Open Access Article

This Open Access Article is licensed under a Creative Commons Attribution-Non Commercial 3.0 Unported Licence

Mohammad Reza
Zamani Kouhpanji
^{ab} and
Bethanie J. H.
Stadler
*^{ac}
^{a}Department of Electrical and Computer Engineering, University of Minnesota Twin Cities, Minneapolis, MN 55455, USA
^{b}Department of Biomedical Engineering, University of Minnesota Twin Cities, Minneapolis, MN 55455, USA
^{c}Department of Chemical Engineering and Materials Science, University of Minnesota Twin Cities, Minneapolis, MN 55455, USA. E-mail: stadler@umn.edu; Tel: +1 612 626 1628

Received
4th November 2020
, Accepted 8th December 2020

First published on 9th December 2020

Magnetic nanowires (MNWs) rank among the most promising multifunctional magnetic nanomaterials for nanobarcoding applications owing to their safety, nontoxicity, and remote decoding using a single magnetic excitation source. Until recently, coercivity and saturation magnetization have been proposed as encoding parameters. Herein, backward remanence magnetization (BRM) is used to decode unknown remanence spectra of MNWs-based nanobarcodes. A simple and fast expectation algorithm is proposed to decode the unknown remanence spectra with a success rate of 86% even though the MNWs have similar coercivities, which cannot be accomplished by other decoding schemes. Our experimental approach and analytical analysis open a promising direction towards reliably decoding magnetic nanobarcodes to expand their capabilities for security and labeling applications.

We believe this technological limitation, inability to decode unknown combinations of magnetic nanobarcodes, is due to the magnetic measurements and a lack of a robust analytical approaches for decoding the measured magnetic signature. For example, the hysteresis loop method has been widely used to measure the saturation magnetization (M_{s}) and coercivity (H_{c}) as magnetic signatures for decoding the magnetic nanobarcodes.^{7,12,14,20,21} Since the hysteresis loop method only provides averaged values for M_{s} and H_{c}, it does not provide any insight for decoding them if there is no prior knowledge about the number and types of magnetic nanobarcodes at the readout. More importantly, even though other magnetic measurements, such as first-order reversal curves (FORC)^{22–25} and magnetic nanoparticles spectroscopy,^{26–28} provide distributions as magnetic signatures, there is still a lack of robust analytical approaches to reliably decode the measured magnetic signatures without prior knowledge about the magnetic nanobarcodes.

To overcome these limitations, we propose to use magnetic remanence measurements as they can provide three main advantages compared to the aforementioned methods. First, the magnetic remanence measurements provide a unique remanence spectrum for the magnetic nanobarcodes that is related to the H_{c}, the H_{c} standard deviation, interaction fields (H_{u}), and their correlations, in contrast to the hysteresis loop measurements only provide a single value for decoding. Second, similar to the FORC method, the remanence measurement provides remanence distributions instead of single values, and it is significantly faster than FORC measurement. Lastly yet most importantly, the remanence measurements measure the magnetization at zero applied field, so there is no background signal to reduce its sensitivity. Indeed, the standard remanence measurements, isothermal remanence magnetization (IRM) and DC demagnetization (DCD) measurements, have been widely used in rock magnetism for predicting the unknown magnetic phases in natural minerals.^{29–34} According to the Stoner–Wohlfarth model, the IRM and DCD spectra are identical for non-interacting MNWs (H_{u} = 0), but they will yield different results when H_{u} is not negligible. Since H_{u} dictates the initial and final magnetization states, the initial magnetization state of an array of MNWs plays an important role in their magnetic remanence. Therefore, to further enhance the reliability and reproducibility decoding using the remanence spectra, we propose to use a modified remanence measurement that is different from the standard remanence measurements. We reported our protocol for this method, named “backward remanence magnetization (BRM)”, in our previous work.^{35} Details about the BRM measurement and how it is beneficial compared to the IRM and DCD measurements are fully discussed in ESI† using experimental comparison. An expectation algorithm was used to decode unknown magnetic remanence spectra that will be fully described in the experimental and analytical section below.

More specifically, in this study, we measure the magnetic remanence spectra of several MNWs arrays inside polycarbonate templates using the BRM measurement to investigate the capability of the BRM method and the expectation algorithm for decoding unknown combinations of MNWs arrays. The BRM spectra of magnetic nanobarcodes were engineered by composition and geometrical parameters, e.g. diameter and interwire distance—characterized by the filling factor of the polycarbonate templates. Next, we prepared several combinations of magnetic nanobarcodes and re-measured their magnetic remanence spectra. We show that the ∼86% of all ‘unknown’ combinations of two magnetic nanobarcodes can be reliably decoded using our algorithm. We also discuss the fundamentals of our experimental and analytical approaches to improve the decoding rate when there are more than two magnetic nanobarcodes in unknown combinations.

Fig. 1 A schematic depicting magnetic nanobarcodes synthesis procedure, dimensions are not drawn to scale. (a) The ion irradiation of a raw polycarbonate foil (grey), (b) chemical etching (light blue) of the ion irradiated polycarbonate, (c) evaporation of Ti (blue) then Au (yellow) for a back contacts, (d) electrodeposition of MNWs (pink), and (e) a combination wet and dry etch process to remove the back contacts. (f) The chemical structure of polycarbonate. Further details are in ESI.† |

To realize the strength of remanence spectra for encoding, we systematically measured the remanence spectra using the backward remanence magnetization (BRM) technique, see ESI† for more details. Briefly, samples are saturated using a positive field before applying and removing a negative field. The positive saturation is repeated before applying and removing progressively more negative fields. Each time the negative field is removed (H = 0), the remanent moment is measured and normalized by the sample's remanent saturation magnetization (M_{sr}), as shown in Fig. 2. M_{sr} is the amplitude of each BRM spectrum, and it is equal to the hysteresis loop remanence, or the amount of the magnetic material in each magnetic nanobarcode that is magnetically stable once the field (H) is removed after saturation.

To utilize the BRM spectra for decoding, we combined at least two different types of magnetic nanobarcodes and measured the BRM spectra of each combination, labeled as “Exp. data” in Fig. 2. Then the BRM spectra of these combinations were compared with the linear summation of the weighted BRM spectra of the known individual magnetic nanobarcodes in the combination, labeled as “Recreated curve” in Fig. 2. As can be seen in Fig. 2, a very good agreement between the “Exp. data” and the “Recreated curve” was achieved. The weights were also used to calculate the volume ratio of each magnetic nanobarcode; the details and results are given in ESI.† The good agreement between the “Exp. data” and the “Recreated curve” indicates that the MNWs from one nanobarcode do not cross-talk with the MNWs of the other nanobarcodes in the combination. Therefore, the BRM spectra can be linearly summed for decoding. For practical applications, where two different subjects are labeled using magnetic nanobarcodes, the physical distance between the magnetic nanobarcodes will be significantly larger than the interwire distances used here (∼500 nm), see SEM images in ESI.† This observation is our hypothesis for using a cumulative log-normal distribution, which was proposed by ref. 39 and 40 for unmixing unknown mixtures of natural minerals with complex geometries in rock magnetism. This cumulative log-normal distribution is

(1) |

(2) |

For an unknown BRM spectrum, in addition to M_{sr}^{i}, H_{1/2}^{i}, and σ^{i}; the number of magnetic nanobarcodes (N) is also unknown. Undoubtedly, increasing the N improves the fitting quality (defined as the root mean square (RMS) error between eqn (2) and the experimentally measured BRM spectra) leading to boundless values for N. However, after a certain increase in N, the RMS error does not improve, and it converges to a constant value. Therefore, to determine number of the magnetic nanobarcodes in an unknown combination, we use the expectation algorithm similar to what was proposed and developed by ref. 42–44. Briefly, we assume there are N types of unknown magnetic nanobarcodes in an unknown combination. We fit eqn (2) to the BRM spectra of the unknown combination to calculate the BRM parameters for each N types of magnetic nanobarcodes. Then, we construct the BRM spectra of the combination using eqn (1) and (2) and we calculate the RMS error between the constructed BRM spectra and the experimental BRM spectra. Next, we increase the number of nanobarcodes to N + 1 and repeat the fitting to find the new set of the remanence parameters and recalculate the RMS error of the new fit. At this point, the new RMS error is compared with the previous RMS error to evaluate how much the RMS error is improved. If the RMS improvement is not significant, we stop the procedure and the number of magnetic nanobarcodes is equal to the N of the previous step. However, if the RMS error improvement is significant, we increase the number of magnetic nanobarcodes to N + 2 and repeat the procedure. The “significant” improvement of the RMS error depends on the number of nanobarcodes and how unique their BRM spectra are. For example, we found a 50% hierarchy improvement is sufficient to correctly decode unknown combinations, as discussed below. Eventually, these N BRM spectra were individually compared with the BRM spectrum of each individual magnetic nanobarcode to find the best matches, as discussed below and tabulated in Table 1.

Best matches | Ni30:FeCo30 | Ni30:FeCo50 | Ni30:FeCo100 | Ni30:FeCo200 | ||||

RMS for best matches (%) | 6 | 22 | 6 | 24 | 15 | 10 | 4 | 4 |

Best matches | Ni50:FeCo30 | Ni50:FeCo50 | Ni50:FeCo100 | Ni50:FeCo200 | ||||

RMS for best matches (%) | 10 | 25 | 12 | 17 | 6 | 24 | 15 | 11 |

Best matches | Ni100:FeCo30 | Ni100:FeCo50 | Ni100:FeCo100 | Ni100:FeCo200 | ||||

RMS for best matches (%) | 14 | 26 | 10 | 12 | 10 | 11 | ||

Best matches | Ni200:FeCo30 | Ni200:FeCo50 | Ni200:FeCo100 | Ni200:FeCo200 | ||||

RMS for best matches (%) | 5 | 7 | 12 | 13 | 10 | 13 | ||

Best matches | Ni30:Ni50 | Ni50:Ni100 | FeCo30:FeCo50 | FeCo50:FeCo100 | ||||

RMS for best matches (%) | 8 | 17 | 15 | 16 | 7 | 15 | ||

Best matches | Ni30:Ni100 | Ni50:Ni200 | FeCo30:FeCo100 | FeCo50:FeCo100 | ||||

RMS for best matches (%) | 3 | 4 | 20 | 23 | 7 | 18 | ||

Best matches | Ni30:Ni200 | Ni100:Ni200 | FeCo30:FeCo200 | FeCo100:FeCo200 | ||||

RMS for best matches (%) | 3 | 3 | 9 | 18 | 6 | 9 | 10 | 27 |

H
_{1/2} and σ are correlated to the magnetic moment (M_{s}), H_{c}, relative strength of H_{c} and H_{u}, and the standard deviation of H_{c}, which is related to MNWs compositions and the pore size distributions of the templates. Since the BRM spectra are measured at zero applied field, the total field experienced by each MNW inside the nanobarcode is simply the H_{u} among the MNWs. If the H_{u} is larger than the MNWs H_{c}, it causes some MNWs to switch until an equilibrium state is reached, where the magnetization at the equilibrium state is called the remanence, or remenant magnetization. Since H_{u} is maximum after saturation of MNWs, analyzing the squareness (M_{sr}/M_{s}) is the simplest way to understand the relative strength of the H_{u} and H_{c} for the MNWs in this study. Thus, the H_{c}, H_{1/2}, σ, and squareness of our magnetic nanobarcodes are given in Fig. 3. H_{c} is a function of the shape anisotropy and crystal anisotropy. Both Ni and FeCo MNWs have the same H_{c} and it reduces as the MNWs diameter increases, Fig. 3a. Since the Ni and FeCo MNWs have similar dimensions, it can be concluded that the shape anisotropy is the dominant term determining the H_{c}. Note H_{1/2} is dominantly determined by H_{c}, so they are the same for our MNWs, see Fig. 3a. Furthermore, since FeCo has higher magnetic moment (∼2.45 T) compared to Ni (∼0.63 T), the FeCo MNWs have larger H_{u} compared to the Ni MNWs for the same diameter and filling factor. Therefore, the FeCo MNWs have smaller squareness compared to the Ni MNWs, see Fig. 3b, and the squareness decreases as the diameter increases due to the fact that the H_{c} decreases. Even though the H_{c} decreases as the diameter increases, σ is fairly constant. Since all templates were prepared in the same way, this indicates that the pore size distributions were the same for all of our magnetic nanobarcodes and it profoundly impacts σ compared to H_{u}. Indeed, because the FeCo magnetic nanobarcodes had higher H_{u} (due to larger M_{s}), they have larger σ which shows that H_{u} only causes a vertical shift in σ. Note, the MNWs with 30 nm diameter are non-interacting, H_{u} ≈ 0, thus they have similar σ values. Therefore, it can be realized that the curvature of the BRM spectra, see Fig. 2, can be tailored using the MNWs composition and diameter to achieve unique BRM spectra, which we will explain how to decode in the discussion section.

Fig. 3 Magnetic parameters that were tailored to produce unique BRM spectra, (a) coercivity (H_{c}) and half BRM field (H_{1/2}), (b) squareness (M_{sr}/M_{s}), and (c) the broadening parameter (σ). See ESI† for hysteresis loops (H_{c}, M_{sr}, M_{s}) and Fig. 2b for BRM spectra (H_{1/2}, σ). The maximum error bar (n = 2) for subfigures (a–c) are shown. |

Fig. 4 shows the results for determining the number (N) of magnetic nanobarcodes in unknown combinations of magnetic nanobarcodes. The important feature in these curves is the N at which the RMS error reaches a plateau, which is why the curves were normalized with respect to the RMS error found by matching the experimental data to N = 1. According to our procedure in the experimental and analytical approaches section above, we first assume the unknown sample is made up of only one magnetic nanobarcode (N = 1). Therefore, we fit its BRM spectrum to a single BRM(H) function given in eqn (1) and calculate the RMS error. For example, the curve labeled by a star in Fig. 4b was fit to one BRM curve with M_{sr}, H_{1/2} and σ proportional to 7.8, 6.2, and 0.9 of the actual values, respectively, but the RMS^{1} error (the superscript “1” indicates N = 1) for was 22%. Also, these values do not match any of the samples in Fig. 3.

Next, we assume there were two magnetic nanobarcodes (N = 2) forming the unknown BRM spectrum. In this case, two BRM(H) function were linearly added according to eqn (2) and the resulting function was fitted to the unknown BRM spectrum. In this case, the RMS^{2} error (the superscript “2” indicates N = 2) was found to be 9% for the same labeled curve in Fig. 4b. By comparing RMS^{1} and RMS^{2}, the RMS^{2} error was decreased by a factor of 2.4 (∼40% of the RMS^{1} error). Next, we increased the number of magnetic nanobarcodes to three (N = 3) and recalculated the RMS^{3} error, which was 7%, or 22% of RMS^{2}, see N = 3 point for the starred curve in Fig. 4b. RMS^{4} error for this same sample was calculated as 7% again, which means zero improvement in the RMS, so we stopped the decoding procedure.

To determine the number of magnetic nanobarcodes in this unknown combination, it is critical to answer “what reduction percentage is optimum?” For unknown combinations of our magnetic nanobarcodes, we found a 50% reduction of the RMS error in each step (RMS^{i+1} ≤ 50% of RMS^{i}, superscript “i” indicates i^{th} step of decoding) is sufficient to determine the number of magnetic nanobarcodes in unknown BRM spectra. Choosing 50% reduction, we managed to successfully decode 24 out of 28 (∼86% success rate) unknown combinations of two magnetic nanobarcodes. If the chosen reduction percentage is too small, the determined number of magnetic nanobarcodes will be incorrectly large. In contrast, choosing a large reduction percentage can cause the determined number of the magnetic nanobarcodes to be incorrectly small. This latter case is observed in Fig. 4 (arrows) where all of the samples contained only 2 barcodes (N = 2). Moreover, we can improve the decoding rate further in at least two ways. First, we can design and synthesize multi-segmented or multi-component MNWs to engineer the magnetic anisotropy for minimal overlap in nanobarcode signatures.^{45–48} Second, more advanced decoding algorithms based on machine learning can be implemented.

Once the number of the magnetic nanobarcodes in an unknown combination is determined, we compare the fitting results (M_{sr}, H_{1/2}, and σ) to those of our magnetic nanobarcodes to find the best match indicating the type of magnetic nanobarcodes in the unknown combination. The best match was found by calculating the lowest RMS error between the BRM spectra of the individual magnetic nanobarcodes and the fitting results. Table 1 shows the best matches and the corresponding RMS errors.

As mentioned in experimental and analytical section, we intentionally chose the Ni and FeCo because they have roughly the same crystal anisotropy, so H_{c} is only a function of the MNWs diameter except for the smallest diameter (30 nm), see Fig. 3a. In the 30 nm diameter case, the mechanism of reversal most likely changes for the low moment Ni nanowires to coherent rotation vs. vortex walls, which are likely the mechanism in all other cases based on simulations.^{48} Furthermore, the templates for electrodeposition of the MNWs were prepared in the same way thus the σ is only a function of composition, Ni or FeCo, and not diameter except for the 30 nm sample, see Fig. 3c. The 30 m samples have the lowest interactions between nanowires (lowest filling factor templates), which is likely why σ of FeCo matches Ni in this case.

Therefore, it can be realized that indeed there was only one encoding parameter when decoding the magnetic nanobarcodes with the same composition or the same diameter. These results indicate the unprecedented advantage of the BRM measurement compared to the current state of the art, such as the hysteresis loop measurement or magnetic nanoparticle spectroscopy.^{1} Specifically, hysteresis loops only provide a single value for H_{c} and M_{s} that cannot be used to distinguish between the different types of magnetic nanobarcodes in an unknown combinations. Moreover, since the BRM measurement is able to decode the magnetic nanobarcodes with only one distinct encoding, it can boost the number of codes.

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## Footnote |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/d0na00924e |

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