From spectral analysis to hysteresis loops: a breakthrough in the optimization of magnetic nanomaterials for bioapplications

Magnetite (Fe₃O₄) nanoparticles were prepared by a non-hydrolytic sol-gel (NHSG) process in the presence of benzyl alcohol (BzOH) following the synthesis procedure described elsewhere [51]. The Fe(AcAc)₃:BzOH ratio was fixed at 0.03. The resulting Fe₃O₄ NPs are characterized by an average size of about 7 nm with a narrow dispersion ($\lt$1 nm) around the mean value, and an aspect ratio of 1.13 [51].

Co_0.84Zn_0.16Fe₂O₄ ferrite powders were prepared by a sol-gel auto-combustion method utilizing citrate-nitrate precursors at low temperature ($\leqslant$110 ^∘C) without any post-preparation heat treatments, as described elsewhere [52]. The morphology of the synthesized Co-Zn ferrite is a quite complex grain arrangement consisting in a micrometer-sized agglomerate with surface similar to a beehive composed of nanosized crystals (grains). X-ray diffraction (XRD) reveals a grain mean diameter of 34.9 nm [52].

An amorphous ribbon of composition Fe_62.50 Ni_10.00 Si_11.05Cr_2.23B_11.14C_3.09 was obtained by the melt-spinning technique. The pre-alloy was prepared and then inductively re-melted in a quartz tube with a nozzle (0.8 mm diameter) under vacuum atmosphere, and subsequently injected onto a rotating copper wheel by applying a high-purity Ar pressure. The thickness of the produced ribbons is about 40 µm.

Au-coated permalloy (Ni₈₀Fe₂₀) nanodiscs (diameter: 630 nm) were produced by combining polystyrene nanosphere lithography and sputtering [53]. A metallic trilayer comprised of a 30 nm permalloy film sandwiched between two 5 nm Au films was deposited on a Si substrate coated by a resist layer. The Au coatings were deposited with the purpose of making the nanodisc surface biocompatible and apt to be bio-functionalized [53].

By dissolving the sacrificial resist underlayer in acetone, the nanodiscs were detached from the substrate and dissolved in water. The disc were magnetically characterized both before detachment and in water solution.

Quasi-static (dc) magnetization curves were measured at room temperature by means of a Lakeshore 7200 vibrating sample magnetometer (VSM) in the magnetic field range ±17 kOe. The rate of the driving magnetic field ($\mathrm{d}H/\mathrm{d}t$) was kept constant in order to get a triangular H waveform during the measurement of the hysteresis loops.

Room-temperature dynamic magnetization curves were measured by two custom-built hysteresis loop tracers based on the inductive technique [54]. In both setups, a harmonic driving field capable of magnetizing the sample was generated by a RF coil placed inside one of the two counter-wound pick-up coils designed to measure the inductive magnetic signal. Each hysteresis loop tracer operates in a range of driving field amplitudes and at a frequency appropriate to the specific magnetic material under study. In particular, one setup generates a magnetic field with a frequency of 69 kHz and an amplitude in the range 100–470 Oe and was used to analyse the high-frequency behaviour of typical iron-oxide nanoparticles; the other one operates at a frequency of 20–100 Hz with an amplitude of 1.78 Oe and was used to measure soft magnetic materials such as amorphous ribbons.

VSM setups and custom-built hysteresis loop tracers are typically designed to directly give the M(H) curve. In the measurements done for this work, the sweeping time t was also picked up in order to get the H(t) and M(t) waveforms.

In frequency measurements, the two signals were recorded for a time corresponding to 100 periods of the driving field and processed in frequency by using the discrete Fast Fourier Transform algorithm implemented in the commercial GNU Octave software. In this way the line frequency spectrum of the magnetization was obtained. The number of periods was such that the spectrum was virtually not affected by the finite time length of the signal.

Quasi-static VSM measurements are typically characterized by an overall time of the experiment of about two hours (the field's period T being of the order of 7200 s). During our measurements the magnetic field was linearly varied with time from the positive maximum field amplitude to the negative one and back, resulting in one full period of a triangular field wave. In order to compare the results obtained using the VSM with the ones obtained from the standard frequency measurements described above, the total sweeping time of the VSM was considered to correspond to the first period of a virtual succession of 100 identical periods, and the variations with time of both M and H recorded during the actual measurement were replicated 100 times. In this way it was possible to implement the harmonic analysis of the M(t) signal under the triangular field wave by making use of the same Fast Fourier Transform algorithm as in actual frequency measurements. Such a method allowed us to obtain a line spectrum of magnetization also in this case.

In the experimental practice, the frequency analysis of periodic magnetization signals characterized by a period of the order of thousands of seconds (as in the case of VSM measurements) is not typically performed, although frequencies as low as about 10 µHz can be investigated by dedicated digital instruments [55]. The procedure developed in this paper was applied to the results of VSM measurements in order to highlight features and flexibility of the proposed mathematical treatment over a wide frequency range.

We consider a harmonic waveform of the driving field first; with the present choice of the time basis, H(t) is written as:

$$\begin{equation} H(t) = H_V \cos(\omega t) \end{equation} \tag{ 1 }$$

where H_V is the maximum field amplitude (also called the vertex field of the corresponding hysteresis loop) and $\omega = 2\pi f$, f being the operating frequency. The resulting magnetization M(t) is in general expressed as a Fourier series with only odd harmonics:

$$\begin{equation} M(t)= \sum_{n=0}^{\infty} M_{2n+1} \cos \left[(2n+1) \omega t + \phi_{2n+1}\right]. \end{equation} \tag{ 2 }$$

Here, the $M_{2n+1}$ and $\phi_{2n+1}$ parameters are the amplitudes (moduli) and the phases of the harmonics, as directly derived from spectral analysis. The φ's are defined as the phase shifts between each Fourier component of the magnetization spectrum and the sinusoidal H(t) wave. The index n takes all integer values from zero to $+\infty$. Equation (2) can be cast in the equivalent form:

$$\begin{equation} M(t)= \sum_{n=0}^{\infty} P_{2n+1} \cos \left[(2n+1) \omega t\right] - \sum_{n=0}^{\infty} Q_{2n+1} \sin \left[(2n+1) \omega t\right] \end{equation} \tag{ 3 }$$

where

$$\begin{equation} \begin{split} P_{2n+1} = M_{2n+1} \cos \phi_{2n+1} \\ Q_{2n+1} = M_{2n+1} \sin \phi_{2n+1}. \end{split} \end{equation} \tag{ 4 }$$

The quantities $P_{2n+1}$ and $Q_{2n+1}$ will play a central role in the discussion. An equivalent expression of both parameters is reported in the appendix in the case when it is the frequency spectrum of the voltage induced in a pickup coil which is measured.

It is easy to see that the term $\omega t$ of the argument of the sine and cosine functions in equation (3) can be written in each half period of the sinusoidal wave as:

$$\begin{equation*} \begin{split} \omega t & =\arccos(H/H_V) \quad\left[NT \leqslant t \leqslant \left(N+\frac{1}{2}\right)\, T \right] \\[8pt] \omega t & =2\pi-\arccos(H/H_V) \quad \left[\left(N+\frac{1}{2}\right)\, T \leqslant t \leqslant NT \right] \end{split} \end{equation*}$$

where $T = 2\pi/\omega$ is the period and N is an integer positive, negative or zero. Therefore the magnetization M can be expressed as a function of H as:

$$\begin{align} M(H)&= \sum_{n=0}^{\infty} P_{2n+1} \cos \Big[(2n+1) \arccos(H/H_V)\Big ] \nonumber\\[8pt] &\quad - \sum_{n=0}^{\infty} Q_{2n+1} \sin \Big[(2n+1) \arccos(H/H_V)\Big] \end{align} \tag{ 5 }$$

in the upper branch of the hysteresis loop, i.e. when H takes decreasing values from $+H_V$ to $-H_V$ (corresponding to the first half-period of the driving field), and as:

$$\begin{align} M(H)&= \sum_{n=0}^{\infty} P_{2n+1} \cos \Big[(2n+1) \big(2\pi - \arccos(H/H_V)\big)\Big] \nonumber \\[8pt] & \quad - \sum_{n=0}^{\infty} Q_{2n+1} \sin \Big[(2n+1) \big(2\pi-\arccos (H/H_V)\big) \Big] \end{align} \tag{ 6 }$$

in the lower branch, i.e. when H is increased from $-H_V$ to $+H_V$, corresponding to the second half period of H(t).

A triangular H waveform of period $T = \frac{2\pi}{\omega}$ and maximum amplitude H_V , starting from H_V , can be expressed as:

$$\begin{equation} H(t) = \frac{8}{\pi^2}H_V \sum_{m=0}^{\infty} \frac{1}{(2m+1)^2}\cos \big[(2m+1)\omega t \big] \end{equation} \tag{ 7 }$$

The resulting magnetization is always given by equation (3); by elementary geometric considerations the term $\omega t$ in each half period of the triangular wave is expressed as a function of $H/H_V$ as:

$$\begin{equation} \begin{split} \omega t = \frac{\pi}{2} \enspace (1-H/H_V) \qquad &\left[NT \leqslant t \leqslant \left(N+\frac{1}{2}\right)T \right] \\[8pt] \omega t = \frac{\pi}{2} \enspace (3+H/H_V) \qquad &\left[\left(N+\frac{1}{2}\right)T \leqslant t \leqslant NT \right] \end{split} \end{equation} \tag{ 8 }$$

N being an integer positive, negative or zero. Therefore, the magnetization M can be expressed as a function of the field H as:

$$\begin{align} M(H)&= \sum_{n=0}^{\infty} P_{2n+1} \cos \left[\left(n+\frac{1}{2}\right) \pi \, \left(1-H/H_V\right) \right] \nonumber \\[8pt] &\quad - \sum_{n=0}^{\infty} Q_{2n+1} \sin \left[\left(n+\frac{1}{2}\right) \pi \, \left(1-H/H_V\right) \right] \end{align} \tag{ 9 }$$

in the upper branch of the hysteresis loop, and as:

$$\begin{align} M(H)&= \sum_{n=0}^{\infty} P_{2n+1} \cos \left[\left(n+\frac{1}{2}\right) \pi \, (3+ H/H_V) \right] \nonumber \\ & \quad - \sum_{n=0}^{\infty} Q_{2n+1} \sin \left[\left(n+\frac{1}{2}\right) \pi \, (3+ H/H_V) \right] \end{align} \tag{ 10 }$$

in the lower branch.

Equations (5) and (6) or (9) and (10) relate M to H for the two commonest driving field waveforms through simple analytic expressions (albeit not cast in closed form) where the coefficients of the double series are derived from the harmonics of the frequency spectrum.

As a matter of fact, determining the relation between M and H in a ferromagnetic material is a difficult task as well as a fascinating mathematical problem which has been approached at different levels of complexity [36]. In the majority of cases no simple analytic expression of the hysteretic M(H) curve can be produced, and the loop properties can be obtained through complex mathematical methods, often involving numerical solutions, such as the Jiles-Atherthon model and the Preisach approach [36, 57]. Only a few exceptions involving explicit analytical solutions exist (e.g. the Rayleigh region of very low fields [58], or the magnetic loops of single-domain particles described by the Stoner-Wohlfarth law at well-defined angles between the easy axis and the magnetic field [59]). The present analysis shows that a simple analytical representation of the loop branches (equations (5) and (6) or (9) and (10)) emerges when the spectral harmonics of the magnetization signal are experimentally known. It should be explicitly noted that the information extracted from the harmonics must be complete, i.e. involving not only the amplitude but also the phase of all significant harmonics of the spectrum. Another analytic formula for M(H) in a loop generated by a sinusoidal driving field, alternative and equivalent to equations (5) and (6), is reported in the appendix. It should be remarked that when a constant term H₀ is added to the oscillating field (equations (1) and (7)), such as for instance in the case of thin films or nanoparticles where a constant exchange-bias field [60] is present, all the previous expressions for the M(H) curves in terms of the spectral harmonics keep their validity with the substitution $H \rightarrow (H-H_0)$.

The area enclosed by a hysteresis loop plays a central role in biomedical applications, being related to the energy absorbed by the particles from the applied field in one period, which acts to increase their temperature and finally results in a net transfer of heat to the environment. In some cases, as in magnetic hyperthermia [61], such a heat is instrumental to the effectiveness of the therapy and has to be maximized and/or controlled; on the contrary the release of heat can be an unnecessary side effect or even a nuisance in other applications, such as MPI [62], so that in these cases the loop area should be minimized.

The Fourier expressions derived in section 3 can be used to link A_L to the spectral harmonics through simple transformation formulas. When M is plotted as a function of H, the loop's area is by definition:

$$\begin{equation*} A_L = \oint M\mathrm{d}H. \end{equation*}$$

where the integral is on a closed path in the (H,M) plane. Different expressions for A_L apply for sinusoidal or triangular H waveforms.

4.1.1. Harmonic field

When the ac field is harmonic (equation (1)), using time as a parameter and taking into account that $\mathrm{d}H = -\omega H_V \sin \omega t\mathrm{d}t$, one gets

$$\begin{equation*} A_L= \oint M\mathrm{d}H = -\omega H_V \int_0^T M(t) \sin\omega t \mathrm{d}t. \end{equation*}$$

where M(t) is given by the double Fourier series of equation (3). Exploiting the orthogonality of the Fourier basis functions and calling $x = \omega t$ (whence the upper integration limit becomes 2π) it turns out that only the integral $\int_0^{2\pi}\sin(2n+1)x \sin x \, \mathrm{d}x$ with $n =$ 0 has a nonzero value, so that the loop area turns out to be simply expressed by:

$$\begin{equation} A_L= H_V Q_1 \int_0^{2 \pi} \sin^2 x \, \mathrm{d}x = \pi H_V Q_1. \end{equation} \tag{ 11 }$$

where $Q_1 = M_1 \sin \phi_1$, M₁ and φ₁ being the amplitude and phase of the first harmonic, respectively. Therefore, for a sinusoidal H waveform modulus and phase of the first harmonic of M obtained by spectral analysis are enough to get the loop's area.

4.1.2. Triangular field

When the ac field has triangular waveform (equation (7)), the differential element $\mathrm{d}H$ can be written as:

$$\begin{equation} \mathrm{d}H = -\frac{8}{\pi^2}H_V \omega \sum_{m=0}^{\infty} \frac{1}{(2m+1)}\sin \big[(2m+1)\omega t \big] \, \mathrm{d}t, \end{equation} \tag{ 12 }$$

so that using again equation (3) and exploiting the orthogonality of the Fourier basis function, one gets:

$$\begin{equation} \begin{split} A_L&= \frac{8}{\pi^2}H_V \omega \sum_{n=0}^{\infty} \frac{Q_{2n+1}}{(2n+1)} \int_0^T \sin^2 \big[(2m+1)\omega t \big] \mathrm{d}t \\ &= \frac{8}{\pi^2}H_V \sum_{n=0}^{\infty} \frac{Q_{2n+1}}{(2n+1)} \int_0^{2 \pi} \sin^2 \big[(2m+1) x \big] \mathrm{d}x\\ &=\frac{8}{\pi}H_V \sum_{n=0}^{\infty} \frac{Q_{2n+1}}{(2n+1)}= \frac{8}{\pi}H_V \left[\frac{Q_1}{1}+\frac{Q_3}{3}+\frac{Q_5}{5} + \ldots \right].\\ \end{split} \end{equation} \tag{ 13 }$$

Therefore, all odd harmonics obtained by spectral analysis are needed to determine the loop's area in the case of triangular field waveform.

It should be explicitly noted that both transformation formulas for A_L (equations (11) and (13)) involve the quantities $Q_{2n+1} (\propto \sin\phi_{2n+1}$) only. This is explained recalling that a magnetic hysteresis loop appears when the harmonics of M are shifted by a generic phase angle with respect to the driving field. When all harmonics of M are either in phase (φ = 0) or in opposition ($\phi = \pi)$ with the driving field, the loop area is exactly equal to zero, i.e. the M(H) curve displays no hysteresis, as in the very special (and unlikely) case of superparamagnetic particles whose magnetization can still be described by a Langevin function at the operating frequency [38, 40].

• The magnetization at the positive vertex field (M_V ) is immediately derived from equation (3) by simply taking t = 0. Therefore,
  $$\begin{equation} M_V = \sum_{n=0}^{\infty} P_{2n+1} = \Big[ P_1 + P_3 + P_5 + \ldots\Big]. \end{equation} \tag{ 14 }$$
  Of course the magnetization at the negative vertex of a symmetric loop is just $-M_V$. This transformation formula applies to both harmonic and triangular waveform.
• The magnetic remanence (M_R ) is the value taken by the magnetization at $H =$ 0. For both considered driving field waveforms, H is equal to zero when $t = (N+1/4)\pi$ and $t = (N+3/4) \pi$ (where N is again an integer positive, negative or zero). Therefore, the positive/negative magnetic remanence are immediately obtained from equation (3) by taking into account that in both cases all cosine terms in the double series are equal to zero and the sine terms alternatively take the value ±1, resulting in the following transformation formula:
  $$\begin{equation} \pm M_R = \pm \sum_{n=0}^{\infty} (-1)^{2n} Q_{2n+1} =\pm \Big[ Q_1 -Q_3 +Q_5 - \ldots\Big]. \end{equation} \tag{ 15 }$$
• The negative/positive coercive fields (i.e. the fields where the hysteretic magnetization is equal to zero) are obtained by requiring that the right-side members of equations (5) and (6) (sinusoidal H waveform) and equations (9) and (10) (triangular H waveform) be equal to zero. When the measured spectrum includes harmonics up to the $(2n+1)$th order, it can be checked that the solution for H_C is obtained by solving an algebraic equation of degree $(2n+1)$ in $(H/H_V)^2$. As a consequence, no analytical solution can be found for odd n > 1, and the equation has to be numerically solved. In contrast, for nearly elliptical minor hysteresis loops where all harmonics with n > 1 (i.e. beyond the third harmonic) are negligible H_C can be obtained in closed form, as shown in the appendix.

The ability of equations (5), (6) or (9), (10) to reconstruct the hysteresis loop branches on the basis of the knowledge of the harmonics of the M(t) signal is checked in figure 1 for a dried powder of magnetite nanoparticles.

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Panel (a) shows the magnetic behaviour of the sample submitted to a harmonic magnetic field of frequency 69 kHz and maximum amplitude 470 Oe. These field parameters are of the same order of magnitude as the ones used in present-day techniques of magnetic hyperthermia, imaging and spectroscopy [20, 21, 63]. The top left frame shows one full period of both H(t) and M(t) signals, as generated and measured by a hysteresis loop tracer. The amplitudes $M_{2n+1}$ of the harmonics obtained from spectral analysis of the M(t) signal are reported in the centre left frame, while the bottom left frame shows the corresponding phases $\phi_{2n+1}$. The results are plotted as functions of the harmonic number $n_h = 2n+1$; n_h takes all positive odd integer values from 1 to $+\infty$. In the present case, the M(t) signal is not much deformed from a sinusoidal curve, so that only five harmonics of the frequency spectrum are significant (i.e. distinctly above the background noise). Note also that the phases are in general different from either zero or π, indicating a hysteretic behaviour of the cyclic magnetization process. This is actually shown in the right frame, where the measured M(H) loop obtained from the time-resolved signals is reported (open symbols). It should be explicitly noted that this is a typical example of frequency-sustained magnetic hysteresis, as it will become apparent by comparison with the results in dc conditions. The red line shows the loop as reconstructed from the harmonics obtained by frequency analysis, using equations (5) and (6) for the upper/lower branch of the loop, respectively. The $P_{{}2n+1}$ and $Q_{2n+1}$ coefficients were generated from the $M_{2n+1}$ and $\phi_{2n+1}$ values reported in the left-side frames. The agreement between the two curves is excellent.

Panel (b) shows the cyclic magnetization of the same sample measured in dc conditions by a VSM setup. In this case, the magnetic field waveform is triangular instead of sinusoidal and its frequency is basically zero. The maximum field amplitude was set to $1 \times 10^4$ Oe in order to approach magnetic saturation. The top left frame shows again one full period of the H(t) wave generated by the VSM as well as the resulting M(t) wave, which turns out to be severely deformed from the triangular wave. The amplitudes and phases of the harmonics obtained from spectral analysis of the M(t) signal are again reported in the centre and bottom left frames. In the present conditions the number of significant harmonics turns out to be much higher than in the previous measurement (panel (a)). All phases are now either zero or π, indicating that the M(t) signal—albeit deformed—in still in phase with H(t), so that the magnetization is expected to be an anhysteretic function of H. This is checked in the right frame where the experimental M(H) curve is reported (open symbols); the blue line shows the reconstruction done using equations (9) and (10) appropriate to the triangular H waveform; the curves for the two branches turn out to be exactly superimposed. Once again theoretical curves and experimental data are perfectly matching.

The inset in the right frame of panel (b) shows the region around zero field of the M(H) curve, measured by VSM with a much lower vertex field ($H_V \approx 470$ Oe). The absence of magnetic hysteresis in both theoretical and experimental curves is confirmed, indicating that the magnetite nanoparticles measured in this work can be properly considered as SPION [41], being superparamagnetic at room temperature and characterized by a M(H) curve which follows the Langevin law [56]. The inset can be compared with the right frame of panel (a), showing the M(H) behaviour in the very same field range but at high frequency: the comparison shows that magnetic hysteresis can emerge in SPION submitted to a driving field of sufficiently high frequency. When the magnetization is analysed in the frequency domain, the insurgence of frequency-sustained hysteresis can be easily and quickly checked by looking at the behaviour of the phases of the spectral harmonics.

Figure 2 shows other two examples of reconstruction of the hysteresis loop, done using the same scheme as in figure 1. The magnetic materials were chosen because the are markedly different in size and properties from typical SPION. In panel (a) the loop reconstruction method is applied to a dried powder of Co-Zn ferrite grains (composition: Co_0.84Zn_0.16Fe₂O₄), a material with large coercive field and loop area, magnetically harder [59] than iron-oxide nanoparticles (both hysteretic and SPIONs); panel (b) refers to a macroscopic piece cut from an amorphous metallic ribbon (composition: Fe_62.50 Ni_10.00 Si_11.05Cr_2.23B_11.14C_3.09), which is an extremely soft magnetic material with coercive field almost three orders of magnitude lower than the one of hysteretic iron-oxide NPs, and can be used in sensors for bioapplications.

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The measurements on Co-Zn ferrite were done in dc conditions using a VSM and a triangular H wave with maximum value of 7.5 kOe; the resulting hysteresis loop (open symbols in the right frame of panel (a)) is a major loop, i.e. it is characterized by a vertex field well above the so-called closure field, which is where the two loop's branches become superimposed and the magnetization process becomes reversible. On the other hand, the measurements on the amorphous ribbon were done at 20 Hz using a sinusoidal H field of maximum amplitude 1.78 Oe: such a value is enough to bring the material close to saturation and to obtain a major hysteresis loop as well (open symbols in the right frame of panel (b)). In both cases, the number of significant harmonics in the frequency spectrum of M is high (centre left frames in both panels) and the phases are in general different from zero or π (bottom left frames). The reconstructed loops (lines in colour in the two frames to the right) are perfectly superimposed to the experimental curves.

In spite of the strong differences existing among the magnetic materials studied in this section as well as among the measuring setups and the experimental conditions, hysteresis loop reconstruction from the spectral harmonics is always fully satisfactory.

The simple transformation formulas of section 4, relating basic magnetic properties and spectral harmonics, are shown here to precisely reproduce the experimental data. An example is given in figure 3, where experimental values taken from the hysteresis loops are compared with the expressions obtained from spectral analysis for the samples containing magnetite nanoparticles and Co-Zn ferrite grains.

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In magnetic materials the four quantities A_L , M_V , M_R , H_C are monotonically increasing functions of the vertex field from zero up to the closure field [58, 59]. A set of hysteresis loops were produced by properly increasing H_V in the two considered samples without changing the driving-field frequency.

The experimental behaviour of A_L (open symbols in panels (a) and (b)) is almost linear in the H_V range 150–470 Oe and at $f =$ 69 kHz (sinusoidal wave), i.e. in the typical conditions of MPS/MPI and magnetic hyperthermia (left-side panel); for the Co-Zn ferrite characterized by VSM with triangular H wave, the range of field amplitude values was much wider ($1\times 10^2$ to $7\times 10^3$ Oe), and A_L approaches the asymptotic maximum at high fields. The experimental results for $A_L(H_V)$ are well reproduced by the relevant transformation formulas (equations (11) and (13), respectively; full symbols in colour).

The coercive field is a particularly important parameter in applications because it gives a quick estimate of the amplitude of the magnetic field needed to switch the magnetization of a magnetic material. The notable agreement between measured coercive field and the values obtained from equations (5)–(10) is shown in panels (c) and (d) (open / full symbols, respectively).

A similar agreement is shown in panels (e) and (f) for vertex magnetization (M_V ) and remanence (M_R ). Both properties play important roles in many recent applications of magnetic materials: the vertex magnetization is proportional to the amplitude of the induced voltage signal and can therefore be used to determine, for a given magnetic nanomaterial, the minimum concentration needed to get a measurable signal [64]; on the other hand, magnetic remanence of nanoparticles is very sensitive to physical and chemical effects such as dipole-dipole interactions among distant particles, magnetic contact interactions and magnetic coherence length [65] in aggregated particles, interfacial coupling between particles and non-magnetic entities, such as molecules decorating the particle surface and/or functionalization agents [66], and can therefore be used to monitor changes with time of nanoparticle concentration, degree of aggregation, surface coverage both in vitro and in vivo.

The present results confirm that spectral analysis of the M(t) signal permits to accurately derive the most useful parameters of the hysteresis loop by applying the transformation formulas. As a practical example of the interplay between time and frequency representations of the magnetization process, the following section is devoted to a study of biocompatible ferromagnetic nanodiscs.

The frequency spectrum of the voltage V(t) induced in a pickup coil can be directly used to get information on the magnetic properties without the need of integrating the picked-up signal. Keeping in mind that

$$\begin{equation*} V(t) = - S \frac{\mathrm{d}B}{\mathrm{d}t} \approx - S \frac{\mathrm{d}M}{\mathrm{d}t} \end{equation*}$$

where S is the effective cross section of the pickup coil, the expansion for V(t) equivalent to equation (2) is:

$$\begin{equation} V(t)= \sum_{n=0}^{\infty} V_{2n+1} \cos \left[(2n+1) \omega t + \phi^{(V)}_{2n+1}\right]. \end{equation} \tag{ 16 }$$

where the $V_{2n+1}$'s and $\phi^{(V)}_{2n+1}$'s are the amplitudes (moduli) and the phases of the harmonics of the induced-voltage spectrum, related to the equivalent quantities for the magnetization by:

$$\begin{equation} \begin{split} V_{2n+1} = (2n+1) \, \omega S\, M_{2n+1} \\ \phi^{(V)}_{2n+1} = \phi_{2n+1}-\frac{\pi}{2}.\kern27.7pt \end{split} \end{equation} \tag{ 17 }$$

Therefore the $P_{2n+1}, Q_{2n+1}$ coefficients entering all transformation formulas used in this work are directly expressed in terms of the induced-voltage harmonics as:

$$\begin{equation} \begin{split} P_{2n+1} = - \, \frac{1}{(2n+1) \, \omega S} \, V_{2n+1} \, \cos \phi^{(V)}_{2n+1}\\ Q_{2n+1} = - \, \frac{1}{(2n+1) \, \omega S} \, V_{2n+1} \, \sin \phi^{(V)}_{2n+1}. \end{split} \end{equation} \tag{ 18 }$$

Equations (5) and (6) describe the upper/lower branch of a M(H) hysteresis loop under a sinusoidal driving field. An alternative expression involving increasing powers of the field is derived here. Using $\cos (\omega t) = h$ where $h = H/H_V$ and exploiting the representation of trigonometric functions of multiples of the argument in terms of powers of these functions, equations (5) and (6) can be recast as:

$$\begin{align} M(t)& = P_1h \pm Q_1 (1-h^2)^{1/2} + P_3 \, [4h^3-3h] \pm Q_3 \, [3(1-h^2)^{1/2}- 4(1-h^2)^{3/2}] \nonumber \\ & \quad + P_5[16h^5-20h^3-5h] \pm Q_5[5(1-h^2)^{1/2}- 20(1-h^2)^{3/2}+16(1-h^2)^{7/2}] + \ldots \end{align} \tag{ 19 }$$

where the plus/minus signs refer to the upper/lower loop branch. Equation (19) can be further modified by grouping all terms containing the same power of $h = H/H_V$ and $(1-h^2)^{1/2}$:

$$\begin{equation} M(H)= \sum_{n=0}^{\infty} C_{2n+1} (H/H_V)^{2n+1} \pm \sum_{n=0}^{\infty} D_{2n+1} [1-(H/H_V)^2]^{2n+1} \end{equation} \tag{ 20 }$$

where the plus/minus signs again refer to the upper/lower loop branch and the general expressions of the $C_{2n+1}$, $D_{2n+1}$ coefficients in terms of the quantities $P_{2n+1}$ and $Q_{2n+1}$ are:

$$\begin{equation} C_{2n+1}=2^{2n} \left[P_{2n+1}+\frac{1}{(2n+1)!}\sum_{k=n+1}^{\infty}(-1)^{k-n}\frac{2k+1}{k-n} \frac{(k+n)!}{(k-n-1)!}P_{2k+1}\right] \end{equation} \tag{ 21 }$$

and:

$$\begin{equation} D_{2n+1}=(-1)^n \enspace 2^{2n} \left[Q_{2n+1}+\frac{1}{(2n+1)!}\sum_{k=n+1}^{\infty}\frac{2k+1}{k-n} \frac{(k+n)!}{(k-n-1)!}Q_{2k+1}\right] \end{equation} \tag{ 22 }$$

Equation (20), fully equivalent to equations (5) and (6), is an analytical expression of the hysteretic M(H) curve of a generic magnetic material submitted to a harmonic driving field of maximum amplitude H_V and arbitrary frequency in terms of powers of the field, involving only quantities obtained from frequency analysis of magnetization.

When all harmonics of M(t) beyond the first one are completely negligible, the corresponding hysteresis loop has a perfectly elliptical shape. Actual minor loops measured on magnetic nanomaterials using fields of low amplitude are often characterized by a shape slightly deformed from the ellipse; in such an instance, only the first and the third harmonic are present all higher-order terms being negligible.

In this case that the coercive field can be obtained in closed form from equation (20) by solving the cubic equation $ax^3+bx^2+cx+d = 0$ for $x = h^2 = (H/H_V)^2$ where:

$$\begin{equation} \begin{split} &a = C_3^2+D_3^2\\ &b = \big[2C_1C_3-D_3^2-2D_3(D_1+D_3)\big]\\ & \kern1.3pt c=\big[C_1^2+2D_3(D_1+D_3)+(D_1+D_3)^2\big]\\ & \kern-0.3pt d=-(D_1+D_3)^2,\\ \end{split} \end{equation} \tag{ 23 }$$

the $C_{1,3}$ and $D_{1,3}$ quantities being obtained from equations (21) and (22). In particular:

$$\begin{equation} \begin{split} &C_1 = P_1-3P_3\\ &C_3=-4P_3\\ &D_1=Q_1+3Q_3\\ &D_3=4Q_3.\\ \end{split} \end{equation} \tag{ 24 }$$

The $P_{1,3}$ and $Q_{1,3}$ parameters were defined in equation (4). It can be easily checked that the cubic equation for x has one (and only one) real positive root. Finally, the positive and negative coercive fields are obtained by taking the square root of the solution.