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Single reconstructed Fermi surface pocket in an underdoped single-layer cuprate superconductor

Nature Communications volume 7, Article number: 12244 (2016) | Download Citation


The observation of a reconstructed Fermi surface via quantum oscillations in hole-doped cuprates opened a path towards identifying broken symmetry states in the pseudogap regime. However, such an identification has remained inconclusive due to the multi-frequency quantum oscillation spectra and complications accounting for bilayer effects in most studies. We overcome these impediments with high-resolution measurements on the structurally simpler cuprate HgBa2CuO4+δ (Hg1201), which features one CuO2 plane per primitive unit cell. We find only a single oscillatory component with no signatures of magnetic breakdown tunnelling to additional orbits. Therefore, the Fermi surface comprises a single quasi-two-dimensional pocket. Quantitative modelling of these results indicates that a biaxial charge density wave within each CuO2 plane is responsible for the reconstruction and rules out criss-crossed charge stripes between layers as a viable alternative in Hg1201. Lastly, we determine that the characteristic gap between reconstructed pockets is a significant fraction of the pseudogap energy.


The identification of broken symmetry states, particularly in the pseudogap region, is essential for understanding the cuprate phase diagram. The surprising discovery of a small Fermi surface from quantum oscillations (QOs) in underdoped YBa2Cu3O6+x (Y123) (ref. 1) motivated proposals for a crystal lattice symmetry-breaking order parameter2,3,4,5,6,7,8,9 that reconstructs either the large Fermi surface identified in overdoped cuprates10,11,12 or the Fermi arcs of the pseudogap state13,14,15.

The spectrum of QOs is, in principle, a distinct probe of the Fermi surface morphology and is thus a signature of the broken symmetry state2,3,4,5,6,7,8,9. The ubiquity of short-range charge density wave (CDW) in underdoped cuprates16,17,18,19,20,21,22 make CDW a natural choice as the order responsible for Fermi surface reconstruction. However, despite the availability of exquisitely detailed QO studies in Y123 (refs 8, 23, 24, 25), its complicated multi-frequency spectrum has prevented a consensus on the exact model for reconstruction8,9,24,25,26,27,28. Part of the difficulty stems from the crystal structure of Y123, in particular the bilayer splitting of the elementary Fermi pockets due to the two CuO2 planes per unit cell. Different models are sensitive to the magnitude, symmetry and momentum dependence of the bilayer coupling8,26,27,28, which are controversial. Furthermore, neither diffraction nor QO experiments in the cuprates have yet been able to address the crucial question as to whether the two orthogonal CDW vectors spatially coexist in the same CuO2 plane or whether stripes alternate in a criss-cross manner on consecutive CuO2 planes29,30.

Apart from the Y-based bilayer compounds1,31, HgBa2CuO4+δ (Hg1201) is the only other hole-doped cuprate for which QOs have been detected32 in the pseudogap regime. Importantly, in addition to featuring a very high-Tc (≈97 K at optimal doping), Hg1201 has a tetragonal crystal symmetry consisting of only one CuO2 plane per primitive unit cell. This means that the analysis of the experimental data on this compound is free from complications associated with bilayer coupling and orthorombicity.

Here we show that high-resolution measurements of up to ten cycles of the QOs in Hg1201 permit a resolution of the reconstructed electronic structure. Using pulsed magnetic fields extending to 90 T combined with contactless resistivity measurements, we find the QOs in Hg1201 to be remarkably simple featuring: a single oscillation frequency with a monotonic magnetic field dependence characteristic of a single Fermi surface pocket. We find quantitative agreement between the observed single QO frequency and that from a diamond-shaped electron pocket resulting from biaxial CDW reconstruction18,33. There are no signatures of the predicted additional small hole-like pocket9 reported for Y123 (ref. 25). This could be due to the antinodal states, which constitute these hole pockets, being gapped out or strongly supressed by the pseudogap phenomena. We also determine a very small c axis transfer integral for Hg1201, which precludes a model based on an alternating criss-cross pattern of uniaxial charge stripes on consecutive CuO2 planes30. The absence of signatures of magnetic breakdown tunnelling to neighbouring sections of the Fermi surfaces (such as the putative small hole pockets9) provides a lower bound estimate of ≈20 meV for the relevant gap. Importantly, this is a significant fraction of the anti-nodal pseudogap energy34. Overall, our results point to biaxial CDW reconstruction acting on the short nodal Fermi arcs produced by the pseudogap phenomena.


QO measurements in pulsed magnetic fields

The typical sample quality and magnetic field requirements for observing QOs is exponentially dependent on the condition ωcτ≈1, where ωc=eB/m* is the cyclotron frequency and 1/τ is the scattering rate. For the Hg1201 samples studied here, ωcτ≈0.35 at B=45 T on average. Thus, compared with Y123, which has ωcτ≈1.2 (ref. 35) at the same field, QO measurements in Hg1201 are much more challenging. To overcome this, we use a high-sensitivity contactless resistivity method, wherein the sample forms part of a proximity detector oscillator circuit36, and extremely large magnetic fields. Changes Δf in the proximity detector oscillator circuit frequency f in an applied magnetic field B are directly related to the changes in the complex penetration depth of the sample37 and hence the in-plane resistivity. We focus on hole doping where a plateau in the Tc dome occurs, which is also the region where detailed QO measurements in Y123 have focused, as indicated in Fig. 1. Figure 2a shows Δf/f for an underdoped Hg1201 sample UD71 (Tc=71 K) in an applied magnetic field. The large increase in Δf/f at B≈35 T corresponds to the transition from the superconducting to the resistive state. A derivative of the data with respect to magnetic field clearly reveals QOs in the resistive state, without the need for background removal (see Fig. 2b).

Figure 1: Hole concentration of Hg1201 and Y123 samples.
Figure 1

Superconducting temperature, Tc(p), as a function of hole concentration p for Hg1201 (ref. 57) (black line, bottom x axes) and Y123 (ref. 58) (red line, top x axes). Tc(p) of Hg1201 UD71 and UD74 highlighted in the current study of the topology of Fermi-surface reconstruction are indicated by blue and black circles, respectively. The doping of the Y123 sample studied here is indicated by the red circle. Although QOs for Y123 have been reported over a wider range of hole concentrations59,60, detailed studies of the spectra have focused on the narrow range indicated by the thick red line8,23,24,25,61, corresponding to the plateau on the Tc(p) dome and where the amplitude of oscillations is largest.

Figure 2: Observation of QOs in Hg1201 with contactless resistivity.
Figure 2

(a) Evolution of the proximity detector oscillator (PDO) circuit frequency coupled to Hg1201 UD71 with applied magnetic field B along the c axis of the sample at T=1.8 K. The sample undergoes a transition from superconducting (SC, black-shaded region) to normal (blue region) at B35 T. (b) Derivative of the raw data with respect to magnetic field reveals QOs in the normal state. As described in the text, a non-oscillatory polynomial background is subtracted from the raw data to extract the QOs. The derivative of the background is shown as the dashed black line. (c) QOs after the polynomial background has been removed. The dashed black line is a fit to the Lifshitz–Kosevitch form discussed in Methods.

Single QO frequency in Hg1201

Figure 2c shows QOs after removing the background as described in Methods. The dashed line is a good fit of the data (solid line) to the expected QO waveform for a single quasi-two-dimensional Fermi surface with no warping and no magnetic breakdown tunnelling (see equation (1) in Methods). Warping and magnetic breakdown introduces other frequency components manifest as a beat or non-monotonic amplitude modulation, which are absent in our data. In Fig. 3a, we show high-resolution data obtained by averaging multiple magnetic field shots for UD71 and an additional Hg1201 sample of slightly higher Tc (Tc=74 K, labelled UD74). Seven and ten full oscillations are resolved for UD71 and UD74, respectively. For both samples, the observed QOs are well captured by the single frequency fit (dotted lines in Fig. 3a), which yields oscillation frequencies of F=847(15) T and 893(15) T for UD71 and UD74, respectively.

Figure 3: Spectrum of QOs in Hg1201 and Y123.
Figure 3

(a) Percentage change of the frequency Δf/f as a function of inverse applied field 1/B for proximity detector oscillator (PDO) circuits coupled to Y123 with x=0.58 (Tc=60 K, red), Hg1201 UD74 (Tc=74 K; black) and Hg1201 UD71 (Tc=71 K; blue). The small amplitude oscillations for Hg1201 UD74 at low fields (large 1/B) are magnified × 8 for clarity. Single-frequency fits to the data with equation (1) (see Methods) are indicated by dashed lines. (b) The residual after subtracting the single-frequency fit from the data in a. The smaller vertical scale of the axis is noteworthy. (ce) Fourier transform for Y123 with x=0.58, Hg1201 UD74 and Hg1201 UD71. The solid lines are FFT of the data and the dark-shaded regions represent the FFT of the single-frequency fits in a. The light-shaded regions are the FFT of the residual in b.

To set limits on the amplitude of additional QO frequencies, we determine the residuals by subtracting the single-frequency fits from the data in Fig. 3b. The residuals for both Hg1201 samples do not show evidence for additional oscillatory components. On further comparing the Fourier transform of the data with the Fourier transform of the fit (dark-shaded regions in Fig. 3d,e), both can be seen to have the same line shape, thereby providing further evidence for the absence of additional QO frequencies. The Fourier transform of the residuals (light-shaded regions in Fig. 3d,e) are devoid of prominent peaks, consistent with it representing the noise floor of the experiment. From the noise floor, we can infer that in order for additional QO frequencies to go undetected, they must fall below ≈8% of the Fourier amplitude of the observed QO frequency. It is tempting to attribute the weak modulating background we observe in the derivative of the raw data in sample UD71 (Fig. 2b), to the additional low frequency oscillation previously reported for Y123 (ref. 25). However, as discussed in Supplementary Note 1 and Supplementary Figs 1 and 2, the lack of such a modulation in sample UD74 and the disappearance of the modulation in UD71 at slightly elevated temperatures (in contradiction with the light mass of the small pocket reported for Y123 (ref. 25)), leads us to conclude that this is a background effect unrelated to slow QOs.

Our finding of a single frequency in Hg1201 contrasts significantly with the multiple frequencies present in Y123 (refs 23, 39) evident from the data with the same number of oscillation periods as for Hg1201 UD74 (see Fig. 3). In contrast to the residuals obtained for Hg1201, the residual for Y123 (see Fig. 3b), again obtained on subtracting a fit to the dominant QO frequency (of F≈530 T), reveals a distinctive beat pattern resulting from the interference between the two remaining QO components whose amplitudes are ≈40 and 50% of the dominant frequency (FT of the residual shown Fig. 3c).

Limits on the Fermi surface warping and c axis hopping

In quasi-two-dimensional metals, the inter-plane hopping leads to warping of the cylindrical Fermi surface, yielding two oscillation frequencies originating from minimum and maximum extremal cross-sections. Although our observation of a single QO frequency rules out very large warping, small warping can manifest in observable nodes in the magnetic field-dependent QO amplitude. This is represented by an additional amplitude factor, Rw, which is parametrized by the separation between the two frequencies 2ΔFc (see Methods). To illustrate this point, in Fig. 4 we fit the data with several different fixed values of ΔFc in Rw. The absence of nodes in the experimental data enables us to make an upper bound estimate of ΔFc<16 T. Using m*≈2.7me (see Fig. 5) and , we obtain a c axis hopping of t<0.35 meV for Hg1201, revealing it to be at least 1,000 times smaller than the nearest-neighbour hopping (t=460 meV40) within the CuO2 planes. Our upper bound is also 25 times smaller than the bare value determined from local density approximation (LDA) calculations (t=10 meV40), reflecting a large quasiparticle re-normalization.

Figure 4: Effect of warping on the QOs.
Figure 4

Influence of two-dimensional Fermi surface warping on fitting QOs in Hg1201. (a) Fit (blue line) of the QO data (grey line) for UD74 with no warping (ΔFc=0). The dotted line represents the 1/B dependence in arbitary units of the warping term (see Methods). The degree of warping is increased for b and c, yielding nodes in the QO spectra corresponding to 1/B values where Rw changes sign. The small amplitude oscillations at large 1/B, bordered by the grey box, is magnified by a factor of 15 in all three panels and shown as insets.

Figure 5: Determination of the effective mass.
Figure 5

Relative proximity detector oscillator (PDO) circuit frequency change Δf/f as a function of field B for sample Hg1201 UD71 at the temperatures indicated above each curve. Solid lines are simultaneous fits with equation (1) (see Methods) to the total data set, where all parameters are constrained to be temperature independent. For the single-frequency spectrum of Hg1201, this method produces a more robust determination of the effective mass than examining the temperature dependence of the FFT amplitude. The effective mass is extracted do be , where me is the bare electron mass. This is slightly larger than that determined for a sample with almost the same doping32.

Our ability to set a firm upper bound estimate for t in Hg1201 contrasts with the situation in Y123, where estimates of the c axis warping are challenging to separate from the effects of bilayer coupling. Estimates for ΔFc in Y123 range from ≈15 to 90 T depending on whether the observed beat pattern originates from the combined effects of bilayer-splitting and magnetic breakdown tunnelling8 or Fermi surface warping23,24,25.

Fermi surface reconstruction by biaxial CDW

The simple crystalline structure of Hg1201 makes it the ideal system for relating the k-space area of the observed Fermi surface pocket to prior photoemission41 and X-ray scattering18 measurements. The former constrains the unreconstructed Fermi surface, whereas the later provides the magnitude of the reconstruction wave vector. Following Allais et al.9, we require that the large unreconstructed hole-like Fermi surface of area AUFS accommodate 1+p carriers, where p is the hole doping defined relative to the half-filled band. We then proceed to translate the Fermi surface multiple times by the wavevectors (QCDW,0) and (0,QCDW), and their combinations in Fig. 6a. Here we have assumed a biaxial reconstruction scheme. Further details of the calculation is described in the Methods section.

Figure 6: Fermi surface reconstruction by CDW.
Figure 6

Unreconstructed (grey lines) and reconstructed (colored lines) Fermi surface of HgBa2CuO4+δ (p=0.095) (a) and YBa2Cu3O6+x (p=0.11) (b). The single-frequency QO we observe for Hg1201 is in agreement with the area of the electron-like diamond-shaped pocket (solid red), whereas there are no signatures of the small hole-like pocket (dashed dark blue) in our data. This might be due to the lack of quasiparticle weight in the pseudogapped antinodal regions of the Fermi-surface determined from angle-resolved photoemission13,14,15. For the reconstruction, we assume biaxial CDW wavevectors (QCDW,0) and (0,QCDW). QCDW=0.275 r.l.u. for Hg1201 (Tc=72 K) was taken from ref. 18. QCDW=0.325 for Y123 is estimated based on measurements on Y123 with x=0.54 (ref. 62) and Y123 with x=0.55 (ref. 63). The colour plots in the upper right edges of the panels are photoemission data showing the Fermi surface map for Hg1201 (p≈0.12) (adapted with permission from ref. 41, copyrighted by the American Physical Society) and Y123 (p=0.11) (adapted by permission from Macmillan Publishers Ltd: Nature (ref. 12), copyright (2008)). The points on top of the angle-resolved photoemission data in a are Fermi surface crossings. The tight-binding hopping parameters were determined by fitting the photoemission data, while constraining the area of the Fermi surface to match the quoted hole concentrations. Dashed blue lines indicate the antiferromagnetic zone boundaries.

The biaxial CDW reconstruction (shown in Fig. 6) yields a diamond-shaped electron pocket (depicted in red) flanked by smaller hole pockets (depicted in blue) accompanied by additional open Fermi surface sheets (shown in Supplementary Fig. 3). Although the parameters for Hg1201 are slightly different than for Y123, the topology of the reconstructed Fermi surface is essentially the same.

Using the Onsager relation , where Ae is the area of the electron pocket and Ah is the area of the hole pocket, the calculated QO frequencies are Fe=885 T and Fh=82 T for the electron and hole pocket, respectively. Fe is remarkably close to that observed for Hg1201: 847 and 893 T for UD71 and UD74, respectively. However, Fh is not observed in our experiment (see Supplementary Note 1 and Supplementary Figs 1 and 2).

If we instead consider a purely uniaxial reconstruction, that is, stripe CDW order, our calculation with the same parameters for Hg1201 yields an oval-shaped hole-like pocket at the anti-nodal regions of the original Fermi surface with a frequency of 590 T and no futher closed pockets (see Supplementary Fig. 4). This is in much poorer agreement to the measured QO frequency and furthermore disagrees with Hall effect measurements, which imply the existence of a predominantly electron-like reconstructed Fermi surface38.

In Fig. 6b we show the same calculation for Y123 with x=0.58 (p≈0.106). The agreement between the measured dominant frequency (534 T) and calculated electron-pocket frequency Fe=380 T is less satisfactory. However, here (similar to Allais el al.9) we have neglected some of the complications in Y123 such as the bilayer coupling and orthorhombic crystal structure. The photoemission data are also made more difficult to interpret in Y123, owing to the necessity of surface K-doping to reach the desired hole doping of p≈11% (ref. 12). These uncertainties highlight the utility of studying the structurally simpler Hg1201.

Limits on magnetic breakdown tunnelling across band gaps

It was recently proposed by Allais et al.9 that a sufficiently large magnetic field enables magnetic breakdown tunnelling to occur between the electron and hole pockets shown in Fig. 6, providing a possible explanation for one or more of the observed cluster of three frequencies in Y123 (ref. 25). The probability of magnetic breakdown tunnelling increases with magnetic field, giving rise to new orbits and a reduction in the elementary QO amplitudes by . Here, and are the tunnelling and reflection amplitudes, whereas np and nq are the number of breakdown tunnelling and Bragg reflection events encountered en route around the orbit, respectively42. The magnetic breakdown probability is given by P=exp{−B0/B}, where is the characteristic breakdown field, in which is the approximate Fermi energy and Eg is the band gap separating adjacent sections of Fermi surface.

Magnetic breakdown can manifest itself in two ways in our data. The first is by way of a reduction of the primary QO amplitude at higher magnetic fields. For the diamond-shaped electron pocket in Fig. 6, the amplitude is reduced by to account for the four Bragg reflection points at the tips of the diamond. In a Dingle plot of the magnetic field dependence of the QO amplitude (Fig. 7), this should be discerned as deviations from a straight line for small 1/B. Accordingly, we fit the Dingle plot to where (solid lines in Fig. 7b), yielding B0≈200 T and ≈250 T for UD74 and UD71, respectively. These large values for B0 are consistent with no observable effects of magnetic breakdown (that is, a straight-line Dingle plot); therefore, we take the fitted B0 as lower bound values.

Figure 7: Dingle plot.
Figure 7

Dingle plot of the QO amplitude normalized by RT as a function of 1/B for UD71 and UD74. Amplitude of maxima and minima are taken from the data in Fig. 3a. Solid lines are fits to the data as described in the text.

The second manifestation of magnetic breakdown is through the appearance of new QO frequencies corresponding to sums and differences of the areas of the Fermi surfaces involved in the tunnelling process. Magnetic breakdown between the electron and hole pocket9,25 in our reconstruction model results in additional QO frequencies of the form Fe-h,n=FenFh, in which n is an integer. Based on our modelling, Fh≈80 T, meaning that the frequencies Fe−h,n are sufficiently distinct from Fe to be discernible in the raw and Fourier-transformed data in Fig. 3b,d,e. The noise floor of ≈8% of the dominant oscillation amplitude Ae provides an upper limit for the amplitude Ae−h,1, the leading magnetic breakdown frequency (n=1). Using the inequality

we obtain a second lower bound of B0≈200 T. Here we have assumed a similar m* and scattering rate values for the various combination orbits, whereas the factor of two for Ae−h accounts for the two possible orbits involving one of the two hole-like pockets in Fig. 6a.

We have shown above that both the Dingle plot and the absence of additional Fourier peaks above the noise floor provide mutually consistent large lower bound estimates for B0. The most conservative of these (that is, B0 200 T) enables a lower bound estimate of Eg 20 meV to be made for the band gap between the observed electron and presumed hole pockets in Fig. 6.


Our observation of a simple monotonic waveform of a single QO frequency in Hg1201 and a single Fermi surface cross-sectional area that is compatible with photoemission and X-ray scattering measurements are essential for resolving issues relating to the nature of the CDW ordering. One of these concerns is whether the two charge-ordering wavevectors (QCDW,0) and (0,QCDW) coexist in the same CuO2 plane9,28,33 or whether stripes alternate in a criss-cross manner on consecutive CuO2 planes29,30. In the absence of a coupling between CuO2 planes, criss-cross stripes lead to open Fermi surface sheets running in orthogonal directions on adjacent planes. The effect of the inter-plane coupling is to introduce a hybridization30. Whereas a strong coupling in the range 10–100 meV occurs within the bilayers in Y123 and Y124 (ref. 43), no such coupling occurs in single-layer Hg1201 and only a very weak coupling provided by the interlayer c axis hopping determined here to be tc<0.4 meV can exist. In the context of criss-cross stripe order, the effect of such a weak c axis hopping is to introduce a very small gap of order tc in magnitude between the electron and hole pockets in Fig. 6, which would then have a very small characteristic magnetic breakdown field of B00.1 T (several orders of magnitude smaller than the lower bound constraint on B0 determined from our experimental results). The magnetic breakdown amplitude reduction factor RMB for B0=0.1 T would be so small that it would render the electron pocket not observable in experimentally relevant magnetic fields of  40 T. Our observations of a single electron pocket and small c axis hopping therefore rule out criss-cross stripes as a viable route for creating observable Fermi surface pockets in Hg1201 at high magnetic fields.

Although the CDW correlations detected with X-ray scattering are a natural candidate for the cause of Fermi-surface reconstruction, an open question concerns whether the correlation length is sufficiently large to support QOs. A small correlation length of the order parameter can manifest as additional damping of the QO amplitude in the Dingle term (see Methods), thus suppressing the effective mean free path l44. For Y123, the CDW correlation length at Tc and B=0 T is ξCDW≈65 Å16. The effective mean free path, l≈200 Å35 obtained from QO measurements is of the same order of magnitude as ξCDW. For Hg1201, both ξCDW and l are similarly reduced compared with Y123: ξCDW≈20 Å18 at TTc and l=85 Å (average of UD71 and UD74 and consistent with prior Hg1201 results32). Thus, it appears that the QO effective mean free path might be correlated with the CDW domain size. Alternatively, both l and ξCDW could be similarly affected by disorder or impurities. The relatively small l for Hg1201 indicates that the CDW need not be long ranged, even at low temperatures and high magnetic fields, to yield the reconstructed Fermi surface observed here. Although ξCDW in Y123 increases at low temperatures and high magnetic fields, it remains rather small (≈100–400 Å)45,46.

Another issue concerns the origin of the Eg20 meV gap separating the diamond-shaped electron pocket from adjacent sections of Fermi surface in Hg1201. There are two possible CDW Fermi surface reconstruction scenarios that have been discussed in the literature. One of these involves the folding of the large Fermi surface9, as shown in Fig. 6, which is expected to produce small hole pockets and open sheets in addition to the observed electron pocket. In such a scenario, Eg would then simply correspond to the CDW gap 2ΔCDW. The alternative scenario is that the reconstructed Fermi surface occurs by connecting the tips of Fermi arcs produced by a pre-existing or coexisting pseudogap state18,20. In this scenario, we would expect the small hole pockets to be gaped out by the pseudogap causing Eg, then to correspond to the pseudogap energy. Two observations suggest the latter scenario to be more applicable to the underdoped cuprates. First, we find no evidence for QOs originating from the hole pocket, either by direct observation or by way of magnetic breakdown combination frequencies. Second, the Fermi arc, which refers to the region in momentum space over which the photoemission spectral weight is strongest, is seen to be very similar in length to the sides of the electron pocket in Fig. 6a,b (for both Hg1201 and Y123). The spectral weight drops off precipitously beyond the tips of the pocket. We note that although low-frequency QOs in Y123 have been attributed to small hole pockets, this low frequency could also originate from Stark quantum interference effects associated with bilayer splitting8. Alternatively, the pseudogap phenomena could also menifest as strong scattering at the antinodal regions, thus preventing an observation of such pockets in Hg1201. Recent high-temperature normal-state transport measurements in Hg1201 have also been interpreted in terms of Fermi-liquid-like47,48 Fermi arcs49.

Our findings in Hg1201 have direct implications for the interpretation of QO measurements made in other cuprate materials. If we assume a similar gap size between Hg1201 and Y123, the large Eg suggests that magnetic breakdown combination frequencies involving the electron and small hole pocket9,25 cannot be responsible for the complicated beat pattern associated with closely spaced frequencies in Y123 (ref. 8, 23) and Y124 (ref. 50). The splitting of the main frequency into two or more components must therefore be the consequence of the bilayer coupling in those systems8,28,50 or a stronger interlayer c axis hopping.

The biaxial reconstruction confirmed here for Hg1201 has also been proposed for Y123 (refs 8, 26, 28), which is supported by ultrasound measurements in high fields51. However, X-ray measurements show apparent local-stripe CDW domains at high temperatures52, which presumably become long ranged and possibly arranged in a criss-cross pattern of stripes at low temperatures and high fields27,30. Recent X-ray measurements on Y123 show a new magnetic field induced three-dimensional CDW centred at c axis wave vector L=1 r.l.u.45 only along the CuO chain directions46, which breaks the mirror symmetry of the CuO2 bilayers. The role of bilayer coupling and CuO chains for this stripe-like ordering tendency is still an open question and its relevance for Hg1201, which features neither, is unclear. The stripe picture is attractive because of natural analogies to single-layered La-based cuprates53 and its implications for the role of nematicity (broken planar rotational symmetry) for the cuprate phase diagram54. However, neutron-scattering experiments have found that the typical signatures of spin stripes are absent in the magnetic excitations of Hg1201 (ref. 55). Despite the appearance of a new uniaxial three-dimensional order in Y123, the CDW wavevector, with a smaller c axes correlation length, is still clearly observed in both planar directions in magnetic fields up to 17 T (ref. 46). For Hg1201, we have shown here that the CDW that causes the Fermi-surface reconstruction is biaxial. It thus remains an open question as to whether electronic nematicity is generic to the cuprates, particularly in tetragonal Hg1201.



Hg1201 single crystals were grown using a self-flux method56. As grown crystals have Tc≈80 K. Post-growth heat treatment in N2 atmosphere at 400 °C and 450 °C was used to achieve Tc=74 K (hole concentration p=0.097) and Tc=71(2) K (p=0.09), respectively. Tc was determined from constant-field (DC) susceptibility measurements. The 95% level transition width of both samples is 2 K. The hole concentration p is determined based on the phenomenological Seebeck coefficient scale57.

The YBCO crystal was flux grown and heat treated to obtain oxygen content x=0.58 with Tc=60 K and hole doping p=0.106 at the University of British Columbia, Canada58.

Pulsed field measurements

High magnetic field measurements were performed at the Pulsed-Field Facility at Los Alamos National Laboratory. The magnet system used consists of an inner and outer magnet. The outer magnet is first generator driven relatively slowly (3 s total width) between 0 and 37 T, followed by a faster (15 ms) capacitor bank driven pulse to 90 T.

Fitting QOs

We fit the field dependence to , where the first term is a polynomial representing the non-oscillatory background and Aosc is the oscillatory component. In the case of a single Fermi surface cylinder, the QOs are described by the Lifshitz–Kosevitch form42

where F is the frequency of QOs, γ is the phase and A0 is a temperature- and field-independent pre-factor. Here, RT, RD, RS, RMB and RW are the thermal, Dingle, spin, magnetic breakdown and warping damping factors, respectively8,24. RT=αT/[Bsinh(αT/B)] where accounts for the thermal broadening of the Fermi–Dirac distribution relative to the cyclotron energy and m*=2.7 me, determined for one of our samples as shown in Fig. 5, is the quasiparticle effective mass (me being the free electron mass). Meanwhile, RD=exp(−πlc/l), where is the cyclotron radius and l is the mean free path. To lowest order, warping of a cylindrical Fermi surface leads to an amplitude reduction factor of the form Rw=J0(2πΔFc/B) in which J0 is a zeroth-order Bessel function and is the difference in frequency between the minimum and maximum cross-sections of the warped cylinder. As our experiments are performed at fixed angle (that is, B || c), we neglect RS by setting it to unity. As discussed in the main text, our data shows no signatures of magnetic breakdown tunnelling or warping; thus, we also set RMB and RW to unity. Limits on these two terms are discussed in the Results section.

Calculation of reconstructed Fermi surface

The unreconstructed Fermi surface is calculated with the dispersion where the tight-binding parameters are 40 for Hg1201 and (0.35, −0.112, 0.007 and 0) eV for YBCO. μ is the chemical potential, and and where kx and ky are the planar wavevectors. We required that the tight-binding parameters produce a Fermi surface in agreement with the photoemission data and have carrier number 1+p where p=0.12 and 0.11 for the Hg1201 and Y123 samples, respectively, on which the photoemission data were taken. Hence, 1+p=2AUFS/AUBZ, where AUFS and AUBZ are the areas of the unreconstructed Fermi surface and Brillouin zone, respectively. Before calculating the reconstructed Fermi surface, only μ is adjusted to match the hole doping p=0.095 and p=0.106 on which the QO data were taken for Hg1201 and Y123, respectively.

Following ref. 33, the reconstructed Fermi surface is determined by diagonalizing a Hamiltonian considering translations of the biaxial CDW wavevector , where nx and ny are the number of translations in the planar directions. Strictly speaking, reconstruction by observed incommensurate CDW wavevectors requires an infinite number of terms in the Hamiltonian to obtain all the bands. However, as Δ<<t, the inclusion of high-order terms in the Hamiltonian gives rise to a hierarchy of higher-order gaps that are exponentially small and thus do not effect the primary closed orbits resulting from our calculation, which we restrict to nine terms. Supplementary Fig. 3 shows all the bands resulting from our reconstruction calculation.

We use ΔCDW/t=0.1 for the ratio of the CDW order parameter magnitude to the in-plane hopping9. This implies ΔCDW=46 meV, based on band structure determination of t (ref. 40), which is larger than the lower bound value determined from our analysis of magnetic breakdown tunnelling in the main text, but sufficiently small that it does not adversely affect the sizes of the pockets. Reducing the ratio to zero increases the area of the reconstructed pockets by only ≈3%.

Data availability

The data that support the findings of this study are available from the corresponding author upon request.

Additional information

How to cite this article: Chan, M. K. et al. Single reconstructed Fermi surface pocket in an underdoped single-layer cuprate superconductor. Nat. Commun. 7:12244 doi: 10.1038/ncomms12244 (2016).


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The work performed at Los Alamos National Laboratory, was supported by the US Department of Energy BES ‘Science at 100T’ grant number LANLF100. The National High Magnetic Field Laboratory - PFF facility is funded by the National Science Foundation Cooperative Agreement Number DMR-1157490, the State of Florida and the U.S. Department of Energy. Work at the University of Minnesota was supported by the Department of Energy, Office of Basic Energy Sciences, under Award Number DE-SC0006858. N.B. acknowledges the support of FWF project P2798. We thank Ruixing Liang, W.N. Hardy and D.A. Bonn at UBC, Canada, for generously supplying the Y123 crystal measured as part of this work. We aknowledge fruitful discussion with S.E. Sebastian. We also thank the Pulsed Field Facility, Los Alamos National Lab engineering and technical staff for experimental assistance.

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  1. Pulsed Field Facility, National High Magnetic Field Laboratory, Los Alamos National Laboratory, Mail Stop E536, Los Alamos, New Mexico 87545, USA

    • M. K. Chan
    • , N. Harrison
    • , R. D. McDonald
    • , B. J. Ramshaw
    •  & K. A. Modic
  2. School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA

    • M. K. Chan
    • , N. Barišić
    •  & M. Greven
  3. Technische Universität Wien, Wiedner Haupstrasse 8-10, 1040 Vienna, Austria

    • N. Barišić


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M.K.C., N.H., R.D.M., B.J.R., K.A.M. and N.B. performed contactless resistivity measurements in pulsed magnetic fields on Hg1201 and Y123. M.K.C. designed the experiment and analyzed the data. M.K.C. synthesized and prepared the Hg1201 samples. N.H. supervised the work at Los Alamos National Lab. M.G. supervised the work at the University of Minnesota. M.K.C. and N.H. wrote the manuscript with critical input from all authors.

Competing interests

The authors declare no competing financial interests.

Corresponding authors

Correspondence to M. K. Chan or N. Harrison.

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