One of the main challenges in understanding high T C superconductivity is to disentangle the rich variety of states of matter that may coexist, cooperate, or compete with d-wave superconductivity. At center stage is the pseudogap phase, which occupies a large portion of the cuprate phase diagram surrounding the superconducting dome 1 . Using scanning tunneling microscopy, we find that a static, non-dispersive, "checkerboard"-like electronic modulation exists in a broad regime of the cuprate phase diagram and exhibits strong doping dependence. The continuous increase of checkerboard periodicity with hole density strongly suggests that the checkerboard originates from charge density wave formation in the anti-nodal region of the cuprate Fermi surface. These results reveal a coherent picture for static electronic orderings in the cuprates and shed important new light on the nature of the pseudogap phase. Author ContributionsWDW, MCB and KC shared equal responsibility for all aspects of this project from instrument construction through data collection and analysis. TK grew the samples and helped refine the STM. TT and HI contributed to sample growth. YW contributed to analysis and writing of the manuscript. EWH advised.

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The unclear relationship between cuprate superconductivity and the pseudogap state remains an impediment to understanding the high transition temperature (T(c)) superconducting mechanism. Here, we used magnetic field-dependent scanning tunneling microscopy to provide phase-sensitive proof that d-wave superconductivity coexists with the pseudogap on the antinodal Fermi surface of an overdoped cuprate. Furthermore, by tracking the hole-doping (p) dependence of the quasi-particle interference pattern within a single bismuth-based cuprate family, we observed a Fermi surface reconstruction slightly below optimal doping, indicating a zero-field quantum phase transition in notable proximity to the maximum superconducting T(c). Surprisingly, this major reorganization of the system's underlying electronic structure has no effect on the smoothly evolving pseudogap.

Particle-wave duality suggests we think of electrons as waves stretched across a sample, with wavevector k proportional to their momentum. Their arrangement in 'k-space', and in particular the shape of the Fermi surface, where the highest-energy electrons of the system reside, determine many material properties. Here we use a novel extension of Fourier-transform scanning tunnelling microscopy to probe the Fermi surface of the strongly inhomogeneous Bi-based cuprate superconductors. Surprisingly, we find that, rather than being globally defined, the Fermi surface changes on nanometre length scales. Just as shifting tide lines expose variations of water height, changing Fermi surfaces indicate strong local doping variations. This discovery, unprecedented in any material, paves the way for an understanding of other inhomogeneous characteristics of the cuprates, such as the pseudogap magnitude, and highlights a new approach to the study of nanoscale inhomogeneity in general.That high-temperature superconductors should show nanoscale inhomogeneity is unsurprising. In correlated electron materials, Coulomb repulsion between electrons hinders the formation of a homogeneous Fermi liquid, and complex real-space phase separation is ubiquitous 1 (Bi-2212; refs 3-5).This intrinsic inhomogeneity poses challenges to the interpretation of bulk or spatially averaged measurements. For example, angle-resolved photoemission spectroscopy (ARPES) is a powerful technique for studying k-space structure in the cuprates 6 . However, ARPES can provide only spatially averaged results, and uniting these with the nanoscale disordered electronic structure measured by STM remains a formidable task.Our approach to addressing this issue originates from discoveries by Fourier-transform STM (FT-STM), which has emerged as an important tool for studying the cuprates. These studies begin with the collection of a spectral survey, in which differential conductance spectra, proportional to local density of states (LDOS), are measured at a dense array of locations, creating a three-dimensional dataset of LDOS as a function of energy and position in the plane. By Fourier transforming constant-energy slices of these surveys, referred to as LDOS or conductance maps, FT-STM enables the study of two phenomena linked to the cuprate Fermi surface (FS) (Fig. 1b). First, non-dispersive wavevectors of the checkerboardlike charge order observed in many cuprates 7-10 are probably connected to the FS-nesting wavevectors near the antinodal (π,0) Brillouin zone boundary (see, for example, the arrow in Fig. 1b) 11 . Second, dispersive quasiparticle interference (QPI) patterns 12-14 originate from elastic scattering of quasiparticles on the FS near the nodal (π, π) direction 15 . Taken together, these phenomena provide complementary information about the cuprate FS. However, because these phenomena were previously characterized using Fourier transforms of large LDOS maps containing a wide range of energy gaps and spectra, previous FT-STM mapping of the FS was still sp...

We investigate the topological surface state properties at various surface cleaves in the topological insulator Bi2Se3, via first principles calculations and scanning tunneling microscopy/spectroscopy (STM/STS). While the typical surface termination occurs between two quintuple layers, we report the existence of a surface termination within a single quintuple layer where dangling bonds form with giant spin splitting owing to strong spin-orbit coupling. Unlike Rashba split states in a 2D electron gas, these states are constrained by the band topology of the host insulator with topological properties similar to the typical topological surface state, and thereby offer an alternative candidate for spintronics usage. We name these new states "topological dangling-bond states". The degree of the spin polarization of these states is greatly enhanced. Since dangling bonds are more chemically reactive, the observed topological dangling-bond states provide a new avenue for manipulating band dispersions and spin-textures by adsorbed atoms or molecules.