Transcript Lecture 4: Crystal Vibration and Phonon

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
[email protected]
Outline
Reciprocal Lattice
• Crystal Vibration
• Phonon
•Reading: 1.3 in Tien et al
•References: Ch3, Ch4 in Kittel
2
Reciprocal Lattice
K’: wavevector of refracted X ray
Real
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 ray
Construction refraction occurs only when
DKK’-K=ng1+mg2
Diffraction
pattern or
reciprocal
lattice
3
Reciprocal Lattice
Points
4
Reciprocal lattice & K-Space
1-D lattice
Lattice constant
a
Periodic potential wave function:
 x    n exp in 2x a 
n
 x  a    n exp in 2x a   exp i 2n    x 
n
Wave vector or reciprocal lattice vector:
G or g = 2n/a, n = 0, 1, 2, ….
K-space or reciprocal lattice:
First Brillouin Zone
G/2
G
0
2/a
4/a
6/a
5
Reciprocal Lattice in 1D
a
Real lattice
x
/a
-/a
Reciprocal lattice
-6/a -4/a -2/a
0
2/a 4/a
k
The 1st Brillouin zone:
Weigner-Seitz primitive cell in the reciprocal lattice
6
Reciprocal Lattice of a 2D Lattice
Kittel pg. 38
7
FCC in Real Space
•Kittel, P. 13
•Angle between a1, a2, a3: 60o
8
Reciprocal Lattice of the FCC Lattice
Kittel pg. 43
9
Special Points in the K-Space for the FCC
Kz
X
W
L
U

X
X
Ky
K
Kx
1st Brillouin Zone
10
BCC in Real Space
•Kittel, p. 13
•Rhombohedron primitive cell
0.53a
109o28’
•Primitive Translation Vectors:
11
1st Brillouin Zones of FCC, BCC, HCP
Real: FCC
Reciprocal: BCC
Real: FCC
Reciprocal: BCC
Real: HCP
12
Crystal Vibration
Interatomic Bonding
Spring constant (C)
Energy
Parabolic Potential of
Harmonic Oscillator
ro
Eb
Distance
s-1
s
s+1
x
Mass (M)
Transverse wave:
13
Crystal Vibration of a Monoatomic Linear Chain
Longitudinal wave of a 1-D Array of Spring Mass System
Spring constant, g
Mass, mM
Equilibrium
Position
a
Deformed
Position
xun-1
s-1
usx n
x n+1
us+1
us: displacement of the sth atom from its equilibrium position
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Solution of Lattice Dynamics
s-1
s
s+1
Same M
Wave solution:
u(x,t) ~ uexp(-iwt+iKx)
Time dep.:
w: frequency
K: wavelength
= uexp(-iwt)exp(isKa)exp(iKa)
cancel
Identity:
Trig:
15
w-K Relation: Dispersion Relation
K = 2/l
lmin  2a
Kmax = /a
-/a<K< /a
l: wavelength
2a
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Polarization and Velocity
w 2 M  C2  exp  iKa   exp iKa   2C 1  cos Ka 
1
2C
1  cos Ka 2
w
M
Group Velocity:
Frequency, w
dw
vg 
dK
Speed of Sound:
dw
vs  lim
K 0 dK
0
Wave vector, K
/a
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Two Atoms Per Unit Cell
M2
Lattice Constant, a
M1
yn-1
xn
2
M1
M2
d xn
dt
2
d 2 yn
dt
2
yn
 f  yn  yn1  2 xn 
xn+1
f: spring constant
 f xn1  xn  2 yn 
Solution:
Ka
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Acoustic and Optical Branches
Ka
1/µ = 1/M1 + 1/M2
What is the group velocity of the optical branch?
What if M1 = M2 ?
K
19
Polarization
TA & TO
Lattice Constant, a
LA & LO
yn-1
xn
xn+1
yn
Total 6 polarizations
Optical
Vibrational
Modes
Frequency, w
LO
TO
LA
0
TA
Wave vector, K /a
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Dispersion in Si
21
Dispersion in GaAs (3D)
LO
Frequency (1012 Hz)
8
LO
TO
TO
6
LA
LA
4
2
0
L
TA
0.4
0.2
0
(111) Direction 
TA
0.2
Ka/
0.4
0.6
(100) Direction
0.8
1.0
X
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Allowed Wavevectors (K)
A linear chain of N=10 atoms
with two ends jointed
a
Solution: us ~uK(0)exp(-iwt)sin(Kx), x =sa
B.C.:
us=0 = us=N=10
x
K=2n/(Na), n = 1, 2, …,N
Na = L
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
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Allowed Wave Vectors in 3D
2 4
N
K x , K y , K z  0;
;
;...; 
L
L
L
N3: # of atoms
Kz
Ky
Kx
2/L
24
Phonon
Energy
•The linear atom chain can only have
N discrete K  w is also discrete
Distance
• The energy of a lattice vibration mode at
frequency w was found to be
1

u   n  w
2

hw
• where ħw 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:
1
n 
 w 
  1
exp 
 k BT 
Equilibrium distribution
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Total Energy of Lattice Vibration
El  
p


1

  n w K , p  2 w K , p
K
p: polarization(LA,TA, LO, TO)
K: wave vector
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