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Observation of broken time-reversal symmetry in the heavy-fermion superconductor UPt3

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Science  11 Jul 2014:
Vol. 345, Issue 6193, pp. 190-193
DOI: 10.1126/science.1248552

Optically probed superconductor

The exotic superconductor UPt3 has two superconducting phases that appear at different temperatures, but their nature remains unclear. Schemm et al. shone circularly polarized light on a crystal of UPt3 and studied its reflection (see the Perspective by van der Marel and Sawatzky). In the low-temperature phase, the pairs of electrons that make the material superconducting have a handedness to them. The finding narrows down the possible descriptions of the electron-pair wave function.

Science, this issue p. 190; see also p. 138

Abstract

Models of superconductivity in unconventional materials can be experimentally differentiated by the predictions they make for the symmetries of the superconducting order parameter. In the case of the heavy-fermion superconductor UPt3, a key question is whether its multiple superconducting phases preserve or break time-reversal symmetry (TRS). We tested for asymmetry in the phase shift between left and right circularly polarized light reflected from a single crystal of UPt3 at normal incidence and found that this so-called polar Kerr effect appears only below the lower of the two zero-field superconducting transition temperatures. Our results provide evidence for broken TRS in the low-temperature superconducting phase of UPt3, implying a complex two-component order parameter for superconductivity in this system.

The heavy-fermion metal UPt3 (1) is one of only a handful of unconventional superconductors (2, 3) exhibiting multiple superconducting phases (46). In the normal state, strong hybridization between itinerant platinum 5d electrons and localized uranium 5f moments results in an effective mass that is ~50 times that of free electrons (2). Below the Néel temperature TN ~ 5 K, the local U moments order antiferromagnetically in the a-b plane (7). In zero magnetic field, two peaks in the specific heat at Tc+ ~ 550 mK and Tc– ~ 480 mK indicate the presence of two superconducting states of differing symmetry, called the A and B phases, respectively (46). Pressure studies suggest that these two phases couple to, and are stabilized by, the antiferromagnetic order parameter (8). In finite magnetic fields, three distinct vortex phases are also observed (911). The phase diagram of UPt3 therefore presents a particular challenge for models of unconventional superconductivity.

In the absence of a detailed understanding of the microscopic origins of unconventional superconductivity, theoretical and experimental efforts center on identifying the structure of the macroscopic superconducting order parameter—the pair wavefunction, or gap. In the case of UPt3, acceptable candidate order parameters should respect the D6h point-group symmetry of the underlying crystal lattice and should therefore transform under one or more representations of this group. In this framework, many—but not all—experimental studies of the superconducting states (1) favor an E2u odd-parity triplet representation in which the gap is given byEmbedded Image (1)in a coordinate system where Embedded Imagec. Here, Embedded Imageis the real component of the superconducting order parameter, marking the onset of the A phase at Tc+, whereas Embedded Imageintroduces an additional imaginary component in the B phase at Tc– (12, 13). An order parameter of this form first breaks gauge symmetry in the A phase, in which it also exhibits fourfold rotational symmetry in the a-b plane distinct from the hexagonal symmetry of the crystal lattice. In the B phase, the order parameter becomes isotropic in the a-b plane as T → 0 K, and the phase difference between the real and imaginary components imparts an overall angular momentum to the pair wave function. Hence, time-reversal symmetry (TRS) is broken in this phase, with the sign of the imaginary component determining the orientation (chirality) of the internal angular momentum of the pair along ± Embedded Image.

The results of Josephson interferometry experiments (13, 14) are consistent with the spatial symmetries of the E2u order parameter of Eq. 1. However, attempts to observe TRS-breaking (TRSB) in UPt3 via muon spin relaxation measurements have yielded conflicting results (15, 16). Moreover, recent thermal conductivity data (17) have been interpreted to support a gap function belonging to an E1u representation that precludes TRSB in the B phase. Thus, the unresolved question of whether TRS is indeed broken in the B phase has become critical to determining the symmetry and hence the proper classification of the superconducting order parameter of UPt3.

A general consequence of a TRSB order parameter (with a net moment oriented along the c axis) in the presence of particle-hole asymmetry is the appearance of a finite difference between the complex indices of refraction for right (nR) and left (nL) circularly polarized light, resulting in ellipticity and circular birefringence (12). In particular, the polar Kerr angle, which measures the degree of rotation of linearly polarized light at normal incidence, is related to the off-diagonal terms of the optical conductivity (18) byEmbedded Image (2)where ñ is the average complex index of refraction of the material, d is a microscopic length scale, and ω is the frequency of the incident light. The off-diagonal term of the conductivity tensor, Embedded Image, is nonzero if TRS is broken (19), and the second equality holds for weak absorption (21). In the simplest estimate, σxy will have the natural scale of the Hall conductivity e2/h, where h is Planck’s constant, reduced by a factor that reflects the large difference between the superconducting gap and probe energies: Δ0 << ℏω in the optical regime, where Δ0 is the magnitude of the order parameter at the Fermi level and ℏ is h divided by 2π. Because the current response to the incident electric field involves a product of the wave function and its gradient, the conductivity must be proportional to Embedded Image, yielding Embedded Image (21). As a result, the degree of optical rotation expected from TRSB in unconventional superconductors is expected to be on the order of 1 microradian (μrad) or less (12). Measurements of such small rotations pose challenges to experiment; nevertheless, high-resolution measurements of polar Kerr effect (PKE) have been used to place limits on anyon superconductivity in the high-Tc cuprates (22) and to establish TRSB in the order parameter of the spin-triplet superconductor Sr2RuO4 (23).

The requirement to resolve optical rotation below 1 μrad without application of an external TRSB field (such as a magnetic field) prevents us from using the modulation techniques commonly used to detect small signals in magnetic materials. Thus, to measure Kerr rotation in UPt3 we have constructed a fiber-based zero-area loop Sagnac interferometer operating at 1550 nm (24, 25) with a focused spot size of 10.6 μm. The interferometer is designed and biased to reject optical signals attributable to nonreciprocal effects (such as linear birefringence and backscattering off of interfaces), allowing us to resolve changes in Kerr angle on the order of 50 nrad with a minimum of optical power incident on the sample. This latter quality is of particular relevance in this study because spot heating in the temperature range of interest (300 to 600 mK) can be substantial even for thermally well-anchored samples with high thermal conductivities. To optimize the signal-to-noise ratio (SNR), we typically applied 20 μW of optical power to the sample, of which ~40% is expected to have been absorbed (26). Lower incident powers (with inferior SNR) were also used to verify our results (27).

Our experiments were performed on a 4.5- by 3.3- by 0.91-mm single crystal of UPt3 with residual resistivity ratio (RRR) of ~850 measured along the c axis (Fig. 1). The crystal was mounted with GE varnish to a copper stage attached to the cold finger of a helium-3 cryostat, with the optical probe beam incident on the a-b plane of the crystal. The sample was then cooled down to the cryostat’s base temperature, and Kerr angle was measured in ambient magnetic field (Hext < 0.3 Oe) as the sample was slowly warmed through Tc– and Tc+. In addition to measuring the Kerr angle, we installed a split-coil mutual inductance assembly (28) operating at 100 kHz with a drive amplitude of ~8.5 mG. Mutual inductance measurements of the sample could then be conducted under the same experimental conditions as the PKE measurements in order to verify the upper transition temperature of ~550 mK (Fig. 2).

Fig. 1 Material information and experimental setup.

(A) Crystal structure of UPt3. In the image, the c axis is pointing out of the page. (B) Photograph of the single crystal used in this study. Length scale on ruler is in 1/64ths of an inch. (C) Experiment geometry of the Kerr rotation and mutual inductance measurements.

Fig. 2 Measurement of Kerr effect and Tc+ in UPt3.

Kerr angle (red, left axis) and the real part of the mutual inductance (blue, right axis) are plotted as a function of temperature for a UPt3 single crystal. The onset of superconductivity at Tc+ is well removed from the onset of TRSB at TKerr ~ 460 mK ~ Tc–. The gray bands indicate uncertanties in Tc+ and Tc– from the effects of optical heating. Error bars on the Kerr data are statistical. The solid line is a guide to the eye of the form Embedded Image.

The main experimental result is summarized in Fig. 2. On the left-hand axis, we plot the Kerr angle versus temperature obtained in a typical zero-field warmup after a zero-field cooldown. A temperature-independent background, consisting of electrical and optical offsets in the instrument, has been averaged between 600 and 700 mK and subtracted from the plotted data as in (23); the equivalent magnitude of this offset varied between ~0.2 and ~3 μrad over all of the data taken on UPt3, although for most experimental runs, the equivalent offset was of order 0.3 μrad (27). Above TKerr ≈ 460 mK, the measured Kerr signal remains unchanged from the background. Below this temperature, however, a finite change in Kerr angle appears, which reaches ~400 nrad at 350 mK and falls to zero at TKerr. Finite element analysis using the thermal conductivity and heat capacity data of (29, 30) allows us to estimate the degree of sample heating to be no more than 50 mK at the spot of illumination. The resulting uncertainty in the transition temperature in the limit of zero optical heating is indicated by the thick gray bar in Fig. 2.

Theoretical treatments of the optical response of chiral superconductors suggest that the Kerr effect varies as the product of the real and imaginary components of the order parameter (31). The Kerr effect may arise either via impurity scattering (32, 33) or interband coupling (34), although for our measurements, the latter mechanism is more likely given the high purity of our sample and the involvement of five bands in the Fermi surface (1). Because the real and imaginary components of the proposed order parameter for UPt3 have different transition temperatures, the phenomenological expression for θK involves both Tc+ and Tc–Embedded Image (3)This functional form is plotted as a solid line in Fig. 2 and is consistent with measurement, although the scatter in the data is too large to exclude the possibility of a more complex relationship between θK, temperature, and the gap structure of UPt3.

To compare the observed optical Kerr transition with the onset of A-phase superconductivity in UPt3, a baseline magnetic susceptibility curve, taken with no incident light from the Kerr probe, is shown in Fig. 2 on the right-hand axis of the plot. We identify the upper superconducting transition temperature Tc+ ~ 550 mK with the departure of the susceptibility from the normal state linear background, as in (35). The absorption of light from the Kerr measurement results in bulk sample heating that depresses the measured mutual inductance Tc from the 0-μW curve by no more than ~8.5 mK for the optical powers used in this study (fig. S3). This offset is indicated by the thin gray band in Fig. 2. The transitions at Tc+ and TKerr are clearly separated in temperature; in the absence of any other structural or electronic phase transitions in this temperature range, we ascribe the additional broken symmetry at TKerr to the onset of B phase superconductivity at Tc–.

A defining characteristic of spontaneous symmetry-breaking is the sensitivity of the order parameter to alignment by a small symmetry-breaking field as the transition temperature is crossed. In the case of UPt3, the TRSB state is expected to couple to magnetic fields applied along the c axis (12); therefore, it should be possible to train the sign of the TRSB Kerr signal with an arbitrarily small external field oriented in this direction. We show the results of field-training measurements of the Kerr effect in Fig. 3. For these experiments, the sample was cooled to base temperature in an applied field of +50 or –50 G. At base temperature, the field was switched off, and Kerr angle was again measured upon warmup. The residual magnetic field with the magnet off was identical to that of the zero-field cooled measurements to within the resolution of a commercial Hall sensor installed in the sample space.

Fig. 3 Magnetic field training of the Kerr effect.

Solid lines are guides to the eye of the form Embedded Image. (A) Zero-field warmup data after cooling the sample through Tc– in a +50 G field (red) and in zero field (gray, replotted from Fig. 2). (B) Zero-field warmup data after cooling the sample down in a –50 G field, showing complete reversal of the TRSB signal.

There are several features in the data that are of note. First, the signal tracks the direction of the training field, indicating that the TRSB order parameter couples to magnetic fields as expected. Second, the maximum size of the field-trained signal matches that of the zero-field measurements to within our experimental resolution. Although in general one would expect TRSB domains to form with random alignments, leading to partial or even complete cancellation of Kerr signal in the area sampled by the probe beam, our repeated observation of maximal Kerr signal even in the absence of field training is consistent with observations elsewhere that UPt3 tends to spontaneously form very large superconducting domains, possibly spanning the entire crystal (13, 14, 17). Last, the absence of a discernable additional Kerr signal in the field-cooled measurements implies that the TRSB signal originates from the superconducting order parameter itself, rather than from vortices induced by an external field.

To properly analyze broken TRS in the superconducting state, one must also consider the potential influence of magnetic effects already present at higher temperatures. In UPt3, antiferromagnetic (AF) correlations associated with magnetic moments on the uranium sites appear at 20 K, with static in-plane AF order developing below TN ~ 5 K (7, 36). TRS is therefore broken well above Tc+ and Tc–. However, one can still examine the superconducting condensate for additional TRSB that is ferromagnetic in character, that is, displaying a two-state degeneracy because of the presence of an imaginary component in the gap function, as discussed above. This additional out-of-plane signal can be considered independently from the in-plane TRSB arising from antiferromagnetism, although the AF order may still couple to superconductivity by acting as, for example, a symmetry-breaking field (7, 12). We can verify the independence of these two sources of TRSB by measuring Kerr rotation through TN. As expected for in-plane ordering with no net moment, we find no signature (within ±100 nrad) of the AF transition in the Kerr effect under both zero-field and field-cooled conditions (fig. S4), despite the enhancement provided by the large spin-orbit coupling present in the material (12). The null result at TN suggests that the signal observed in the B phase is a true property of superconductivity in UPt3, rather than a secondary effect of the background AF ordering in this system.

The appearance of broken TRS in the B phase carries with it several implications for the theory of superconductivity in UPt3. Because a general multicomponent superconducting order parameter can break TRS only if it belongs to a multidimensional representation (37), our results taken alone imply that the superconducting order parameter belongs to one of the four two-dimensional representations of D6h. In principle, the data are consistent with any choice of basis functions with relatively complex coefficients within these representations. However, the preponderance of other experimental evidence (1, 911, 1315, 30) narrows the possibilities further, leaving the E2u order parameter of Eq. 1 as the likely candidate to describe the superconducting phases of UPt3. Although a microscopic theory for PKE in UPt3 is not available at present, the temperature dependence of our data appears to be consistent with the phenomenology developed in (34) to account for a finite Kerr rotation in a multiband super-conductor with interorbital coupling ε12 and intraband pairing: θK ∝ ε12η1η2. This in turn suggests that Kerr effect measurements can more generally be used to track the temperature dependence of the gap function in TRSB superconductors.

Supplementary Materials

www.sciencemag.org/content/345/6193/190/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S4

References and Notes

  1. Other mechanisms, such as chiral crystal structure or gyrotropic order, may also give rise to a finite Kerr effect; however, such sources of Kerr rotation do not exhibit the same magnetic field training behavior observed in TRS-breaking materials (20).
  2. Materials and methods are available as supplementary materials on Science Online.
  3. Acknowledgments: Stimulating discussions with S. Kivelson, A. Huxley, and D. Agterberg; sample characterization by K. Avers (Northwestern); and instrument design assistance from G. Burkhard (Stanford) are greatly appreciated. This work was supported by the U.S. Department of Energy Office of Basic Energy Science, Division of Materials Science and Engineering, at Stanford under contract DE-AC02-76SF00515 and at Northwestern under contract DE- FG02-05ER46248. Construction of the Sagnac apparatus was partially funded by the Stanford Center for Probing the Nanoscale (NSF NSEC 0425897). E.R.S. received additional support from a Gabilan Stanford Graduate Fellowship and the DARE Doctoral Fellowship Program.
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