A functional renormalization group application to the scanning tunneling microscopy experiment

Figures & data

Figure 1. STM tip coupled to a host surface with an impurity. Notes: The tunneling matrix elements $t_{12},t_{13}$ and $t_{23}$ represent the couplings tip-impurity, tip-surface and impurity-surface, respectively. The tip-impurity lateral distance is denoted by r.

Figure 2. The Green functions of the impurity $g(\mathit{\omega })$, substrate $g_k(\mathit{\omega })$, and tip $g_p(\mathit{\omega })$. Notes: Feynman diagrams for the hybridizations $t_{23}$, $t_{12}$ and $t_{13}$. The self-energy ${\mathrm{\Sigma }}_{tun}$ include two tunneling events: an electron can tunnel from the impurity to the substrate and from the impurity to the tip. The interaction ${\tilde{t}}_{12}$ contains the physical event that an electron from the tip jump to the surface and then to the impurity.

Figure 3. Flow equations. Notes: The dots over the vertices indicate a derivative with respect to the flow parameter $\mathrm{\Lambda }$, the lines connecting different vertices are different Green’s functions in the Keldysh formalism. Each of the Green’s functions have a contribution from the self-energy in Figure 2 which is denoted by a contribution $\mathrm{\Gamma }$ to the Green’s function.

Figure 4. The differential conductance for $r=0$ in units of $k_F^{-1}$. A negative differential conductance develops over the whole range.

Figure 5. The differential conductance for $r=2$ in units of $k_F^{-1}$. A positive Lorentzian is developed when the STM tip is away from the impurity.