Abstract
Scattering rates arising from the interactions of electrons with bulk longitudinal optical (LO) phonon modes in a hollow cylinder are calculated as functions of the inner radius and the uniform axial applied magnetic field. Now, the specific nature of electron-phonon interactions mainly depends on the character of the energy spectrum of electrons. As is well known, in cylindrical quantum wires, the application of a parallel magnetic field lifts the double degeneracy of the non-zero azimuthal quantum number states; \(m \not = 0\); irrespective of all electron’s radial quantum number l states. In fact, this Zeeman splitting is such that the \(m < 0\) electron’s energy subbands initially decrease with the increase of the parallel applied magnetic field. In a solid cylinder, the lowest-order; {\(l = 1;\,m = 0\)} subband is always the ground state. In a hollow cylinder, however, as the axial applied magnetic field is increased, the electron’s energy subbands take turns at becoming the ground state; following the sequence \(\lbrace m=0,-1,-2...~ -N\rbrace\) of azimuthal quantum numbers. Furthermore, in a hollow cylinder, in general, the electron’s energy separations between any two subbands are less than the LO phonon energy except for exceptionally high magnetic fields, and some highest-order quantum number states. In view of this, the discussion of the energy relaxation here is focused mainly on intrasubband scattering of electrons and only within the lowest-order {\(l = 1;\,m = 0\)} electron’s energy subband. The intrasubband scattering rates are found to be characterized by shallow minima in their variations with the inner radius, again, for a fixed outer radius. This feature is a consequence of a balance between two seemingly conflicting effects of the electron’s confinement by the inner and outer walls of the hollow cylinder. First; increased confinement of the charge carriers generally leads to the enhancement of the rates. Second; the presence of a hole in a hollow cylinder leads to a significant suppression of the scattering rates. The intrasubband scattering rates also show a somewhat parabolic increase in their variations with the applied magnetic field; an increase which is more pronounced in a relatively thick hollow cylinder.
Highlights
The rapid advancement in fabrication technologies (Liang et al, 2019; Fu et al, 2020) as well as experimental design (Beenakker and van Houten, 1991; Kim and Olendski, 1996) has led to a great deal of renewed interest in the interactions of confined electrons with the various crystal excitations, especially in semiconductors (Peeters and De Boeck, 1999)
Electron-longitudinal optical (LO)-phonon scattering rates of a hollow cylinder were calculated as functions of the inner radius as well as the axial applied magnetic field, assuming bulk LO-phonon modes
The nature of this Zeeman splitting in a hollow cylinder is such that the {l = 1, m ≤ 0} electron’s energy subbands take turns at being the ground state, as the magnetic field is increased; following the sequence {m = 0, − 1, − 2...} of the azimuthal quantum numbers
Summary
The rapid advancement in fabrication technologies (Liang et al, 2019; Fu et al, 2020) as well as experimental design (Beenakker and van Houten, 1991; Kim and Olendski, 1996) has led to a great deal of renewed interest in the interactions of confined electrons with the various crystal excitations, especially in semiconductors (Peeters and De Boeck, 1999). The overall potential of low-dimensional systems falls into two broad categories: a) Confinement of the charge carriers may be due to the intrinsic socalled electric potential of the fabricated nanostructures (Tonucci et al, 1992; Goldberger et al, 2003) This form of the potential depends on the specific topology or even the geometrical symmetry of the quantum structure (Santos Jr. et al, 2005; Filikhin et al, 2006). The specific character of the electron’s spectrum itself depends on the overall confinement as well as the geometrical symmetry of the system considered In this regard, of particular interest here, at least qualitatively, is how the topological evolution from a solid to a hollow cylinder influences the ground-state intrasubband scattering rates
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