Below, I describe some highlights of my research over the past several years focusing on quantum electronic matter. Additionally, I have been looking for atomic scale signatures of quantum criticality in heavy fermion compounds, exploring correlated topological materials, and working on new scanning probe technologies including fractional-charge imaging and scanned Josephson microcopy to directly visualize superfluid density. Recently, I have begun using molecular-beam-epitaxy in conjunction with SI-STM to study the single layer FeSe compound believed to exhibit properties of high temperature superconductivity. Please see the Publications Page for a complete list of my published and upcoming papers.

  • Correlated Topological Materials

    Correlated Topological Materials

    We search for the tell-tale signature of strong correlations generating a topological electronic structure and find heavy Dirac fermions. CLICK TO EXPAND

    (Under Review) Correlated topological matter is a frontier in the search for exotic quantum phases. Theory now predicts that heavy fermion systems, fertile grounds for discovery in strongly interacting electronic materials, may host such novel phases in the form of topological Kondo insulators. Within these systems, a correlation-driven gap protected by a bulk topological invariant is predicted to harbor emergent surface modes that are entangled with f-electrons, spawning heavy Dirac fermions. In stark contrast to conventional surface states of the non-interacting topological insulators, heavy Dirac fermions are expected to give rise to exotic Dirac liquid states, non-Abelian quantum statistics and topological order. In search of these strongly interacting topological states SmB6 has recently emerged as the most promising candidate. However, no experiments have directly observed the correlated ground state and its emergent heavy Dirac fermions. We used heavy fermion quasiparticle interference imaging and co-tunneling spectroscopy to resolve the topological nature of SmB6. On cooling through TΔ* ≈ 35 K we observed the opening of a Kondo insulator gap that expands to Δ* ≈ 10 meV at 2 K, in agreement with transport studies. Within the gap, momentum space imaging reveals flatly dispersing Dirac surface states with effective masses reaching m* = (330 ±20)me. Collectively, our observations demonstrate existence of a strongly correlated topological phase hosting the heaviest known Dirac fermions. The prodigious density of the Dirac states observed near zero energy magnifies their susceptibility to novel orders anticipated for interacting topological matter.

  • Direct Visualization of a Modulated Superfluid

    Direct Visualization of a Modulated Superfluid

    The pair density wave has been proposed to unify the various electronic orderings in the cuprates . Searching for this new phase of matter has led us to develop Scanned Josephson Tunneling Microscopy (SJTM) to directly visualize modulations in the superfluid density. CLICK TO EXPAND

    Complete details can be found at  Nature 532, 343 (2016). A superconductor is a homogeneous quantum condensate of Cooper pairs, each formed by binding two electrons into a zero-spin, zero-momentum state. In 1964, Fulde-Ferrel-Larkin-Ovchinnikov (FFLO) proposed that in the presence of a magnetic field an alternative ground state wavefunction of Cooper pairs forms, carrying momentum Q and inducing superfluid density modulations with wavelength 2π/Q. The last 40 years have seen a proliferation of novel and exotic superconductors. But, despite decades of effort, FFLO-like states have not been imaged in any material. Recently the challenge of detecting modulated superfluids has become particularly urgent: the discovery of a charge density wave state in the cuprates has motivated several contemporary microscopic theories in which the pseudogap phase contains a self-assembled (zero magnetic field) FFLO state known as a Pair Density Wave. To search for a pair density wave state in the cuprates we developed a nanometer resolution scanned Josephson (Cooper-pair) tunneling microscope (SJTM) [3]. We imaged Cooper-pair tunneling from a d-wave superconducting STM tip at millikelvin temperatures to the electronic superfluid state of underdoped Bi2Sr2CaCu2O8. Images of the Cooper-pair condensate with nanometer resolution through large fields of view reveal clear density modulations oriented along the Cu-O bond directions. By using Fourier analysis we detect the direct signature of a Cooper-pair density modulation at wavevectors QP≈(0.25,0)2π/a0;(0,0.25)2π/a0. While the wavelength of the condensate modulation closely matches that of the charge density wave, the Cooper-pair density wave is of different symmetry and more coherent. Our new technique for imaging Cooper-pair condensates, furthermore, opens the prospect of condensate visualizations in other cuprates, iron-based and unconventional superconductors, and will be especially advantageous in the study of topological superconductors.

  • Field Induced Coherent Density Wave in Cuprates

    Field Induced Coherent Density Wave in Cuprates

    Using energy and momentum resolved phase registered visuzalization methods, we demonstrate how the high-field density wave state of cuprates begins to emerge coherently within vortex cores. CLICK TO EXPAND

    The magnetic field dependence of the density wave (DW) in the cuprates has revealed some intriguing insights into its relationship with superconductivity. The graph below shows x-ray data measuring DW amplitude in YBCO as function of temperature and magnetic field. xray_DW The striking result is that as the magnetic field is increased, the amplitude of the DW increases for temperatures only below Tc.  Naturally, one is lead to ask how and through what mechanism the DW recovers strength with increasing field.  Our most recent SI-STM studies have found the answer by directly visualizing vortex core states in BSCCO. The results show that at only the vortex sites, puddles of  strong d-form factor density wave (dFF-DW) are induced,  directly accounting for total increase in DW amplitude. The high field state of the cuprates has been shown to host a long range ordered density wave, in contrast to the very short range order at low fields.  The path from which local puddles of induced dFF-DW inside the vortex cores evolves to a phase coherent long range density wave remains contested.  However, our determination of the phase and directionality of the dFF-DW inside the vortex cores sheds light on the matter.  The left panel below shows the local extraction of the density wave modulation in a 65nm field of view in BSCCO (shown as blue and yellow lines), with lines of constant phase for a constant consine function shown in grey.  The right panel is a histogram showing the difference in phase between the cosine function and the phase of the vortices. vortex_phase_reg The histogram has a very narrow distribution indicating that all the vortices are globally phase coherent in the field of view.  Remarkably, the puddles of dFF-DW at the vortex site which are far apart must be linked though some mechanism which locks their phases. Recently, there has been growing theoretical work that the pseudogap (PG) state in the cupates, the parent state to superconductivity in the underdoped regime, is a composite order state.  

  • Discovery of d-Form Factor Density Wave in Cuprates

    Discovery of d-Form Factor Density Wave in Cuprates

    By developing phase registered visualization methods, we discover a new form of electronic organization in the cuprates. CLICK TO EXPAND

    Complete details can be found in PNAS, 111, E3026 (2014) & Nat. Phys 12, 150-156 (2016) Stripes, a specific form of uni-directional charge and spin density wave, were first discovered by neutron scattering measurements in the La-based cuprates. STM was next to detect and visualize low energy electronic state modulations at the same Q-vector in Bi-based and oxchloride cuprates. Recently, NMR and resonant x-ray experiments have extended the observation to YBCO. Even though there is consensus about the presence of a density wave (DW) involving the electronic degrees of freedom new questions arise: 1) is the DW related to the PG state, 2) what is the driving instability of the modulation, 3) is the DW truly disordered in all cuprates,  4) how does it relate to d-symmetry superconductivity, 5) what is the mechanism that drives the magnetic field changes. A detailed microscopic view of the density wave was achieved by SI-STM from the Davis Lab at Cornell. Spectroscopic energy resolved images near the pseudogap (PG) energy showed beautiful arrangements of a ladder pattern with a bright backbone and an approximate wavelength of 4 unit cells (see image below). dFF_zoom The initial interpretation of this pattern was that they were consistent with stripes, similar to those observed by neutron measurements on La-cuprates. However, during the the last 5 years we have been developing a new set of experimental and analysis tools to perform sub-unit-cell resolved measurements to determine the additional degrees of freedom essential to the structure of the density wave. Having the capability to robustly resolve the local density of states between the copper and two distinct oxygen sites (see CuO figure in last section) within each and every unit cell we demonstrated that the ladder patterns were not stripes but a d-form factor density wave (dFF-DW). The distinction arises exactly at the sub-unit-cell scale: for stripes, the density at the two different oxygen sites within each unit cell modulates in phase whereas in a dFF-DW they are out of phase by π. A schematic of a dFF-lattice, left panel below, shows the density at every oxygen site along the x-direction (Ox) has the opposite sign as the ones along y, Oy. Modulating this pattern along the x-direction, as shown in the right panel, results in a dFF-DW, where the phase between the sublattice of Ox and Oy are π out of phase. dFF_DW_cartoon This form of density wave, never before identified in nature, arranges itself into small unidirectional domains with long range order. The figure below shows our determination of this domain structure using Fourier methods. The orange patches show the density wave running along the x-direction whereas the blue patches show modulations in the y-direction. The white areas in between are regions of coexistence in which the ladder patterns are frustrated. dFF_domans While X-ray measurements, for example, determine the onset of the density wave at temperatures well below that of the pseudogap state, our energy resolved tracking of the dFF-DW actually reveals that the energies at which the modulations are most intense are those of the pseudogap.  In analogy to a classic density wave gap opening in a metal, the energies of maximal modulation typically determine the characteristic gap or energy scale of the ordering. The origin of density waves, especially in more complex materials, is a difficult problem to tackle.  While on the ends of the spectrum either lattice instability or momentum space susceptibility can be primary drivers, in reality both effects must play a role.  Our momentum resolved measurements determine that the Q-vector associated with the dFF-DW evolves with doping along the same monotonic trajetory as the vector connecting the tips of Fermi arcs.  While this may indicate the the wavelength of the density wave is determined by scattering between k-space hot spots, the highly spatially disordered structure may also indicate that the evolution is a result of domain structure changes. However, one piece of evidence strongly supporting that Fermiology must play a key role is the observation that the dFF-DW modulation on the filled and empty sides are exactly out of phase as would be expected in a picture where a momentum space instability at hot spots opens a gap across the chemical potential.

  • Cuprate k-Space Topological Transition

    Cuprate k-Space Topological Transition

    The cuprate electronic structure is shown to undergo a topological transition at a critical hole doping which is simultaneously accompanied by the disappearance of the d-form factor density wave. CLICK TO EXPAND

    Complete details can be found in Science, 344, 612 (2014) The parent state of the cuprates is a charge-transfer insulator with a superexchange process that drives the system into an antiferromagnetic state. The figure below shows single spins pinned to each copper site with the oxygen orbital the mediators of the superexchange process. Whether the explicit inclusion of oxygen degrees of freedom is required in theoretical models is still a matter of great debate. CuO_spins The insulator can be weakened by hole doping (p): removing electrons from the copper sites restored charge mobility. However, the resulting electronic reorganization is a great mystery leading to the phase diagram below. From an antiferromagnetic insulator (AFI) emerges a pseudogap (PG) state, d-wave superconductivity (d-SC), and a density wave (DW) region. Eventually with enough hole doping the system becomes metallic. cuprate_phase_diagram2 The relationship between all of these phases is a matter of active research. For example, 1) does the PG phase compete with d-SC or can it be considered a precursory phase, 2) could antiferromagnetic fluctuations be responsible for Cooper pairing, and 3) is the DW state the ground state of the PG? Our recent results and scientific endeavors attempt to address some of these questions. From the perspective of momentum space structure (insets to figure above), in the PG region, photoemission measurements reveal a mysterious and exotic Fermi surface comprised of open arcs. In the metallic phase, on the other hand, a conventional metallic Fermi surface is recovered. The nature of this transition was elucidated by our SI-STM studies using simultaneously obtained real and momentum space data. The analysis shows an abrupt transition from arcs to closed contours at p ≈ 0.19, coincident with the disappearance of the density wave order. The left panel in the figure below shows the growing arcs as a function of doping with an abrupt jump between p=0.17 and p=0.20, representing a topological change in the momentum space structure. The right panel tracks the strength of the DW amplitude which plummets down at the same doping. FS_topology The result points to an intimate relationship between the exotic Fermi surface and the translational broken symmetry generating the density wave. In recent work (see following sections), we have shown that the density wave is strongly tied to the PG energy scale further solidifying the link between Fermi surface topology and the enigmatic PG state.

  • Visualizing Individual Kondo Holes and Heavy Fermion Destruction

    Visualizing Individual Kondo Holes and Heavy Fermion Destruction

    By visualizing invidivual Kondo holes, the local absence of magnetic atoms necessary for heavy fermion formation, intense hybridization disorder is observed across the entire sample. CLICK TO EXPAND

    Complete details can be found in PNAS, 108, 18233 (2011) The presence of magnetic interactions in heavy fermion compounds stems from the presence of a lattice of magnetic moment carrying atoms. It is the interaction of a background Fermi sea with this magnetic lattice that generates the new heavy state as the temperature is lowered. Very soon after the discovery of heavy fermion compounds a small number of careful bulk effect experiments focused on better understanding the atomic scale interactions by introducing a small concentration of nonmagnetic impurities into the magnetic lattice, termed Kondo Holes. The observations led to a series of hypotheses regarding how the local microscopic perturbations to the heavy fermion structure were affecting the bulk properties. Amongst them were disorder in the hybridization strength between the magnetic lattice and the delocalized electrons, bound states at the defect sites, and a weakening of the heavy fermion effect. Furthermore, motivated by STM methods, recent numerical calculations predicted a spatial oscillation of the heavy fermion hybridization strength around a Kondo Hole, an effect that cannot be studied by bulk probes.

Nonmagnetic Th atoms in the sample Th0.01U0.99Ru2Si2 substitute for magnetic U atoms and are believed to act as Kondo Holes. By developing the ‘hybridization gapmap’ technique, a novel spatial visualization of hybridization strength became possible as a way to study the effect of Kondo Holes. The top left panel in the figure below shows such a hybridization gapmap extracted from SI-STM data. It is highly disordered but the Fourier transform, shown next to it, clearly highlights some spatially periodic signature with wavevector Q*. By imaging hybridization oscillations around a single Kondo hole, visualized in the lower panel of the figure below, it became evident that the disordered image was actually generated by the interference of periodic hybridization oscillations emanating from a random distribution Kondo holes. Kondo_hole The analysis further revealed the unexpected result that the wavevector Q* of the oscillations was actually set by the high temperature electronic structure, a finding which unmistakably distinguishes it from a Friedel type response. More definitively, the visualization of the hybridization strength made clear that the disorder due to Kondo Hole dilution is in fact not just a local effect but one in which the impurities add texture to the electronic structure across the whole sample.

  • First Visualization of Heavy Fermion Electronic Structure

    First Visualization of Heavy Fermion Electronic Structure

    The long predicted heavy fermion electronic structure is visualized for the first time using SI-STM. CLICK TO EXPAND

    Complete details can be found in Nature, 465, 570-576 (2010) Heavy fermion compounds constitute a large class of strongly correlated materials that exhibit a rich variety of electronic states including magnetic ordering, density waves, unconventional superconductivity, and quantum criticality. Their branding as ‘heavy’ is due to the hybridization between the localized f-electrons and delocalized electrons of Fermi sea which can renormalize the carrier mass up to 1000 times the bare mass. The basic implications of such hybridization for the microscopic electronic structure have long been known within the context of the Anderson Lattice Model (ALM) and experiments over the past 30 years have been consistent with those predictions. However, a direct observation of the hybridization process and the resulting heavy band formation, the central result of the theoretical framework, was not achieved until the introduction of spectroscopic imaging STM (SI-STM) techniques to heavy fermion materials.  The band structure for the 1-band spin-½ ALM generating the heavy bands is shown below. By imaging the real space and energy resolved quasiparticle interference of heavy quasiparticle with SI-STM, we began studies of the heavy fermion problem in which hybridization processes between conduction and localized electrons generate the heavy fermion electronic state and open a hybridization gap near the chemical potential. The first imaging of the opening of a hybridization gap and of heavy band formation in a heavy fermion system was achieved using the compound URu2Si2 and its alloy Th0.01U0.99Ru2Si2.  The 3-panel movie presents energy dependent quasiparticle interference imaging by SI-STM at T=2K: 1) the left panel is data of spatially resolved conductance, 2) the middle panel, the amplitude of the Fourier transform of the same data set, and 3) the right panel, the same data as panel 2 but with image processing code identifying the q-space position of the dispersing bright bands.

    The figure below is an extraction from the data above along the directions (1,0) and (1,1).  Clearly present is the fast dispersion of the band structure near the chemical potential (E = 0 mV).  While the material URu2Si2 is far from a 1-band spin-½ Anderson Lattice, there is astonishingly good agreement between experiment and theory. heavy_bands