Single reconstructed Fermi surface pocket in an underdoped single layer cuprate superconductor
M. K. Chan, N. Harrison, R. D. McDonald, B. J. Ramshaw, K. A. Modic, N. Barisic, M. Greven
aa r X i v : . [ c ond - m a t . s up r- c on ] J un Single reconstructed Fermi surface pocket in an underdoped single layer cupratesuperconductor
M. K. Chan,
1, 2, ∗ N. Harrison, ∗ R. D. McDonald, B. J. Ramshaw, K. A. Modic, N. Bariˇsi´c, and M. Greven Mail Stop E536, Pulsed Field Facility, National High Magnetic Field Laboratory,Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA Technishce Universit¨at Wein, Wiedner Haupstr. 8-10, 1040, Vienna, Austria
The observation of a reconstructed Fermi surface via quantum oscillations in hole-doped cupratesopened a path towards identifying broken symmetry states in the pseudogap regime. However, suchan identification has remained inconclusive due to the multi-frequency quantum oscillation spectraand complications accounting for bilayer effects in most studies. We overcome these impedimentswith high resolution measurements on the structurally simpler cuprate HgBa CuO δ (Hg1201),which features one CuO plane per unit cell. We find only a single oscillatory component withno signatures of magnetic breakdown tunneling to additional orbits. Therefore, the Fermi surfacecomprises a single quasi-two-dimensional pocket. Quantitative modeling of these results indicatesthat biaxial charge-density-wave within each CuO plane is responsible for the reconstruction, andrules out criss-crossed charge stripes between layers as a viable alternative in Hg1201. Lastly, wedetermine that the characteristic gap between reconstructed pockets is a significant fraction of thepseudogap energy. I. INTRODUCTION
The identification of broken symmetry states, particu-larly in the pseudogap region, is essential for understand-ing the cuprate phase diagram. The surprising discoveryof a small Fermi surface from quantum oscillations (QOs)in underdoped YBa Cu O x (Y123) [1] motivated pro-posals for a crystal-lattice-symmetry-breaking order pa-rameter [2–9] that reconstructs either the large Fermisurface identified in overdoped cuprates [10–12] or theFermi arcs of the pseudogap state [13–15].The spectrum of quantum oscillations is, in principle,a distinct probe of the Fermi surface morphology andis thus a signature of the broken symmetry state [2–9].The ubiquity of short-range charge-density-wave (CDW)in underdoped cuprates [16–22] make CDW a naturalchoice as the order responsible for Fermi surface recon-struction. However, despite the availability of exquisitelydetailed QO studies in Y123 [8, 23–25], its complicatedmulti-frequency spectrum has prevented a consensus onthe exact model for reconstruction [8, 9, 24–28]. Part ofthe difficulty stems from the crystal structure of Y123,particularly the bilayer splitting of the elementary Fermipockets due to the two CuO planes per unit cell. Thedifferent models are sensitive to the magnitude, sym-metry, and momentum dependence of the bilayer cou-pling [8, 26–28], which are controversial. Furthermore,neither diffraction nor QO experiments in the cuprateshave yet been able to address the crucial question as towhether the two orthogonal CDW vectors spatially coex-ist in the same CuO plane or whether stripes alternate ina criss-cross fashion on consecutive CuO planes [29, 30]. ∗ Correspondence to [email protected] and [email protected]
Apart from the Y-based bilayer compounds [1, 31],HgBa CuO δ (Hg1201) is the only other hole-dopedcuprate for which QOs have been detected [32] in thepseudogap regime. Importantly, in addition to featur-ing a very high- T c ( ≈
97 K at optimal doping), Hg1201has a tetragonal crystal symmetry consisting of only oneCuO plane per unit cell. This means that the analy-sis of the experimental data on this compound is freefrom complications associated with bilayer coupling andorthorombicity.Here we show that high resolution measurements ofup to 10 cycles of the QOs in Hg1201 permit a reso-lution of the reconstructed electronic structure. Usingpulsed magnetic fields extending to 90 T combined withcontactless resistivity measurements, we find the QOsin Hg1201 to be remarkably simple: a single oscilla-tion frequency exhibiting a monotonic magnetic field de-pendence characteristic of a single Fermi surface pocket.We find quantitative agreement between the observedsingle QO frequency and that from a diamond-shapedelectron pocket resulting from biaxial CDW reconstruc-tion [18, 33]. There are no signatures of the predicted ad-ditional small hole-like pocket [9] reported for Y123 [25].This could be due to the antinodal states, which con-stitute these hole pockets, being gapped out or stronglysupressed by the pseudogap phenomena. We also de-termine 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 consecutiveCuO planes [30]. The absence of signatures of magneticbreakdown tunneling to neighboring sections of the Fermisurfaces (such as the putative small hole pockets [9]) pro-vides a lower bound estimate of ≈
20 meV for the relevantgap. Importantly, this is a significant fraction of the anti-nodal pseudogap energy [34]. Overall, our results pointto biaxial CDW reconstruction acting on the short nodalFermi arcs produced by the pseudogap phenomena.
II. RESULTS
Quantum oscillation measurements in pulsedmagnetic fields.
The typical sample quality and mag-netic field requirements for observing QOs is exponen-tially dependent on the condition ω c τ &
1, where ω c = eB/m ⋆ is the cyclotron frequency and 1 /τ is the scatter-ing rate. For the Hg1201 samples studied here, ω c τ ≈ .
35 at B = 45 T on average. Thus, compared to Y123,which has ω c τ ≈ . f in the PDOcircuit frequency f in an applied magnetic field B are di-rectly related to the changes in the complex penetrationdepth of the sample [37] and hence the in-plane resis-tivity. We focus on hole-doping where a plateau in the T c dome occurs , which is also the region where detailedQO measurements in Y123 have focused, as indicated inFig. 1. Fig. 2(a) shows ∆ f /f for an underdoped Hg1201sample UD71 ( T c = 71 K) in an applied magnetic field.The large increase in ∆ f /f at B ≈
35 T corresponds tothe transition from the superconducting to the resistivestate. A derivative of the data with respect to magneticfield clearly reveals QOs in the resistive state, withoutthe need for background removal (see Fig. 2(b)).
Single quantum oscillation frequency inHg1201.
Figure 2(c) shows QOs after removing thebackground as described in Methods. The dashed lineis a good fit of the data (solid line) to the expectedQO waveform for a single quasi-two-dimensional Fermisurface with no warping and no magnetic breakdowntunneling (see Eq .1 in Methods). Warping and magneticbreakdown introduces other frequency components man-ifest as a beat or non-monotonic amplitude modulation,which are absent in our data. In Fig. 3a, we show highresolution data obtained by averaging multiple magneticfield shots for UD71 and an additional Hg1201 sample ofslightly higher T c ( T c = 74 K, labeled UD74). Seven andten full oscillations are resolved for UD71 and UD74,respectively. For both samples, the observed QOs arewell captured by the fit (dotted lines in Fig. 3(a)), whichyields oscillation frequencies of F = 847(15) T and893(15) T for UD71 and UD74.In order to set limits on the amplitude of additionalQO frequencies, we determine the residuals by subtract-ing the single frequency fits from the data in Fig. 3b. Theresiduals for both Hg1201 samples do not show evidencefor additional oscillatory components. On further com-paring the Fourier transform of the data with the Fouriertransform of the fit (dark shaded regions in Fig. 3d and3e), both can be seen to have the same line shape, thereby Hg1201
Hg1201 Hole Concentration T e m pe r a t u r e ( K ) Y123
Y123 Hole Concentration
FIG. 1.
Hole concentration of Hg1201 and Y123 sam-ples.
Superconducting temperature , T c ( p ), as a function ofhole concentration p for Hg1201 [38] (black line, bottom x-axes) and Y123 [39] (red line, top x-axes). T c ( p ) of Hg1201UD71 and UD74 highlighted in the current study of the topol-ogy of Fermi-surface reconstruction are indicated by blue andblack circles respectively. The doping of the Y123 samplestudied here is indicated by the red circle. Although quantumoscillations for Y123 have been reported over a wider rangeof hole concentrations [40, 41], detailed studies of the spectrahave focused on the narrow range indicated by the thick redline [8, 23–25, 42], corresponding to the plateau on the T c ( p )dome and where the amplitude of oscillations is largest. providing further evidence for the absence of additionalquantum oscillation frequencies. The Fourier transformof the residuals (light shaded regions in Fig. 3d and 3e)are devoid of prominent peaks, consistent with it repre-senting the noise floor of the experiment. From the noisefloor, we can infer that in order for additional quantumoscillation frequencies to go undetected, they must fallbelow ≈
40 60 8000.10.20.3 ∆ f ( M H z ) HgBa CuO δ a SC Normal50 60 70 80 90050010001500 d ∆ f/ d B ( H z T - ) b
50 60 70 80 90 (cid:1) B (T) ∆ f/f ( % ) c FIG. 2.
Observation of quantum oscillations inHg1201 with contactless resistivity. (a) Evolution ofthe PDO circuit frequency coupled to Hg1201 UD71 withapplied magnetic field B along the c axis of the sample at T = 1 . B ∼
35 T. (b) Derivative of the raw data with respectto magnetic field reveals quantum oscillations in the normalstate. As described in the text, a non-oscillatory polynomialbackground is subtracted from the raw data to extract thequantum oscillations. The derivative of the background isshown as the dashed black line. (c) Quantum oscillations afterthe polynomial background has been removed. The dashedblack line is a fit to the Lifshitz-Kosevitch form discussed inMethods.
Figs. 3). In contrast to the residuals obtained for Hg1201,the residual for Y123 (see Fig. 3b), again obtained onsubtracting a fit to the dominant quantum oscillation fre-quency (of F ≈
530 T), reveals a distinctive beat patternresulting from the interference between the two remain-ing 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 -axishopping . In quasi-two-dimensional metals, the inter-plane hopping leads to warping of the cylindrical Fermisurface, yielding two oscillation frequencies originatingfrom minimum and maximum extremal cross-sections.While our observation of a single quantum oscillationfrequency rules out very large warping, small warpingcan manifest in observable nodes in the magnetic field-dependent QO amplitude. This is represented by an ad-ditional amplitude factor, R w , which is parametrized bythe separation between the two frequencies 2∆ F c (seeMethods). To illustrate this point, in Fig. 4 we fit thedata with several different fixed values of ∆ F c in R w .The absence of nodes in the experimental data enablesus to make an upper bound estimate of ∆ F c <
16 T.Using m ∗ ≈ . m e (see Fig.5) and 2∆ F c ≈ t ⊥ m ∗ / ( ~ e ),we obtain a c -axis hopping of t ⊥ < .
35 meV for Hg1201,revealing it to be at least 1000 times smaller than thenearest neighbor hopping ( t = 460 meV [45]) within theCuO planes. Our upper bound is also 25 times smallerthan the bare value determined from LDA calculations( t ⊥ = 10 meV [45]), reflecting a large quasiparticle re-normalization.Our ability to set a firm upper bound estimate for t ⊥ in Hg1201 contrasts with the situation in Y123, where es-timates of the c -axis warping are challenging to separatefrom the effects of bilayer coupling. Estimates for ∆ F c in Y123 range from ≈
15 to 90 T depending on whetherthe observed beat pattern originates from the combinedeffects of bilayer-splitting and magnetic breakdown tun-neling [8] or Fermi surface warping [23–25].
Fermi surface reconstruction by biaxial CDW.
The simple crystalline structure of Hg1201 makes it theideal system for relating the k -space area of the observedFermi surface pocket to prior photoemission [46] and x-ray scattering [18] measurements. The former constrainsthe unreconstructed Fermi-surface while the later pro-vides the magnitude of the reconstruction wave-vector.Following Allais et al. [9], we require that the large un-reconstructed hole-like Fermi surface of area A UFS ac-commodate 1 + p carriers, where p is the hole dopingdefined relative to the half filled band. We then pro-ceed to translate the Fermi surface multiple times by thewavevectors ( Q CDW ,
0) and (0 , Q
CDW ) and their combi-nations in Fig. 6a. Here we have assumed a biaxial re-construction scheme. Further details of the calculation isdescribed in the Methods section.The biaxial CDW reconstruction (shown in Fig. 6)yields a diamond-shaped electron pocket (depicted inred) flanked by smaller hole pockets (depicted in blue)accompanied by additional open Fermi surface sheets(shown in Supplementary Figure 3). While the parame-ters for Hg1201 are slightly different than for Y123, thetopology of the reconstructed Fermi surface is essentiallythe same.Using the Onsager relation F e , h = ~ πe A e , h , where A e is the area of the electron pocket and A h is the area of (T -1 ) ∆ f/f ( % ) Hg1201 UD74x 8Hg1201 UD71 a YBCO6.58 ∆ f/f ( % ) (T -1 ) 0.012 0.017 0.022-0.100.10.20.3 (T -1 ) ∆ f/f ( % ) Hg1201 UD74Hg1201 UD71 b Residual ∆ f/f ( % ) YBCO6.58 (T -1 ) 012 c YBCO6.58 A m p li t ude ( a r b ) d Hg1201 UD74 e Hg1201 UD71
FIG. 3.
Spectrum of quantum oscillations in Hg1201 and Y123 (a) Percentage change of the resonance frequency∆ f/f as a function of inverse applied field 1 /B for PDO circuits coupled to Y123 with x = 0.58 ( T c = 60 K, red), Hg1201UD74 ( T c = 74 K; black) and Hg1201 UD71 ( T c = 71 K; blue). The small amplitude oscillations for Hg1201 UD74 at low fields(large 1 /B ) are magnified × x = 0.58, Hg1201 UD74, and Hg1201 UD71. The solid lines are FFT of the dataand the dark shaded regions represent the FFT of the single-frequency fits in (a). The light shaded regions are the FFT of theresidual in (b). the hole pocket, the calculated QO frequencies are F e =885 T and F h = 82 T for the electron and hole pocket. F e is remarkably close to that observed for Hg1201: 847 Tand 893 T for UD71 and UD74, respectively. However, F h is not observed in our experiment (see SupplementaryNote 1 and Supplementary Figures 1&2).If we instead consider a purely uniaxial reconstruction,i.e. stripe CDW order, our calculation with the sameparameters for Hg1201 yields an oval-shaped hole-likepocket at the anti-nodal regions of the original Fermi sur-face with a frequency of 590 T and no futher closed pock-ets (see Supplemental Figure 4) . This is in much pooreragreement to the measured QO frequency, and further-more, disagrees with Hall effect measurements which im-ply the existence of a predominantly electron-like recon-structed Fermi surface [43].In Fig. 6(b) we show the same calculation for Y123with x = 0.58 ( p ≈ . F e = 380 T is less satisfactory.However, here (like Allais el al. [9]) we have neglectedsome of the complications in Y123 such as the bilayer-coupling and orthorhombic crystal structure. The pho- toemission data are also made more difficult to interpretin Y123 owing to the necessity of surface K doping toreach the desired hole doping of p ≈
11 % [12]. Theseuncertainties highlight the utility of studying the struc-turally simpler Hg1201.
Limits on magnetic breakdown tunneling acrossband gaps.
It was recently proposed by Allais et al. [9]that a sufficiently large magnetic field enables magneticbreakdown tunneling to occur between the electron andhole pockets shown in Fig. 6, providing a possible expla-nation for one or more of the observed cluster of threefrequencies in Y123 [25]. The probability of magneticbreakdown tunneling increases with magnetic field, giv-ing rise to new orbits and a reduction in the elementaryQO amplitudes by R MB = ( i √ P ) n p ( √ − P ) n q . Here, i √ P and √ − P are the tunneling and reflection am-plitudes respectively, while n p and n q are the numberof breakdown tunneling and Bragg reflection events en-countered en route around the orbit [49]. The magneticbreakdown probability is given by P = exp {− B /B } ,where B = ( m ∗ / ~ e )( E /E F ) is the characteristic break-down field, in which E F ≈ ~ eF e /m ∗ is the approximateFermi energy and E g is the band gap separating adjacent (cid:0) ∆ f/f ( % ) a R w ∆ F = 0 T (cid:2) ∆ f/f ( % ) b ∆ F = 18 T (cid:3) (T (cid:4) ) ∆ f/f ( % ) c ∆ F = 25 T
FIG. 4.
Effect of warping on the quantum oscilla-tions.
Influence of 2D Fermi surface warping on fitting QOsin Hg1201 (a) Fit (blue line) of the QO data (grey line) forUD74 with no warping (∆ F c = 0). The dotted line repre-sents the 1 /B dependence in arbitary units of the warpingterm R w = J (2 π ∆ F c B ) (see Methods). The degree of warpingis increased for (b) and (c) yielding nodes in the QO spectracorresponding to 1 /B values where R w changes sign . Thesmall amplitude oscillations at large 1 /B , bordered by thegrey box, is magnified by a factor of 15 in all three panels andshown as insets. sections of Fermi surface.Magnetic breakdown can manifest itself in two waysin our data. The first, is by way of a reduction of theprimary QO amplitude at higher magnetic fields. Forthe diamond-shaped electron pocket in Fig. 6, the am-plitude is reduced by R MB = [ p − exp {− B /B } ] toaccount for the four Bragg-reflection points at the tipsof the diamond. In a Dingle plot of the magnetic fielddependence of the QO amplitude (Fig. 7), this shouldbe discerned as deviations from a straight line for small
50 55
Hg1201 UD71 B (T) ∆ f/f ( % ) FIG. 5.
Determination of the effective mass.
Rel-ative PDO circuit frequency change ∆ f/f as a function offield B for sample Hg1201 UD71 at the temperatures indi-cated above each curve. Solid lines are simultaneous fits withEq.(1) (see Methods) to the total data set where all parame-ters are constrained to be temperature independent. For thesingle frequency spectrum of Hg1201, this method produces amore robust determination of the effective mass than examin-ing the temperature dependence of the FFT amplitude. Theeffective mass is extracted do be m ⋆ /m e = 2 . ± .
1, where m e is the bare electron mass. This is slightly larger than thatdetermined for a sample with almost the same doping [32]. /B . Accordingly, we fit the Dingle plot to a − b (1 /B ) +2ln[1 − exp {− B /B } ] where b = π ( p ~ F/e ) /l (solid linesin Fig. 7b), yielding B ≈
200 T and ≈
250 T for UD74and UD71, respectively. These large values for B areconsistent with no observable effects of magnetic break-down (i.e. a straight-line Dingle plot), therefore, we takethe fitted B as lower bound values.The second manifestation of magnetic breakdown isthrough the appearance of new QO frequencies corre-sponding to sums and differences of the areas of theFermi surfaces involved in the tunneling process. Mag-netic breakdown between the electron and hole pocket inour reconstruction model [9, 25], results in additional QOfrequencies of the form F e − h ,n = F e − nF h , in which n is an integer. Based on our modeling, F h ≈
80 T, mean-ing that the frequencies F e − h ,n are sufficiently distinctfrom F e to be discernible in the raw and Fourier trans-formed data in Fig. 3b,d,e. The noise floor of ≈ A e in Fig. 3 providesan upper limit for the amplitude A e − h , the leading mag-netic breakdown frequency ( n = 1). Using the inequality A e − h A e = 2 R e − h MB R e MB = 2( i √ P ) ( √ − P ) ( √ − P ) . . , we obtain a second lower bound of B ≈
200 T. Here, we k x (r.l.u.) k y (r . l . u . ) q CDW
HgBa CuO δ a k x (r.l.u.) q CDW
YBa Cu O b FIG. 6.
Fermi surface reconstruction by charge-density-wave
Unreconstructed (grey lines) and reconstructed (coloredlines) Fermi surface of HgBa CuO δ ( p = 0 . Cu O x ( p = 0 .
11) (b). The single-frequency QO we observe forHg1201 is in agreement with the area of the electron-like diamond shaped pocket (solid red), while there are no signatures of thesmall hole-like pocket (dashed dark blue) in our data. This might be due to the lack of quasiparticle weight in the pseudogappedantinodal regions of the Fermi-surface determined from angle-resolved photoemission [13–15]. For the reconstruction, we assumebiaxial CDW wavevectors ( Q CDW ,
0) and (0 , Q
CDW ). Q CDW = 0 .
275 r.l.u. for Hg1201 ( T c = 72 K) was taken from Ref.[18]. Q CDW = 0 .
325 for Y123 is estimated based on measurements on Y123 with x = 0.54[47] and Y123 with x = 0.55 [48]. Thecolor plots in the upper right edges of the panels are photoemission data showing the Fermi surface map for Hg1201 ( p ≈ . p = 0 .
11) (adapted bypermission from Macmillan Publishers Ltd: Nature (Ref. [12]), copyright (2008)). The points on top of the angle-resolvedphotoemission data in (a) are Fermi surface crossings. The tight-binding hopping parameters were determined by fitting thephotoemission data while constraining the area of the Fermi surface to match the quoted hole concentrations. Dashed bluelines indicate the antiferromagnetic AF zone boundaries. -(cid:6) (T (cid:7)(cid:8) ) l n [ A * B s i nh ( α m ∗ T B )] UD74UD71
FIG. 7.
Dingle plot.
Dingle plot of the QO amplitude nor-malized by R T as a function of 1 /B for UD71 and UD74.Amplitude of maxima and minima are taken from the datain Fig. 3a. Solid lines are fits to the data as described in thetext. have assumed a similar m ∗ and scattering rate values forthe various combination orbits, while the factor of twofor A e − h accounts for the two possible orbits involvingone 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 floorprovide mutually consistent large lower bound estimatesfor B . The most conservative of these (i.e. B &
200 T)enables a lower bound estimate of E g &
20 meV to bemade for the band gap between the observed electron andpresumed hole pockets in Fig. 6.
III. DISCUSSION
Our observation of a simple monotonic waveform of asingle QO frequency in Hg1201, and a single Fermi sur-face cross-sectional area that is compatible with photoe-mission and X-ray scattering measurements, is essentialfor resolving issues relating to the nature of the CDWordering. One of these concerns whether the two chargeordering wavevectors ( Q CDW ,
0) and (0 , Q
CDW ) coexistin the same CuO plane [9, 28, 33] or whether stripesalternate in a criss-cross fashion on consecutive CuO planes [29, 30]. In the absence of a coupling betweenCuO planes, criss-cross stripes lead to open Fermi sur-face sheets running in orthogonal directions on adjacentplanes. The effect of the inter-plane coupling is to intro-duce a hybridization [30]. Whereas a strong coupling inthe range ∼
10 to 100 meV occurs within the bilayersin Y123 and Y124 [50], no such coupling occurs in sin-gle layer Hg1201 and only a very weak coupling providedby the interlayer c -axis hopping determined here to be t c < c -axis hopping is to intro-duce a very small gap of order t c in magnitude betweenthe electron and hole pockets in Fig. 6, which would thenhave a very small characteristic magnetic breakdown fieldof B ∼ B determined from ourexperimental results). The magnetic breakdown ampli-tude reduction factor R MB for B = 0.1 T would be sosmall that it would render the electron pocket not observ-able 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 aviable route for creating observable Fermi surface pock-ets in Hg1201 at high magnetic fields.Although the CDW correlations detected with X-rayscattering are a natural candidate for the cause of Fermi-surface reconstruction, an open question concerns if thecorrelation length is sufficiently large to support quan-tum oscillations. A small correlation length of the orderparameter can manifest as additional damping of the QOamplitude in the Dingle term (see Methods), thus sup-pressing the effective mean free path l [51]. For Y123, theCDW correlation length at T c and B = 0 T is ξ CDW ≈
65 ˚A [16].The effective mean free path, l ≈
200 ˚A [35]obtained from QO measurements is of the same order ofmagnitude as ξ CDW . For Hg1201 both ξ CDW and l aresimilarly reduced compared to Y123: ξ CDW ≈
20 ˚A [18]at T ∼ T c and l = 85 ˚A (average of UD71 and UD74 andconsistent with prior Hg1201 results [32]). Thus, it ap-pears that the QO effective mean free path might be cor-related with the CDW domain size. Alternatively, both l and ξ CDW could be similarly affected by disorder or im-purities. The relatively small l for Hg1201 indicates thatthe CDW need not be long-ranged, even at low tempera-tures and high magnetic fields, to yield the reconstructedFermi surface observed here. Although ξ CDW in Y123 in-creases at low temperatures and high magnetic fields, itremains rather small ( ≈
100 ˚A −
400 ˚A) [52, 53].Another issue concerns the origin of the E g &
20 meVgap separating the diamond-shaped electron pocket fromadjacent sections of Fermi surface in Hg1201. There aretwo possible CDW Fermi surface reconstruction scenar-ios that have been discussed in the literature. One ofthese involves the folding of the large Fermi surface [9],as shown in Fig. 6, which is expected to produce smallhole pockets and open sheets in addition to the observedelectron pocket. In such a scenario, E g would then sim-ply correspond to the CDW gap 2∆ CDW . The alternativescenario is that the reconstructed Fermi surface occursby connecting the tips of Fermi arcs produced by a pre-existing or coexisting pseudogap state [18, 20]. In thisscenario, we would expect the small hole pockets to begaped out by the pseudogap causing E g then to corre-spond to the pseudogap energy. Two observations sug-gest the latter scenario to be more applicable to the un-derdoped cuprates. First, we find no evidence for quan-tum oscillations originating from the hole pocket, eitherby direct observation or by way of magnetic breakdown combination frequencies. Second, the Fermi arc, whichrefers to the region in momentum space over which thephotoemission spectral weight is strongest, is seen to bevery similar in length to the sides of the electron pocketin Figs. 6a and 6b (for both Hg1201 and Y123). Thespectral weight drops off precipitously beyond the tipsof the pocket. We note that while low frequency QOs inY123 have been attributed to small hole pockets, this lowfrequency could also originate from Stark quantum inter-ference effects associated with bilayer splitting [8]. Alter-natively, the pseudogap phenomena could also menifestas strong scattering at the antinodal regions, thus pre-venting an observation of such pockets in Hg1201. Re-cent high-temperature normal-state transport measure-ments in Hg1201 have also been interpreted in terms ofFermi-liquid-like [54, 55] Fermi arcs [56].Our findings in Hg1201 have direct implications forthe interpretation of QO measurements made in othercuprate materials. If we assume a similar gap size be-tween Hg1201 and Y123, the large E g suggests that mag-netic breakdown combination frequencies involving theelectron and small hole pocket [9, 25] cannot be respon-sible for the complicated beat pattern associated withclosely spaced frequencies in Y123 [8, 23] and Y124 [57].The splitting of the main frequency into two or morecomponents must therefore be the consequence of the bi-layer coupling in those systems [8, 28, 57], or a strongerinterlayer c -axis hopping.The biaxial reconstruction confirmed here for Hg1201has also been proposed for Y123[8, 26, 28], which is sup-ported by ultrasound measurements in high-fields [58].However, x-ray measurements show apparent local-stripeCDW domains at high temperatures [59], which pre-sumably become long-ranged and possibly arranged ina criss-cross pattern of stripes at low temperatures andhigh-fields [27, 30]. Recent X-ray measurements on Y123show a new magnetic field induced three-dimensionalCDW centered at c -axis wave-vector L = 1 r.l.u. [52]only along the CuO chain directions [53] which breaksthe mirror symmetry of the CuO bilayers. The role ofbilayer coupling and CuO chains for this stripe-like or-dering tendency is still an open question, and its rele-vance for Hg1201 which features neither is unclear. Thestripe picture is attractive because of natural analogiesto single-layered La-based cuprates [60] and its implica-tions for the role of nematicity (broken planar rotationalsymmetry) for the cuprate phase diagram [61]. However,neutron scattering experiments have found that the typ-ical signatures of spin stripes are absent in the magneticexcitations of Hg1201 [62]. Despite the appearance of anew uniaxial 3D order in Y123, the CDW wavevector,with a smaller c -axes correlation length, is still clearlyobserved in both planar directions in magnetic fields upto ∼
17 T [53]. For Hg1201, we have shown here thatthe CDW that causes the Fermi-surface reconstruction isbiaxial. It thus remains an open question as to whetherelectronic nematicity is generic to the cuprates, particu-larly in tetragonal Hg1201.
IV. METHODS
Samples.
Hg1201 single crystals were grown using aself-flux method [63]. As grown crystals have T c ≈
80 K.Post-growth heat treatment in N atmosphere at 400 o Cand 450 o C was used to achieve T c = 74 K (hole concen-tration p = 0 . T c = 71(2) K ( p = 0 .
09) respec-tively. T c was determined from DC susceptibility mea-surements.The 95% level transition width of both sam-ples is 2 K. The hole concentration p is determined basedon the phenomenological Seebeck coefficient scale [38].The YBCO crystal was flux grown and heat treated toobtain oxygen content x = 0 .
58 with T c = 60 K and holedoping p = 0 .
106 at the University of British Columbia,Canada [39].
Pulsed field measurements.
High magnetic fieldmeasurements were performed at the Pulsed-Field Fa-cility at Los Alamos National Laboratory. The magnetsystem used consists of an inner and outer magnet. Theouter magnet is first generator driven relatively slowly( ∼ ∼
15 ms) capacitor bank driven pulse to 90 T.
Fitting quantum oscillations.
We fit the field de-pendence to ∆ f /f = ( a + a B + a B + . . . )(1 + A osc ),where the first term is a polynomial representing the non-oscillatory background and A osc is the oscillatory compo-nent. In the case of a single Fermi surface cylinder, theQOs are described by the Lifshitz-Kosevitch form [49] A osc = A R T R D R S R MB R W cos h π (cid:16) FB − γ (cid:17)i , (1)where F is the frequency of QOs, γ is the phase and A is a temperature and field independent pre-factor.Here, R T , R D , R S , R MB and R W are the thermal, Dingle,spin, magnetic breakdown and warping damping factors,respectively [8, 24]. R T = αT / [ B sinh( αT /B )] where α = 2 π k B m ∗ / ( e ~ ) accounts for the thermal broaden-ing of the Fermi-Dirac distribution relative to the cy-clotron energy and m ∗ = 2.7 m e , determined for one ofour samples as shown in Fig. 5, is the quasiparticle effec-tive mass ( m e being the free electron mass). Meanwhile, R D = exp( − πl c /l ), where l c = p ~ F/e/B is the cy-clotron radius and l is the mean free path. To lowest or-der, warping of a cylindrical Fermi surface leads to an am-plitude reduction factor of the form R w = J (2 π ∆ F c /B )in which J is a zeroth order Bessel function and 2∆ F c ≈ t ⊥ m ∗ / ( ~ e ) is the difference in frequency between theminimum and maximum cross-sections of the warpedcylinder. Since our experiments are performed at fixedangle (i.e. B k c ), we neglect R S by setting it to unity.As discussed in the main text, our data shows no signa-tures of magnetic breakdown tunneling or warping, thuswe also set R MB and R W to unity. Limits on these twoterms are discussed in the Results section. Calculation of reconstructed Fermi surface.
Theunreconstructed Fermi surface is calculated with the dis-persion ǫ k = − t ( φ x + φ y ) − t ′ φ x φ y − t ′′ ( φ x + φ y ) − t ′′ ( φ x φ y + φ yφ x ) − µ where the tight-binding parame-ters are ( t, t ′ , t ′′ , t ′′′ ) = (0 . , − . , . , − .
02) eV [45]for Hg1201 and (0 . , − . , . ,
0) eV for YBCO. µ is the chemical potential and φ nx = cos( nk x ) and φ ny = cos( nk y ) where k x and k y are the planar wavevec-tors. We required that the tight-binding parametersproduce a Fermi surface in agreement with the pho-toemission data and have carrier number 1 + p where p = 0 .
12 and 0 .
11 for the Hg1201 and Y123 sampleson which the photoemission data were taken. Hence,1 + p = 2 A UFS /A UBZ , where A UFS and A UBZ are theareas of the unreconstructed Fermi surface and Brillouinzone respectively. Before calculating the reconstructedFermi surface, only µ is adjusted to match the hole dop-ing p = 0 .
095 and p = 0 .
106 on which the QO data wastaken for Hg1201 and Y123 respectively.Following Ref. [33], the reconstructed Fermi surfaceis determined by diagonalizing a Hamiltonian consider-ing translations of the biaxial CDW wavevector k → k + n x Q CDW ˆx + n y Q CDW ˆy , where n x and n y are thenumber of translations in the planar directions. Strictlyspeaking, reconstruction by observed incommensurateCDW wavevectors requires an infinite number of terms inthe Hamiltonian to obtain all the bands. However, since∆ ≪ t , the inclusion of high order terms in the Hamilto-nian gives rise to a hierarchy of higher order gaps that areexponentially small, and thus do not effect the primaryclosed orbits resulting from our calculation, which we re-strict to nine terms. Supplementary Figure 3 shows allthe bands resulting from our reconstruction calculation.We use ∆ CDW /t = 0.1 for the ratio of the CDW orderparameter magnitude to the in-plane hopping [9]. Thisimplies ∆ CDW = 46 meV, based on band structure deter-mination of t [45], which is larger than the lower boundvalue determined from our analysis of magnetic break-down tunneling in the main text, but sufficiently smallthat it does not adversely affect the sizes of the pock-ets. Reducing the ratio to zero increases the area of thereconstructed pockets by only ≈ Acknowledgments
This work, performed at Los Alamos National Lab, wassupported by the US Department of Energy BES “Sci-ence at 100 T” grant no. LANLF100. The National HighMagnetic Field Laboratory - PFF facility is funded by theNational Science Foundation Cooperative Agreement No.DMR-1157490, the State of Florida, and the U.S. Depart-ment of Energy. Work at the University of Minnesota wassupported by the Department of Energy, Office of BasicEnergy Sciences, under Award No. de-sc0006858. N.B.acknowledges the support of FWF project P2798. Wethank Ruixing Liang, W. N. Hardy and D. A. Bonn atUBC, Canada for generously supplying the Y123 crystalmeasured as part of this work. We aknowledge fruitfuldiscussion with S.E. Sebastian. We also thank the PulsedField Facility, Los Alamos National Lab engineering andtechnical staff for experimental assistance.
Correspondence
Correspondence should be addressed to M.K.C.([email protected]) and N.H. ([email protected]).
Author Contributions
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