1. Multi-scale Heat Conduction Quantum Size Effect on the Specific Heat
Nov. 1st, 2011
Multi-scale Heat Conduction Quantum Size Effect on the Specific Heat
Hong goo, Kim
1st year of M.S. course

2. Contents Lattice Vibrational Waves Lattice Specific Heat
Density of States
Thin Films
Nanocrystals
Carbon Nanostructures

3. Lattice Wave Lattice Waves Dispersion Relation : ω = ω(k)
- Systemic motions of atoms in periodic lattice structure
- Periodicity is assumed → Fourier series of harmonic function L
Dispersion Relation : ω = ω(k)
- 1-to-1 correspondence between frequency and wavevector for each polarization of lattice vibrational waves
- Can be derived from atomic force constant and lattice geometry
- Slope dω/dk = vg : Group velocity

4. Lattice Wave Harmonic Plane Wave Fourier Series
- Displacement of atoms
- Phase velocity vp
Fourier Series
- Superposition of harmonic waves
- Spatial period : L

5. Boundary Condition Harmonic Lattice Waves Fixed B.C.
- To determine the wavevectors of Fourier series, boundary condition is required
Fixed B.C.
- End nodes(x = 0, L) are fixed → Standing wave solution
Periodic B.C. (Born−von Kármán)
- Simulates the physics of macroscopic periodicity better than the fixed B.C.

6. Periodic B.C. Discretized Wavevector (1-D) Independency of Wavevectors
- Number of atoms = Nx + 1
- Odd Number of Nx = 2M + 1
±(Nx − 3)π/Lx
±(Nx − 1)π/Lx
- Even Number of Nx = 2M
±(Nx − 2)π/Lx
±Nx π/Lx
Independency of Wavevectors
- Number of independent wavevectors are restricted to Nx
- Upper limit can be defined : Cut-off wavenumber KD

7. Total Number of Modes Dependence of wavevectors
- Odd Number : Nx = 2M + 1
2π/Lx
(Nx − 1)π/Lx
(Nx − 3)π/Lx
Nx modes
− (Nx − 1)π/Lx
(Nx + 1)π/Lx
- Even Number : Nx = 2M

8. Total Number of Modes Example : Dependence of wavevectors (Nx = 10)
k2 = 2π/Lx = 2π
k1 = (Nx + 1)2π/Lx = 22π
k2 = 2π/Lx = 2π
k1 = (Nx + 1)2π/Lx = 22π

9. Lattice Vibrational Energy
Energy for each wave mode (quantum state)
- Phonon energy levels are quantized by
- Dispersion relation assigns one quantum state(KP) to one energy level( ) for each polarization(P)
→ Degeneracy for energy level is one ; gP, K = 1
- Phonons obey Bose-Einstein distribution
- Therefore, energy of each quantum state KP is

10. Lattice Specific Heat Total Lattice Vibrational Energy
Lattice Specific Heat (Discrete)
Lattice Specific Heat (Integral)

11. Density of States Distribution of Quantum States 3 2 1 ω ω1 ω2 ω3 ω4ω5 ω6 ω7 ω8 ω9 g ω ω1 ω2 ω3 ω4 ω5 ω6 ω7 ω8 ω9

12. Density of States Density of States as a Continuous Function D(ω) D(ω)ω1 ω2 ω3 ω4 ω5 ω6 ω7 ω8 ω9

13. Spherical shell in K3-space Obtained from Periodic B.C
Density of States : 3-D
3-D Case ky kx kz
K3-space
Spherical shell in K3-space
2π/Ly
2π/Lx
2π/Lz
Linear
dispersion relation
ω/K = dω/dK = va
Obtained from Periodic B.C

14. Density of States : 2-D, 1-D
2-D Case
1-D Case
LK = 2K
K1-space kx K

15. Thin Film : Concept Thin Film − Confined in 1-D
- Fabricated microstructure with thickness much smaller than lateral dimenstions
- Application : thermal barrier, optical/electrical device
- Thickness : 1 nm ~ 100 μm
Assumption (Example 5-4)
. Monatomic solid thin silicon film
. Confined in z-direction
. Infinite in x-, y-direction
. Film thickness L
. Number of monatomic layers q ( = L / L0 )
. Acoustic speed va : independent of T

16. Thin Film : Specific Heat
Specific Heat of Thin Film

17. Thin Film : kD Debye Wavevector kD
- Upper limit of absolute value of wavevector which includes all the vibrational modes in the 1st Brillouin zone
- Total number of modes = Total number of atoms N
- Number of modes in z-direction = Number of 1-atom layers q

18. Thin Film : kD
Debye Wavevector kD : 3-D Bulk kz kD kx
2π/Lx
2π/Lz

19. Thin Film : kD Debye Wavevector kD : Thin Film kD kz kz kx Δkz=2π/Lz
Δkx=2π /Lx
Δkz=2π/Lz

20. Thin Film : Quantum Size Effect
Specific heat, cv(T)
Temperature, T [K]
Bulk
( ~T3 )
Single layer ( ~T2 )
q = 1
q = 2
....
q = 7
....
q = 20

21. Thin Film : Quantum Size Effect
T 2 Dependence of Specific Heat x
- Quantum size effect becomes more significant at lower temperature and for smaller film thickness q
- Specific heat for thinner film increases due to q−1 dependence and contribution of planar modes (kz = 0)
- Specific heat at lower temperature converges to zero slowly due to T 2 instead of T 3 dependence

22. Nanocrystal Cubic Solid L3 (L = qL0) - Debye wavevector
- Fraction of planar modes ~ q -1
Δkx=2π /qL0
K3-space kD
Δkz=2π /qL0
- Fraction of axial modes ~ q -2

23. Nanocrystal Quantum Size Effect of Nanocrystals
- Quantum size effect of nanocrystal becomes significant as size parameter q decreases
- At low temperature, planar mode ( ~T 2) contribution increases
kx = 0 or ky = 0 or kz = 0
- At lower temperature, axial mode ( ~T 1) contribution increases
kx = ky = 0 or ky = kz = 0 or kz = kx = 0
- Temperature dependence of specific heat (general form)

24. Nanocrystal Second Size Effect − Extremely Low T
- Only the lowest vibrational modes are excited
- Results in reduction of specific heat
Converges to ‘0’ faster than T3

25. Departure from bulk solid specific heat
Nanocrystal
Second Size Effect − Lead Grains
R. Lautenschläger (1975)
Departure from bulk solid specific heat

26. Carbon : Graphite / Graphene
- Layers of hexagonal plane (graphene) structure
- Covalent bond of neighboring atoms within a layer
- Weakly bonded between layers: Van der Waals bond
- Lattice vibrational modes have 2-D characteristics
- T 2 dependence of specific heat (Debye Theory)
Graphene
- T 1 dependence at low temperature due to dominant contribution of out-of-plane(perpendicular) mode: ω ~ k2
→ Transition to T 2 dependence at higher temperature (2-D)

27. Carbon Nanotube Specific Heat Twisting Mode
M. S. Dresselhaus and P. C. Ecklund(2000)
~T 2
- T 1 dependence of specific heat at low temperature
- Bounded between graphene and graphite
~T 1
Twisting Mode
- Rigid rotation around nanotube axis
~T 2.3
- Coupling of in-plane and out-of-plane modes due to curvature by rolling up the graphene sheet

28. Carbon Nanotube Coupling of In-Plane and Out-of-Plane Modes
M. S. Dresselhaus and P. C. Ecklund(2000)

29. Conclusion Lattice Vibrational Waves Density of States
- Spatial periodicity of lattice
- Superposition of harmonic waves
- Periodic boundary condition : Discretized wavevectors
Density of States
- Dimensionality of the crystal structure should be considered
Lattice Specific Heat
- Lattice vibrational energy : Summation of phonon energy over quantum states
Quantum size effect in thin films
- Departure from bulk behavior at low temperature and thickness (T 2 dependency )
Quantum size effect in nanocrystals

31. Thin Film : Quantum Size Effect
- T 2 dependence :
- At low temperature :
- Specific heat