# The Performance of the Magneto-Impedance Effect for the Detection of Superparamagnetic Particles

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

_{max}is obtained for an applied field similar in magnitude to the anisotropy field H

^{k}, when the transverse permeability reaches its maximum value µ

_{max}. The minimum impedance Z

_{min}is obtained when the sample is magnetically saturated and the permeability µ

_{min}is close to µ

_{0}.

^{k}. For instance, the sensitivity of a multilayered planar sample can reach a value of 27 kΩ/T, occurring at an applied field of µ

_{0}H = 200 µT (H = 180 A/m) and measured at 23 MHz [15].

_{max}. Wang and collaborators [16] have compiled the results from quite a number of works that use the MI effect to detect magnetic particles. Overall, the comparison of the results seems to be utterly inconsistent, because some authors report an increase in the MI response when increasing the concentration of particles (for example, Devkota and collaborators [17]), while others describe a decreasing tendency (for example, Yang and collaborators [18]). As the MI materials and the nature of the particles differ between studies, it becomes necessary to adopt a systematic approach to evaluate the sensitivity of the MI effect to quantify the concentration of MNPs. The present work attempts to shed light on this issue from a theoretical point of view, using a numerical procedure (finite element method) to solve Maxwell’s equations and calculate the impedance in the conditions of the experiments described in the literature.

## 2. Numerical Calculation Procedure

_{dc}is the dc resistance, $j=\sqrt{-1}$, the imaginary unit, and $\theta =a\sqrt{2\pi f\sigma \mu}=\sqrt{2}a/\delta $. Similar expressions are obtained for the case of cylindrical samples (wires).

_{max}, corresponding to the maximum value of transverse permeability (and the impedance Z

_{max}) and µ

_{min}= µ

_{0}, corresponding to the saturated state (and the impedance Z

_{min}) as depicted in Figure 1.

_{P}. As explained before, the permeability of the particles is the only relevant magnetic property in this approach. There is no attempt to model the individual magnetic behavior of the particles. To reproduce the conditions of the experiments found in the literature, the MI response is evaluated as a function of the concentration of the nanoparticles. If we assume that the nature of the particles is always the same, the mean permeability of the sample of magnetic nanoparticles is simply proportional to its concentration. That is, we can calculate the evolution of the MI sensor’s response in the presence of a sample of magnetic nanoparticles with different concentrations by simply modifying the value of µ

_{P}—the permeability of the MNPs system.

_{P}. As previously discussed, increasing values of µ

_{P}correspond to increasing concentrations of MNPs.

^{5}S/m) with a constant permeability µ. No definite magnetization process is assumed, and the MI is calculated as the relative ratio between the values of the impedance obtained with a high value of permeability µ

_{max}and with a value of µ

_{min}= µ

_{0}. The geometry of the MI sample is kept constant through all the simulations: 1 mm wide and 20 µm thick (the length is not relevant for the 2D problem, and the net value of the impedance is calculated for a sample 1 m long).

^{7}S/m, µ = µ

_{0}) with a thickness of 35 µm. The 0.8 mm thick dielectric presents no conductivity, and µ = µ

_{0}.

_{max}and µ

_{P}. For each configuration, the MI is calculated as a function of the frequency, in a range from 0 to 150 MHz or 0 to 1 GHz, depending on the case. To accommodate the intensive computational resources needed, we have made use of the XFEMM implementation of the software [26], which is run in a computer cluster.

## 3. Results and Discussion

_{max}= 5000 µ

_{0}used in the simulation, the maximum value of MI ratio is MI

_{max}= 568%. The inset in Figure 3 shows the experimentally measured MI ratio of an amorphous ribbon composed of Co

_{65}Fe

_{4}Ni

_{2}Si

_{15}B

_{14}. We can observe that the shapes of both curves are essentially similar, although the MI experimental values are significantly lower, indicating that the permeability µ

_{max}= 5000 µ

_{0}used in the simulation is largely overestimated. Nevertheless, this case, which corresponds to a very sensitive MI sensor, is considered the starting point in our goal of studying the relevance of MI to detect magnetic nanoparticles.

_{P}). The simulation results are divided into two plots for better clarity. Figure 4a determines that the MI response decreases when the permeability of the MNPs system increases from 2 µ

_{0}to 15 µ

_{0}. However, when µ

_{P}is increased further, Figure 4b shows that the MI response changes tendency and start to increase. In both plots, the large red and black dots indicate the position of MI

_{max}for the different values of µ

_{P}. Figure 5 compiles these results, plotting the evolution of MI

_{max}for the whole range of tested µ

_{P}values.

_{P}< 15) or an increasing (for µ

_{P}> 20) behavior. This can certainly explain the discrepancies found in the compiled works [16] indicated in the introduction: the situation in the works where a decrease in the MI ratio is reported when increasing the concentration of MNPs, according to the results shown in Figure 5, can correspond to situations in which the permeability of the nanoparticle ensemble is low (due to a low intrinsic permeability of the nanoparticles or low concentrations). In contrast, the situation in the works reporting an increasing MI response with concentration can correspond to cases in which the permeability of the MNPs system is large.

_{max}is calculated as a quotient as displayed in Equation (2), the negative slope of the curve in Figure 5 for µ

_{P}< 15 indicates that Z

_{min}increases more than (Z

_{max}− Z

_{min}) in this range. This is easily explained considering that the increase in the impedance is due to the presence of a magnetic medium (the particles) near the MI conductor. It is the same phenomenon occurring when a soft ferrite increases the impedance of a wire in a RF choke. When the permeability of the medium is very low compared with the permeability of the conductor itself, as in Z

_{max}, the influence of the magnetic medium is low. However, when the permeability of the conductor is low, as in Z

_{min}, even a surrounding medium with a low permeability produces a change in the impedance.

_{max}= 5000 µ

_{0}have been performed for sensors with µ

_{max}/µ

_{0}= 500, 100, and 10—that is, with a decreasing intrinsic MI response (intrinsic MI

_{max}values are 156%, 46%, and 5.5%, respectively). We introduce a new parameter, η, to compare the capacity of the different sensors to detect MNPs with increasing concentrations, defined as

_{max,woP}is the maximum value of the MI ratio in the sensor without particles (the intrinsic MI

_{max}value). The parameter η therefore quantifies the sensitivity of an MI sensor to the presence of MNPs, expressed as the change in the MI ratio experienced by the sensor when a drop of MNPs is placed on top of it. Figure 6 plots the results obtained for η in the sensors with different MI performance (represented by their values of µ

_{max}). Note that the curve for the sensor with µ

_{max}= 5000 µ

_{0}is the same as the one plotted in Figure 5.

## 4. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Sketch of the dependence of the impedance on the applied magnetic field in a soft magnetic sample with transverse anisotropy (in-plane easy axis, perpendicular to the current flow and the applied field).

**Figure 2.**(

**a**) Illustration of the experiment simulated by FEM: a drop containing a certain concentration of magnetic nanoparticles (MNPs) is placed on the MI material, which is inserted in a microstrip line to determine its impedance. (

**b**) Layout (not at real-scale) of the two-dimensional problem solved by FEM, which corresponds to the middle plane of the setup as indicated in (

**a**). Only half of the plane needs to be simulated due to symmetry.

**Figure 3.**Magneto-impedance ratio MI as defined in Equation (2), calculated using FEM for the case of a planar sample 1 mm wide and 20 µm thick, with the permeability µ

_{max}= 5000 µ

_{0}. For comparison, the inset shows the MI ratio experimentally measured in an amorphous ribbon.

**Figure 4.**Variation of the MI curves calculated for a sensor with µ

_{max}= 5000 µ

_{0}when a drop with increasing MNPs concentration is placed on it. (

**a**) for concentrations up to µ

_{P}= 15 µ

_{0}, the magnitude of MI decreases. (

**b**) For larger concentrations, from µ

_{P}= 20 µ

_{0}, the magnitude of MI increases. Large red and black dots indicate the position of MI

_{max}for each curve.

**Figure 5.**Maximum MI ratio of a sensor with µ

_{max}= 5000 µ

_{0}when a drop of MNPs is placed on it, as a function of the concentration of nanoparticles (expressed as increasing permeability µ

_{P}values).

**Figure 6.**Relative change of the MI

_{max}response, experienced by sensors with different MI performance (different values of µ

_{max}) when a system of MNPs with increasing concentration is placed on top.

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García-Arribas, A.
The Performance of the Magneto-Impedance Effect for the Detection of Superparamagnetic Particles. *Sensors* **2020**, *20*, 1961.
https://doi.org/10.3390/s20071961

**AMA Style**

García-Arribas A.
The Performance of the Magneto-Impedance Effect for the Detection of Superparamagnetic Particles. *Sensors*. 2020; 20(7):1961.
https://doi.org/10.3390/s20071961

**Chicago/Turabian Style**

García-Arribas, Alfredo.
2020. "The Performance of the Magneto-Impedance Effect for the Detection of Superparamagnetic Particles" *Sensors* 20, no. 7: 1961.
https://doi.org/10.3390/s20071961