Advances in Condensed Matter Physics

Volume 2015, Article ID 531498, 10 pages

http://dx.doi.org/10.1155/2015/531498

## Two-Dimensional Metallicity with a Large Spin-Orbit Splitting: DFT Calculations of the Atomic, Electronic, and Spin Structures of the Au/Ge(111)- Surface

Institut für Theoretische Physik und Astrophysik, Universität Würzburg, Am Hubland, 97074 Würzburg, Germany

Received 10 November 2014; Accepted 26 January 2015

Academic Editor: Victor V. Moshchalkov

Copyright © 2015 Andrzej Fleszar and Werner Hanke. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Density functional theory (DFT) is applied to study the atomic, electronic, and spin structures of the Au monolayer at the Ge(111) surface. It is found that the theoretically determined most stable atomic geometry is described
by the conjugated honeycomb-chained-trimer (CHCT) model, in a very good agreement with experimental data. The calculated electronic structure of the system, being in qualitatively good agreement with the photoemission measurements, shows fingerprints
of the many-body effects (self-interaction corrections) beyond the LDA or GGA approximations. The most interesting property of this surface system is the large spin splitting of its metallic surface bands and the undulating spin texture along the hexagonal Fermi contours, which highly resembles the spin texture at the Dirac state of the topological insulator Bi_{2}Te_{3}. These properties make this system particularly interesting from both fundamental and technological points of view.

#### 1. Introduction

Two-dimensional metallic systems are of particular theoretical and technological interests for the richness of physical phenomena they display and the possibility of their tuning in a controlled way. Recently, special attention has been paid to spin properties of such systems which are relevant to the potential applicability in the spintronics devices. In the surface systems the space inversion symmetry is broken. This fact together with the spin-orbit interaction leads to the momentum-spin locking, which protects against backscattering by potentials preserving the time-reversal symmetry. In this perspective, surfaces of the 3-dimensional topological insulators display the desired properties in their topological surface states [1, 2]. From the point of view of the integrability with the electronic technology, however, it would be specially desirable to find a semiconductor surface displaying both a good metallicity and a large spin splitting of its metallic surface bands. These properties have been recently found in the Ge(111) surface covered with a monolayer of a heavy metal. The -phase of ()-Pb/Ge(111) [3] and the ()-Au/Ge(111) [4–6] systems possess metallic surface bands which are split by the strong spin-orbit interaction.

In this paper, we present results of detailed calculations based on the density functional theory (DFT) [7, 8] done for the -Au/Ge(111) surface. We address the questions of the stability of the possible atomic configurations at this surface and the electronic and spin structures of the stable phase. We find that although the standard local-density approximation (LDA) and the gradient corrections to it (GGA) reproduce very well the experimental data on the geometry of this surface, that is, a ground-state property, the calculated electronic structure shows some characteristic deviations from the photoemission results, which point at the importance of the many-body effects for electronic excitations in this system, that is, quasiparticle excitations. We find that the self-interaction corrections (SIC) applied in a semiempirical way to the Au- states significantly improve the electronic structure: they partly correct the depth and the effective mass of the surface-state parabola and enhance the amount of the hexagonal warping of the Fermi contour, bringing its shape closer to the experimental photoemission results. An important outcome of our calculations is the spin texture of the surface states split by the spin-orbit coupling (SOC). It turns out that, in addition to a very large value of the spin splittings of the surface bands, their spin texture shows the characteristic and searched features of the helical (keyword: Rashba [9]) and radial (keyword: Dresselhaus [10]) fingerprints, very similar to the recently theoretically predicted spin texture of the Dirac states of the topological insulator Bi_{2}Te_{3} [11, 12].

In the next section, we present the methodology of calculations. In Section 3, the results of the theoretical determination of the surface geometry are discussed. Section 4 is devoted to the analysis of the electronic structure of the most stable geometry and the effect of the self-interaction corrections on the band structure. In Section 5, we discuss the spin texture at the Fermi contours.

#### 2. Computational Details

The calculations in this work have been done within the density functional theory using the local-density approximation [7, 8] and the generalized gradient approximation (GGA) of the Perdew-Wang (PW91) type [13]. Two types of codes were applied. The plane-wave code of the QUANTUM ESPRESSO suite [14] has been used in studies of surface energies for various reconstructions models of the Au/Ge(111) system. For the detailed calculations of the electronic structure and spin texture of the most stable* conjugate-honeycomb trimer* (CHCT) model [15], our own mixed-basis codes were used, where, apart from plane waves in the basis, also -type Gaussians were present in order to represent accurately the Au- orbitals. In both codes, the norm-conserving pseudopotentials have been employed. The surface was modelled using the periodic-slab construction, where a large number of germanium layers are terminated on one side with the Au adsorbates and on the opposite side the dangling bonds are saturated by the hydrogen atoms. Using asymmetric slabs is specially convenient when spin splittings are a goal of investigations. The whole slab is repeated periodically in the perpendicular-to-surface direction including a vacuum spacer between slabs. In the calculations of the atomic relaxations for different models of the surface the typical number of Ge-monolayers in the slab was 7. In the calculations of the electronic and spin structure for the most stable CHCT model, larger slabs, up to 25 Ge-monolayers, have been employed in order to disentangle the real surface features from the artificial quantization effects produced by the slab geometry. All calculations were done within the lateral unit cell with respect to the unit cell of the unreconstructed Ge(111) surface. The lattice parameter of this hexagonal cell was fixed at the experimental value of 6.93 Å. In the self-consistency iterations, we have applied the (6,6,4) division of the Monkhorst and Pack method [16] and the “Gaussian smearing” technique of Fu and Ho [17] in determining the Fermi energy in metallic systems.

#### 3. Results of Geometry Optimizations

Experimentally, two different studies done for the ()-Au/Ge(111) system, the surface-X-ray-diffraction (XRD) of Howes et al. [18] and the low-energy electron-diffraction (LEED) of Over et al. [19], agree well with the CHCT model proposed for the Au deposition on the Si(111) surface [15] and also agree with each other in the values of determined atomic positions. A confirmation of the validity of the CHCT model brings also the recent STM study for this surface [20]. Correctness of the CHCT model might question, however, a certain disagreement between the electronic structure calculated on the basis of this model and results of the angular-resolved-photoemission (ARPES) measurements for this surface [4, 5, 20]. Although calculations done within LDA or GGA theory reproduce the metallic surface state seen in experiments, the theoretically determined bottom of this surface state is energetically too high, the electron-like Fermi surface contour is too spherical (shows less hexagonal warping compared with experiment), and the -diameter of this contour is too small. The purpose of this section is to verify if the CHCT structural model describes indeed the most stable geometry of the Au/Ge(111) system and if some other model of this surface does not give a better agreement with the electronic structure determined by ARPES measurements.

In the search of an appropriate model, we limited ourselves to models of possibly high symmetry, compatible with the periodicity and showing a trimerization of Au or Ge atoms seen in experiments. With these “boundary conditions” there is still place for variation of some parameters, such as the precise stoichiometry, the subsurface geometry (missing or not of the Ge-surface layer), or the point of placement of the trimers at the surface and their nature (Au or Ge trimers). We have considered altogether 22 geometrical models differing in the coverage of Au adatoms, in missing or not of the top Ge-surface layer (MTL or TL cases), in the placement of the characteristic structure (trimer) on top of , , or positions in the surface, and in the geometrical orientation of the apices of the trimer. The Au coverage was varied between the 1/3 one (1 Au atom in the unit cell) and the 2/3, 3/3 (i.e., one Au monolayer), and 4/3 coverage. The atomic positions in each structural model were relaxed within the LDA approach and the surface energy was obtained according to the following formula: where is the total energy of the slab; and are chemical potentials and number of atoms of the th species in the slab. As the chemical potentials for Ge and Au, we have taken the energy per atom in the solid-state phase of these materials. It turned out that all structures with the Ge-TL configuration, as well as those with the 1/3 Au coverage, have much higher surface energies and could be therefore ruled out. Figure 1 shows top views of 9 most stable structures, all with the Ge-MTL configuration, together with the excess energy per surface unit cell of each structure over the most stable CHCT model.