Existence of topological nontrivial surface states in strained transition metals: W, Ta, Mo, and Nb

We show that a series of transition metals with strained body-centered cubic lattice—W, Ta, Nb, and Mo—hosts surface states that are topologically protected by mirror symmetry and, thus, exhibits nonzero topological invariants. These findings extend the class of topologically nontrivial systems by topological crystalline transition metals. The investigation is based on calculations of the electronic structures and of topological invariants. The signatures of a Dirac-type surface state in W(110), e.g., the linear dispersion and the spin texture, are verified. To further support our prediction, we investigate Ta(110) both theoretically and experimentally by spin-resolved inverse photoemission: unoccupied topologically nontrivial surface states are observed.

Figure 1

Surface electronic structure of W(110). (a) Topological surface state (marked TSS). The spectral density of the topmost layers is shown as normalized color scale for the $\overline{\mathrm{\Gamma }}-\overline{H}$ line of the surface Brillouin zone. (b) As (a), but spin resolved with respect to the Rashba component of the spin polarization ($\pm 1$ is fully spin polarized: $-1$ spin-down, $+1$ spin-up). (c) Brillouin zone of a bcc lattice and its projection onto the (110) surface. Red lines indicate an irreducible part. $H$ and $H^{\prime }$ become inequivalent under strain. (d) Schematic illustration of the surface-state dispersion from (a), with spin polarization indicated by colors. Recall that bulk states show up also in the dark green area around $\overline{\mathrm{\Gamma }}$ because they are projected onto the surface Brillouin zone.

Figure 2

Geometry for the spin- and angle-resolved inverse-photoemission experiments.

Figure 3

Band inversion and band gap opening in W. Tight-binding band structures are shown along those lines in the bulk Brillouin zone that are relevant for the topological phase transition. (a) Cubic W without (gray) and with (black) spin-orbit coupling. The inset shows a zoom to the topological “hot spot” of the dispersion (b) body-centered cubic crystal, with lattice sites represented by spheres; the green area visualizes the unit cell of the (110) surface spanned by the lattice vectors ${\mathit{a}}_1$ and ${\mathit{a}}_2$. The lattice vector between the $\left(110\right)$ planes is ${\mathit{a}}_3$ which is altered upon strain $\mathbf{\sigma }$. The two mirror planes—with surface normal in $\left(001\right)$ and $\left(\overline{1}10\right)$—for which the Chern numbers have been computed are displayed in blue and red. (c) Tensile and compressively strained W with spin-orbit coupling. Insets show band dispersions that are relevant for the topology of W under strain at $H$ (red) and $H^{\prime }$ (blue). $H^{\prime }$ and $N^{\prime }$ are defined in Fig. 1.

Figure 4

Unoccupied electronic structure of Ta(110) for the $\overline{\mathrm{\Gamma }}-\overline{H}$ line of the surface Brillouin zone. (a) Spectral density of the topmost surface layer of unstrained Ta calculated with the tight-binding method. The bulk bands that form the boundary of the $(E,\mathit{k}$)-dependent band gap are shown in green and red. (b) As (a) but resolved with respect to the Rashba spin component. (c) and (d) As (a) and (b) but calculated by the ab initio KKR method. The symbols result from spin-resolved inverse-photoemission experiments and indicate peak positions derived from spectra shown in Fig. 8. The topological surface states are marked TSS1 and TSS2.

Figure 5

Spin-resolved surface electronic structure of Nb(110) (a) and Mo(110) (b), analogous to Fig. 4.

Figure 6

Bulk band structure of cubic W without spin-orbit coupling calculated with vasp (black) and with the tight-binding approach (red). $E_{\mathrm{F}}$ is the Fermi energy.

Figure 7

Bulk band inversion in W, obtained from KKR (left panel) and TB (right panel). The band inversion is mediated by strain of $-3\%$ (compressive strain; black line), $0\%$ (red line), and $+3\%$ (tensile strain; blue line). Arrows and dashed lines indicate the band gap inversion associated with compressive strain.

Figure 8

Spin-integrated (a) and spin-resolved (b) IPE spectra for Ta(110) for various angles of incidence $\theta$ in the $\overline{\mathrm{\Gamma }}\overline{H}$ mirror-symmetry direction. In (b), spin-up (spin-down) is marked by red (blue) triangles. The shape of the background intensity is indicated by solid lines below the surface-state emissions. The precise procedure for determining the energy positions of TSS1 and TSS2 is described in detail in Ref. [35].