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

lishi@mail.utexas.edu

2

Outline

Reciprocal Lattice

• Crystal Vibration

• Phonon

•Reading: 1.3 in Tien et al

•References: Ch3, Ch4 in Kittel

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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

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Reciprocal Lattice Points

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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, ….

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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

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Kittel pg. 38

Reciprocal Lattice of a 2D Lattice

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FCC in Real Space

•Angle between a1, a2, a3: 60o

•Kittel, P. 13

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Kittel pg. 43

Reciprocal Lattice of the FCC Lattice

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X

L

K

X

X

U

W

Kz

Ky

Kx

Special Points in the K-Space for the FCC

1st Brillouin Zone

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BCC in Real Space

•Primitive Translation Vectors:

•Rhombohedron primitive cell

0.53a

109o28’

•Kittel, p. 13

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Real: FCC Reciprocal: BCC

1st Brillouin Zones of FCC, BCC, HCP

Real: HCP

Real: FCC Reciprocal: BCC

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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

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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

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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

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-K Relation: Dispersion Relation

K = 2/minaKmax = /a-/a<K< /a

2a: wavelength

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Polarization and VelocityPolarization and Velocity

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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

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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

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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

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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

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Dispersion in Si

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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)

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Dispersion in GaAs (3D)Dispersion in GaAs (3D)

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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

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Kx

Ky

Kz

2/L

Allowed Wave Vectors in 3D

L

N

LLKKK zyx

;...;

4;

2;0,,

N3: # of atoms

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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

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Total Energy of Lattice VibrationTotal Energy of Lattice Vibration

pKpKp

l nE ,, 2

1

K

p: polarization(LA,TA, LO, TO)K: wave vector