Self-energy and lifetime of Shockley and image states on Cu(100) and Cu(111): Beyond the $GW$ approximation of many-body theory

We report many-body calculations of the self-energy and lifetime of Shockley and image states on the (100) and (111) surfaces of Cu that go beyond the $GW$ approximation of many-body theory. The self-energy is computed in the framework of the $GW\Gamma$ approximation by including short-range exchange-correlation (XC) effects both in the screened interaction $W$ (beyond the random-phase approximation) and in the expansion of the self-energy in terms of $W$ (beyond the $GW$ approximation). Exchange-correlation effects are described within time-dependent density-functional theory from the knowledge of an adiabatic nonlocal XC kernel that goes beyond the local-density approximation.

Figure 1

The $\Gamma L$ projected bulk band structure (shaded areas) of the (111) and (100) surfaces of Cu. The solid lines represent Shockley $(n=0)$ and image-potential $(n=1)$ surface-state bands.

Figure 2

Imaginary part of the screened interaction $W(z,z^{\prime };\mathbf{k},\omega )$ and the effective screened interaction $\tilde{W}(z,z^{\prime };\mathbf{k},\omega )$, as a function of $z=z^{\prime }$ and for fixed values of $k$ and $\omega$ ($k=0.5{\mathrm{\AA }}^{-1}$ and $\omega =0.5\mathrm{eV}$), in the vicinity of the (100) and (111) surfaces of Cu. ALDA calculations of $\mathrm{Im}\left[\tilde{W}(z,z^{\prime };\mathbf{k},\omega )\right]$ are represented by thick solid lines. RPA and ALDA calculations of $\mathrm{Im}\left[W(z,z^{\prime };\mathbf{k},\omega )\right]$ are represented by thin solid and dotted lines, respectively.

Figure 3

Imaginary part of the effective screened interaction $\tilde{W}(z,z;\mathbf{k},\omega )$, as a function of $z=z^{\prime }$ and for fixed values of $q$ and $\omega$ ($q=0.5{\mathrm{\AA }}^{-1}$ and $\omega =0.5\mathrm{eV}$), in the vicinity of the (111) surface of Cu. ALDA, refined ALDA, and ANLDA calculations are represented by thin solid, dashed, and thick solid lines, respectively.

Figure 4

$G^0{W}^0$ ($G^{0}W$-RPA), $G^{0}W$, and $GW\Gamma$ calculations of the imaginary part of the $n=0$ surface-state self-energy $\Sigma (z,z^{\prime };\mathbf{k}=0,{\epsilon }_{\mathbf{k}})$, versus $z$, in the vicinity of the (111) surface of Cu. The solid circle represents the value of $z^{\prime }$ in each case. $GW\Gamma$ calculations (as obtained with the use of our ANLDA XC kernel) are represented by dashed lines. $G^{0}W$ (also using our ANLDA XC kernel) and $G^0{W}^0$ calculations are represented by thin and thick solid lines, respectively. ALDA calculations, which nearly coincide with ANLDA calculations, are not plotted in this figure.

Figure 5

$G^0{W}^0$ reciprocal lifetimes of Shockley and image states on the (100) and (111) surfaces of Cu, as a function of the parameter $\eta$ that accounts for the imaginary part of the complex frequencies entering the evaluation of the noninteracting density-response function ${\chi }^0(z,z^{\prime };\mathbf{k},\omega )$.