Holstein Polarons Near Surfaces

We study the effects of a nearby surface on the spectral weight of a Holstein polaron, using the inhomogeneous momentum average approximation which is accurate over the entire range of electron-phonon ($e\mathrm{\text{−}}\mathrm{ph}$) coupling strengths. The broken translational symmetry is taken into account exactly. We find that the $e\mathrm{\text{−}}\mathrm{ph}$ coupling gives rise to a large additional surface potential, with strong retardation effects, which may bind surface states even when they are not normally expected. The surface, therefore, has a significant effect and bulk properties are recovered only very far away from it. These results demonstrate that interpretation in terms of bulk quantities of spectroscopic data sensitive only to a few surface layers is not always appropriate.

Figure 1

$A(k=0,i,\omega )$ vs $\omega /t$ in various layers, for $\Omega =0.5t$, $g=\sqrt{2}t$, $U=0$, $\eta =0.004t$.

Figure 2

Left: Comparison between full ${\mathrm{MA}}^{(1)}$ solution (solid line) and solution if $v_1(j,\omega )=0$ (dashed line). Right: Real part of the surface potential $v_1(i=0,\omega )$ in units of $t$ vs energy. The upper panels are for $U/t=0$ and the lower are $U/t=0.5$, while $\Omega =0.5t$, $g=\sqrt{2}t$, $\eta =0.004t$.

Figure 3

3D spectral weight $A(k_x,k_y=0,i,\omega )$ vs $k_x$ and $\omega$, for (a) $i=0$, (b) $i=1$, and (c) bulk. Note the very different scales for these panels. (d) Cut at $k_x=0.1\pi$. Parameters are $U=t$, $\Omega =0.5t$, $g=\sqrt{3}t$, $\eta =0.001t$.