Temperature effect on the 002 structure factor of ternaryGa1−xInxAscrystals

Abstract

In this work, we study the effect of relaxation of the atomic positions and lattice parameter of mixed Ga 1−x In x As systems on the 002 electron atomic scattering amplitudes ͑ASA͒. The ASAs are shown to vary strongly with lattice parameter but to be indistinguishable for different In concentrations when the lattice parameter is held fixed. The resulting relationship between ASA and lattice parameter is then used in a finite temperature, zero pressure Monte Carlo relaxation of the mixed crystal. The consequence of the use of this atomic scattering amplitude is a reduction of the structure factor, leading to an increase of the In concentration at which the 002 structure factor vanishes. This method directly includes the effect of temperature on the ASA and on the structure factors and therefore does not require an explicit knowledge of the Debye-Waller factor.It was recently demonstrated that for scattering vectors lower than some critical values, electron diffraction is more sensitive to charge fluctuations in crystals than x-ray diffraction, 1 thus supporting the use of transmission electron microscopy ͑TEM͒ for quantification purposes. The quantification of Ga 1−x In x As crystals using TEM requires an accurate knowledge of the structure factors ͑SFs͒ which are obtained from the atomic positions and the scattering amplitudes ͑ASAs͒ of the atoms of the system. Several sets of ASAs have been considered ranging from the isolated atom values 2 to those obtained from density functional theory ͑DFT͒ calculations on the binary crystals. 3 The latter set of parameters are more accurate since it accounts for the charge redistribution in the system as a result of bonding. For example, it is well known that the charge density distribution along the Ga-As bond shows a slight shift of charge towards the As atom. 4 Also, the static atomic displacement which is present in these mixed crystals as a result of atoms with different atomic radii sharing a mixed sublattice strongly influences the SF. 5 These atomic displacements were first observed by Mikkelsen and Boyce in 1983 ͑Ref. 6͒ using the extended x-ray absorption fine structure ͑XAFS͒ and has been further confirmed by other workers. 7 Its importance in the quantification of III-V nitrides has been recently demonstrated. 8 By incorporating static atomic displacements via the Keating 9 valence force field ͑VFF͒, Rosenauer et al. 3 used DFT-based modified atomic scattering amplitudes ͑MASA͒ to calculate the 002 SF of strained and bulk Ga 1−x In x As crystals for various In concentrations. These weak reflections are composition sensitive and are therefore useful for composition analysis via the high resolution transmission electron microscopy ͑HRTEM͒. 10 So far little or no attempt has been made to include temperature effects in the ASAs. This effect can be included in the SF through the Debye-Waller factor. 11 The knowledge of this factor for III-V binary and mixed crystals is very scanty. Even if these factors are known accurately for the binary systems, they can not be use...

Modified atomic scattering amplitudes and size effects on the 002 and 220 electron structure factors of multiple Ga1−xInxAs/GaAs quantum wells

“…8 whereby linear fits of the MASA over relevant lattice parameters are performed. During the MC data analysis cycles each atom is considered, and the local lattice parameter defined as the radius of the fourth coordination shell around the atom is calculated and by linear interpolation the corresponding MASA is deduced.…”

“…Recently we used this approach to probe the temperature effect on the 002 SF of bulk Ga 1−x In x As. 8 However, the MASAs that have so far been used are those obtained from DFT calculations based on the binary zinc-blende crystals. The effect on the MASAs of the static atomic displacements present in the ternary and quaternary systems has never been investigated.…”

“…However, we have recently shown that for temperatures lower than 500 K, the 002 SF of bulk Ga 1−x In x As systems can be approximated by the average SF. 8 We do not expect the temperature behavior to change for the heterostructures; hence for fixed In content it is only the static atomic displacement ͑and/or the strain͒ that may influence the intensity. The presence of strain or static displacements in the heterostructures supports the idea of a SDWF ͑Refs.…”