Simultaneous transitions in cuprate momentum-space topology and electronic symmetry breaking
In recent years, several experimental probes have reported broken symmetries In underdoped cuprates of almost linearly decreasing onset temperatures with doping. However, how the real-space ordering and fluctuations are related to the momentum space electronic structure that hosts high temperature superconductivity and its apparent topology change is not well established. It is primarily because the techniques used are single-sided; scattering experiments can only see periodic arrangements in real space, while photoemission reveals electronic bands with no direct access to real space. In contrast, SI-STM can, and only SI-STM can simultaneously investigate both real and momentum space from a single measurement. The doping evolution of the broken symmetry as well as the Fermi surface transition are studied here with Bi2Sr2CaCu2O8+δ by spanning the entire range of superconductivity from heavily underdoped to highly overdoped region.
 Momentum Space: Fermi surface
Out of the seven distinct quasi-particle interference (QPI) peaks connecting the octet regions of high density of states, q4 is unique in that its trajectory by definition is simply a replica of the Fermi surface rescaled by a factor of two. The q4 loci at different energies are conveniently integrated into a single curve by adding the Fourier-transform of individual energy layers up to the superconducting gap edge. The Fermi surface thus extracted clearly shows an abrupt transition from truncated 'arc' to fully closed contour centered at (π, π) near p=19%.
[Figure Caption] A, Schematic of d-wave superconducting energy gap Δ(kF) on two opposing segments of the Fermi surface. The q4 wavevector for the octet scattering process q4(E=Δ(kF))=2kF that always passes through the Brillouin zone symmetry point (0, 0) is indicated for several Δ(kF) by colored arrows B, By focusing on q4(E) one can then determine the location of the Fermi surface because it is the set of wavevectors k=q4(0<E<Δ0)/2. This is demostrated using the correspondence between the colored arrows q4(E) in A and the colored arrows k(Δ)=q4(E=Δ(kF))/2. Here we show the resulting kF that we measure for a sample with p=0.23. C, The measured Λ(q), Z(q, E) integrated up to the SC energy scale for underdoped p=0.14 sample. No complete contour for q4 can be detected; instead, the coherent Bogoliubov QPs are found to be restricted to four arcs terminating at AFZB. D, The measured Λ(q) for overdoped p=0.23 sample. A complete closed contour for q4 surrounding (±1,±1)Δ/a0 can be identified immediately. E, The measured doping dependence of the k-space topology of coherent BG QP in Bi2212 using q4 technique. The transition from arcs terminating at AFZB to complete hole-pockets surrounding (π,π) points at p~0.19 is strikingly clear.
 Real space: broken symmetry
Electronic intra-unit-cell rotational symmetry breaking and incommensurate charge modulation are best characterized in real space. The former is captured with the inequivalence of electronic density of states at the two oxygen sites lying along x and y direction off a Cu atom. Statistics on the spatial distribution of the nematic order parameter clearly exhibits monotonously vanishing average magnitude along with increasing spread (more disordered) toward ~19% doping. The domain sizes of opposite ising nematic order parameter become comparable beyond ~19% leading to zero average, i. e., disappearance of global nematicity.
[Figure Caption] A, Measured Z(r, E~Δ1) for p=0.08. Incommensurate bidirectional modulations are clearly seen. B, Measured Z(r, E~Δ1) for p=0.23. No specific q-vector for modulations is seen although the QPI signature of Bogoliubov quasiparticles does produce a jumbled standing wave pattern. C, Measured Z(r, E~Δ1) for p=0.08 from A. The two wavevectors universally reported in SI-STM experiments for the broken-symmetry states, q1* and q5* are indicated using violet and orange circles, respectively. The two Bragg peaks are shown using red circles. D, Measured Z(r, E~Δ1) for p=0.08 from B. No specific broken-symmetry state wavevectors are apparent, while the residual dispersive effects of Bogoliubov quasiparticles are seen. E, Measured Q=0 C4 broken-symmetry order parameter ONq(r, E~Δ1) for p=0.06; the whole field of view is a single color indicating that long range Q=0 intra-unit-cell C4 breaking exists. F, Measured intra-unit-unit cell broken C4 symmetry ONq(r, E~Δ1) for p=0.22; long-range order has been lost but nanoscale domains of opposite nematicity persist. Broken circle represents the spatial resolution of the analysis. G, Measured intensity of incommensurate modulation with wavevector q5*: I(q5*). The incommensurate modulations do not show long-range order in contrast to the intra-unit-cell nematicity. Instead, the modulations show severe spatial fluctuation in phase and amplitude; such fluctuations are reflected in how broad the modulation peaks are in C. Nevertheless, the intensity of the modulation peaks initially increase upon doping peaking near p~1/8, and then diminish to reach zero at p~0.19. H, Filled square symbols indicate measured spatial average value of the Q=0 C4 broken symmetry |ONq(r, E~Δ1)| which diminishes steadily with increasing p, to reach zero at p=0.19. Triangles indicate measured standard deviation of the Q=0 C4 broken symmetry ONq(r, E~Δ1) which increases as the spatial range of fluctuations increases.
Both spatial ordering phenomena disappear between 17% and 20% doping at which the Fermi surface undergoes abrupt transition from arc to closed full contour. The simultaneous transitions of real and momentum space electronic structures implies that the broken symmetries are closely linked with the Fermi surface topology. Moreover, the fact that all transitions occur at around 19% doping points toward quantum critical point concealed underneath the superconducting dome.
[Figure Caption] A, Schematic of k-space locus of states generating Bogoliubov QPI with increasing hole density in Bi2212. An abrupt transition occurs at p~0.19. B, The k-space area proportional to hole-density p is observed between the arc and the AFZB. This transitions with the appearance of the closed FS at p~0.19, to a diminishing area of electron count 1-p (as would be expected conventionally for uncorrelated electrons). C, The wavevectors k of states at which Bogoliubov quasiparticle scattering interference disappears qE/2 (closed circles), and those of the quasistatic broken-symmetry modulations with q1/2 and (2π-q5*)/2 (closed squares). It is unknown why all these wavevectors, characteristic of both termination in the Bogoliubov QPI and the Q≠0 broken symmetry states, are associated so closely with the AFZB. D, T-matrix scattering interference simulation for Λ(q) for band structure with a complete Fermi surface upon which a d-wave gap opens and for conventional (time-reversal preserving) scattering. E, Measured Λ(q) for p=0.14 showing the combined effects of all detectable Bogoliubov QPI. In stark contrast to the simulation result in D, the energy-integrated QPI data reveals finer features near the nodal direction resulting from the dispersion of "octet" peaks characteristic of underdoped cuprates, while the set of scattering q-vectors dispersed unexpectedly with energy are limited to a set of four arcs. Conventional Bogoliubov scattering on the complete Fermi surface (D) definitely fails to reproduce all these notable hallmarks. F, For p=0.23, on the contrary, Λ(q) conforms very nicely to the conventional d-wave Bogoliubov scattering scheme in that the FS is fully closed and that the large nodal 'blobs' are unresolvable. The clear distinction between E and F, and the excellent correspondence between D and F, strongly suggests that p~0.19 is also associated with the critical change of the underlying QP scattering mechanism.
These results were published in Science 344, 612.
Intra-unit-cell electronic nematicity of the high-Tc copper-oxide pseudogap states
A route to find the mechanism of cuprate superconductivity is to understand the physics of pseudogap states. We use SI-STM to probe the electronic property of pseudogap states in underdoped Bi2Sr2CaCu2O8+δ, and directly visualize the broken symmetries and their interaction.
[Figure Caption] A, A schematic phase diagram of cuprate. Pseudogap states emerge at T* much higher than Tc, and T* decreases with increasing hole density. B, Observed cuprate pseudogap states by SI-STM. Inset shows its Fourier transform indicating two different kinds of electronic modulations in the pseudogap states; electronic modulations commensurate to the lattice (Qx and Qy) and incommensurate to the lattice (Sx and Sy).
We have discovered an evidence for electronic nematicity of the states close to the pseudogap energy of strongly underdoped Bi2Sr2CaCu2O8+δ. Moreover, we demonstrated directly that broken symmetry arises from electronic inequivalence at the two oxygen sites within each unit cell. If the characteristics of the pseudogap seen here and by other techniques all have the same microscopic origin, this phase involves weak magnetic states at the O sites that break 90°-rotational symmetry within every CuO2 unit cell. These results were published in Nature 466, 374 (2010).
[Figure Caption] A, Topographic image. Inset shows real part of Bragg peak intensitiy for Cu-O bond directions x, and y. Same height indicates that crystal structure preserve 90°-rotational (C4) symmetry.
B, Image of pseudogap states. Inset shows an inequivalent Bragg peak height indicating the C4 symmetry breaking in the electronic structure.
C, C4 symmetry breaking function at different energies, which is proportional to the difference of Bragg peak height between x and y direction. C4 symmetry breaking becomes evident at pseudogap energy scale e~1.
D, Topographic image in the area indicated by box in Fig. 1A. Cu, Ox and Oy are located.
E, Image of pseudogap states at corresponding FOV to Fig. 2D overlapped with Cu, Ox, and Oy locations. Broken symmetry primarily arises at oxygen sites.
F, Intensity at Ox and Oy in the image of pseudogap states are directly compared as a function energy. C4 symmetry breaking becomes evident at e~1.
Topological defects coupling smectic modulations to intra-unit-cell nematicity in cuprates
We have visualized the spatial components of coexisting smectic modulations and intra-unit-cell nematicity in the pseudogap states of underdoped Bi2Sr2CaCu2O8+δ, and identified 2π topological defects throughout the phase-fluctuating smectic states. Imaging the locations of large numbers of these topological defects simultaneously with the fluctuations in the intra-unit-cell nematicity revealed strong empirical evidence for a coupling between them.
[Figure Caption] A, B Image of pseudogap states is fourier filtered in terms of incommensurate modulations Sx(A) and Sy(B). Dislocations are seen in the pseudogap states as highlighted by box. C, D phase shift of the incommensurate modulations Sx(C) and Sy(D) of pseudogap states. At the locations of dislocation, 2π topological defects are found, which are marked by dots (phase winding directions: black-clockwise, white-counterclockwise).
E, spatial fluctuation of C4 symmetry breaking component together with topological defects. Topological defects tend to be located at zero fluctuation lines indicating an interplay between commensurate and incommensurate modulations in the pseudogap states.
F, a schematic image of interaction between commensurate and incommensurate modulations, which is proposed to explain our experimental results. Topological defects create spatial fluctuation of C4 symmetry breaking function through .
Strongly underdoped double-layer BSCCO study in Brookhaven