Atomic frustrated impurity states in Weyl metals

We theoretically analyze the effect of the inversion symmetry breaking on the structure of the impurity molecular states in Weyl metals. We show that for the case of a highly noncentrosymmetric Weyl metallic host, the standard picture of the alternating bonding and antibonding orbitals breaks down, and a qualitatively different frustrated atomic state emerges. This is a consequence of the pseudogap closing and related delicate Fano interplay between intra- and interimpurity scattering channels.

Figure 1

(a) Sketch of the considered system, consisting of a pair of impurities placed inside a Weyl metal close to its interface. The positions of the impurities are characterized by the vectors ${\mathbf{R}}_{1,2}$. The impurity molecular states can be probed on the surface of the host by an STM tip, whose location is characterized by the vector $\mathbf{r}$. (b) Sketch of the dispersion, characteristic for a Dirac semimetal with two degenerated Dirac cones. The pseudogap is formed around the Dirac point, with the host density of states (DOS) $\rho \left(\varepsilon \right)=0$. (c) Sketch of the dispersion, characteristic for the Weyl metal. The degeneracy of the Dirac cones is lifted due to the breaking of the inversion symmetry, and a pair of Weyl nodes vertically shifted with respect to each other appear. The pseudogap is closed due to the lifting of the degeneracy of the Weyl nodes. (d) DOS $\rho \left(\varepsilon \right)$ of a Dirac semimetal. (e) DOS $\rho \left(\varepsilon \right)$ of a Weyl metal. The plus and minus signs identify the DOS resolved in opposite chiralities.

Figure 2

(a) Spatial profile of the LDOS, corresponding to the antibonding state of a pair of impurities, placed inside a Dirac semimetal ($Q_0=0$). (b) Spatial profile of the LDOS for a pair of impurities, placed inside a Weyl metal with moderate value of $Q_0=0.25D$. (c) Spatial profile of the LDOS, corresponding to the frustrated atomic state, for a pair of impurities, placed inside a Weyl metal with large value of $Q_0=0.4D$.

Figure 3

Impurity-induced contributions to the density of states $\delta {\rho }_{jl}$ as a function of energy. Position of the STM tip is fixed at $\mathbf{r}=(1,1,1)\text{nm}$. (a) The case of a Dirac semimetal host, $Q_0=0$. One clearly sees two well-resolved pairs of peaks in $\delta {\rho }_{jj}$, centered around ${\varepsilon }_d$ and ${\varepsilon }_d+U$ and corresponding to bonding (green arrow) and antibonding (red arrow) molecular orbitals. (b) The case of a Weyl metal host with small value of $Q_0=0.1D$. The peaks corresponding to the molecular states become broadened but are still clearly resolved. (c) The case of a Weyl metal host with moderate value of $Q_0=0.25D$. Intermediate Fano structures with merged peaks and dips appear. (d) The case of a Weyl metal host with large value of $Q_0=0.4D$. A broad plateau in the density of states flanked by a pair of the merged peaks or dips is formed around $\varepsilon =0$. A transition to the regime of atomic frustrated state occurs, as seen in Fig. 2.

Figure 4

(a) Induced LDOS term $\delta {\rho }_{j\overline{j}}$ ($j=1,\overline{j}=2$ and $j=2,\overline{j}=1$) for $Q_0=0.4D$ with STM tip at $\mathbf{r}=(1,1,1)\text{nm}$ as a function of energy for several values of ${\mathbf{R}}_{12}.$ For sake of clarity, we present each case vertically shifted, thus making explicit that the increasing of ${\mathbf{R}}_{12}$ leads to the LDOS $\delta {\rho }_{j\overline{j}}$ vanishing. Hence, such a quenching reveals the crossover from the profile with two Hubbard bands, characteristic of the atomic frustrated state, towards that completely flat, for the uncorrelated pair of atoms situation. (b) Amplitude of $\delta {\rho }_{j\overline{j}}$ evaluated at the black-dashed line cut $\varepsilon \approx -0.07D$ marked in panel (a) as a function of $|{\mathbf{R}}_{12}|$, which exhibits an exponential-like decay (crossed points in black). In particular, it is fitted by $\delta {\rho }_{j\overline{j}}(\varepsilon \approx -0.07D)=1.96exp(-0.41|{\mathbf{R}}_{12}\left|\right)$ (red line).

Figure 5

(a) The total LDOS of the system consisting of two impurities placed inside a Weyl metal host with $Q_0=0.4D$, corresponding to the regime of the formation of an atomic frustrated state. Position of the STM tip is fixed at $\mathbf{r}=(1,1,1)\text{nm}$. (b) Phase diagram, showing the total DOS as function of the energy $\varepsilon$ and the parameter $Q_0$. With an increase of $Q_0$ one clearly observes the crossover from the regime of standard bonding (green arrow) and antibonding (red arrow) molecular orbitals, characterized by four well-resolved Hubbard bands, to the frustrated atomic state regime.