Magnetic field screening in hydrogen-rich high-temperature superconductors

In the last few years, the superconducting transition temperature, Tc, of hydrogen-rich compounds has increased dramatically, and is now approaching room temperature. However, the pressures at which these materials are stable exceed one million atmospheres and limit the number of available experimental studies. Superconductivity in hydrides has been primarily explored by electrical transport measurements, whereas magnetic properties, one of the most important characteristic of a superconductor, have not been satisfactory defined. Here, we develop SQUID magnetometry under extreme high-pressure conditions and report characteristic superconducting parameters for Im-3m-H3S and Fm-3m-LaH10—the representative members of two families of high-temperature superconducting hydrides. We determine a lower critical field Hc1 of ∼0.82 T and ∼0.55 T, and a London penetration depth λL of ∼20 nm and ∼30 nm in H3S and LaH10, respectively. The small values of λL indicate a high superfluid density in both hydrides. These compounds have the values of the Ginzburg-Landau parameter κ ∼12–20 and belong to the group of “moderate” type II superconductors, rather than being hard superconductors as would be intuitively expected from their high Tcs.


INTRODUCTION
Current search for room-temperature superconductivity has progressed due to advances in high-pressure studies.A decade ago, superconductivity was as assumed to be a solely low-temperature phenomenon with the highest Tc of approximately 130 K found in mercury 4 -and thallium 5 -containing cuprate superconductors.The discovery of superconductivity in H3S with a Tc of 200 K 1 changed this paradigm and paved the way for higher Tcs in conventional superconductors for which BCS theory [6][7][8] was established.Thereafter, Tcs up to 250 K were found in transition metal superhydrides YH9 9 and LaH10 10- 12 .These experimental discoveries were anticipated by accurate theoretical predictions [13][14][15] , which became feasible owing to the continuous development of ab-initio methods for prediction of crystal structure and computation of superconducting properties [16][17][18] .
Although extreme pressure conditions render these record high-Tc superconductors unsuitable for technological applications, the study of high-temperature superconductivity in various phases of hydrogen-rich compounds has highlighted the fundamental problem of the ultimate value of Tc possible in a superconductor.Latest theoretical predictions suggest that very high multi-megabar pressures are required for superconductivity substantially above room temperature 19,20 .Another promising direction of high-pressure studies includes investigating effective methods for the stabilization of high-Tc phases at lower pressure values, ideally at ambient conditions, for example by doping [21][22][23] .
However, high pressures necessitate tiny micrometer-sized samples and drastically limit the available techniques for experimental study of superconductivity.When cooled below its critical temperature, a superconductor exhibits two characteristic properties: it abruptly loses electrical resistance and generates superconducting current loops that cancel the imposed magnetic field within its interior (Meissner effect).These two fundamental effects should be observed experimentally for the ultimate evidence of superconductivity in a newly discovered material.
Electrical transport measurements have been successfully adopted in the a multi-megabar pressure range and have become the primary experimental technique to detect superconductivity [24][25][26] .They provide data on Tc and its dependence on pressure, and the estimates of the upper critical magnetic field 0Hc2 and coherence length  of superconductors, as they are performed in high external magnetic fields.Other key parameters of superconductivity, such as the lower critical field Hc1 and London penetration depth L, can be obtained from magnetic susceptibility measurements.This type of measurements is crucial for understanding the complex behavior of type-II superconductors in a magnetic field, particularly for the study of vortices.
Two techniques are currently used for magnetic susceptibility measurements at high pressures: a double modulation ac technique with a system of coils 27 , and a superconducting quantum interference device (SQUID) 28 .The first technique provides only a qualitative indication of superconducting transition and is not widely distributed due to its complexity and low sensitivity 29,30 .Conversely, static dc measurements in SQUID are simpler, and the results are well-interpreted because they comprise absolute values of magnetic susceptibility.Initially, such measurements were limited to a pressure of approximately 10 GPa, and the registered signal was severely distorted by the magnetic background of bulky diamond anvil cell (DAC) 31,32 .The development of a miniature DAC capable of generating pressure values above 150 GPa 1,28 and recent improvements in the background subtraction procedure 33 have greatly extended the scope of the method.Nevertheless, these measurements are at the limit of their experimental capability.Herein, we dramatically improved studies of the magnetic properties of superconductivity at high pressures by introducing the trapped flux method.This method is based on the detection of the magnetic flux, that originates from a conjunctive action of several properties/effects in superconductors (the persistent dissipation-free currents flow, quantum split of magnetic field into Abrikosov vortices, the pinning of the vortices by structural defects, etc.) and traps in a sample after switching off an external magnetic field.These measurements have several advantages over typical magnetic susceptibility measurements, as they are performed in zero magnetic field and thus, the huge magnetic background signal from the DAC is cancelled.The signal from the trapped flux can be approximately one-two orders higher than the response from the zero-field cooled (ZFC) field-screened state at low magnetic fields, and therefore, significantly smaller samples can be studied.This method is particularly beneficial for the study of superconductors having large pinning (which has been observed in several recently discovered superconductors), and is effective at distinguishing between different phases in multiphase samples, probing critical current densities jc in a wide temperature range, and is more precise at determining Hc1.

RESULTS
Im-3m-H3S sample pressurized at P 155±5 GPa showed pronounced superconducting transition in ZFC m(T) magnetization measurements with a Tc 195 K, 18 months after its synthesis (see Figure 1a).At lower temperatures, the sample underwent the second superconducting transition with a Tc 15 K (seen as a separation of the ZFC and field-cooled (FC) portions of m(T) data on the background of the DAC signal in Figure 1b).The detected low-temperature superconductivity most probably corresponded to elemental sulfur, which was extruded on bevels of diamond anvils during the pressurization of the sandwiched sample of NH3BH3+S and remained intact after the pulsed laser-assisted synthesis (see inset in Figure 1b).The superconducting transition in S was not sharp and extended in the temperature range of 12-18 K, which could be attributed to the substantial pressure gradients on the beveled slope of the diamond anvils.According to the universal diamond edge Raman scale 34 , the pressure dramatically decreased from 140 GPa at the edges of the anvil culet to 95 GPa at the outer edge of the ring-shaped sample of sulfur.The measured Tc was in good agreement with previous observations of superconductivity in sulfur at high pressures obtained from electrical transport 35 and ac magnetic susceptibility 29 measurements.
1. Trapped flux measurements.The magnetization of a type-II superconductor is irreversible due to the pinning of vortices if the superconductor contains defects, such as dislocations, precipitates, and phase boundaries, which interact with the flux lines penetrating the superconductor above Hc1.We created a trapped flux in Im-3m-H3S under ZFC and FC conditions (run 1 and 2).ZFC mode consisted of cooling the sample at zero field from TTc to the target magnetization temperature TMTc, switching on an applied magnetic field HM and gradually decreasing of the magnetic field to 0 T (see details in Methods section).In FC mode, the sample was cooled from TTc at applied magnetic field HM, which was gradually decreased to 0 T at the target lowest temperature TM.
The temperature-dependencies of the trapped magnetic moment mtrap(T) generated in H3S under ZFC and FC conditions at different HM and measured between 10 and 250 K at zero field clearly showed a superconducting transition with a Tc ~195 K (see Figure 1c, d), and further revealed a steep upturn below 15 K, which originated from pure superconducting sulfur.The entry of magnetic flux into sulfur occurred at 0HM = 30 mT in ZFC mode and already at 0.5 mT in FC mode, which were the lowest applied magnetic fields in two runs.This indicated that the metallic sulfur belonged either to type-II superconductors or dirty type-I superconductors 36,37 .To exclude the contribution of sulfur to the measured magnetic response, only a portion of the mtrap(T) data collected above 30 K was used to interpret the temperature-dependence of the trapped flux in H3S.
Figure 2a shows the magnetic moment trapped in the sample under ZFC and FC conditions at different HM and measured at 30 K. In FC mode even weak applied magnetic fields as low as 0.5 mT resulted in positive non-zero magnetic response of the trapped flux in H3S.In this mode the magnetic flux penetrated the sample in the normal state at TTc and became trapped at TTc due to very strong pinning (see details below).
Contrary to the FC mode, the ZFC magnetization provided valuable data for estimating Hp, when the applied magnetic field starts to penetrate the sample [38][39][40] (Figure 1c).No trapped flux in ZFC H3S was observed up to 0HM = 45 mT.In this range, the external magnetic field was completely repelled from the superconducting sample.We estimated 0Hp(10 K) = 423 mT from the data measured at 45 mT  HM  125 mT (see inset in Figure 2a).At higher HM, magnetic flux penetrated the superconductor from the outer edges of the disk-shaped sample and gradually propagated towards its center and filled the sample.The entry of magnetic flux led to a reduction in the Meissner currents, and being trapped to the appearance of superconducting current loops and a corresponding increase in the sample magnetization at zero field.The absence of steps in the mtrap(HM) data and smooth increase in mtrap with the increase in HM implied the absence of a weak link network in the superconducting sample 41 .
The observed trapped magnetic moment in the superconducting Im-3m-H3S phase saturated under ZFC and FC conditions at different values of HM.We qualitatively interpreted this behavior of magnetization process in terms of the classical critical state Bean model 42,43 , while considering the reversible part of magnetization in type-II superconductors 38,44 .According to the model, the flux trapping saturates if the applied magnetic field reaches the value HM = 2H * + Hp in ZFC mode and HM = H * in FC mode, where H * is the "full penetration field" (Figure 2b).
The trapped magnetic moment became saturated ( ) at 0HM 1.7 T under ZFC conditions and at 0HM 0.8 T under FC conditions, and did not vary with further increase in 0HM up to 6 T, which was the highest applied magnetic field in the experiment (see Figure 1c and 2a).Thus, 0H * = 80050 mT.
The observed saturation is defined by the maximum value of the trapped magnetic field in the sample and corresponds to the critical current density jc that can be achieved in a superconductor.We also estimated Hc1, which is proportional to Hp using the relation  =  .The demagnetization factor N was derived from the absolute value of m, which was the difference in ZFC m(T) magnetization measurements between the normal and superconducting states.For the thin diskshaped Im-3m-H3S sample having d 85 µm and h 2.8 µm (estimated lower and upper limits of h were 2.1 µm and 3.1 µm), the demagnetization correction was 8.5 (7.7-11.4 for the estimated approximate limits of h, see details in Methods section).Thus, 0Hc1(10 K) was approximately 0.36 T (0.32-0.48 T).
The obtained value of 0Hp 42 mT (and consequently, the retrieved value of Hc1) was noticeably lower than 95 mT derived from the virgin curves of hysteretic magnetization loops in classical m(H) measurements 28,33 .We considered the trapped flux method to be more sensitive to the determination of the onset of magnetic flux entry because a large linear contribution of the initial portion of m(H) was cancelled.The advantage of this method for type-II superconductors having strong pinning was illustrated in ref 44 .Additionally, the signal of the trapped magnetic moment at zero field did not contain the field-dependent magnetic background arising from the DAC, which is unavoidable in typical m(H) magnetization measurements.It should be noted that the presence of tiny, fragmented parts of the superconducting phase and the ragged edges of the bulky sintered sample (see photo of sample in the inset in Figure 1b) could influence the observed Hp.The contribution of these imperfections to the total measured magnetic moment was almost negligible in m(H) experiments because the magnetic flux penetrated these areas at significantly low applied magnetic fields due to the extremely large demagnetization factor.In the trapped flux method these areas conversely accommodated magnetic flux at lower applied magnetic fields and provided magnetic response before the magnetic flux entered into the massive sample.We expected the true value of Hp and the derived values of L and the Ginsburg- to lie approximately between the values observed in the two types of measurements.Using the data of electrical resistance measurements under high magnetic fields 45 , which provided a coherence length (10 K) of approximately 1.85 nm, we obtained L(10 K) of approximately 37 nm (31-40 nm within the estimated limits of h), and (10 K) 20 (17-22).The corresponding values estimated from the m(H) magnetization measurements 33 were L(0) 22 nm (18-23 nm) and (0) 12 (10-13).
2. Critical current density.In addition to the determination of fundamental characteristics Tc and L, we deduced another important characteristics of a superconductor from measured  (  , ) data -the critical current density over a wide temperature range.According to the Bean critical state model 42,43 , a concentric screening current pattern having a current density jc averaged over a sample thickness creates a magnetic moment expressed as  =  ℎ . .
The ultimate critical current density is limited by the depairing currents 49 as , where  is the magnetic flux quantum.Substituting the estimated values of L and  we obtained jd 410 13 A m -2 .The ratio between jc(0) and jd provided additional information.The values of jc(0)/jd being approximately in the order of 10 -2 -10 -1 is conventionally observed in low-temperature superconductors, which characterized by relatively low Ginzburg-Landau parameter  of 20 and strong pinning that stems from the interaction of vortices with extended defects 50 .However, lower values of jc(0)/jd of approximately 10 -3 -10 -2 are characteristic for high-temperature superconductors, for example, the cuprate family in which the Ginzburg-Landau parameter is large ( 100) and relatively weaker pinning that originated from point defects, such as oxygen vacancies 51 .To improve the in-field critical current performance of cuprates, different types of artificial pinning centers were introduced into the sample structure.The synthesized sample of H3S was characterized by strong pinning (see below), but the ratio of jc(0)/jd was quite small (approximately 10 -3 ).This suggested that the density of defect centers in the sintered sample was not maximal and could further be enhanced, for example, by modifying the synthesis protocol.
The different factors influencing the flux pinning in type-II superconductors are divided into two main types: δTc and δl 50,52 .The δl-type pinning arises from a spatial variation in the mean free path of charge carriers, and the defects are small and point sized, e.g.hydrogen vacancies in hydrides.The δTc-type pinning is caused by a spatial variation of the Ginzburg-Landau parameter due to fluctuations in Tc, and the defects are larger than , such as dislocations, grain boundaries and deviations of the chemical composition.Pressure gradients can also cause variations in Tc in case of samples in DACs.We compared the temperature-dependence of the normalized critical current density jc(T)/jc(0) in H3S sample with the simulated curves for δTc and δl types of pinning (see Figure 3).Both theoretical curves for pure δTc and δl types of pinning deviate from experimental data; however, the resulting fit of the data by the combination of two models showed that the δTc-type of pining dominates over the δl-type.
3. Pinning and thermally activated motion of vortices.The kinetics of the trapped magnetic moment demonstrates extremely slow rate of flux creep even at temperatures in vicinity of Tc, at which thermal fluctuations must be significantly higher (see Figure 4c).The estimated creep rate  = , in Im-3m-H3S was approximately 0.002 (or 500 ppm/h) at 165 K and approximately 0.005 (or 2000 ppm/h) at 185 K, which was comparable to the lowest values of S measured in type-II superconductors 53 , including the extremely slow creep rate in high-jc MgB2 films at substantially lower T/Tc  0.5 54 .The scale of fluctuations responsible for vortex melting 55 and vortex creep 53 in a superconductor can be quantified using the Ginzburg-Levanyuk number  = , were kB is the Boltzmann constant.Gi ~710 -6 in H3S was substantially lower than the values reported for cuprate (~10 -2 ) and iron-based (~10 -5 -10 -2 ) high-temperature superconductors 53,55 , and comparable to those reported in lowtemperature superconductors (~10 -9 -10 -6 ) and MgB2 (~10 -6 ).The observed slow creep rate S in H3S was close to the theoretical limit demarcated by the approximate  line 53 and was in good agreement with the lower values of L and  (larger values of L would lead to the theoretical limit of S being higher than the observed value).Additionally, electrical transport measurements of Im-3m-H3S at high magnetic fields 45 showed weak vortex fluctuations and narrow vortex liquid regions due to the low values of  and Gi.
On account of strong pinning of vortices in H3S sample the amount of the trapped flux solely depended on HM at which the flux was generated and the maximum temperature at which the sample was exposed after the magnetization step (see Figure 4).For example, the trapped magnetic moment created at 0HM = 1 T by three different protocols merged at 100 K and above upon subsequent warming (brown and orange curves of cycle II, purple curve of cycle III, and green curve of cycle IV in Figure 4a).The behavior of the trapped magnetic moment was found to be temperature-independent if the sample was warmed to a certain temperature below Tc and cooled down again (see horizontal trend of  () in Figure 4a and b).No pronounced hysteresis was observed for the magnetic moment during the cooling/warming cycles.This memory effect, which allows one to control the amount of trapped flux in the superconducting sample, stems from the very strong pinning of the vortices.
4. Trapped flux in lanthanum superhydrides.The trapped magnetic flux was further probed in the lanthanum-hydrogen system.The ZFC curve of m(T) data measured at an applied magnetic field of 10 mT demonstrate an extended transition with the onset of Tc 200 K at 120±5 GPa (Figure 5a).The observed Tc was in good agreement with the values of the C2/m-LaH10 phase at similar pressures measured by electrical transport measurements 56 .The transition at Tc 200 K was significantly broader than that measured in the initial sample of Fm-3m-LaH10 at 130±8 GPa with a Tc 231 K 33 , thus indicating the poorer quality of the superconducting phase.Likely, the high-temperature superconducting Fm-3m-LaH10 phase sustained structural distortions into the C2/m-LaH10 phase and partial decomposition into the hydrogen-deficient phases due to the unexpected drop in pressure during transportation from the synchrotron (see Methods section).
Contrarily, the trapped flux method demonstrates more pronounced superconducting transitions in this sample.In addition to the detection of the transition at 200 K in the C2/m-LaH10 phase, it revealed a superconducting state of another phase below 70 K. Superconductivity with a Tc 70 K was previously observed in several samples prepared from La and H2 (taken in a large deficiency) at approximately 150-178 GPa by four-probe electrical transport measurements and could be related to LaHx (where 3  x  10) 10 .We were unable to perform measurements of the trapped flux generated at different HM as we did for the sample of H3S.At magnetization of the sample at 0HM = 4 T, one of the diamonds cracked, and pressure dropped below 10 GPa.However, this allowed us to measure the background response of the DAC body induced after sweeping off a high magnetic field of 4 T.No remnant nonlinear magnetic background of the DAC body or other anomalies in the reference  () curve were detected in the DAC at 10 GPa when the sample was evidently not superconducting (cycle III in Figure 5b).
Furthermore, we clearly demonstrated that the observed superconducting transitions in the  () data collected at 1205 GPa (cycles I and II in Figure 5b) belonged to the sample and not to the highpressure assembly (DAC, gasket, diamonds, etc.).

Discussion. The Meissner effect is one of the most fundamental visualization of superconductivity
and it has been considered as the proof for the bulk superconductivity.However, a magnitude of the Meissner effect in type-II superconductors substantially varies in experiments from almost complete expulsion in pinning-free conventional type-II superconductors 57,58 , to practically no expulsion in iron pnictides 59 , or even to magnetic moment enhancement in various materials with extreme sensitivity to disorder [60][61][62][63] .The specific features of the Meissner effect in high-temperature superconductors and its strong suppression by external magnetic fields in the mixed state are associated with the strong pinning which prevents the vortices inside the sample from shifting towards the sample surface even after crossing the Hc1(T) line 64,65 .The subtle or scarcely observable Meissner effect (in FC mode) in H3S 33 agrees with very strong pinning observed in the present work in H3S even at temperatures near Tc.Under FC conditions, the strong pinning leads to the trapping of magnetic flux in the superconducting H3S already at 0.5 mT, i.e. much lower than Hp.Such strong pinning of vortices may also be promoted by the high-pressure conditions.The absence of significant Meissner effect in elemental sulfur and LaH10 supports this hypothesis, though systematic measurements of different high-pressure superconductors are required to prove it.
In contrast to standard m(T) magnetization measurements at low magnetic fields, which demonstrate the shielding of an external magnetic field in the Meissner state, the registered value of the trapped magnetic moment is substantially larger (see Figure 1).The maximum value of a trapped magnetic moment  16.510 -9 A m 2 in Im-3m-H3S at 20 K was approximately 40 times larger than the value of m 4.510 - 10 A m 2 observed in ZFC m(T) magnetization measurements at 0H = 4 mT.Importantly, the response from the trapped flux at zero magnetic field was not perturbed by the magnetic background stemming from the bulky body of the DAC, including diamond anvils and a rhenium gasket.This significantly simplifies measurements of superconducting samples at high pressures in SQUID because the large background from the DAC is eliminated.Additionally, unlike electrical transport measurements, the trapped flux method is sensitive to a particular superconducting phase in mixtures because it does not require the specific arrangement of different phases relative to electrical leads.The analysis of the  () data provides estimations for important characteristics of superconductivity, such as Hc1, L, pinning of vortices, and especially jc, which can be probed in a wide temperature range.We believe that the trapped flux method, which shows great benefit for high-pressures studies, has also been underestimated for the study of superconductivity at ambient pressure.It can be a powerful tool for the routine screening of new superconducting materials, the study of multiphase, contaminated samples, or samples with a low superconducting fraction.

Samples
We used the same samples, which were studied in our previous work 33 .The superconducting Im-3m-H3S and Fm-3m-LaH10 phases were synthesized from sandwiched samples S+NH3BH3 and LaH3+NH3BH3 pressurized in miniature DACs, which were specially designed for a standard commercial SQUID magnetometer 1,33 .The details of the preparation of DACs, chemical synthesis of the samples, estimation of pressure values, and characterization of the initial reactants and final products can be found in Ref. 33 The DAC with Im-3m-H3S sample retained the same pressure value of P = 155±5 GPa throughout all measurements, whereas the DAC with Fm-3m-LaH10 sample did not survive the transportation from APS synchrotron in Argonne, USA.The pressure decreased from P = 130±8 GPa to P = 120±5 GPa.We attempted to restore the sample and pressurized the DAC to P = 125±5 GPa and heated the sample to 2000 K by a pulsed YAG laser.However, the sample demonstrated a lower Tc 200 K and likely corresponded to the structurally distorted C2/m phase of LaH10, which formed from Fm-3m-LaH10 upon decompression at pressures below 130 GPa 56

Magnetization measurements
Magnetization measurements m(T) were performed using the S700X SQUID magnetometer by Cryogenic Limited, and a miniature DAC was attached to a 140-mm-long straw made of Kapton polyimide film, which was specially designed to minimize end effects.The position of the sample relative to the pickup coil of SQUID magnetometer was determined using the ferromagnetic signal from a small steel piece with a size of approximately 14010025 m 3 attached directly to the rhenium gasket surrounding the sample.The precision of the centering procedure is approximately 0.2 mm.The superconducting transition in H3S and LaH10 was probed by ZFC and FC m(T) measurements at 4 mT and 10 mT, respectively.A Tc 195 K in the Im-3m-H3S phase, which was determined as the offset of the diamagnetic transition on the ZFC curves of m(T), was in good agreement with the values estimated in our previous measurements 33 .The much broader superconducting transition in the sample with LaH10 and the lower Tc 200 K indicated the deterioration of the superconducting phase of LaH10 after altering the pressure in the DAC, which likely sustained the monoclinic structural distortions 56 .
The trapped flux was generated in two different protocols under ZFC (run 1) and FC (run 2) conditions. The

Estimation of sample size and demagnetization correction
The diameter and thickness of the thin disk-shaped Im-3m-H3S sample were estimated from optical microscopy and X-ray diffraction data as 85 µm 3 and 2.8 µm 3 , respectively (theoretical lower and upper limits of h were 2.1 µm and 3.1 µm).Only the geometry of the ideal disk-shaped sample yielded high values of the demagnetization correction of approximately 20 using the proposed equation for the effective demagnetization factor in ref 66 .However, the real shape of sample, particularly the alteration of the thickness, can deviate from the ideal disk.Therefore, we further considered the absolute value of m, the difference in m(T) between a normal metal state (above Tc) and a superconducting state (below Tc), which includes geometrical imperfections in the sample shape.This approach gives more reasonable and reliable values of demagnetization correction 8.5 (7.7-11.4 for the estimated limits of h), which we use for the estimation of Hc1 (see details in Methods in ref 33 ).
. One of the diamond anvils cracked, and the pressure dropped to 10 GPa during the magnetization step at 0HM = 4 T after two successful measurements of the trapped flux generated at 1 T and 2 T. The subsequent measurements of the DAC with the decomposed non-superconducting sample magnetized at 0HM = 4 T demonstrated the linear background signal of the DAC with no anomalies in  () curve over the entire temperature range of 10-280 K.
typical ZFC protocol included cooling of the sample at zero field from TTc to the desired temperature TM below the corresponding Tcs, namely TM = 10 K for Im-3m-H3S sample and at TM = 4 K for C2/m-LaH10 sample.The magnetic flux was also trapped in H3S at TM = 4 K in cycle I, TM = 100 K in cycles II and IV, and TM = 8 K and 25 K in cycles V a and V b .A typical magnetization cycle in ZFC mode consisted of a gradual increase in the applied magnetic field HM perpendicular to the sample surface at the lowest temperature point TM, standing at the target value of HM for an hour, and a gradual decrease in the magnetic field to 0 T. In FC mode the sample was cooled from its normal state (TTc) at the applied magnetic field HM to the target temperature TMTc.Then HM was gradually decreased to 0 T.After removing the applied magnetic field HM in both ZFC and FC modes the SQUID magnetometer was allowed to stand at 0 T for approximately 5 hours to prevent successive measurements of  () from outliers associated with jumps of flux in a superconducting magnet.An applied magnetic field of magnetization 0HM ranged from 30 mT to 6 T.  () measurements were performed upon warming (cooling) of the sample with a temperature step of 0.3-3 K (a smaller step in the vicinity of Tc) and 3-4 iterations at each temperature point.Additionally, the reference  () data were measured for Im-3m-H3S sample by skipping the magnetization cycle (0HM = 0 T).The trapped magnetic moment was determined as the difference between the measured magnetic moment after magnetization cycle and the residual magnetic moment, which arose from the body of the miniature DAC above the corresponding Tc (see Supplementary Figure1 and 2).Before each cycle of magnetization, the sample was converted to the normal state by warming to ambient temperature, and the superconducting magnet of SQUID was degaussed to eliminate the remaining fields.Measurements of the trapped flux created in the sample of H3S in run 1 and 2 were separated in time of 6 months.The small discrepancy of the value of  between two runs likely aroused from the slightly different positions of the superconducting sample relative to the pickup coil of SQUID magnetometer, which were found at the centering procedure, and not from the different conditions of the creation of the trapped flux (FC or ZFC).The fact that the same values of  measured in run 1 under ZFC (2, 3 and 6 T) and FC (4 T) conditions support this explanation.

Figure 2 .
Figure 2. Trapped magnetic moment in Im-3m-H3S sample at 30 K. a) Dependence of a trapped magnetic moment at zero field on magnetization field 0HM measured in ZFC and FC modes (run 1 and 2).Circles correspond to the experimental data, magenta and grey curves are guides for the eye.Trapped flux was created in ZFC (black circles) and FC (red circle) modes at several applied magnetic fields HM.The inset with the enlarged plot shows the entry of magnetic field into the sample at low HM.b) The profile of magnetic field in the disk-shaped sample in applied magnetic field HM = 2H * + Hp (ZFC mode) and HM = H * (FC mode) (area below blue lines) and after removing the applied magnetic field (hatched red area).

Figure 3 .
Figure 3.The temperature-dependence of the normalized zero field critical current density jc(T)/jc(0) in Im-3m-H3S sample.Black circles are the experimental data (trapped flux is created at TM = 10 K and 0HM = 3 T).Dashed blue and green curves correspond to the temperature dependence of critical current density limited by δTc-and δl-type pinning; red dashed curve is the fit of experimental data by δTc + δl model.

Figure 4 .
Figure 4. Vortex pinning in H3S.a) Temperature-dependence of a trapped magnetic moment created at different temperatures TM and magnetic fields HM of magnetization.Arrows show the change of temperature in different warming/cooling cycles.b) Temperature-dependence of a trapped magnetic moment in H3S generated at 0HM = 2 T and TM = 8 K (black data, cycle V a ).The trapped flux can be restored if the sample is magnetized again (red data, cycle V b ).c) Creep of the trapped flux in H3S at several temperatures near a Tc (the trapped flux was generated at 0HM = 1 T and TM = 165 K).

Figure 5 .
Figure 5. Magnetic measurements of sample containing C2/m-LaH10 and LaHx (3  x  10).a) ZFC and FC m(T) data of C2/m-LaH10 and LaHx at 120±5 GPa.Photo of the sample is shown in the inset.b) Temperature-dependence of a trapped magnetic moment at zero field.Red and black open circles and fitting curves of different colors correspond to temperature-dependence of the trapped flux created under ZFC conditions at a magnetic field 0HM = 1 T and 2 T, respectively.Blue open circles are the data of the same sample at 10 GPa after magnetization at 0HM = 4 T, showing the absence of a non-linear magnetic background from the DAC without superconducting sample.