EMS172-2009-Homework1_Solutions

EMS172-2009-Homework1_Solutions - UNIVERSITY OF CALIFORNIA,...

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EMS 172 Profs. Y. Takamura and A. Moulé, 2009 UNIVERSITY OF CALIFORNIA, DAVIS Department of Chemical Engineering and Materials Science EMS 172: Electronic, Magnetic, and Optical Properties of Materials Homework 1 - Solutions Due Date: Tuesday, Oct. 6 th , 2009, in class Note: You must show all the steps you used to obtain your answer in order to receive full credit. 1. Energy Levels [9 pts] Draw a block band diagram of the energy levels for (a) free electron, (b) strongly bound electron, (c) electron in a periodic potential (see Hummel Fig. 4.3 as an example). What are the boundary conditions that lead to each of these different band schemes? For (c), word descriptions are sufficient. (a) Free electron: all energy levels are allowed forming an energy continuum since there are no boundary conditions on the Schrödinger equation. [2 pts] (b) Strongly bound electron: only discrete energy levels are allowed because the boundary conditions are such that the electron can move freely between two infinitely high potential barriers, ψ (x=0)=0= ψ (x=a ). [2 pts] (c) Electron in periodic potential: allowed energy bands are formed because the potential consists of periodic barriers with a finite strength. ψ and d ψ /dx are continuous and periodic at the boundaries between regions I and II. [2 pts] [1 pt for each diagram] 2. Electron in a periodic potential [30 pts] (a) Draw the Kronig-Penney model (potential,
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EMS172-2009-Homework1_Solutions - UNIVERSITY OF CALIFORNIA,...

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