note05 - ECE 4060: Introduction to Quantum and Statistical...

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1 ECE 4060: Introduction to Quantum and Statistical Mechanics Handout 5 Schrodinger Equation with Piecewise-Constant Potentials Textbook Reading: Hagelstein, Senturia and Orlando, Chaps. 7 and 8 (pp. 111 – 162) and Appendix D (pp. 703 -704). 5.1 The E-k dispersion relations in piecewise-constant potentials In this Chapter, we will use the time-independent Schrödinger equation to treat simple but useful piece-wise constant potentials which are important in the electronics and optoelectronics applications. If the potential energy is constant within a region (however, there is some other region with different constant potentials), then the time-independent Schrödinger equation will be: () () () () () () x ψ V E x ψ x m x ψ E x ψ V x ψ x m x ψ H ) ( 2 2 ˆ 0 2 2 2 0 2 2 2 = = + = h h (5.1) We can see that the eigenfunction will still be the plane waves e ikx , but the dispersion relation for the eigenvalue E(k) now becomes: m k V E 2 2 2 0 h = or ( ) 2 0 2 h V E m k = (5.2) Remember that the eigenvalue E represents the total energy or the stationary states. For E < V 0 , k will become imaginary and e ikx becomes a real number that is exponentially decaying or growing. We often write: () 2 0 2 h E V m ik = = α (5.3) The function e - x is often called the evanescent function , where the wave function has no classical mechanics counterpart since the total energy is lower than the potential energy (sometime referred to as the classically forbidden region ). 5.2 The interface condition for the wave function Since we will be dealing with the piece-wise constant potentials, we need to see how the wave functions can be stitched together at the point where the potential energy is discontinuous. If V(x) remains finite, though with a discontinuity, from Eq. (5.1), we know that the second
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2 derivative of ) ( x ψ is a finite number, which implies that ) ( x and its first derivative must be continuous when the mass m is constant. In crystalline solids, the mass is often expressed as the effective mass m* and may however be quite different in various materials at the material interface. In those systems, if we integrate Eq. (5.1) in space for once we will obtain: () ) ( ) ( 2 ) ( 1 ) ( 1 2 0 0 0 * 0 * 2 x x V E x x x dx d m x x dx d m Δ = Δ Δ + h (5.4) f(x) df(x)/dx d 2 f(x)/dx 2 Continuous and Finite Continuous and Finite Continuous and Finite Continuous and Finite Dicontinuous and Finite Infinite Discontinuous and Finite Infinite Table 5.1. Disontinuity and infinity possibilities for a function and its derivatives. From observing Eq. (5.4), we can see if V(x 0 ) can be infinite, then only (x) needs to be continuous and there is no constraint on d (x)/dx . If V(x 0 ) is finite although contains a discontinuity, we will have two interface conditions, called the Ben-Daniels boundary condition , to be met: ) ( 1 ) ( 1 ) ( ) ( * 2 * 1 x x dx d m x x dx d m x x x x Δ + = Δ Δ + = Δ
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This note was uploaded on 11/26/2010 for the course ECE 3060 at Cornell.

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note05 - ECE 4060: Introduction to Quantum and Statistical...

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