In optically thin AlGaAs/GaAs/AlGaAs double heterostructures the radiative recombination coefficient BB of electron–hole pairs is not purely a material constant: it is controlled by the optical-mode availability of the surrounding media. Yablonovitch, Gmitter and Bhat (1988) established this experimentally and derived approximate scaling laws (1/τR∝next/nint1/\tau_R\propto n_{\mathrm{ext}}/n_{\mathrm{int}} for sub-half-wavelength films and 1/τR∝(next2+1)/(2nint2)1/\tau_R\propto(n_{\mathrm{ext}}^2+1)/(2n_{\mathrm{int}}^2) for thicker optically thin films) using statistical, flux-matching arguments rather than an exact solution of Maxwell's equations, and they explicitly left open several issues: the exact electromagnetic-mode computation for real film geometries, the role of localized (guided) modes that do not radiate to infinity, a proof of the assertion that totally reabsorbed localized modes leave the net decay unchanged, absolute BB determinations with reduced uncertainty, and regimes with next>nintn_{\mathrm{ext}}>n_{\mathrm{int}} or with structured (photonic-crystal) surroundings. Later work addresses the individual components: exact planar-multilayer quantization (Khosravi and Loudon 1992; Ho, McCall and Slusher 1993; Creatore and Andreani 2008) closes the ideal-geometry mode-density question and shows the exact answer is richer than the averaged laws; the photon-recycling literature (Lush 1992; Gilliland 1993; Patkar 1995; Steiner 2013; Walker 2015; Cheng and O'Carroll 2021) reframes and quantifies the localized-mode/reabsorption question; and modern absolute measurements (Niemeyer 2019) still disagree with detailed-balance values by factors of several. The surviving open core is the unification the 1988 paper called for: a wave-optically exact, uncertainty-quantified account of the net radiative decay of optically thin GaAs films — exact Green-function mode densities and internal fields for the actual absorbing, nonideal geometries, coupled to photon-recycling kinetics, validated against absolute BB measurements and extended to high-index and periodically structured surroundings.

Background

Spontaneous emission is the decay of an excited electronic state by photon emission. In quantum electrodynamics its rate is governed by the local density of optical states (LDOS) — the density of electromagnetic modes per unit frequency at the emitter position — together with the strength of the field that couples to the material dipole. Purcell's insight that the spontaneous-emission rate of a dipole is set by the density of resonant electromagnetic modes at its position and frequency (the Purcell effect), and the parallel ideas of Yablonovitch (1987) and John (1987) on photonic band gaps — frequency ranges in which a periodic dielectric structure forbids light propagation — imply that structuring or index-matching the dielectric environment can inhibit or enhance spontaneous emission. In a direct-gap semiconductor such as GaAs the corresponding macroscopic quantity is the radiative recombination coefficient BB, defined by the rate equation −dN/dt=BN2-dN/dt=BN^{2} for the electron and hole densities NN (bimolecular band-to-band recombination), usually measured from the decay of a photoexcited carrier population. The van Roosbroeck–Shockley detailed-balance relation fixes BB from the absorption spectrum: at thermal equilibrium the radiative recombination rate equals the generation rate by blackbody radiation, giving BNi2=∫0∞(8πnint2ν2/c2)α(ν)e−hν/kT dνBN_i^{2}=\int_0^{\infty}(8\pi n_{\mathrm{int}}^{2}\nu^{2}/c^{2})\alpha(\nu)e^{-h\nu/kT}\,d\nu (photon occupation approximated by the Boltzmann factor e−hν/kTe^{-h\nu/kT}), with NiN_i the intrinsic carrier density and α\alpha the absorption coefficient. An AlGaAs/GaAs/AlGaAs double heterostructure is an epitaxial sandwich in which GaAs (refractive index nint≈3.6n_{\mathrm{int}}\approx3.6) is clad by wider-gap AlGaAs that confines carriers; 'optically thin' means the GaAs layer (here about 500 nm) is thin compared with the reabsorption length, so most internally emitted photons traverse the film. Because nintn_{\mathrm{int}} exceeds the index of air and of typical substrates, much of the internally emitted light undergoes total internal reflection and is trapped in guided (localized) optical modes that propagate in the film plane and do not couple to plane waves at infinity; photons in such modes are eventually reabsorbed and re-emitted, a process called photon recycling. Yablonovitch, Gmitter and Bhat (Phys. Rev. Lett. 61, 2546 (1988)) transferred nominally identical GaAs double heterostructures by epitaxial liftoff (chemical release of the grown film from its substrate, followed by bonding to a new host) onto SiO2_2 (next=1.5n_{\mathrm{ext}}=1.5), TiO2_2 (next=2.6n_{\mathrm{ext}}=2.6) and the original GaAs substrate (next=3.6n_{\mathrm{ext}}=3.6) and extracted BB from the decay of the photoexcited carrier density measured with a radio-frequency (rf) inductive-probe technique. They found BB scaling linearly with (next2+1)(n_{\mathrm{ext}}^{2}+1), with a measured value of about 5.5×10−10 cm3s−15.5\times10^{-10}\ \mathrm{cm^3s^{-1}} on GaAs versus 13×10−10 cm3s−113\times10^{-10}\ \mathrm{cm^3s^{-1}} from detailed balance for index-matched surroundings. Their theory was not an exact solution of Maxwell's equations: for sub-half-wavelength films they set the internal one-photon field operator Eint†E_{\mathrm{int}}^{\dagger} (the operator creating the electric field acting on the material dipole) equal to the external vacuum field using tangential-field continuity, giving 1/τR∝next/nint1/\tau_R\propto n_{\mathrm{ext}}/n_{\mathrm{int}}; for thicker films they equated internal and external energy fluxes (a Kirchhoff-type statistical averaging over angles, polarizations, positions, bandwidth and thickness fluctuations), giving 1/τR∝next2/nint21/\tau_R\propto n_{\mathrm{ext}}^{2}/n_{\mathrm{int}}^{2} for symmetric immersion and, for the experimental air-on-one-side geometry, 1/τR∝(next2+1)/(2nint2)1/\tau_R\propto(n_{\mathrm{ext}}^{2}+1)/(2n_{\mathrm{int}}^{2}); and they asserted that localized modes that are 'totally absorbing' within the film cannot carry energy away and so do not change the net decay, without a quantitative electromagnetic calculation. The paper itself acknowledged as unfinished: a rigorous electromagnetic-mode computation for the actual geometries (coherent multiple reflections, full angular and polarization averaging, finite emission bandwidth, thickness fluctuations and surface roughness) delivering an exact Eint†E_{\mathrm{int}}^{\dagger} and mode density; the explicit treatment of localized internal modes and how they partition emission between escaping, nonradiative and reabsorbed channels; a quantitative account of when localized modes materially change the net decay; reduction of experimental uncertainties in absolute BB; and the regimes next>nintn_{\mathrm{ext}}>n_{\mathrm{int}} and multilayer or photonic-crystal surroundings.

Problem Statement

For optically thin AlGaAs/GaAs/AlGaAs double heterostructures, in which the radiative recombination coefficient BB of electron–hole pairs is controlled by the optical-mode availability of the surrounding media rather than being a fixed material constant, the problem is to replace the statistical and flux-matching approximations of Yablonovitch, Gmitter and Bhat (1988) by a rigorous and experimentally validated electromagnetic account of the net radiative decay. Specifically: (1) compute, for the actual film geometries — GaAs layer of order half to a few optical wavelengths with AlGaAs claddings, air on one side and a substrate of refractive index nextn_{\mathrm{ext}} on the other, including absorption, coherent multiple reflections, full angular and polarization dependence, finite emission bandwidth, thickness fluctuations and surface roughness — the exact optical-mode density and internal field operator (equivalently, the full electromagnetic Green function or local density of optical states) that determines the spontaneous-emission rate, in place of the averaged proportionalities 1/τR∝next/nint1/\tau_R\propto n_{\mathrm{ext}}/n_{\mathrm{int}} and 1/τR∝(next2+1)/(2nint2)1/\tau_R\propto(n_{\mathrm{ext}}^{2}+1)/(2n_{\mathrm{int}}^{2}); (2) determine how the emission is partitioned among modes that escape to infinity, localized (guided) modes confined to the film, and modes reabsorbed in the film or claddings, and establish rigorously under what conditions localized modes change the net carrier decay — the 1988 assertion that localized modes which are totally absorbing within the film do not change the net decay was made without quantitative calculation; (3) reduce the experimental uncertainties of absolute BB determinations sufficiently to test quantitatively whether measured values agree with the van Roosbroeck–Shockley detailed-balance prediction after the correct electromagnetic geometry corrections and photon-recycling corrections are applied, and thereby explain the several-fold scatter of literature BB values for GaAs; and (4) establish the corresponding exact laws for regimes with surrounding index next>nintn_{\mathrm{ext}}>n_{\mathrm{int}} and for multilayer or periodically structured (one- or three-dimensional photonic-crystal) surroundings, where the simple proportionalities are expected to break down. The single unifying objective is a quantitative, self-consistent theory — exact electromagnetics of the mode structure coupled to recombination and reabsorption kinetics — that predicts the net radiative decay of optically thin semiconductor films in any specified optical environment and is validated against controlled absolute measurements on identical films in differing environments.

References

  1. E. Yablonovitch, T. J. Gmitter, and R. Bhat, 'Inhibited and Enhanced Spontaneous Emission from Optically Thin AlGaAs/GaAs Double Heterostructures', Phys. Rev. Lett. 61, 2546 (1988). DOI: 10.1103/PhysRevLett.61.2546, https://doi.org/10.1103/PhysRevLett.61.2546
  2. H. Khosravi and R. Loudon, 'Vacuum field fluctuations and spontaneous emission in a dielectric slab', Proc. R. Soc. Lond. A 436, 373 (1992). DOI: 10.1098/rspa.1992.0024, https://doi.org/10.1098/rspa.1992.0024
  3. S. T. Ho, S. L. McCall, and R. E. Slusher, 'Spontaneous emission from excitons in thin dielectric layers', Opt. Lett. 18, 909 (1993). DOI: 10.1364/OL.18.000909, https://doi.org/10.1364/OL.18.000909
  4. C. Creatore and L. C. Andreani, 'Quantum theory of spontaneous emission in multilayer dielectric structures', Phys. Rev. A 78, 063825 (2008). DOI: 10.1103/PhysRevA.78.063825, https://doi.org/10.1103/PhysRevA.78.063825
  5. V. I. Pipa, 'Radiative lifetimes of electrons and holes in a thin semiconductor layer', Semiconductors 40, 665 (2006). DOI: 10.1134/S1063782606060108, https://doi.org/10.1134/S1063782606060108
  6. G. B. Lush, M. R. Melloch, M. S. Lundstrom, D. H. Levi, R. K. Ahrenkiel, and H. F. MacMillan, 'Microsecond lifetimes and low interface recombination velocities in moderately doped n-GaAs thin films', Appl. Phys. Lett. 61, 2440 (1992). DOI: 10.1063/1.108190, https://doi.org/10.1063/1.108190
  7. G. D. Gilliland, D. J. Wolford, T. F. Kuech, J. A. Bradley, and H. P. Hjalmarson, 'Minority-carrier recombination kinetics and transport in "surface-free" GaAs/AlxGa1-xAs double heterostructures', J. Appl. Phys. 73, 8386 (1993). DOI: 10.1063/1.353407, https://doi.org/10.1063/1.353407
  8. M. P. Patkar, M. S. Lundstrom, and M. R. Melloch, 'Characterization of photon recycling in thin crystalline GaAs light emitting diodes', J. Appl. Phys. 78, 2817 (1995). DOI: 10.1063/1.360081, https://doi.org/10.1063/1.360081
  9. I. Schnitzer, E. Yablonovitch, C. Caneau, and T. J. Gmitter, 'Ultrahigh spontaneous emission quantum efficiency, 99.7% internally and 72% externally, from AlGaAs/GaAs/AlGaAs double heterostructures', Appl. Phys. Lett. 62, 131 (1993). DOI: 10.1063/1.109348, https://doi.org/10.1063/1.109348
  10. M. Boroditsky, R. Vrijen, T. F. Krauss, R. Coccioli, R. Bhat, and E. Yablonovitch, 'Spontaneous emission extraction and Purcell enhancement from thin-film 2-D photonic crystals', J. Lightwave Technol. 17, 2096 (1999). DOI: 10.1109/50.803000, https://doi.org/10.1109/50.803000
  11. T. Szkopek, 'The fine structure constant determines spontaneous emission rates from semiconductors', Appl. Phys. Lett. 98, 211117 (2011). DOI: 10.1063/1.3591171, https://doi.org/10.1063/1.3591171
  12. M. A. Steiner, J. F. Geisz, I. Garcia, D. J. Friedman, A. Duda, and S. R. Kurtz, 'Optical enhancement of the open-circuit voltage in high quality GaAs solar cells', J. Appl. Phys. 113, 123109 (2013). DOI: 10.1063/1.4798267, https://doi.org/10.1063/1.4798267
  13. A. Walker, O. Hohn, D. Micha, B. Blasi, A. Bett, and F. Dimroth, 'Impact of Photon Recycling on GaAs Solar Cell Designs', IEEE J. Photovoltaics 5, 1636 (2015). DOI: 10.1109/JPHOTOV.2015.2479463, https://doi.org/10.1109/JPHOTOV.2015.2479463
  14. Z. Cheng and D. M. O'Carroll, 'Photon Recycling in Semiconductor Thin Films and Devices', Adv. Sci. 8, 2004076 (2021). DOI: 10.1002/advs.202004076, https://doi.org/10.1002/advs.202004076
  15. M. Niemeyer, P. Kleinschmidt, A. W. Walker, L. E. Mundt, C. Timm, R. Lang, T. Hannappel, and D. Lackner, 'Measurement of the non-radiative minority recombination lifetime and the effective radiative recombination coefficient in GaAs', AIP Advances 9, 045034 (2019). DOI: 10.1063/1.5051709, https://doi.org/10.1063/1.5051709
  16. E. Yablonovitch, 'Inhibited Spontaneous Emission in Solid-State Physics and Electronics', Phys. Rev. Lett. 58, 2059 (1987). DOI: 10.1103/PhysRevLett.58.2059, https://doi.org/10.1103/PhysRevLett.58.2059
  17. S. John, 'Strong localization of photons in certain disordered dielectric superlattices', Phys. Rev. Lett. 58, 2486 (1987). DOI: 10.1103/PhysRevLett.58.2486, https://doi.org/10.1103/PhysRevLett.58.2486
  18. W. van Roosbroeck and W. Shockley, 'Photon-Radiative Recombination of Electrons and Holes in Germanium', Phys. Rev. 94, 1558 (1954). DOI: 10.1103/PhysRev.94.1558, https://doi.org/10.1103/PhysRev.94.1558