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

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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 acoustic 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:
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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
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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
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This note was uploaded on 11/26/2010 for the course ECE 3060 at Cornell University (Engineering School).

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

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