Experimental Observation of Dirac Nodal Links in Centrosymmetric Semimetal 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 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 $\Gamma$-$K$ direction, composing a nodal-link configuration. The simple electronic structure with Dirac nodal links mainly constituting the Fermi surfaces suggests TiB$_2$ as a remarkable platform for studying and applying the novel physical properties related to nodal-line fermions.


Introduction
Topological materials with symmetry protected nodes have recently attracted great attentions in condensed matter physics. Perturbation that preserves a certain symmetry cannot remove the nodes by opening a full direct gap in these materials. When such nodes are close to the Fermi level (EF), the low-energy quasiparticle excitations are drastically different from the usual Schrödinger-type fermions. These nodes can be classified by their dimensionality [1,2]. The zero-dimensional ones include Dirac points, Weyl points, 3-fold degenerate points and other higher-fold degenerate nodal points. With the 4-or 2-fold degenerate Dirac or Weyl points, respectively, Dirac and Weyl semimetals have been theoretically predicted and experimentally confirmed [3][4][5][6][7][8][9][10][11][12]. More recently, 3-fold degenerate fermions, which conceptually lie between Dirac fermions and Weyl fermions, have been demonstrated to exist in the ZrTe family of compounds [13,14].
Nodal ring, nodal link, and nodal chain belong to the one-dimensional (1D) nodalline systems whose nodes extend along 1D lines instead of discrete points in the threedimensional (3D) Brillouin zone (BZ) [2,[15][16][17][18][19]. Protected by the combination of spatial-inversion symmetry and time-reversal symmetry (the P•T symmetry) or certain crystalline symmetry [20,21], the nodal line may either take the form of an extended line running across the BZ, whose ends meet at the BZ boundary, or wind into a closed loop inside the BZ, or even form a chain consisting of several connected loops. 3 Nontrivial Berry phase around the nodal lines would shift the Landau-level index by 1/2 [17] and lead to drumhead surface states (SSs) [18,19].
Although band theory has predicted the existence of nodal-line fermions in some materials [19], only very few candidates have been experimentally studied by angleresolved photoemission spectroscopy (ARPES), i.e., CaAgAs, PbTaSe2, and the ZrSiS family [20,[22][23][24][25][26]. Very recently, two AlB2-type diborides TiB2 and ZrB2 have been proposed to possess nodal-line configurations coexisting with a pair of the triply degenerate nodal points (TNPs) with negligible spin-orbit coupling (SOC) [27,28]. The formation of the TNPs was well uncovered by calculations, but these nodes cannot be observed by ARPES due to locating above EF. On the other hand, the nodal lines could be clearly observed with the crossing located below EF. In this work, we unambiguously identify two groups of Dirac nodal rings in TiB2 by means of ARPES and firstprinciples calculations. The nodal rings embedded in the mirror plane h (the -K-M plane) around K points compose one group and those in the mirror plane v (the -A-H plane) around  point compose the other. Without interfering with other bands, the dispersions forming the nodal rings exhibit linearly in a wide energy range of ~2 eV.
These two groups of nodal rings linked together along the-K direction forming a nodal-link configuration, which goes beyond the isolated nodal line configuration in other systems. The compelling evidences of the nodal-line fermions existing just below EF and the nodal links mainly constituting the Fermi surfaces (FSs) provide an ideal system for further investigations and potential applications on transport phenomena of nodal lines.

Results and Discussion
TiB2 has a simple AlB2-type centrosymmetric structure with the space group P6/mmm (No. 191) [29]. As shown in Fig. 1a, titanium and boron atoms lie in planar close-packed hexagonal layers alternately. Figure 1b shows the x-ray diffraction (XRD) pattern of a TiB2 crystal, indicating the measured surface is the (00l) plane. The mirrorlike (001) surface is illustrated in the inset of Fig. 1b. In Fig. 1c, the zero-field 4 resistivity of TiB2 exhibits a metallic behavior in the measured temperature range from 3 K to 300 K, and the magnetic field dependence of Hall resistivity at T = 3 K is displayed in the inset of Fig. 1c. Based on the nonlinear feature of extending to high fields, electrons and holes are believed to coexist near EF in TiB2. Thus, we perform a quantitative analysis by fitting with the two-carrier model (superimposed on the original data) [30]. The extracted carrier densities of TiB2 are ne = 2.74(4) 10 21 cm -3 and nh = 2.63(4) 10 21 cm -3 , respectively. They are much higher than those of Dirac semimetals like Na3Bi (~10 17 cm -3 [31]) and Cd3As2 (~10 18  CaAgAs (~73 meV) [22] and ZrSiTe (~60 meV) [26], and comparable to that in ZrSiS and ZrSiSe (~25-35 meV) [20,26], indicating the weak SOC effect on the bulk electronic structure of TiB2. The r2 nodal ring in the v plane formed by  and needs to be further demonstrated by the photon-energy-dependent studies, which will be discussed later. Here, we extract the Fermi wave vectors of r1 according to the ARPES data and plot them in Fig. 2e with superimposed calculation contours. The size and shape of the calculated ones are well consistent with those determined by ARPES.
To quantitatively study the feature of r1 nodal ring embedded in the h plane, we record the ARPES spectra near EF along the M-K and -K directions, as indicated by cuts 1 and 2 in Fig. 2f. The intensity plots and corresponding second derivative plots 6 are presented in Figs. 3a-3d. The overall band structures agree well with the appended theoretical calculations. We also notice that some band dispersions near  along the -K direction are not reproduced by bulk calculations. These bands most likely come from the contributions of bulk states in other kz planes due to the kz broadening effect [24,35].
To avoid the interference with other bands, we use the photon energy of 110 eV, in principle corresponding to the -K-M plane, to record the intensity. As shown in Figs.    Table S1 of the SM. With a Z2 index as (0; 001), the SSs of TiB2 predicted in Refs. [27] and [28] are not topologically protected, a small perturbation on the surface could hinder the observation of SSs in our experiments.

Conclusion
By systematically mapping out the 3D electronic structure both in experiment and theory, we unambiguously demonstrate the existence of nodal-line fermions in TiB2, hosting Dirac nodal links formed by two groups of nodal rings with negligible SOC effect, under the protection of P•T symmetry and certain mirror reflection symmetries. 8 The fact that the nodal-line fermions existing just below EF combined with the nodal links mainly constituting the FSs must have significant contribution to the low-energy excitations, which are favorable to the emergence of associated novel transport properties. The simple electronic structure composed of large energy range of linearly dispersive bands offers a good platform for further studies into Dirac physics.