In 2015, Drozdov et al. reported the discovery of high-temperature superconductivity in sulfur hydride under high pressure [5]. Two subsequent experimental papers claimed to demonstrate unambiguously that sulfur hydride has the magnetic behavior expected for a superconductor: (a) In 2016, Troyan et al. reported [2] that the material excludes applied magnetic fields from its interior, by using a nuclear resonant scattering (NRS) technique that detected the presence or absence of magnetic field in a thin Sn foil inserted in the interior of the sample; (b) in 2021, Minkov et al. reported [3, 4] magnetization measurements on sulfur hydride with improved samples and improved measurement technique and claimed that they provide “definitive evidence of the Meissner effect.”

We have recently pointed out that the reported NRS experimental results [2] are inconsistent with the properties of standard superconductors under the assumption that the superconductor is in thermodynamic equilibrium in the London-Meissner state [6]. We have also recently argued that the magnetization results reported in [3, 4] indicate that the material is not a superconductor [7].

Recently, Prozorov and Bud’ko (hereafter denoted PB) reexamined the NRS experiment in great detail [1]. The NRS technique is based on the Mössbauer effect and uses resonant X-ray scattering from synchrotron radiation [8], allowing the study of very small samples. PR concluded [1] that the results [2] are inconsistent with the system being in the London-Meissner state, in agreement with [6], but that the observed results are fully consistent with the system being in the Bean critical state, where magnetic flux partially penetrates but is prevented from fully penetrating by vortex pinning. In this Comment, I point out that the PB analysis implies qualitatively different behavior for magnetization versus magnetic field than what was reported by Minkov et al. [3, 4].

The sample used in the NRS experiment was a thin disk of diameter \(30 \mu m\), and we consider the geometry where the applied magnetic field is perpendicular to the plane of the disk. Ref. [2] reported that an applied magnetic field of magnitude 0.68T was fully excluded from the interior region of the sample occupied by a Sn thin foil of diameter \(15 \mu m\) for temperatures below 60K.

Fig. 1
figure 1

Behavior of an applied magnetic field in the interior of the sample according to the Bean model and ref. [1]. The blue lines are given in Fig. 5 of ref. [1] corresponding to an applied magnetic field 0.68T. The brown lines correspond to an applied magnetic field 0.2T

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Figure 1 shows the behavior of the applied magnetic field 0.68T in the interior of the sample according to PB, assuming a critical current value that leaves the interior of the sample of diameter \(15 \mu m\) field free, as required to be consistent with the NRS experiment. The critical current density \(J_c\) determines the slope of the magnetic field inside the sample according to the Bean model: \(tan \alpha =\mu _0J_c\). Assuming the same critical current, the brown lines show the expected behavior for an applied magnetic field 0.2T. The actual values of the magnetic field at the edge of the sample are of course larger than the applied fields 0.68T and 0.2T due to demagnetization, as is shown schematically in Fig. 1. As can be seen in Fig. 1, an applied field of 0.2T leaves a portion of the sample of diameter \(27 \mu m\) field free.

Figure 2 shows the measured magnetic moments as a function of magnetic field reported in ref. [3, 4]. Focusing on the curve for temperature 20K, it shows linear behavior reaching maximum magnitude \(m\sim 12.5\times 10 ^{-6} emu\) at an applied field value \(H_p\sim 0.1T\), which is interpreted to be the point where the applied magnetic field starts to penetrate into the sample [3, 4]. At that point, the magnetization curve turns around sharply, and reaches a value of approximately \(m\sim 6\times 10 ^{-6} emu\) for applied field 0.2T.

Fig. 2
figure 2

Magnetic moment vs. applied magnetic field for sulfur hydride, from ref. [3, 4], and expected behavior of the moment at \(T=20K\) according to the physics shown in Fig. 1 implied by the NRS experiment

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However, we would expect qualitatively different behavior according to the scenario shown in Fig. 1. The magnetic moment of a flat disk of radius r under an applied magnetic field H is given approximately by [5] \(m \sim 0.2 r^3 H\). When the magnetic field is increased from 0.1 to 0.2T, if we assume the magnetic moment originates only from the field-free part of the sample, it should increase for the sample of Fig. 1 by a factor \(0.9^3\times 2=1.46\). In reality, the outer part of the sample where the field partially penetrates also contributes to the magnetic moment, so a more accurate estimate is an enhancement by a factor \(0.95^3\times 2=1.71\). So the measured magnetic moment for applied field 0.2T should be approximately \(21.4\times 10 ^{-6} emu\), and certainly larger than the lower bound \(18.25 \times 10 ^{-6} emu\), instead of the measured \(m\sim 6\times 10 ^{-6} emu\). Both these estimates are shown in Fig. 2, labeled “expected.” Similarly, for the curves for temperatures 40K and 60K, we would expect the magnetic moment to continue decreasing rather than to start increasing beyond \(H=0.1T\) as Fig. 2 shows, given that in the NRS experiment the region occupied by the Sn foil remains field free for those temperatures also for applied field 0.68T.

In fact, because the sample used in the Minkov et al. experiment [3, 4] was considerably larger than the one used in the NRS experiment [2] (diameter \(85 \mu m\) vs. \(30 \mu m\)), the discrepancy between observed and expected behavior is even larger than what is shown in Fig. 2. The value of the critical current estimated to be compatible with the experimental results in refs. [1] and [3, 4] was comparable, \(4.4\times 10^{10} A/m^2\) and \(7\times 10^{10} A/m^2\).

We conclude that the measurements reported in ref. [2] analyzed in detail in ref. [1], and the measurements reported in ref. [3, 4], both interpreted to unambiguously demonstrate superconductivity in sulfur hydride, are incompatible with each other. Either the behavior found in ref. [2], or the behavior found in ref. [3, 4], or both are not indicative of superconductivity. This casts doubt on the conclusion that sulfur hydride under pressure is superconducting, in agreement with the analysis of ref. [7].