Abstract
Nonsymmorphic crystals are generating great interest as they are commonly found in quantum materials, like ironbased superconductors, heavyfermion compounds, and topological semimetals. A new type of surface state, a floating band, was recently discovered in the nodalline semimetal ZrSiSe, but also exists in many nonsymmorphic crystals. Little is known about its physical properties. Here, we employ scanning tunneling microscopy to measure the quasiparticle interference of the floating band state on ZrSiSe (001) surface and discover rotational symmetry breaking interference, healing effect and halfmissingtype anomalous Umklapp scattering. Using simulation and theoretical analysis we establish that the phenomena are characteristic properties of a floating band surface state. Moreover, we uncover that the halfmissing Umklapp process is derived from the glide mirror symmetry, thus identify a nonsymmorphic effect on quasiparticle interferences. Our results may pave a way towards potential new applications of nanoelectronics.
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Introduction
Research into surface states has been conducted for several decades and has recently begun to flourish again due to the discovery of topologically nontrivial materials^{1,2}. Several topological surface states have been uncovered with prominent examples including the spinmomentum locked Dirac cones in topological insulators^{3,4} and the disconnected Fermi arcs in Weyl semimetals^{5,6}_{.} Many characteristic phenomena, e.g., the prohibition of electron back scattering on a topological insulator surface^{7,8}, the tunable mass acquisition of surface fermions in a topological crystalline insulator^{9} and the electronic sink effect in a Weyl semimetal^{10,11,12,13} have been discerned through quasiparticle interference (QPI) approaches. These have all been proven advances in the understanding of the unconventional twodimensional electron gases. Therefore, the search for new classes of surface states with intriguing physical consequences is an invaluable endeavor in condensed matter physics.
ZrSiSe is a newly discovered nonsymmorphic topological Dirac nodalline semimetal^{14,15,16,17,18,19} and is part of the class of materials which includes ZrSiS, ZrSiSe, and ZrSiTe. Its bulk band features a linear dispersion in the energy range as broad as 2 eV, much larger than other known Dirac materials and presents the ZrSiSe class of materials as an ideal candidate to target new related physics^{17,18,19}. Indeed, a high electron mobility and a butterfly magnetoresistance was discovered by transport measurements^{20}. More importantly, a very recent study revealed an unconventional floating band surface state on ZrSiS but which is also applicable to ZrSiSe and ZrSiTe. Its origin is directly derived from the nonsymmorphic symmetry of the crystal and is distinct from the wellknown Shockley type or danglingbond type surface state^{21}. As demonstrated in Fig. 1a, ZrSiSe is a layered material and crystallizes into a tetragonal lattice with a space group P4/nmm (#129), which is shared with a broad variety of quantum materials, e.g., nematic Febased superconductor NaFeAs^{22,23} and heavyfermion compound with antiferromagnetism (AFM) CeRuSiH_{1.0}^{24}. In the electronic band structure of ZrSiSe (Fig. 1b), nonsymmorphic symmetry enforces the bulk bands to be doubly (quadruply if considering spin degrees of freedom) degenerate along entire XM line; in other words, there exists a Dirac nodal line on the Brillouin Zone (BZ) boundary. On its (001) surface, the symmetry breaking splits a twodimensional electronic state from the bulk Dirac band, termed as a floating band. Figure 1c presents the first principle calculation result where the floating band is highlighted. Obviously, this previously unknown surface state exists in a wide range of P4/nmm symmetric crystals which goes beyond topologically nontrivial materials. However, other than the identification of its origin, little is known about this surface state.
Among all surface sensitive measurements, QPI which is acquired via scanning tunneling microscopy (STM) may be the most direct method to reveal the unique physics of surface states. An ordinary QPI map measures surface standing wave induced by a number of (usually various types of) point defects. However, the local geometrical and chemical structures of different types of defects carry distinctive information.
Here, we apply a singledefect induced QPI (sQPI) approach, which is of both experimental and theoretical challenge, to directly measure the interferences at both single Sidefect and Zrdefect sites on the ZrSiSe (001) surface. In addition to the previously insightful QPI discoveries on ZrSiS^{25,26} and ZrSiSe^{27}, we directly identify the characteristic properties of a floating band surface state. Moreover, our theoretical analysis reveals the observed anomalous Umklapp process to in principle exist in a broad class of nonsymmorphic crystals.
Results
Rotational symmetry breaking feature
An overview of our lowtemperature STM images, spectroscopy and dI/dV map on a ZrSiSe (001) surface is shown in Fig. 1. The atomically resolved STM image in Fig. 1d clearly shows the square lattice of our high quality ZrSiSe sample, in which the C_{4v} symmetry and the measured lattice constant of 0.37 nm confirms the cleaved surface to be the (001) orientation. The measured local density of state from the dI/dV spectrum (Fig. 1e) exhibits nonvanishing intensity at the Fermi level, revealing the (semi)metallic nature of our sample. Interestingly, our STM image (Fig. 1f) and dI/dV map (Fig. 1g) acquired at an energy near the Fermi level demonstrates an unusual ripple pattern. The pattern contains two orthogonal features, each clearly breaking the C_{4v} symmetry of the crystal surface, and which were not observed on dI/dV maps measured on a cousin material ZrSiS^{25,26}.
Healing effect
In order to reveal the unique properties of the floating band surface state, we performed systematic sQPI measurements on ZrSiSe(001). Three characteristics stand out in the voltagedependent dI/dV maps and their corresponding fast Fourier transforms (FFTs). First, in contrast to the C_{4v} pattern expected from the crystalline symmetry, defects showing C_{2v} symmetric pattern are also found. Figure 2a shows a clear standing wave pattern around the point defect located at the center of the image in the voltage range starting from −50 mV. The wavelength shrinks with elevated bias voltage, thus proving that the surface quasiparticle possesses an electron like band. Clearly, from the map taken at an energy close to Fermi level, i.e., 50 mV, the wave only propagates along one direction. Second, from both Fig. 2a, b, one can discern that the rotational symmetry breaking phenomenon gradually disappears at a higher bias around 400 meV, indicating a healing effect occurring in the sample.
Anomalous Umklapp process
Third, figure 3 shows the expected C_{4v} symmetric sQPI patterns (see Supplementary Fig. 1 for STM images of such defects) in which the standing waves propagate equally along two orthogonal directions. Unexpectedly, from the FFT maps in Fig. 3b, one can note that the QPI features around a Bragg point (inside the dotted circle) do not resemble the central pockets (solid circle). This appears to violate the ordinary theoretical understanding of Umklapp scattering and indicates an anomalous structure in the Umklapp process.
Discussion
It happens that the nonsymmorphic symmetry and the special crystal structure of the ZrSiSe class of materials are what lead to the observed QPI features. One unique feature in the structure of its P4/nmm lattice is the layer dependent location of the rotational axis. The ZrSiSe crystal is formed by alternatively stacked Se–Zr–Si–Zr–Se atomic layers. Each layer itself comprises of a square mesh of atoms which preserves global C_{4v} symmetry. However, the Si layer forms the glide mirror plane, and thus loses C_{4v} symmetry locally at each Si atom site in order to fulfill the \(\left( {\left. {M_z} \right\frac{1}{2}\frac{1}{2}0} \right)\) operation. The bulk symmetry gives rise to a (001) surface atomic structure (Fig. 1d) where the Se atoms are located at the corners of the square surface unit cell, the Zr atom sits at the face center, and the Si atoms occupy the edge centers. This naturally leads to the appearance of two inequivalent Si positions, each exhibiting only hidden local C_{2v} symmetry. In addition to single Si defect, multiple C_{2v} symmetric Si defects may arrange into multidirectional (Fig. 1g) or unidirectional (Supplementary Fig. 2) configurations. The latter case, if artificially controllable, will give rise to anisotropic scattering of electrons at the Fermi level and consequently lead to a twofold resistance in ZrSiSe based nanostructures.
In contrast to the crystal structure analysis, we find that first principle simulations and theoretical analysis are crucial to interpret the healing effect occurring in the sample restoring to C_{4v} symmetry and the anomalous Umklapp process. We begin the discussion by considering sQPIs on a Zr defect. Having carefully identified the floating band from the other states (see details in Supplementary Fig. 3), we can now draw a schematic constant energy contour (CEC) which only exhibits such bands (Fig. 4a). The floating band pockets manifest as four large rings enclosing the corners of the first BZ. Near a X point, two floating band contours exist approximately parallel to each other, which gives rise to a significantly enhanced nesting vector, i.e., Q_{1} in Fig. 4a. Q_{1} and its C_{4v} rotational partner Q_{2} together constitute the bright central square in the sQPI pattern, which is shown in Fig. 4b. In addition to these intrafirst BZ scatterings (normal processes), interBZ scatterings (Umklapp processes) usually also contribute to the QPI. For example, a normal scattering vector Q_{1} followed by a unit reciprocal vector (G_{x} or G_{y}) produces a typical Umklapp process, which should preserve the crystal symmetry. The C_{4v} point group in ZrSiSe should in principle result in the vector Q_{1} + G_{x} or Q_{1} + G_{y} generating a QPI contour with the exact shape as Q_{1}, which appears as replica squares at the four Bragg points as shown in Fig. 4d. However, our observation clearly contradicts this ordinary Umklapp process. Concretely, the QPI feature near a Bragg point manifests as a doubleparallel arc (Fig. 4b, c) rather than a square. We name this phenomenon, not previously understood, to be an anomalous halfmissing Umklapp process, as exactly half of the expected Umklapp pattern (a square) is absent in the observation.
We carry out a Tmatrix based Green’s function approach to simulate the sQPI patterns of a single Zr defect based on three different assumptions. We first consider a band structure simulation without considering any band unfolding or form factor effects. Figure 4d, e show such sQPI patterns which respectively allow and forbid interBZ scatterings. While both results are able to capture the squareshaped central feature, they both fail to reproduce the halfmissing patterns around the Bragg points. In contrast in Fig. 4f, by including a sublattice induced form factor effect and forbidding interBZ scatterings, we are able to successfully and quantitively simulate the entire sQPI pattern, and, in particular, the double arcs around Bragg points.
Based on the above analysis, we now understand that the origin of these anomalous halfmissing Umklapp processes is actually a direct consequence of the nonsymmorphic effect on the energy band structure in a P4/nmm crystal. In fact, many important effects induced by glide mirror symmetries have been established based on Febased superconductors research^{22,23}, but the conclusions also apply to the all materials which contain \(\left( {\left. {M_z} \right\frac{1}{2}\frac{1}{2}0} \right)\). We invoke the relevant discovery here to interpret the halfmissing Umklapp scattering. Namely, the glide mirror symmetry \(\left( {\left. {M_z} \right\frac{1}{2}\frac{1}{2}0} \right)\) splits the lattice into two sublattices, where the A and B sublattices are glide mirror partners to each other. This induces a particular type of nontrivial form factor, which must be accounted for in a first principle band calculation. Atomic orbitals can be divided into even or odd parity under the mirror M_{z} operation. By adding a minus sign (a form factor) to odd orbitals on B sublattice but keep all other orbitals intact, the M_{z} is effectively absorbed by the wavefunctions. In the new wavefunction basis, the fractal translation \(\left( {\frac{1}{2}\frac{1}{2}0} \right)\) becomes a good symmetry, which effectively reduces the original unit cell by half and thus expands the area of the first BZ by exactly two folds. In the reconstructed first BZ (dashed line in Fig. 4a), Q_{1} + G_{x} changes from an Umklapp to a normal process, which has no reason to be weak or absent. On the other side, Q_{1} + G_{y} is now terminated out of the new first BZ and becomes a real interBZ scattering, i.e., a real Umklapp process. The suppression or absence of this feature exactly leads to the halfmissing Umklapp feature. More profoundly, it indicates that the floating band surface state contains fewer atomicscale ripples in contrast to the behavior from a danglingbond derived surface state. The floating band state is thus believed to be weakly bounded to the surface, analogous to a Fermiarc surface state on a Weyl semimetal^{28}.
By introducing the glide mirror symmetry \(\left( {\left. {M_z} \right\frac{1}{2}\frac{1}{2}0} \right)\) enforced particular type of form factor into our energy band simulations, we can now reproduce the measurements on both C_{4v} and C_{2v} types of defects by considering different Tmatrices which are directly derived from the first principle simulations on a single defect. In Fig. 5a, we present a set of voltagedependent experimental C_{2v} symmetric QPI patterns around unidirectional Si defects (Supplementary Fig. 2). The adjacent panels in Fig. 5b show the simulated patterns which also reproduce the healing effect. The line cuts in Fig. 5c (experimental) and Fig. 5d (theoretical) demonstrate that our simulation corroborates the measurement across a wide energy range, thus proving the robustness of our theory. The healing effect can be understood by analyzing the scattering channels. Namely, at a Sidefect, both a direct scattering channel between Sip orbits and an indirect scattering channel through pd orbits coupling (Zr–Si interaction) coexist. Near the Fermi level, the major signal in a C_{2v} sQPI pattern is dominated by the direct scattering. However, away from the Fermi level, the signal from Zrd orbits, which carry strong C_{4v} symmetry, becomes enhanced (see Supplementary Figs. 5–7 for details), and the interference begins to appear less C_{2v} symmetric. In principle, this healing effect could exist in the entire ZrSiSefamily of topological nodalline semimetals.
In summary, we systemically combined experimental and theoretical sQPI techniques to directly visualize the unconventional floating band type of surface state on a topological Dirac nodal line semimetal ZrSiSe, which features a nonsymmorphic space group P4/nmm. Three effects, namely a rotational symmetry violation, a healing effect, and a halfmissing type anomalous Umklapp process are identified as characteristic properties of a floating band. Moreover, the halfmissing Umklapp process can be understood as a nonsymmorphic effect, which theoretically exists in a broad class of materials whose lattices contain glide mirrors \(\left( {\left. {M_z} \right\frac{1}{2}\frac{1}{2}0} \right)\). One may potentially be able to deduce the \(\left( {\left. {M_z} \right\frac{1}{2}\frac{1}{2}0} \right)\) symmetry induced phase shift of the electron wavefunction by using the atomic manipulation technique to arrange an array of adatoms into a particular geometry^{29}. Furthermore, the revealed anisotropic charge carrier scattering behavior may provide insights in the development of new nanoelectronics. Therefore, we believe our results here are of both fundamental and applicational importance.
Methods
Sample growth
The single crystalline ZrSiSe samples were synthetized by a standard chemical vapor transport method. A stoichiometric mixture of Zr, Si, and Se powders and the transport agent I_{2} (5 mg cm^{−3}) were placed at the end of a quartz tube. The quartz tube was then evacuated, sealed and loaded into a horizontal tube held at high temperature. The occupied end, which contained the reaction powders, and empty end of the quartz tube were maintained at the high temperature 950 °C and low temperature 850 °C respectively. The temperature gradient of tube furnace was maintained for two weeks. The square and rectangular shaped ZrSiSe crystals were formed at the cold end.
STM measurement
The STM/STS measurements were carried out in a scanning tunneling microscope (USM1600, Unisoku) with an ultrahigh vacuum (base pressure~1 × 10^{−10} torr). The samples were cleaved in situ at 80 K and then transferred into the STM head immediately. All the measurements were performed at T = 4.8 K using platinum iridium tips treated with in situ electronbeam cleaning. dI/dV signals were acquired by a lockin amplifier with modulation of 20 mV at 991 Hz. All presented Fourier transformed maps are raw data.
DFT calculations
The firstprinciples calculations were based on the generalized gradient approximation^{30} (GGA) using the fullpotential projector augmentedwave method^{31,32} as implemented in the VASP package^{33,34}. The electronic structure of bulk ZrSiSe were calculated using a 20×20×10 MonkhorstPack kmesh over the Brillouin zone (BZ). We also conducted the calculations of 30layer ZrSiSe slab using a 20×20×1 MonkhorstPack kmesh. The vacuum thickness was larger than 2 nm to ensure the separating of the slabs. The spin–orbit coupling was included. We used Zr s, p, and d orbitals, Si s and p orbitals, and Se p orbitals to construct Wannier functions without performing the procedure for maximizing localization. We combined the bulk Wannier functions and the surface part of slab Wannier functions to simulated the surface spectral weight via a semiinfinite Green’s function method.
In a ZrSiSe crystal, the glide mirror \(\left( {\left. {M_z} \right\frac{1}{2}\frac{1}{2}0} \right)\) guarantees the existence of AB sublattice and enforce the sublattice to precisely locate in the middle of a surface unit cell. The nonsymmorphic effect induces a nontrivial structure factor which must be considered in a first principle simulation. In our calculation, we construct a unitary matrix U(k):
where k is the wavevector, r_{i} is the real space coordinates of ith atom in one ZrSiSe unit cell. By acting this unitary matrix with the Hamiltonian, i.e., U(k) H(k) U^{+}(k), we are able to simulate the nonsymmorphic effect. In contrast, a direct diagonalization of H(k) gives rise to the simulated band structure without considering nonsymmorphic effect, which fails to capture the experimental results
Simulation of sQPI patterns using the Tmatrix approach for ZrSiSe
To simulate the interference patterns, we adopted the Tmatrix approach, which has been widely used in the QPI studies for the surface states on topological materials^{35,36,37,38}. The retarded surface Green’s function of the system can be written as
where E = ω + iη with ω representing energy and η being a small broadening factor and \(H_{\mathrm{s}}^{{\mathrm{eff}}}\left( {\mathbf{k}} \right)\) is an effective surface Hamiltonian calculated by semiinfinite Green’s function method.
When the interference due to the presence of a single nonmagnetic impurity is considered, the Fourier transformed impurityinduced local density of states at a given scattering wavevector q and energy ω can be derived as
where the impurityinduced electronic Green’s function gives rise to g_{imp}(k, q, ω) = Tr(G(k, ω)T(k, k + q, ω)G(k + q, ω)) − Tr(G(k, ω)T(k, k − q,ω)G(k − q,ω))^{*}.
The Tmatrix, T(k, k′,ω), can be expressed as
Note that the impurity potential matrix V_{imp}(k, k′) is induced by an impurity or a vacancy on the surface and carries kdependent matrix elements, where k(k′) indicates outgoing (incoming) wavevector. We considered two types of vacancies on the top most surface: (1) Zr and (2) Si vacancies. We modeled a vacancy by removing all hopping terms associated with the vacancy site. The impurity potential of vacancy for α atom can be expressed in real space as following:
where H_{ij} denotes the hopping amplitude between two orbitals i and j. The summation over any one of i or j basis belonging to α atom site is claimed to ensure that no interactions between the vacancy site α and the surroundings. V_{imp}(k, k′) was obtained from a Fourier transform. Our final results are obtained by extracting the signal of Se orbitals on the topmost surface from ρ_{imp}(q, ω) by assuming the tunneling currents are only from atoms on the topmost surface in scanning tunneling microscopy measurement.
Data availability
All relevant data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
We thank S. Zhang for the helpful discussions. We acknowledge the financial support from National Natural Science Foundation of China (Grant Nos. 11674226, 11790313, 11521404, 11634009, U1632102, 11504230, 11674222, 11574202, 11674226, 11574201, U1632272, U1732273, U1732159, 11655002, 1674220 and 11447601), the National Key Research and Development Program of China (Grant Nos. 2016YFA0300403, 2016YFA0301003, 2016YFA0300500 and 2016YFA0300501), and Technology Commission of Shanghai Municipality (Grant Nos. 15JC402300 and 16DZ2260200). This work is supported in part by the Key Research Program of the Chinese Academy of Sciences (Grant No. XDPB082), the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDB28000000). S.M.H. is supported by the Ministry of Science and Technology in Taiwan under Grant No. 1052112 M110014MY3. T.R.C. is supported from Young Scholar Fellowship Program by Ministry of Science and Technology (MOST) in Taiwan, under MOST Grant for the Columbus Program MOST1072636M006004, National Cheng Kung University, Taiwan, and National Center for Theoretical Sciences (NCTS), Taiwan.
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H. Z. and J.F. J. oversaw the project. Z.Z. conducted the STM measurement with the help of X.A. N, X.Z. W, D.D. G, S. W, Y.Y. L, C. L. and D. Q. T.R. C and C.Y. H. performed the simulations with S.Y. X, S.M.H and H.L. H.P and F. S grow the crystals. Z.T. J and W.K did the theoretical analysis. All authors discussed the result and contributed to the paper writing.
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Zhu, Z., Chang, TR., Huang, CY. et al. Quasiparticle interference and nonsymmorphic effect on a floating band surface state of ZrSiSe. Nat Commun 9, 4153 (2018). https://doi.org/10.1038/s41467018066619
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DOI: https://doi.org/10.1038/s41467018066619
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