# handout18 - Handout 18 Lattice Waves (Phonons) in 2D...

This preview shows pages 1–4. Sign up to view the full content.

1 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Handout 18 Lattice Waves (Phonons) in 2D Crystals: Monoatomic Basis and Diatomic Basis In this lecture you will learn: • Lattice waves (phonons) in a 2D crystal with a monoatomic basis • Lattice waves (phonons) in a 2D crystal with a diatomic basis • Dispersion of lattice waves • LA and TA acoustic phonons • LO and TO optical phonons ECE 407 – Spring 2009 – Farhan Rana – Cornell University 1 a r x 2 1 a m a n R nm r r r + = Phonons in a 2D Crystal with a Monoatomic Basis y 2 a r y a n x a n y a n x a n ˆ ˆ ˆ ˆ 4 3 2 1 = = = = r r r r General lattice vector: Nearest-neighbor vectors: Atomic displacement vectors: () ( ) = t R u t R u t R u nm y nm x nm , , , r r r r Atoms, can move in 2D therefore atomic displacements are given by a vector: y a x a p y a x a p y a x a p y a x a p ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 4 3 2 1 = = + = + = r r r r Next nearest-neighbor vectors:

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
2 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Vector Dynamical Equations m m m R R m r r r r r = = ˆ 1 2 ( ) t R u , 1 r r ( ) t R u , 2 r r () ( ) [] m m t R u t m R u dt t R u d M ˆ ˆ . , , , 1 1 2 1 2 r r r r r r r + = α Vector dynamical equation: If the nearest-neighbor vectors are known then the dynamical equations can be written easily. Component dynamical equation: To find the equation for the x-component of the atomic displacement, take the dot-product of the above equation on both sides with x ˆ ( ) x m m t R u t m R u dt t R u d M x ˆ . ˆ ˆ . , , , 1 1 2 1 2 r r r r r r + = ECE 407 – Spring 2009 – Farhan Rana – Cornell University Vector Dynamical Equations for a 2D Crystal 1 a r x y 2 a r 2 1 a m a n R nm r r r + = y a n x a n y a n x a n ˆ ˆ ˆ ˆ 4 3 2 1 = = = = r r r r General lattice vector: Nearest-neighbor vectors: y a x a p y a x a p y a x a p y a x a p ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 4 3 2 1 = = + = + = r r r r Next nearest-neighbor vectors: + + + = = = 4 , 3 , 2 , 1 2 4 , 3 , 2 , 1 1 2 2 ˆ ˆ . , , ˆ ˆ . , , , j j j nm j nm j j j nm j nm nm p p t R u t p R u n n t R u t n R u dt t R u d M r r r r r r r r r r r r summation over 4 nn summation over 4 next nn 1 2
3 ECE 407 – Spring 2009 – Farhan Rana – Cornell University ( ) () ( ) [] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , , 2 , , 2 , , 2 , , 2 , , 2 , , 2 , , 2 , , 2 , , , , , 4 2 4 2 3 2 3 2 2 2 2 2 1 2 1 2 3 1 1 1 2 2 t p R u t R u t p R u t R u t p R u t R u t p R u t R u t p R u t R u t p R u t R u t p R u t R u t p R u t R u t n R u t R u t n R u t R u dt t R u d M nm y nm y nm x nm x nm y nm y nm x nm x nm y nm y nm x nm x nm y nm y nm x nm x nm x nm x nm x nm x nm x r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r + + + + + + + + + + + + = α Dynamical Equations + + + = = = 4 , 3 , 2 , 1 2 4 , 3 , 2 , 1 1 2 2 ˆ ˆ . , , ˆ ˆ . , , , j j j nm j nm j j j nm j nm nm p p t R u t p R u n n t R u t n R u dt t R u d M r r r r r r r r r r r r If we take the dot-product of the above equation with we get: x ˆ ECE 407 – Spring 2009 – Farhan Rana – Cornell University ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , , 2 , , 2 , , 2 , , 2 , , 2 , , 2 , , 2 , , 2 , , , , , 4 2 4 2 3 2 3 2 2 2 2 2 1 2 1 2 4 1 2 1 2 2 t p R u t R u t p R u t R u t

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 04/04/2009 for the course ECE 4070 taught by Professor Rana during the Spring '08 term at Cornell University (Engineering School).

### Page1 / 12

handout18 - Handout 18 Lattice Waves (Phonons) in 2D...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online