Discovery of Lorentz-violating type II Weyl fermions in LaAlGe

Abstract

In quantum field theory, Weyl fermions are relativistic particles that travel at the speed of light and strictly obey the celebrated Lorentz symmetry. Their low-energy condensed matter analogs are Weyl semimetals, which are conductors whose electronic excitations mimic the Weyl fermion equation of motion. Although the traditional (type I) emergent Weyl fermions observed in TaAs still approximately respect Lorentz symmetry, recently, the so-called type II Weyl semimetal has been proposed, where the emergent Weyl quasiparticles break the Lorentz symmetry so strongly that they cannot be smoothly connected to Lorentz symmetric Weyl particles. Despite some evidence of nontrivial surface states, the direct observation of the type II bulk Weyl fermions remains elusive. We present the direct observation of the type II Weyl fermions in crystalline solid lanthanum aluminum germanide (LaAlGe) based on our photoemission data alone, without reliance on band structure calculations. Moreover, our systematic data agree with the theoretical calculations, providing further support on our experimental results.

Keywords: Topological Materials; Weyl semimetals.

Fig. 1. Topology and BZ symmetry of LaAlGe.

(A) Body-centered tetragonal structure of LaAlGe, with space group I4₁md (109). The structure consists of stacks of La, Al, and Ge layers, and along the (001) direction, each layer consists of only one type of element. (B) The bulk and (001) surface BZ. (C) First-principles band structure calculations along high-symmetry directions without spin-orbit coupling (SOC). (D) Momentum space configuration of the four nodal lines (two on the k_x = 0 and two on the k_y = 0 mirror planes) denoted by the rings, as well as the four spinless pairs of Weyl nodes denoted as W₃ on the k_z = 0 plane, in the absence of SOC. Blue and red colors indicate positive and negative chiralities, respectively. (E) Configuration of the 40 Weyl nodes in the bulk BZ created upon the inclusion of SOC. The nodal lines are gapped out by SOC, and 24 Weyl nodes emerge in the vicinity of the nodal lines. In addition, each spinless W₃ Weyl node splits into two spinful Weyl nodes of the same chirality, which we denote as W₃′ and W₃″. Hence, the eight W₃ without SOC evolve into eight W₃′ and eight W₃″ Weyl nodes with SOC. Therefore, in total, there are 40 Weyl nodes. For the 24 Weyl nodes that emerge from the gapping of the nodal line, we denote the 8 Weyl nodes that are near the boundaries of k_z = 0 plane as W₁ and the other 16 that are away from the k_z = 0 plane as W₂. The W₃′ and W₃″ are also on the k_z = 0 plane, but they are near the diagonal lines. (F) Projection of the Weyl nodes on the (001) surface BZ in one quadrant. (G) Schematics comparing the three types of Weyl nodes appearing upon the inclusion of SOC. The W₂ nodes are type II Weyl nodes, whereas the W₁, W₃′, and W₃″ nodes are type I. W₂ Weyl nodes are located almost exactly at the Fermi level, whereas W₁, W₃′, and W₃″ Weyl nodes are about 60, 110, and 130 meV above the Fermi level, respectively. (H) Core level measurement of the studied samples, which clearly shows the expected La, Al, and Ge peaks. a.u., arbitrary units.

Fig. 2. Observation of type II Weyl nodes in LaAlGe.

(A) High-energy soft x-ray ARPES (SX-ARPES) measurement and (B) first-principles calculation of k_z − k_x Fermi surface maps at k_y = 0. The dashed green hexagon in (A) and the black hexagon in (B) correspond to the boundary of the first BZ on the k_y = 0 plane. The photon energies corresponding to the k_z values in (A) are between 320 and 600 eV. (C) SX-ARPES–measured k_x − k_y Fermi surface map at k_y corresponding to the W₂ Weyl nodes. (D) Zoomed-in version of the measured and (E) calculated k_x − k_y Fermi surface maps in the region marked by the green rectangle in (C). In the first-principles calculations, blue lines correspond to hole-like pockets, whereas red lines indicate electron-like pockets. (F and G) Same as (D) and (E) for a constant energy contour at 25 meV below the Fermi level. The photon energy used in the measurements presented in (C), (D), and (F) is 542 eV. (G and H) First-principles–calculated k_x − k_y Fermi surface map at k_z = 0 (location of W₁, W₃′, and W₃″ Weyl nodes), which shows the shapes of the pockets resulting from W1, W₃′, and W₃″ and other trivial pockets. (I) SX-ARPES–measured k_x − k_y Fermi surface map at k_z = 0 in the region of the BZ denoted by the green dashed rectangle in (H). This experimentally observed Fermi surface map is in qualitative agreement with the calculated results presented in (H). The photon energy used here is 478 eV. (J) Calculated Fermi surface over the first bulk BZ. The pockets arising from the Weyl fermion cones are marked. Other pockets are trivial bulk bands. The red and blue colors denote electron- and hole-like pockets. Particularly, it can be seen that the W₂ node arises from the touching between electron and hole pockets, confirming their type II nature.

Fig. 3. Type II Weyl fermions in LaAlGe.

(A) The k_x − k_y Fermi surface map revealing a pair of type II W2 Weyl nodes in the form of touching points between electron and hole pockets. (B) Measured and (C) calculated E − k_y dispersion maps along the cut y direction denoted in (A), which clearly resolve the W₂ type II Weyl node at the crossing of two bands with the same sign of Fermi velocity. (D) Measured and (E) calculated E − k_x dispersion maps for the direction along the cut x in (A). Here, the two type II W2 Weyl cones with opposite chirality nodes are resolved simultaneously. The photon energy used in the measurements presented in (A), (B), and (D) is 542 eV. (F) Measured and (G) calculated E − k_z dispersion maps along cut z, showing that the W₂ Weyl cone also disperses linearly along the out-of-plane k_z direction. (H and I) ARPES dispersion maps showing the difference between the type II Weyl fermions of LaAlGe and type I Weyl fermions of TaAs, respectively. (L) The Fermi surface of a type II Weyl semimetal where electron and hole pockets touch to form the type II Weyl node. (J) The projected Fermi surface shows a crossing between the projected electron and hole pockets. (M) We have a completely different scenario in the bulk. The electron and hole pockets are separated at different k_z values. (K) However, on the surface, their projections can still touch. This example clearly shows that the observation of a crossing in the projected band structure on surface does not mean a crossing in the bulk. The red and blue ellipsoids represent the electron and hole Fermi surfaces in the bulk BZ, respectively. The red and blue ellipses are the surface projection of the bulk Fermi surfaces.

Fig. 4. Berry curvature contributions from the Weyl nodes.

(A) Projection of the Weyl nodes of LaAlGe on the (001) surface BZ in one quadrant. (B and C) Berry curvature magnitude in k_x − k_y space at two different k_z values (k_z = 0 and k_z = W₂) summed over a wide energy range. Specifically, we summed up Berry curvature contributions of all the valence bands, from the lowest valence band near the Fermi level to the highest valence band far from the Fermi level. (E and F) Same as (B) and (C) but only for energies very close to the Fermi level at an E_F of a ±10-meV window. (D) A zoomed-in view of the Berry curvature magnitude near the energy of the W₂ Weyl nodes. (G) Berry curvature magnitude in k_x − k_y space summed over a wide energy range in WTe₂. The black lines show the Fermi surface contours. We see significant Berry curvature contribution due to the existence of Weyl nodes. (H) A zoomed-in view of (G) in the k range highlighted by the orange box. (I) Same as (H) for energies close to the Fermi level in an E_F of a ±10-meV window window. Because the Weyl nodes are far (50 meV) away from the Fermi level, when we only consider the Fermi level, we do not see any significant Berry curvature.