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Abstract
The mathematics for this tool are described in the following recently published article:
https://authors.elsevier.com/a/1ThTa50WDMY7~ (This is an open link to the full article, which will remain valid until October 29, 2016.)
http://dx.doi.org/10.1016/j.jcrysgro.2016.08.050
This program calculates the required strain compensation (SC) thickness for quantum dots (QD) and their associated wetting layer (WL) based on Continuum Elasticity Theory (CET) or a modified version thereof. It can also be used to calculate SC thickness for quantum wells (QW), or to calculate QD volume or effective QD material coverage based on QD properties.
The user selects the material (Mat) and composition for the Substrate (Sub), QD, and SC systems. The program works with binary, ternary, or quaternary, or higher order compounds for each of these three categories.
The user then enters the known (through AFM or some other method) average Diameter, Height, Density, and Wetting Layer thickness of the QDs. If the user is only interested in calculating SC thickness for QWs, the QW thickness should be entered as a QD Height (and the other parameters Diameter, Density, and Wetting Layer will not affect the calculation)
Lattice constants (lc) and elastic stiffness (C11, C21) constants for the Sub, QD, and SC materials are displayed as outputs. If the materials are binary (e.g.: GaAs, InAs, etc.) the material parameters are taken from Vurgaftman, et al., and denoted 'i' [1]. If they are ternaries, quaternaries, or higher order alloys, the lattice constant and Literature/Interpolated (denoted 'i', for interpolated, though in the case of binary compounds it is simply the Vurgaftman values) values are determined using a weighted average (Vegard's law) of their binary components [2]. An alternative empirical method of calculating stiffness constants (denoted 'e', for Empirically Calculated) based on material lattice constant is also used for comparison presented by Adachi [3].
QD volumes are displayed as outputs. Three methods are used to calculate QD volume. The first is as a spherical cap (Sph. Cap), as described by Leonard, et al. [4]. This is value is not used in any subsequent calculations and is included only for reference. The second volume is of a cylinder (Cylinder), as described by Bailey, et al. [5]. This value is used for the calculation of necessary SC using Modified CET (QD as Cylinder). Finally, the volume of a QD as a oblate hemispheroid (Obl. Sph.) as described by Polly, et al. [6]. This value is used for the calculation of necessary SC using Modified CET (QD as Oblate-Hemispheroid).
The final output is the SC layer thicknesses (Req. SC Thick) calculated using several methods previously mentioned. The one not yet referenced is CET (QD Height as QW), which was developed by Ekins-Daukes et al. [7], which treats the QD heights as a 1-D QW. These values are calculated using both the interpolated (Lit (i)) and empirically derived (Calc (e)) stiffness constants. Additionally, the effective coverage of QD material (Eff QD+WL Thick) is shown.
The input values are printed below all calculations to simplify usage across multiple simulations.
Sponsored by
This work was supported by the National Science Foundation under Grant DMR-0955752 and the Air Force Research Laboratory, Project FA9453-11-C-0253
References
[1] I. Vurgaftman, J. R. Meyer, and L. R. Ram-Mohan, "Band parameters for III-V compound semiconductors and their alloys", Journal of Applied Physics, vol. 89, no. 11, pp. 5815-5875, Jun. 1, 2001, doi:10.1063/1.1368156.
[2] S. Adachi, "Band gaps and refractive indices of AlGaAsSb, GaInAsSb, andInPAsSb: key properties for a variety of the 2-4-µm optoelectronic device applications", Journal of Applied Physics, vol. 61, no. 10, pp. 4869-4876, May 15, 1987, doi: 10.1063/1.338352.
[3] S. Adachi, Properties of group-IV, III-V and II-VI semiconductors. Chichester, England; Hoboken, NJ: John Wiley & Sons, 2005, isbn: 9780470090329. doi: 10.1002/0470090340.
[4] D. Leonard, K. Pond, and P. M. Petroff, "Critical layer thickness for self-assembled InAs islands on GaAs", Physical Review B, vol. 50, no. 16, pp. 11687-11692, Oct. 15, 1994. doi: 10.1103/PhysRevB.50.11687.
[5] C. G. Bailey, S. M. Hubbard, D. V. Forbes, and R. P. Raffaelle, "Evaluation of strain balancing layer thickness for InAs/GaAs quantum dot arrays using high resolution x-ray diffraction and photoluminescence", Applied Physics Letters, vol. 95, no. 20, Nov. 2009, doi: 10.1063/1.3264967.
[6] S. J. Polly, C. G. Bailey, A. J. Grede, D. V. Forbes, and S. M. Hubbard, “Calculation of strain compensation thickness for III–V semiconductor quantum dot superlattices,” Journal of Crystal Growth, vol. 454, pp. 64–70, Nov. 2016.
[7] N. J. Ekins-Daukes, K. Kawaguchi, and J. Zhang, "Strain-balanced criteria for multiple quantum well structures and its signature in x-ray rocking curves", Crystal Growth & Design, vol. 2, no. 4, pp. 287?292, Jul. 1, 2002, doi: 10.1021/cg025502y.
Publications
S. J. Polly, C. G. Bailey, A. J. Grede, D. V. Forbes, and S. M. Hubbard, “Calculation of strain compensation thickness for III–V semiconductor quantum dot superlattices,” Journal of Crystal Growth, vol. 454, pp. 64–70, Nov. 2016. http://dx.doi.org/10.1016/j.jcrysgro.2016.08.050
Cite this work
Researchers should cite this work as follows:
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S. J. Polly, C. G. Bailey, A. J. Grede, D. V. Forbes, and S. M. Hubbard, “Calculation of strain compensation thickness for III–V semiconductor quantum dot superlattices,” Journal of Crystal Growth, vol. 454, pp. 64–70, Nov. 2016. http://dx.doi.org/10.1016/j.jcrysgro.2016.08.050