Observation of broken time-reversal symmetry in the heavy-fermion superconductor UPt 3 E. R. Schemm et al. Science 345, 190 (2014); DOI: 10.1126/science.1248552
This copy is for your personal, non-commercial use only.
Permission to republish or repurpose articles or portions of articles can be obtained by following the guidelines here. The following resources related to this article are available online at www.sciencemag.org (this information is current as of July 11, 2014 ): Updated information and services, including high-resolution figures, can be found in the online version of this article at: http://www.sciencemag.org/content/345/6193/190.full.html Supporting Online Material can be found at: http://www.sciencemag.org/content/suppl/2014/07/09/345.6193.190.DC1.html A list of selected additional articles on the Science Web sites related to this article can be found at: http://www.sciencemag.org/content/345/6193/190.full.html#related This article cites 33 articles, 2 of which can be accessed free: http://www.sciencemag.org/content/345/6193/190.full.html#ref-list-1 This article has been cited by 1 articles hosted by HighWire Press; see: http://www.sciencemag.org/content/345/6193/190.full.html#related-urls This article appears in the following subject collections: Physics http://www.sciencemag.org/cgi/collection/physics
Science (print ISSN 0036-8075; online ISSN 1095-9203) is published weekly, except the last week in December, by the American Association for the Advancement of Science, 1200 New York Avenue NW, Washington, DC 20005. Copyright 2014 by the American Association for the Advancement of Science; all rights reserved. The title Science is a registered trademark of AAAS.
Downloaded from www.sciencemag.org on July 11, 2014
If you wish to distribute this article to others, you can order high-quality copies for your colleagues, clients, or customers by clicking here.
R ES E A RC H | R E PO R TS
10. 11. 12. 13.
14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.
27. 28. 29. 30. 31. 32. 33.
P. A. Lee, N. Nagaosa, X.-G. Wen, Rev. Mod. Phys. 78, 17–85 (2006). J. Chaloupka, G. Khaliullin, Phys. Rev. Lett. 100, 016404 (2008). B. J. Kim et al., Phys. Rev. Lett. 101, 076402 (2008). B. J. Kim et al., Science 323, 1329–1332 (2009). G. Jackeli, G. Khaliullin, Phys. Rev. Lett. 102, 017205 (2009). J. Kim et al., Phys. Rev. Lett. 108, 177003 (2012). S. Fujiyama et al., Phys. Rev. Lett. 108, 247212 (2012). F. Wang, T. Senthil, Phys. Rev. Lett. 106, 136402 (2011). H. Watanabe, T. Shirakawa, S. Yunoki, Phys. Rev. Lett. 110, 027002 (2013). M. Ge et al., Phys. Rev. B 84, 100402 (2011). O. B. Korneta et al., Phys. Rev. B 82, 115117 (2010). M. A. Hossain et al., Nat. Phys. 4, 527–531 (2008). Note that the positive sign of next-nearest hopping t′ renders a FS topology with a large electron pocket around the G point, as opposed to a hole pocket centered at (p, p) in cuprates with a negative t′ (8). A. Damascelli, Z. Hussain, Z.-X. Shen, Rev. Mod. Phys. 75, 473–541 (2003). T. M. Rice, AIP Conf. Proc. 816, 92–99 (2006). K. M. Shen et al., Science 307, 901–904 (2005). R. C. Dynes, V. Narayanamurti, J. P. Garno, Phys. Rev. Lett. 41, 1509–1512 (1978). K. Okazaki et al., Science 337, 1314–1317 (2012). T. Yoshida, M. Hashimoto, I. M. Vishik, Z.-X. Shen, A. Fujimori, J. Phys. Soc. Jpn. 81, 011006 (2012). A. Kanigel et al., Nat. Phys. 2, 447–451 (2006). T. J. Reber et al., Nat. Phys. 8, 606–610 (2012). C. Martins, M. Aichhorn, L. Vaugier, S. Biermann, Phys. Rev. Lett. 107, 266404 (2011). R. Arita, J. Kuneš, A. V. Kozhevnikov, A. G. Eguiluz, M. Imada, Phys. Rev. Lett. 108, 086403 (2012). D. Hsieh, F. Mahmood, D. H. Torchinsky, G. Cao, N. Gedik, Phys. Rev. B 86, 035128 (2012). Q. Li et al., Sci. Rep. 3, 3073 (2013). A. Yamasaki et al., Bulk nature of layered perovskite iridates beyond the Mott scenario: An approach from bulk sensitive photoemission study. http://arxiv.org/abs/1310.7160. D. Sénéchal, A.-M. S. Tremblay, Phys. Rev. Lett. 92, 126401 (2004). G. Ghiringhelli et al., Science 337, 821–825 (2012). J. M. Tranquada, B. J. Sternlieb, J. D. Axe, Y. Nakamura, S. Uchida, Nature 375, 561–563 (1995). P. Abbamonte et al., Nat. Phys. 1, 155–158 (2005). T. Hanaguri et al., Nature 430, 1001–1005 (2004). T. Yoshida et al., Phys. Rev. Lett. 103, 037004 (2009). D. L. Feng et al., Phys. Rev. Lett. 88, 107001 (2002).
ACKN OW LEDG MEN TS
We thank J. H. Shim, K. Haule, G. Kotliar, C. Kim, M. Norman, and G. Khaliullin for helpful discussions. Work at the University of Michigan was supported by the U.S. NSF under grant no. DMR-07-04480. B.J.K. acknowledges the Institute for Complex Adaptive Matter for a travel grant that enabled a visit and helpful discussions at Rutgers University. Work in the Materials Science Division of Argonne National Laboratory (sample preparation and characterization) was supported by the U.S. Department of Energy (DOE) Office of Science, Basic Energy Sciences, Materials Science and Engineering Division. The Advanced Light Source is supported by the Director, Office of Science, Office of Basic Energy Sciences of the U.S. DOE under contract no. DE-AC02-05CH11231. Y.K.K. is supported through the National Research Foundation (grant no. 20100018092).
SUPPLEMENTARY MATERIALS
www.sciencemag.org/content/345/6193/187/suppl/DC1 Materials and Methods Supplementary Text Figs. S1 and S2 References (34–36) 22 January 2014; accepted 28 May 2014 10.1126/science.1251151
190
11 JULY 2014 • VOL 345 ISSUE 6193
E. R. Schemm,1,2,3* W. J. Gannon,4† C. M. Wishne,4 W. P. Halperin,4 A. Kapitulnik1,2,3,5 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.
T
he heavy-fermion metal UPt3 (1) is one of only a handful of unconventional superconductors (2, 3) exhibiting multiple superconducting phases (4–6). 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 (4–6). 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 (9–11). 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. 1
Department of Physics, Stanford University, Stanford, CA 94305, USA. 2Geballe Laboratory for Advanced Materials, Stanford University, Stanford, CA 94305, USA. 3Stanford Institute for Materials and Energy Sciences (SIMES), SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025, USA. 4Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208, USA. 5Department of Applied Physics, Stanford University, Stanford, CA 94305, USA. *Corresponding author. E-mail: [email protected] †Present address: Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794, USA.
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 by DðkFÞ ¼ z ½h1 ðT Þðk2x − k2yÞkz þ − 2i ⋅ h2 ðT Þkx ky kz ð1Þ
in a coordinate system where z ‖ c. Here, h1 ðT Þº qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1−ðT =Tcþ Þ2 is the real component of the superconducting order parameter, marking the onset of the A phase at Tc+, whereas h2 ðT Þº qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1−ðT =Tc− Þ2 introduces 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 T z. 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 ˇ
1. 2. 3. 4. 5. 6. 7. 8. 9.
Observation of broken time-reversal symmetry in the heavy-fermion superconductor UPt3
ˇ
RE FE RENCES AND N OT ES
SUPERCONDUCTIVITY
ˇ
on the other hand, Fermi arcs are a precursor signature of d-wave superconductivity, Sr2IrO4 may superconduct at lower temperatures. Further inquiries into these questions will shed new light on the long-sought connection between the pseudogap and HTSC.
sciencemag.org SCIENCE
RE S EAR CH | R E P O R T S
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) by nR − nL 4p s″xy ðwÞ ð2Þ ≈ qK ¼ −= ⋅ nR nL − 1 wd n˜ ðn˜ 2 − 1Þ where ñ is the average complex index of refraction of the material, d is a microscopic length scale, and w is the frequency of the incident light. The off-diagonal term of the conductivity tensor, sxy ¼ s′xy þ is″xy , is nonzero if TRS is broken (19), and the second equality holds for weak absorption (21). In the simplest estimate, sxy 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: D0
This copy is for your personal, non-commercial use only.
Permission to republish or repurpose articles or portions of articles can be obtained by following the guidelines here. The following resources related to this article are available online at www.sciencemag.org (this information is current as of July 11, 2014 ): Updated information and services, including high-resolution figures, can be found in the online version of this article at: http://www.sciencemag.org/content/345/6193/190.full.html Supporting Online Material can be found at: http://www.sciencemag.org/content/suppl/2014/07/09/345.6193.190.DC1.html A list of selected additional articles on the Science Web sites related to this article can be found at: http://www.sciencemag.org/content/345/6193/190.full.html#related This article cites 33 articles, 2 of which can be accessed free: http://www.sciencemag.org/content/345/6193/190.full.html#ref-list-1 This article has been cited by 1 articles hosted by HighWire Press; see: http://www.sciencemag.org/content/345/6193/190.full.html#related-urls This article appears in the following subject collections: Physics http://www.sciencemag.org/cgi/collection/physics
Science (print ISSN 0036-8075; online ISSN 1095-9203) is published weekly, except the last week in December, by the American Association for the Advancement of Science, 1200 New York Avenue NW, Washington, DC 20005. Copyright 2014 by the American Association for the Advancement of Science; all rights reserved. The title Science is a registered trademark of AAAS.
Downloaded from www.sciencemag.org on July 11, 2014
If you wish to distribute this article to others, you can order high-quality copies for your colleagues, clients, or customers by clicking here.
R ES E A RC H | R E PO R TS
10. 11. 12. 13.
14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.
27. 28. 29. 30. 31. 32. 33.
P. A. Lee, N. Nagaosa, X.-G. Wen, Rev. Mod. Phys. 78, 17–85 (2006). J. Chaloupka, G. Khaliullin, Phys. Rev. Lett. 100, 016404 (2008). B. J. Kim et al., Phys. Rev. Lett. 101, 076402 (2008). B. J. Kim et al., Science 323, 1329–1332 (2009). G. Jackeli, G. Khaliullin, Phys. Rev. Lett. 102, 017205 (2009). J. Kim et al., Phys. Rev. Lett. 108, 177003 (2012). S. Fujiyama et al., Phys. Rev. Lett. 108, 247212 (2012). F. Wang, T. Senthil, Phys. Rev. Lett. 106, 136402 (2011). H. Watanabe, T. Shirakawa, S. Yunoki, Phys. Rev. Lett. 110, 027002 (2013). M. Ge et al., Phys. Rev. B 84, 100402 (2011). O. B. Korneta et al., Phys. Rev. B 82, 115117 (2010). M. A. Hossain et al., Nat. Phys. 4, 527–531 (2008). Note that the positive sign of next-nearest hopping t′ renders a FS topology with a large electron pocket around the G point, as opposed to a hole pocket centered at (p, p) in cuprates with a negative t′ (8). A. Damascelli, Z. Hussain, Z.-X. Shen, Rev. Mod. Phys. 75, 473–541 (2003). T. M. Rice, AIP Conf. Proc. 816, 92–99 (2006). K. M. Shen et al., Science 307, 901–904 (2005). R. C. Dynes, V. Narayanamurti, J. P. Garno, Phys. Rev. Lett. 41, 1509–1512 (1978). K. Okazaki et al., Science 337, 1314–1317 (2012). T. Yoshida, M. Hashimoto, I. M. Vishik, Z.-X. Shen, A. Fujimori, J. Phys. Soc. Jpn. 81, 011006 (2012). A. Kanigel et al., Nat. Phys. 2, 447–451 (2006). T. J. Reber et al., Nat. Phys. 8, 606–610 (2012). C. Martins, M. Aichhorn, L. Vaugier, S. Biermann, Phys. Rev. Lett. 107, 266404 (2011). R. Arita, J. Kuneš, A. V. Kozhevnikov, A. G. Eguiluz, M. Imada, Phys. Rev. Lett. 108, 086403 (2012). D. Hsieh, F. Mahmood, D. H. Torchinsky, G. Cao, N. Gedik, Phys. Rev. B 86, 035128 (2012). Q. Li et al., Sci. Rep. 3, 3073 (2013). A. Yamasaki et al., Bulk nature of layered perovskite iridates beyond the Mott scenario: An approach from bulk sensitive photoemission study. http://arxiv.org/abs/1310.7160. D. Sénéchal, A.-M. S. Tremblay, Phys. Rev. Lett. 92, 126401 (2004). G. Ghiringhelli et al., Science 337, 821–825 (2012). J. M. Tranquada, B. J. Sternlieb, J. D. Axe, Y. Nakamura, S. Uchida, Nature 375, 561–563 (1995). P. Abbamonte et al., Nat. Phys. 1, 155–158 (2005). T. Hanaguri et al., Nature 430, 1001–1005 (2004). T. Yoshida et al., Phys. Rev. Lett. 103, 037004 (2009). D. L. Feng et al., Phys. Rev. Lett. 88, 107001 (2002).
ACKN OW LEDG MEN TS
We thank J. H. Shim, K. Haule, G. Kotliar, C. Kim, M. Norman, and G. Khaliullin for helpful discussions. Work at the University of Michigan was supported by the U.S. NSF under grant no. DMR-07-04480. B.J.K. acknowledges the Institute for Complex Adaptive Matter for a travel grant that enabled a visit and helpful discussions at Rutgers University. Work in the Materials Science Division of Argonne National Laboratory (sample preparation and characterization) was supported by the U.S. Department of Energy (DOE) Office of Science, Basic Energy Sciences, Materials Science and Engineering Division. The Advanced Light Source is supported by the Director, Office of Science, Office of Basic Energy Sciences of the U.S. DOE under contract no. DE-AC02-05CH11231. Y.K.K. is supported through the National Research Foundation (grant no. 20100018092).
SUPPLEMENTARY MATERIALS
www.sciencemag.org/content/345/6193/187/suppl/DC1 Materials and Methods Supplementary Text Figs. S1 and S2 References (34–36) 22 January 2014; accepted 28 May 2014 10.1126/science.1251151
190
11 JULY 2014 • VOL 345 ISSUE 6193
E. R. Schemm,1,2,3* W. J. Gannon,4† C. M. Wishne,4 W. P. Halperin,4 A. Kapitulnik1,2,3,5 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.
T
he heavy-fermion metal UPt3 (1) is one of only a handful of unconventional superconductors (2, 3) exhibiting multiple superconducting phases (4–6). 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 (4–6). 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 (9–11). 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. 1
Department of Physics, Stanford University, Stanford, CA 94305, USA. 2Geballe Laboratory for Advanced Materials, Stanford University, Stanford, CA 94305, USA. 3Stanford Institute for Materials and Energy Sciences (SIMES), SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025, USA. 4Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208, USA. 5Department of Applied Physics, Stanford University, Stanford, CA 94305, USA. *Corresponding author. E-mail: [email protected] †Present address: Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794, USA.
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 by DðkFÞ ¼ z ½h1 ðT Þðk2x − k2yÞkz þ − 2i ⋅ h2 ðT Þkx ky kz ð1Þ
in a coordinate system where z ‖ c. Here, h1 ðT Þº qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1−ðT =Tcþ Þ2 is the real component of the superconducting order parameter, marking the onset of the A phase at Tc+, whereas h2 ðT Þº qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1−ðT =Tc− Þ2 introduces 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 T z. 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 ˇ
1. 2. 3. 4. 5. 6. 7. 8. 9.
Observation of broken time-reversal symmetry in the heavy-fermion superconductor UPt3
ˇ
RE FE RENCES AND N OT ES
SUPERCONDUCTIVITY
ˇ
on the other hand, Fermi arcs are a precursor signature of d-wave superconductivity, Sr2IrO4 may superconduct at lower temperatures. Further inquiries into these questions will shed new light on the long-sought connection between the pseudogap and HTSC.
sciencemag.org SCIENCE
RE S EAR CH | R E P O R T S
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) by nR − nL 4p s″xy ðwÞ ð2Þ ≈ qK ¼ −= ⋅ nR nL − 1 wd n˜ ðn˜ 2 − 1Þ where ñ is the average complex index of refraction of the material, d is a microscopic length scale, and w is the frequency of the incident light. The off-diagonal term of the conductivity tensor, sxy ¼ s′xy þ is″xy , is nonzero if TRS is broken (19), and the second equality holds for weak absorption (21). In the simplest estimate, sxy 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: D0
