lattice vibrations l & t modes ~ compression & shear waves...
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Lattice vibrations general results:
1st Brillouin Zone: N k-points (N = # cells)
3m modes/point (m = # in basis).
• Group velocity ( ):
Generally zero at high symmetry points on Z.B. (“standing waves”)• 3D case:
Acoustic – always 3 of these (in 3D cases)
Optical: the remaining 3m-3 branches (only for lattice with basis).
k¶¶w
wkgv Ñ=!!
LA
TA
L & T modes ~ compression & shear waves (best for large l; for general k displacement orientations may be more complicated)
Mg2Ca and Al2Ca(hexagonal structure) (Zhang et al., Intermetallics 22, 17, 2012)(note density of modes D(w) at bottom.)
Ge: Weber, Phys. Rev. B 15 (1977) 4789
Pd: Stewart, New J. Phys. 10, 043025 (2008).
FCC (with and without basis)
Typical plots: dispersion curves shown alongselected paths in k-space
Ba8Al16Ge30Nenghabi and Myles, J Phys. Cond. Mat. 20, 415214 (2008)
54 atoms / cell (x-ray, same structure)
Y. Li et al., Phys. Rev. B 75, 054513 (2007).
(from a talk posted by A. Kirk, McGill Univ.)
(theoretical result, material = Si)
metamaterials: Artificial crystallinity can be used to tailor wave propagation.“Phononic crystals” refers to larger scale periodic structures for sound waves.
Phonon Density of Modes:• Recall density of k-points = V/(2p)3
• Per k-point, 3Nm branches (3 ´ # cells ´ number in basis)
• For quantities depending only on energy: density of modes,D(w) = (# states between w, w + dw)
ò Ñ=w
wpw
@
2
3)2()(
shell k
kdVDdk
dLD wp
w2
)( =1D 3D,
per branch
úúú
û
ù
êêê
ë
é -´=
)
()2()( 3
unitstoconverted
volumespacekV
ddD
wp
wwFor later
use:
In Debyeapproximation: D(ω ) =
3V2π 2
ω 2
c3
Note, D(w) is same as density of quantizedmodes (phonons)
dwcorrection →
Debye approximation:• Assume w = kc for all modes.• Assume 3 branches, cut off at a sphere
containing # k-points = # atoms.• “Debye wavevector” etc
D(ω ) = 3V2π 2
ω 2
c3
kD = 6π 2n3
3 26 ncD pw =
ΘD = ωD / kB = (c / kB ) 6π2n3
Debye approximation: Commonly used as measure of phonon behavior (even when “real” behavior can be obtained)
from “The Specific Heat of Matter at Low Temperatures” [Tari, 2003].
X Zheng et al. Phys. Rev. B 85, 214304 (2012) [my lab]:
Specific heat of thermoelectric crystal.
Quantized Modes (phonons):
å -= iikxik eu
NQ ˆ1
åå ÷øö
çèæ -+Þ÷
øö
çèæ ×F×+=
ji jii
ji jijii uuK
MpuuM
pH,
22
,
2)ˆˆ(22ˆˆ2
!
(1D)N atoms
Convert to sum over N wave-vectors k (appendix C of Kittel)
å += iikxik ep
NP 1 (note N k vectors in
1st BZ make complete set)
( )åå +=÷øö
çèæ +=Þ --
n nnnnn ww
, 21
,
2
22 k ktkkk
kkkkk aaQQMM
PPH !
(branch)
Can show:
Formally equivalent to a sum of 3N independent harmonic oscillators.
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