Lecture 4.0 Properties of Metals

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

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

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

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

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

Energy Bands

Partially Filled

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

Fermi Energy

Work Function

Fermi-Dirac Probability Distribution

Density of States- 3D Schrodinger Eq.

Electron Filling in Band- density of occupied states

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

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

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

Thermal Expansion

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 )

Heat Capacity of Electrons

Heat Capacity

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

Conductivities

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

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

Thermal Conductivity

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

Phonon Interactions