ACOLS'98 December 1998 % 19981023 26 27 1213 ------------------------------------------------------ Introduction: get to the point 1 1 1 1 1 1 1 ------------------------------------------------------ * semiconductor microstructures * stacks of different semiconductor layers, with the different materials having different conduction band energies, so the electrons see a stepped potential profile -- eg QW, SL, etc. * imagine a layer of GaAs sandwiched between AlGaAs - this is a square quantum well, where the bound states are subbands with 2D of in-plane freedom -- we can use an effective mass approximation, and assume parabolic subbands with energy proportional to square of (crystal) momentum: E prop k^2 ------------------------------------------------------ Motivation 2 2 2 2 2 2 2 ------------------------------------------------------ * Helm, England, Colas, DeRosa, Allen (PRL 1989) et al produced emission in the mid-IR. * 60 period SL under bias, charge redistribution flattens QW bottoms * active regions separated by (simple) tunnel barriers -- e- tunnel through barriers; scatter down, perhaps emitting photon, then tunnel out (into next well). * radiative emission is weak, but gets relatively more likely for smaller inter-band separations (LO cutoff) -- many periods helps * Progress: Faist, Capasso, Sivco, Sirtori, Hutchinson, Cho (Science 1994) demonstrated an inter-subband laser; then further progress with room temperature devices Sirtori, Faist, Sivco, Hutchinson, Cho (IEEE PL 1997) * basic concept: 25 periods of active regions and relaxation regions -- U is fed by electrons tunneling from the (LH) minibands, then we get emission on U-L, L-G, and electron tunnel into (RH) miniband, where they scatter down, and then tunnel into the next active region (U-L-G) * Read points on slide * before we answer the question "Can we", we need to understand the non-radiative processes which dominate the electron dynamics: eLO and ee. ------------------------------------------------------ Scattering: An overview 3 3 3 3 3 3 3 ------------------------------------------------------ * As I have said, the non-radiative scattering processes are the most important, and dominate the carrier dynamics -- however I do not have time to go through them in detail. * Scattering; radiative vs non-radiative (e-phonon and e-electron), in a square QW, parabolic subbands E vs k (see RH diagrams) * Radiative scattering: this is the process we want to promote to get good THz emission. * its rate is proportional to E, and it is much weaker than the non-radiative -- and the smaller the subband separation, the slower the radiative emission (this is bad!). * emission: the electron drops from a higher subband to a lower, emitting the energy difference as a photon. * it is "vertical" -- the phonon momentum is negligible compared to the electron momentum, so we get a delta-function-like emission, which is collision limited, symmetric, and relatively insensitive to thermal broadening (cf to emission by e-h recombination) * figs 1: * Non-radiative scattering (1) * Electron-LO phonon scattering is one of the main non-radiative processes. * here the absorption or emission of a phonon changes the state of an electron * LO phonons are effectively dispersionless -- same energy for any k, so eLO can transfer momentum (k), but the amount so transferred depends on the band separation and (assumed dispersionless) LO phonon energy. The higher the k, the slower the scattering rate. * E < E_{LO}: For band separations less than the LO phonon energy, the scattering rate is highly temperature dependent. + ie. zero unless the electron has some in-plane k; this can occur due to the electron population having some thermalised distribution. + And the higher the temperature, the less of a cut-off. * E = E_{LO}, a high "resonant" rate due to low-k transfer * E > E_{LO}, its rate is proportional to 1/E * DCM phonons -- each layer in the structure has its own LO (LC) phonon modes, with differing phonon energies; and there are additional interface phonons that extend across the entire structure, and have dispersion. * figs 2: * Non-radiative scattering (2) * Electron--electron scattering is the other main non-radiative processes. * e bounces off e with total E and total k conserved * many different rates ... 4^N of ijfg * rates in general proportional to 1/E (graph of 2211 rate) * intra-band rates are typically very fast * inter-band rates vary according to inter-band energy separations and wavefunction overlaps. * two-fermions -- exchange! * Symmetric systems only have "symmetric" processes (eg LH diagram), asymmetric ones have "auger-type" processes as well (eg RH diagram) * Scattering and Pauli exclusion ... an electron cannot scatter into an already filled state. * Sum up: * Note the complex behaviour or scattering rates - eLO peaks at 30um and are temperature sensitive for small subband separations * ee increases for small subband separations * radiativeproportional to 1/E * we can make progress despite these difficulties ------------------------------------------------------ Electron - Phonon Scattering 4 4 4 4 4 4 4 ------------------------------------------------------ * Bulk LO phonons - "usually" assumed dispersionless, for GaAs ELO=36meV * Each phonon mode has an associated electric potential, and this perturbs the electrons in the system and causes them to scatter * bulk mode potentials are like exp(ik.r) * for calculations we assume the electrons have a thermalised Fermi-Dirac distribution, due to fast ee intra-subband scattering * then we average the detailed rates over the distributions to get the net rates * Briefly on DCM: * Dielectric Continuum Phonons (DCM) are split into two types * confined LO (LC), which have sinusoidal potentials that are completely confined to each individual layer, and do not interact with adjacent layers * LC are usually assumed to be dispersionless * AlAs or GaAs layers have phonons at their bulk phonon energies, but alloy layers (AlGaAs) have both AlAs-like and GaAs-like LC phonons, whose energies depend on the alloy composition * the number of LC phonons in a structure is determined by the numbers of layers. * interface phonons (IF) potentials decay or increase exponentially, and extend across the entire structure, and are matched up at the interfaces. * IF phonons have dispersion (eg about 2meV in AlGaAs AQWs) * the number of IF phonons in a structure is determined by the number of differing materials it is made up of. (?) ------------------------------------------------------ Electrons - Electron Scattering 5 5 5 5 5 5 5 ------------------------------------------------------ * scattering is due to the coulomb interaction * energy and momentum are conserved * temperature dependent static screening * ignore (for now) plasmons, which are collective effects at higher electron densities * rate increases with density (low densities) * rate increases with decreasing subband spacing + (diagrams) energy (symmetric and Auger), momentum conservation * labelling ijfg * form factors due to wavefunction overlap * Exchange --> momentum diagram * Pauli exclusion -- cannot scatter into a filled state * for calculations we assume the electrons have a thermalised Fermi-Dirac distribution, due to fast ee intra-subband scattering * then we average the detailed rates over the distributions for each of the pair of electrons to get the net rates ------------------------------------------------------ Electrically Injected Emitters 6 6 6 6 6 6 6 ------------------------------------------------------ * for 1-10 THz emission, need subband separation delta_E ~ 4-41meV * Superlattice: regular series of QWs, with period L. * design the SL so that the right E field changes the usually delocalised mini-band states into ones localised in the region of one (or perhaps 2) wells. * Then the energy of each state is evenly spaced in a Stark Ladder, and the energy difference between each adjacent state is proportional to the applied field F (NB avoid domain formation and/or charge build-up) * This gives us tunable emission (within design limits) * delta_E = E_{n+1}-E_n = eFL * example SL cf Donovan, Harrison, Kelsall J.App.Phy. * AlGaAs barriers 50 angstroms, GaAs wells 50 angstroms * eLO rate is 50 thousand times faster than the radiative rate - a tiny quantum efficiency of 2E-5. * 100 periods ... rate is better. * doping N ~ 10^{11} cm^{-2}, E=34kV/cm * for a level-lifetime of T, the current density will be J=Ne/T (2kA/cm^2) * phoons emitted per well is N/T_{rad} * read slide ... 1mW of output power * Scamarcio, Capasso et al PRB1998 ~200meV emission (~50THz) mid-IR ------------------------------------------------------ Optically excited AQW 7 7 7 7 7 7 7 ------------------------------------------------------ * Optical excitation simplifies design * laser - needs stimulated emission and population inversion -- we need a fast depopulation of the lower laser level L. * Berger 3 level, Harrison-Kelsall 3&4 level * population inversion -- population ratio, from a ratio of scattering rates. * population ratio is the ratio of the upper-to-lower-level rate (1/T_{UL}) to the rate out of the lower-level (1/T_{Lx}) -- the faster the Lx rate, the faster the lower laser level is depopulated, and the better the inversion. But this needs to be faster than the lower level repopulation from the upper level (rate UL). * * 3 level (filled symbols): ok at very low temps and low carrier densities (reduced ee) -- due to strong LO rate cut-off at low temperatures * 4 level (open symbols): resonant eLO depopulation from lower-laser level enhances the population ratio; this gives better performance at higher temperatures, and even the possibility of room temperature inversion! ------------------------------------------------------ Triple Quantum Well 8 8 8 8 8 8 8 ------------------------------------------------------ * triple QW, one level per well * designed to have levels in wells 2 and 3 (U,L; or L,U) that are close enough in energy so by applying a bias voltage across the structure we get an anticrossing. * wavefunctions are localised in the wells, except near the anticrossing wher we get symetric and antisymetric combinations in wells 2 and 3. * triple well "active region", designed to have an anticrossing between levels 1 and 2 at field strength F0 -- NB increase field, increase bias, increase slope on diagram * the anticrossing is used to maximise the 2->1 rate, but it means that we can no longer ignore the 1->2 rate. As a first approximation, we use the nett rate (ie the difference) * we get the maximum eLO and ee (2->1) rates at the anticrossing * at 300K * results: population ratio as a function of applied field. the shaded region is the "practical" region as for lower fields (ie below the anticrossing) the U and L wavefunctions are localised at opposite ends of the structure, thus killing off the radiative emission between them. * if we narrow the barrier between the central and RH well, E2-E1 increases (wider anticrossing), so ee scattering decreases but eLO increases (more thermal electron LO emission), and the results look like ------------------------------------------------------ Conclusions 9 9 9 9 9 9 9 ------------------------------------------------------ * read the slide