Ever since Kamerlingh Onnes' 1911 discovery of superconductivity in mercury 1 , scientists have searched for materials with higher transition temperatures, initially in the elements, and then progressing to more complex systems. Although largely curiosity driven, each discovery of a material with higher critical field and temperature addresses clear technological needs.
Fully fifty-five elements are now known to be superconducting at ambient or high pressure 2 , with hydrogen being a notable exception. In 1968 Ashcroft 3 proposed that atomic metallic hydrogen at sufficiently high density could be a very high T c Bardeen-Cooper-Schrieffer (BCS) superconductor 4 . In view of the high pressures required to reach this proposed state of hydrogen, Carlsson and Ashcroft 5 later suggested alternative routes to effectively produce superconducting atomic hydrogen, including the incorporation of other elements in the structure. This prediction prompted the experimental search for such compounds and alloys 6 . Ashcroft 7 later extended and recast the above considerations in terms ofâȂŸchemical pre-compressionâȂŹ, a proposal in which H 2 molecules in dense structures might be expected to dissociate at pressures well below those required for pure hydrogen. Advances in crystal structure prediction methods then began to provide experimentalists specific targets for higher T c BCS superconductivity 8, 9, 10, 11, 12, 13 while at the same time theoretical studies have focused on understanding the superconducting mechanism 14, 15, 16, 17, 18, 19, 20 . Notably, this effort led to the prediction 21,22 and, independently, to the experimental discovery 23 of superconductivity in the H-S system with a T c of 203 K, including an isotope study that pointed to BCS behavior. Subsequent simulations 24,25 guided the discovery of new hydrides with T c approaching room temperature, the highest being LaH 10 with a critical temperature of at least 260 K, 26, 27,28 and somewhat lower values of 227-243 K reported for related YH 6 /YH 9 phases 29, 30 .
What had once been proposed as a goal in and of itself, superconductivity at room temperature now appears to be the stepping off point for still higher T c 's. Much like when the observation of superconductivity in cuprates led to a flurry of discoveries and engineering applications that pushed the critical temperatures in these materials from 40 to 164 K over the course of a decade 31, 32 , hydrogen-based materials hold out great promise. Current calculations predict that crystalline atomic metallic hydrogen 3,33 has a T c near room temperature at 500 GPa, which increases to 420 K above 3 TPa 34 . Other calculations predict that hydrogen could be a superconducting superfluid at comparable conditions 35 . Emerging systematics for the binary hydrides indicate maximum T c in the vicinity of room temperatures at megabar pressures 20 , and there is now a focus on the possibility of higher critical temperatures in more chemically complex hydride systems. Indeed, recent theoretical studies predict a T c of 473 K in Li 2 MgH 16 near 250 GPa 36 .
The present work was motivated by a desire to extend the P-T-H phase diagram of previous measurements of the magnetic field dependence of the lanthanum-based superhydrides to fields approaching 100 T 28, 24, 37 . We also sought to examine lower pressure phases such as LaH 6 which is more experimentally accessible. In addition we hoped to better understand the variable T c in experiments in which LaH 10 superhydride was synthesized using ammonia borane (NH 3 BH 3 ) as the hydrogen source 27 . For this purpose, we developed metallic DACs designed for superhydride studies in DC magnetic fields that are also small enough to fit into pulsed magnets. Made from the highly resistive non-magnetic superalloy NiCrAl [Pascalloy, Tevonics] to limit the eddy-current heating, these DACs can be used to access temperatures down to at least 30 K in pulsed fields.
We coupled this design with robust Pt electrodes created by Focused Ion Beam (FIB) techniques to withstand the extreme P-T conditions required to synthesize the superhydrides 38 . Attempts to characterize in situ by x-ray diffraction were unsuccessful due to an insufficient downstream opening in the DAC. We find that the La-based superhydride initially synthesized by laser-heating beginning at 160 GPa had a T c of 294 K. Most remarkably, subjecting the sample to subsequent thermal cycling shifted the T c to higher temperatures in a fortuitous progression that reached well above 500 K.
We loaded two diamond-anvil cells (DACs) with pieces of 99% La and ammonia borane (AB, NH 3 BH 3 ), the hydrogen source and pressure medium (see Methods and SI). Both cells were initially as identical as experimentally possible, with the same FIB-patterned electrodes on one anvil and an MP35N gasket with cBN insert and AB on the other. The first cell (B002) was loaded initially at 160 GPa for LaH 10 synthesis, whereas the second cell (B003) was loaded at 120 GPa to generate the lower stoichiometry superhydride, LaH 6 ( Fig. 1 ). Both cells were then laser-heated at the HPCAT beamline of the APS. A 20 µm diameter laser spot was rastered across the sample at fixed positions to promote the dissociation of AB into cBN and the hydrogen, that reacts with the La. Diffraction patterns were collected at each spot after laser heating. The sample in B002 received about 45 laser pulses (4 to 5 pulses at each of nine points on a 10 x 10µm grid). The synthesis for B003 was stopped prematurely after a few pulses for fear that a catastrophic failure of the diamond anvils had occurred (see SI for details). The electrical resistivity was then measured on the samples at NHMFL-Tallahassee, first using a Quantum Design 16 T PPMS. Figure 2 shows the first cool down trace from 300 to 230 K for B002 using four of the six available electrodes. A resistance drop is clearly evident at 294 K with no additional transition observed upon cooling to 230 K, the temperature range in which T c has been previously observed for LaH X systems.
In an attempt to measure the resistance of the sample in B002 further into the normal state, we subjected the sample to a series of thermal cycles at 0 T. Surprisingly, these successive higher temperatures excursions pushed the onset of the transition to higher and higher temperatures. We had to stop temporarily at 390 K as this is the maximum temperature possible in our PPMS. Figure 3 compares the initial and final traces, with the 0 T onset appearing near 357 K in the final thermal cycle in the PPMS.
Four thermal cycles between 370 K and 290 K were then performed at 0, 2, 10 and 16 T (Fig. 4) . The traces are shifted vertically for clarity, but collapse onto one another below 310 K. All curves show hysteresis. The warming curve has a lower onset temperature than the cooling curve, indicating either the first order nature of the transition or that the sample is still evolving upon successive thermal cycles. The warm up traces show additional changes which we believe points to a continuing synthesis, as discussed below.
To analyze these data, three temperature points are identified in each trace within 
yields an H c (0 K) of 1500 T, 230 T, and 130 T for T1, T2, and T3, respectively. However the error is large for H c (0 K) as we had access to a limited field range.
To further investigate the shift in the superconducting transition with field, B002 was then measured in a 41.5 T DC resistive magnet at the NHMFL-Tallahassee. All six working electrodes were used in order to measure an additional resistance channel. We present in the main text only the results from the same configuration as that measured in the PPMS, but the other one yielded similar results (see SI). Successive thermal cycles carried out to yet higher temperatures in order to establish the clear signature of the normal state resulted again in a clear shift of the transition to higher temperature. The stability of epoxies internal to the DAC limited our measurements to below 580 K. Figure 5 shows the cool down traces starting from various initial maximum temperatures. The cell remained at each maximum temperature for at least 30 minutes to allow the synthesis to stabilize as realized by a steady sample resistance, and was then cooled at 0.5 K/min. With each higher temperature excursion T c rises further, eventually reaching a maximum value nearing 560 K. The amplitude of the transition also dramatically increases with each excursion to higher temperatures. Similar shifts in transition temperature with repeated thermal cycling has been documented for other superconducting hydrides under pressures 28, 40 . The resistance within the superconducting state is almost identical for all the curves, the non-zero state being attributed to the additional unreacted lanthanum between the electrodes, as this same behaviour is seen in B003 (see SI).
Temperature sweeps in static high magnetic fields were performed for initial maximum temperatures of 400 K, 430 K, 445 K, 503 K and 530 K. Only cool down traces were performed in field to remain within the allocated energy budget of a magnet time run. Figure  6 shows the result of the run at 503 K, since the data at 530 K indicated either an electrical contact degradation or the development of a touch between the cryostat and the magnet, adding vibrations in the signal which degraded the signal-to-noise. The temperatures for the traces obtained in field are corrected for the magnetoresistance of the thermometer, and a quadratic background in the superconducting state was subtracted over the whole temperature range (see SI).
Although the onset of the transition shifts from 492 K at 0 T to 486 K at 40 T, the width of the transition at 0 T, as well as the additional transitions appearing in the 20 T and 33 T curves make the interpretation difficult. The incomplete nature of the chemical transformation precludes fixed-field temperature sweeps within a family of curves above 370 K. Indeed, the shift in the transition with each increasing temperature excursion (Fig. 5) suggests that the chemical state of the material is still evolving.
We report the observation of superconductivity in a La-based superhydride sample beginning at room temperature that shifts up in a controlled fashion with thermal excursions to a value close to 560 K with a notable concomitant increase in the amplitude. The observation of the onset at 294 K is consistent with preliminary observations of T c above 260 K that were reported previously 27 . The initial increase in T c from 294 K to 370 K may be related to the enhancement of T c on repeated thermal cycling observed in the simpler binary hydride H 3 Se 40 and described in the preliminary reports of the synthesis of superconducting H 3 S 41 . However, this is unlikely, as the maximum predicted transition temperature for the binary LaH 10/11 system is 288 K 24,25 . Rather, the higher temperature transitions point to additional chemical transformations induced by pressure, shear, temperature, potentially magnetic field, and possibly molten hydrogen, which at room temperature exists above 200
Mbar 35, 42 . In addition to B and N from the hydrogen source (NH 3 BH 3 ) and/or the composite gasket insert (cBN) and the carbon from the epoxy binder, C and Ga from the Pt electrodes also make contact with the La-H and could react with this binary system to form a ternary or higher order system. One might speculate that the initial laser synthesis gen- FIBed electrodes to form a cold weld. The same plastic transfer anvil is used to initially align this anvil during the assembly of the DAC and the visible impression that remains of that anvilâȂŹs culet helps ensure that the La is positioned over the electrodes in this second step. To prevent the reaction between La and air, we loaded the sample and sealed the DAC within 30 minutes of extracting the 3-6 µm samples from the freshly exposed metal.
The assembled DAC was then taken to the desired pressure for synthesis, using the B003. Both show the superconducting transition due to unreacted La or a lower stoichiometry hydride at temperatures below 5 K and a superconducting transition at 12.5 K in B002 and 9 K in B003 that we attribute to a platinum hydride 50 . The transition in B003 is at a lower temperature, consistent with B003 being at a lower pressure. It is also not as fully developed, which we attribute to the small number of laser pulses that B003 was subjected to. B002 has the additional transition at temperatures greater than 365 K. Both B002 and B003 have a non-zero resistance down to 1.9 K. The resistance of B003 is almost a factor of 10 higher than B002, which is attributed to the thinness of the B003 sample. is the temperature at which the hysteresis of the cool down and warm up traces closes. This run used the 2nd configuration of electrodes (see SI). The traces were taken sequentially from 0 T to 16 T. All four traces collapse onto one another below 315 K. The T c (90/10) is approximately 11 K after a background subtraction is done. Note that the transition in warming curve for each field is lower in temperature than the cooling.
Although it is possible that this is due to a lag between the thermometer and the sample, the hysteresis is most likely real as the temperature was swept at 0.2 K/min and the separation is different for the four fields.
b) Bottom : evolution of T1, T2, and T3, at various magnetic fields, and corresponding fits to the points. The difference in signal over noise ratio between the zero-field and applied field traces arises from the vibrations generated by the cooling water flowing through the resistive magnet when in operation. The data acquisition software crashed during the 33 T cool down and resulted in an incomplete curve between 470 and 480 K, and the 20 T cool down was shortened by the COVID -19 shutdown. It is not possible to perform a G-L fit on this data set with any confidence, since the sample continues to evolve with each temperature excursion.