On the integrated reflectivity of perfect crystals in extremely asymmetric Bragg cases of X-ray diffraction

Härtwig, J.

doi:10.1107/s0567739481001770

Cited by 35 publications

(5 citation statements)

References 8 publications

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“…[9][10][11][12][13][14][15][16][17][18][19][20]28 In this section, we will consider some examples at high incidence angles, where our GDT is always valid, in contrast to other theoretical models. [9][10][11][12][13][14][15][16][17][18][19][20]28 As a first example we may consider a short-period AlAs/GaAs superlattice ͑SL͒ coherently grown on a ͓100͔oriented GaAs substrate crystal. We assume that the SL is made by 66 periods with a period length of Pϭ1.2 nm, while the thicknesses of the two layers constituting one SL period are equal, i.e., 0.56 nm.…”

Section: Comparison With the Conventional Dynamical X-ray Theorymentioning

confidence: 99%

“…7,8 However, in some cases the above approximations are not valid: ͑i͒ for highly asymmetric x-ray diffractions, where simultaneously x-ray diffraction and reflection occur, [9][10][11][12][13][14][15][16][17][18] ͑ii͒ for grazing-incidence diffraction, 19,20 ͑iii͒ for Bragg angles that are very close to /2, 21-23 and ͑iv͒ for diffraction patterns, where the angular distance between the Bragg peaks is large ͑of the order of degrees͒. 24 Experimental evidences of the failure of the conventional x-ray dynamical theory ͑CDT͒ were also reported for extremely asymmetric Bragg x-ray diffractions on silicon ͑001͒ crystals.…”

Section: Introductionmentioning

confidence: 99%

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Generalized Laue dynamical theory for x-ray reflectivity at low and high incidence angles on strained multilayers

Giannini

Tapfer

1997

Phys. Rev. B

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In this work, we present a generalized dynamical theory for strained multilayers within the Laue formalism valid for x-ray Bragg diffraction. In the generalized theory we take into account all the four solutions of the secular equation, the asymptotic sphericity of the dispersion surface, the difference between electric and displacement fields, and the boundary conditions of continuity at the heterointerfaces for the tangential components of the electric and magnetic fields. Only the following approximations are made: to consider a two-beam case, and to neglect quadratic terms of the dielectric susceptibility. Our general equations give the correct x-ray reflectivity in the whole angular range between 0 and /2, i.e., also very far from the Bragg peaks. Great differences between the predictions of the conventional and generalized theories are obtained in several cases, for important features of the rocking curves, like angular position, peak intensity, and full width at half maximum of superlattice satellite or epitaxial layer peaks. We discuss in detail the following cases for which only the generalized theory describes correctly the x-ray scattering: coplanar diffraction ͑i͒ very far from the Bragg condition, ͑ii͒ near to the Bragg condition for strong asymmetric reflections, and ͑iii͒ for highly mismatched heterostructures.

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Section: Comparison With the Conventional Dynamical X-ray Theorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Generalized Laue dynamical theory for x-ray reflectivity at low and high incidence angles on strained multilayers

Giannini

Tapfer

1997

Phys. Rev. B

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“…1͒: ͑i͒ The coplanar extremely asymmetric diffraction ͑EAD͒ is realized when the diffraction planes make the Bragg angle with the crystal surface and either the incident or exit x-ray wave is grazing. [5][6][7][8][9][10][11][12] ͑ii͒ Surface or ''grazing-incidence'' diffraction 13 ͑GID͒ is the geometry where the Bragg planes are perpendicular to the surface and both the x-ray waves are grazing. [14][15][16][17][18][19][20][21][22][23] ͑iii͒ Finally, grazing Bragg-Laue diffraction ͑GBL͒ is a combination of the EAD and GID.…”

Section: Introductionmentioning

confidence: 99%

“…X-ray diffraction at grazing incidence and/or exit can be treated with the help of either an extended kinematical theory ͑often called the ''distorted wave Born approximation''͒, 16,19,41,49,55 or extended dynamical theory. 5,6,8,9,11,14,17,22,45,48,51,[56][57][58][59] Both approaches take into account refraction and specular reflection effects for grazing x rays at crystal surfaces and interfaces. As with ordinary Bragg diffraction, the kinematical theory is applicable to mosaic crystals, to the tails of the Bragg peaks, and to the diffraction from layers thinner than the x-ray extinction depth.…”

Section: Introductionmentioning

confidence: 99%

Dynamical x-ray diffraction of multilayers and superlattices: Recursion matrix extension to grazing angles

et al. 1998

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A generalized dynamical theory has been developed that extends previous models of x-ray diffraction from crystals and multilayers with vertical strains to the cases of grazing incidence and/or exit below the critical angle for total specular reflection. This provides a common description for extremely asymmetric diffraction, surface ͑''grazing-incidence''͒, and grazing Bragg-Laue diffraction, thus providing opportunities for the applications of grazing geometries to the studies of thin multilayers. The solution, obtained in the form of recursion formulas for ͑2ϫ2͒ scattering matrices for each individual layer, eliminates possible divergences of the ͑4ϫ4͒ transfer-matrix algorithm developed previously. For nongrazing x-ray diffraction in the Bragg geometry and for grazing-incidence x-ray specular reflection out of the Bragg diffraction conditions, the matrices are reduced to scalars and the recursion formulas become equivalent to the earlier recursion formulas by Bartels et al. ͓Acta Cryst. A 42, 539 ͑1986͔͒ and Parratt ͓Phys. Rev. 95, 359 ͑1954͔͒, respectively. The theory has been confirmed by an extremely asymmetric x-ray-diffraction experiment with a strained AlAs/GaAs superlattice carried out at HASYLAB. A solution to the difficulties due to dispersion encountered in extremely asymmetric diffraction measurements has been demonstrated. Finally, the validity of Ewald's expansion for thin layers and the relation of the matrix method to the Darwin theory, as well as the structure of x-ray standing waves in multilayers are discussed. ͓S0163-1829͑98͒05408-3͔

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“…only the points close to the dispersion surface intersections are considered. However this second-order description of the dispersion surface is an assumption and the actual shape in figure 2.3.5 is of fourthorder, Härtwig (1981), Holý and Fewster (2003);…”

Section: Combination Of Specular Scattering and One Diffracted Wavementioning

confidence: 99%

The Theory of X-Ray Scattering

Fewster¹

2015

X-Ray Scattering From Semiconductors and Other Materials

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This chapter presents the scattering theories applicable to analyzing thin film and polycrystalline materials. The 2-beam dynamical theory is derived, because of its usefulness in simulating scattering from epitaxial semiconductors. The limitations of this model are discussed along with extensions and alternatives that are more exact, including an approach that generates all the appropriate reflections making it a truly multiplebeam dynamical theory. The kinematic model is discussed and its applicability to thin films and imperfect materials, and how the optical theory is adequate for specular reflectivity. An alternative theory for Xray diffraction is also presented that considers the scattering throughout space, which is particularly relevant for studying polycrystalline materials. It will also become obvious how the whole process is intricately linked to the instrument. The distorted-wave Born Approximation is discussed along with applications in modeling diffuse scattering.

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On the integrated reflectivity of perfect crystals in extremely asymmetric Bragg cases of X-ray diffraction

Cited by 35 publications

References 8 publications

Generalized Laue dynamical theory for x-ray reflectivity at low and high incidence angles on strained multilayers

Generalized Laue dynamical theory for x-ray reflectivity at low and high incidence angles on strained multilayers

Dynamical x-ray diffraction of multilayers and superlattices: Recursion matrix extension to grazing angles

The Theory of X-Ray Scattering

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