Modeling the compositional dependence of electron diffraction in dilute GaAs- and GaP-based compound semiconductors

The change in the intensity of the (200) electron-beam reflection induced by the incorporation of isovalent impurities in GaAs and GaP is studied theoretically. Calculations are performed in the framework of the kinematical scattering theory. The structure factor of random alloys was obtained using atomic form factors calculated with the density-functional theory and an empirical-potential valence force-field model for the structure relaxation. The calculations include the effect of redistribution of the electron density on the electron-scattering amplitudes as well as the effect of the local lattice distortions associated with the impurity sites. We propose a way to calculate these distortions analytically and to introduce them in a simple form to the expression for the structure factor. This method is an alternative to the simulations, which invoke demanding computations for atomic relaxation, and enables quantitative prediction of the compositional variation of the structure factor for dilute alloys taking into account static atomic displacements. The implications of the results for quantification of composition fluctuations in heterostructures using dark-field transmission electron microscopy are discussed. The effect of nitrogen and boron incorporation on the intensity of the (200) reflection is found to be partly compensated by the static atomic displacements they cause. Neglecting this effect would lead to an underestimation of the impurity content by approximately a factor of two. The redistribution of the electron density is found to be less crucial for the evaluation of the chemical composition leading to a relative error in the (200) scattering amplitude of about 16%.

Figure 1

Compositional dependence of the (200) electron-beam intensity in ternary III-V semiconductor alloys calculated using the virtual-crystal approximation (◻) and including static atomic displacements (SAD) for large supercells (◼). Experimental results for ${\text{GaAs}}_{1-x}{\text{N}}_x/\text{GaAs}$ quantum wells (Ref. 10) are shown by open triangles on panel (a). The solid line shows results of Eqs. (12, 13), which include the effect of atomic displacements in a simplified way. Atomic form factors calculated using DFT are taken from Table . Results of the ab initio calculations for a single impurity in the 64-atom supercell are shown by crosses. Results of the dynamical theory for the acceleration voltage of 300 kV and the sample thickness of 50 nm obtained assuming the virtual-crystal approximation are shown on panels (a) and (b) by dashed lines. The compositional variation of the relative intensities, which additionally takes into account diffuse scattering, are shown on panels (a) and (b).

Figure 2

Local bond configuration of isolated substitutional atom C in zinc-blende AB host crystal. Dotted lines correspond to the unrelaxed bond configuration.

Figure 3

Compositional bowing parameter for the (200) electron-scattering intensity in various III-V ternary alloys. Results are calculated using Eq. (15) assuming different input parameters for the atomic scattering amplitudes (ASA) from Table . In the calculations, the local bond strain $ϵ$ associated with the static atomic displacements (SAD) is either determined using Eq. (13) or assumed to be zero in the case labeled as “no SAD.”