First-principles study on the atomic and electronic structures of the Au/Si(111)- α ( √ 3 × 3 )R30 surface

We have investigated the atomic and electronic structures of the Au/Si(111)- α ( √ 3 × √ 3)R30 ◦ surface using the ﬁrst-principles calculations within the density functional theory in the generalized gradient approximation, focusing on honeycomb conﬁgurations. The energetically favored honeycomb model has a geometry that two Au atoms are located at H 3 site on a Si substrate with missing top layer conﬁguration. The formation energy of this model is less than that of the conjugate honeycomb-chained trimer (CHCT) one proposed by Ding et al . [Surf. Sci., 275 , L691 (1992)] and that of the most stable conﬁguration having the Au coverage of 1/3 ML in a certain range of chemical potential of Au. The observed energy band structure is not completely reproduced in our calculation using the honeycomb conﬁguration as well as the CHCT one. On the other hand, the scanning tunneling microscope image simulated for the stable honeycomb model agrees with the observed ones. Our results, together with the recent experimental evidence that the Au coverage of the surface is less than 1 ML, suggest that the honeycomb conﬁguration is more plausible than the CHCT one, which has been considered to be most suitable for the atomic structure of the commensurate region in the surface. [DOI: 10.1380/ejssnt.2004.146]


I. INTRODUCTION
Over a few decades, the Au/Si(111) system has attracted great interest as one of the systems of noble metals on semiconductor surfaces. Among reconstructed surfaces of this system, the Au/Si(111)-α( √ 3× √ 3)R30 • (hereafter, referred as α √ 3) surface is known as its complexity of surface morphology [1][2][3][4][5][6][7]. This complex feature can be seen in the images of scanning tunneling microscope (STM) as coexistence of two kinds of regions: One is commensurate regions where the protrusions array in the √ 3× √ 3 periodicity, and the other is zigzag incommensurate regions called Domain Walls (hereafter, referred as DWs) separating the anti-phase commensurate ones.
The atomic structure of this complex surface have been investigated using various diffraction techniques such as surface X-ray diffraction [8], transmission electron diffraction [9], low energy electron diffraction [10], reflection high energy electron diffraction [11], and so on. The results obtained by these techniques suggest that the atomic arrangement of the surface is explained by a trimer model, which is also supported by a theoretical investigation [12]. It is worth noting that most of the experiments were performed with the coverage of about 1.0 ML [1 monolayer (ML) = 1 atom / (1×1 unit cell) = 7.8 × 10 14 atoms/cm 2 ] where the diffraction spots arising from the √ 3 super- * Corresponding author: kado@cello.t.u-tokyo.ac.jp structure are sharpest [7,13,14].
However, the recent STM observation by Nagao et al. has revealed that the feature of the surface appeared at 1.0 ML is different from that of the α √ 3 one [7]. To distinguish between the two surfaces, they named the former surface β √ 3. Furthermore, their experiment has also shown that the DWs increase with increasing the Au coverage from 0.76 ML to 0.96 ML. In addition, Takami et al. have also estimated the Au coverage of the commensurate phase to be 0.5 -0.6 ML using STM [5]. These results suggest that the Au coverage of the commensurate phase is less than 0.76 ML, and smaller than that of DWs. These facts are difficult to explain by the trimer models having the coverage of 1.0 ML, and thus leads us to the speculation that the configuration of the commensurate region is different from the trimer one. Among possible configurations having a coverage lower than 0.76 ML, the honeycomb configurations [15,16] have a potential to explain these results.
In this study, we examine the atomic and electronic structures of the Au/Si(111)-α( √ 3 × √ 3)R30 • surface, focusing mainly on the honeycomb configurations using first-principles calculations within the density functional theory in the generalized gradient approximation. The results of formation energies, energy band structures and simulated STM images show that the honeycomb configuration is a promising candidate for the structure of the commensurate region of the α √ 3 surface, although the agreement between the present calculation and experimental data is not perfect.

II. CALCULATIONS
The theoretical calculations were performed with TAPP (Tokyo Ab initio Program Package) [17]. The total energy calculations were performed within the density functional theory [18] in the generalized gradient approximation [19,20], using the scalar-relativistic [21,22] ultrasoft [23][24][25][26] and norm-conserving [27] pseudopotentials for Au and Si, respectively. We introduced a partial core correction to the Au pseudopotential in order to consider a nonlinear effect for the exchange-correlation term [28]. We adopted a supercell geometry, which has a slab consisting of six atomic layers of Si, adlayers of Si and Au, and a vacuum region corresponding to eight atomic layers in thickness. The bottom surface of the slab was terminated with hydrogen atoms to eliminate artificial dangling bonds and prevent them from coupling with the states near the front surface of the slab. The wave functions were expanded in a plane-wave basis set with an energy cutoff of 16.0 Ry. Sixteen special k points were employed to sample the Brillouin zone for the √ 3 × √ 3 unit cell. Both electronic and ionic degrees of freedom were optimized using the conjugate gradient method. STM images were simulated using the Tersoff-Hamann formalism [29].
We examined honeycomb configurations shown in Fig. 1. The models (a) and (b) in this figure have the missing top layer (MTL) configuration, while the models (c) and (d) have the bulk-truncated Si(111) surface configuration. As for the location of Au atoms, it is the T 4 site in (a) and (c), while the H 3 site in (b) and (d). To examine the relative stability of the honeycomb configurations, we also studied configurations having Au coverages different from 2/3 ML: The most plausible trimer configuration called "Conjugate Honeycomb Chained Trimer (CHCT)" having the Au coverage of 1 ML, and adatom configurations having the coverage of 1/3 ML. In the latter configurations, a Au adatom is located on the H 3 or T 4 site, and the topmost layer of Si substrate forms the MTL or the bulk-truncated Si(111) surface configuration, as well as the case of the honeycomb models.
To examine relative stability of models, we compare the formation energies defined as and δµ Au are the total energy of the slab X, the number of atom species i (Au or Si) in the slab, the chemical potential of bulk for the atomic species i and the difference of chemical potential from the bulk state, respectively. The estimation of E X form using eq.(1) contains contributions from both the top and bottom surfaces. We, however, do not worry about the effects of the bottom surface, because the atomic configurations of the bottom surfaces are identical in all the models.

III. RESULTS AND DISCUSSION
The structural parameters optimized for the honeycomb configurations are shown in Table I  and first Si layers, and z is the vertical distance between the Au and Si layers. All the configurations keep the Au atoms as the topmost layer, and have the threefold surface symmetry which has been confirmed by experimental observations [30]. As for the formation energy, the evaluated values are +0.06 eV, 0.00 eV, +1.62 eV and +1.46 eV for the models (a), (b), (c) and (d), respectively, which are measured from the most energetically favored model (b) (hereafter, we refer this configuration as H 3 -MTL).
For configurations having the Au coverage different from 2/3 ML, results of geometry optimization are as follows: For the 1/3 ML configurations, the most energetically favored configuration is the one in which the Au atom locates at the T 4 site on the MTL. The structural parameters of this configuration are also shown in Table  I. For the CHCT configuration (1 ML), our optimized geometry agrees extremely well with the configuration of Ding et al. as shown in Table I.
After obtaining the optimized geometries for all the configurations, we examine the relative stability of these configurations. Fig. 2 shows the formation energies calculated for the configurations employed as a function of the chemical potential of Au. The upper and lower limits of the chemical potential for Au at the surface are set to those for the fcc bulk and an isolated atom, respectively. As seen in this figure, there is a range in which the H 3 -MTL  configuration is more stable than both the CHCT and the most stable adatom one between these limits, which shows that the H 3 -MTL is one of the stable geometries of this surface.
Next, we examine the energy band structures shown in Fig. 3 for the CHCT (Fig. 3(a)) and H 3 -MTL (Fig. 3(c)) configurations, together with the one observed at the Au coverage of ∼0.9 ML by angle-resolved photoelectron spectroscopy (ARPES) [31] (Fig. 3(b)). The observed energy bands have the following features: The dominant ones are two partially occupied, dispersive surface bands (S 1 ) near the Fermi level. These bands have features similar to the surface state found for the Ag/Si(111)-( √ 3× √ 3) surface [32]. The other surface bands are named S 2 , S 3 and S 4 . The S 2 and S 3 become degenerate at an energy of −1.4 eV below E F around the Γ point. The energy of the S 2 band increases from the Γ point to the M one at which this band locates about −0.1 eV below E F , while that of the S 3 band decreases from the Γ point, takes the minimum of −1.5 eV around the middle of ΓM line, and then increases to the M point at the energy of −0.7 eV. Finally, the S 4 band is observed around the upper edge of the bulk bands at the Γ point. It is noted that the Au coverage where the ARPES data were observed is in the middle of the coverage range where only the α √ 3 surface exists. Therefore, coexistence of the commensurate and the DW regions is expected in the observed surface corresponding to the energy bands shown in Fig. 2(b).
To compare the calculated results with the observed ones, we picked up surface states marked by filled diamonds in Figs. 3(a) and (c). Here, we define the surface states as the states whose norm within a sphere centered at a Au atom is larger than a certain threshold C 0 . The radius of the sphere and the value of C 0 for the CHCT configuration is set to be 1.42Å and 0.05, respectively, while the values for the H 3 -MTL configuration is set to be 1.92Å and 0.075, respectively. Although the values of the radius and C 0 may have some ambiguity, qualiative features of the surface states defined in this way are not sensitive to the small changes in these values. First, comparing the CHCT's energy band (Fig. 3(a)) with the observed one ( Fig. 3(b)), we find that the S 1 and S 3 bands are reproduced in our calculation, especially at the point of the energy of S 1 band at the Γ point and the dispersive behavior of them. However, the calculated energy band for the CHCT configuration is not consistent with the observed features, namely the behavior of the S 2 band around the M point and the number of the bands crossing the Fermi level. In contrast, these features are well reproduced in the energy band of H 3 -MTL configuration (Fig. 3(c)), although agreement with the observed S 1 and S 3 bands is worse in the H 3 -MTL energy bands than in the CHCT's. From these results, we can say that the CHCT configuration is not the best one for the atomic structure of the commensurate region.
As mentioned before, coexistence of the commensurate and DW regions is expected in the observed surface corresponding to the energy bands shown in Fig. 2(b). On the other hand, our calculations assume a surface consisting of only a commensurate region, having either the CHCT or H 3 -MTL configuration. Therefore, it is natural that quantitative agreements between the calculated and observed energy bands are not sufficient. To achieve the quantitative agreements, we should consider that the observed ARPES spectra contains the information not only from the commensurate region but also from the DW region and the bonding states between the commensurate and DW regions. To examine the effects of coexistence is an interesting future task. Concerning this point, we would like to propose experimentalists to examine the coverage dependence of the ARPES spectra.
Finally, we discuss the STM images of the α √ 3 sur- face. Up to now, two patterns are observed in the STM images for empty states of the commensurate region in the surface. One is a hexagonal pattern [1,[3][4][5][6], and the other is a honeycomb one obtained recently in higher resolution images [7,33]. As can be seen from the simulated STM image for the H 3 -MTL configuration shown in Fig 4, we successfully reproduce such a honeycomb pattern. On the other hand, if we use the CHCT configuration, honeycomb patterns are not reproduced at any bias voltages. As for the hexagonal patterns, the CHCT configuration reproduces them, while the H 3 -MTL not. We, however, suspect that the hexagonal patterns may be caused by extrinsic origins that are not considered in the present study, judging from the fact that they appear only in low resolution images. Therefore, we consider that in the light of STM images the H 3 -MTL configurations is more plausible as a structural model of the commensurate phase in the α √ 3 surface than the CHCT one.

IV. CONCLUSION
In summary, the atomic and electronic structures of the Au/Si(111)-α( √ 3 × √ 3)R30 • surface have been investigated using first-principles calculations. Among honeycomb configurations, the one having two Au atoms at the H 3 sites on the missing top layer is energetically favored. The formation energy of the honeycomb configuration is found to be smaller than those of the conjugate honeycomb chained trimer (CHCT) one with the Au coverage of 1 ML and the most stable adatom one with 1/3 ML in a certain range of the chemical potential of Au. Further, a honeycomb pattern observed in STM is reproduced using the honeycomb configuration, but not done using the CHCT one. From these results, we can say that the honeycomb configuration is more plausible than the CHCT one, which has been considered to be most suitable model, for the atomic structure of the commensurate region in the α √ 3 surface. However, concerning the energy band structure the agreement between the calculated and observed ones is not satisfactory for the H 3 -MTL and CHCT configurations, though most of the observed features are reproduced qualitatively. Further studies on the structure of the surface is definitely needed.