Energy Levels in Nanowires and Nanorods with a Finite Potential Well

)e 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.)e 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.)e 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. )e results were obtained for the InP/ InAs heterostructures.


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]. e optical properties of core-multishell nanowires based on InP/InAs/InP were experimentally studied in [5][6][7][8]. e 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. e 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. e influence of electronic, polaron, and coulomb interactions on energy states in quantum nanowires was studied [13][14][15][16].
e I-V characteristic of nanowires was considered taking into account the tunneling of electrons in quantum states [17,18]. e effect of temperature on the energy levels [19,20] and optical absorption [21] of quantum dots are studied. e energy levels of InAs/InP quantum dots have been studied [22]. In [23][24][25][26][27][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. e effect of an electric field on the energy spectrum of a rectangular quantum wire is investigated [29]. e authors of [30][31][32][33] obtained analytical solutions of the Schrödinger equation for cylindrical quantum wires with a finite potential and a parabolic dispersion law. e 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. e relationship between the effective mass of charge carriers and the radius of a quantum wire is determined.

An Analytical Method for Calculating the Electron Energy in a Cylindrical Quantum
Wire with a Finite Height of the Potential Well Figure 1 shows the geometric and potential diagram of a cylindrical quantum wire with a finite potential well. e potential energy of an electron of a cylindrical quantum wire with a finite depth has the form: e Schrödinger equation in a cylindrical coordinate system is as follows: We seek a solution to this equation in the following form: Here, the parameters k x and l are independent of coordinates. erefore, equation (2) will be solved for the radial wave function. us, equation (2) in region 0 ≤ ρ ≤ R for the radial wave function ψ(ρ) takes the following form: where In this case, equation (4) takes the following form: e general solution to this equation is in the form of a linear combination of the Bessel function J l (ξ) and the Neumann function N l (ξ) of the l-th order [37]: Equation (2) for the radial wave function ψ(ρ) in region ρ > R takes the following form: e solution of equation (8) gives a linear combination of the imaginary argument I l (ζ) of the Bessel function and the Macdonald function K l (ζ) of the l-th order [37]: erefore, ψ(ρ) for a radial wave function is appropriate for the following: 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: Here, A 1 and A 2 are constant values. We select relation A 1 /A 2 in such a way that the following boundary conditions are satisfied: From (12) and (13), we obtain the following transcendental equation: Solving the transcendental equation (14), we can determine the electron energy in a cylindrical quantum wire.
When the argument is too small under condition l � 0, expanding the imaginary argument of the Bessel function I l (ζ) and MacDonald K l (ζ) in a row, we get the first terms and an analytical formula for calculating the energy: 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 5 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:

The Analytical Method for Calculating the Electron Energy in a Cylindrical Quantum Nanorod with a Finite Height of the Potential Well
e Schrödinger equation in a cylindrical coordinate system in this case is as follows: e solution to the radial part of the equation will be 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: For the continuity of the wave function, the following conditions must be met: Applying boundary conditions (20) to expression (19), we obtain the following system of equations: Solving the system of equation (21), we obtain the following transcendental equation: Transcendental equation (22) allows us to calculate the energy of electrons and holes in cylindrical nanolayers with a finite potential well.

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: 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: Using (23)-(26), we obtain the following: From here, we get If the values of ψ(ρ − δρ) and ψ(ρ) are known for wave functions, then using (28), it is possible to determine the value of ψ(ρ + 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:  Figure 6: e band diagram of the InP/InAs heterostructure.

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]. is 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][41][42][43]; we will use the following expression for the effective mass: where i � e, hh, lh. Table 1 shows the necessary parameters for InAs and InP obtained by various authors. For our calculations, we will choose the parameters shown in Table 2.
us, the nonparabolicity of the system at small radii is more noticeable than at large radii. e resulting expression (22) allows us to calculate the energy in cylindrical nanorod with inner and outer radii R 1 and R 2 , respectively. e 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 R 1 is zero, and the error in this case is 0.01 meV. e 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 R 1 , i.e., with an increase in R 1 , a slight increase in particle energy is observed.

Conclusion
e 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. e 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 radius R 1 , that is, with an increase in R 1 , a slight increase in the particle energy is observed.
Data Availability e data that support the findings of this study are available on request from the corresponding author (Kh.N. Juraev).

Conflicts of Interest
e authors declare that they have no conflicts of interest. Nonparabolic Parabolic E (meV) Figure 15: e dependence of the particle energy on the inner radius with a constant layer thickness of a cylindrical quantum nanorod and with a finite potential.