# Energy Levels in Nanowires and Nanorods with a Finite Potential Well

Authors: G. Gulyamov; A. G. Gulyamov; A. B. Davlatov; Kh. N. Juraev
Journal: Advances in Condensed Matter Physics
 (2020)

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## Abstract

The energy of electrons and holes in cylindrical quantum wires with a finite potential well was calculated by two methods. An analytical expression is approximately determined that allows one to calculate the energy of electrons and holes at the first discrete level in a cylindrical quantum wire. The electron energy was calculated by two methods for cylindrical layers of different radius. In the calculations, the nonparabolicity of the electron energy spectrum is taken into account. The dependence of the effective masses of electrons and holes on the radius of a quantum wires is determined. An analysis is made of the dependence of the energy of electrons and holes on the internal and external radii, and it is determined that the energy of electrons and holes in cylindrical layers with a constant thickness weakly depends on the internal radius. The results were obtained for the InP/InAs heterostructures.


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## Body

## 1. Introduction

In recent years, a lot of work has been done on calculating the energy of electrons and holes in quantum wells based on InP/InAs/InP heterostructures, due to the fact that today, for the creation of new-generation devices, it largely depends on semiconductor nanostructures.In [1, 2], various technologies for growing quantum wires were presented, and nanowires of various sizes were obtained. In [3], core-shell nanowires were experimentally obtained, and it was shown that the cross-sectional area is in the shape of a hexagon. Optical properties of InAs/InP-based quantum wires were studied by photoluminesce spectroscopy [4]. The optical properties of core-multishell nanowires based on InP/InAs/InP were experimentally studied in [5–8]. The relaxation time of electron spin in semiconductor quantum wires has been experimentally investigated.In type III-V semiconductors and in heterostructures based on them, electron dispersion is strongly nonparabolic; the Kane model was used to study the spectrum of charge carriers of these materials [9]. In [10], the energy spectrum of an electron with a nonparabolic dispersion law in quantum wires with a rectangular cross-sectional shape was theoretically investigated. The authors of [11, 12] also theoretically studied the energy spectrum of an electron with a nonparabolic dispersion law in quantum wires, but with a hexagonal and triangular cross-sectional shape. The influence of electronic, polaron, and coulomb interactions on energy states in quantum nanowires was studied [13–16]. The I-V characteristic of nanowires was considered taking into account the tunneling of electrons in quantum states [17, 18]. The effect of temperature on the energy levels [19, 20] and optical absorption [21] of quantum dots are studied. The energy levels of InAs/InP quantum dots have been studied [22]. In [23–28], to determine the energy spectrum and wave function of an electron in quantum wires with a rectangular cross-sectional shape, solutions of the Schrödinger equation were obtained using various mathematical methods. The effect of an electric field on the energy spectrum of a rectangular quantum wire is investigated [29].The authors of [30–33] obtained analytical solutions of the Schrödinger equation for cylindrical quantum wires with a finite potential and a parabolic dispersion law. The solution of the Schrödinger equation is obtained by the finite difference method (shooting method) for rectangular [34] and cylindrical [35, 36] quantum wires.In this paper, two methods are used to calculate the energy of electrons and holes for a cylindrical quantum wire and a quantum nanorod with a finite potential well. In this case, the nonparabolicity of the dispersion of electrons and holes is taken into account. The relationship between the effective mass of charge carriers and the radius of a quantum wire is determined.


## 2. An Analytical Method for Calculating the Electron Energy in a Cylindrical Quantum Wire with a Finite Height of the Potential Well

Figure1 shows the geometric and potential diagram of a cylindrical quantum wire with a finite potential well. The potential energy of an electron of a cylindrical quantum wire with a finite depth has the form:(1)Ur=Uρ=0,0≤ρ≤R,W,ρ>R.Figure 1
Geometric and potential diagram of a cylindrical quantum wire with a finite potential well.The Schrödinger equation in a cylindrical coordinate system is as follows:(2)−ℏ221ρ∂∂ρρmρ∂∂ρ+1ρ2∂2∂ϕ2+∂2∂z2fr+Urfr=Efr.We seek a solution to this equation in the following form:(3)fr=eikzzeilϕψρ,Here, the parameterskx and l are independent of coordinates. Therefore, equation (2) will be solved for the radial wave function. Thus, equation (2) in region 0≤ρ≤R for the radial wave function ψρ takes the following form:(4)∂2ψρ∂ρ2+1ρ∂ψρ∂ρ+kA2−l2ρ2ψρ=0,0≤ρ≤R,where(5)ξ=kAρ,kA=2mAℏ2E−kz2.In this case, equation (4) takes the following form:(6)ξ2∂2ψρ∂ξ2+ξ∂ψρ∂ξ+ξ2−l2ψρ=0.The general solution to this equation is in the form of a linear combination of the Bessel functionJlξ and the Neumann function Nlξ of the l-th order [37]:(7)ψ1ρ=A1JlkAρ+B1NlkAρ,0≤ρ≤R.Equation (2) for the radial wave function ψρ in region ρ>R takes the following form:(8)ζ2∂2ψ∂ζ2+ζ∂ψ∂ζ+−ζ2−M2ψ=0,ρ>R,where(9)ζ=γBρ,γB=2mBℏ2W−E+kz2.The solution of equation (8) gives a linear combination of the imaginary argument Ilζ of the Bessel function and the Macdonald function Klζ of the l-th order [37]:(10)ψ2ρ=A2KlγBρ+B2IlγBρ,ρ>R.Therefore, ψρ for a radial wave function is appropriate for the following:(11)ψρ=A1JlkAρ+B1NlkAρ,0≤ρ≤R,A2KlγBρ+B2IlγBρ,ρ>R.If we take into account that the wave function inside the cylinder is finite and equal to zero at an infinite distance from the center of the cylinder, expression (11) takes the following form:(12)ψρ=A1JlkAρ,0≤ρ≤R,A2KlγBρ,ρ>R.Here,A1 and A2 are constant values. We select relation A1/A2 in such a way that the following boundary conditions are satisfied:(13)ψ1ρ|ρ=R=ψ2ρ|ρ=R,1mAdψ1ρdρ|ρ=R=1mBdψ2ρdρ|ρ=R.From (12) and (13), we obtain the following transcendental equation:(14)Jl′kARKlγBRJlkARKl′γBR=mAmB.Solving the transcendental equation (14), we can determine the electron energy in a cylindrical quantum wire.When the argument is too small under conditionl=0, expanding the imaginary argument of the Bessel function Ilζ and MacDonald Klζ in a row, we get the first terms and an analytical formula for calculating the energy:(15)E0,1=4ℏ2mAR22mBW/ℏ2RK02mBW/ℏ2R+2K12mBW/ℏ2R2mBW/ℏ2RK02mBW/ℏ2R+4K12mBW/ℏ2R.If we take into account (5) and (9), expression (15) gives very close results to the exact results of equation (14) when calculating the energy of electrons and holes in cylindrical nanowires with larger radii (Figures 2–4).Figure 2
The dependence of energy on the radius of a cylindrical nanowire for electrons and holes with the parabolic and nonparabolic dispersion law.Figure 3
The dependence of the energy of a heavy hole on the radius of a cylindrical quantum wire for the parabolic and nonparabolic dispersion law.Figure 4
The dependence of the energy of a light hole on the radius of a cylindrical quantum wire for the parabolic and nonparabolic dispersion law.


## 3. The Analytical Method for Calculating the Electron Energy in a Cylindrical Quantum Nanorod with a Finite Height of the Potential Well

Figure5 shows the geometric and potential diagram of a cylindrical quantum nanorod with a finite potential well. Let the potential energy of an electron in a cylindrical quantum nanorod with a finite depth be given by the following expression:(16)Ur=Uρ=W,0<ρ<R1,0,R1≤ρ≤R2,W,R2<ρ.Figure 5
The geometric and potential scheme of a cylindrical quantum nanorod with a finite potential well.The Schrödinger equation in a cylindrical coordinate system in this case is as follows:(17)−ℏ221ρ∂∂ρρmρ∂∂ρ+1ρ2∂2∂ϕ2+∂2∂z2fr+Urfr=Efr,The solution to the radial part of the equation will be(18)ψρ=A1KlγBρ+B1IlγBρ,0<ρ<R1,A2JlkAρ+B2NlkAρ,R1≤ρ≤R2,A3KlγBρ+B3IlγBρ,R2<ρ.If we take into account that the wave function inside a cylindrical quantum nanorod is finite and equal to zero at an infinite distance from the center of the quantum nanorod, then expression (18) takes the following form:(19)ψρ=B1IlγBρ,0<ρ<R1,A2JlkAρ+B2NlkAρ,R1≤ρ≤R2,A3KlγBρ,R2<ρ.For the continuity of the wave function, the following conditions must be met:(20)ψ1ρ|ρ=R1=ψ2ρ|ρ=R1,1mBdψ1ρdρ|ρ=R1=1mAdψ2ρdρ|ρ=R1,ψ2ρ|ρ=R2=ψ3ρ|ρ=R2,1mAdψ2ρdρ|ρ=R2=1mBdψ3ρdρ|ρ=R2.Applying boundary conditions (20) to expression (19), we obtain the following system of equations:(21)B1IlγBR1=A2JlkAR1+B2NlkAR1,1mBB1Il′γBR1=1mAA2Jl′kAR1+B2Nl′kAR1,A2JlkAR2+B2NlkAR2=A3KlγBR2,1mAA2Jl′kAR2+B2Nl′kAR2=1mBA3Kl′γBR2.Solving the system of equation (21), we obtain the following transcendental equation:(22)mA/mBJlkAR1Il′γBR1−Jl′kAR1IlγBR1mA/mBJlkAR2Kl′γBR2−Jl′kAR2KlγBR2=Nl′kAR1IlγBR1−mA/mBNlkAR1Il′γBR1Nl′kAR2KlγBR2−mA/mBNlkAR2Kl′γBR2.Transcendental equation (22) allows us to calculate the energy of electrons and holes in cylindrical nanolayers with a finite potential well.


## 4. The Shooting Method for Calculating the Electron Energy in a Cylindrical Quantum Wire and in a Quantum Nanorod with a Finite Height of a Potential Well

Given the effective mass, the Schrödinger equation will be as follows:(23)−ℏ22ρ∂∂ρρmρ∂∂ρψρ+Vρψρ=Eψρ.Equation (23) is used for the parabolic dispersion law, and it is solved by the finite difference method—the Shooting method [38]. We will try to solve equation (23) by the same method for the nonparabolic dispersion law. In this case, we represent the wave function and the derivative of the effective mass in the following form:(24)dψdρ≈ψρ+δρ−ψρ−δρ2δρ,(25)d2ψdρ2≈ψρ+δρ−2ψρ+ψρ−δρδρ2,(26)dmdρ≈mρ+δρ−mρ−δρ2δρ.Using (23)–(26), we obtain the following:(27)ρmρψρ+δρ−2ψρ+ψρ−δρδρ2−ψρ+δρ−ψρ−δρ2δρρmρ+δρ−mρ−δρ2δρ−mρ,=2ρmρ2ℏ2Vρ−Eψρ.From here, we get(28)ψρ+δρ=8δρmρ2/ℏ2Vρ−E+mρ2mρ2+δρ/ρ−mρ+δρ+mρ−δρψρ−2mρ2−δρ/ρ+mρ+δρ−mρ−δρ2mρ2+δρ/ρ−mρ+δρ+mρ−δρψρ−δρ.If the values ofψρ−δρ and ψρ are known for wave functions, then using (28), it is possible to determine the value of ψρ+δρ for an arbitrary energy. In order to calculate the wave function and energy of electrons and holes, it is necessary to take into account the following three boundary conditions:(29)ψ∞⟶0,ψ0=1,ψδρ=1.


## 5. Analysis of the Results

It is known that the lattice constant for InP is 0.5869 nm, and this value is close to the lattice constant for InAs 0.6058 nm [39]. This allows you to get the perfect heterostructures using these materials. Figure 6 shows the band diagram of the InP/InAs based heterostructures. To take into account the nonparabolicity of the zone, we use the Kane model [9]. Various approximations for the effective mass are given in [10, 40–43]; we will use the following expression for the effective mass:(30)mInPiE=m0/InPi1+αInPiE−Vρ,αInPi=1EgInP1−mim02,αInAsi=1EgInAs1−mim02,where i=e,hh,lh.Figure 6
The band diagram of the InP/InAs heterostructure.Table1 shows the necessary parameters for InAs and InP obtained by various authors. For our calculations, we will choose the parameters shown in Table 2.Table 1
Material parameters of InAs and InP.
ParameterInAsInPEgeV0.35[39]1.35[39,44,45]0.36[44,46]1.424[47]0.417[47,48]1.423[48]0.42[48]1.42[48,49]0.418[49]me/m00.022[39]0.077[39,47,50]0.023[44,47,48,50]0.08[44]0.024[48]0.079[48,49]mhh/m00.41[39,48]0.6[39]0.6[44]0.85[44]0.34[50]0.65[48]0.472[50]mlh/m00.026[39]0.12[39,48]0.027[44,50]0.098[44]0.025[48]0.096[50]χeV4.92[39]4.38[39]4.9[45]4.4[45]2m0P2/ℏ2eV22.2[39]20.4[39]21.5[47,48]20.7[47,48]2mnP2/ℏ2eV0.5[48]1.46[48]Δ0eV0.38[39,47,49]0.11[39,49]0.39[48]0.108[47,48]Table 2
Material parameters of InAs and InP.
Material parametersEgeVmem0mhhm0mlhm0χeVαe1/eVαhh1/eVαlh1/eVΔ0eVΔECeVΔEVeVInAs0.360.0230.410.0264.912.650.972.630.380.520.47InP1.350.0770.650.0964.390.630.090.60.11Eg is the band gap, me/m0 is the effective mass of the electron in the conduction band, mlh/m0 and mhh/m0 are the effective masses of light and heavy holes in the valence band, χ is the electron affinity, αi is the nonparabolicity coefficient, and Δ0 is spin-orbit interaction.A numerical solution of transcendental equation (14) determines the wave functions and the energy of electrons and holes. By an approximate solution of this equation, equation (15) is obtained. For large radii, the solution of equation (15) gives results that are close to the exact results of equation (14). The energy and wave function of the particles are calculated using expression (28), which is obtained by the finite difference method. The calculations were performed taking into account the nonparabolicity of the zones using expression (30). The dependence of the energy on the radius of a cylindrical nanowire for electrons and holes is shown taking into account the parabolic and nonparabolic zones (Figures 2–4). Figure 7 shows the radial wave function of the electron. Usually, the electron wave function as ρ⟶∞ is equal to zero. If at the size of the potential well R, we choose that the wave function at a distance inside the barrier is R/2, and then the error in calculating the energy was 0.01 meV compared to ρ⟶∞. The dependence of the effective masses of electrons and holes on the radius of a cylindrical nanowire was determined (Figures 8–10). The results of the graphs show that the effective mass of charge carriers decreases with increasing radius of the nanowires and, at large radii, approaches m0. Thus, the nonparabolicity of the system at small radii is more noticeable than at large radii.Figure 7
The dependence of the radial wave function of an electron on the radius of a cylindrical quantum wire with a finite potential.Figure 8
The dependence of the effective electron mass on the radius of a cylindrical quantum wire with a finite potential.Figure 9
The dependence of the effective mass of a heavy hole on the radius of a cylindrical quantum wire with a finite potential.Figure 10
The dependence of the effective mass of a light hole on the radius of a cylindrical quantum wire with a finite potential.The resulting expression (22) allows us to calculate the energy in cylindrical nanorod with inner and outer radii R1 and R2, respectively. The results are compared by expression (22) and expression (28) (Figures 11–13). Figure 14 shows the radial wave function of the electron. Here, we assumed that the wave function inside the barrier at a distance R1 is zero, and the error in this case is 0.01 meV. The results shown in Figure 15 allow us to draw the following conclusion: for a constant thickness of cylindrical nanorod, there is a weak dependence of the particle energy on the internal radius R1, i.e., with an increase in R1, a slight increase in particle energy is observed.Figure 11
The dependence of the electron energy on the thickness of the layer of a cylindrical quantum nanorod with a finite potential.Figure 12
The dependence of the energy of a heavy hole on the thickness of a layer of a cylindrical quantum nanorod with a finite potential.Figure 13
The dependence of the energy of a light hole on the layer thickness of a cylindrical quantum nanorod with a finite potential.Figure 14
The dependence of the radial wave function of an electron on the radius of a cylindrical quantum nanorod with a finite potential.Figure 15
The dependence of the particle energy on the inner radius with a constant layer thickness of a cylindrical quantum nanorod and with a finite potential.


## 6. Conclusion

The Schrödinger equation for cylindrical nanowires and for cylindrical nanorods was solved by two methods. An approximate equation is obtained that determines the first energy level in cylindrical quantum wires with large radii. When solving the Schrödinger equation, the change in the effective masses of electrons and holes was taken into account. Graphs of the effective mass of electrons and holes versus nanowire radius are presented. The energy of electrons and holes in cylindrical nanorods was calculated, and the effects of parabolicity and nonparabolicity of the zones were compared. It is shown that, for a constant thickness of cylindrical nanorods, there is a weak dependence of the particle energy on the change in the internal radiusR1, that is, with an increase in R1, a slight increase in the particle energy is observed.



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