ME 381R Fall 2003

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ME 381R Fall 2003 Micro-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
78712 www.me.utexas.edu/lishi lishi_at_mail.utexas.
edu
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Outline

• Reciprocal Lattice
• Crystal Vibration
• Phonon

• Reading 1.3 in Tien et al
• References Ch3, Ch4 in Kittel

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Reciprocal Lattice
K wavevector of refracted X ray
K wavevector of Incident X ray
Real lattice
Construction refraction occurs only when
DK?K-Kng1mg2

• 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

Diffraction pattern or reciprocal lattice
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Reciprocal Lattice Points
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Reciprocal lattice K-Space
Lattice constant
1-D lattice
Periodic potential wave function
Wave vector or reciprocal lattice vector G or g
2n?/a, n 0, 1, 2, .
K-space or reciprocal lattice
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Reciprocal Lattice in 1D
a
Real lattice
x
-?/a
?/a
Reciprocal lattice
k
4?/a
0
2?/a
-2?/a
-4?/a
-6?/a
The 1st Brillouin zone Weigner-Seitz primitive
cell in the reciprocal lattice
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Reciprocal Lattice of a 2D Lattice
Kittel pg. 38
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FCC in Real Space

• Kittel, P. 13

• Angle between a1, a2, a3 60o

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Reciprocal Lattice of the FCC Lattice
Kittel pg. 43
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Special Points in the K-Space for the FCC
1st Brillouin Zone
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BCC in Real Space

• Kittel, p. 13

• Rhombohedron primitive cell

0.5?3a
109o28
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1st Brillouin Zones of FCC, BCC, HCP
Real FCC Reciprocal BCC
Real FCC Reciprocal BCC
Real HCP
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Crystal Vibration
Interatomic Bonding
Spring constant (C)
s-1
s
s1
x
Mass (M)
Transverse wave
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Crystal Vibration of a Monoatomic Linear Chain
Longitudinal wave of a 1-D Array of Spring Mass
System
M
us-1
us
us1
us displacement of the sth atom from its
equilibrium position
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s-1
s
s1
Solution of Lattice Dynamics
Same M
Wave solution u(x,t) uexp(-iwtiKx)
Time dep.
w frequency K wavelength
uexp(-iwt)exp(isKa)exp(?iKa)
cancel
Identity
Trig
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w-K Relation Dispersion Relation
l wavelength
K 2?/l lmin 2a Kmax ?/a -?/altKlt ?/a
2a
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Polarization and Velocity
Group Velocity
Speed of Sound
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Two Atoms Per Unit Cell
M2
M1
f spring constant
Solution
Ka
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Acoustic and Optical Branches
Ka
optical branch
1/µ 1/M1 1/M2
acoustic branch
What is the group velocity of the optical branch?
What if M1 M2 ?
K
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Polarization
TA TO
LA LO
Optical Vibrational Modes
Total 6 polarizations
LO
TO
Frequency, w
TA
LA
Wave vector, K
0
p/a
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Dispersion in Si
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Dispersion in GaAs (3D)
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Allowed Wavevectors (K)
A linear chain of N10 atoms with two ends
jointed
x
a
Solution us uK(0)exp(-iwt)sin(Kx), x sa B.C.
us0 usN10
K?2np/(Na), n 1, 2, ,N Na L
Only N wavevectors (K) are allowed (one per
mobile atom)
K -8p/L -6p/L -4p/L -2p/L 0
2p/L 4p/L 6p/L 8p/L p/aNp/L
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Allowed Wave Vectors in 3D
N3 of atoms
Kz
Ky
Kx
2p/L
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Phonon

• The linear atom chain can only have
• N discrete K ? w is also discrete
• The energy of a lattice vibration mode at
• frequency w was found to be

• where hw 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 w, and follows the Bose-Einstein
  statistics

Equilibrium distribution
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Total Energy of Lattice Vibration
p polarization(LA,TA, LO, TO) K wave vector