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T.S. Fisher, Purdue University 1
Lattice Dynamics
Timothy S. FisherPurdue University
School of Mechanical Engineering, andBirck Nanotechnology Center
ME 595MMay 2007
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Phonon Heat Conduction• Phonons are quantized
lattice vibrations• Govern thermal properties
in electrical insulators and semiconductors
• Can be modeled to first order with spring-mass dynamics
• Wave solutions♦ wave vector K=2π/λ♦ phonon energy=ħω♦ dispersion relations gives
ω = fn(K)
ω
K
optical branch
acoustic branch
sound speed(group velocity)
springconstant
g
atom,mass
m
a
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T.S. Fisher, Purdue University 3
Heat Conduction Through Thin Films• Experimental results
for 3-micron silicon films
• Non-equilibrium scattering models work fairly well
• Crystalline structure often has largerimpact than film thickness
3 micron
Asheghi et al., 1999
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T.S. Fisher, Purdue University 4
Heat Conduction Through Multiple Thin Films• Fine-pitch 5 nm
superlattices
• Cross-thickness conductivity measurement
• Measured values are remarkably close to bulk alloy values (nearly within measurement error)
• Expected large reduction in conductivity not observed
5nm
Cahill et al., 2003
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T.S. Fisher, Purdue University 5
Lattice Vibrations
• Consider two neighboring atoms that share a chemical bond
• The bond is not rigid, but rather like a spring with an energy relationship such as…
r0
U
r
r0
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T.S. Fisher, Purdue University 6
Lattice Vibrations, cont’d
• Near the minimum, the energy is well approximated by a parabola
♦ u = r – r0 and g = spring constant• Now consider a one-dimensional chain of
molecules
212
U gu=
springconstant
g
atom,mass
m
a
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T.S. Fisher, Purdue University 7
Lattice Energy and Motion
• Harmonic potential energy is the sum of potential energies over the lattice
• Equation of motion of atom at location u(na)
• Simplified notation
{ }21 [ ] [( 1) ]2
harm
n
U g u na u n a= − +∑
[ ] [ ]{ }2
2
( ) 2 ( ) ( 1) ( 1)( )
harmd u na UF m g u na u n a u n adt u na
∂= = − = − − − − +
∂
{ }2
1 12 2nn n n
d um g u u udt − += − − −
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T.S. Fisher, Purdue University 8
Lattice Motion, cont’d
• Seek solutions of the form
• Boundary conditions♦ Born-von Karman: assume that the ends of the chain are
connected• uN+1 = u1
• u0 = uN
( ){ }( ) ~ expnu t i Kna t− ω
12
N
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T.S. Fisher, Purdue University 9
Lattice Motion, cont’d
• Then the boundary conditions become
• Let λ be the vibration wavelength, λ = aN/n
• Minimum wavelength, λmin = 2a = 2(lattice spacing)
( ){ }[ ]{ }[ ]
1
1
~ exp 1
~ exp
1 exp 2 , where is an integer
Nu i K N a t
u i Ka t
iKNa KNa nn
+ + − ω⎡ ⎤⎣ ⎦
− ω
→ = → = π
2 2nKaNπ π
= =λ
K = wave vector
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T.S. Fisher, Purdue University 10
Solution to the Equations of Motion
• Substitute exponential solution into equation of motion
• Solve for ω
• This is the dispersion relation for acoustic phonons♦ relates phonon frequency (energy) to wave vector
(wavelength)
( ) ( )
( ) ( )
2 2
2 1 cos
i Kna t i Kna tiKa iKa
i Kna t
m e g e e e
g Ka e
−ω −ω−
−ω
⎡ ⎤− ω = − − −⎣ ⎦
= − −
12
2 (1 cos )( ) 2 sin( )g Ka gK Kam m−
ω = =
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Dispersion Curve• Changing K by 2π/a leaves u unaffected
♦ Only N values of K are unique♦ We take them to lie in -π/a < K < π/a
ω(K)
K
π/a- π/a
2(g/m)1/2
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Wave Velocities
• Phase velocity: c = ω/K• Group velocity: vg = ∂ω/∂K = a(g/m)1/2cos(Ka/2)• For small K:
• Thus, for small K (large λ), group velocity equals phase velocity (and speed of sound)
• We call these acoustic vibration modes
0
0
lim
lim
K
gK
ga Km
gv a cm K
→
→
ω=
ω→ = = =
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Notes on Lattice Vibrations
• For K = ±π/a, the group velocity is zero♦ why?
♦ neighbors are 180 deg out of phase
• The region -π/a < K < π/a is the first Brillouin zone of the 1D lattice
• We must extrapolate these results to three dimensions for bulk crystals
{ } { }1 exp exp cos sin 1n
n
u iKa i iu+ = = π = π + π = −
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T.S. Fisher, Purdue University 14
Density of Phonon States• Consider a 1D chain of total length L carrying M+1 particles
(atoms) at a separation a♦ Fix the position of atoms 0 and M♦ Each normal vibrational mode of polarization p takes the form of a
standing wave
♦ Only certain wavelengths (wavevectors) are allowedλmax=2L (Kmin=π/L), λmin=2a (Kmax=π/a=Mπ/L)
♦ In general, the allowed values of K are
0 1 MM-1
L
a
~ sin( ) exp( )n Kpu nKa i t− ω
2 3 ( 1), , ,..., MKL L L Lπ π π − π
=
Note: K=Mπ/L is not included because it implies no atomic motion, i.e., sin(nMπa/L)=sin(nπ)=0.
See Kittel, Ch5, Intro to Solid-StatePhysics, Wiley 1996
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T.S. Fisher, Purdue University 15
Density of States, cont’d
• Thus, we have M-1 allowed, independent values of K♦ This is the same number of particles allowed to move♦ In K-space, we thus have M-1 allowable wavevectors♦ Each wavevector describes a single mode, and one mode exists in
each distance π/L of K-space♦ Thus, dK/dN = π/L, where N is the number of modes
π/(M-1)a π/aπ/L
π/aDiscrete K-space representation
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T.S. Fisher, Purdue University 16
Density of States, cont’d
• The phonon density of states gives the number of modes per unit frequency per unit volume of real space
♦ The last denominator is simply the group velocity, derived from the dispersion relation
11 1 1 1( )
/dN dN dKDd L dK d d dKLα=
ω = = =ω ω π ω
11 1( ) cos ( )( ) 2
Note singularity for /g
gD a K av m
K a
−⎡ ⎤⎛ ⎞ω = = π ω⎢ ⎥⎜ ⎟π ω ⎝ ⎠⎣ ⎦
= π
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T.S. Fisher, Purdue University 17
Periodic Boundary Conditions
• For more generality, apply periodic boundary conditions to the chain and find
♦ Still gives same number of modes (one per particle that is allowed to move) as previous case, but now the allowed wavevectors are separated by ΔK = 2π/L
♦ Useful in the study of higher-dimension systems (2D and 3D)
2 40, , ,..., MKL L Lπ π π
= ± ±
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T.S. Fisher, Purdue University 18
2D Density of States• Each allowable wavevector
(mode) occupies a region of area (2π/L)2
• Thus, within the circle of radius K, approximately N=πK2/ (2π/L)2 allowed wavevectors exist
• Density of states
K-space
π/a
2π/L
K
21 1 ( ) 1( )
2 ( )g
dN dN dK KDd V dK d vLα=
ωω = = =
ω ω π ω
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T.S. Fisher, Purdue University 19
3D Density of States• Using periodic boundary conditions in 3D, there is
one allowed value of K per (2π/L)3 volume of K-space
• The total number of modes with wavevectors of magnitude less than a given K is thus
• The 3D density of states becomes
3 33
24
2 3 6L VKN K⎛ ⎞ ⎛ ⎞= π =⎜ ⎟ ⎜ ⎟π⎝ ⎠ ⎝ ⎠ π
2
3 21 1 ( ) 1( )
( )2 g
dN dN dK KDd V dK d vLα=
ωω = = =
ω ω ωπ
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Glossary for Lattice Descriptions and Lattice Dynamics
T.S. FisherPurdue University
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Lattice Structure
• a = lattice constant• Common crystal structures
• Body centered (bcc)• Face centered (fcc)• Diamond (dia)
• Bravais lattice: an infinite array of discrete points whose position vectors can be expressed as:
a
a
a
1 1 2 2 3 3
i
where are PRIMITIVE VECTORSand n are integers
i
R n a n a n a
a
= + +
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Primitive Unit Cells
• A primitive unit cell is a volume of real space that , when translated through all R, just fills all space without overlaps or voids and contains one lattice point
• 2D examples
Jarrell (2) Fig2
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T.S. Fisher, Purdue University 23
Lattice with a Basis• Often, we need to
describe a crystalline material’s structure by placing a primary atom at each lattice point and one or more basis atoms relative to it♦ For compound
materials (eg CuO2), this is an obvious requirement
♦ Also applies to some monoatomic crystals (eg Si)
Jarrell (2) Fig3
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T.S. Fisher, Purdue University 24
Reciprocal Lattice
• More convenient to express spatial dependencies in terms of wave vectors, instead of wavelengths
• Reciprocal lattice (RL) is like the inverse of a Bravais lattice
• G is the vector that satisfies
♦ mi are integers, and
1 1 2 2 3 3G m b m b m b= + +
( ) ( ) ( )2 3 3 1 1 2
1 2 31 2 3 1 2 3 1 2 3
2 ; 2 ; 2a a a a a ab b ba a a a a a a a a
× × ×= π = π = π
× × ×i i i
2 integer• = π×G R
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T.S. Fisher, Purdue University 25
Primitive Cells & Miller Indices
• Primitive Cell: A region of space that is closer to one point than any others
• 1st Brillouin Zone: The primitive cell of the reciprocal lattice
• For a given lattice plane, Miller indices are coordinates of the shortest reciprocal lattice vector normal to the plane♦ A plane with Miller indices (hkl) is perpendicular to the
vector
1 2 3G hb kb lb= + +
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T.S. Fisher, Purdue University 26
Dispersion Curves
Animated chains from http://www.chembio.uoguelph.ca/educmat/chm729/Phonons/movies.htm
• Phase velocity: c = ω/q• Group velocity: vg = ∂ω/∂q• Acoustic phonons: determine the speed of sound in a solid
and are characterized by ω ~ q for q 0• Optical phonons: occur for lattices with more than one atom
per unit cell and are characterized by flat dispersion curves with relatively high frequencies
• Branch: acoustic or optical• Polarization: defines the direction of oscillation of
neighboring atoms of a given dispersion curve♦ Longitudinal: atomic displacements aligned with wave direction
♦ Transverse: atomic displacements perpendicular to wave direction
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Phonons• Phonon: a quantized lattice vibration (i.e., one that
can take on only a discrete energy, ħω)• Normal mode: a lattice wave that is characterized
by a branch, polarization, wave vector, and frequency
• Occupation number (or excitation number) nKp: the number of phonons of a given wave vector (K) and branch/polarization p♦ Note that nKp depends on frequency, which in turn
depends on wave vector and branch/polarization as defined by the dispersion curve
♦ Note also that, in this context, the term p implies both branch and polarization
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T.S. Fisher, Purdue University 28
Overview of Phonon Simulation Tools
• Boltzmann Transport Equation (BTE)♦ Requires boundary scattering models♦ Requires detailed understanding of phonon scattering and
dispersion for rigorous inclusion of phonon physics• Molecular Dynamics (MD)
♦ Computationally expensive♦ Not strictly applicable at low temperatures♦ Handling of boundaries requires great care for links to larger
scales and simulation of functional transport processes• Atomistic Green’s Function (AGF)
♦ Efficient handling of boundary and interface scattering♦ Straightforward links to larger scales♦ Inclusion of anharmonic effects is difficult