Bogoliubov Quasiparticle Interference Research Articles

Magic angle of Sr2RuO4 : Optimizing correlation-driven superconductivity

Understanding of unconventional superconductivity is crucial for engineering materials with specific order parameters or elevated superconducting transition temperatures. However, for many materials, the pairing mechanism and symmetry of the order parameter remain unclear: reliable and efficient methods of predicting the order parameter and its response to tuning parameters are lacking. Here, we investigate the response of superconductivity in Sr2RuO4 to structural distortions via the random phase approximation (RPA) and functional renormalization group (FRG), starting from realistic models of the electronic structure. Our results suggest that RPA misses the interplay of competing fluctuation channels. FRG reproduces key experimental findings. We predict a magic octahedral rotation angle, maximizing the superconducting Tc and a dominant dx2−y2 pairing symmetry. To enable experimental verification, we provide calculations of the phase-referenced Bogoliubov Quasiparticle Interference imaging. Our work demonstrates a designer approach to tuning unconventional superconductivity with relevance and applicability for a wide range of quantum materials. Published by the American Physical Society 2024

Superconducting gap symmetry from Bogoliubov quasiparticle interference analysis on Sr2RuO4

The nature of the superconducting order parameter in ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$ has generated intense interest in recent years. Since the superconducting gap is very small, high-resolution methods such as scanning tunneling spectroscopy might be the best chance to directly resolve the gap symmetry. Recently, a Bogoliubov quasiparticle interference imaging (BQPI) experiment has suggested that the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ gap symmetry is appropriate for ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$. In this work, we use a material-specific theoretical approach based on Wannier functions of the surface of ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$ to calculate the continuum density of states as detected in scanning tunneling microscopy experiments. We examine several different proposed gap order parameters and calculate the expected BQPI pattern for each case. Comparing to the available experimental data, our results suggest that a ${s}^{\ensuremath{'}}+i{d}_{xy}$ gap order parameter is the most probable state, but the measured BQPI patterns still display features unaccounted for by the theory for any of the states currently under discussion.

Orbital-Selective High-Temperature Cooper Pairing Developed in the Two-Dimensional Limit.

For multiband superconductors, the orbital multiplicity yields orbital differentiation in normal-state properties and can lead to orbital-selective spin-fluctuation Cooper pairing. The orbital-selective phenomenon has become increasingly pivotal in clarifying the pairing "enigma", particularly for multiband high-temperature superconductors. Meanwhile, in one-unit-cell (1-UC) FeSe/SrTiO3, since the standard electron-hole Fermi pocket nesting scenario is inapplicable, the actual pairing mechanism is subject to intense debate. Here, by measuring high-resolution Bogoliubov quasiparticle interference, we report observations of highly anisotropic magnetic Cooper pairing in 1-UC FeSe. Theoretically, it is important to incorporate orbitally selective effects of electronic correlations within a spin-fluctuation pairing calculation, where the dxy orbital becomes coherence-suppressed. The resulting pairing gap is compatible with the experimental findings, which suggests that high-Tc Cooper pairing with orbital selectivity applies to 2D-limit 1-UC FeSe. Our findings imply the general existence of orbital selectivity in iron-based superconductors and the universal significance of electron correlations in high-Tc superconductors.

Topological Shiba bands in artificial spin chains on superconductors

A major challenge in developing topological superconductors for implementing topological quantum computing is their characterization and control. It has been proposed that a p-wave gapped topological superconductor can be fabricated with single-atom precision by assembling chains of magnetic atoms on s-wave superconductors with spin-orbit coupling. Here, we analyze the Bogoliubov quasiparticle interference in atom-by-atom constructed Mn chains on Nb(110) and for the first time reveal the formation of multi-orbital Shiba bands using momentum resolved measurements. We find evidence that one band features a topologically non-trivial p-wave gap as inferred from its shape and particle-hole asymmetric intensity. Our work is an important step towards a distinct experimental determination of topological phases in multi-orbital systems by bulk electron band structure properties only.

Coexistence of checkerboard and quasiparticle interference modulations in Bi2Sr2CaCu2O8+x studied by scanning tunneling microscopy/spectroscopy

We performed scanning tunneling microscopy/spectroscopy measurements in the superconducting state of underdoped ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}\mathrm{Ca}{\mathrm{Cu}}_{2}{\mathrm{O}}_{8+x}$, and demonstrated that the checkerboard (CB) and Bogoliubov quasiparticle interference (QPI) modulations, which are considered to be characteristic features for the low-energy states around the antinodal and nodal regions of the Fermi surface, respectively, coexist in identical regions of the Cu-O plane. In the present work, these electronic superstructures are successfully separated from each other, and the local amplitude and phase are evaluated for the CB modulation and the ${\mathbit{q}}_{1}$ component of QPI, which are characterized by similar wave vectors. The ${\mathbit{q}}_{1}$-QPI component tends to be in-phase and antiphase with the CB modulation for positive and negative bias voltages, respectively. The significant spatial phase relationship is confirmed in regions where the local amplitude of CB modulation is comparatively large. This finding implies an interplay between the QPI and CB modulations.

Atomic-scale electronic structure of the cuprate pair density wave state coexisting with superconductivity

The defining characteristic of hole-doped cuprates is d-wave high temperature superconductivity. However, intense theoretical interest is now focused on whether a pair density wave state (PDW) could coexist with cuprate superconductivity [D. F. Agterberg et al., Annu. Rev. Condens. Matter Phys. 11, 231 (2020)]. Here, we use a strong-coupling mean-field theory of cuprates, to model the atomic-scale electronic structure of an eight-unit-cell periodic, d-symmetry form factor, pair density wave (PDW) state coexisting with d-wave superconductivity (DSC). From this PDW + DSC model, the atomically resolved density of Bogoliubov quasiparticle states [Formula: see text] is predicted at the terminal BiO surface of Bi2Sr2CaCu2O8 and compared with high-precision electronic visualization experiments using spectroscopic imaging scanning tunneling microscopy (STM). The PDW + DSC model predictions include the intraunit-cell structure and periodic modulations of [Formula: see text], the modulations of the coherence peak energy [Formula: see text] and the characteristics of Bogoliubov quasiparticle interference in scattering-wavevector space [Formula: see text] Consistency between all these predictions and the corresponding experiments indicates that lightly hole-doped Bi2Sr2CaCu2O8 does contain a PDW + DSC state. Moreover, in the model the PDW + DSC state becomes unstable to a pure DSC state at a critical hole density p*, with empirically equivalent phenomena occurring in the experiments. All these results are consistent with a picture in which the cuprate translational symmetry-breaking state is a PDW, the observed charge modulations are its consequence, the antinodal pseudogap is that of the PDW state, and the cuprate critical point at p* ≈ 19% occurs due to disappearance of this PDW.

Momentum-resolved superconducting energy gaps of Sr2RuO4 from quasiparticle interference imaging

Sr2RuO4 has long been the focus of intense research interest because of conjectures that it is a correlated topological superconductor. It is the momentum space (k-space) structure of the superconducting energy gap [Formula: see text] on each band i that encodes its unknown superconducting order parameter. However, because the energy scales are so low, it has never been possible to directly measure the [Formula: see text] of Sr2RuO4 Here, we implement Bogoliubov quasiparticle interference (BQPI) imaging, a technique capable of high-precision measurement of multiband [Formula: see text] At T = 90 mK, we visualize a set of Bogoliubov scattering interference wavevectors [Formula: see text] consistent with eight gap nodes/minima that are all closely aligned to the [Formula: see text] crystal lattice directions on both the α and β bands. Taking these observations in combination with other very recent advances in directional thermal conductivity [E. Hassinger et al., Phys. Rev. X 7, 011032 (2017)], temperature-dependent Knight shift [A. Pustogow et al., Nature 574, 72-75 (2019)], time-reversal symmetry conservation [S. Kashiwaya et al., Phys. Rev B, 100, 094530 (2019)], and theory [A. T. Rømer et al., Phys. Rev. Lett. 123, 247001 (2019); H. S. Roising, T. Scaffidi, F. Flicker, G. F. Lange, S. H. Simon, Phys. Rev. Res. 1, 033108 (2019); and O. Gingras, R. Nourafkan, A. S. Tremblay, M. Côté, Phys. Rev. Lett. 123, 217005 (2019)], the BQPI signature of Sr2RuO4 appears most consistent with [Formula: see text] having [Formula: see text] [Formula: see text] symmetry.

Calorimetric evidence of nodal gaps in the nematic superconductor FeSe

Superconductivity in FeSe has recently attracted a great deal of attention because it emerges out of an electronic nematic state of elusive character. Here we study both the electronic normal state and the superconducting gap structure using heat-capacity measurements on high-quality single crystals. The specific-heat curve, from 0.4 K to Tc = 9.1 K, is found to be consistent with a recent gap determination using Bogoliubov quasiparticle interference [P. O. Sprau et al., Science 357, 75 (2017)], however only if nodes are introduced on either the electron or the hole Fermi-surface sheets. Our analysis, which is consistent with quantum-oscillation measurements, relies on the presence of only two bands, and thus the fate of the theoretically predicted second electron pocket remains mysterious.

Spatial variation of the two-fold anisotropic superconducting gap in a monolayer of FeSe0.5Te0.5 on a topological insulator

We present our investigations on the superconducting properties of monolayers of FeSe$_{0.5}$Te$_{0.5}$ grown on the 3D topological insulator Bi$_{2}$Se$_{1.2}$Te$_{1.8}$ using low temperature scanning tunneling spectroscopy (STS). While the morphology and the overall transition temperature resemble those of similarly doped bulk crystals, the spatially resolved spectroscopic data at 1.1K shows a much larger spatial inhomogeneity in the superconducting energy gaps. Despite the gap inhomogeneity all the spectra can be fitted with a two-fold anisotropic s-wave gap function. The two-fold nature of the gap symmetry is evident from the Bogoliubov quasiparticle interference (QPI) pattern which shows distinct C$_{2}$ symmetric scattering intensities. We argue that the gap inhomogeneity emerges as a result of intrinsic disorder in our system similar to disordered conventional superconductors. Even though most of our findings clearly differ from the current understanding of the corresponding bulk system, it provides an ideal platform to study unconventional superconductivity in Fe chalcogenides thinned down to a single layer and in close proximity to a topological insulator.

Quasiparticle scattering image in hidden order phases and chiral superconductors

The technique of Bogoliubov quasiparticle interference (QPI) has been successfully used to investigate the symmetry of unconventional superconducting gaps, also in heavy fermion compounds. It was demonstrated that QPI can distinguish between the d-wave singlet candidates in CeCoIn5. In URu2Si2 presumably a chiral d-wave singlet superconducting (SC) state exists inside a multipolar hidden order (HO) phase. We show that hidden order leaves an imprint on the symmetry of QPI pattern that may be used to determine the essential question whether HO in URu2Si2 breaks the in-plane rotational symmetry or not. We also demonstrate that the chiral d-wave SC gap leads to a crossover to a quasi-2D QPI spectrum below Tc which sharpens the HO features. Furthermore we investigate the QPI image of chiral p-wave multigap superconductor Sr2RuO4.

Imaging Cooper pairing of heavy fermions in CeCoIn5

The Cooper pairing mechanism of heavy-fermion superconductors, while long hypothesized as due to spin fluctuations, has not been determined. It is the momentum space (k-space) structure of the superconducting energy gap delta(k) that encodes specifics of this pairing mechanism. However, because the energy scales are so low, it has not been possible to directly measure delta(k) for any heavy-fermion superconductor. Bogoliubov quasiparticle interference (QPI) imaging, a proven technique for measuring the energy gaps of high-Tc superconductors, has recently been proposed as a new method to measure delta(k) in heavy-fermion superconductors, specifically CeCoIn5. By implementing this method, we immediately detect a superconducting energy gap whose nodes are oriented along k||(+-1, +-1)pi/a0 directions. Moreover, we determine the complete k-space structure of the delta(k) of a heavy-fermion superconductor. For CeCoIn5, this novel information includes: the complex band structure and Fermi surface of the hybridized heavy bands, the fact that highest magnitude delta(k) opens on a high-k band so that gap nodes occur at quite unanticipated k-space locations, and that the Bogoliubov quasiparticle interference patterns are most consistent with dx2-y2 gap symmetry. The availability of such quantitative heavy band- and gap-structure data will be critical in identifying the microscopic mechanism of heavy fermion superconductivity in this material, and perhaps in general.

Electronic structure of the cuprate superconducting and pseudogap phases from spectroscopic imaging STM

We survey the use of spectroscopic imaging scanning tunneling microscopy (SI-STM) to probe the electronic structure of underdoped cuprates. Two distinct classes of electronic states are observed in both the d-wave superconducting (dSC) and the pseudogap (PG) phases. The first class consists of the dispersive Bogoliubov quasiparticle excitations of a homogeneous d-wave superconductor, existing below a lower energy scale E=Δ0. We find that the Bogoliubov quasiparticle interference (QPI) signatures of delocalized Cooper pairing are restricted to a k-space arc, which terminates near the lines connecting k=±(π/a0,0) to k=±(0,π/a0). This arc shrinks continuously with decreasing hole density such that Luttinger's theorem could be satisfied if it represents the front side of a hole-pocket that is bounded behind by the lines between k=±(π/a0,0) and k=±(0,π/a0). In both phases, the only broken symmetries detected for the |E|<Δ0 states are those of a d-wave superconductor. The second class of states occurs proximate to the PG energy scale E=Δ1. Here the non-dispersive electronic structure breaks the expected 90°-rotational symmetry of electronic structure within each unit cell, at least down to 180°-rotational symmetry. This electronic symmetry breaking was first detected as an electronic inequivalence at the two oxygen sites within each unit cell by using a measure of nematic (C2) symmetry. Incommensurate non-dispersive conductance modulations, locally breaking both rotational and translational symmetries, coexist with this intra-unit-cell electronic symmetry breaking at E=Δ1. Their characteristic wavevector Q is determined by the k-space points where Bogoliubov QPI terminates and therefore changes continuously with doping. The distinct broken electronic symmetry states (intra-unit-cell and finite Q) coexisting at E∼Δ1 are found to be indistinguishable in the dSC and PG phases. The next challenge for SI-STM studies is to determine the relationship of the E∼Δ1 broken symmetry electronic states with the PG phase, and with the E<Δ0 states associated with Cooper pairing.

Checkerboard superconducting order and antinodal Bogoliubov quasiparticle interference

Numerical study of momentum-dependent gap function is presented to make clear the origin of superconductivity in copper oxides. We claim that antinodal region with pronounced nesting feature of the Fermi contour gives rise to superconducting pairing with large momentum under screened Coulomb repulsion. Such a pairing results in both spatial checkerboard pattern of the superconducting state below Tc and a gapped state of incoherent pairs in a broad temperature range above Tc. We explain the momentum dependence of the coherent spectral weight detected in angle-resolved photoemission spectroscopy and predict antinodal Bogoliubov quasiparticle interference other than observed in the nodal region.

Pair-density-wave pseudogap and superconducting states in cuprates

Bogoliubov quasiparticle interference and localized high-energy excitations observed in cuprates in nodal and antinodal regions of the momentum space, respectively, would lead to a conclusion that only the nodal region might give rise to superconductivity whereas the antinodal one might be associated with the pseudogap. We argue that both pseudogap and superconducting states arise exactly in the antinodal region with pronounced nesting feature of the Fermi contour as spatially inhomogeneous incoherent and coherent states of pairs with large momentum. The nodal region gives rise to conventional superconducting pairing with zero momentum which, together with the pairing with large momentum in the antinodal region, forms a biordered superconducting state in the whole of the Brillouin zone. This coherent state with complicated momentum dependence of the order parameter manifests itself as a pair-density wave that can exist without any driving insulating order. We believe that quasiparticle interference, other than observed in the nodal region, should be observable in the antinodal region as well.