Materials Science and Engineering - Electronic and Mechanical Properties of Materials Part 1 doc

Fitzgerald-1999 Response of Material to Applied Potential I V e-V I Linear, Ohmic Rectification, Non-linear, Non-Ohmic V=IR V=fI Metals show Ohmic behavior microscopic origin?. 3.225

3.225 1

Electronic Materials

Silicon Age:

• Communications

• Computation

• Automation

• Defense

• ………

Factors:

• Reproducibility/Reliability

• Miniaturization

• Functionality

• Cost

• …………

© H.L Tuller-2001

Pervasive technology

What Features Distinguish Different Conductors?

• metal; semiconductor; insulator

• Carrier type:

• electrons vs ions;

• negative vs positive

• Mechanism:

• wave-like

• activated hopping

• Field Dependence:

• Linear vs non-linear

varies by over 25 orders of m

3.225 3

How Do We Arrive at Properties That We Want?

• Crystal Structure:

• diamond vs graphite

• Composition

• silicon vs germanium

• Doping

• n-Si:P vs p-Si:B

• Microstructure

• single vs polycrystalline

• Processing/Annealing Conditions

• Ga1+xAs vs Ga1-xAs

© H.L Tuller-2001

• Interconnect

• Resistor

• Insulator

• Non-ohmic device

– diode, transistor

• Thermistor

• Piezoresistor

• Chemoresistor

• Photoconductor

• Magnetoresistor What is the Application?

3.225 5

Origin of Conduction Range of Resistivity

Why?

© E.A Fitzgerald-1999

Response of Material to Applied Potential

I

V

e-V

I

Linear, Ohmic

Rectification, Non-linear, Non-Ohmic

V=IR

V=f(I)

Metals show Ohmic behavior microscopic origin?

R

3.225 7

Microscopic Origin: Can we Predict Conductivity of Metals?

• Drude model: Sea of electrons

– all electrons are bound to ion atom cores except valence electrons

– ignore cores

– electron gas

© E.A Fitzgerald-1999

Schematic model of a crystal of sodium metal

From: Kittel, Introduction to Solid State Physics, 3rd

Ed., Wiley (1967) p 198 C

Does this Microscopic Picture of Metals Give us Ohm’s Law?

F=-eE

E

F=ma

m(dv/dt)=-eE

v =-(eE/m)t

v,J,σ,I

t

t

E

No, Ohm’s law can not be only from electric force on electron!

Constant E gives ever-increasing v

3.225 9

Equation of Motion - Impact of Collisions

Assume:

• probability of collision in time dt = dt/τ

• time varying field F(t)

v(t+dt) = (1- dt/τ) {v(t) +dv} = (1- dt/τ) {v(t) + (F(t)dt)/m}

≈ v(t) + (F(t)dt)/m - v(t) dt/τ (for small dt)

⇒ dv(t)/dt + v(t)/τ = F(t)/m

Note: erm proportional to velocity corresponds to

frictional damping term

© H.L Tuller-2001

T

Hydrodynamic Representation of e- Motion

dp t

dt

p t

F t F t

( ) ( )

Response (ma)

p=momentum=mv

Drag Driving Force Restoring Force

dp t

dt

p t eE

( ) ≈ − ( )−

τ

Add a drag term, i.e the electrons have many collisions during drift

1/τ represents a ‘viscosity’ in mechanical terms

2

3.225 11

In steady state, dp t

dt ( ) = 0

t

( ) = ∞ ( 1 − τ )

p ∞ = − τ E

p

t

-eE τ

τ

If the environment has a lot of collisions,

mvavg=-eE τ vavg=-eE τ /m

µ = e τ

m

© E.A Fitzgerald-1999

E

µ

−

= Define v

Mean-free Time Between Collisions, Electron Mobility

−

e

vd

E

j = I/A A

dx

What is the Current Density ?

n (#/vol)

• # electrons crossing plane in time dt = n(dxA) = n(vddtA)

• # charges crossing plane per unit time and area = j

• Ohm’s Law:

Dimensional analysis: (A/cm 2 )/(V/cm)=A/(V-cm)= (ohm-cm) -1 = Siemens/cm-(S/cm)

( )v dtA( ) e dtA ne v ( ne m E

n

j = d − =− d = 2τ

( ne m j E E

j = σ ⇒ σ = 2 τ =

) )

3.225 13

Energy Dissipation - Joule Heating

Frictional damping term leads to energy losses:

• Power absorbed by particle from force F:

P = W/t = (F•d)/t = F•v

• Electron gas: P/vol= n(-eE)•(-eτE/m)

= ne2τE2/m = σ E2

= jE = (I/A)(V/l) = IV/vol

• Total power absorbed: 2/R = I2R

How much current does a 100 W bulb draw?

I = 100W/115V = 0.87A

© H.L Tuller-2001

P = IV = V

Predicting Conductivity using Drude

ntheory from the periodic table (# valence e- and the crystal structure)

ntheory=AVZ ρ m /A,

where AV is 6.023x10 23 atoms/mole

ρ m is the density

Z is the number of electrons per atom

A is the atomic weight

For metals, ntheory~10 22 cm -3

If we assume that this is correct, we can extract τ

3.225 15

• τ~10-14 sec for metals in

Drude model

© E.A Fitzgerald-1999

Thermal Velocity

• So far we have discussed drift velocity vD and scattering time τ

related to the applied electric field

• Thermal velocity vth is much greater than vD

kT

mv th

2

3

2

1 2

=

m

kT

v th = 3

Thermal velocity is much greater than drift velocity

x

x

x L=vDτ

3.225 17

Resistivity/Conductivity Pessimist vs Optimist

L

V

t

R = ρ L/Wt = ρ L/A ⇒ ρ(οhm-cm)

σ = 1/ρ ⇒ σ (οhm-cm)-1 ⇒ σ (Siemens/cm)

(Test your dimensions: σ=E/j=neµ)

Ohms/square ⇒ Note, if L=W, then R= ρ /t independent

of magnitude of L and W Useful for working with films of

thickness, t

R R R

© H.L Tuller-2001

R=V/I;

How to Make Resistance Measurements

Rs

Rc1

Rc2

I

V

V/I = Rc1 + Rs + Rc2

I s >> Rc1 + Rc2 ; no problem

II For Rs ≤ Rc1 + Rc2 ; major problem ⇒ 4 probes

For R

3.225 19

How to Make Resistance Measurements

Rs

Rc4

Rc1

I

V14

v23

c3

4 probe method: Essential feature - use of high impedance voltmeter to measure V23 ⇒ no current flows through Rc2

& Rc3 ⇒ therefore no IR contribution to V23

Rs(2-3) = v23 /I = σ-1 (d23/A) = ρ (d23/A)

(Note: ρ -resistivity is inverse of σ− conductivity)

© H.L Tuller-2001

How to Make Resistance Measurements - Wafers

R

R+dR

x

j = I/2πR2 ; V = IR = Iρd/A = jρd

V23 = ⌠2d (I/2πR2 ) ρ dR = (- Iρ/ 2πR) 2d = Iρ/4πd

ρ = (2πd/I) V ; ρ = (π/ln2) V/I for d x

Si

I

d