INTRODUCTION Time and Length Scales Energy Transport in Microelectronics Ultra Short Pulse Laser Processing Transport Laws and Thermal Transport Modeling

Typical atom, One AngstromElectron radiusAnimal cell mean diameter, Fine dustLength ScalesHuman hairmicronanopicoDiameter of AIDS virusfemtoattomilliMean-free path of phonon in Si at T = 300 K

0Time, s10-1810-1510-610-3Time ScalesmicronanopicofemtoattomilliX-Rays ignition time in nuclear fusion reactorPhonon intercollision period in Si at T = 300 KOscillation period of lattice vibration (acoustic phonons) of aluminum oxideLight travels one micrometer10-12Thermal diffusion10-9

101 () 1020 () 102 () 1024 ()103 ()1028 () 104 () 1032 () 108 () 1036 () 1012 () 1040 () 1016 () 1044 () 1024 Yotta 1027 Taxo Numbering Systems (1)

10-6 Micro10-7 10-8 10-9 Nano10-10 10-11 10-12 Pico10-13 10-14 10-15 Femto10-16 10-17 10-18 Atto10-19 10-20 10-21 ZeptoNumbering Systems (2)

Significance of Length Scales (1)Pond skaterWater stride

Significance of Length Scales (2)

Molecule-Based GearSignificance of Length Scales (3)

Body force to Surface tension Ratio Scaling law: Reynolds numberSignificance of Length Scales (4)

SOI TransistorChannel length current: 65 nmphonon mean free path in Si at 300 K: O(100 nm)

dphononLphononPhonon-Boundary ScatteringLSi@300K = 300 nm

Thermal Conductivity of Silicon

4.2 J/cm2 @ 3.3 nsSteel foil100 mm in thicknessNanosecond Machining Processsurface debrisrecast layerejected molten dropslong pulse laser beamdamage to adjacent structure

0.5 J/cm2 @ 200 fsno surface debrisno recast layerultrafast laser pulsesno damage to adjacent structuresplasma plumeno melt zoneno microcracksno shock wavehot, dense ion /electron soupno heat transfer to surrounding materialFemtosecond Machining Process

Non-equilibrium PhenomenonCarriers (Electron-Hole Pairs)Silicon Lattice (Phonons)Laser Irradiation Electron relaxation time ~ 100 fs Within this time period, electrons do not lose energy to phonons. tC-L ~ 0.5 ps

1) tc : collision time or duration of collision smallest time scale on the order of the wavelength of the carrier divided by the propagation speed for phonons 100 fs (Si at 300 K)Time Scales (1)2) t : average time between collisions or mean free time For time scales t < t, carriers travel ballistically and the evolution of the system depends strongly on the details of the initial state. not relaxation time since it takes several collisions to reach equilibrium generally t >> tc

3) tr : relaxation time (collision induced equilibrium) associated with local thermodynamic equilibrium equilibrium achieved in 5 to 20 collisions, tr > tTime Scales (2)

Length Scales2) L : mean free path associated mean free time between collision, L = t3) lr : associated with relaxation time the characteristic size of a volume over which local thermodynamic equilibrium can be defined typically l < L < lr 1) l : wavelength of the energy carrier associated with the collision process shortest length scale

1) When L ~ l Wave phenomena such as diffraction, tunneling, and interference are important. Photon : wave optics based on Maxewlls equation Electrons and phonons: quantum transport lawsTransport Laws (1)2) When L ~ L, lr and t >> t, tr Transport is ballistic in nature and local thermodynamic equilibrium cannot be defined. This transport is nonlocal in space. One has to resort to time-averaged statistical particle transport equations.Eq. Phonon Radiative Transfer

3) When L >> L, lr and t ~ t, tr Time-dependent terms cannot be averaged. Approximation of local thermodynamic equilibrium can be assumed over space. The nonlocality is in time but not in space.4) When both L ~ L, lr and t ~ t, tr Statistical transport equations in full form should be used. No spatial or temporal averages can be made.Transport Laws (2)Hyperbolic Heat Eq.BTE Based Eq.

5) When both L >> L, lr and t >> t, tr Local thermodynamic equilibrium can be applied over space and time, leading to macroscopic transport laws such as the Fourier law.Transport Laws (3)

Thermal Transport ModelingrEq. Phonon Radiative TransferFouriers LawHyperbolicHeat Eq.Boltzmann Transport Eq.Molecular DynamicslrL lrLength Scalet r cTime Scale

fspsms1 nm10 nm100 nm1 mm10 mmClassical MD QMD continuum nsMD simulation records Limitation of MD

Boltzmann Transport Equation (BTE)Drift termScattering termAcceleration term (~ 0 for phonons)BTE applies to all ensembles of particles : electrons, ions, phonons, photons, gas molecules

Equilibrium distribution Relaxation-time approximation Relaxation Time Approximation Maxwell-Boltzmann for gas molecules Fermi-Dirac for electrons Bose-Einstein for photons or phonons

TitleUltrashort Pulse Laser ProcessingCarriers (Electron-Hole Pairs)Silicon Lattice (Phonons)Laser Irradiation

Carrier Number DensityCarrier TemperatureLattice Temperaturewhere,Auger recombinationCarrier/lattice energy transportThermal diffusionLaser absorptionTwo-Temperature Equation

Auger RecombinationECEv123 1st carrier and 2nd carrier of same type collide instantly annihilating the electron-hole pair (1st and 3rd carriers) The energy lost in the annihilation process is given to the 2nd carrier 2nd carrier gives off a series of phonons until its energy returns to equilibrium energy (E = EC)Very important at high carrier concentrationNon-radiative, thermal processRelated to the two-peak structure of carrier temperature

Maximum values of carrier and lattice temperatures, and carrier number density for different laser pulses when = 790 nm and J = 3.82 mJ/cm2 for case 1Maximum Temperatures

Transient behaviors of the carrier and lattice temperatures,and the carrier number density for different laser pulses when = 530 nm and J = 50.0 mJ/cm2 for case 2Two peaksOne peakAuger recombinationLaser Pulse Effect

Small is more powerful !!!

Paradigm Shift

The mean free path, L, of the phonon energy carriers in silicon is approximately 300 nm at room temperature and is comparable to the 200 nm characteristic length scales of current generation transistors and is larger than the 10 nm 100 nm heat deposition region because of device scaling. In this case, the classical diffusion theory fails to accurately predict temperature distributions in transistors. The Phonon Boltzmann Transport Equation (BTE) shown below must be solved to capture the microscale heat transport effects.

BTE is a general transport equationNavier-Stokes, Fourier, Ohm, are special cases of simplified BTE