Abstract
In underdoped cuprate superconductors, the Fermi surface undergoes a reconstruction that produces a small electron pocket, but whether there is another, as yet, undetected portion to the Fermi surface is unknown. Establishing the complete topology of the Fermi surface is key to identifying the mechanism responsible for its reconstruction. Here we report evidence for a second Fermi pocket in underdoped YBa_{2}Cu_{3}O_{y}, detected as a small quantum oscillation frequency in the thermoelectric response and in the caxis resistance. The fieldangle dependence of the frequency shows that it is a distinct Fermi surface, and the normalstate thermopower requires it to be a hole pocket. A Fermi surface consisting of one electron pocket and two hole pockets with the measured areas and masses is consistent with a Fermisurface reconstruction by the charge–density–wave order observed in YBa_{2}Cu_{3}O_{y}, provided other parts of the reconstructed Fermi surface are removed by a separate mechanism, possibly the pseudogap.
Introduction
The phase diagram of cuprate superconductors is shaped by ordered states, and their identification is essential for understanding hightemperature superconductivity. Evidence for a new state with broken symmetry in cuprates recently came from two major developments. The observation of quantum oscillations in underdoped YBa_{2}Cu_{3}O_{y} (YBCO)^{1} and HgBa_{2}CuO_{4+d} (Hg1201) (ref. 2), combined with negative Hall^{3,4,5} and Seebeck^{5,6,7} coefficients, showed that the Fermi surface contains a small closed electron pocket and is therefore reconstructed at low temperature, implying that translational symmetry is broken. The detailed similarity of the Fermisurface reconstruction in YBCO and La_{1.6−x}Eu_{0.4}Sr_{x}CuO_{4} (EuLSCO)^{6,7,8} revealed that YBCO must host a density–wave order similar to the stripe order of EuLSCO^{9}. More recently, charge–density–wave (CDW) modulations were observed directly, first by NMR in YBCO^{10} and then by Xray diffraction in YBCO^{11,12,13} and Hg1201 (ref. 14). In YBCO, a thermodynamic signature of the CDW order was detected in the sound velocity at low temperature and finite magnetic field^{15}. These CDW modulations are reminiscent of the checkerboard pattern previously observed by STM on Bi_{2}Sr_{2}CaCu_{2}O_{8+d} (refs 16, 17), for instance.
Fermisurface reconstruction and CDW modulations are therefore two universal signatures of underdoped cuprates, which begs the following question: is the Fermi surface seen by quantum oscillations compatible with a reconstruction by the observed CDW modulations? This issue requires a detailed knowledge of the Fermi surface, to be compared with Fermi surface calculations based on the measured parameters of the CDW order, in the same material at the same doping. In this Article, we report quantum oscillations measurements that reveal an additional, holelike Fermi pocket in underdoped YBCO. As we discuss below, a Fermi surface consisting of one electron and two hole pockets of the measured sizes and masses is consistent with a reconstruction by the observed CDW.
Results
We have measured quantum oscillations in the thermoelectric response and caxis resistance of underdoped YBCO. Our samples were chosen to have a doping p=0.11–0.12, at which the amplitude of quantum oscillations is maximal^{18}. In the dopingtemperature phase diagram, this is also where the CDW modulations are strongest^{19,20} (Fig. 1a) and where the critical magnetic field B_{c2} needed to suppress superconductivity is at a local minimum^{21} (Fig. 1b). In the T=0 limit, the Seebeck (S) and Nernst (ν=N/B) coefficients are inversely proportional to the Fermi energy^{22,23} and are therefore expected to be enhanced for small Fermi surfaces. In Fig. 2a, we show isotherms of S and N at T=2 K measured up to B=45 T in a YBCO sample with p=0.11. Above B_{c2}=24 T (ref. 21), both S and N are negative; the fact that S<0 is consistent with an electron pocket dominating the transport at low temperature^{6,7}. The normalstate signal displays exceptionally large quantum oscillations, with a main frequency F_{a}=540 T and a beat pattern indicative of other, nearby, frequencies. In Fig. 2, we also show the caxis resistance of two YBCO samples at p=0.11 and 0.12, measured in pulsed fields up to 68 T. The overall behaviour of the caxis magnetoresistance at p=0.11 is consistent with previous reports^{24,25}. Quantum oscillations are clearly visible and the three distinct frequencies F_{a1}=540 T, F_{a2}=450 T and F_{a3}=630 T in the Fourier spectrum at p=0.11 (Fig. 1d) agree with reported values^{26}.
With increasing temperature, the amplitude of these ‘fast’ oscillations decreases rapidly and above T~10 K we are left with a slowly undulating normalstate signal, clearly seen in the raw Seebeck data (Figs 2b and 3a). In Fig. 3b, the oscillatory part of that signal, obtained by subtracting a smooth background, is plotted as a function of inverse magnetic field. Although the ‘slow oscillations’ at 18 K are 20 times weaker than the fast oscillations at 2 K, they are clearly resolved and periodic in 1/B. After their discovery in the Seebeck signal, the slow oscillations were also detected in the caxis resistance, as shown in Fig. 3c. In both the Seebeck and caxis resistance data, the frequency of these slow oscillations is F_{b}=95±10 T (p=0.11). Similar oscillations were also detected in the caxis resistance of a sample at p=0.12 (Fig. 3d), with F_{b}=120±15 T. In Fig. 4a, we show the derivative dR_{c}/dB, which unambiguously reveals F_{b}, without the need for a background subtraction. (Note that in the caxis resistance data, the amplitude of F_{b} is about 0.1% of the total signal and is more sensitive to the background subtraction.) This slow frequency persists up to 30 K and its amplitude follows the usual Lifshitz–Kosevich formula (Fig. 4b), with a small effective mass m*=0.45±0.1 m_{0}, where m_{0} is the free electron mass.
Using the caxis resistance, we have measured the dependence of F_{b} on the angle θ at which the field is tilted away from the c axis. In Fig. 4c, the oscillatory part of the c axis resistance for p=0.11 at T=15 K is plotted versus 1/Bcos(θ), and the angular dependence of F_{b} is displayed in Fig. 4d. F_{b}(θ) varies approximately as 1/cos(θ), indicating that the Fermi surface associated with F_{b} is a warped cylinder along the c axis, as expected for a quasitwodimensional system.
Discussion
The slow frequency F_{b}~100 T reported here bears the key signatures of quantum oscillations and in the following discussion we argue that it comes from a small holelike Fermi surface, distinct from the larger electronlike Fermi pocket responsible for the main frequency F_{a1}=540 T.
We note that the frequency F_{b} is nearly equal to the difference between the main frequency of the electron pocket F_{a1} and its satellites F_{a2} and F_{a3}. While the identification of the multiple F_{a} frequencies is not definitive, it is likely that two of them are associated with the two separate Fermi surfaces that come from the two CuO_{2} planes (bilayer) in the unit cell of YBCO. The third frequency could then either come from magnetic breakdown between these two Fermi surfaces^{27} or from a warping due to caxis dispersion^{26,28}. In layered quasitwodimensional materials, slow quantum oscillations can appear in the caxis transport as a result of interlayer coupling^{29,30}. Two observations allow us to rule out this scenario in the present context. First, F_{b} is observed in the inplane Seebeck coefficient, which does not depend on the caxis conductivity. Second, at a special fieldangle θ, called the Yamaji angle, where the caxis velocity vanishes on average along a cyclotron orbit, one should see a vanishing F_{b}. This is not seen in our fieldangle dependence of F_{b}, which, if anything, only deviates upward from a cylindrical 1/Bcos(θ) dependence (Fig. 4d).
Quantum interference from magnetic breakdown between two bilayersplit orbits could in principle produce a difference frequency close to F_{b}. In this scenario, however, the amplitudes of the two nearby frequencies F_{a2} and F_{a3} should be identical, irrespective of the field range, in disagreement with torque^{26} and caxis resistance measurements (see Fig. 1d). Furthermore, in a magnetic breakdown scenario, we would expect F_{b} to scale with F_{a}, since both frequencies originate from the same cyclotron orbits. This is not what we observe in our thermoelectric data: as seen in Fig. 2a, the amplitude of F_{a} is larger in the Nernst effect than in the Seebeck effect, yet F_{b} is only detected in the latter. This is strong evidence that F_{a} and F_{b} do not involve cyclotron orbits on the same Fermi surface. We therefore conclude that F_{b} must come from a distinct Fermi pocket, in contrast with the interpretation of ref. 31 in terms of quantum interference.
For a number of reasons, we infer that this second pocket in the reconstructed Fermi surface of YBCO is holelike. The first reason is the strong dependence of resistivity ρ and Hall coefficient R_{H} on magnetic field B, as observed in YBCO and in YBa_{2}Cu_{4}O_{8} (ref. 3), a closely related material with similar quantum oscillations^{32,33}. For instance, R_{H}(B) goes from positive at low field to negative at high field^{3} and ρ(B) exhibits a significant magnetoresistance^{25}. These are natural consequences of having both electron and hole carriers. In YBa_{2}Cu_{4}O_{8}, the Hall and resistivity data were successfully fit in detail to a twoband model of electrons and holes^{34}.
A second indication that both the electron and hole carriers are present in underdoped YBCO is the fact that quantum oscillations are observed in the Hall coefficient^{35,36}. In an isotropic singleband model, the Hall coefficient is simply given by R_{H}=1/ne, where n is the carrier density and e the electron charge. Quantum oscillations in R_{H} appear via the scattering rate, which enters R_{H} either when two or more bands of different mobility are present or when the scattering rate on a single band is strongly anisotropic. At low temperatures, however, where impurity scattering dominates, the latter scenario is improbable.
The most compelling evidence for the presence of holelike carriers in underdoped YBCO comes from the magnitude of the Seebeck coefficient. In the T=0 limit and for a single band, it is given by^{22}:
where k_{B} is Boltzmann’s constant, T_{F} is the Fermi temperature and ζ=0 or −1/2 depending on whether the relaxation time or the mean free path is assumed to be energy independent, respectively. The sign of S/T depends on whether the carriers are holes (+) or electrons (−). This expression has been found to work very well in a variety of correlated electron metals^{22}. We stress that S/T (in the T=0 limit) is governed solely by T_{F}, which allows a direct quantitative comparison with quantum oscillation data, with no assumption on pocket multiplicity. This contrasts with the specific heat, which depends on the number of Fermi pockets (see below).
In Fig. 5a, we reproduce normalstate Seebeck data in YBCO at four dopings, plotted as S/T versus T (from ref. 7). S/T goes from positive at high T to negative at low T, in agreement with a similar sign change in R_{H}(T) (refs 3, 4), both evidence that the dominant carriers at low T are electron like. Extrapolating S/T to T=0 as shown by the dashed lines in Fig. 5a, we obtain the residual values and plot them as a function of doping in Fig. 5b (S_{measured}, red squares). We see that the size of the residual term is largest (that is, is most strongly negative) at p=0.11, and that it decreases on both sides.
This dopingdependent S/T is to be compared with the Fermi temperature directly measured by quantum oscillations via:
where F is the frequency, m* the effective mass and assuming a parabolic dispersion. For YBCO at p=0.11, the electron pocket gives F_{a1}=540±20 T and m*=1.76m_{0}, so that T_{F}=410±20 K and hence S_{e}/T=–1.0 μV K^{−2} (–0.7 μV K^{−2}), for ζ=0 (–1/2). In Fig. 5a, the measured S/T extrapolated to T→0 gives a value of −0.9 μV K^{−2}. The electron pocket alone therefore accounts by itself for essentially the entire measured Seebeck signal at p=0.11. From quantum oscillation measurements at different dopings^{24,37,38}, we know the values of F_{a1} and m* from p=0.09 to p=0.13, and can therefore determine the evolution of S_{e}/T in that doping interval. The result is plotted as blue dots in Fig. 5b, where we see that the calculated S_{e}/T increases by a factor 2.5 between p=0.11 and p=0.09. This is because the mass m* increases strongly as p→0.08 (ref. 38), while F_{a1} decreases only slightly^{18}. This strong increase in the calculated S_{e}/T is in stark contrast with the measured value of S/T, which decreases by a factor of 3 between p=0.11 and p=0.09 (Fig. 5b). To account for the observed doping dependence of the thermopower in YBCO, we are led to conclude that there must be a holelike contribution to S/T. We emphasize that the Hall coefficient R_{H} (ref. 4) measured well above B_{c2} (ref. 21) displays the same domelike dependence on doping as the Seebeck coefficient^{7}, which further confirms the presence of a holelike Fermi pocket.
In a twoband model, the total Seebeck coefficient is given by
where the hole (h) and electron (e) contributions are weighted by their respective conductivities σ_{h} and σ_{e}. As shown in Fig. 5c, we can account for the measured S/T at T→0 by adding a holelike contribution, S_{h}, which we estimate from F_{b}=95±10 T and m*=0.45m_{0}, giving T_{F}=280±80 K. Assuming for simplicity that S_{h} is doping independent leaves the ratio of conductivities, σ_{e}/σ_{h}, as the only adjustable parameter in the above twoband expression (equation (3)). In Fig. 5d, we plot the resulting σ_{e}/σ_{h} as a function of doping, and we see that it peaks at p=0.11 and drops on either side. This is consistent with the fact that the amplitude of the fast quantum oscillations is largest at p=0.11, and much smaller away from that doping^{18}, direct evidence that the mobility of the electron pocket is maximal at p=0.11. This is clearly seen in the resistance data in Fig. 2c,d, where the amplitude of the quantum oscillations of the electron pocket is strongly reduced when going from p=0.11 to 0.12. In contrast, Fig. 3c,d shows that the amplitude of the oscillations from the hole pocket remains nearly constant: at T=4.2 K and H=68 T, their relative amplitude is ΔR_{c}/R_{c}=0.036% at p=0.11 and 0.03% at p=0.12. The change in conductivity ratio therefore comes mostly from a change in σ_{e}.
To summarize, in addition to the twoband description of transport data in YBa_{2}Cu_{4}O_{8} (ref. 34), the doping dependence of the Seebeck^{7} and Hall^{4} coefficients in YBCO is firm evidence that the reconstructed Fermi surface of underdoped YBCO (for 0.08<p<0.18) contains not only the wellestablished electron pocket^{4}, but also another holelike surface (of lower mobility). We combine this evidence with our discovery of an additional small Fermi surface to conclude that this new pocket is hole like.
From the measured effective mass m*, the residual linear term γ in the electronic specific heat C_{e}(T) at T→0 can be estimated through the relation^{39}
where n_{i} is the multiplicity of the i^{th} type of pocket in the first Brillouin zone (this expression assumes an isotropic Fermi liquid in two dimensions with a parabolic dispersion). For a Fermi surface containing one electron pocket and two hole pockets per CuO_{2} plane, we obtain a total mass of (1.7±0.2)+2 (0.45±0.1)=2.6±0.4 m_{0}, giving γ=7.6±0.8 mJ K^{−2} mol (for two CuO_{2} planes per unit cell). Highfield measurements of C_{e} at T→0 in YBCO at p~0.1 yield γ=5±1 mJ K^{−2} mol (ref. 39) at B>B_{c2}=30 T (ref. 21). We therefore find that the Fermi surface of YBCO can contain at most two of the small hole pockets reported here, in addition to only one electron pocket. No further sheet can realistically be present in the Fermi surface.
There is compelling evidence that the Fermi surface of YBCO is reconstructed by the CDW order detected by NMR and Xray diffraction. In particular, Fermisurface reconstruction^{4,7} and CDW modulations^{19,20} are detected in precisely the same region of the temperaturedoping phase diagram. Because the CDW modulations are along both the a and b axes, the reconstruction naturally produces a small closed electron pocket along the Brillouin zone diagonal, at the socalled nodal position^{40,41}. Given the wavevectors measured by Xray diffraction, there will also be small closed holelike ellipses located between the diamondshaped nodal electron pockets. An example of the Fermi surface calculated^{42} using the measured CDW wavevectors is sketched in Fig. 1c. It contains two distinct closed pockets: a nodal electron pocket of area such that F_{e}~430 T and a holelike ellipse such that F_{h}~90 T p=0.11 (ref. 42). Note that a reconstruction by a commensurate wavevector q=1/3 π/a, very close to the measured value, yields one electron and two hole pockets per Brillouin zone, as assumed in our calculation of γ above.
If, as indeed observed in YBCO at p=0.11 (ref. 20), the CDW modulations are anisotropic in the a–b plane, the ellipse pointing along the a axis will be different from that pointing along the b axis^{42}. If one of the ellipses is close enough to the electron pocket, that is, if the gap between the two is small enough, magnetic breakdown will occur between the hole and the electron pockets, and this could explain the complex spectra of multiple quantum oscillations seen in underdoped YBCO (Fig. 1d).
In most models of Fermisurface reconstruction by CDW order, the size of the Fermi pockets can be made to agree with experiments using a reasonable set of parameters. For example, a similar Fermi surface (with one electron pocket and two hole pockets) is also obtained if one considers a ‘crisscrossed’ stripe pattern instead of a checkerboard^{43}. At this level, there is consistency between our quantum oscillation measurements and models of Fermisurface reconstruction by the CDW order. However, in addition to the electron and hole pockets, the folding of the large Fermi surface produces other segments of Fermi surface whose total contribution to γ greatly exceeds that allowed by the specific heat data. Consequently, there must exist a mechanism that removes parts of the Fermi surface beyond the reconstruction by the CDW order. A possible mechanism is the pseudogap. The loss of the antinodal states caused by the pseudogap would certainly remove parts of the reconstructed Fermi surface.
Further theoretical investigations are needed to understand how pseudogap and CDW order are intertwined in underdoped cuprates. An important point in this respect is the fact that, unlike in Bi_{2}Sr_{2−x}La_{x}CuO_{6+d} (ref. 44), the CDW wavevector measured by Xray diffraction in YBCO and in Hg1201 (ref. 14) does not connect the hot spots where the large Fermi surface intersects the antiferromagnetic Brillouin zone (Fig. 1c), nor does it nest the flat antinodal parts of that large Fermi surface (Fig. 1c).
Methods
Samples
Single crystals of YBCO with y=6.54, 6.62 and 6.67 were obtained by flux growth at UBC^{45}. The superconducting transition temperature T_{c} was determined as the temperature below which the zerofield resistance R=0. The hole doping p is obtained from T_{c} (ref. 46), giving p=0.11 for y=6.54 and 6.62, and p=0.12 for y=6.67. The samples with y=6.54 and 6.62 have a high degree of orthoII oxygen order, and the sample with y=6.67 has orthoVIII order. The samples are detwinned rectangular platelets, with the a axis parallel to the length (longest dimension) and the b axis parallel to the width. The electrical contacts are diffused evaporated gold pads with a contact resistance less than 1 Ω.
Thermoelectric measurements
The thermoelectric response of YBCO with y=6.54 (p=0.11) was measured at the National High Magnetic Field Laboratory (NHMFL) in Tallahassee, Florida, up to 45 T, in the temperature range from 2 to 40 K. The Seebeck and Nernst coefficients are given by S≡−∇V_{x}/∇T_{x} and ν≡N/B≡(∇V_{y}/∇T_{x})/B, respectively, where ∇V_{x} (∇V_{y}) is the longitudinal (transverse) voltage gradient caused by a temperature gradient ∇T_{x}, in a magnetic field Bz. A constant heat current was sent along the a axis of the single crystal, generating a temperature difference ΔT_{x} across the sample. ΔT_{x} was measured with two uncalibrated Cernox chip thermometers (Lakeshore), referenced to a third, calibrated Cernox. The longitudinal and transverse electric fields were measured using nanovolt preamplifiers and nanovoltmeters. All measurements were performed with the temperature of the experiment stabilized within ±10 mK and the magnetic field B swept at a constant rate of 0.4–0.9 T min^{−1} between positive and negative maximal values, with the heat on. The field was applied normal to the CuO_{2} planes (Bzc).
Since the Seebeck coefficient S is symmetric with respect to the magnetic field, it is obtained by taking the mean value between positive and negative fields:
where ΔV_{x} is the difference in the voltage along x measured with and without thermal gradient. This procedure removes any transverse contribution that could appear due to slightly misaligned contacts. The longitudinal voltages and the thermal gradient being measured on the same pair of contacts, no geometric factor is involved.
The Nernst coefficient N is antisymmetric with respect to the magnetic field; therefore, it is obtained by the difference:
where L and w are the length and width of the sample, respectively, along x and y and V_{y} is the voltage along y measured with the heat current on. This antisymmetrization procedure removes any longitudinal thermoelectric contribution and a constant background from the measurement circuit. The uncertainty on N comes from the uncertainty in determining L and w, giving typically an error bar of ±10%.
Resistance measurements
The caxis resistance was measured at the Laboratoire National des Champs Magnétiques Intenses (LNCMI) in Toulouse, France, in pulsed magnetic fields up to 68 T. Measurements were performed in a conventional fourpoint configuration, with a current excitation of 5 mA at a frequency of ~60 kHz. Electrical contacts to the sample were made with large current pads and small voltage pads mounted across the top and bottom so as to short out any inplane current. A highspeed acquisition system was used to digitize the reference signal (current) and the voltage drop across the sample at a frequency of 500 kHz. The data were analysed with software that performs the phase comparison. θ is the angle between the magnetic field and the c axis, and measurements were done at θ=0° up to 68 T and at various angles θ up to 58 T. The uncertainty on the absolute value of the angle is about 1°.
Additional information
How to cite this article: DoironLeyraud, N. et al. Evidence for a small hole pocket in the Fermi surface of underdoped YBa_{2}Cu_{3}O_{y}. Nat. Commun. 6:6034 doi: 10.1038/ncomms7034 (2015).
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Acknowledgements
We thank A. Allais, K. Behnia, A. Carrington, S. Chakravarty, B. Fauqué, A. Georges, N. E. Hussey, M.H. Julien, S.A. Kivelson, M.R. Norman, S. Raghu, S. Sachdev, A.M. Tremblay and C. Varma for fruitful discussions. We thank J. Béard and P. Frings for their assistance with the experiments at the LNCMI. The work in Toulouse was supported by the French ANR SUPERFIELD, the EMFL and the LABEX NEXT. A portion of 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. R.L., D.A.B. and W.N.H. acknowledge support from NSERC. L.T. acknowledges support from the Canadian Institute for Advanced Research and funding from NSERC, FRQNT, the Canada Foundation for Innovation and a Canada Research Chair.
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N.D.L., S.R.d.C. and J.H.P. performed the Seebeck and Nernst measurements at the NHMFL in Tallahassee. N.D.L., F.L. and E.H. analysed the Seebeck data. S.B., S.L., D.L.B., D.V., B.V. and C.P. performed and analysed the resistance measurements at the LNCMI in Toulouse. B.J.R., R.L., D.A.B. and W.N.H. prepared the YBCO single crystals at UBC (crystal growth, annealing, detwinning and contacts). L.T. and N.D.L. supervised the thermoelectric measurements. C.P. supervised the pulsedfield measurements. N.D.L., L.T. and C.P. wrote the manuscript with input from all authors.
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DoironLeyraud, N., Badoux, S., René de Cotret, S. et al. Evidence for a small hole pocket in the Fermi surface of underdoped YBa_{2}Cu_{3}O_{y}. Nat Commun 6, 6034 (2015). https://doi.org/10.1038/ncomms7034
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DOI: https://doi.org/10.1038/ncomms7034
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