Unconventional Magnetization below 25 K in Nitrogen-doped Diamond provides hints for the existence of Superconductivity and Superparamagnetism

The magnetization of nitrogen-doped single crystalline diamond bulk samples shows unconventional field and temperature hysteresis loops at T \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\lesssim }}$$\end{document}≲ 25 K. The results suggest the existence of superparamagnetic and superconducting regions in samples with nitrogen concentration <200 ppm. Both phases vanish at temperatures above 25 K where the samples show diamagnetic behavior similar to undoped diamond. The observation of superparamagnetism and superconductivity is attributed to the nitrogen doping and to the existence of defective regions. From particle-induced X-ray emission with ppm resolution we rule out that the main observations below 25 K are due to magnetic impurities. We investigated also the magnetic properties of ferromagnetic/high-temperature superconducting oxide bilayers. The magnetization results obtained from those bilayers show remarkable similarities to the ones in nitrogen-doped diamond.


I. MAGNETIZATION OF AN UNDOPED DIAMOND SAMPLE
In the main text we have mentioned that the magnetization of pure, undoped diamond behaves as a diamagnetic material in the temperature range of 2 K to 300 K. The results of the field hysteresis at two temperatures and also as a function of temperature are shown in Fig. S1. These results indicate that after subtraction of the diamagnetic contribution the small hysteresis loop (see inset Fig. S1(a)) is of the order of 10 −7 emu, i.e. about three orders of magnitude smaller than the hysteresis found in N-doped samples below ∼ 20 K. The resolution of the used SQUID device is ∼ 2 × 10 −8 emu [1]. The magnetic moment shows a weak temperature dependence above 50 K. At lower temperatures, it shows a weak paramagnetic contribution probably due to natural defects present in the sample; the estimated concentration of paramagnetic centres obtained from this dependence is extremely small (< 0.01 ppm). The small differences of the order of 10 −7 emu between the zero field cooled (ZFC) and field cooled (FC) curves (see inset in Fig. S1(b)) are orders of magnitude smaller than in N-doped samples. From all these results we conclude that pure diamond without nitrogen doping shows basically a diamagnetic behavior in the whole temperature range, without any signs of magnetic or superconducting order within experimental resolution.

II. TIME DEPENDENT MAGNETIZATION
The magnetization as a function of time was measured in the N-doped diamond sample ND-S1 at different temperatures. The corresponding results are plotted in Fig. S2. A clear time dependence below T 25 K is observed. The results (a), (c) and (e) were measured after the magnetic field was applied. The results in (b), (d) and (f) were measured after the magnetic field was reduced from the previous field, e.g., the data at 4 T were obtained immediately after the field was reduced from 7 T. Above 25 K, the N-doped sample behaves as in the case of undoped diamond, without any time dependence within experimental error. Fig. S 3 shows the parameters t 0 and β as a function of temperature obtained from the fits to the stretched exponential function of sample ND-S1 (see Fig. 5 in the main text).
The time dependence of the magnetization below ∼ 25 K influences to some extent the shape of the hysteresis loop, as the results obtained at different waiting times shown in Fig. S4 indicate. This time relaxation is not taken into account in the fits of the field hysteresis. At this stage it is better to minimize the number of free parameters to convince the reader on the main contributions that influence the field hysteresis loops.

III. TEMPERATURE AND FIELD DEPENDENCE OF THE MAGNETIC MOMENT OF THE DIAMOND SAMPLES WITH DIFFERENT NITROGEN CONTENT
In the present work a total of 8-samples with different N-content were investigated, seven of them were obtained by the Japanese company Sumitomo and one was delivered from the Russian Institute. The results of the six samples not shown in the main text can be seen in Fig. S5. The samples in (a) to (e) are labeled according to their original identification number from the Sumitomo company. As the company indicates, the N-content in the diamond samples is between 10 ppm to 100 ppm. The results presented in Fig. S5 are the raw data, very similar to the one presented in the main text. The differences are related to the N-content and not to the different masses of the samples, i.e. the samples with high N-content show a larger absolute hysteresis (e.g. samples CD2016 and CD2318-01) than the ones with less N-content (e.g. sample CD1512-02). Figure 7 of the main text shows the correlation of the magnetization at 2 K with the N-content of all samples.

IV. MAGNETIC MOMENT OF THE YBCO FILM AND THE BILAYERS LSMO/YBCO AND Ni/YBCO
To support our interpretation regarding the coexistence of superconductivity and superparamagnetism in the Ndoped diamond samples, we have measured bilayers with a superconductor and a ferromagnetic material. Two different ferromagnetic materials and the high temperature superconductor YBa 2 Cu 3 O 7−x (YBCO) were used. One investigated bilayer had ferromagnetic La 2/3 Sr 1/3 MnO 3 (LSMO) and the second sample nickel (Ni). In the first sample both materials were deposited by pulsed laser deposition (PLD) [2]. In the second sample YBCO was deposited first by PLD and the Ni film afterwards by thermal evaporation. Because of the growth conditions, the sample LSMO/YBCO was made in one process, but in the case of the sample Ni/YBCO, we had the chance to measure initially the YBCO sample before depositing the Ni. The results of the YBCO film alone are shown in Fig whereas for the diamagnetic term we used a linear in field m dia (H) = −a(T )H, where a(T ) is a sample and temperature dependent diamagnetic slope. For the FM/SPM contribution we used the phenomenological model proposed by Jiles et al. [3,4] for the case of an isotropic material in which the magnetization has essentially no preferred direction. The with m 0 the saturation magnetic moment, the fit parameters α and a (as well as dM/dH) are described in [4]. The main differences in the simulation of the FM and SPM contributions is that in the last no field hysteresis is assumed and the low field susceptibility depends strongly on temperature.
For the SC contribution we have used the theory developed by Irie and Yamafuji [5], which is a further development of the Bean model, and the equation is as follow: with H p = B p /µ 0 the penetration field, and γ a free pinning dependent parameter (γ = 1 corresponds to the original Bean model) and c(H) a complex function, see [5]. Using the above equations for each contribution in Eq. (1) we have modeled the experimental results of the bilayers and of the N-doped diamond samples (assuming for simplicity γ = 1 in this last case).

A. Results of the FM/SC bilayers
The first results we present in Fig. S11 are of the LSMO/YBCO sample at T = 65 K, where the continuous line is the result of the fit obtained from the addition of the three contributions shown in (b) to (d). Fig. S12 shows the field hysteresis loops at four different temperatures and the corresponding fits for the same bilayer. Fig. S13 shows the results of the Ni/YBCO bilayer. The fitting curves describe the experimental results rather well and at all temperatures. For both bilayers we observe that the obtained penetration field B p = µ 0 H p from the fits, see Fig. S14, follows the empirical equation B p (T ) = B 0 (1−(T /T c ) 2 ) with critical temperatures similar to that obtained from the magnetization measurements of the YBCO layer alone, see Fig. S7. The penetration field B p (T ) from the fits is of the same order as the one we obtain from the single YBCO layer. We note that the values of B p (T ) obtained from the YBCO alone and shown in Fig. S14 were multiplied by a constant factor 1.5. This change in the values of B p can be partially attributed to some influence of the FM layer on the vortex pinning in the YBCO layer.

B. Results of the N-doped diamond samples
In Fig. S 15 we have plotted all the field hysteresis and the fit results obtained at T = 2 K of the investigated N-doped diamond samples. The fit results describe well the experimental results, supporting our interpretation that the N-doped diamond samples contain regions with SPM and SC properties. Some of the differences between the experimental results and the results of the model can be due to the time dependence due to creep, which is not consider in the calculations.

VI. INFRARED SPECTROSCOPY
Infrared (IR) spectra are collected with a resolution of 1 cm −1 using a Varian FTS6000 Fourier Transform Infrared spectrometer equipped with an UMA600 microscope. Data were corrected with a linear baseline between 980 cm −1 and 1400 cm −1 (Fig. S 16 (e1)). The concentration of the A-defects (c A ) was determined according to the absorption coefficient in cm −1 at 1282 cm −1 : c A = [16.5 ± 2]ppmcmα 1282 ppm [6,7]. The concentration of the C-(P 1 − ) defects in ND-S1 sample as determined from EPR (86 at ppm) is used to calculate the molar absorption coefficient, C = 44.6 ppm cm, of the C-defects [8] at 1344.5 cm −1 : c C = C α 1344.5 .
To the best of our knowledge, no molar absorption coefficients for B-type defects are reported for type IaAB diamonds [9], and apparently no one has yet managed to measure a real concentration of nitrogen in diamond [7]. We therefore approximate the concentration of B-defects, c B , by the absorption of the B-specific peak (ν B ) at 1332.3 cm −1 . In this spectral region also A defects add some absorbance [6], which needs to be subtracted before. For all samples, especially those without ν B (Fig. S 16 (e1)), a step in absorption coefficient, α step , exists between 1400 cm −1 and 1316 cm −1 , and its height is proportional to c A : Consequently, we assign α step , to the A-defects, and subtract it before determining c B (Fig. S 16 (e2)).
The molar absorption coefficient of the B-defects, B at 1332.3 cm −1 , is estimated from the total amount of dispersed nitrogen c tot = c A + c B + c C . For ND-S1 no B-defects are found, and hence c tot = c A + c C = 100 ppm. This value is in agreement with the manufacturers information. Assuming the same c tot for sample 2318-01 yields: where B is strongly dependent on c tot . For example, for c tot = 150 ppm one finds B = 210 ppm cm. But in this case c tot exceeds 200 ppm for sample 1512-03, a concentration twice as high as given by the manufacturer.
In summary, we can determine the concentration of A-and C-defects with an accuracy about ±10% based on published studies. We propose a new approach to determine the concentration of B-type defects in type IaAB diamonds, which suffers from the uncertainty of the total nitrogen concentration. But the approach allows for a quantitative comparison of c B between different samples. Furthermore, no indications of boron [9] or nitrogenplatelets [8] (area defect) were found in the IR-spectra.

VII. ELECTRON PARAMAGNETIC RESONANCE AND PHOTOLUMINESCENCE
Electron paramagnetic resonance results shown in the inset of Fig. S17 confirm the existence of the well-known P1 centre in our samples. Photoluminescence spectroscopy measurements were done with two different wavelengths,     (e) IR-spectra and analysis: After subtraction of a straight baseline (black dashed lines) the peak at 1344 cm −1 (νC) is fitted with a Lorentzian (in red). To analyze the peak at 1335 cm −1 (related to B-defects), a scaled spectrum of a B-defect free sample (αstep: magenta) is subtracted before fitting the peak with a Lorentzian. The residuum of fit and baseline is shown as a cyan line. Spectra are shifted vertically for easy comparison. Panel (e2) is a magnification of (e1), and the spectrum of the sample 1512-02 has been scaled by a factor of 2. Photoluminescence spectrum measured at room temperature using two different wavelengths on sample ND-S1. The inset shows the EPR spectrum measured at room temperature that shows the existence of the so-called P1 centre due to the nitrogen doping.