Reciprocal Lattice Rods Research Articles

Quantitative Strain and Topography Mapping of 2D Materials Using Nanobeam Electron Diffraction.

It is known that 2D materials can exhibit a nonflat topography, which gives rise to an inherent strain. Since local curvature and strain influence mechanical, optical, and electrical properties, but are often difficult to distinguish from each other, a robust measurement technique is needed. In this study, a novel method is introduced, which is capable of obtaining quantitative strain and topography information of 2D materials with nanometer resolution. Relying on scanning nanobeam electron diffraction (NBED), it is possible to disentangle strain from the local sample slope. Using the positions of the diffraction spots of a specimen at two different tilts to reconstruct the locations and orientations of the reciprocal lattice rods, the local strain and slope can be simultaneously retrieved. We demonstrate the differences to strain measurements from a single NBED map in theory, simulation, and experiment. MoS2 monolayers with different shapes are used as simulation test structures. The slope and height information are recovered, as well as tensile and angular strain which have an absolute difference of less than 0.2% and 0.2° from the theoretical values. An experimental proof of concept using a freely suspended WSe2 monolayer together with a discussion of the accuracy of the method is provided.

Sources of error in Debye–Waller-effect measurements relevant to studies of photoinduced structural dynamics

We identify and quantify several practical effects likely to be present in both static and ultrafast electron-scattering experiments that may interfere with the Debye-Waller (DW) effect. Using 120-nm thick, small-grained, polycrystalline aluminum foils as a test system, we illustrate the impact of specimen tilting, in-plane translation, and changes in z height on Debye-Scherrer-ring intensities. We find that tilting by less than one degree can result in statistically-significant changes in diffracted-beam intensities for large specimen regions containing > 105 nanocrystalline grains. We demonstrate that, in addition to effective changes in the field of view with tilting, slight texturing of the film can result in deviations from expected DW-effect behavior. Further, we find that in-plane translations of as little as 20 nm also produce statistically-significant intensity changes, while normalization to total image counts eliminates such effects arising from changes in z height. The results indicate that the use of polycrystalline films in ultrafast electron-scattering experiments can greatly reduce the negative impacts of these effects as compared to single-crystal specimens, though it does not entirely eliminate them. Thus, it is important to account for such effects when studying thin-foil specimens having relatively short reciprocal-lattice rods.

Observation of in‐situ reciprocal lattice evolution of AlGaN/InGaN on Si (111) through GaN and AlN interlayers by RHEED and reflectance

AbstractDue to the wave‐particle duality of high energy electron beam, RHEED describes a thin film surface through the interaction of reciprocal lattice rod (RLR) of the film. The RLR spacing of the crystal has been computed using RHEED streak spacing on a fluorescent screen. Present evolution study of RLR spacing has been performed for AlGaN/InGaN heterostructure on thick GaN buffer layer during plasma‐assisted molecular beam epitaxy. Effect of composition, strain and temperature on the crystal has been identified as the function of lattice spacing during the growth. The calibrated reflectivity of LED signal has also been employed to map the thickness of different growing epilayers. RLR spacing decreases for interlayer GaN as compared to AlN. Initially, the RLR spacing of GaN buffer decreases more as compared to interlayer GaN owing to its increased growth temperature after the interlayer AlN. The RLR spacing of GaN again gradually increases with decrement of growth temperature up to the InGaN channel layer. The InGaN RLR spacing decreases again which attributes to the relaxation of InGaN on GaN. Additionally, AlGaN reveals strained state to InGaN in combination with effect of high growth temperature. The phenomena result in additional decrement of lattice distance in reciprocal space. (© 2015 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Diffraction features due to ordered distribution of twin boundaries in orthorhombic Ni–Mn–Ga crystals

The diffraction intensity distributions for orthorhombic crystals containing the ordered twin boundaries of the system (110), 〈1{\overline 1}0〉 were calculated in the kinematic approach using the Monte Carlo computer simulation technique. A simple model of short-range order in the arrangement of the twin boundaries was assumed to generate the nano-twinned structure. Analysis of the intensity distributions showed that such a twin boundary distribution results in the formation of extra peaks along the reciprocal lattice rods that are perpendicular to the close-packed planes. These extra peak intensities and positions are dependent on the most probable distances between the twin boundaries and on the degree of short-range order.

Reflection high-energy electron diffraction study of molecular beam epitaxy growth of Pr 2O 3 on Si(0 0 1)

Thin Pr 2O 3 films have been grown by molecular beam epitaxy on Si(0 0 1) and investigated by in situ reflection high-energy electron diffraction (RHEED). The epitaxial growth results in [1 0 1]-oriented films of cubic Pr 2O 3 with low interface and surface roughness, as confirmed by subsequent X-ray diffraction and reflectivity measurements. During nucleation of Pr 2O 3 on double-domain Si(0 0 1) (2×1)-reconstructed surfaces, a diffuse RHEED pattern with weak streaks evolves. Subsequently, a pattern with modulated streaks appears. According to the analysis of the conventional RHEED patterns and the in-plane symmetry of the reciprocal lattice rods obtained by RHEED azimuthal scans, the modulated streaks are attributed to the reflections resulting from two orthogonal [1 0 1]-oriented domains of cubic Pr 2O 3. The [0 1 0] direction of one domain and the [ 1 0 1 ¯ ] of the other are parallel to Si[1 1 0]. The RHEED patterns resulting from good epitaxy deteriorate when the layer thickness reaches 18 nm, due to the formation of hexagonal Pr 2O 3 in the films. The structural features of this phase dominate thereafter. This observation indicates an epitaxial stabilization of the cubic phase at the interface.

Three-dimensional diffractive imaging for crystalline monolayers with one-dimensional compact support

The use of a compact support constraint along the beam direction is considered as a solution to the phase problem for diffraction by two-dimensional protein crystals. Specifically we apply the iterative Gerchberg–Saxton–Fienup algorithm to simulated three-dimensional transmission electron diffraction data from monolayer organic crystals. We find that oversampling along the reciprocal-lattice rods (relrods) normal to the monolayer alone does not solve the phase problem in this geometry in general. However, based on simulations for a crystalline protein monolayer (lysozyme), we find that convergence is obtained in three dimensions if phases are supplied from a few high resolution electron microscope images recorded at small tilts to the beam direction. In the absence of noise, amplitude-weighted phase residuals of around 5°, and a cross-correlation coefficient of 0.96 between the true and estimated potential are obtained if phases are included from images at tilts of up to 15°. The performance is almost as good in the presence of noise at a level that is comparable to that commonly observed in electron crystallography of proteins. The method should greatly reduce the time and labor needed for data acquisition and analysis in cryo-electron microscopy of organic thin crystals by avoiding the need to record images at high tilt angles.

Surface Step Information of Vicinal Si(001) in EF-RHEED Pattern.

For a vicinal Si(001) 2×1 surface, a row of extra spots has been clearly observed along the reciprocal lattice rod in energy-filtered reflection high-energy electron diffraction (ER-RHEED) pattern. The glancing angle dependence of these extra spots along (0 0), (1/2 0) and (1 0) rods was simulated by kinematical analysis based on a monatomic regular step surface. Behavior of these extra spots was reproduced well by the calculations and the mean terrace length along the [110] direction was estimated. It has been found that this method can evaluate very small vicinal angle near 0.1o, which is superior to low energy electron diffraction (LEED) method in this point.

Surface X-ray diffraction study on the initial oxidation of NiAl(100)

The oxidation behavior of NiAl(100) surfaces was studied using surface X-ray diffraction. After oxidation of the preferentially Ni-terminated surface at 950°C for 500 s at 2×10 −6 mbar O 2, the formation of epitaxial Al 2O 3 in a (2×1) superstructure was observed. Along the (2×1) superstructure reciprocal lattice rods, two different modulation periods could be detected. The shorter period arises from the growth of 2 nm high θ-Al 2O 3 oxide islands in monoclinic structure, the longer period from a 0.76 nm thick Al 2O 3 layer. This layer exhibits a θ-Al 2O 3-like structure with strongly modified Al ion positions at the surface and at the metal/oxide interface.

RHEED study of atomic steps on ZnSe surface growing on vicinal GaAs substrates

Atomic steps of ZnSe surface growing on vicinal GaAs(1 0 0) substrate are investigated with in-situ reflection high-energy electron beam diffraction. The diffraction patterns indicating 1 ML-height step with uniform terrace width are clearly observed as the growth proceeds. The uniformity of step array on growing surface becomes stable by ZnSe growth of 500 Å. The terrace width and substrate vicinal angle measured by the diffraction patterns agree well with the substrate nominal values. The observed diffraction patterns are compared with simulation results based on kinematical diffraction theory to discuss the fluctuation of surface step structure. The improvement of uniformity of step array is anisotropic with regard to the substrate vicinal direction. The insertion of CdSe layer does not cause broadening of full width at half maximum of reciprocal lattice rod, which indicates that the uniformity of step array is unchanged.

Surface x-ray diffraction onK/Si(001)(2×1)andCs/Si(001)(2×1)

The geometric structures of $\mathrm{K}/\mathrm{S}\mathrm{i}(001)(2\ifmmode\times\else\texttimes\fi{}1)$ and $\mathrm{C}\mathrm{s}/\mathrm{S}\mathrm{i}(001)(2\ifmmode\times\else\texttimes\fi{}1)$ formed at room temperature have been investigated using surface x-ray diffraction. The analysis of the intensity distribution along both half-order and integer-order reciprocal-lattice rods (135 and 113 independent reflections in total for K and Cs, respectively) indicates that at saturation coverage in both cases the alkali metal $(A)$ adsorbs in two different sites. The average adsorption sites on the clean Si surface are at the dimer bridge site on top of the dimer rows, and at the valley bridge site in the groove between the Si dimers. Actually, the dimerization is (at least partially) lifted upon $A$ adsorption. There is no evidence for a large $z$ disorder of the top-layer Si atoms, which would be characteristic of an asymmetry of the Si-Si dimer. We derive K-Si bonding lengths in the range between 2.90 and 3.58 \AA{}, and Cs-Si bonding lengths between 3.33 and 3.50 \AA{}. The minimum bond lengths are at most 11% lower than values found in bulk KSi and CsSi. This can be interpreted by the formation of a weak covalent bond. In order to obtain good fits to the data, up to four layers of the Si substrate have to be included into the refinement. The Si-Si distances do not deviate by more than 5--10 % from the bulk Si-Si bond length (2.35 \AA{}). The only significant difference between the K and Cs adsorption structures is found for the alkali saturation coverage. For K we find a coverage of 1.2 ML, whereas only 0.3 ML is found for Cs.

Direct Mapping of Reciprocal Lattice Rods of Metal-Induced Facet on Si(001) Surface

On Si(001) surfaces annealed at 700°C after the deposition of 1.5 monolayer (ML) of Ba at room temperature, facet spots were observed in addition to the 2×1 pattern of the Si(001) clean surface. These patterns were recorded using a TV camera and stored in a personal computer as an image. Line profiles were obtained on the line along which the facet spot moved. From these profiles the peak position of facet spots was extracted and superposed on a reciprocal lattice space to construct reciprocal lattice rods. From the angle of these rods to those of the Si(001) substrate, the facet is indicated to be a metal-induced Si(113) face.

Use of x-ray reflectivity for determining the Si(111) sqrt 3 x sqrt 3-Bi surface structures.

Absolute reflectivities of x-rays from the Si(111)\ensuremath{\surd}3\ifmmode\times\else\texttimes\fi{}\ensuremath{\surd}3-Bi surface were measured along the 00 and 12\ifmmode\bar\else\textasciimacron\fi{} reciprocal-lattice rods. Two phases of the surface, a low-coverage phase and a high-coverage phase, were recognized from the reflectivity measurement. The Bi coverages of the two phases were determined as 1 ML and 1/3 ML. The 1-ML phase takes the milk stool structure. For the 1/3-ML phase the adsorbed Bi atom is found to be on the ${\mathit{T}}_{4}$ site and the distance between the Bi layer and the first Si layer is 1.60 \AA{}. For data analysis, expressions of the absolute reflectivities obtained by a newly developed dynamical theory are used.

Darwin spherical-wave theory of kinematic surface diffraction

In 1912, von Laue first described X-ray diffraction by approximating as plane waves the spherical waves radiated by atoms in a crystal. Darwin recognized that this approximation is valid only in the limit of very small crystals, and published in 1914 the more general spherical-wave theory based on the reflectivity of individual atomic planes. The Darwin theory is extended here to surface Bragg diffraction from a single-crystalline monolayer, including the rederivation of a `surface' Bragg's law and the reflected intensity versus \varphi, the angle of incidence. This more general theory demonstrates how the intensity along a reciprocal-lattice rod associated with diffraction from a semi-infinite crystal is modified by the \varphi dependence of the length over which atoms in a plane emit spherical waves that constructively interfere at the detector. The diffracted amplitude is not proportional to the Fourier transform of the charge density. The plane- and spherical-wave models yield identical results for the integrated intensity across a finite detector area, even for incident angles far from a Bragg reflection.

Bloch wave origin of surface resonance scattering in RHEED

Surface resonance scattering in RHEED is shown to result from Bloch wave degeneracy, which gives singular Bloch wave excitations and an anomaly in the surface reflected beam amplitudes. The minimum condition for the resonance to occur is that two reciprocal lattice rods are involved, and the involvement of bound states of atomic planes parallel to the surface is not necessary. The concept of Bloch wave degeneracy has been applied to explain resonance scattering occurring either in the selvedge or/and in the substrate. When the diffraction condition has been so chosen that the diffraction processes are dominated by two reciprocal lattice rods, the total electron wave function is shown to be highly localized at the surface. In a general case when many reciprocal lattice rods are involved, the total electron wave function may or may not be localized, depending on the diffraction condition

Grazing incidence X-ray diffraction on Cs/Ge(001)(2 × 1): evidence for cave site adsorption

Grazing incidence X-ray diffraction has been used to investigate the atomic structure of Cs adsorbed on Ge(001)(2 × 1) at room temperature. For about half monolayer coverage (1 monolayer = 6.25 × 10 14 atoms cm 2 for Ge(001)) the analysis of the data set consisting of 12 symmetry independent in plane ( Q Z = 0) reflections and of four fractional order reciprocal lattice rods (62 independent reflections in total) gave clear evidence for the occupation of the cave site which is in the groove between the dimer rows and above the 3rd layer Ge-atom. The Cs atoms are found to be strongly disordered along the surface normal. Within a static disorder model this is accounted for by splitting the Cs leading to different height levels, d ⊥ 1 = 1.73(11) Å and d ⊥ 2 = 0.44(11) Å above the Ge dimer level. Minimum Cs-Ge neighbor distances of R = 3.14(21) Å and 3.50(16) Å for the split atoms to nearest Ge atoms might indicate a mixed strength of interaction between adsorbate and substrate. A dynamical model allowing an unsplitted Cs atom to strongly vibrate along the surface normal by a root mean square amplitude of 〈u 33 2〉 1 2 = 0.93 Å leads to an equilibrium distance of R = 3.56(20) Å to second layer Ge also suggesting some charge transfer to the substrate. Whereas the Ge dimer bonding length is found not to be strongly affected by Cs adsorption ( R D = 2.45(11) Å), extensive relaxations are observed down to the 3rd substrate layer.

The Ge(001) (2 × 1) reconstruction: asymmetric dimers and multilayer relaxation observed by grazing incidence X-ray diffraction

Grazing incidence X-ray diffraction has been used to analyze in detail the atomic structure of the (2 × 1) reconstruction of the Ge(001) surface involving far reaching subsurface relaxations. Two kinds of disorder models, a statistical and a dynamical were taken into account for the data analysis, both indicating substantial disorder along the surface normal. This can only be correlated to asymmetric dimers. Considering a statistical disorder model assuming randomly oriented dimers the analysis of 13 symmetrically independent in-plane fractional order reflections and of four fractional order reciprocal lattice rods up to the maximum attainable momentum transfer q z = 3c ∗ ( c ∗ = 1.77 × 10 −1 A ̊ −1 ) indicates the formation of asymmetric dimers characterized by R> D = 2.46(5) A ̊ as compared to the bulk bonding length of R = 2.45 A ̊ . The dimer height of Δ z = 0.74(15) A ̊ corresponds to a dimer buckling angle of 17(4)°. The data refinement using anisotropic thermal parameters leads to a bonding length of R D = 2.44(4) A ̊ and to a large anisotropy of the root mean-square vibration amplitudes of the dimer atoms ( 〈u 11 2〉) 1 2 = 0.25 A ̊ , ( 〈u 22 2〉) 1 2 = 0.14 A ̊ , ( 〈u 33 2〉) 1 2 = 0.50 A ̊ ). We have evidence for lateral and vertical disp tenth layer below the surface.

Direct methods in electron crystallography - structure analysis of boric acid

Direct phasing methods based on the probabilistic estimate of triplet and quartet structure invariants are used to determine the crystal structure of boric acid from room-temperature electron diffraction data published by Cowley [Acta Cryst. (1953), 6, 522–529] and also new low-temperature data collected at or below 128 K. In either instance, the resulting structure is the monolayer found earlier and the bond distances and angles for heavy atoms computed from the atomic coordinates agree well with three-dimensional X-ray and neutron diffraction determinations. H-atom positions are not accurately defined, however. A reason for the monolayer structure resulting from the analysis of electron diffraction data is suggested, given the continuous reciprocal-lattice rods observed in diffraction patterns from tilted crystals also seen earlier at room temperature.

A systematic RHEED study of regular and random steps on GaAs(001) surfaces

A detailed understanding of step structures is becoming increasingly important in the growth of low-dimensional semiconductor structures such as quantum wells and quantum wires and also in the study of thin-film growth dynamics. One of the main techniques to have been employed is reflection high-energy electron diffraction (RHEED), mostly within the framework of a kinematic treatment of the observed diffraction effects. In this paper we report the results of a systematic study of diffracted intensity distribution along and perpendicular to reciprocal lattice rods from both the regular step structures present on a vicinal GaAs(001) surface and the irregular structures resulting from submonolayer film deposition. The validity of using a kinematic treatment is examined critically and an alternative model which includes various forms of incoherent scattering is proposed.

Influence of surface disorder on RHEED patterns from GaAs(001)−2 × 4 surfaces

Reflection high-energy electron diffraction measurements have been carried out for a nearly perfect and a disordered GaAs(001)−2 × 4 surface in the [ 1 10] azimuth. The surfaces were prepared in-situ by molecular beam epitaxy and the measurements were made at a substrate temperature of 560 °C under different flux conditions, giving the different degrees of surface ordering. Diffracted intensity distributions along various reciprocal surface lattice rods were measured for different angles of incidence using a primary beam energy of 15 keV. Drastic differences with surface order were found for both the (quasi) elastically diffracted beams (rocking curves) and the diffuse beams. The rocking curves can be interpreted largely by Bragg reflection from the bulk crystal and a number of causes for the intensity changes are given. In the case of the disordered surface the intensity distributions along the reciprocal lattice rods are dominated by broad diffuse peaks which occur at the intersection of reciprocal rods and Kikuchi lines. It is shown that while a kinematical model is unable to explain the experimental observations, a previously published model provides a very satisfactory interpretation.

A review of the geometrical fundamentals of reflection high-energy electron diffraction with application to silicon surfaces

Reflection high-energy electron diffraction (RHEED) is an experimentally simple technique, and yet a powerful one for examining the structure of a substrate surface and for monitoring the surface crystal structure and the crystallographic orientation of thin films during their growth. However, it can be difficult to learn to interpret the RHEED patterns of new materials, because a practical and adequately detailed introduction to the technique is not generally available. To address this need, we develop the geometrical principles of RHEED; using the kinematic approximation, we show how a particular point of the sample surface’s reciprocal net gives rise to a diffraction maximum at a particular location on the RHEED viewing screen. We explain the origins of ‘‘reciprocal lattice rods,’’ RHEED streaks, and Laue rings. We show how to calculate the streak spacing, and clarify the basic effect on the RHEED pattern of using a nonzero angle of incidence for the incident beam. Crystalline nets, reciprocal nets, and their RHEED patterns are derived for both (001) and (111) silicon surfaces and are compared to experimentally obtained patterns.