Lattice Dynamics Physical properties of solids determined by electronic structure related to...
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Transcript of Lattice Dynamics Physical properties of solids determined by electronic structure related to...
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