APPLICATION OF NEWTON RAPHSON METHOD TO A FINITE BARRIER QUANTUM WELL (FBQW) SYSTEM

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1 APPLICATION OF NEWTON RAPHSON METHOD TO A FINITE BARRIER QUANTUM WELL (FBQW) SYSTEM AJAYI Jonathan Olanipekun, ALABI Akinniyi Michael & ADEDOKUN Oluwaseun Department of Pure and Applied Physics, Ladoke Akintola University of Technology, P.M.B. 4000, Ogbomosho, Oyo state, Nigeria. ABSTRACT Quantum wells are important in semiconductor lasers because they allow some degree of freedom in the design of the emitted wavelength through adjustment of the energy levels within the well by careful consideration of the well width. Many realistic model in Physics requires numerical methods since these models cannot always be solved analytically i.e. in closed form. In this paper, a simple model of the energy levels in a quantum well was considered, with the adoption of Newton Raphson method (due to its rapid convergence) to a special case in which one of the parameters of the transcendental equations of the finite barrier quantum well equals four. We have been careful enough with the choice of initial estimate, to obtain the results for the eigenstates of this system which compares favourably well (with only marginal error) with other results obtained using graphical approach. Keywords: Transcendental, convergence, eigenstates, bound states. 1. INTRODUCTION Numerical analysis is invaluable as well indispensable in solving model equations e.g. systems of differential equations, matrices, tensors e.t.c all which cannot always be solved analytically or exactly. These models equations are used in physics to investigate some areas of the physical world. Of course, mathematical models serve the purpose of isolating the important part of particular situation. It as well pinpoints accurately the problem to be solved [1]. Considering this fact, numerical analysis aids more realistic models to be treated. One of such realistic models is the finite barrier quantum well which is the part of the application of quantum mechanics [].This model is of great application in laser diodes, including red lasers for DVDS, infra red lasers in fiber optic transmitters, and as well in the theory of conduction and insulation in solids. In this paper, we basically apply a type of numerical analysis, the Newton Raphson method to solve the transcendental equations of a finite quantum well, whose eigenstates solutions are determined until now with the graphical method. THEORETICAL CONSIDERATIONS AND CALCULATIONS Numerical iteration involves a method which aids the solving of nonlinear and transcendental equations by numerical method [3]. Of all numerical methods, Newton Raphson method [1] remains one of the mostly used methods due to its rapid convergence to the required root (the initial values being sufficiently close though). To use this method, we start with an initial value (say x 0 ), which is not far from a root, x 1, extrapolation along the tangent to its intersection with the x-axis is obtained, the value so obtained is taken as the next approximation. The process is repeated until either the successive x-values are sufficiently close or the value of the function is sufficiently near zero. The calculation process is actually aimed at solving, g(x) =0 1 Considering the Taylor series expansion of g(x) about some point x = x 0 g(x) = g (x o ) + (x-x 0 ) g'(x) + ½ (x-x 0 ) g"( x 0 ) + 1/3(x - x 0 ) 3 g"' (x 0 ) + Setting the quadratic and higher forms to zero and solving the linear approximation of g(x) = 0 for x gives, x 1 = x o - g(x 0) 3 g (x 0 ) Subsequent iterations are defined in a similar manner as, x n+1 = x n g(x n ) g (x n ), n= 0, 1,, 3 4 Equation (4) is the Newton-Raphson formula and it converges quadratically because the error is, ε n+1 = ½ε n g"(x ) g (X ) + O(ε n 3 ) 5 One would notice that each error is roughly proportional to the square of the previous error [4, 5]. Lets us divert our attention to the finite barrier quantum well depicted in Fig 1. 88

2 i ii iii V(x) V 0 V 0 -r/ 0 +r/ Fig.1. Schematics of the assumed structure used in the numerical calculation [7] Starting with the time-independent Schrodinger equation, ħ m d dx Ψ + (E-V) Ψ = 0 6 With V (Ψ) = V 0 x< -r/.region 1 0 -r/< x < r/..region II V 0 x > r/.region III Ψ 1 = Ce px + De px, p = m (V 0 E) ħ....region I 7 Ψ = Fsin kx+ecos kx, k = me ħ....region II 8 Ψ 3 = Qe px +Ne px, p = m (V 0 E) ħ... Region III 9 Note in Region I as x -, values of D will be infinite, also in Region III as x, the values of Q will be infinite, so we reject them as they do not satisfy the boundary condition (that Ψ must be infinite). Rewriting them, Ψ 1 = Ce px 10 Ψ = Fsin kx+ecos kx 11 Ψ 3 = Ne px 1 Since Ψ must be continuous and differentiable, also applying the condition that For the even case F = 0, C = N For the odd case E = 0, C = -N X=-b/ We have for the even case p = k tan kr/ 13 Similarly for the odd case p = kcot kr/ 14 Equation (13) and (14) cannot be solved analytically, unless a graphical or numerical approach is used. But we note from the definition of p and k that, p = m (V 0 E) ħ, k = me ħ m (V 0 E) p = ħ Multiplying both sides by r/ pr = m (V 0 E) ħ (pr/) r/, squaring both sides, we have = m (V 0 E) ħ r /4 p r = m V 0r - mer 4 4h 15 4ħ Introducing dimensionless variable 89

3 ε,, ρ ε me r =, ρ = m V 0r 4ħ 4ħ = p r 4 One would notice ε = kr, = pr me, since k = 4 ħ So equation (15) becomes = ρ - ε 16 So equation (13) becomes = ε tan ε 17 And equation (14) becomes = ε cot ε 18 Equation (17) and (18) cannot as well be solved analytically. Since = ρ - ε So equation (17) and equation (18) becomes simultaneously, ρ ε = ε tan ε - Even or Symmetric case - ρ ε = ε cot ε - Odd or Antisymmetric case We thus solve for a special case of ρ = 4, applying N-R method. Now for the even or symmetric case, we have ε n+1 = ε n - ε n tan ε n ρ ε n tan ε n ε n tan ε n + 1 +ε n ( 1+(ρ ε n ) ½ ) Picking initial values a= 1.3, b = 1. ε 0 = a+b = = 1.5 After about five iterations, we obtain, ε a = Still within range, picking initial values a=3.5, b=3.6 ε 0 = a+b = = 3.55 After about nine iterations, we obtain ε b = Now, for the Odd or Antisymmetric case ε n+1 = ε n + ε n cot ε n + ρ ε n cot ε n ε n cot ε n 1 +ε n ( 1+(ρ ε n ) ½ ) Also, picking initial values a=.5, b =.4 ε o =.5+.4 =.45 After seven iterations, we have ε c = TABLES OF VALUES ε Ø ε ε tan ε ε ε cot ε ε ε tan ε

4 7 εtanε εcotε 6 16 ε 5 Variable Variable ε Fig.. A graphical solution of the transcendental equation of a finite barrier quantum well 91

5 4. RESULTS AND DISCUSSION For even case ε a , ε b and for the odd case ε c the normalization constants C, D, F, E, Q and N as well as the values of other parameters ( ρ,, p, k ) are obtained by simple algebraic manipulation of ε and this was found to compare favourably with the values obtained graphically with ε a 1.5, ε b 3.58, ε c.47. For a given quantum well width b, depth V o and particle mass m, equation (16) describes a circle of radius ρ in Cartesian ε space. The intersections of this circle with the graph of equation (17) and equation (18), determines the eigen energy corresponding to even and odd eigenstate. In particular, the value of ρ used allows the quantum well to be used as a scatterer within allowable arguments, also specific cases of ρ helps in determining the bound states for even and odd cases. 5. CONCLUSION The solution of the transcendental equation of the finite barrier quantum well has been examined using Newton- Raphson (N-R) method; the values obtained also compares favourably well with other works using graphical method, with marginal error of about 0.188% ( ) in ε a, 0.48% ( ) in ε b and 0.185% ( ) in ε c. REFERENCES [1]. Ralston. A (1965) A First course in numerical analysis, McGraw-Hill, New York. []. Liboff, R.L. (1980) Applied Numerical Analysis Addison-Wesley Publishing Comp., Philippines. [3]. Wilkes, M.V. (1966) A Short Introduction to Numerical Analysis,University Press [4]. Lambert, J.D. and McLead, R.J.Y. (1986) Numerical methods for Phase-plane problems in ordinary differential equation. Springer Lecture Notes in Math, Springer-Verlag, pp [5]. Stephenson Gasiorowicz (003) Quantum Physics, third edition, John Wiley & Sons. [6]. B. S. Rajput (1965) Mathematical Physics, Pragati Prakashan. [7]. Eisberg, R.M. (1961) Fundamentals of Modern Physics, Jon Wiley and sons Ltd,New York. [8]. Florian Scheck (1965) Quantum Physics, Springer Berlin Heidelberg, New York. [9]. K. A. Stroud (1986) Advanced Engineering Mathematics, Palgrave Macmillan, New York. [10]. Moler (004) Numerical Computing with MATLAB, SIAM. [11]. Stanoyevitch (005) Introduction to MATLAB with Numerical Priliminaries, John Wiley & Sons, Inc. 9