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Lecture 13 Phonons: thermal properties Lattice contribution to the thermal properties of solids, in 3-D Aims: Thermal properties of a crystalline solid: Heat capacity: Debye treatment T3 law for low temperature heat capacity Thermal conductivity: Phonon scattering Mean-free path Debye, Debye, TT33 Law Law cV = December 06 Lecture 13 ∂U ∝T3 ∂T V 1 Lattice heat-capacity Heat capacity Follows from differentiating the internal energy (as usual). No. of phonons Internal energy in dω at ω U= ∞ 0 ω exp( ω kT ) − 1 g (ω )dω Energy per phonon (Planck formula, Lecture 5) Density of modes g(ω). Einstein Approximation: all modes have the same frequency, ωE. (See lecture 5) Debye approximation: In the low temperature limit acoustic modes, with small q, dominate. So assume ω = vs q. Exact: calculate g(ω), numerically, from the phonon dispersion curves Einstein approximation gave the correct hightemperature behaviour (C = 3NkT) and gave C 0 as T 0 though the exact temperature dependence was inaccurate (the reason, we can now understand from the phonon dispersion curves) December 06 Lecture 13 2 Debye Model Density of states Assume ω = vs q. i.e. dispersionless waves Result is therefore similar to that for photons in a 3-D cavity (black body radiation), except for a numerical constant. 3 a 2 g (q) d q = 3 . q dq 2 2π g (q ) d q = g (ω ) d ω 33 acoustic acoustic modes modes a3 2 g (ω ) d ω = 3 2 3 ω d ω for for each each qq π vs where ω = vs q has been substituted in RHS. Energy becomes: Upper Upper limit limit on on integral integral guaranguaran- tees tees the the correct, correct, total total number number of of modes modes (3N). (3N). ItIt isis known known as as the the Debye Debye frequency. frequency. U= Vol, Vol, VV==aa33.. ωD 0 3Vω 2 1 ω dω 2 3 2π vs exp( ω kT ) − 1 3V = 2 3 2π vs ωD 0 ω3 1 exp( ω kT ) − 1 dω Formula gives the full T dependence. We are interested in the behaviour at low T. December 06 Lecture 13 3 heat capacity at low-temperture Limiting behaviour as T 0. At low temperature the higher frequency modes are not excited. Thus contributions to the integral for large ω (~ωD) can be ignored and ωD replaced by ∞. 3V U= 2 3 2π vs ∞ 0 ω 3V kT = 2 3 2π vs 3 1 exp( ω kT ) − 1 4 ∞ 0 dω x 3 (exp( x ) − 1) dω Vπ 2 k 4 4 U= T 3 10(vs ) −1 Integral Integral == ππ44/15 /15 Differentiating gives the heat capacity as cV = ∂U ∝T3 ∂T V 3 Debye, Debye, TT3 Law Law gives the correct, observed dependence at low temperatures. Recall the Einstein model gave an exponential dependence at low T. December 06 Lecture 13 4 Measured density of states Example: Aluminium (shows common features) Measured density of states compared with Debye approximation. Both measured and Debye density of states are similar at low ω, as expected (ω∝q). Debye frequency chosen to give same total number of modes (i.e. equal area under both curves) Largest deviations where phonon modes approach zone boundary. Measured curve is complex because the 3-D zone has a relatively complicated shape, and the transverse and longitudinal modes have different dispersions (as we have seen earlier) December 06 Lecture 13 5 Thermal conductivity Phonons and thermal conductivity Phonons have energy and momentum and, therefore, can conduct heat. Kinetic theory gives the thermal conductivity l z θ ∆ z = -l cos θ Excess temperature of phonons crossing plane dT dT ∆T = ∆z = − l cos θ dz dz Excess energy of each phonon dT c ph ∆T = −c ph l cos θ dz heat heat capacity capacity of of aa phonon phonon December 06 Lecture 13 6 conductivity.......... Number density of phonons, n number number with with speed speed cc to to c+dc c+dc n f (c )dc fraction fraction with with angles angles θθ to to θθ+d +dθθ 2π sin θ dθ θ dθ 2π sin θ dθ 4π = sin θ dθ 2 speed speed normal normal to to plane plane net net heat heat per per phonon phonon Heat flux across plane H= π ∞ 0 0 [nf (c )dc sin θ dθ 2][c cosθ ][− c phl cosθ dT dz ] ∞ 1 dT π 2 H = − c ph nl sin θ cos θ dθ c f (c )dc 0 2 dz 0 π 1 dT c H = + c ph nl c cos 2 θ d (cos θ ) 0 2 dz 1 dT dT H = + c ph nl c =κ 3 dz dz κ = c ph nl c 3 Definition Definition of of thermal thermal conductivity conductivity Thermal conductivity 1 κ= C c l 3 Mean Mean free free path path Average Average speed speed Heat Heat capacity capacity per per unit unit vol vol December 06 Lecture 13 7 Temperature dependence of thermal conductivity Mean free path – limited by scattering processes With many scattering processes: 1 l = 1 l 1 + 1 l2 + Thus, the shortest mean free path dominates “Geometric” scattering: Sample boundaries (only significant for purest samples at low temperatures). Impurities: scattering rate ~ independent of T. Phonon-phonon scattering: Phonons, being normal modes, should not affect each other. However, in an anharmonic lattice, they can scatter. Free path ~ 1/T. Insulators (no contribution from electrons). -1 -1 CC,, and and hence hence κκ~T ~T33 December 06 Lecture 13 Ge purified K/(Wm K ) 1 10 In pure crystalline form the conductivity can be very high (larger than metals) N.B. Non-crystalline systems (eg glass) have much lower conductivity. l ~ local order 10 100 T/K 8