Lecture1 Fixed

8/6/2019 Lecture1 Fixed

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3.225 1

Electronic Materials

Silicon Age:• Communications

• Computation

• Automation

• Defense

• ………..

Factors:

• Reproducibility/Reliability

• Miniaturization

• Functionality

• Cost

• …………..

© H.L. Tuller-2001

Pervasive technology

3.225 2

What Features Distinguish Different Conductors?

• Magnitude: agnitude!

• metal; semiconductor; insulator

• Carrier type:

• electrons vs ions;

• negative vs positive

• Mechanism:

• wave-like

• activated hopping

• Field Dependence:

• Linear vs non-linear

© H.L. Tuller-2001

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

3.225 4

• Interconnect

• Resistor

• Insulator

• Non-ohmic device

– diode, transistor

• Thermistor

• Piezoresistor

• Chemoresistor

• Photoconductor

• Magnetoresistor

What is the Application?

© H.L. Tuller-2001



3.225 5

Origin of Conduction Range of Resistivity

Why?

© E.A. Fitzgerald-1999

3.225 6

Response of Material to Applied Potential

I

V

e-V

I

Linear,

OhmicRectification,

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


Schematic model of a crystal of sodium

metal.

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

Ed., Wiley (1967) p. 198.

C.

3.225 8

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

3.225 10

Hydrodynamic Representation of e- Motion

dp t

dt

p t F t F t

( ) ( )( ) ( ) ...= − + + +

τ 1

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

p t p et

( ) ( )= ∞

− 1 τ

p E ∞ = − τ

p

t

-eEτ

τ

If the environment has a lot of collisions,

mvavg

=-eEτ vavg

=-eEτ/m

µ τ = e

m


E µ−=Define v

Mean-free Time Between Collisions, Electron Mobility

−

e

3.225 12

vd

E

j = I/A

Adx

What is the Current Density ?

n (#/vol)

© H.L. Tuller-2001

• # 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/cm2)/(V/cm)=A/(V-cm)= (ohm-cm)-1 = Siemens/cm-(S/cm)

( )( ) ( E mnevnedtAedtAvn jd d τ

2=−=−=

( E jmne E j ==⇒= τσσ 2

)

)



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

3.225 14

Predicting Conductivity using Drude

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

ntheory=AVZρm/A,

where AV is 6.023x1023 atoms/mole

ρm is the density

Z is the number of electrons per atom

A is the atomic weight

For metals, ntheory~1022 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

Extracting Typical τ for Metals


3.225 16

Thermal Velocity

• So far we have discussed drift velocity vD and scattering time τrelated to the applied electric field

• Thermal velocity vth is much greater than vD

kT mvth2

3

2

1 2=

m

kT vth

3=

Thermal velocity is much greater than drift velocity

x

x

xL=vDτ




3.225 17

Resistivity/Conductivity-- Pessimist vs Optimist

L

WI

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;

3.225 18

How to Make Resistance Measurements

R s

R c1R c2

I

V

V/I = R c1 + R s + R c2

I.s

>> R c1

+ R c2

; no problem

II. For R s ≤ R c1 + R c2 ; major problem ⇒ 4 probes

© H.L. Tuller-2001

For R



3.225 19

How to Make Resistance Measurements

R s

R c4R c1

I

V14

v23

R c2 R c3

4 probe method: Essential feature - use of high impedance

voltmeter to measure V23 ⇒ no current flows through R c2

& R c3 ⇒ therefore no IR contribution to V23

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

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

© H.L. Tuller-2001

3.225 20

How to Make Resistance Measurements - Wafers

IV

d d

R

R+dR

x

j = I/2πR 2 ; V = IR = Iρd/A = jρd

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

⌡d d

ρ = (2πd/I) V23 ; ρ = (π/ln2) V/I for d >>x

Si

© H.L. Tuller-2001

Id



3.225 21

Example: Conductivity Engineering

• Objective: increase strength of Cu but keep conductivity high

τ τ

µ τσ

v

m

e

m

ne

= ==

l 2

Scattering length

connects scattering time

to microstructure

Dislocation

(edge)

l decreases, τ decreases, σ decreases

e-


3.225 22

• Can increase strength with second phase particles

• As long as distance between second phase< l, conductivity marginally effected


L

S

L+S

Sn Cu

L

X Cu

α β

α+L β+L

α+β

Smicrostructure

Material not strengthened, conductivity decreases

α

β dislocation

LL>l

Dislocation motion inhibited by second phase;

material strengthened; conductivity about the same




- - - - - - - - -

3.225 23

• Scaling of Si CMOS includes conductivity engineering

• One example: as devices shrink…

– vertical field increases

– τ decreases due to increased scattering at SiO2/Si interface

– increased doping in channel need for electrostatic integrity: ionized

impurity scattering

– τSiO2<τimpurity if scaling continues ‘properly’


Evert

Ionized impurities

(dopants)

S D

GSiO2


3.225 24

Determining n and µ: The Hall Effect

Vx, Ex

I, Jx

Bz

+ + + + + + + + + + +

Bvq E q F rrrr

×+= z D y Bev F −=

Ey

y y eE F −= In steady state,

H Z DY E Bv E == , the Hall Field

Since vD=-Jx/en,

Z X H Z x H B J R B J ne

E =−= 1

ne R H

1−=

µσ ne= © E.A. Fitzgerald-1999



3.225 25

Experimental Hall Results on Metals

• Valence=1 metals look like

free-electron Drude metals

• Valence=2 and 3, magnitude

and sign suggest problems


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