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Dipolar interaction in arrays of magnetic nanotubes
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 026001 (http://iopscience.iop.org/0953-8984/26/2/026001) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 134.99.128.41 This content was downloaded on 05/12/2013 at 01:27
Please note that terms and conditions apply.
Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 026001 (5pp)
doi:10.1088/0953-8984/26/2/026001
Dipolar interaction in arrays of magnetic nanotubes ´ ´ 1 , J M Mart´ınez-Huerta1 , J De La Torre Medina2 , Y Velazquez-Galv an 3 ´ , L Piraux3 and A Encinas1 Y Danlee 1
Instituto de F´ısica, Universidad Aut´onoma de San Luis Potos´ı, Avenida Manuel Nava 6, Zona Universitaria, 78290 San Luis Potos´ı, SLP, Mexico 2 Facultad de Ciencias F´ısico Matem´aticas, Universidad Michoacana de San Nicol´as de Hidalgo, Avenida Francisco J. M´ujica S/N, 58060, Morelia, Michoac´an, Mexico 3 Institute of Condensed Matter and Nanosciences, Universit´e Catholique de Louvain, Place Croix du Sud 1, B-1348, Louvain-la-Neuve, Belgium E-mail: [email protected] Received 5 July 2013, in final form 3 November 2013 Published 4 December 2013 Abstract
The dipolar interaction field in arrays of nickel nanotubes has been investigated on the basis of expressions derived from the effective demagnetizing field of the assembly as well as magnetometry measurements. The model incorporates explicitly the wall thickness and aspect ratio, as well as the spatial order of the nanotubes. The model and experiment show that the interaction field in nanotubes is smaller than that in solid nanowires due to the packing fraction reduction in tubes related to their inner cavity. Finally, good agreement between the model and experiment is found for the variation of the interaction field as a function of the tube wall thickness. (Some figures may appear in colour only in the online journal)
Arrays of nanotubes (NTs), either continuous or in multilayers, have recently attracted a lot of interest since they are promising for future applications and devices [1–3]. Compared to the case for homogeneous nanowires (NWs), the presence of the inner cavity and the possibility of changing the wall thickness provide an additional geometric parameter that can be used to further control or tailor their magnetic properties. Cylindrical nanotubes are expected to show a behavior close to that of NWs; however differences between them are known and they relate to the demagnetizing factor and their volume—in particular, the volume difference changes the effective packing fraction and thus the value of the dipolar interaction field [4, 5]. Magnetic properties such as the coercive field and its angular variation as well as reversal mechanisms and the magnetic configuration in single NTs have been extensively studied [6–8]; however little has been reported regarding the dipolar interaction field (DIF) in arrays of NTs [9, 10], which, as is known from numerous studies on assemblies of magnetic particles, including NWs, plays an important role in the magnetic properties of the array. 0953-8984/14/026001+05$33.00
In arrays of parallel, high aspect ratio, columnar particles, the DIF is antiferromagnetic and it tends to demagnetize, or reverse the magnetization of, the individual particles, as shown in both NTs and NWs [9, 11]. As the packing fraction is increased, the DIF results in a decrease of the effective shape anisotropy of individual particles, resulting in less stable magnetic configurations. However, as shown both theoretically for the case of two NTs [12] and experimentally for arrays [10], the DIF is weaker as compared to that for nanowires having the same outer diameter, which is especially interesting for densely packed magnetic memories. Furthermore, the possibility of controlling the NT wall width suggests that the DIF value can be varied and this could be useful for adjusting the sensitivity of magnetoresistive magnetic field sensors [13]. Theoretically, the dipolar interaction between two NTs has been considered in several reports, and fully analytical and approximated solutions have been determined for the DIF [12, 14, 15]; yet from these solutions it is not possible to derive the effective demagnetizing field and incorporate the effects of 1
c 2014 IOP Publishing Ltd Printed in the UK
J. Phys.: Condens. Matter 26 (2014) 026001
Y Velazquez-Galv an ´ ´ et al
Figure 1. SEM micrographs of the Ni nanotubes after dissolution of the P = 9.5% polycarbonate template. Scale bar: 200 nm.
Isothermal remanent magnetization (IRM) and DC demagnetizing (DCD) remanence curves have been obtained using an alternating gradient magnetometer at room temperature with the field applied parallel to the NT (NW) axis, as shown in figure 2. The IRM curve is measured after the application and removal of an increasingly positive field with the sample initially demagnetized, as indicated by the arrow in figure 2. The DCD curve is measured starting from the remanent state obtained after having saturated the sample with a large positive applied field and then by application of increasing negative demagnetizing fields, as shown by the arrow in figure 2. From these measurements, the 1M plots have been elaborated. The 1M plots are used in an analysis method based on the Wohlfarth relation that provides information about the interactions in an assembly of particles [18]. In particular, rewriting the Wohlfarth relation [19], md = 2mr − 1, 1M is defined as
the packing fraction and its dependence on the wall thickness for an array containing a large number of tubes. In this contribution a mean field model is presented, for describing the effective demagnetizing field, that includes the shape anisotropy and the DIF in arrays of magnetic NTs. From this model, expressions for the effective field in the case of 2D arrays are derived, and the effect of the NT wall thickness is analyzed. Finally the model is validated by the measurement of the dipolar interaction field from magnetometry measurements for both NTs and NWs, which are then used to determine the wall thickness, with a very good agreement found with scanning electron microscopy (SEM) measurements. Arrays of magnetic nanotubes and nanowires have been grown by electrodeposition into the pores of 21 µm thick lab-made track-etched polycarbonate membranes with a 150 nm pore diameter and porosities P of 5.2 and 9.5%, in which the pores are parallel to each other but randomly distributed [16]. Prior to electrodeposition, a Cr5 nm /Au300 nm layer was evaporated on one side of the membranes in order to cover the pores and enable its use as a cathode. Ni NWs were grown at a constant potential of −1.1 V from a 1 M NiSO4 ·6H2 O + 0.5 M H3 BO3 electrolyte, while Ni NTs were fabricated following the process reported by Wang et al [17] where Ni/Cu wires are grown at a constant reduction potential using a 0.4 M Ni(H2 NSO3 )2 · 4H2 O + 0.05 M CuSO4 · 5H2 O + 0.1 M H3 BO3 electrolyte, followed by the electrochemical etching of Cu at an oxidation potential of +0.2 V. Figure 1 shows a typical SEM image of the NTs obtained after the dissolution of the membrane. The average wall thickness of the nanotubes was determined from a statistical study carried out on all the nanotubes displayed in several SEM images for each sample. Three NT samples were prepared with the P = 9.5% membrane, two of them using V = −0.9 V, with wall thickness of t = 20 ± 1.5 nm, while the third one was grown at V = −1 V and had a larger average wall thickness of t = 25 ± 1.4 nm, in agreement with previous work by Wang et al [17]. One NT sample was grown at V = −1 V in the P = 5.2% membrane with t = 30 ± 2.1 nm.
1M = 2mr − md − 1
(1)
where it follows that for the assembly of non-interacting particles, 1M = 0 for all field values, and the deviations from zero show the presence of an interaction which can be of antiferromagnetic type (negative deviation) or of ferromagnetic type (positive deviation). Finally, the DIF coefficient αz was obtained from the IRM and DCD remanence curves using αz = 2(Hr0.5 − Hd0 )
(2)
where Hr0.5 is the field value at which the normalized IRM curve is equal to 0.5, while Hd0 is the field value at which the DCD curve is zero [20]. To describe magnetostatic effects in an infinite 2D planar array of nanotubes without magnetocrystalline or magnetoelastic anisotropy contributions, a mean field model is used which is based on the D = M N . In this approach effective demagnetizing field Heff s eff the particles, NTs or NWs, are assumed to be homogeneously magnetized along their easy axis, i.e. the cylinder long axis. As shown in [21], the effective demagnetizing field for an 2
J. Phys.: Condens. Matter 26 (2014) 026001
Y Velazquez-Galv an ´ ´ et al
Figure 2. Isothermal remanent magnetization (IRM) and DC
Figure 3. 1M plots measured in NT and NW arrays grown on the
demagnetizing (DCD) remanence curves measured for the P = 5.2% Ni NTs.
P = 5.2% template.
nanowires is obtained as a particular case. Indeed, the above expressions reduce to those for NWs when the inner radii vanish. In particular, when β = 0, we have Pt = Pw and equation (10) reduces to the known expression for the effective anisotropy of an array of infinitely long NWs [11]. The second term in equation (10) corresponds to the effective DIF which is proportional to the packing fraction, so from equation (4) it follows that if β > 0, then Pt < Pw . Thus, the dipolar interaction field is lower in NTs than in NWs, and this is only related to the volume (or packing fraction) difference between them. In figure 3 the 1M curves for NTs and NWs grown on the same template (P = 5.2%) are compared and, as can be seen, both show negative 1M values, consistent with an antiferromagnetic DIF. Moreover, |1MNTs | < |1MNWs |, which clearly shows that the DIF is lower in NTs than in NWs, in agreement with very recent results obtained by Proenca et al [10]. As a consequence, one can see from equation (10) that the total magnetostatic anisotropy field is higher for NTs than for NWs. In order to compare the model with experimental results, the DIF coefficient αz has been determined from the IRM and DCD remanence curves using equation (2) [20]. Moreover, in a recent study, the experimental value of αz has been related to the magnetization dependent part of the interaction field component along the easy axis [21]. In particular it was shown that αz corresponds to the dipolar term of the easy axis component in the saturated state divided by 2. From equation (8), αz = 4πMs Pt /2, and the axial component of the configuration dependent interaction field is H = αz m, where −1 ≤ m ≤ 1 is the normalized magnetization, so αz can be written as
assembly of particles in the saturated states is expressed as D = Ms N + (N + − N)Ms Pt , Heff
(3)
where Ms is the saturation magnetization, and N = {Nx , Ny , Nz } is the self-demagnetizing factor of a cylindrical tube where, by symmetry, Nx = Ny . The second term accounts for the DIF that is expressed in terms of N and the demagnetizing factor of the volume which contains the array—for this case, an infinite thin film, N + = {0, 0, 4π }. Pt is the packing fraction of the NTs, which is related to the packing fraction of the nanowires Pw by Pt = Pw (1 − β 2 )
(4)
where β = r1 /r2 is the ratio of the internal and external radii of the NT. From equation (3), the components of the effective z x , to , and perpendicular, Heff demagnetizing field parallel, Heff the NT axis and the effective or total anisotropy field, defined T = H Dx − H Dz , are as HA eff eff Dz = Ms Nz + (4π − Nz )Ms Pt Heff
(5)
Dx Heff = Ms Nx − Nx Ms Pt
(6)
T HA = Ms (Nx − Nz ) − (Nx − Nz + 4π )Ms Pt .
(7)
For NTs of arbitrary height L, the components of N can be calculated in terms of the demagnetizing factor of a circular cylindrical using Nz = Nzw (1 − β 2 ) [5]. The axial demagnetizing factor of a single cylinder, Nzw , can be computed in terms of the aspect ratio τ = L/2r2 [22]. However, the NTs considered in the present study have a length of ∼20 µm and an external diameter of ∼150 nm, so they can be considered as infinite; in this case, Nzwire = 0 and the demagnetizing factor of the tube is N = {2π, 2π, 0}, and equations (5)–(7) reduce to Dz Heff = 4π Ms Pt
(8)
Dx Heff T HA
= 2π Ms − 2π Ms Pt
(9)
= 2π Ms − 6π Ms Pt .
(10)
αz = 2π Ms Pw (1 − β 2 ).
(11)
This expression gives the dependence of the DIF on the packing fraction of the template Pw , Ms and β, so the variation of the interaction field coefficient with β can be calculated using known values of Ms (Ni) = 485 emu cm−3 and the template porosity. Figure 4 compares the calculated (continuous lines) and measured (circles) variations of αz as a function of β, for the two templates considered. The experimental values of αz are plotted using the values of β determined by SEM. Overall, a good agreement is observed between the measured and calculated values of αz , which
These expressions for arrays of NTs provide a description of the demagnetizing field and the effective magnetostatic anisotropy in the saturated state, where now the case of 3
J. Phys.: Condens. Matter 26 (2014) 026001
Y Velazquez-Galv an ´ ´ et al
Figure 4. Dipolar interaction field as a function of the ratio between
the inner and outer radii of the tubes β. Continuous lines correspond to the model, while dots correspond to the measured values. Table 1. Measured dipolar interaction coefficient αz and values of
the NT wall thickness obtained from the value of the interaction field as well as from SEM. P (%)
β
αz (Oe)
tα (nm)
tSEM (nm)
9.5 9.5 9.5 5.2
0.73 0.73 0.66 0.60
126 ± 10 146 ± 10 154 ± 10 104 ± 10
19 ± 3.5 21 ± 3.7 24 ± 3.8 31 ± 8.1
20 ± 1.5 20 ± 1.5 25 ± 1.4 30 ± 2.1
Figure 5. (a) αz as a function of the reduced reciprocal center to
center distance for a 2D hexagonal array of Ni NTs with a fixed aspect ratio of τ = 5 and for different ratios between the inner and outer radii β. (b) αz for Ni NTs as a function of their aspect ratio for different ratios between the inner and outer radii β; in all cases the template porosity or Pw is 10%.
in terms of Nx as
provides validation and support for the mean field approach. However, a small discrepancy is observed for the NWs made on the P = 9.5% template, which is attributed to the lower remanence of this sample (mr /ms = 0.6) which increases the uncertainty of the method used to determine αz [20]. Regarding the dependence of the interaction field on β, the results show that, as expected, as the tube wall thickness decreases, the magnitude of the interaction field decreases as a result of the reduction of the packing fraction. So even for infinitely tall nanotubes, the wall thickness provides a controllable parameter for tailoring the magnetic properties of the assembly. Finally, from equation (11) it is possible to express the tube wall thickness, t = r2 − r1 , as a function of αz and use the measured value of the interaction field to determine t; indeed, r αz t = r2 1 − 1 − . (12) 2π Ms Pw
αz = Nx Ms Pw (1 − β 2 ),
(13)
where now Nx = Nx (τ ) can be calculated using the expressions for circular cylinders given in [22], and Nz = Nzw (1 − β 2 ) [5]. The spatial arrangement of the NTs can be taken into account through a proper definition of the packing fraction. In this sense, figure 5(a) shows the DIF coefficient αz as a function of the reciprocal reduced distance de /D for a 2D hexagonal array of NTs, where de is the external diameter of the NTs and D the center to center distance. The aspect ratio is τ = 5 and different values of the ratio between the inner and outer radii, β, are considered. As expected, the interaction field tends to zero when the distance between tubes becomes large, and increases gradually when the distance is reduced. From the figure it is clear that as the NT wall thickness decreases, β increases and the interaction field becomes weaker, and this is simply related to the decrease in the packing fraction. In this sense, from the results it follows that for a given external diameter de the interaction field is lower in NTs than in NWs, in agreement with the results reported recently by Proenca et al, for Co NTs [10]. In figure 5(b) the variation of the DIF coefficient αz with the NT aspect ratio is shown for different values of the ratio between the inner and outer radii β, keeping the template packing fraction Pw at a constant value of 10%. As seen in the figure, the dipolar interaction field decreases as the NT aspect ratio is reduced and vanishes as it tends to zero, since in this limit Nx → 0. On the other hand, when the aspect ratio increases, the interaction field increases and tends asymptotically to the value of the very tall NTs or NWs, and, as seen in the figure, the limiting value decreases as the ratio of the internal and external radii β increases.
Table 1 shows the calculated and measured NT wall thicknesses as functions of the value of the interaction field αz . Here the uncertainty in the experimental values of αz is considered to be 10 Oe, which is the smallest field increment used in these measurements. From these results a very good agreement is observed between these values. If now the condition of the NT infinite length is removed, the model can be used to determine the variation of the DIF as a function of the NT aspect ratio, packing fraction and, as shown before, with β. For NTs of arbitrary aspect ratio τ , the corresponding expression follows from the second term in equation (5) divided by 2. Moreover, since the cross section of the NTs is circular, the demagnetizing factors are such that 2Nx + Nz = 4π and the DIF coefficient can be expressed just 4
J. Phys.: Condens. Matter 26 (2014) 026001
Y Velazquez-Galv an ´ ´ et al
Finally, it is important to underline that the experimental values of the DIF determined from the remanence curves correspond to equation (11) for infinitely tall NTs, and to equation (13) when the NTs have a finite height, which provides a direct relation between the model and experiment. In conclusion, the effective demagnetizing field has been derived for a 2D array of magnetic NTs, and has been validated by comparing the calculated and measured values of the configuration dependent dipolar interaction field for the particular case of very tall NTs. The model incorporates explicitly the wall thickness of the NTs and their aspect ratio, as well as their spatial order, and provides results on their effect on the magnetic properties of a single nanotube as well as the effects associated with the dipolar interaction in the assembly. In particular, the dipolar interaction field (magnetic anisotropy) decreases (increases) as the wall thickness is reduced, which shows that the inner cavity can serve as a parameter for controlling the magnetic properties of the NTs and could be of potential interest in applications where the dipolar interaction needs to be minimized.
[2] Nielsch K, Casta˜no F J, Matthias S, Lee W and Ross C A 2005 Adv. Eng. Mater. 7 217 [3] Pitzschel K, Bachmann J, Moreno J, Gorlitz J D and Nielsch K 2012 Nanotechnology 23 495718 [4] Beleggia M, Vokoun D and De Graef M 2009 J. Magn. Magn. Mater. 321 1306 [5] Nam B, Kim J and Hyeon J J 2012 J. Appl. Phys. 111 07E347 [6] Sharif R, Shamaila S, Ma M, Yao L D, Yu R C, Han X F and Khaleeq-ur-Rahman M 2008 Appl. Phys. Lett. 92 032505 [7] Landeros P, Allende S, Escrig J, Salcedo E, Altbir D and Vogel E E 2007 Appl. Phys. Lett. 90 102501 [8] Han X F, Shamaila S, Sharif R, Chen J Y, Liu H R and Liu C P 2009 Adv. Mater. 21 4619 [9] Zhou D, Wang T, Guao M G, Li W and Li F S 2011 J. Magn. 16 413 [10] Proenca M P, Sousa C T, Escrig J, Ventura J, Vazquez J and Araujo J P 2013 J. Appl. Phys. 113 093907 [11] Encinas-Oropesa A, Demand M, Piraux L, Huynen I and Ebels U 2001 Phys. Rev. B 63 104415 [12] Escrig J, Allende S, Altbir D and Bahia M 2008 Appl. Phys. Lett. 93 23101 [13] Pant B B 1996 J. Appl. Phys. 79 6123 [14] Suarez O J, Vargas P and Vogel E E 2009 J. Magn. Magn. Mater. 321 3658 [15] Morales-Concha C, Ossad´on M, Pereira A, Altbir D and Escrig J 2012 J. Appl. Phys. 111 07D131 [16] Ferain E and Legras R 1997 Nucl. Instrum. Methods Phys. Res. B 131 97 [17] Wang Q, Wang G, Han X, Wang X and Hou J G 2005 J. Phys. Chem. B 109 23326 [18] Kelly P E, O’Grady K, Mayo P I and Chantrell R W 1989 IEEE Trans. Magn. 25 3881 [19] Wohlfarth E P 1958 J. Appl. Phys. 29 595 [20] Martinez Huerta J M, De La Torre Medina J, Piraux L and Encinas A 2012 J. Appl. Phys. 111 083914 [21] Martinez Huerta J M, De La Torre Medina J, Piraux L and Encinas A 2013 J. Phys.: Condens. Matter 25 226003 [22] Tandon S, Beleggia M, Zhu Y and De Graef M 2004 J. Magn. Magn. Mater. 271 9
Acknowledgments
Financial support was provided by CONACYT M´exico projects CB-105568, CB-177896 and 2013 CIC-UMSNH, as well as scholarships 306252 (Y Vel´azquez-Galv´an) and 201754 (J M Mart´ınez-Huerta). Y Danl´ee acknowledges the Research Science Foundation of Belgium (FRS-FNRS) for financial support (an FRIA grant). The authors also thank V Antohe for his valuable help with the experimental work. References [1] Son S J, Reichel J, He B, Schuchman M and Lee S B 2005 J. Am. Chem. Soc. 127 7316
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Dipolar interaction in arrays of magnetic nanotubes
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 026001 (http://iopscience.iop.org/0953-8984/26/2/026001) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 134.99.128.41 This content was downloaded on 05/12/2013 at 01:27
Please note that terms and conditions apply.
Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 026001 (5pp)
doi:10.1088/0953-8984/26/2/026001
Dipolar interaction in arrays of magnetic nanotubes ´ ´ 1 , J M Mart´ınez-Huerta1 , J De La Torre Medina2 , Y Velazquez-Galv an 3 ´ , L Piraux3 and A Encinas1 Y Danlee 1
Instituto de F´ısica, Universidad Aut´onoma de San Luis Potos´ı, Avenida Manuel Nava 6, Zona Universitaria, 78290 San Luis Potos´ı, SLP, Mexico 2 Facultad de Ciencias F´ısico Matem´aticas, Universidad Michoacana de San Nicol´as de Hidalgo, Avenida Francisco J. M´ujica S/N, 58060, Morelia, Michoac´an, Mexico 3 Institute of Condensed Matter and Nanosciences, Universit´e Catholique de Louvain, Place Croix du Sud 1, B-1348, Louvain-la-Neuve, Belgium E-mail: [email protected] Received 5 July 2013, in final form 3 November 2013 Published 4 December 2013 Abstract
The dipolar interaction field in arrays of nickel nanotubes has been investigated on the basis of expressions derived from the effective demagnetizing field of the assembly as well as magnetometry measurements. The model incorporates explicitly the wall thickness and aspect ratio, as well as the spatial order of the nanotubes. The model and experiment show that the interaction field in nanotubes is smaller than that in solid nanowires due to the packing fraction reduction in tubes related to their inner cavity. Finally, good agreement between the model and experiment is found for the variation of the interaction field as a function of the tube wall thickness. (Some figures may appear in colour only in the online journal)
Arrays of nanotubes (NTs), either continuous or in multilayers, have recently attracted a lot of interest since they are promising for future applications and devices [1–3]. Compared to the case for homogeneous nanowires (NWs), the presence of the inner cavity and the possibility of changing the wall thickness provide an additional geometric parameter that can be used to further control or tailor their magnetic properties. Cylindrical nanotubes are expected to show a behavior close to that of NWs; however differences between them are known and they relate to the demagnetizing factor and their volume—in particular, the volume difference changes the effective packing fraction and thus the value of the dipolar interaction field [4, 5]. Magnetic properties such as the coercive field and its angular variation as well as reversal mechanisms and the magnetic configuration in single NTs have been extensively studied [6–8]; however little has been reported regarding the dipolar interaction field (DIF) in arrays of NTs [9, 10], which, as is known from numerous studies on assemblies of magnetic particles, including NWs, plays an important role in the magnetic properties of the array. 0953-8984/14/026001+05$33.00
In arrays of parallel, high aspect ratio, columnar particles, the DIF is antiferromagnetic and it tends to demagnetize, or reverse the magnetization of, the individual particles, as shown in both NTs and NWs [9, 11]. As the packing fraction is increased, the DIF results in a decrease of the effective shape anisotropy of individual particles, resulting in less stable magnetic configurations. However, as shown both theoretically for the case of two NTs [12] and experimentally for arrays [10], the DIF is weaker as compared to that for nanowires having the same outer diameter, which is especially interesting for densely packed magnetic memories. Furthermore, the possibility of controlling the NT wall width suggests that the DIF value can be varied and this could be useful for adjusting the sensitivity of magnetoresistive magnetic field sensors [13]. Theoretically, the dipolar interaction between two NTs has been considered in several reports, and fully analytical and approximated solutions have been determined for the DIF [12, 14, 15]; yet from these solutions it is not possible to derive the effective demagnetizing field and incorporate the effects of 1
c 2014 IOP Publishing Ltd Printed in the UK
J. Phys.: Condens. Matter 26 (2014) 026001
Y Velazquez-Galv an ´ ´ et al
Figure 1. SEM micrographs of the Ni nanotubes after dissolution of the P = 9.5% polycarbonate template. Scale bar: 200 nm.
Isothermal remanent magnetization (IRM) and DC demagnetizing (DCD) remanence curves have been obtained using an alternating gradient magnetometer at room temperature with the field applied parallel to the NT (NW) axis, as shown in figure 2. The IRM curve is measured after the application and removal of an increasingly positive field with the sample initially demagnetized, as indicated by the arrow in figure 2. The DCD curve is measured starting from the remanent state obtained after having saturated the sample with a large positive applied field and then by application of increasing negative demagnetizing fields, as shown by the arrow in figure 2. From these measurements, the 1M plots have been elaborated. The 1M plots are used in an analysis method based on the Wohlfarth relation that provides information about the interactions in an assembly of particles [18]. In particular, rewriting the Wohlfarth relation [19], md = 2mr − 1, 1M is defined as
the packing fraction and its dependence on the wall thickness for an array containing a large number of tubes. In this contribution a mean field model is presented, for describing the effective demagnetizing field, that includes the shape anisotropy and the DIF in arrays of magnetic NTs. From this model, expressions for the effective field in the case of 2D arrays are derived, and the effect of the NT wall thickness is analyzed. Finally the model is validated by the measurement of the dipolar interaction field from magnetometry measurements for both NTs and NWs, which are then used to determine the wall thickness, with a very good agreement found with scanning electron microscopy (SEM) measurements. Arrays of magnetic nanotubes and nanowires have been grown by electrodeposition into the pores of 21 µm thick lab-made track-etched polycarbonate membranes with a 150 nm pore diameter and porosities P of 5.2 and 9.5%, in which the pores are parallel to each other but randomly distributed [16]. Prior to electrodeposition, a Cr5 nm /Au300 nm layer was evaporated on one side of the membranes in order to cover the pores and enable its use as a cathode. Ni NWs were grown at a constant potential of −1.1 V from a 1 M NiSO4 ·6H2 O + 0.5 M H3 BO3 electrolyte, while Ni NTs were fabricated following the process reported by Wang et al [17] where Ni/Cu wires are grown at a constant reduction potential using a 0.4 M Ni(H2 NSO3 )2 · 4H2 O + 0.05 M CuSO4 · 5H2 O + 0.1 M H3 BO3 electrolyte, followed by the electrochemical etching of Cu at an oxidation potential of +0.2 V. Figure 1 shows a typical SEM image of the NTs obtained after the dissolution of the membrane. The average wall thickness of the nanotubes was determined from a statistical study carried out on all the nanotubes displayed in several SEM images for each sample. Three NT samples were prepared with the P = 9.5% membrane, two of them using V = −0.9 V, with wall thickness of t = 20 ± 1.5 nm, while the third one was grown at V = −1 V and had a larger average wall thickness of t = 25 ± 1.4 nm, in agreement with previous work by Wang et al [17]. One NT sample was grown at V = −1 V in the P = 5.2% membrane with t = 30 ± 2.1 nm.
1M = 2mr − md − 1
(1)
where it follows that for the assembly of non-interacting particles, 1M = 0 for all field values, and the deviations from zero show the presence of an interaction which can be of antiferromagnetic type (negative deviation) or of ferromagnetic type (positive deviation). Finally, the DIF coefficient αz was obtained from the IRM and DCD remanence curves using αz = 2(Hr0.5 − Hd0 )
(2)
where Hr0.5 is the field value at which the normalized IRM curve is equal to 0.5, while Hd0 is the field value at which the DCD curve is zero [20]. To describe magnetostatic effects in an infinite 2D planar array of nanotubes without magnetocrystalline or magnetoelastic anisotropy contributions, a mean field model is used which is based on the D = M N . In this approach effective demagnetizing field Heff s eff the particles, NTs or NWs, are assumed to be homogeneously magnetized along their easy axis, i.e. the cylinder long axis. As shown in [21], the effective demagnetizing field for an 2
J. Phys.: Condens. Matter 26 (2014) 026001
Y Velazquez-Galv an ´ ´ et al
Figure 2. Isothermal remanent magnetization (IRM) and DC
Figure 3. 1M plots measured in NT and NW arrays grown on the
demagnetizing (DCD) remanence curves measured for the P = 5.2% Ni NTs.
P = 5.2% template.
nanowires is obtained as a particular case. Indeed, the above expressions reduce to those for NWs when the inner radii vanish. In particular, when β = 0, we have Pt = Pw and equation (10) reduces to the known expression for the effective anisotropy of an array of infinitely long NWs [11]. The second term in equation (10) corresponds to the effective DIF which is proportional to the packing fraction, so from equation (4) it follows that if β > 0, then Pt < Pw . Thus, the dipolar interaction field is lower in NTs than in NWs, and this is only related to the volume (or packing fraction) difference between them. In figure 3 the 1M curves for NTs and NWs grown on the same template (P = 5.2%) are compared and, as can be seen, both show negative 1M values, consistent with an antiferromagnetic DIF. Moreover, |1MNTs | < |1MNWs |, which clearly shows that the DIF is lower in NTs than in NWs, in agreement with very recent results obtained by Proenca et al [10]. As a consequence, one can see from equation (10) that the total magnetostatic anisotropy field is higher for NTs than for NWs. In order to compare the model with experimental results, the DIF coefficient αz has been determined from the IRM and DCD remanence curves using equation (2) [20]. Moreover, in a recent study, the experimental value of αz has been related to the magnetization dependent part of the interaction field component along the easy axis [21]. In particular it was shown that αz corresponds to the dipolar term of the easy axis component in the saturated state divided by 2. From equation (8), αz = 4πMs Pt /2, and the axial component of the configuration dependent interaction field is H = αz m, where −1 ≤ m ≤ 1 is the normalized magnetization, so αz can be written as
assembly of particles in the saturated states is expressed as D = Ms N + (N + − N)Ms Pt , Heff
(3)
where Ms is the saturation magnetization, and N = {Nx , Ny , Nz } is the self-demagnetizing factor of a cylindrical tube where, by symmetry, Nx = Ny . The second term accounts for the DIF that is expressed in terms of N and the demagnetizing factor of the volume which contains the array—for this case, an infinite thin film, N + = {0, 0, 4π }. Pt is the packing fraction of the NTs, which is related to the packing fraction of the nanowires Pw by Pt = Pw (1 − β 2 )
(4)
where β = r1 /r2 is the ratio of the internal and external radii of the NT. From equation (3), the components of the effective z x , to , and perpendicular, Heff demagnetizing field parallel, Heff the NT axis and the effective or total anisotropy field, defined T = H Dx − H Dz , are as HA eff eff Dz = Ms Nz + (4π − Nz )Ms Pt Heff
(5)
Dx Heff = Ms Nx − Nx Ms Pt
(6)
T HA = Ms (Nx − Nz ) − (Nx − Nz + 4π )Ms Pt .
(7)
For NTs of arbitrary height L, the components of N can be calculated in terms of the demagnetizing factor of a circular cylindrical using Nz = Nzw (1 − β 2 ) [5]. The axial demagnetizing factor of a single cylinder, Nzw , can be computed in terms of the aspect ratio τ = L/2r2 [22]. However, the NTs considered in the present study have a length of ∼20 µm and an external diameter of ∼150 nm, so they can be considered as infinite; in this case, Nzwire = 0 and the demagnetizing factor of the tube is N = {2π, 2π, 0}, and equations (5)–(7) reduce to Dz Heff = 4π Ms Pt
(8)
Dx Heff T HA
= 2π Ms − 2π Ms Pt
(9)
= 2π Ms − 6π Ms Pt .
(10)
αz = 2π Ms Pw (1 − β 2 ).
(11)
This expression gives the dependence of the DIF on the packing fraction of the template Pw , Ms and β, so the variation of the interaction field coefficient with β can be calculated using known values of Ms (Ni) = 485 emu cm−3 and the template porosity. Figure 4 compares the calculated (continuous lines) and measured (circles) variations of αz as a function of β, for the two templates considered. The experimental values of αz are plotted using the values of β determined by SEM. Overall, a good agreement is observed between the measured and calculated values of αz , which
These expressions for arrays of NTs provide a description of the demagnetizing field and the effective magnetostatic anisotropy in the saturated state, where now the case of 3
J. Phys.: Condens. Matter 26 (2014) 026001
Y Velazquez-Galv an ´ ´ et al
Figure 4. Dipolar interaction field as a function of the ratio between
the inner and outer radii of the tubes β. Continuous lines correspond to the model, while dots correspond to the measured values. Table 1. Measured dipolar interaction coefficient αz and values of
the NT wall thickness obtained from the value of the interaction field as well as from SEM. P (%)
β
αz (Oe)
tα (nm)
tSEM (nm)
9.5 9.5 9.5 5.2
0.73 0.73 0.66 0.60
126 ± 10 146 ± 10 154 ± 10 104 ± 10
19 ± 3.5 21 ± 3.7 24 ± 3.8 31 ± 8.1
20 ± 1.5 20 ± 1.5 25 ± 1.4 30 ± 2.1
Figure 5. (a) αz as a function of the reduced reciprocal center to
center distance for a 2D hexagonal array of Ni NTs with a fixed aspect ratio of τ = 5 and for different ratios between the inner and outer radii β. (b) αz for Ni NTs as a function of their aspect ratio for different ratios between the inner and outer radii β; in all cases the template porosity or Pw is 10%.
in terms of Nx as
provides validation and support for the mean field approach. However, a small discrepancy is observed for the NWs made on the P = 9.5% template, which is attributed to the lower remanence of this sample (mr /ms = 0.6) which increases the uncertainty of the method used to determine αz [20]. Regarding the dependence of the interaction field on β, the results show that, as expected, as the tube wall thickness decreases, the magnitude of the interaction field decreases as a result of the reduction of the packing fraction. So even for infinitely tall nanotubes, the wall thickness provides a controllable parameter for tailoring the magnetic properties of the assembly. Finally, from equation (11) it is possible to express the tube wall thickness, t = r2 − r1 , as a function of αz and use the measured value of the interaction field to determine t; indeed, r αz t = r2 1 − 1 − . (12) 2π Ms Pw
αz = Nx Ms Pw (1 − β 2 ),
(13)
where now Nx = Nx (τ ) can be calculated using the expressions for circular cylinders given in [22], and Nz = Nzw (1 − β 2 ) [5]. The spatial arrangement of the NTs can be taken into account through a proper definition of the packing fraction. In this sense, figure 5(a) shows the DIF coefficient αz as a function of the reciprocal reduced distance de /D for a 2D hexagonal array of NTs, where de is the external diameter of the NTs and D the center to center distance. The aspect ratio is τ = 5 and different values of the ratio between the inner and outer radii, β, are considered. As expected, the interaction field tends to zero when the distance between tubes becomes large, and increases gradually when the distance is reduced. From the figure it is clear that as the NT wall thickness decreases, β increases and the interaction field becomes weaker, and this is simply related to the decrease in the packing fraction. In this sense, from the results it follows that for a given external diameter de the interaction field is lower in NTs than in NWs, in agreement with the results reported recently by Proenca et al, for Co NTs [10]. In figure 5(b) the variation of the DIF coefficient αz with the NT aspect ratio is shown for different values of the ratio between the inner and outer radii β, keeping the template packing fraction Pw at a constant value of 10%. As seen in the figure, the dipolar interaction field decreases as the NT aspect ratio is reduced and vanishes as it tends to zero, since in this limit Nx → 0. On the other hand, when the aspect ratio increases, the interaction field increases and tends asymptotically to the value of the very tall NTs or NWs, and, as seen in the figure, the limiting value decreases as the ratio of the internal and external radii β increases.
Table 1 shows the calculated and measured NT wall thicknesses as functions of the value of the interaction field αz . Here the uncertainty in the experimental values of αz is considered to be 10 Oe, which is the smallest field increment used in these measurements. From these results a very good agreement is observed between these values. If now the condition of the NT infinite length is removed, the model can be used to determine the variation of the DIF as a function of the NT aspect ratio, packing fraction and, as shown before, with β. For NTs of arbitrary aspect ratio τ , the corresponding expression follows from the second term in equation (5) divided by 2. Moreover, since the cross section of the NTs is circular, the demagnetizing factors are such that 2Nx + Nz = 4π and the DIF coefficient can be expressed just 4
J. Phys.: Condens. Matter 26 (2014) 026001
Y Velazquez-Galv an ´ ´ et al
Finally, it is important to underline that the experimental values of the DIF determined from the remanence curves correspond to equation (11) for infinitely tall NTs, and to equation (13) when the NTs have a finite height, which provides a direct relation between the model and experiment. In conclusion, the effective demagnetizing field has been derived for a 2D array of magnetic NTs, and has been validated by comparing the calculated and measured values of the configuration dependent dipolar interaction field for the particular case of very tall NTs. The model incorporates explicitly the wall thickness of the NTs and their aspect ratio, as well as their spatial order, and provides results on their effect on the magnetic properties of a single nanotube as well as the effects associated with the dipolar interaction in the assembly. In particular, the dipolar interaction field (magnetic anisotropy) decreases (increases) as the wall thickness is reduced, which shows that the inner cavity can serve as a parameter for controlling the magnetic properties of the NTs and could be of potential interest in applications where the dipolar interaction needs to be minimized.
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Acknowledgments
Financial support was provided by CONACYT M´exico projects CB-105568, CB-177896 and 2013 CIC-UMSNH, as well as scholarships 306252 (Y Vel´azquez-Galv´an) and 201754 (J M Mart´ınez-Huerta). Y Danl´ee acknowledges the Research Science Foundation of Belgium (FRS-FNRS) for financial support (an FRIA grant). The authors also thank V Antohe for his valuable help with the experimental work. References [1] Son S J, Reichel J, He B, Schuchman M and Lee S B 2005 J. Am. Chem. Soc. 127 7316
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