# Modelling and Measurement of Magnetically Soft Nanowire Arrays for Sensor Applications

^{1}

^{2}

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^{*}

## Abstract

**:**

## 1. Introduction

- The sensing direction is defined by the direction of the sensor core and not by the direction of the pickup coil. This allows the construction of highly stable gradiometers of the Foerster type [3].
- The high shape anisotropy reduces the Crossfield error (a non-linearity response to fields perpendicular to the primary sensing direction) [6].
- The low demagnetization of the sensor core increases sensitivity and thus allows miniaturization of sensors.

## 2. Nanowire Fabrication

^{2+}ions. During pulsed electrochemical deposition, the composition of the nanowires’ material is controlled by the parameters of the voltage pulses. This technique has been shown to achieve homogeneous growth of the wires [47]. The delay time between pulses ensures constant material transport through the pores and helps to renew the concentration of the metal ions at the pore-electrolyte interface. All deposition processes were carried out by using VersaStat4 (Princeton Applied Research, Oakridge, TN, USA), since this instrument allows for very precise adjustment of the required parameters. For the pulsed deposition, an aqueous electrolyte was developed which consists of the following components: 300 g/L NiSO

_{4}.6H

_{2}O, 45 g/L NiCl

_{4}.6H

_{2}O, 43.27 g/L FeSO

_{4}.7H

_{2}O, 45 g/L H

_{3}BO

_{3}[48].

_{depo}) of 10 ms and a delay time (t

_{delay}) of 100 ms were applied, with voltage amplitude of −1.2 V and −0.7 V, respectively (Figure 1f), which resulted in uniform nanowire growth. The morphology and the size of the nanowires were characterized by scanning electron microscopy (SEM) (Phenom ProX), and the micrographs are shown in Figure 2. The chemical composition of the wires was determined to Ni

_{80}Fe

_{20}by energy dispersive X-ray spectroscopy (EDX) attached to the SEM. An EDX spectrum of the Ni

_{80}Fe

_{20}nanowires grown in the PC template is shown in Figure 3 (see the red line).

_{delay}= 100 ms to t

_{delay}= 50 ms, which resulted in a reduction in iron content of 10%, leading to Ni

_{90}Fe

_{10}), see Figure 3. In both samples, no signatures of oxygen contamination were found, as they were kept under a constant nitrogen atmosphere. The nanowire lengths were controlled by monitoring the current during deposition and adjusting the deposition time, as the length increased linearly with time. In addition to the nanowire arrays depicted in Figure 2, we fabricated nanowires with diameters of 30 nm and 400 nm by employing the same process. The nanowire length was always 20 µm. In summary, we have presented a method for fabrication of permalloy nanowires that allows the geometry and the composition of the nanowires to be controlled. The quality of the membrane is very important, as it determines the density and the geometry of the wires.

## 3. Nanowire Magnetic Characterization

_{┴}= 8.5 kA/m, while the coercivity in the longitudinal direction is only Hc

_{║}= 2.6 kA/m (Figure 5, Table 1). Also, the relative permeability in the perpendicular direction is μ

_{┴}= 5.4 and in the longitudinal direction it is μ

_{║}= 3.3, which indicates that the easy direction is already perpendicular to the wire axis.

## 4. Modelling Nanowire Arrays

#### 4.1. Demagnetization, Apparent Permeability and Amplification Factor

_{a}, which is defined as:

_{mean}is the average value of magnetic flux density within the wire volume that was inserted into the homogeneous field with intensity H

_{0}.

_{r}is relative permeability of the material.

_{air}is the coil flux with and without the core, respectively.

_{a}; this is not valid for wire cores with large coil area.

_{air}and core cross-section A

_{core}. This formula was derived by Primdahl for fluxgate sensors and it is commonly used in literature [51]:

_{w}/A

_{air}is array density.

#### 4.2. 2D Model for Single Wire

_{r}= 500 inside a 20 μm long pick-up coil as a function of the coil diameter d. As the system is rotationally symmetrical, the calculation was made by 2D FEM.

_{a}= 361. Figure 7 shows that high values of the amplification factor can be achieved only when thin coil is fabricated tightly around the magnetic nanowire core. Even though, the achievable value of amplification factor is only 230, which is significantly lower value than the apparent permeability. When increasing the coil diameter, the amplification factor decreases rapidly; for 500 nm internal coil diameter and 50 nm coil thickness, the amplification factor calculated by FEM is only 50, while using Equation (4) the expected value would be a = 81. The reason of this behavior is that the magnetic flux density B around the magnetic wire core is weaker than the measured homogenous B

_{0}. The profile of B in the wire midplane is shown in Figure 8. When going from the wire center in radial direction, magnetic flux density B steeply drops upon crossing the boundary of the high-permeability core and air. The magnified part outside the wire shows that the field in the wire vicinity is weaker because the field lines are concentrated in the high-permeability region and this shielding effect is decreasing with distance.

#### 4.3. 3D Model for Small Wire Array

_{r}= 500. The wire array diameter is D = 20 µm, and the single-turn pickup coil has internal diameter of 22 µm, length of 20 μm and thickness of 1 μm or 50 nm. The apparent is decreasing with decreasing wire distance due to increasing magnetostatic coupling. For very small density the coupling is minimum and apparent permeability is approaching its maximum value of μ

_{a}= 361 for single wire. The minimum value of μ

_{a}= 4 is reached for 100% density, i.e., for solid permalloy cylinder with diameter of 20 μm, and length of 20 μm (Figure 9a). If we calculate the amplification factor using the simplified Equation (4), we obtain maximum amplification for 12% wire density (500 nm wire pitch). However, more accurate results obtained by FEM modelling show monotonous increase of the amplification factor with array wire density. The maximum value for both calculation methods is a = 3 for solid cylinder. The amplification factor only slightly depends on the coil thickness (Figure 9b).

#### 4.4. Equivalent 2D Model for Large Wire Arrays

_{w}, length L, and distance d

_{w}. Figure 10b shows a similar square lattice model.

_{c}, is calculated according to Equations (1) and (2), which are based on the assumption that the circle with radius R

_{c}has the same area as the red color hexagon/square shown in Figure 11a,b. The red color lines connect the centers of the wires into a single hexagonal/square “shell” of wires. The thickness, t

_{c}, of the hollow cylinders is calculated based on the assumption that the volume of the hollow cylinder must be equal to the volume of the wires that belong to the same shell.

_{h}

_{−c}and R

_{h}

_{−i}are the outer (circumference) radius of the hexagon and the inner radius of the hexagon, respectively. R

_{r}

_{−c}and R

_{r}

_{−i}are the outer (circumference) radius of the square and the inner radius of the square, respectively. The distance, d

_{c}, between the hollow cylinders is the same between all cylinders, as it is proportional to the distance of the wires, d

_{w}, as mentioned in Equations (1) and (2).

#### 4.5. Verification of the 2D Model on Arrays of Thousands of Wires

#### 4.6. Using the 2D Model on Very Large Arrays

_{w}= 275 nm, the apparent permeability is 2.25 for a hexagonal lattice and 2.6 for a square lattice. For the permeability 3.7 measured on Sample 1 (Table 1) the corresponding distance between wires is d

_{w}= 350 nm, which is only slightly lower value than the mean distance estimated from micrographs (500 nm).

## 5. Conclusions

_{a}= 3 measured by SQUID magnetometer for an array of millions of 200 nm diameter wires.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Fabrication of nickel-iron nanowires using nanoporous membranes: (

**a**) insulation layer (photoresist) on top of sputtered Cu electrode, (

**b**) membrane placed, (

**c**) Filling of the holes by electordeposition, (

**d**) growing nanowires, (

**e**) optional dissolving of the membrane not used in this study, (

**f**) waveform used for electrodeposition.

**Figure 2.**SEM micrographs of a NiFe wire array with a wire diameter of 200 nm, length of 20 µm, and an aspect ratio of 100. (

**a**) top view and (

**b**) lateral view.

**Figure 3.**Energy dispersive X-ray spectroscopy (EDX) spectrum of the Ni80Fe20 and Ni90Fe10 nanowire arrays. Differences in the composition were achieved by using different delay times in the pulsed deposition process.

**Figure 5.**Low-field hysteresis loops of (

**a**) 3 mm diameter circular arrays of Permalloy wires with different diameters. The wire length is always 20 µm. The inset shows the low-field details of the same loops. (

**b**) 1 mm diameter array of 30 nm diameter wires also showing the virgin curve.

**Figure 6.**Low-field hysteresis loop of the array of 200 nm diameter 20 μm long permalloy wires measured in the direction perpendicular to wires.

**Figure 7.**Amplification factor of a single 20 μm long permalloy wire with a diameter of 200 nm inside a 20 μm long pick-up coil with two different thicknesses as a function of the coil diameter d (FEM simulation). The wire material relative permeability is μ

_{r}= 500 and apparent permeability μ

_{a}= 361.

**Figure 8.**Magnetic flux density B in the midplane of a 200 nm diameter wire as a function of radial distance from the wire center: (

**a**) inside and outside the wire (

**b**) the outside part enlarged to show decrease in B close to the wire surface.

**Figure 9.**Apparent permeability (

**a**) and amplification factor (

**b**) of a small wire array as a function of wire density (FEM simulation). The array diameter D = 20 µm is the same as the wire length.

**Figure 10.**Part of the model of the wire array (

**a**) with a hexagonal arrangement and (

**b**) with a square arrangement.

**Figure 11.**Equivalent hollow cylinders (

**a**) for a hexagonal arrangement and (

**b**) for a square arrangement of the wires.

**Figure 12.**Apparent permeability versus wire distance with hexagonal and square arrangements–Membrane diameter, D

_{m}= 1 mm. Calculated by FEM using the 2D-equivalent model.

**Figure 13.**Amplification factor of a 1 mm diameter array of nanowires as a function of wire density–calculated by the approximate Equation (4).

**Figure 14.**Amplification factor of a 1 mm diameter array of nanowires as a function of wire density–calculated by FEM.

**Table 1.**Measured coercivity and calculated permeability values of arrays of 20 μm long permalloy nanowires.

# | Sample | Hc [kA/m] | μ_{a} |
---|---|---|---|

1 | A1-1mm 30 nm | 31 | 20 |

2 | A1-3mm 30 nm | 29 | 18 |

3 | A2-3mm 30 nm | 39 | 16 |

4 | A4-1mm 30 nm | 31 | 14 |

5 | B2-1mm 400 nm | 3.9 | 3.3 |

6 | B2-3mm 400 nm | 4.5 | 3.4 |

7 | C1-1mm 200 nm | 4.6 | 3.7 |

8 | C1-3mm 200 nm | 2.6 | 3.3 |

Case-µ_{r} = 500 D_{w} = 200 nm, L_{w} = 20 µm | µ_{a} 3D | µ_{a} 2D | Rel. Diff. (%) |
---|---|---|---|

2791 wires–Hexagon, d_{w} = 1.6 µm | 85.75 | 88.39 | 3.1 |

2791 wires–Hexagon, d_{w} = 1.2 µm | 59.40 | 60.24 | 1.4 |

2791 wires–Hexagon, d_{w} = 0.8 µm | 34.91 | 34.94 | 0.1 |

2791 wires–Hexagon, d_{w} = 0.6 µm | 24.13 | 23.97 | −0.7 |

2791 wires–Hexagon, d_{w} = 0.4 µm | 14.72 | 14.51 | −1.4 |

2791 wires–Hexagon, d_{w} = 0.275 µm | 9.68 | 9.49 | −2.0 |

2521 wires–Square, d_{w} = 1.6 µm | 97.06 | 98.98 | 2.0 |

2521 wires–Square, d_{w} = 1.2 µm | 67.13 | 67.90 | 1.1 |

2521 wires–Square, d_{w} = 0.8 µm | 39.84 | 39.52 | −0.8 |

2521 wires–Square, d_{w} = 0.6 µm | 27.83 | 27.13 | −2.5 |

2521 wires–Square, d_{w} = 0.4 µm | 16.92 | 16.40 | −3.1 |

2521 wires–Square, d_{w} = 0.275 µm | 11.13 | 10.70 | −3.9 |

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Ripka, P.; Grim, V.; Mirzaei, M.; Hrakova, D.; Uhrig, J.; Emmerich, F.; Thielemann, C.; Hejtmanek, J.; Kaman, O.; Tesar, R.
Modelling and Measurement of Magnetically Soft Nanowire Arrays for Sensor Applications. *Sensors* **2021**, *21*, 3.
https://doi.org/10.3390/s21010003

**AMA Style**

Ripka P, Grim V, Mirzaei M, Hrakova D, Uhrig J, Emmerich F, Thielemann C, Hejtmanek J, Kaman O, Tesar R.
Modelling and Measurement of Magnetically Soft Nanowire Arrays for Sensor Applications. *Sensors*. 2021; 21(1):3.
https://doi.org/10.3390/s21010003

**Chicago/Turabian Style**

Ripka, Pavel, Vaclav Grim, Mehran Mirzaei, Diana Hrakova, Janis Uhrig, Florian Emmerich, Christiane Thielemann, Jiri Hejtmanek, Ondrej Kaman, and Roman Tesar.
2021. "Modelling and Measurement of Magnetically Soft Nanowire Arrays for Sensor Applications" *Sensors* 21, no. 1: 3.
https://doi.org/10.3390/s21010003