Theoretical Considerations About the Generation and Properties of Narrow   Electron Flows in Solid State Structures
  A method for the evaluation of the angular width of an electron beam generated by a nanoconstriction is proposed and demonstrated. The approach is based on analysis of a narrow-width electron flow, that quantizes into modes inside a confining constriction which is described in the adiabatic approximation, evolving into a freely propagating electronic state after exiting the constriction. The method that we developed allows us to find the parameters and the shape of the constriction that are optimal for generation of extremely narrow electron beams. In the case of a constriction characterized by a linear widening shape an asymptotically exact solution for the injection problem is found. That solution verifies semi-quantitative results related to the angular characteristics of the beam, and it opens the way for determination of the distribution function of the electrons in the beam. We have found the relationship between the angular distribution of the electron density in the beam and the quantum states of the electrons inside the constriction. Such narrow electron beams may be employed in investigations of electronic systems and in data manipulations in electronic and spintronic devices. 
R. N. Gurzhi, A. N. Kalinenko, A. I. Kopeliovich, A. V. Yanovsky, E.   N. Bogachek and Uzi Landman