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Phonons II: Thermal properties specific heat of a crystal density of states Einstein model

Dept of Phys. M.C. Chang. Phonons II: Thermal properties specific heat of a crystal density of states Einstein model Debye model anharmonic effect thermal conduction. Specific heat approaches a constant 3R (per mole) at high temperature (Dulong-Petit law).

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Phonons II: Thermal properties specific heat of a crystal density of states Einstein model

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  1. Dept of Phys M.C. Chang • Phonons II: Thermal properties • specific heat of a crystal • density of states • Einstein model • Debye model • anharmonic effect • thermal conduction

  2. Specific heat approaches a constant 3R (per mole) at high temperature (Dulong-Petit law) Specific heat drops to zero at low temperature After rescaling the temperature by (Debye temperature), which differs from material to material, a universal behavior emerges: Specific heat: experimental fact

  3. Debye temperature In general, a harder material has a higher Debye temperature

  4. For a crystal in thermal equilibrium, the average phonon number is determined by (see Kittel, p.107) Specific heat is nothing but the change of U(T) w.r.t. to T: Specific heat: theoretical framework Internal energyU of a crystalis the summation of vibrational energies (consider an insulator so there’s no electronic energies) Therefore, we have

  5. Density of states D() (DOS,態密度) Connection between summation and integral D()d is the number of states within the constant- surfaces  and +d f(x) a b x For example, assume N=16, then there are 22=4 states within the intervald Once we know the DOS, we can reduce the 3-dim k-integral to a 1-dim  integral. Flatter (k) curve, higher DOS.

  6. DOS: 3-dim (assume(k)= (k) is isotropic) There is no use to memorize the result, just remember the way to derive it. DOS: 1-dim a simple change of variable Ex: Calculate D() for the 1-dim string with (k)=M|sin(ka/2)|

  7. 3 dim  number of atoms Einstein model of calculating the specific heat CV (1907) Assume that each atom vibrates independently of each other, and every atom has the same vibration frequency 0 (Activation behavior)

  8. Debye model (1912) In actual solids, because of the atomic bonds, the atoms vibrate collectively in a wave-like fashion. Debye assumed a simple dispersion relation :  = vsk. Therefore,Ds()=(L3/22vs3) 2 Actual density of states for silicon

  9. = 4/15 as T 0 Also, the range of integration is slightly changed. The 1st BZ is approximated by a sphere with the same volume for 3 identical branches

  10. Comparison between the two models At low T, Debye’s curve drops slowly because, in Debye’s model, long wave length vibration can still be excited. Energy dispersion solid Argon (=92 K)

  11. kD kT A simple explanation for the T3 behavior: Suppose that 1. All the phonons with wave vector k<kT are excited with thermal energy kT. 2. All the modes between kT and kD are not excited. Then the fraction of excited modes = (kT/kD)3 = (T/)3. energy U kT3N(T/)3, and the heat capacity C 12Nk(T/)3

  12. Surface +d =const dk dS Surface =const More discussion on DOS (general case)

  13. Thermal properties of crystals • specific heat of a crystal • density of states • Einstein model • Debye model • anharmonic effect • thermal conduction

  14. V(r) r • Anharmonic effect in crystals If there is no anharmonic effect, then • There is no thermal expansion • There is no phonon-phonon interaction • A crystal would vibrate forever • Thermal conductivity would be infinite • ...

  15. Thermal conductivity • Thermal current density (Fourier’s law, 1807) • 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 23, 24

  16. Heat conductivityK=1/3 Cv where C is the specific heat, v is the velocity, and is the mean free path of the phonon. (Kittel p.122) • A phonon can be scattered by an electron, a defect (including a 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.

  17. Normal process(正常過程):totalmomentum 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

  18. T-dependence of the mean free path • At low T, for a crystal with few defects, a phonon does not scatter frequently with other phonons and the defects. The mean free path is limited mainly by the boundary of the sample. • At high T, C is T-independent. The number of phonons are proportional to T, therefore the mean free path ~ 1/T T-dependence of the thermal conductivity (K=1/3 Cv) : • At low T, K~C~T3 • At high T, K~ ~1/T

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