Statistical Diffraction Theory and High Resolution X-ray Diffraction

Contact Dr. K.M. Pavlov

(Al,Ga,In)-nitrides are important materials for photonic and electronic applications. The physical properties of the epitaxial layers depend on structural features like phase purity, mosaicity, strains and chemical composition.

The evaluation of a layer's defect structure can be done from HRXRD Reciprocal Space Maps (RSM). Crystal defects lead to changes in the width and shape of X-ray diffraction peaks. The amount and orientation of the "peak broadening" depends on the nature of the crystal defect and will, in general, be different for different reflections. Very often the broadening of a "rocking curve" is used to quantify e.g., dislocation densities. However, such rocking curves represent convoluted, projected and experimentally distorted images of the actual situation in Reciprocal Space and should be looked upon with suspicion unless the nature of the crystal defects present is well known.

Depending on the growth parameters (Al,Ga,In)N films can show "natural" ordering of the group-III-elements (see Figure). This chemical ordering is kinetic in nature and a consequence of the large differences of the Al – N, Ga – N and In – N bond strength. Ordering is of substantial interest since it is known to influence optical and electronic properties of (Al,Ga,In)-nitride solid solutions. The formation of a superstructure is indicated by 'super-structure' reflections in diffraction experiments. In general, the formation of a superstructure is not homogeneous throughout the bulk volume and more and less ordered regions are commonly observed on a sub-micron scale.

The approach used in our papers (see References below) allows information about the spatial distribution of the mosaic block parameters to be obtained. For this purpose the Takagi equations (Takagi, 1969) were transformed to a new form which allows a simulation of the distribution of the diffracted intensity in Reciprocal Space within the framework of the dynamical diffraction theory. The parameters used for the reconstruction comprise the size of the blocks, their average rotation angle and spatial distribution of some components of the microdistortion tensor.

PhD Project :This project will contribute to the development of statistical diffraction theory as mean of characterisation of multilayer semiconductor structures. The experimental part of the project is planned to be done in collaboration with IAF-FhG (Freiburg, Germany).


Figure 6. Model of chemical antiphase domain boundaries in ordered Al0.5Ga0.5N layers.


1) K.M. Pavlov, D.M. Paganin, D.J. Vine, and L. Kirste. Wide angle X-ray dynamical diffraction by deformed crystals: recurrence relations (2007) Physica Status Solidi A, 204(8), 2613-2619.
2) L. Kirste, K.M. Pavlov, S.T. Mudie, V.I. Punegov and N. Herres (2005). Analysis of the mosaic structure of an ordered (Al,Ga)N layer. J. Appl. Cryst. 38, 183-192.
3) Pavlov, K.M., Punegov, V.I. (2000): Statistical dynamical theory of X-ray diffraction in the Bragg case: application to triple-crystal diffractometry. Acta Cryst. Sec. A56 , Pages 227-234.
4) Pavlov, K., Faleev, N., Tabuchi, M., Takeda, Y. (1999): Specific aspects of X-ray diffraction on statistically distributed QDs in perfect crystal matrix. Jpn. J. Appl. Phys. Vol.38 Suppl. 38-1 P.269-272.
5) Faleev, N., Pavlov, K., Tabuchi, M., Takeda, Y. (1999): Influence of long-range lateral ordering in structures with quantum dots on the spatial distribution of diffracted X-ray radiation. Jpn. J. Appl. Phys. ( 15 February 1999), V.38, Part 1, No. 2A, P. 818-821.
6) Pavlov, K.M., Punegov, V.I.. (1998): The equations of the statistical dynamical theory of X-ray diffraction for deformed crystals. Acta Cryst. A54, 214-218.
7) N. Herres, F. Fuchs, J. Schmitz, K.M. Pavlov, J. Wagner, J.D. Ralston, P. Koidl, C. Gadaleta and G. Scamarcio: "Effect of interfacial bonding on the structural and vibrational properties of InAs/GaSb superlattices". Physical Review B, Volume 53, Number 23 (15 June 1996), Pages 15688 - 15705. (Phys. Rev. B 53, 15688 (1996))