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nanoHUB.org online simulations and more T.S. Fisher, Purdue University 1 Lattice Dynamics Timothy S. Fisher Purdue University School of Mechanical Engineering, and Birck Nanotechnology Center [email protected] ME 595M May 2007

Lattice Dynamics - Purdue Engineeringtsfisher/ME595M/Lattice...Lattice Dynamics Timothy S. Fisher Purdue University School of Mechanical Engineering, and Birck Nanotechnology Center

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Page 1: Lattice Dynamics - Purdue Engineeringtsfisher/ME595M/Lattice...Lattice Dynamics Timothy S. Fisher Purdue University School of Mechanical Engineering, and Birck Nanotechnology Center

nanoHUB.org online simulations and more

T.S. Fisher, Purdue University 1

Lattice Dynamics

Timothy S. FisherPurdue University

School of Mechanical Engineering, andBirck Nanotechnology Center

[email protected]

ME 595MMay 2007

Page 2: Lattice Dynamics - Purdue Engineeringtsfisher/ME595M/Lattice...Lattice Dynamics Timothy S. Fisher Purdue University School of Mechanical Engineering, and Birck Nanotechnology Center

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T.S. Fisher, Purdue University 2

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

Page 3: Lattice Dynamics - Purdue Engineeringtsfisher/ME595M/Lattice...Lattice Dynamics Timothy S. Fisher Purdue University School of Mechanical Engineering, and Birck Nanotechnology Center

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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

Page 4: Lattice Dynamics - Purdue Engineeringtsfisher/ME595M/Lattice...Lattice Dynamics Timothy S. Fisher Purdue University School of Mechanical Engineering, and Birck Nanotechnology Center

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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

Page 5: Lattice Dynamics - Purdue Engineeringtsfisher/ME595M/Lattice...Lattice Dynamics Timothy S. Fisher Purdue University School of Mechanical Engineering, and Birck Nanotechnology Center

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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

Page 6: Lattice Dynamics - Purdue Engineeringtsfisher/ME595M/Lattice...Lattice Dynamics Timothy S. Fisher Purdue University School of Mechanical Engineering, and Birck Nanotechnology Center

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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

Page 7: Lattice Dynamics - Purdue Engineeringtsfisher/ME595M/Lattice...Lattice Dynamics Timothy S. Fisher Purdue University School of Mechanical Engineering, and Birck Nanotechnology Center

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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 − += − − −

Page 8: Lattice Dynamics - Purdue Engineeringtsfisher/ME595M/Lattice...Lattice Dynamics Timothy S. Fisher Purdue University School of Mechanical Engineering, and Birck Nanotechnology Center

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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

Page 9: Lattice Dynamics - Purdue Engineeringtsfisher/ME595M/Lattice...Lattice Dynamics Timothy S. Fisher Purdue University School of Mechanical Engineering, and Birck Nanotechnology Center

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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

Page 10: Lattice Dynamics - Purdue Engineeringtsfisher/ME595M/Lattice...Lattice Dynamics Timothy S. Fisher Purdue University School of Mechanical Engineering, and Birck Nanotechnology Center

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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 = =

Page 11: Lattice Dynamics - Purdue Engineeringtsfisher/ME595M/Lattice...Lattice Dynamics Timothy S. Fisher Purdue University School of Mechanical Engineering, and Birck Nanotechnology Center

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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

Page 12: Lattice Dynamics - Purdue Engineeringtsfisher/ME595M/Lattice...Lattice Dynamics Timothy S. Fisher Purdue University School of Mechanical Engineering, and Birck Nanotechnology Center

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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→ = = =

Page 13: Lattice Dynamics - Purdue Engineeringtsfisher/ME595M/Lattice...Lattice Dynamics Timothy S. Fisher Purdue University School of Mechanical Engineering, and Birck Nanotechnology Center

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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 = −

Page 14: Lattice Dynamics - Purdue Engineeringtsfisher/ME595M/Lattice...Lattice Dynamics Timothy S. Fisher Purdue University School of Mechanical Engineering, and Birck Nanotechnology Center

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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

Page 15: Lattice Dynamics - Purdue Engineeringtsfisher/ME595M/Lattice...Lattice Dynamics Timothy S. Fisher Purdue University School of Mechanical Engineering, and Birck Nanotechnology Center

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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

Page 16: Lattice Dynamics - Purdue Engineeringtsfisher/ME595M/Lattice...Lattice Dynamics Timothy S. Fisher Purdue University School of Mechanical Engineering, and Birck Nanotechnology Center

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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

Page 17: Lattice Dynamics - Purdue Engineeringtsfisher/ME595M/Lattice...Lattice Dynamics Timothy S. Fisher Purdue University School of Mechanical Engineering, and Birck Nanotechnology Center

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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

= ± ±

Page 18: Lattice Dynamics - Purdue Engineeringtsfisher/ME595M/Lattice...Lattice Dynamics Timothy S. Fisher Purdue University School of Mechanical Engineering, and Birck Nanotechnology Center

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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

Page 19: Lattice Dynamics - Purdue Engineeringtsfisher/ME595M/Lattice...Lattice Dynamics Timothy S. Fisher Purdue University School of Mechanical Engineering, and Birck Nanotechnology Center

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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

Page 20: Lattice Dynamics - Purdue Engineeringtsfisher/ME595M/Lattice...Lattice Dynamics Timothy S. Fisher Purdue University School of Mechanical Engineering, and Birck Nanotechnology Center

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Glossary for Lattice Descriptions and Lattice Dynamics

T.S. FisherPurdue University

Page 21: Lattice Dynamics - Purdue Engineeringtsfisher/ME595M/Lattice...Lattice Dynamics Timothy S. Fisher Purdue University School of Mechanical Engineering, and Birck Nanotechnology Center

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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

= + +

Page 22: Lattice Dynamics - Purdue Engineeringtsfisher/ME595M/Lattice...Lattice Dynamics Timothy S. Fisher Purdue University School of Mechanical Engineering, and Birck Nanotechnology Center

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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

Page 23: Lattice Dynamics - Purdue Engineeringtsfisher/ME595M/Lattice...Lattice Dynamics Timothy S. Fisher Purdue University School of Mechanical Engineering, and Birck Nanotechnology Center

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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

Page 24: Lattice Dynamics - Purdue Engineeringtsfisher/ME595M/Lattice...Lattice Dynamics Timothy S. Fisher Purdue University School of Mechanical Engineering, and Birck Nanotechnology Center

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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

Page 25: Lattice Dynamics - Purdue Engineeringtsfisher/ME595M/Lattice...Lattice Dynamics Timothy S. Fisher Purdue University School of Mechanical Engineering, and Birck Nanotechnology Center

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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= + +

Page 26: Lattice Dynamics - Purdue Engineeringtsfisher/ME595M/Lattice...Lattice Dynamics Timothy S. Fisher Purdue University School of Mechanical Engineering, and Birck Nanotechnology Center

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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

Page 27: Lattice Dynamics - Purdue Engineeringtsfisher/ME595M/Lattice...Lattice Dynamics Timothy S. Fisher Purdue University School of Mechanical Engineering, and Birck Nanotechnology Center

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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

Page 28: Lattice Dynamics - Purdue Engineeringtsfisher/ME595M/Lattice...Lattice Dynamics Timothy S. Fisher Purdue University School of Mechanical Engineering, and Birck Nanotechnology Center

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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