Suppression of Spectral Gap and Flat Bands on a Cuprate Superconductor Side-Surface

Abstract

Side surfaces of cuprate superconductors are expected to display a suppressed $d$-wave order parameter and zero-energy topological flat bands with a large density of states, making them susceptible to symmetry broken orders. Yet such surfaces have never been investigated with momentum-resolved, surface-sensitive probes, because high-temperature superconductors rarely cleave along them. Using focused-ion-beam milling to define a controlled breaking point, we expose pristine (110) side surfaces of overdoped La$_{2-x}$Sr$_x$CuO$_4$ ($x=0.22$) suitable for angle-resolved photoemission. We observe the suppression of the superconducting spectral gap within our energy resolution ($\sim 4~\mathrm{meV}$), and surprisingly, the expected zero-energy flat band peak is also suppressed, despite the high topographic quality of the surface. Self-consistent Bogoliubov--de~Gennes calculations show that the measured geometric roughness of the cleaved surface is too weak to eliminate these modes. The calculations further demonstrate that bulk inhomogeneities characteristic of high-temperature superconductors, modelled as moderate Anderson-type disorder, can broaden the flat-band states beyond detectability. Our results provide the first momentum-resolved view of the electronic structure on a cuprate side surface and reveal disorder as the key factor currently preventing appearance of flat bands and their associated correlated orders.

Figures (13)

• Figure 1: Edge states in clean $d_{x^{2}-y^{2}}$-wave superconductors.a) Illustration of the realization of novel phenomena at the intersection of topology, strong correlations and superconductivity, ranging from topological superconductivity (TSC), unconventional superconductivity (USC) and correlated topological matter (CTM). b) Illustration of the mapping defined by the field $\bm{d}_{k_{\parallel}}$ from one-dimensional submanifolds of the Brillouin zone, which is homeomorphic to a 2-Torus, into the group $\mathbb{R}^{2}\setminus\{0\}$, which is homotopy equivalent to $S^1$. c) Illustration of the vector field $\bm{d}_{\bm{k}}$ for a $d_{x^{2}-y^{2}}$-wave superconductor in the first two Brillouin zones of a square lattice, so that $k_{\perp} = k_{[11]}$ and $k_{\parallel} = k_{[1\bar{1}]}$. The red horizontal line corresponds to a loop in momentum space as in the previous panel. The superconducting nodes of winding number $+1$ and $-1$ are shown in red and blue, respectively. The right inset shows the resulting winding numbers $\nu_{k_{\parallel}}$ computed along $k_{\perp}$ and color coded based on their sign, which also identifies their chirality. d) Band structures for different edge orientations, with the two eigenvalues closest to the center of the spectrum highlighted in red and blue according to their chirality eigenvalues. The figure on the left corresponds to a [11] edge, in the same configuration as in Fig. \ref{['figure_1']}c, where the non-trivial values of the winding numbers lead to the formation of zero-energy flat bands connecting the projections of the superconducting nodes of opposite winding numbers. On the other hand, the figure on the right corresponds to a [10] edge, where the winding number is trivial for all $k_{\parallel} = k_{[10]}$. In all the figures we defined $\delta = a\sqrt(2)$, with $a$ the lattice constant of the square lattice. e) Exponential decay of the edge states away from the edge ($r_{[1\bar{1}]} = 0$) and their delocalization in the bulk in correspondence of the projection of the superconducting nodes at $k_{[11]} = -k_{F}$, $0$, $k_{F}$. The analytical expression of the edge state wavefunction is derived in the Supplementary section \ref{['sup_sub_sec_1']}.
• Figure 2: Sample preparationa) Illustration showing the different stages of the sample preparation. b) Temperature dependence of the magnetic susceptibility measured by SQUID magnetometry for a piece of the original rod (red) and of a small part of the milled region of the pillar (blue). Measurements were performed under zero-field-cooled conditions with an applied magnetic field of 5 Oe (red) and 2 Oe (blue), showing no change in the critical temperature during sample preparation. c) Scanning Electron Microscope image of the sample after the mechanical cleave. d) Topography gradient map of the cleaved surface obtained by Atomic Force Microscope and (e) its corresponding height distribution showing a roughness of only $1.8\ \text{\AA}$ (for comparison the in-plane lattice constant of LSCO is $3.78\ \text{\AA}$). The roughness is defined as the root-mean-square deviation of the height profile.
• Figure 3: ARPES spectra of a (110) surface of LSCO.a) Calculated three-dimensional Fermi surface of LSCO showing the contours at $k_{z} = 0$ (red) as in panel b and the nodal cut (blue) as in panel c. b) Varying the photon energy from a (110) surface yields a cut of the well known (001) Fermi surface of LSCO shown in the left plot. The right plot shows the calculated (001) Fermi surface at the $\Gamma X \Sigma$ plane, for comparison. c) On the other hand, varying the analyzer deflector angle allows to directly obtain the dispersion of the bands along the [001] direction. Here the photon energy was 79 eV, corresponding to the nodal cut shown also in the next panel. d) Dispersion of the bands at the superconducting nodes taken with a photon energy of 79 eV, and (e) at the antinodes taken with a photon energy of 95 eV. Notice that due to $k_{z}$ broadening, the small dispersion of the bands at the antinodes is not recognizable as the bands become smeared out along the [110] direction, along which they have a large dispersion. This results in the strong intensity observed between the two visible bands. All the calculations include a $k_{z}$ broadening of $\Delta k_{[110]} = 0.2\ \text{\AA}^{-1}$, based on the typical inelastic mean-free path of photoelectrons with energies in the Vacuum Ultraviolet (VUV) region, which is about $5\ \text{\AA}$. All measurements were performed at 6 K.
• Figure 4: Boundary effects at a (110) surface of a $d_{x^{2}-y^{2}}$-wave superconductor.a) Simulated spectral function of the [11] edge at the Fermi level. b) $d_{x^{2}-y^{2}}$ pairing amplitude versus distance from the [11] edge for a clean system (C), AFM-derived geometric edge roughness (GE${\mathrm{AFM}}$), probabilistic roughness (GE${0.5}$), and GE${\mathrm{AFM}}$ combined with bulk Anderson disorder (BA). The BA disorder is Gaussian with a standard deviation of $100~\mathrm{meV}$. c) Real-space expectation value of edge-state eigenstates for AFM-derived geometric edge roughness. States are selected by their eigenvalues and participation ratios. Dotted lines indicate periodic boundary conditions. d) Simulated antinodal spectral functions for C (left) and GE$_{\mathrm{AFM}}$ + BA disorder (right), extracted along the red line in a. e) Simulated antinodal EDCs for different disorder realizations and $k{[11]}$; purple, black, and orange arrows mark the surface-state peak, DOS dip, and coherence peak, respectively. f) ARPES spectra measured with 95 eV photons and at 6 K (see Fig. \ref{['figure_2']}b). g) Antinodal EDCs ($h\nu = 95~\mathrm{eV}$) normalized by total intensity at different temperatures, compared with the gold DOS obtained at 6 K. The integration ranges are indicated by the colored lines above d and f (see also the Supplementary section \ref{['Supplementary:EDC_boundary']}). Reported values are mean intensities within these ranges, normalized to the total intensity. For the ARPES data, a background subtraction following Ref. Matt_2018 was applied prior to the EDC extraction. All calculations are performed at $0~\mathrm{K}$.
• Figure 5: Let $X \coloneq [-\pi, \pi] \times [-\pi, \pi]$ be a square Brillouin zone with the equivalence relation $(k_{\parallel},-\pi) \sim (k_{\parallel},\pi)$ and $(-\pi,k_{\perp}) \sim (\pi,k_{\perp})$. Then the quotient space $X/\sim$ is a $T^{2}$ torus. Then paths with a constant momentum component are topologically equivalent to non-contractable $S^{1}$. Each map $d_{k}$ is then connecting the elements of these loops to the elements of another $S^{1}$ loop.