Importance of interactions for the band structure of the topological Dirac semimetal ${\mathrm{Na}}_3\mathrm{Bi}$

We experimentally measure the band dispersions of topological Dirac semimetal ${\mathrm{Na}}_3\mathrm{Bi}$ using Fourier-transform scanning tunneling spectroscopy to image quasiparticle interference on the (001) surface of molecular-beam epitaxy-grown ${\mathrm{Na}}_3\mathrm{Bi}$ thin films. We find that the velocities for the lowest-lying conduction and valence bands are $1.6\times {10}^6\mathrm{m}{\mathrm{s}}^{-1}$ and $4.2\times {10}^5\mathrm{m}{\mathrm{s}}^{-1}$ respectively, significantly higher than previous theoretical predictions. We compare the experimental band dispersions to the theoretical band structures calculated using an increasing hierarchy of approximations of self-energy corrections due to interactions: generalized gradient approximation (GGA), meta-GGA, Heyd-Scuseria-Ernzerhof exchange-correlation functional (HSE06), and $GW$ methods. We find that density functional theory methods generally underestimate the electron velocities. However, we find significantly improved agreement with an increasingly sophisticated description of the exchange and interaction potential, culminating in reasonable agreement with experiments obtained by the $GW$ method. The results indicate that exchange-correlation effects are important in determining the electronic structure of this ${\mathrm{Na}}_3\mathrm{Bi}$, and are likely the origin of the high velocity. The electron velocity is consistent with recent experiments on ultrathin ${\mathrm{Na}}_3\mathrm{Bi}$ and also may explain the ultrahigh carrier mobility observed in heavily electron-doped ${\mathrm{Na}}_3\mathrm{Bi}$.

Figure 1

Topographic and spectroscopic characterization of ${\mathrm{Na}}_3\mathrm{Bi}$ thin film. (a) $300\mathrm{nm}\times 300\mathrm{nm}$ topographic STM image (bias voltage $V=-3\mathrm{V}$ and tunnel current $I=50\mathrm{pA}$) showing the overall topography of the sample; inset: atomically resolved image taken on a separate 20-nm ${\mathrm{Na}}_3\mathrm{Bi}$ film, where the white arrows mark the 5.45-Å lattice constant. (b) Point scanning tunneling spectrum $dI/dV$ vs bias on 20-nm ${\mathrm{Na}}_3\mathrm{Bi}$ film (bias modulation rms amplitude is 5 mV). The bias voltage position of minimum $dI/dV$, visible in the inset, is labeled $E_{\mathrm{D}}$ and identified as the Dirac point energy.

Figure 2

Topography and quasiparticle interference mapping of the conduction band of a ${\mathrm{Na}}_3\mathrm{Bi}$ thin film. (a) $60\mathrm{nm}\times 60\mathrm{nm}$ topographic STM image (bias voltage $V=0.375\mathrm{V}$ and tunnel current $I=600\mathrm{pA}$) showing an atomically flat region of 20-nm ${\mathrm{Na}}_3\mathrm{Bi}$ on Si(111). (b), (c) $dI/dV$ maps of the same area as shown in Fig. 1 obtained at $V=0.7$ and 0.9 V, and $I=700$, 850 pA respectively. (d) 3D Schematic of the binding energy vs ${\mathbf{k}}_{\mathbf{x}}$–${\mathbf{k}}_{\mathbf{y}}$ on the (001) plane of ${\mathrm{Na}}_3\mathrm{Bi}$, with the red circle representing a constant-energy contour for states around the Γ point, and the intravalley wave-vector scattering process denoted by q. (e), (f) Two-dimensional Fourier transform (2D FT) of (b) and (c) respectively where the scale bar represents $2.1\mathrm{n}{\mathrm{m}}^{-1}$. (g) Radial averaged intensity profiles of the 2D FT acquired at $V=0.45$, 0.55, 0.65, 0.75, and 0.85 V (2D FT not shown) plotted as a function of wave vector $q$. Curves are offset for clarity. Pink arrows denote the position of the main dispersing feature as a guide for the eye.

Figure 3

Quasiparticle interference mapping of the valence bands of a ${\mathrm{Na}}_3\mathrm{Bi}$ thin film. (a), (b) $dI/dV$ maps of the same area as shown in Fig. 1 obtained at $V=-0.25$ and −0.30 V respectively; $I=450\mathrm{pA}$ in both cases. (c), (d) Two-dimensional Fourier transform (2D FT) of (a) and (b) respectively where the scale bar represents $2.1\mathrm{n}{\mathrm{m}}^{-1}$. (e) Radial averaged intensity profiles of the 2D FT shown in (d)–(f) (along with −0.25, −0.35, −0.4, −0.55 and −0.65 V, 2D FT not shown) plotted as a function of wave vector q. Curves are offset for clarity, vertical arrows follow the dispersion of the main features as a guide for the eye. (f) Exemplary peak fitting for the radial averaged intensity profile of the 2D FT for the mappings collected at $V=-0.35\mathrm{V}$ (top panel) and $V=0.7\mathrm{V}$ (bottom panel).

Figure 4

Comparison between experimental results and calculation for the band structure of ${\mathrm{Na}}_3\mathrm{Bi}$. (a) Calculated band structure for ${\mathrm{Na}}_3\mathrm{Bi}$ along the $\tilde{\mathrm{\Gamma }}\text{−}\tilde{K}$ direction in the plane containing the Dirac point. Markers are the measured data from QPI, linear fittings to evaluate the Fermi velocities are superimposed on the data. (b) Electron velocity evaluated with DFT and $GW$ methods (green dots) and experiment (grey star). In each case the velocity is obtained from a linear fit to $E\left(k\right)$ over the range $450\mathrm{meV}\le E\le 950\mathrm{meV}$. (c) Comparison of the calculated band structures for ${\mathrm{Na}}_3\mathrm{Bi}$ with GGA (solid blue lines) and $GW$ (red dashed lines) approaches along the main symmetry directions.