Hot electrons under quantization conditions .3. Analytical results and new nonlinear regimes

It is shown that the kinetic behaviour of a one-dimensional electron system is qualitatively different at low and high lattice temperatures. At low lattice temperatures the interaction has a strong inelastic character for the majority of electrons. As a result the electron distribution function is to be found from the integro-differential equation. This equation was solved analytically and we obtain the new distribution functions. We have shown that the current-voltage characteristic obeys a sublinear behaviour for warm and hot electrons. Within a wide range of the external electric field E the distribution function for the hot electrons has a sharp anisotropic shape corresponding to the electron streaming regime. The electric-field-dependences of the hot-electron mobility and the mean energy are E-(5/6) and E(1/2), respectively. With increasing E the electron-acoustic-phonon interaction becomes quasi-elastic an the electron distribution function, which is quasi-isotropic, is described by a differential equation of the Fokker-Planck type. No runaway effect arises in strong electric fields, the electron mobility does not depend on E (the 'second ohmic regime') and the mean energy increases as E(4). In the opposite case of high lattice temperatures the electron-acoustic-phonon interaction is always quasi-elastic for a majority of the electrons. The scattering rate decreases when the energy of the electron increases. This results in a runaway effect for hot electrons in a quantum wire and superlinear behaviour of the current-voltage characteristic. To stabilize the one-dimensional electron system it is necessary to take into account the transition of electrons to the continuous energy spectrum for thick quantum wires or interaction with optical phonons for thin quantum wires. We have derived general expressions for the distribution functions under different conditions which are of experimental interest. The theory we have developed can be generalized for a two- or three-dimensional electron gas subjected to an arbitrary quantizing potential, as well as to incorporate other scattering mechanisms.