1 Introduction

Following the paper “Conventional superconductivity at 203 kelvin at high pressures in the sulfur hydride system”, published in 2015 [2], several other hydrogen-rich materials under high pressure have been reported in recent years to be high-temperature superconductors based on observed drops in resistance versus temperature [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]. Many more such materials have been determined to be conventional high-temperature superconductors based on theoretical evidence [20,21,22,23,24,25]. However, little magnetic evidence has so far been provided in support of the claims of superconductivity [2, 26,27,28,29], and what evidence does exist has been strongly called into question [30,31,32,33].

In particular, these materials show no trace of magnetic flux expulsion, i.e., the Meissner effect, when cooled in the presence of a magnetic field [2, 26, 27]. They also apparently are able to screen very large applied magnetic fields [28]. This has been interpreted as indicating that the materials are “hard superconductors” with very strong pinning centers that prevent both flux penetration and flux expulsion [26,27,28]. We have argued that if that is the case the materials should also trap large magnetic fields [34], and that observation of flux trapping would provide definitive evidence that the materials can sustain persistent currents, hence are indeed superconductors [34].

Experiments aimed at detecting flux trapping were recently performed by Minkov et al. and the results analyzed and reported in Ref. [1]. Ref. [1] interprets the measured data as clearly indicating that the materials are superconductors. Instead, we analyze here the information presented in Ref. [1] and conclude that it proves the absence of superconductivity in these materials.

2 Experimental Protocols

The flux trapping experiments on sulfur hydride (\(H_3S\)) [1] were performed under zero-field-cooling (ZFC) conditions for 13 values of applied field ranging from 0 to 6T, and under field cooling (FC) conditions for one field value only, 4T. The results for both protocols for field 4T were reported to agree [1]. In the ZFC protocol, the sample was cooled to low temperatures in zero magnetic field, a magnetic field was then applied and gradually increased to reach value \(H_M\); then after 1 h, the external field was gradually decreased to zero; then, the resulting magnetic moment was measured with a SQUID magnetometer.

Fig. 1
figure 1

Trapped magnetic moment for \(H_3S\) at 30K, from Ref. [1]. The points are experimental data [35]; the lines are a fit to the data performed in Ref. [1]

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Figure 1 shows the experimental data and a theoretical fit to the data given in Ref. [1]. Note in particular that the measured magnetic moment rises linearly from zero when the applied field exceeds the threshold value \(H_p\) both for the experimental data and for the theoretical fit.

The experimental results were reportedly analyzed in Ref. [1] assuming the Bean model [36] controls the behavior of fields and currents in the material. From the experimental results, Ref. [1] inferred the parameters:

\(H_p=0.042T\)=threshold value of the applied field where it begins to penetrate the sample at low temperatures. Assuming demagnetization \(1/(1-N)=8.5\), this implies a lower critical field value \(H_{c1}=0.36T\).

\(H^*=0.835T\)= minimum applied field that reaches the center of the sample (called “full penetration field”), with assumed sample diameter and height \(d=85 \mu m\), \(h=2.5 \mu m\).

Fig. 2
figure 2

Magnetic fields and currents predicted by the Bean model under field-cooled (FC) and zero-field-cooled (ZFC) protocols. Here we assume \(H_p=0\) for simplicity

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The measured moment was found to increase with magnetic field \(H_M\) up to a maximum value of approximately \(m_s=15.9\times 10^{-9}Am^2\) for \(T=30K\) when the applied magnetic field was \(\sim 1.7T\equiv H_M^{sat}\) or larger. Following the Bean model, Ref. [1] concluded that

$$\begin{aligned} H_M^{sat}=2H^*+H_p \end{aligned}$$
(1)

from which the value of \(H^*\) was extracted.

The theoretical fit performed in Ref. [1] assumed the magnetic moment is given by (with \(j_c\) the critical current)

$$\begin{aligned} m=\int _{r}^{d/2} \pi r'^2 j_c h dr'=m_s[1-(\frac{r}{d/2})^3] \end{aligned}$$
(2)
$$\begin{aligned} r=r(H_M)=\frac{d}{2}(1-\frac{H_M-H_p}{2H^*}) \end{aligned}$$
(3)

so that \(r(H_p)=d/2\), \(r(2H^*+H_p)=0\).

3 Our Analysis

Just as Ref. [1], we assume the validity of the Bean model. However, we disagree that Eqs. (2) and (3) used by the authors of [1] are the proper way to calculate the trapped magnetic moment under ZFC conditions. Instead, we argue that Eq. (2) is the proper way to calculate the trapped moment under FC conditions, provided Eq. (3) is replaced by

$$\begin{aligned} r=r(H_M)=\frac{d}{2}(1-\frac{H_M-H_p}{H^*}) \end{aligned}$$
(4)

for \(H_M<H^*+H_p\), \(r=0\) for \(H_M>H^*+H_p\), with \(H_p = 0\). This is illustrated in the left panels of Fig. 2. For ZFC conditions instead, the diagrams shown in the right panels of Fig. 2 apply. For that case, the magnetic moment is given by

$$\begin{aligned} m=m_s[1-2(\frac{r_1}{d/2})^3+(\frac{r_2}{d/2})^3] \end{aligned}$$
(5)

where, for \(H_M<H^*+H_p\)

$$\begin{aligned} r_1=\frac{d}{2}(1-\frac{H_M-H_p}{2H^*}) \end{aligned}$$
(6)
$$\begin{aligned} r_2=\frac{d}{2}(1-\frac{H_M-H_p}{H^*}) . \end{aligned}$$
(7)

For \(H^*+H_p<H_M<2H^*+H_p\), \(r_1\) is given by Eq. (6) and \(r_2=0\), and for \(H_M>2H^*+H_p\), \(r_1=r_2=0\).

Fig. 3
figure 3

Expected trapped magnetic moment versus magnetization field \(H_M\) for the parameters assumed in Ref. [1] under FC and ZFC protocols. For small \(H_M\) the dependence is linear (quadratic) for FC (ZFC) protocols. The experimental points are also shown

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Figure 3 shows what these expressions predict for the trapped magnetic moment versus magnetization field \(H_M\) for the parameters assumed [37] in Ref. [1]. Most importantly, the moment rises from zero linearly under FC conditions and quadratically for ZFC conditions. As seen in the inset, for small fields, the ZFC moment is very much smaller than the FC moment and in stark disagreement with the experimental observations.

The experimental results of Ref. [1] are actually well fit by our FC calculation for all values of the magnetization field \(H_M\) if we take the value of \(H^*\) to be twice as large as inferred in Ref. [1], i.e. \(H^*=1.67 T\). This is shown in Fig. 4. We conclude that this agreement is accidental, since the experimental protocol was ZFC for all but one experimental point [1].

Fig. 4
figure 4

Expected trapped magnetic moment versus magnetization field \(H_M\) assuming \(H^*\) is twice the value inferred in Ref. [1], i.e., \(H^*=1.67 T\), compared to the experimental points. Remarkably, the experimental points obtained with the ZFC protocol are actually fitted by the calculation assuming FC

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In order to try to fit the low-field ZFC experimental data to the ZFC calculation, we would have to take a much smaller value of \(H^*\). Figure 5 shows the results for \(H^*=0.2 T\), chosen to fit as well as possible the low-field data. In addition to not fitting the low-field data very well, the higher field data deviate strongly from the theoretical ZFC curve. For this assumed value of \(H^*\) the trapped moment saturates for \(H_M^{sat}=0.44T\) (Eq. (1)), in clear contradiction with the experimental data that show no saturation until \(H_M>1T\).

Fig. 5
figure 5

Trapped magnetic moment versus magnetization field \(H_M\) assuming \(H^*=0.2T\) to fit the low-field experimental data. The ZFC curve saturates at \(H_M^{sat}=0.44T\), well before the experimental data

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4 Discussion

Is it possible that under the ZFC protocol of the experiment with the field \(H_M\) applied for 1 h, the field could penetrate sufficiently so as to mimic the FC protocol? It is not possible, because Ref. [1] also measured the rate of flux creep and there was negligible flux creep over a 1-h period even at temperatures as high as 165K. Also, according to the NRS experiment [28] the flux didn’t penetrate over times substantially larger than 1 h.

Therefore, the experimental results of Ref. [1] shown in Fig. 1 of this paper are incompatible with the interpretation that the magnetic moment observed originates in flux trapping. If the magnetic moment had originated in flux trapping, it would rise quadratically from zero as function of the magnetization field \(H_M\) under the ZFC conditions of the experiment, not linearly as observed. Therefore, the experiment indicates that there is no flux trapping in this material, \(H_3S\). As argued in Refs. [32, 34], if the material doesn’t trap flux, and in addition it does not expel flux, then the material is not a superconductor.

The question then arises, what is the origin of the magnetic moments measured in Ref. [1] shown in Fig. 1? We suggest they are either experimental artifacts associated with the experimental apparatus used (SQUID magnetometer) or magnetic moments of localized spins originating either in the sample or in the diamond anvil cell environment (gasket, etc.). It is also possible that the measurements could signal unexpected collective magnetic behavior of hydrogen-rich materials under high pressure, as suggested in Ref. [38].

To confirm the results of our analysis, we suggest that it would be of interest to repeat the measurements of Ref. [1] under FC conditions. We expect that the results will be similar to the results under ZFC conditions, in contradiction with what is expected from trapped flux shown in Figs. 3 and 4, namely a marked difference between FC and ZFC behavior, and consistent with the hypothesis that the origin of the magnetic moments measured is localized spins rather than delocalized supercurrents. We suggest that it would also be informative to perform these experiments using FC and ZFC protocols for a known hard superconductor and verify the expected qualitatively different behaviors shown in Figs. 3 and 4.

Finally, we would like to point out that the interpretation of the measurements of magnetic moment of Ref. [1] as originating in flux trapping with \(H^*\sim 0.8T\) appears to be in contradiction with the magnetic moment measurements presented in Ref. [26]. For example, according to the former (see our Fig. 2 top right panel) for an applied field \(H\sim H^*/4=0.2T\), the magnetic field should still be excluded from more than \(75 \%\) of the sample even at temperature \(T\sim 100K\) (see Fig. 1c of [1]). Instead, the magnetic moment measurements shown in Fig. 3a of Ref. [26] indicate that the diamagnetism has essentially disappeared at that point.