Crystal Vibration and Phonon-Solid State Physics-Handouts, Lecture notes of Solid State Physics

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ME 381R Fall 2003 Micro-Nano Scale Thermal-Fluid Science and Technology

Lecture 4:

Crystal Vibration and Phonon

2

Outline

Reciprocal Lattice

• Crystal Vibration
• Phonon
• Reading: 1.3 in Tien et al
• References: Ch3, Ch4 in Kittel

4

Reciprocal Lattice

Points

5

Reciprocal lattice & K-Space

a

   

 (^) x a   in (^) x a   i (^) n    x

x in x a

n

n

n

n

    

  

   







exp 2 exp 2

exp 2

0^2^  /a^4^  /a^6^  /a

G

G/

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 = 2 n / a , n = 0, 1, 2, ….

7

Kittel pg. 38

Reciprocal Lattice of a 2D Lattice

8

FCC in Real Space

• Angle between a 1 , a 2 , a 3 : 60o
  • Kittel, P. 13

10

X



L

K

X X

U

W

Kz

Ky

Kx

Special Points in the K-Space for the FCC

1 st^ Brillouin Zone

11

BCC in Real Space

• Primitive Translation Vectors:
• Rhombohedron primitive cell

0.53a

109 o 28 ’

• Kittel, p. 13

13

Crystal Vibration

s-1 s s+

Mass (M)

Spring constant (C)

x

Transverse wave:

Energy

ro Distance

Parabolic Potential of Harmonic Oscillator

Eb

Interatomic Bonding

14

Crystal Vibration of a Monoatomic Linear Chain

a

Spring constant, g Mass, m

xn-1 xn xn+

Equilibrium Position

Deformed Position

Longitudinal wave of a 1-D Array of Spring Mass System

u s: displacement of the sth^ atom from its equilibrium position

us-1 (^) us (^) u s+

M

16

w - K Relation: Dispersion Relation

K = 2  / l

l min  2 a

Kmax =  /a

-  /a<K<  /a

2a

l: wavelength

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17

Polarization and Velocity

      

2  1 cos  12

(^22) exp exp 2 1 cos

Ka M

C

M C iKa iKa C Ka

 

     

w

w

Frequency,

w

(^0) Wave vector, K /a

Group Velocity:

dK

d vg

w 

Speed of Sound:

dK

d v s K

w 0

lim 

19

1/μ = 1/M 1 + 1/M 2

What is the group velocity of the optical branch? What if M 1 = M 2?

Acoustic and Optical Branches

K

Ka

20

Lattice Constant, a

yn-1 xn yn xn+

Polarization

Frequency,

w

(^0) Wave vector, K /a

LA TA

LO TO

Optical Vibrational Modes

LA & LO

TA & TO

Total 6 polarizations

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