for the full long range magnetostatic energy interaction for a pair of wire-tube nanostructures. It is well
˜int of two tubes (η = 0) is always below the known that E corresponding value of two wires (η = 1). Noteworthy for wire-tubes nanostructures a different behavior can be re˜int shows a non-monotonically behavior ported, in fact E with respect to β, decreasing for intermediate β values and increasing for large β values. Moreover depending of the size of the hole of the tube part, the interaction energy of wire-tube nanostructures can be greater than the corresponding to two wires (this effect is more notorious when the tube segment predominates over the wire part). We have also performed numerical simulations in order to investigate the effect of the interaction over the magnetic properties of these nanostructures. We have reported a completely different behavior of the hysteresis curves of highly interacting wire-tube nanostructures respect to weakly interacting nanostructures. The number of Barkhausen jumps exhibited in the hysteresis curves showed in Fig. 6 allow to conclude that the interaction energy is predominant for wire-tube nanostructures with aspect ratio 0 < η < 0.3 which is in perfect agreement with our analytical results. ACKNOWLEDGMENTS
In Chile, the authors acknowledge financial support from the Fondecyt Grant 1150952, DICYT Grant 041631EM-POSTDOC from VRIDEI-USACH, and the Financiamiento Basal para Centros Cient´ıficos y Tecnol´ ogicos de Excelencia, under project FB0807. In Brazil, the authors acknowledge financial support from the Brazilian funding agency CNPq, CENAPAD-SP for the use of SGI/Altix ICE and computer facilities, under project proj588 dsalazar. Appendix A: Magnetostatic potential of a wire-tube nanostructure
In order to calculate the magnetostatic potential (Eq. (1)) for a wire-tube nanostructure (e.g. for nanostructure 1) with uniform magnetization along its axis of symmetry, we first notice that the volumetric contribution is zero. In addition, the surface integral is non-zero only on three surfaces of the nanostructure (upper surface (z = L), lower surface (z = 0) and inner surface (z = h)). After expanding |r − r |−1 in cylindrical coordinates (r, φ, z), we obtain, M0 σ1 2π im(φ−φ ) ∞ U1 (r) = dφ e dkJm (kr) × 4π m 0 0 R R dr r Jm (kr )e−k|L−z| − dr r Jm (kr )e−k|z| + (A1) a
a 0
dr r Jm (kr ) e
0
−k(h−z)
Θ(−z + h) + e
−k(z−h)
Θ(z − h)
where the first, second and third term within the parenthesis corresponds to the surface contribution of the up-
,
5 per, lower and inner surface of the nanostructure, respectively. Also, because of the integral over φ , only the term for m = 0 is nonzero. After calculating the integrals on r , we obtain the integral expression for the magnetostatic potential of one of the wire-tube nanostructures investigated (see Eq. (2)).
Appendix B: Magnetostatic self-energy produced by a wire-tube nanostructure
Since the magnetization is uniform in the z-direction for any of the nanostructures considered, then the magnetostatic self-energy of these nanostructures (e.g. (1) for nanostructure 1) can be written as Eself = ∂U1 (r) (μ0 M0 σ1 /2) ∂z dV , where the volumetric integral is on the volume of the nanostructure, while U1 (r) was calculated in Appendix A. If we note that the volumetric integral can be written in the form dV = R h R L 2π dφ 0 drr 0 dz + a drr h dz , and after calcu0 lating the integrals on φ, r, z (by a straightforward calculation), we will have
Appendix C: Magnetostatic interaction between two wire-tube nanostructures
In this appendix we calculate the magnetostatic interaction between two wire-tube nanostructures separated by a distance d, each of which with a uniform magnetization along its axis of symmetry. In this way, we will have r1 ) 1 ( Eint = μ0 σ2 dV2 ∂U∂z , where U1 is the magnetostatic potential of the first nanostructure, while the volumetric integral is performed on the volume of the second nanostructure. After performing the partial derivative, the interaction energy can be written as ∞ μ0 M02 σ1 σ2 dV2 dkJ0 (kr1 ) { Eint = 2 0 [RJ1 (kR) − aJ1 (ka)]e−k(L−z1 ) + RJ1 (kR)e−kz1 +
[RJ1 (kR) − aJ1 (ka)][RJ1 (kR) − 2aJ1 (ka)](1 − e−k(L−h) )
aJ1 (ka)[e−k(h−z1 ) Θ(−z1 + h) − e−k(z1 −h) Θ(z1 − h)] aJ1 (ka) + δ(z1 − h)[e−k(z1 −h) − e−k(h−z1 ) ] } , (C1) k where the integral over dV2 should be performed by obtaining r1 = r1 (r2 ) and z1 = z1 (r2 ). Since both nanostructures areat same height, then z1 = z2 . Furthermore, r1 = r22 + d2 − 2r2 d cos(φ 2 ). After performing the cylindrical expansion of J0 (k r22 + d2 − 2r2d cos φ2 ) and rewriting the volumetric integral by dV2 = 2π h L R R dφ2 0 dr2 r2 0 dz2 + a dr2 r2 h dz2 , then the 0 integrals on φ2 , r2 , z2 can be calculated by a straightforward calculation, allowing to write Eint in the form ∞ dk J0 (kd) { Eint = πμ0 M02 σ1 σ2 k2 0
This expression is the same as that obtained in Eq. (3), once the change of variable q = kL in the integral is made, defining γ = R/L, η = h/L, β = a/R, and after (1) having divided Eself by μ0 M02 V .
Finally, using the same change of variable as that done in Appendix B, and dividing Eint by μ0 M02 V , we obtain the integral expression for the magnetostatic interaction (in units of μ0 M02 V ) between the two wire-tube nanostructures given in the Eq. (5).
(1) Eself
πμ0 M02 = 2
∞ 0
dk { k2
−k(L−h) ) + RJ1 (kR)[RJ1 (kR) − aJ1 (ka)](e−k(L−h) + e−kh − 2e−kL ) [RJ1 (kR) − aJ1 (ka)][RJ1 (kR) − 2aJ1 (ka)](1 − e
−k(L−h) −kh −kL +e − 2e ) + RJ1 (kR)[RJ1 (kR) − aJ1 (ka)](e + RJ1 (kR)[RJ1 (kR) + aJ1 (ka)](1 − e−kh ) .
−kh ) . (B1) + RJ1 (kR)[RJ1 (kR) + aJ1 (ka)](1 − e (C2)
[1] Muhammad Shahid Arshad, Darja Pecko, Saso Sturm, Juan Escrig, Matej Komelj, Paul J. McGuiness, Spomenka Kobe and K. Zuzek Rozman, IEEE Trans. Magn. 50, 2302904 (2014). [2] R. F. Neumann, M. Bahiana, N. M. Vargas, D. Altbir, S. Allende, D. Gorlitz and K. Nielsch, Appl. Phys. Lett. 102, 202407 (2013). [3] D. Salazar-Aravena, R. M. Corona, D. Goerlitz, K. Nielsch and J. Escrig, J. Magn. Magn. Mater. 346, 171174 (2013). [4] Diego Salazar-Aravena, Juan Luis Palma and Juan Escrig, Mater. Res. Express 1, 026112 (2014).
[5] A. P. Espejo, N. Vidal-Silva, J. A. Lopez-Lopez, D. Goerlitz, K. Nielsch and J. Escrig, Appl. Phys. Lett. 106, 132405 (2015). [6] M. G. Cottam, J. Phys. C: Solid State Phys. 4, 2658 (1971). [7] M. G. Cottam, J. Phys. C: Solid State Phys. 4, 2673 (1971). [8] M. G. Cottam, J. Phys. C: Solid State Phys. 5, 2205 (1972). [9] D. Salazar-Aravena, J. L. Palma and J. Escrig, J. Appl. Phys. 117, 193905 (2015).
6 [10] A. Aharoni, Introduction to the Theory of Ferromagnetism (Clarendon, Oxford, 1996). [11] L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Addison-Wesley: Reading, MA, 1960).
[12] J. Escrig, S. Allende, D. Altbir and M. Bahiana, Appl. Phys. Lett. 93, 023101 (2008). [13] M. Beleggia, S. Tandon, Y. Zhu, and M. De Graef, J. Magn. Magn. Mater. 278, 270 (2004). [14] D. Laroze, J. Escrig, P. Landeros, D. Altbir, M. Vazquez, and P. Vargas, Nanotechnology 18, 415708 (2007). [15] M. J. Donahue, D. G. Porter, OOMMF User’s Guide, Version 1.2a3 (2002) http://math.nist.gov/oommf
www.elsevier.com/locate/jmmm
PII: DOI: Reference:
S0304-8853(16)31985-0 http://dx.doi.org/10.1016/j.jmmm.2016.12.127 MAGMA62335
To appear in: Journal of Magnetism and Magnetic Materials Received date: 30 August 2016 Revised date: 21 November 2016 Accepted date: 23 December 2016 Cite this article as: A. Riveros, D. Salazar-Aravena and J. Escrig, Theoretical investigation on the magnetostatic interaction between two wire-tube nanostructures, Journal of Magnetism and Magnetic Materials, http://dx.doi.org/10.1016/j.jmmm.2016.12.127 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.
Theoretical investigation on the magnetostatic interaction between two wire-tube nanostructures A. Riveros1 , D. Salazar-Aravena2,3 , and J. Escrig1,4
1
Departamento de F´ısica, Universidad de Santiago de Chile (USACH), Av. Ecuador 3493, 9170124 Santiago, Chile 2 Instituto de F´ısica Gleb Wataghin (IFGW), Universidade Estadual de Campinas (Unicamp), Campinas, SP 13083-859, Brazil 3 Departamento de F´ısica, Facultad de Ciencias, Universidad de Tarapac´ a, 1000007 Arica, Chile 4 Center for the Development of Nanoscience and Nanotechnology (CEDENNA), 9170124 Santiago, Chile In this paper we have calculated analytically the magnetostatic interaction between two wire-tube nanostructures as a function of their magnetic and geometric parameters. As expected, the interaction energy increases as the nanostructures approach, but interestingly when the nanostructures are close enough, a non-monotonic behavior with the wire-tube portions is reported. Besides, we investigate the hysteresis loop for two interacting Ni81 Fe19 wire-tube nanostructures by micromagnetic simulations in order to study how the interaction affects the magnetic properties of these nanostructures. This expression allows for the study of magnetostatic interactions between wire-tube nanostructures that have been proposed as an interesting alternative to store information or even perform logic functions, because to their ability to pin a domain wall.
I.
INTRODUCTION
In recent years, the nanometric systems have received considerable attention mainly due to the fact that their sizes are in the range of many physical phenomena. Of course magnetic nanostructures are no exception, and therefore their magnetic properties depend strongly on their size and shape. Thus, it is important to consider that the implementation of these nanostructures in potential applications will require a thorough understanding of how their magnetic properties change as a function of their magnetic and geometric parameters. Among the different forms that can acquire magnetic nanostructures, the pseudo-one-dimensional systems have been widely investigated because their shape anisotropy generates an axis of easy magnetization, producing square hysteresis curves when an external field is applied parallel to axis of the structure. Recently, Shahid et al. [1] have synthesized wire-tube nanostructures consisting of a pair of nanotube and nanowire segments and investigated systematic changes in the angular dependence of the coercivity behavior due to the change in the length of the nanowire segment. Leveraging the advantages of both the nanowires and nanotubes, these complex nanostructures that combine both structures may be useful for potential applications [1–5]. An interesting property of this nanostructure is that because the nanowires reverse their magnetization to a different field to which reverse the nanotubes, it is expected that the hysteresis curve of this complex nanostructure exhibits two clearly defined Barkhausen jumps (corresponding to the reversal of each of the segments), producing a step in the hysteresis curve, where one can expect two head-to-head domains with a vortex domain wall pinned in the center of the nanostructure [2, 3]. Interestingly, it has been shown that the step size changes as a function of angle in which the external magnetic field is applied [4]. Additionally, because these nanostructures may pin or release a domain wall as a function of their magnetic and geometric parameters,
these nanostructures have been proposed as an alternative to perform logic functions and even as information storage systems [5]. To increase information storage density in magnetic recording systems is necessary to decrease the size of the nanostructures that comprise the system, while decrease the distance between them. However, this approach between the nanostructures can produce changes on the magnetic properties of the system due to the effect of magnetostatic interactions between nanostructures [6– 8]. As mentioned above, the shape anisotropy of magnetic nanowires and nanotubes favors that the magnetization aligns with the axis of the structure. Furthermore, the magnetostatic interaction between two parallel nanostructures will favor an antiferromagnetic coupling, causing the reversal of one of the structures. In the case of a structure with complex geometry, such as a wire-tube nanostructure, it is expected that magnetostatic interactions also play a fundamental role changing their magnetic behavior. In a previous work [9] we obtained numerically that the hysteresis loop of a hexagonal cell with seven wire-tube nanostructures evolves from displaying a single plateau when the interaction vanishes, to several plateaus or even no plateaus as the interaction increases. However, in that study we studied a particular case, so that the influence of the interaction as a function of the geometric and magnetic parameters of the nanostructures is not easily identifiable, and then a theoretical study can shine light on this problem. The purpose of this work is to calculate analytically the magnetostatic interaction between two wire-tube nanostructures (see Fig. 1) exploring the possibility of varying the magnetic coupling as a function of their geometric and magnetic parameters. II.
ANALYTICAL MODEL AND RESULTS
The geometry of the wire-tube nanostructures is characterized by their external and internal radii, R and a,
2 After some manipulation, the integral expression for the scalar potential (defined in Eq. (1)) can be written as (details of the calculations can be found in Appendix A) M0 σ1 ∞ dk J0 (kr) [RJ1 (kR)− aJ1 (ka)] e−k|L−z| U1 (r, z) = 2 0 k
z a
d
L h
x
−RJ1 (kR)e−k|z| + aJ1 (ka)[e−k(h−z) Θ(−z + h) +e−k(z−h) Θ(z − h)] , (2)
y
FIG. 1. (Color online) Geometric parameters of wire-tube nanostructures.
respectively, and the total length, L, which includes the magnetic wire, h, and tube, L − h, segments. The separation between the nanostructures is written in terms of the interaxial distance, d. It is convenient to define the ratios γ = R/L, η = h/L, β = a/R and δ = 2R/d. We adopt a simplified description of the system, in which the discrete distribution of magnetic moments is replaced with a continuous one characterized by a slowly vary (r) [10]. The total magnetization ing magnetization M 2 (r), where M 1 (r) 1 (r) + M can be written as M (r) = M and M2 (r) are the magnetization of wire-tube nanostructures 1 and 2, respectively. In this case, the magnetostatic potential U (r) splits up into two components, U1 (r) and U2 (r), associated with the magnetization of each individual nanostructure. Then, the magnetostatic energy of two interacting magnetic wire-tube nanostructures may be written in terms of their magnetizations and the fields generated by each one. The general expression, after using the reciprocity theorem [10, 11] is Ed = (1) (2) (i) i (r)dVi i (r)· ∇U Eself +Eself +Eint . The Eself = (μ0 /2) M terms correspond to the self-energy of the ith structure, 1 (r)dV2 is the interaction en 2 (r) · ∇U and Eint = μ0 M ergy between two wire-tube nanostructures, which is the focus of this paper. To continue, we first need to calculate the magnetostatic potential U (r) of a single nanostructure. For this purpose, it is necessary to specify the functional form of the magnetization for each wire-tube nanostructure. We consider nanostructures with an axial magnetization i (r) = M0 σi zˆ, where M0 is the saturation defined by M magnetization of each nanostructure, zˆ is the unit vector parallel to the axis of the nanostructure, and σi takes the value ±1, allowing the magnetization of structure i to point up (σi = +1) or down (σi = −1) along zˆ. The magnetostatic potential produced by the structure 1 with 1 (r) is given by volume and surface conmagnetization M tributions and can be written as [10] 4πU1 (r) = −
dV V1
M 1 (r ) ∇ + |r − r |
dS S1
1 (r ) n ˆ · M . (1) |r − r |
here Θ(x) is the well-known heaviside step function and Jn (x) is the nth order Bessel function of first kind. The magnetostatic field produced by such structure is given (1) 1 = Hr(1) (r, z)ˆ 1 = −∇U r + Hz (r, z)ˆ z . Figure 2 by H shows this magnetostatic field in the space region (r, z) for a tube (a), a wire-tube nanostructure (b) and a wire (c). In this figure we also show the strength of U1 (r, z) by a density color plot. From this figure we show that the magnetostatic field produced by a wire-tube nanostructure is very different than that produced by a homogeneous nanowire or nanotube. a )
b)
c )
6 0 0
z
R
4 0 0 2 0 0 0
2 0 0 2 0 0
1 0 0
0
1 0 0
r
2 0 02 0 0
1 0 0
0
1 0 0
r
2 0 02 0 0
1 0 0
0
1 0 0
r
2 0 0
FIG. 2. (Color online) Magnetostatic field in the (r, z) region produced by (a) a tube, (b) a wire-tube nanostructure and (c) a wire with L = 500 nm, R = 50 nm, a = 40 nm, and with a magnetization pointing in +z.
Now it is possible to calculate the dipolar self-energy produced by a wire-tube nanostructure using the magnetostatic field experienced by one nanostructure due to itself. For notation we will use a tilde for the energies ˜ = E/μ0 M 2 V . The final rein units of μ0 M02 V , i.e.: E 0 sult reads as (details of the calculations can be found in Appendix B) ∞ dq 1 ˜self = E gγηβ (q). (3) 2 2 (1 − β + β η) 0 q 2 Here V = (1 − β 2 + β 2 η)πR2 L is the volume of a wiretube nanostructure and the function gγηβ (q) is defined by: gγηβ (q) = J1 (qγ) [J1 (qγ) + βJ1 (qβγ)] (1 − e−qη ) + J1 (qγ) [J1 (qγ) − βJ1 (qβγ)] (e−q(1−η) + e−qη − 2e−q ) + [J1 (qγ) − βJ1 (qβγ)] [J1 (qγ) − 2βJ1 (qβγ)] (1 − e
(4)
−q(1−η)
˜self as a function of η = h/L, In Fig. 3 we have plotted E for different β values. It can be seen that if the tube
).
3 segment predominates over the wire η 0.5, then as ˜self becomes larger, where the maximum β increases E occurs at β → 1. As the wire segment increases in length, ˜self decreases until a constant value (independent of β) E for wires (η → 1) as should be.
β = 0.4
β = 0.95
δ
R a h 1
η
η
FIG. 3. (Color online) Dipolar self-energy (in units of μ0 M02 V ) of a wire-tube nanostructure with R = 50 nm and L = 500 nm due to its own magnetostatic field as a function of the tube-wire aspect ratio η, for different β values.
The magnetostatic interaction energy between two identical wire-tube nanostructures using the magnetostatic field experienced by one of the structures due to the other is given by (details of the calculations can be found in Appendix C) ∞ dq σ1 σ 2 ˜ J0 (2γq/δ) gγηβ (q) (5) Eint = 2 2 (1 − β + β η) 0 q 2 By taking the limit η → 0 in Eq. (5), we obtain the interaction energy of two tubes separated by a distance d [12]. By the other hand, putting η → 1, the magnetostatic interaction energy of two nanowires [13, 14] is recovered. To study the behaviour of the magnetostatic interaction we consider wire-tube nanostructures with aligned magnetization (σ1 = σ2 = 1). Since the interaction energy is proportional to σ1 σ2 , the behaviour of this energy for wire-tube nanostructures with antiparallel magnetization is only changed by an overall −1 factor. In Fig. 4 using a density color plot we show the behavior of the interaction (Eq. (5)) in the (η, δ) region, for β = 0.4 and 0.95 (see Figs. 4a and 4b, respectively). We have used γ = 0.1. As can be seen when the tube segment is made by a small hole (Fig. 4a), the change ˜int as the structures approach each other is slow rate of E and quite similar for any tube-wire aspect ratio η. As ˜int the wire segment decreases until disappears (η = 0), E slightly decreases as compared to the corresponding behavior of two wires (η = 1). This monotonic behaviour as the nanostructures approach (with respect to η) completely changes as the hole size grows (see Fig. 4b) in ˜int is greater for wire-tube nanostructures with which E predominating tube segment, and can be disregarded for η = 0. For completeness, we show by a line plot the interaction energy as a function of η for different δ values for β = 0.8 (see Fig. 5a) and as a function of β for different η values, when both nanostructures are joined, δ = 1
η
FIG. 4. (Color online) Magnetostatic interaction energy of two wire-tube nanostructures (with aspect ratio γ = R/L = 0.1) in the (η = h/L, δ = 2R/d) region when (a) β = a/R = 0.4 and (b) β = a/R = 0.95.
(see Fig. 5b). As can be seen in Fig. 5a the interaction energy increases for η values between 0 < η < 0.3, and then remains approximately constant for other values of η. As expected, the interaction energy increases as the nanostructures approach. Furthermore, in Fig. 5b, it can be noted that the interaction energy remains almost constant for small β values (independent of the aspect ratio η). Moreover after this plateau regime, for tubes (η = 0), the interaction energy decreases as β increases. Instead, for wire-tube nanostructures, a change in the curvature depending of the magnitude of β can be observed (which is consistent with the behaviors reported in the previous figures), decreasing for intermediate β values and increasing for large β values. This effect is more notorious when the tube segment predominates over the wire (small η values). b)
a )
~ Eint
~ Eself
L
R a
η
β
˜int (for β = 0.8) as a function FIG. 5. (Color online) (a) E of the tube-wire aspect ratio η = h/L, for different values of ˜int for two joined wire-tube nanostructures δ = 2R/d. (b) E as a function of β for different η values. Here γ = R/L = 0.1.
III.
NUMERICAL SIMULATIONS AND DISCUSSION
In the previous section we have investigated analytically how the interaction energy between two nanostruc-
4 tures changes as a function of their geometric parameters. In order to show how the interaction affects the magnetic properties of these nanostructures, we have performed the hysteresis loop for two interacting Ni81 Fe19 wire-tube nanostructures using a micromagnetic simulator [15]. We use the material parameters M0 = 796 × 103 A m−1 and a stiffness constant A = 13 × 10−12 J m−1 . Since the structures are polycrystalline, we neglected the crystalline anisotropies. Besides, to mimic analytical results we used an outer radii of R = 50 nm, β = 0.8 and a total length of L = 500 nm, and chose cubic cells of 2 × 2 × 5 nm3 . Besides, we have considered the damping constant to be equal to 0.5. Figure 6 shows the hysteresis curves for two wiretube nanostructures for (a) δ → 0, (b) δ = 0.5 and (c) δ = 1 as a function of the parameter η when the magnetic field is applied parallel to the z-axis. From these hysteresis curves we observed that indeed the number of Barkhausen jumps depends on both the η parameter and the distance between the nanostructures. In particular, we note that for both isolated (Fig. 6a) and weakly interacting (Fig. 6b) nanostructures, the number of Barkhausen jumps exhibited by hysteresis curves follows a nonmonotonic behavior with η. That is, for η = 0.25 and 0.5 we obtain three Barkhausen jumps, but for η = 0.75 the hysteresis curve is simplified, exhibiting a single Barkhausen jump. However, the hysteresis curves for strongly interacting nanostructures (Fig. 6c) exhibit a completely different behavior. In this case we have that for η = 0.25 hysteresis curves exhibit two Barkhausen jumps, while for η > 0.25, hysteresis curves exhibit three Barkhausen jumps, which means a significant change from what was observed for weakly interacting systems. This is in perfect accordance with our analytical calculations, because as we saw in Figs. 5a and 5b, the interaction energy increases for η values between 0 < η < 0.3, and then remains approximately constant for other values of η. 1.0
c)
b)
a)
M/M0
0.5 0.0 η = 0.25 η = 0.5 η = 0.75
-0.5 -1.0 -200
-100
0
100
200 -200
Applied field (mT)
η = 0.25 η = 0.5 η = 0.75
η = 0.25 η = 0.5 η = 0.75
-100
0
100
200 -200
Applied field (mT)
-100
0
100
200
Applied field (mT)
FIG. 6. (Color online) Hysteresis curves for two wire-tube nanostructures for (a) δ → 0, (b) δ = 0.5 and (c) δ = 1 as a function of the parameter η when the magnetic field is applied parallel to the z-axis.
IV.
CONCLUSIONS
In conclusion, we have obtained an analytical expression for the full long range magnetostatic energy interaction for a pair of wire-tube nanostructures. It is well
˜int of two tubes (η = 0) is always below the known that E corresponding value of two wires (η = 1). Noteworthy for wire-tubes nanostructures a different behavior can be re˜int shows a non-monotonically behavior ported, in fact E with respect to β, decreasing for intermediate β values and increasing for large β values. Moreover depending of the size of the hole of the tube part, the interaction energy of wire-tube nanostructures can be greater than the corresponding to two wires (this effect is more notorious when the tube segment predominates over the wire part). We have also performed numerical simulations in order to investigate the effect of the interaction over the magnetic properties of these nanostructures. We have reported a completely different behavior of the hysteresis curves of highly interacting wire-tube nanostructures respect to weakly interacting nanostructures. The number of Barkhausen jumps exhibited in the hysteresis curves showed in Fig. 6 allow to conclude that the interaction energy is predominant for wire-tube nanostructures with aspect ratio 0 < η < 0.3 which is in perfect agreement with our analytical results. ACKNOWLEDGMENTS
In Chile, the authors acknowledge financial support from the Fondecyt Grant 1150952, DICYT Grant 041631EM-POSTDOC from VRIDEI-USACH, and the Financiamiento Basal para Centros Cient´ıficos y Tecnol´ ogicos de Excelencia, under project FB0807. In Brazil, the authors acknowledge financial support from the Brazilian funding agency CNPq, CENAPAD-SP for the use of SGI/Altix ICE and computer facilities, under project proj588 dsalazar. Appendix A: Magnetostatic potential of a wire-tube nanostructure
In order to calculate the magnetostatic potential (Eq. (1)) for a wire-tube nanostructure (e.g. for nanostructure 1) with uniform magnetization along its axis of symmetry, we first notice that the volumetric contribution is zero. In addition, the surface integral is non-zero only on three surfaces of the nanostructure (upper surface (z = L), lower surface (z = 0) and inner surface (z = h)). After expanding |r − r |−1 in cylindrical coordinates (r, φ, z), we obtain, M0 σ1 2π im(φ−φ ) ∞ U1 (r) = dφ e dkJm (kr) × 4π m 0 0 R R dr r Jm (kr )e−k|L−z| − dr r Jm (kr )e−k|z| + (A1) a
a 0
dr r Jm (kr ) e
0
−k(h−z)
Θ(−z + h) + e
−k(z−h)
Θ(z − h)
where the first, second and third term within the parenthesis corresponds to the surface contribution of the up-
,
5 per, lower and inner surface of the nanostructure, respectively. Also, because of the integral over φ , only the term for m = 0 is nonzero. After calculating the integrals on r , we obtain the integral expression for the magnetostatic potential of one of the wire-tube nanostructures investigated (see Eq. (2)).
Appendix B: Magnetostatic self-energy produced by a wire-tube nanostructure
Since the magnetization is uniform in the z-direction for any of the nanostructures considered, then the magnetostatic self-energy of these nanostructures (e.g. (1) for nanostructure 1) can be written as Eself = ∂U1 (r) (μ0 M0 σ1 /2) ∂z dV , where the volumetric integral is on the volume of the nanostructure, while U1 (r) was calculated in Appendix A. If we note that the volumetric integral can be written in the form dV = R h R L 2π dφ 0 drr 0 dz + a drr h dz , and after calcu0 lating the integrals on φ, r, z (by a straightforward calculation), we will have
Appendix C: Magnetostatic interaction between two wire-tube nanostructures
In this appendix we calculate the magnetostatic interaction between two wire-tube nanostructures separated by a distance d, each of which with a uniform magnetization along its axis of symmetry. In this way, we will have r1 ) 1 ( Eint = μ0 σ2 dV2 ∂U∂z , where U1 is the magnetostatic potential of the first nanostructure, while the volumetric integral is performed on the volume of the second nanostructure. After performing the partial derivative, the interaction energy can be written as ∞ μ0 M02 σ1 σ2 dV2 dkJ0 (kr1 ) { Eint = 2 0 [RJ1 (kR) − aJ1 (ka)]e−k(L−z1 ) + RJ1 (kR)e−kz1 +
[RJ1 (kR) − aJ1 (ka)][RJ1 (kR) − 2aJ1 (ka)](1 − e−k(L−h) )
aJ1 (ka)[e−k(h−z1 ) Θ(−z1 + h) − e−k(z1 −h) Θ(z1 − h)] aJ1 (ka) + δ(z1 − h)[e−k(z1 −h) − e−k(h−z1 ) ] } , (C1) k where the integral over dV2 should be performed by obtaining r1 = r1 (r2 ) and z1 = z1 (r2 ). Since both nanostructures areat same height, then z1 = z2 . Furthermore, r1 = r22 + d2 − 2r2 d cos(φ 2 ). After performing the cylindrical expansion of J0 (k r22 + d2 − 2r2d cos φ2 ) and rewriting the volumetric integral by dV2 = 2π h L R R dφ2 0 dr2 r2 0 dz2 + a dr2 r2 h dz2 , then the 0 integrals on φ2 , r2 , z2 can be calculated by a straightforward calculation, allowing to write Eint in the form ∞ dk J0 (kd) { Eint = πμ0 M02 σ1 σ2 k2 0
This expression is the same as that obtained in Eq. (3), once the change of variable q = kL in the integral is made, defining γ = R/L, η = h/L, β = a/R, and after (1) having divided Eself by μ0 M02 V .
Finally, using the same change of variable as that done in Appendix B, and dividing Eint by μ0 M02 V , we obtain the integral expression for the magnetostatic interaction (in units of μ0 M02 V ) between the two wire-tube nanostructures given in the Eq. (5).
(1) Eself
πμ0 M02 = 2
∞ 0
dk { k2
−k(L−h) ) + RJ1 (kR)[RJ1 (kR) − aJ1 (ka)](e−k(L−h) + e−kh − 2e−kL ) [RJ1 (kR) − aJ1 (ka)][RJ1 (kR) − 2aJ1 (ka)](1 − e
−k(L−h) −kh −kL +e − 2e ) + RJ1 (kR)[RJ1 (kR) − aJ1 (ka)](e + RJ1 (kR)[RJ1 (kR) + aJ1 (ka)](1 − e−kh ) .
−kh ) . (B1) + RJ1 (kR)[RJ1 (kR) + aJ1 (ka)](1 − e (C2)
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