Phonons Handout

Phonons Handout - University of California, Berkeley EE230...

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- 15 - University of California, Berkeley EE230 - Solid State Electronics Prof. J. Bokor Phonons (Classical theory) (Read Kittel ch. 4) Classical theory. Consider propagation of elastic waves in cubic crystal, along [100], [110], or [111] directions. Entire plane vibrates in phase in these directions we will treat this as a 1D problem. 1 atom per primitive cell Assume nearest neighbor interations only, atomic mass, M s-1 ss + 1 s+2 s+3 longitudinal wave equilibrium positions U s-1 U s+1 U s+2 U s+3 U i : displacements from equilibrium s-1 s s+1 s+2 transverse wave 2 modes - why? (2 polarizations) U s+1 U s+2 U s+3 U s-1 F s CU s 1 + U s () = s 1 U s +
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- 16 - University of California, Berkeley EE230 - Solid State Electronics Prof. J. Bokor look for harmonic solutions then and so, only k values inside 1 st BZ are physically significant. Why? long wavelength limit (also continuum limit): ka << 1 cos ka 1- ( ka ) 2 /2 U s Ai ω tk s a () [] exp = ω 2 2 C M ------ 1 ka cos = ω 4 C M ------ 2 ----- ⎝⎠ ⎛⎞ sin = 1 st Brillouin zone π a π a ω
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- 17 - University of California, Berkeley EE230 - Solid State Electronics Prof. J. Bokor so, ω = v sound k where the velocity of sound, v sound now consider 2 atoms per primitive basis let atom displacements { v } atom displacements { u } again, take travelling wave solutions Plug into equations and obtain dispersion relation where is the reduced mass, a u s v s u s 1 + v s 1 + M 1 M 2 M 1 v ·· s Cu s u s 1 + 2 v s + () = M 2 u s Cv s 1 v s 2 u s + = u s v s ⎩⎭ ⎪⎪ ⎨⎬ ⎧⎫ u v i ω ts k a [] exp = ω 2 C μ --- C μ ⎝⎠ ⎛⎞ 2 4 C 2 M 1 M 2 -------------- 2 sin ka 2 ----- ± = μ
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- 18 - University of California, Berkeley EE230 - Solid State Electronics Prof. J. Bokor One way to see the origin of the optical and acoustic modes: Check Same as before. The cos curve duplicates the sin curve. Each pt. on the cos curve repre- sents a mode on the sin curve (extended zone representation). Now let differ from . Primitive cell now larger . Brillouin zone boundary at M 1 M 2 case ( a ' a 2 -- == ω 2 C M ---- ka ' 2 ------ ⎝⎠ ⎛⎞ cos ' 2 sin ⎩⎭ ⎪⎪ ⎨⎬ ⎧⎫ for + for - = k π a ' π 2 a ' π 2 a ' π a ' --- 2 C M 2 C M M 1 M 2 2 a ' ()
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- 19 - University of California, Berkeley EE230 - Solid State Electronics Prof. J. Bokor . Gaps open at zone boundaries. for optic modes, the two atoms vibrate opposite to each other, for acoustic-vibrate together π 2 a ' ± Acoustic branch optical branch ω 2 C μ ------ 2 C M 2 2 C M 1 π a ' π 2 a ' π 2 a ' π a ' --- New BZ acoustic mode optical mode
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- 20 - University of California, Berkeley EE230 - Solid State Electronics Prof. J. Bokor if binding is partly ionic, then optic mode has a dipole moment interacting with radiation. Optic mode can be excited by light. For p atoms in primitive cell, there are 3p branches 3 acoustic, 3p-3 optical Phonons in “real” crystal • long range forces. In metals, the coduction electrons tend to modulate forces at long dis- tances. In ionic crystals, coulomb forces from the charged ions also act at long distance.
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Phonons Handout - University of California, Berkeley EE230...

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