Dirac Node Lines in Pure Alkali Earth Metals

Beryllium is a simple alkali earth metal, but has been the target of intensive studies for decades because of its unusual electron behavior at surfaces. The puzzling aspects include (i) severe deviations from the description of the nearly free-electron picture, (ii) an anomalously large electron-phonon coupling effect, and (iii) giant Friedel oscillations. The underlying origins for such anomalous surface electron behavior have been under active debate, but with no consensus. Here, by means of first-principles calculations, we discover that this pure metal system, surprisingly, harbors the Dirac node line (DNL) that in turn helps to rationalize many of the existing puzzles. The DNL is featured by a closed line consisting of linear band crossings, and its induced topological surface band agrees well with previous photoemission spectroscopy observations on the Be (0001) surface. We further reveal that each of the elemental alkali earth metals of Mg, Ca, and Sr also harbors the DNL and speculate that the fascinating topological property of the DNL might naturally exist in other elemental metals as well.

Figure 1

Lattice structure and electronic band structure of hcp beryllium. (a) Crystal structure (here a $3\times 3\times 5$ supercell) (no. 194 $P63/mmc$ space group) with Be atom at the $2c$ site ($1/3$, $2/3$, $1/4$) in a perspective projection along the $c$ axis. The layers consist of green and pink atoms alternatively stack along the $c$ axis. (b) Brillouin zone (BZ) of the bulk phase and projected surface BZ of the (0001) surface. (c) Calculated electronic band structure along the high-symmetry points for the hcp Be bulk phase.

Figure 2

Fermi surface, DNL, and six invariant $S_{abcd}$ surfaces in the 3D BZ of hcp Be. (a) The Fermi surface in bulk hcp Be exhibits two-band features. (b) Illustration of the DNL, visualized by the two-band crossings. (c)–(h) Shaded regions represent three topologically nontrivial invariant planes (c)–(e) and three topologically trivial invariant planes (f)–(h). In (c)–(h), the red circle sketches the DNL.

Figure 3

The evolution of thickness-dependent surface electronic band structures of the Be (0001) surface. (a)–(e) The derived surface electronic structure is shown as a function of the slab thickness from 5 unit cells (ucs) to 25 ucs along the $c$ axis with an increase of 5 ucs thickness in every image. (f) Surface electronic band structure with a slab thickness of 100 ucs, together with the experimentally measured data (squares [26]).

Figure 4

Fermi surface of the Be (0001) surface. (a) DFT-derived Fermi surface highlighting a surface state of a fully closed circle around the centered $\overline{\mathrm{\Gamma }}$ point in the Brillouin zone. The other parts are the projections of bulk bands. (b),(c) The experimentally measured Fermi surface (directly taken from Ref. [46]) cuts with $hv=32.5$ and 86 eV, respectively, showing the difference between horizontal and vertical polarizations measured by the ARPES experiments.

Figure 5

The giant Friedel oscillation for the Be (0001) surface. The screened potential $\phi (r)/Z$ is given as a function of the distance $r$. The red and blue solid curves denote the Friedel oscillation for the Be (0001) surface of the cases with and without DNL, respectively. The dashed red and blue curves denote the amplitude of $\phi (r)/Z$.