Thermal Properties of Materials Li Shi Department of Mechanical Engineering & Center for Nano and Molecular Science and Technology, Texas Materials Institute The University of Texas at Austin Austin, TX 78712 www.me.utexas.edu/~lishi lishi@mail.utexas.edu

Outline  Macroscopic Thermal Transport Theory– Diffusion -- Fourier’s Law -- Diffusion Equation Microscale Thermal Transport Theory – Particle Transport -- Kinetic Theory of Gases -- Electrons in Metals -- Phonons in Insulators -- Boltzmann Transport Theory Thermal Properties of Nanostructures -- Thin Films and Superlattices -- Nanowires and Nanotubes -- Nano Electromechanical System (NEMS)

Fourier’s Law for Heat Conduction Q (heat flow) Hot Th Cold Tc L Thermal conductivity

Heat Diffusion Equation 1st law (energy conservation) Heat conduction = Rate of change of energy storage Specific heat Conditions: t >> t  scattering mean free time of energy carriers L >> l  scattering mean free path of energy carriers Breaks down for applications involving thermal transport in small length/ time scales, e.g. nanoelectronics, nanostructures, NEMS, ultrafast laser materials processing…

Length Scale l 1 km 1 m 1 mm 1 mm 100 nm 1 nm Aircraft Automobile Human Computer Butterfly 1 mm Fourier’s law Microprocessor Module MEMS Blood Cells 1 mm Wavelength of Visible Light Particle transport l MOSFET, NEMS 100 nm Nanotubes, Nanowires 1 nm Width of DNA

Outline Macroscopic Thermal Transport Theory– Diffusion -- Fourier’s Law -- Diffusion Equation  Microscale Thermal Transport Theory– Particle Transport -- Kinetic Theory of Gases -- Electrons in Metals -- Phonons in Insulators -- Boltzmann Transport Theory Thermal Properties of Nanostructures -- Thin Films and Superlattices -- Nanowires and Nanotubes -- Nano Electromechanical System (NEMS)

Mean Free Path for Intermolecular Collision for Gases D D Total Length Traveled = L Average Distance between Collisions, mc = L/(#of collisions) Total Collision Volume Swept = pD2L Mean Free Path Number Density of Molecules = n Total number of molecules encountered in swept collision volume = npD2L s: collision cross-sectional area

Mean Free Path for Gas Molecules kB: Boltzmann constant 1.38 x 10-23 J/K Number Density of Molecules from Ideal Gas Law: n = P/kBT Mean Free Path: Typical Numbers: Diameter of Molecules, D  2 Å = 2 x10-10 m Collision Cross-section: s  1.3 x 10-19 m Mean Free Path at Atmospheric Pressure: At 1 Torr pressure, mc  200 mm; at 1 mTorr, mc  20 cm

Effective Mean Free Path Wall b: boundary separation Wall Effective Mean Free Path:

Kinetic Theory of Energy Transport u: energy Net Energy Flux / # of Molecules u(z+z) z + z qz q  z through Taylor expansion of u u(z-z) z - z Integration over all the solid angles  total energy flux Thermal conductivity: Specific heat Velocity Mean free path

Questions Kinetic theory is valid for particles: can electrons and crystal vibrations be considered particles? If so, what are C, v,  for electrons and crystal vibrations?

Free Electrons in Metals at 0 K Fermi Energy – highest occupied energy state: Vacuum Level F: Work Function Fermi Velocity: EF Energy Fermi Temp: Band Edge

Effect of Temperature Fermi-Dirac equilibrium distribution for the probability of electron occupation of energy level E at temperature T 1 E F Electron Energy, Occupation Probability, f Work Function, Increasing T = 0 K k T B Vacuum Level

Number and Energy Densities Number density: Energy density: Density of States -- Number of electron states available between energy E and E+dE in 3D

Electronic Specific Heat and Thermal Conductivity in 3D Mean free time: te = le / vF Thermal Conductivity Electron Scattering Mechanisms Defect Scattering Phonon Scattering Boundary Scattering (Film Thickness, Grain Boundary) e Temperature, T Defect Scattering Phonon Scattering Increasing Defect Concentration Bulk Solids

Thermal Conductivity of Cu and Al Matthiessen Rule: Electrons dominate k in metals

Afterthought Since electrons are traveling waves, can we apply kinetic theory of particle transport? Two conditions need to be satisfied: Length scale is much larger than electron wavelength or electron coherence length Electron scattering randomizes the phase of wave function such that it is a traveling packet of charge and energy

Crystal Vibration Interatomic Bonding Equation of motion with nearest neighbor interaction Solution 1-D Array of Spring Mass System

Dispersion Relation Group Velocity: Frequency, w Speed of Sound: p/a Wave vector, K p/a Longitudinal Acoustic (LA) Mode Transverse Acoustic (TA) Mode Group Velocity: Speed of Sound:

Two Atoms Per Unit Cell Optical Vibrational Modes LO TO Frequency, w Lattice Constant, a xn yn yn-1 xn+1 Optical Vibrational Modes LO TO Frequency, w TA LA Wave vector, K p/a

Phonon Dispersion in GaAs

Energy Quantization and Phonons Total Energy of a Quantum Oscillator in a Parabolic Potential n = 0, 1, 2, 3, 4…; w/2: zero point energy Phonon: A quantum of vibrational energy, w, which travels through the lattice Phonons follow Bose-Einstein statistics. Equilibrium distribution: In 3D, allowable wave vector K:

Lattice Energy in 3D p: polarization(LA,TA, LO, TO) K: wave vector Dispersion Relation: Energy Density: Density of States: Number of vibrational states between w and w+dw in 3D Lattice Specific Heat:

Debye Model Debye Approximation: Debye Density of States: Frequency, w Wave vector, K p/a Debye Approximation: Debye Density of States: Specific Heat in 3D: Debye Temperature [K] In 3D, when T << qD,

Phonon Specific Heat 3kBT Diamond Each atom has a thermal energy of 3KBT Specific Heat (J/m3-K) C  T3 Classical Regime Temperature (K) In general, when T << qD, d =1, 2, 3: dimension of the sample

Phonon Thermal Conductivity Phonon Scattering Mechanisms Kinetic Theory Boundary Scattering Defect & Dislocation Scattering Phonon-Phonon Scattering Decreasing Boundary Separation l Increasing Defect Concentration Phonon Scattering Defect Boundary 0.01 0.1 1.0 Temperature, T/qD

Thermal Conductivity of Insulators Phonons dominate k in insulators

Drawbacks of Kinetic Theory Assumes local thermodynamics equilibrium: u=u(T) Breaks down when L  ; t  t Assumes single particle velocity and single mean free path or mean free time. Breaks down when, vg(w) or t(w) Cannot handle non-equilibrium problems Short pulse laser interactions High electric field transport in devices Cannot handle wave effects Interference, diffraction, tunneling

Boltzmann Transport Equation for Particle Transport Distribution Function of Particles: f = f(r,p,t) --probability of particle occupation of momentum p at location r and time t Equilibrium Distribution: f0, i.e. Fermi-Dirac for electrons, Bose-Einstein for phonons Non-equilibrium, e.g. in a high electric field or temperature gradient: Relaxation Time Approximation t Relaxation time

Energy Flux Energy flux in terms of particle flux carrying energy: q v Energy flux in terms of particle flux carrying energy: dk q k f Vector Integrate over all the solid angle: Scalar Integrate over energy instead of momentum: Density of States: # of phonon modes per frequency range

Continuum Case BTE Solution: Quasi-equilibrium Direction x is chosen to in the direction of q Energy Flux: Fourier Law of Heat Conduction: t(e) can be treated using Callaway method (Phys. Rev. 113, 1046) If v and t are independent of particle energy, e, then  Kinetic theory:

At Small Length/Time Scale (L~l or t~t) Define phonon intensity: From BTE: Equation of Phonon Radiative Transfer (EPRT) (Majumdar, JHT 115, 7): Heat flux: Acoustically Thin Limit (L<<l) and for T << qD Acoustically Thick Limit (L>>l)

Outline Macroscopic Thermal Transport Theory – Diffusion -- Fourier’s Law -- Diffusion Equation Microscale Thermal Transport Theory – Particle Transport -- Kinetic Theory of Gases -- Electrons in Metals -- Phonons in Insulators -- Boltzmann Transport Theory  Thermal Properties of Nanostructures -- Thin Films and Superlattices -- Nanowires and Nanotubes -- Nano Electromechanical System

Thin Film Thermal Conductivity Measurement 3w method (Cahill, Rev. Sci. Instrum. 61, 802) Metal line Thin Film L 2b V I ~ 1w T ~ I2 ~ 2w R ~ T ~ 2w V~ IR ~3w I0 sin(wt) Substrate

Silicon on Insulator (SOI) Ju and Goodson, APL 74, 3005 IBM SOI Chip Lines: BTE results Hot spots!

Thermoelectric Cooling No moving parts: quiet and reliable No Freon: clean

Thermoelectric Figure of Merit (ZT) Coefficient of Performance where Seebeck coefficient Electrical conductivity Thermal conductivity Temperature Bi2Te3 Freon TH = 300 K TC = 250 K

ZT Enhancement in Thin Film Superlattices SiGe superlattice (Shakouri, UCSC) Increased phonon-boundary scattering decreased k + other size effects  High ZT = S2sT/k Si Barrier Ge Quantum well (QW) Ec E Ev x

Thermal Conductivity of Si/Ge Superlattices k (W/m-K) Bulk Si0.5Ge0.5 Alloy Circles: Measurement by D. Cahill’s group Lines: BTE / EPRT results by G. Chen Period Thickness (Å)

Superlattice Micro-coolers Ref: Venkatasubramanian et al, Nature 413, P. 597 (2001)

Nanowires p 22 nm diameter Si nanowire, P. Yang, Berkeley Increased phonon-boundary scattering Modified phonon dispersion  Suppressed thermal conductivity Ref: Chen and Shakouri, J. Heat Transfer 124, 242 Hot p Cold

Thermal Measurements of Nanotubes and Nanowires Themal conductance: G = Q / (Th-Ts) Suspended SiNx membrane Long SiNx beams I Q Pt resistance thermometer Kim et al, PRL 87, 215502 Shi et al, JHT, in press

(Berkeley Device group) Si Nanowires Si Nanotransistor (Berkeley Device group) Gate Source Drain Nanowire Channel D. Li et al., Berkeley Symbols: Measurements Lines: Modified Callaway Method Hot Spots in Si nanotransistors!

ZT Enhancement in Nanowires Top View Nanowire Al2O3 template Nanowires based on Bi, BiSb,Bi2Te3,SiGe k reduction and other size effects  High ZT = S2sT/k Bi Nanowires Ref: Phys. Rev. B. 62, 4610 by Dresselhaus’s group

Nanotube Nanoelectronics TubeFET (McEuen et al., Berkeley) Nanotube Logic (Avouris et al., IBM)

Thermal Transport in Carbon Nanotubes Hot p Cold Few scattering: long mean free path l Strong SP2 bonding: high sound velocity v  high thermal conductivity: k = Cvl/3 ~ 6000 W/m-K Below 30 K, thermal conductance  4G0 = ( 4 x 10-12T) W/m-K, linear T dependence (G0 :Quantum of thermal conductance) Heat capacity

Thermal Conductance of a Nanotube Mat Linear behavior 25 K Ref: Hone et al. APL 77, 666 Estimated thermal conductivity at 300K: ~ 250 << 6000 W/m-K  Junction resistance is dominant Intrinsic property remains unknown

Thermal Conductivity of Carbon Nanotubes CVD SWCN CNT An individual nanotube has a high k ~ 2000-11000 W/m-K at 300 K k of a CN bundle is reduced by thermal resistance at tube-tube junctions The diameter and chirality of a CN may be probed using Raman spectroscopy

Nano Electromechanical System (NEMS) Thermal conductance quantization in nanoscale SiNx beams (Schwab et al., Nature 404, 974 ) Quantum of Thermal Conductance Phonon Counters?

Summary Macroscopic Thermal Transport Theory – Diffusion -- Fourier’s Law -- Diffusion Equation Microscale Thermal Transport Theory – Particle Transport -- Kinetic Theory of Gases -- Electrons in Metals -- Phonons in Insulators -- Boltzmann Transport Theory Thermal Properties of Nanostructures -- Thin Films and Superlattices -- Nanowires and Nanotubes -- Nano Electromechanical System (NEMS)