Experimental Observation of Dirac Nodal Links in Centrosymmetric Semimetal ${\mathrm{TiB}}_2$

The topological nodal-line semimetal state, serving as a fertile ground for various topological quantum phases, where a topological insulator, Dirac semimetal, or Weyl semimetal can be realized when the certain protecting symmetry is broken, has only been experimentally studied in very few materials. In contrast to discrete nodes, nodal lines with rich topological configurations can lead to more unusual transport phenomena. Utilizing angle-resolved photoemission spectroscopy and first-principles calculations, here, we provide compelling evidence of nodal-line fermions in centrosymmetric semimetal ${\mathrm{TiB}}_2$ with a negligible spin-orbit coupling effect. With the band crossings just below the Fermi energy, two groups of Dirac nodal rings are clearly observed without any interference from other bands, one surrounding the Brillouin zone (BZ) corner in the horizontal mirror plane ${\sigma }_h$ and the other surrounding the BZ center in the vertical mirror plane ${\sigma }_v$. The linear dispersions forming Dirac nodal rings are as wide as 2 eV. We further observe that the two groups of nodal rings link together along the $\mathrm{\Gamma }‐K$ direction, composing a nodal-link configuration. The simple electronic structure with Dirac nodal links mainly constituting the Fermi surfaces suggests ${\mathrm{TiB}}_2$ as a remarkable platform for studying and applying the novel physical properties related to nodal-line fermions.

Popular Summary

Topological materials have attracted a lot of attention recently because of the novel ways in which they transport charge, which promises intriguing applications in quantum computers and spintronic devices. The specifics of these transport properties depend on how the conduction band intersects the valence band. These band crossings can be classified by their dimensionality. While pointlike crossings have been confirmed in a number of materials, nodal lines have not been experimentally confirmed until now. We present compelling experimental evidence for nodal lines in the semimetal ${\mathrm{TiB}}_2$, which suggests that this material is an ideal platform for further studies of unusual transport behavior.

Using angle-resolved photoemission spectroscopy and first-principles calculations, we find a nodal-link structure composed of two groups of nodal rings, which goes beyond the insulated nodal lines reported in other materials. Our work provides the first conclusive experimental evidence of fermions that have this nodal-line crossing type without any interference from other bands. Additionally, our work provides an important step toward studying the nodal-line properties in weakly spin-orbit coupled systems such as borides.

We believe that the simple electronic structure and the advantages in realizing the nodal-line fermions in ${\mathrm{TiB}}_2$ offer an ideal platform for further studies of Dirac nodal-line fermions, a crucial step for both fundamental science and future applications.

Figure 1

Single-crystal and wide electronic structure of ${\mathrm{TiB}}_2$. (a) Crystal structure of ${\mathrm{TiB}}_2$ with space group $P6/mmm$ (No. 191). (b) XRD pattern of a ${\mathrm{TiB}}_2$ single crystal. Inset: Picture of a ${\mathrm{TiB}}_2$ crystal against the 1-mm scale. (c) Temperature dependence of the resistivity ${\rho }_{xx}$ at $B=0\mathrm{T}$. Inset: Hall resistivity ${\rho }_{xy}$ (black curve) as a function of magnetic field ($B\parallel [001]$) at $T=3\mathrm{K}$, with the superimposed fitting result (red curve) using the two-carrier model. (d) 3D bulk BZ with marked high-symmetry points and two mirror planes, ${\sigma }_h$ ($\mathrm{\Gamma }‐K‐M$ plane) and ${\sigma }_v$ ($\mathrm{\Gamma }‐A‐H$ plane). (e),(f) ARPES intensity plot and corresponding second derivative plot along the $\mathrm{\Gamma }‐K‐M$ direction. $a’=\sqrt{3}a/2$ ($a=3.0335\AA$). The red curves in (f) represent the calculated bulk bands with SOC. The band-crossing features along the $M‐K$ and $K‐\mathrm{\Gamma }$ directions are denoted as $\alpha$ and $\beta$, respectively.

Figure 2

FSs and calculated band structure of ${\mathrm{TiB}}_2$. (a),(b) Intensity plots at $E_F$ and 0.5 eV below $E_F$ taken with 80-eV photons. The red dashed lines indicate high-symmetry directions and the first BZ projected on the (001) surface. Two groups of Dirac nodal rings $r_1$ and $r_2$ [projected on the (001) surface] are resolved. The link points (denoted as $J$ points) of $r_1$ and $r_2$ are indicated by black dashed circles. (c),(d) Calculated bulk FSs in the 3D BZ and the top view of the FSs, respectively. The $r_1$ nodal ring surrounding the $K$ point is embedded in the $\mathrm{\Gamma }‐K‐M$ plane. The $r_2$ nodal ring surrounding the $\mathrm{\Gamma }$ point is embedded in the $\mathrm{\Gamma }‐A‐H$ plane. (e) Comparison of the ARPES data and calculations for $r_1$. The green curves are calculated $r_1$ FSs. The red solid circles represent the experimental data, which are determined as follows: extracting the Fermi wave vectors from $1/3$ (marked by a blue arrow) of the $r_1$ nodal ring in (a); then, rotating it to a whole ring around the $K$ point; and last, symmetrizing to obtain other rings in the BZ. (f) Schematic 2D BZs with marked cuts 1 and 2 indicating the momentum locations of the measured bands in Fig. 3. (g) Calculated bulk band structure along high-symmetry lines including SOC. Four near-$E_F$ band-crossing features are denoted as $\alpha$, $\beta$, $\gamma$, and $\delta$, respectively. Inset: The enlarged view of $\alpha$ and $\beta$, small gaps of 22 and 25 meV open at the nodes, respectively, when SOC is taken into consideration.

Figure 3

Electronic structure of the $r_1$ nodal ring. (a),(b) Intensity plot and corresponding second derivative plot along the $M‐K$ direction [cut 1 in Fig. 2] recorded at $h\nu =120\mathrm{eV}$ ($k_z\sim 0.4\pi /c$). (c),(d) Same as (a),(b), but along the $\mathrm{\Gamma }‐K$ direction [cut 2 in Fig. 2]. (e),(f) Same as (a),(b), but recorded along the $\mathrm{\Gamma }‐K‐M$ direction with 110-eV photons ($k_z\sim 0$). The band-crossing features along the $M‐K$ and $K‐\mathrm{\Gamma }$ directions, corresponding to the $r_1$ nodal ring surrounding the $K$ point, are denoted as $\alpha$ and $\beta$ in (f), respectively. The calculated bands with SOC are plotted on the second derivative plots in (b),(d), and (f). (g),(h) MDCs plots of $\alpha$ and $\beta$, respectively. The black dashed lines indicate the linear dispersions. (i) EDC taken at the center (approximately $-0.81\pi /a’$) of $\beta$. Inset: Zoomed in EDC near $E_F$ indicated by the green dashed rectangle. By fitting the EDC with two Lorentzian profiles, shown as green curves, an opening gap of 25 meV can be determined. The blue curve is the superimposed fitting result.

Figure 4

Photon-energy-dependent dispersions of ${\mathrm{TiB}}_2$. (a) Intensity plot in the $h\nu ‐k_{//}$ plane at $E_F$, with $k_{//}$ oriented along the $\mathrm{\Gamma }‐K$ ($A‐H$) direction. The high-symmetry points and nodal-link point are plotted. The $r_2$ nodal ring embedded in the $\mathrm{\Gamma }‐A‐H$ plane surrounding the $\mathrm{\Gamma }$ point is indicated by open circles. (b) Projections of calculated bulk FSs on the $\mathrm{\Gamma }‐A‐H$ plane. (c) ARPES spectra taken along the $H‐A$ line with 86-eV photons ($k_z\sim \pi$). The appended red curves are bulk band calculations with SOC. (d, e) Constant-energy ARPES images at $E_F$ and 0.5 eV below $E_F$ obtained by 110-eV photons ($k_z\sim 0$), respectively. (f),(g) Same as (d),(e), but taken with 120-eV photons ($k_z\sim 0.4\pi /c$).