Abstract

GaAs nanomembranes are thin crystalline GaAs semiconductor structures that can be bent or otherwise elastically deformed from their natural shape. We present a microscopic theory of the linear optical response of such deformed structures. Our approach combines conventional structural analysis (based on the theory of elasticity), the valence band Hamiltonians (Luttinger and Pikus–Bir) for III–V semiconductors, and the semiconductor Hamiltonian including Coulomb interaction. We formulate the general equation of motion for the interband polarization for thin elastically deformed nanomembranes. A simple limiting case results from the single-subband approximation and the averaged-strain approximation. Within this approximation scheme, we present numerical results for excitonic spectra for a cylindrically deformed membrane.

© 2012 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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  37. M. Wraback, H. Shen, J. Pamulapati, M. Dutta, P. G. Newman, M. Taysing-Lara, and Y. Lu, “Femtosecond studies of excitonic optical non-linearities in GaAs/AlxGa1−xAs multiple quantum wells under in-plane uniaxial strain,” Surf. Sci. 305, 238–242(1994).
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  39. M. Wraback, H. Shen, S. Liang, C. R. Gorla, and Y. Lu, “High contrast, ultrafast optically addressed ultraviolet light modulator based upon optical anisotropy in ZnO films grown on R-plane sapphire,” Appl. Phys. Lett. 74, 507–509 (1999).
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  53. We neglect the electron-hole exchange effect, which is on the order of tens of micro electron volts and thus small compared to typical values of the broadening of exciton resonances in quasi-two-dimensional systems. Under certain circumstances, not considered in this paper, it can become important or even dominant in excitonic spectra, especially in quantum dots (see, e.g., Ref. [68]).
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    [CrossRef]

2011 (1)

N. W. Sinclair, J. K. Wuenschell, Z. Vörös, B. Nelsen, D. W. Snoke, M. H. Szymanska, A. Chin, and J. Keeling, “Strain-induced darkening of trapped excitons in coupled quantum wells at low temperature,” Phys. Rev. B 83, 245304 (2011).
[CrossRef]

2010 (3)

C. Deneke, A. Malachias, S. Kiravittaya, M. Benyoucef, T. Metzger, and O. Schmidt, “Strain states in a quantum well embedded into a rolled-up microtube: x-ray and photoluminescence studies,” Appl. Phys. Lett. 96, 143101 (2010).
[CrossRef]

Y. Mei, S. Kiravittaya, S. Harazim, and O. G. Schmidt, “Principles and applications of micro and nanoscale wrinkles,” Mat. Sci. Eng. R 70, 209–224 (2010).
[CrossRef]

M. Hossein-Zadeh and K. J. Vahala, “An optomechanical oscillator on a silicon chip,” IEEE J. Sel. Top. Quantum Electron. 16, 276–287 (2010).
[CrossRef]

2009 (2)

Y. Mei, D. J. Thurmer, C. Deneke, S. Kiravittaya, Y.-F. Chen, A. Dadgar, F. Bertram, B. Bastek, A. Krost, J. Christen, T. Reindl, M. Stoffel, E. Coric, and O. G. Schmidt, “Fabrication, self-assembly, and properties of ultrathin AIN/GaN porous crystalline nanomembranes: tubes, spirals, and curved sheets,” ACS Nano 3, 1663–1668 (2009).
[CrossRef]

P. Cendula, S. Kiravittaya, Y. Mei, C. Deneke, and O. Schmidt, “Bending and wrinkling as competing relaxation pathways for strained free-hanging films,” Phys. Rev. B 79, 085429 (2009).
[CrossRef]

2008 (4)

L. Zhang, L. Dong, and B. J. Nelson, “Bending and buckling of rolled-up SiGe/Si microtubes using nanorobotic manipulation,” Appl. Phys. Lett. 92, 243102 (2008).
[CrossRef]

H. C. Ko, M. P. Stoykovich, J. Song, V. Malyarchuk, W. M. Choi, C.-J. Yu, J. B. Geddes, J. Xiao, S. Wang, Y. Huang, and J. A. Rogers, “A hemispherical electronic eye camera based on compressible silicon optoelectronics,” Nature 454, 748–753 (2008).
[CrossRef]

A. Bernardi, S. Kiravittaya, A. Rastelli, R. Songmuang, D. Thurmer, M. Benyoucef, and O. Schmidt, “On-chip Si/SiOx microtube refractometer,” Appl. Phys. Lett. 93, 094106 (2008).
[CrossRef]

L. Duggen, M. Willatzen, and B. Lassen, “Crystal orientation effects on the piezoelectric field of strained zinc-blende quantum-well structures,” Phys. Rev. B 78, 205323 (2008).
[CrossRef]

2006 (4)

G. Bester, X. Wu, D. Vanderbilt, and A. Zunger, “Importance of second-order piezoelectric effects in zinc-blende semiconductors,” Phys. Rev. Lett. 96, 187602 (2006).
[CrossRef]

S. Seidl, M. Kroner, A. Högele, and K. Karrai, “Effect of uniaxial stress on excitons in a self-assembled quantum dot,” Appl. Phys. Lett. 88, 203113 (2006).
[CrossRef]

Y. Sun, E. Menard, and J. A. Rogers, “Gigahertz operation in flexible transistors on plastic substrates,” Appl. Phys. Lett. 88, 183509 (2006).
[CrossRef]

S. Mendach, R. Songmuang, S. Kiravittaya, A. Rastelli, M. Benyoucef, and O. Schmidt, “Light emission and wave guiding of quantum dots in a tube,” Appl. Phys. Lett. 88, 111120 (2006).
[CrossRef]

2005 (1)

Y. Sun, S. Kim, I. Adesida, and J. A. Rogers, “Bendable GaAs metal-semiconductor field-effect transistors formed with printed GaAs wire arrays on plastic substrates,” Appl. Phys. Lett. 87, 083501 (2005).
[CrossRef]

2004 (1)

N. Ohtani, K. Kishimoto, K. Kubota, S. Saravanan, Y. Sato, S. Nashima, P. Vaccaro, T. Aida, and M. Hosoda, “Uniaxial-strain-induced transition from type-II to type-I band configuration of quantum well microtubes,” Physica E 21, 732–736 (2004).
[CrossRef]

2001 (1)

T. Chung, G. Walter, and J. N. Holonyak, “Coupled strained-layer InGaAs quantum-well improvement of an InAs quantum dot AlGaAs-GaAs-InGaAs-InAs heterostructure laser,” Appl. Phys. Lett. 79, 4500–4502 (2001).
[CrossRef]

2000 (2)

J. Coleman, “Strained-layer InGaAs quantum-well heterostructure lasers,” IEEE J. Sel. Top. Quantum Electron. 6, 1008–1013(2000).
[CrossRef]

D. W. Snoke, V. Negoita, and K. Eberl, “Energy shifts of indirect excitons in coupled quantum wells,” J. Luminesc. 87–89, 157–161 (2000).
[CrossRef]

1999 (3)

M. Wraback, H. Shen, S. Liang, C. R. Gorla, and Y. Lu, “High contrast, ultrafast optically addressed ultraviolet light modulator based upon optical anisotropy in ZnO films grown on R-plane sapphire,” Appl. Phys. Lett. 74, 507–509 (1999).
[CrossRef]

G. Khitrova, H. M. Gibbs, F. Jahnke, M. Kira, and S. W. Koch, “Nonlinear optics of normal-mode coupling semiconductor microcavities,” Rev. Mod. Phys. 71, 1591–1639 (1999).
[CrossRef]

B. Lane, Z. Wu, A. Stein, J. Diaz, and M. Razeghi, “InAsSb/InAsP strained-layer superlattice injection lasers operating at 4.0 μm grown by metal-organic chemical vapor deposition,” Appl. Phys. Lett. 74, 3438–3440 (1999).
[CrossRef]

1998 (2)

S. Nakamura, M. Senoh, S. Nagahama, N. Iwasa, T. Yamada, T. Matsushita, H. Kiyoku, Y. Sugimoto, T. Kozaki, H. Umemoto, M. Sano, and K. Chocho, “InGaN/GaN/AlGaN-based laser diodes with modulatioin-doped strained-layer superlattices grown on an epitaxially laterally overgrown GaN substrate,” Appl. Phys. Lett. 72, 211–213 (1998).
[CrossRef]

Z. Liu, S. Pau, K. Syassen, and J. Kuhl, “Photoluminescence and reflectance studies of exciton transitions in wurtzite GaN under pressure,” Phys. Rev. B 58, 6696–6699 (1998).
[CrossRef]

1997 (1)

S.-H. Park and D. Ahn, “Many-body effects on optical gain in strained hexagonal and cubic GaN/AlGaN quantum well lasers,” Appl. Phys. Lett. 71, 398–400 (1997).
[CrossRef]

1996 (1)

S. L. Chuang and C. S. Chang, “Effective-mass Hamiltonian for strained wurtzite GaN and analytical solutions,” Appl. Phys. Lett. 68, 1657–1659 (1996).
[CrossRef]

1995 (3)

D. Ahn, S. Yoon, S. Chuang, and C.-S. Chang, “Theory of optical gain in strained-layer quantum wells within the 6×6 Luttinger-Kohn model,” J. Appl. Phys. 78, 2489–2497 (1995).
[CrossRef]

G. D. C. Kuiken, “The symmetry of the stress tensor,” Ind. Eng. Chem. Res. 34, 3568–3572 (1995).
[CrossRef]

M. Wraback, H. Shen, J. Pamulapati, P. G. Newman, and M. Dutta, “Polarization dependent excitonic optical nonlinearities in GaAs/AlGaAs multiple quantum wells under anisotropic in-plane strain,” Phys. Rev. Lett. 74, 1466–1469 (1995).
[CrossRef]

1994 (4)

W. Bardyszewski and D. Yevick, “Electroabsorption and electrorefraction effects in quantum-well modulators,” Phys. Rev. B 49, 5368–5378 (1994).
[CrossRef]

D. Ahn and S. L. Chuang, “The theory of strained-layer quantum-well lasers with bandgap renormalization,” IEEE J. Quantum Electron. 30, 350–365 (1994).
[CrossRef]

H. Shen, J. Pamulapati, M. Wraback, M. Taysing-Lara, M. Dutta, H. C. Kuo, and Y. Lu, “High contrast optical modulator based on electrically tunable polarization rotation and phase retardation in uniaxially strained (100) multiple quantum wells,” IEEE Photon. Technol. Lett. 6, 700–702 (1994).
[CrossRef]

M. Wraback, H. Shen, J. Pamulapati, M. Dutta, P. G. Newman, M. Taysing-Lara, and Y. Lu, “Femtosecond studies of excitonic optical non-linearities in GaAs/AlxGa1−xAs multiple quantum wells under in-plane uniaxial strain,” Surf. Sci. 305, 238–242(1994).
[CrossRef]

1993 (3)

H. Shen, M. Wraback, J. Pamulapati, M. Dutta, P. G. Newman, A. Ballato, and Y. Lu, “Normal incidence high contrast multiple quantum well light modulator based on polarization rotation,” Appl. Phys. Lett. 62, 2908–2910 (1993).
[CrossRef]

H. Shen, M. Wraback, J. Pamulapati, P. G. Newman, M. Dutta, Y. Lu, and H. C. Kuo, “Optical anisotropy in GaAs/AlxGa1−xAs multiple quantum wells under thermally induced uniaxial strain,” Phys. Rev. B 47, 13933–13936 (1993).
[CrossRef]

C.-K. Sun, H. K. Choi, K. A. Wang, and J. G. Fujimoto, “Femtosecond gain dynamics in InGaAs/AIGaAs strained-layer single-quantum-well diode lasers,” Appl. Phys. Lett. 63, 96–98 (1993).
[CrossRef]

1992 (1)

D. Nichols and P. Bhattacharya, “Differential gain in InP-based strained layer multiple quantum well lasers,” Appl. Phys. Lett. 61, 2129–2131 (1992).
[CrossRef]

1991 (2)

S. L. Chuang, “Efficient band-structure calculations of strained quantum wells,” Phys. Rev. B 43, 9649–9661 (1991).
[CrossRef]

S. Offsey, L. Lester, W. Schaff, and L. Eastman, “High-speed modulation of strained-layer InGaAs-GaAs-AlGaAs ridge waveguide multiple quantum well lasers,” Appl. Phys. Lett. 58, 2336–2338 (1991).
[CrossRef]

1990 (2)

T. Chen, L. Eng, B. Zhao, Y. Zhuang, S. Sanders, and H. Morkoc, “Submilliamp threshold InGaAs-GaAs strained layer quantum-well laser,” IEEE J. Quantum Electron. 26, 1183–1190 (1990).
[CrossRef]

S. W. Corzine, R. H. Yan, and L. A. Coldren, “Theoretical gain in strained InGaAs/AlGaAs quantum wells including valence-band mixing effects,” Appl. Phys. Lett. 57, 2835–2837 (1990).
[CrossRef]

1989 (2)

T. C. Chong and C. G. Fonstad, “Theoretical gain of strained-layer semiconductor lasers in the large strain regime,” IEEE J. Quantum Electron. 25, 171–178 (1989).
[CrossRef]

E. P. O’Reilly, “Valence band engineering in strained-layer structures,” Semicond. Sci. Technol. 4, 121–137 (1989).
[CrossRef]

1988 (1)

D. Smith and C. Mailhiot, “Piezoelectric effects in strained-layer superlattices,” J. Appl. Phys. 63, 2717–2719 (1988).
[CrossRef]

1987 (2)

R. Eppenga, M. F. H. Schuurmans, and S. Colak, “New k·p theory for GaAs/Ga1−xAlxAs-type quantum wells,” Phys. Rev. B 36, 1554–1564 (1987).
[CrossRef]

D. Nolte, W. Walukiewicz, and E. Haller, “Band-edge hydrostatic deformation potentials in III–V semiconductors,” Phys. Rev. Lett. 59, 501–504 (1987).
[CrossRef]

1986 (1)

K. S. Chan, “The effects of the hole subband mixing on the energies and oscillator strengths of excitons in a quantum well,” J. Phys. C: Solid State Phys. 19, L125–L130 (1986).
[CrossRef]

1982 (2)

G. Bastard, E. E. Mendez, L. L. Chang, and L. Esaki, “Exciton binding energy in quantum wells,” Phys. Rev. B 26, 1974–1979(1982).
[CrossRef]

J. Blakemore, “Semiconducting and other major properties of gallium arsenide,” J. Appl. Phys. 53, R123–R181(1982).
[CrossRef]

1971 (1)

L. D. Laude, F. H. Pollak, and M. Cardona, “Effects of uniaxial stress on the indirect exciton spectrum of silicon,” Phys. Rev. B 3, 2623–2636 (1971).
[CrossRef]

1968 (1)

F. H. Pollak and M. Cardona, “Piezo-electroreflectance in Ge, GaAs and Si,” Phys. Rev. 172, 816–837 (1968).
[CrossRef]

1963 (2)

H. Hasegawa, “Theory of cyclotron resonance in strained silicon crystals,” Phys. Rev. 129, 1029–1040 (1963).
[CrossRef]

J. Hensel and G. Feher, “Cyclotron resonance experiments in uniaxially stressed silicon: valence band inverse mass parameters and deformation potentials,” Phys. Rev. 129, 1041–1062 (1963).
[CrossRef]

1961 (2)

G. Pikus and G. Bir, “Cyclotron and paramagnetic resonance in strained cyrstals,” Phys. Rev. Lett. 6, 103–105 (1961).
[CrossRef]

G. Whitfield, “Theory of electron-phonon interactions,” Phys. Rev. 121, 720–734 (1961).
[CrossRef]

1958 (1)

J. L. Birman, “Theory of piezoelectric effect in the zincblende structure,” Phys. Rev. 111, 1510–1514 (1958).
[CrossRef]

1956 (1)

J. Luttinger, “Quantum theory of cyclotron resonance in semiconductors: general theory,” Phys. Rev. 102, 1030–1041 (1956).
[CrossRef]

Adesida, I.

Y. Sun, S. Kim, I. Adesida, and J. A. Rogers, “Bendable GaAs metal-semiconductor field-effect transistors formed with printed GaAs wire arrays on plastic substrates,” Appl. Phys. Lett. 87, 083501 (2005).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal and N. K. Dutta, Semiconductors Lasers, 2nd ed. (Kluwer Academic, 1993).

Ahn, D.

S.-H. Park and D. Ahn, “Many-body effects on optical gain in strained hexagonal and cubic GaN/AlGaN quantum well lasers,” Appl. Phys. Lett. 71, 398–400 (1997).
[CrossRef]

D. Ahn, S. Yoon, S. Chuang, and C.-S. Chang, “Theory of optical gain in strained-layer quantum wells within the 6×6 Luttinger-Kohn model,” J. Appl. Phys. 78, 2489–2497 (1995).
[CrossRef]

D. Ahn and S. L. Chuang, “The theory of strained-layer quantum-well lasers with bandgap renormalization,” IEEE J. Quantum Electron. 30, 350–365 (1994).
[CrossRef]

Aida, T.

N. Ohtani, K. Kishimoto, K. Kubota, S. Saravanan, Y. Sato, S. Nashima, P. Vaccaro, T. Aida, and M. Hosoda, “Uniaxial-strain-induced transition from type-II to type-I band configuration of quantum well microtubes,” Physica E 21, 732–736 (2004).
[CrossRef]

Ballato, A.

H. Shen, M. Wraback, J. Pamulapati, M. Dutta, P. G. Newman, A. Ballato, and Y. Lu, “Normal incidence high contrast multiple quantum well light modulator based on polarization rotation,” Appl. Phys. Lett. 62, 2908–2910 (1993).
[CrossRef]

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W. Bardyszewski and D. Yevick, “Electroabsorption and electrorefraction effects in quantum-well modulators,” Phys. Rev. B 49, 5368–5378 (1994).
[CrossRef]

Bastard, G.

G. Bastard, E. E. Mendez, L. L. Chang, and L. Esaki, “Exciton binding energy in quantum wells,” Phys. Rev. B 26, 1974–1979(1982).
[CrossRef]

Bastek, B.

Y. Mei, D. J. Thurmer, C. Deneke, S. Kiravittaya, Y.-F. Chen, A. Dadgar, F. Bertram, B. Bastek, A. Krost, J. Christen, T. Reindl, M. Stoffel, E. Coric, and O. G. Schmidt, “Fabrication, self-assembly, and properties of ultrathin AIN/GaN porous crystalline nanomembranes: tubes, spirals, and curved sheets,” ACS Nano 3, 1663–1668 (2009).
[CrossRef]

Benyoucef, M.

C. Deneke, A. Malachias, S. Kiravittaya, M. Benyoucef, T. Metzger, and O. Schmidt, “Strain states in a quantum well embedded into a rolled-up microtube: x-ray and photoluminescence studies,” Appl. Phys. Lett. 96, 143101 (2010).
[CrossRef]

A. Bernardi, S. Kiravittaya, A. Rastelli, R. Songmuang, D. Thurmer, M. Benyoucef, and O. Schmidt, “On-chip Si/SiOx microtube refractometer,” Appl. Phys. Lett. 93, 094106 (2008).
[CrossRef]

S. Mendach, R. Songmuang, S. Kiravittaya, A. Rastelli, M. Benyoucef, and O. Schmidt, “Light emission and wave guiding of quantum dots in a tube,” Appl. Phys. Lett. 88, 111120 (2006).
[CrossRef]

Bernardi, A.

A. Bernardi, S. Kiravittaya, A. Rastelli, R. Songmuang, D. Thurmer, M. Benyoucef, and O. Schmidt, “On-chip Si/SiOx microtube refractometer,” Appl. Phys. Lett. 93, 094106 (2008).
[CrossRef]

Bertram, F.

Y. Mei, D. J. Thurmer, C. Deneke, S. Kiravittaya, Y.-F. Chen, A. Dadgar, F. Bertram, B. Bastek, A. Krost, J. Christen, T. Reindl, M. Stoffel, E. Coric, and O. G. Schmidt, “Fabrication, self-assembly, and properties of ultrathin AIN/GaN porous crystalline nanomembranes: tubes, spirals, and curved sheets,” ACS Nano 3, 1663–1668 (2009).
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D. Nichols and P. Bhattacharya, “Differential gain in InP-based strained layer multiple quantum well lasers,” Appl. Phys. Lett. 61, 2129–2131 (1992).
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Bir, G.

G. Pikus and G. Bir, “Cyclotron and paramagnetic resonance in strained cyrstals,” Phys. Rev. Lett. 6, 103–105 (1961).
[CrossRef]

Birman, J. L.

J. L. Birman, “Theory of piezoelectric effect in the zincblende structure,” Phys. Rev. 111, 1510–1514 (1958).
[CrossRef]

Blakemore, J.

J. Blakemore, “Semiconducting and other major properties of gallium arsenide,” J. Appl. Phys. 53, R123–R181(1982).
[CrossRef]

Cardona, M.

L. D. Laude, F. H. Pollak, and M. Cardona, “Effects of uniaxial stress on the indirect exciton spectrum of silicon,” Phys. Rev. B 3, 2623–2636 (1971).
[CrossRef]

F. H. Pollak and M. Cardona, “Piezo-electroreflectance in Ge, GaAs and Si,” Phys. Rev. 172, 816–837 (1968).
[CrossRef]

Cendula, P.

P. Cendula, S. Kiravittaya, Y. Mei, C. Deneke, and O. Schmidt, “Bending and wrinkling as competing relaxation pathways for strained free-hanging films,” Phys. Rev. B 79, 085429 (2009).
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K. S. Chan, “The effects of the hole subband mixing on the energies and oscillator strengths of excitons in a quantum well,” J. Phys. C: Solid State Phys. 19, L125–L130 (1986).
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S. L. Chuang and C. S. Chang, “Effective-mass Hamiltonian for strained wurtzite GaN and analytical solutions,” Appl. Phys. Lett. 68, 1657–1659 (1996).
[CrossRef]

Chang, C.-S.

D. Ahn, S. Yoon, S. Chuang, and C.-S. Chang, “Theory of optical gain in strained-layer quantum wells within the 6×6 Luttinger-Kohn model,” J. Appl. Phys. 78, 2489–2497 (1995).
[CrossRef]

Chang, L. L.

G. Bastard, E. E. Mendez, L. L. Chang, and L. Esaki, “Exciton binding energy in quantum wells,” Phys. Rev. B 26, 1974–1979(1982).
[CrossRef]

Chen, T.

T. Chen, L. Eng, B. Zhao, Y. Zhuang, S. Sanders, and H. Morkoc, “Submilliamp threshold InGaAs-GaAs strained layer quantum-well laser,” IEEE J. Quantum Electron. 26, 1183–1190 (1990).
[CrossRef]

Chen, Y.-F.

Y. Mei, D. J. Thurmer, C. Deneke, S. Kiravittaya, Y.-F. Chen, A. Dadgar, F. Bertram, B. Bastek, A. Krost, J. Christen, T. Reindl, M. Stoffel, E. Coric, and O. G. Schmidt, “Fabrication, self-assembly, and properties of ultrathin AIN/GaN porous crystalline nanomembranes: tubes, spirals, and curved sheets,” ACS Nano 3, 1663–1668 (2009).
[CrossRef]

Chin, A.

N. W. Sinclair, J. K. Wuenschell, Z. Vörös, B. Nelsen, D. W. Snoke, M. H. Szymanska, A. Chin, and J. Keeling, “Strain-induced darkening of trapped excitons in coupled quantum wells at low temperature,” Phys. Rev. B 83, 245304 (2011).
[CrossRef]

Chocho, K.

S. Nakamura, M. Senoh, S. Nagahama, N. Iwasa, T. Yamada, T. Matsushita, H. Kiyoku, Y. Sugimoto, T. Kozaki, H. Umemoto, M. Sano, and K. Chocho, “InGaN/GaN/AlGaN-based laser diodes with modulatioin-doped strained-layer superlattices grown on an epitaxially laterally overgrown GaN substrate,” Appl. Phys. Lett. 72, 211–213 (1998).
[CrossRef]

Choi, H. K.

C.-K. Sun, H. K. Choi, K. A. Wang, and J. G. Fujimoto, “Femtosecond gain dynamics in InGaAs/AIGaAs strained-layer single-quantum-well diode lasers,” Appl. Phys. Lett. 63, 96–98 (1993).
[CrossRef]

Choi, W. M.

H. C. Ko, M. P. Stoykovich, J. Song, V. Malyarchuk, W. M. Choi, C.-J. Yu, J. B. Geddes, J. Xiao, S. Wang, Y. Huang, and J. A. Rogers, “A hemispherical electronic eye camera based on compressible silicon optoelectronics,” Nature 454, 748–753 (2008).
[CrossRef]

Chong, T. C.

T. C. Chong and C. G. Fonstad, “Theoretical gain of strained-layer semiconductor lasers in the large strain regime,” IEEE J. Quantum Electron. 25, 171–178 (1989).
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Chow, W. W.

W. W. Chow and S. W. Koch, Semiconductor-Laser Fundamentals: Physics of the Gain Materials, 2nd ed. (Springer, 1999).

Christen, J.

Y. Mei, D. J. Thurmer, C. Deneke, S. Kiravittaya, Y.-F. Chen, A. Dadgar, F. Bertram, B. Bastek, A. Krost, J. Christen, T. Reindl, M. Stoffel, E. Coric, and O. G. Schmidt, “Fabrication, self-assembly, and properties of ultrathin AIN/GaN porous crystalline nanomembranes: tubes, spirals, and curved sheets,” ACS Nano 3, 1663–1668 (2009).
[CrossRef]

Chuang, S.

D. Ahn, S. Yoon, S. Chuang, and C.-S. Chang, “Theory of optical gain in strained-layer quantum wells within the 6×6 Luttinger-Kohn model,” J. Appl. Phys. 78, 2489–2497 (1995).
[CrossRef]

S. Chuang, Physics of Optoelectronic Devices (Wiley-Interscience, 1995).

Chuang, S. L.

S. L. Chuang and C. S. Chang, “Effective-mass Hamiltonian for strained wurtzite GaN and analytical solutions,” Appl. Phys. Lett. 68, 1657–1659 (1996).
[CrossRef]

D. Ahn and S. L. Chuang, “The theory of strained-layer quantum-well lasers with bandgap renormalization,” IEEE J. Quantum Electron. 30, 350–365 (1994).
[CrossRef]

S. L. Chuang, “Efficient band-structure calculations of strained quantum wells,” Phys. Rev. B 43, 9649–9661 (1991).
[CrossRef]

Chung, T.

T. Chung, G. Walter, and J. N. Holonyak, “Coupled strained-layer InGaAs quantum-well improvement of an InAs quantum dot AlGaAs-GaAs-InGaAs-InAs heterostructure laser,” Appl. Phys. Lett. 79, 4500–4502 (2001).
[CrossRef]

Colak, S.

R. Eppenga, M. F. H. Schuurmans, and S. Colak, “New k·p theory for GaAs/Ga1−xAlxAs-type quantum wells,” Phys. Rev. B 36, 1554–1564 (1987).
[CrossRef]

Coldren, L. A.

S. W. Corzine, R. H. Yan, and L. A. Coldren, “Theoretical gain in strained InGaAs/AlGaAs quantum wells including valence-band mixing effects,” Appl. Phys. Lett. 57, 2835–2837 (1990).
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Coleman, J.

J. Coleman, “Strained-layer InGaAs quantum-well heterostructure lasers,” IEEE J. Sel. Top. Quantum Electron. 6, 1008–1013(2000).
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Coric, E.

Y. Mei, D. J. Thurmer, C. Deneke, S. Kiravittaya, Y.-F. Chen, A. Dadgar, F. Bertram, B. Bastek, A. Krost, J. Christen, T. Reindl, M. Stoffel, E. Coric, and O. G. Schmidt, “Fabrication, self-assembly, and properties of ultrathin AIN/GaN porous crystalline nanomembranes: tubes, spirals, and curved sheets,” ACS Nano 3, 1663–1668 (2009).
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Corzine, S. W.

S. W. Corzine, R. H. Yan, and L. A. Coldren, “Theoretical gain in strained InGaAs/AlGaAs quantum wells including valence-band mixing effects,” Appl. Phys. Lett. 57, 2835–2837 (1990).
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Dadgar, A.

Y. Mei, D. J. Thurmer, C. Deneke, S. Kiravittaya, Y.-F. Chen, A. Dadgar, F. Bertram, B. Bastek, A. Krost, J. Christen, T. Reindl, M. Stoffel, E. Coric, and O. G. Schmidt, “Fabrication, self-assembly, and properties of ultrathin AIN/GaN porous crystalline nanomembranes: tubes, spirals, and curved sheets,” ACS Nano 3, 1663–1668 (2009).
[CrossRef]

Deneke, C.

C. Deneke, A. Malachias, S. Kiravittaya, M. Benyoucef, T. Metzger, and O. Schmidt, “Strain states in a quantum well embedded into a rolled-up microtube: x-ray and photoluminescence studies,” Appl. Phys. Lett. 96, 143101 (2010).
[CrossRef]

P. Cendula, S. Kiravittaya, Y. Mei, C. Deneke, and O. Schmidt, “Bending and wrinkling as competing relaxation pathways for strained free-hanging films,” Phys. Rev. B 79, 085429 (2009).
[CrossRef]

Y. Mei, D. J. Thurmer, C. Deneke, S. Kiravittaya, Y.-F. Chen, A. Dadgar, F. Bertram, B. Bastek, A. Krost, J. Christen, T. Reindl, M. Stoffel, E. Coric, and O. G. Schmidt, “Fabrication, self-assembly, and properties of ultrathin AIN/GaN porous crystalline nanomembranes: tubes, spirals, and curved sheets,” ACS Nano 3, 1663–1668 (2009).
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Diaz, J.

B. Lane, Z. Wu, A. Stein, J. Diaz, and M. Razeghi, “InAsSb/InAsP strained-layer superlattice injection lasers operating at 4.0 μm grown by metal-organic chemical vapor deposition,” Appl. Phys. Lett. 74, 3438–3440 (1999).
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L. Zhang, L. Dong, and B. J. Nelson, “Bending and buckling of rolled-up SiGe/Si microtubes using nanorobotic manipulation,” Appl. Phys. Lett. 92, 243102 (2008).
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M. Wraback, H. Shen, J. Pamulapati, P. G. Newman, and M. Dutta, “Polarization dependent excitonic optical nonlinearities in GaAs/AlGaAs multiple quantum wells under anisotropic in-plane strain,” Phys. Rev. Lett. 74, 1466–1469 (1995).
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M. Wraback, H. Shen, J. Pamulapati, M. Dutta, P. G. Newman, M. Taysing-Lara, and Y. Lu, “Femtosecond studies of excitonic optical non-linearities in GaAs/AlxGa1−xAs multiple quantum wells under in-plane uniaxial strain,” Surf. Sci. 305, 238–242(1994).
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H. Shen, J. Pamulapati, M. Wraback, M. Taysing-Lara, M. Dutta, H. C. Kuo, and Y. Lu, “High contrast optical modulator based on electrically tunable polarization rotation and phase retardation in uniaxially strained (100) multiple quantum wells,” IEEE Photon. Technol. Lett. 6, 700–702 (1994).
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H. Shen, M. Wraback, J. Pamulapati, M. Dutta, P. G. Newman, A. Ballato, and Y. Lu, “Normal incidence high contrast multiple quantum well light modulator based on polarization rotation,” Appl. Phys. Lett. 62, 2908–2910 (1993).
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H. Shen, M. Wraback, J. Pamulapati, P. G. Newman, M. Dutta, Y. Lu, and H. C. Kuo, “Optical anisotropy in GaAs/AlxGa1−xAs multiple quantum wells under thermally induced uniaxial strain,” Phys. Rev. B 47, 13933–13936 (1993).
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T. Chen, L. Eng, B. Zhao, Y. Zhuang, S. Sanders, and H. Morkoc, “Submilliamp threshold InGaAs-GaAs strained layer quantum-well laser,” IEEE J. Quantum Electron. 26, 1183–1190 (1990).
[CrossRef]

Eppenga, R.

R. Eppenga, M. F. H. Schuurmans, and S. Colak, “New k·p theory for GaAs/Ga1−xAlxAs-type quantum wells,” Phys. Rev. B 36, 1554–1564 (1987).
[CrossRef]

Esaki, L.

G. Bastard, E. E. Mendez, L. L. Chang, and L. Esaki, “Exciton binding energy in quantum wells,” Phys. Rev. B 26, 1974–1979(1982).
[CrossRef]

Feher, G.

J. Hensel and G. Feher, “Cyclotron resonance experiments in uniaxially stressed silicon: valence band inverse mass parameters and deformation potentials,” Phys. Rev. 129, 1041–1062 (1963).
[CrossRef]

Fonstad, C. G.

T. C. Chong and C. G. Fonstad, “Theoretical gain of strained-layer semiconductor lasers in the large strain regime,” IEEE J. Quantum Electron. 25, 171–178 (1989).
[CrossRef]

Fujimoto, J. G.

C.-K. Sun, H. K. Choi, K. A. Wang, and J. G. Fujimoto, “Femtosecond gain dynamics in InGaAs/AIGaAs strained-layer single-quantum-well diode lasers,” Appl. Phys. Lett. 63, 96–98 (1993).
[CrossRef]

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H. C. Ko, M. P. Stoykovich, J. Song, V. Malyarchuk, W. M. Choi, C.-J. Yu, J. B. Geddes, J. Xiao, S. Wang, Y. Huang, and J. A. Rogers, “A hemispherical electronic eye camera based on compressible silicon optoelectronics,” Nature 454, 748–753 (2008).
[CrossRef]

Gibbs, H. M.

G. Khitrova, H. M. Gibbs, F. Jahnke, M. Kira, and S. W. Koch, “Nonlinear optics of normal-mode coupling semiconductor microcavities,” Rev. Mod. Phys. 71, 1591–1639 (1999).
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Y. Mei, S. Kiravittaya, S. Harazim, and O. G. Schmidt, “Principles and applications of micro and nanoscale wrinkles,” Mat. Sci. Eng. R 70, 209–224 (2010).
[CrossRef]

Hasegawa, H.

H. Hasegawa, “Theory of cyclotron resonance in strained silicon crystals,” Phys. Rev. 129, 1029–1040 (1963).
[CrossRef]

Hensel, J.

J. Hensel and G. Feher, “Cyclotron resonance experiments in uniaxially stressed silicon: valence band inverse mass parameters and deformation potentials,” Phys. Rev. 129, 1041–1062 (1963).
[CrossRef]

Högele, A.

S. Seidl, M. Kroner, A. Högele, and K. Karrai, “Effect of uniaxial stress on excitons in a self-assembled quantum dot,” Appl. Phys. Lett. 88, 203113 (2006).
[CrossRef]

Holonyak, J. N.

T. Chung, G. Walter, and J. N. Holonyak, “Coupled strained-layer InGaAs quantum-well improvement of an InAs quantum dot AlGaAs-GaAs-InGaAs-InAs heterostructure laser,” Appl. Phys. Lett. 79, 4500–4502 (2001).
[CrossRef]

Hosoda, M.

N. Ohtani, K. Kishimoto, K. Kubota, S. Saravanan, Y. Sato, S. Nashima, P. Vaccaro, T. Aida, and M. Hosoda, “Uniaxial-strain-induced transition from type-II to type-I band configuration of quantum well microtubes,” Physica E 21, 732–736 (2004).
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M. Hossein-Zadeh and K. J. Vahala, “An optomechanical oscillator on a silicon chip,” IEEE J. Sel. Top. Quantum Electron. 16, 276–287 (2010).
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Huang, Y.

H. C. Ko, M. P. Stoykovich, J. Song, V. Malyarchuk, W. M. Choi, C.-J. Yu, J. B. Geddes, J. Xiao, S. Wang, Y. Huang, and J. A. Rogers, “A hemispherical electronic eye camera based on compressible silicon optoelectronics,” Nature 454, 748–753 (2008).
[CrossRef]

Iwasa, N.

S. Nakamura, M. Senoh, S. Nagahama, N. Iwasa, T. Yamada, T. Matsushita, H. Kiyoku, Y. Sugimoto, T. Kozaki, H. Umemoto, M. Sano, and K. Chocho, “InGaN/GaN/AlGaN-based laser diodes with modulatioin-doped strained-layer superlattices grown on an epitaxially laterally overgrown GaN substrate,” Appl. Phys. Lett. 72, 211–213 (1998).
[CrossRef]

Jahnke, F.

G. Khitrova, H. M. Gibbs, F. Jahnke, M. Kira, and S. W. Koch, “Nonlinear optics of normal-mode coupling semiconductor microcavities,” Rev. Mod. Phys. 71, 1591–1639 (1999).
[CrossRef]

Karrai, K.

S. Seidl, M. Kroner, A. Högele, and K. Karrai, “Effect of uniaxial stress on excitons in a self-assembled quantum dot,” Appl. Phys. Lett. 88, 203113 (2006).
[CrossRef]

Keeling, J.

N. W. Sinclair, J. K. Wuenschell, Z. Vörös, B. Nelsen, D. W. Snoke, M. H. Szymanska, A. Chin, and J. Keeling, “Strain-induced darkening of trapped excitons in coupled quantum wells at low temperature,” Phys. Rev. B 83, 245304 (2011).
[CrossRef]

Khitrova, G.

G. Khitrova, H. M. Gibbs, F. Jahnke, M. Kira, and S. W. Koch, “Nonlinear optics of normal-mode coupling semiconductor microcavities,” Rev. Mod. Phys. 71, 1591–1639 (1999).
[CrossRef]

Kim, S.

Y. Sun, S. Kim, I. Adesida, and J. A. Rogers, “Bendable GaAs metal-semiconductor field-effect transistors formed with printed GaAs wire arrays on plastic substrates,” Appl. Phys. Lett. 87, 083501 (2005).
[CrossRef]

Kira, M.

G. Khitrova, H. M. Gibbs, F. Jahnke, M. Kira, and S. W. Koch, “Nonlinear optics of normal-mode coupling semiconductor microcavities,” Rev. Mod. Phys. 71, 1591–1639 (1999).
[CrossRef]

Kiravittaya, S.

C. Deneke, A. Malachias, S. Kiravittaya, M. Benyoucef, T. Metzger, and O. Schmidt, “Strain states in a quantum well embedded into a rolled-up microtube: x-ray and photoluminescence studies,” Appl. Phys. Lett. 96, 143101 (2010).
[CrossRef]

Y. Mei, S. Kiravittaya, S. Harazim, and O. G. Schmidt, “Principles and applications of micro and nanoscale wrinkles,” Mat. Sci. Eng. R 70, 209–224 (2010).
[CrossRef]

P. Cendula, S. Kiravittaya, Y. Mei, C. Deneke, and O. Schmidt, “Bending and wrinkling as competing relaxation pathways for strained free-hanging films,” Phys. Rev. B 79, 085429 (2009).
[CrossRef]

Y. Mei, D. J. Thurmer, C. Deneke, S. Kiravittaya, Y.-F. Chen, A. Dadgar, F. Bertram, B. Bastek, A. Krost, J. Christen, T. Reindl, M. Stoffel, E. Coric, and O. G. Schmidt, “Fabrication, self-assembly, and properties of ultrathin AIN/GaN porous crystalline nanomembranes: tubes, spirals, and curved sheets,” ACS Nano 3, 1663–1668 (2009).
[CrossRef]

A. Bernardi, S. Kiravittaya, A. Rastelli, R. Songmuang, D. Thurmer, M. Benyoucef, and O. Schmidt, “On-chip Si/SiOx microtube refractometer,” Appl. Phys. Lett. 93, 094106 (2008).
[CrossRef]

S. Mendach, R. Songmuang, S. Kiravittaya, A. Rastelli, M. Benyoucef, and O. Schmidt, “Light emission and wave guiding of quantum dots in a tube,” Appl. Phys. Lett. 88, 111120 (2006).
[CrossRef]

Kishimoto, K.

N. Ohtani, K. Kishimoto, K. Kubota, S. Saravanan, Y. Sato, S. Nashima, P. Vaccaro, T. Aida, and M. Hosoda, “Uniaxial-strain-induced transition from type-II to type-I band configuration of quantum well microtubes,” Physica E 21, 732–736 (2004).
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N. Ohtani, K. Kishimoto, K. Kubota, S. Saravanan, Y. Sato, S. Nashima, P. Vaccaro, T. Aida, and M. Hosoda, “Uniaxial-strain-induced transition from type-II to type-I band configuration of quantum well microtubes,” Physica E 21, 732–736 (2004).
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[CrossRef]

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C. Deneke, A. Malachias, S. Kiravittaya, M. Benyoucef, T. Metzger, and O. Schmidt, “Strain states in a quantum well embedded into a rolled-up microtube: x-ray and photoluminescence studies,” Appl. Phys. Lett. 96, 143101 (2010).
[CrossRef]

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[CrossRef]

A. Bernardi, S. Kiravittaya, A. Rastelli, R. Songmuang, D. Thurmer, M. Benyoucef, and O. Schmidt, “On-chip Si/SiOx microtube refractometer,” Appl. Phys. Lett. 93, 094106 (2008).
[CrossRef]

S. Mendach, R. Songmuang, S. Kiravittaya, A. Rastelli, M. Benyoucef, and O. Schmidt, “Light emission and wave guiding of quantum dots in a tube,” Appl. Phys. Lett. 88, 111120 (2006).
[CrossRef]

Schmidt, O. G.

Y. Mei, S. Kiravittaya, S. Harazim, and O. G. Schmidt, “Principles and applications of micro and nanoscale wrinkles,” Mat. Sci. Eng. R 70, 209–224 (2010).
[CrossRef]

Y. Mei, D. J. Thurmer, C. Deneke, S. Kiravittaya, Y.-F. Chen, A. Dadgar, F. Bertram, B. Bastek, A. Krost, J. Christen, T. Reindl, M. Stoffel, E. Coric, and O. G. Schmidt, “Fabrication, self-assembly, and properties of ultrathin AIN/GaN porous crystalline nanomembranes: tubes, spirals, and curved sheets,” ACS Nano 3, 1663–1668 (2009).
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S. Nakamura, M. Senoh, S. Nagahama, N. Iwasa, T. Yamada, T. Matsushita, H. Kiyoku, Y. Sugimoto, T. Kozaki, H. Umemoto, M. Sano, and K. Chocho, “InGaN/GaN/AlGaN-based laser diodes with modulatioin-doped strained-layer superlattices grown on an epitaxially laterally overgrown GaN substrate,” Appl. Phys. Lett. 72, 211–213 (1998).
[CrossRef]

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M. Wraback, H. Shen, S. Liang, C. R. Gorla, and Y. Lu, “High contrast, ultrafast optically addressed ultraviolet light modulator based upon optical anisotropy in ZnO films grown on R-plane sapphire,” Appl. Phys. Lett. 74, 507–509 (1999).
[CrossRef]

M. Wraback, H. Shen, J. Pamulapati, P. G. Newman, and M. Dutta, “Polarization dependent excitonic optical nonlinearities in GaAs/AlGaAs multiple quantum wells under anisotropic in-plane strain,” Phys. Rev. Lett. 74, 1466–1469 (1995).
[CrossRef]

M. Wraback, H. Shen, J. Pamulapati, M. Dutta, P. G. Newman, M. Taysing-Lara, and Y. Lu, “Femtosecond studies of excitonic optical non-linearities in GaAs/AlxGa1−xAs multiple quantum wells under in-plane uniaxial strain,” Surf. Sci. 305, 238–242(1994).
[CrossRef]

H. Shen, J. Pamulapati, M. Wraback, M. Taysing-Lara, M. Dutta, H. C. Kuo, and Y. Lu, “High contrast optical modulator based on electrically tunable polarization rotation and phase retardation in uniaxially strained (100) multiple quantum wells,” IEEE Photon. Technol. Lett. 6, 700–702 (1994).
[CrossRef]

H. Shen, M. Wraback, J. Pamulapati, P. G. Newman, M. Dutta, Y. Lu, and H. C. Kuo, “Optical anisotropy in GaAs/AlxGa1−xAs multiple quantum wells under thermally induced uniaxial strain,” Phys. Rev. B 47, 13933–13936 (1993).
[CrossRef]

H. Shen, M. Wraback, J. Pamulapati, M. Dutta, P. G. Newman, A. Ballato, and Y. Lu, “Normal incidence high contrast multiple quantum well light modulator based on polarization rotation,” Appl. Phys. Lett. 62, 2908–2910 (1993).
[CrossRef]

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N. W. Sinclair, J. K. Wuenschell, Z. Vörös, B. Nelsen, D. W. Snoke, M. H. Szymanska, A. Chin, and J. Keeling, “Strain-induced darkening of trapped excitons in coupled quantum wells at low temperature,” Phys. Rev. B 83, 245304 (2011).
[CrossRef]

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D. Smith and C. Mailhiot, “Piezoelectric effects in strained-layer superlattices,” J. Appl. Phys. 63, 2717–2719 (1988).
[CrossRef]

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N. W. Sinclair, J. K. Wuenschell, Z. Vörös, B. Nelsen, D. W. Snoke, M. H. Szymanska, A. Chin, and J. Keeling, “Strain-induced darkening of trapped excitons in coupled quantum wells at low temperature,” Phys. Rev. B 83, 245304 (2011).
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H. C. Ko, M. P. Stoykovich, J. Song, V. Malyarchuk, W. M. Choi, C.-J. Yu, J. B. Geddes, J. Xiao, S. Wang, Y. Huang, and J. A. Rogers, “A hemispherical electronic eye camera based on compressible silicon optoelectronics,” Nature 454, 748–753 (2008).
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A. Bernardi, S. Kiravittaya, A. Rastelli, R. Songmuang, D. Thurmer, M. Benyoucef, and O. Schmidt, “On-chip Si/SiOx microtube refractometer,” Appl. Phys. Lett. 93, 094106 (2008).
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Y. Sun, E. Menard, and J. A. Rogers, “Gigahertz operation in flexible transistors on plastic substrates,” Appl. Phys. Lett. 88, 183509 (2006).
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S. Mendach, R. Songmuang, S. Kiravittaya, A. Rastelli, M. Benyoucef, and O. Schmidt, “Light emission and wave guiding of quantum dots in a tube,” Appl. Phys. Lett. 88, 111120 (2006).
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H. Shen, M. Wraback, J. Pamulapati, M. Dutta, P. G. Newman, A. Ballato, and Y. Lu, “Normal incidence high contrast multiple quantum well light modulator based on polarization rotation,” Appl. Phys. Lett. 62, 2908–2910 (1993).
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Surf. Sci. (1)

M. Wraback, H. Shen, J. Pamulapati, M. Dutta, P. G. Newman, M. Taysing-Lara, and Y. Lu, “Femtosecond studies of excitonic optical non-linearities in GaAs/AlxGa1−xAs multiple quantum wells under in-plane uniaxial strain,” Surf. Sci. 305, 238–242(1994).
[CrossRef]

Other (11)

S. Chuang, Physics of Optoelectronic Devices (Wiley-Interscience, 1995).

W. W. Chow and S. W. Koch, Semiconductor-Laser Fundamentals: Physics of the Gain Materials, 2nd ed. (Springer, 1999).

G. P. Agrawal and N. K. Dutta, Semiconductors Lasers, 2nd ed. (Kluwer Academic, 1993).

Membranes can be formally defined as very thin plates without flexural resistance [62], but we will use the term in a loose way denoting thin plates, and we allow for thin plates sandwiched by other materials to be included in the definition of membranes.

A. Love, A Treatise on the Mathemathical Theory of Elasticity, 4th ed. (Dover, 1944).

S. Timoshenko and J. Goodier, Theory of Elasticity, 2nd ed. (McGraw-Hill, 1951).

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D. R. Lovett, Tensor Properties of Crystals (Institute of Physics, 1989).

We neglect the electron-hole exchange effect, which is on the order of tens of micro electron volts and thus small compared to typical values of the broadening of exciton resonances in quasi-two-dimensional systems. Under certain circumstances, not considered in this paper, it can become important or even dominant in excitonic spectra, especially in quantum dots (see, e.g., Ref. [68]).

F. H. Pollak, “Effects of homogenous strain on the electronic and vibrational levels in semiconductors,” in Strained-Layer Superlattices: Physics, T. T. Pearsall, ed., Vol. 32 of Semiconductors and Semimetals (Academic, 1990), pp. 17–53.

R. Szilard, Theories and Applications of Plate Analysis, 1st ed. (Wiley, 2004).

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Figures (4)

Fig. 1.
Fig. 1.

Cylindrically deformed membrane (with a cylinder radius of 5.5 μm). We assume the membrane’s left face to be bent without being elongated, i.e., we apply pure displacement boundary conditions on the left face. The boundary conditions on all other faces are taken to be vanishing surface forces.

Fig. 2.
Fig. 2.

Dominant strain tensor elements, averaged over z at y=0.

Fig. 3.
Fig. 3.

Dominant stress tensor elements, averaged over z at y=0.

Fig. 4.
Fig. 4.

Deformation-induced differential transmittance in the vicinity of the heavy-hole (hh) and light-hole (lh) exciton resonance, showing a redshift of the hh exciton and a blueshift of the lh exciton.

Equations (48)

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H^L=22m[(γ1+52γ2)k22γ2(kx2J^x2+ky2J^y2+kz2J^z2)γ3({kx,ky}{J^x,J^y}+c.p.)],
H^s=-av(ϵxx+ϵyy+ϵzz)b[(J^x213J^2)ϵxx+c.p.]13d[{J^x,J^y}ϵxy+c.p.],
H^=d3rsψ(r,z,s){222m+VL(r,z)+h^SO+h^}ψ(r,z,s)+12d3rd3rssψ(r,z,s)ψ(r,z,s)Vc(r-r,zz)ψ(r,z,s)ψ(r,z,s)+d3rsψ(r,z,s)emcA·p^ψ(r,z,s).
φνk(r,z,s)=1Lzkzξνk(kz)1Aeik·reikzzuνkkz(r,z,s),
uνkkz(r,z,s)=νWνν(k,kz)uν00(r,z,s).
φνk(r,z,s)=1Aeik·rνW˜νν(k,z)uν00(r,z,s),
νHνν(k,iz)W˜νν()(k,z)=ενkW˜νν()(k,z),
Hνν(k,iz)=(εν0+2k22m2z22m)δνν+mk·pνν+mpννz(iz)+hνν(z).
νdzW˜νν()*(k,z)W˜νν()(k,z)=δννδ.
v{HvvL(k,iz)+Vc(z)δvv+Hvvs(z)}Wvv()(k,z)=εvkWvv()(k,z),
(Eg+2k22me2z22me+Vc(z)+ac(ϵxx(z)+ϵyy(z)+ϵzz(z)))W˜cc()(k,z)=εckW˜cc()(k,z),
H0=νkενkaνkaνk,
Hc=ν1ν414kk1AqVqc×dzdzμ1W˜μ1ν1(1)*(k+q,z)W˜μ1ν4(4)(k,z)×eq|zz|μ2W˜μ2ν2(2)*(kq,z)W˜μ2ν3(3)(k,z)×aν1,k+q,1aν2,kq,2aν3k3aν4k4,
HR=cvk12Mc,v(k)·E0eiω0tackavk+h.c.
Mc,v(k)=cvdzW˜cc()*(k,z)W˜vv()(k,z)dcv.
Pc,v(k)=avkack
iddtPc,v(k)=(εckεvkiγ)Pc,v(k)1Akc˜v˜VkkcFcvcv˜˜(k,k)Pc˜,v˜(k)12Ωc,v(k),
Fcvcv˜˜(k,k)=dzdzcW˜cc()*(k,z)W˜cc(˜)(k,z)×e|kk||zz|vW˜vv(˜)*(k,z)W˜vv()(k,z),
Ωc,v(k)=Mc,v(k)·E0eiωot.
j^(z)=1AkccvvW˜cc()*(k,z)W˜vv()(k,z)×empcvackavk+h.c.
P(z)=1AkcvM˜c,v*(k,z)Pc,v(k)+c.c.,
M˜c,v*(k,z)=cvW˜cc()*(k,z)W˜vv()(k,z)dcv*.
P(z)=δ(zzmem)1AkcvMc,v*(k)Pc,v(k)+c.c.
{ω02c2n+2z2}E()=4πc2ω02P(),
E0,T=E0,in+4πiω0cnP,
E0,R=4πiω0cnP,
dzcW˜cc()*(k,z)W˜cc(˜)(k,z)=δccδ,1δ˜,1
dzvW˜vv(˜)*(k,z)W˜vv()(k,z)=δ,1δ˜,1vWvv(˜)*(k)Wvv()(k).
Fcvcv˜˜(k,k)=δccδ,1δ˜,1δ,1δ˜,1vWvv(˜)*(k)Wvv()(k).
Mc,v(k)=δccδ,1δ,1vWvv(1)(k)dcv,
vvdzW˜vv()*(k,z)Hvvv(k,iz)W˜vv()(k,z)=εvkδvvδ,
vvW˜vv(1)*(k)H¯vvv(k)W˜vv(1)(k)=εv,k,1δvv,
H¯vvv(k)=dzξ*(z)Hvvv(k,iz)ξ(z).
Pcvnd(k)=vWvv(1)*(k)Pcv(k),
iddtPcvnd(k)={ϵckiγ}Pcvnd(k)-vH¯vvv(k)Pcvnd(k)1AkVk-kcPcvnd(k)12dcv·E0eiω0t.
σijxj+Fi=0
ϵk=12(ukx+uxk)
σij=Cijkϵk,
12j,k,Cijk(2ukxjx+2uxjxk)+Fi=0.
[σxxσyyσzz]=[C11C12C12C12C11C12C12C12C11][ϵxxϵyyϵzz],
H32,32L=22m{(γ1+γ2)(kx2+ky2)+(γ12γ2)kz2},
H32,12L=22m3{(γ2(kx2ky2)2iγ3kxky},
H32,12L=22m23γ3(kxiky)kz,
H12,12L=22m{(γ1γ2)(kx2+ky2)+(γ1+2γ2)kz2}.
H32,32s=av(ϵxx+ϵyy+ϵzz)+12b(ϵxx+ϵyy2ϵzz),
H32,12s=32b(ϵxx-ϵyy)+idϵxy,
H32,12s=d(ϵzxiϵyz),
H12,12s=av(ϵxx+ϵyy+ϵzz)12b(ϵxx+ϵyy2ϵzz).

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