Lecture_6

Lecture_6 - 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 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) Transition rate in a monochromatic ±eld Transition rate for absorption between an initial state, i (assumed occupied), with energy E i , and a ±nal state, f (assumed unoccupied), with energy, E f , in the presence of an oscillating perturbation of angular frequency ω , is This is derived using time dependent perturbation theory and is commonly known as Fermi ʼ s Golden Rule (especially when rate is summed over a continuum of initial and ±nal states) δ -function because, in ideal case, we need exactly the right photon energy (Derived in Appendix 4 of Bhattacharya) Alternatively, replace δ -function with, e.g., Lorentzian line of the same area, with some ±nite linewidth; ±nal results will be similar. (3.1) W abs = 2 π f H ' r ( ) i 2 E f E i ( ) Direct gap optical absorption - transition rate
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2 Prof. J. S. Harris 3 EE243. Semiconductor Optoelectronic Devices (Winter 2010) Assume we are dealing with a single electron, so f H ' r ( ) i = ψ f * r ( ) H ' r ( ) i r ( ) d 3 r H ' r , t ( ) = H ' r ( ) e i ω t + H ' + r ( ) e i t H ' r , t ( ) ≅ − e m o A p i h A = ˆ e A o 2 exp i k op r t ( ) [ ] + A o 2 exp i k op r t ( ) [ ] Where i ( r ) and f ( r ) are, respectively, the wave functions of the electron initial and Fnal states H ʼ ( r ) is spatial part of the time-dependent perturbing Hamiltonian [ H ʼ + ( r ) is the Hermitian adjoint of H ʼ ( r ) ]. for electron in an electromagnetic Feld, usual form for H ʼ ( r ) where p is the momentum operator, & A is the electromagnetic vector potential corresponding to a wave of (angular) frequency . (3.2) (3.3) (3.4) (3.5) Direct gap optical absorption-transition rate (2) Prof. J. S. Harris 4 EE243. Semiconductor Optoelectronic Devices (Winter 2010) k op is the wave vector of the optical Feld inside the material and the Feld is to be linearly polarized with its electric vector in the direction, ê . With these choices for H ʼ ( r ,t) , we have H ' r ( ) = eA o exp i k op r ( ) 2 m o ˆ e p (3.6) Direct gap optical absorption-transition rate (3)
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3 Prof. J. S. Harris 5 EE243. Semiconductor Optoelectronic Devices (Winter 2010) We need to know ψ i ( r ) and f ( r ) The initial and Fnal states, i and f , are not “single-electron” states They are states of the entire crystal before ( i ) and after ( f ) absorption of a photon, therefore multi-electron states are possible (important for excitons,which are discussed later) Presume, for the moment, i ( r ) and f ( r ) are “single-electron” Bloch states in valence and conduction bands, respectively, in the “non-excitonic” approximation.
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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_6 - 3 Optical absorption emission and refraction...

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