MIT OpenCourseWare http://ocw.mit.edu 3.23 Electrical, Optical, and Magnetic Properties of MaterialsFall 2007For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.Homework # 8 November 6, 2007 1 Propagation equation in a non-dispersive medium. Consider the following propagation equation in an infinite, one-dimension medium: ∂2y 1 ∂2y ∂x2 − v2 ∂t2 = 0 where y is a real function and v is called the phase velocity. a) Prove that any solution is the superposition of two waves propagating in opposite directions at the sp e ed v, in other words any solution y can be written y(x, t) = f(x + vt) + g(x − vt) where f and g can be of arbitrary shape (twice differentiable though). Show the details of the derivation. Hint: You can define new variables α = x + vt and β = x − vt and rewrite the differential equation with these new variables. b) What can you conclude about the shape of a wave propagating in a non-dispersive medium? Now, we look for stationary solutions, that is solutions for which space and time dependences can b e separated: y(x, t) = f (x)g(t), where f and g must be real functions. c) Find equations for f and g, and solve them. Use physical arguments to elimi-nate some classes of solutions. For a finite medium, what else would you have to know to completely solve the equation? How does that introduce quantization? Give an example and draw a parallel with a quantum-mechanical system that you have studied. d) Since the result of a) still holds, prove that any stationary solution is in-deed the superposition of two waves propagating in opposite directions. Where does that happen in the first Brillouin z one of an infinite crystal? 2 Propagation in a dispersive medium. For a dispersive medium, the phase velocity v must be defined as a function of an angular frequency ω. Therefore, we first consider waves whose time dependence 1 Nicolas Poilvert & Nicola Marzari� � is imposed by an oscillatory source, leading us to write: y(x, t) = Re e−iω tg(x) , or, to simplify the notation (provided we remember to take the real part at the end), y(x, t) = e−iω tg(x) (this is not a priori a stationary solution as we de-fined it earlier because taking the real part may have the effect of mixing time and space dependences). An example of such functions are the well-known de Broglie waves e−i(ωt−kx), where k is called the wave number. a) Derive an equation for g(x), and introducing a wave number k, find the ωdispersion relation = v(ω). When is the solution a de Broglie wave? k Now we want to study the propagation of any type of wave function. At t=0, the space dependence is not eikx anymore, but a more complicated function f(x). b) How would you proc ee d to solve the problem by means of de Broglie waves only? Let us take an example. Assume that the initial space depe ndence f(x) can be decomposed into an infinite number of harmonics eikx with k spanning the [k0 − Δk; k0 + Δk] interval with uniform sp e ctral density A. In other words, � Δk f(x) = A e ikxdx −Δk c) Calculate f(x) explicitly. Can you comment on the space width of the wave with respect to the wave number domain width? This is a very general result, that gives rise, in quantum mechanics, to the Heisenberg uncertainty principle. d) Now, at t > 0, express the solution in the form of an integral. To calculate the integral that you found in d), we will consider the Taylor expansion, dω 1 d2ω ω = ω0 + (k − k0) + (k − k0)2 dk 2 dk2 where ω0 = ω(k0). e) Assuming that ω is a linear function of k, calculate the solution at any dωtime t. Can you see why is called the group velocity? dk d2ω f) If is not equal to zero, give a qualitative discussion of what will happen. dk2 Justify the expression dispersive medium. Propagation of light in a conducting medium. To solve Maxwell’s equations, one has to know how to re late the charge density ρ and the current density J to the fields E and B. In this problem, we treat the case of an uncharged, non-polarizable metal. The latter assumption means that we neglect polarization due to bound electrons. The electrons of the conduction band, however, are free to move under the influence of an electromagnetic field. 2 3� In the classical Drude model, the movement of a conduction electron in a field (E,B) is desc ribed by the equation, dp p= − + f(t) dt τ where p is the electron momentum, 1/τ a damping coefficient representing the collisions of the e lec tron with the atoms of the lattice, and f(t) is the force exerted by the fields on the electron. a) Show that for a conduction electron in the field of an electromagnetic wave, one can neglect the magnetic force and write f(t) = −eE. b) If ω is the electromagnetic wave angular frequency, upon what condition on the product ωτ can we reasonably set dp = 0 in the e quation of motion? dt For τ ∼ 1 − 10 fs, for what wave numbers is this condition satisfied? For the rest of the problem, suppose that we work in the spectral range of far infrared wavelengths. c) Find the expression of the electric conductivity σ, defined by, J = σE d) Using Maxwell’s equations, derive the propagation equation for E in the metal. e) Looking for a solution of the form ei(ωt−kx), where k is a priori a com-plex number, find an equation for k. Express the complex refractive index n as a function of σ, ω and the electric permittivity of free space �0. f) Prove that the relation you just found can be simplified into, σ n =2ω�0 (1 − i) g) Conclude that the intensity of the light beam propagating in the metal follows the law, I(x) = I0e−x/δ(ω) where I0 is the intensity of the beam at the surface of the metal, x measures the distance from the surface, and δ(ω) is a typical penetration length called skin depth. Show that, � 1 δ(ω) = 2σωµ0 Interband absorption. Let us consider a direct gap semiconductor. Around the origin, the highest va-h2¯ k2 lence band is approximated by the parabola Ev(k) = − 2mh , where mh is the 3 4effective mass of a hole at the top of the valence band, and the lowest conduction h2k2¯band is represented by the parabola Ec(k) = , where me is the effective 2me mass of an electron at the