Simulation of reciprocal space maps from elastic strain field in periodical nanostructures

L. Horak, J. Matejova

Department of Condensed Matter Physics, Faculty of Mathematics and Physics, Charles University in Prague, Ke Karlovu 5, 121 16 Praha, Czech Republic

horak@karlov.mff.cuni.cz

Laterally periodical nanostructures epitaxially grown on substrates, such as quantum dot arrays and wires, are extensively used in electronic and optoelectronic applications. The strain field in the nanostructure, caused by an epitaxial mismatch, strongly affects the material properties, e.g. electronic band structure or magnetic anisotropy. This strain field can be investigated by means of High Resolution X-ray diffraction (HRXRD).

The strain induced shifts of the atoms from their bulk-lattice positions together with the shape of the objects are manifested in the distribution of the diffracted intensity in the reciprocal space in the vicinity of the Bragg diffraction maxima. The interpretation of measured reciprocal space maps is not direct, the strain field has to be determined by a comparison of the measured data and the numerical simulation of the diffraction experiment. The shape of the objects can be obtained by complementary methods, e.g. scanning electron microscopy (SEM), and this known information is usually included in the model for the computation of the strain field.

We will present the experimental technique (coplanar HRXRD) for the measurement of reciprocal space maps, which can be performed with a standard laboratory high-resolution diffractometer. The simulation of the diffraction maps is based on the simple kinematic x-ray scattering theory, which transforms the problem to the numerical computation of Fourier-like integrals producing numerical difficulties to be treated.

The strain field entering into the computation of the diffraction maps can be constructed in several ways depending on specified problem we want to solve. From many of them, three particular cases will be presented:

Firstly, one can be interested in the shape of crystalline core of the nanostructure, while this information is not accessible by SEM. If the composition of the nanostructure material is homogenous and the elastic properties are known, the strain field can be calculated using theory of elasticity by means of finite-element method (FEM). The strain field is fully defined just by the object-core shape, which has to be optimized.

Secondly, the nanostructure of a known shape has inhomogeneous composition, which locally determines the unstrained lattice parameter. The unknown distribution of occupancy enters into the strain computation. The resulting strain field is given by the solution of elasticity equations solved be FEM. If we have a model predicting the occupancy distribution, e.g. based on atomic diffusion, we can optimize its parameters to find agreement with the measured data.

Thirdly, the periodical nanostructures, made of identical material to the substrate, are covered by an additional film, which induces the strain in the objects. The task is to evaluate just the strain field in the nanostructures, although there is no reliable model for computation via the theory of elasticity. The strain field is parameterized and directly optimized to get agreement of the experimental and simulated diffraction maps.

Figure 1: Reciprocal space maps near Bragg maxima (004) and (224) measured and simulated in order to determine the shape of the (Ga,Mn)As micro-wires on GaAs substrate. The most intensive point is the substrate peak of unstrained GaAs, the intensity distributed around this peak is related to the strained parts of the substrate in the vicinity of the wires. The intensity from partially relaxed (Ga,Mn)As wires is concentrated in the intense streak bellow the substrate peak, the fringes in between are the thickness oscillations from the wires.

Figure 2: Measured (left) and simulated (right) diffraction map from oxidized Si-wires for Ge nanoheteroepitaxy. The intensity is present only along the truncation rods as a consequence of the lateral periodicity of the wires. The lateral period is very small in this case, therefore the distance of the truncation rods in the reciprocal space is large enough to see the individual satellites.