introduction to thermal transport - icmr · introduction to thermal transport ... ¾radiative heat...
TRANSCRIPT
![Page 1: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/1.jpg)
Introduction to Thermal Transport
Simon PhillpotDepartment of Materials Science and Engineering
University of FloridaGainesville FL 32611
![Page 2: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/2.jpg)
Objectives
•Identify reasons that heat transport is important
•Describe fundamental processes of heat transport, particularly conduction
•Apply fundamental ideas and computer simulation methods to address significant issues in thermal conduction in solids
![Page 3: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/3.jpg)
•Patrick Schelling (Argonne National Lab U. Central Florida)
•Pawel Keblinski (Rensselaer Polytechnic Institute)
•Robin Grimes (Imperial College London)
•Taku Watanabe (University of Florida)
•Priyank Shukla (U. Florida)
•Marina Yao (Dalian University / U. Florida)
Many Thanks to …
![Page 4: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/4.jpg)
Part 1: Fundamentals
![Page 5: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/5.jpg)
Thermal Transport: Why Do We Care?
![Page 6: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/6.jpg)
Apollo 10 Heat Shield
Cross-section of Mercury Heat Shield
Heat Shield: Disposable space vehicles
Ablation of polymeric system
![Page 7: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/7.jpg)
Heat Shield: Space Shuttle
![Page 8: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/8.jpg)
http://www.mse.eng.ohio-state.edu/fac_staff/faculty/padture/padturewebpage/padture/turbine_blade.jpg
Thermal Barrier Coatings for Turbines
![Page 9: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/9.jpg)
SOURCE
DRAIN
GATE
20 nm FINS
Y-K. Choi et al., EECS, U.C. Berkeley
• Scaling → Localized heating → Phonon hotspot• Impact on ESD, parasitic resistances ?
Heat Generation in NanoFETs
DRAIN
ELECTRON RELAXATION
HOTSPOT EDGE
BALLISTICTRANSPORT
D ~ 20 nmQ”’ ~ 1010 eV/nm3/s
![Page 10: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/10.jpg)
n-ty
pe
p-ty
pe
cold junction
hot junctions
Thermoelectrics
![Page 11: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/11.jpg)
Heat Transfer Mechanisms
• Convection
• Conduction
• Radiation
Convection is a mass movement of fluids (liquid or gas) rather than a real heat transfer mechanism (heat transfer is with convection rather than by convection)
Radiative heat transfer is important at high temperature
Conduction is heat transfer by molecular or atomic motionHeat conduction dominates in solids
Three fundamental mechanisms of heat transfer:
![Page 12: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/12.jpg)
http://csep10.phys.utk.edu/astr162/lect/light/radiation.html
RadiationPlanck’s Law
Stefan-Boltzmann Law
E = σΤ4 σ= 5.67 x 10-8 Watts m-2 K-4 Stefan-Boltzmann constant
Wien’s Law
λmax=3x106 /T
![Page 13: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/13.jpg)
Convection
Transport of energy by motion of atoms
Liquids and Gases
![Page 14: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/14.jpg)
Thermal Conduction
Why does his tongue stick to a metal pole?
Would it stick to a wooden pole?
Dumb and Dumber
![Page 15: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/15.jpg)
What is “Ice”?
The Hot Ice Caper
with
Sam Spade
![Page 16: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/16.jpg)
http://wholesale-scales.com
Diamond vs. Cubic Zirconia
Which is which?
Diamond Cubic Zirconia
![Page 17: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/17.jpg)
http://www.sei.co.jp/RandD/itami/e-tool/gif/variety.gif
Tools
![Page 18: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/18.jpg)
Phenomenology of Thermal Conductivity
heatsource
T
x
J = - κ dT/dx
Fourier’s Law
Heat current Thermal conductivity
![Page 19: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/19.jpg)
Thermal Conductivity of Solids
![Page 20: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/20.jpg)
1
YSZ
Isotropic polymers
Thermal conductivity(W/mK)
Phonons/vibrations
10 100
AluminaOriented polymers
phonons
Diamond
phonons
1000
Copper
electrons
Mechanisms of Thermal Conductivity
Electrical conductivity
σ(Cu ) ~ 5 105 (Ω cm)-1
σ(diamond) ~ 10-10 (Ω cm)-1
![Page 21: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/21.jpg)
Electronic Conductivity and Wiedemann-Franz Law
k/σT = L0
k – thermal conductivityσ – electrical conductivityT – temperatureL0 – Lorentz constant
L0 = 2.45 10-8 WΩ/K2
Element k k/σT(W/mK)
Li 71 2.22 10-8Na 138 2.23In 88 2.58Bi 9 3.53Al 238 2.14
Electrons carry energy as they move transport of heat
![Page 22: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/22.jpg)
Crystalline Materials: From Solids to Springs
Heat transport from atomic vibrations
Vibration of spring system similar to vibrations in solids
![Page 23: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/23.jpg)
Harmonic Model
E = ½ kx2
Simple harmonic solid with one and two atoms in the basis acoustic and optical phonons
![Page 24: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/24.jpg)
Long Wavelength Longitudinal Acoustic Phonon
![Page 25: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/25.jpg)
Short Wavelength Longitudinal Acoustic Phonon
![Page 26: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/26.jpg)
Longitudinal Optical Phonon
![Page 27: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/27.jpg)
Acoustic vs. Optical
Which has lower energy?Why?
![Page 28: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/28.jpg)
Transverse Phonons
![Page 29: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/29.jpg)
Longitudinal vs. Transverse Phonons
Which has lower energy?Why?
![Page 30: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/30.jpg)
Schematic dispersion curves for diamond
http://physics.ucsc.edu/groups/condensed/moseley/simulations
Phonons
![Page 31: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/31.jpg)
Phonons and Thermal Conductivity
Dominant phonon – Debye solid
fD = 2.82 kT/h
f [THz]J(
f)
0.00000005
0.0000001
0 2 4 6 8 10 12 14 16 18 20
10K
0.00002
0.00004
0.00006
0 2 4 6 8 10 12 14 16 18 20
100K
f [THz]
J[f]
![Page 32: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/32.jpg)
If phonons did not scatter the thermal conductivity would be infinite?
Why is the Thermal Conductivity Finite?
![Page 33: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/33.jpg)
Phonon-defectPhonon-phonon Phonon-electron
Macroscale ***** *** *
Phonon-boundary
Phonon Scattering Mechanisms
![Page 34: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/34.jpg)
Phonon-phonon Scattering
• Usually l < Λ < L• New physics arises when L is reduced to nanometers
LΛ
λ
Diamond at 300KΛ ~300nmλD~30nm
Blakemore, Solid State Physics
Sample-size dependent thermal conductivity of single crystal LiF
L~7mm
L~1mm
![Page 35: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/35.jpg)
Temperature Dependence
κ ~ 1/3 cv v λ
Low T
Quantum Solid
Cv ~ T3
κ ~ T3
High T:
Phonon-phonon
scattering
λ~ T-α
α ∼ 1
κ ~ T−α
![Page 36: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/36.jpg)
Thermal Conductivity of Solids
![Page 37: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/37.jpg)
Thermal Conductivity and Thermal Diffusivity
k = ρ cp DT
ρ – densityCp – heat capacityDT – thermal diffusivity
![Page 38: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/38.jpg)
Thermal Expansion and Thermal Conductivity
• Thermal expansionα =0 for a harmonic solid
– Thermal expansion ismeasure of anharmonicity
• Thermal Conductivityκ = ∞ for harmonic solid
– Anharmonicity finite κ
![Page 39: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/39.jpg)
Anharmonicity: Thermal Expansion and Thermal Conductivity
Anharmonicitylow high
thermal conductivity
thermal expansion
r E(r)
smaller α
larger α
r0
![Page 40: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/40.jpg)
Thermal Conductivity and Thermal Expansion
Reasonable correlation between thermal conductivity and thermal expansion
![Page 41: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/41.jpg)
What About Glasses?
No long-ranged order no phonons How is heat transported?
We will look at amorphous materials later
![Page 42: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/42.jpg)
Water 0.6Ethylene Glycol 0.25PTFE 0.2Wood 0.2 – 0.4Engine Oil 0.15Fiberglass 0.04Air 0.03Snow 0.05 – 0.25 (T < 0C)Silica Aerogel 0.003
Solids have Much High Thermal Conductivities than Liquids
Low k materials W/mK
Liquid Na - 72 W/mK
![Page 43: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/43.jpg)
Nanofluids
http://www.anl.gov/Media_Center/News/2004/nanofluidsbig.html
![Page 44: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/44.jpg)
Summary
•Radiation vs. Convection vs. Conduction
•Electronic vs. Phonon Conduction
•Physics of Phonons
•Basic Phenomenology of Thermal Conductivity
![Page 45: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/45.jpg)
Part 2: Case Studies in Thermal Conductivity
![Page 46: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/46.jpg)
Preamble: Introduction to Molecular-Dynamics Simulation
![Page 47: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/47.jpg)
• Treat atoms as structureless spheres
• Write down a form by which the atoms interact, V(r)
• Solve Newton’s Second Law:
• F= ma
•Where F = -∇V(r)
![Page 48: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/48.jpg)
Irsee, August 20, 2003
Case Study #1:Thermal Barrier Coatings
![Page 49: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/49.jpg)
http://www.mse.eng.ohio-state.edu/fac_staff/faculty/padture/padturewebpage/padture/turbine_blade.jpg
![Page 50: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/50.jpg)
D. R. Clarke, UCSB
![Page 51: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/51.jpg)
Thermal Conductivity of Oxides
![Page 52: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/52.jpg)
• Non-equilibrium: Add and remove heat
Approach to determination κ
J=-κ (dT/dz)
Heat source Heat sink
[001]
![Page 53: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/53.jpg)
20mol% YSZ
Pure ZrO2
Steady-State Temperature Profile
N
J - -κ T
![Page 54: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/54.jpg)
Temperature and Concentration Dependence of κ
Temperature (K)
Ther
mal
Con
duct
ivity
(W/m
K)
ZrO2
4 mol%
8 mol%
12, 16, 20 mol%
Sim. Expt.(W/mK) (W/mK)
ZrO2(300Κ) κ=8.92 κ=8.22
YSZ κ= 3.0 κ= 2.3 (>500K)
![Page 55: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/55.jpg)
x-polarization
y-po
lari
zatio
n
Phonon Polarization
![Page 56: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/56.jpg)
f= 3THzf=1.6THzHighly polarized - propagon Random polarization - diffusion
![Page 57: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/57.jpg)
Normal Modes in Amorphous Materials
k Propagon
Diffuson
Locon
• Propagons transport heat efficiently
• Diffusons less efficient, no directionality
• Energy in locon mode remains localized unless scattered. True insulating state.
Feldman and Allen (PRB48, 1993):
![Page 58: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/58.jpg)
Si Single crystal
(001) Si GB Superlattice
Amorphous Si
Chemical vs. Structural Disorder
A. Bodapati et al., APL 88 141908 (2006)
![Page 59: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/59.jpg)
ObjectiveIdentify candidate materials for new thermal barrier coatings
CurrentYttria-stabilized zirconia (κ~2 W/m-K)
Performance criteriaLow thermal conductivityhigh thermal expansion coefficient (match alloy substrate)Elastically soft / mechanically compliant (match alloy)Chemical compatibilityMechanical integrity
New Materials for Thermal Barrier Coatings
![Page 60: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/60.jpg)
Unoccupied 8a site
Fluorite vs. Pyrochlore
FluoriteComplete unit cell
Pyrochlore1/8 unit cell
MO2 A2B2O7A2O3-(BO2)2
M 4+O 2-
B 4+A 3+O 2-
ZrO2 Gd2Zr2O7
![Page 61: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/61.jpg)
Comparison with Experiment - Thermal Conductivity
![Page 62: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/62.jpg)
Thermal Conductivity @ 1500K
• varies by factor of 2.2
![Page 63: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/63.jpg)
Analysis
κ =1/3 Cv v λ
κ- thermal conductivityCv - specific heat = 3kB/a3
v - speed of soundλ - mean free path
λ/a = κa2/88kBv
![Page 64: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/64.jpg)
a2
![Page 65: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/65.jpg)
Inverse Speed of Sound
![Page 66: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/66.jpg)
Mean Free Path
• varies by ~30%
![Page 67: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/67.jpg)
Effect of Cation and Oxygen Disorder
Pyrochlore YSZNo disorder Anion and cation disorder
Y2Zr2O7 (ZrO2)2 - Y2O3
κ = 2.27 W/m-K κ =1.96 W/m-K
Disorder decreases thermal conductivity by ~10%
![Page 68: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/68.jpg)
• B ion radius largely determine κA ion radius not as important
- Use B ion to determine thermal conductivity
- Use A ion to tune to other design criteria- chemical compatibility- mechanical stability
Engineering Better Thermal Barriers?
![Page 69: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/69.jpg)
Large B is good forlower thermal conductivitylower bulk modulushigher thermal expansion
Engineering Better Thermal Barriers?
Large B is bad forphase stability
Thermal expansion coefficient Bulk modulus
![Page 70: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/70.jpg)
Pyrochlores for Inert Matrix Fuel
Sickafus, Minervini, Grimes et al. Science 2000
Schelling, Phillpot and Grimes, Phil Mag. Lett , 2004
Zirconates have low antisiteenergies good radiation tolerance
However, zirconates have low thermal conductivity
composite with higher k material
![Page 71: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/71.jpg)
Can Pyrochlores Be Made Even Better Thermal Barriers?
• How low can thermal conductivity go?
- κ =1/3 C v λ- λ ~ 0.31 - 0.45 a- nearest neighbor distance ~0.22a- further reduction of κ by 1/3 - 1/2 possible
• How can this be accomplished?
- alloying to increase cation and oxygen disorder- graded compositions, composites- microstructural control (interfacial resistance)
![Page 72: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/72.jpg)
Case Study #2:Interfacial Thermal Conductivity
![Page 73: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/73.jpg)
Phonon-defectPhonon-phonon Phonon-electron
Macroscale ***** *** *
Phonon-boundary
Phonon Scattering Mechanisms
Nanoscale * *** *****
![Page 74: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/74.jpg)
Interfacial (Kapitza) Thermal ResistanceTe
mpe
ratu
re
distance
dTgb
To
• Kapitza resistance results in temperature discontinuities at interfaces
J = GK ∆T
Three grain boundaries:
• (001) θ=43.6° Σ29 – high angle, high energy
(001) θ=11.4° Σ101 – high angle, high energy
(111) θ=42.1° Σ31 – high angle, high energy
![Page 75: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/75.jpg)
Kapitza Conductance in YSZ
H.-S. Yang et al., Acta Mater. (2002)
k =ko
1 + ko Rk
d
• Rk and ko are obtained by a 2-parameter fit to k(d)
• Gk = 1/Rk
Kapitza Conductance
Kapitza Resistance
![Page 76: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/76.jpg)
Cahill, Goodson, and Majumdar, Jn. Heat Transfer, 124, 223, (2002)
YSZ Yang et al. (2002)
Kap
itza
c ond
u cta
n ce
(MW
/m2 -K
)
• data for nanocrystalline YSZ similar to that for several heterophase systems
Kapitza Conductance
Three Si grain boundaries:
• (001) θ=43.6° Σ29 – high angle, high energy
(001) θ=11.4° Σ101 – high angle, high energy
(111) θ=42.1° Σ31 – high angle, high energy
![Page 77: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/77.jpg)
Egb = 0.64 J/m2
Low-Energy, High-Angle GB: (111) θ=42.10° Σ31
GK = 1.53GW/m2K
Coordination at GBC3: 0%C4: 94%C5: 6%
J = GK ∆T
![Page 78: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/78.jpg)
Typical High-Energy GB: (001) θ=43.60° (Σ29)
Egb = 1.32 J/m2
Coordination at GBC3:8%C4: 82%C5: 10%
GK = 0.85GW/m2K
![Page 79: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/79.jpg)
(001) Σ29 θ=43.60ºEgb = 1.32 J/m2
GΚ = 0.85GW/m2K
(001) θ=11.42° Σ101 Egb = 0.91 J/m2
GΚ = 1.71GW/m2K
(111) Σ31 θ=42.10°Egb = 0.64 J/m2
GΚ = 1.53GW/m2K
Maiti et al:(310) STGBGK = 0.9GW/m2K @575K
(510) STGBGK = 0.8 GW/m2K @ 575K
J = GK ∆T
Interface Conductance
![Page 80: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/80.jpg)
![Page 81: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/81.jpg)
![Page 82: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/82.jpg)
![Page 83: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/83.jpg)
(100) θ=43° Σ29
LA
kz =0.35
High-Frequency LA mode
Diffuse scatteringAcoustic scattering
![Page 84: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/84.jpg)
Frequency Dependence of Phonon Scattering
Acoustic Mismatch Model
Diffuse Mismatch Model
reflection
transmission
![Page 85: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/85.jpg)
Kapitza conductance using MD results for α(ω)
σΚ (T) = (1/Ω) Σ hωλk ∂n(ωλk,T)/∂T ∂ωλk/∂kz αλk
σK (MD) σK (computed) computed/MDGW/m2K
Σ29 0.855 0.457 0.534Σ101 1.710 0.904 0.528
1. Computed results only includes LA, TA modes
2. The Σ101 computed result assumes α(ω) same for TA modes as LA
3. Effects of non-normal incidence not included
![Page 86: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/86.jpg)
S.R.Phillpot and A.J.H. McGaughey, Materials Today, June 2005
300K
~2200
![Page 87: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/87.jpg)
Ultrananocrystalline Diamond (Experiment)
Measured κ = 12 W/mK
for grain size 3~5 nm at 310K
→ GK = 3 GW/m2K
Angadi et al. JAP 99, 114301 (2006)
![Page 88: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/88.jpg)
A Typical Temperature Profile
Σ29(001) θ=43.6°46400 atoms<T>=1000K
∆T=22.5KGK = 9.7 GW/m2K
900
920
940
960
980
1000
1020
1040
1060
1080
1100
0 20 40 60 80 100 120 140 160z [nm]
T [K
]
∆T ∆T
Tersoff potential: 3-body interactions and naturally incorporates sp3, sp2
hybridization
![Page 89: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/89.jpg)
Ultrananocrystalline Diamond (Simulation)
• Average grain size, 3 nm, thickness 2a0, mean temperature 300K
• Thermal conductivity 13.8 W/mK
2
Using 3 ,4.6K
d nmG GW m K
=
∴ =
![Page 90: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/90.jpg)
Comparison of Experiment and Simulation
8.526Simulation(Grain Boundary)
4.614Simulation(Polycrystalline)
31-12Experiment
GK
[GW/m2K]κ
[W/mK]
• Experimental and simulation results compare within the factor of 2.
• κ/κo ~ 1%
• GK of diamond is much higher than that of any other materials.
Note: d = 3 nm, κo=2200 W/mK at room temperature
Cahill et al. JAP 90 5 (2003)
![Page 91: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/91.jpg)
• GK remains fairly constant over a wide range of angle (~20° to 70°)
→ Coordination of atoms in the GBs is very similar
•Higher conductance at low angle (<15°) because of the sparse distribution of dislocation
• Slight increase near 90°because of the symmetry of diamond structure
Kapitza Conductance and Twist Angle
0
5
10
15
20
25
0 10 20 30 40 50 60 70 80 90 100
Twist angle [degree]
Con
duct
ance
[GW
/m2 K
] <T>=1000K
![Page 92: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/92.jpg)
Bonding and Interfacial Conductance: Diamond vs. Si
(001) θ=43.60° (Σ29)
Silicon Diamond
Better bonding across the silicon GB should lead to better thermal transport
3.264.02<C>
0%10%5
18%82%4
80%8%3
2%0%2
CSiCoordination
![Page 93: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/93.jpg)
Diamond vs. Silicon
(001) Σ29 Grain Boundary
Si: (001) Egb = 1.32 J/m2 GK = 0.85GW/m2K
C: (001) Egb = 6.09 J/m2 GK = 8.85 GW/m2K
Kapitza length (equivalent thickness of perfect crystal):
λK = k/GK
Conclude: Diamond GBs 10X better conductors of heat than Si GBs
But: κ (diamond) = 2000 W/m-Kκ(Si) = 150 W/m-K
![Page 94: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/94.jpg)
Normalize Kapitza Length to Lattice Parameter
Final conclusion: Silicon GBs are better conductors of heat than diamond GBs!
3.264.02<C>
0%10%5
18%82%4
80%8%3
2%0%2
CSiCoordination
0
50
100
150
200
250
0 10 20 30 40 50 60 70 80 90
θ [degree]
l K [a
0]
![Page 95: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/95.jpg)
0
50
100
150
200
250
0 10 20 30 40 50 60 70 80 90
θ [degree]
lK [a
0]
0
50
100
150
200
250
0.0 2.0 4.0 6.0 8.0EGB [J/m2]
l K [a
0]Kapitza Length
Kapitza length proportional to GB energy
Extended Read-Shockley fit to Kapitza length
![Page 96: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/96.jpg)
Case Study #3Point Defects
![Page 97: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/97.jpg)
Motivation
• Phonons play a dominant role in thermal transport of electric insulators e.g. silicon and many oxides
• Two most important processes:– Phonon-phonon scattering– Phonon-point defect scattering
• We focus on the phonon scattering from point defects
![Page 98: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/98.jpg)
Classic formula-isotopic effect
P. G. Klemens, Proc. R. Soc. London, Ser. A 68, 1113 (1955)
1/t ~ c1/t ~ f 4
Question:• in what range of frequency• at what concentration of dopantthe formula is applicable ?
t-1 – phonon scattering rateA – material constant ∆m – mass differencec - isotope concentrationf – frequency
The formula was derived from weak scattering.
⋅∆
⋅=− 21 )()]([mmAft C · f 4
![Page 99: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/99.jpg)
Simulation system
• Simulation cell:Maximum used:2×2×6000 unit cellsLz=6000 unit cells (3.268µm)
• Simulation time step size:dt = 0.55 fsec
• Diamond-structured Si • Stillinger-Weber potential
Thickness of doped region ∆z ∆z = z2-z1 (20 - 300 unit cell)
Dopant concentration c0.078 - 1.56 atom%
Phonon wave packet frequency f:1.5 -13THz
z = −Lz/2 z = Lz/2z1 z2
[001]
dopedregion
![Page 100: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/100.jpg)
Simulation system
• Dopants randomly distributed in doped region
• Differ from Si in mass Mdop= 4MSi =112 amu (atomic mass unit)larger than MGe (72.59amu)
– To understand the effects of mass defect– To compare with previous simulations of
superlattices
![Page 101: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/101.jpg)
Calculation procedure
Four steps:• structure creation• initial phonon wave
packet generation– well-defined
longitudinal acoustic phonon
• MD simulation• energy analysis
dopedregion
![Page 102: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/102.jpg)
Phonon propagation and scattering description at atomic scale
• Incident phonon frequency: 2.96THz
• 1.56% dopants in doped region
•Δz = 200 unit cell
![Page 103: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/103.jpg)
3000
t=0
t=26.3 ps
t=60.1 ps
t=201.3 ps
-3000 -100 100
z [a]
Snapshots
Energy trapped in the defect region becomes negligible by ~200 ps
![Page 104: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/104.jpg)
Energy analysis
coefficient of reflected energy R = Ereflect/Etotal
coefficient of trapped energy L = (Etotal - Ereflect - Etran) /Etotal
coefficient of transmitted energy T = Etran/Etotal
R + L + T = 1
R T
L
![Page 105: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/105.jpg)
R T
L
![Page 106: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/106.jpg)
Time evolution of trapped energy
The higher concentration, the longer time to scatter out.
Physically reasonable results
00.10.20.30.40.50.60.70.80.9
1
0 50 100 150 200 250
Time, ps
Frac
tion
of tr
appe
d en
ergy
1.56%
0.25% 2.96THz
![Page 107: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/107.jpg)
Concentration effect
1/t ~ c
2.96THzΔz = 200 unit cell
Basically linear in concentration, but unexplained peak in reflection at 1%-old description
⋅∆
⋅=− 21 )()]([mmAft C · f 4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Atomic fraction of dopants (%)
fract
ion
of e
nerg
y
transmittedreflectedtrapped
well linear when the concentration is less than 0.3%.
![Page 108: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/108.jpg)
Mass effect
50dopants-200a (0.78%)
00.10.20.30.40.50.60.70.80.9
11.1
0 1 2 3 4 5 6 7 8 9 10 11 12 13
Frequency, THz
Frac
tion
of tr
ansm
itted
Ene
rgy
4Msi2Msi
⋅∆
⋅=− 21 )()]([mmAft C · f 4
![Page 109: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/109.jpg)
Frequency effect
T drops quickly with f decreasing. ( f < 3.5 THZ)
T reaches minimum around 3.5 THz
4Msi - 0.78% dopants
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6 7 8 9 10 11 12 13
Frequency, THz
T, R
& L
TRL
![Page 110: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/110.jpg)
Frequency effect
α = 3.82 for Δz=200
α = 5.23for ∆z=40
compared with classic resultα= 4
(assumes Debye DOS)
1/t ~ f α ⋅∆
⋅=− 21 )()]([mmAft C · f 4
-6
-5
-4
-3
-2
-1
00.0 0.5 1.0 1.5
ln f
ln R
1.5 ~ 3.24 THz
∆z = 40
∆z = 200
![Page 111: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/111.jpg)
Phonondensity of state
for pure Si
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
THZ
DO
Stotal LAkxyLAzero LOTA TO
0.78%dopant in 2*2*200 unit cell
00.10.20.30.40.5
0.60.70.80.9
1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Frequency, THz
Tran
smis
sion
coe
ffici
ent,
TTT-LAT-TA
![Page 112: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/112.jpg)
Conclusions
• Combining phonon wave-packet dynamics and MD simulation can provide an effective approach to developatomic-level view of the scattering of phonons from point defects.
• The energy transmission and reflection coefficients are approximately linear in the dopant concentration.
• The transmission (reflection) coefficient is strongly affected by phonon frequency. At low frequency the order of power is around 4.
⋅∆
⋅=− 21 )()]([mmAft C · f 4
![Page 113: Introduction to Thermal Transport - ICMR · Introduction to Thermal Transport ... ¾Radiative heat transfer is important at high temperature ... Heat conduction dominates in solids](https://reader031.vdocument.in/reader031/viewer/2022021820/5aebedf37f8b9a90318da687/html5/thumbnails/113.jpg)
Summary
•Radiation vs. Convection vs. Conduction
•Electronic vs. Phonon Conduction
•Physics of Phonons
•Basic Phenomenology of Thermal Conductivity
•Use of Simulation to Develop Understanding of Phonon-mediated Thermal transport