CN6-motion - Crystal Momentum, Velocity and Acceleration of...

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1 1 Crystal Momentum, Velocity and Acceleration of Electrons and “Holes” within an Energy Band 2 Outline ± Introduction ± Group velocity ± Crystal momentum ± Effective mass ± Holes revisited ± Effective mass Schrödinger equation ± Summary
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2 3 Introduction Electron behavior inside a semiconductor can be quite a bit different from what one would expect of a free-space carrier (which should be no surprise) … … but often not so different (which, given the complexity of the system, perhaps should be a surprise). This set of notes provides both a general discussion of how, in particular, electron velocities and accelerations due to applied fields are determined from the bandstructure (which can seem a bit strange) … … and how under limiting but also common conditions these determinations can be made to seem more familiar (provided you are familiar with Newtonian mechanics) through the introduction of concepts such as “ effective masses ” and (again, but more formally) “ holes .” 4 This section of notes will then provide part of the foundation for determining things like mobility, conductivity and resistively in a subsequent set of notes (CN9-transport) … … which in turn provides a foundation for device analysis … (… which provides a foundation for circuit analysis, and so it goes).
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3 5 Current density Current density is given by where = = states filled ) ( ) ( ) ( ) ( ) )( ( k k k k v k v k k v k j q f q q f y probabilit occupation ) ( = k f velocity (group) electron ) ( = k v somewhat inexact notation 6 (Group) velocity (The average value of the*) electron velocity is given by … as for any system obeying wave mechanics such as light and sound, where E ( k )= ħω ( k ) here. ) ( 1 ) ( k k v k E = h ) ( k k ω = *The electron velocity, which is momentum divided by mass, associated with Bloch functions ( energy eigenstates of a periodic potential or, sufficiently for this purpose, of a non-uniform potential ) is inherently uncertain . ) ( ) ˆ ˆ ˆ ( 1 k z y x E k k k z y x + + = h
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4 7 Example Question: What is the velocity vs. k relation for a free particle? Answer: [100] k E free-space [111] k m m E 2 2 2 k p 2 k h = = r k k r = i e Vol 1 ) ( ψ m E k 2 1 1 ) ( 2 k k v k k h h h = = m k k k k k k z y x z y x 2 ) ˆ ˆ ˆ ( 1 2 2 2 2 + + + + = h h z y x m k k k z y x ) ˆ ˆ ˆ ( z y x + + = h m m p k = = h 8 However, within a semiconductor the velocity need not be proportional to k or even parallel to it .
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5 9 Example Question: What are the directions and approximate relative magnitudes of of electron velocities (in real-space) for electrons at Points A-F (in k -space)? Use arrows and their relative lengths to indicate direction and relative magnitude, respectively. Figure 3—10 (modified) Conduction and valence bands in Si along [111] and [100] (From Chelikowsky and Cohen, Phys. Rev. B14, 556, 1976).
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This note was uploaded on 02/03/2009 for the course EE 339 taught by Professor Banjeree during the Spring '08 term at University of Texas at Austin.

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CN6-motion - Crystal Momentum, Velocity and Acceleration of...

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