Phonons II: Thermal properties specific heat of a crystal density of state Einstein mode Debye model anharmonic effect thermal conduction A technician holding a silica fibre thermal insulation tile at 100 Celsius Dept of Phys M.C. Chang
Specific heat: experimental fact Specific heat approaches R (per mole) at high temperature (Dulong-Petit law) Specific heat drops to zero at low temperature After rescaling the temperature by θ (Debye temperature), which differs from material to material, a universal behavior emerges:
Debye temperature In general, a harder material has a higher Debye temperature
Specific heat: theoretical framework Internal energy U of a crystal is the summation of vibrational energies (consider an insulator so there s no electronic energies) UT ( ) = ( n + 1/2) ω ks, ks, ks, where s sums over different phonon branches (L/T, A/O). For a crystal in thermal equilibrium, the average phonon number is (see Kittel, p.107) 1 n ks =, ks, / kt e ω Therefore, we have ω ω UT ( )= + ks, / kt ks, e ω 1 2, Bose-Einstein distribution 1 Specific heat is nothing but the change of U(T) w.r.t. to T: V ( / ) C = U T ks, ks, V
important Connection between summation and integral f(x) b a dxf ( x) = lim Δx f ( x ), or Δ x 0 i i a b x or i dx f( xi ) f( x). Δ x Generalization to -dim: x k b a dx f( x) f( x) Δ x dk f( k) f( k) Δ k in solid state
important Density of states D(ω) (DOS, 態密度 ) D(ω)dω is the number of states within the surfaces of constant ω and ω+dω D( ω) dω = dk shell Δ k, Δ k = 2π L Alternative definition: dk D( ω ) = ( ) k k δ ω Δ ω For example, assume N=16, then there are 2 2=4 states within the interval dω k f dk ( ω ) f( ω ) k Δ k k = dωd( ω) f( ω) does not work for f(k) Once we know the DOS, we can reduce the -dim k-integral to a 1-dim ω integral. Flatter ω(k) curve, higher DOS.
DOS: 1-dim dk dk / dω D( ω) dω = 2 = 2 dω Δk Δk L 1 for ω ωm D( ω) = π dω/ dk 0 otherwise Ex: Calculate D(ω) for the 1-dim string with ω(k)= ω M sin(ka/2) DOS: -dim (assume ω(k)= ω(k) is isotropic) 2 dk 4π 2 L k D( ω) dω = = k dk = dω Shell 2 Δ k Δ k 2 π dω/ dk 2 2 for example, if ω = vk, then D( ω) = Vω /2π v There is no use to memorize the result, just remember the way to derive it.
Einstein model (1907) Assume that each atom vibrates independently of each other, and every atom has the same vibration frequency ω 0 Note: the DOS can be written as D( ω) = Nδ ω ω ( ) 0 dim number of atoms 1 ω0 ω0 U = N n + ω0 = N + N 2 exp( ω / kt ) 1 2 2 ω0 / kt ω0 e = ( / ) V = CV U T Nk kt 0 as 0 ω0 / kt ( e 1) ω0 / kt e T K ( Activated behavior) 2
Debye model (1912) Atoms vibrate collectively in a wave-like fashion. UT ( ) = n ω ( ω /2 neglected) ks, s= 1 ks, ks, ks, = dωd ( ω) s ω 1 / kt e ω Debye assumed a simple dispersion relation: 2 2 ω = v s k. Therefore, D ( ω) = Vω / 2 π v (quadratic) s Also, the 1st BZ is approximated by a sphere with the same total number of states. Cut-off frequency: s s= 1 dωd ( ω) = N Vω = N D 2 s= 1 6π vs 1 v s= 1 s ω = D v s 2 1/ v(6 π n), n= N / V Debye frequency ω D
UT ( ) = ωd V 2 ω dω ω 2 ω/ kt B s 1 2π v = s e 1 0 4 xd V kt B x 2 dx x 2π 1 0 ωd θ =, xd = =, kbθ v e k T T T =9NkBT θ xd 0 x dx e x 1 = π 4 /15 as T 0 (x D ) B Debye temperature ω D 4 12π T CV = NkB T 5 θ as T 0 (Debye T law) solid Argon (θ=92 K) At low T, Debye s curve drops slowly because long wavelength vibration can still be excited.
A simple explanation of the T behavior: Suppose that 1. All the phonons with wave vector k<k T are excited, each with thermal energy k B T. 2. All the modes between k T and k D are not excited. k D k T Then the fraction of excited modes = (k T /k D ) = (T/θ). Energy U ~ k B T N(T/θ) Heat capacity C ~ 12Nk B (T/θ)
DOS for general dispersion important Surface ω+dω =const ds ω dk Surface ω=const L D( ω) dω = d k 2π Shell If v g = ω = 0, then there is k "van Hove singularity" (195) d k = dω = dk = ds dk k ω ω dk ds ω dω k ω D( ω) L dsω = 2π ω k Group velocity
Dispersion relation and DOS Giannozzi et al, PRB 4, 721 (1991)
specific heat of a crystal density of states Einstein model Debye model anharmonic effect thermal conduction
Anharmonic effect in crystals If there is no anharmonic effect, then Inter-atomic potential v(r) There is no thermal expansion r A crystal would vibrate forever There is no phonon-phonon interaction Thermal conductivity would be infinite... Shift of equilibrium position
Thermal conductivity Thermal current density (Fourier s law, 1807) j = K T U In metals, thermal current is carried by both electrons and phonons. In insulators, only phonons can be carriers. The collection of phonons are similar to an ideal gas Ashcroft and Mermin, Chaps 2, 24
Thermal conductivity K=1/ Cvl where C is the specific heat, v is the velocity, and l is the mean free path of the phonon. (Kittel p.122) A phonon can be scattered by defects, boundary, and other phonons. Such scattering will shorten the mean free path. Phonon-phonon scattering is a result of the anharmonic vibration Modulation of elastic const. (~ acoustic grating) F = kx+ ck' x = ( k ck' x) x k ( x) = k ck' x eff 2
Phonon-phonon scattering Normal process: total momentum of the 2 phonons remains the same before and after scattering, no resistance to thermal current! Umklapp process ( 轉向過程, Peierls 1929): 1st BZ
T-dependence of the phonon mean free path l Low T: For a crystal with few defects, a phonon does not scatter frequently with other phonons and defects. The mean free path is limited mainly by the boundary of the sample. High T: C is T-independent. The number of phonons are proportional to T. The mean free path ~ 1/T x (X=1~2) T-dependence of the thermal conductivity K (=1/ Cvl) Low T: K ~ C ~ T High T: K ~ l ~ 1/T x mean free path
Debye temp Melting temp Glassbrenner and Slack, Phys Rev 1964