Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-10T13:18:04.918Z Has data issue: false hasContentIssue false
  • English
  • Français

Computational Methods for Electromechanical Fields in Self-Assembled Quantum Dots

Published online by Cambridge University Press:  20 August 2015

D. Barettin ,
S. Madsen ,
B. Lassen  and
M. Willatzen
D. Barettin*
Affiliation:
Alsion 2, Mads Clausen Institute, University of Southern Denmark, DK-6400 Sønderborg, Denmark
S. Madsen*
Affiliation:
Alsion 2, Mads Clausen Institute, University of Southern Denmark, DK-6400 Sønderborg, Denmark
B. Lassen*
Affiliation:
Alsion 2, Mads Clausen Institute, University of Southern Denmark, DK-6400 Sønderborg, Denmark
M. Willatzen*
Affiliation:
Alsion 2, Mads Clausen Institute, University of Southern Denmark, DK-6400 Sønderborg, Denmark
*
Email:daniele@mci.sdu.dk
Email:sma@mci.sdu.dk
Corresponding author.Email:benny@mci.sdu.dk
Email:willatzen@mci.sdu.dk
Get access

Abstract

A detailed comparison of continuum and valence force field strain calculations in quantum-dot structures is presented with particular emphasis to boundary conditions, their implementation in the finite-element method, and associated implications for electronic states. The first part of this work provides the equation framework for the elastic continuum model including piezoelectric effects in crystal structures as well as detailing the Keating model equations used in the atomistic valence force field calculations. Given the variety of possible structure shapes, a choice of pyramidal, spherical and cubic-dot shapes is made having in mind their pronounced shape differences and practical relevance. In this part boundary conditions are also considered; in particular the relevance of imposing different types of boundary conditions is highlighted and discussed.

Keywords

82-0862.20.-x68.65.Hb77.65.LyQuantum dotselectromechanical fieldscontinuum modelvalence force fieldboundary conditions
Type
Research Article
Information
Communications in Computational Physics , Volume 11 , Issue 3 , March 2012 , pp. 797 - 830
Copyright
Copyright © Global Science Press Limited 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Yu, P. Y. and Cardona, M., Fundamentals of Semiconductors-Physics and Materials Properties, Third Ed., Springer Berlin, 2005.Google Scholar
[2]Ando, T., Fowler, A. and Stern, F., Electronic properties of two-dimensional systems, Rev. Mod. Phys., 54 (1992), 437.CrossRefGoogle Scholar
[3]Stranski, I. N. and Krastanow, L., Sitzungsberichte der Akademie der Wissenschaften in Wien, Abteilung IIb, 146 (1937), 797.Google Scholar
[4]Ya Prinz, V., Grutzmacher, D., Beyer, A., David, C. and Ketterer, B., A new technique for fabricating three-dimensional micro- and nanostructures of various shapes, Nanotechnology, 12 (2001), 399–402.Google Scholar
[5]Schmidt, O. G. and Eberl, K., Nanotechnology: thin solid films roll up into nanotubes, Nature, 410 (2001), 168.Google Scholar
[6]Tanda, S., Tsuneta, T., Okajima, Y., Inagaki, K., Yamaya, K. and Hatakenaka, N., Crystal topology: a Mbius strip of single crystals, Nature, 417 (2002), 397.Google Scholar
[7]Hens, Z., Vanmaekelbergh, D., Stoffels, E. J. A. J. and van Kempen, H., Effects of crystal shape on the energy levels of zero-dimensional PbS quantum dots, Phys. Rev. Lett., 88 (2002), 236803.Google Scholar
[8]Duan, X., Niu, C., Sahi, V., Chen, J., Parce, J. W., Empedocles, S. and Goldman, J. L., Highperformance thin-film transistors using semiconductor nanowires and nanoribbons, Nature, 425 (2003), 274.Google Scholar
[9]Gao, P. X., Ding, Y., Mai, W., Hughes, W. L., Lao, C. and Wang, Z. L., Conversion of zinc oxide nanobelts into superlattice-structured nanohelices, Science, 309 (2005), 1700.Google Scholar
[10]Huang, M. H., Mao, S., Feick, H., Yan, H., Wu, Y., Kind, H., Weber, E., Russo, R. and Yang, P., Room-temperature ultraviolet nanowire nanolasers, Science, 292 (2001), 1897.CrossRefGoogle ScholarPubMed
[11]Lal, S., Link, S. and Halas, N. J., Nano-optics from sensing to waveguiding, Nature Photonics, 1 (2007), 641–648.Google Scholar
[12]Guy, L., Muensit, S. and Goldys, E. M., Electrostriction in gallium nitride, Appl. Phys. Lett., 75 (1999), 3641–3643.CrossRefGoogle Scholar
[13]Newnham, R. E., Sundar, V., Yimnirun, R., Su, J. and Zhang, Q. M., Electrostriction: nonlinear electromechanical coupling in solid dielectrics, J. Phys. Chem. B, 101 (1997), 10141.Google Scholar
[14]Willatzen, M. and Lew Yan Voon, L. C., Static and dynamic effects due to electrostriction in GaN/AlN, J. Phys. Cond. Matter, 19 (2007), 506202.Google Scholar
[15]Kornev, I., Willatzen, M., Lassen, B. and Lew Yan Voon, L. C., Electrostriction coefficients of GaN, AlN, MgO and ZnO in the wurtzite structure from first-principles, AIP Conf. Proc., 1199 (2010), 71.Google Scholar
[16]Willatzen, M., Lassen, B. and Lew Yan Voon, L. C., Nonlinearities and piezoelectric fields in AlN/GaN wurtzite heterostructures, J. Appl. Phys., 100 (2006), 124309.Google Scholar
[17]Lew Yan Voon, L. C. and Willatzen, M., Electromechanical phenomena in semiconductor nanostructures, J. Appl. Phys., 109 (2011), 031101.Google Scholar
[18]Saito, T. and Arakawa, Y., Electronic structure of piezoelectric In0.2Ga0.8N quantum dots in GaN calculated using a tight-binding method, Phys. E, 15 (2002), 169.Google Scholar
[19]Lassen, B., Willatzen, M., Barettin, D., Melnik, R. V. N. and Lew Yan Voon, L. C., Electromechanical effects in electron structure for GaN/AlN quantum dots, J. Phys. Conf. Series, 107 (2008), 012008.Google Scholar
[20]Pan, E., Elastic and piezoelectric fields in substrates GaAs (001) and GaAs (111) due to a buried quantum dot, J. Appl. Phys., 91 (2002), 6379.Google Scholar
[21]Chu, H. J., Pan, E., Ramsey, J. J., Wang, J. and Xue, C. X., A general perturbation method for inhomogeneities in anisotropic and piezoelectric solids with applications to quantum-dot nanostructures, Int. J. Solids Struct., 48 (2011), 673–679.Google Scholar
[22]Musgrave, M. J. P. and Pople, J. A., A general valence force field for diamond, Proc. Roy. Soc. London Ser. A, 268 (1962), 474.Google Scholar
[23]Nusimovici, M. A. and Birman, J. L., Lattice dynamics of wurtzite: CdS, Phys. Rev., 156 (1967), 925.Google Scholar
[24]Keating, P. N., Effect of invariance requirements on the elastic strain energy of crystals with application to the diamond structure, Phys. Rev., 145 (1966), 637–645.Google Scholar
[25]Martin, R. M., Dielectric screening model for lattice vibrations of diamond-structure crystals, Phys. Rev., 186 (1969), 871–884.Google Scholar
[26]Martin, R. M., Elastic properties of ZnS structure semiconductors, Phys. Rev. B, 1 (1970), 4005–4011.Google Scholar
[27]Fonoberov, V. A. and Balandin, A. A., Excitonic properties of strained wurzite and zinc-blende GaN/AlxGa1-xN quantum dots, J. Appl. Phys., 94 (2003), 7178–7186.Google Scholar
[28]Davies, John H., Elastic and piezoelectric fields around a buried quantum dot: a simple picture, J. Appl. Phys., 84 (1998), 1358.Google Scholar
[29]Ipatova, I. P., Malyshkin, V. G. and Shchukin, V. A., On spinodal decomposition in elastically anisotropic epitaxial films of IIIV semiconductor alloys, J. Appl. Phys., 74 (1993), 7198.Google Scholar
[30]Jogai, B., Albrecht, J. D. and Pan, E., Effect of electromechanical coupling on the strain in AlGaN/GaN heterojunction field effect transistors, J. Appl. Phys., 94 (2003), 3984.Google Scholar
[31]Willatzen, M., Lassen, B., Lew Yan Voon, L. C. and Melnik, R. V. N, Dynamic coupling of piezo-electric effects, spontaneous polarization, and strain in lattice-mismatched semiconductor quantum-well heterostructures, J. Appl. Phys., 100 (2006), 024302.Google Scholar
[32]Lassen, B., Melnik, R. V. N. and Willatzen, M., Spurious solutions in the multiband effective mass theory applied to low dimensional nanostructures, Commun. Comput. Phys., 6 (2009), 699–729.Google Scholar
[33]Fry, P. W., Itskevich, I. E., Mowbray, D. J., Scolnick, M. S., Finley, J. J., Barker, J. A., O’Reilly, E. P., Wilson, L. R., LArkin, I. A., Maksym, P. A., Hopkinson, M., Al-Khafaji, M., Davis, J. P. R., Cullis, A. G., Hill, G. and Clark, J. C., Inverted electron-hole alignment in InAs-GaAs self-assembled quantum dots, Phys. Rev. Lett., 84 (2000), 733–736.Google Scholar
[34]Zhang, Y., Motion of electrons in semiconductors under inhomogeneous strain with application to laterally confined quantum wells, Phys. Rev. B, 49 (1994), 14352–14366.Google Scholar
[35]Schliwa, A., Winkelnkemper, M. and Bimberg, D., Impact of size, shape, and composition on piezoelectric effects and electronic properties of In(Ga)AsGaAs quantum dots, Phys. Rev. B, 76 (2007), 205324.CrossRefGoogle Scholar
[36]Stier, O., Grundmann, M. and Bimberg, D., Electronic and optical properties of strained quantum dots modeled by 8-band k• p theory, Phys. Rev. B, 59 (1999), 5688–5701.Google Scholar
[37]Bimberg, D., Grundmann, M. and Ledentsov, N. N., Quantum Dot Heterostructures, John Wiley & Sons, 1999.Google Scholar
[38]Pryor, C., Pistol, M-E. and Samuelson, L., Electronic structure of strained InP/Ga0.51In0.49P quantum dots, Phys. Rev. B, 56 (1997), 10404.Google Scholar
[39]Andreev, A. D. and O’Reilly, E. P., Theory of the electronic structure of GaN/AlN hexagonal quantum dots, Phys. Rev. B, 62 (2000), 15851.Google Scholar
[40]Jogai, B., Free electron distribution in AlGaN/GaN heterojunction field-effect transistors, J. Appl. Phys., 91 (2002), 3721–3729.Google Scholar
[41]Piprek, J., Nitride Semiconductor Devices-Principles and Simulation, Wiley-VCH, 2007.Google Scholar
[42]Barker, J. A. and O’Reilly, E. P., Theoretical analysis of electron-hole alignment in InAs-GaAs quantum dots, Phys. Rev. B, 61 (2000), 13840–13850.Google Scholar
[43]Cornet, C., Schliwa, A., Even, J., Dor, F., Celebi, C., Ltoublon, A., Mac, E., Paranthon, C., Simon, A., Koenraad, P. M., Bertru, N., Bimberg, D. and Loualiche, S., Electronic and optical properties of InAs/InP quantum dots on InP(100) and InP(311)B substrates: theory and experiment, Phys. Rev. B, 74 (2006), 035312.Google Scholar
[44]Inoue, T., Wada, O., Konno, M., Yaguchi, T. and Kamino, T., Electron tomography of embedded semiconductor quantum dots, Appl. Phys. Lett., 92 (2008), 031902.Google Scholar
[45]Landau, L. D. and Lifshitz, E. M., Theory of Elasticity, Course of Theoretical Physics, Vol. 7, Pergamon Press, 1970.Google Scholar
[46]Chuang, S. L., Physics of Photonic Devices, Wiley, 2009.Google Scholar
[47]Biswas, K., Franceschetti, A. and Lany, S., Generalized valence-force-field model of (Ga,In)(N,P) ternary alloys, Phys. Rev. B, 78 (2008), 085212.Google Scholar
[48]Madsen, S. and Jensen, F., Locating seam minima for macromolecular systems, Theor. Chem. Acc., 123(5-6) (2008), 477.Google Scholar
[49]Nocedal, J. and Wright, S. J., Numerical Optimization, Springer-Verlag New York, 1999.Google Scholar
[50]Renka, R., Algorithm 660: QSHEP2D: Quadratic shepard method for bivariate interpolation of scattered data, ACM Trans. Math. Software, 14 (1988), 149–150.Google Scholar
[51]Gullett, P. M., Horstemeyer, M. F., Baskes, M. I. and Fang, H., A deformation gradient tensor and strain tensors for atomistic simulations, Model. Simul. Mater. Sci. Eng., 16 (2008), 015001.Google Scholar
[52]Pan, E., Elastic and piezoelectric fields around a quantum dot: fully coupled or semicoupled model?, J. Appl. Phys., 91 (2002), 3785–3796.Google Scholar
[53]Duggen, L. and Willatzen, M., Crystal-orientation effects on wurtzite quantum-well elec-tromechanical fields, Phys. Rev. B, in Press, 2010.CrossRefGoogle Scholar
[54]Chen, S., Gong, X. G. and Wei, S. H., Ground-state structure of coherent lattice-mismatched zinc-blende A1xBxC semiconductor alloys (x=0.25 and 0.75), Phys. Rev. B, 77 (2008), 073305.Google Scholar
[55]Liu, J. Z., Triamrchi, G. and Zunger, A., Strain-minimizing tetrahedral networks of semiconductor alloy, Phys. Rev. Lett., 99 (2007), 145501.Google Scholar
[56]Lassen, B., Wang, P., Jones, R. and Zhang, X., Fully coupled electromechanical model for dielectric elastomer sheets, IEEE Trans. Mech., in Press, 2010.Google Scholar
[57]Parrinello, M. and Rahman, A., Crystal structure and pair potentials: a molecular-dynamics study, Phys. Rev. Lett., 45 (1980), 1196–1199.Google Scholar
[58]Bir, G. L. and Pikus, G. E., Symmetry and Strain-Induced Effects in Semiconductors, Wiley New York, 1974.Google Scholar
[59]Barettin, D., Houmark, J., Lassen, B., Willatzen, M., Nielsen, T. R., Mørk, J. and Jauho, A.-P., Optical properties and optimization of electromagnetically induced transparency in strained InAs/GaAs quantum dot structures, Phys. Rev. B, 80 (2009), 235304.Google Scholar