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

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

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

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

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.

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

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

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

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

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

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

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

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

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

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

Thin Film : Specific Heat Specific Heat of Thin Film

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

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

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

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

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

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

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)

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

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

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)

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

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

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

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