Abstract
Broken fourfold rotational (C_{4}) symmetry is observed in the experimental properties of several classes of unconventional superconductors. It has been proposed that this symmetry breaking is important for superconducting pairing in these materials, but in the highT_{c} cuprates this broken symmetry has never been observed on the Fermi surface. Here we report a pronounced anisotropy in the angle dependence of the interlayer magnetoresistance of the underdoped high transition temperature (highT_{c}) superconductor YBa_{2}Cu_{3}O_{6.58}, directly revealing broken C_{4} symmetry on the Fermi surface. Moreover, we demonstrate that this Fermi surface has C_{2} symmetry of the type produced by a uniaxial or anisotropic densitywave phase. This establishes the central role of C_{4} symmetry breaking in the Fermi surface reconstruction of YBa_{2}Cu_{3}O_{6+δ }, and suggests a striking degree of universality among unconventional superconductors.
Introduction
Broken C_{4} symmetry is observed in a number of experiments on unconventional superconductors, including transport,^{1,2,3,4} nuclear magnetic resonance (NMR),^{5, 6} neutron scattering,^{7, 8} Xray scattering^{9,10,11} and scanning tunneling microscopy.^{12, 13} In the ironbased superconductors broken C_{4} symmetry is observed directly on the Fermi surface,^{14} and has been taken as an indication that this broken symmetry drives the high T_{c}.^{15} The question of whether the same type of symmetry breaking is relevant in the copperoxide highT_{c} superconductors^{16} has been left open because it has never been observed on the Fermi surface.^{17, 18} Without a link to the Fermi surface, it is hard to make a compelling argument that the experimental observations of broken C_{4} symmetry are relevant to the phenomenon of highT_{c}.
The Fermi surface of a metal is constrained by the symmetry of its electronic environment. A precise determination of Fermi surface geometry, therefore, provides information about symmetrybreaking states of matter that feed back into the electronic structure. In the underdoped highT_{c} cuprates it is now well established that a chargedensity wave (CDW) competes with superconductivity, but a direct experimental connection is still missing between the CDW and the Fermi surface. QO (quantum oscillation) measurements, performed in the highfield state, have been instrumental in determining the presence of a small electron Fermi surface in the underdoped side of the phase diagram.^{17, 19,20,21,22} While QOs provide precise information about the area of the Fermi surface, they are relatively insensitive to its overall shape in twodimensional (2D) metals (e.g., circular vs. square in crosssection). Here, we present the results of a complementary technique, angledependent magnetoresistance (AMR), that determines the geometry and symmetry of the Fermi surface. We identify the shape of the Fermi surface as consistent with certain proposed CDW reconstruction scenarios,^{17, 23,24,25,26,27} and we find that this reconstruction is anisotropic in nature.
Results
Since the 1930s it has been known that a change in resistance with an applied magnetic field—magnetoresistance—provides geometric information about a metallic Fermi surface,^{28} and early magnetoresistance experiments were instrumental in developing the modern quantum theory of metals.^{29} Real metals have nonspherical Fermi surfaces, thus magnetoresistance can be a strong function of the angle between the magnetic field and the crystal axes, giving rise to AMR. The signatures of AMR are particularly strong for quasi2D Fermi surfaces, and thus this technique is well suited for determining the Fermi surface geometry of the highT_{c} cuprates.^{30,31,32,33,34} To map the Fermi surface geometry of YBa_{2}Cu_{3}O_{6.58} we performed interlayer (ρ _{zz}) magnetoresistance measurements in a fixed magnetic field of 45 T even in the absence of interlayercoherence, AMR is still sensitive to the Fermi surface geometry (see ref. 35). The fieldangle dependence of ρ _{zz} was obtained by rotating the sample in situ, sweeping the polar angle θ between the magnetic field and the crystalline ĉaxis for several values of the azimuthal angle ϕ (see the inset of Fig. 1a for angle definitions). Our primary observation is twofold anisotropy of the AMR as a function of ϕ—immediately apparent in Fig. 1—with the AMR increasing much more rapidly on rotating the field towards the crystalline âaxis (ϕ = 0°) than towards the \(\hat{{\rm{b}}}\)axis (ϕ = 90°). As we demonstrate below, this directly indicates that Fermi surface has strongly broken C_{4} symmetry.
In order to understand the connection between AMR and the Fermi surface, we first perform a qualitative analysis of the three most salient features in the data: C_{2} symmetry; negative AMR at low θ near the ϕ = 90° directions; and suppression of AMR along the ϕ = 90° direction at high θ. The semiclassical conductivity of a metal is given by the velocityvelocity correlation function of all quasiparticles on the Fermi surface, averaged over their lifetime τ.^{36, 37} Magnetoresistance arises because the quasiparticle velocity—which is always perpendicular to the Fermi surface—is altered by the Lorentz force, which induces cyclotron motion perpendicular to the magnetic field (see Fig. 2a for a schematic cyclotron orbit). If τ is sufficiently long, orbiting quasiparticles sample a significant portion of the Fermi surface perimeter before scattering. Depending on the specific geometry of the cyclotron orbit, components of the quasiparticle velocity (v _{z}, for example) may average to zero, resulting in a vanishingly small contribution to the conductivity for that field direction. The total magnetoresistance at a particular field orientation is then given by the ensemble of all quasiparticle orbits on the Fermi surface at that angle, and AMR, therefore, encodes the Fermi surface geometry.
Underdoped YBa_{2}Cu_{3}O_{6+δ } contains, at a minimum, two sections of Fermi surface due to its bilayer crystal structure (Fig. 3a). The phenomenon of magnetic breakdown, whereby quasiparticles in a magnetic field can jump between Fermi surfaces separated by small energy gaps,^{38} leads to multiple possible cyclotron orbits for the reconstructed Fermi surface of YBa_{2}Cu_{3}O_{6.58} (refs 17, 39, 40). For the purpose of simulating our AMR, we calculate the conductivity for each breakdown path separately and then sum the conductivities in parallel. In general, the interlayer dispersion of a quasi2D Fermi surface—which determines the interlayer resistivity ρ _{zz}—can be expanded in cylindrical harmonics, three of which are shown in Fig. 2b. The simplest harmonic cos k _{z} produces magnetoresistance that increases with θ for all ϕ. The harmonic sin 2ϕcosk _{z} produces fourfold symmetric AMR that decreases with increasing θ, with a weaker effect along the diagonal directions. The simplest harmonic to break C_{4} symmetry in the interlayer tunneling is sinϕcosk _{z}, which has twofold symmetry and gives AMR that decreases with increasing θ along one ϕ direction and very weak AMR along the perpendicular direction. The measured twofold anisotropy of ρ _{zz} in YBa_{2}Cu_{3}O_{6.58}, along with the negative AMR at low θ near the ϕ = 90° directions, therefore, suggests that there is a significant contribution from a section of Fermi surface with sin ϕcosk _{z} symmetry. The most obvious reason for the existence of such a symmetrybreaking Fermi surface inYBa_{2}Cu_{3}O_{6.58} is if the Fermi surface reconstruction itself breaks C_{4} symmetry—the unreconstructed Fermi surface of YBa_{2}Cu_{3}O_{6+δ } is only slightly distorted from C_{4} symmetry by the weakly orthorhombic crystal structure.^{41} While the crystal structure of YBa_{2}Cu_{3}O_{6+δ } inherently breaks C_{4} symmetry, we argue that the small symmetrybreaking strain field arising from the orthorhombicity serves to preferentially align an underlying electronic instability.^{3} Our simulations suggest that anisotropy in the scattering rate; anisotropy in the interlayer hopping that is still finite in all directions; or an inplane anisotropy in the Fermi wavevector (i.e., a Fermi surface elongated along one inplane direction), cannot produce magnetoresistance with the angular structure we observe (see Supplementary Information for details).
Another prominent feature in the data is the suppression of AMR along the ϕ = 45° direction, particularly above ϕ = 50°. In addition to the interlayer velocity, AMR is responsive to the inplane geometry of the Fermi surface.^{30,31,32,33,34, 42} For a cylindrical surface with simple cos k _{z} warping and an isotropic Fermi radius k _{F} (see Fig. 2b), the AMR evolves with field angle θ as \({\rho }_{{\rm{zz}}}(\theta )\propto \mathrm{1/}{({J}_{0}({k}_{{\rm{F}}}c\tan \theta ))}^{2}\), where c is the ĉaxis lattice constant (see footnote for more complicated warping geometries the actual form of ρ _{zz}(θ) is different, but it is still the product k _{F} c that sets the angular scale over which the maxima in ρ _{zz} appear). The AMR shows maxima wherever k _{F} c tan (θ) equals a zero of the Bessel function J _{0}, which happens for certain “Yamaji” angles where the interlayer velocity averages to zero around the cyclotron orbit.^{43} A critical feature of this form of ρ _{zz} is that the product k _{F} c sets the scale in θ over which these maxima in ρ _{zz} appear: a smaller k _{F} pushes the resistance maxima out to higher angles. For quasi2D Fermi surfaces that are noncircular in cross section (anisotropic k _{F}(ϕ)), k _{F} c still sets the angular scale, but to a first approximation it is k _{F}(ϕ _{0})—where ϕ _{0} is the direction in which the magnetic field is being rotated—that determines where θ and ϕ the maxima in ρ _{zz} appear. For YBa_{2}Cu_{3}O_{6.58} we can use the average value of k _{F}≈1.27 nm^{−1} determined by QOs to estimate that the first maximum in ρ _{zz} should appear around θ ≈ 58°. While this means that we cannot reach any AMR maxima before the onset of superconductivity, we are still able to observe the approach to the first maximum. The relatively weak AMR observed when the field is rotated in the ϕ = 45° direction suggests that the Fermi surface has a smaller k _{F} along the diagonals than along the â and \(\hat{{\rm{b}}}\) axes—that its first maximum is pushed to higher θ. This points to a Fermi surface with a square or diamondlike inplane geometry—strikingly similar to the Fermi surface reconstruction predicted to occur via CDW.^{17, 23,24,25,26,27} Two scenarios that could give rise to this type of reconstruction are shown in Fig. 3.
We can now combine our qualitative Fermi surface information—interlayer tunneling with C_{2} symmetry on at least one Fermi surface section, and an inplane diamond shape—and quantitatively model our data by numerically solving the Boltzmann transport equation^{36} (see Supplementary Information for details). To avoid overparametrization of the data we model only the three F = 530 T “breakdown” surfaces (plus their symmetryrelated copies), which are known to dominate the ĉaxis conductivity.^{44, 45} We fix the crosssectional area of the orbits to the value of \({A_k}=2\pi e/\hslash \cdot \mathrm530\ {\rm{T}}\approx 5.1\,{\rm{nm}}^{2}\) obtained from QO measurements on YBa_{2}Cu_{3}O_{6.58} (ref. 44), and allow a single value of τ to vary as a free parameter for all three surfaces. We find that the AMR is best modeled by a sum of 41% of a surface with sin ϕ cosk _{z} symmetry, 27% of a surface with cosk _{z} symmetry, and 32% of a surface with sin 2ϕ cosk _{z} symmetry. These proportions are reasonably in line with what is expected from the number of possible breakdown orbits and their magnetic breakdown probabilities.^{17} In Fig. 4 we show that the most important features of the AMR which we first identified on qualitative grounds—C_{2} symmetry, negative AMR near ϕ = 90°, and suppression along the (ϕ = 45°) direction at high θ—are captured by this model. It is important to note that resistivity is not a linear function of the interlayer hopping parameters: combining all three symmetry warpings onto a single section of Fermi surface cannot reproduce the data.
The consistency between our Fermi surface model and other experiments can now be checked. The quasiparticle lifetime we extract from the simulations is τ = 0.24 ± 0.05 ps—in agreement with the 0.27 ps reported from QOs.^{46} Zerofield resistivity measurements at this doping find an inplane resistive anisotropy that is collapsing towards one at low temperatures: this behavior is attributed to the conductivity of the 1D copperoxide chains freezing out at low temperature.^{2} Our model, which breaks C_{4} symmetry only in the ĉaxis dispersion of the Fermi surface, naturally preserves the nearisotropy of the â\(\hat{{\rm{b}}}\)plane conductivity. If the 1D copper oxide chain layer were clean enough to produce AMR—and there are is evidence from spectroscopy to believe that this is not the case^{47,48,49}—then the resultant inplane resistivity would be highly anisotropic: such inplane anisotropy is not observed experimentally.^{2} AMR from an open Fermi surface produced by the chain layer would also not produce the observed upturn in the AMR we observe above θ = 40° when ϕ = 90°(see Supplementary Information for details). AMR measurements can detect sections of Fermi surface that are invisible to QOs, such as open sheets and surfaces with higher scattering rates (shorter τ). AMR, therefore, has the potential to observe sections of Fermi surface in YBa_{2}Cu_{3}O_{6.58} formed by CDW reconstruction that have remained unobserved by QOs. Our simulation reveals, however, that the AMR can be fully accounted for by the same bilayersplit electron pocket that appears in QO experiments, with no additional sheets or pockets. This is consistent with previous suggestions that there is only a single Fermi pocket in the Brillouin zone of underdoped YBa_{2}Cu_{3}O_{6+δ } (refs 50, 51). We note that the small hole pocket reported at this doping^{52} has a crosssection (k _{F}) that is too small to contribute any significant AMR below θ≈75°, thus we do not include the possibility of this surface in our model.
It is important to note that our model relies on the validity of the singleparticle picture and Boltzmann transport equations. While these assumptions may be valid for the small Fermi pocket we observe—and the observation of the same pocket via QOs strongly suggests this to be true^{53}—there is much experimental evidence that large portions of the Fermi surface exhibit decidedly nonquasiparticle behavior.^{54} This includes the linearintemperature zerofield resistivity above the pseudogap temperature.^{55,56,57} One possibility is that high magnetic fields suppress the pseudogap and restore the entire Fermi surface to a Fermiliquid state—the possible Fermi surface reconstruction presented in Fig. 3d, e would require such a scenario for the pocket near the antinodal region to be ungapped. This possibility suffers from the fact that no other pieces of Fermi surface besides the small electron pocket are observed in high magnetic fields. Despite the apparent ability of singleparticle Boltzman equations to model the highfield magneotransport, it remains an important open question as to how to connect the zerofield metallic state to that observed in high magnetic fields.
Discussion
In order to obtain a Fermi surface with C_{2} symmetry from the C_{4} symmetric unreconstructed Fermi surface,^{41} the mechanism of Fermi surface reconstruction must itself break C_{4} symmetry. We suggest that the diamondlike shape we measure with AMR strongly constrains the reconstruction scenario to one of densitywave origin. There is experimental evidence for broken C_{4} symmetry in the CDW: NMR measurements to 28.5 T are consistent with a highfield unidirectional CDW (“stripe”) phase,^{5} and recent Xray measurements indicate that the CDW modulated along the \(\hat{{\rm{b}}}\)axis —which has the same wavevector as the CDW modulated along the âaxis in zero field—acquires a new ĉaxis component above 15 T (refs 11, 58). We suggest that this asymmetry in the ĉaxis components of the wavevectors persists to at least 45 T, and that these CDWs are responsible for the Fermi surface reconstruction, as has been proposed theoretically.^{17, 23,24,25,26,27} A second route to a diamondlike pocket with C_{2} symmetry in the ĉaxis tunneling requires a strong nematic distortion of the Fermi surface (see Fig. 3d, e).^{59} In high magnetic fields this surface is then reconstructed by a unidirectional CDW. This scenario only requires the single CDW wavevector that acquires threedimensional coherence in high fields.^{11, 58}
Independent of the specific model, our analysis shows that Fermi surface reconstruction in YBa_{2}Cu_{3}O_{6.58} is anisotropic, and that the entire AMR signature can be accounted for by the same Fermi surface that is responsible for QOs and a negative Hall coefficient in high magnetic fields. We have also established that although the CDW in the cuprates is relatively weak and disordered compared to more traditional CDW metals,^{60} its symmetry and wavevector directly determine Fermi surface properties. This is similar to the iron pnictide superconductors, where the nematic state is directly observable on the Fermi surface,^{14} and suggests that the physics underlying both the superconductivity and the quantum criticality in these two classes of materials share a similar origin. Our Fermi surface model predicts that for tetragonal HgBa_{2}CuO_{4}, where both shortrange CDW order and a small Fermi surface have been observed,^{61, 62} sufficient uniaxial strain should favour anisotropic CDW formation and give rise to anisotropic AMR similar to what we have observed in YBa_{2}Cu_{3}O_{6.58}.
Methods
Detwinned single crystals of YBa_{2}Cu_{3}O_{6.58} were grown and prepared as described in Liang et al.^{63} Caxis contacts were prepared as described in Ramshaw et al.^{44} Resistivity measurements were conducted in the 45 Tesla hybrid magnet at the National High Magnetic Field Lab in Tallahassee using a 2axis rotator. The resistance was measured with a 4point contact geometry using an SRS 830 lockin amplifier: the sample was driven with I = 500 μA at a frequency of f = 37.7 Hz.
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Acknowledgements
We would like to thank A. Damascelli and I. Elfimov for discussions on the bandstructure of YBa_{2}Cu_{3}O_{6+δ }. We would also like to thank S.A. Kivelson, S. Lederer, A.V. Maharaj, and L. Nie for helpful discussions. P.A.G. and S.G. thank the EPSRC for support (grant EP/H00324X/1). S.E.S. acknowledges support from the Royal Society, the Winton Programme, and the European Research Council under the European Union’s Seventh Framework Programme (grant number FP/20072013)/ERC Grant Agreement number 337425. D.B., W.N.H, and R.L were supported by the Canadian Institute for Advanced Research, the Natural Science and Engineering Research Council, and the Stewart Blusson Quantum Matter Institute. This work was performed at the National High Magnetic Field Laboratory, which is supported by the National Science Foundation Cooperative Agreement No. DMR1157490, the State of Florida, and the U.S. Department of Energy. N.H. and B.J.R acknowledge funding by the U.S. Department of Energy Office of Basic Energy Sciences “Science at 100 T” program. Data presented in this paper resulting from the UK effort are available at http://wrap.warwick.ac.uk/84954/.
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S.G., P.A.G., N.H., B.J.R, and S.E.S. performed the 45 Tesla AMR measurements at the NHMFL in Tallahassee, Florida. B.J.R and K.A.M. performed additional sample characterization at the NHMFL in Los Alamos, New Mexico. D.A.B, W.N.H, R.L., and B.J.R. grew and prepared the YBa_{2}Cu_{3}O_{6.58} samples. N.H. and B.J.R. performed the data analysis and wrote the manuscript with input from all coauthors.
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Ramshaw, B.J., Harrison, N., Sebastian, S.E. et al. Broken rotational symmetry on the Fermi surface of a highT_{c} superconductor. npj Quant Mater 2, 8 (2017). https://doi.org/10.1038/s415350170013z
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DOI: https://doi.org/10.1038/s415350170013z
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