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Energy Levels in Nanowires and Nanorods with a Finite Potential Well

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Energy Levels in Nanowires and Nanorods with a Finite Potential Well Hindawi Advances in Condensed Matter Physics Volume 2020, Article ID 4945080, 12 pages https://doi.org/10.1155/2020/4945080 Research Article Energy Levels in Nanowires and Nanorods with a Finite Potential Well G. Gulyamov ,1 A. G. Gulyamov ,2 A. B. Davlatov ,2 and Kh. N. Juraev 2 1Namangan Engineering Construction Institute, 12 Islam Karimov Street, Namangan 160103, Uzbekistan 2Physical-Technical Institute, Uzbek Academy of Sciences, Chingiz Aytmatov Street, 2 “B”, Tashkent 100084, Uzbekistan Correspondence should be addressed to Kh. N. Juraev; [email protected] Received 19 July 2020; Revised 15 October 2020; Accepted 19 October 2020; Published 7 November 2020 Academic Editor: Kamakhya Prasad Ghatak Copyright © 2020 G. Gulyamov et al. )is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. )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. 1. Introduction wires with a rectangular cross-sectional shape was theo- retically investigated. )e authors of [11, 12] also theoret- In recent years, a lot of work has been done on calculating ically studied the energy spectrum of an electron with a the energy of electrons and holes in quantum wells based on nonparabolic dispersion law in quantum wires, but with a InP/InAs/InP heterostructures, due to the fact that today, for hexagonal and triangular cross-sectional shape. )e influ- the creation of new-generation devices, it largely depends on ence of electronic, polaron, and coulomb interactions on semiconductor nanostructures. energy states in quantum nanowires was studied [13–16]. In [1, 2], various technologies for growing quantum )e I-V characteristic of nanowires was considered taking wires were presented, and nanowires of various sizes were into account the tunneling of electrons in quantum states obtained. In [3], core-shell nanowires were experimentally [17, 18]. )e effect of temperature on the energy levels obtained, and it was shown that the cross-sectional area is in [19, 20] and optical absorption [21] of quantum dots are the shape of a hexagon. Optical properties of InAs/InP-based studied. )e energy levels of InAs/InP quantum dots have quantum wires were studied by photoluminesce spectros- been studied [22]. In [23–28], to determine the energy copy [4]. )e optical properties of core-multishell nanowires spectrum and wave function of an electron in quantum wires based on InP/InAs/InP were experimentally studied in with a rectangular cross-sectional shape, solutions of the [5–8]. )e relaxation time of electron spin in semiconductor Schrodinger¨ equation were obtained using various mathe- quantum wires has been experimentally investigated. matical methods. )e effect of an electric field on the energy In type III-V semiconductors and in heterostructures spectrum of a rectangular quantum wire is investigated [29]. based on them, electron dispersion is strongly nonparabolic; )e authors of [30–33] obtained analytical solutions of the Kane model was used to study the spectrum of charge the Schrodinger¨ equation for cylindrical quantum wires with carriers of these materials [9]. In [10], the energy spectrum of a finite potential and a parabolic dispersion law. )e solution an electron with a nonparabolic dispersion law in quantum of the Schrodinger¨ equation is obtained by the finite 2 Advances in Condensed Matter Physics 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 InP case, the nonparabolicity of the dispersion of electrons and holes is taken into account. )e relationship between the InAs R effective mass of charge carriers and the radius of a quantum wire is determined. 0 2. An Analytical Method for Calculating the ρ Electron Energy in a Cylindrical Quantum WirewithaFiniteHeightofthePotentialWell 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: 0; 0 ≤ ρ ≤ R; U(r) � U(ρ) �( (1) W; ρ > R: )e Schrodinger¨ equation in a cylindrical coordinate system is as follows: Z2 1 z ρ z 1 z2 z2 − + + !f(r) z m( ) z 2 2 2 2 ρ ρ ρ ρ ρ zϕ zz (2) + U(r)f(r) � Ef(r): We seek a solution to this equation in the following form: U f(r) � eikzzeilϕψ(ρ); (3) ΔEc Here, the parameters kx and l are independent of co- ordinates. )erefore, equation (2) will be solved for the radial wave function. )us, equation (2) in region 0 ≤ ρ ≤ R 0 ρ for the radial wave function ψ(ρ) takes the following form: 2 2 z ψ(ρ) 1 zψ(ρ) 2 l 0 ρ + + kA − ! !ψ(ρ) � 0; 0 ≤ ρ ≤ R; zρ2 ρ zρ ρ2 (4) ΔE where V ξ � kAρ; s���������� (5) Figure 2m 1: Geometric and potential diagram of a cylindrical k � A E − k2: quantum wire with a finite potential well. A Z2 z In this case, equation (4) takes the following form: z2ψ(ρ) zψ(ρ) ξ2 + ξ +�ξ2 − l2 �ψ(ρ) � 0: (6) Equation (2) for the radial wave function ψ(ρ) in region 2 zξ zξ ρ > R takes the following form: )e general solution to this equation is in the form of a 2 2z ψ zψ 2 2 linear combination of the Bessel function Jl(ξ) and the ζ + ζ +� − ζ − M �ψ � 0; ρ > R; (8) 2 z Neumann function Nl(ξ) of the l-th order [37]: zζ ζ ψ1(ρ) � A1JlkAρ � + B1NlkAρ �; 0 ≤ ρ ≤ R: (7) where Advances in Condensed Matter Physics 3 350 ζ � cBρ; s���������������� (9) 300 2mB 2 cB � 2 (W − E) + kz: Z 250 )e solution of equation (8) gives a linear combination 200 of the imaginary argument Il(ζ) of the Bessel function and (meV) the Macdonald function Kl(ζ) of the l-th order [37]: E 150 ψ (ρ) � A2KlcBρ � + B2IlcBρ �; ρ > R: (10) 2 100 )erefore, ψ(ρ) for a radial wave function is appropriate for the following: 50 A1JlkAρ � + B1NlkAρ �; 0 ≤ ρ ≤ R; 56789101112131415 ψ(ρ) �( (11) R (nm) A2KlcBρ � + B2IlcBρ �; ρ > R: Transcendental equation, nonparabolic-e1 If we take into account that the wave function inside the Transcendental equation, parabolic-e1 cylinder is finite and equal to zero at an infinite distance Approximate, nonparabolic-e1 from the center of the cylinder, expression (11) takes the Approximate, parabolic-e1 following form: Shooting equation, nonparabolic-e1 Shooting equation, parabolic-e1 A1JlkAρ �; 0 ≤ ρ ≤ R; ψ(ρ) �( (12) Figure A K c ρ �; ρ > R: 2: )e dependence of energy on the radius of a cylindrical 2 l B nanowire for electrons and holes with the parabolic and non- parabolic dispersion law. Here, A1 and A2 are constant values. We select relation A1/A2 in such a way that the following boundary conditions are satisfied: Solving the transcendental equation (14), we can ψ1(ρ)|ρ�R � ψ2(ρ)|ρ�R; determine the electron energy in a cylindrical quantum (13) wire. 1 dψ1(ρ) 1 dψ2(ρ) When the argument is too small under condition jρ�R � !jρ�R: mA dρ mB dρ l � 0, expanding the imaginary argument of the Bessel function Il(ζ) and MacDonald Kl(ζ) in a row, we get the From (12) and (13), we obtain the following transcen- first terms and an analytical formula for calculating the dental equation: energy: J′k R �K c R � m l A l B � A: (14) JlkAR �K′lcBR � mB q���������� q���������� q���������� 2 2 2 2 �m W Z �RK � �m W Z � R� + K � �m W Z � R� 4Z 2 B / 0 2 B / 2 1 2 B / E0;1 � q���������� q���������� q���������� : (15) 2 2 2 2 mAR �2mBW/Z �RK0� �2mBW/Z � R� + 4K1� �2mBW/Z � R� If we take into account (5) and (9), expression (15) gives the potential energy of an electron in a cylindrical quantum very close results to the exact results of equation (14) when nanorod with a finite depth be given by the following calculating the energy of electrons and holes in cylindrical expression: nanowires with larger radii (Figures 2–4). 3. The Analytical Method for Calculating the 8> W; 0 < ρ < R1; Electron Energy in a Cylindrical Quantum <> Nanorod with a Finite Height of the U(r) � U(ρ) �> 0;R1 ≤ ρ ≤ R2; (16) :> Potential Well W; R2 < ρ: Figure 5 shows the geometric and potential diagram of a )e Schrodinger¨ equation in a cylindrical coordinate cylindrical quantum nanorod with a finite potential well. Let system in this case is as follows: 4 Advances in Condensed Matter Physics R (nm) 5 6 7 8 9 101112131415 –2 –4 –6 –8 –10 –12 –14 (meV) –16 E –18 –20 –22 –24 –26 Transcendental equation, nonparabolic-hh1 Transcendental equation, parabolic-hh1 Approximate, nonparabolic-hh1 Approximate, parabolic-hh1 Shooting equation, nonparabolic-hh1 Shooting equation, parabolic-hh1 Figure 3: )e dependence of the energy of a heavy hole on the radius of a cylindrical quantum wire for the parabolic and nonparabolic dispersion law.
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