Resonant electron tunneling spectroscopy of antibonding states in a Dirac semimetal

Recently, it was shown both theoretically and experimentally that certain three-dimensional (3D) materials have Dirac points in the Brillouin zone, thus being 3D analogs of graphene. Moreover, it was suggested that under specific conditions a pair of localized impurities placed inside a three-dimensional Dirac semimetal may lead to the formation of an unusual antibonding state. In the meantime, the effect of vibrational degrees of freedom which are present in any realistic system has avoided attention. In this work, we address the influence of phonons on characteristic features of (anti)bonding state, and discuss how these results can be tested experimentally via local probing, namely, inelastic electron tunneling spectroscopy curve obtained in STM measurements.

Figure 1

Schematic representation of the model system: two impurities, positioned at ${\mathbf{R}}_1$ and ${\mathbf{R}}_2$ with energy levels ${\varepsilon }_{1\sigma }$ and ${\varepsilon }_{2\sigma }$, are embedded into a three-dimensional Dirac semimetal. We account for the vibrational degrees by coupling a phonon mode of the coupling strength $\lambda$ and energy ${\omega }_0$ to each of the impurities.

Figure 2

Electronic properties of the system in the valence band with ${\varepsilon }_{j\sigma }=-0.10D,U_j=0.20D,v_0=0.20D, {\omega }_0=0.01D$, and $\lambda =0.7{\omega }_0$ for $T=10$ K (left panels) and $T=40$ K (right panels). Impurities density of states presenting the electronic and polaronic levels (a) and (d). The profiles of $\delta {\rho }_{j{j}^{\prime }}\left(\omega \right)$ at ${\mathbf{R}}_m=(1,1,1)\text{nm}$ for $j^{\prime }=j$ (solid green curve) and $j^{\prime }=\overline{j}$ (dashed red curve) revealing the interferences between diagonal and mixed electronic waves scattering (b) and (e). Total LDOS (c), described in Eq. (13), displaying a pure electronic ground state formed by a destructive interference shown in panel (b). Total LDOS revealing the emergence of a lowest-energy polaronic mode due to the thermal excitation (f). The insets display the entire spectrum of each panel, i.e., valence and conduction bands.

Figure 3

The $d\mathrm{LDOS}\left(\varepsilon \right)/d\varepsilon =0$ profile on the valence bands with the parameters: ${\varepsilon }_{j\sigma }=-0.10D,U_j=0.20D,v_0=0.20D,{\omega }_0=0.01D$, and $\lambda =0.7{\omega }_0$ for $T=10$ K (a) and $T=40$ K (b). The vertical dashed lines mark the energetic position of the ground states.

Figure 4

LDOS topography of the surface of 3D-DSM $[{\mathbf{R}}_m=(x,y,1)\text{nm}$ plane]. For $T=10$ K (a), at the ground state energy $\varepsilon =-0.10D$, an antibonding profile emerges due to the destructive interference between the waves scattered by the impurities. For higher temperature $T=40$ K (b), the ground state at $\varepsilon =-0.11D$ features bonding characteristics. The orbital characteristics of the state $\varepsilon =-0.11D$ for $T=10$ K (c) and $T=30$ K (d), respectively, revealing smooth crossover in the nature of the ground state.