Chung Koo Kim

Intra-unit-cell electronic nematicity of the high-Tc copper-oxide pseudogap states

In the high-transition-temperature (high-Tc) superconductors the pseudogap phase becomes predominant when the density of doped holes is reduced. Within this phase it has been unclear which electronic symmetries (if any) are broken, what the identity of any associated order parameter might be, and which microscopic electronic degrees of freedom are active. Here we report the determination of a quantitative order parameter representing intra-unit-cell nematicity: the breaking of rotational symmetry by the electronic structure within each CuO2 unit cell. We analyse spectroscopic-imaging scanning tunnelling microscope images of the intra-unit-cell states in underdoped Bi2Sr2CaCu2O8 + δ and, using two independent evaluation techniques, find evidence for electronic nematicity of the states close to the pseudogap energy. Moreover, we demonstrate directly that these phenomena arise from electronic differences 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.

Simultaneous Transitions in Cuprate Momentum-Space Topology and Electronic Symmetry Breaking

Under the Dome The superconducting transition temperature Tc of copper oxides has a dome-shaped dependence on chemical doping. Whether there is a quantum critical point (QCP) beneath the dome, and whether it is related to the enigmatic pseudogap, has been heavily debated. Two papers address this question in two different families of Bi-based cuprates. In (Bi,Pb)2(Sr,La)2CuO6+δ, He et al. (p. 608) found that the Fermi surface (FS) undergoes a topological change as doping is increased, which points to the existence of a QCP at a doping close to the maximum in Tc, seemingly uncorrelated with the pseudogap. Fujita et al. (p. 612) studied a range of dopings in Bi2Sr2CaCu2O8+δ to find an FS reconstruction simultaneous with the disappearance of both rotational and translational symmetry breaking, the latter of which has been associated with the pseudogap. These findings point to a concealed QCP. Scanning tunneling microscopy is used to provide evidence for a quantum critical point beneath the superconducting dome. The existence of electronic symmetry breaking in the underdoped cuprates and its disappearance with increased hole density p are now widely reported. However, the relation between this transition and the momentum-space (k→-space) electronic structure underpinning the superconductivity has not yet been established. Here, we visualize the Q→=0 (intra–unit-cell) and Q→≠0 (density-wave) broken-symmetry states, simultaneously with the coherent k→-space topology, for Bi2Sr2CaCu2O8+δ samples spanning the phase diagram 0.06 ≤ p ≤ 0.23. We show that the electronic symmetry-breaking tendencies weaken with increasing p and disappear close to a critical doping pc = 0.19. Concomitantly, the coherent k→-space topology undergoes an abrupt transition, from arcs to closed contours, at the same pc. These data reveal that the k→-space topology transformation in cuprates is linked intimately with the disappearance of the electronic symmetry breaking at a concealed critical point.

Direct phase-sensitive identification of a d-form factor density wave in underdoped cuprates

Significance High-temperature superconductivity emerges when holes are introduced into the antiferromagnetic, insulating CuO2 plane of the cuprates. Intervening between the insulator and the superconductor is the mysterious pseudogap phase. Evidence has been accumulating that this phase supports an exotic density wave state that may be key to its existence. By introducing visualization techniques that discriminate the electronic structure at the two oxygen sites with each CuO2 unit cell, we demonstrate that this density wave consists of periodic modulations maintaining a phase difference of π between every such pair of oxygen sites. Therefore, the cuprate pseudogap phase contains a previously unknown electronic state—a density wave with a d-symmetry form factor. The identity of the fundamental broken symmetry (if any) in the underdoped cuprates is unresolved. However, evidence has been accumulating that this state may be an unconventional density wave. Here we carry out site-specific measurements within each CuO2 unit cell, segregating the results into three separate electronic structure images containing only the Cu sites [Cu(r)] and only the x/y axis O sites [Ox(r) and Oy(r)]. Phase-resolved Fourier analysis reveals directly that the modulations in the Ox(r) and Oy(r) sublattice images consistently exhibit a relative phase of π. We confirm this discovery on two highly distinct cuprate compounds, ruling out tunnel matrix-element and materials-specific systematics. These observations demonstrate by direct sublattice phase-resolved visualization that the density wave found in underdoped cuprates consists of modulations of the intraunit-cell states that exhibit a predominantly d-symmetry form factor.

Imaging Dirac-mass disorder from magnetic dopant atoms in the ferromagnetic topological insulator Crx(Bi0.1Sb0.9)2-xTe3.

Significance Surface states of topological insulators (TIs) should exhibit extraordinary electronic phenomena when a ‘Dirac-mass gap’ is opened in their spectrum, typically by creating a ferromagnetic state. However, our direct visualization of the Dirac-mass gap Δ(r) in a ferromagnetic TI reveals its intense disorder at the nanoscale. This is correlated with the density of magnetic dopant atoms n(r), such that Δ(r)∝n(r) as anticipated for surface-state–mediated ferromagnetism. Consequent new perspectives on ferromagnetic TI physics include that the quantum anomalous Hall effect occurs in this environment of extreme Dirac-mass disorder and that paths of associated chiral edge states must be tortuous. To achieve all the exotic physics expected of ferromagnetic TIs, greatly improved control of dopant-induced Dirac-mass gap disorder is therefore required. To achieve and use the most exotic electronic phenomena predicted for the surface states of 3D topological insulators (TIs), it is necessary to open a “Dirac-mass gap” in their spectrum by breaking time-reversal symmetry. Use of magnetic dopant atoms to generate a ferromagnetic state is the most widely applied approach. However, it is unknown how the spatial arrangements of the magnetic dopant atoms influence the Dirac-mass gap at the atomic scale or, conversely, whether the ferromagnetic interactions between dopant atoms are influenced by the topological surface states. Here we image the locations of the magnetic (Cr) dopant atoms in the ferromagnetic TI Cr0.08(Bi0.1Sb0.9)1.92Te3. Simultaneous visualization of the Dirac-mass gap Δ(r) reveals its intense disorder, which we demonstrate is directly related to fluctuations in n(r), the Cr atom areal density in the termination layer. We find the relationship of surface-state Fermi wavevectors to the anisotropic structure of Δ(r) not inconsistent with predictions for surface ferromagnetism mediated by those states. Moreover, despite the intense Dirac-mass disorder, the anticipated relationship Δ(r)∝n(r) is confirmed throughout and exhibits an electron–dopant interaction energy J* = 145 meV·nm2. These observations reveal how magnetic dopant atoms actually generate the TI mass gap locally and that, to achieve the novel physics expected of time-reversal symmetry breaking TI materials, control of the resulting Dirac-mass gap disorder will be essential.

Atomic-scale electronic structure of the cuprate d-symmetry form factor density wave state

A detailed scanning tunnelling microscopy study of the cuprate superconductor Bi2Sr2CaCu2O8+x reveals the microscopic origin of the d-symmetry form factor density wave that exists in the pseudogap phase of this material.

Spectroscopic Imaging STM: Atomic-Scale Visualization of Electronic Structure and Symmetry in Underdoped Cuprates

Atomically resolved spectroscopic imaging STM (SI-STM) has played a pivotal role in visualization of the electronic structure of cuprate high temperature superconductors. In both the d-wave superconducting (dSC) and the pseudogap (PG) phases of underdoped cuprates, two distinct types of electronic states are observed when using SI-STM. The first consists of the dispersive Bogoliubov quasiparticles of a homogeneous d-wave superconductor existing in an energy range \(\vert {}E\vert {} \le {}\varDelta _{0}\) and only upon an arc in momentum space (k-space) that terminates close to the lines connecting k \(=\) \(\pm {}(\pi {}/a_{0},0)\) to k \(=\) \(\pm {}(0, \pi {}/a_{0})\). This ‘nodal’ arc shrinks continuously as electron density increases towards half filling. In both phases, the only broken symmetries detected in the \(\vert E\vert \le \varDelta _{0}\) states are those of a d-wave superconductor. The second type of electronic state occurs near the pseudogap energy scale \(\vert E\vert \sim \varDelta _{1}\) or equivalently near the ‘antinodal’ regions k \(=\) \(\pm (\pi /a_{0},0)\) and k \(=\) \(\pm (0, \pi /a_{0})\). These states break the expected 90\(^{\circ }\)-rotational (C\(_{4}\)) symmetry of electronic structure within each CuO\(_{2}\) unit cell, at least down to 180\(^{\circ }\)-rotational (C\(_{2}\)), symmetry. This intra-unit-cell symmetry breaking is interleaved with the incommensurate conductance modulations locally breaking both rotational and translational symmetries. Their wavevector S is always found to be determined by the k-space points where Bogoliubov quasiparticle interference terminates along the line joining \(\mathbf k =(0,\pm \pi /a_{0})\) to \(\mathbf k =(\pm \pi /a_{0},0)\), and thus diminishes continuously with doping. The symmetry properties of these \(\vert E\vert \sim \varDelta _1\) states are indistinguishable in the dSC and PG phases. While the relationship between the \(\vert E\vert \sim \varDelta _1\) broken symmetry states and the \(\vert E\vert \le \varDelta _{0}\) Bogoliubov quasiparticles of the homogeneous superconductor is not yet fully understood, these two sets of phenomena are linked inextricably because the k-space locations where the latter disappears are always linked by the modulation wavevectors of the former.

Underpinning Data: Atomic-scale Electronic Structure of the Cuprate d-Symmetry Form Factor Density Wave State

Research on high-temperature superconducting cuprates is at present focused on identifying the relationship between the classic ‘pseudogap’ phenomenon1,2 and the more recently investigated density wave state3–13. This state is generally characterized by a wavevector Q parallel to the planar Cu–O–Cu bonds4–13 along with a predominantly d-symmetry form factor14–16 (dFF-DW). To identify the microscopic mechanism giving rise to this state17–29, one must identify the momentum-space states contributing to the dFF-DW spectral weight, determine their particle–hole phase relationship about the Fermi energy, establish whether they exhibit a characteristic energy gap, and understand the evolution of all these phenomena throughout the phase diagram. Here we use energy-resolved sublattice visualization14 of electronic structure and reveal that the characteristic energy of the dFF-DW modulations is actually the ‘pseudogap’ energy ∆1. Moreover, we demonstrate that the dFF-DW modulations at E=−∆1 (filled states) occur with relative phase π compared to those at E=∆1 (empty states). Finally, we show that the conventionally defined dFF-DW Q corresponds to scattering between the ‘hot frontier’ regions of momentum-space beyond which Bogoliubov quasiparticles cease to exist30–32. These data indicate that the cuprate dFF-DW state involves particle–hole interactions focused at the pseudogap energy scale and between the four pairs of ‘hot frontier’ regions in momentum space where the pseudogap opens. A conventional ‘Peierls’ charge density wave (CDW) in a metal results from particle–hole interactions which open an energy gap at specific regions of k-space that are connected by a common wavevector Q. This generates a modulation in the density of free charge at Q along with an associated modulation of the crystal lattice parameters. Such CDW states are now very well known33. In principle, a density wave modulating at Q can also exhibit a ‘form factor’ (FF) with different possible symmetries34,35 (see Supplementary Section 1). This is relevant to the high-temperature superconducting cuprates because numerous researchers have recently proposed that the ‘pseudogap’ regime1,2 (PG in Fig. 1a) contains an unconventional density wave with a d-symmetry form factor17–29. The basic phenomenology of such a state is that intraunit-cell (IUC) symmetry breaking renders the Ox and Oy sites within each CuO2 unit-cell electronically inequivalent, and that this inequivalence is then modulated periodically at wavevector Q parallel to (1,0);(0,1). The real-space (r-space) schematic of such a d-symmetry FF density wave (dFF-DW) atQx , as shown in Fig. 1b, exemplifies periodic modulations at the Ox sites that are π out of phase with those at the Oy sites. Such a state is then described by A(r)=D(r)cos(φ(r)+φ0(r)), where A(r) represents whatever is the modulating electronic degree of freedom, φ(r)=Qx ·r is the DW spatial phase at location r, φ0(r) represents disorder related spatial phase shifts, and D(r) is the magnitude of the d-symmetry form factor14,21,23. To distinguish between the various microscopic mechanisms proposed for the Q=(Q, 0);(0,Q) dFF-DW state of cuprates17–29, it is essential to establish its atomic-scale phenomenology, including the momentum space (k-space) eigenstates contributing to its spectral weight, the relationship (if any) between modulations occurring above and below the Fermi energy, whether the modulating states in the DW are associated with a characteristic energy gap, and how the dFF-DW evolves with doping. To visualize such phenomena directly as in Fig. 1c, we use sublattice-phase-resolved imaging of the electronic structure14 of the CuO2 plane. Both the scanning tunnelling microscope (STM) tip–sample differential tunnelling conductance dI/dV (r,E=eV)≡g (r,E) and the tunnel-current I(r, E) are measured at bias voltage V = E/e and with sub-unitcell spatial resolution. Because the density of electronic states N (r, E) is related to the differential conductance as g (r, E) ∝ [eIs/ ∫ eVs 0 N (r,E )dE ]N (r, E), with Is and Vs being arbitrary parameters and the denominator ∫ eVs 0 N (r,E )dE ′ unknown, valid imaging of N (r,E) is challenging (Supplementary Section 2). However, one can suppress these serious systematic ‘set-point’ errors by using R(r, E) = I(r, E)/I(r, −E) or Z(r,E)=g (r,E)/g (r,−E); this allows distances, wavelengths and spatial phases of electronic structure to be measured accurately. The unprocessed g (r, E) acquired for and analysed in this paper are measured over very large fields of view (to achieve high phase precision in Fourier analysis), simultaneously maintain deeply sub-unit-cell precision measurements in r (to achieve high precision in sublattice segregation), and are taken over a wide range of energies E with fine energy spacing, so that energy dependences of d-symmetry FF modulations may be accurately determined. We then calculate each sublattice-phase-resolved Z(r,E) image and separate it into three: the first, Cu(r), contains