# Optical Properties of GaAs Nanowires with an Electric Potential That Varies Inversely with the Square of the Radial Distance

Authors: Moletlanyi Tshipa
Journal: Advances in Condensed Matter Physics
 (2019)

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## Abstract

A theoretical investigation of optical properties of a cylindrical quantum wire (CQW) is presented. The properties studied were optical absorption coefficient (AC) and change in refractive index (CRI) of the quantum wire. In particular, effect of an inverse parabolic potential on the optical properties of CQWs was investigated. This was done by solving the Schrödinger equation within the effective mass approximation to obtain the wave functions. The inverse parabolic potential reduces transition energies and therefore redshifts peaks of the AC, as well as the anomalous dispersion region of the dependence of change in refractive index on the photon energy. The inverse parabolic potential also has effect on the magnitudes of these optical quantities, reducing the AC and enhancing the CRI. These properties of the inverse parabolic confining electric potential can have a wide range of applications in nanodevice technology, some details of which are discussed.


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## Body

## 1. Introduction

The emergence of nanofabrication technology has ushered in an era in which quantum effects can be controlled for various applications and equally importantly, for probing natural phenomena on the nanoscale. Nanotechnology has enabled the scientific community to have at its disposal a plethora of nanostructures of various sizes and geometries [1–3]. These nanostructures have applications in a wide variety of disciplines. In medicine, they have been utilized in a broad spectrum of branches like osteology, for example, using a combination of nanowires and nanoparticles to repair bones with defects [4]. Silicon nanoribons have also been used as biosensors to detect carcinoembryonic antigen [5]. The potential of titanate nanowires to remove Uranium (VI) from the environment has been investigated [6]. Some nanowires have shown great potential to be used as photocatalysts as in the case of zinc oxide nanowires [7]. Wang et al. successfully demonstrated that double barrier nanostructures in conjunction with quantum dots can be used as photodetectors [8]. Other nanostructures have upconversion emission abilities which would be very useful inin vivo optical imaging [9], while others have been utilized to improve the efficiency of solar cells [10].Most of these applications depend on optoelectronic properties of constituent nanostructures. It is thus vital to investigate optical properties of the constituent nanostructures. This has prompted researchers to turn their attention towards comprehending optical properties of nanostructures. Kumari et al. studied the dependence of change in refractive indices of nanostructures on both optical intensity and length of the nanowire [11]. Optical properties of laterally coupled quantum wires were studied, which revealed that the linear and nonlinear absorption coefficients become blueshifted when the nanowires get closer to each other [12]. Other studies on optical properties of nanowires have shown that external magnetic field and electric fields modify absorption coefficient of the nanostructures [13, 14]. Effect of a tilted electric field on optical absorption of quantum wires has also been reported on [15]. The effect of noise on optical properties of impurity doped quantum dots has also been comprehensively studied [16–18]. Another way of manipulating optical properties is through spatially variant confining electric potentials, for example, the Tietz potential [19], potential steps [20], power exponential potential [21], and the inverse parabolic potential [22]. Spatially variant confining electric potentials are crucial as they can be used to modify properties of quantum structures without having to meddle with the sizes of the structures.In this communication, the effect of an inverse parabolic potential on optical properties of nanowires is investigated. The inverse parabolic potential is superimposed on an infinite cylindrical square well (ICSW). The Schrödinger equation is solved for within the effective mass approximation. This report has the following organizational structure: Section2 presents the theoretical treatment of the problem, Section 3 deals with results and discussions, and concluding remarks can be found in Section 4.


## 2. Theoretical Model

The envisaged system is a cylindrical quantum wire (CQW) of radiusR and very long length, which may be, for example, a gallium arsenide (GaAs) nanowire embedded in a glass matrix or may be a free standing cylindrical nanowire. The nanowire is envisioned to have a negatively charged strand (of radius ρ0) lying along the axis of the nanowire, which through nanopatterning could be achieved by varying the lattice concentration. For ρ0→0, the confinement electric potential can be modelled as (1)Vρ=12μω02R2R2ρ2-1ρ<Rand infinity elsewhere, where μ is the effective mass of the electron and ω0 is the angular frequency associated with the classical harmonic oscillator. Considering the geometry of our system and due to the separability of the Hamiltonian, the wave function will be cast in the form ψρ,ϕ,z=Cmlχρexp⁡ikzzexp⁡imϕ, in cylindrical coordinates. Here,m is azimuthal quantum number which quantifies angular momentum,l is radial quantum number, and Cml is normalization constant. kz is the axial wave number while χρ is the radial component of the wave function which satisfies the Schrödinger equation(2)1ρddρρχρdρ+2μℏ2Eml-Vρ-m2ρ2χρ=0with Eml being the radial confinement energy. The solution to the above second-order differential equation is a linear combination of the BesselJ andY functions. The BesselY diverges at the origin and therefore has to be discarded as a solution for a solid cylindrical quantum wire, leaving the BesselJ function being the only surviving solution:(3)χρ=Jνlκρ.Here, κ=μ2Eml+μω02R2/ℏ2 and ν=m2+μω02R4/ℏ2. Applying the boundary condition at interfaces concerning the continuity of the wave function, one arrives at the expression for the energy spectrum of an electron confined in a CQW with an inverse parabolic potential as(4)ETot=Eml+Ez=ℏ2joml22μR2-12μω02R2+ℏ2kz22μwhere joml are the roots of the BesselJ function and Ez is the axial contribution to the total energy of the electron.


### 2.1. Optical Properties

Consider a circularly polarized monochromatic electromagnetic radiation of angular frequencyω incident on a cylindrical quantum wire. Absorption of the incident radiation is only possible if the energy of the radiation, ℏω, is coincident with the energy difference between different states between which transitions are possible. First-order and third-order absorption coefficient of a crystal can be evaluated by utilizing the density matrix approach in conjunction with perturbation expansion method via [11–14](5)α1ℏω=4παFSσVnce2ℏωMfi2δ∆Efi-ℏω(6)α3ℏω,Iω=-32π2αFS2σVIωnc2e4ℏΓfiℏωMfi4δ2∆Efi-ℏω1-Ξαwhere (7)Ξα=Mff-Mii24Mfi2ℏω-∆Efi2-ℏΓfi2+2∆Efi∆Efi-ℏω∆Efi2+ℏΓfi2.In the above, αFS is the fine structure constant, σV the electron volume density, and nc the refractive index of the crystal. Iω is intensity of the incident radiation and e is the electronic charge while ∆Efi=Ef-Ei are transition energies, with Ei and Ef being energies of the initial i and final f states. Mfi=f∣eρ∣i are the matrix elements coupling one state i to the other f while Γfi are the linewidths associated with the different states indicated by the subscripts i and f. The total AC is calculated according to(8)αℏω,Iω=α1ℏω+α3ℏω,Iω.The energy conserving delta function has been replaced with the Lorentzian(9)δ∆Efi-ℏω→ℏΓfiπ∆Efi-ℏω2+ℏΓfi2.It is also useful to scrutinise the behaviour of refractive indices of quantum structures if they are to be effectively utilized in opto-electro-nanodevices. The first- and third-order contributions to the refractive index change are given by [13](10)∆nc1ℏωnc=σV2εcMfi2∆Efi-ℏω∆Efi-ℏω2-ℏΓfi2and (11)∆nc3ℏω,Iωnc=-μ0cIω4εcnc2σVMfi2∆Efi-ℏω2-ℏΓfi22Ξnwhere (12)Ξn=4Mfi2∆Efi-ℏω-Mff-Mii2∆Efi∆Efi-ℏω2-ℏΓfi23∆Efi-2ℏω∆Efi2+ℏΓfi2.In the above, c is the speed of light in vacuum, εc the permittivity of the CQW, and μ0 the permeability of the material. Similarly, the summed CRI is given by(13)∆nnc=∆nc1ℏωnc+∆nc3ℏω,Iωnc.


## 2.1. Optical Properties

Consider a circularly polarized monochromatic electromagnetic radiation of angular frequencyω incident on a cylindrical quantum wire. Absorption of the incident radiation is only possible if the energy of the radiation, ℏω, is coincident with the energy difference between different states between which transitions are possible. First-order and third-order absorption coefficient of a crystal can be evaluated by utilizing the density matrix approach in conjunction with perturbation expansion method via [11–14](5)α1ℏω=4παFSσVnce2ℏωMfi2δ∆Efi-ℏω(6)α3ℏω,Iω=-32π2αFS2σVIωnc2e4ℏΓfiℏωMfi4δ2∆Efi-ℏω1-Ξαwhere (7)Ξα=Mff-Mii24Mfi2ℏω-∆Efi2-ℏΓfi2+2∆Efi∆Efi-ℏω∆Efi2+ℏΓfi2.In the above, αFS is the fine structure constant, σV the electron volume density, and nc the refractive index of the crystal. Iω is intensity of the incident radiation and e is the electronic charge while ∆Efi=Ef-Ei are transition energies, with Ei and Ef being energies of the initial i and final f states. Mfi=f∣eρ∣i are the matrix elements coupling one state i to the other f while Γfi are the linewidths associated with the different states indicated by the subscripts i and f. The total AC is calculated according to(8)αℏω,Iω=α1ℏω+α3ℏω,Iω.The energy conserving delta function has been replaced with the Lorentzian(9)δ∆Efi-ℏω→ℏΓfiπ∆Efi-ℏω2+ℏΓfi2.It is also useful to scrutinise the behaviour of refractive indices of quantum structures if they are to be effectively utilized in opto-electro-nanodevices. The first- and third-order contributions to the refractive index change are given by [13](10)∆nc1ℏωnc=σV2εcMfi2∆Efi-ℏω∆Efi-ℏω2-ℏΓfi2and (11)∆nc3ℏω,Iωnc=-μ0cIω4εcnc2σVMfi2∆Efi-ℏω2-ℏΓfi22Ξnwhere (12)Ξn=4Mfi2∆Efi-ℏω-Mff-Mii2∆Efi∆Efi-ℏω2-ℏΓfi23∆Efi-2ℏω∆Efi2+ℏΓfi2.In the above, c is the speed of light in vacuum, εc the permittivity of the CQW, and μ0 the permeability of the material. Similarly, the summed CRI is given by(13)∆nnc=∆nc1ℏωnc+∆nc3ℏω,Iωnc.


## 3. Results and Discussion

The values used in these computations are0.067me (relevant to GaAS crystals), me being the mass of a free electron, σV=3×1022m-3, nc=3.2, εc=12.5, and ℏΓfi=3.294meV [11]. Throughout this communication, the radius of the quantum wire has been held constant at R=100Å and intensity of radiation fixed at 1200MW/m2.Figure1 depicts the dependence of absorption coefficient on the photon energy for a quantum wire of radius R=100Å. The two graphs have been generated for an ICSW η1=ℏω0=0meV and for a quantum wire with an inverse parabolic potential of strength η2=ℏω0=10meV. Results for the ℏω0=0meV case concur with those in the literature [14, 23]. As can be seen from the figure, the inverse parabolic potential redshifts peaks of the AC. This is due to the fact that this potential decreases transition energies as it increases in strength [24]. Transition energies are differences in the energies of states between which transitions occur. Also, the inverse parabolic potential diminishes the magnitude of the AC, which is due to the fact that the potential closes the gap between the matrix elements Mii and Mff; that is, Mff-Mii decreases monotonically with increase in the inverse parabolic potential. The double peaks arise due to the (negative) third-order contribution of the absorption coefficient, and merge into a single peak for very low radiation intensities Iω≈0. Enhancement of the double peak in Figure 1 is due to the effect of the inverse parabolic potential on both the first- and the third-order contributions to the AC. This has been graphically illustrated in Figures 2(a) and 2(b), which depict the variation of the first (a) and third (Figure 2(b)) contributions to peaks of AC with the inverse parabolic potential. The three curves in Figures 2(a) and 2(b) have been generated for the first three radial quantum numbers in a cylindrical quantum wire of radius R=100Å, for the m=0→m=1 transitions. First, this potential drastically reduces magnitudes of the peaks of first-order AC at resonance, regardless of the azimuthal and the radial quantum numbers involved. Second, peaks of the third-order AC initially decrease with increasing inverse parabolic potential strength and start increasing asymptotically after reaching a certain minimum for the l=1 transitions. For the l>1 cases, variation of the third-order AC is characterized by regions where the inverse parabolic potential decreases the magnitude of the third-order AC. For example, for the nanowire of radius R=100Å, this region is around ℏω0=82.6meV for l=2 and ℏω0=35.1meV and ℏω0=231.5meV for l=3. Despite these slight variations, as the inverse parabolic potential increases, the third-order AC at resonance does not vary much, and for this case it is in the vicinity of ~−4×104/m. This is in contrast to the first-order AC at resonance, which gets drastically reduced with increase of the inverse parabolic potential. Thus, for the l=1 AC, the inverse parabolic potential promotes the negative third-order contribution while it reduces the (positive) first-order contribution, consequently reducing AC at resonance. It is thus expected that the inverse parabolic potential will have reduced efficacy to enhance the development of double peaks for the l>1 transitions; for example in this case, potential strengths of ℏω0=82.6meV (for l=2) and ℏω0=35.1meV and ℏω0=231.5meV (for l=3) will not promote the development of the double peaks.Figure 1
Dependence of thel=1 absorption coefficient, involving the ground state, on the incident photon energy for a CQW of radius R=100Å. The two graphs correspond to an ICSW η1=ℏω0=0meV and to η2=ℏω0=10meV.Figure 2
(a) Variation of first-order contribution to AC (at resonance),αp(1), with the inverse parabolic potential for a CQW of radius R=100Å. The three curves have been generated for the first three radial quantum numbers l=1,2and3, for the m=0→m=1 transition. (b) Same as for (a), but for the third-order contribution of AC, αp(3).
(a) (b)Figure3 shows the variation of AC with the inverse parabolic potential for a quantum wire of radius R=100Å, for the different values of the photon energy: ϖ1=ℏω=20meV,ϖ2=ℏω=25meV, and ϖ3=ℏω=30meV. Peaks of the AC shift to high values of the inverse parabolic potential as the photon energy decreases. Absorption occurs when the photon energy coincides with transition energies. For an infinite cylindrical square well (ICSW) η1=ℏω0=0meV of radius =100Å, transition energy between the ground state m=0 and the first excited state m=1 is around ΔE10=50.6meV. Thus, the value of the energy of radiation used in this figure is less than the m=0,m=1 transition energy. This means that electrons in an ICSW of this radius will neither absorb nor emit radiation having the selected energies in Figure 3 due to a transition, in this particular case, the m=0,m=1 transition. Since the inverse parabolic potential reduces transition energies, it can be used to tune nanowires in cases where the sizes of the nanowires are to have specific values. When photon energy decreases, higher values of the potential will be required to make the transition energies equal the photon energy, bringing the system into resonance.Figure 3
Variation of thel=1 AC (involving the ground state) with the inverse parabolic potential for a quantum wire of radius R=100Å for the three photon energies, viz., ϖ1=ℏω=20meV,ϖ2=ℏω=25meV, and ϖ3=ℏω=30meV.Figure4 illustrates AC (involving the ground state) as a function of strength of the inverse parabolic potential for the radial quantum numbersl = 1, 2 andl = 3. Here, the quantum wire of radius R=100Å is irradiated with an electromagnetic radiation of energy ℏω=20meV. Transition energies associated with higher radial quantum numbers are generally larger than those associated with the lower radial quantum numbers. Consequently, for fixed photon energy, transitions of electrons with higherl values require greater values of the inverse parabolic potential to bring the system to resonance.Figure 4
Dependence of AC (involving the ground state) on the inverse parabolic potential for three first three radial quantum numbers. The incident radiation has energyℏω=20meV and radius of the quantum wire is R=100Å.Figure5 depicts the scaled change in refractive index (CRI), Δn/nc, of a cylindrical quantum wire of radius R=100Å as a function of the photon energy for different values of the inverse parabolic potential varying in steps of 2meV from ℏω0=0meV (thick plot marked 0) to ℏω0=30meV (plot marked 30). At low radiation field intensities, variation of the Δn/nc is characterized by regions where Δn/nc increases with increasing energy of the electromagnetic radiation (called the normal dispersion region) and a region where it decreases with increasing energy of the radiation (called the anomalous dispersion region) [25]. As already mentioned, the inverse parabolic potential decreases transition energies as it intensifies. This manifests as the redshifting of the CRI as the inverse parabolic potential increases. An additional effect of the inverse parabolic potential is enhancing the magnitude of the CRI. This is attributable to the fact that this potential “expels” electron away from the axis of the wire, signified by shifting of peaks of electron wave functions towards the walls of the nanowire [24]. This enhances the overlap between wave functions of the initial and final states.Figure 5
CRI as a function of the photon energy in a cylindrical nanowire of radiusR=100Å with an inverse parabolic potential of values from ℏω0=0meV (thick plot marked 0) to ℏω0=30meV (plot marked 30) in increments of 2meV. Note the almost parabolic enhancement of the magnitude of maxima (and minima) of the CRI as the inverse parabolic potential increases in strength.Figure6 illustrates the dependence the scaled CRI of a cylindrical quantum wire on the strength of the inverse parabolic potential for the first three radial quantum numbers. Here, electromagnetic radiation of energy ℏω=20meV is incident upon a cylindrical quantum wire of radius R=100Å. Transition energies associated with greater radial quantum numbers are usually greater than those associated with lower radial quantum numbers. This is also depicted in the inset of this figure, where the scaled CRI has been plotted as a function of photon energy for the first three radial quantum numbers: The inset shows that the anomalous dispersion region is shifted towards higher energy as thel values increase. As such, for a given photon energy, higher values of the inverse parabolic potential are required to bring the system to resonance. Another feature of the dependence of the CRI on the inverse parabolic potential is the emergence of local peaks which are a consequence of the effect of this potential on the electron’s radial wave function [24].Figure 6
Scaled change in refractive index (CRI) of a CQW as a function of the inverse parabolic potential for the first three radial quantum numbers. The radius isR=100Å and the photon energy remains fixed at ℏω=20meV. The inset shows the scaled CRI of the cylindrical quantum wire as a function of the photon energy for the first three radial quantum numbers.


## 4. Conclusions

Optical properties of a cylindrical quantum wire with an inverse parabolic potential superimposed on an infinite cylindrical square well have been studied. The wave functions were obtained by solving the Schrödinger equation within the effective mass approximation. Results show that the inverse parabolic potential redshifts both the peaks of the absorption coefficient (AC) and the anomalous region of change in refractive index (CRI) of the cylindrical nanowire. Additionally, the inverse parabolic potential diminishes the magnitude of the AC while it enhances the magnitude of the CRI. These properties of the inverse parabolic potential make it a very attractive tool for manipulating optical properties of nanostructures, especially where the sizes the nanostructures are to have specific values. This can be very beneficial to the design of nanodevices.



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