Abstract
When a pure material is tuned to the point where a continuous phasetransition line is crossed at zero temperature, known as a quantum critical point (QCP), completely new correlated quantum ordered states can form^{1,2,3,4,5,6,7}. These phases include exotic forms of superconductivity. However, as superconductivity is generally suppressed by a magnetic field, the formation of superconductivity ought not to be possible at extremely high field^{8}. Here, we report that as we tune the ferromagnet, URhGe, towards a QCP by applying a component of magnetic field in the material’s easy magnetic plane, superconductivity survives in progressively higher fields applied simultaneously along the material’s magnetic hard axis. Thus, although superconductivity never occurs above a temperature of 0.5 K, we find that it can survive in extremely high magnetic fields, exceeding 28 T.
Main
The formation of superconducting states close to magnetic quantum critical points (QCPs) first came to prominence in studies of the antiferromagnets CeCu_{2}Ge_{2} (ref. 1), CeIn_{3} and CePd_{2}Si_{2} (ref. 2) and more recently CeRhIn_{5} (refs 5,6). Theoretically, on tuning a material towards a QCP by application of pressure or magnetic field, the strength of the magnetic fluctuations that potentially bring about superconductivity increases. On approaching a ferromagnetic QCP, longitudinal magnetic fluctuations promote the formation of unconventional spintriplet superconductivity^{9,10,11}. Theories predict dwave superconductivity close to QCPs involving antiferromagnetic states. The evolution of the superconductivity as a material is tuned more closely to the field or pressure of the underlying QCP depends on the balance between the weights of fluctuations that are pair forming and pair breaking^{12}. The critical temperature for superconductivity, T_{s}, is predicted to saturate^{9} or may even decrease^{10}. Experimentally, superconducting states in antiferromagnetic systems have been more extensively studied than in ferromagnets and measurements show T_{s} to have a domeshaped pressure dependence crossing the underlying QCP^{2,5}. However, recent work indicates that the pressure–temperature phase diagrams of antiferromagnetic systems might be more complex than previously thought^{5,6,13}. As it is difficult to vary pressure continuously at low temperature, other properties of the superconducting state, such as the critical field to suppress superconductivity in CePd_{2}Si_{2} (ref. 14), lack measurements at a sufficient number of pressures to discern how they vary approaching a QCP. For URhGe, the QCP is ferromagnetic rather than antiferromagnetic and can be approached by applying a magnetic field, which can be swept continuously.
In zero magnetic field, URhGe undergoes a ferromagnetic transition at T_{Curie}=9.5 K. Below this temperature, the ordered magnetic moment is aligned parallel or antiparallel to the crystallographic c axis^{15}. Magnetic fields that correspond to opposite directions of the caxis moment are separated in low magnetic field by a plane of firstorder transitions that is crossed when the caxis component of the magnetic field changes sign. The temperature above which this firstorder transition plane ceases defines a line along which the phase transition is continuous. The schematic temperature–field phase diagram for fields throughout the b c plane is shown in Fig. 1. Close to a field of 12 T, the plane of firstorder transitions bifurcates into two spurs. There is a corresponding tricritical point (TCP) at low, but nonzero temperature, where the continuoustransition line splits. The points where the continuoustransition line falls to zero temperature are QCPs^{4}. URhGe is also close to a quantum TCP; such a point would occur if the TCP could be suppressed to zero temperature by varying some further tuning parameter. The phase diagram in Fig. 1 resembles that predicted theoretically for a ferromagnet driven towards a QCP (ref. 16) with the magnetic field applied along the b axis in the present case replacing pressure in the theoretical diagram.
A consequence of the lowtemperature TCP is that fluctuations develop with a similar strength at very low temperature at all points along the length of the firstorder spurs. Experimentally, this is manifested by the peaks in the temperaturedependent part of the resistivity having comparable amplitudes crossing the weak firstorder transition at different points^{4}. Closer than 0.25 T to the spurs, there is an additional sharp peak in the normalstate resistivity, which is independent of temperature (up to 800 mK). We have attributed this effect to extra scattering of charge carriers from magnetic domains that form close to the firstorder transition^{4}. As the temperature is reduced to 50 mK, superconductivity occurs both in low applied magnetic fields below 2 T and at high applied fields engulfing the QCPs and the regions where the firstorder transition is weak (Fig. 1). The magnitude of the fields at which superconductivity occurs relative to the superconducting transition temperature in both the lowfield and highfield pockets implies that equal spins must be paired in contrast to opposite spin pairing in conventional superconductivity^{4,17}.
Figure 2 shows our measurements with magnetic fields applied at different angles, γ, from the b axis towards the aaxis direction. The persistence of the sharp peak in the resistivity at 500 mK shows that the magnetic transition remains weakly first order as γ is increased. The field at which the peak occurs, H_{R}(γ), is well modelled by the relation H_{R}(γ)=H_{R}(0)/cos(γ). For fields in the a b plane, the transition field is therefore crossed when the component of the field along the b axis is equal to H_{R}(0) (henceforth abbreviated to H_{R}) irrespective of the aaxis field. Application of a magneticfield component parallel to the a axis for a constant baxis field therefore enables us to determine the critical field for superconductivity at a constant offset from the QCP.
Within our experimental resolution, the angular dependence of the critical field for the lowfield pocket of superconductivity at low temperature (denoted as H_{sc1}) follows the standard form for an anisotropic superconductor (with an orbitally limited critical field),
H_{sc1a}=φ_{0}/2πξ_{c}ξ_{b}=2.53 T, H_{sc1b}=φ_{0}/2πξ_{c}ξ_{a}=2.07 T and H_{sc1c}=φ_{0}/2πξ_{b}ξ_{a}=0.69 T are the critical fields along each of the crystal axes for the sample studied, determined in separate more accurate measurements (ξ_{a},ξ_{b} and ξ_{c} are the coherence lengths along the different crystal axes and φ_{0} is the flux quantum). An equation of the same form as equation (1) is derived from the Ginzburg–Landau theory and is applicable in general close to the zerofield superconducting transition temperature, T_{s}. For URhGe, we can specialize to the case of no Pauli limiting^{17}. In the absence of Pauli limiting, although the coherence lengths depend on temperature, for simple ellipsoidal Fermi surfaces (which can account for an anisotropic electronic effective mass) and isotropic swave superconductivity, the anisotropy of the coherence lengths and the angular dependence of the critical field is unchanged at low temperature from that close to T_{s} (ref. 18). Different factors can cause the anisotropy to be temperature dependent or modify the angular dependence of the critical field at low temperature^{19}. (1) Strong coupling gives only minor modifications in case studies where detailed calculations have been made^{20}. (2) Anisotropy of the superconducting order parameter can give changes of anisotropy with temperature of the order of 20% (refs 17,21). However, numerical calculations for different nonconventional superconductors^{18} suggest that equation (1) continues to provide a good phenomenological description of the angular dependence of the critical field at low temperature. (3) Special choices of Fermisurface geometry can lead to major modifications of the form of the angular dependence at low temperature^{19}. As the angular dependence we have measured is standard, the Fermi surface in URhGe cannot have such a special geometry in low magnetic field.
The present sample has a residual resistance ratio, RRR=50. RRR is the ratio of the electrical resistance at room temperature divided by the normalstate resistance at low temperature and is proportional to the defectlimited electronic meanfreepath. The critical field, H_{sc1}, of the present sample is larger than that for a previously studied lower quality sample with RRR=20(ref. 17). The ratio H_{sc1c}(RRR=50)/H_{sc1c}(RRR=20)≈H_{sc1a}(RRR=50)/H_{sc1a}(RRR=20)=1.34 is approximately equal to the ratio of the square of the superconducting transition temperatures of the two samples in zero magnetic induction (T_{s}(RRR=50)/T_{s}(RRR=20))^{2}=(0.278 K/0.237 K)^{2}=1.37 (ref. 17). This result is the theoretical expectation for a nonconventional superconducting state. The ratio H_{sc1b}(RRR=50)/H_{sc1b}(RRR=20)=1.70 is 30% larger than the above ratio; this is discussed further below in the light of our study of the highfield superconductivity.
The angular dependence of the critical fields, H_{sc2} and H_{sc3}, bounding the pocket of superconductivity at high magnetic field is quite different from the angular dependence of the critical field for the lowfield superconducting pocket. The field window over which the highfield superconductivity occurs extends to either side of H_{R}(γ) up to the highest applied field of 28 T (Fig. 2b). The main result of our study is that the component of the critical field parallel to the a axis plotted as a function of H_{b} (that serves to tune the material through the underlying magnetic transition) shows a strong upward curvature as H_{b} approaches H_{R} (Fig. 2c).
The measured angular dependence of H_{sc1} shows that the Fermisurface geometry does not give rise to a stronger anisotropy of the critical field than that which can be accounted for with an anisotropic effective mass. Assuming that the Fermisurface geometry does not change radically with field, the strong field dependence of the critical fields H_{sc2} and H_{sc3} visible in Fig. 2 indicates that the magnitude of the coherence length changes as H_{b} is tuned through H_{R}. The geometric average of the coherence length, ξ=(ξ_{a}ξ_{b}ξ_{c})^{1/3}, deduced from the measured H_{sc1},H_{sc2} and H_{sc3}, assuming a fixed coherence length anisotropy, is shown in Fig. 3 (see the Methods section). Remarkably, the lowfield and highfield superconducting regions seem to lie on the same smoothly varying curve. Furthermore, the stronger than T_{s}^{2} increase of H_{sc1b} with sample quality can now be understood to result from an increase of φ_{0}/2πξ^{2} as the lowfield pocket of superconductivity extends to cover higher baxis magnetic fields in better samples, reinforced by a reduction in the effectiveness of pair breaking by impurities owing to a shortening of the coherence length.
Theoretically, 1/ξ≈k_{B}T_{s}/ℏv_{f}, where T_{s} is the hypothetical superconducting temperature in the absence of magnetic field and v_{f} is the Fermi velocity, inversely proportional to the effective electron quasiparticle mass for a fixed Fermi surface (ℏ is the Planck constant divided by 2π and k_{B} is the Boltzmann constant). The increase of 1/ξ as the baxis field approaches H_{R} requires that the effective mass and/or T_{s} increase strongly. For field applied along the b axis, the measured critical temperature for superconductivity has an upward curvature when plotted against field approaching H_{R} (ref. 4). Furthermore, the measured critical temperature reaches a higher value in the highfield superconducting pocket than found at zero field^{4}. These measurements suggest that most of the increase of 1/ξ approaching H_{R} over the field range measured can be accounted for by an increase of T_{s}. Theoretically, T_{s} depends exponentially on different contributions to the effective mass^{12}, so this is not inconsistent with more modest changes of the effective mass. At a QCP driven by local physics, the effective quasiparticle mass diverges, as observed in YbRh_{2}Si_{2} (refs 22,23). An increase of the effective mass is also predicted for a QCP in an itinerant ferromagnet^{24} and observed approaching magneticfieldinduced transitions in several materials^{25,26}.
In other materials where QCPs have been studied experimentally under high magnetic field, including URu_{2}Si_{2} (ref. 3) and Sr_{3}Ru_{2}O_{7} (refs 7,27), new states that are not superconducting have been found. Our work on URhGe shows that superconductivity can, however, survive to much higher magnetic fields close to a QCP than previously thought likely^{8}.
Methods
The measurements were made with the M9 resistive magnet at the Grenoble High Magnetic Field Laboratory (GHMFL) in a toploading dilution refrigerator equipped with a lowtemperature rotator. The sample studied is the same as that used for resistivity measurements for fields in the b c plane^{4} and is a small bar aligned with its c axis parallel to the axis of rotation (within 1^{∘}). Current was passed along the length of the sample (b axis). The position of 90^{∘} rotation with the field parallel to the a axis was determined so that H_{R}(γ) changed symmetrically for positive and negative rotations of γ away from this direction.
The fixed value for the coherence length anisotropy used to estimate the geometrically averaged coherence length from the measured values of the critical fields H_{sc1},H_{sc2} and H_{sc3} was chosen to give the measured critical field anisotropy of H_{sc1} of the sample with RRR=20 with a constant value of φ_{0}/2πξ^{2}. Symmetry imposes that the leading field dependence of φ_{0}/2πξ^{2} is quadratic around zero field. The lower quality sample has a lower value of H_{scb1} than the higher quality sample and the change of φ_{0}/2πξ^{2} from its zerofield value at H_{scb1} is small enough to be neglected.
References
 1
Jaccard, D., Behnia, K. & Sierro, J. Pressure induced heavy fermion superconductivity of CeCu2Ge2 . Phys. Lett. A 163, 475–480 (1992).
 2
Mathur, N. D. et al. Magnetically mediated superconductivity in heavy fermion compounds. Nature 394, 39–43 (1998).
 3
Harrison, N., Jaime, M. & Mydosh, J. A. Reentrant hidden order at a metamagnetic quantum critical end point. Phys. Rev. Lett. 90, 96402 (2003).
 4
Lévy, F., Sheikin, I., Grenier, B. & Huxley, A. D. Magnetic fieldinduced superconductivity in the ferromagnet URhGe. Science 309, 1343–1346 (2005).
 5
Park, T. et al. Hidden magnetism and quantum criticality in the heavy fermion superconductor CeRhIn5 . Nature 440, 65–68 (2006).
 6
Knebel, G., Aoki, D., Braithwaite, D., Salce, B. & Flouquet, J. Coexistence of antiferromagnetism and superconductivity in CeRhIn5 under high pressure and magnetic field. Phys. Rev. B 74, 020501R (2006).
 7
Borzi, R. A. et al. Formation of a nematic fluid at high fields in Sr3Ru2O7. Science 315, 214–217 (2007)
 8
Coleman, P. & Schofield, A. J. Quantum criticality. Nature 433, 226–299 (2005).
 9
Roussev, R. & Millis, A. J. Quantum critical effects on transition temperature of magnetically mediated pwave superconductivity. Phys. Rev. B 63, 140504 (2001).
 10
Fay, D. & Appel, J. Coexistence of pstate superconductivity and itinerant ferromagnetism. Phys. Rev. B 22, 3173–3182 (1980).
 11
Monthoux, P. & Lonzarich, G. G. p and dwave superconductivity in quasitwodimensional metals. Phys. Rev. B 59, 14598–14605 (2001).
 12
Millis, A. J., Sachdev, S. & Varma, C. M. Inelastic scattering and pair breaking in anisotropic and isotropic superconductors. Phys. Rev. B 37, 4975–4986 (1988).
 13
Kawasaki, S. et al. New superconducting and magnetic phases emerge on the magnetic criticality in CeIn3 . J. Phys. Soc. Japan 73, 1647–1650 (2004).
 14
Sheikin, I. et al. Superconductivity, upper critical field and anomalous normal state in CePd2Si2 near the quantum critical point. J. Low Temp. Phys. 122, 591–604 (2001).
 15
Aoki, D. et al. Coexistence of superconductivity and ferromagnetism in URhGe. Nature 413, 613–616 (2001).
 16
Belitz, D., Kirkpatrick, T. R. & Rollbühler, J. Tricritical behavior in itinerant quantum ferromagnets. Phys. Rev. Lett. 94, 24706 (2005).
 17
Hardy, F. & Huxley, A. D. pwave superconductivity in the ferromagnetic superconductor URhGe. Phys. Rev. Lett. 94, 247006 (2005).
 18
Prohammer, M. & Carbotte, J. P. Upper critical field of s and dwave superconductors with anisotropic effective mass. Phys. Rev. B 42, 2032–2040 (1990).
 19
Kita, T. & Arai, M. Ab initio calculations of Hc2 in typeII superconductors: Basic formalism and model calculations. Phys. Rev. B 70, 224522 (2004).
 20
Werthamer, N. R. & McMillan, W. L. Temperature and purity dependence of the superconducting critical field, Hc2. IV. Strong coupling effects. Phys. Rev. 158, 415–417 (1967).
 21
Scharnberg, K. & Klemm, R. A. Upper critical field in pwave superconductors with broken symmetry. Phys. Rev. Lett. 54, 2445–2448 (1985).
 22
Gegenwart, P. et al. Magneticfield induced quantum critical point in YbRh2Si2 . Phys. Rev. Lett. 89, 056402 (2002).
 23
Custers, J. et al. The breakup of heavy electrons at a quantum critical point. Nature 424, 524–527 (2003).
 24
Lonzarich, G. G. in Electron, a Centenary Volume (ed. Springford, M.) 109–147 (Cambridge Univ. Press, Cambridge, 1997).
 25
Paulsen, C. et al. Low temperature properties of the heavyfermion compound CeRu2Si2 at the metamagnetic transition. J. Low Temp. Phys. 81, 317–339 (1990).
 26
Borzi, R. A. et al. de Haasvan Alphen effect across the metamagnetic transition in Sr3Ru2O7 . Phys. Rev. Lett. 92, 216403 (2004).
 27
Grigera, S. A. et al. Disordersensitive phase formation linked to metamagnetic quantum criticality. Science 306, 1154–1157 (2004).
Acknowledgements
Financial support was provided for work at the GHMFL from the European Commission. A.H. gratefully acknowledges support from the Royal Society, UK.
Author information
Affiliations
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Rights and permissions
About this article
Cite this article
Lévy, F., Sheikin, I. & Huxley, A. Acute enhancement of the upper critical field for superconductivity approaching a quantum critical point in URhGe. Nature Phys 3, 460–463 (2007). https://doi.org/10.1038/nphys608
Received:
Accepted:
Published:
Issue Date:
Further reading

Extreme magnetic fieldboosted superconductivity
Nature Physics (2019)

125TeNMR Study on a Single Crystal of Heavy Fermion Superconductor UTe2
Journal of the Physical Society of Japan (2019)

Enhancement of superconductivity by pressureinduced critical ferromagnetic fluctuations in UCoGe
Physical Review B (2019)

Review of Ubased Ferromagnetic Superconductors: Comparison between UGe2, URhGe, and UCoGe
Journal of the Physical Society of Japan (2019)

Thermodynamic Investigation of Metamagnetism in Pulsed High Magnetic Fields on Heavy Fermion Superconductor UTe2
Journal of the Physical Society of Japan (2019)