Symmetry-selective quasiparticle scattering and electric field tunability of the ZrSiS surface electronic structure

A detailed ZrSiS crystal and electronic structure is summarized in figure 1. Figure 1(A) shows the Brillouin zones (BZ) for the 3D bulk and its 2D surface projection (k_z = 0) (white dashed lines), highlighting the position of the nodal lines in momentum space, the high symmetry directions, and lattice constants a, b, and c. Closed nodal loops at bulk and surface ensue from the crossings of bands with different orbital character, and hence, different symmetry eigenvalues [45, 46, 62]. The relationship between symmetry eigenvalues to the Dirac bands and possible scattering wave vectors are illustrated in figures 1(B), (C). Figure 1(B) shows simplified schematics of the nodal dispersion along $\bar{{\rm{\Gamma }}}-\bar{M},$c while figure 1(C) shows the corresponding constant energy contours at an energy above a DNL. According to density functional theory (DFT) calculations [46], the 'inner' branch/square of the nodal dispersion (closer to $\mathop{{\rm{\Gamma }}}\limits^{̅}$) is composed of the Zr d_xz /d_yz orbital, while the 'outer' branch/square stems from the Zr d_x ²–_y ² orbital. Furthermore, the distinct orbital character of these bands can be described by distinct symmetry eigenvalues (ν = 1 for the d_xz /d_yz orbital and ν = 0 for the d_x ²–_y ² orbital), which impose stringent selection rules for quasiparticle scattering [46].

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The electronic band dispersion as obtained from our DFT calculation of a 3-layer slab is shown in figure 1(D), clearly displaying Dirac nodes close to the Fermi level (E = 0) in the band manifolds along the $\mathop{{\rm{\Gamma }}}\limits^{̅}-\mathop{M}\limits^{̅}$ and $\mathop{{\rm{\Gamma }}}\limits^{̅}-\mathop{X}\limits^{̅}$ directions. Green-coloured bands carry a large spectral weight at the ZrSiS surface, which the STM is predominantly sensitive to. While the bulk band structure including orbital symmetry are reflected at the surface, breaking of the non-symmorphic symmetry across the vertical glide-mirror planes shifts the bulk bands (BB) in energy, including the k_z = 0 nodal loop. Symmetry breaking at the surface further lifts the fourfold band degeneracy of the bulk Dirac node near the $\mathop{X}\limits^{̅}$ point [46, 48, 49, 51–53, 55], shifting the surface bands downward in energy by approximately ∼1 eV. This gives rise to a saddle-like state with a large spectral weight in the surface (green lines in figure 1(D)) which extends across $\mathop{{\rm{\Gamma }}}\limits^{̅}-\mathop{X}\limits^{̅}-\mathop{M}\limits^{̅}.$ As this saddle-like state is completely unpinned from corresponding bulk bands at certain energies and wave vectors, it has been termed a FBSS [45, 53, 54], previously identifiable in ARPES [45, 48], QPI [50–55] and transport [9, 53, 57, 58] experiments.

The FBSS 'saddle' with concomitant band flattening near $\mathop{X}\limits^{̅}$ is reflected (figure 1(E)) in an enhancement of the LDOS near −300 meV (red arrow), measured in STM/STS at T = 4.5 K, and in good agreement with the projected density of states calculated. The expected double-peak structure in the LDOS, owing to the two inflection points near the FBSS saddle, is more clearly observed with increased spectral resolution in figure 3(B) (discussion below). The minimum at E = −21 meV reflects the reduction in the measured LDOS (black arrow) for energies near the DNL. Energies E > 136 meV corresponds to the onset of parabolic bulk bands, reflected as a strong enhancement of the LDOS.

We then probe the surface states of ZrSiS, including the FBSS using QPI spectroscopy [50–55, 66–71, 75–81] (figures 1(G)–(I)). A comparison of an atomic-resolution STM image of the S-terminated ZrSiS surface after cleaving and its corresponding differential conductance (dI/dV) map at $(E-{E}_{F})$ = +500 mV is shown in figures 1(F), (G). The dI/dV map has been processed by Fourier filtering of the atomic lattice to highlight QPI induced charge density modulations with wave vector $\vec{q}$ = $\vec{k}$ _f —$\vec{k}$ _i . The dominant scattering vectors $\vec{q}$ and hence the likelihood of scattering between different initial and final states can be obtained from a 2D fast Fourier transform (FFT), which are then compared with the corresponding surface spectral weight at a similar energy in figures 1(H), (I). We find that scattering within the surface floating band [52, 54, 55] gives rise to a square pattern at small wave vector ($\vec{q}$ ₁) (figure 1(I)). Transitions between states within the concentric squares in the $\mathop{{\rm{\Gamma }}}\limits^{̅}-\mathop{M}\limits^{̅}$ direction are most prominently observed in $\vec{q}$ _4, $\vec{q}$ ₅ and, to a lesser extent, $\vec{q}$ ₃. The vector $\vec{q}$ ₆ corresponds to scattering between the surface and bulk [55] bands in the $\mathop{{\rm{\Gamma }}}\limits^{̅}-\mathop{X}\limits^{̅}$ direction. Notably, scattering with small momentum transfer between the concentric squares (figure 1(B)), and hence between states of different symmetry eigenvalues, is strongly suppressed as inferred from the absence of spectral feature corresponding to scattering vector $\vec{q}$ ₂ (dashed red square in figure 1(I) [52]). This indicates that native atomic point defect [54] and defect ensembles do not allow for scattering between states of different orbital symmetry eigenvalues [46, 55]. This finding is consistent with the lack of any report of a clear, unambiguous signature of this scattering vector across all QPI studies published so far [51–55] and confirms that orbital symmetry is effective in forbidding these states from quasiparticle scattering.

After investigating QPI with point defects as scatterers, we then perform QPI measurements at an atomic step edge of our ZrSiS sample. Step edges have a different spatial symmetry from point defects given that the former break translational invariance with a strong short range scattering potential and impose a hardwall boundary on surface-confined state. Indeed, atomic step edges have been extensively investigated for the presence of 1D electronic edge states [77–84] as well as scattering/transmission of 2D surface states, e.g. on the surfaces of 3D topological insulators (TIs) [81, 84], Weyl [85] and Dirac semimetals [84], as well as higher-order topological insulators [77, 86]. For 3D TI surface states, it has been shown that translational symmetry breaking within the surface [87] faciliates scattering of the otherwise topologically protected 2D helical state.

In figure 2(A), we show the topography around a ZrSiS surface line defect atomic step, terminated along the [110] direction of the crystal, allowing us to probe the $\mathop{{\rm{\Gamma }}}\limits^{̅}-\mathop{M}\limits^{̅}$ direction in reciprocal space (see inset). An STM height profile and a corresponding series of point spectra along a line (arrow in figure 2(A)) across the edge are shown in figures 2(B) and (C). To highlight the charge density modulations due to quasi-particle scattering and interference, we have subtracted a smooth background reference spectrum, taken from the upper terrace at a distance 10 nm from the edge, where the modulations subside (supplementary figure 1). We find a rich QPI pattern on the upper ZrSiS terrace (figure 2(C)), which is nearly absent on the lower terrace. To identify the dominant scattering vectors, we compare 1D FFTs for both the lower and the upper terraces (figures 2(E), (F), and figure S2 of the supplementary material), with the corresponding DFT dispersion in the $\mathop{M}\limits^{̅}-\mathop{{\rm{\Gamma }}}\limits^{̅}-\mathop{M}\limits^{̅}$ direction (figure 2(D)). Expected scattering vectors are indicated by white arrows, and are in excellent agreement with those determined in figures 1(H), (I). The absence of any prominent features in the lower terrace FFT confirms that scattering by the step edge is strongly suppressed here, with only indications of one scattering vector ($\vec{q}$ ₇) resolved within the noise, which reduces to $|\vec{q}|$ = 0 at E ≈ −200 meV, reflecting the DNL in the ZrSiS bulk. The complex QPI pattern on the upper terrace allows us to identify as many as four scattering vectors ($\vec{q}$ ₂, $\vec{q}$ ₄, $\vec{q}$ ₅ and $\vec{q}$ ₇), with $\vec{q}$ _2,4,5 being in excellent agreement with those predicted in figure 1(H), and $\vec{q}$ _4,5 with previous reports [50–55].

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However, in stark contrast to the absence of scattering vector $\vec{q}$ ₂ on terraces without symmetry eigenvalue mixing (figures 1(H), (I), 2(E), as well as previous reports [50–55]), we find that $\vec{q}$ ₂ is a very noticeable scattering vector at the upper terrace near the step edge (figure 2(F)), with comparable prominence to $\vec{q}$ ₄ and $\vec{q}$ ₅ (see also figure S2 of the supplementary material). The vector $\vec{q}$ ₂ corresponds to small momentum scattering between states of different symmetry eigenvalues, i.e. the step edge effectively mixes the Zr d_xz /d_yz (ν = 1) and d_x ²–_y ² (ν = 0) orbitals. To our knowledge, this is the first time [46, 55] that this scattering vector $\vec{q}$ ₂ has been unambiguously identified in ZrSiS. Previously, there were faint signatures of $\vec{q}$ ₂ in ZrSiSe [55], but the observation has not been discussed in the context of symmetry eigenvalues. Additionally, to the best of our knowledge, there have not been other reports of QPI scattering in ZrSiX-type compounds in which $\vec{q}$ ₂ is identified.

The scattering vector $\vec{q}$ ₂ further allows us to confirm the presence of the Dirac nodal dispersion at the surface and extract the nodal line energy of E = 140 meV in the $\mathop{{\rm{\Gamma }}}\limits^{̅}-\mathop{M}\limits^{̅}$ direction, where $|{\vec{q}}_{2}|$ reduces to zero. We note that, although vectors $\vec{q}$ ₂ and $\vec{q}$ ₄ represent scattering between states of different symmetry eigenvalues, there are important differences. The vector $\vec{q}$ ₂ corresponds to a small wave vector scattering on the same side of the Brillouin zone, whereas $\vec{q}$ ₄ corresponds to a large wave vector scattering across the opposite sides of the Brillouin zone [52–55]. The non-dispersive features in figure 2(E) may have originated from symmetry eigenvalue mixing that could occur at energies far above or below the nodal line even in clean terraces [50, 52, 55]. Furthermore, since the vectors $\vec{q}$ ₄ (which mixes different symmetry eigenvalues) and $\vec{q}$ ₅ (which does not mix different symmetry eigenvalues) are closely separated in wave vector (by ∼$0.1\times \frac{2\pi }{a}$) and can be difficult to distinguish (compare with figure 1(I)), $\vec{q}$ ₂ is the only unambiguous proof of a QPI scattering that mixes different symmetry eigenvalues.

Following identification of the ZrSiS surface nodal dispersion, we proceed to show that the ZrSiS surface electronic structure can be effectively tuned by applying a strong vertical electric field locally via the STM tip [88]. Local field-effect tuning of semiconductor surfaces, often referred to as tip-induced bend bending, is a well-established phenomenon on metal–insulator–semiconductor junctions [89–91], but more recently has also been used to field-tune the band structure of atomically thin layers of the 3D Dirac semimetal Na₃Bi [88]. In [88], tip-induced electric fields have been shown to Stark shift electronic bands near the Fermi energy to induce a topological phase transition [88]. As shown schematically in figure 3(A), tip-induced band bending occurs when the STM tip is placed in proximity to a sample of a different work function. The difference in work function Δϕ combined with the applied bias then causes a built-in electric field ${F}_{z}=-\frac{\partial U}{\partial z}\,={\rm{\Delta }}{\phi }/z$ at the junction, where $U$ and z are the electric potential and the tip height, respectively.

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Figure 3(B) shows the measured tunnelling LDOS as a function of the tip-sample separation Δz = z – z₀ , measured at position z relative to the initial tip height z₀ , similar to the approach of [88]. For each spectral feature identified in figure 3(B), there is a monotonic shift in energy as the tip approaches the sample surface (i.e. Δz becomes more negative). Of particular interest are two LDOS peaks found at −158 meV and −323 meV (arrows) when Δz = 0, which have previously been attributed to arise from the flattened bands at the inflection points of the FBSS saddle [48–51, 53]. These FBSS shift lower in energy by as much as 200 meV as the tip approaches the surface, which we interpret as evidence for electric field tunability of the floating band state. This interpretation is indeed supported by our DFT calculations (figure 3(B), righthand side, blue lines) for zero and finite (F_z = 1 V Å⁻¹) vertical electric field (red lines), respectively, which reproduce the observed surface band shifts (black arrows in figure 3(B)) well. Our calculations further confirm that at a strong vertical electric field breaks the degeneracy of the FBSS between top (bottom) surfaces, whereby the surface bands shift down (up) in energy, respectively. The polarity of the electric field is also confirmed through the dependence of surface-concentrated parabolic bulk bands (BB) between $\mathop{{\rm{\Gamma }}}\limits^{̅}-\mathop{X}\limits^{̅}$ on the applied electric field, where the bands shift upward in energy in both experiment and DFT calculations.

We obtain a rough estimate of the built-in electric field strength by considering the work function difference at the junction, following the procedure in [88]. As the work function of ZrSiS is unknown, we first extract an average work function of tip (t) and sample (s) $\mathop{\phi }\limits^{̅}=\tfrac{1}{2}\left({\phi }_{{\rm{s}}}+{\phi }_{{\rm{t}}}\right)$ by fitting $I(z)={I}_{0}{e}^{-2\kappa z}$ to the exponential dependence of the tunneling current in figure 3(B), where $\kappa =\sqrt{2m\mathop{\phi }\limits^{̅}}/{\rm{ \hbar }}$ and m the free electron mass. We find $\kappa =1.04$ ${\mathring{\rm A} }^{-1}$ and $\mathop{\phi }\limits^{̅}=$ 4.16 eV. Further assuming the work function of a gold-coated Pt/Ir tip, (ϕ_tip = 5.3 eV [90]), we estimate ϕ_s = 3.02 eV for ZrSiS, comparable with the work functions of both Zr and Si [92]. Hence, for a typical tip-sample separation of $z$ = 1 nm and work function difference Δϕ = ϕ_s – ϕ_t = 2.28 eV, we estimate an electric field of ${F}_{z}{\rm{\approx }}2$ V nm⁻¹ applied at zero bias. We do note that this electric field estimate is substantially smaller compared to the assumptions made in the theoretical calculations. However, this discrepancy is consistent with previous demonstrations of the tip-induced electric field effect [88]. Also, our simplified estimate of the built-in electric field relies on the assumption of a planar tip-sample junction and should thus be considered a lower bound to the field strength. Indeed, the atomically sharp probe of the STM may be expected to lead to a considerable enhancement of the electric field strength in the region directly underneath the tip's apex where the LDOS is probed [91].

Finally, we focus on the narrow suppression of the LDOS observed in figure 3(B) near E = E_F. A close-up of this feature and its tip-height dependence is shown in figure 3(C). The position of the LDOS minimum V_c of the suppression and its width ${\rm{\Delta }}$ are plotted in figure 3(D), respectively, as extracted from Gaussian fits (solid lines in figure 3(C)). We observe a clear shift of V_c and widening of the LDOS suppression as the tip approaches the surface (solid and dashed lines are guides to the eye). Comparing to our DFT model (figure 3(B)), we interpret the feature observed in figure 3(C) as a spin–orbit induced avoided crossing that causes the formation of a soft energy gap to near the $\mathop{X}\limits^{̅}$-point (see inset to figure 3(B) comparing DFT with and without SOC). We find that ${\rm{\Delta }}$ can be tuned between 30 meV and 85 meV, nearly tripling over the range of tip heights probed. For the asymptotic limit at a large tip-sample distance (and hence zero electric fields), we extract a spin–orbit gap ${\rm{\Delta }}\,$= 20 meV, in good agreement with our DFT predictions (${\rm{\Delta }}\,$= 11.6 meV), as well as with previous ARPES-based measurements (${\rm{\Delta }}\,$= 10 meV) [48, 49], confirming that the change in the spin–orbit gap in our sample is indeed due to the applied electric field.