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.