The mixed FEM for analysis of quantum-dot systems based on gradient theory

by

The mixed FEM for analysis of quantum-dot systems based on gradient theory

Jan Sladek1, Vladimir Sladek1, Miroslav Repka1, Slavomir Hrcek2

1Institute of Construction and Architecture, Slovak Academy of Sciences, 84503 Bratislava, Slovakia.

2Faculty of Mechanical Engineering, University of Zilina, 01026 Zilina, Slovakia.

DOI:

https://doi.org/10.7494/cmms.2018.3.0616

Abstract:

The QD nanostructures are analyzed under a thermal load. The dimensions of the QDs are of the same order as the material length scale. Therefore, the gradient elasticity theory should be applied to account for the size-dependent behavior of such nano-sized QDs. Since governing equations contain higher order derivatives than in conventional approaches the C1-elements are required for approximation of primary fields in the FEM. The mixed FEM are developed here, where C0 continuous interpolation is applied independently for displacement and displacement gradients. The kinematic constraints between strains and displacements are satisfied by collocation at some cleverly chosen internal points in elements. A unit cell of Indium Arsenide QD in a finite sized Gallium Arsenide (GaAs) substrate is analysed.

Cite as:

Sladek, J., Sladek, V., Repka, M., Hrcek, S. (2018). The mixed FEM for analysis of quantum-dot systems based on gradient theory. Computer Methods in Materials Science, 18(3), 81 – 89. https://doi.org/10.7494/cmms.2018.3.0616

Article (PDF):

Keywords:

Mixed FEM, Gradient theory, Thermo-piezoelectricity, Quantum dots, Thermal shock

References:

Aifantis, E., 1984, On the microstructural origin of certain inelasticmodels, ASME J. Eng. Mater. Technol., 106, 326-330.

Altan, S., Aifantis, E., 1992, On the structure of the mode IIIcrack-tip in gradient elasticity, Scripta Metall. Mater., 26,319-324.

Askes, H., Aifantis, E.C., 2011, Gradient elasticity in statics anddynamics: An overview of formulations, length scale identificationprocedures, finite element implementations andnew results, Int. J. Solids Struct., 48, 1962-1990.

Benabbas, T., Francois, P., Androussi, Y., Lefebvre, A., 1996,Stress relaxation in highly strained InAs/GaAs structuresas studied by finite element analysis and transmission electronmicroscopy, J. Appl. Phys., 80, 2763-2767.

Bimberg, D., Grundmann, M., Ledentsov, N.N., 1998, QuantumDot Heterostructures, Wiley, New York.Bishay, P.L., Atluri, S.N., 2012, High performance 3D hybrid/mixed, and simple 3D Voronoi cell finite elements,for macro- & micro-mechanical modeling of solids, withoutusing multi-field variational principles, CMES: ComputerModeling in Engineering & Sciences, 84, 41-97.

Chen, X.O., Dong, B., Lei, X.L., 2008, Thermal rectificationeffect of and interacting quantum dot, Chin. Phys. Lett.,25, 3032-3035.

Davies, J.H., Larkin, I.A., 1994, Theory of potential modulationin lateral surface superlattices, Phys. Rev. B, 49, 4800.

Davies, J.H., 1998, Elastic and piezoelectric fields around aburied quantum dot: A simple picture, J. Appl. Phys., 84,1358-1365.

DiVincenzo, D.P., 1986, Dispersive corrections to continuumelastic theory in cubic crystals, Phys. Rev. B., 34, 5450-5465.

Dong, L., Atluri, S.N., 2011, A simple procedure to developefficient & stable hybrid/mixed elements, and Voronoi cellfinite elements for macro- & micromechanics, CMC:Computers, Materials & Continua, 24, 61-104.

Giazotto, F., Heikkil, T.T., Luukanen, A., Savin, A.M., Pekola,J.P., 2006, Opportunities for mesoscopics in thermometryand refrigeration: Physics and Applications, Rev. Mod.Phys., 78, 217-274.

Gitman, I., Askes, H., Kuhl, E., Aifantis, E., 2010, Stress concentrationsin fractured compact bone simulated with aspecial class of anisotropic gradient elasticity, InternationalJournal of Solids and Structures, 47, 1099-1107.

Glazov, V.M., Pashinkin, A.S., 2000, Thermal expansion andheat capacity of GaAs and InAs, Inorganic Materials, 36,225-231.

Grundmann, M., Stier, O., Bimberg, D., 1995, InAs/GaAs pyramidalquantum dots: Strain distribution, optical phonons,and electronic structure, Phys. Rev. B, 52, 11969.

Jogai, B., 2001, Three-dimensional strain field calculations inmultiple InN/AlN wurtzite quantum dots, J. Appl. Phys.,90, 699.

Liang, X., Shen, S.P., 2013, Size-dependent piezoelectricity andelasticity due to the electric field-strain gradient couplingand strain gradient elasticity, Int. J. Appl. Mech., 5,1350015.

Liao, X.Z., Zhou, J., Cockayne, D.J.H., Leon, R., Lobo, C.,1999, Indium Segregation and Enrichment in CoherentInxGa1-x As/GaAs Quantum Dots, Phys. Rev. Lett., 82,5148-5151.

Liu, Y.M., Yu, Z.Y., Ren, X.M., Xu Z.H., 2008, Self-organizedGaN/AlN hexagonal quantum-dots: strain distribution andelectronic structure, Chinese Physics B, 17, 3471-3478.

Majdoub, M.S., Sharma, P., Cagin, T., 2008, Enhanced sizedependentpiezoelectricity and elasticity in nanostructuresdue to the flexoelectric effect, Physics Review B, 77,125424.

Mindlin, R.D., 1964, Micro-structure in linear elasticity, Arch.Ration. Mech. Anal., 16, 51-78.

Pan, E., 2002a, Elastic and piezoelectric fields around a quantumdot: Fully coupled or semicoupled model, J. Appl.Phys., 91, 3785-3796.

Pan, E., 2002b, Elastic and piezoelectric fields in substratesGaAs (001) and GaAs (111) due to a buried quantum dot,J. Appl. Phys., 91, 6379-6387.

Parton, V.Z., Kudryavtsev, B.A., 1988, Electromagnetoelasticity,Piezoelectrics and Electrically Conductive Solids, Gordonand Breach Science Publishers, New York.Patil, S.R., Melnik R.V.N., 2009, Thermoelectromechanicaleffects in quantum dots, Nanotechnology, 20, 125402.

Pryor, C., Pistol, M.E., Samuelson, L., 1997, Electronic structureof strained InP/Ga0.51In0.49P quantum dots, Phys.Rev. B, 56, 10404-10411.

Sladek, J., Sladek, V., Wunsche, M., Tan, C.L., 2017, Crackanalysis of size-dependent piezoelectric solids under athermal load, Engineering Fracture Mechanics, 182, 187-200.

Toupin, R.A., Gazis, D.C., 1963, Surface effects and initialstress in continuum and lattice models of elastic crystals.In: Wallis, R.F. (Ed.), Proc. Int. Conf. on Lattice Dynamics,Copenhagen, 597-605.

Yaghoubi, S.T., Mousavi, S.M., Paavola, J., 2017. Buckling ofcentrosymmetric anisotropic beam structures within straingradient elasticity, International Journal of Solids andStructures, 109, 84-92.