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Lattice Dynamics
Physical properties of solids
determined by electronic structurerelated to movement of atomsabout their equilibrium positions
•Sound velocity •Thermal properties: -specific heat -thermal expansion -thermal conductivity (for semiconductors)
•Hardness of perfect single crystals (without defects)
Reminder to the physics of oscillations and waves:
Harmonic oscillator in classical mechanics:
Example: spring pendulum
Hooke’s law
2
2
1xDEpot
x
springFxm
Equation of motion:
0 xDxm or 0 x~m
Dx~
where ))t(x~Re()t(x
Solution with tieA~)t(x~
)tcos(A)t(x
where m
D
X=A sin ωt
X
Dx
m
D
Traveling plane waves: )kxt(cosA)t(y
X0
Y
X=0: tcosA)t(y
t=0: kxcosA)x(y
Particular state of oscillation Y=const
0 in particular
or )kxt(ieA~)t(y~
)kxt(cosA)t(y
travels according
0 .constdt
dkxt
dt
d
kvx
/2
2v
)kxt(ieA~)t(y~ 2
2
2
2
2
1
x
y
t
y
v
solves wave equation
Transverse wave
Longitudinal wave
Standing wave
)tkx(ieA~y~ 1
)tkx(ieA~y~ 2
)tkx(i)tkx(is eeA~y~y~y~ 21
titiikx eeeA~ tcoseA~ ikx 2
Re( ) 2 cos coss sy y A kx t
Large wavelength λ 02
k
Crystal can be viewed as a continuous medium: good for m810
λ>10-8m
10-10m
Speed of longitudinal wave:
sBv where Bs: bulk modulus with
compressibilityBs determines elastic deformation energy density 2
2
1 sBU
dilation V
V
(ignoring anisotropy of the crystal) sB
1
sB
v
E.g.: Steel
Bs=160 109N/m2
ρ=7860kg/m3 s
m
m/kg
m/Nv 4512
7860
101603
29
(click for details in thermodynamic context)
< interatomic spacing continuum approach fails
In addition: phononsvibrational modes quantized
Vibrational Modes of a Monatomic Lattice
Linear chain:
Remember: two coupled harmonic oscillators
Superposition of normal modes
Symmetric mode Anti-symmetric mode
generalization Infinite linear chain
How to derive the equation of motion in the harmonic approximation ?n n+1 n+2n-1n-2
un un+1 un+2 un-1un-2
un un+1 un+2 un-1un-2
fixed
D
1 nnln uuDF
1 nnrn uuDF
a
Total force driving atom n back to equilibrium
11 nnnnn uuDuuDF
n n nnn uuuD 211
equation of motion nn Fum
nnnn uuum
Du 211
Solution of continuous wave equation )tkx(ieAu
approach for linear chain )tkna(in eAu
)tkna(in eAu 2 ika)tkna(i
n eeAu 1 ika)tkna(i
n eeAu 1, ,
? Let us try!
22 ikaika eem
D kacosm
D 122
)/kasin(m
D22
)/kasin(m
D22
Continuum limit of acoustic waves:
m
D2
k
02
k
.../ka/kasin 22 kam
D a
m
Dv
k
Note: here pictures of transversal wavesalthough calculation for the longitudinal case
k
)t)k(nak(ieAnu
ahkk
2
)k()k(
)tnak(ieA
, here h=1
)tna)a
hk((ieA
2nhie)tnak(ieA 2 )tnak(ieA
12 nhie
))k(,k(nu))k(,k(nu
ahkk
2 1-dim. reciprocal
lattice vector Gh
ak
a
Region is called
first Brillouin zone
Brillouin zones
We saw: all required information contained in a particular volume in reciprocal space
first Brillouin zone 1d:a
xeannr xea
hhG
2
mnrhG 2 where m=hn integer
a
2
1st Brillouin zone
In general: first Brillouin zone Wigner-Seitz cell of the reciprocal lattice
Vibrational Spectrum for structures with 2 or more atoms/primitive basis
Linear diatomic chain:
2n 2n+1 2n+22n-12n-2
u2n u2n+1 u2n+2 u2n-1u2n-2
D a
2a
nununum
Dnu 2212122 Equation of motion for atoms on even positions:
Equation of motion for atoms on even positions: 12222212 nununuM
Dnu
)tkna(ieAnu 22Solution with:
)tka)n((ieBnu 12
12and
A)ikaeikae(B
m
DA 22
B)ikaeikae(A
M
DB 22
kacosBm
D
m
DA 222
kacosAM
D
M
DB 222
22
2
mD
kacosB
m
DA
kacosMm
D
M
D
m
D 22
42222
kacosMm
D
m
D
M
D
Mm
D 22
4422222
4
0212
4224
kacos
Mm
D
M
D
m
D
kasin2
Mm
kasin
MmD
MmD
24211112
1 12D
m M
22
M1
m1
DM1
m1
D
mD
2 , MD
2
mD
2
MD
2
2 2
•Click on the picture to start the animation M->m note wrong axis in the movie
:a
k2
Ato
mic
Dis
plac
emen
t
Optic Mode
M
mkA
B0
Ato
mic
Dis
plac
emen
t
Acoustic Mode10 kA
B
Click for animations
Dispersion curves of 3D crystals
•Every additional atom of the primitive basis
•3D crystal: clear separation into longitudinal and transverse mode only possible in particular symmetry directions
•Every crystal has 3 acoustic branches sound waves of elastic theory1 longitudinal
2 transverseacoustic
further 3 optical branches
again 2 transvers 1 longitudinal
p atoms/primitive unit cell ( primitive basis of p atoms):
3 acoustic branches + 3(p-1) optical branches = 3p branches
1LA +2TA (p-1)LO +2(p-1)TO
Intuitive picture: 1atom 3 translational degrees of freedom
3+3=6 degrees of freedom=3 translations+2rotations
+1vibraton
Solid: p N atoms
no translations, no rotations
3p N vibrations
x
yz
# of primitive unit cells
# atomsin primitivebasis
diamond lattice: fcc lattice with basis
(0,0,0)),,(4
1
4
1
4
1
Longitudinal Acoustic
Longitudinal Optical
Transversal Acoustic
degenerated
Part of the phonon dispersion relation of diamond
Transversal Opticaldegenerated
P=2
2x3=6 branches expected
2 fcc sublattices vibrate against one anotherHowever, identical atoms no dipole moment
Phonon spectroscopy
Inelastic interaction of light and particle waves with phonons
Constrains: conservation law of
momentum energy
Condition for elastic scattering
hklGkk 0
in
± q
incoming wave scattered wave
Reciprocal lattice
vector
phonon wave vector
hklGqkk 0
00 )q(
elastic sattering in
“quasimomentum”
02
20
2
2
22 )q(
nM
k
nM
k
for neutrons
for photonscattering
0
)q(0k
k
q
Triple axis neutron spectrometer
@ ILL in Grenoble, France
Lonely scientist in the reactor hall
Very expensive and involved experiments
Table top alternatives ?
Yes, infra-red absorption and inelastic light scattering (Raman and Brillouin)
However only 0q accessible
see homework #8