ME 381R Fall 2003 Micro-Nano Scale Thermal-Fluid Science and Technology Lecture 4: Crystal Vibration...
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Transcript of ME 381R Fall 2003 Micro-Nano Scale Thermal-Fluid Science and Technology Lecture 4: Crystal Vibration...
ME 381R Fall 2003Micro-Nano Scale Thermal-Fluid Science and Technology
Lecture 4:
Crystal Vibration and Phonon
Dr. Li Shi
Department of Mechanical Engineering The University of Texas at Austin
Austin, TX 78712www.me.utexas.edu/~lishi
2
Outline
Reciprocal Lattice
• Crystal Vibration
• Phonon
•Reading: 1.3 in Tien et al
•References: Ch3, Ch4 in Kittel
3
Reciprocal Lattice
• The X-ray diffraction pattern of a crystal is a map of the reciprocal lattice.
• It is a Fourier transform of the lattice in real space
• It is a representation of the lattice in the K space
K: wavevector of Incident X rayReal lattice
Diffraction pattern or reciprocal lattice
K’: wavevector of refracted X ray
Construction refraction occurs only when KK’-K=ng1+mg2
5
Reciprocal lattice & K-Space
a
xniaxinax
axinx
nn
nn
2exp2exp
2exp
0 2/a 4/a 6/a
G
G/2
First Brillouin Zone
1-D lattice
K-space or reciprocal lattice:
Lattice constant
Periodic potential wave function:
Wave vector or reciprocal lattice vector: G or g = 2n/a, n = 0, 1, 2, ….
6
Reciprocal Lattice in 1D
a
The 1st Brillouin zone: Weigner-Seitz primitive cell in the reciprocal lattice
Real lattice
Reciprocal lattice
k0 2/a 4/a-2/a-4/a-6/a
x
-/a /a
11
BCC in Real Space
•Primitive Translation Vectors:
•Rhombohedron primitive cell
0.53a
109o28’
•Kittel, p. 13
12
Real: FCC Reciprocal: BCC
1st Brillouin Zones of FCC, BCC, HCP
Real: HCP
Real: FCC Reciprocal: BCC
13
Crystal Vibration
s-1 s s+1
Mass (M)
Spring constant (C)
x
Transverse wave:
Energy
Distancero
Parabolic Potential of Harmonic Oscillator
Eb
Interatomic Bonding
14
Crystal Vibration of a Monoatomic Linear Chain
a
Spring constant, g Mass, m
xn xn+1xn-1
Equilibrium Position
Deformed Position
Longitudinal wave of a 1-D Array of Spring Mass System
us: displacement of the sth atom from its equilibrium position
us-1 us us+1
M
15
Solution of Lattice Dynamics
Identity:
Time dep.:
cancel
Trig:
s-1 s s+1
Same MWave solution:u(x,t) ~ uexp(-it+iKx)
= uexp(-it)exp(isKa)exp(iKa)
frequency K: wavelength
17
Polarization and VelocityPolarization and Velocity
21
cos12
cos12expexp22
KaM
C
KaCiKaiKaCM
Fre
que
ncy,
Wave vector, K0 /a
Longitudinal A
cousti
c (LA) M
ode
Transverse
Acousti
c (TA) M
ode
Group Velocity:
dK
dvg
Speed of Sound:
dK
dv
Ks
0
lim
18
Lattice Constant, a
xn ynyn-1 xn+1
nnnn
nnnn
yxxfdt
ydM
xyyfdt
xdM
2
2
12
2
2
12
2
1
Two Atoms Per Unit CellTwo Atoms Per Unit Cell
Solution:
Ka
M2 M1
f: spring constant
19
1/µ = 1/M1 + 1/M2
What is the group velocity of the optical branch? What if M1 = M2 ?
Acoustic and Optical Branches
K
Ka
20
Lattice Constant, a
xn ynyn-1 xn+1
PolarizationPolarization
Fre
que
ncy,
Wave vector, K0 /a
LATA
LO
TO
OpticalVibrationalModes
LA & LO
TA & TO
Total 6 polarizations
22
0 0.2 0.4 0.6 0.8 1.00.20.40
2
4
6
8
(111) Direction (100) Direction XL Ka/
LA
TATA
LA
LO
TO
LO
TO
Freq
uenc
y (
10
Hz)
12
Dispersion in GaAs (3D)Dispersion in GaAs (3D)
23
Allowed Wavevectors (K)
Solution: us ~uK(0)exp(-it)sin(Kx), x =saB.C.: us=0 = us=N=10
K=2n/(Na), n = 1, 2, …,NNa = L
a
A linear chain of N=10 atoms with two ends jointed x
Only N wavevectors (K) are allowed (one per mobile atom):
K= -8/L -6/L -4/L -2/L 0 2/L 4/L 6/L 8/L /a=N/L
25
PhononPhonon
h
Energy
Distance
Equilibrium distribution1exp
1
Tk
n
B
• where ħ can be thought as the energy of a particle called phonon, as an analogue to photon
• n can be thought as the total number of phonons with a frequency and follows the Bose-Einstein statistics:
2
1nu
•The linear atom chain can only have N discrete K is also discrete
• The energy of a lattice vibration mode at frequency was found to be