Metallic surface of a bipolaronic insulator

We investigate the possibility that the surface of a strongly coupled electron-phonon system behaves differently from the bulk when the relevant parameters are inhomogeneous due to the presence of the interface. We consider parameter variations which make the surface either more metallic or more insulating than the bulk. While it appears impossible to stabilize a truly insulating surface when the bulk is metallic, the opposite situation can be realized. A metallic surface can indeed be decoupled from a bipolaronic insulator realized in the bulk.

Figure 1

Geometry of the (001) surface of a simple-cubic lattice. The index $\alpha$ (horizontal axis) labels the layers parallel to surface. $\alpha =1$ refers to topmost layer. In the embedding approach for DMFT the system is divided into a surface region of $N$ layers and a semi-infinite substrate. The physical properties in the substrate (e.g., electron self-energy) are not layer-dependent and coincide with those of the bulk.

Figure 2

Layer-dependent quasiparticle weight $z_{\alpha }$ (top panel) and layer-dependent double occupancy $d_{\alpha }$ (bottom panel) as a function of modified intralayer surface hopping $t_{11}$. We show results for the first three layers of the semi-infinite Holstein model with simple cubic (001) surface geometry and for the bulk quasiparticle weight. $\alpha =1$ indicates the topmost surface layer. The solid lines show bulk calculations. The insets show the imaginary part of the surface self-energy $\text{Im}{\Sigma }_1\left(i{\omega }_n\right)$ on the discrete mesh of the imaginary energies ${\omega }_n=\left(2n+1\right)\pi /\tilde{\beta }$ $\left(\tilde{\beta }=400\right)$ for $t_{11}/t=0.1$ and $t_{11}/t=0.5$. $\text{Im}{\Sigma }_{\alpha =1}\left(i{\omega }_n\right)$ shows an upturn at small frequencies, compatible with a Fermi-liquid behavior.

Figure 3

$g$ dependence of quasiparticle weights $z_{\alpha }$ of semi-infinite Holstein model for simple-cubic lattice in the (001) orientation for enhanced intralayer surface hopping. Surface transition at $g=g_{c,surface}$. Bulk transition at $g=g_{c,bulk}$.

Figure 4

Phonon probability-distribution function for the first three layers of the semi-infinite Holstein model with nonuniform model parameters and the bulk phonon probability-distribution function for two different values of e-ph coupling strength, $g$. We used $t_{11}=1.5t$.