In 2015, Eremets and coworkers reported high temperature superconductivity in sulfur hydride (hereafter \(H_3S\)) under pressure [4], starting the hydride superconductivity epoch. Since then to the present, considerable evidence for superconductivity in various pressurized hydrides has been presented based on resistance measurements [5]; however, little magnetic evidence for superconductivity has been reported so far. In the original paper [4] some magnetic evidence based on SQUID measurements was presented. After a 7 year hiatus, new magnetic evidence was presented by Minkov et al. in [1]. That evidence is the focus of this paper. We discuss here the magnetic measurements reported for sulfur hydride (\(H_3S\)), but exactly the same considerations apply to the same measurements reported for lanthanum hydride (\(LaH_{10}\)) in Ref. [1], the only other hydride material for which magnetic measurements have been reported to date.

We raised the issues discussed in this paper with the authors of Ref. [1] through emails on repeated occasions beginning October 2022, with no response from the authors. We also submitted a Comment on Ref. [1] to Nature Communications raising these issues in November 2022. Five months later the journal decided that the technical details which explain the discrepancies that we point out will not be of sufficient interest to the wide readership of Nature Communications. Therefore we are bringing here these issues to the attention of researchers in the field that we believe could benefit from this information.

Figure 1 top left and right panels reproduce Fig. 3a and Fig. 3e of ref. [1] respectively. To the best of our understanding, according to the description provided in the paper, both panels show in their light blue and blue curves respectively the same quantity: magnetic moment versus magnetic field, for the same sample at the same temperature (100K) and same pressure (155 GPa). The center blue curve in the top right panel is the virgin curve, which starts (when properly shifted vertically, as shown in Fig. S10 of [1]) with zero moment for zero applied field. It should be the same as the light blue curve labeled 100K on the top left panel. Yet the curves look very different. The left panel curve shows an upturn for magnetic field beyond 95mT while the right panel curve show no upturn. When plotting both curves on the same scale in the bottom panel in Fig. 1 it is apparent that they are very different in magnitude and shape.

Fig. 1
figure 1

a Magnetic moment versus applied field for \(H_3S\) under pressure, from Fig. 3a of Ref. [1]. b Magnetic moment versus applied field in a hysteresis cycle, from Fig. 3e of Ref. [1]. The middle blue curve in (b) is presumably the virgin curve, which should be identical to the light blue curve in (a) labeled 100K. c Quantitative comparison of the virgin curves for 100K from a (Fig. 3a of Ref. [1]) and b (Fig. 3e of Ref. [1])

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It should also be noted that the rapid decrease in the magnitude of the magnetic moments beyond the minimum points of the curves shown in Fig. 1 top left panel is inconsistent with what is expected for a type II superconductor with very large upper critical field [6], estimated in Ref. [1] to be \(H_{c2}(T=0)\sim 97T\). For example, at \(T=100K\) \(H_{c2}(T)\) should be above 60T. When corrected for demagnetization factor estimated as \(1/(1-N)\sim 8.5\) in Ref. [1], it implies that the curve labeled \(T=100K\) should evolve smoothly from its value attained at \(H \sim 95mT\) to reach zero at \(H_{c2}(T)(1-N) \sim 7T\). This is qualitatively inconsistent with the behavior seen in Fig. 1 top left panel that shows that the magnetic moment magnitude has already decreased to less than \(15\%\) of its maximum value for field as small as \(H\sim 0.2T\sim H_{c2}(T)(1-N)/35\).

Figure 2 shows as a green curve the magnetic moment versus magnetic field for a hysteresis cycle at temperature 100K for the same sample at the same pressure, reported in Fig. 4a of Ref. [2]. In the same figure we show the magnetic moment versus magnetic field at the same temperature from the left panel of Fig. 1, i.e., Fig. 3a of Ref. [1]. The blue curve on the left panel of Fig. 2 should be the virgin curve for this hysteresis cycle, joining smoothly the green curve, as is universally seen in such measurements for superconductors. One such typical example is shown on the right panel of Fig. 2, from Ref. [7]. It can be seen that the blue curve on the left panel shows no hint of joining the green curve. In other words, these measured results on the same sample for the same temperature and pressure measured in the same laboratory are completely incompatible with each other under the assumption that they arise from superconductivity in the sample. Note that the results from the hysteresis cycles from Fig. 4a of Ref. [2], for temperatures \(T=100K, 140K, 160K, 180K\) were used to infer the values of critical current versus temperature plotted in Fig. S5 of Ref. [1] (and Fig. 4c of Ref. [2]). For these higher temperatures the magnetic moment curves of Fig. 3a of Ref. [1] are equally incompatible with the hysteresis cycle curves of Fig. 4a of Ref. [2].

Fig. 2
figure 2

Green curve, left panel: hysteresis cycle for magnetic moment of \(H_3S\) at 100K, from Fig. 4a of Ref. [2]. The blue curve on the left panel shows the magnetic moment versus magnetic field for 100K from the light blue curve on the top left panel of Fig. 1, which is Fig. 3e of Ref. [1]. Right panel: a typical hysteresis cycle for a type II hard superconductor, from Ref. [7]. The virgin curve starting at the origin smoothly joins the hysteresis loop curve

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In Fig. 3 we consider the magnetic moment measurements of Ref. [1] at lower temperature in relation with the flux trapping results under zero field cooling (ZFC) reported in Ref. [3] for \(T=30K\). We show the curve for magnetic moment from Fig. 1 for \(T=20K\), the curve for \(T=40K\) it is very similar, as seen in Fig. 1. The ZFC flux trapping data at low fields, shown in the inset of Fig. 3, show onset of field trapping at field \(H_p^t=42 mT\). That necessarily implies that the sample is allowing the magnetic field to penetrate. Yet the magnetic moment behavior shown in Fig. 3 main panel shows that the diamagnetic moment at 20K continues to grow linearly up to \(H_p=95mT\), indicating that the field is not penetrating. \(H_p\) was identified in Ref. [1] as the field corresponding to \(H_{c1}\) when corrected for demagnetization, where the magnetic field starts to penetrate the sample. However, there can be no flux trapping for fields below the lowest field for which the field starts to penetrate, namely \(H_p\). Hence \(H_p^t<H_p\) is an impossibility. Thus, these reported measurements are in clear contradiction with each other.

Fig. 3
figure 3

Red curve: magnetic moment versus field from Fig. 1 left panel (Fig. 3a of Ref. [1]) at \(T=20K\). Inset: trapped field under ZFC (zero field cooling) and FC (field cooling) protocols at T = 30K, from Ref. [3]. \(H_p^t=0.042T\) is the threshold field for onset of trapping under ZFC conditions. The curve shows the behavior of the induced moment expected from the Bean model, which was used to interpret the field-trapping results in Ref. [3]. It also shows (blue curve) the qualitative behavior expected for an ideal type II superconductor with no pinning, where the moment would reach zero at the upper critical field (corrected for demagnetization)

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Furthermore, the ZFC trapped moment measurements found a saturation moment \(m^s\sim 16\times 10^{-9} Am^2\) [8]. This is larger that the largest diamagnetic moment induced under an applied field, \(-12.5 \times 10^{-9} Am^2\), as shown in Fig. 3. It is usually the case that the remnant (trapped) moment obtained after a magnetic field is applied and then removed is smaller than the largest diamagnetic moment generated while the field is applied, as seen, e.g., in Fig. 2 right panel. While the data for the diamagnetic moment are shown only up to field 200mT, it is clear from the behavior seen in Fig. 3 that the induced diamagnetic moment certainly would not increase again for larger field (if it did, it would be completely inconsistent with standard superconductivity).

The trapped flux measurements of Ref. [3] were interpreted according to the Bean model, where the applied field penetrates partially when it exceeds the lower critical field and is prevented from further penetrating due to strong pinning centers. The model assumes that a critical current \(j_c\) flows that is independent of field magnitude, estimated in Ref. [3] to be \(j_c\sim 7.3\times 10^{10} A/m^2\). When the external field is removed, Faraday’s law implies that a reverse current is induced, and field remains trapped due to the same pinning. The maximum trapped moment under ZFC conditions was found in Ref. [3] to be \(m_s\sim 16 \times 10^{-9}Am^2\) attained for applied field \(H_M\sim 1.7T\), from which it was inferred that the field magnitude necessary for the applied field to reach the center of the sample was \(H^*\sim 0.8T\). With those parameters, the Bean model predicts [9] the behavior of magnetic moment versus applied field shown in Fig. 3, qualitatively different from the measured behavior. Allowing for variations in the critical current with magnetic field would somewhat change this behavior [11], e.g., to what is shown in Fig. 2 right panel, but its magnitude would never fall below what is expected for an ideal type II superconductor, shown as the blue curve in Fig. 3. Thus, the reported behavior of magnetic moment versus field shown as the red curve in Fig. 3 is inconsistent with the ZFC trapped flux results reported in Ref. [3]. We have previously also pointed out that the reported ZFC linear behavior of moment versus field shown in the inset of Fig. 3 is inconsistent with what is expected for a superconductor [9].

Ref. [1] uses a background subtraction procedure that involves “recent improvements in the background subtraction procedure (that) have greatly expanded the scope of the method” [3]. However, neither is the background signal given in Ref. [1] nor is the improved procedure explained. Perhaps, information on the data processing that has been performed, together with the raw data obtained in the measurements [10], would help explain some of the anomalies pointed out above. But even with such clarification we believe that the above analysis indicates that the various reported magnetic measurements are incompatible with one another under the assumption that they originate in superconductivity. Instead, we suggest that they must originate in localized magnetic moments associated with the samples, the diamond anvil cell environment, and/or the measuring apparatus.

The signature property of superconductors, that cannot be mimicked by localized magnetic moments, is the Meissner effect, the ability to expel magnetic fields when cooled in a field (FC). In Ref. [1], the authors claim to find “subtle Meissner effect in FC measurements at 2 mT” indicated by the light blue smoothed curve shown in Fig. 4 middle panel. However, in the absence of the blue curve no such evidence is apparent, as Fig. 4 left panel shows. In addition, before laser heating the precursor sample is not expected to be superconducting, yet the same measurements show an unexpected divergence between FC and ZFC moments around the supposedly critical temperature, as shown on the right panel of Fig. 4. Similar behavior as in Fig. 4 is shown also for a field of 4mT in Ref. [1].

Fig. 4
figure 4

Moment versus temperature for FC and ZFC with magnetic field 2mT. The left panel is from Ref. [2] Fig. SI1, left middle panel; the middle panel is from Ref. [1] Fig. S1, left middle panel; the right panel is from Ref. [1] Fig. S1, left bottom panel. The figure caption for the middle panel reads in Ref. [1] “Light blue smoothed curve shows the subtle Meissner effect in FC measurements at 2 mT”

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While for some standard superconductors with strong pinning the percentage of flux expulsion (Meissner fraction) can be very small for larger fields, it rapidly increases for small fields, as shown, e.g., in Refs. [12,13,14,15]. The Meissner fraction is expected to depend on the ratio \(H/H_{c1}\) [16], and for \(H_3S\) \(H_{c1}\) is estimated to be 0.82T [1], which is more than an order of magnitude larger than lower critical fields for standard superconductors with high \(T_c\) such as cuprates and pnictides. So the field 2mT of Fig. 4 is equivalent to a field of less than 2 Oe for those materials, for which a sizable Meissner fraction is found [12,13,14,15]. Additionally, the Meissner fraction is expected to increase as the thickness of the sample decreases [14, 17], and the samples used in these high pressure experiments are rather thin. The absence of any evidence of flux expulsion for hydride materials under pressure, contrary to all other known superconductors, is incompatible with the claim that these materials are superconductors.

In summary, the considerations in this paper, together with other analysis of magnetic evidence for superconductivity in sulfur hydride and lanthanum hydride discussed recently [9, 18,19,20,21], strongly indicates that the magnetic measurements interpreted to show superconductivity in hydrides under pressure do not originate in superconductivity.