1. © 2010 Eric Pop, UIUCECE 598EP: Hot Chips Nanoelectronics: –Higher packing density  higher power density –Confined geometries –Poor thermal properties –Thermal resistance at material boundaries Where is the heat generated? –Spatially: channel vs. contacts –Spectrally: acoustic vs. optical phonons, etc. Power Dissipation in Semiconductors 1

2. © 2010 Eric Pop, UIUCECE 598EP: Hot Chips Simplest Power Dissipation Models Resistor: P = IV = V 2 /R = I 2 R Digital inverter: P = fCV 2 Why? 2 R

3. © 2010 Eric Pop, UIUCECE 598EP: Hot Chips Revisit Simple Landauer Resistor 3 µ1µ1 µ2µ2 E µ 1 -µ 2 = qV Ballistic µ1µ1 µ2µ2 E Diffusive Q: Where is the power dissipated and how much? ? I = q/t P = qV/t = IV

4. © 2010 Eric Pop, UIUCECE 598EP: Hot Chips Continuum View of Heat Generation Lumped model: Finite-element model: More complete finite-element model: 4 µ1µ1 µ2µ2 E (phonon emission) (recombination) be careful: radiative vs. phonon-assisted recombination/generation?!

5. © 2010 Eric Pop, UIUCECE 598EP: Hot Chips Most Complete Heat Generation Model Lindefelt (1994): “the final formula for heat generation” 5 Lindefelt, J. Appl. Phys. 75, 942 (1994)

6. © 2010 Eric Pop, UIUCECE 598EP: Hot Chips Computing Heat Generation in Devices Drift-diffusion:  Does not capture non-local transport Hydrodynamic:  Needs some avg. scattering time  (Both) no info about generated phonons  Monte Carlo:  Pros: Great for non-equilibrium transport  Complete info about generated phonons:  Cons: slow (there are some short-cuts) x (m) y (m) H (W/cm 3 ) 6

7. © 2010 Eric Pop, UIUCECE 598EP: Hot Chips Freq (Hz) Energy (meV) Details of Joule Heating in Silicon High Electric Field Hot Electrons (Energy E) Source Gate Drain IBM Heat Conduction to Package  ~ 1 ms – 1 s Wave vector qa/2 E < 50 meV Acoustic Phonons  ~ 0.1ps acoustic (v ac ~ 5-9000 m/s) 10 20 30 40 50 60 Optical Phonons  ~ 10 ps E > 50 meV  ~ 0.1ps optical (v op ≤ 1000 m/s) 7

8. © 2010 Eric Pop, UIUCECE 598EP: Hot Chips Self-Heating with the Monte Carlo Method Electrons treated as semi- classical particles, not as “fluid” Drift (free flight), scatter and select new state Must run long enough to gather useful statistics Main ingredients:  Electron energy band model  Phonon dispersion model  Device simulation: Impurity scattering, Poisson equation, boundary conditions Must set up proper simulation grid 8

9. © 2010 Eric Pop, UIUCECE 598EP: Hot Chips Energy E (eV) Analytic band Full band acoustic optical 20 meV 50 meV Monte Carlo Implementation: MONET Analytic electron energy bands + analytic phonon dispersion First analytic-band code to distinguish between all phonon modes Easy to extend to other materials, strain, confinement Typical MC codes E. Pop et al., J. Appl. Phys. 96, 4998 (2004) Wave vector qa/2 Phonon Freq.  (rad/s) Density of States (cm -3 eV -1 ) OK to use Our analytic approach 9

10. © 2010 Eric Pop, UIUCECE 598EP: Hot Chips Inter-Valley Phonon Scattering in Si Six phonons contribute –well-known: phonon energies –disputed: deformation potentials What is their relative contribution? Rate : Include quadratic dispersion for all intervalley phonons  (q)  = v s q-cq 2 2.0 (TO, 59 meV) 2.0 (LA/LO, 50 meV) 0.3 (TA, 19 meV) 11.0 (LO, 63 meV) 0.8 (LA, 18 meV) 0.5 (TA, 10 meV) g-type f-type Deformation Potentials D p ( 10 8 eV/cm ) g f Jacoboni, 1983 1.5 (TO, 57 meV) 3.0 (LA/LO, 51 meV) 7.0 (LO, 64 meV) 1.5 (LA, 19 meV) Pop, 2004

11. © 2010 Eric Pop, UIUCECE 598EP: Hot Chips Intra-Valley Acoustic Scattering in Si  = angle between phonon k and longitudinal axis Averaged values: D LA =6.4 eV, D TA =3.1 eV, v LA =9000 m/s, v TA =5300 m/s longitudinal (  TA /v TA ) 2 (  LA /v LA ) 2 Herring & Vogt, 1956 Pop, 2004 Yoder, 1993 Fischetti & Laux, 1996 11 SKIP

12. © 2010 Eric Pop, UIUCECE 598EP: Hot Chips Scattering and Deformation Potentials Herring & Vogt, 1956 This work Yoder, 1993 Fischetti & Laux, 1996 Intra-valley Inter-valley Average values: D LA = 6.4 eV, D TA = 3.1 eV (Empirical  u = 6.8 eV,  d = 1eV) Phonon type Energy (meV) Old model * This work (x 10 8 eV/cm) f-TA190.30.5 f-LA5123.5 ** f-TO5721.5 g-TA100.50.3 g-LA190.81.5 ** g-LO63116 ** * old model = Jacoboni 1983 ** consistent with recent ab initio calculations (Kunikiyo, Hamaguchi et al.) (isotropic, average over  ) E. Pop et al., J. Appl. Phys. 96, 4998 (2004) 12 SKIP

13. © 2010 Eric Pop, UIUCECE 598EP: Hot Chips Mobility in Strained Si on Si 1-x Ge x Conduction Band splitting + repopulation Less intervalley scattering Smaller in-plane m t <m l  Larger μ=q  /m * !!! 22 44  E s ~ 0.67x 44 22 66 Bulk Si Strained Si Strained Si on Relaxed Si 1-x Ge x biaxial tension Various Data (1992-2002) Simulation 13

14. © 2010 Eric Pop, UIUCECE 598EP: Hot Chips Computed Phonon Generation Spectrum Complete spectral information on phonon generation rates Note: effect of scattering selection rules (less f-scat in strained Si) Note: same heat generation at high-field in Si and strained Si E. Pop et al., Appl. Phys. Lett. 86, 082101 (2005) 14

15. © 2010 Eric Pop, UIUCECE 598EP: Hot Chips Phonon Generation in Bulk and Strained Si Doped 10 17 Bulk Si Strained Si x=0.3, E=0.2 eV strained Si bulk Si Bulk (all fields) and high-field strained Si Low-field strained Si TA< 0.03 0.02 LA 0.32 0.08 TO 0.09< 0.01 LO 0.56 0.89 Longitudinal optical (LO) phonon emission dominates, but more so in strained silicon at low fields (90%) Bulk silicon heat generation is about 1/3 acoustic, 2/3 optical phonons E. Pop et al., Appl. Phys. Lett. 86, 082101 (2005) 15

16. © 2010 Eric Pop, UIUCECE 598EP: Hot Chips 1-D Simulation: n+/n/n+ Device qV V N+ i-Si (including Poisson equation and impurity scattering) 16

17. © 2010 Eric Pop, UIUCECE 598EP: Hot Chips 1-D Simulation Results 17 L=500 nm 20 nm 100 nm Potential (V) Medici MONET MONET vs. Medici (drift-diffusion commercial code):  “Long” (500 nm) device: same current, potential, nearly identical  Importance of non-local transport in short devices (J. E method insufficient)  MONET: heat dissipation in DRAIN (optical, acoustic) of 20 nm device Heat Gen. (eV/cm 3 /s) LL MONET Medici Error: L/L = 0.10L/L = 0.38L/L = 0.80 MONET Medici

18. © 2010 Eric Pop, UIUCECE 598EP: Hot Chips Heat Generation Near Barriers 18 Lake & Datta, PRB 46 4757 (1992) Heating near a single barrierHeating near a double-barrier resonant tunneling structure

19. © 2010 Eric Pop, UIUCECE 598EP: Hot Chips Heat Generation in Schottky-Nanotubes Semiconducting nanotubes are Schottky-FETs Heat generation profile is strongly influenced by barriers +Quasi-ballistic transport means less dissipation 19 Ouyang & Guo, APL 89 183122 (2006)

20. © 2010 Eric Pop, UIUCECE 598EP: Hot Chips Are Hot Phonons a Possibility?! Hot phonons: if occupation (N) >> thermal occupation Why it matters: added impact on mobility, leakage, reliability Longitudinal optical (LO) phonon “hot” for H > 10 12 W/cm 3 Such power density can occur in drain of L ≤ 20 nm, V > 0.6 V device sourcedrain V = 0.2, 0.4, 0.6, 0.8, 1.0 V L = 20 nm where and 20

21. © 2010 Eric Pop, UIUCECE 598EP: Hot Chips Last Note on Phonon Scattering Rates Note, the deformation potential (coupling strength) is the same between phonon emission and absorption The differences are in the phonon occupation term and the density of final states What if k B T >> ħω (~acoustic phonons)? What if k B T << ħω (~optical phonons)? Sketch scattering rate vs. electron energy: 21

22. © 2010 Eric Pop, UIUCECE 598EP: Hot Chips Sketch of Scattering Rates vs. Energy 22 Γ=1/τ EEħωħω emission ≈ absorption emission absorption k B T » ħω  N q « 1 Γ ~ g(E) ~ E 1/2 in 3-D, etc. k B T « ħω  N q » 1 Γ ~ N q g(E± ħω) ~ (E ± ħω) 1/2 in 3-D Note emission threshold E > ħω