lecture12 - 2.57 Nano-to-Macro Transport Processes Fall...

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2.57 Fall 2004 – Lecture 12 1 2.57 Nano-to-Macro Transport Processes Fall 2004 Lecture 12 5.1 Plane waves & their interface reflection (continue) For the above problem, we have obtained 1 () itk x i Ae ω −− Ψ= (incoming wave), 1 x r Be −+ (reflected wave), 1 2 2 mE k = = 2 x t Ce (transmitted wave), 2 2 2( ) mE u k = = . The boundary conditions are applied 00 | ir t xx == Ψ+Ψ , (' ' ) ' | t + = = , which yields 12 ;( ) ABC kAB kC += − = or 1 2 ; kk k BC AkkAkk ++ . The incoming flux term is (note A and k 1 can be complex) * Re ii i i J m ⎛⎞ Ψ ⎜⎟ ⎝⎠ = * 11 ( ) ** 10 Re ( ) | x x x i Ae A ik e m ωω = =− = * 2 * Re | ik k x x Ak e m = = = 2 * 1 Re A k m = = 2 1 m = = , where we use the fact that k 1 is a real number in the last step. Similarly, we have Energy barrier u Incoming wave Reflection wave Transmission wave x

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2.57 Fall 2004 – Lecture 12 2 2 1 r JB k m =− = (negative sign indicating the direction of reflection), () 2 * 2 Re t JCk m = = . The reflectivity is 2 2 2 12 / ri kk B EE u RJ J Ak k u −− = = = + +− , and transmittivity is * 2 2 1 Re / ti k C TJJ == . For E>u, the equation gives reflectivity 0 R in both cases shown below, which is inconsistent with classical mechanics. When E<u, the equations yield R=1, T=0, which is reasonable from classical viewpoint. However, the wavefunction is 2 2 0 mu E it x t Ce ω Ψ= = in this case. The wave will decay rapidly from the barrier and is thus called “evanescent wave.” Electromagnetic (EM) waves In this chapter, we will see that the wave reflection, interference, and tunneling phenomena can occur for all the three types of carriers (phonons, electrons, photons) and the descriptions of these phenomena are also similar. An electromagnetic wave in vacuum is characterized by an electric field vector , E JG , and a magnetic field vector , H JJG . Consider a pair of charged particles placed in an electrical field. The field will attract the positively charged particle in one direction and repel the other particle in the same direction. Consequently, the particles are distorted and form an electrical dipole, whose moment is p ed = ( d is the separation distance). x x e e E E e e d
2.57 Fall 2004 – Lecture 12 3 A measure of the capability of the material responding to incoming electric field is the electric polarization per unit volume, or the dipole moment per unit volume, P [C m -2 ], which is related to the electric field through the electric susceptibility, χ , P = ε o χ E, where ε o is the vacuum electric permittivity, ε o =8.85x10 -12 [C 2 N -1 m -2 ], and the electric susceptibility is nondimensional. The electric susceptibility χ describes the extent to which positive and negative charges are displaced in a dielectric material under an applied electric field.

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This note was uploaded on 01/12/2011 for the course ME 305 taught by Professor Wright,j during the Spring '10 term at Birla Institute of Technology & Science, Pilani - Hyderabad.

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lecture12 - 2.57 Nano-to-Macro Transport Processes Fall...

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