Alloy broadening

Alloy broadening is a mechanism by which the spectral lines of an alloy are broadened by the random distribution of atoms within the alloy.^[1] It is one of a number of spectral line broadening mechanisms.

Alloy broadening occurs because the random distribution of atoms in an alloy causes a different material composition at different positions. In semiconductors and insulators the different material composition leads to different band gap energies. This gives different exciton recombination energies. Therefore, depending on the position where an exciton recombines the emitted light has a different energy. The alloy broadening is an inhomogeneous line broadening, meaning that its shape is Gaussian.^{[citation needed]}

Binary alloy

In the mathematical description it is assumed that no clustering occurs within the alloy. Then, for a binary alloy of the form ${\displaystyle {\ce {A_{1-x}B_{x}}}}$, e.g. ${\displaystyle {\ce {Si_{1-x}Ge_{x}}}}$, the standard deviation of the composition is given by:^[2]

${\displaystyle \Delta x={\sqrt {\frac {x\cdot (1-x)}{N}}}}$,

where ${\displaystyle N}$ is the number of atoms within the excitons' volume, i.e. ${\displaystyle N=V_{exc}\cdot n}$ with ${\displaystyle n}$ being the atoms per volume. In general, the band gap energy ${\displaystyle E_{g}}$ of a semiconducting alloy depends on the composition, i.e. ${\displaystyle E_{g}}$. The band gap energy can be considered to be the fluorescence energy. Therefore, the standard deviation in fluorescence is:^[3]

${\displaystyle \Delta E={\frac {\mathrm {d} E_{g}}{\mathrm {d} x}}\cdot {\sqrt {x\cdot {\frac {1-x}{N}}}}}$

As the alloy broadening belongs to the group of inhomogeneous broadenings the line shape of the fluorescence intensity ${\displaystyle I(E)}$ is Gaussian:^{[why?][citation needed]}

${\displaystyle I(E)\sim \exp \left(-{\frac {(E-E_{0})^{2}}{2\cdot \Delta E^{2}}}\right)}$

References

1. Bahaa E. A. Saleh, Malvin Carl Teich: Grundlagen der Photonik. Wiley-VCH, ISBN 978-3-527-40677-7, Seite 825 ([1], p. PA825, at Google Books).
2. J. Weber, M. I. Alonso (1989), "Near-band-gap photoluminescence of Si-Ge alloys", Phys. Rev. B, vol. 40, no. 8, pp. 5683–5693, Bibcode:1989PhRvB..40.5683W, doi:10.1103/PhysRevB.40.5683, PMID 9992606, retrieved 2018-06-15, Format: PDF, KBytes: 2005
3. Schubert, E. F.; Göbel, E. O.; Horikoshi, Y.; Ploog, K.; Queisser, H. J. (1984-07-15). "Alloy broadening in photoluminescence spectra of Al_xGa_1−xAs". Physical Review B. 30 (2). American Physical Society (APS): 813–820. Bibcode:1984PhRvB..30..813S. doi:10.1103/physrevb.30.813. ISSN 0163-1829.