Optical transition rates in a cylindrical quantum wire with a parabolic and inverse parabolic electric confining potential in a magnetic field

Electron transition rates due to interaction with circularly polarized light incident along the axis of a free-standing solid cylindrical nanowire are evaluated in the dipole approximation. The electric confinement potential of the nanowire is modeled as a superposition of two parts, in general, of different strengths; viz; parabolic and inverse parabolic in the radial distance. Additional confinement of the charge carriers is through the vector potential of the axial applied magnetic field. In systems with cylindrical symmetry, the electronic states are in part characterized by azimuthal quantum numbers: m=0,±1,±2..,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m=0,\pm 1,\pm 2..,$$\end{document} which in the absence of the axial applied magnetic field are doubly degenerate. In the dipole approximation and for circularly polarized light the selection rules are such that optical transitions are allowed between electronic states whose azimuthal quantum numbers differ by unity. Transition rates are characterized by peaks whenever the energy of the incident electromagnetic radiation matches transition energies for states between which transitions occur. The parabolic potential blue shifts peaks of transition rates while the inverse parabolic potential redshifts the peaks. Results also indicate that transition rates are higher in nanowires of smaller radii. The homogeneous magnetic field lifts the double-degeneracies of electrons with opposite angular momenta, which leads to the emergence of two branches of the transition rates.


Introduction
Confinement of charge carriers is crucial in the design of nanodevices because of inherent quantum effects. Quantum confinement is possible due to nanofabrication techniques like molecular beam epitaxy Gómez et al. 2019), lithography (Williams et al.

Theory
The system studied here is a very long solid cylinder of length L z and radius R in a uniform magnetic field applied parallel to the axis of the wire. The vector potential of the corresponding applied magnetic field is taken in the gauge = (0, 1 2 B , 0). The intrinsic confining potential of the quantum wire is taken in the form of a linear combination of the parabolic and inverse parabolic parts given by where op and oi are the angular frequencies corresponding to the appropriate factors. These frequencies may be regarded as measures of strengths the corresponding forms of the potential. The Hamiltonian for the system considered here is cylindrically symmetric thence the wavefunction is taken in the general form: ( , , z) = C ml ( )e ik z z e m , where i = √ −1 is the unitary complex number, k z is the axial wavenumber, C ml the normalization constant of the corresponding azimuthal and radial quantum numbers m and l, respectively. The radial part of the electron's wave function satisfies the following differential equation: where E ml is the electron's subband energy and c = eB∕ is the electron's cyclotron frequency. The general solution of Eq. (2) is found in terms of a linear combination of the confluent hypergeometric functions M and U. The function U, however, has a divergent behavior at the origin = 0 of the wire and is therefore discarded as a solution. To be more precise, the solution of equation that is well behaved for 0 ≤ ≤ R is given by where and Imposing the boundary condition of continuity of the wave function at = R gives the equation for the determination of the energy eigenvalues as where a 0 is the value of a which satisfies the explicit boundary condition of continuity of the wavefunction at = R given by M(a 0 , b; R ) = 0 with R = √ 2 c + 4 2 0p × ( R 2 ∕2ℏ).

Transition rates
We consider circularly polarized electromagnetic radiation of energy (ℏ ) incident along the axis of the quantum wire. An electron can absorb an incident photon and transition from an initial state to a final state, provided that the energy of the photon is equal to the difference in energies of the initial and final states. The transition rate of the electron from an initial state i with energy E i to a final state f with energy E f is given by the Fermi Golden rule (Hashimzade et al. 2005) The electron-photon interaction H int is given by where is the photon field wave vector, the electron position vector, ̂ is the unitary polarization vector of the radiation field and A 0 = √ (N q ℏ)∕(2 0 m V) is the amplitude of the vector potential. N q is the number of photons in volume V of the wire of dielectric constant m and 0 is the permittivity of free space. Now, for circularly polarized light incident along the axis of the cylindrical wire, = (0, 0, q z ) and ̂⋅ = (cos ± sin ), where the + (-) is for right (left) circular polarization. This gives the absorption and emission transition rates as and respectively, where is the interaction integral and x = ∕R . n z is the photon linear density. For circularly polarized light and in the dipole approximation, the allowed transitions are only those for which the azimuthal quantum numbers of the initial and the final states differ by unity. These selection rules of optical transitions apply only to the azimuthal quantum numbers and with no restriction on the radial quantum numbers. Now, for purposes of numerical calculations, the Dirac delta function has been replaced with a Lorentzian factor in which is the so-called linewidth of resonance. The actual total transition rate of the quantum system involves a summation of all the individual transition rates, of course, taking into account the transition probabilities of all the allowed transitions. This is can be calculated from the following equations (Renk 2012) and (8) for the initial and final state respectively. Here, E F is the Fermi energy of the crystal at temperature T and K B is the Boltzmann's constant. The above equations can be used to evaluate rates of excitonic transitions, and with a number of factors taken into account. i) The band gap between the valence and the conduction band must be taken into account when evaluating the transition energies. ii) Appropriate wave functions should be used: those of the holes in the valence band and those of electrons in the conduction band under the Coulomb interaction, with proper effective masses for the electrons and holes (Singh and Oh 2005). These values are of the order of picoseconds, for example, Citrin found ≈ 150 ps for a cylindrical GaAs quantum wire of radius 100 Å (Citrin 1993).

Results and discussions
This section is concerned with the discussions mainly of the influence of the different variations of the electric confinement potential and the applied magnetic field on the optical transition rates. It has to be said that the two parts of the intrinsic potentials tend to have opposite influences on the optical transition rates. For this reason, the influence of these parts of the intrinsic confinement potential on the transition rates is evaluated separately. The discussions, however, are limited to transitions only between the lowest electron's energy subbands; viz; {= 1;m = 0} → { � = 1;m � = ±1} . Furthermore, only the results for the absorption process are shown for, brevity, since the functional forms of the absorption and the emission processes are very similar. The key parameters used for the computations here, and relevant to the system of GaAs are = 0.67m e , m = 12.5, E F = 3 meV, = 1 meV and n z = 4.634 × 10 9 m −1 . Figure 1 shows some of the lowest quantum number { ; ± m} = {1, 0, ±1, ±2} electron's energy subbands in a solid cylindrical quantum wire of radius R = 300Å with the parallel applied magnetic field. Here, the electric confinement potential of the quantum wire is taken as a special form of that given by equation (1), with the parabolic part 'switched off', that is, ℏ op = 0 but ℏ oi = 2 meV. As is well known, in systems with cylindrical symmetry, the parallel applied magnetic field lifts the double degeneracy of the m ≠ 0 energy subbands. The m ≥ 0 electron's energy subbands increase with the increase of the magnetic field. The m < 0 energy subbands, however, initially decrease with the increase of the magnetic field and attain minima in their variations with the applied magnetic field. An additional feature of the electron's energy spectrum in a solid cylindrical quantum wire wherein the electric potential has an inverse parabolic variation with the radial distance. Such an electric potential tends to infinity near the axis of the quantum wire which results in a forbidden innermost region for the motion of electrons. Thus, for a wire with this type of potential, as the field increases, the m < 0 states take turns at becoming the ground state (that is, m = −1, −2, −3, … ). This is rather akin to the effect of the inner wall in cylindrical shell in a magnetic field (Masale et al. 1992) and electron-electron repulsion of confined electrons in a magnetic field (Wagner et al. 1951).  The dependence of transition energies on the energy of the magnetic field in a quantum wire of radius R = 300Å . The pair of plots marked a corresponds to an infinite cylindrical square well (ICQW) ( ℏ 0 = 0 ), b corresponds to a parabolic potential ( ℏ 0p = 5 meV) superimposed on an ICQW while c is associated with an inverse parabolic potential ( ℏ 0i = 2 meV) superposed on an ICQW these curves with the same intercepts correspond to the different specific variations of the electric confining potential as follows: (a) Infinite Cylindrical Quantum Well (ICQW), corresponding to ℏ oi = ℏ op = 0; (b) parabolic potential solely (ℏ oi = 0) of strength such that ℏ op = 5 meV and (c) inverse parabolic potential only (ℏ op = 0) of strength ℏ oi = 2 meV. Note that the inverse parabolic potential tends to dilate the electron's wavefunction and, in a sense, 'push' it towards the outer regions of the quantum wire. This is in contrast to the effect of the parabolic potential, which just like the vector potential of the magnetic field, tends to compress the electron's wavefunction towards the inner regions of the quantum wire. In the case of the inverse parabolic potential, the relative increase of the electron's energy with the increase of the magnetic field of the m = 0 subband is thus more pronounced than that of the m = ±1 subband. The opposite is true in the case of the parabolic potential alone. As such, the parabolic potential enhances transitions energies while the inverse parabolic potential reduces the transition energies (Tshipa 2014). Now, by inference from Fig. 1, the transition energies corresponding to the ground state and the first excited state ΔE = |E −m − E −m−1 | initially close but then open up again as the parallel applied magnetic field is increased. The absolute values of these transition energies therefore pass through a minimum of zero in their variations with the applied magnetic field at the respective cross-over points whenever the −(m + 1)th energy subband now becomes the ground state. Figure 3 depicts the zero-magnetic field lowest-order {m = 0 → |m| = 1} optical transition rates of an ICQW of three different radii: r 1 = R = 200Å, r 2 = R = 250Å and r 3 = R = 300Å, as indicated in the inset. Each curve, for a given radius, is characterized by a peak. As stated earlier, a peak of a transition rate occurs whenever the photon energy matches the transition energy. Here, determines the width of the curves for the optical transition rates. Note that maxima occur at lower photon energies in quantum wires of larger radii. It is also worth noting that stronger transitions occur in a quantum wire of smaller radius. This is a feature consistent with Sakaki's (1980) prediction of increased speeds of operation in nano-devices as a result of reduced dimensions. Maxima of the transition rates is found to occur at higher values of the photon energy for smaller radii of the quantum wire. Figure 4 depicts the { m = 0 → m = ±1 } transition rates as functions of photon energy for ℏ c = 0 meV (the middle curve) and for ℏ c = 5 meV for the other two curves. The additional relevant parameters are R = 300Å and ℏ oi = ℏ op = 0. It is seen that the variation of the transition rates with ℏ is similar to that shown in Fig. 3. In fact, the middle curve here is exactly the curve corresponding to r 3 = R = 300Å in Fig. 3, and is included here for reference. The key feature of this fig. is that the application of the axial magnetic field splits the zero-field curve into the two branches flanking it. These branches of the transition rates emerge as a consequence of the lifting of the double-degeneracy of the m ≠ 0 electron's energy subbands by the parallel applied magnetic field in a cylindrical quantum wire (Masale 2000(Masale , 2002(Masale , 2008. Note also that the transition energies as well as the rates are higher for transitions involving m = +1 than those involving m = −1. Figure 5 shows variations of the lowest-order {m = 0 → |m| = 1} optical transition rates with the photon energy in a quantum wire of radius R = 300Å in the absence of an applied magnetic field. The other relevant parameters are (a) ℏ oi = ℏ op = 0, (b) ℏ oi = 0; ℏ op = 10 meV and (c) ℏ oi = 2 meV; ℏ op = 0. Since the parabolic potential increases transition energies, it blueshifts the peaks of transition rates. The inverse parabolic potential on the other hand deceases transition energies and hence redshift peaks of transition rates. Both potentials decrease the magnitude of the transition rates, with the inverse parabolic potential causing more decrease. Figure 6 shows the variations of optical transition rates with the photon energy in the absence of an applied magnetic field in a solid cylindrical wire of radius R = 300 Å. The additional parameters are the corresponding values of temperature used: T = 100 K, 200 K and 300 K, as indicated. The solid curves are transition rates evaluated by considering other subbands (according to Eq. 12) while the dash-dotted curve represents the { m = 0 → m = ±1 } transition rates as calculated using Eq. 9. As temperature decreases, peaks of transition rates reduce in magnitude. This reduction is more apparent in peaks  The variation of { m = 0 → m = ±1 , l = l � = 1 } transition rates with the energy of the incident radiation field in the absence of the magnetic field. Radius of the quantum wire is R = 300Å . The pair of plots marked a corresponds to an infinite cylindrical square well (ICQW) ( ℏ 0 = 0 ), b corresponds to a parabolic potential ( ℏ 0p = 10 meV) superimposed on an ICQW while c is associated with an inverse parabolic potential ( ℏ 0i = 2 meV) superposed on an ICQW Fig. 6 The total scattering rates (solid plot) in a cylindrical quantum wire of radius 300Å at temperatures T 1 = T = 100 K, T 2 = T = 200 K and T 3 = T = 300 K, as functions of energy of the incident EM radiation, as calculated according to Eq. 12. The dash-dotted are rates for transitions from the m = 0 state to the m = ±1 states, as calculated using Eq. 9. Here, ℏ 0p = ℏ 0i = ℏ c = 0 'a' correspond to an ICQW ( ℏ 0p = ℏ 0i = 0 ), those labelled 'b' correspond to a parabolic potential ( ℏ 0p = 5 meV, ℏ 0i = 0 ) superimposed on an ICQW, while 'c' is associated with the inverse parabolic potential ( ℏ 0i = 2 meV, ℏ 0p = 0 ), superimposed on an ICQW. In Fig. 7, the energy of incident radiation is 1 = ℏ = 4 meV, which is less than zero-field transition energies while for Fig. 8 it is 2 = ℏ = 6 meV, which is greater than the zero-field transition energies. In the case where the beam energy is less than zero-field transition energies, increase in the magnetic field will induce transitions involving only the The situation is the other way around when the photon energy is greater than the zero-field transition energies. In that case, an increase of the magnetic field induces transitions involving only the m = 0 and m = +1 electron's states. With reference to Fig. 2, even though 1 is less than zero-field transition energies of an ICQW, it is greater than the zero-field transition energies for an ICQW with a superimposed inverse parabolic potential. This implies that in this case, as the magnetic field energy increases, only { m = 0 → m = 1 , l = l � = 1 } transitions will be possible. 2 , which is greater than the zero-field transition energies for an ICQW, is nevertheless less than the zero-field transition energies for an ICQW with a superimposed parabolic potential. As such, in this case, the magnetic field can induce only the { m = 0 → m = −1 , l = l � = 1 } transitions.
The values of transition rates obtained here are comparable to those obtained experimentally. For example, Bellesa et al. deduced excitonic lifetimes of 350 picoseconds (ps) from photoluminescence measurements in GaAs∕Al x Ga 1−x As quantum boxes of length L = 40 nm formed on a V-grooved GaAs substrate, at a temperature of 6 K (Bellessa et al. 1998). Lifetimes are inverses of transition rates. Since the geometry is different from the one studied here, to compare the two we consider the radius of a quantum wire that would have the same cross-sectional area as the QB considered by Bellesa et al., that is R = L∕ √ ≈ 226 Å . With this value of the radius, using Eq. 12 in conjunction with the full width at half maximum of = 8 meV, we obtained peaks of transition rates of 0.00264/ ps corresponding to lifetimes of 379.3 ps. Akiyama et al. carried out photoluminescence measurements in GaAs rectangular quantum wires grown by molecular beam epitaxy (Akiyama et al. 2005). In the paper, the authors investigate the effect of temperature on radiative lifetimes. Similar to comparison with the QB above, we consider a radius that will give same cross-sectional area as the one studied by Akiyama et al., that where L x = 9 nm and L y = 18 nm are the dimensions of the quantum wire. Using a broadening of = 10 meV and a temperature of T = 77 K, we obtained peaks of transition rates of 0.00169/ ps (corresponding to lifetimes of 591 ps), compared with the value of 600 ps obtained experimentally Akiyama et al. Melliti et al. obtained lifetimes in InAs/GaAs quantum dots (Melliti et al. 2003) of around 800 ps. It is worth noting that the rates obtained in these experiments were for excitonic transitions.

Conclusions
Optical transition rates of a cylindrical quantum wire immersed in an axial applied magnetic field were evaluated within the effective mass and the dipole approximations. The radiation field was taken as that of circularly polarized light incident along the axis of the wire. The investigations here also included the influence of each of the two intrinsic confinement potential forms of the quantum wire on optical transition rates; parabolic or inverse parabolic in the radial distance. In systems with cylindrical symmetry, the angular momentum of an electron is quantized in integral multiples of m, the azimuthal quantum number. For circularly polarized light, and in the dipole approximation, the allowed optical transitions are only those for which the azimuthal quantum numbers of the initial and final states of the electron differ by unity. The investigations carried out here were limited mainly to evaluation of the transition rates only between the lowest pairs of the azimuthal quantum numbers: {m = 0 → m = ±1}. Overall, optical transitions rates were found to posses a characteristic peak in their variations with any of the relevant parameters. These resonance peaks occur whenever the photon energy matches exactly the electron's energy separations of the subbands involved in the optical transitions. The specific character of the electron's energy subbands, and therefore the energy separation between these, is determined by the overall confinement potential of the system considered. The parabolic potential has a tendency to blueshift peaks of optical transition rates, while the inverse parabolic potential redshifts peaks of optical transition rates. The axial applied magnetic field lifts the double degeneracy of the m ≠ 0 electron's states, and consequently the emergence of the two optical transition rates corresponding to the m = −1 and m = +1 electron's final states. Results also indicate that transition rates diminish as temperature decreases, and this reduction is more pronounced for transitions involving higher states. The results of these investigations suggest that the overall confinement of a nanosystem and temperature can be useful in the tuning of nanodevices in the regime of appropriate wavelengths.
Funding None.

Conflict of Interest
The authors declare that they have no conflict of interest.