Lattice Dynamics related to movement of atoms about their equilibrium positions determined by electronic structure Physical properties of solids Sound velocity Thermal properties: -specific heat -thermal expansion -thermal conductivity (for semiconductors) Hardness of perfect single crystals (without defects) Reminder to the physics of oscillations and waves: Harmonic oscillator in classical mechanics: Equation of motion: Hookes law m x Fspring Example: spring pendulum m x D x 0 or where x D ~ ~ x 0

m x( t ) Re( ~ x( t )) ~ x( t ) A ei t Solution with ~ x( t ) A cos(t ) where D m Dx X=A sin t 1 Epot D x 2 2 X x D m y( t ) A cos (t kx ) ~

~ y( t ) A ei( t kx ) Traveling plane waves: or Y 0 in particular y( t ) A cos (t kx ) 0 X X=0: y( t ) A cos t t=0: y( x ) A cos kx Particular state of oscillation Y=const travels according ~ ~ y( t ) A e solves wave equation i( t kx ) 1 2 y 2 y

2 v t 2 x 2 d t kx d const. 0 dt dt 2 x v v k 2 / Transverse wave Longitudinal wave Standing wave ~ ~ y1 A ei(kx t ) ~ ~ y 2 A ei(kx t ) ~

~ y s ~ y1 ~ y 2 A ei(kx t ) ei(kx t ) ~ ~ A eikx eit e it 2 A eikx cos t ys Re( y s ) 2 A cos kx cos t Large wavelength k 2 0 >10-8m 10-10m 8 Crystal can be viewed as a continuous medium: good for 10 m Speed of longitudinal wave: v (ignoring anisotropy of the crystal)

Bs where Bs: bulk modulus with 1 2 Bs determines elastic deformation energy density U Bs2 (click for details in thermodynamic context) dilation compressibility V V E.g.: Steel v Bs Bs=160 109N/m2 =7860kg/m3 v 160 109 N / m2 7860 kg / m3

1 Bs 4512 m s < interatomic spacing continuum approach fails In addition: vibrational modes quantized phonons Vibrational Modes of a Monatomic Lattice Linear chain: Remember: two coupled harmonic oscillators Symmetric mode Anti-symmetric mode Superposition of normal modes generalization Infinite linear chain How to derive the equation of motion in the harmonic approximation n-2 D un-2

n-1 a un-1 n n+1 un un+1 n+2 un+2 Fnr Dun un1 Fnl Dun un 1 un-2 un-1 un fixed un+1 un+2 ?

Total force driving atom n back to equilibrium Fn Dun un 1 Dun un1 n n Dun1 un 1 2un n Fn mu D n un1 un 1 2un u m equation of motion Solution of continuous wave equation u A ei(kx t ) i(kna t ) approach for linear chain un A e n 2 A ei(kna t ) , u 2 D ika e e ika 2 m

un1 A ei(kna t )eika 2 2 D 1 cos ka m ? , Let us try! un 1 A ei(kna t )e ika D 2 sin(ka / 2) m D 2 sin(ka / 2) m 2 D m Note: here pictures of transversal waves although calculation for the longitudinal case k Continuum limit of acoustic waves:

sin ka / 2 ka / 2 ... k 2 0 D ak m D v a k m k k h k 2 , here h=1 a 2 i (( k h

) na t ) a un A ei(k na (k )t ) A ei(k na t ) A e A ei(k na t )ei2 h n A ei(k na t ) (k ) (k ) k k h 2 a 1-dim. reciprocal lattice vector Gh Region un (k , (k)) un (k, (k )) k a a ei2 h n 1 is called first Brillouin zone

Brillouin zones We saw: all required information contained in a particular volume in reciprocal space a first Brillouin zone 1d: r n n a e x Gh r n 2m 2 a In general: first Brillouin zone 2 Gh h e a x where m=hn integer 1st Brillouin zone Wigner-Seitz cell of the reciprocal lattice Vibrational Spectrum for structures with 2 or more atoms/primitive basis Linear diatomic chain: 2n-2 D 2n-1 a

2n 2n+1 2n+2 2a u2n-2 u2n u2n-1 u2n+1 2n Equation of motion for atoms on even positions: u u2n+2 D u2n1 u2n 1 2u2n m 2n1 Equation of motion for atoms on even positions: u i( 2kna t ) Solution with: u2n A e and D u2n2 u2n 2u2n1 M

u2n1 B ei(( 2n1)ka t ) D A2 B(eika e ika ) 2 A m D B2 A(eika e ika ) 2B M D D A 2 2 2 B cos ka m m A 2 D B cos ka m D 2 2 m D D B 2 2 2 A cos ka M M Click on the picture to start the animation M->m D the movie

1 1 note wrong axis2 in 2D m m M 2 D 2 2 D 2 4 D cos 2 ka 2 m M Mm D2 D 2 D 2 D2 4 4 2 2 4 cos2 ka Mm M m Mm 2

D M D D D2 4 2 2 2 4 1 cos ka 0 Mm m M 2 2 sin2 ka k : 2a 2 1 1 4 sin2 ka 1 1 2 D D Mm m M

m M 1 1 1 1 D D m M m M 2 2 D m , 2 D M 2 Atomic Displacement B m k 0 A M Optic Mode

B k 0 1 A Atomic Displacement Click for animations Acoustic Mode Dispersion curves of 3D crystals 3D crystal: clear separation into longitudinal and transverse mode only possible in particular symmetry directions sound waves of elastic theory Every crystal has 3 acoustic branches 1 longitudinal 2 transverse acoustic Every additional atom of the primitive basis further 3 optical branches again 2 transvers 1 longitudinal p atoms/primitive unit cell ( primitive basis of p atoms): 3 acoustic branches + 3(p-1) optical branches = 3p branches 1LA +2TA

(p-1)LO +2(p-1)TO z y Intuitive picture: 1atom 3 translational degrees of freedom x 3+3=6 degrees of freedom=3 translations+2rotations +1vibraton # atoms in primitive basis # of primitive unit cells Solid: p N atoms 3p N vibrations no translations, no rotations Part of the phonon dispersion relation of diamond 2 fcc sublattices vibrate against one another However, identical atoms no dipole moment Longitudinal Optical Transversal Optical degenerated Longitudinal Acoustic

(0,0,0) 1 1 1 ( , , ) 4 4 4 diamond lattice: fcc lattice with basis Transversal Acoustic degenerated P=2 2x3=6 branches expected Phonon spectroscopy Inelastic interaction of light and particle waves with phonons Constrains: conservation law of energy momentum Condition for in elastic scattering in elastic sattering k k 0 q Ghkl phonon wave vector sc att ere d

incoming wave 0 (q) 0 for photon scattering ic e t t a l cal o r p i Rec vector wa ve 2k 2 2k 0 2 (q) 0 2Mn 2Mn quasimomentum k k 0 q Ghkl for neutrons k

0 (q) k 0 q Triple axis neutron spectrometer @ ILL in Grenoble, France Very expensive and involved experiments Table top alternatives ? Yes, infra-red absorption and inelastic light scattering (Raman and Brillouin) However only q 0 accessible see homework #8 Lonely scientist in the reactor hall