Magnetic Field-Induced Superconductivity in the Ferromagnet URhGe

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Science  26 Aug 2005:
Vol. 309, Issue 5739, pp. 1343-1346
DOI: 10.1126/science.1115498


In several metals, including URhGe, superconductivity has recently been observed to appear and coexist with ferromagnetism at temperatures well below that at which the ferromagnetic state forms. However, the material characteristics leading to such a state of coexistence have not yet been fully elucidated. We report that in URhGe there is a magnetic transition where the direction of the spin axis changes when a magnetic field of 12 tesla is applied parallel to the crystal b axis. We also report that a second pocket of superconductivity occurs at low temperature for a range of fields enveloping this magnetic transition, well above the field of 2 tesla at which superconductivity is first destroyed. Our findings strongly suggest that excitations in which the spins rotate stimulate superconductivity in the neighborhood of a quantum phase transition under high magnetic field.

The discovery of fundamentally new correlated electronic phases is rare in condensed matter physics. A promising parameter region in which to prospect for the emergence of such states is, however, found when a material is tuned through a continuous magnetic phase transition at very low temperature. As a material is tuned through such a transition, magnetic fluctuations become soft in energy and have large quantum amplitudes. Under such circumstances it becomes easier to deform the electronic spin system to adopt new ground state configurations potentially brought about by the large amplitudes of the fluctuations themselves. This description appears to apply at the pressures where antiferromagnetic order is suppressed in CeIn3 and CePd2Si2; a new superconducting phase is induced in a pocket of pressures at low temperature surrounding the critical pressure at which the antiferromagnetism is destroyed (1). For a magnetic transition between two different itinerant ferromagnetic phases, the correlated states that can emerge are potentially quite different. Equal spins as opposed to opposite spins must be paired for superconductivity to occur, owing to the magnetic polarization of the electronic band structure. UGe2 (2) and URhGe (3) have already attracted attention because of the recent discoveries that bulk superconductivity coexists with ferromagnetism in these materials well below their ferromagnetic transition temperatures. The inference that equal spins are paired in the superconducting state is additionally supported for URhGe by measurements of the critical field necessary to suppress the superconductivity (4). We report here that in URhGe a magnetic transition can be induced by applying a much larger magnetic field than that at which superconductivity is first destroyed and that at low temperature a new pocket of superconductivity emerges surrounding this transition.

URhGe is ferromagnetic below a Curie temperature, TC, of 9.5 K, with a spontaneous moment aligned to the c axis of its orthorhombic crystal structure. The field-induced magnetization is smaller for fields applied along the a axis than the other crystal axes. We therefore restrict the discussion to fields in the bc plane. The first indication that a magnetic transition occurs under magnetic field in URhGe can be found in (5), where a jump in the field-induced magnetization was seen at high field and low temperature. We find that such a jump occurs when the field is applied close to the b axis direction. We show below that this transition corresponds to a sudden rotation of the moment in the bc plane.

Measurements of the torque acting on a single crystal of URhGe (Fig. 1A) show that the component of the magnetization parallel to the c axis, Mc, collapses abruptly to zero when a magnetic field of magnitude, HR, is applied along the b axis, with μ0HR = 11.7 T (μ0 is the permeability of a vacuum). Measurements, both directly in a magnetometer and by neutron scattering (Fig. 1B), show that the magnetization parallel to the b axis also increases suddenly at this field, but that the total moment above HR is close to the value extrapolated from fields well below HR. Therefore, the main change in the magnetization across the transition is a rotation of the moment toward the b axis. For fields aligned at an angle, θ, from the b axis, a rapid rotation of the moment occurs at higher fields, HR(θ). The moment no longer aligns perfectly to the field direction immediately above the transition, and the transition becomes progressively broader.

Fig. 1.

Field-induced moment rotation and superconductivity in URhGe. (A) The magnetic field dependence of the torque acting on a single crystal of URhGe at low temperature (0.1 K). This is shown for different angles, θ, of the field from the crystal b axis in the bc plane. The arrows indicate measurements made increasing and decreasing the field for θ < 0.1°. In zero field the spontaneous magnetization is parallel to the c axis. When θ is close to zero, hysteresis is seen at low fields due to the polarization of the ferromagnetic domain structure by the small field component parallel to the c axis. Only a very slight rotation of the field in the bc plane is necessary for this component to exceed the coercive field necessary to drive the sample monodomain. The torque, τ, divided by μ0 H is then proportional to the saturated ferromagnetic moment perpendicular to the applied field. The sudden drop in the torque at high field corresponds to a sudden reduction of this component of the magnetization. (B) The total magnetic moment and the component of the moment parallel to the b axis (in units of Bohr magnetons per URhGe) for θ = 0, measured by neutron scattering at 2 K (the lines are to guide the eye) (23). The moment parallel to b increases rapidly at HR (the slightly smaller value of HR and broader transition compared with the torque data are explained by the larger temperature). Error bars show the estimated standard deviation of each measurement. (C) The electrical resistance of the sample over the same field range at temperatures of 40 mK and 500 mK. At 500 mK the sample is in the normal state and a clear peak in the resistance is seen at HR. At 40 mK the resistance is zero for a range of fields about HR. This pocket of field induced superconductivity occurs in addition to that observed below 2 T [reported previously (3, 4)].

In the normal state, there is a clear peak in the electrical resisitivity as a function of the applied field close to HR(θ) (Fig. 1C). The peak in the resistivity can be described as the sum of two parts: (i) a very narrow δ-function-like peak that is almost temperature-independent (in the temperature range from just above the superconducting transition up to 0.8 K) and (ii) a broader asymmetric peak whose amplitude increases with temperature. The former feature has a width of 0.5 T whereas the latter has a characteristic width of about 5 T. The amplitude of the δ function is smaller for larger θ, and it is completely absent for θ ≥ 5°. It follows from its temperature independence that it represents an enhancement of the residual normal-state resistivity. This might be caused by the presence of a finely divided domain structure close to a first-order transition. The amplitude of the broader peak, (ii), depends only weakly on θ. Its field dependence is different from the field dependence of the residual normal-state resistivity; the two can be compared directly in a low quality sample that does not become superconducting. It therefore corresponds in part to an increase in the dynamic scattering rate of the conduction electrons close to HR(θ) and is not due simply to a change in the electronic density of states. A low-temperature phase diagram for fields in the bc plane that describes all the above results is shown schematically in Fig. 2B. A discontinuous change in the moment orientation occurs across the thick line in the figure. The change becomes continuous at the point where the line ends. In the limit of zero temperature, this point is referred to as a quantum critical end point (QCEP).

Fig. 2.

The low temperature resistivity and magnetic phase diagram for fields in the crystallographic bc plane. (A) The measured resistance for fields in the bc plane. The resistance at 40 mK is represented by the color (top scale). The black areas are regions where the sample has zero resistance and is superconducting. Contour lines depict the resistance at 500 mK (bottom scale). The area where superconductivity occurs at low temperature is seen to correspond to the region over which the resistance is peaked at higher temperature. (B) A representation of the magnetic phase diagram at low temperature. The thin lines are contours of constant angle, ϕ, of the magnetic moment from the b axis. The thick line denotes a first order transition across which ϕ changes discontinuously. The first order line ends at a QCEP. Beyond this point a sharp crossover behavior still occurs in the field dependence of the moment orientation. The definition of ϕ is illustrated in the sketch at the right, with arrows depicting the direction of the magnetization, M, and of the components of the applied field, Hb and Hc.

In good quality samples with a small normal-state residual resistivity, ρ0, equivalent to a large residual resistivity ratio (RRR, the resistance at 300 K divided by ρ0), superconductivity occurs below 2 T as previously reported (3). Our finding is that superconductivity reappears over a wide range of fields about HR below 400 mK. A lower quality sample with RRR = 5, however, was not superconducting at any field, thus suggesting superconductivity in both field ranges is unconventional; that is, the phase of the superconducting order parameter has an intrinsic directional dependence. Measurements on a high-quality single crystal of RRR ≈ 50 are shown in Figs. 1, 2, 3. The maximum transition temperature, Ts, in the high-field superconducting state is almost 50% greater than Ts observed in zero field. The large field range over which superconductivity occurs around HR shows that superconductivity exists in regions in which the sample is monodomain and that superconductivity is not confined to domain walls.

Fig. 3.

The field-temperature phase diagram for applied fields parallel to the b axis. The color represents the resistivity. Superconductivity occurs throughout the black region where the resistivity is zero. The maximum transition temperature corresponds to the field, HR, at which the resistivity has a sharp maximum at higher temperature. The blue solid lines show the position at which the resistance is half its normal-state value (for the data at low field, this was determined more precisely in separate measurements). (Inset) The resistance as a function of field at several temperatures corresponding to horizontal cuts through the main figure.

Field-induced superconductivity has been observed before in other materials, for example in the Chevrel phase compound Eu3/4Sn1/4 Mo6S7.2Se0.8 (6) and more recently in several organic superconductors including k-(BETS)2 FeBr4 (7) and k-(BETS)2FeCl4 [in the latter, field induced superconductivity is observed when the field is accurately aligned to two-dimensional (2D) structural planes (8, 9)]. These cases can be explained by a compensation of the applied field by an internal field produced by the polarization of magnetic ions, resulting in a total effective field that is actually small in high applied fields [the Jaccarino-Peter effect (10)]. For all these materials the induced moment is parallel to the applied field. In contrast, for URhGe superconductivity occurs when the ordered moment is inclined over a range of angles from 30° to 55° to the applied field direction. It is unlikely that an exchange field due to magnetic moments could completely cancel the effect of the applied field over such a wide range of angles. Further, in the Chevrel phase and organic superconductors the field induced superconductivity is distinct from other phase transitions seen at different fields. The originality of the behavior observed in URhGe is that the maximum Ts occurs exactly at the transition field HR (Fig. 3). The correlation between superconductivity and the moment rotation transition is confirmed by comparing the dependence of HR(θ) with the angle dependence of the superconducting phase boundary (Fig. 2). The superconductivity at high field is thus intimately connected with the moment rotation.

Theoretically, in a ferromagnet, the orientation of the magnetic moment, Math, in a magnetic field of induction, Math, is determined from the balance between the magnetic energy Math, which seeks to align the magnetic moment parallel to the field, and the energy required to rotate the moment away from its preferred orientation with respect to the crystal structure. For a fixed direction of the applied magnetic field, the moment orientation usually changes in a continuous manner as the magnitude of the applied field is changed, but a first order process in which the moment rotates discontinuously at some field is possible. It can occur for appropriate values of the crystal field anisotropy and spin-orbit coupling with a fixed magnitude of the magnetic moment. For URhGe the magnitude of the moment depends on the field, and a more complete description is given by a Landau expansion of the free energy, Math. A first-order transition occurs in this case when the term bxy is greater than a critical value (determined by the values of the other coefficients, ax, ay, bx, and by). For URhGe the various coefficients can be determined from the initial differential susceptibility parallel to the b axis and Arrott plots of the magnetization for fields parallel to the c axis and for fields H > HR parallel to the b axis. The condition for a first order transition for fields close to the b axis is found to be satisfied, and the computed phase diagram based on the above expression for the free energy is qualitatively compatible with that shown in Fig. 2B.

The theory of superconductivity mediated by the exchange of spin fluctuations is most often considered close to a ferromagnetic-paramagnetic quantum critical point where the longitudinal differential susceptibility diverges at low energy and wave vectors. Only this region of energy–wave vector space then has to be considered (11, 12). Under these conditions, a large value of the uniform differential susceptibility parallel to the magnetization favors the formation of Cooper pairs with equal spins, whereas a large value of the differential susceptibility perpendicular to the magnetization breaks such pairs. The situation is modified well inside the ferromagnetic state, because the transverse excitations no longer have the same form; they are collective spin waves rather than incoherent overdamped modes. In an isotropic ferromagnet, they can lead to an enhancement of the longitudinal susceptibility due to mode coupling that outweighs their pair-breaking effect (13). For URhGe this same mechanism could be active in a modified form. An important aspect not considered in previous theory is the anisotropy of the spin fluctuation spectrum for different directions of the wave-vector transfer, Math. For example, a magnetic-field energy is incurred when Math is not perpendicular to the change in magnetization associated with an excitation (14). For spin rotation excitations in the bc plane, this energy would be absent for wave-vector transfers along the a axis, and excitations propagating in this direction would consequently have a lower energy than along other directions of Math. This could favor a polar superconducting order parameter oriented along the a axis. It is noteworthy that such a state can explain the critical field of the low field superconductivity (4). Theoretically, the symmetry of such a state would also be consistent with the crystal structure and ferromagnetism with the moments aligned along the b axis (15).

Over recent years, the application of a magnetic field at very low temperature has been established to be an effective tuning parameter to drive a number of materials to a quantum critical point or QCEP [examples are YbRh2Si2 (16), Sr3Ru2O7 (17), and URu2Si2 (18)]. In the limit of zero temperature, the divergence of the differential magnetic susceptibility at this point implies a diverging amplitude for zero-point motion (quantum fluctuations) that can destabilize the system relative to other forms of order (19). The behavior in URhGe can be compared with that of the almost-2D material Sr3Ru2O7, where a new as-yet incompletely identified ground state appears in high quality samples enveloping a QCEP at 7.8 T. For Sr3Ru2O7 it has been argued that superconductivity is not viable because of the large field at which the QCEP occurs (17). For opposite-spin pairing both paramagnetic limitation and orbital limitation restrict the maximum field up to which superconductivity can survive. For equal-spin pairing only the second limit applies. This requires that the superconducting coherence length, ξ0, is small enough to satisfy the relation ϕ0/(2πξ02) > B0 is the flux quantum and B the magnetic induction); a value ξ0 < 50 Å would be compatible with the high field superconducting phase of URhGe.

It appears that the high field superconductivity in URhGe, like the superconductivity at low field in UGe2, is not directly driven by fluctuations associated with a quantum critical point or QCEP separating ferromagnetism from paramagnetism. In both materials superconductivity is instead associated with a magnetic transition between two strongly polarized states, although the transitions differ; in URhGe there is a large change in the transverse moment at the transition, whereas in UGe2 only the longitudinal moment changes. For UGe2 the superconducting coupling strength and transition temperature increase as the magnetic transition is approached by tuning the pressure (20). The magnetic transition is, however, first order (21), and UGe2 has not yet been studied under the conditions necessary to drive it to a QCEP. The apparent relationship of high field superconductivity to a field-induced quantum critical point in URhGe established here, however, reinforces the general notion that new strongly correlated electron ground states emerge close to quantum critical transitions between apparently simpler magnetic phases. An interesting possibility is that the low field superconductivity in URhGe might also be related to the same quantum critical point that we now outline. Superconductivity occurs when the upper critical field for the superconducting state, Hc2, exceeds the total magnetic field acting on the electrons. For URhGe, as the applied field is reduced from HR moving the material away from the QCEP, Hc2 is expected to fall rapidly. Superconductivity would disappear when Hc2 falls below the applied field (for simplicity, the small internal field in the sample due to its magnetization, μ0M≈ 0.1 T, can be ignored). However, if Hc2 is still finite at low fields, the condition for superconductivity (with a much weaker coupling strength) would once again be fulfilled when the applied field is reduced to zero.

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