Classical Theory Expectations • Equipartition: 1/2kBT per degree of freedom • In 3-D electron gas this means 3/2kBT per electron • In 3-D atomic lattice this means 3kBT per atom (why?) • So one would expect: CV = du/dT = 3/2nekB + 3nakB • Dulong & Petit (1819!) had found the heat capacity per mole for most solids approaches 3NAkB at high T Molar heat capacity @ high T 25 J/mol/K © 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 32 Heat Capacity: Real Metals CV bT aT 3 due to electron gas due to atomic lattice Cv = bT • • • • So far we’ve learned about heat capacity of electron gas But evidence of linear ~T dependence only at very low T Otherwise CV = constant (very high T), or ~T3 (intermediate) Why? © 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 33 Heat Capacity: Dielectrics vs. Metals Cv = bT • Very high T: CV = 3nkB (constant) both dielectrics & metals • Intermediate T: CV ~ aT3 both dielectrics & metals • Very low T: © 2010 Eric Pop, UIUC CV ~ bT metals only (electron contribution) ECE 598EP: Hot Chips 34 Phonons: Atomic Lattice Vibrations Graphene Phonons [100] 200 meV CO2 molecule vibrations transverse small k transverse max k=2p/a Frequency ω (cm-1) 160 meV 100 meV ~63 meV Si optical phonons 26 meV = 300 K u(r, t ) A exp[ i (k r it )] k • Phonons = quantized atomic lattice vibrations ~ elastic waves • Transverse (u ^ k) vs. longitudinal modes (u || k), acoustic vs. optical • “Hot phonons” = highly occupied modes above room temperature © 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 35 A Few Lattice Types • Point lattice (Bravais) – 1D – 2D – 3D © 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 36 Primitive Cell and Lattice Vectors • Lattice = regular array of points {Rl} in space repeatable by translation through primitive lattice vectors • The vectors ai are all primitive lattice vectors • Primitive cell: Wigner-Seitz © 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 37 Silicon (Diamond) Lattice • Tetrahedral bond arrangement • 2-atom basis • Each atom has 4 nearest neighbors and 12 next-nearest neighbors • What about in (Fourier-transformed) k-space? © 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 38 Position Momentum (k-) Space Sa(k) k • The Fourier transform in k-space is also a lattice • This reciprocal lattice has a lattice constant 2π/a © 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 39 Atomic Potentials and Vibrations • Within small perturbations from their equilibrium positions, atomic potentials are nearly quadratic • Can think of them (simplistically) as masses connected by springs! © 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 40 Vibrations in a Discrete 1D Lattice 2u x 2u x 2 EY 2 t x • Can write down wave equation • Velocity of sound (vibration propagation) is proportional to stiffness and inversely to mass (inertia) 0 EY Wave vector, K p/a vs k See C. Kittel, Ch. 4 or G. Chen Ch. 3 © 2010 Eric Pop, UIUC vs Frequency, 2u x 1 2u x x 2 vs2 t 2 ECE 598EP: Hot Chips 41 Two Atoms per Unit Cell Lattice Constant, a xn xn+1 yn d 2 xn m1 2 k yn yn 1 2 xn dt d 2 yn m2 k xn 1 xn 2 yn 2 dt LO Frequency, yn-1 TO LA 0 © 2010 Eric Pop, UIUC Optical Vibrational Modes ECE 598EP: Hot Chips TA Wave vector, K p/a 42 Energy Stored in These Vibrations • Heat capacity of an atomic lattice • C = du/dT = • Classically, recall C = 3Nk, but only at high temperature • At low temperature, experimentally C 0 • Einstein model (1907) – All oscillators at same, identical frequency (ω = ωE) • Debye model (1912) – Oscillators have linear frequency distribution (ω = vsk) © 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 43 • All N oscillators same frequency • Density of states in ω (energy/freq) is a delta function E Frequency, The Einstein Model g 3N ( E ) 0 Wave vector, k 2p/a • Einstein specific heat du df ( ) CE g d dT dT © 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 44 Einstein Low-T and High-T Behavior • High-T (correct, recover Dulong-Petit): CE (T ) 3Nk B E 2 T 1 1 1 E T E T 2 3Nk B Einstein model OK for optical phonon heat capacity • Low-T (incorrect, drops too fast) CE (T ) 3 Nk B 3 Nk B © 2010 Eric Pop, UIUC E 2 k BT E 2 k BT e E / k BT e E / k BT 2 e E / k BT ECE 598EP: Hot Chips 45 • Linear (no) dispersion with frequency cutoff • Density of states in 3D: Frequency, The Debye Model vs k 2 g 2 3 2p vs (for one polarization, e.g. LA) 0 (also assumed isotropic solid, same vs in 3D) Wave vector, k 2p/a • N acoustic phonon modes up to ωD • Or, in terms of Debye temperature vs D 6p 2 N kB © 2010 Eric Pop, UIUC kD roughly corresponds to max lattice wave vector (2π/a) 1/3 ωD roughly corresponds to max acoustic phonon frequency ECE 598EP: Hot Chips 46 Annalen der Physik 39(4) p. 789 (1912) Peter Debye (1884-1966) © 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 47 Website Reminder © 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 48 Frequency, The Debye Integral • Total energy u (T ) D f ( ) g ( ) d vs k 0 • Multiply by 3 if assuming all polarizations identical (one LA, and 2 TA) • Or treat each one separately with its own (vs,ωD) and add them all up • C = du/dT © 2010 Eric Pop, UIUC 0 Wave vector, k 2p/a people like to write: (note, includes 3x) T CD (T ) 9 Nk B D ECE 598EP: Hot Chips 3 /T D 0 x 4 e x dx (e x 1) 2 49 Debye Model at Low- and High-T • At low-T (< θD/10): CD (T ) T 12p Nk B 5 D 4 3 • At high-T: (> 0.8 θD) CD (T ) 3NkB • “Universal” behavior for all solids • In practice: θD ~ fitting parameter to heat capacity data • θD is related to “stiffness” of solid as expected © 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 50 Experimental Specific Heat 3 3 (J/m Heat Specific Specific Heat, C (J/m -K) -K) 10 7 C 3 kB 4.7 10 6 10 6 J 3NkBT m3 K Diamond Each atom Diamondhas a thermal energy of 3KBT 10 5 10 4 TT3 CC 3 10 3 Classical Regime 10 2 D 1860 K 10 1 1 10 2 10 10 Temperature, T (K) 3 10 4 Temperature (K) In general, when T << θD © 2010 Eric Pop, UIUC uL T d 1 , CL T d ECE 598EP: Hot Chips 51 Phonon Dispersion in Graphene Maultzsch et al., Phys. Rev. Lett. 92, 075501 (2004) Optical Phonons Yanagisawa et al., Surf. Interf. Analysis 37, 133 (2005) © 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 52 Heat Capacity and Phonon Dispersion • Debye model is just a simple, elastic, isotropic approximation; be careful when you apply it • To be “right” one has to integrate over phonon dispersion ω(k), along all crystal directions • See, e.g. http://www.physics.cornell.edu/sss/debye/debye.html © 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 53 Thermal Conductivity of Solids thermal conductivity spans ~105x (electrical conductivity spans >1020x) © 2010 Eric Pop, UIUC how do we explain the variation? ECE 598EP: Hot Chips 54 Kinetic Theory of Energy Transport u(z+z) qz λ z + z z Net Energy Flux / # of Molecules 1 q' z v z u z z u z z 2 through Taylor expansion of u u(z-z) z - z q ' z v z z du du cos 2 v dz dz Integration over all the solid angles total energy flux 1 du dT dT qz v k 3 dT dz dz Thermal conductivity: © 2010 Eric Pop, UIUC 1 k Cv 3 ECE 598EP: Hot Chips 55 Simple Kinetic Theory Assumptions • Valid for particles (“beans” or “mosquitoes”) – Cannot handle wave effects (interference, diffraction, tunneling) • Based on BTE and RTA • Assumes local thermodynamic equilibrium: u = u(T) • Breaks down when L ~ _______ and t ~ _________ • Assumes single particle velocity and mean free path – But we can write it a bit more carefully: © 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 56 Phonon MFP and Scattering Time • Group velocity only depends on dispersion ω(k) • Phonon scattering mechanisms – Boundary scattering – Defect and dislocation scattering – Phonon-phonon scattering 1 1 2 k Cv Cv 3 3 Decreasing Boundary Separation l kl Increasing Defect Concentration Increasing Defect Concentration kl T d Phonon Scattering Defect Boundary 0.01 Boundary Defect 0.1 1.0 0.01 Temperature, T/D © 2010 Eric Pop, UIUC ECE 598EP: Hot Chips Phonon Scattering 0.1 Temperature, T/D 1.0 57 Temperature Dependence of Phonon KTH T low T 3Nk B high T d C ph ph 1 e n ph / kT 1 low T high T kT C λ low T Td nph 0, so λ , but then λ D (size) Td high T 3NkB 1/T 1/T © 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 58 Ex: Silicon Film Thermal Conductivity Thermal Conductivity (W m-1K-1) McConnell, Srinivasan, and Goodson, JMEMS 10, 360-369 (2001) 10 4 bulk 1000 100 single crystal films size effect doped 10 1 undoped 10 polycrystal 100 Temperature (K) 1 A k (d G , n) Cv 1 A2 ni 3 dG © 2010 Eric Pop, UIUC Undoped single-crystal film: Asheghi et al. (1998) d = 3 mm Doped single-crystal film: Asheghi et al. (1999) d = 3 mm n = 1·1019 cm-3 boron undoped doped Bulk single-crystal silicon: Touloukian et al. (1970) d = 0.44 cm 1 ECE 598EP: Hot Chips Doped polysilicon film: McConnell et al. (2001) d = 1 mm dg = 350 nm n = 1.6·1019 cm-3 boron Undoped polysilicon film: Srinivasan et al. (2001) d = 1 mm dg = 200 nm 59 Ex: Silicon Nanowire Thermal Conductivity Nanowire diameter • Recall, undoped bulk crystalline silicon k ~ 150 W/m/K (previous slide) Li, Appl. Phys. Lett. 83, 2934 (2003) © 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 60 Ex: Isotope Scattering isotope ~impurity ~T3 ~1/T © 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 61 Why the Variation in Kth? • A: Phonon λ(ω) and dimensionality (D.O.S.) • Do C and v change in nanostructures? (1D or 2D) • Several mechanisms contribute to scattering – Impurity mass-difference scattering 1 phi 2 nV M 4 i 2 4p vs M 2 – Boundary & grain boundary scattering 1 phb vs D – Phonon-phonon scattering 1 ph ph © 2010 Eric Pop, UIUC A T exp B k T B ECE 598EP: Hot Chips 62 Surface Roughness Scattering in NWs What if you have really rough nanowires? • Surface roughness Δ ~ several nm! • Thermal conductivity scales as ~ (D/Δ)2 • Can be as low as that of a-SiO2 (!) for very rough Si nanowires smooth Δ=1-3 Å rough Δ=3-3.25 nm © 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 63 Data and Model From… Data… Model… (Hot Chips class project) © 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 64 What About Electron Thermal Conductivity? • Recall electron heat capacity du df Ce E g E dE dT 0 dT Ce p 2 kBT at most T in 3D ne kB 2 EF • Electron thermal conductivity Mean scattering time: e = _______ ke e e Bulk Solids Increasing Defect Concentration Defect Scattering Phonon Scattering Electron Scattering Mechanisms • Defect or impurity scattering • Phonon scattering • Boundary scattering (film thickness, grain boundary) Temperature, T © 2010 Eric Pop, UIUC Grain ECE 598EP: Hot Chips Grain Boundary 65 Ex: Thermal Conductivity of Cu and Al • Electrons dominate k in metals Matthiessen Rule: 1 1 1 1 Thermal Conductivity, k [W/cm-K] 10 3 e 1 Copper 10 2 e 1 1 1 defect boundary 1 boundary phonon 1 phonon Aluminum 10 1 Phonon Scattering Defect Scattering 10 defect 0 10 0 10 1 10 2 10 3 T emperature, T [K} © 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 66 Wiedemann-Franz Law 1p2 2 T e kB n 3 2 EF recall electrical conductivity taking the ratio 2 vF where EF q 2 qm n n m ke • Wiedemann & Franz (1853) empirically saw ke/σ = const(T) • Lorenz (1872) noted ke/σ proportional to T © 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 67 Lorenz Number Experimentally L = /T 10-8 WΩ/K2 e p 2 kB2 L T 3q 2 8 L 2.45 10 WΩ/K 2 This is remarkable! It is independent of n, m, and even ! Metal 0°C 100 °C Cu 2.23 2.33 Ag 2.31 2.37 Au 2.35 2.40 Zn 2.31 2.33 Cd 2.42 2.43 Mo 2.61 2.79 Pb 2.47 2.56 Agreement with experiment is quite good, although L ~ 10x lower when T ~ 10 K… why?! © 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 68 Amorphous Material Thermal Conductivity a-SiO2 GeTe a-Si Amorphous (semi)metals: both electrons & phonons contribute © 2010 Eric Pop, UIUC Amorphous dielectrics: K saturates at high T (why?) ECE 598EP: Hot Chips 69 Summary • Phonons dominate heat conduction in dielectrics • Electrons dominate heat conduction in metals (but not always! when not?!) • Generally, C = Ce + Cp and k = ke + kp • For C: remember T dependence in “d” dimensions • For k: remember system size, carrier λ’s (Matthiessen) • In metals, use WFL as rule of thumb © 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 70