1ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityHandout 29Optical Transitions in Solids, Optical Gain, and Semiconductor LasersIn this lecture you will learn:• Electron-photon Hamiltonian in solids• Optical transition matrix elements• Optical absorption coefficients • Stimulated absorption and stimulated emission• Optical gain in semiconductors• Semiconductor heterostructure lasersECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityInteractions Between Light and Solids The basic interactions between light and solids cover a wide variety of topics that can include:• Interband electronic transitions in solids• Intraband electronic transitions and intersubband electronic transitions• Plasmons and plasmon-polaritons• Surface plasmons• Excitons and exciton-polaritons• Phonon and phonon-polaritons• Nonlinear optics• Quantum optics• Optical spintronics2ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityFermi’s Golden Rule: A ReviewNow suppose a time dependent externally applied potential is added to the Hamiltonian:titioeVeVHHˆˆˆˆConsider a Hamiltonian with the following eigenstates and eigenenergies:integerˆ mEHmmmoSuppose at time t = 0 an electron was in some initial state k:pt 0Fermi’s golden rule tells that the rate at which the electron absorbs energy from the time-dependent potential and makes a transition to some higher energy level is given by:pmpmmEEVpW2ˆ2The rate at which the electron gives away energy to the time-dependent potential and makes a transition to some lower energy level is given by:pmpmmEEVpW2ˆ2EEECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityOptical Transitions in Solids: Energy and Momentum ConservationkFor an electron to absorb energy from a photon energy conservation implies:infmkEkEFinal energyInitialenergyPhotonenergyMomentum conservation implies:qkkif Initial momentumFinal momentumPhotonmomentumIntraband photonic transitions are not possible: For parabolic bands, it can be shown that intraband optical transitions cannot satisfy both energy and momentum conservation and are therefore not possibleIntrabandIntrabandInterbandNote that the momentum conservation principle is stated in terms of the crystal momentum of the electrons. This principle will be derived later. E3ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityElectromagnetic Wave BasicsConsider an electromagnetic wave passing through a solid with electric field given by:trqEntrEo.sinˆ,The vector potential associated with the field is:ttrAtrE,, trqAntrqEntrAoo.cosˆ .cosˆ,The power per unit area or the Intensity of the field is given by the Poynting vector:  2ˆ2ˆ,,,222ooAqEqtrHtrEtrSI nooThe photon flux per unit area is:22oAIF EHqThe divergence of the field is zero:0,.,.  trAtrEECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityConsider electrons in a solid. The eigenstates (Bloch functions) and eigenenergies satisfy:knnknokEH,,ˆrVmPPHlatticeoˆ2ˆ.ˆˆElectron-Photon Hamiltonian in Solidswhere:In the presence of E&M fields the Hamiltonian is:    PnmeeAeHPtrAmeHPtrAmetrAPmeHtrAtrAmePtrAmetrAPmerVmPPrVmtrAePHtirqitirqioooolatticelatticeˆˆ2ˆ ˆ.,ˆˆ ˆ.,ˆ2,ˆ.ˆ2ˆ ,ˆ.,ˆ2ˆ.,ˆ2,ˆ.ˆ2ˆ2ˆ.ˆ ˆ2,ˆˆˆˆ.ˆ.22Assume smallProvided:0,.  trAtrqAntrAo.cosˆ,4ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversitykInterbandEOptical Interband Transitions in SolidsknnknokEH,,ˆSuppose at time t = 0 the electron was sitting in the valence band with crystal momentum :ikvt,0nPmeeAeHHtirqitirqiooˆ.ˆ2ˆˆˆ.ˆ.ikThe transition rate to states in the conduction band is given by the Fermi’s golden rule:ivfckvkckikEkEVkWiff2,,ˆ2The summation is over all possible final states in the conduction band that have the same spin as the initial state. Energy conservation is enforced by the delta function.ikComparison with:nPmeAeVrqioˆ.ˆ2ˆˆ.nPmeAeVrqioˆ.ˆ2ˆˆ.titioeVeVHHˆˆˆˆgives:ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityOptical Matrix Element ivfckvrqikckoivfckvkckikEkEnPemeAkEkEVkWiffiff2,ˆ.,22,,ˆ.ˆ22 ˆ2Now consider the matrix element: rnPerrdnPeififkvrqikckvrqikc,.,*3,ˆ.,ˆ.ˆˆ.ˆkInterbandEkInterbandE ruenPeruerdiiffkvrkirqikcrki,..,*.3ˆ.ˆ runPeeruerdrunkPeeruerdiiffiiffkvrkirqikcrkikvirkirqikcrki,..,*.3,..,*.3ˆ.ˆˆ.ˆ0Rapidly varying in spaceSlowly varying in space5ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityOptical Matrix ElementifkvrqikcnPe,ˆ.,ˆ.ˆkInterbandEkInterbandE  jifjfiiiffRkvkcjRkqkikvrkirqikcrkirunPrurderunPeeruerd,,*cell primitive th-3.,..,*.3ˆ.ˆˆ.ˆ  