Quasiparticle Interference Patterns Research Articles

Surface states and quasiparticle interference in Bernal and rhombohedral graphite with and without trigonal warping

We use an exact analytical technique [Phys. Rev. B \textbf{101}, 115405 (2020), Phys. Rev. B \textbf{102}, 165117 (2020)] to recover the surface Green's functions for Bernal (ABA) and rhombohedral (ABC) graphite. For rhombohedral graphite we recover the predicted surface flat bands. For Bernal graphite we find that the surface state spectral function is similar to the bilayer one, but the trigonal warping effects are enhanced, and the surface quasiparticles have a much shorter lifetime. We subsequently use the T-matrix formalism to study the quasiparticle interference patterns generated on the surface of semi-infinite ABA and ABC graphite in the presence of impurity scattering. We compare our predictions to experimental STM data of impurity-localized states on the surface of Bernal graphite which appear to be in a good agreement with our calculations.

Robust topological state against magnetic impurities observed in the superconductor PbTaSe2

Magnetic impurities deposited on topological superconductor candidate PbTaSe2 can introduce a non-splitting zero-energy state inside the superconducting gap, which has been proposed as a field-free platform for topological zero modes. However, it is still unclear how robust the topological state in PbTaSe2 is against magnetic impurities, which is related to the topological nature of the zero-energy state as well as its potential for quantum computation. In this work, we use scanning tunneling microscopy (STM) to study the topological surface state in the normal state of PbTaSe2 under the perturbation of magnetic impurities. We visualize the quasi-particle interference (QPI) arising from the topological surface state. We then deposit Fe impurities on the surface to form atomic Fe adatoms. We find that each Fe adatom sits at a unique interstitial position on the surface and features a local state at high energies, both of which are consistent with our first-principles calculation that further reveals its large magnetic moment. Our systematic Fe deposition and subsequent measurements show that the arc-like QPI pattern at the Fermi energy is robust with up to 3% Fe coverage where the atomic nature of Fe adatoms still holds. Our results provide evidence that the topological surface state at the Fermi energy in PbTaSe2 is robust against dilute magnetic impurities.

Molecular Beam Epitaxy Growth and Electronic Structures of Monolayer GdTe3 Supported by the National Natural Science Foundation of China (Grant Nos. 11934001 and 11874233), and the Science Challenge Project (Grant No. TZ2016004).

GdTe3 is a layered antiferromagnetic (AFM) metal with charge density wave (CDW). We grew monolayer (ML) GdTe3 on graphene/6H-SiC(0001) substrates by molecular beam epitaxy. The electronic and magnetic structures are studied by scanning tunneling microscopy/spectroscopy, quasi-particle interference (QPI) and first-principles calculations. Strong evidence of CDW persisting at the two-dimensional (2D) limit is found. Band dispersions and partially gapped energy bands near the Fermi surface are revealed by the QPI patterns. By density functional theory +U calculations, AFM order with stripe pattern is found to be the magnetic ground state for ML GdTe3. These results provide fundamental understanding and pave the way for further investigation of GdTe3 at the 2D limit.

General, Strong Impurity-Strength Dependence of Quasiparticle Interference

Quasiparticle interference (QPI) patterns in momentum space are often assumed to be independent of the strength of the impurity potential when compared with other quantities, such as the joint density of states. Here, using the $T$-matrix theory, we show that this assumption breaks down completely even in the simplest case of a single-site impurity on the square lattice with an $s$ orbital per site. Then, we predict from first-principles, a very rich, impurity-strength-dependent structure in the QPI pattern of TaAs, an archetype Weyl semimetal. This study thus demonstrates that the consideration of the details of the scattering impurity including the impurity strength is essential for interpreting Fourier-transform scanning tunneling spectroscopy experiments in general.

Multi-atom quasiparticle scattering interference for superconductor energy-gap symmetry determination

Complete theoretical understanding of the most complex superconductors requires a detailed knowledge of the symmetry of the superconducting energy-gap {mathrm{{Delta}}}_{mathbf{k}}^alpha, for all momenta k on the Fermi surface of every band α. While there are a variety of techniques for determining |{mathrm{{Delta}}}_{mathbf{k}}^alpha |, no general method existed to measure the signed values of {mathrm{{Delta}}}_{mathbf{k}}^alpha. Recently, however, a technique based on phase-resolved visualization of superconducting quasiparticle interference (QPI) patterns, centered on a single non-magnetic impurity atom, was introduced. In principle, energy-resolved and phase-resolved Fourier analysis of these images identifies wavevectors connecting all k-space regions where {mathrm{{Delta}}}_{mathbf{k}}^alpha has the same or opposite sign. But use of a single isolated impurity atom, from whose precise location the spatial phase of the scattering interference pattern must be measured, is technically difficult. Here we introduce a generalization of this approach for use with multiple impurity atoms, and demonstrate its validity by comparing the {mathrm{{Delta}}}_{mathbf{k}}^alpha it generates to the {mathrm{{Delta}}}_{mathbf{k}}^alpha determined from single-atom scattering in FeSe where s± energy-gap symmetry is established. Finally, to exemplify utility, we use the multi-atom technique on LiFeAs and find scattering interference between the hole-like and electron-like pockets as predicted for {mathrm{{Delta}}}_{mathbf{k}}^alpha of opposite sign.

Mapping Dirac fermions in the intrinsic antiferromagnetic topological insulators (MnBi2Te4)(Bi2Te3)n ( n=0,1 )

Topological surface states with intrinsic magnetic ordering in the MnBi$_2$Te$_4$(Bi$_2$Te$_3$)$_n$ compounds have been predicted to host rich topological phenomena including quantized anomalous Hall effect and axion insulator state. Here we use scanning tunneling microscopy to image the surface Dirac fermions in MnBi$_2$Te$_4$ and MnBi$_4$Te$_7$. We have determined the energy dispersion and helical spin texture of the surface states through quasiparticle interference patterns far above Dirac energy, which confirms its topological nature. Approaching the Dirac point, the native defects in the MnBi$_2$Te$_4$ septuple layer give rise to resonance states which extend spatially and potentially hinder the detection of a mass gap in the spectra. Our results demonstrate that regulating defects is essential to realize exotic topological states at higher temperatures in these compounds.

Surface Green's functions and quasiparticle interference in Weyl semimetals

We use the exact analytical technique introduced in Phys. Rev. B 101, 115405 to recover the surface Green's functions and the corresponding Fermi-arc surface states for various lattice models of Weyl semimetals. For these models we use the T-matrix formalism to calculate the quasiparticle interference patterns generated in the presence of impurity scattering. In particular, we consider the models introduced in Phys. Rev. B 93, 041109(R) (A) and Phys. Rev. Lett. 119, 076801 (B), and we find that, as opposed to observations previously obtained via joint density of states and spin-dependent scattering probability, the inter-arc scattering in the quasiparticle interference features is fully suppressed in model A, and very small in model B. Our findings indicate that these models may not correctly describe materials such as MoTe2, since for such materials inter-arc scattering is clearly visible experimentally, e.g., in Nature Phys. 12, 1105-1110 (2016). We also focus on the minimal models proposed by McCormick et al. in Phys. Rev. B 95, 075133, which indeed recover significant inter-arc scattering features.

Exact lattice-model calculation of boundary modes for Weyl semimetals and graphene

We provide an exact analytical technique to obtain within a lattice model the wave functions of the edge states in zigzag- and bearded-edge graphene, as well as of the Fermi-arc surface states in Weyl semimetals described by a minimal bulk model. We model the corresponding boundaries as an infinite scalar potential localized on a line, and respectively within a plane. We use the T-matrix formalism to obtain the dispersion and the spatial distribution of the corresponding boundary modes. Furthermore, to demonstrate the power of our approach, we write down the surface Green’s function of the considered Weyl semimetal model, and we calculate the quasiparticle interference patterns originating from an impurity localized at the respective surface.

Quasiparticle interference of spin momentum locked surface states at step edges on Re(0001)

Quasiparticle interference patterns formed by a surface state on the Re(0001) surface were investigated using scanning tunneling spectroscopy. The energy dispersion is inferred from Fourier-transformed differential conductivity maps for occupied and unoccupied states. The band dispersion for occupied states agrees with earlier published results obtained by angle-resolved photoemission spectroscopy. An analysis of the phase of interference patterns at step edges reveals a drastic change in the effective energy barrier for backscattering above and below the Fermi level. The attenuation of the interference pattern with increasing distance indicates interband scattering is the dominant scattering mechanism. Step decoration by Ni has a negligible influence on the pattern, excluding spin-flip scattering as a dominant contribution. The one-dimensional Re/Au line interface, however, reverses the scattering barrier behavior, indicating a coupling of surface states.

Distinct Topological Surface States on the Two Terminations of MnBi4Te7

The recent discovered intrinsic magnetic topological insulator MnBi2Te4 have been met with unusual success in hosting emergent phenomena such as the quantum anomalous Hall effect and the axion insulator states. However, the surface-bulk correspondence of the Mn-Bi-Te family, composed by the superlattice-like MnBi2Te4/(Bi2Te3)n (n = 0, 1, 2, 3 ...) layered structure, remains intriguing but elusive. Here, by using scanning tunneling microscopy (STM) and angle-resolved photoemission spectroscopy (ARPES) techniques, we unambiguously assign the two distinct surface states of MnBi4Te7 (n = 1) to the quintuple-layer (QL) Bi2Te3 termination and the septuple-layer (SL) MnBi2Te4 termination, respectively. A comparison of the experimental observations with theoretical calculations reveals the diverging topological behaviors, especially the hybridization effect between magnetic and nonmagnetic layers, on the two terminations: a gap on the QL termination originating from the topological surface states of the QL hybridizing with the bands of the beneath SL, and a gapless Dirac-cone band structure on the SL termination with time-reversal symmetry. The quasi-particle interference patterns further confirm the topological nature of the surface states for both terminations, continuing far above the Fermi energy. The QL termination carries a spin-helical Dirac state with hexagonal warping, while at the SL termination, a strongly canted helical state from the surface lies between a pair of Rashba-split states from its neighboring layer. Our work elucidates an unprecedented hybridization effect between the building blocks of the topological surface states, and also reveals the termination-dependent time-reversal symmetry breaking in a magnetic topological insulator, rendering an ideal platform to realize the half-integer quantum Hall effect and relevant quantum phenomena.

Gapped Dirac cones and spin texture in thin film topological insulator

The protected surface states of topological insulators (TIs) form gapless Dirac cones corresponding non-degenerate eigenstates with helical spin polarisation. The presence of a warping term deforms the isotropic cone of the most simple model into snowflake Fermi surfaces as in Bi2Se3 and Bi2Te3. Their features have been identified in STM quasiparticle interference (QPI) experiments on isolated surfaces. Here we investigate the QPI spectrum for the TI thin-film geometry with finite tunnelling between the surface states. This leads to a dramatic change of spectrum due to gapping and a change in spin texture that should leave distinct signatures in the QPI pattern. We consider both normal and magnetic exchange scattering from the surface impurities and obtain the scattering t-matrix in Born approximation as well as the general closed solution. We show the expected systematic variation of QPI snowflake intensity features by varying film thickness and study, in particular, the influence on backscattering processes. We predict the variation of the QPI spectrum for Bi2Se3 thin films using the observed gap dependence from ARPES results.

Visualization of Multifractal Superconductivity in a Two-Dimensional Transition Metal Dichalcogenide in the Weak-Disorder Regime.

Eigenstate multifractality is a distinctive feature of noninteracting disordered metals close to a metal-insulator transition, whose properties are expected to extend to superconductivity. While multifractality in three dimensions (3D) only develops near the critical point for specific strong-disorder strengths, multifractality in 2D systems is expected to be observable even for weak disorder. Here we provide evidence for multifractal features in the superconducting state of an intrinsic, weakly disordered single-layer NbSe2 by means of low-temperature scanning tunneling microscopy/spectroscopy. The superconducting gap, characterized by its width, depth, and coherence peaks' amplitude, shows a characteristic spatial modulation coincident with the periodicity of the quasiparticle interference pattern. The strong spatial inhomogeneity of the superconducting gap width, proportional to the local order parameter in the weak-disorder regime, follows a log-normal statistical distribution as well as a power-law decay of the two-point correlation function, in agreement with our theoretical model. Furthermore, the experimental singularity spectrum f(α) shows anomalous scaling behavior typical from 2D weakly disordered systems.

Electronic States and Magnetic Response of MnBi2Te4 by Scanning Tunneling Microscopy and Spectroscopy.

Exotic quantum phenomena have been demonstrated in recently discovered intrinsic magnetic topological insulator MnBi2Te4. At its two-dimensional limit, the quantum anomalous Hall effect and axion insulator state were observed in odd and even layers of MnBi2Te4, respectively. Here, we employ low-temperature scanning tunneling microscopy to study the electronic properties of MnBi2Te4. The quasiparticle interference patterns indicate that the electronic structures on the topmost layer of MnBi2Te4 are different from those of the expected out-of-plane A-type antiferromagnetic phase. The topological surface states may be embedded in deeper layers beneath the topmost surface. Such novel electronic structure is presumably related to the modification of crystalline structure during sample cleaving and reorientation of the magnetic moment of Mn atoms near the surface. Mn dopants substituted at the Bi site on the second atomic layer are observed. The electronic structures fluctuate at atomic scale on the surface, which can affect the magnetism of MnBi2Te4.

Quasiparticle interference testing the possible pairing symmetry in Sr2RuO4**Project supported by the National Natural Science Foundataion of China (Grant Nos. 11604168, 11604166, and 11604303). WSW also acknowledges the supports by K. C. Wong Magna Fund in Ningbo University.

The quasiparticle interference (QPI) patterns of the superconducting state in Sr2RuO4 are theoretically studied by taking into account the spin–orbital coupling and two different pairing modes, chiral p-wave pairing and equal d-wave pairing, in order to propose an experimental method to test them. Both of the QPI spectra for the two pairing modes have clearly peaks evolving with energy, and their locations can be determined from the tips of the constant energy contour. But the number, location, and evolution of these peaks with energy are different between the two pairing modes. The different behaviors of the QPI patterns in these two pairing modes may help to resolve whether Sr2RuO4 is a chiral p-wave or d-wave superconductor.

BaHgSn: A Dirac semimetal with surface hourglass fermions

We proposed that BaHgSn is a Dirac semimetal (DSM) which can host hourglass-like surface states (HSSs) as protected by nonsymmorphic glide symmetry. Compared to KHgSb, an isostructural topological crystalline insulator with the same HSSs, BaHgSn has an additional band inversion at $\Gamma$ point. This band inversion is induced by the stronger interlayer coupling among Hg-Sn honeycomb layers than that among Hg-Sb-layers in KHgSb, which leads to bulk Dirac nodes in BaHgSn along the layer stacking direction $\Gamma$-$A$. In addition, the mirror Chern number $C_{i}$ protected by the mirror plane $\overline{M}_{z}$ ($k_z$=0) changes from 2 in KHgSb to 3 in BaHgSn. Therefore, when a compressive uniaxial strain is applied along the $y$ axis to break the rotation symmetry protecting the DSM state, BaHgSn becomes a strong topological insulator with $Z_{2}$ indices of $(1;000)$ and the topological surface Dirac cone co-exists with HSSs on the (010) surface. The Wilson-loop spectra have been calculated to verify these topological features. The calculated surface states, the Fermi surfaces and their quasiparticle interference patterns are ready to be compared with experimental measurements.

Nodal plane and persistent spin texture in a Weyl semimetal without mirror symmetry

By utilizing symmetry analysis and electronic structure calculations, we investigated the low-temperature orthorhombic phase of Ag$_{2}$Se in ${\cal SG}$~17. In addition to the discovery of a nodal plane at $k_{z}=\pi$ protected by the joint operation of time-reversal (${\cal T}$) and the 2-fold screw rotation $S_{2z}$, we found 24 Weyl points mainly residing at the $k_{y}=0$ plane with notable Fermi arc and large quasiparticle interference pattern (QPI). Due to the absence of mirror symmetry, a pair of Weyl points with opposite chirality reside at different binding energies, which makes this system an excellent material candidate for realizing the novel chiral anomaly related phenomenon, such as the quantized circular photogalvanic and the chiral magnetic effects. Furthermore, we also reveal the striking spin textures at $k_{z}=\pi$ plane which demonstrates, in a large region of the surface Brillouin Zone, a direction-selective spin polarization, which has a strong implication to spintronic applications.

Superconductivity of Topological Surface States and Strong Proximity Effect in Sn1- x Pbx Te-Pb Heterostructures.

Superconducting topological crystalline insulators are expected to form a new type of topological superconductors to host Majorana zero modes under the protection of lattice symmetries. The bulk superconductivity of topological crystalline insulators can be induced through chemical doping and the proximity effect. However, only conventional full gaps are observed, so the existence of topological superconductivity in topological crystalline insulators is still controversial. Here, the successful fabrication of atomically flat lateral and vertical Sn1- x Pbx Te-Pb heterostructures by molecular beam epitaxy is reported. The superconductivity of the Sn1- x Pbx Te-Pb heterostructures can be directly investigated by scanning tunneling spectroscopy. Unconventional peak-dip-hump gap features and fourfold symmetric quasiparticle interference patterns taken at the zero energy in the superconducting gap support the presence of the topological superconductivity in superconducting Sn1- x Pbx Te. Strong superconducting proximity effect and easy preparation of various constructions between Sn1- x Pbx Te and Pb make the heterostructures to be a promising candidate for topological superconducting devices to detect and manipulate Majorana zero modes in the future.

Quasiparticle interference and symmetry of superconducting order parameter in strongly electron-doped iron-based superconductors

Motivated by recent experimental reports of significant spin–orbit coupling (SOC) and a sign-changing order-parameter in the Li1−xFex(OHFe)1−yZnySe superconductor with only electron pockets present, we study the possible Cooper-pairing symmetries and their quasiparticle interference (QPI) signatures. We find that each of the resulting states—s-wave, d-wave and helical p-wave—can have a fully gapped density of states consistent with angle-resolved photoemission spectroscopy experiments and, due to SOC, are a mixture of spin singlet and triplet components leading to intra- and inter-band features in the QPI signal. Analyzing predicted QPI patterns we find that only the spin-triplet dominated even parity A1g (s-wave) and B2g (d-wave) pairing states are consistent with the experimental data. Additionally, we show that these states can indeed be realized in a microscopic model with atomic-like interactions and study their possible signatures in spin-resolved STM experiments.

Tunable ferromagnetic Weyl fermions from a hybrid nodal ring

Realization of nontrivial band topology in condensed matter systems is of great interest in recent years. Using first-principles calculations and symmetry analysis, we propose an exotic topological phase with tunable ferromagnetic Weyl fermions in a half-metallic oxide CrP2O7. In the absence of spin–orbit coupling (SOC), we reveal that CrP2O7 possesses a hybrid nodal ring. When SOC is present, the spin-rotation symmetry is broken. As a result, the hybrid nodal ring shrinks to discrete nodal points and forms different types of Weyl points. The Fermi arcs projected on the (100) surface are clearly visible, which can contribute to the experimental study for the topological properties of CrP2O7. In addition, the calculated quasiparticle interference patterns are also highly desirable for the experimental study of CrP2O7. Our findings provide a good candidate of ferromagnetic Weyl semimetals, and are expected to realize related topological applications with their attracted features.

Direct visualization of sign-reversals±superconducting gaps inFeTe0.55Se0.45

In many unconventional superconductors, the pairing of electrons is driven by the repulsive interaction, which leads to the sign reversal of superconducting gaps along the Fermi surfaces (FS) or between them. However, to measure this sign change is not easy and straightforward. It is known that, in superconductors with sign reversal gaps, non-magnetic impurities can break Cooper pairs leading to the quasiparticle density of states in the superconducting state. The standing waves of these quasiparticles will interfere each other leading to the quasiparticle interference (QPI) pattern which carries the phase message reflecting also the superconducting gap structure. Based on the recently proposed defect-bound-state QPI technique, we explore the applicability of this technique to a typical iron based superconductor FeTe$_{0.55}$Se$_{0.45}$ with roughly equivalent gap values on the electron and hole pockets connected by the wave vector q_2=(0,\pi). It is found that, on the negative energy side, with the energy slightly below the gap value, the phase reference quantity $|g(q,-E)|\cos(\theta_{q,+E}-\theta_{q,-E}) becomes negative and the amplitude is strongly enhanced with the scattering vector q_2, but that corresponding to the scattering between the electron-electron pockets, namely q_3=(\pi,\pi), keeps all positive. This is well consistent with the theoretical expectation of the s^+- pairing gap and thus serves as a direct visualization of the sign reversal gaps. This experimental observation is also supported by the theoretical calculations with the Fermi surface structure and s^+- pairing gap.