Atomic-scale interpretation of the quantum oscillations in cuprate superconductors

Cuprate superconductors display unusual features in both k space and real space as the superconductivity is suppressed—a broken Fermi surface, charge density wave, and pseudogap. Contrarily, recent transport measurements on cuprates under high magnetic fields report quantum oscillations (QOs), which imply rather a usual Fermi liquid behavior. To settle the disagreement, we investigated Bi2Sr2CaCu2O8+δ under a magnetic field in an atomic scale. A particle-hole (p–h) asymmetrically dispersing density of states (DOSs) modulation was found at the vortices on a slightly underdoped sample, while on a highly underdoped sample, no trace of the vortex was found even at 13 T. However, a similar p–h asymmetric DOS modulation persisted in almost an entire field of view. From this observation, we infer an alternative explanation of the QO results by providing a unifying picture where the aforementioned seemingly conflicting evidence from angle-resolved photoemission spectroscopy, spectroscopic imaging scanning tunneling microscopy, and magneto-transport measurements can be understood solely in terms of the DOS modulations.

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One of the most challenging aspects in the study of the cuprate superconductors is the conflicting observations and their interpretations from different experimental results. As the superconductivity in cuprates disappears above T c , the angle-resolved photoemission spectroscopy (ARPES) studies revealed a Fermi arc, or a broken Fermi surface [1][2][3][4]. Opposingly, the magneto-transport measurements on underdoped YBa 2 Cu 3 O 6+δ, (YBCO) and HgBa 2 Cu 4 O 8+δ (HBCO) under an extremely high magnetic field show oscillatory behaviors in magnetization [5], in-plane [6][7][8] and out-of-plane resistivity [9], contactless resistivity [10] as well as in a Seebeck coefficient [11]. These quantum oscillation (QO) results combined with the Hall measurement [12] results suggest the existence of a closed electron-like Fermi surface in hole-doped cuprates in the underdoped regime [6]. Meanwhile, spectroscopic imaging scanning tunneling microscopy results on underdoped Bi 2 Sr 2 CaCu 2 O 8+δ (BSCCO) show a clear broken symmetry in real space: local density of states (LDOS) modulations showing checkerboard-like patterns that seem getting stronger as the superconductivity weakens [13,14], while a pair density wave (PDW) [15,16] is observed by scanning Josephson tunneling microscopy in the superconducting phase [17]. Under a magnetic field, the formation of vortices accompanied by four unit cell patterns on nearly optimally-doped BSCCO was observed by STM studies [18][19][20][21]. However, there is no spectroscopic study on highly underdoped (HUD) cuprates under a high magnetic field, which can provide a clue to resolve the ostensible discrepancy.
We performed an STM experiment on a slightly underdoped (SUD) BSCCO sample with T c = 91 K and a HUD BSCCO with T c = 40 K with varying magnetic fields at 4.2 K. l) show differential conductance maps g (r, E, B) at E = 8 meV. Superconducting vortices are discernible by the well-known four unit cell checkerboard-like core structures [18] on the maps in low energies (|E| < 12 meV) under the magnetic fields (figure S2). As the magnetic field is increased, the number of vortices also increases. We confirmed that each vortex contains a superconducting flux quantum through a large FOV scanning (figure S3). Wu et al [22] suggested that the vortex core structure of 4a 0 periodicity [18] comes from a static charge order while Machida et al [21] reported that the vortex core structures are the vortex-enhanced quasiparticle interference (QPI). We find that the vortex core structure is not static but varies as the sample bias changes (figure S2), and the period of this pattern is about 4a 0 at E = 8 meV (figures 1(j)-(l)).
To analyze the energy-dependent variation of the vortex core patterns further, we masked out the vortex (VR) and non-vortex regions (NVRs) of the conductance map at 5 T separately (see masking method and detailed description in supplementary materials). We performed a two-dimensional Fourier transformation (2DFT) to each masked conductance map, and the results are shown in figures 2(a)-(d). In the VR, q 1 * peaks have the strongest intensity at the low energy (red circle in figure 2(c)) while in the NVR, q 1 * peaks are much weaker (figure 2(d)). Two-dimensional image plots of the linecut along the q x direction (figures 2(e) and (f)) show that the vortex core pattern associated with q 1 * is neither a nondispersing static order nor an enhanced QPI. The q 1 * from the VR disperses from −12 meV to 12 meV, where the vortex checkerboard is observable. Remarkably, instead of showing a p-h symmetric dispersion as in the QPI following an octet model in the superconducting phase [23], the dispersion showed a p-h asymmetric behavior. On the other hand, the q 1 * in the NVR shows a dispersion similar to the q 1 * 's in the 0 T field: a p-h symmetric QPI (figure S5).
It was reported that the density modulations in the HUD BSCCO above T c show a similar dispersion as q 1 * in the VR in our result [24]. Moreover, Lee et al [25] predicted that the pseudogap states could be observed inside the vortices as well. To verify, we conducted a magnetic field dependent scan on a HUD BSCCO (T c = 40 K). Figure 3 shows spectroscopic imaging results on a HUD BSCCO sample at 0 T and 13 T at the same FOV at 4.2 K. Since each vortex has a superconducting magnetic flux quantum (figure S3), one can expect about 4 or 5 vortices within the FOV. Unlike the SUD experiment, however, no vortex was found within our instrumental limit. Furthermore, on the HUD BSCCO, the checkerboardlike pattern spanned to an extended area and showed a similar p-h asymmetric dispersion in both measurements with and without magnetic fields (figures 3(c) and (d)). We subtracted the g map at 0 T from the g map at 13 T at the Fermi energy and 8 meV respectively (figure 3(g)), and we found that the relative intensity of not only q 1 * but q 5 * was higher in 13 T than in 0 T. To check if the dispersions in the VR and HUD are related, we plotted the q 1 * in different doping and magnetic field conditions in figure 3(h). All three q 1 * except in the NVR showed an electron-like dispersion (figure 3(h)). Our result shows the similar p-h asymmetrically dispersing (figure 3(h)) density of state (DOS) modulations both in VR on the SUD BSCCO and in an extended area on the HUD BSCCO's case at 13 T as well as 0 T without a trace of vortices, which have a striking resemblance to the dispersing patterns reported on the underdoped BSCCO above T c [24]. Only NVR shows a ph symmetric QPI dispersion following an octet model [23] at q 1 location. As the magnetic field is increased and the doping level is lowered, the p-h asymmetric DOS modulation became stronger and expanded in an area rather than localized at the vortex core regions. By integrating the LDOS maps g (r, E) from −30 meV to 0 meV, we obtained modulated features in all three cases: VR on HUD BSCCO at 0 T and HUD BSCCO at 13 T (figure S9) with a q 1 * (in the unit of 2π a0 ) value of about 0.21-0.225, which is a conventional way to obtain a charge distribution near the Fermi energy. This result suggests that the The average gaps are from 43 meV to 47 meV (figure S1). (i)-(l), differential conductance maps g (r, E, B) at E = 8 meV of the same FOV's as a-d, respectively. Except for the zero field image, superconducting vortices and its core structures are clearly presented in all images under magnetic fields. As the magnetic field increases, the number of vortices also increases. The radius of each vortex varies from 8 nm to 10 nm, but has no dependence on the energy. (e)-(f), two-dimensional images of the energy-dependence of the line-cut data along the qx direction. In the VR, q 1 * shows a p-h asymmetric and electron-like dispersion near the Fermi energy. In the NVR, q 1 * shows a p-h symmetric dispersion, following the octet model Bogoliubov QPI [23]. Each value of the line-cut is divided by the maximum value in e and the average value in f to show the dispersion clearly (raw images in figure S6). Schematic plots of a p-h asymmetric and symmetric dispersion curves in k space are presented ((e)-(f), inset).
p-h asymmetric DOS modulation we present in this paper and the bilateral charge density wave (CDW) reported in a wide range of cuprates [26][27][28][29], can be of the same origin. From this observation, we developed a model that explains how the existence of such DOS modulation can have an impact on transport measurements, especially the QO results on underdoped cuprates [6].
Under a much higher magnetic field and at a low temperature, the DOS in conductors rearranges into a series of Landau levels (LL's), as shown in figures 4(a) and (b). In such circumstances, many phenomena, including quantum Hall effect [30], de Haas van-Alphen effect [31], and Aharonov-Bohm effect [32], can emerge. Recently, QO has been reported in cuprates, including underdoped YBCO [6], HBCO [8], overdoped TBCO [33], and electron-doped PCCO [34] under extremely high magnetic fields. The oscillatory behavior was observed in in-plane and out-of-plane resistivity, contactless resistivity [6][7][8][9][10] as well as magnetic torque measurements [5]. Previously, these QO phenomena in underdoped p-type cuprates were explained in terms of a reconstructed Fermi surface and conventional Fermi liquid theory [5]. Still, such explanations are in debate due to a lack of evidence of a closed Fermi surface in underdoped cuprates at the pseudogap phase. ARPES measurements report a 'Fermi arc' [1][2][3][4], which makes a simple Fermi liquid interpretation challenging. Here, we suggest a simple but novel approach to explain the QO phenomena entirely in terms of the nano-scale real space features such as the DOS modulations we presented in this paper. There is an example of the magnetic field dependent oscillations that originated from a real space modulation. Weiss oscillations or commensurability oscillations [35][36][37] is due to the interplay between the charge carriers' cyclotron orbit and the periodic potential, usually under a weaker magnetic field far before the formation of LL's [38]. However, such oscillations cannot explain the oscillations in magnetization observed in QO results on cuprates.
Once the Landau quantization sets in the cuprates, which are non-superconducting at a high magnetic field as in QO experiments, one needs to consider a new set of sum rules. In low temperatures, the charge carriers or quasiparticles fill up the LL's starting from the lowest LL. At a given magnetic field B and a sample size of A, each LL has the number of occupiable states of AB Φ 0 , where Φ 0 = h e is a flux quantum. Then the total number of the quasiparticles N should satisfy where n is the index of the highest occupied LL and m is an integer satisfying − AB 2Φ 0 ⩽ m (B) ⩽ AB 2Φ 0 . That is, the occupation of the LL's is completely determined by two integers: n and m. Rearranging equation (1), we derive where A nm ≡ A N−m represents roughly an area per quasiparticle which changes as B changes. Let the actual size of an orbit of a quasiparticle in n th LL A n , and it is clear that A n will be proportional to A nm and A n changes as n and m changes according to the variation of B (figures 4(c) and (d)). For a simplicity, we will use A nm instead of A n as an estimated orbit size in n th LL in the following argument. If the orbit or circumference of the A nm corresponds to a specific loop where LDOS is high, the number of orbiting quasiparticles will increase greatly, resulting in an increase of the relevant magneto-transport quantities, including magnetization and in-plane conductivity. Let the area of such a high LDOS loop be A reson . Varying n and B, we have a new resonance condition of A reson = A nm , or where we define a magneto frequency . From this relation, we can estimate the size of A reson . For example, using the value of F ∼ 530 T reported from the QO result on an underdoped YBCO [6], we obtain an area 7.8 × 10 −18 m 2 for the A reson . This value is comparable to about an area of 8a 0 × 8a 0 ( (2 × 4a 0 ) 2 ∼ 9.2 × 10 −18 m 2 ), where the lattice constant (Cu-Cu distance) of the CuO 2 plane a 0 = 0.38nm in cuprates. This area corresponds to red squares in figure 4(e) in our data on BSCCO and the length of the squares' sides is twice the commonly known four-unit cell modulation period which is related to our observed q 1 * as 2π / q 1 * ∼ 4a 0 . If we assume the CDW's q vector ∼0.3 reported in YBCO [29, S46] as q 1 * , A reson_YBCO = ( 2 × 1 0.3 a 0 ) 2 ∼ 6.4 × 10 −18 m 2 which is even closer to our estimated value of the A reson . The estimated A reson size fits remarkably well to a DOS modulation mesh, which is omnipresent in energies near the Fermi energy even at the magnetic field of 13 T in the HUD BSCCO, as shown in figure 3(b). The resonance between the high DOS area in the modulations and the Landau orbits is illustrated in figures 4(c) and (d), which show the orbit of a quasiparticle in the highest LL containing a Fermi level (black square) coinciding with (figure 4(c)), and non-coinciding (figure 4(d)) with a given 8a 0 × 8a 0 pattern (red square). Our simple model captures almost all of the features of the QO results, including but not limited to a dominating ∆ , an increase of the oscillation amplitude (due to the increase of DOS of LL with increased B) without an assumption of a well-defined closed Fermi surface. Even a cause of the reported small side oscillations [5,9,10] can be speculated ( figure S8). As shown in figure 3(h), quasiparticles, which can cause such resonance, should originate from the electron-like dispersing modulations, which also agrees with the reported Hall probe results on the holedoped cuprates under magnetic fields where QO was observed [12]. Another STM study on VR reports a pronounced periodicity of 8a 0 × 8a 0 , which also seems to substantiate our model, although their results are only in a vicinity of vortices, and they argued that the modulation is related to the  (2)) at a fixed B. Since the orbit area is inversely proportional to the magnitude of the magnetic field, the higher the magnetic field, the smaller the orbit for a fixed n. (c). When a Landau orbit (black square, Anm) coincides the resonance orbit in the real space (red square, Ares), the physical quantities related to the electron DOS can diverge resulting in a local maximum of those quantities ((c), inset). (d). As the magnetic field continues to grow, the n th Landau orbit moves inward from the resonance orbit, and the next (n − 1) th Landau orbit approaches the resonance orbit. When the resonance orbit is located between two Landau orbits, the value of the physical quantities is also located between the local maxima ((d), inset). (e), As the magnetic field goes beyond H c2 , the superconducting state changes to the normal state, and the patterns in vortices cover the entire map. Then the resonance orbit appears throughout the map. Resonance orbits are indicated as red squares on a differential conductance map g (r, E = 14 meV, B = 13 T) of the HUD BSCCO.
PDW [15,16,39]. Theoretically, a period-8 d-density wave order was also suggested regarding the origin of the QO in underdoped YBCO [40]. In our model, however, the 8a 0 × 8a 0 area is not a uniquely fixed value, unlike the fixed pocket size assumed in the Fermi pocket picture. Indeed, there is a report on QO magneto frequency of about 270 T [41], which corresponds to a 12a 0 × 12a 0 area from equation (3), which contains 9 plaquettes of the area 4a 0 × 4a 0 (figure S9). In this argument, our model not only explains the multiple QO frequencies but also predicts more frequencies unreported experimentally.
To estimate the quasiparticle density, we can inverse the A reson ∼ 2.8 nm × 2.8 nm yielding a quasiparticle areal density n qp ∼ 1.3 × 10 13 cm −2 , which matches remarkably well with the electron density n e ∼ 2.6 × 10 13 cm −2 (0.038 electron per Cu atom) estimated from the QO result [6,12] considering a factor of 2 due to the bi-layer nature of the CuO 2 planes. This comparison can be understood in the sense that the area of the electron pocket suggested in a conventional QO interpretation can be translated into an inverse of the real space area A reson from our model multiplied by (2π ) 2 . In conclusion, we discovered a p-h asymmetric dispersion of the DOS modulations inside the superconducting vortices on an SUD BSCCO, and the dispersion was electronlike, unlike a conventional CDW. On a HUD BSCCO, we could not find any sign of vortices up to 13 T but we found a globally expanded DOS modulations displaying the same ph asymmetric, electron-like dispersion as the patterns inside the vortices in the SUD BSCCO. Due to the lack of direct spectroscopic evidence of small electron-like pockets on hole underdoped cuprates and based on our observations, we propose an alternative model of the QO results on hole underdoped cuprates in a simple picture of the commensurate resonance between the quasiparticle orbit of the highest Landau level due to the magnetic field and the DOS modulations, which does not require a Fermi surface reconstruction nor a closed Fermi surface as proposed previously. Our result implies that the dispersive DOS modulations can impose a significant impact on magneto-transport properties as well as spectroscopic features. Furthermore, our result implies that these modulated LDOS features can be the basis of the ground states just before the superconducting ground states appear. Two important questions still remain. How is this DOS modulation formed? How does high-T c superconductivity emerge from such an unusual electronic structure?

Data availability statement
The data cannot be made publicly available upon publication because they are not available in a format that is sufficiently accessible or reusable by other researchers. The data that support the findings of this study are available upon reasonable request from the authors.