Phonons II - hermal Properties - Continued (Kittel Ch. 5) Low High Outline Anharmonicity Crucial for hermal expansion other changes with pressure temperature Gruneisen Constant hermal Heat ransport Phonon Heat Conductivity Low High behaviors (Read Kittel Ch 5) Physics 460 F 2006 Lect 11 1 Physics 460 F 2006 Lect 11 2 Consequences of Anharmonicity If we exp the energy beyond the harmonic order: E = E 0 + (1/2) Σ ij R i. Dij. Rj + (1/6) Σ ijk D ijk. Ri R j R k +... Difficult Messy he problem is fundamentally changed: No longer exactly soluble Consequences: here is thermal expansion other changes with temperature he heat capacity is not constant at high Phonons decay in time wo phonons can interact to create a third Phonons can establish thermal equilibrium conduct heat like a gas of particles Physics 460 F 2006 Lect 11 3 Energies of Crystal hermal expansion - I he general shape applies for any type of binding Increases more rapidly at short distances hermal Expansion - Average distance increases as vibration amplitude increases Absolute Minimum Distance Between Atoms Physics 460 F 2006 Lect 11 4 hermal expansion - II here is no thermal expansion In the harmonic approximation! Why? In the harmonic approximation the magnitude of the force is symmetric around the minimum (F = -Cx). herefore, the vibrations are symmetric the average position <x> is always the center, i.e., <x> = 0 there is no thermal expansion Energies of Crystal Harmonic Approximation E = ½ C x 2 Minimum Distance Between Atoms Physics 460 F 2006 Lect 11 5 hermal expansion - III Anharmonicity causes thermal expansion. Why? If the potential energy curve is not symmetric, the vibrations are not symmetric the average position <x> changes as the magnitude of the vibration increases, i.e., <x> increases there is thermal expansion Energies of Crystal Anharmonic potential energy E = ½ C x 2 + (1/6) A x 3 Minimum Distance Between Atoms Physics 460 F 2006 Lect 11 6 1
Gruneisen Constant γ Anharmonicity is very complicated because there are all the possible terms involving 3 atoms! A simple characterization requires finding a typical measurable quantity. his is given by the dimensionless Gruneisen constant γ defined as an average value of the change of vibration frequency per unit dilation: ω /ω = γ ( V/V) = 3 γ ( R/R) where L is a characteristic near-neighbor distance. hus γ = (1/3)( ω /ω)/( R/R) On the next slide we consider a simple case that shows the relation to anharmonicity Physics 460 F 2006 Lect 11 7 Gruneisen Constant γ From previous slide: he Gruneisen constant γ defined as an average value of the change of vibration frequency per unit dilation: ω /ω = γ ( V/V) = 3 γ ( R/R) where L is a characteristic near-neighbor distance. An example is a nearest neighbor interaction with E = ½ C R 2 + (1/6) A R 3 hen the force is given by F = -de/dr = -C( R) ½ A ( R) 2 + = -C [1 + ½ A R/C] R Since ω 2 ~ C/M ω /ω = ½ C/C = ½ A R / C hus γ = (1/3)( ω /ω)/( R/R) = (1/6) (A/C) R is a measure of anharmonicity Physics 460 F 2006 Lect 11 8 Other effects of Anharmonicity Anharmonicity Crucial for hermal expansion other changes with pressure temperature Gruneisen Constant hermal Heat ransport Phonon Heat Conductivity Low High behaviors (Read Kittel Ch 5) Physics 460 F 2006 Lect 11 9 hermal Energy (Heat) ransport First consider heat transport in an ordinary gas Molecules in the gas have average energy ½m v 2 = 3[ ½ k B ] move with average speed v Factor of 3 in 3 dimensions wo limiting cases Ordinary case where molecules scatter from one another very often as they move he molecules diffuse thermalize with one another his is the usual case for gases like the air in this room Cases where the molecules hit the walls of the container without hitting one another the molecules transport energy directly from one wall to another Physics 460 F 2006 Lect 11 10 ransport of heat in an ordinary gas Molecules move in all directions scatter so that they come to local thermal equilibrium in each region. How can rom motion of molecules cause heat flow in one direction? On average, in ter regions molecules have more kinetic energy. A molecule that moves from a ter region to a er region brings energy above the local average. he opposite for a molecule moving from a er to a ter region. Either way, there is transport of energy from ter to er regions. Physics 460 F 2006 Lect 11 11 Heat ransport due Molecular Motion Definition: j = heat flow (energy per unit area per unit time ) = - K d/dx If a particle moves from a region with local temperature to one with local temperature -, it supplies excess energy c, where c = heat capacity per particle. (Note can be positive or negative). On the average for a particle with average thermal speed v in the x direction: = (d/dx) v x τ, where τ = mean time between collisions hen j = - n v x c v x τ d/dx = - n c v x2 τ d/dx Density Flux Physics 460 F 2006 Lect 11 12 2
Heat ransport due Molecules - II his can be simplified in an isotropic case, since averaging over directions gives ( v x 2 ) average = (1/3) v 2 his leads to j = - (1/3) n c v x2 τ d/dx Finally we can define the mean free path: L = v τ C = nc = total heat capacity per unit volume hen j = - K d/dx = - (1/3) C v L d/dx K = (1/3) C v L = thermal conductivity his is a general result for thermal conductivity due to thermal motion of particles - Eq. 42 in Kittel Physics 460 F 2006 Lect 11 13 Heat ransport due Molecules - III For molecules in a gas we can use classical expressions for C v, we can underst some facts about L Heat capacity: C = 3Nk B Average speed: v determined by ½ Mv 2 = (3/2) k B Mean free path L is distance between collisions increases with density temperature independent hen K = (1/3) C v L = Nk B (3k B /M) 1/2 L Increases with for fixed density because v increases Physics 460 F 2006 Lect 11 14 Heat ransport due Molecules - IV What happens if the mean free path is long compared to the size of the container L ~ dimension of container the thermal conductivity is given by K = (1/3) C v L And the basic idea are still the same What causes heat transport in a solid? Motion of the atoms is thermal energy But the atoms vibrate around their sites do not move through the solid How does energy moves from to? he amplitude of the vibrational motion is transferred through the solid ter atoms transfer energy to er atoms How do we describe this? Physics 460 F 2006 Lect 11 15 Physics 460 F 2006 Lect 11 16 From Before Inelastic Scattering Fourier Analysis Scattering of neutrons, x-rays,. Inelastic diffraction occurs for k in - k out = G ± k phonon ω in - ω out = ± ω phonon or Ε n - Ε out = ± hω phonon Quantum Mechanics Scattering of Phonons - I he same idea applies to phonons. One phonon can scatter to create two. We can say one phonon is destroyed two are created. his can occur if k in phonon = k out phonon 1 + k out phonon 2 ± G ω in phonon = ω out phonon 1 +ω out phonon 2 k in ω in k out ω out K in phonon ω in phonon K out phonon 1 ω out phonon 1 k phonon ω phonon Physics 460 F 2006 Lect 11 17 Anharmonic Interaction K out phonon 2 ω out phonon 2 Physics 460 F 2006 Lect 11 18 3
Scattering of Phonons - II Also a phonon can absorb (destroy) another phonon create a different phonon. hus two phonons are destroyed one is created which can occur if k out phonon = k in phonon 1 + k in phonon 2 ± G ω out phonon = ω in phonon 1 +ω in phonon 2 Phonons also act like a gas A phonon is a particle - a quantum of vibration It carries energy just like a molecule. Phonon can come to equilibrium by scattering just like molecules Scattering is due to defects anharmonicity. Leads to heat transport just as for molecules Anharmonic Interaction K in phonon 1 ω in phonon 2 Kout phonon ω out phonon K = (1/3) C v L = thermal conductivity his is proportional to the average occupation of phonons in state 2 the probability K in phonon 2 ω in phonon 2 vanishes if there are no phonons to destroy! Physics 460 F 2006 Lect 11 19 Physics 460 F 2006 Lect 11 20 Phonons also act like a gas What is different about phonons gas molecules? Speed = v group is largest for low energy (v sound ) smaller for high energy he heat capacity ~ 3 at low, ~ 3Nk B at high he density of phonons <n> increases with. Scattering increases ~ <n> since each phonon has scatters from other phonons. herefore we expect the mean free path to decrease with Phonon Heat ransport he same logic applies for phonons as for molecules j = heat flow (energy per unit area per unit time ) = - K d/dx If a phonon moves from a region with local temperature to one with local temperature -, it supplies excess energy c, where c = heat capacity per phonon. hen j = - (1/3) C v L d/dx K = (1/3) C v L = thermal conductivity where L = mean free path, C = nc = total heat capacity (a function of temperature) Physics 460 F 2006 Lect 11 21 Physics 460 F 2006 Lect 11 22 Phonon Heat ransport - continued What determines mean free path L = v τ? At low temperature, the thermal phonons are sound waves that have long mean free paths - L ~ sample size At high temperature, phonons scatter from other phonons. he density of other phonons is ~, so L ~ 1/ At intermediate temperature, phonon scatter from defects other phonons. he latter decreases very rapidly as decreases because ONLY Umklapp scattering limits the energy flow. Physics 460 F 2006 Lect 11 23 Phonon Heat ransport - continued Behavior in an excellent quality crystal: hermal conductivity K Maximum controlled by defects 100 10 1 Low K = CvL/3 ~ 3 since C ~ 3 (v L are ~ constant) 1 10 K decreases as Umklapp scattering increases rapidly High K = CvL/3 ~ 1/ since L~1/ (C v are ~ constant) Physics 460 F 2006 Lect 11 24 4
Phonon Heat ransport - continued Low K ~ 3 - increases because density of phonons increases with roughly constant v L High K ~ 1/ - decreases as scattering increases Low Umklapp Scattering What is Umklapp scattering? Scattering that changes total crystal momentum by a reciprocal lattice vector. Recall: scattering can occur for k in phonon = k out phonon 1 + k out phonon 2 ± G ω in phonon = ω out phonon 1 +ω out phonon 2 K in phonon ω in phonon K out phonon 1 ω out phonon 1 High Physics 460 F 2006 Lect 11 25 Anharmonic Interaction K out phonon 2 ω out phonon 2 Unless G 0, the scattering does not change the total phonon momentum or energy. herefore only Umklapp scattering limits the heat flow. Physics 460 F 2006 Lect 11 26 Other effects Maximum value of K affected by all scattering mechanisms How did the collapse of the Soviet Union affect thermal conductivity? Impurities scatter phonons Isotope effects Isotopes do not affect chemical properties, but they do change phonon frequencies cause scattering of phonons - limits the maximum value of K in high quality crystals Physics 460 F 2006 Lect 11 27 Physics 460 F 2006 Lect 11 28 Availability of pure isotopes from Soviet nuclear program has made possible growth of large crystals of pure isotope materials New record for heat conductivity for all crystals - isotopically pure diamond - better than any metal at room temperature Stimulated new research to create better diamond heat sinks for lasers, etc. hermal conductivity K Record Value Physics 460 F 2006 Lect 11 29 Are there special aspects of thermal conductivity in nanosystems? Ball stick model of a 1-atom wide gold wire actually made seen in an electron microscope An electrical current leads to very high current densities, heating, An electron can fall into the well only if it can give up energy to phonons the rate is reduced because the phonons have only quantized energies e More later Quantum well for electrons in a semiconductor used to contain electrons for lasers, electronic devices,. Physics 460 F 2006 Lect 11 30 5
Active Lecture 11 - Phonons II - hermal Prop. Continued Are there special aspects of thermal conductivity in nanosystems? Ball stick models of a Buckyball a nanotube ube axis c ube axis A nanotube is a sheet of graphite rolled up into a long thin tube 6a 1 (6,1) a 2 Buckyball with 60 carbon atoms now made in quantity Perhaps later we can say a little about electrons in nanotubes Work now to use nanotubes as wires in semiconductor circuits An electrical current in the nanotube leads to very high current densities, heating, phonons scatter very little in the tube good for removing heat --- but the mismatch with other materials makes it hard for the heat to excape burn in the middle Physics 460 F 2006 Lect 11 31 Summary Anharmonicty crucial for certain effects hermal expansion,. Phonon scattering Gruneisen Constant γ Heat transport due to thermal motion of particles j = - K d/dx, K = (1/3) C v L K = thermal conductivity C = heat capacity v = mean velocity L = mean free path Gas of molecules K ~ = Nk B (3k B /M) 1/2 L Phonons act like particles in a gas (We treat heat transport due to electrons later) Low K ~ 3 (C ~ 3 v, L ~ constant) K ~ 1/ (C ~ 3Nk B, v ~ constant L ~ 1/) he same ideas are used later for both thermal electrical conductivity of electrons! Physics 460 F 2006 Lect 11 32 Next time Start next Part of Course: Electronic Properties of Solids Free electron Fermi Gas Summary of Part I his is an incomplete list of concepts, topics, equations, examples! Energy levels in one dimension three dimensions Heat Capacity (Read Kittel Ch 6) Remember: EXAM MONDAY, OCOBER 9 Physics 460 F 2006 Lect 11 33 Physics 460 F 2006 Lect 11 34 Summary of Part I --- 1 Solids are defined by their solidity - i.e., to ability to resist shear stress keep a shape A crystal is truly solid (as opposed to a glass which is like a slow liquid ) A crystal has a true long range crystalline order Crystalline order is defined by the regular positions of the nuclei Crystal Structure = Lattice + Basis Bravais lattice of points in real space Examples: line, square, simple cubic, hexagonal, fcc, hcp, bcc Primitive Cell Parallelpiped in 3d Wigner-Seitz cell Examples of crystals: line, square, simple cubic fcc, hcp, bcc, NaCl, ZnS, diamond, graphene Physics 460 F 2006 Lect 11 35 Summary of Part I --- 2 Diffraction experimental studies of crystal structures Examples: X-ray, neutron diffraction Appropriate wavelength (k = 2π/λ) Bravais lattice on points in real space Reciprocal Lattice Reciprocal lattice Brillouin Zone diffraction occurs only on boundary of BZ Diffraction experimental studies of crystals Bragg Law - easier to describe in terms of reciprocal lattice Elastic scattering : k in = k out + G Examples: line, square, simple cubic, hexagonal, fcc, hcp, bcc Structure factor example of diamond Here it is the idea that is important not the details Physics 460 F 2006 Lect 11 36 6
Summary of Part I --- 3 Binding of Crystals 5 basic types Simplest examples: pair potentials φ(r) Van der Waals. Attractive Repulsive terms Coulomb interactions in ionic crystals For pair potentials, binding is sum of over pairs Count each interaction as ½ for each atom in the pair Binding curve Minimum energy equilibrium structure Bulk modulus Curvature B = Vd 2 E/dV 2 Elastic properties Stress strain relations - elastic constants C ij (he ideas that are most important not the details) Example: cubic crystals C 11, C 12, C 44 B= (C 11 +2C 12 )/3 Sound waves - v s = (C ij /ρ) 1/2, C ij = appropriate elastic constant, ρ = mass density Physics 460 F 2006 Lect 11 37 Summary of Part I --- 4 Vibrations of atoms Harmonic approximation Exact solution for waves in a crystal Dispersion curves ω k,m where k is the wavevector in Brillouin Zone m = 1,, 3N Counting k states he number of k states = number of cells in crystal N Example in 1 dimension Same ideas in 2 3 dimensions Result Each k state corresponds to a volume of reciprocal space = (2π) 3 /V = V BZ /N (2π/L in 1 dimension) (1/N) Σk inside BZ fl (1/V BZ ) BZ dk Quantization of vibrations For each vibration with frequency ω, the quantum is hω Phonons act like particles Observed in inelastic scattering experiments (neutrons, ) Inelastic scattering is a central idea that is used again in transport Physics 460 F 2006 Lect 11 38 Summary of Part I --- 5 hermal properties Fundamental law of probabilities exp(-e/k B ) Planck distribution for phonons 1 <n> = exp ( hω/ k B ) - 1 Debye Einstein models Debye captures critical features of acoustic modes Heat Capacity C Low : C ~ 3 High : C ~ constant hermal conductivity K Low : K ~ 3 High : K~ 1/ Maximum as function of Summary of Part I Looking Forward What will be important in describing electrons? In order to underst: Metals, insulators, semiconductors,. We need: he structure of the crystal Lattice, basis, Recip. Lattice Electronic states in in periodic crystal Electrons act like waves/particles he same Brillouin zone as for phonons he same counting of states in the BZ Density of states D(E) Conductivity transport of charge he same ideas as for thermal energy j = σ E = σ dv/dx, σ = (1/3) e 2 v L (Ohm s law!) Scattering - essential for resistance ( for Ohm s law) is due to phonons, impurities, And we need to underst the crucial differences between electrons phonons! Specific heat ideas the same but very different result Conductivity. Physics 460 F 2006 Lect 11 39 Physics 460 F 2006 Lect 11 40 7