Lecture contents. NNSE 618 Lecture #11

Transcription

1 Lecture contents Couped osciators Dispersion reationship Acoustica and optica attice vibrations Acoustica and optica phonons Phonon statistics Acoustica phonon scattering NNSE 68 Lecture #

2 Few concepts rom Soid State Physics. Adiabatic approximation When vaence and core ectrons are separated, genera Schrödinger equation or a condensed medium without spin = L + e ( R, r) E( R, r) Mass o ions > (or most semiconductors > times greater than mass o eectrons Ion veocities > times sower Eectrons adjust instantaneousy to the positions o atoms ( R, r) ( r, R ) ( R) L ( R) E ( R) ( r, R) E ( r, R) e L e Separate ion and eectron motion (accuracy ~m/m) NNSE 68 Lecture #

3 NNSE 68 Lecture # 3 Few concepts rom Soid State Physics. Phonons anhar m m m m m L U u u C R R U M p,,, exp ) ( T n t i ir e u u, amitonian or attice motion (harmonic osciations) : Dispacements show up as pane waves with wea interaction via anharmonicity: Phonon dispersion reation in GaAs,, n E Energy in a mode: Equiibrium distribution (Bose Einstein):

4 Lattice vibrations 4 Lattice amitonian: L p M, m U R R m Binding energy vs. interatomic distance in a crysta Expanding binding energy around the equiibrium position R : Linear term is zero at minimum Negecting anharmonic terms: U( R) U( R ) C R with a orce constant C NNSE 68 Lecture #

5 Diatomic chain 5 Let s consider diatomic chain to demonstrate acoustica and optica dispersion branches Masses are connected by springs with equa spring constants, C, or simpicity Force = - C. R With u and v, the dispacements o respective atoms, we can write down cassica equation o motion (second Newton aw) The soution or dispacements in the chain wi be searched as traveing waves: d us Cvs v s M u dt d vs M Cu s us v dt u s ue v s ve isait isait s s From Singh, 3 NNSE 68 Lecture #

6 Soution or diatomic chain 6 NNSE 68 Lecture #

7 Dispersion reations or diatomic chain 7 Soutions or sma : Soutions or the edge o Briouin zone =p/a : From Singh, 3 NNSE 68 Lecture #

8 Acoustica and optica waves 8 For acoustica branch in ong waveength imit (at sma ): u v or u s v s Sound veocity: v s d d a C M av For optica branch at = : (Two atoms vibrate against their center o masses) u M M v NNSE 68 Lecture #

9 Dispersion curves in semiconductor crystas 9 For each wavevector there are ongitudina mode and transverse modes The requencies are determined by orce constants Usuay ongitudina mode (LA) is stier Energy scaes (or simiar crystas) as M -/ Atomic vibrations are in Tz range Exampe: she mode Si GaAs InAs From Singh, 3 NNSE 68 Lecture #

10 Anisotropy o phonon dispersion curves Experimenta (points) and cacuated phonon dispersion curves or Si From Yu, Cordona, NNSE 68 Lecture #

11 Quantum harmonic osciator: amitonian Quantum harmonic osciator p M Cx M C M Soution gives resonance requency (as in cassica mechanics) C M E And quantum osciation spectrum: (n may be considered as number o quasipartices ) E n n x NNSE 68 Lecture #

12 Quantization o attice vibrations: phonons For a singe osciator the requency is ixed, but when many osciators interact we have a number o modes (norma modes) Each mode is occupied by n phonons E n For a D chain states are determined as: Occupancy o modes is given by Bose-statistics: n( ) n p ; or n,,... N Na exp T Bose-Einstein distribution unction NNSE 68 Lecture #

13 Optica phonons: Raman scattering 3 Ineastic ight scattering = Raman scattering gives inormation on opticay active vibrations in a materia Wavevector o photons is SMALL Stoes (creation o vibration) and anti-stoes (emission o vibration) Symmetry and seection rues: Raman scattering intensity depends on geometry and poarization GaAs From Yu and Cordona, 3 NNSE 68 Lecture #

14 NNSE 68 Lecture # 4 Lattice scattering rate cacuation 3 3,, p d W W t co Goa: cacuation o the scattering integra or reaxation time: Step. Determine scattering potentia Step. Cacuate matrix eements rom to Step 3. Cacuate transition rate rom to using goden Fermi rue Step 4. Cacuate state reaxation time Step 5. Average reaxation time t i iqr e r d V 3 * p ) ( ) ( ), ( E E W ), ( ), ( ) ( W W t co ()