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Submitted on 1 Jan 1987
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INHIBITED SPONTANEOUS EMISSION IN THREE DIMENSIONALLY MODULATED PERIODIC
DIELECTRIC STRUCTURES
E. Yablonovitch
To cite this version:
E. Yablonovitch. INHIBITED SPONTANEOUS EMISSION IN THREE DIMENSIONALLY MOD-
ULATED PERIODIC DIELECTRIC STRUCTURES. Journal de Physique Colloques, 1987, 48 (C5),
pp.C5-615-C5-616. �10.1051/jphyscol:19875132�. �jpa-00226717�
JOURNAL DE PHYSIQUE
Colloque C5, supplkment au n o l l , Tome 48, novembre 1987
INHIBITED SPONTANEOUS EMISSION IN THREE DIMENSIONALLY MODULATED PERIODIC DIELECTRIC STRUCTURES
E. YABLONOVITCH
Bell Communications Research, Navesink Research Center, Red Bank, NJ 07701, U.S.A.
Spontaneous radiative recombination of electrons and holes, occurring for example in semiconductor lasers, has long been regarded as unavoidable. Until recently the same may have been said about the spontaneous radiative decay of excited atoms. I t is now becoming recognized'-3 that spontaneous emission is not necessarily a fixed and immutable property of the coupling between matter and space, but, that it can be controlled by modifying the properties of the radiation field.
The essential point is that the density of final states for spontaneous emission is actually the density of electromagnetic modes available to the emitted photon If there are no electromagnetic modes available to the emitted photon then spontaneous emission is inhibited. Recently there have been experiments in which the spontaneous emission of Rydberg atoms2 and of Penning trapped electrons3 h3s been inhibited in a microwave cavity which had no electromagnetic modes at the transition frequency.
Such a situation would be quite desirable in the solid state, where spontaneous emission plays a fundamental role in limiting the performance of semiconductor lasers, heterojunction bipolar transistors, and solar cells. If a solid medium has a three dimensionally periodic dielectric structure in which the index of refraction is modulated, then a gap can appear in the electromagnetic mode density analogous to the electronic band gap caused by the periodic potential of the electrons. It makes sense then to speak of the band structure of photons and of their reciprocal space which has a Brillouin zone approximately 1000 times smaller than the Brillouin zone of the electrons. If the dielectric constant is periodically modulated in all three dimensions then it is possible to have an electromagnetic band gap which overlaps the electronic band edge and for spontaneous electron-hole recombination to be rigorously forbidden
For direct gap semiconductors it is commonplace4 for the internal spontaneous cmission quantum efficiency of electron-hole recombination to exceed 90% even at room temperature. Thercfore we suggest that the direct gap semiconductors are a natural physical system to the study the phenomenon of inhibited spontaneous emission
A starting point for our discussion is the observation by Einstein that spontancous emission must inescapably co-exist with absorption and stimulated emission It will be neither feasible nor desirable to entirely eliminate spontaneous emission if the function of the semiconductor happens to be the emission or the absorption of light itself. Examples of this would be the scmiconductor laser and the solar cell respectively. In those cases the god would be to restrict the spontaneous emission to only those electromagnetic modes which are absolutely neccssary. In other functions such as the heterojunction bipolar transistor, where the radiation field would prefcrably be absent, all spontancous emission should be suppressed if possible.
At the lasing threshold spontaneous emission represents the major parasitic current sink.
Periodic structures have been playing an increasingly important role in semiconductor lasers. For example, in distributed feedback laser$ the index of refraction is periodically modulated along the laser axis In wave optics, layered structures (interference coatings) in which the index of refraction alternates from high to low every quarter wavelength have also become indispensable. The periodic spatial modulation opens up a forbidden gap in the electromagnetic dispersion relation, at least for light propagating perpendicular to the layers The prohibition of wave propagation in the forbidden gap makes these periodic dielectric structures valuable as highly ideal reflective mirrors, for example, for use in Fabry-Perot resonators. We can anticipate then, that full three dimensional spatial periodicity of A/2 in the refractive index can result in a forbidden gap in the electromagnetic spectrum near the wavelength A irrespective of propagation direction, just as the electronic spectrum has a band gap in crystals. If the electromagnetic band gap overlaps the electronic band edge by at least a few kT in energy, then electron-
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19875132
C5-616 JOURNAL DE PHYSIQUE
hole radiative recombination will be severely inhibited Such an inhibition can be much more complete than in a metallic waveguide2 since dielectric structures can in principle be lossless. Therefore it would be very interesting to create periodic three dimensional dielectric structures in which there exists an electromagnetic band gap.
There are two important differences between electromagnetic band structure and electronic band structure. (1)The underlying dispersion relation for electrons is parabolic while the underlying dispersion relation for photons is linear. (2)The polarization property of light means that there will be different dispersion relations for the two polarizations while for electrons this occurs only in a magnetic field Nevertheless the analogies are very strong. In particular, if the three dimensional dielectric modulation follows a common checker-board pattern in real space then the resulting band structure in reciprocal space will be that of a face centered cubic Brillouin zone. To assure an actual gap in the electromagnetic density of states rather than merely a minimum our calculations show that a finite modulation depth in the index of refraction will be required The minimum permissible modulation in the index of refraction n for an fcc Brillouin zone will be An=: 0.21n.
In conclusion the interplay between the band structure of light and that of electrcps results in some important new physical effccts. These effects can be employed in solid state electronics to reduce the threshold currents of semiconductor lasers.
I. "Spontaneous Emission Probabilities at Radio Frequencies", E. M. Purcell, Phys. Rev. 69, 681 (1946).
2. "Inhibited Spontaneous Emission by a Rydberg Atom", R G. Hulet, E S. Hilfer, and D. Kleppner, Phys. Rev. Lett 55, 2137 (1985)
3. "Observation of Inhibited Spontaneous Emission", G. Gabrielse and H. Dehmelt, Phys. Rev. Lett. 55, 67 (1985)
4. "Minority Carrier Lifetime and Internal Quantum Efficiency of Surface-Free GaAs", R J. Nelson and R G. Sobers, J. Appl. Phys. 49,6103 (1978)
5. "Coupled Wave Theory of Distributed Feedback Lasers", H. Kogelnik and C. V. Shank, J. of Appl.
Phys 43,2328 (1972).
