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Ronggui Yang Department of Mechanical Engineering ECME 136, 427 UCB University of Colorado Boulder, CO 80309-0427 Tel:

Nano-to-Macroscale Transport Processes (Microscale Heat Transfer) A Quick Review. Ronggui Yang Department of Mechanical Engineering ECME 136, 427 UCB University of Colorado Boulder, CO 80309-0427 Tel: (303) 735-1003, Fax: (303) 492-3498 Email: Ronggui.Yang@Colorado.Edu

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Ronggui Yang Department of Mechanical Engineering ECME 136, 427 UCB University of Colorado Boulder, CO 80309-0427 Tel:

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  1. Nano-to-Macroscale Transport Processes (Microscale Heat Transfer)A Quick Review Ronggui Yang Department of Mechanical Engineering ECME 136, 427 UCB University of Colorado Boulder, CO 80309-0427 Tel: (303) 735-1003, Fax: (303) 492-3498 Email: Ronggui.Yang@Colorado.Edu http://spot.colorado.edu/~yangr

  2. Nano-to-Macroscale Transport Processes (Microscale Heat Transfer) This course focuses on understanding thermal energy transport across all scales and particularly when device dimensions approaches the fundamental lengths-scales of the charge and energy carriers in nanostructures. The course will address size effects on thermal and fluid transport in nanostructures and how to possibly engineer novel effective transport properties. Moreover, the current state of the art developments in the microscale thermal transport field will be reviewed. Topics include the energy levels, the statistical behavior and internal energy, energy transport in the forms of waves and particles, scattering and energy conversation processes, Boltzmann equation and derivation of classical laws, deviation from classical laws at nanoscale and their appropriate descriptions, with applications in nanotechnology and microtechnology. Textbook Gang Chen, Nanoscale Energy Transport and Conversion: A Parallel Treatment of Electrons, Molecules, Phonons, and Photons. New York: Oxford University Press, 2005. ISBN: 019515942X. Grading: Bi-weekly homework 30%, midterm 25%, final exam 45% Learning Goals Understand, Analyze, Innovate

  3. Course Objectives • Students in this course will: • Gain an understanding of the fundamental elements of solid-state physics. • Develop skills to derive continuum physical properties from sub-continuum principles. • Apply statistical and physical principles to describe energy transport in modern small-scale materials and devices.

  4. No Pains, No Gains! Required Textbook Gang Chen, Nanoscale Energy Transport and Conversion: A Parallel Treatment of Electrons, Molecules, Phonons, and Photons. New York: Oxford University Press, 2005. ISBN: 019515942X. Recommended Books C. Kittel, Introduction to Solid State Physics, 7th Ed., Wiley, 1996. N.W. Ashroft and N.D. Mermin, Solid State Physics, Brooks Cole, 1976 C. Kittel and H. Kroemer, Thermal Physics, 2nd Ed., Freeman and Company, 1980. M. Lundstrom, Fundamentals of Carrier Transport, 2nd Ed, Cambridge University Press, 2000. Z.M. Zhang, Nano/Microscale Heat Transfer, Wiley, 2007. Lecture Notes References MIT Courses: 2.57 Nano-to-Macroscale Transport Processes 6.728 Quantum Mechanics and Statistical Mechanics 6.730 Solid State Physics 6.720 Physics of Semiconductor Devices 6.772 Physics of Semiconductor Compounds and Devices Lecture Notes by colleagues in other universities

  5. red blood cell ~5 m (SEM) diatom 30 m DNA proteins nm Simple molecules <1nm bacteria 1 m nm m mm 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 SOI transistor width 0.12m semiconductor nanocrystal (CdSe) 5nm Circuit design Copper wiring width 0.2m Nanometer memory element (Lieber) 1012 bits/cm2 (1Tbit/cm2) IBM PowerPC750TM Microprocessor 7.56mm×8.799mm 6.35×106 transistors Length Scales

  6. Revolution and Evolution in Electronics

  7. Biology Nano-Bio Interface Nanoelectronics Conventional Spintronics Molectronics Nanotubes NANO Nanosciences Nanostructures Nanomaterials Nanophotonics Photonic Crystals Plasmonic Photonics Energy Conversion, Storage, and Transportation Nanotechnology Landscape

  8. carriers wavelength electrons phonons photons air molecules 10-100 nm 1 nm 0.1-10 0.01 nm Characteristic Lengths of Energy Carriers Room Temperature

  9. Microscopic Picture of Thermal Transport Th Tc Air molecules Phonon gas ε = nhν Free electron model Th Th Tc Tc Free electron Atom core Conduction -- random motion of energy carriers Gas Molecules Electrons Phonons

  10. qx Hot Cold x vxt x Simple Kinetic Theory Taylor Expansion: , local thermodynamics equilibrium: u=u(T)

  11. Heat Conduction in Solids • Heat is conducted by electrons and phonons. • k is determined by electron-electron, phonon-phonon, and electron-phonon collisions. Hot Hot Cold p - Cold To understand transport and energy conversion, we need to know: How much energy/momentum can a particle have? How many particles have the specified energy E? How fast do they move? How far can they travel? How do they interact with each other?

  12. Microstructure of solids • Atomic bonds • Crystalline, polycrystalline, amorphous materials • Bravais lattice, reciprocal lattice, Miller indices

  13. U: Potential Energy a+b x Energy Repulsion = ¥ U Harmonic Force Approximation Interatomic Distance Attraction Equilibrium Position Transmission wave Reflection wave Energy barrier U0 ENERGY AND Incoming wave WAVEFUNCTION n=3 n=2 n=1 x U=0 u r Quantum Mechanics 101 Problem 2.10 Next Lecture

  14. Vibrations in solids • Crystal vibrations, dispersion relations • Quantization and phonons • Phonon branches and modes • Lattice specific heat • Thermal expansion • Phonon scattering • Heat conduction

  15. Lattice Constant, a xn+1 yn-1 xn yn 1-D Lattice with Diatomic Basis Consider a linear diatomic chain of atoms (1-D model for a crystal like NaCl): In equilibrium: Applying Newton’s second law and the nearest-neighbor approximation to this system gives a dispersion relation with two “branches”:

  16. 1-D Lattice with Diatomic Basis: Results -(k)   0 as k  0 acoustic modes (M1 and M2 move in phase) +(k)   max as k  0 optical modes (M1 and M2 move out of phase) These two branches may be sketched schematically as follows: optical gap in allowed frequencies acoustic

  17. 3-D Solids: Phonon Dispersion In a real 3-D solid the dispersion relation will differ along different directions in k-space. In general, for a p atom basis, there are 3 acoustic modes and p-1 groups of 3 optical modes, although for many propagation directions the two transverse modes (T) are degenerate.

  18. Density of States wD Phonon Frequency w wE 3-D Solids: Heat Capacity Einstein Model Debye Model (a) Optical Phonons Acoustic Phonons Density of States Phonon Frequency w

  19. Thermal Expansion • If the curve is not symmetric, the average position in which the atom sits shifts with temperature. • Bond lengths therefore change (usually get bigger for increased T). • Thermal expansion coefficient is nonzero.

  20. k = C v l 1 3 Thermal Conductivity C ~ constant l Phonon mfp k lum ~eQ/ T Specific heat Phonon group velocity C ~ T d l= vt Matthiessen Rule: lst ~ lum lboundary ~ constant limpurity ~ weak dependance on T T Low T: High T: Umklapp phonon scattering: lum ~ eQ/ T If T > Q, C ~ constant If T << Q, C ~ T d (d: dimension) Specific heat :

  21. Electrons in solids • Free electron theory of metals • Fermi-Dirac statistics • Electron structure and quantization • Band structures of metals, semiconductors, and insulators • Electron scattering and transport

  22. The Free Electron Gas Model Plot U(x) for a 1-D crystal lattice: Simple and crude finite-square-well model: U U = 0 Can we justify this model? How can one replace the entire lattice by a constant (zero) potential?

  23. Properties of the FEG By using periodic boundary conditions for a cubic solid with edge L and volume V = L3, we define the set of allowed wave vectors: This shows that the volume in k-space per solution is: And thus the density of states in k-space is: Since the FEG is isotropic, the surface of constant E in k-space is a sphere. Thus for a metal with N electrons we can calculate the maximum k value (kF) and the maximum energy (EF). ky kx kz Fermi sphere

  24. Density of States N(E) We often need to know the density of electron states, which is the number of states per unit energy, so we can quickly calculate it: The differential number of electron states in a range of energy dE or wavevector dk is: This allows: Now using the general relation: we get:

  25. Heat Capacity of the Quantum-Mechanical FEG Quantum mechanics showed that the occupation of electron states is governed by the Pauli exclusion principle, and that the probability of occupation of a state with energy E at temperature T is: where  = chemical potential  EF for kT << EF

  26. Heat Capacity of Metals: Theory vs. Expt. at low T Very low temperature measurements reveal: Results for simple metals (in units mJ/mol K) show that the FEG values are in reasonable agreement with experiment, but are always too high: The discrepancy is “accounted for” by defining an effective electron mass m* that is due to the neglected electron-ion interactions

  27. Energy Bands and Energy Gaps in a Periodic Potential Recall the electrostatic potential energy in a crystalline solid along a line passing through a line of atoms: bare ions solid Along a line parallel to this but running between atoms, the divergences of the periodic potential energy are softened: U x A simple 1–D mathematical model that captures the periodicity of such a potential is:

  28. Metals, Insulators, and Semiconductors • Metals are solids with incompletely filled energy bands • Semiconductors and insulators have a completely filled or empty bands and an energy gap separating the highest filled and lowest unfilled band. Semiconductors have a small energy gap (Eg < 2.0 eV).

  29. Energy Bands in 3-D Solids 6 Si Cu 5 GaAs 4 3 2 Energy (eV) Direct Band Gap Indirect Ban Gap 1 Eg=1.42 eV Eg=1.12 eV E f 0 -1 -2 -3 -4 G G G L [111] [100] X [111] [100] X [111] [100] X L L Fermi Level

  30. v v - d k e E = dt h Dynamics of Electrons in a Band Now we see that the external electric field causes a change in the k vectors of all electrons: E If the electrons are in a partially filled band, this will break the symmetry of electron states in the 1st BZ and produce a net current. But if they are in a filled band, even though all electrons change k vectors, the symmetry remains, so J = 0. When an electron reaches the 1st BZ edge (at k = /a) it immediately reappears at the opposite edge (k = -/a) and continues to increase its k value. kx v As an electron’s k value increases, its velocity increases, then decreases to zero and then becomes negative when it re-emerges at k = -/a!! kx Thus, an AC current is predicted to result from a DC field! (Bloch oscillations)

  31. Physical Meaning of the Band Effective Mass The effective mass is inversely proportional to the curvature of the energy band. Near the bottom of a nearly-free electron band m* is approximately constant, but it increases dramatically near the inflection point and even becomes negative (!) near the zone edge.

  32. Photoelectric Effect Blackbody Radiation Emissive Power Light Continuum Theory Metal Electrodes Experiments Wavelength . Current Gas Absorption Spectrum de Broglie’s Materials Wave 656.5 nm 486.3 nm 364.7 nm 434.2 nm Wave vs. Particle?

  33. qx Hot Cold x Q1->2 vxt x 1 2 Q2->1 x How wave-like energy carriers transport energy? Generalizing the simple kinetic theory Landauer formulism Net Energy Flux Energy Storage (J/m3) The key is how to calculate energy transmissivity:

  34. Reflection and Refraction: EM Waves n2<0? The Snell’s law indicated momentum conservation in x-direction If n2<0, n1>0, the refraction light will be bended as the right figure. Metamaterials Reflectivity Transmissivity What happens if n1>n2?

  35. Low-Dimensional Electrons and Phonons Reflection and Transmission DENSITY OF STATES n=2 r12f1 t12f1 n=1 f1 r21f2 t21f2 ENERGY f2 (a) Electrons in Quantum Well Single Interface Multiple Interfaces: Wave Effects (b) Phonons in Superlattice Mode Coupling – Multiple Carriers

  36. carriers wavelength electrons phonons photons air molecules 10-100 nm 1 nm 0.1-10 0.01 nm Characteristic Lengths of Energy Carriers

  37. Transition from Quantum to Classical: Nanowires Wavelength vs. Diameter Typical Nanowire Specular Wavelength vs. Roughness Specularity (Ziman) Diffuse scattering  Incoherence Diffuse 10-3 10-2 10-1 100 10+1 Roughness / Wavelength

  38. Statistical Transport Theories • Time and length scales • Boltzmann transport equation • Carrier scattering • Moments of the BTE

  39. Modeling Approaches

  40. From Equilibrium To Nonequilibrium Boltzmann Transport Equation • The distribution function can change by… • a spatial inflow/outflow of carriers (recall that the distribution function describes the probability of finding a carrier at a particular spatial location r, among other things) • an inflow/outflow of carriers in momentum space by means of a force (recall also that the distribution function describes the probability of finding a carrier with a particular momentum p) • scattering of carriers into or out of a particular location in position-momentum space • the presence of a source or sink of carriers

  41. Forms of the Boltzmann Transport Equation (BTE) • One dimension • General, multi-dimensional • PDE in 7 dimensions (3 spatial, 1 time, 3 momentum—but wait! there’s more)

  42. Scattering Term • Scattering can increase the distribution function f(p,…) by in-scattering from p’ to p • It can also decrease the distribution function by out-scattering from p to p’ • Net result • Normally, f(p)<<1. Thus, • S(p’,p) represents the probability per unit time that a carrier of momentum p’ will scatter in a state with momentum p • The scattering term’s sum can be converted to an integral • Thus, the BTE is a 7-dimensional integro-differential equation!

  43. 1.0 0.01 0.1 Phonon Scattering and Mean Free Path Phonon Scattering Mechanisms • Boundary Scattering • Defect & Dislocation Scattering • Phonon-Phonon Scattering Decreasing Boundary Separation l Increasing Defect Concentration PhononScattering Defect Boundary Temperature, T/qD

  44. Carrier Scattering • Carrier Scattering Mechanisms • Defect Scattering • Phonon Scattering • Boundary Scattering (Film Thickness, • Grain Boundary) Electronic Bandstructure • Intra-valley • Inter-valley • Inter-band

  45. Momentum and Energy Relaxation Time • Some scattering events only slightly perturb the incident particle’s direction • Thus, several collisions may be necessary to randomize the particle’s direction • where  is the polar angle between incident and scattered momentum vectors • Some scattering events (called elastic) alter momentum without altering energy • Thus, many scattering events may be needed to relax a particle’s energy

  46. Moments of the Distribution Function • We generally seek to determine macroscopic quantities from the statistical distribution function • carrier number density • electric current • average energy • These averages are obtained via weighting the distribution function and summing over all states • where n is a general averaged quantity that depends on the form of 

  47. Balance Equations • Carrier concentration • Carrier momentum • Carrier energy

  48. Transport Processes in Transistors

  49. Transport Regime Maps in Transistors

  50. Thermoelectrics COLD SIDE I I Phonons L=10-100 nm l=1 nm - Electrons L=10-100 nm l=10-50 nm + N P I INSULATOR SEMICONDUCTOR HOT SIDE SEMIMETAL S METAL s ZT k Reducing k In Bulk Materials Alloy, 1950s (Ioffe) Nanostructured Materials Wanted: Phonon Glass / Electron Crystal • Interfaces scatter phonons to reduce thermal conductivity • Quantum effects to improve electron behaviors carrier concentration

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