Linear density response function in the projector augmented wave method: Applications to solids, surfaces, and interfaces

We present an implementation of the linear density response function within the projector-augmented wave method with applications to the linear optical and dielectric properties of both solids, surfaces, and interfaces. The response function is represented in plane waves while the single-particle eigenstates can be expanded on a real space grid or in atomic-orbital basis for increased efficiency. The exchange-correlation kernel is treated at the level of the adiabatic local density approximation (ALDA) and crystal local field effects are included. The calculated static and dynamical dielectric functions of Si, C, SiC, AlP, and GaAs compare well with previous calculations. While optical properties of semiconductors, in particular excitonic effects, are generally not well described by ALDA, we obtain excellent agreement with experiments for the surface loss function of graphene and the Mg(0001) surface with plasmon energies deviating by less than 0.2 eV. Finally, the method is applied to study the influence of substrates on the plasmon excitations in graphene.

Figure 1

The imaginary part of the dielectric function (a) and energy-loss function (b) of graphene at $q=0.046$ Å${}^{-1}$ along $\mathrm{\Gamma ̄}-\mathrm{M̄}$ direction of its surface Brillouin zone. The LDA wave functions and energies entering $\chi {}^0$ have been obtained using a 3D uniform grid (black solid line) and a localized atomic orbital (LCAO) containing single $\zeta$ (red dotted), single $\zeta$ with polarization (green dashed), and double $\zeta$ with polarization (purple dashed-dotted), respectively.

Figure 2

The energy-loss function of Mg(0001) surface at $q=0.07$ Å${}^{-1}$ along $\mathrm{\Gamma ̄}-\mathrm{M̄}$ direction of its SBZ calculated with 3D uniform grid (GRID, black solid line) and localized atomic orbital (LCAO) using dzp (red dashed line) basis. The dzp basis used here includes double-$\zeta$ orbitals of 3$s$ and 3$p$ atomic orbitals as well as one $d$-type Gaussian polarization function.

Figure 3

Schematic illustration of the applied parallelization scheme. Each box represents a single CPU. (a) The calculation of $S_{G{G}^{\prime }}^0(q,\omega )$ is performed in parallel over wave vectors $k$ (or bands $n$, for large cells) and frequencies $\omega$. (b) The Hilbert transform is parallelized over $G$. (c) Finally the Dyson equation is solved by parallelizing over the frequencies.

Figure 4

Imaginary part of the dynamical dielectric function of bulk silicon. The arrows indicate the absorption onset and the position of main and secondary peaks, respectively, as extracted from Ref. 30. Calculations have been performed including local-field effects (dotted and dashed) and exchange-correlation effects at the ALDA level (dashed).

Figure 5

Surface loss function of the Mg(0001) surface along the $\mathrm{\Gamma ̄}-\mathrm{M̄}$ direction of the surface BZ calculated using RPA (a) and TDLDA (b). In both cases $\left|q\right|$ increases from bottom to top. (c) Surface plasmon dispersion for both the $\mathrm{\Gamma ̄}-\mathrm{M̄}$ and $\mathrm{\Gamma ̄}-\mathrm{K̄}$ directions. Results from this work (filled dots) compare well with other calculations (hollow dots, Ref. 38) and experiments, Ref. 45.

Figure 6

Atomic structure of graphene adsorbed on SiC(0001) (a),(b), and Al(111) (d),(e). The lateral unit cells are indicated by red lines in the top panels. The LDA band structures of the surfaces are shown in the lower panels. Also shown is the band structure of free-standing graphene (red dots). The Fermi level is set to zero.

Figure 7

Loss function of free-standing graphene (a) and graphene on SiC substrate (b) as a function of $q$. The loss functions, from bottom to top (solid lines), correspond to increasing $q$ at an interval of $0.046$ Å${}^{-1}$. The dashed line corresponds to the loss function of the substrate at $q=0.092$ Å${}^{-1}$. (c) Dispersion relations for the $\pi$ plasmons of free-standing graphene (red filled circles) and graphene on SiC (black filled squares). They are compared with earlier ab initio calculation on free-standing graphene (blue hollow circles) and experiments on single-wall carbon nanotubes (green hollow squares) (Ref. 51) as well as experiments on graphene / SiC(0001) (purple hollow diamonds) (Ref. 52). Lines are added to guide the eye.

Figure 8

Surface loss functions for graphene on Al(111) as a function of the adsorption distance $d$ for $\left|q\right|=0.046$ Å${}^{-1}$. The surface loss function of free-standing graphene is shown as black dots. Inset: sketch of graphene on Al substrate.