# Structure and Strain Field of Surface Dislocations on Gold Determined by Surface X-Ray Diffraction

Vincent Repain, Julien Durinck, Yves Garreau, Alessandro Coati, Yann Girard, Sylvie Rousset, Christophe Coupeau, Amandine Bellec

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Vincent Repain, Julien Durinck, Yves Garreau, Alessandro Coati, Yann Girard, et al., Structure and Strain Field of Surface Dislocations on Gold Determined by Surface X-Ray Diffraction. Physical Review Letters, 2025, 134 (23), pp.236201. 10.1103/PhysRevLett.134.236201. hal-05237549

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# Structure and strain field of surface dislocations on gold determined by surface x-ray diffraction

Vincent Repain,¹,* Julien Durinck,² Yves Garreau,³,¹ Alessandro Coati,³

Yann Girard,¹ Sylvie Rousset,¹ Christophe Coupeau,² and Amandine Bellec¹

¹Université Paris Cité, CNRS, Laboratoire Matériaux et Phénomènes Quantiques, 10, rue A. Domon et L. Duquet, 75013 Paris, France

²Institut P', UPR 3346 CNRS, Université de Poitiers-ENSMA-CNRS, Poitiers, France

³Synchrotron SOLEIL, L'Orme des Merisiers, Saint-Aubin, 91192 Gif sur Yvette, France

(Dated: May 7, 2025)

Au(111) with its famous surface reconstruction has led to many studies but its subsurface structure has never been determined. Here, we study the Au(677) surface which is a vicinal of Au(111) with steps and surface reconstruction, showing an array of surface dislocations. By using surface x-ray diffraction and atomistic simulations, we are able to quantitatively determine the complete structure and the strain fields associated with the surface reconstruction described in terms of surface dislocations, especially a stair-rod dislocation lying parallel to the surface step. The stair-rod dislocation strain field adds up to the one due to the step dipole, explaining the extraordinary regularity of the step array. We can also give new insights on the strain field associated to the Au(111) surface discommensuration lines (22×√3 reconstruction) and find that it penetrates deeply into the bulk, in contrast with commonly used surface reconstruction models.

Surface or interface dislocations are of prime importance to understand the growth and strain relief of crystalline layers [1–6]. The Au(111) surface is particularly famous, due to its natural herringbone reconstruction, giving rise to an array of surface dislocations which is prototypical for a number of other heteroepitaxial metallic systems [1, 5]. Despite several scanning tunneling microscopy (STM) studies [4, 6, 7], the exact structure and therefore the energetics of such dislocations are not very well-known, especially their associated subsurface strain fields. The measurement of strain fields is rather delicate and requires a very high precision on the atomic positions. Although STM can give some information, limited to the relative displacements of surface atoms [7–9], the best technique to measure surface and subsurface strain fields is surface x-ray diffraction [10–12]. A key prerequisite is that the surface structure should be periodic, the better the periodicity, the better the determination of the atomic positions. The Au(111) herringbone reconstruction does not show a long-range periodicity, which prevents a detailed surface diffraction study of its structure [13]. Instead, vicinal surfaces of Au(111) have a lower symmetry and can display periodic dislocations with a long-range order. An extensive measurement of the diffracted intensities along the reciprocal space allows us to determine by comparison with atomistic simulations the surface structure and subsurface strain fields of such dislocations. Taking into account those subsurface displacements is of prime importance to quantitatively determine the energetic of the surface structures, in contrast with what is generally believed.

Fig. 1a shows a sketch of the Au(677) surface with discommensuration lines (dotted lines) running perpendicu-

lar to the step edges. The sample surface was cleaned and measured under ultrahigh vacuum (10⁻¹⁰ mbar) on the SixS beamline of the synchrotron SOLEIL with 18.43 keV x-rays. Further experimental details and STM images of the surface are given in section I of the Supplemental Information (SI). Fig. 1b shows an in-plane map of the diffracted intensity with a grazing incidence geometry of the incident x-ray beam allowing to be sensitive to the sample surface and subsurface over a thickness around 15 layers. Fig. 1c is a zoom on the area delimited by the dotted rectangle. Both the periodicities of step edges and discommensuration lines lead to very well-defined diffraction rods. The measured periodicities from the positions of the diffracted spots are 3.26±0.04 nm per-

<figure><img src="image_2.png" /><figcaption>FIG. 1. a) Sketch of the Au(677) surface with $\vec{A}$ defining the period between step edges and $\vec{B}$ the period of the Au reconstruction (dotted lines represent the discommensuration lines). The corresponding reciprocal vectors $\vec{A}^*$ and $\vec{B}^*$ are shown in c). b) In-plane map integrated between $L = 2.25$ and $L = 3.5$. The color scale is representing the number of photons per pixel per 0.3 s integration time. c) Zoom around $H = -20$ and $K = 25$ (area delimited by the dotted rectangle in the large scale map).</figcaption></figure>

<footnote>* vincent.repain@u-paris.fr</footnote>
<figure><img src="image_3.png" /><figcaption>FIG. 2. Structure factor (square-root of the integrated intensity) as a function of L for different values of H and K. The data are displayed as black disk, the error bars are calculated as the standard deviation of measurements recorded for equivalent rods. The model based on the calculated atomic displacements is shown as a full red line. a) Crystal Truncation Rod $H = 0$, $K = 50$. Bragg at $L = 0$. b) Crystal Truncation Rod $H = -16$, $K = 0$. Bragg at $L = 26$. c) Crystal Truncation Rod $H = -28$, $K = 50$. Bragg at $L = 12$. d) Crystal Truncation Rod $H = -22$, $K = 25$. Bragg at $L = 19$. e) Fractional Rod $H = -16$, $K = 1$. f) Fractional Rod $H = -9$, $K = 26$. g) Fractional Rod $H = 19$, $K = 24$. h) Crystal Truncation Rod $H = 21$, $K = 25$. Bragg at $L = -9$.</figcaption></figure>

perpendicular to the steps and 7.34±0.07 nm along the steps, corresponding to 25.5±0.25 atomic units for the period of the discommensuration lines, slightly more than the 22 atomic units measured on Au(111). To index the spots, we have therefore chosen the following surface unit cell (cf. Fig. 1a), i.e. $\vec{A}=[\bar{7}33] (||\vec{A}||= 3.34$ nm), $\vec{B} = \frac{25}{2}[0\bar{1}1] (||\vec{B}||= 7.21$ nm) and $\vec{C} = \frac{1}{2}[6\bar{7}7] (||\vec{C}||= 2.36$ nm). Those three vectors are used to calculate the associated reciprocal space (cf. 1c) with indices ($HKL$). Bragg spot positions can be found (section II of SI for details) at $H = 14i-j+7k$, $K = 25k$ and $L = -6i+10j-3k$ with $i,j,k \in \mathbb{Z}$. The very long-range order of both steps and discommensuration lines allowed an impressive collect of diffracted data. In total, 319 rods have been processed, with 224 non equivalent, corresponding to more than 14000 independent experimental structure factors (cf. section III of SI for the full dataset), giving access to the refined surface and subsurface structure of this prototypical reconstruction.

The atomic displacements, both at the surface and in the subsurface, modifies the intensity variations along Crystal Truncation Rods (CTR, passing through bulk Bragg spots) and Fractional Rods (FR, not associated to a bulk periodicity). Fig. 2 shows eight typical rods (square-root of the integrated intensity in Log scale as a function of L). The first striking observation is that out of bulk Bragg spots, constructive or destructive interference patterns are the signature of surface induced displacements. The sharp variations with L (typically on a scale around 0.5 in Fig. 2a, e and g) indicate that those displacements occur on a depth of few tens of layers.

In order to figure out the arrangement of atoms at the reconstructed and stepped Au(111) surface, these data were compared to configurations obtained by atomic simulations using an interatomic Modified Embedded Atom Method (MEAM) potential [14, 15]. This potential was modified to correct the underestimated tensile surface stress so that it better matches the value obtained by ab initio calculations and makes the Au(111) reconstruction energetically favorable [16]. It was also used to model the interaction of the surface reconstruction with the step edges, in good agreement with experiments [17]. It is worth noting that machine learning methods have also allowed recently to better describe the reconstruction of metal surfaces [18–20]. The Au(677) surface can be conceptualized as the Au(111) surface that would have been sheared by perfect dislocations gliding in regularly spaced slip planes, thereby leaving a periodic array of atomic steps. As detailed in the section IV of SI, a reconstructed Au(677) configuration with a thickness of 67 atomic layers and a surface reconstruction with the same periodicity as that observed in the experiments has been generated.

The atomic positions of this Au(677) reconstructed surface, described in details in the following, have been used to calculate structure factors that are plotted as full red lines in Fig. 2. As can be seen in the eight rods and more generally in all the measured ones (cf. section III of SI), the agreement between the data and the model is excellent ($\chi^2=2.0$) with a single fitting parameter which is a scaling factor. In particular, all the sharp interferences related to displacements deep in the bulk are very
<figure><img src="image_4.png" /><figcaption>FIG. 3. (a) 3D view of the relaxed configuration 1 with a reconstructed Au(677) surface. The Burgers vectors of both subsurface Shockley dislocations $A\delta$ and $\delta C$ are represented by arrows. (b) Top view of the configuration for which surface atoms of the upper terrace ($p_1$ plane) have been removed (see atoms in red at the bottom of the figure). Dislocation lines are represented by thick solid lines and lie between the first and the second atomic planes from the surface. A Burgers circuit is drawn with dashed lines oriented along (110) directions. The lengths of the circuit edges are given in lattice units. For (a) and (b), the atoms are colored according to the number of their atom neighbors in a cutoff radius of 3.65 Å (12 neighbors for red, 11 for light red, 10 for white, 9 for light blue and 8 or 7 for darker blue).</figcaption></figure>

Burgers dislocations are given using the Thompson's tetrahedron notation (see inset figures).

well reproduced. As discussed in the section VII of SI, the experimental structure factors cannot be reproduced with a non reconstructed surface, i.e. atomic displacements only due to step edges. The surface structure and its associated strain field described in the following can therefore be considered as a reliable representation of the experimental atomic positions.

The relaxed surface configuration is shown in figure 3a, with three terraces on which the atoms of the $p_{-1}$, $p_0$ and $p_1$ (111) planes are visible. Atoms are colored according to the number of their neighbors, illustrating the positions of step edges (dark blue atoms) and the commensuration lines (white atoms). These discommensuration lines can also be described as subsurface dislocations with Burgers vectors $A\delta$ and $\delta C$ (according to the Thompson's tetrahedron notation), separated by an hexagonal close packed (hcp) stacking fault between the first $p_i$ and the second $p_{i-1}$ atomic planes from the surface [16]. The sole distinction between the reconstructed

Au(677) surface and the aforementioned Au(111) $22\times\sqrt{3}$ surface is the presence of a stacking fault pinch on each terrace near the step with the upper one. This particular configuration of the surface reconstruction has also been observed in the STM image of Fig. S1 and predicted by Density Functional Theory (DFT) calculations [21]. Figure 3b, which shows a top view of the surface following the removal of atoms from the $p_1$ plane at the upper terrace, thus uncovering atoms in the $p_0$ plane, provides further details regarding the dislocation structure in the vicinity of and below the step. It can be observed that the $p_0$ plane is divided into two distinct parts. One part corresponds to the reconstructed terrace (in blue), comprising 26 atoms along the step direction. The other part comprises atoms (in red) in their bulk configuration, with 25 atoms along the step direction. As illustrated by the Burgers vector circuit shown in figure 3b, the discrepancy between the two parts is accommodated by an AC dislocation with a line of one atomic plane length oriented along $z$.

In other words, the reconstruction of the Au(677) surface can be conceptualized as that of an Au(111) surface that has been sheared by dislocations, thereby generating a periodic array of terraces and steps, as previously mentioned. In this perspective, it comes that the hcp stacking fault is no longer continuous for the Au(677) surface and must close on itself, forcing the two partial dislocations that delimit it to recombine according to the reaction $A\delta + \delta C \rightarrow AC$ just below the step. This threading AC dislocation, whose line extends vertically from below the $p_0$ plane to the surface, is similar to the one at the elbows of the well-known herringbone reconstruction. The only difference is that it dissociates into two other partial dislocations. The characterization of these dislocations can be performed through the calculation of the disregistry vector and the distribution of the Burgers vector (refer to Section V of SI for further details). It is thus shown that the AC dislocation dissociates following the reaction $AC \rightarrow A\beta + \beta C$, leaving beneath the surface a stair-rod dislocation $\delta\beta$ whose line is parallel to the direction of the step. The attractive tension line force of this dislocation is responsible for the observed constriction near the surface step, as has been previously described theoretically for extended unit jogs in bulk face-centered cubic (fcc) crystals [22]. This dislocation configuration is depicted in the Figure 3b, with dislocations $A\delta$, $\delta C$, and $\delta\beta$ lying between planes $p_0$ and $p_{-1}$ at the middle terrace, while dislocations $A\beta$ and $\beta C$ emerge at the surface through the plane $p_0$.

In the following, a particular emphasis is placed on the displacement field associated with the surface structure. The displacement vector $\vec{u}$ representing the displacement of atoms from their initial to their relaxed positions has been calculated for the reconstructed Au(677) configuration in a $(x, z)$ plane located halfway between the two dislocations $A\delta$ and $\delta C$, at the center of the hcp zone (Fig. 4a). Additionally, configuration with a non-reconstructed Au(677) surface have been relaxed, and the
resulting displacement fields are presented in Fig. 4b. The step edge can be viewed as a force dipole in the $(x, z)$ plane, which generates a displacement field with a typical vortex as observed for the non-reconstructed Au(677) configuration in Fig. 4b. The reconstructed Au(677) configuration exhibits a comparable displacement field, including a characteristic vortex, but with displacements $u_z$ of greater magnitude than those of the non-reconstructed Au(677) configuration, even at distances far from the surface. As discussed in the section VI of SI, this extra strain field and the increase of displacement magnitudes in the vortex cannot arise from the discommensuration lines which generate a strain field independent of $x$ (cf. Fig. S8), but is more due to the $\delta\beta$ stair-rod dislocation, previously identified, situated just below the step edge. This large displacement field gives rise to an extraordinary strong step-step interaction, which explains the very narrow terrace width distribution on such stepped reconstructed surfaces [6].

We have also taken advantage of the information about this strain field in three dimensions to look at cross section in $(y, z)$ plane. Fig. 5a shows a cut in the middle of the terrace, in order to minimize the displacements due to the stair-rod dislocation and mainly focus on the strain field generated by the discommensuration lines. For sake of comparison, Fig. 5b shows the same cut on a reconstructed Au(111) surface. The difference between both is mainly due to the out-of-plane displacements generated by the step dipole (cf. Fig. S8). This first determination of the subsurface strain field of the well-known $22 \times \sqrt{3}$ reconstruction sheds a new light on the understanding of this structure. Indeed, many theoretical models have

<figure><img src="image_5.png" /><figcaption>FIG. 4. Atomic relaxations with respect to the bulk positions in a $(x, z)$ plane located at the center of the hcp region (see inset) for (a) the reconstructed Au(677) (fit of Fig. 2), (b) the non reconstructed Au(677) (calculated). Only the first 34 atomic (111) planes are represented. The vectors magnitude are increased by a factor 50.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>FIG. 5. Atomic relaxations with respect to the bulk positions in a $(y, z)$ plane located halfway of two successive surface steps (see insets of the figures) for (a) the reconstructed Au(677) (fit of Fig. 2), (c) the reconstructed Au(111) (calculated with the same potential parameters used for (a)). Only the first 34 atomic (111) planes are represented. The vectors magnitude are increased by a factor 50.</figcaption></figure>

perfectly described this reconstruction as surface solitons arising from adapted Frenkel-Kontorova model for surface adsorbates [23]. In such models or in most theoretical calculations on surface structures by DFT [24], and more generally in the textbook understanding of surface reconstructions, the subsurface is rigid, or with small out-of-plane relaxations in the first subsurface layers. In the framework of the Frenkel-Kontorova model and in conjunction with DFT calculations, Bartelt and Thürner [7] confirmed that the rigid substrate hypothesis overestimated the elbows' energy and, consequently, the periodicity of the herringbone structure. They demonstrated that this could be partially corrected by allowing at least relaxation of the second atomic layer below the surface. In light of these findings, it is shown in Fig. 5 that discommensuration lines generate a strong in-plane strain field with a vortex structure located around the third subsurface layer, and with significant displacements at least on the first ten layers. This pattern of strain field demonstrates that the discommensuration lines of the $22 \times \sqrt{3}$ reconstruction (corresponding to $A\delta$ and $\delta C$ subsurface dislocations) can be viewed as extended force dipoles, generating such vortices and long-range strain field in self-organized surfaces [25]. As discussed quantitatively in the section VI-B of SI, taking into account those subsurface displacements is very important to calculate correctly the energy of the surface reconstruction. Indeed, a substantial error around 2 eV per slab is done when only the surface atoms are allowed to relax. Truncating the displacement field at the 10th atomic plane results still in an error around 0.1 eV, while at the 35th atomic plane, it reduces to 1 meV.
In conclusion, the strain field generated by the gold re-
construction on a stepped surface has been quantified for
the first time. A high degree of correlation has been ob-
served between the measured and simulated atomic-scale
models, which enabled the determination of the subsur-
face dislocation arrangement that describes the surface
reconstruction. These dislocations, in conjunction with
the step edges, generate a strain field that extends far
from the surface, at distances deeper than the second
atomic layer below the surface, as it is often considered
in more demanding DFT calculations. In particular, a
stair-rod dislocation located just below the step edge in
the hcp region has been identified as the result of the con-
comitant presence of both the discommensuration lines of
the surface reconstruction and the step edge. This dislo-
cation has been found to make a significant contribution
to the whole displacement field in the bulk, explaining
the regularity of the step array. Moreover, we have mea-
sured significant deep subsurface in-plane displacements
generated by the famous discommensuration lines. These
findings show that the Au(111) surface, and in general
any surface discommensurations arising from heteroepi-
taxy, have not to be treated as a single layer over a rigid
substrate, as it has already been considered in standard

Frenkel-Kontorova models [7, 18]. Instead, the elastic
perturbation over tens of atomic layers towards the bulk
has to be taken into account, which is quite challenging
for DFT calculations. The semi-empirical potential used
in this study has been demonstrated to reproduce the
experimental determination of the structure with an ex-
treme degree of accuracy, thereby instilling confidence in
its ability to describe any Au(111) surface configuration,
including those with or without steps, and to propose
valuable relaxed slabs for further electronic structure cal-
culations.

ACKNOWLEDGMENTS

ANR (Agence Nationale de la Recherche) and CGI
(Commissariat Général à l'Investissement) are grate-
fully acknowledged for their financial support of this
work through LabEx SEAM (Science and Engineering for
Advanced Materials and devices), ANR-10-LABX-0096
and ANR-18-IDEX-0001 and LabEx Interactifs (ANR-
11-LABX-0017-01) of Poitiers University. We acknowl-
edge Benjamin Voisin for technical assistance on the SiXS
beamline. We thank Geoffroy Prévot for providing the
binoviewer software for data analysis.

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