Lecture 5Lecture 5

Electrical Conductivity of Metals

Why are metals important (& interesting) for electronics?

Metal interconnects are used in microelectronics to wire the devices within the chip, the intergraded circuit.

Why are metals important (& interesting) for electronics?

Metal interconnects are used in microelectronics to wire the devices within the chip, the intergraded circuit.

Why are metals important (& interesting) for electronics?

Metal interconnects are used in microelectronics to wire the devices within the chip, the intergraded circuit.

Why are metals important (& interesting) for electronics?

Metal interconnects are used in microelectronics to wire the devices within the chip, the intergraded circuit.

Why are metals important (& interesting) for electronics?

Metal interconnects are used in microelectronics to wire the devices within the chip, the intergraded circuit.

Why are metals important (& interesting) for electronics?

Metal interconnects are used in microelectronics to wire the devices within the chip

1 μm

Metal interconnects are used in microelectronics to wire the devices within the chip, the intergraded circuit.

Multilevel interconnects are used for implementing the necessary interconnections.

• Metals comprise 2/3 of all elements on the periodic table

• Accounting for the striking features of metals (including their high electrical conductivity, high luster, and ductility) gave the

f h d h f dstarting impetus for the modern theory of solids

• 1897: J.J. Thomson discovered the electronJ J

• 1890: Drude applied kinetic molecular theory of gases to a metal:Metal = “gas of electrons”Metal = gas of electrons

Drude Model of Conductivity: Assumptions

0) Assume metals are composed of a lattice of positively charged ion cores

+ + ++ + +

+++Each ion core has a

charge eZa(Za=atomic number)

+ + +( a )

• These core each have Za electrons of total charge –eZa• A few of these electrons, Z, are the relatively weakly bound valence

electronselectrons

• The remaining Za-Z electrons are relatively bound to the nucleus and do not participate in conduction

• The Z valence electrons give a gas of electrons, each with charge e and mass me

Drude Model of Conductivity: Assumptions

0) Assume metals are composed of a lattice of positively charged ion cores

+ + ++ + +

+++Each ion core has a

charge eZa(Za=atomic number)

+ + +( a )

• The density of this electron gas is calculated via:

z m231002261022 e-/cm3

Axn m 310022.6

n=# electrons/cm3 ρ =Density of metal A = Atomic mass of metal

typical

n # electrons/cm ρm Density of metal (g/cm3)

A Atomic mass of metal

Drude Model of Conductivity: Assumptions

1) Electrons travel in straight lines unless they collide with each other of atoms in the lattice

2) Collisions are instantaneous events that abruptly alter the velocity of 2) Collisions are instantaneous events that abruptly alter the velocity of an electron (Drude only considered electron-ion collisions)

3) An electron will travel for a time τ before it’s next collision. An electron ff ll h b b l 1/suffers a collision with a probability per unit time 1/ τ

τ : relaxation time

dt/ τ : probability of a collision in time dt dt/ τ : probability of a collision in time dt

4) Collisions maintain or restore thermal equilibrium

Electrons emerge from collisions with a random velocity determined by Electrons emerge from collisions with a random velocity determined by the local temperature at the collision site

Deriving the Drude Model

We want to relate the applied electric field to the current density, Jxflowing in the wire.

Deriving the Drude Model

The electric current density is the net amount of charge flowing across an area per unit time:

The total number of electrons crossing A in time ∆t is:The total number of electrons crossing A in time ∆t is:

xnAq xnAe

l d b# electrons per unit volume

Distance traversed by electrons in ∆t = vx∆t

Drude Model of Conductivity

Drift of electrons in a conductor in the presence of an applied electric field Electrons drift with an average velocity v in the x direction (E is field. Electrons drift with an average velocity vdx in the x-direction. (Ex is

the electric field, n is the electron concentration.)

Drude Model of Conductivity

Now, relate vx to the applied electric field.

Consider a typical electron at time zero. Let t be the time elapsed since its last collision. Its velocity at time zero will be its velocity voits last collision. Its velocity at time zero will be its velocity vo

immediately after that collision, plus the additional velocity acquired after the collision.

xeEmvmaF

meEv x

Em

eEvtv xo

)( Averaged over all electrons, =0

Drude Model of Conductivity

xx EneenvJ

2

m

Ef lvdx = drift velocity, d = drift mobility, Ex = applied field

vdx = dExDrift Velocity

Drift Mobility d em

d = drift mobility, e = electronic charge, = mean scattering time (mean time between collisions) = relaxation time, me = mass of an

me

( ) eelectron in free space.

Drude Model of Conductivity

xxx EEneenvJ

2

m

end e2nm

Conductivity

= conductivity, e = electronic charge, n = number of electrons per unit volume, d = drift velocity, = mean scattering (collision) time =

me

, d y, g ( )relaxation time, me = mass of an electron in free space.

How far do electrons in Cu travel between collisions?

115 cm10x95 mol/g563M

Given:

cm10x9.5 Cu

cm/g96.8Cu

mol/g5.63AM

s/m10x5.1 6v

Temperature Dependence of Conductivity

A vibrating metal atom

Scattering of an electron from the thermal vibrations of the atoms. The electron travels a mean distance = u between collisions. Sinc th c tt ring cro ction l r i S in th olum SSince the scattering cross-sectional area is S, in the volume S

there must be at least one scatterer, Ns (Su ) = 1.

Mean Free Time Between Collisions

1

1SuNs

= mean free time, u = mean speed of the electron, Ns = concentration of scatterers, S = cross-sectional area of the scatterer

Resistivity Due to Thermal Vibrations of the Crystal

Mean Free Time Between Collisions

1

1SuNs

= mean free time, u = mean speed of the electron, Ns = concentration of scatterers, S = cross-sectional area of the scatterer

Resistivity Due to Thermal Vibrations of the Crystal

AT

T = resistivity of the metal A = temperature independent constant T

T = AT

T resistivity of the metal, A temperature independent constant, T= temperature

Multiple Scattering Processes

Two different types of scattering processes involving scattering f i iti l d f th l ib ti lfrom impurities alone and from thermal vibrations alone.

Matthiessen’s Rule

= T + I

= effective resistivity, T = resistivity due to scattering by thermal vibrations only, I = resistivity due to scattering of electrons from

impurities only.

More generally:

= T + R

g y

= overall resistivity, T = resistivity due to scattering from thermal vibrations, R = residual resistivity (impurities, dislocations, grain

b d )boundaries, etc)

Temperature Coefficient of Resistivity:fractional change in resistivity with temperature

1

= TCR (temperature coefficient of resistivity) = change in the

oTToo T

o = TCR (temperature coefficient of resistivity), = change in the resistivity, o = resistivity at reference temperature To , T = small

increase in temperature, To = reference temperature

Temperature Dependence of Resistivity

[1 + (T T )]

= resistivity, o = resistivity at reference temperature, 0 = TCR

[1 + o(TTo)]

(temperature coefficient of resistivity), T = new temperature, T0 = reference temperature

The resistivity of various metals as a function of temperature above 0 °C.

Tin melts at 505 K whereas nickel and iron go through a magnetic to non-magnetic g g g g(Curie) transformations at about 627 K and 1043 K respectively. The theoretical

behavior ( ~ T) is shown for reference.

Electron mean free path ~40 nm

Electron mean free path ~107 nm

The resistivity of copper from lowest to highest temperatures (near melting temperature, 1358 K) on a log-log plot. Above about 100 K, T, whereas at low temperatures,) g g p , , p , T 5 and at the lowest temperatures approaches the residual resistivity R. The inset shows the vs. T behavior below 100 K on a linear plot (R is too small on this scale).

Resistivity change between seasons in Canada

In northern Canada, temperatures in the summer and winter vary between 20oC and -30oC.

What is the percentage change in the resistance of a pure metal W at t e pe e tage a ge t e e ta e a pu e eta wire?

R ~ ~ T

)2 330()2 320()27320(

)27330()27320(

summer

wintersummer

summer

wintersummer

TTT

RRR

A 17% change in resistivity!