First-principles Green's-function method for surface calculations: A pseudopotential localized basis set approach

We present an efficient implementation of a surface Green's-function method for atomistic modeling of surfaces within the framework of density functional theory using a pseudopotential localized basis set approach. In this method, the system is described as a truly semi-infinite solid with a surface region coupled to an electron reservoir, thereby overcoming several fundamental drawbacks of the traditional slab approach. The versatility of the method is demonstrated with several applications to surface physics and chemistry problems that are inherently difficult to address properly with the slab method, including metal work function calculations, band alignment in thin-film semiconductor heterostructures, surface states in metals and topological insulators, and surfaces in external electrical fields. Results obtained with the surface Green's-function method are compared to experimental measurements and slab calculations to demonstrate the accuracy of the approach.

Figure 1

Illustration of a typical semi-infinite surface configuration. The Hamiltonians of the surface region (${\mathbf{H}}_{\mathrm{S}}$) and the principal layer of the semi-infinite bulk region (${\mathbf{H}}_{\mathrm{B}}$), are coupled through the coupling terms ${\mathbf{V}}_{\mathrm{SB}}$. A Dirichlet boundary condition is used at the boundary between the bulk and surface regions. In the vacuum region, a Neumann or Dirichlet boundary conditions can be used. $L_{\mathrm{S}}$ ($L_{\mathrm{B}}$) is the length of the surface region (the bulk region principal layer) along the out-of-surface-plane ($Z$) direction.

Figure 2

CPU time per self-consistent (SCF) cycle as function of the length of the surface region $L_{\mathrm{S}}$ (see Fig. 1) for a surface configuration (red filled circles, solid line), compared to the CPU time per SCF cycle of a slab configuration (blue filled diamond, dashed line) with the slab thickness equivalent to $L_{\mathrm{S}}$. The system considered is a $2\times 2$ unreconstructed Si(100) surface.

Figure 3

The macroscopic in-plane averaged electrostatic difference potential $\langle \delta V_{\mathrm{E}}\rangle$ [59] calculated throughout the surface and slab structures for the SGF model (blue solid line) and the ATK slab model (red solid line), respectively. For both surface models, the $\langle \delta V_{\mathrm{E}}\rangle$ potential is calculated for 14-monolayer Ag(001) slab or surface region, and is projected onto the $Z$ axis (normal to the surface plane). Boundary conditions of Dirichlet or Neumann type are indicated for both SGF and slab models of the surface.

Figure 4

The Ag(001) work function calculated as a function of a number of atomic monolayers in the surface region, using the ATK-SGF method (red filled circles), the ATK (blue filled diamonds), and vasp (gray filled squares) slab models.

Figure 5

DFT-calculated vs experimental work functions. The work functions are calculated using the ATK-SGF method (red filled circles) and the VASP plane-wave slab method (gray filled squares). The measured work functions are taken from Ref. [60].

Figure 6

(a) Local density of states (LDOS) calculated for a slab configuration (blue) or a surface configuration (red) with thickness $L=3a$, where the thickness $L$ is given in units of the lattice constant of bulk Ge, $a$. (b) Same as (a), but for $L=7a$. The dashed vertical lines indicate the extremities of the band gap of bulk Ge.

Figure 7

The conduction-band minimum and valence-band maximum calculated for bulk Ge (black dotted lines), the Ge(001) surface (red solid lines, filled circles) modeled with the SGF approach, and the Ge slabs (blue dashed lines, filled diamonds) with different slab thicknesses $L$ given in units of the lattice constant of bulk Ge, $a$.

Figure 8

(a) The Ge(001)$|\mathrm{Si}$ interface structure and the macroscopic in-plane averaged Hartree difference potential, $\langle \delta V_{\mathrm{H}}\rangle$, calculated for the Si film on the $p$-doped Ge substrate ($p={10}^{20}{\mathrm{cm}}^{-3}$). The green solid, black dotted and dark magenta solid lines indicate the regions of $\langle \delta V_{\mathrm{H}}\rangle$ corresponding to the Ge(001) surface, the Ge(001)$|\mathrm{Si}$ interface, and the Si film, respectively. Note that for visualization purposes the structure has been repeated in the $XY$ plane, while only a single lateral unit cell is used for the actual calculations. (b) The macroscopic in-plane averaged Hartree difference potential, $\langle \delta V_{\mathrm{H}}\rangle$, along the $Z$ direction for the $p$-doped (blue), and $n$-doped (red, green) Ge substrate.

Figure 9

Band diagrams showing the density of states (DOS) across the Ge(001)$|\mathrm{Si}$ heterostructure for (a) $p$-doped Ge(001) substrate with $p={10}^{20}{\mathrm{cm}}^{-3}$, and $n$-doped Ge(001) substrate with (c) $n={10}^{20}{\mathrm{cm}}^{-3}$ and (d) ${10}^{21}{\mathrm{cm}}^{-3}$. Notice that $\langle \delta V_{\mathrm{H}}\rangle$ in the Si region has been shifted to match the energy gap of the Si film. (b) Band diagram of the unsupported Si film (Si slab) showing the $\langle \delta V_{\mathrm{H}}\rangle$ superimposed on the DOS.

Figure 10

(a) Field-induced difference in the macroscopic in-plane averaged Hartree potential in the Ag(111) surface region, $\mathrm{\Delta }{V}_{\mathrm{H}}$, plotted in the out-of-surface-plane ($Z$) direction, for different external electric fields, $E_z$. The interface between the semi-infinite bulk Ag region and surface region is at $Z=0\AA$. The black dashed vertical line at $Z_{\mathrm{surf}}=64.68\AA$ indicates the position of the Ag(111) top monolayer. Ag atoms are shown as gray spheres. (b) $100\times \mathrm{zoom}$ of the inset (a) for $E_z=\pm 0.27\mathrm{V}/\AA$.

Figure 11

(a) Band structure of the Ag(111) surface along the $\mathrm{M}\to \mathrm{\Gamma }\to \mathrm{K}k$-point path of the 2D Brillouin zone in the vicinity of the Fermi energy and close to the $\mathrm{\Gamma }$ point, for applied fields of $-0.27\mathrm{V}/\AA$ (left inset), $0.0\mathrm{V}/\AA$ (center inset) and $+0.27\mathrm{V}/\AA$ (right inset). The horizontal solid (dashed) lines indicate the lowest energy of the surface states (highest occupied bulk states) at the $\mathrm{\Gamma }$ point. (b) Stark energy shift ($\mathrm{\Delta }\varepsilon$) of the Ag(111) surface state against the $E_z$-electric field applied. Filled circles and black solid line correspond to the surface state (SS) and a linear fit of the calculated data, respectively. The filled squares correspond to lower band edge (LBE).

Figure 12

(a) Band structure of the Se-terminated ${\mathrm{Bi}}_2{\mathrm{Se}}_3$(111) surface along the $M\to \mathrm{\Gamma }\to Kk$-point path in the 2D Brillouin zone in the vicinity of the Fermi energy, close to the $\mathrm{\Gamma }$ point. (b) Density of states at the $\mathrm{\Gamma }$ point. The red horizontal lines in both insets indicate the position of the Fermi energy.

Figure 13

Zero-field potential-energy curve (solid lines, filled circles) for atomic iodine adsorbed on the Pt(111) surface, calculated using the SGF method. Cubic interpolation of the PEC near the minimum yields an adsorption energy of $-1.45$ eV at an equilibrium I–Pt(111) separation distance ($d_0$) of 2.28 Å. The measured and plane-wave DFT-calculated adsorption energies (dotted lines) are taken from Ref. [87].

Figure 14

(a) Finite-field potential-energy curves for atomic iodine adsorbed on the Pt(111) surface, for electrostatic fields ranging from $+6$ V/Å (red) over zero field (green) to $-6$ V/Å (black). For all the curves, zero energy is chosen at the equilibrium position of the adsorbate. (b) Equilibrium I–Pt(111) separation distance $d_0$ (solid line) and Mulliken population (dashed line) on the iodine atom as a function of the applied field. Positive Mulliken population means accumulation of electrons on the iodine atom.

Figure 15

Band structures of bulk Si and Ge, calculated with the PPS-PBE method. Note that the fundamental $\mathrm{\Gamma }-L$ band gap of bulk Ge calculated using the PBE functional is zero, in contradiction with experiment, whereas the PPS-PBE calculated band gap and other band energies agree with the experimental data in a semi-quantitative manner. The fundamental gaps for bulk Si and Ge are $E_{\mathrm{g}}^{\mathrm{Si}}=1.14$ eV and $E_{\mathrm{g}}^{\mathrm{Ge}}=0.64$ eV, respectively.