Frequency limits of terahertz radiation generated by optical-phonon transit-time resonance in quantum wells and heterolayers

A universal description of the OPTTR-assisted generation band of THz radiation is developed in the framework of a 3D transport in bulk materials and 2D transport in QWs and HLs. On the one hand, in the framework of an ideal streaming motion we have carried out an analytical investigation based on phonon scattering rates: namely, elastic deformation acoustic phonons in the passive energy region below the optical-phonon energy 0 and emission of polar optical phonons in the active energy region, 0. On the another hand, quantitative estimations of the qualitative constraints given by Eqs. 1 and 2 for the streaming conditions are obtained on the basis of numerical calculations of the OPTTR DNDM by the Monte Carlo method. It is found that i for Eq. 1 the average momentum relaxation time in the passive region must satisfy the condition −1–2E and ii for Eq. 2 carrier penetration into the active region must occurr at a level less than 14%–15%. In the framework of such a model we have provided simple analytical expressions that estimate the low- and high-frequency limits of the generation band determined by, respectively, the average relaxation time − see Eq. 25 and the carrier penetration into the active region see Eq. 33. Having in mind that for THz radiation generation the high-frequency limit is of the most interest, for the highfrequency cutoff of the DNDM we have found that i for 3D bulk materials the relevant physical quantities are the carrier effective mass, the optical-phonon energy, and the polaroptical coupling strength; ii in passing from 3D to 2D vertical transport, for the same material the influence of 2D transport on the OPTTR is characterized entirely by the dimensionless parameter k0d related to both the radius of the optical-phonon sphere in wave-vector space k0 and the effective width of the electron localization d associated with the lowest miniband of the QW/HL structure; iii in going from 3D to 2D transport, the change of the energy dependence of the density of states is responsible for an extension of the maximum generation frequency for up to a factor of 5 times. In essence, such a model gives the “upper” estimation of the generation band limits determined primarily by the parameters of a bulk material and a 2D structure. Any other scatterings not incorporated directly into our model— namely, impurity, electron-electron, interface roughness, etc.—act mainly on the low-frequency limit since one can speak about the upper limit only in the case when an electron runaway from the regime of low-energy scatterings in the passive region takes place. Of course, the presence of additional low-energy scatterings will increase the low-frequency limit and eventually destroy the generation. Nevertheless, additional scattering mechanisms can be simply incorporated into the model by taking into account their contribution to the average momentum relaxation time in the passive region − as 1/−=1/DA+1/imp +1/ee. For example, such a procedure was used to estimate − in the experimental observation of the OPTTR generation in bulk InP.20 These estimations give relaxation times, which is reasonably close to the values estimated above.