Martin Köhl

Improved success rate and stability for phase retrieval by including randomized overrelaxation in the hybrid input output algorithm

In this paper, we study the success rate of the reconstruction of objects of finite extent given the magnitude of its Fourier transform and its geometrical shape. We demonstrate that the commonly used combination of the hybrid input output and error reduction algorithm is significantly outperformed by an extension of this algorithm based on randomized overrelaxation. In most cases, this extension tremendously enhances the success rate of reconstructions for a fixed number of iterations as compared to reconstructions solely based on the traditional algorithm. The good scaling properties in terms of computational time and memory requirements of the original algorithm are not influenced by this extension.

The photonic Wannier function approach to photonic crystal simulations: status and perspectives

Photonic crystals represent a novel platform for the realization of integrated photonic circuits with numerous applications. This article reviews the present status of a computational approach dedicated to these systems that is based on photonic Wannier functions. The workings of the Wannier function approach are illustrated by way of selected examples. In addition, we discuss the advantages and limitations of this approach and sketch potential future developments.

Polytypism in GaAs nanowires: determination of the interplanar spacing of wurtzite GaAs by X-ray diffraction

In GaAs nanowires grown along the cubic [111]c direction, zinc blende and wurtzite arrangements have been observed in their stacking sequence, since the energetic barriers for nucleation are typically of similar order of magnitude. It is known that the interplanar spacing of the (111)c Ga (or As) planes in the zinc blende polytype varies slightly from the wurtzite polytype. However, different values have been reported in the literature. Here, the ratio of the interplanar spacing of these polytypes is extracted based on X-ray diffraction measurements for thin GaAs nanowires with a mean diameter of 18-25 nm. The measurements are performed with a nano-focused beam which facilitates the separation of the scattering of nanowires and of parasitic growth. The interplanar spacing of the (111)c Ga (or As) planes in the wurtzite arrangement in GaAs nanowires is observed to be 0.66% ± 0.02% larger than in the zinc blende arrangement.

Monitoring the thin film formation during sputter deposition of vanadium carbide

The theoretical description and the experimental realisation of in situ X-ray reflectivity measurements during thin film deposition of polycrystalline vanadium carbide coatings are presented.

Retrieval of the atomic displacements in the crystal from the coherent X-ray diffraction pattern.

The retrieval of spatially resolved atomic displacements is investigated via the phases of the direct(real)-space image reconstructed from the strained crystals coherent X-ray diffraction pattern. It is demonstrated that limiting the spatial variation of the first- and second-order spatial displacement derivatives improves convergence of the iterative phase-retrieval algorithm for displacements reconstructions to the true solution. This approach is exploited to retrieve the displacement in a periodic array of silicon lines isolated by silicon dioxide filled trenches.

Retrieving the displacement of strained nanoobjects: the impact of bounds for the scattering magnitude in direct space

Coherent X-ray diffraction imaging (CXDI) of the displacement field and strain distribution of nanostructures in kinematic far-field conditions requires solving a set of non-linear and non-local equations. One approach to solving these equations, which utilizes only the objects geometry and the intensity distribution in the vicinity of a Bragg peak as a priori knowledge, is the HIO+ER-algorithm. Despite its success for a number of applications, reconstruction in the case of highly strained nanostructures is likely to fail. To overcome the algorithms current limitations, we propose the HIO(O(R))(M)+ER(M)-algorithm which allows taking advantage of additional a priori knowledge of the local scattering magnitude and remedies HIO+ERs stagnation by incorporation of randomized overrelaxation at the same time. This approach achieves significant improvements in CXDI data analysis at high strains and greatly reduces sensitivity to the reconstructions initial guess. These benefits are demonstrated in a systematic numerical study for a periodic array of strained silicon nanowires. Finally, appropriate treatment of reciprocal space points below noise level is investigated.

Asymmetric skew X-ray diffraction at fixed incidence angle: application to semiconductor nano-objects

A procedure for obtaining three-dimensionally resolved reciprocal-space maps in a skew X-ray diffraction geometry is described. The geometry allows tuning of the information depth in the range from tens of micrometres for symmetric skew diffraction down to tens of nanometres for strongly asymmetric skew geometries, where the angle of incidence is below the critical angle of total external reflection. The diffraction data are processed using a rotation matrix formalism. The whole three-dimensional reciprocal-space map can be measured by performing a single azimuthal rotation of the sample and using a two-dimensional detector, while keeping the angle of incidence and the X-ray information depth fixed (FIXD method). Having a high surface sensitivity under grazing-incidence conditions, the FIXD method can be applied to a large variety of Bragg reflections, particularly polar ones, which provide information on strain and chemical composition separately. In contrast with conventional grazing-incidence diffraction, the FIXD approach reveals, in addition to the lateral (in-plane) components, the vertical (out-of-plane) component of the strain field, and therefore allows the separation of the scattering contributions of strained epitaxial nanostructures by their vertical misfit. The potential of FIXD is demonstrated by resolving the diffraction signal from a single layer of InGaN quantum dots grown on a GaN buffer layer. The FIXD approach is suited to the study of free-standing and covered near-surface nano-objects, as well as vertically extended multilayer structures.

Cluster coherent potential approximation for disordered photonic crystals using photonic Wannier functions.

We present an investigation of disordered photonic crystals (PhCs) based on the combination of photonic Wannier functions with the concept of the coherent potential approximation (CPA). In particular, we provide the theoretical foundation of a real-space cluster CPA that is causal, enforces the proper symmetries of the effective medium, and includes effects of multiple scattering of the same and nearby defects, which is essential for strong defects. Based on this, we present results for the density of states of disordered PhCs for different types of disorder. Our results are thus relevant to such diverse areas as random lasing and the analysis of fabricational imperfections in PhCs.

Perspectives and limitations of symmetric X-ray Bragg reflections for inspecting polytypism in nanowires

X-ray diffraction, possibly time-resolved during growth or annealing, is an important technique for the investigation of polytypism in free-standing nanowires. A major advantage of the X-ray diffraction approach for adequately chosen beam conditions is its high statistical significance in comparison with transmission electron microscopy. In this manuscript the interpretation of such X-ray intensity distribution is discussed, and is shown to be non-trivial and non-unique given measurements of the [111]c or [333]c reflection of polytypic nanowires grown in the (111)c direction. In particular, the diffracted intensity distributions for several statistical distributions of the polytypes inside the nanowires are simulated and compared. As an example, polytypic GaAs nanowires are employed, grown on a Si-(111) substrate with an interplanar spacing of the Ga (or As) planes in the wurtzite arrangement that is 0.7% larger than in the zinc blende arrangement along the (111)c direction. Most importantly, ambiguities of high experimental relevance in the case of strongly fluctuating length of the defect-free polytype segments in the nanowires are demonstrated. As a consequence of these ambiguities, a large set of deviations from the widely used Markov model for the stacking sequences of the nanowires cannot be detected in the X-ray diffraction data. Thus, the results here are of high relevance for the proper interpretation of such data.

Disordered photonic crystals: a cluster coherent potential approach using photonic Wannier functions

We present a cluster coherent potential approach for disordered photonic crystals (PhCs) that is based on maximally localized Wannier functions. In particular, the Wannier basis facilitates an efficient representation of the photonic band structure of a defect-free PhC and the Green’s function thereof, which we discuss first. Moreover, the Wannier basis allows for adapting elaborate approaches developed for the analysis of disorder in electronic and phononic systems to PhCs. Our detailed studies reveal that neither the virtual crystal approximation nor the single-site coherent potential approximation (CPA) provides a reliable description of the effects of disorder in PhCs. Rather, we demonstrate that a cluster based CPA yields qualitatively and quantitatively useful results even for strong disorder. In all these approaches, we include multiple bands, enforce the proper translational properties of the effective medium, and incorporate interaction beyond nearest neighbor coupling. We also discuss the efficient evaluation of the corresponding system of equations. Our results establish a firm basis for various extensions, such as to the theory of Anderson localization or random lasing in disordered PhCs.