Surface Floating 2D Bands in Layered Nonsymmorphic Semimetals: ZrSiS and Related Compounds

In this work, we present a model of the surface states of nonsymmorphic semimetals. These are derived from surface mass terms that lift the high degeneracy imposed on the band structure by the nonsymmorphic bulk symmetries. Reflecting the reduced symmetry at the surface, the bulk bands are strongly modified. This leads to the creation of two-dimensional floating or unpinned bands, which are distinct from Shockley states, quantum well states, or topologically protected surface states. We focus on the layered semimetal ZrSiS to clarify the origin of its surface states. We demonstrate an excellent agreement between density functional theory calculations and angle-resolved photoemission spectroscopy measurements and present an effective four-band model in which similar surface bands appear. Finally, we emphasize the role of the surface chemical potential by comparing the surface density of states in samples with and without potassium coating. Our findings can be extended to related compounds and generalized to other crystals with nonsymmorphic symmetries.

Popular Summary

Depending on their electronic properties, materials can be divided into three groups: metals, semiconductors, and insulators. This classification becomes more complicated if one considers the surface of such materials. Topological insulators, for example, are metallic at the surface, while the interior is insulating. There are many reasons for the occurrence of different electronic states on the surface of materials in general. However, there is no explanation for their appearance in a class of compounds known as “nonsymmorphic square-net compounds,” which incorporate translational symmetry elements in their crystal structure. We focus on one of these compounds, the semimetal zirconium silicon sulfide (ZrSiS), and we show both experimentally and theoretically that the surface states arise from the breaking of the nonsymmorphic symmetry at the surface.

Using angle-resolved photoemission spectroscopy (ARPES) and density-functional-theory calculations, we rule out explanations based on other known types of surface states. We then develop a theory that explains that a loss of nonsymmorphic symmetry at the surface unpins bands that are forced to be degenerate in the bulk, resulting in a surface state that occupies a region not usually accessible for the bulk states. We thus classify a new type of surface state, which is not restricted to ZrSiS and other square-net materials but can appear in many different nonsymmorphic compounds.

These surface states differ in their properties from other surface states and will be important for experimental techniques besides ARPES that are sensitive to surface properties.

Figure 1

Schematic of the surface states possible in nonsymmorphic materials, as occurrent in ZrSiS. The symmetry enforcement of the band degeneracy at $\overline{X}$ is relaxed at the surface, creating a sizable energy separation between the surface band (red line) and bulk bands (black lines). With minimal coupling between them, the two-dimensional surface band can be seen as “floating” in an energy region not accessible to bulk states.

Figure 2

Visualization of the surface states in ZrSiS. Panels (a) and (b) show dispersion plots along $\overline{\mathrm{\Gamma }}\overline{X}\overline{M}$. The path through the Fermi surface is shown in red in Fig. 3. (a) ARPES data of a bare surface, presenting the surface states (SS and ${\mathrm{SS}}^{\prime }$) as high intensity features crossing the bulk bands (BS). (b) DFT slab calculations where the surface state character is indicated by the size of the red circles. (c) Crystal structure of ZrSiS. The unit cell is shown in black. The cleavage plane ($CP$) is indicated by a black arrow. (d) Isosurfaces of the charge density of the surface state at different $k$ points. The arrows indicate the corresponding independent particle states in the band structure. Only the topmost layer of the supercell is shown because the charge density vanishes in the bulk. At the $\overline{X}$ point the surface state is mostly composed of Zr $d_{z^2}$ and S $p$ states. We find that the dispersion of the surface state towards $\overline{\mathrm{\Gamma }}$ and $\overline{M}$ is linked to a change in Zr $d_{z^2}$, $d_{xz}$, and $d_{yz}$ character. Panels (e) and (f) show dispersion plots along $\overline{\mathrm{\Gamma }}\overline{X}\overline{M}$ for a potassium covered surface. Again, the red circles show the states arising from the surface termination in the calculations. The green squares in (f) show states that originate from an ordered array of potassium atoms on both sides of the slab. In (f), the relaxed $z$ distance between K and the surface is 2.74 Å. The potassium mixes the orbital character of the surface bands, resulting in a gapping of the surface related bands (SS1 and SS2) in (e), which is reproduced, but overestimated, in the DFT calculations. For easier visibility, the red dashed line in (e) shows the band connectivity.

Figure 3

Constant energy plots at the Fermi level for photon energies (a) $h\nu =26\mathrm{eV}$, (b) $h\nu =248\mathrm{eV}$, (c) $h\nu =374\mathrm{eV}$, and (d) $h\nu =700\mathrm{eV}$. (a) The bare ZrSiS surface shows the right quadrant of the diamond-shaped Fermi surface as well as the surface-derived pockets around $\overline{X}$. The path for the dispersion data in Fig. 2 is shown in red. Panels (b)–(d) show a similar surface for photon energies of 248, 374, or 700 eV, respectively. (e) Dispersion along $\overline{\mathrm{\Gamma }}\overline{X}$ for $h\nu =700\mathrm{eV}$ [green line in (d)]. The pockets around $\overline{X}$ are clearly visible in the Fermi surface and the dispersion, showing the considerable weight of these surface states even for high photon energies.

Figure 4

(a) Schematic of the symmetries of space group $P4/nmm$. Colored planes, lines, and points indicate the symmetry-protected degeneracies in the absence of spin-orbit coupling. Solid colors identify momenta where all states are doubly degenerate due to a combination of a nonsymmorphic symmetry and time reversal; dashed lines represent allowed crossings of bands with different symmetry eigenvalues. At the (001) surface the symmetry is reduced to the symmorphic group $P4mm$. (b) Surface density of states for the effective model in Eqs. (1) and (2), superposed with the bulk band structure at $k_z=0$ (dashed white line). (c) Surface density of states for a 26-band tight-binding model fitted to the DFT band structure of ZrSiS. The Kramers degeneracy at $\overline{X}$ is lifted at the surface since the symmetry group is reduced to a symmorphic one. This creates a two-dimensional surface band, since the degeneracy at $X$ is lifted. Here, we call such a band floating band. The effect is mostly visible in bands with large $k_z$ dispersion, in contrast to planar orbitals, such as $d_{x^2-y^2}$.

Figure 5

(a) DFT bulk calculation of ZrSiS. The size of the red colored circles here stands for the overall Zr character in the unit cell, not just of the topmost one, as shown when considering the surface in Figs. 2 and 2. (b) Orbital character of the bulk bands at the $X$ point shown through the isosurfaces of the charge density. The cleavage plane ($CP$) is shown as a dashed line in the unit cells. The color scale indicates the affiliation to the atoms (green, Zr; blue, Si; yellow, S). For the ZrSiS bulk material, the conduction band minimum at $X$ shows a considerable DOS in the gap between the sulfur atoms, while the valence band maximum is much less affected by the introduction of a surface.