introduction of thermal transport

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Doreen McKinney
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1 Subgroup meeting introduction of thermal transport members: 王虹之. 盧孟珮

introduction of thermal transport Phonon effect Electron effect Lattice

lattice Scattering mechanism Dispersion relation Optical and Acoustic

scattering Phonon-phonon scattering Heat capacity The contribution of

2 introduction of thermal transport Phonon effect Electron effect Lattice vibration phonon Debye model of lattice vibration K space, Reciprocal lattice Scattering mechanism Dispersion relation Optical and Acoustic phonons Bose-Einstein model Dulong-petit model Brillouin zone Boundary scattering Phonon-phonon scattering Heat capacity The contribution of electron of heat capacity Normal Process Debye temperature Umklapp process

3 Outline Dispersion relation (review) Sepcific heat (review) Different temperature ranges Low temperature range Debye temperature

4 Disparsion relation ω axis Optical branch Electromagnetic wave ω/k=c c:light speed c is constant linear dispersion relation Acoustic branch At lower energy (lower k) Ka << π (near zero) The dispersion relation is regard as linear 0 п/a K axis

5 Disparsion relation ω axis Optical branch Acoustic branch 0 п/a K axis We get linear part of dispersion relation.

6 Disparsion relation ω axis Optical branch Standing wave k= π/a 2a=λ (k is fixed, ω is independent from k) Acoustic branch 0 п/a K axis Group velocity vg = 0 at k=π/a Energy does not propagate in the medium (standing wave)

7 Disparsion relation Classical mechanism : Harmonic oscillator (conservative force and wave) We get dipersion relation by only classical assumption

8 Specific heat Low temperature Intermediate High temperature There are three segments of the experiment curve of specific heat (Cv).

wave factor Quatum theory of Harmonic solid

9 Specific heat General form of specific heat s: types of phonons k: wave factor Frequencies are functions of wave factor Quatum theory of Harmonic solid Bose-Einstein distribution function Stationary states of vibration

10 Specific heat Low temperature Intermediate High temperature Different approximationa of temperature ranges: Low temperature: Debye model Intermediate temperature: Debye & Einstein High temperature: Dulong-Petit

11 Specific heat at Low temperature D(ω) ωd Debye model approximation ω Low temperature Debye model The angular frequencies are diverse values. The maximum allowed value is cut-off frequency ωd

12 Debye model Specific heat Density of states Energy per phonon Bose-Einstein distribution Cut-off frequency

13 Debye model Density of states D(ω) D(ω) ωd ds (ω) Δk1 ω dω/dk1 > dω/dk2 dk ω+dω constant surface Δk2 ω surface The difference between constant ω surface is Δω K space The spacing between constant frequency surface depends on (group velocity) dω/dk For simple case that vp=vg ω/k=constant=sound velocity We can get approximation D(ω) of Debye model (low temperature)

14 Specific heat at Low temperature ω axis Optical branch Replacing by approximation Acoustic branch - п/a 0 п/a K axis Dispersion relation Ignored optical branches Replace acoustic branch with liner branches

15 Specific heat at Low temperature ω axis Acoustic branch Optical branch - п/a 0 п/a K axis When T is small High frequencies result in large The contributions of high frequencies for LOW temperature are negligible.

16 Specific heat at Low temperature

Acoustic modes are thermally excited For actual crystals

17 Specific heat at Low temperature At very low temperature Specific heat Which is Debye law Condition assumption: Only Acoustic modes are thermally excited For actual crystals Temperature may necessary to be below T=θ/50 θ :Debye temperature

18 Debye temperature Determine a cut-off frequency of specimen D(ω) ωd ω E=kB θ KT KD E=kBT Define Debye temperature θ in terms of cut-off frequency

19 Debye temperature Compare boundary conditions: Brillion Zone (atom spacing) L: length of specimen N: number of ions in specimen (total vibration modes) Cut off frequency (atom packing) For ideal simple cubic specimen n/ L KD are different with different kinds of lattice packing

20 Debye temperature What are the factors that determine Debye temperature? Theoretical : v(ρ,k,g): sound velocity in the specimen N/V: atom concentration (packing density) Experimental: The Debye temperature were determined by fitting the observed specific heats Cv formula at the point of ½ Dulong-Petit value (3NkB/2)

21 Debye temperature Element Li Na K Rb Cs Crystal structure Bcc Bcc Bcc Bcc Bcc Concentration 10^ /cm^3 Sound speed m/s ~ Debye temperature Element Be Mg Ca Sr Ba Crystal structure Hcp Hcp Fcc Fcc Bcc Concentration 10^ /cm^3 Sound speed m/s ~ 1620 Debye temperature

22 Future work Intermediate region High temperature region Electron Effect

23 Thanks for your attention

Phonons II: Thermal properties

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