SUPPLEMENTARY MATERIAL
We have shown in the main body of the article the experimental validity of the D-D-SI technique
to find and separate the various contributions present in a measured hysteresis loop and to isolate
the sole intrinsically hysteretic contribution of the ferroic material both for ferromagnetic and
ferroelectric hysteresis loops. In this section we would like to further validate the choice of a
Voigt distribution as the most appropriate fitting function, by studying its shape and its ability to
adequately fit the function used in the already cited Ising and Langevin models15.
As a reminder, a Voigt distribution is the convolution between a Gaussian and a Lorentz
∞
distribution, of the form 𝑉(𝑥; 𝜎, 𝛾) = ∫−∞ 𝐺(𝑥 ′ ; 𝜎)𝐿(𝑥 − 𝑥 ′ ; 𝛾)𝑑𝑥′), where 𝐺(𝑥; 𝜎) ≡
the centered Gaussian profile, and 𝐿(𝑥; 𝛾) ≡
𝛾
𝜋(𝑥 2 +𝛾2 )
2
2
𝑒 −𝑥 /2𝜎
𝜎√2𝜋
is
is the centered Lorentzian profile.
As shown in Figure 1, the first derivative of these two functions, Ising and Langevin, are
bounded by the Gaussian and Lorentz distributions over the complete range of real numbers.
This suggests that the Voigt distribution may indeed be a good fitting function for the first
derivative of the two components of the split hysteresis loop.
Figure 1: Comparison of the first derivative of the functions used in the Ising and Langevin models together with Gaussian and
Lorentz distributions. First derivatives are upper bounded by the Lorentz distribution and lower bounded by the Gaussian
everywhere except between both half maximum, where the Gaussian and the Lorentz distribution “switch places” becoming the
upper and lower bound respectively. The fact that the values of the Ising and Langevin functions are always between the values
of the Gauss and Lorentz distributions is a hint of how the Voigt is possibly a good fitting curve, being the convolution of the two.
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For further validation, we performed a detailed study of the ability of each distribution (Gauss,
Lorentz and Voigt) to fit split hysteresis loop first derivatives, using simulated curves calculated
using Ising and Langevin models. Relative errors values on each fitting parameters have been
chosen as criteria to determine the reliability of a function (Figure 2); the relative error was
calculated by taking the peak amplitude, the peak position (Shift), the full width at half
maximum (FWHM) and the offset of the fitting function and calculating the relative difference
of each of them with the parameters of the function to fit (the simulated curves calculated using
Ising and Langevin models). Note that the fit of Ising using Ising and Langevin using Langevin
gives no relative error.
Figure 2: Relative errors on fit parameters obtained after fitting an Ising model (a) and Langevin model (b) -based hysteresis
loops with several functions.
We conclude that a Voigt function can represent fairly well the Ising model and the Langevin
model, since the error for each parameter is minimum when using a Voigt function, compared to
pure Gauss or Lorentz functions.
In order to further test how robust a fitting function is the Voigt distribution, we introduced white
noise in the simulated curves calculated using the Ising and Langevin models, and studied its
effect on the fit parameters. In Figure 3, we present the relative error on each parameter
depending on the noise level characterized by the Signal-to-Noise Ratio (SNR) and presented for
decreasing values of the SNR (i.e. increasing noise levels). Our results show that using a Voigt
function as fitting function gives very reasonable and similar relative errors down to very low
SNR.
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Figure 3: Relative errors on fit parameters obtained after fitting a simulated noisy hysteresis loops based on the Ising model (a)
and a simulated noisy hysteresis loops based on the Langevin model (b) with a Voigt distribution.
As mentioned earlier, another very useful feat of the technique presented is the possibility of
obtaining the main characteristics of the hysteresis loop directly from the resulting fitting
parameters, namely:



The total of the diamagnetic, paramagnetic, dielectric, and paraelectric contributions is
given by the constant offset since such effects vary linearly with the applied field;
The instrumental contribution, and leakage currents for ferroelectric measurements, are
represented by a polynomial function;
The values of the coercive fields are given by the peak position.
The values of the remanent and saturation magnetization cannot be found directly from the
fitting parameters, given the complicated nature of the Voigt distribution, but an upper and lower
bound can be found (by solving the Gauss and Lorentzian functions), and they will be a function
of the amplitude, of the peak position and of the FWHM of the Voigt distribution. However, the
remanent and saturation magnetization can be easily found graphically from the hysteresis loops
calculated by integrating the Voigt distributions used for the fit.
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