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# Fermi surface reconstruction in electron-doped cuprates without antiferromagnetic long-range order

Contributed by Zhi-xun Shen, December 25, 2018 (sent for review September 19, 2018; reviewed by and Peter Armitage and Louis Taillefer)

## Significance

Fermi surface (FS) topology is a fundamental property of metals and superconductors. In electron-doped cuprate Nd_{2−x}Ce_{x}CuO_{4}, an unexpected FS reconstruction has been observed in optimal- and overdoped regime (*x* = 0.15 − 0.17) by quantum oscillation measurements (QOM). This is puzzling because neutron scattering suggests that the antiferromagnetic long-range order, which is believed to reconstruct the FS, vanishes before *x* = 0.14. Here, we report angle-resolved photoemission evidence of FS reconstruction. The observed FSs are in quantitative agreement with QOM, suggesting an intrinsic FS reconstruction without field. Furthermore, the energy gap of the reconstruction decreases rapidly near *x* = 0.17 like an order parameter, echoing the quantum critical doping in transport. The totality of the data points to a mysterious order between *x* = 0.14 and 0.17.

## Abstract

Fermi surface (FS) topology is a fundamental property of metals and superconductors. In electron-doped cuprate Nd_{2−x}Ce_{x}CuO_{4} (NCCO), an unexpected FS reconstruction has been observed in optimal- and overdoped regime (*x* = 0.15–0.17) by quantum oscillation measurements (QOM). This is all the more puzzling because neutron scattering suggests that the antiferromagnetic (AFM) long-range order, which is believed to reconstruct the FS, vanishes before *x* = 0.14. To reconcile the conflict, a widely discussed external magnetic-field–induced AFM long-range order in QOM explains the FS reconstruction as an extrinsic property. Here, we report angle-resolved photoemission (ARPES) evidence of FS reconstruction in optimal- and overdoped NCCO. The observed FSs are in quantitative agreement with QOM, suggesting an intrinsic FS reconstruction without field. This reconstructed FS, despite its importance as a basis to understand electron-doped cuprates, cannot be explained under the traditional scheme. Furthermore, the energy gap of the reconstruction decreases rapidly near *x* = 0.17 like an order parameter, echoing the quantum critical doping in transport. The totality of the data points to a mysterious order between *x* = 0.14 and 0.17, whose appearance favors the FS reconstruction and disappearance defines the quantum critical doping. A recent topological proposal provides an ansatz for its origin.

- high-temperature superconductors
- angle-resolved photoemission
- quantum critical point
- topological order
- strongly correlated electrons

Fermi surface topology is the starting point to understand various emergent quantum phenomena in metals, including high-temperature superconductivity. With both momentum and energy resolution, angle-resolved photoemission (ARPES) is an ideal tool to directly reveal the Fermi surface (FS) topology of a material. However, in electron-doped cuprates, a direct understanding of the ARPES results has been limited by the data quality (1⇓⇓–4). This is primarily due to the lack of a large high-quality surface area in a material that is difficult to cleave. Utilizing a recently developed ARPES beam line at Stanford Synchrotron Radiation Lightsource (SSRL) with a small beam spot, we have managed to probe intrinsic electronic structures from a small but uniform region on the cleaved sample surface. This technical advancement leads to a significant improvement on the experimental data quality (*SI Appendix*, Fig. S1) that enables us to quantitatively investigate the FS topology in electron-doped cuprates.

When an FS reconstruction takes place, the energy band is folded with respect to the antiferromagnetic zone boundary (AFMZB) and an energy gap opens up, giving rise to a back-bending behavior of the band at the AFMZB (5) (see Fig. 1 *A–C* for a schematic diagram). If the gap is below Fermi level (E_{F}) (Fig. 1*B*), then the E_{F} cuts through the conduction band, resulting in an electron-like pocket (e.g., antinodal region in Fig. 1*A*). Conversely, if the gap is above E_{F} (Fig. 1*C*), a hole-like pocket appears on the FS (e.g., nodal region in Fig. 1*A*). On the other hand, when the FS reconstruction is absent (Fig. 1*D*), the electron band disperses continuously, regardless of the AFMZB (Fig. 1 *E* and *F*). Neither band folding nor gap opening is expected.

Earlier ARPES measurements on underdoped samples have revealed the AFM gap (2⇓⇓⇓⇓–7), hints of the folded band (8), and disconnected segments on the FS (2⇓⇓⇓⇓–7), supporting the reconstruction scenario in underdoped regime (2). However, things become more complicated with electron doping (4, 6, 9⇓–11). Photoemission constant-energy map at E_{F} of the optimal-doped Nd_{2−x}Ce_{x}CuO_{4} (NCCO, *x* = 0.15) seems to suggest a large FS centered at (π, π) (4). But, a spectral weight analysis of the nodal dispersion favors a reconstructed FS for the optimal-doped Sm_{2−x}Ce_{x}CuO_{4−δ} (SCCO, *x* = 0.15) (10). While slight variations between different material families have been discussed (11), a direct understanding of the FS topology requires a better resolution of the key features––band folding and gap opening at the AFMZB.

## Results

With the improved precision of data, our measurements on the optimal-doped NCCO (*x* = 0.15) clearly reveal both band folding and gap opening at the AFMZB (momentum cut near the “hotspot” where the FS intersects AFMZB, Fig. 1 *G–I*), demonstrating the existence of a reconstructed electron-like pocket. The FS reconstruction is also supported by the nodal dispersion (Fig. 1 *J–L*), where the Fermi level crosses the hole band, forming a hole-like pocket near the node. The possible gap opening above E_{F} cannot be seen by ARPES, but the back-bending of the hole band (folded band) is discernible (Fig. 1 *J–L*). A reconstructed FS should also be accompanied by a hotspot between the electron-like and hole-like pockets, where a gap exists at the Fermi level (Fig. 1*A*). This is also observed in our experiment (see *SI Appendix*, Fig. S2 for the FS mapping and the momentum cut through the hotspot).

After establishing the FS reconstruction in optimal-doped NCCO, we quantitatively investigate the associated energy gap. Surprisingly, the gap shows a strong momentum dependence. An ∼80-meV gap appears near the hotspot (Fig. 2*B*), but it vanishes at the AFMZB near the antinode (Fig. 2 *C* and *H* and *SI Appendix*, Fig. S3). This is distinct from the mean-field band-folding picture, where a momentum-independent constant energy gap is expected (Fig. 2 *E–G*). To understand the differences, we note that a moderate energy gap can be smeared out on the photoelectron spectra when an enhanced scattering rate takes place (*SI Appendix*, Fig. S4). The measured electron scattering rate, represented by the width of the momentum distribution curves (MDC) near E_{F} (Fig. 2*I*), does show a substantial momentum dependence (Fig. 2*J*). The enhanced scattering rate near the antinode, when combined with its deep binding energy where the band crosses the AFMZB, could give rise to the folded band without a resolvable gap opening (*SI Appendix*, Fig. S4). As such, the gap itself is likely isotropic and the apparent momentum dependence of the gap in Fig. 2*H* can be attributed to the scattering rate difference. Although the absolute value of scattering rate deduced from ARPES is different from that in transport, the strong momentum dependence coincides with the fact that only the hole-like pocket near the node has been observed in QOM, while the expected electron-like pocket near the antinode is absent (12⇓⇓⇓⇓–17). The origin of the momentum-dependent scattering rate is yet to be understood, where the electron correlation might be at play.

Next, we study the doping dependence of the FS reconstruction. Both the back-bending behavior (folded band) and gap opening have been observed at all doping levels we measured (*x* = 0.11, *x* = 0.15, *x* = 0.16), demonstrating that the FS reconstruction persists to the overdoped regime. This is consistent with QOM (12⇓⇓⇓⇓–17), and the well-defined gap suggests that a magnetic breakdown in QOM is less likely below *x* = 0.16. However, the gap decreases rapidly near *x* = 0.16 (Fig. 3*F*), which is consistent with our inability to observe a gap beyond *x* = 0.16. Such a behavior of gap closing near *x* = 0.16∼0.17 is consistent with a quantum critical point (QCP) in that doping range, as suggested by an *x* to 1−*x* density transition in transport experiments (13, 18, 19). The size of the reconstructed Fermi pockets can also be measured. One way is to directly estimate the area of the pockets via FS mapping (Fig. 3*E* and *SI Appendix*, Fig. S5). Alternatively, one can also estimate the pocket size using the reconstructed band dispersion and the gap size (see *SI Appendix* for details). Both methods give similar results, which are quantitatively consistent with quantum oscillation results as in Fig. 3*G*. The similar Fermi pockets observed by our measurements and QOM indicate the existence of a robust FS reconstruction, regardless of the external magnetic field.

## Discussion

One scenario for the ARPES data is a remnant short-range order (8, 20⇓–22). Strictly speaking, a short-range order does not break the global translational symmetry of the crystal. However, it might provide a scattering channel with a wave vector Q ± ΔQ, where ΔQ is proportional to the inverse of the correlation length, and thus an approximate “FS folding.” Nevertheless, with the small AFM correlation length in overdoped NCCO (*x* = 0.16) of ∼7 planar lattice constant (23), a weak coupling mean-field simulation does not capture the experimental observations (*SI Appendix*, Fig. S6). Our data, on the other hand, leave room for a strong coupling picture with short-range AFM fluctuation, as those suggested by the Hubbard model calculations. Here the momentum folding remains commensurate with the local correlation at Q = (π, π), and the gap magnitude is also dominated by the local interaction and thus remains similar in different regions (*SI Appendix*, Fig. S7). Our experimental energy and momentum width allow such a picture. However, it is unclear whether the quantum critical doping, as seen by the rapid decrease of the energy gap near *x* = 0.16 (Fig. 3*F*) and the corresponding transport data (13, 18, 19, 24⇓–26), can be understood by such a purely local picture without invoking long-range order.

The totality of ARPES, QOM, and transport data suggests the presence of an intrinsic long-range order that persists up to the critical doping near *x* = 0.16–0.17. Long-range AFM order can naturally explain this phenomenology. However, neutron-scattering data from the rod-like magnetic scattering in the bulk indicate the lack of coexistence between long-range AFM order and superconductivity beyond *x* = 0.14 (23). A similar conclusion is drawn by the muon spin rotation measurements on another electron-doped cuprate La_{2−x}Ce_{x}CuO_{4−δ}, where the static magnetism and superconductivity do not coexist (27). One is therefore left with a puzzle on the origin of the long-range order in our superconducting NCCO samples with *x* = 0.15–0.16 doping.

Charge order has been reported in NCCO (28), but the associated wave vector does not match the observed FS reconstruction. Another possible way out involves topological order that exploits the topological character of the Luttinger theorem (29⇓–31). Without breaking the translational symmetry, the existence of topological order in a state with short-range AFM order can still reconstruct the FS with respect to the AFMZB (29⇓–31) (also see *SI Appendix*, Fig. S8). In such gauge theory, the gauge-dependent Higgs field cannot be directly observed, but can play a role similar to an order parameter. Its presence could have observable consequences like the opening of the gap, which has magnitude related to the local magnetic order and its closing defines a QCP. It would be instructive to have deeper understanding of the transport behavior near a topological QCP for refined comparison with experiment to further test the validity of such ansatz. It would also be interesting to explore whether the same basic scenario can be at play in hole-doped cuprates.

Through much improved experiment, our data have established the intrinsic doping dependence of FS topology in NCCO, and provided a microscopic underpinning for QOM without the need to assume magnetic-field–induced long-range AFM order in the optimal- and overdoped regime. The rapid closing of the gap near *x* = 0.16–0.17 reveals the likely order parameter of the quantum critical doping in transport experiments (13, 18, 19, 24⇓–26). Confronted by the neutron conclusion of an absence of long-range AFM order beyond *x* = 0.14 (23), a correlation driven topological order provides an ansatz to reconcile the dilemma.

## Materials and Methods

### Samples.

Single crystals of NCCO (*x* = 0.11, 0.15, and 0.16) were grown by the traveling-solvent floating-zone method in O_{2} and annealed in Ar. The doping levels were determined by electron probe microanalysis (EPMA).

### ARPES.

ARPES measurements were carried out at beam line 5–2 of the Stanford Synchrotron Radiation Lightsource of SLAC National Accelerator Laboratory with a total energy resolution of ∼12 meV and a base pressure better than 5 × 10^{−11} Torr. The data were collected with 53-eV photons at ∼20 K. The Fermi level was referenced to that of a polycrystalline Au film in electrical contact with the sample. The smallest beam spot size at the beam line is ∼40 μm (horizontal) × 10 μm (vertical). For our experiments, a beam spot of ∼40 μm (horizontal) × 80 μm (vertical) was chosen, which optimized the photoelectron counts on a single uniform surface region.

## Acknowledgments

We thank S. Sachdev, D. J. Scalapino, D.-H. Lee, P. J. Hirschfeld, S. A. Kivelson, J. Zaanen, N. Nagaosa, A. Georges, C. M. Varma, and S. Uchida for useful discussions. The work at SLAC and Stanford University is supported by the US Department of Energy (DOE), Office of Basic Energy Science, Division of Materials Science and Engineering. SSRL is operated by the Office of Basic Energy Sciences, US DOE, under Contract DE-AC02-76SF00515. M.S.S. acknowledges support from the German National Academy of Sciences Leopoldina through Grant LPDS 2016-12. This research used resources of the National Energy Research Scientific Computing Center (NERSC), a US DOE Office of Science User Facility operated under Contract DE-AC02-05CH11231.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: zxshen{at}stanford.edu.

Author contributions: J.H. and Z.-x.S. designed research; J.H., C.R.R., M.S.S., Y.H., M.H., K.-J.X., Y.W., E.W.H., T.J., S.C., B.M., D.L., Y.S.L., and T.P.D. performed research; J.H., Y.H., and Z.-x.S. analyzed data; J.H. led the experiment; Z.-x.S. was responsible for the overall project management; and J.H. and Z.-x.S. wrote the paper with input from all authors.

Reviewers: P.A., Johns Hopkins University; and L.T., University of Sherbrooke.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1816121116/-/DCSupplemental.

Published under the PNAS license.

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