Lecture 13 Phonons: thermal properties

Transcription

1 Leture 1 Phonon: thermal propertie Lattie ontribution to the thermal propertie of olid, in -D Aim: Thermal propertie of a rytalline olid: Heat apaity: Debye treatment T law for low temperature heat apaity Thermal ondutivity: Phonon attering Mean-free path Debye, T Law V U T V T Deember 06 Leture 1 1

2 Lattie heat-apaity Heat apaity Follow from differentiating the internal energy (a uual). Internal energy U ω 0 exp 1 ( ω kt ) ( ω) Denity of mode g(ω). Eintein Approximation: all mode have the ame frequeny, ω E. (See leture 5) Debye approximation: In the low temperature limit aouti mode, with mall q, dominate. So aume ω v q. Exat: alulate g(ω), numerially, from the phonon diperion urve Eintein approximation gave the orret hightemperature behaviour (C NkT) and gave C0 a T0 though the exat temperature dependene wa inaurate (the reaon, we an now undertand from the phonon diperion urve) Deember 06 Leture 1 g No. of phonon in at ω Energy per phonon (Plank formula, Leture 5)

3 Debye Model Denity of tate Aume ω v q. i.e. diperionle wave Reult i therefore imilar to that for photon in a -D avity (blak body radiation), exept for a g ( q)d q g( Vol, V a.. numerial ontant. g( q)d q ω) g( ω) a π a π ω d q aouti mode for eah q where ω v q ha been ubtituted in RHS. Energy beome: Upper limit on integral guarantee the orret, total number of of mode (N). It It i i known a the Debye frequeny. U ω D V π v Vω π v 0 ω 0 D Formula give the full T dependene. We are intereted in the behaviour at low T. Deember 06 Leture 1. v ω exp ω exp 1 q ( ω kt ) 1 ( ω kt ) 1 1

4 heat apaity at low-temperture Limiting behaviour a T0. At low temperature the higher frequeny mode are not exited. Thu ontribution to the integral for large ω (~ω D ) an be ignored and ω D replaed by. V 1 U ω π v 0 exp ω kt 1 U Vπ k 10 4 ( v ) V π v kt T 4 4 ( ) ( exp( x) 1) Differentiating give the heat apaity a V U T 0 x give the orret, oberved dependene at low temperature. Reall the Eintein model gave an exponential dependene at low T. 1 Integral π 4 /15 T Debye, T Law V Debye, T Law Deember 06 Leture 1 4

5 Meaured denity of tate Example: Aluminium (how ommon feature) Meaured denity of tate ompared with Debye approximation. Both meaured and Debye denity of tate are imilar at low ω, a expeted (ω q). Debye frequeny hoen to give ame total number of mode (i.e. equal area under both urve) Larget deviation where phonon mode approah zone boundary. Meaured urve i omplex beaue the -D zone ha a relatively ompliated hape, and the tranvere and longitudinal mode have different diperion (a we have een earlier) Deember 06 Leture 1 5

6 Thermal ondutivity Phonon and thermal ondutivity Phonon have energy and momentum and, therefore, an ondut heat. Kineti theory give the thermal ondutivity l z z - l o θ θ heat apaity of of a phonon Exe temperature of phonon roing plane T dt dz z dt dz l oθ Exe energy of eah phonon ph T ph dt dz l oθ Deember 06 Leture 1 6

7 ondutivity... Number denity of phonon, n number with peed to to +d fration with angle θ to to θ+dθ n f ( )d Heat flux aro plane H π dt π H phnl in o d dz θ θ θ dt π H + phnl o d( o ) dz θ θ 0 1 dt dt H + phnl κ dz dz κ phnl Thermal ondutivity κ 1 C Definition of of thermal ondutivity Deember 06 Leture 1 7 l Mean free path Average peed Heat apaity per unit vol θ π in θ dθ [ nf ( ) dinθ dθ ][ oθ ][ l oθ dt dz] f ph ( ) dθ π inθ d θ 4π inθ dθ peed normal to to plane net heat per phonon d

8 Temperature dependene of thermal ondutivity Mean free path limited by attering proee With many attering proee: 1 l 1 l1 + 1 l + Thu, the hortet mean free path dominate Geometri attering: Sample boundarie (only ignifiant for puret ample at low temperature). Impuritie: attering rate ~ independent of T. Phonon-phonon attering: Phonon, being normal mode, hould not affet eah other. However, in an anharmoni lattie, they an atter. Free path ~ 1/T. Inulator (no ontribution from eletron). In pure rytalline form the ondutivity an be very high (larger than metal) N.B. Non-rytalline ytem (eg gla) have muh lower ondutivity. l ~ loal order C, C, and hene κ~t -1-1 K/(Wm K ) T/K Ge purified Deember 06 Leture 1 8