Lecture_9

Lecture_9 - 3. Optical absorption, emission, and refraction...

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1 Prof. J. S. Harris 1 EE243. Semiconductor Optoelectronic Devices (Winter 2010) Direct gap optical absorption Indirect gap optical absorption Kramers-Kronig relationships Excitons Free carrier absorption Optical emission-Einstein A and B coef±cients Stimulated emission Spontaneous emission Optical Absorption-Einstein A and B coef±cients Optical refraction Reading-Ch 3 Notes, Bhattacharya pp 114-150 3. Optical absorption, emission, and refraction processes Prof. J. S. Harris 2 EE243. Semiconductor Optoelectronic Devices (Winter 2010) Even though derived from a basis of thermal equilibrium, the conclusions are valid more generally The A and B coef±cients are not changed whether in thermal equilibrium or not, hence we can use results for non-equilibrium situations, with the extreme example being the laser. Ratio between stimulated and spontaneous radiation from Eq. (3.45) for stimulated transition rate and Eq. (3.46) for spontaneous rate, and substituting using Eqs. (3.51) and (3.55), we have Ratio of the stimulated to spontaneous emission is the number of photons per mode, which is the Bose-Einstein Distribution . R 21 stim R 21 spon = B 21 f 2 (1 f 1 ) n phot ( E 21 ) A 21 f 2 f 1 ) = B 21 N phot n ph A 21 = n ph (3.63) Einstein A and B coefFcients - general applicability
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2 Prof. J. S. Harris 3 EE243. Semiconductor Optoelectronic Devices (Winter 2010) Note: in Eq. (3.63), we do not need to assume thermal equilibrium f 1 and f 2 need not be Fermi-Dirac distributions with the same Fermi energy--they can be different quasi-Fermi energies They could even be in another distribution altogether and n ph need not be Bose-Einstein distribution (certainly would not be a laser). Eq. (3.63) says threshold of laser (if de±ned as when stimulated emission exceeds spontaneous emission) is when there is one photon per mode. Note: can also prove equivalence of B 12 and B 21 (Eq. (3.54)) directly from perturbation theory, and relation equivalent to Eq. (3.62) appears naturally if the optical ±eld is quantized. Einstein A and B coefFcients - general applicability (2) Prof. J. S. Harris 4 EE243. Semiconductor Optoelectronic Devices (Winter 2010) Suppose in non-excitonic model that, with negligible carriers present, at some photon energy, , exceeding the bandgap energy, the absorption coef±cient is Hence when illuminated with light with some speci±c intensity at angular frequency ω , transition rate R vco from valence to conduction band is α o ( ) Suppose now that there are holes in valence band and electrons in conduction band, hence for a particular set of k -states involved, the probability of ±nding hole is f h and probability of ±nding electron is f e . The rate of transitions from valence to conduction band will be reduced, weighted by the probabilities of valence state being full and conduction state empty.
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This note was uploaded on 06/05/2010 for the course EE 243 taught by Professor Harris,j during the Winter '10 term at Stanford.

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Lecture_9 - 3. Optical absorption, emission, and refraction...

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