Imaging nanoscale Fermi-surface variations in an inhomogeneous superconductor

Abstract

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...

Universal disorder in Bi2Sr2CaCu2OAlldredge1, Fujita2, Eisaki3 et al. 2013Phys. Rev. B

“…Overall  cross is not dependent on the sample doping/ Fermi surface, but changes throughout each sample in a manner consistent with a continuously changing PAF-zone boundary intersection driven by Fermi surface that varies from patch to patch. This agrees with the measurement of the local variance in q1* measured in Bi-2201 50 . This continuous change suggests that each patch can be modeled as having the equivalent electronic structure as a homogeneous sample consisting of a single  1 value and doping.…”

“…Each patch with the same  1 value will have the same  cross and its value should scale with local effective doping. The 2 nd scenario is supported by previous work that shows the q 1 * modulations wavelength varies locally and that its wavelength depends on the local  1 value of a patch in Bi-2201 50 . Although here we have shown that the q 1 * modulation is confined to a specific energy range.…”

Competition between spin density wave order and superconductivity in the underdoped cuprates

“…2b, but with their components determined in a 'rotating reference frame' set by the local orientation of ϕ. This idea of Fermi surfaces correlated with the local order is supported by the recent STM observations of Wise et al 32 . Focussing only on the electron pocket components, we can write the physical electron operators c α as 7,8 c…”

Quasiparticle interference and the interplay between superconductivity and density wave order in the cuprates

“…1,8,9 STS has revealed that the cuprates have a spatially inhomogeneous electronic structure, including modulations in the LDOS and superconducting gap magnitude. [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] In the d-wave superconducting phase, the LDOS modulations can arise from quasiparticle interference (QPI), due to the scattering of wave-like quasiparticles off impurities. 12,15,22,[25][26][27] The wavevectors of the modulations can be determined from the Fourier transform of the LDOS.…”