handout17 - Handout 17 Lattice Waves (Phonons) in a 1D...

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1 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Handout 17 Lattice Waves (Phonons) in a 1D Crystal: Monoatomic Basis and Diatomic Basis In this lecture you will learn: • Equilibrium bond lengths • Atomic motion in lattices • Lattice waves (phonons) in a 1D crystal with a monoatomic basis • Lattice waves (phonons) in a 1D crystal with a diatomic basis • Dispersion of lattice waves • Acoustic and optical phonons ECE 407 – Spring 2009 – Farhan Rana – Cornell University The Hydrogen Molecule: Equilibrium Bond Length 0 x ( ) r V r d -d B E A E s E 1 : 2 A E : 1 B E : 1 σ ss V 2 The equilibrium distance between the two hydrogen atoms in a hydrogen molecule is set by the balance among several different competing factors: • The reduction in electronic energy due to co-valent bonding is 2 V ss . If the atoms are too far apart, V ss becomes to small • If the atoms are too close, the positively charged nuclei (protons) will repel each other and this leads to an increase in the system energy • Electron-electron repulsion also plays a role ss s B V E E = 1 ss s A V E E + = 1 () φ ss s s V x d r H x d r + ˆ ˆ ˆ 1 1 r r
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2 ECE 407 – Spring 2009 – Farhan Rana – Cornell University A Mass Attached to a Spring: A Simple Harmonic Oscillator x 0 x 0 u Equilibrium position Stretched position Potential Energy: () 2 2 1 u k u V PE = = Kinetic Energy: 2 2 = dt du M KE M M spring constant = k Dynamical Equation (Newton’s Second Law): u k du dV dt u d M = = 2 2 Restoring force varies linearly with the displacement “ u ” of the mass from its equilibrium position Solution: () ( ) ( ) t B t A t u o o ω sin cos + = where: M k o = PE varies quadratically with the displacement “ u ” of the mass from the equilibrium position ECE 407 – Spring 2009 – Farhan Rana – Cornell University A 1D Crystal: Potential Energy Consider a 1D lattice of N atoms: x a a ˆ 1 = r x 1 a n R n r r = • The potential energy of the entire crystal can be expressed in terms of the positions of the atoms. The potential energy will be minimum when all the atoms are at their equilibrium positions. • Let the displacement of the atom at the lattice site given by from its equilibrium position be • One can Taylor expand the potential energy around its minimum equilibrium value: ()()() () [] ()() ∑∑ + + = kj k j EQ k j j j EQ j EQ N R u R u R u R u V R u R u V V R u R u R u R u V r r r v v v v r r r 2 3 2 1 2 1 .. .......... , , n R r ( ) n R u r 0 Potential energy varies quadratically with the displacements of the atoms from their equilibrium positions Atoms can move only in the x-direction
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3 ECE 407 – Spring 2009 – Farhan Rana – Cornell University A 1D Crystal: Potential and Kinetic Energies A1D lattice of N atoms: x a a ˆ 1 = r x 1 a n R n r r = ()() () ( ) ∑∑ + = + = kj k j k j EQ k j EQ k j EQ t R u t R u R R K V t R u t R u R u R u V V V , , , 2 1 , , 2 1 2 r r r r r r r v Potential Energy: ( ) = j j dt t R du M KE 2 , 2 r Kinetic Energy:
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handout17 - Handout 17 Lattice Waves (Phonons) in a 1D...

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