Energy of Electrons in a Nanowire Subject to Spin-orbit Interaction

Masthead Logo The Fordham Undergraduate Research Journal Volume 1 | Issue 1 Article 9 December 2013 Energy of Electrons in a Nanowire Subject to Spinorbit Interaction Ryan Brennan FCRH '11 Fordham University, Sheehan Ahmed FCRH '11 Fordham University, Antonios Balassis Fordham University, Vassilios Fessatidis Fordham University, Follow this and additional works at: https://fordham.bepress.com/furj Part of the Physics Commons Recommended Citation Brennan, Ryan FCRH '11; Ahmed, Sheehan FCRH '11; Balassis, Antonios; and Fessatidis, Vassilios (2013) "Energy of Electrons in a Nanowire Subject to Spin-orbit Interaction," The Fordham Undergraduate Research Journal: Vol. 1 : Iss. 1 , Article 9. Available at: https://fordham.bepress.com/furj/vol1/iss1/9 This Article is brought to you for free and open access by DigitalResearch@Fordham. It has been accepted for inclusion in The Fordham Undergraduate Research Journal by an authorized editor of DigitalResearch@Fordham. For more information, please contact . Energy of Electrons in a Nanowire Subject to Spin-orbit Interaction Cover Page Footnote Ryan Brennan, FCRH 2011, is from Long Island, New York. He is a graduating physics major. Ryan conducted research on energy bands in quantum nanowires and is currently studying the properties of graphene. After graduation, Ryan plans to attend graduate school in the hopes of attaining a Ph.D in physics. This article is available in The Fordham Undergraduate Research Journal: https://fordham.bepress.com/furj/vol1/iss1/9 C ommunication s FURJ | Volume 1 | Spring 2011 Brennan et al.: Energy of Electrons in a Nanowire www.fordham.edu/fcrh/furj Energy of Electrons in a Nanowire Subject to Spin-orbit Interaction Ryan Brennan, FCRH ’11; Sheehan Ahmed, FCRH ’11; Dr. Antonios Balassis; Dr. Vassilios Fessatidis The Physics PHYSICS The Hamiltonian for a particle subject to spin-orbit interaction is more complicated than that of a free particle, containing terms corresponding to the electric dipole and Thomas precession processes. For a thin quantum wire in the x-y plane, a non-zero electric field perpendicular to the plane of the wire gives rise to yet another process of spin-orbit interaction called the Rashba spin-orbit interaction. The contribution of this Rashba mechanism is dictated by a parameter α which is proportional to the perpendicular electric field. Additionally, a strong potential well within the x-y plane may be associated with an electric field, which is not negligible compared to the field that causes the α -interaction. In this case of planar, as well as perpendicular confinement, there is one more contribution to the Hamiltonian and the spin-orbit interaction, this time corresponding to the parameter β , which is dictated by the width and potential depth of the nanowire. Typical values of β are about one tenth of α .1 Our goal was to write a program that would compute the eigenenergies of an electron in the nanowire. The Problem Electrons are confined to a long, thin nanowire in the x-y plane. They are subject to the Rashba α -coupling due to an electric field in the z-direction. In addition, the particles are confined along the x-direction by the sides of the wire, only able to move between x=0 and x=W, which gives rise to the β spin-orbit coupling. The total wave function of an electron within the nanowire has the form1 the particle. Applying the Hamiltonian containing all of the SOI (spin-orbit interaction) terms to the above wave function gives two coupled differential equations. The Hamiltonian and the two differential equations are as follows: (2) (3) (4) where the Hamiltonian is comprised of the free particle contribution, the α contribution that arises from the asymmetry of the quantum well (Rashba mechanism), and the β contribution that arises from the lateral confining electric field. m* is the reduced mass of the electron,  is the reduced Planck’s constant, ε is the eigenenergy of the particle and  W   ( x − W )2   x 2   − F ( x ) =  exp − exp  − 2   2 l 2 l  0  0    2l0   . F(x) is  related to the lateral confinement field in the x-y plane, where l0 is a measure of the steepness of the potential at the edges of the nanowire. A small value of l0 means that the the particle hits a very steep potential at the edge of the wire. We needed to find ε -, the energies that satisfy these two equations simulataneously. Solving the Problem (1) where ky is the wave number in the y direction, Ly is a normalization length used to set the probability of finding the particle somewhere in space equal to one, and ψ A ( x ) and ψ B ( x ) are two different spin states of Our first task was to cast the equations in dimensionless form. We made our unit of length dimensionless x by changing x to X = , and adjusting the derivaW tives and the function F(x) accordingly. We introduced Direct all correspondence to Ryan Brennan at . 161 Published by DigitalResearch@Fordham, 2011 1 C ommunication s FURJ | Volume 1 | Spring 2011 The Fordham Undergraduate Research Journal, Vol. 1 [2011], Iss. 1, Art. 9 lα = 2 2 2 m ∗ w 2ε K k W E = = l = β y y and 2 2m∗α , 2m ∗ β , . Finally, we made the lα , lβ and l0 parameters dimensionless by setting τ α = W W W , τβ = and τ 0 = . lα lβ l0 Equations 1 and 2 became: (5) X = δ , ψ A ( X − δ ) and ψ B ( X − δ ) would be ψ A (0 ) andψ B (0 ) and could be set equal to 0. At X = 1 − δ the same could be done for ψ A ( X + δ ) and ψ B ( X + δ ) and in this way the boundary conditions at both sides were addressed. Several equations of this kind can be written at once as one matrix equation in the following way: (7) x + 3y - 4z = Ex 2x - 7y + 6z = Ey y - z = Ez (6) where and typical values were , and . It became clear that while these equations did not look very daunting, they could not be solved by elementary functions. We used a central difference approximation to replace the derivatives in the two equations: (8) PHYSICS becomes  x 1 3 − 4   x  2 − 7 6   y  = E  y        z  0 1 − 1   z  or M ⋅ r = Er. In our case, M was a large matrix of mostly zeroes, since for any given equation, only the current, previous and next values of the wave functions were present, and had non-zero coefficients. The matrix r became , a long column matrix containing the two wave functions at each x coordinate between 0 and 1: (9) where δ is a small distance in the x-direction. The derivative of the wave function at each point can be approximated by using the values of the function around the point of interest. The smaller the increment, the better the approximation. We first thought that we could write the two equations at X=0, setting ψ A/B ( X − δ ) = ψ A/B ( X ) = 0 , and solve for ψ A/B ( X + δ ). We would then plug these values into the two equations as the current values of the wave functions and solve for the wave functions at the next step of X. We soon realized that the v (...truncated)