Lecture 12: Waves in periodic structures

Transcription

1 Lecture : Waves in periodic structures Phonons: quantised lattice vibrations of a crystalline solid is: To approach the general topic of waves in periodic structures fro a specific standpoint: Lattice vibrations Modes of a -diensional, haronic chain Monatoic chain with nearest neighbour forces ore general interactions 3-D case Diatoic chain lattice with a basis origin of optical and acoustic odes Phonons as quantised vibrations May 04 Lecture

2 Waves in crystals Periodic structures Fro the waves and optics course in I Physics we are failiar with waves passing through a periodic structure (e.g. a diffraction grating) We can think of the process as an input wavevector, k i, being converted to any output wavevectors defined by k f (y) = k i (y) +ng (y) (in general, a vector equation); n is an integer. The lattice syetry (periodicity) induces a syetry (periodicity) in any wave propagating through the lattice. G is a reciprocal lattice vector, as we shall see. May 04 Lecture

3 Lattice vibrations -D D haronic chain The effects of diffraction in a periodic structure are siilar for all waves. These effects eerge naturally fro a discussion of any one syste. We will discuss lattice vibrations. Take identical asses,, connected by springs (spring constant, α) This is a odel liited to nearest-neighbour interactions. Equation of otion for the nth ato is u u n n {( un+ un ) ( un un ) } ( u + u u ) n+ n We have N coupled equations (for N atos) n May 04 Lecture 3

4 Noral odes Look for travelling wave solutions Wave of angular frequency, ω, and wavevector, q (q is the conventional choice for phonons) u n = u exp 0 { i( nqa ωt) } Substitute into equation of otion ω u ω ω ( q) = 0 exp { i( nqa ωt) } iqa iqa ( e + e ) u exp i( nqa ωt) ( cos qa) 4α sin qa = 0 4α sin { } qa Dispersion relation for phonon odes These are the noral odes for the syste of coupled atos. Note: the continuu odel for copressive waves ( Physics, Oscillations, waves and optics course) gave dispersionless solutions, which are the sae as the above in the liit of q 0, (i.e. the long wavelength liit). May 04 Lecture 4

5 Phonon dispersion Dispersion curves ω versus q gives the wave dispersion Key points The periodicity in q (reciprocal space) is a consequence of the periodicity of the lattice in real space (c.f. diffraction on slide ). Thus the phonon at soe wavevector, say, q is the sae as that at q +ng, for all integers n, where G=π/a (a reciprocal lattice vector). In the long wavelength liit (q 0) we expect the atoic character of the chain to be uniportant. May 04 Lecture 5

6 Liiting behaviour Long wavelength liit dispersion forula (p. 4) ω = sin leads to the continuu result (see I waves course) ω q 0 4α qa sin ( qa ) qa q αa a ω = q These are conventional sound-waves. Short wavelength liit ; q 0 toic character is evident as the wavelength approaches atoic diensions q π/a. λ=a is the shortest, possible wavelength Y ρ Continuu result Y - Young s odulus ρ - density Here we have a standing wave ω/ q=0 ωax = 4α May 04 Lecture 6

7 st rillouin zone Periodicity: ll the physically distinguishable odes lie within a single span of π/a. First rillouin zone (Z) choose the range of q to lie within q < π/a. This is the st Z. st st rillouin zone (shaded) Nuber of odes (ust equal the nuber of atos, N, in the chain) the allowed q values are discrete. Each ode (at particular q) is a quantised, siple-haronic oscillator E = n + ω ; n = exp( ω kt ) with a particulate character (bosons). Energy= ω, Mo.= q, Velocity=v g = ω/ q. May 04 Lecture 7

8 Diatoic lattice Technically a lattice with a basis proceeding as before. Equations of otion are: u u n n+ Trial solutions: u u n n = U exp substituting gives ( un+ + un un ) ( u + u u ) n+ n+ hoogeneous equations require deterinant to be zero giving a quadratic equation for ω. + 4 sin qa May 04 Lecture 8 n { i( nqa ω t) } { i( ( n + ) qa ω t) } + = U exp ( ω α ) U + ( α cos qa) U = 0 ( α cos qa) U + ( ω α ) U = 0 ω [( + ) {( ) } ] ± Two solutions for each q

9 coustic and Optic odes Solutions ω= (α/ ) q 0: Optic ode (higher frequency) ω = α ( + ) α µ coustic ode (lower frequency) ω ω α ( + ) αa + = [( + ) 4 q ( ) ( qa ) ] + Effective ass µ ω= (α/ ) Periodic: all distinguishable odes lie in in q <π/a May 04 Lecture 9

10 Origin of optic and acoustic branches Effect of periodicity The odes of the diatoic chain can be seen to arise fro those of a onatoic chain. Diagraatically: Monatoic chain, period a period in in q is is π/a for diatoic chain coustic Optical and and acoustic optical odes May 04 Lecture 0

11 Displaceent patterns Displaceents shown as transverse to ease visualisation. coustic odes: Neighbouring atos in phase Optical odes: Neighbouring atos out of phase Zone-boundary odes q=π/a; λ=π/q=4a (standing waves) Higher energy ode only light atos ove Lower energy ode only heavier atos ove May 04 Lecture

12 coustic odes: Diatoic chain: suary correspond to sound-waves in the longwavelength liit. Hence the nae. ω 0 as q 0 Optical odes: In the long-wavelength liit, optical odes interact strongly with electroagnetic radiation in polar crystals. Hence the nae. Strong optical absorption is observed (Photons annihilated, phonons created). ω finite value as q 0 Optical odes arise fro folding back the dispersion curve as the lattice periodicity is doubled (halved in q-space). Zone boundary: ll odes are standing waves at the zone boundary, ω/ q = 0: a necessary consequence of the lattice periodicity. In a diatoic chain, the frequency-gap between the acoustic and optical branches depends on the ass difference. In the liit of identical asses the gap tends to zero. May 04 Lecture

13 p. Corrections forula for k f corrected: k f (y) = k i (y) +ng (y) May 04 Lecture 3