Visualization of the local dipole moment at the Si(111) surface using DFT calculations

Abstract

A method to simulate the dipole moment mode of the scanning nonlinear dielectric microscope (SNDM) has been developed. This method has been applied to the so-called ${\rm Si(111)-(7 \times 7)}$ dimer-adatom-stacking-fault (DAS) structure and a ${\rm Si(111)-(2\times2)}$ surface with one adatom and one restatom, which are the main motifs of the DAS structure. It has been revealed that a local upward dipole moment is observed at the adatom site, consistent with the SNDM experiments. Differences in the local atomic arrangements of the adatom and restatom correlate with the amount of charge transfer between adatoms and restatoms, which varies in concert with the magnitude of the local dipole moment. This method will provide information on local dielectric properties necessary for the interpretation of various surface probe microscopy images.

Introduction

Recently, an atomic-resolution microscope technique called the scanning nonlinear dielectric microscope (SNDM)^1,2,3 has been developed. SNDM is one of very sensitive capacitance microscopy, capable of detecting very small capacitance changes of $10^{-22}$ $\textrm{F}/\sqrt{\textrm{Hz}}$ at the highest sensitivity and of capturing even very small nonlinear components of dielectric constant. SNDM has been applied to ferroelectric polarization distribution measurement⁴, charge distribution measurement in flash memory^5,6,7, carrier distribution measurement in semiconductor devices^8,9,10,11, depletion layer measurement⁹, and next-generation ultra-high-density ferroelectric recording^4,12. Non-Contact SNDM (NC-SNDM) has also been proposed¹³, which detects the nonlinear dielectric signal for controlling the non-contact condition. NC-SNDM can illuminate atomic-scale information on the surface topography and polarization simultaneously. The Si(111)-($7\times 7$) dimer-adatom-stacking fault (DAS) structure, a typical reconstructed surface of Si, has been observed using NC-SNDM^14,15. In the topographic mode, the adatoms, which are characteristic of the ($7\times 7$) DAS structure, have been observed as bright spots, similar with previous researches, e.g., NC-atomic force microscope¹⁶, and scanning tunneling microscope (STM)¹⁷. In the dipole moment image, upward dipole moments have been observed at the position corresponding to the adatoms. Negative electrons arising from covalent bonds between the adatom and its neighboring atoms and the positive nucleus of the adatom have been understood to be the origin of the upward dipole moment¹⁵.

Theoretical simulations of images for surface probe microscopes, e.g., STM, have played a decisive role in interpreting experimental results^{18,19,20,21,22,23,24}. For example, for the surface atomic arrangement of Si(111)-($\sqrt{3}\times \sqrt{3}$)-Ag surface, different atomic arrangement models had been proposed using different experimental methods^25,26, but STM simulations have settled the controversy^27,28,29. SNDM has revealed various dielectric properties of surfaces, but no theoretical simulations of them, especially regarding in-plane local dipole moments, have been found. In this study, we have developed a method to visualize the in-plane local dipole moment for surfaces. We have applied this method to the Si(111)-($2\times 2$) and the Si(111)-($7\times 7$) DAS surfaces. The Si(111)-($2\times 2$) surface is composed of one adatom and one restatom, characteristic motifs of the Si(111)-($7\times 7$) DAS structure. Our calculations have shown that the adatom (restatom) has an upward (almost zero) dipole moment in both models, being consistent with the experimental images by SNDM¹⁵. The occurrence (disappearance) of the dipole moment is attributed to the surface charge transfer between the adatom and the restatom.

Computational methods

Structural optimization and total charge density calculations were performed using first-principles calculations based on the density functional theory³⁰. The computational package, VASP 6^31,32, was used with the GGA-PBE type exchange-correlation term^33,34. The cutoff energy of plane-wave expansion was taken to 400 eV for Si(111)-($2 \times 2$) and 250 eV for Si(111)-($7 \times 7$). Note that the difference in cutoff energy is not an issue in comparing (2 × 2) and (7 × 7), since the dipole moment value changed by only 0.7% when (2 × 2) was calculated with a cutoff energy of 250 eV. The Si(111)-($2 \times 2$) surface and Si(111)-($7 \times 7$) were modeled using the supercell method with a Si(111) slab and a vacuum layer of 16 Å. The k-point sampling was set to ($21 \times 21 \times 1$) per ($1\times 1$) for the structural optimization and ($34\times 34 \times 1$) per ($1\times 1$) for the band calculations. A dipole correction for the supercell approach³⁵ was applied to eliminate an artificial electric field. The atomic structures were optimized so that the forces acting on atoms were less than 0.01 eV/Å. We utilized the following tools for our calculations: VESTA³⁶, Fermisurfer³⁷, and VASPKIT³⁸.

In this study, first, we targeted the Si(111)-($2 \times 2$) surface having motifs characteristic of the ($7 \times 7$) DAS structure, the adatom and the restatom to elucidate the relationship between the origin of the surface stabilization mechanism and surface dipole moment. Furthermore, we applied our method to the Si(111)-($7 \times 7$) DAS structure to compare with the SNDM observations. Figure S1 shows the ($2 \times 2$) model used for the calculations.To check the slab thickness dependence, we prepared the slab models with various layer thicknesses from 3 to 15 bi-layers (BLs). The optimized lattice constant for the bulk Si was 5.47 Å, 0.7% larger than the experimental value³⁹. The back side of the slab was terminated with hydrogen atoms to eliminate unsaturated bonds. The initial atomic positions were set at the ideal bulk ones: the adatom was initially arranged so that the distances between the nearest neighbor atoms were the same as in the bulk. Atomic positions in the lower side of the slab, from about half of the slab to the bottom, were fixed to the bulk position during the structural optimization process; e.g., three layers were fixed for the 6 BLs model, four layers for the 9 BLs model, and six layers for both 12 BLs and 15 BLs models.

We defined the in-plane surface dipole moment at the surface as the following formula:

$$\begin{aligned} \mu ' {(x,y)} = \int ^{z_\textrm{max}}_{z_\textrm{min}} \rho {(x,y,z')} \cdot (z' - z_{0}) dz' \end{aligned}$$

(1)

The right side is integrated from $z_\textrm{min}$ to $z_\textrm{max}$ along the z-axis, perpendicular to the surface. $\mu ^{'}(x,y)$ is the dipole moment at the position (x, y), $\rho {(x,y,z')}$ represents the total charge density at each mesh, and $z_{0}$ is the origin coordinate for the dipole moment. The dipole moment is induced not only at the surface but also at the hydrogen-terminated backside. However, the dipole moment at the backside is an artifact caused by the slab approximation and, thus, should be eliminated. In this study, $z_\textrm{min}$ was taken at the mid point of 3rd and 4th BLs from the top surface because the change in electron density near the surface extends from the surface to about 3 BL as shown in Fig. S2. The origin, $z_{0}$, is set as the weighted center of the ionic charge within the integration range. Note that even if the weighted center of the electronic charges was taken at the origin, the distribution in dipole moment hardly changed. The total charge density, $\rho {(x,y,z')}$, is derived from the electronic, negative, and ionic, positive charges. The electron density is obtained from the probability density of the wave function. On the other hand, the positive charges are represented as ionic shells with finite radii by the pseudopotential approximation. Therefore, in our calculations, we applied a Gaussian filter to the point charge of valence at each atomic site. In this study, Gaussian half-widths at half maximum of 0.582 Å and 1.01 Å were used for H and Si atoms, respectively.

Results and discussion

Figure S1 shows the initial and optimized structures for the ($2 \times 2$) surface. Table S1 lists the bond lengths near the surface for each model. No change in surface bond length is observed beyond 9 BLs, indicating that 9 BLs is necessary and sufficient for the structural optimization. Hereafter, we present the results for models with a layer thickness of 12 BLs for Si(111)-($2 \times 2$) and 9 BLs for Si(111)-($7 \times 7$) DAS, respectively.

Fig. 1

(Color Online) Band dispersions of (a) the initial and (b) the optimized structures for the ($2 \times 2$) surface. The Fermi energy and the highest energy of the valence band are set to 0.0 eV for the initial and the optimized structures, respectively. Probability densities for the optimized structure at the K-point of (c) the conduction band and (d) the valence band edges.

As depicted in Fig. S1, a characteristic feature of the ($2\times 2$) surface is that the adatom (restatom) is displaced towards the bulk (vacuum) side from its bulk position. Structural optimization results in a total energy reduction of 1.08 eV/($2 \times 2$). Given that the adatom and restatom are each accompanied by three nearest-neighbor Si atoms, one dangling bond remains per atom. We calculated the band dispersion before and after structural optimization to examine the electronic state of the surface. Figure 1a and b illustrate the band dispersions before and after optimization. When all Si atoms are positioned ideally in the bulk, two bands cross the Fermi energy in the bulk energy gap, indicating a metallic nature. Post-optimization, two states persist in the bulk band gap: one completely filled and the other entirely empty, signifying the elimination of two surface dangling bonds and the emergence of a semiconducting nature with a finite energy gap. These two bands are relatively flat, suggesting spacial localization. Figure 1c and d show the probability density of wave functions for the conduction and valence band edges at the K-point for the optimized structure, respectively. Clearly, the fully occupied valence and fully unoccupied conduction edges are associated with states localized at the restatom and adatom, respectively. Such disappearance of dangling bonds is attributed to surface atom displacement. Post-optimization, the backbond angle (see Fig. S1) of the restatom increase to 118.7 degrees, significantly larger than the bulk value of 109.5 degrees, altering the s-character of the surface dangling bond. It has been shown that the orbital electronegativity of a surface state increases with greater s-character^40,41,42. As a result, an electron from the dangling bond of the adatom transfers to the restatom, forming a lone pair at the restatom. To verify such a lone pair formation at the restatom, we checked the electron localization function (ELF), which measures electron pairing.

Fig. 2

(Color Online) Distribution of ELF for the optimized ($2\times 2$) surface. The ELF value is defined between 0 and 1. The closer the value is to 1, the more pairs of electrons are formed.

Figure 2 shows the distribution of ELF at the surface. It can be seen that the electron pair forms just above the restatom.

Figure 3a and b show the in-plane distribution of the dipole moment and its line profile in a cross-section through both the restatom and the adatom for the Si(111)-($2 \times 2$) surface.

Fig. 3

(Color Online) Normalized in-plane dipole moment distribution of (a) the Si(111)-($2\times 2$) and (b) the Si(111)-($7\times 7$) surfaces. Gray circles represent the locations of atoms of top 1BL. The corresponding normalized line profiles along the red dashed lines in (a, c) are shown in (b, d), respectively.

A pronounced, positive peak is observed at the adatom site, whereas the dipole moment at the restatom site is nearly zero. At the restatom site, the surface orbital forms a pair of electrons, namely a lone pair. This electron configuration at the restatom site is similar to that around Si atoms forming covalent bonds in the bulk, resulting in a zero dipole moment at the restatom site. At the adatom site, on the other hand, the surface orbital becomes empty due to the charge transfer associated with the structural shift from the bulk position, causing the weighted center of the electron distribution to move towards the bulk side. Consequently, a distinct positive dipole moment is induced at the adatom site.

Figure 3c and d show the dipole moment distribution and its line profile for the Si(111)-($7\times 7$) structure. Similar to the Si(111)-($2\times 2$) model, upward dipole moments are observed at the adatom site, while the dipole moment at the restatom sites is nearly zero, as also seen in the Si(111)-($2\times 2$) model. Similar results have been obtained for the Si(111)-($7\times 7$) DAS structure using the dipole moment mode of SNDM¹⁵. The agreement between our calculations and experimental results suggests that the charge transfer occurs between the adatom and restatom with dangling bonds formed in the ($7\times 7$) DAS structure.

Table 1 Summed dipole moment obtained by integrating values within each square pillar region.

In Table 2, we observe a noteworthy trend in the the back-bond angles of restatoms, which systematically decrease in the order ($2\times 2$) > ($7\times 7$) unfaulted half > ($7\times 7$) faulted half. To clarify the relationship between this trend and the local dipole moment, we estimated the dipole moment for each configuration. Table 1 shows the integrated dipole moment values within a circular region with a radius of 0.53 Å centered at each adatom. This integration radius corresponds to one-quarter of the in-plane horizontal distance to the nearest neighboring atom of the adatom.

Fig. 4

Optimized surface structures for (a) Si(111)-($2 \times 2$) and (b) Si(111)-($7\times 7$). For the ($7\times 7$) structure, the left and right halves correspond to unfaulted and faulted halves, respectively. The positions of typical adatoms and restatoms are indicated by arrows.

Reviewing Table 1, we observe the dipole moment magnitude of adatoms in the order ($2\times 2$) < ($7\times 7$) unfaulted half < ($7\times 7$) faulted half, inversely correlated with the previously noted back-bond angle trend of restatoms. Table 3 also highlights a tendency in adatom back-bond angles, where the order is ($2\times 2$) < ($7\times 7$) unfaulted half < ($7\times 7$) faulted half.

Table 2 Back bond angles of restatoms.

Table 3 Back bond angles of adatoms.

This correlation can be explained as follows: larger back-bond angles in adatoms lead to an increased upward dipole moment due to a larger s-character in the dangling bond, shifting the average position of electrons towards the bulk side. A larger s-character of the adatom reduces the outflow of electrons, resulting in smaller back-bond angles for restatoms in the faulted half. Consequently, charge transfer in the faulted half is relatively smaller compared to the unfaulted half. Furthermore, similar trends are observed when comparing corner and center adatoms within the faulted and unfaulted halves of the Si(111)-($7\times 7$) structure. Near the corner hole, the back-bond angles of adatoms increase, coinciding with a higher upward dipole moment. In other words, variations in back-bond angles reflect differences in local adatom-to-restatom charge transfer, which correspondingly influences changes in the dipole moment.

The differences in local dipoles between the ($7\times 7$) faulted and ($7\times 7$) unfaulted halves, as well as between corner and center adatoms, are consistent with previously observed STM mesurements^43,44,45,46. In negative sample bias measurements, the adatoms in the faulted half have been reported to appear brighter than those in the unfaulted half one. Furthermore, within faulted half, the corner adatoms have been observed to appear brighter than the center adatoms. These experimental results suggest that more electrons are left at adatoms in the faulted half than in the unfaulted half, and that more electrons are left at corner adatoms than at center adatoms in the faulted half. These observations align with the trend shown in the Table 3: an increase in the backbond angle of adatoms is associated with a more s-like orbital character, resulting in reduced charge transfer from adatoms to restatoms due to increased orbital electronegativity of the adatoms.

The systematic differences in the backbond angles of the Si(111)-($2\times 2$) and Si(111)-($7\times 7$) surfaces are also consistently explained in their different electronic states. On the Si(111)-($2\times 2$) surface, each unit cell contains one adatom and one restatom, resulting in complete charge transfer from the adatom to the restatom. In contrast, the Si(111)-($7\times 7$) DAS surface includes 12 adatoms, 6 restatoms, and 1 cornerhole per unit cell. This structural difference leads to incomplete charge transfer. In fact, it is well known that the Si(111)-($7\times 7$) DAS surface has finite density of states at the Fermi level^34,47. This fact agrees with the trends of the backbond angles: in ($2\times 2$), the restatom angle is larger, and the adatom angle is reduced compared to the Si(111)-($7\times 7$) DAS. This implies more significant charge transfer on the ($2\times 2$) surface.

Conclusions

In summary, we have simulated the in-plane distribution of the dipole moment for the Si(111)-($2 \times 2$) structure with the adatom and restatom and the Si(111)-($7\times 7$) DAS structure. An upward dipole moment is observed at the adatom sites, while the restatom sites exhibit almost zero dipole moment on both surfaces. This behavior of the dipole moment in each structural motif can be explained by the charge transfer associated with changes in surface structure. The increase in back bond angle enhances the orbital electronegativity of the restatoms, resulting in charge transfer from the adatom to the restatom. In the ($7 \times 7$) DAS structure, SNDM observations have shown local upward dipole moments at the adatom sites. Although the ($7\times 7$) surface is metallic rather than semiconducting, it has been suggested that surface charge transfer between adatoms and restatoms occurs similarly to that on the ($2\times 2$) surface. However, differences in the atomic arrangement of local region of adatoms such as faulted, unfaulted, or corner and center on the ($7\times 7$) surface produce differences in the aspect of dipole moment, and our calculation results can explain the previous STM observations consistently. Using our simulation method in combination with experimental methods such as SNDM will facilitate the identification of surface polarization structures at the atomic level. The surface dipole visualization technique we have developed will provide unique insights for the analysis and prediction of surface polarization states at the atomic level.