Band Structure Dependent Physical Properties of Quantized Systems
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G. Gulyamov, A. G. Gulyamov, A. B. Davlatov, Kh. N. Juraev, "Energy Levels in Nanowires and Nanorods with a Finite Potential Well", Advances in Condensed Matter Physics, vol. 2020, Article ID 4945080, 12 pages, 2020. https://doi.org/10.1155/2020/4945080
Energy Levels in Nanowires and Nanorods with a Finite Potential Well
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.
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 newgeneration 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], coreshell nanowires were experimentally obtained, and it was shown that the crosssectional area is in the shape of a hexagon. Optical properties of InAs/InPbased quantum wires were studied by photoluminesce spectroscopy [4]. The optical properties of coremultishell 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 IIIV 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 crosssectional 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 crosssectional shape. The influence of electronic, polaron, and coulomb interactions on energy states in quantum nanowires was studied [13–16]. The IV 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 crosssectional 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
Figure 1 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:
The 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. Therefore, equation (2) will be solved for the radial wave function. Thus, equation (2) in region for the radial wave function takes the following form:where
In this case, equation (4) takes the following form:
The general solution to this equation is in the form of a linear combination of the Bessel function and the Neumann function of the lth order [37]:
Equation (2) for the radial wave function in region takes the following form:where
The solution of equation (8) gives a linear combination of the imaginary argument of the Bessel function and the Macdonald function of the lth order [37]:Therefore, 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, and are constant values. We select relation 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 , expanding the imaginary argument of the Bessel function and MacDonald 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).
3. The Analytical Method for Calculating the Electron Energy in a Cylindrical Quantum Nanorod with a Finite Height of the Potential Well
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 Schrödinger equation in a cylindrical coordinate system in this case is as follows:
The 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.
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:
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 and are known for wave functions, then using (28), it is possible to determine the value of 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:
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:where
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.

 
E_{g} is the band gap, m_{e}/m_{0} is the effective mass of the electron in the conduction band, m_{lh}/m_{0} and m_{hh}/m_{0} are the effective masses of light and heavy holes in the valence band, χ is the electron affinity, α_{i} is the nonparabolicity coefficient, and Δ_{0} is spinorbit 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 m_{0}. Thus, the nonparabolicity of the system at small radii is more noticeable than at large radii.
The resulting expression (22) allows us to calculate the energy in cylindrical nanorod with inner and outer radii R_{1} and R_{2}, 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 R_{1} 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 R_{1}, i.e., with an increase in R_{1}, a slight increase in particle energy is observed.
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 radius R_{1}, that is, with an increase in R_{1}, a slight increase in the particle energy is observed.
Data Availability
The data that support the findings of this study are available on request from the corresponding author (Kh.N. Juraev).
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was financially supported by the Program for Fundamental Research (grant number BAFAF2005).
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