Extreme magnetic field-boosted superconductivity
Sheng Ran, I-Lin Liu, Yun Suk Eo, Daniel J. Campbell, Paul Neves, Wesley T. Fuhrman, Shanta R. Saha, Christopher Eckberg, Hyunsoo Kim, Johnpierre Paglione, David Graf, John Singleton, Nicholas P. Butch
EExtreme magnetic field-boosted superconductivity
Sheng Ran , , ∗ , I-Lin Liu , , , Yun Suk Eo , Daniel J. Campbell , Paul Neves ,Wesley T. Fuhrman , Shanta R. Saha , , Christopher Eckberg , Hyunsoo Kim ,Johnpierre Paglione , , David Graf , John Singleton , & Nicholas P. Butch , , ∗ Center for Nanophysics and Advanced Materials, Department of Physics,University of Maryland, College Park, MD 20742, USA NIST Center for Neutron Research,National Institute of Standards and Technology, Gaithersburg, MD 20899, USA Department of Materials Science and Engineering,University of Maryland, College Park, MD 20742, USA National High Magnetic Field Laboratory,Florida State University, Tallahassee, FL 32313, USA National High Magnetic Field Laboratory,Los Alamos National Laboratory, Los Alamos, NM 87545, USA Department of Physics, The Clarendon Laboratory,University of Oxford, Oxford OX12JD, United Kingdom (Dated: May 14, 2019)
Abstract
Applied magnetic fields underlie exotic quantum states, such as the fractional quantum Hall ef-fect and Bose-Einstein condensation of spin excitations . Superconductivity, on the other hand, isinherently antagonistic towards magnetic fields. Only in rare cases can these effects be mitigatedover limited fields, leading to reentrant superconductivity. Here, we report the unprecedented co-existence of multiple high-field reentrant superconducting phases in the spin-triplet superconductorUTe . Strikingly, we observe superconductivity in the highest magnetic field range identified forany reentrant superconductor, beyond 65 T. These extreme properties reflect a new kind of ex-otic superconductivity rooted in magnetic fluctuations and boosted by a quantum dimensionalcrossover . a r X i v : . [ c ond - m a t . s up r- c on ] M a y t is a basic fact that magnetic fields are destructive to superconductivity. The maximummagnetic field in which superconductivity survives, the upper critical field H c , is restrictedby both the paramagnetic and orbital pair-breaking effects: electron spin polarization dueto the Zeeman effect, and electron cyclotron motion due to the Lorentz force, respectively.In a few very rare cases, however, magnetic fields can do the opposite and actually stabilizesuperconductivity . In these cases, the applied magnetic field is most often compensatedby an internal field produced by ordered magnetic moments through exchange interactions,resulting in a reduced total effective field . A different set of circumstances involving uncon-ventional superconductivity occurs in the ferromagnetic superconductor URhGe , wherefield-induced superconductivity is instead attributed to very strong ferromagnetic fluctua-tions that emanate from a quantum instability of a ferromagnetic phase, strengthening spintriplet pairing .Here we report the presence of two independent high-field superconducting phases in therecently discovered triplet superconductor UTe , for a total of three superconducting phases(Fig. 1). This is the first example of two field-induced superconducting phases existing inone system, one of which has by far the highest lower and upper limiting fields of anyfield-induced superconducting phase, more than 40 T and 65 T, respectively. Both of thefield-induced superconducting phases are stabilized by ferromagnetic fluctuations that areinduced when the magnetic field is applied perpendicular to the preferred direction of theelectron spins. Interestingly, the high-field superconducting phase exists exclusively in amagnetic field polarized state, unique among these superconductors. This discovery providesan excellent platform to study the relation between ferromagnetic fluctuations, spin tripletsuperconducting pairing, and dimensionality in the quantum limit.UTe crystallizes in an anisotropic orthorhombic structure, with the a -axis being themagnetic easy axis along which spins prefer to align in low magnetic fields. The supercon-ducting upper critical field H c is strongly direction-dependent, and exceedingly large alongthe b -axis, with an unusual upturn in its temperature dependence above 15 T. The H c is ex-traordinarily sensitive to the alignment of the magnetic field along the b -axis , and accuratemeasurements require the use of a specialized two-axis rotator (Fig. 1b). When the magneticfield is perfectly aligned along the b -axis, superconductivity persists up to 34.5 T at 0.35 K(Fig. 2a). A small misalignment from the b -axis, less than 5 ◦ , decreases the H c value byover half, to 15.8 T. However, even this misaligned superconductivity is resilient, and upon2urther increasing the magnetic field, superconductivity reappears between 21 T and 30 T.Our measurements show that this reentrant phase does not persist beyond misalignmentgreater than 7 ◦ .Although UTe is closely related to the ferromagnetic triplet superconductors , the obser-vation of reentrant superconductivity in UTe resembles neither that of URhGe , whichis completely separated from the low-field portion, nor the sharp H c cusp in UCoGe in an-gle dependence . The angle-dependence of the superconducting phase boundary suggeststhat the reentrant portion of the UTe superconductivity, SC RE , may have a distinct orderparameter from the lower-field superconductivity, SC PM . However, unlike the case for bothURhGe and UCoGe , there is no normal-state change in the underlying magneticorder in UTe that would drive a change in the superconducting order parameter symmetry.We discuss the magnetic interactions that stabilize this unusual behavior after an excursionto even higher field.The upper field limit of SC RE of 35 T coincides with a dramatic magnetic transition intoa field polarized phase (Fig. 2c). The magnetic moment along the b -axis jumps from 0.35 to0.65 µ B discontinuously, due to a spin rotation from the easy a -axis to the orthogonal b -axis.The abrupt change in moment direction is accompanied by a jump in magnetoresistance anda sudden change of frequency in the proximity detector oscillator (PDO) circuit (Fig. 2b).The critical field H m of this magnetic transition has little temperature dependence up to10 K, but increases as magnetic field rotates away from the b -axis to either the a or c -axis(Fig. 1c). Meanwhile, the magnitude of the jump in magnetic moment, 0.3 µ B , appears tobe direction-independent (Supplemental material Fig. 5). This magnetic field scale appearsto represent a general energy scale for correlated uranium compounds: weak anomalies areobserved in ferromagnetic superconductor UCoGe , whereas a large magnetization jumpoccurs in the hidden order compound URu Si .As H m limits the SC RE phase, it gives rise to an even more startling form of superconduc-tivity. Sweeping magnetic fields through the angular range of θ = 20 - 40 ◦ from the b towardsthe c -axis reveals an unprecedented high-field superconducting phase, SC FP (Fig. 3). Theonset field of the SC F P phase precisely follows the angle dependence of H m , while the uppercritical field goes through a dome, with the maximum value exceeding 65 T, the maximumfield possible in our measurements. This new superconducting phase significantly exceedsthe magnetic field range of all known field-induced superconductors . Due to its shared3hase boundary with the magnetic transition, this superconducting phase tolerates a ratherlarge angular range of offsets from the b - c rotation plane. However, it does not appear whenthe field is rotated from the b - to the a -axis.Having established the field limits and angle dependence of the SC FP phase, we turn to itstemperature stability (Fig. 4). The onset field has almost no temperature dependence, againfollowing H m , while the upper critical field of the SC FP phase disappears near 1.6 K, similarto the zero-field superconducting critical temperature. This suggests that even though it isstabilized at remarkably high field, the new superconducting phase involves a similar pairingenergy scale to the zero-field superconductor.Such a large magnetic field and temperature stability of the SC FP phase begs the questionof what the mechanism is. A natural candidate is the Jaccarino-Peter effect used to describeother reentrant superconductors . This antiferromagnetic type of exchange interaction canlead to an internal magnetic field that is opposite to the external magnetic field, resulting ina total magnetic field that is much smaller. This compensation mechanism has successfullyexplained the field induced superconductivity in Chevrel phase compounds and organicsuperconductors , but it is very unlikely to apply to the SC FP phase of UTe , which lacksthe requisite localized atomic moments. Further, SC FP persists over a wider field-angle rangethan typical of the compensation effect .The temperature dependence of the SC FP phase, as well as its close relation to the mag-netic transition, is reminiscent of the field induced superconducting phase in URhGe, whichhas been attributed to ferromagnetic spin fluctuations associated with the competition ofspin alignment between two weakly anisotropic axes. In URhGe, a magnetic field transverseto the direction of the ordered magnetic moments leads to the collapse of the Ising ferro-magnetism and this instability enhances ferromagnetic fluctuations, which in turn inducesuperconductivity .UTe , however, is not ferromagnetic. Nevertheless, the overall similarities between UTe and the ferromagnetic superconductors, with regard to the relationship between the pre-ferred magnetic axis and the direction of high H c , suggest that strong transverse spinfluctuations play a central role in these superconducting phases . The H c values and di-rectionality in UTe can thus be understood in the following manner. Starting from zeromagnetic field, superconductivity is most resilient to magnetic field applied along the b -axis,which is perpendicular to the easy magnetic a -axis. Magnetic field applied along the b -axis4hus induces spin fluctuations that stabilize superconductivity against field-induced pair-breaking. At 34.5 T, however, a magnetic phase transition occurs, and magnetic momentsrotate from the a to the b -axis. In the high-field polarized phase, magnetic field along the b -axis no longer induces transverse spin fluctuations, and superconductivity is suppressedcompletely. However, it is possible once again to induce transverse spin fluctuations by nowapplying a magnetic field along the c -axis. When viewed as a vector sum of fields along the b and c -axes (Fig. 3), it is clear that H b stabilizes the magnetic phase, while a range of H c strength stabilizes superconductivity, with the highest reentrant magnetic field values yetobserved.This ferromagnetic fluctuation scenario is qualitatively consistent with the whole pictureof field-induced superconducting phases in UTe , yet a very important distinction existsbetween the SC FP phase and the field induced superconducting phase in URhGe: the SC FP phase only exists in the field polarized state. This greatly challenges the current theoryproposed for URhGe, which allows superconductivity to exist on both sides of the phaseboundary .A compelling resolution to this conundrum is that the SC FP phase is the realization of aspin-triplet superconductor in the 1-dimensional quantum limit . This exotic superconductorrequires spin-triplet pairing and is predicted to occur at very high magnetic fields appliedtransverse to the axis of a quasi-1-dimensional chain. The field-induced lower-dimensionalityis both field-angle dependent and facilitates the recovery of the zero-field superconductingcritical temperature, as we observe in UTe (Figs. 3 and 4). Further, this mechanism permitssuperconductivity in a pure material to survive in any magnetic field, making UTe anexciting playground for further testing the limits of high field-boosted superconductivity.The exclusive existence of the SC FP phase in the field polarized state, and in such highmagnetic field, guarantees that the superconducting state has odd parity, with time reversalsymmetry breaking. Odd parity is the cornerstone of topological superconductivity , and itis certain that the SC FP phase has non-trivial topology. Since time reversal symmetry is alsobroken, a special topological superconducting state, such as chiral superconductivity , ishighly likely, which hosts Majorana zero modes, the building block for topological quantumcomputing . 5 . METHODS Single crystals of UTe were synthesized by the chemical vapor transport method usingiodine as the transport agent. Crystal orientation was determined by Laue x-ray diffrac-tion performed with a Photonic Science x-ray measurement system. Magnetoresistancemeasurements were performed at the National High Magnetic Field Laboratory (NHMFL),Tallahassee, using the 35-T DC magnet, and at the NHMFL, Los Alamos, using the 65-Tshort-pulse magnet. Proximity Detector Oscillator and magnetization measurements wereperformed at the NHMFL, Los Alamos, using the 65-T short-pulse magnetIn order to perfectly align the magnetic field along the b -axis in the DC magnet, a singlecrystal of UTe was fixed to a home-made sample mount on top of an Attocube ANR31 piezo-actuated rotation platform (Fig. 1b). Thin, copper wires were fixed between the probe androtation platform to measure the sample and two, orthogonal Toshiba THS118 hall sensors.All three measurements were performed using a conventional four-terminal transport setupwith Lake Shore Cryotronics 372 AC resistance bridges. Adjustments to the θ angle weremade using a low friction apparatus to find the center of the range where the sampleresistance was zero at B = 25.5 T. With a lower field of 0.5 T, small changes were thenmade to the φ orientation while monitoring the Hall sensors. With the field aligned near tothe b -axis, the magnetic field was swept to 34.5 T.The contactless-conductivity was measured using the proximity detector oscillator (PDO)circuit described in Refs. , which has been used to study field stabilized superconductingphase . A coil comprising 6-8 turns of 46-gauge high-conductivity copper wire is woundabout the single-crystal sample; the number of turns employed depends on the cross-sectionalarea of the sample, with a larger number of turns being necessary for smaller samples. Thecoil forms part of a PDO circuit resonating at 22-29 MHz. A change in the sample skindepth or differential susceptibility causes a change in the inductance of the coil, whichin turn alters the resonant frequency of the circuit. The signal from the PDO circuit ismixed down to about 2‘MHz using a double heterodyne system . Data are recorded at20 M samples/s, well above the Nyquist limit. Two samples in individual coils coupled toindependent PDOs are measured simultaneously, using a single-axis, worm-driven, cryogenicgoniometer to adjust their orientation in the field.The pulsed field magnetization experiments used a 1.5 mm bore, 1.5 mm long, 1500-6urn compensated-coil susceptometer, constructed from 50 gauge high-purity copper wire .When a sample is within the coil, the signal is proportional to dM/dt . Numerical integra-tion is used to evaluate magnetization M. The sample is mounted within a 1.3 mm diameterampoule that can be moved in and out of the coil. Accurate values of M are obtained bysubtracting empty coil data from that measured under identical conditions with the sam-ple present. These results were calibrated against results from Quantum Design MagneticProperty Measurement System.The data that support the results presented in this paper and other findings of thisstudy are available from the corresponding authors upon reasonable request. Identificationof commercial equipment does not imply recommendation or endorsement by NIST. II. ADDENDUM
We acknowledge Fedor Balakirev for assistance with the experiments in the pulsed fieldfacility. We also acknowledge helpful discussions with Andrei Lebed and Victor Yakovenko.Wesley T. Fuhrman is grateful for the support of the Schmidt Science Fellows program,in partnership with the Rhodes Trust. Research at the University of Maryland was sup-ported by the US National Science Foundation (NSF) Division of Materials Research AwardNo. DMR-1610349, the US Department of Energy (DOE) Award No. DE-SC-0019154(experimental investigations), and the Gordon and Betty Moore Foundations EPiQS Initia-tive through Grant No. GBMF4419 (materials synthesis). Research at the National HighMagnetic Field Laboratory (NHMFL) was supported by NSF Cooperative Agreement DMR-1157490, the State of Florida, and the US DOE. A portion of this work was supported bythe NHMFL User Collaboration Grant Program.[Competing Interests] The authors declare no competing interests.[Correspondence] Correspondence and requests for materials should be addressed toS.R. (email: [email protected]) and N.P.B. ([email protected]).
III. AUTHOR CONTRIBUTION
N. P. Butch directed the project. S. Ran, W. T. Fuhrman and S. R. Saha synthesized thesingle crystalline samples. S. Ran, I. Liu, and J. Singleton performed the magnetoresistance,7DO and magnetization measurements in pulsed field. Y. S. Eo, D. J. Campbell, P. Nevesand D. Graf performed the magnetoresistance measurements in DC field. C. Eckberg andH. Kim performed magnetoresistance measurements in low magnetic fields. S. Ran and N.P. Butch wrote the manuscript with contributions from all authors. H. L. Stormer, RMP , 875 (1999). V. Zapf, M. Jaime, and C. D. Batista, RMP , 563 (2014). H. W. Meul, C. Rossel, M. Decroux, . Fischer, G. Remenyi, and A. Briggs, PRL , 497 (1984). S. Uji, H. Shinagawa, T. Terashima, T. Yakabe, Y. Terai, M. Tokumoto, A. Kobayashi,H. Tanaka, and H. Kobayashi, Nature , 908 (2001). T. Konoike, S. Uji, T. Terashima, M. Nishimura, S. Yasuzuka, K. Enomoto, H. Fujiwara,B. Zhang, and H. Kobayashi, PRB , 094514 (2004). S. Ran, C. Eckberg, Q.-P. Ding, Y. Furukawa, T. Metz, S. R. Saha, I.-L. Liu, M. Zic, H. Kim,J. Paglione, and N. P. Butch, Science, accepted (2019). V. P. Mineev, PRB , 014506 (2015). A. G. Lebed and O. Sepper, PRB , 024510 (2014). V. Jaccarino and M. Peter, PRL , 290 (1962). F. L`evy, I. Sheikin, B. Grenier, and A. D. Huxley, Science , 1343 (2005). F. L`evy, I. Sheikin, and A. Huxley, Nature Physics , 460 (2007). D. Aoki, A. Nakamura, F. Honda, D. Li, Y. Homma, Y. Shimizu, Y. J. Sato, G. Knebel,J.-P. Brison, A. Pourret, D. Braithwaite, G. Lapertot, Q. Niu, M. Valika, H. Harima, andJ. Flouquet, J. Phys. Soc. Jpn. , 043702 (2019). D. Aoki, T. D. Matsuda, V. Taufour, E. Hassinger, G. Knebel, and J. Flouquet, J. Phys. Soc.Jpn. , 113709 (2009). A. D. Huxley, S. J. C. Yates, F. Lvy, and I. Sheikin, J. Phys. Soc. Jpn. , 051011 (2007). T. Hattori, K. Karube, K. Ishida, K. Deguchi, N. K. Sato, and T. Yamamura, J. Phys. Soc.Jpn. , 073708 (2014). W. Knafo, T. D. Matsuda, D. Aoki, F. Hardy, G. W. Scheerer, G. Ballon, M. Nardone, A. Zi-touni, C. Meingast, and J. Flouquet, PRB , 184416 (2012). F. R. De Boer, J. J. M. Franse, E. Louis, A. A. Menovsky, J. A. Mydosh, T. T. M. Palstra,U. Rauchschwalbe, W. Schlabitz, F. Steglich, and A. De Visser, Physica B+C , 1 (1986). L. Balicas, J. S. Brooks, K. Storr, S. Uji, M. Tokumoto, H. Tanaka, H. Kobayashi, A. Kobayashi,V. Barzykin, and L. P. Gor’kov, PRL , 067002 (2001). K. Hattori and H. Tsunetsugu, PRB , 064501 (2013). Y. Sherkunov, A. V. Chubukov, and J. J. Betouras, PRL , 097001 (2018). M. Sato and Y. Ando, Reports on Progress in Physics , 076501 (2017). C. Kallin and J. Berlinsky, Reports on Progress in Physics , 054502 (2016). S. D. Sarma, M. Freedman, and C. Nayak, Npj Quantum Information , 15001 (2015). T. Karzig, C. Knapp, R. M. Lutchyn, P. Bonderson, M. B. Hastings, C. Nayak, J. Alicea,K. Flensberg, S. Plugge, Y. Oreg, C. M. Marcus, and M. H. Freedman, PRB , 235305(2017). E. C. Palm and T. P. Murphy, Review of Scientific Instruments , 237 (1999). M. M. Altarawneh, C. H. Mielke, and J. S. Brooks, Review of Scientific Instruments , 066104(2009). S. Ghannadzadeh, M. Coak, I. Franke, P. A. Goddard, J. Singleton, and J. L. Manson, Reviewof Scientific Instruments , 113902 (2011). J. Singleton, J. A. Symington, M.-S. Nam, A. Ardavan, M. Kurmoo, and P. Day, Journal ofPhysics: Condensed Matter , L641 (2000). P. A. Goddard, J. Singleton, P. Sengupta, R. D. McDonald, T. Lancaster, S. J. Blundell, F. L.Pratt, S. Cox, N. Harrison, J. L. Manson, H. I. Southerland, and J. A. Schlueter, New Journalof Physics , 083025 (2008). S C
P M H S C
P M H S C
R E H S C
F P H m H (T) (cid:1) F P
F PS C
P M a - a x i s c - a x i s H (T) (cid:2) S C
F P b - a x i s S C
R E ( c ) ( b ) H (cid:2) (cid:1) Hb c aH
H a l ls e n s o r a b s a m p l e c (cid:2) (cid:1) ( a ) FIG. 1. Magnetic field-induced superconducting and polarized phases of UTe . (a) Sketch of howthe magnetic field is applied with respect to the three crystallographic axes of UTe . (b) Top viewof the sample platform with two-axis rotator used in DC field measurements to achieve the bestalignment. (c) Magnetic field - angle phase diagram showing three superconducting phases. Themagnetic field is rotated within the ab and bc -plane. The critical field values of the SC PM and SC RE phases are based on DC field measurements, and those of the SC FP and field polarized phases arebased on pulsed field measurements. (cid:2) = 0 o R (arb. units) H ( T ) o o o o T = 0 . 3 5 K ac b H (cid:1) (cid:1) = ( c )( b ) R (arb. units) H ( T ) ( a ) R e s i s t a n c e P D O R (arb. units) T = 0 . 4 5 K f (10-6Hz) S C
P M
S C
R E H l l b M ( m B/f.u) H ( T ) H l l b FIG. 2. Reentrance of superconductivity in UTe . (a) Field dependence of magnetoresistance ofUTe at T = 0.35 K measured in the DC field. The magnetic field is rotated from b towards a axis. Zero resistance persists up to 34.5 T when the magnetic field is perfectly along b -axis. Thesame data set is plotted in logarithmic scale in the inset. Reentrance of superconductivity canbe clearly seen when the magnetic field is applied slightly off the b -axis. (b) Magnetoresistanceand the contactless-conductivity measurements using the PDO circuit (see Methods section for thetechnical details) of UTe at T = 0.45 K in the pulsed field, with the magnetic field applied alongthe b -axis. (c) Magnetization measurements UTe at T = 0.45 K and 1.7 K in the pulsed field,with the magnetic field applied along the b -axis. For the measurements in pulsed field, the two-axisrotator is not compatible. There is likely a slight angle offset along the perpendicular direction.The field induced superconducting phase SC RE is not observed in these measurements. R (arb. units) H ( T ) o o o o o (cid:1) = T = 0 . 5 5 K F PS C
F P Hc (T) H b ( T ) f ( 1 0 H z ) (cid:1) = 33 o R (arb. units) H ( T ) o o o o o o T = 0 . 5 K (cid:2) = R ( a r b . u n i t s ) F P Ha (T) H b ( T ) ( f )( e ) ( c )( d ) F PS C
F P (cid:1) = 23.7 o Hc (T) H b ( T ) R ( a r b . u n i t s ) ( b ) f (106 Hz) H ( T ) o o o o o T = 0 . 5 K (cid:2) = ( a ) FIG. 3. Angle dependence of the field-induced superconducting and polarized phases of UTe .When the magnetic field is applied at an angle from b towards c -axis ( a -axis), it is equivalent totwo applied magnetic fields: one along the b -axis, H b = H cos θ or H cos φ , and the other along the c -axis, H c = H sin θ ( a -axis, H a = H sin φ ). It is more convenient to visualize the magnetoresistanceand PDO data as a function of these two components of the applied magnetic fields. Such colorcontour plots are shown in (a) - (c). The blue dots are the critical fields for the field polarizedstate and the red dots are the critical fields for the high-field superconducting phase SC FP . Thecorresponding data as a function of the applied magnetic fields are shown in (d) - (f) at selectedangles. The dashed lines on the (a) and (b) indicates the directions where measurements were alsoperformed at different temperatures, as shown in next figure. . 5 1 . 0 1 . 5 2 . 002 04 06 00 H (T) T ( K ) R ( a r b . u n i t s ) (cid:1) = 23.7 o S C
F P R (arb. units) H ( T ) (cid:1) = 2 3 . 7 o f (106 Hz) H ( T ) (cid:1) = 3 3 o ( d )( a ) ( c ) S C
F P (cid:1) = 33 o H (T) T ( K ) f ( 1 0 H z ) ( b )
FIG. 4. Temperature dependence of the high-field superconducting phase SC FP of UTe . Colorcontour plots of (a) magnetoresistance and (b) frequency of PDO measurements as a function oftemperature and magnetic field, at θ = 23.7 ◦ and 33 ◦ , respectively. The corresponding data as afunction of the applied magnetic fields are shown in (c) and (d) at selected temperatures. UPPLEMENTARY TEXTA. SC RE phase in bc plane We did not observe the SC RE phase for θ larger than 3.9 ◦ in the bc -plane. It is very likelythat this is the angle limit of the SC RE phase. The magnetoresistance data show a slopechange between 20 and 30 T for θ = 3.9 ◦ (Fig. 5), indicating it is on the edge of SC RE phase. B. Angle offest in pulsed field measurements
In order to detect the magnetic transition at higher angles, we had to perform experimentsin pulsed field, where a probe compatible with a two-axis rotator is not available. Therefore,when the magnetic field rotates in one plane (e.g., ab -plane), there generally is a small angleoffset in the perpendicular plane (e.g., bc -plane). This is probably why the field range ofSC P M looks smaller in pulsed field compared to the value in our previous paper, and SC RE is not observed, in both magnetoresistance and PDO measurements. In addition, the basetemperature in pulsed field, 0.5 K, is higher than that of the DC field, 0.35 K. For thesereasons, pulsed field data are not used to characterize the SC P M and SC RE phases.The field polarized state and SC F P phase extends to very high fields, beyond the limitof DC field, and therefore the pulsed field measurements are the only choice to charac-terize both phases, which inevitably gives some angle offset. This angle offset, of order afew degrees, explains the slight difference between the phase diagrams based on PDO andmagnetoresistance measurements.
C. hysteresis in PDO and magnetoresistance
The high field induced superconducting phase SC
F P can be seen in both PDO and mag-netoresistance measurements upon down-sweep of the magnetic field. PDO measurementsshow a sudden increase of frequency, corresponding to the decrease of the sample resistance.However, the frequency decreases instead upon the up-sweep of the magnetic field, leading toan unusual, large hysteresis loop between up and down sweep. Similarly, magnetoresistanceshows a sudden increase upon the up-sweep, leading to large hysteresis. On the other hand,the hysteresis in magnetoresistance decreases with increasing angle from b towars c -axis and14lmost disappears for θ = 24 ◦ .Most of the hysteresis is known to be caused by heating due to dissipative vortex motionin the mixed state during the rapid up-sweep of the field. The rise time to peak field uponthe up-sweep is about 9 ms, and the down-sweep time is about 90 ms, hence dB/dt is muchlarger as the field increases. Therefore, the sample is relatively hot upon up-sweep as it exitsthe vortex state. Based on the temperature dependence of the SC F P phase, it turns intofield polarized state above the critical temperature. In such a scenario, what we observedupon up-sweep and down-sweep corresponds to the magnetic transition and superconductingreentrance, respectively. However, once the samples are in the normal state, there is knownto be little heating due to the changing field. This is consistent with the fact that thehysteresis is only observed for the superconducting phase, not the field polarized phase.Sample sizes are kept (i) small to present very little cross-sectional area to the field(thereby minimizing eddy-current heating) and (ii) thin to provide a large surface area-to-volume ratio to maximize cooling. In addition, rapid thermalization is assisted by using arelatively high pressure of He exchange gas. Finally, as mentioned above, during the down-sweep, dB/dt is significantly smaller than during the up-sweep, further reducing any residualeddy-current heating. Hence, the sample is essentially in equilibrium with the thermometerwhen it enters the vortex state on the way down, leading to an accurate measurement of thetransition. Therefore, we use the down-sweep data for determining the phase diagrams.There are other possible sources for the hysteresis. It might be that the SC
F P phaseis unusual, and that the magnetic susceptibility, rather than the conductivity, dominatesthe PDO response. The hysteresis loop we observed can be due to the irreversibility field ofsuperconductivity. Also, the vortices in the SC
F P phase can cause high-frequency dissipationduring the up-sweep, which is distinct from the heating mentioned above. Finally, the spatialdistribution of the screening currents within the sample could be different on the up-sweepand down-sweep. This could affect the samples ability to screen radio frequency fields,leading to hysteresis in the PDO measurement.In PDO measurements, we also observed hysteresis for θ = 0 ◦ . It is not clear whetherthe increase in frequency upon the up-sweep is related to the SC RE phase or the magnetictransition. 15 . criteria to determine the critical fields In order to construct the phase diagram with consistent critical field values, the followingcriteria are used to extrapolate the critical fields of various phases: for magnetoresistancemeasurements, we use the field at which the maximum slope of the resistance data that goesto zero resistance extrapolates to zero resistance (Fig. 8a and b); for PDO measurements,we use the field at which the maximum in derivative occurs (Fig. 8c and d).
E. Magnetization measurements
The magnetization measurement was performed with the magnetic field applied at θ =35 ◦ from b towards c -axis, where the SC F P phase was observed in magnetoresistance andPDO measurements. Similar to what seen for field along b -axis, the magnetic moment jumpsfrom 0.4 to 0.7 µ B , indicating a field polarized state in the high magnetic field.16 (cid:1) = o R (arb. units) H ( T ) T = 0 . 3 5 K (cid:1) = o (cid:1) = o FIG. 5. Field dependence of magnetoresistance of UTe at T = 0.35 K measured in the DC field.The magnetic field is rotated from b towards c axis. Zero resistance persists up to 34.5 T when themagnetic field is perfectly along b -axis. Reentrance of superconductivity is not observed when themagnetic field is rotated from b towards the c -axis. . 8 01 . 8 51 . 9 01 . 9 52 . 0 01 . 8 01 . 8 51 . 9 01 . 9 52 . 0 0 0 2 0 4 0 6 01 . 8 01 . 8 51 . 9 01 . 9 52 . 0 0 0 2 0 4 0 6 0 f (106 Hz) u p s w e e p d o w n s w e e p (cid:1) = 0 o ( a ) ( b )( c ) ( d )( e ) ( f ) (cid:1) = 20 o f (106 Hz) (cid:1) = 25 o (cid:1) = 35 o f (106 Hz) H ( T ) (cid:1) = 40 o H ( T ) (cid:1) = 50 o FIG. 6. PDO measurements of UTe at T = 0.45 K in the pulsed field, for magnetic field appliedat various angles from b towards the c -axis . 0 00 . 0 50 . 1 00 . 1 50 . 0 00 . 0 50 . 1 00 . 1 5 0 2 0 4 0 6 00 . 0 00 . 0 50 . 1 00 . 1 5 0 2 0 4 0 6 0 ( f )( e ) ( d )( c )( a ) R (arb. units) u p s w e e p d o w n s w e e p (cid:1) = 14.5 o (cid:1) = 19 o (cid:1) = 24 o (cid:1) = 22 o (cid:1) = 29 o (cid:1) = 34 o ( b ) R (arb. units) R (arb. units) H ( T ) H ( T ) FIG. 7. Magnetoresistance measurements of UTe at T = 0.45 K in the pulsed field, for magneticfield applied at various angles from b towards the c -axis. (cid:2) = 25 o H ( T ) f (106 Hz) d f /dT (arb. units) R (arb. units) H ( T ) (cid:2) = 19.3 o ( d )( c )( b ) H ( T ) f (106 Hz) - 1 . 0- 0 . 50 . 00 . 5 (cid:2) = 20 o d f /dT (arb. units) (cid:1) = 4.7 o R (arb. units) H ( T ) ( a ) FIG. 8. Selected magnetoresistance (a nd b) and PDO (c and d) measurements to show the criteriaused to extrapolate the critical field values for various phases. M ( m B/f.u) H ( T ) (cid:1) = 3 5 o (cid:1) = 3 5 o M ( m B/f.u) H ( T ) FIG. 9. Magnetization measurements UTe at T = 0.95 K and 1.5 K in the pulsed field, with themagnetic field applied at θ = 35 ◦ from b towards c -axis.-axis.