1. Lecture 4.0 Properties of Metals

2. Importance to Silicon Chips Metal Delamination –Thermal expansion failures Chip Cooling- Device Density –Heat Capacity –Thermal Conductivity Chip Speed –Resistance in RC interconnects

3. Electrical Current Flow of Charged Particles due to applied voltage –Solids Ions/holes are large and slow electrons are small and fast –Electrons are often responsible for conduction

4. Ohm's Law Current density, J=I/A=  =  /  –  =electric field[V/cm] –  =Conductivity,  [=1/  ] =Resistivity –  =n  e,  =mobility, e=electron charge, n=#/vol. Resistance, R=  L/A V=IR

5. Metal Conduction Drude’s theory –electron scattering by lattice Mobility,  e  /m e –  = average time between collisions of electron with ions Bloch’s Quantum theory –no electron scattering in perfect lattice only in a imperfect lattice Scattering –lattice vibrations –impurities –dislocations

6. Remember Molecular Orbitals New Energy –Bonding –Anti Bonding 1s  

7. Energy Bands

8. Partially Filled

9. Distribution of Electrons in Band Fermi-Dirac distribution Probability, –F(E)=1/(exp{[E-E f ]/k B T}+1) –E f is the Fermi Energy

10. Fermi Energy

11. Work Function

12. Fermi-Dirac Probability Distribution

13. Density of States- 3D Schrodinger Eq.

14. Electron Filling in Band- density of occupied states

15. Eletrical Conductivity  =n  e  =mobility, e=electron charge, n=#/vol.  =(N/V) F(E)G(E) e 2  /m e,

16. Thermal Properties - Chapter 7 Thermal Conductivity Thermal Expansion Heat Capacity Thermoelectric effect –thermocouple

17. Thermal Properties - Chapter 7 Thermal Vibrations-phonons –Displacement, x max =(3k B T/Ya o ) 1/2 – Y a o is the spring constant Thermal Expansion –  (  l/l o )(1/  T), also volume->(  V/V o )(1/  T) Heat Capacity –C p =1/2 k B T per degree of freedom –6 degrees of freedom per ion,  C p =3R kinetic and potential Variation of Conductivity with Temp. d  /dT

18. Thermal Expansion

19. Heat Capacity -Effect of Phonons/electrons Einstein Model Debye Model Electrons –density of occupied states E n =(n+1/2)h  = h  /(exp(h  /k B T)-1) g(  )=  2 V/(2  2 v 3 )

20. Heat Capacity of Electrons

21. Heat Capacity

22. Thermal Conduction Transport of Phonons (vibrations) k thermal /(  T)=constant –thermal conductivity scales with electrical conductivity k thermal =k electrons + k phonons

23. Conductivities

24. Thermal Conductivity-Phonon k phonons = N e C p ph V ph /3 –N e number e - /volume, –C p =heat capacity of atoms =3k B – ph =mean free path, –V ph =velocity

25. Thermal Conductivity - Electron k e = N e C e e V e /3 –N e number e - /volume, –C e =heat capacity of electrons – e =mean free path, –V e =velocity

26. Thermal Conductivity

27. Phonon Interactions With other phonons With impurities –depends upon phonon wavelength With imperfections in Crystal –depends upon phonon wavelength Phonons travel at speed of sound

28. Phonon Interactions