Quasiparticle interference of the Fermi arcs and surface-bulk connectivity of a Weyl semimetal

See allHide authors and affiliations

Science  11 Mar 2016:
Vol. 351, Issue 6278, pp. 1184-1187
DOI: 10.1126/science.aad8766

Sinking into the bulk of a Weyl semimetal

A recently discovered class of topological materials, Weyl semimetals, have surface states in the form of so-called Fermi arcs. Inoue et al. used a high-resolution scanning tunneling microscope to explore the properties of these states in the material TaAs. They mapped the scattering of electrons off impurities on the surface of the material and compared the data to the predictions of density functional theory. The data could be reconciled with the theory only if electrons associated with Ta orbitals on the Fermi arcs sank into the bulk of the material.

Science, this issue p. 1184


Weyl semimetals host topologically protected surface states, with arced Fermi surface contours that are predicted to propagate through the bulk when their momentum matches that of the surface projections of the bulk’s Weyl nodes. We used spectroscopic mapping with a scanning tunneling microscope to visualize quasiparticle scattering and interference at the surface of the Weyl semimetal TaAs. Our measurements reveal 10 different scattering wave vectors, which can be understood and precisely reproduced with a theory that takes into account the shape, spin texture, and momentum-dependent propagation of the Fermi arc surface states into the bulk. Our findings provide evidence that Weyl nodes act as sinks for electron transport on the surface of these materials.

Understanding the exotic properties of quasiparticles at the boundaries of topological phases of electronic matter is at the forefront of condensed-matter physics research. Examples include helical Dirac fermions on the surface of time-reversal invariant topological insulators (1), which have been demonstrated to be immune to backscattering (24), or the emergent Majorana fermions at the edge of a topological superconductor (57). Topological properties of phases of matter are, however, not limited to gapped systems, and recent theoretical efforts have uncovered the possibility of topologically protected metallic phases. Topological Dirac semimetals (8, 9)—three-dimensional analogs of graphene with band crossings protected by crystalline symmetry—have been recently realized experimentally (1012). Breaking of the inversion or time-reversal symmetry splits the Dirac crossings of the band structure into Weyl points, which are singularities in the Berry curvature characterized by a Chern number or more colloquially monopole charge (13, 14). These semimetals host bulk quasiparticles that are chiral Weyl fermions. The topologically protected boundary modes of Weyl semimetals are surface states with a disjointed two-dimensional Fermi surface (15, 16). These so-called Fermi arcs connect surface projections of bulk Weyl points of opposing Chern numbers. A notable property of these Fermi arc states is that they can become delocalized into the bulk through the projected Weyl points. Ideally, in the absence of disorder, an electron on the Fermi arcs of one surface can sink through the bulk and appear on the arcs of the opposing surface.

Recent work has shown strong evidence that inversion symmetry–breaking transition-metal compounds (TaAs, TaP, NbAs, NbP) are Weyl semimetals (1721). Angle-resolved photoemission spectroscopy (ARPES) measurements have confirmed that the surface electronic structure of these compounds has a disconnected arclike topology connecting the surface projection of 24 Weyl crossings in the bulk band structure. Here we used the scanning tunneling microscope (STM) to directly visualize the surface states of the Weyl semimetal TaAs and examine their scattering and quantum-interference properties. Our experimental approach follows similar STM studies that showed that the spin texture (and time-reversal symmetry) protects surface states on topological insulators from backscattering (24, 22). In contrast to these previous studies, we find that the momentum-dependent delocalization of the Fermi arcs into the bulk, which is a unique property of these surface states, determines the coherent interference properties of these surfaces states. The surface-bulk connectivity examined here underlies several other novel electronic phenomena, such as nonlocal transport that can occur in Weyl semimetals (2327). A recent magnetotransport study has observed quantum oscillations that may be associated with the nonlocal nature of transport, with electron orbits traversing through bulk Weyl nodes and surface states (28).

To probe the properties of the Weyl semimetal’s Fermi arc states, we have carried out STM studies (at 40 K) of in situ cleaved surfaces of TaAs single crystals that show atomically ordered terraces and tunneling density of states spectra consistent with a semimetal (Fig. 1). The analysis of the atomic step edge heights in STM topographs shows only one type of surface for the cleaved samples, which is likely terminated by As atoms [see below and (29)]. Although the surface atomic structure visualized in the STM images has fourfold symmetry, the underlying electronic structure of this compound only has C2v symmetry, which is evident from its overall crystal structure shown in Fig. 1A. A C4v nonsymmorphic symmetry is broken by the surface. The anisotropic local shape of electronic signatures caused by native surface defects shown in STM topography (Fig. 1, B and D), as well as the scattering of surface quasiparticles around such defects over long length scales displayed in STM conductance maps (Fig. 1E), both show a clear C2v symmetry.

Fig. 1 STM topography and dI/dV spectroscopy of Weyl semimetal TaAs.

(A) Illustration of the crystal structure of TaAs with (001) As termination. One unit cell contains four layers of As and Ta. The lattice parameters are a = b = 3.43 Å and c = 11.64 Å. (B) STM topographic image (Vbias = 500 mV, Isetpoint = 100 pA) of the cleaved (001) surface of TaAs. Magnified views on some of the pronounced impurities (right panels) show apparent C2v symmetric deformation of the electronic structure. (C) Spatial variation of the differential conductance measurements along a line of 30 ‎Å, shown as a yellow line in (B), was measured with lock-in techniques employing 3-mV excitations at 707 Hz. The orange curve shows the spatially averaged dI/dV spectra. An atomically periodic modulation of the spectra is visible. (D) Topographic image and (E) conductance map at 120 meV (Vbias = –340 mV, Isetpoint = 80 pA) on the same area, where the C2v symmetric nature of the surface electronic states is clearly visible.

Detailed information about the electronic properties of the surface states can be obtained from measurements of the quasiparticle interference (QPI) patterns in large-area STM conductance maps. The Fourier transform of such STM conductance maps identifies scattering wave vectors q, which connect ki and kf states on the contours of constant-energy surface in momentum space (30). QPI measurements can be used not only to follow the evolution of the Fermi surface probed near the surface but also to determine whether some scattering wave vectors are forbidden as a result of discrete symmetries (2, 23, 24). Information on quasiparticle lifetime can also be extracted from QPI; however, such effects do not typically result in momentum-specific changes in the QPI, as we discuss here.

Experimentally, the detection of scattering wave vectors in QPI measurements is limited by instrumental resolution, which is determined by the stability of the instrument and the maximum averaging time possible during each map. In Fig. 2, we show QPI measurements on the TaAs surface obtained with a high-resolution, home-built STM instrument capable of averaging for up to 6 days while maintaining atomic register. The Fourier transform of these QPI measurements (Fig. 2) allows us to resolve the rich array of scattering wave vectors on this compound. Such high-resolution measurements are required to resolve the entire set of allowed scattering wave vectors in this compound. In Fig. 2, Q to S, we also show measurements of the QPI features as a function of energy along specific directions in q space, displaying continuous dispersion of these features with energy, as is typical of QPI features. Recent measurements on a related Weyl compound (NbP) at lower resolution have yielded a subset of QPI signals (3 out of 10 wave vectors) reported here (31).

Fig. 2 Quasiparticle interference of TaAs surface states.

(A to D and I to L) Spatially resolved dI/dV conductance maps at different energies obtained on the area shown in Fig. 1B (Vbias = 240 meV, Isetpoint = 120 pA). (E to H and M to P) Symmetrized and drift-corrected Fourier transforms of the dI/dV maps (QPI maps). Red dot indicates the (2π, 2π) point in the reciprocal space in the unit of 1/a; PSD denotes power spectral density. In the color bar, m corresponds to the mean value of the map and σ to the standard deviation. (Q to S) Energy-momentum structure of the dI/dV maps (Vbias = 600 meV, Isetpoint = 80 pA) along the high-symmetry directions shown in (M).

QPI measurements from surface states of most materials can be understood by starting from a model of contours of constant energy in momentum space, consistent with ARPES-measured band structure, that can be used to calculate a joint density of states (JDOS) probability for scattering as a function of momentum difference q (32). If the Fermi surface is spin textured, as in the case of helical Dirac surface states or strongly spin-orbit–coupled systems, the experimental results can be compared with the spin-dependent scattering probability (SSP) maps that also take into account the influence of relative spin orientations of the initial and final states on the QPI measurements (2, 29). Following such an approach, we use density functional theory (DFT) methods to calculate the surface states for TaAs. We reproduce both the shape (17, 20, 21) and spin texture (3335) of the surface Fermi contours (Fig. 3A) recently measured with ARPES on this compound and calculate the SSP map near the chemical potential, which can then be compared to our experimental results at the same energy. In addition to the Fermi arcs, both ARPES measurements and the DFT calculations capturing them may include some features that are in fact caused by trivial surface states (29). This approach results in an SSP (Fig. 3B) that resembles the overall symmetry of the experimentally measured QPI pattern (Fig. 3F); however, it produces many more scattering wave vectors than seen experimentally (such as those in the middle of the QPI zone highlighted in Fig. 3B that are missing in Fig. 3F). This approach or a related one proposed recently (36) for understanding the data on NbP (31) also fails to capture the QPI data on TaAs at other energies (29).

Fig. 3 Fermi arcs and quasiparticle interference.

(A) DFT-calculated Fermi arc contour of constant energy in first Brillouin zone (BZ) at +40 meV calculated projecting the DFT-calculated spectral density to the top unit cell. Green dots indicate the projected positions of the Weyl nodes, and arrows show the direction of the spin on the Fermi arcs. The combination of time-reversal symmetry and C2v symmetry implies vanishing out-of-plane component of the spin. FS, Fermi surface. (B) SSP derived from (A) marked by regions of strong scattering in the middle of the BZ, which is missing in the QPI data. (C) The integrated spectral density over the full BZ for As and Ta separately as a function of layer index shows the fast decay of As orbital states. (D) Weighted Fermi surface (WFS) calculated by projecting the electronic states only to the topmost As layer. The Q vectors indicate the scattering wave vectors expected. (E) SSP based on (D) with red boxes showing the scattering wave vectors mapped in (F). (F) QPI map at 40 meV (same as in Fig. 2G) and experimentally observed groups of Q vectors. FT-STS, Fourier transform–scanning tunneling spectroscopy.

An accurate model of our experimental results over a wide energy range can be achieved if we consider not only the shape, the density of states, and the spin texture of the Fermi arcs but also their momentum-dependent delocalization into the bulk of the sample. The key conceptual idea is that the QPI of the Fermi arcs is dominated by initial and final momentum states that do not strongly leak into the bulk. The degree of connectivity between Fermi arc surface states and the bulk is a property that depends on their momentum approaching the projection of Weyl points on the surface. It is also related to the atomic character of the Weyl nodes. Our DFT simulation of the TaAs electronic structure shows that the majority (~90%) of electronic states associated with the bulk Weyl nodes are based on Ta atomic orbitals (29). Consistent with this information, we also find from the DFT calculation that the electronic states close to the projected Weyl points on the surface arcs have a large component of such orbitals. This results in the slow decay of the Fermi arcs’ spectral weight associated with Ta orbitals into the bulk, as compared to that associated with As orbitals (Fig. 3C) (29). This picture suggests that to understand the QPI data on TaAs, we should project out the Fermi arc states that are associated with Ta and focus only on states with the As-orbital characteristics, which have the weakest connectivity to the bulk states and hence can interfere coherently to produce the QPI patterns. The Ta orbitals in the first layer only weakly contribute to the QPI signal: An electron that scatters from the As site to a Ta site in the surface layer is more likely to sink into the bulk Weyl states (Fig. 3C). The presence of topologically trivial surface states [likely the inner bowtie feature in Fig. 3, A and D; see also (29)] does not alter the projection of bulk Weyl points at the surface, and all surface states at the Fermi level with these momenta would still be strongly delocalized into the bulk.

To confirm this idea, we consider a weighted Fermi arc contour for TaAs, in which we project the Fermi arc states onto the As orbitals at the topmost surface layer (Fig. 3D). This projection results in changes that are strongly dependent on the distance (in k space) from the Weyl point. Taking into account the delocalization into the bulk, we find strong fading of the innermost spoon-shaped Fermi arcs in the weighted Fermi surface, as compared to that shown in Fig. 3A. These short spoon-shaped sections of the Fermi arcs are close to their corresponding Weyl points, which are expected to act as sinks for electron propagation on the surface (because of their dominant Ta orbital character). In contrast, the prominent features of this weighted Fermi surface come from longer bowtie-shaped arcs at the Brillouin zone boundary, which lie far away from their corresponding Weyl point, with the least probability of sinking into the bulk (owing to their dominant As orbital character). An electron in the weighted Fermi arcs shown in Fig. 3D (or a similar one that also includes the As projection in the second layer; see the supplementary materials) is in a subset of states that do not efficiently propagate away into the bulk and remain near the surface.

Without any further computation, we can understand the various wave vectors seen in the QPI measurements by simply considering scattering around the weighted Fermi arc contours (Fig. 3D). Based on the length and orientation of the possible scattering wave vectors on the weighted Fermi surface, we identify 10 different groups of scattering wave vectors, Q1 to Q10, that are seen experimentally. More detailed calculations of SSP (Fig. 3E) near the chemical potential based on these weighted Fermi arcs also compare favorably to the experimentally measured QPI at the same energy (Fig. 3F). In making this comparison, we note that although the spin texture of the Fermi arcs and the absence of backscattering between time-reversed pairs of states play a role in the scattering data (a point to which we return below), the differences between JDOS and SSP for the Fermi arcs are relatively minor and not critical to understanding the QPI data [(29) and see below]. Contrasting results of the SSPs using weighted (Fig. 3E) and unweighted (Fig. 3B) Fermi arcs with the experimental data (Fig. 3F) demonstrates that at some momenta, the Fermi arc surface electrons have a strong probability of sinking into the bulk state and hence are not part of the QPI process. We further test this physical picture by a more exhaustive comparison of theoretical model calculations based on the weighted Fermi arcs and the experimental data over a wide range of energies (Fig. 4, A to L). As shown in this figure, our model SSP calculations for the weighted Fermi arc can reproduce the finer features of the large body of QPI data obtained in our studies. In addition, isolating the contribution from the As or Ta atomic sites to the QPI signal also can be used to confirm our theoretical identification of the major contribution of each atomic orbital to different sections of Fermi arcs [bowtie and spoon features dominated by As and Ta, respectively (29)]. The agreement between theory and QPI measurements demonstrates the importance of accounting for delocalization of the TaAs surface states caused by the Weyl nature of its band structure.

Fig. 4 Comparison of the FT-STS maps with the DFT calculation at various energies.

(A to C and G to I) QPI maps (same as in Fig. 2) and (D to F and J to L) the corresponding SSP derived from the projection of the spectral density to the topmost As. The obtained data and the calculations are in agreement over a wide energy range. (M to O) Enlargement of the Q9 and Q3 vectors in the calculated JDOS, SSP, and QPI data at different energies. Whereas strong triple-arc structures are visible in JDOS, SSP shows a reduced number of arcs, which is consistent with the single arcs in the QPI data.

We now return to the role of spin texture in determining the scattering properties of the Fermi arcs. Focusing on some of the finer features of the QPI data, in particular, scattering wave vectors Q3 and Q9, and their comparison to the theory, we can also resolve the subtle influence of spin texture on the STM data. This comparison (Fig. 4, M to O) demonstrates that some of the duplicate features in the JDOS maps caused by scattering between arc states that have opposing spins are suppressed in the SSP maps. Although almost at the limit of our experimental resolution, the correspondence between the finer predicted features in the SSP based on our model and those in the experimental QPI data qualitatively confirms that spin does play a role, albeit minor, in our QPI measurements. There are a few special wave vectors that are strictly prohibited, owing to the presence of time-reversal or mirror symmetry (23, 24). However, these signatures of protected scatterings are unfortunately obscured by many allowed overlapping wave vectors with similar lengths and are hard to resolve experimentally. Future experiments at higher resolution or on Weyl semimetal with simpler structures may better resolve this protection and other universal features of Fermi arcs that are predicted theoretically (37, 38).

Our work reveals that the absences of, or restriction on, coherent scattering at certain wave vectors in both theory and experiments are not a consequence of a symmetry-related protection for the Fermi arc surface states, but rather follow from the topological connection between the surface and bulk states. This connection can also be seen in geometries in which the sample thickness is less than the bulk’s scattering mean free path, where the top and bottom surface of a Weyl semimetal would be strongly linked through the Weyl points (23, 2628).

Supplementary Materials

Materials and Methods

Supplementary Text

Figs. S1 to S8

References (3941)

References and Notes

  1. Materials and methods are available as supplementary materials on Science Online.
Acknowledgments: Work at Princeton was supported by Army Research Office–Multidisciplinary University Research Initiative (ARO-MURI) program W911NF-12-1-0461, Gordon and Betty Moore Foundation as part of Emergent Phenomena in Quantum System initiative (GBMF4530), by NSF–Materials Research Science and Engineering Centers programs through the Princeton Center for Complex Materials DMR-1420541, NSF-DMR-1104612, NSF CAREER DMR-0952428, Packard Foundation, and Keck Foundation. This project was also made possible through use of the facilities at Princeton Nanoscale Microscopy Laboratory supported by grants through ARO-W911NF-1-0262, ONR-N00014-14-1-0330, ONR-N00014-13-10661, U.S. Department of Energy–Basic Energy Sciences (DOE-BES)Defense Advanced Research Projects Agency–U.S.Space and Naval Warfare Systems Command Meso program N6601-11-1-4110, LPS and ARO-W911NF-1-0606, and Eric and Wendy Schmidt Transformative Technology Fund at Princeton. Work at University of California–Los Angeles was supported by the DOE-BES (DE-SC0011978).
View Abstract

Stay Connected to Science

Navigate This Article