Progress in surface X-ray crystallography and the phase problem

There exist various kinds of techniques to investigate surface structures. Electron diffraction, SXRD, and X-ray standing wave are mainly used to obtain the averaged structure in a long-range order. On the other hand, photoelectron diffraction, ion scattering, surface sensitive X-ray absorption fine structures, scanning probe microscopy, and electron microscopy are used to obtain the local structures.

Among these methods, diffraction is the most reliable method to determine the atomic structure as proven by crystal structure analysis. Indeed, LEED, which was observed in the first experiment by Davisson and Germer¹⁴⁾ on the wave property of electrons in 1927, is still a leading method for structure determination. An astounding increase in the amount of experimental LEED work since the 1960s was sparked in particular by the renewed efforts of Germer et al.¹⁵⁾ and the contributions of Lander et al.¹⁶⁾ on technical advances, i.e. the post-acceleration display LEED system, which allows the entire spot diffraction pattern to be viewed on a fluorescent screen.¹⁷⁾

SXRD became a major tool in the 1980s when brilliant SR sources appeared because of the very small number of atoms that contribute to the scattering compared to that in bulk, in addition to the intrinsic feature of X-rays that the interaction with matter is weak. In 1979, Marra et al.¹⁸⁾ performed X-ray diffraction experiments on an Al/GaAs interface in the so-called grazing incidence geometry, in which the grazing angles of incident and exit X-rays are kept small, and demonstrated that this geometry is sufficiently surface sensitive. This is because the penetration depth becomes a few nanometers, resulting in suppression of the scatterings from the inner crystal if the incident angles become close to the critical angle of X-ray total reflection. Eisenberger and Marra¹⁹⁾ performed SXRD experiments in 1981 by using ultra-high vacuum (UHV) instruments and an SR source on Ge(001)-2 × 1. Robinson²⁰⁾ reported in 1986 on the crystal truncation rod (CTR) scatterings emanating from the truncation of crystal periodicity, which has a continuous X-ray diffraction intensity distribution perpendicular to the surface. These two diffraction and scattering experiments are the basics of SXRD.

If we cut a crystal parallel to a net plane, the ideal surface that keeps the atomic arrangements the same as that in the crystal is generally unstable. Surface reconstructions or lattice relaxations occur and the total energy decreases. Surface reconstruction also occurs by the adsorption of additional atoms. As a result, in many cases, the surface atoms form a large in-plane periodic structure with a lower in-plane symmetry than that in the crystal. The periodicity along the inward direction of the crystal is lost in several atomic layers.

For the crystal with the surface and the surface superstructure, the diffraction conditions are described as similar to those of a two-dimensional lattice. Let us introduce scattering vector ${\boldsymbol{K}}={{\boldsymbol{k}}}_{f}-{{\boldsymbol{k}}}_{i}$ and reciprocal lattice vector ${{\boldsymbol{G}}}_{{hk}}(l)=h{{\boldsymbol{b}}}_{1}+k{{\boldsymbol{b}}}_{2}+l{{\boldsymbol{b}}}_{3}$ using 1 × 1 fundamental lattice vectors ${{\boldsymbol{b}}}_{1},{{\boldsymbol{b}}}_{2}$, and ${{\boldsymbol{b}}}_{3}$, such that

$$\begin{eqnarray}&&{\boldsymbol{K}}={{\boldsymbol{G}}}_{{hk}}(l).\end{eqnarray} \tag{ 1 }$$

Here, h and k take integer or fractional values, l is continuous, and ${{\boldsymbol{k}}}_{i}$ and ${{\boldsymbol{k}}}_{f}$ denote the incident and exit wave-number vectors, respectively.

In Ewald construction, diffraction conditions are satisfied where reciprocal lattice rods perpendicular to the surface intersect the Ewald sphere in the reciprocal space. Figure 2 shows several diffraction geometries in Ewald construction. When both h and k take integer values, this reciprocal lattice rod is called the CTR, the scattering intensity of which contains contributions from the bulk crystal and surface layers. In contrast, the rod with either h or k as a fractional value is called fractional order rod (FOR).

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The crystal structure factor, $F({\boldsymbol{K}})$, is given by the Fourier transform of charge density $\rho ({\boldsymbol{r}})$ in a unit cell.

$$\begin{eqnarray}&&F({\boldsymbol{K}})={\int }_{{\rm{u}}.{\rm{c}}.}\rho ({\boldsymbol{r}}){{e}}^{2\pi i{\boldsymbol{K}}\cdot {\boldsymbol{r}}}\,d{\boldsymbol{r}},\end{eqnarray} \tag{ 2 }$$

where, ${\boldsymbol{r}}$ represents a position vector in the unit cell.

CTR scattering obtains contributions from both surface layers and inner crystal, whose periodicity perpendicular to a surface is truncated. By using the structure factors of a bulk crystal and surface layers, F_B and F_S, respectively, the absolute reflectivity along CTR is expressed in kinematical approximation in the general form as

$$\begin{eqnarray}\begin{array}{rcl}{I}_{\mathrm{CTR}}(l) & = & g{\left|\sum _{n=-\infty }^{0}{F}_{{hk}}^{{\rm{B}}}(l){{e}}^{i2\pi {nl}}+{F}_{{hk}}^{{\rm{S}}}(l)\right|}^{2},\\ & = & g{\left|\displaystyle \frac{{F}_{{hk}}^{{\rm{B}}}(l)}{1-{{e}}^{-i2\pi l}}+{F}_{{hk}}^{{\rm{S}}}(l)\right|}^{2},\end{array}\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&g={{\Pr }}_{e}^{2}({\lambda }^{2}/{A}_{{\rm{u}}}^{2}){\eta }_{\mathrm{geo}}.\end{eqnarray} \tag{ 4 }$$

Summation index n shows the depth from the surface as measured in a unit cell of the bulk. P, r_e, λ, A_u, and η_geo, respectively represent the polarization factor, classical electron radius, wavelength, area of a unit-cell, and a geometrical factor. The geometrical factor includes the Lorentz factor, area correction, beam correction in a stationary mode, and the intersection correction of the rod in the integration mode.²¹⁾

The contribution from the crystal is explained by the semi-infinite summation of scatterings from the unit cells toward the inside of the crystal. The Bragg conditions are satisfied when l is an integer and the right side of Eq. (3) diverges in the kinematical approximation. The CTR scattering profiles become sensitive to the surface structure, as shown in Fig. 3, as the contributions to CTR scatterings from the crystal and surface become comparable when moving away from the Bragg point. The relative positions of the surface atoms to the crystal atoms, especially the z positions of surface atoms perpendicular to the surface, can be obtained using these characteristics of CTR scatterings.

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FOR originates from a surface superlattice with a large unit cell having a different symmetry from that of the crystal due to surface reconstruction. For FOR scatterings, which contain only contributions from the reconstructed surface layers, the diffraction intensities become

$$\begin{eqnarray}&&{I}_{\mathrm{FOR}}(l)=g{\left|{F}_{{hk}}^{{\rm{S}}}(l)\right|}^{2}.\end{eqnarray} \tag{ 5 }$$

Intensity profile I_FOR along the FORs is only affected by the surface layers. Since I_FOR(l) weakens along the rod owing to the thermal vibration effect, the diffraction intensities at the bottom of the rod (around l = 0.1) are mainly collected.

Therefore, as a scattering vector is directed almost parallel to the surface, the scattering intensity provides information on the in-plane surface structure. To realize this measurement, the so-called grazing incidence X-ray diffraction (GIXD) is employed, in which the grazing angles of both incident and exit X-rays are small, as shown in Fig. 2(b). Many surface and interface studies have reportedly used GIXD, for example, we can find extensive studies on oxide surface.²²⁾ Here, we show the in-plane diffraction intensities from Si(111)-7 × 7 in GIXD geometry in Fig. 4. In-plane diffraction is sensitive to the projected surface structure because of the small l component in the scattering vector. Furthermore, GIXD cannot determine the positional relation between surface and bulk atoms. This requires an analysis of CTR scatterings, reflecting interference of diffraction from the surface and bulk atoms.

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In the kinematical approximation, CTR scattering intensities have no physical meaning at the Bragg points, including the total external reflection region, they diverge. Instead, we must consider the dynamical diffraction theory treating multiple scattering between atomic layers. Reflection coefficient R_B of an ideally terminated single crystal surface derived from Darwin's dynamical theory is given as follows:²³⁾

$$\begin{eqnarray}\begin{array}{rcl}{R}_{{\rm{B}}} & = & -| b{| }^{1/2}\displaystyle \frac{C}{| C| }\displaystyle \frac{{F}_{{\rm{B}}}({\boldsymbol{K}})}{{\left({F}_{{\rm{B}}}({\boldsymbol{K}}){F}_{{\rm{B}}}(\bar{{\boldsymbol{K}}}){e}^{-i2\pi l}\right)}^{1/2}}\\ & & \,\,\,\,\,\,\,\,\,\,\cdot \,\left[\eta \pm {\left({\eta }^{2}-1\right)}^{1/2}\right],\end{array}\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&\eta =\displaystyle \frac{i(\lambda /2\pi d){\gamma }_{0}({e}^{i2\pi l}-1)+\tfrac{1}{2}{\Gamma }{F}_{{\rm{B}}}({\boldsymbol{O}})(1-b)}{| C| | b{| }^{1/2}{\Gamma }{\left({F}_{{\rm{B}}}({\boldsymbol{K}}){F}_{{\rm{B}}}(\bar{{\boldsymbol{K}}}){e}^{i2\pi l}\right)}^{1/2}},\end{eqnarray} \tag{ 7 }$$

where b and d represent the asymmetry factor and d spacing between atomic layers, respectively. Polarization factor C in the dynamical theory is defined as the unity for σ polarization and the cosine of the scattering angle for π polarization, ${\gamma }_{0}=\cos {\theta }_{0}$, where θ₀ is the angle of incidence on a surface, ${\Gamma }={r}_{{\rm{e}}}{\lambda }^{2}/\pi V$, where V represents volume of the unit cell, and η is called the deviation parameter.

Figure 5 shows reflectivities along the CTR (00 rod) near the total reflection and the Bragg point calculated by both dynamical and kinematical theories. In the kinematical approach, the diffraction intensity diverges, whereas the dynamical approach correctly reproduces the finite reflectivity profile, showing the so called silk-hat Darwin curve. At a far tail of the Bragg point, R_B is reduced to the bulk term, ${g}^{1/2}\tfrac{{F}_{{\rm{B}}}}{1-{{e}}^{-i2\pi l}}$ in Eq. (3), which is the same as the result obtained from the kinematical scattering approach shown in Fig. 5. Therefore, except for the diffraction conditions very close to the Bragg points, the kinematical theory is valid for interpretation and analysis of CTR scatterings. For simplicity, although we show here the ideal case in which only the bulk part contributes, the same approach can be applied to situations assumed in deriving Eq. (3) where the surface layers have different structures than that of the bulk crystal by surface reconstructions and/or surface roughness.

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By considering the general formula of the scattering amplitude from an atomic plane²⁴⁾ and multiple scattering within the same atomic plane,²⁵⁾ Eqs. (6) and (7) can be applied to the entire reciprocal space. Based on the dynamical theory, we can execute absolute X-ray reflectivity measurement,²⁶⁾ which is a powerful tool to decide coverages on surfaces.

In SXRD, one can infer an adequate structural model, which reproduces the observed CTR scattering and in-plane diffraction intensities, by using the Patterson map^6,34) or trial and error methods with information obtained from such as scanning tunneling microscopy. Next, the atomic arrangement is refined by the least squares method.

Patterson function $P({\boldsymbol{r}})$ is defined as the Fourier transform of the square of structure factor $| F({\boldsymbol{K}}){| }^{2}$ and is equal to the auto-correlation function of charge density $\rho ({\boldsymbol{r}})$ in the crystal.³⁵⁾ The peaks of the Patterson function give the interatomic vectors in the crystal atoms. The Patterson function can be calculated from the observed data in a straightforward manner.

$$\begin{eqnarray}\begin{array}{rcl}P({\boldsymbol{r}}) & = & \,{ \mathcal F }\left[| {F}({\boldsymbol{K}}){| }^{2}\right]\,\propto \,\rho ({\boldsymbol{r}})\ast \rho (-{\boldsymbol{r}}),\\ & = & {\int }_{{\rm{u}}.{\rm{c}}.}\rho ({{\boldsymbol{r}}}^{{\prime} })\rho ({\boldsymbol{r}}+{{\boldsymbol{r}}}^{{\prime} })d{{\boldsymbol{r}}}^{{\prime} }.\end{array}\end{eqnarray} \tag{ 8 }$$

Here, ∗ represents convolution, and integration is performed over a unit cell. In GIXD, there is a lack of integer rod intensities and calculated Patterson map has both positive and negative regions, contrary to the ideal Patterson map. However, it is possible to interpret that the positive peaks represent the interatomic vectors.^6,7)

The Patterson function for surfaces is a partial Patterson function calculated using only fractional-order reflections. As is obvious in partial Patterson analysis^36–38) performed in the early surface X-ray structure analyses, the direct interpretation that relates peak positions to interatomic vectors becomes invalid when the structure is complex. For n atoms in a unit cell, n(n − 1) peaks should be found in the partial Patterson map. However, an increase in n results in the superpositions of peaks, peak shifts, and even loss of peaks because of the existence of a negative region. Therefore, its interpretation becomes rather trial-and-error and depends on a structural model. Finally, finding a plausible structural model is the most important issue in structure analysis.

There are two methodologies for overcoming aforementioned difficulties, namely the crystallographic direct methods^11,12) and holographic methods using contribution to CTR scatterings from the crystal as the known structure. Crystallographic direct methods introduce a plausible structure model by combining the nonnegativity and atomicity of the charge density in a crystal with information contained in the observed diffraction intensities. We describe these methods for surfaces in detail later.

In conventional SXRD, each diffraction intensity is collected by rocking a sample as shown in Fig. 4 because the number of atoms that contribute to the X-ray scatterings on the surface is predominantly smaller than that in the bulk crystal (by an order of 10⁻⁴ to 10⁻⁵), and it is necessary to discriminate between a weak signal from the surface atoms and the background scattering from the bulk crystal. The quantitativeness of the rocking measurement has been well investigated.²¹⁾ Thus, the reliability of surface-structure determination by SXRD has been established.

In contrast to the bulk reflection, the diffraction conditions are always satisfied in SXRD, as shown in Fig. 2. In the rocking mode, the slits are located in front of the detector, and the integrated intensity from a certain part of the reciprocal lattice rod is obtained by rocking a sample. The amplitudes of the structure factors are derived from the diffraction intensities in the rocking mode by applying the following correction factors considering an instrumental resolution.

Integrated intensity ${I}_{\omega }^{{\rm{int}}}({\boldsymbol{K}})$ at ${\boldsymbol{K}}$ in the reciprocal space obtained by rocking ω circle is related to structure factors $F({\boldsymbol{K}})$ as follows:²¹⁾

$$\begin{eqnarray}&&{I}_{\omega }^{{\rm{int}}}({\boldsymbol{K}})=({{\rm{\Phi }}}_{0}{r}_{e}^{2}A{\lambda }^{2}{\Delta }\gamma /{\omega }_{0}{A}_{u}^{2})| F({\boldsymbol{K}}){| }^{2}{C}_{{\rm{tot}}},\end{eqnarray} \tag{ 9 }$$

with

$$\begin{eqnarray}&&{C}_{{\rm{tot}}}={{PL}}_{\omega }{C}_{{\rm{rod}}}{C}_{{\rm{area}}}{C}_{{\rm{\det }}}{C}_{{\rm{beam}}},\end{eqnarray} \tag{ 10 }$$

where, Φ₀, A, Δγ, ω₀, and C_tot represent the incident photon flux per second per unit area, sample area determined by the incident and exit slits, acceptance of a detector in the γ direction, rotational speed of the ω axis, and total correction factor, respectively. Correction factors ${L}_{\omega },{C}_{{\rm{rod}}},{C}_{{\rm{area}}},{C}_{{\rm{\det }}}$, and ${C}_{{\rm{beam}}}$ represent the Lorentz factor, rod interception, active sample area, detector acceptance, and beam footprint factors, respectively. Rocking curve measurements ensure quantitativeness in both in-plane diffraction and CTR scattering analyses.

The systematic error is estimated by measuring the equivalent reflections in a crystallographic symmetry. Especially, the grazing incidence geometry limits the accuracy of the in-plane diffraction measurement compared to out-of-plane geometry. This is because there exist systematic errors that cannot be corrected by Eq. (9).

X-rays show the total external reflection with an incident angle lower than the critical angle, e.g. 0.1° with 0.1 nm X-rays. In GIXD, where an incident and outgoing angles are near the critical angle, the diffraction intensities are proportional to transmissivity $| { \mathcal T }({\alpha }_{{i}}){| }^{2}| { \mathcal T }({\alpha }_{{f}}){| }^{2}$ owing to a dynamical effect. Here, α_i and α_f represent the incident and exit angles, respectively, and ${ \mathcal T }(\alpha )=2\alpha /(\alpha -\alpha ^{\prime} )$, where $\alpha {{\prime} }^{2}={\alpha }^{2}-{\alpha }_{c}^{2}$ and α_c is the critical angle. Around the critical angle, the surface diffraction is enhanced, and the scattering from a bulk crystal is suppressed because of the small penetration depth. This surface-sensitive feature was actively applied in the early experiments.^18,19) However, this sensitivity strongly depends on the alignment of a surface normal and may result in inevitable systematic errors. To avoid this problem, in-plane diffraction was measured by keeping the incident angle 3–5 times larger than the critical angle.

In addition, the scattering vector in GIXD intrinsically has small but non-negligible l component, which also results in systematic errors, as the projected structure normal to the surface is discussed in GIXD.

The so-called stationary mode, without rocking the sample, realizes higher accuracy than the rocking mode while observing reciprocal rods with large l components in the geometry of a large incident and/or exit angle.^21,49,50) The stationary mode can reduce measurement time because the diffraction conditions are always satisfied as similar to those of LEED and RHEED.

Since Takahashi et al.⁵¹⁾ applied the stationary mode in the analysis of Si(111)-$\sqrt{3}\times \sqrt{3}$-Ag by using a position-sensitive proportional counter, this measurement mode was applied in various sample environments and detector setups. For example, time-effective measurements of SrTiO₃ surface in a stationary geometry by using a novel two-dimensional detector, i.e. pixel array detector realizing single photon counting,⁵²⁾ were performed at the Swiss Light Source. The observations along a total of 27 CTRs including over 1800 measurement points took only approximately 10 h.⁵³⁾

Simultaneous observations of many diffraction spots from surfaces and interfaces by using a two-dimensional detector were also reported.^54–57) Recently, Gustafson et al.⁵⁸⁾ showed time-effective rod data collection in reflection geometry by using high-energy X-rays, clearly showing CTR and FOR intensity distributions originating from the surface in a reciprocal space, as shown in Fig. 8, unlike the conventional technique that shows spot-like reflections from CTR and FOR.

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Regarding the data accuracy, the minimum measurement error reported in the rocking mode was approximately 6% for in-plane diffraction with grazing incident geometry.⁵⁹⁾ In the stationary mode, the minimum error reported is approximately 5% by careful correction of the incident beam flux and effective illuminated sample area.⁵³⁾

To overcome the intrinsic limitations of acquiring precise data and high time consumption in conventional SXRD, we proposed TXD for surfaces⁶⁰⁾ by revisiting the pioneering work of surface TXD in energy dispersive type by Takahashi et al.⁶¹⁾ Recently, TXD was applied to electrochemical liquid–solid interfaces by Reikowski et al.⁶²⁾ In surface TXD, the diffraction conditions are always satisfied, as shown in Fig. 2(c).

As long as we use thick substrate crystals, thermal diffuse scattering (TDS) from a bulk crystal is dominant, as shown in Fig. 9(a). The TDS is the X-ray scattering originating from thermally populated phonons,^63,64) which distributes over the entire reciprocal space; bright spots close to the reciprocal lattice points and a lattice-like distribution are mainly visible. The former appear where the acoustic phonon population is high. By using a thinner substrate (∼5 μm thick), as shown in Fig. 9(b), clear TXD patterns are observed, as shown in Fig. 9(c) for the Si(111)-$\sqrt{3}\times \sqrt{3}$-Ag. The TDS is suppressed and the fractional order reflections originating from Si(111)-$\sqrt{3}\times \sqrt{3}$-Ag around the CTR spots can be seen in a hexagonal manner. For another example, approximately 40 fractional order spots from Si(111)-7 × 7 were collected simultaneously by an image plate. This simultaneous data collection using an area detector accelerates the time-effective measurement.

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In addition, the entire X-ray beam illuminates a selected area on a sample surface, unlike the grazing incidence geometry with the inevitable systematic errors as already mentioned, in which only a part of the incident beam illuminates the sample. Therefore, a focused beam can be easily applied without loosing data accuracy. Besides, in surface TXD, the specific points of just l = 0 in the reciprocal space can be accessed by adjusting the incident angle, as shown in Fig. 2(c), e.g. the red Ewald sphere and vectors meet this condition, resulting in pure in-plane diffraction without the l component. These features reduce systematic errors.

Charge density of crystal $\rho ({\boldsymbol{r}})$ is expanded by the Fourier series of complex structure factors $F({\boldsymbol{K}})=| F({\boldsymbol{K}})| {{e}}^{i\phi ({\boldsymbol{K}})}.$

$$\begin{eqnarray}&&\rho ({\boldsymbol{r}})=\displaystyle \frac{1}{V}\sum _{{\boldsymbol{K}}}| F({\boldsymbol{K}})| {{e}}^{i\phi ({\boldsymbol{K}})}\,{{e}}^{-2\pi i{\boldsymbol{K}}\cdot {\boldsymbol{r}}}.\end{eqnarray} \tag{ 11 }$$

The phase problem, which is the loss of phases of diffracted X-rays in measurement (loss of $\phi ({\boldsymbol{K}})$), has been the most attractive subject in X-ray crystallography for over 50 years. Traditional direct methods for solving crystal structures, which have been successful in bulk crystals, do not work well in SXRD because of matters inherent in SXRD.

In three-dimensional (3D) crystals, powerful solutions known as traditional direct methods have been introduced to solve the phase problem.^11,12) One of the most famous relationship between structure factors of periodic objects utilized in the methods is Sayre's equation.⁶⁷⁾ This equation is derived from the atomicity of crystalline objects, i.e. positivity and localization of charge density distribution. However, in protein crystallography, this atomicity property weakens because the non-zero charge density region widens, and the low-resolution diffraction data makes this preferable property less workable. These features limit the applicable area of the traditional direct methods even in 3D crystals. This situation is similar in SXRD, and there are more inherit limitations in intensity-data acquisition in SXRD.

Several direct methods and holographic approaches for surface include an iterative approach. Here, we introduce the iterative approach briefly. In the traditional direct methods, the phases of observed reflections are inferred from the relations between structure factors in reciprocal lattice space. In contrast, an iterative approach obtains the charge density with the estimation of $\phi ({\boldsymbol{K}})$ in Eq. (11) by executing operations to the charge density iteratively. For example, we can introduce the following relation to charge densities ρ^k and ${\rho }^{k+1}$ after k and k + 1 iterations, respectively.

$$\begin{eqnarray}&&{\rho }^{k+1}={{ \mathcal P }}_{4}{{ \mathcal P }}_{3}{{ \mathcal P }}_{2}{{ \mathcal P }}_{1}{\rho }^{k}.\end{eqnarray} \tag{ 12 }$$

Here, ${{ \mathcal P }}_{1},{{ \mathcal P }}_{2}$, ${{ \mathcal P }}_{3}$, and ${{ \mathcal P }}_{4}$ represent operators such as Fourier transform.

Let us show an example of applying the iterative approach to SXRD analysis. Here, we define ${{ \mathcal P }}_{1},{{ \mathcal P }}_{2}$, and ${{ \mathcal P }}_{3}$ to the Fourier transform, an operator to substitute a set of amplitudes of structure factors to the observed set, and an inverse-Fourier transform, respectively. ${{ \mathcal P }}_{4}$ sets the pixel that has negative value to zero gradually, as follows.

$$\begin{eqnarray}{\rho }^{k+1}=\left\{\begin{array}{ll}{\eta }^{k}, & ({\eta }^{k}\geqslant 0)\\ {\rho }^{k}-\beta {\eta }^{k}, & \mathrm{otherwise}.\end{array}\right.\end{eqnarray} \tag{ 13 }$$

Here, ${\eta }^{k}={{ \mathcal P }}_{3}{{ \mathcal P }}_{2}{{ \mathcal P }}_{1}{\rho }^{k}$, and β is the so-called relaxation parameter (typically between 0.5 and 1.0). In the analysis, as only in-plane data was employed, as shown in Fig. 10(a), the projected structure on the surface was obtained. The objective surface is Si(111)-($\sqrt{3}\times \sqrt{3}$)R30°-Ag, and the surface forms the "honeycomb-chained triangle" structure,⁶⁹⁾ as shown in Fig. 10(b).

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At first, we show an iterative solution by using ideal complete diffraction data. The results are shown in Fig. 10(c), and each pixel represents the final ρ^k. Silver and silicon atoms are clearly visible in the unit cell. Next, we applied this approach to the observed data. Integer-order reflections are excluded as they have contributions from the silicon bulk crystal. As shown in Fig. 10(d), both the topmost silver and first-layer silicon atoms are visible even with the observed data. The second-layer silicon atoms on positions with almost 1 × 1 periodicity disappear. This is because the information on 1 × 1 periodicity lacks in case integer-order reflections are lacking.

Crystallographic direct methods introduce a plausible structure model by combining the non-negativity and atomicity of the charge density in a crystal with information contained in the observed diffraction intensities. The phase problem in surface X-ray crystallography was thought to be untraceable until Marks et al.⁷⁰⁾ showed that the traditional direct methods in X-ray crystallography can be applied to surface-structure analysis to some extent, as SXRD data always have a systematic loss of diffraction intensity data. However, the early results remained unsatisfactory with complex structures, i.e. Si(111)-7 × 7.⁷¹⁾

Then, Marks et al. proceeded to apply multisolution approaches, e.g. combination of a global optimization approach such as genetic algorithm with the Sayre's equation,⁷²⁾ and projection methods with entropy minimization algorithm.^73,74) These methods find a plausible structure among diverse candidate structures by starting from random initial variables and/or charge density distributions. Dozens of surface structures, for example, Ni(111)-($5\sqrt{3}\times 2$)-S, as shown in Fig. 11, and Si(111)-(6 × 6)-Au, were solved by this approach.⁷⁵⁾ A 3D structure analysis of surface structure in combination with the above algorithm and oversampling condition along the reciprocal lattice rod was also discussed.⁷⁶⁾

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Rius et al.⁷⁷⁾ introduced differential charge density δ, including negative charge density region, to establish fundamental equations for solving a superstructure in general, and then derived a tangent formula by using the similarity between δ and δ³. Torrelles et al.^78–80) extended the Rius's direct method by interpreting differential charge density δ as surface charge density so that the method is applicable to surface in-plane diffraction data. For example, C₆₀/Au(110)-p(6 × 5) was solved using Rius's direct methods.^80,81) The direct method analysis with 130 independent in-plane reflections revealed not only the wavy structure of the outermost C₆₀ layer but also large displacement of Au atoms in the interface between C₆₀ layer and Au crystal, as shown in Figs. 12 and 13. This approach is mainly applied to the analysis of in-plane diffraction data.

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Takahashi et al.^83,84) demonstrated that direct imaging of surface atoms is possible with simple interpretation of CTR scatterings based on Gabor's in-line holography,⁸⁵⁾ i.e. CTR scatterings were considered as a hologram of surface atoms by interpreting the contributions of the substrate crystal and surface layers to CTR scatterings as a reference wave and an objective wave, respectively. The reconstructed image of Ge atom on Si(001) are shown in Fig. 14. This method has the advantage of quick characterization of the surface and interface with simple atomic arrangements because it is enough to collect only relatively intense CTR scatterings in a simple arrangement.

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Saldin et al.⁸⁶⁾ established an ab-initio structure analysis based on the holographic interpretation of CTR scatterings with maximum entropy inference. In the direct visualization of surface atoms, the treatment of diffraction contribution from various domains becomes crucial. Saldin et al. also tried ab-initio surface structure analysis with domains by employing Ge(001)-2 × 1 as a test surface⁸⁷⁾ (see Fig. 15).

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Yacoby et al.^88,89) developed the direct determination method of epitaxial structures named coherent Bragg rod analysis (COBRA), which uses an interference between the scatterings from bulk crystal and epitaxial layers, and extracts their phases from the relation of structure factors at two reciprocal points with a small separation along the CTR. Recently, COBRA was utilized in direct imaging of organic layers.⁹⁰⁾ Figure 16 shows several one-dimensional imagings of organic layers. Tiny tilting of oxide tetrahedron in oxide thin films was also directly imaged in three dimension by COBRA, as shown Fig. 17.⁹²⁾

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