Hourglass dispersion and resonance of magnetic excitations in the superconducting state of the single-layer cuprate HgBa{}_{2}CuO{}_{4+\delta} near optimal doping

Hourglass dispersion and resonance of magnetic excitations in the superconducting state of the single-layer cuprate HgBaCuO near optimal doping


We use neutron scattering to study magnetic excitations near the antiferromagnetic wave vector in the underdoped single-layer cuprate HgBaCuO (superconducting transition temperature  K, pseudogap temperature  K). The response is distinctly enhanced below and exhibits a Y-shaped dispersion in the pseudogap state, whereas the superconducting state features an X-shaped (hourglass) dispersion and a further resonance-like enhancement. A large spin gap of about 40 meV is observed in both states. This phenomenology is reminiscent of that exhibited by bilayer cuprates. The resonance spectral weight, irrespective of doping and compound, scales linearly with the putative binding energy of a spin-exciton described by an itinerant-spin formalism.


The dynamic magnetic susceptibility of the hole-doped cuprates exhibits an hourglass-shaped (or X-shaped, upon considering an energy-momentum slice through ) spectrum centered at the two-dimensional antiferromagnetic (AF) wave-vector  Reznik et al. (2004); Tranquada et al. (2004); Pailhès et al. (2004). Although the upper dispersive branch likely results from short-range AF correlations of local moments, the cause of the downward dispersive branch, at energies below the neck of the hourglass, has remained unclear. Results for the two cuprate families most widely studied via neutron scattering, (La,Nd)(Sr,Ba)CuO (La214) and YBaCuO (Y123), support contradictory scenarios. For moderately- to overdoped Y123 (hole concentration ), the low-energy dispersion is accompanied by a magnetic resonance: an increase in scattering at and energy  Rossat-Mignod et al. (1991); Mook et al. (1993). Both features appear in the superconducting (SC) state and can be understood, within an itinerant picture, as a dispersive spin exciton bound below the particle-hole continuum and associated with the -wave SC gap Norman (2001); Eschrig (2006). In contrast, La214 features an hourglass dispersion in both the SC and normal states, and no resonance in the SC state Tranquada et al. (2004); Lipscombe et al. (2009); Fujita et al. (2012). The discovery of static order of spatially segregated localized charge-spin stripes in La214 Tranquada et al. (1995) has motivated an interpretation in terms of fluctuating stripes Chubukov et al. (2007). Reconciliation of these discrepancies has been further complicated by the disparate crystal structures of Y123, a double-layer cuprate (two CuO layers per primitive cell), and La214, a single-layer compound.

The subsequent observation of a magnetic resonance in single-layer TlBaCuO (Tl2201) He et al. (2002) and HgBaCuO (Hg1201) Yu et al. (2010), which feature optimal values of nearly 100 K, more than twice that of La214, raised the prospect of a universal description of the magnetic response. However, detailed results have been difficult to obtain for these single-layer cuprates, and an hourglass dispersion has not been detected. Thus, a connection, or lack thereof, between the hourglass dispersion, the resonance, and superconductivity has not been universally established, rendering a satisfactory description of magnetic excitations of single- and double-layer cuprates elusive.

Figure 1: (a)-(e) Constant-energy images of magnetic susceptibility at  K (left), 100 K (middle), and 250 K (right). Data within a  meV window centered at the indicated energies are averaged, except for  meV, where a 10 meV window was used. White dots (left-most panels): momentum resolution at each energy. (f)-(j) Corresponding constant-energy cuts averaged over and trajectories across . Solid lines: gaussian fits to data convolved with the momentum resolution. (k) Energy dependence of incommensurability at  K. Horizontal error bars: fit uncertainties for . Filled black circles and open squares: data taken with incident energy  meV and 130 meV, respectively. Filled grey region: FWHM of the response. Hatched area: magnetic excitation gap. (l) Energy dependence of incommensurability at  K, with dispersion at  K (dotted line) shown for comparison. Horizontal black bar: experimental momentum resolution at  meV.

A recent study of underdoped Hg1201 (labeled HgUD71,  K) revealed a gapped Y-shaped spectrum both in the pseudogap (PG) and SC states, and no evidence for a resonance. The unusual response in the SC state was attributed to strong competing PG order Chan et al. (2016). Since then, charge-density-wave (CDW) order in Hg1201 was found to be particularly pronounced at this doping level hin (); tab ().

Here we study a Hg1201 sample closer to optimal doping (HgUD88;  K), motivated by early work for optimally-doped Hg1201 that yielded initial evidence for a resonance Yu et al. (2010). First, we confirm the observation for HgUD71 Chan et al. (2016) that the response is enhanced below , and has a gapped, Y-shaped spectrum in the PG state. Whereas the large gap (about 40 meV) is unchanged in the SC state, the response of HgUD88 changes to a distinct hourglass topology and features a resonance-like enhancement at 59 meV. This is reminiscent of the phenomenology established for the bilayer cuprates Bourges et al. (2000); Fauqué et al. (2007). The characteristic resonance energy and spectral weight scale with the particle-hole Stoner continuum threshold energy in a manner consistent with results for other cuprates, and with expectations for a spin-exciton resulting from an itinerant spin formalism.

The sample, prepared following previously described procedures Zhao et al. (2006); Barišić et al. (2008); Chan et al. (2016), consists of approximately coaligned single crystals with a total mass of  g with full-width-at-half-maximum (FWHM) mosaic of 1.5. Similar to ref. Chan et al. (2016), the value  K signifies the transition midpoint obtained by averaging uniform magnetic susceptibility data for the diamagnetic signal of the individual crystals. Measurements were performed on the ARCS time-of-flight (TOF) spectrometer at Oak Ridge National Laboratory Abernathy et al. (2012), with the sample’s crystalline -axis aligned along the incident beam, and incident neutron energies  meV (at  K,  K and 250 K) and  meV ( K). The dynamic magnetic susceptibility, , was determined from the scattering intensity, calibrated to a Vanadium standard, by normalizing by the anisotropic Cu form factor Shamoto et al. (1993) and the Bose population factor. The temperature dependence was measured at meV) at the Laboratoire Léon Brillouin, with the 2T triple-axes-spectrometer, with fixed final energy  meV. We quote the scattering wave-vector  () in reciprocal lattice units (r.l.u.), where Å and Å are the room-temperature values. Constant- data are fit to a gaussian, , convolved with the experimental momentum resolution, where q is the reduced two-dimensional wave vector, , the intrinsic FWHM momentum width, and the incommensurability away from ; see ref. Chan et al. (2016) for further data analysis details.

Figure 2: (a) Energy dependence of magnetic susceptibility at , , determined from fits to data such as those in Fig. 1 (see text). Filled circles:  meV; the data around  meV are contaminated by phonon scattering Chan et al. (2016) and indicated by open circles. Open squares:  meV. Solid lines: guides to the eye. Horizontal bars represent the energy binning (only shown for  K). A large difference in is observed across . In contrast, is nearly the same for HgUD71 Chan et al. (2016) at  K and  K (grey and orange lines in a) and b)). (b) Energy dependence of local susceptibility, . For both HgUD88 and HgUD71 Chan et al. (2016), the magnetic response exhibits large gaps in the PG and SC states.

Figure 1a-j shows for select at , 100 and 250 K. At  K, the gapped spectrum evolves with increasing energy from an incommensurate ring that disperses toward and then outward again, thus exhibiting an archetypical hourglass dispersion (Fig. 1k). At  K ( K), however, the low-energy response is commensurate with (Figs. 1d,i), resulting in the Y-shaped dispersion (Fig. 1l) that is characteristic of the PG state Chan et al. (2016); Bourges et al. (2000); Hinkov et al. (2007). Finally, at K, just above  K Barišić et al. (2013), the response is considerably weaker than deep in the PG state.

The response at  meV (Fig. 1c,h), where the upward dispersion begins (Fig. 1g,k,l), is significantly larger at 5 K than at 100 K. This is reflected in a sharp peak at 5 K in the energy dependence of (Fig. 2a). Detailed measurement of the temperature dependence of at 60 meV (Fig. 3b) shows a distinct increase of scattering below , consistent with the result for HgUD71 Chan et al. (2016). However, contrary to HgUD71 (Fig. 2a, 3b), this is followed by a further increase below . We identify this feature below in HgUD88 as the magnetic resonance Rossat-Mignod et al. (1991); Fong et al. (1999); Yu et al. (2010).

The resonance shows a distinct enhancement in magnetic scattering below in optimally- and over-doped cuprates Pailhès et al. (2004); Dai et al. (2001); Yu et al. (2010). However, it is harder to discern in underdoped samples, which exhibit significant magnetic scattering in the normal state Dai et al. (2001); Hinkov et al. (2007); Chan et al. (2016), because the instrumental energy resolution is large compared to the resonance width. For HgUD88, where and hence the energy resolution of the triple-axis spectrometer are particularly large, the temperature dependence is considerably smoothed (Fig. 3b).

The resonance is better revealed as a peak in (Fig. 3a), centered at  meV, with a width that is not much larger than the experimental resolution of the TOF spectrometer (about 5 meV FWHM). The ratio is the largest value reported for the cuprates Bourges et al. (2005); Yu et al. (2009). Using meV Vishik et al. (2014); Li et al. (2013) for the SC gap amplitude, the ratio is consistent with the value 0.64(4) established for unconventional superconductors Yu et al. (2009).

The present result for HgUD88 bears a striking resemblance to observations for bilayer Y123 Bourges et al. (2000); Hinkov et al. (2007). The hourglass dispersion, particularly the dispersive low-energy branch, is present only below , and thus a characteristic of the SC state. Above , both the resonance and its downward dispersive branch disappear, yielding a Y-shaped spectrum not (). However, for HgUD88 the neck of the hourglass at 5 K is somewhat extended compared to other cuprates (Fig. 1f). Furthermore, the upper dispersion branch extends to slightly lower energy at 100 K than at 5 K. These subtle features, established in the TOF experiment, in combination with the coarse triple-axis energy resolution used to measure the temperature dependence, might further obscure a distinct enhancement of at (Fig. 3b).

The energy-integrated spectral weight of the resonance peak at is defined as . For HgUD88, we find /Cu upon integrating from 51 to 64 meV. can be related to . Within the itinerant picture, the interacting spin susceptibility is computed using the random phase approximation. In the SC state, the resonance at is part of a spin exciton, i.e., a spin-triplet collective mode bound below the threshold of the Stoner continuum,  Millis and Moniens (1996); Eschrig (2006). The weight of the resonance is linearly related to the reduced binding energy, , by , where is the planar interaction that enhances the bare susceptibility and is the Landé factor. The quantities and are related to the hot-spots (hs), defined as Fermi-surface points connected by q: , where and are the Fermi velocities at the hot-spots and is the angle between their directions; at the hot-spots is estimated as  Pailhès et al. (2006), where meV Vishik et al. (2014); Li et al. (2013). As shown in Fig. 3c, upon combining our result for HgUD88 with those for Y123 Pailhès et al. (2006), BiSrCaCuO (Bi2212) Fong et al. (1999) and Tl2201 He et al. (2002), we find remarkably good linear scaling with zero intercept between and the reduced binding energy. The common scaling factor implies universal band-structure and interaction parameters, within the experimental error, for different cuprate families and hole concentrations.

Figure 3: (a) Change of across . Horizontal blue bar: FWHM energy resolution. The large peak at  meV is the magnetic resonance. (b) Temperature dependence of at (black) measured with a triple-axes spectrometer with FWHM energy resolution meV. and (interpolated from planar transport measurements  Barišić et al. (2013)) are indicated by the black dashed vertical lines. In contrast to HgUD88, the magnetic response of HgUD71 (blue) saturates in the SC state (from ref. Chan et al. (2016)); the temperature axis (top) is scaled to match for HgUD71 with for HgUD88. (c) as a function of for numerous cuprates. The linear scaling and zero intercept (dashed line) are consistent with a spin-exciton description of the resonance. Y123 Fong et al. (2000); Pailhès et al. (2006) and Bi2212 Fong et al. (1999) are bilayer cuprates and thus exhibit odd- and even-parity resonances, whereas single-layer Tl2201 He et al. (2002) and Hg1201 (HgUD88, present work) feature only one resonance mode. Labels indicate the hole concentrations corresponding to underdoped (UD), optimally doped (OP), and overdoped (OD) regimes, followed by numbers designating and, if relevant, even (E) and odd (O) resonance modes.

Alternatively, the resonance has been attributed to a redistribution of spectral weight of local spin fluctuations from energies below to energies above a spin gap that appears in the SC state Stock et al. (2004); Chubukov et al. (2007). The gap in HgUD88 is apparent from the lack of low-energy magnetic scattering (Figs. 1e,j). To better determine the gap size, we examine the local susceptibility, (integration over the AF Brillouin zone). As seen from Fig. 2b, HgUD88 features a particularly large gap of about 40 meV in both the PG and SC states. With increasing temperature, the strength of magnetic excitations decreases, yet the gap does not close. Consistent with the result for HgUD71 Chan et al. (2016), the gap thus is a property of the PG and not the SC state Dai et al. (2001); Stock et al. (2004). We thus cannot attribute the resonance to a spectral weight redistribution due to the opening of a gap. Although prior neutron scattering work yielded evidence for a “spin-pseudogap” Rossat-Mignod et al. (1991); Lee et al. (2003); Bourges et al. (2005); Chan et al. (2016), the present result constitutes the clearest and largest manifestation of such a gap.

The spin-exciton scenario can semi-quantitatively account for (i) the magnitude of the resonance and (ii) its connection to a downward dispersing mode in the SC state of HgUD88. However, it fails to explain the absence of both features in HgUD71 Chan et al. (2016). It is interesting to compare the two-particle spectra in the charge and spin sectors, probed by electronic Raman scattering (ERS) and neutron scattering, respectively. In ERS, the hallmark of the SC state is the pair-breaking peak in the channel, which probes the antinodal regions of the Fermi surface that are approximately spanned by q. The magnitude of this peak decreases with decreasing doping. Thus, while the pair-breaking peak is sizable in HgUD88, it is much weaker in HgUD71, and disappears at lower doping LeTacon et al. (2006); Li et al. (2012, 2013). This phenomenon could be ascribed to the vanishing of coherent Bogoliubov quasiparticles, because at lower doping an increasing portion of the Fermi surface is dominated by the PG. Furthermore, CDW order is particularly prominent in underdoped Hg1201 with  K hin (); tab (), which contributes to the destruction of quasiparticle coherence on portions of the Fermi surface connected by the CDW wavevector.

Our results establish that excitations across the Fermi-surface in the presence of either SC and/or PG order should be considered in accounting for the magnetic spectrum in both double- and single-layer cuprates. Recent transport measurements indicate Fermi-liquid behavior in the PG state Chan et al. (2014); Mirzaei et al. (2013); Barišić et al. (2013), which adds further support for the need to pursue such formulations. However, the spin-exiton scenario can in principle only generate a single pole (below the Stoner continuum) at each , and thus this scenario cannot account for both the downward and upward dispersive branches. In addition to the pair-breaking peak in the ERS channel, a two-magnon peak is observed Li et al. (2013), which indicates the persistence of short-range local-moment AF correlations, likely associated with the upward dispersive part of the spectrum. A theoretical approach that incorporates both itinerant and local spins, such as in ref. Eremin et al. (2012), might thus be necessary. Understanding the Y-shaped spectrum and the large spin gap will likely require the consideration of the relationship between the magnetic degrees of freedom and the experimentally-detected broken symmetry states kam (); Fauqué et al. (2006); Li et al. (2008); Xia et al. (2008); Ghiringhelli et al. (2012); tab () in the PG state.

We have established a phenomenology of magnetic excitations in the PG and SC states of the cuprates that is common to single- and double-layer compounds, namely a Y-shaped PG spectrum that evolves into an X-shaped (hourglass) response accompanied by a resonance in the SC state. In the La-based cuprates, the magnetic spectrum does not undergo a sudden X to Y transition at . However, the low-energy incommensurability was found to decrease slowly with increasing temperature, e.g., from at  K to to at  K in LaBaCuO Fujita et al. (2004). It is tempting to attribute this to pre-formed SC pairs, which have been argued to appear at high temperatures in the La-based cuprates Xu et al. (2000). However, the lack of a resonance, the prominence of static stripe correlations Chubukov et al. (2007), and recent experiments indicating a narrow SC fluctuation range above  Cyr-Choiniére et al. (2009); Grbić et al. (2009); Bilbro et al. (2011), indicate the proximity of a stripe instability in this particular cuprate family as the dominating factor determining their low-energy magnetic spectrum.

We thank Andrey Chubukov, Yuan Li and Guichuan Yu for comments on the manuscript. This work was funded by the Department of Energy through the University of Minnesota Center for Quantum Materials, under DE-FG02-06ER46275 and DE-SC-0006858, and through Award No. LANLF100. LLB is supported by UNESCOS (contract ANR-14-CE05-0007) and NirvAna (contract ANR-14-OHRI-0010) of the ANR. ORNL’s SNS is sponsored by the Scientific User Facilities Division, Office of Basic Energy Sciences, US Department of Energy.


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