Experimental Realization of a Three-Dimensional Dirac Semimetal Phase with a Tunable Lifshitz Transition in ${\mathrm{Au}}_2\mathrm{Pb}$

Three-dimensional Dirac semimetals are an exotic state of matter that continue to attract increasing attention due to the unique properties of their low-energy excitations. Here, by performing angle-resolved photoemission spectroscopy, we investigate the electronic structure of ${\mathrm{Au}}_2\mathrm{Pb}$ across a wide temperature range. Our experimental studies on the (111)-cleaved surface unambiguously demonstrate that ${\mathrm{Au}}_2\mathrm{Pb}$ is a three-dimensional Dirac semimetal characterized by the presence of a bulk Dirac cone projected off-center of the bulk Brillouin zone (BZ), in agreement with our theoretical calculations. Unusually, we observe that the bulk Dirac cone is significantly shifted by more than 0.4 eV to higher binding energies with reducing temperature, eventually going through a Lifshitz transition. The pronounced downward shift is qualitatively reproduced by our calculations indicating that an enhanced orbital overlap upon compression of the lattice, which preserves $C_4$ rotational symmetry, is the main driving mechanism for the Lifshitz transition. These findings not only broaden the range of currently known materials exhibiting three-dimensional Dirac phases, but also show a viable mechanism by which it could be possible to switch on and off the contribution of the degeneracy point to electron transport without external doping.

Figure 1

(a)–(c) STM characterization of the pristine ${\mathrm{Au}}_2\mathrm{Pb}$ (111) surface. (a) Atomically resolved $I$-channel STM image of the surface at room temperature (acquisition parameters: 0.3 V, 2 nA). Inset: $Z$-channel topography image (0.3 V, 0.5 nA). (b) Close view of the topography, $I$ channel (0.3 V, 0.5 nA). A structural model is displayed as an overlay [yellow (gray) spheres represent Au (Pb) atoms]. (c) Line profile extracted along the white dashed line in (a). (d) Corresponding LEED patterns of the pristine ${\mathrm{Au}}_2\mathrm{Pb}$ (111) surface acquired at different energies. (e) Sketch of the bulk and surface BZs of ${\mathrm{Au}}_2\mathrm{Pb}$ (111). The red solid line indicates one of the six equivalent directions along which bulk Dirac points are predicted. (f) DFT band structure calculation over all high-symmetry directions of cubic ${\mathrm{Au}}_2\mathrm{Pb}$. The BDC is labeled, and band colors indicate different irreducible representations.

Figure 2

(a)–(e) Band dispersions of the Dirac cone and (f)–(j) corresponding momentum-distribution curves (MDCs) acquired at 300 K along the $k_y$ direction ($k_x=0$ plane) at selected $k_z$. Black solid lines in (f)–(j) are guides to the eye, and electron wave vectors are given relative to the position of the bulk Dirac point. The MDCs shown span an energy range equivalent to that shown in (a)–(e). (k)–(m) Band dispersions acquired along different $k_y$ planes parallel to the $k_x$ direction, and at the $k_z$ value corresponding to panels (c) and (h). (n) Constant-energy contour of the Dirac cone projected into $k_z\text{−}{k}_y$ space ($k_x=0$). (o) Linear dispersions of the Dirac bands along all three momentum directions.

Figure 3

(a) Band dispersions and (b) corresponding second derivative of the photoemission intensities of the Dirac cone at indicated temperatures. Red dashed lines in (a) are fits to the band dispersions tracing the temperature evolution of the Dirac cone and intersect at the bulk Dirac point, which shifts in energy with decreasing lattice temperature as emphasized by the horizontal arrows in (b). (c) Energy-distribution curves (open circles) and corresponding fits (solid lines) across the position of the bulk Dirac point at several temperatures. (d) Dependence of the Dirac point energy and lattice parameter (inset, derived from the data shown in [28]) with temperature. Solid lines are fits to the experimental data.

Figure 4

(a)–(d) DFT band structure calculations showing a significant energy shift of the BDC upon compression of the lattice. Horizontal green lines mark the energy positions of the bulk Dirac point, and the lattice compressions $\mathrm{\Delta }\epsilon$ are (a) 0%, (b) 0.8%, (c) 1.2%, and (d) 2.6%. Band colors indicate different irreducible representations. (e) Calculated dependence of the Dirac point energy with lattice compression.