Diffuse X-ray scattering from graded SiGe layers

J. Endres

Department of Condensed Matter Physics, Faculty of Mathematics and Physics, Charles University Prague, Ke Karlovu 5, 121 16 Prague 2, Czech Republic

e-mail: jan.endres@seznam.cz

Graded SiGe layers are frequently used as virtual substrate for electronic and optoelectronic device applications. The grading makes it possible to make a substrate with lattice constant matched to the device and to decrease the density of threading dislocations crossing the layer and therefore to improve the quality of the layers deposited on the top.

Intensity distribution of scattered X-rays I(Q) is measured as the reciprocal space map of the Q=(Q_x,Q_z) plane or along various direction in this plane (the X-ray beams are assumed to be well collimated in the incidence xz plane). The intensity is concentrated around the reciprocal lattice point so that we consider the wave vector deviation q=Q-Q⁰ (Q⁰ is the nearest reciprocal lattice vector). In the kinematic approximation, the intensity scattered by the layer with thickness h can be represented as Fourier integral [1]

,

(1)

where A is constant, V (V′) is volume of scattering area and

(2)

is correlation function, where u(r) is the displacement at the site r due to randomly distributed dislocations, and the average is performed over their positions. We consider the case of uncorrelated dislocations in two systems, with dislocation lines parallel to x and y axis. Then we can derive [1] approximation of the correlation function. For calculations we need to know the distribution of dislocations in the layer. We can use two models: Tersoff model [2] or model Dodson-Tsao [3].

Tersoff model describes equilibrium distribution of dislocations. The density of dislocations is just enough to exactly cancel the mismatch up to a distance z_c from the interface between layer and substrate and there are no dislocations at all above z_c. The boundary z_c of the dislocation-free region is given by the condition

,

(3)

where f(z) is the mismatch between bulk lattice constant a(z) and substrate lattice constant, λ is energy per unit length of the dislocation, b is length of the Burgers vector of misfit dislocation and c is the appropriate elastic constant. If the layer is graded linearly from the substrate, Equation (3) can be solved analytically and density of dislocations is constant in whole region under z_c.

Dodson-Tsao model describes the kinetics of plastic relaxation in the layer by the equation

,

(4)

where δ_|| is lateral misfit, K is constant, α(σ_exc) is , σ_exc is excess stress, μ is a share modulus and δ₀ is constant, which makes it possible to start plastic relaxation even without presence of misfit dislocations. From the solution of equation (4) the relaxation of the layer r(z)=δ_||(z)/ f(z) is obtained.

1. V. M. Kaganer, R. Köhler, M. Schmidbauer, R. Opitz, X-ray diffraction peaks due to misfit dislocations in heteroepitaxial structures. Physical Review B, 55 (1997) 1793-1810.

2. J. Tersoff, Dislocation and strain relief in compositionally graded layers. Appl. Phys. Lett., 62 (1993) 693-695.

3. J. Y. Tsao, B. W. Dodson, Excess stress and stability of strained heterostructures. Appl. Phys. Lett, 53 (1988) 848-850.