Using the equivalent 2D model for finite element method (FEM) we calculated apparent permeability µa and demagnetization factor D for permalloy nanowire and microwire arrays. The simulation results were verified by 3D FEM for arrays up to 3000 wires and experimentally for very large arrays containing up to 40 million wires. We achieved µa = 3 to 33 and coercivities Hc = 1 to 9 kA/m, which are low values for wire arrays. The µa depends mainly on the array geometry; it can be increased by increasing the distance between wires (pitch) and the wire length-to-diameter ratio L/d.

Arrays of magnetically soft nanowires have a wide potential for applications.1,2 One of the first applications was perpendicular magnetic recording.3 FeNi nanowire arrays developed for the recording exhibited coercivity of 16 kA/m,4 the coercivity for CoNiP was 64 kA/m.5 Possible microwave applications include nanoantennas, tunable filters, and circulators, for GHz to THz frequency range.6,7 Individually functionalized nanowires were used for biomedical applications.8 Magnetic nanowires have been fabricated in several laboratories, but to the best of our knowledge, they have never been used as functional materials for magnetic field sensors, as most of the fabricated wires have high coercivity. For sensor applications, low coercivity is required, which is still a challenge for development. An overview of electrodeposition techniques for the production of FeNi and FeCo nanowires is given in Ref. 9. Systematic study of the magnetization process in nanowires can be found in Ref. 10. The influence of magnetostatic interactions between Ni wires on coercivity has been studied in Ref. 11, but we are not aware of a similar study on effective permeability.

We have reported a nanowire array manufactured by electrodeposition of permalloy into polycarbonate membranes.12 The problem with this type of membrane was that while the pore diameter is well defined, the pores are not ordered and their distance is random. In this paper, we report permalloy nanowire arrays produced using alumina and silicon membranes with hexagonally ordered pores with a constant pitch.

Finite element method (FEM) is being used to solve Maxwell’s equations when analytic solution is not available. In Ref. 13 we used 3D FEM to calculate the apparent permeability and demagnetization factor of single wire and small wire arrays. We have also proposed a simplified 2D equivalent model allowing us to calculate properties of wire arrays very efficiently14 and we have verified the validity of this model by full 3D simulations and also experimentally on arrays of up to 91 permalloy microwires. In this paper, we applied this model to nanowire and microwire arrays containing 105 to 4 · 107 million wires with diameters between 200 nm and 2.5 μm and compared the simulation results to measurements by Superconducting Quantum interference Device (SQUID) magnetometry.

3D FEM calculation of a large number of wires is quite difficult as the number of required mesh elements and simulation time increase very fast. Even if we used symmetry and calculated only part of the array, the task has been very demanding. Our most complex 3D model simulated in this paper represented an array of 2521 wires. Using 8-fold rotational symmetry of the array, we reduced the number of wires to 315, but to achieve reasonable accuracy the required number of elements was 1 661 276 and computational time on personal computer (i7, 3.4 GHz, 8 cores with 32.0 GB RAM) was 43 minutes. Therefore, a 2D equivalent model was proposed to replace the 3D simulation. The model is replacing wires with a system of hollow cylinders, which are axisymmetric. The details of the model are described in Ref. 14.

The calculated values of apparent permeability µa using 3D FEM and equivalent axisymmetric 2D FEM models versus array size (number of wires) are shown in Fig. 1 for relative magnetic permeability µr = 500. The wire diameter was d = 200 nm, length L = 36 µm, which gives aspect ratio L/d = 180. The lattice of the array was either hexagonal or square. Reduction of wire pitch decreases µa and increases demagnetization factor D. The largest difference between the equivalent 2D and 3D models is below 6% for large pitch (Fig. 1a) and it is below 1% for the small pitch of 0.25 µm (Fig. 1b).

FIG. 1.

Apparent permeability µa versus the number of wires and error of the 2D model (with respect to the 3D model) for a pitch (a) dw = 3.2 µm and (b) dw = 0.25 µm. L/d = 180. Both increasing the number of wires and reduction of the pitch decreases µa.

FIG. 1.

Apparent permeability µa versus the number of wires and error of the 2D model (with respect to the 3D model) for a pitch (a) dw = 3.2 µm and (b) dw = 0.25 µm. L/d = 180. Both increasing the number of wires and reduction of the pitch decreases µa.

Close modal

Fig. 2 presents µa versus pitch. The difference of the results for µr = 5000 and µr = 105 is small, which shows that the dominant factor is the wire geometry.

FIG. 2.

Apparent permeability µa versus pitch (wire distance dw) for different µr with hexagonal lattice – The number of wires = 2791. Note that for large µr the apparent permeability depends only on geometry.

FIG. 2.

Apparent permeability µa versus pitch (wire distance dw) for different µr with hexagonal lattice – The number of wires = 2791. Note that for large µr the apparent permeability depends only on geometry.

Close modal

In this section, we calculate µa and D for wire arrays using the 2D equivalent model. The simulations are made for d = 200 and 400 nm, L = 50 µm, and for several values of µr and pitch dw. The dependence of µa and D on the array diameter is exemplified for hexagonal and square lattice in Fig. 3. The pitch is 1.6 µm, the maximum number of nanowires for the 6 mm array is 10 552 501 and 7 035 001 wires for hexagonal and square lattice, respectively. The µa and D are almost constant for membrane diameters larger than 4 mm.

FIG. 3.

Apparent permeability µa (a) and demagnetization factor N (b) versus membrane diameter (outer diameter of wire array). Pitch dw = 1.6 µm and µr = 5000. Notice that for very large arrays both the µa and N are close to the limit values and change only slightly.

FIG. 3.

Apparent permeability µa (a) and demagnetization factor N (b) versus membrane diameter (outer diameter of wire array). Pitch dw = 1.6 µm and µr = 5000. Notice that for very large arrays both the µa and N are close to the limit values and change only slightly.

Close modal

The minimum value of apparent permeability is 17.5 for a 6 mm hexagonal array of 400 nm diameter wires with µr = 500, and it is 20.2 for the square lattice. These permeability values are only slightly higher for µr = 5000.

Reducing a wire diameter from 400 nm to 200 nm causes the minimum value of µa to increase to 63.3 and 72 for hexagonal and square lattice respectively.

The last set of simulations was made for parameters of the produced wire arrays, and the results are given in Sec. III.

This simulation was made to examine the effect of the nonlinear magnetization characteristics. The material model was described in Ref. 15. The array parameters are d = 180 nm, L = 100 µm, pitch = 480 nm, array diameter 3.5 mm. Calculated µa vs. H is shown in Table I.

TABLE I.

Apparent permeability µa as a function of magnetic field H (calculated from the nonlinear model).

H (kA/m) 10 20 50 100 200 
µa 8.77 8.75 8.71 8.4 
H (kA/m) 10 20 50 100 200 
µa 8.77 8.75 8.71 8.4 

The simulation results show that material non-linearity causes a decrease of apparent permeability at higher fields. This refuted the earlier hypothesis of apparent permeability increase by nonlinear shielding effect and propagation of saturated regions from the peripheral wires toward the center of the array.

For the experimental verification of the simulations, we have prepared several types of nanowire and microwire arrays. We have used alumina and silicon membranes manufactured by Smart Membranes. The membranes were covered on one side with a 30 nm layer of sputtered gold to form a conductive seed layer. The additional layer of silver paste was added to completely close the pores and prevent any leakage.

We have been using AC deposition in the following electrolyte:

NiSO46H2O30g+NiCl26H2O4.5g+H3BO34.5g+FeSO47H2O4.3g+H2O=100ml.

During the mixing and deposition, the electrolyte was bubbled with nitrogen to avoid oxidation.

Electrodeposition was made at room temperature using platinum working electrodes, and Ag/AgCl reference electrodes. The process was controlled by potentiostat SP150 by Biologic to achieve the homogeneous composition of the wires.16,17 The parameters of the square pulses were: U = -1.2V…-0.7V, tp = 100ms, td = 50 ms. The deposition time was 25 to 40 minutes.

Magnetic measurements in DC fields at a temperature of 300 K were performed by SQUID magnetometry on a MPMS XL (Magnetic Property Measurement System, Quantum Design, Inc.).

Fig. 4 shows typical examples of the measured hysteresis loops with the red line indicating the slope corresponding to apparent permeability.

FIG. 4.

Hysteresis loops of the nanowire arrays a) d = 180 nm, L = 100 μm pitch = 480 nm. Measured μa = 14, calculated μa = 8.6 (μa = 14 for pitch = 620 nm). Hc = 9 A/m. b) d = 1 μm, L = 200 μm, pitch = 1.5 μm. Measured μa = 7, calculated μa = 3.2 (μa = 7 for pitch = 2.3 μm). Hc = 5 A/mc) d = 2.5 μm, L = 200 μm pitch = 4.2 μm. Measured μa = 5.5, calculated μa = 3.8 (μa = 5.5 for pitch = 5.4 μm), Hc = 2 A/m. Notice the large differences in the loop shapes even though the magnetic material is the same Permalloy.

FIG. 4.

Hysteresis loops of the nanowire arrays a) d = 180 nm, L = 100 μm pitch = 480 nm. Measured μa = 14, calculated μa = 8.6 (μa = 14 for pitch = 620 nm). Hc = 9 A/m. b) d = 1 μm, L = 200 μm, pitch = 1.5 μm. Measured μa = 7, calculated μa = 3.2 (μa = 7 for pitch = 2.3 μm). Hc = 5 A/mc) d = 2.5 μm, L = 200 μm pitch = 4.2 μm. Measured μa = 5.5, calculated μa = 3.8 (μa = 5.5 for pitch = 5.4 μm), Hc = 2 A/m. Notice the large differences in the loop shapes even though the magnetic material is the same Permalloy.

Close modal

Apparent permeability and magnetometric demagnetization factors were analyzed and calculated for very large arrays of cylindrical nanowires using the equivalent 2D model. Increasing wire distance increases apparent permeability and decreases the demagnetization factor. The results were verified by 3D simulations for arrays up to 3000 wires and experimentally for arrays up to 40 million wires. The apparent permeability values calculated by FEM are always smaller than the permeability calculated from the SQUID measurements. Such difference cannot be caused by wrongly estimated material permeability, as the FEM calculated results are practically independent of it. The effect of the geometrical imperfections of the pores is also minor. The most probable source of the deviation is low filling degree of the pores: we have verified by parametric simulations that the apparent permeability value is very sensitive to the average effective pitch, which is increased if some pores are not filled by the magnetic material. It is also possible that our equivalent 2D model, which works very precisely up to 3000 wires, will need some future correction for very large arrays.

For the analyzed large wire arrays the values of measured apparent permeability were from 3 to 33. The achieved coercivity ranged from 1 kA/m to 10 kA/m, which are low values compared to those reported in the literature. In general, the measured coercivity was increasing with reducing the pore diameter.

The authors thank M. Butta and V. Grim for the help with electrodeposition. This study was supported by the Grant Agency of the Czech Republic within the Nanofluxgate project (GACR GA20-27150S). Magnetic experiments were performed in MGML, which is supported by within the program of Czech Research Infrastructures, Ministry of Education, Youth and Sports, Czech Republic (project no. LM2018096).

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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