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Electrical & Thermal Properties

A. Electrical and Thermal Properties of Materials

• Conduction Theory: Drude model

• r(T) – pure metals

– non-pure metals (Matthiessen’s rule)

• r for alloys– 1-phase: Nordheim’s rule

– 2-phase: mixture rules

• r for non-metals– semiconductors, glasses, polymers (plastic)

• special cases for EE– r of amorphous / thin films

– r at high frequencies (skin effect)

– electromigration

• thermal conduction k(T)– metals, non-metals

v.2019.AUG

2102308 1

References:

- S.O. Kasap, 4th edition, Chapter 2

- R.J.D. Tilley, 2nd ed, Chapter 13


2.1* Classical Theory: Drude Model

• Key EE materials: metals, semiconductors

– metals (conductors): transport (drift) of very large number of charge carriers, sea of

electrons (at energy ~EF), under electric field E

– semiconductors: transport (drift, diffusion) of large number of electrons (~EC)

and/or holes (~EV), under electric field E

• E → J : when electric field E (V/cm) is applied, conduction electrons

respond, causing current density J (A/cm2)

• Three important questions

Q1) source of conduction electrons

Q2) their location

Q3) their response under electric field

2102308 2


Q1) Where do (conduction) electrons in metals come from?

Reminders (band theory):

- electron energy levels in atoms split and form bands in

crystals due to Pauli exclusion principle, at equilibrium

distance, some bands overlap

- most important bands: top-most occupied band () and

bottom-most empty (). Electrons in these bands are

valence (VB) or conduction (CB) electrons, the latter

respond to electric fields

- examples: metallic 11Na, 12Mg, 13Al

2102308 3

12Mg

Electrical & Thermal Properties 2102308 4

Q1) Where do (conduction) electrons in metals come from?

- materials can be classified according to the type of band diagrams into three main groups:

insulators, semiconductors, metals

- the conduction electrons in metals arise from the partially filled band ()

- 29Cu: electrons in the half-filled 4s band is responsible for conduction (electrons in the filled 3d

band is shielded by 4s band, does not involve in bonding, thus localized around each Cu atom)

insulators semiconductors metals semi-metals


Q2) Where are the conduction electrons?

A) “throughout” metal, moving (a) without- or (b) with preferred direction depending on the

presence of external electric fields. The conduction electrons (4s1) are said to be delocalized.

In contrast, the core electrons ([Ar] 3d10) are localized.

E = 0 E 0

Q3) How do they respond to electric field (E)?

The short answer (from 2102385): they move, resulting in current (J)

J (E ) ?

– Drude: J = E = nemE

– : conductivity (W-cm)1

– n: electron concentration (/cm3, cm3)

– m : drift mobility (cm2/V-s) , measures the ease with which charge carriers drift


random motion

vd: drift velocity

i/p

o/p: Jx = nevdx

How is vdx related to E ?

Derivation (the long answer)

tA

qJ

t

xv

xAneq

dx

2102308 6


1028 m-3

1022 cm-3

1010 mm-3

10 nm-3

mean free time t (or relaxation time)

- time between successive collisions ()

- with each collision (), the electron is

assumed to lose all its velocity ()

- energy transferred to lattice (heat: P = I2R)

- average free time depends on lattice

vibration, defects, impurities, etc

m/e

ne

EJ

tm

m

v = u + at

:

tm

eEv x

dx

&

classical mechanics

F = ma

terms & units

Material Device

2102308 7

J : current density (A/cm2) I : current (A)

: conductivity (/W.cm) G : conductance (/W)

E : field strength (V/cm) V : voltage (V)

r : resistivity (W.cm) R : resistance (W)

m : mobility (cm2/V.s)

Ex: Cu

FCC, a = 3.6 Å

N = 4/a3 =

8.6 × 1022 /cm3


Ex. 2.2

Calculate drift mobility and mean scattering time of conduction electrons

in Cu at room temperature, given that (Cu) = 5.9105 W-1 cm-1. The

density of Cu is 8.96 g/cm3 and atomic mass is 63.5 (g/mol).

...

/105.1 6

pfm

smv

t

m

given (Ex.4.4 p.299 Fermi speed)

Note: conduction electrons in metals travel at Fermi velocity:

(temperature insensitive)FOF Emv 2

2

1

cf: Cu FCC lattice constant 3.610 Å

37 nm

43.3 cm2/Vs

2.5×10-14 s

2102308 8


2.2 r (T ) for ideal pure metals

%1327340

273027340int

summer

erwsummer

R

RR

TT A)( r

constant

temperature

Pure metal wire:

Tne

aT

tmr

rmt

111

Lattice scattering

2102308 9

l: mean free pathu: mean speed

t: mean free time

scattering centers(host atoms,

impurity, defects,…)

(99.99% Cu)

- drift of conduction electrons limited mainly by lattice scattering (phonon scattering)

- increasing temperature T results in increased scattering cross-section (yellow shaded area a),

decreased mfp mft, mobility, and conductivity, summarized by


2.3 r (T ) for non-pure metals

Impurity scattering

Lattice scattering

2102308 10

- “non-pure” means metals with impurities. When the impurities are unintentional, they are

called contamination. If they are intentional (controlled type and concentration) they are called

dopants (process called doping)

- drift of conduction electrons in non-pure metals are limited by lattice scattering (as in pure

metals) and impurity scattering

- experiments show that scattering rates combine linearly, this is known as Matthiessen’s rule

(next slide)


Overall frequency of independent scattering events:

(number of collisions /unit time)

due to lattice vibration

due to impurity scattering

depends on separation between

impurity atoms (concentration, NI)

In general:

Residual resistivity

[impurities, dislocations, vacancies,

interstitial atoms, grain boundaries…]

IT ttt

111

IT rrr

Matthiessen’s Rule

RT TT rrr

BA Tr [A,B temperature independent]

tr

t

tm

1

11

Drude(slide # 7)

temperature sensitive

microstructure sensitive

2102308 11

“Defects”*(see Appendix Defects)

(intrinsic property, cannot be removed)

(extrinsic property, remove by

careful processing)

Let’s check the validity of Matthiessen’s Rule:

1. overall trends, 2. boundary conditions (0 K), 3. many metals


Q1) How good is M’s rule?

A) Quite good! Measured results (Fig. 2.8) match well with schematic graphs showing increased

resistivities with (Fig. 13.4) temperature (), presence of impurity (), and (Fig. 13.5) linear

dependence with impurity level

BA TTT RT rrr

2102308 12


0Rrfor superconductors

# atoms that vibrate

with sufficient energy

reduced rapidly

RT rr Apredicted

Q2) ... for all temperatures?

A) No, it fails at low T.

Cu

RT rr 5Dexperiments

2102308 13


oTo

ooT

n

oo

T

dT

d

TT

T

T

r

r

rr

r

r

1

1

material npure metal =1

Realitymagnetic ~2

non-magnetic ~1

alloy (NiCr) << 1

(B dominates)

BA Tr

Q3) ... for all metals?

A) No, it depends on the nature of

material

temperature coefficient of

resistivity (TCR):

TCurie

Fe = 1,043 K

Ni = 631 K

TCurie

log:

linear:

2102308 14

IDEAL Tr

nTrREAL


Example 1:

Example 2:

Which is more accurate? See curve p.13

.mn5.118100208

1180C100@In W

r

97.3273100233

114.15:linear

.mn80.4273

1004.15:log

?

16.1

K 100@Cu

W

r


(a) A vacancy in the crystal. (b) A substitutional impurity in the crystal. The

impurity atom is larger than the host atom.

(c) A substitutional impurity in

the crystal. The impurity atom

is smaller than the host atom.

(d) An interstitial impurity in the crystal. It

occupies an empty space between host atoms.

Fig. 1.44: Point defects in the crystal structure. The regions aroundthe point defect become distorted; the lattice becomes strained.

0D: point defects

2102308 16

vacancies are equilibrium

defects (always exist),

concentration in metals is

< 1 in 10,000 atoms

Defects interrupt periodic arrangement of atoms (scatter electrons, reduce mobility, increase resistivity).

Defects have many origins; can be grouped according to geometry/dimension (0D, 1D, 2D, 3D)

Appendix Defects: p.1/3


Edge dislocation line

(a) Dislocation is a line defect. The dislocation shown runs into the paper.

Compression

Tension

(b) Around the dislocation there is a strain field as the atomic bonds have

been compressed above and stretched below the islocation line

From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)

http://Materials.Usask.Ca

Fig. 1.46: Dislocation in a crystal is a line defect which is accompaniedby lattice distortion and hence a lattice strain around it.

2102308 17

extra half plane

Dislocations are non-equilibrium defects

(removable with careful process control);

When present, energies are stored in the

distorted regions.

Dislocation in active area of optical emitter

causes dark-line defects, dark pixels

Dislocation in active area of optical detector

causes dead pixels (always black, white, or

whatever color the pixel is associated with)

1D: line defects



Strained bond

Broken bond (dangling

bond)

Grain boundary

Void, vacancy

Self-interstitial type atom

Foreign impurity

Fig. 1.51: The grain boundaries have broken bonds, voids, vacancies,strained bonds and "interstitial" type atoms. The structure of the grainboundary is disordered and the atoms in the grain boundaries have higherenergies than those within the grains.



Grain Boundaries

width ~2-5 atoms

Edge dislocation line

(a) Dislocation is a line defect. The dislocation shown runs into the paper.

Compression

Tension

(b) Around the dislocation there is a strain field as the atomic bonds have

been compressed above and stretched below the islocation line



Fig. 1.46: Dislocation in a crystal is a line defect which is accompaniedby lattice distortion and hence a lattice strain around it.

2D: area defects

3D: volume defects

3D defects include voids (think air bubbles),

inclusions, precipitates

surfaces

2D defects include surfaces, grain boundaries

(in polycrystalline materials), twins (twin

boundary), twists, stacking faults (pile-up

faults)



Ex. 2.10 (a) Light bulb rated 40W, 120V. Has W filament 0.381 m long with diameter

of 33 mm. Room temperature resistivity of W is 5.51×10-8 Wm. Given that resistivity

of W varies at T1.2, estimate operating temperature of the filament.

120 V

40 W

0.333 A

Fig. 2.9: Power radiated from a light bulb at 2408 °C is

equal to the electrical power dissipated in the filament.

Ans: 2,812 K

2102308 19


2.3.2 Nordheim’s Rule (for solid solutions)

2102308 20

- ‘Pure’ metals rarely used in practice, except pure Cu (99.99%) for low r

- Metals are often mixed (alloyed) to achieve functionality, properties, cost advantages

over pure metals

- Alloys: the simple (brass: Cu70 Zn30 wt%), the complex (Inconel 718: 10 elements)

- Simplest type of alloys: solid solution or single-phase alloys

- Single-phase alloys: atoms mixed at atomic level throughtout solid. This is possible only

if the constituent atoms are completely miscible (similar in size and crystalline structure)

- What determine single-phase alloy resistivities?

Alloying effect of rI , rT

IT rrr still applies

Example 1: NiCr (heater wire)


RTmatrix

matrix

I

XCX

XCX

rrr

rr

r

1

1

IT rrr

C: Nordheim’s coefficient,

effectiveness of impurity atoms in

increasing resistivity

X: atomic % of solute in solvent

solute ตัวละลาย + solvent ตัวท าละลาย = solution สารละลาย

Example 2: Cu-Ni (thermocouple wire)

~ ppm/K

Cu, Ni are both FCC.

2102308 21

Nordheim

Matthiesen

+

(alloys)

(metals)


CX

XCX

I

I

r

r 1

Semi-empirical equation:

for small X:

X

2102308 22


* *

Cu & Au randomly mixed

* orderly structure,

viewed as pure compound

Q) How good is Nordheim’s rule?

A1) Good, if solutions are randomly mixed

A2) Bad (over-estimate), if solutions are ordered

Self-check: r of Cu25%-Au75%

22.8 nW.m

note: wt.% at.%

*

22.8 + 450(0.25)(0.75) = 107.2 nW.m

2102308 23

Cu, Au are both FCC.


TABLE 2.3

Nordheim Coefficient (at 20 C) for Dilute Alloys

Solute in Solvent

(Element in matrix)

Nordheim

coefficient

C (nW m)

Maximum Solubility at

25C at.%

Au in Cu matrix 500 100 Mn in Cu matrix 2900 24

Ni in Cu matrix 1570 100

Sn in Cu matrix 2900 0.6

Zn in Cu matrix 300 30

Cu in Au matrix 450 100

Mn in Au matrix 2410 25

Ni in Au matrix 790 100

Sn in Au matrix 3360 5

Zn in Au matrix 950 15

Data extracted from various sources: Electronics Engineers’ Handbook, Second Edition, ed.

by D.G. Fink and D. Christiansen (McGraw-Hill, New York, 1982), section 6 (Properties of

Materials by I.A. Blech), J.K. Stanley, Electrical and Magnetic Properties of Metals

(American Society for Metals, Metals Park, Ohio, 1963) and Solubility data from

Constitution of Binary Alloys, Second Edition, M. Hansen and K. Anderko (McGraw-Hill,

New York, 1985)

From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002) http://Materials.Usask.Ca

Brass: 70%Cu – 30%Zn

“solubility limit”ขีดจ ำกดัสภำพละลำยได้

2102308 24

miscibleผสมเขำ้กนัได้

immiscibleผสมเขำ้กนัไม่ได้


Phases:physically and chemically distinct material regions ( and b).

Aluminum-

Copper

Alloy

Adapted from

Fig. 9.0,

Callister 3e.

solid solubility limit of

Al in Cu = %

- Single-phase alloys: atoms mixed at atomic level throughtout solid. This is possible only if

the constituent atoms are completely miscible (similar in size and crystalline structure)

- Two-phase alloys: constituent atoms are very different in size and/or have different

crystalline structure. They prefer to phase separate.

Al-Cu

What determine two-phase alloy resistivities?

2102308 25


2.4 Mixture Rules (for 2-phase alloy)

bb

bb

rrr

r

rr

eff

eff

eff

A

L

A

L

A

LR bb eff

b

volume fraction of phase

volume fraction of phase b

(a) Rule 1: (b) Rule 2: (c) Rules 3 & 4:

.

.

.

linear combination

2102308 26

- The three scenarios of two-phase alloys: a) ordered mixture in series, use Rule 1, b) ordered

mixture in parallel, use Rule 2, c) disordered mixture apply linear combination (Rules 3,4…).

- Nature = c (not a, b), hence 2-phase region has a linear change in resisitivity between two end

points, (see Cu-Ag, next slide)


@ X1 Solubility limit

for Ag in Cu reached

2-phase alloys A-BCu-Ag

2102308 27


Electrical materials in power distribution

Ag has high electrical and thermal conductivities best for electrical contacts/switches

(but expensive). Usually use Ag alloy (Ag-Ni preferred over Ag-Pd despite similar

strength because Ag-Ni forms a 2-phase alloy whereas Ag-Pd is a solid solution)

Ag-Pd Ag-Ni

solid solution two-phase alloy

Nordheim’s rule Mixture rule

r r

similar strength

W kg,$

alloy?

r (nW.m)

Ag,16

Ni,70

Pd,100

Criteria:

electrical (low loss)

mechanical (high strength)

chemical (long-term stability)

environmental

Wires: Cu, Al

Plugs: brass (Cu-Zn alloy)

Switches: (Ag-Ni alloy)

2102308 28

… in signal transmission

Criteria: + noise, cross-talk…

hence, twisted pair, shields


> temperature dependency

> conductivity

> transport mechanisms: ? drift-diffusion drift

> material

2.7 Electrical Conductivity of Non-metals- previous sections concerned with metals and alloys

- EEE applications also rely heavily on non-metals

- non-metals conducts poorly (semiconductors) or very poorly (insulators), but they conduct

- this section: electrical properties of semiconductors (next 3 slides) and insulators (11 slides)

nem nem

Superinsulators

= 0 r = 0

TAr ?Tr ?Tr

Note the range: 1027. No other physical property has this range. Age of universe: 1010 years or 1017 s.

2102308 29

nem


Intrinsic semiconductors

- pure, undoped crystals of some elemental (IV), III-V and II-VI compounds

- the greater the size of the component atoms, the smaller the bandgap (see table).

This is due to the orbitals involved are larger, overlap more, easier to break

- the higher the temperature, the more bonds break, the greater number of carriers,

the increased conductivity (see graph)

2102308 30

he pene mm

Reminder (from course 2102385)

- semiconductors: intrinsic vs extrinsic

- carriers origin: intrinsic (thermal) vs extrinsic (doping)

- carriers concentration: equilibrium vs excess (optical, electrical injection)

Semiconductors: key material for electrical/electronic engineer (EEE)

?T


50

100

1000

2000

1015 1016 1017 1018 1019 1020

Dopant Concentration, cm-3

ElectronsHoles



Drif

t M

ob

ilit

y(c

m2 V

-1s-

1)

Fig. 5.19: The variation of the drift mobility with dopant concentration in Si

for electrons and holes

Si Cun = 8.51022 cm-3

m = 43 cm2/Vs

2102308 31

Extrinsic semiconductors

- crystals intentionally doped with small amount of atoms having different valence

- dopant atoms in crystal ionize to donate mobile electron (n-type) or accept valence

electron or equivalently to create mobile hole (p-type)

- though conductivity is lower than metals (conductors), the carriers are more mobile

due to less scattering (see graph and Cu vs Si below)

- temperature-dependent conductivity is a bit more complicated (see next slide), but

can be briefly stated as “it shows the opposite trend to metals”


(T) in extrinsic semiconductors

log(n)

INTRINSIC

EXTRINSIC

IONIZATION

log( )

log( )

mT-3/2

mT3/2

Lattice

scattering

Impurity

scattering

1/TLow TemperatureHigh Temperature

T

Metal

Semiconductor

LO

GA

RIT

HM

ICS

CA

LE

Res

isti

vit

y

Temperature dependence of electrical conductivity for a doped (n-

type) semiconductor.

Tr T/1

2102308 32


Insulators? Which materials exactly? Why we (as an EE) care?

Two examples: plastics, glasses

- Plastics are by-products of petroleum industry (think C,H—light atoms).

Plastics are flexible, when conduct bring unique applications

- Glasses are (nominally) non-conducting ionic solids. Glasses are transparent,

when doped they conduct and bring unique applications

keywords for the (near) future

- plastic electronics

- flexible electronics


Si4+O2-

2102308 34

… glass & ceramics (ionic conduction)

iiiqZn m

- glass: SiO2-framework, depending on purity/manufacturing can be crystalline (quartz) or amorphous (glass)

- example: soda-lime glass 73% SiO2 – 15% Na2O − 7% CaO − 4% MgO − 1% Al2O3

- glass (all solids) framework contains vacancies and interstitials (often ionised or charged)

- VO (oxygen vacancy) introduces an energy level close to the conduction band, thus acts as donor

- glass often contains mobile ions (Na+, for example) from raw materials

- glass EG large (9 eV) hence transparent, for appreciable conduction glass has to be heated up (impractical)

window glass conducts @ > 300C

i : anion, cation

Z: valency

(“drift by diffusion”)

(VO)


… transparent conducting oxides (TCO), or oxide semiconductors

- portable devices demand transparency + electrical conductivity (e.g. touch screen)

- typical metals (Cu), semiconductors (Si) are opaque (visible light cannot go through)

- typical glasses are transparent, but electrical conduction at room temperature is negligible

- TCO: combines electrical conduction (metal-like) & transparency (glass like)

- three important TCO materials: ZnO (Wurtzite), In2O3 (cubic), SnO2 (see table below)

- ionic bonding: anion (O) yields localized 2p band (VB), cation (Zn, In) yields delocalized ns band (CB), see

next page, orbitals Fig. 2.2, electronic band model Fig. 3.4a.

- carriers come from defects: for examples

- VO (oxygen vacancy, anion cavancy) in ZnO acts as donor, Ed close to CB, hence n-type ZnO

- VCu (copper vacancy, cation cavancy) in Cu2O acts as acceptor, Ea close to VB, hence p-type Cu2O

- ideal: n-TCO for electron side, p-TCO for hole side – think light-emitting glass (new industry?)

source: Transparent Oxide Electronics: From Materials to Devices (Barquinha, Martins, Pereira, Fortunato) John Wiley & Sons, 2012.


source: Transparent Oxide Electronics: From Materials to Devices (Barquinha, Martins, Pereira, Fortunato) John Wiley & Sons, 2012.


… polymers / organic / plastic electronics

- “polymers” means repeating unit of monomers or [monomer]n; “organic” means carbon-based

- material advantages: low-cost processings (plastics), and light weight (1H, 6C vs 13Al, 14Si, 29Cu)

- first conducting polymer: polyacetylene (C2H2)n = n units of acetylene (C2H2) (Fig. 13.2)

- electronic conduction possible due to conjugated double and single bond (Figs. 13.2b-c, 13.1)

- conjugated system: a system of atoms covalently bonded (connected p orbitals) with alternating single and

multiple bonds, with delocalized electrons in a molecule, and in general lowers the overall energy of the

molecule and increases stability. Conjugated systems may be cyclic, acyclic, linear or mixed.


plastics/polymers

insulating semiconducting

non-conjugated conjugated: cyclic, acyclic, linear, mixed


-bonding

p-bonding

bonding

anti-bonding

- when two atoms are brought together, bonding will occur if electron in each has opposite spin (Fig. 2.3a,b),

otherwise anti-bonding will occur (Fig. 2.3c,d)

- two main bonding types (name , p represent shape of resulting bond s-like or p-like)

-bond: s—p (Fig. 2.4a), end-on p—p (Fig. 2.4b), electrons fixed (shared between two connecting nuclei)

p-bond: sideways-on p—p (Fig. 2.5), electrons delocalized (not belong to particular atom)

- for large molecules (polymers) need to consider hybridized bonds (sp, sp2, sp3—see next slide)

- polymers that are potentially conducting must have monomers with delocalized p-bondings (Figs. 2.9-2.11)


s1 + p1

s1 + p2

s1 + p3 (or s2 + p2)

bonding between

before

atoms bond (approximately) in such a way that the orbital overlaps are maximum

*concerned mostly with6C: 1s2 2s2 2p2 and 1H: 1s1

orbital hybridization*

after


… polyacetylene (C2H2)n revisited

- in polymer forms, they take either trans- or cis- configuration (Fig. 13.20)

- the two configurations are similar (alternating double [short] and single [long] bonds), but different

(periodicity) different energy gaps (Fig. 13.20)

- doping is possible, with orders of magnitude increase in conductivity (Fig. 13.21) by

- n-type: dope with alkali (Group I)

- p-type: dope with halogens (Group VII)

1.8 eV

2.0 eV


kT

Eo

exp

E = activation energy

(Conductivity is thermally activated)

Corresponds to m ~ 10-10 cm2/V.s

[c.f. Ag: 40 and Si: 1500 cm2/V.s]

soda silica glass

2102308 43

Tg: glass transition temperature(softening point)

borosilicate

Tm 1600 °C

… glasses & polymers

Tg

Tm melting temperature


2.9 Thin Metal Films in IC Interconnects

2102308 45

mfp

r (material property) generally does not depend on dimensions, unless dim ~ mfp

- shrinking electronics results in nm-scale interconnection

- conduction electrons are scattered by the usual mechanisms (lattice scattering,

impurity scattering), plus additional ones (at grain boundaries and surfaces)

reminder: mfp (Cu) 40 nm, slide #6

source: Intel, IEDM Dec 2017

latest MOSFET

(aka FinFET)


Manufacturing method:

• thin films: deposited by evaporation grainy

• bulk: thermal processes, annealing

r (film) > r (bulk) due to:

grain boundary scattering

ttr

rrr

11

//

R

RT

x,y,z mfp

x

y

Matthiessen’s:

r is a function of structure/morphology

d: mean grain size

d

l1

0r

r

mean grain size

mfp in single crystal


Significant in very thin films only, when thickness is comparable to the mean free path

(usually 10 nm or less—note that mfp of electrons in Cu @ room temperature is ~ 37nm)

surface scattering

Thin film (< 10 nm) formed by:

- deposition discontinuous lose conduction (r )

- epitaxy t very resistive (r )materialbulk in path freemean

1.0

B

B

l

l

D

z

Table 2.6 Typical resistivity values for thin metal films (polycrystalline) in microelectronics.

Metal Bulk r (W m)

10-8)

Film r (W m)

10-8)

Comment

Aluminum 2.67 2.8 - 3.3 Vacuum evaporated

Gold 2.2 2.4 Vacuum evaporated or sputtered

Nickel 6.9 12 Vacuum evaporated or sputtered

Molybdenum 5.7 10 Sputtered and annealed

Film thickness (D) vs mean free path.(l)

- on board (PCB), D >> l, use bulk r- in chip (IC), D << l, be careful with r

ttr

rrr

11

//

R

RT

2102308 47

D

l

pr

r1

0

film thickness

film thickness


0 0.05 0.01

1/d (1/micron)

0.015 0.02 0.025

15

20

25

30

35

50

Annealed at 100 C

Annealed at 150 C

As deposited

100 50010510

50

100

rbulk

= 16.7 n m

300

Film thickness (nm)

(a) (b)

(a) rfilm of the Cu polycrystalline films vs. reciprocal mean grain size (diameter), 1/d. Film

thickness D = 250 nm - 900 nm does not affect the resistivity. The straight line is rfilm = 17.8

n m + (595 n m nm)(1/d),

(b) rfilm of the Cu thin polycrystalline films vs. film thickness D. In this case, annealing (heat

treating) the films to reduce the polycrystallinity does not significantly affect the resistivity

because rfilm is controlled mainly by surface scattering.

|SOURCE: Data extracted from (a) S. Riedel et al, Microelec. Engin. 33, 165, 1997 and (b). W. Lim et al,

Appl. Surf. Sci., 217, 95, 2003)

large

grain

size (d) small

surfacegrainT rrrr

improve by annealing improve by increasing thickness

Cu

2102308 48

dominates dominates

varies slightly with deposition technique, substrate, etc…


2.8 Skin Effect

Low inductance

High inductance

Inductance = total magnetic flux threaded per unit currentปริมำณของเสน้แรงแม่เหลก็ต่อหน่ึงหน่วยกระแส

linked to B1 and B2

linked to B2 only

Skin depth is due to the circulating

eddy currents cancelling the current

flow in the center of a conductor and

reinforcing it in the skin.

Lenz's law: the current induced in a

circuit due to a change in the magnetic

field is so directed as to oppose the

change in flux or to exert a mechanical

force opposing the motion.

r is a material property, normally should not depend on excitation (voltage, electric field), except

when excitation is ac with very high frequency

2102308 49

http://upload.wikimedia.org/wikipedia/commons/c/c7/Skineffect_reason.svg


Implications: 1. HF signal Tx should not use solid conductor use pipes (waveguide) to

reduce weight & cost.

2. Ag-coated copper wire for audio applications.

2/

1

m

p

rr

p

pp

aAr

a

aaA

ac2

2

22

(per unit length)

HF signals prefer low impedance path current confined to skin

typermeabili magnetic

ormmm

frequency independent

(dc upto 1/t, mean free time,

see Ex. 2.2)

From solvingMaxwell’s eq

2102308 50


2/

1

m

Q1) what is the skin depth of Cu wire at 50 Hz?

A1) 9.3 mm

Q2) what is the largest diameter of Cu wire available commercially?

A2) Wiki(AWG: American wire gauge)

Q3) materials suitable as magnetic core for LF vs HF applications?

A3) magnets (Fe,….), conductivity

2102308 51


Ex 2.26: What is the change in the dc resistance of a Cu wire of radius 1 mm for

an ac signal at 10 MHz? And at 1 GHz?

[Cu has rdc = 1.7 10-8 W m or dc = 5.9 107 W-1m-1 and a relative permeability

mr ~ 1. Given that mo = 4p 10-7 H/m.]

Soln:

dcac

ac

dc

rr

r

r

/

@ (mm) rac/rdc

10MHz 20.7 24.2

1GHz 2.07 242note: H = Wb/A = V.s/A = W.s

2102308 52


SUMMARY


2.6 Thermal Conduction

• metals conduct electricity

and heat well

• In metals, free electrons

play role in heat conduction

• In nonmetals, heat

conducted by lattice

vibrations (phonons).

Metals

thermal carriers: electrons

T1 T2

heat flow

x

TAQ

x

T

A

k

x

le kkk

2102308 54


Rate of heat flow Rate of charge flow

Fourier’s law of thermal

conduction

Ohm’s law, Drude

(J = E)

x

TA

dt

dQQ

k

x

VA

dt

dqI

note:

I=JA

J=E

2102308 55


In metals, electrons participate in charge () and heat (k) transport

processes and k must be related!

WFL lawLorenz number or

Wiedemann-Franz-Lorenz coefficient

T

TT

?

1

k

r

20C CWFL = 2.44x10-8 WWK-2

In pure metal:

(= p2k2/3e2)

2102308 56


Relatively constant @ > 100K due to

almost constant mean speed of electrons

frozen electrons

electron-phonon

interactions

cf. r(T) of alloy

Fig.2.6 p.15

Note: kalloy < kelement (remember Nordheim?)

2102308 57


Non-metals:

thermal carriers: phonons

Coupling in k (W / m-K)

GOOD Diamonds 1000

POOR Polymers < 1

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Table 2.5 Typical thermal conductivities of various classes of materials at 25 °C.

k (W m-1 K-1)

Pure metal Nb Fe Zn W Al Cu Ag

52 80 113 178 250 390 420

Metal alloys Stainless Steel 55Cu-45Ni 70Ni-30Cu Steel Bronze Brass Dural

12 - 16 19.5 25 50 80 125 147

Ceramics

and glasses

Glass-

borosilicate

Silica-fused

(SiO2)

S3N4 Alumina

(Al2O3)

Saphire

(Al2O3)

Beryllia

(BeO)

Diamond

0.75 1.5 20 30 37 260 ~1000

Polymers Polypropylene PVC Polycarbonate Nylon

6,6

Teflon Polyethylene

low density

Polyethylene

high density

0.12 0.17 0.22 0.24 0.25 0.3 0.5

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1

10

100

1000

10000

100000

1 10 100 1000

Temperature (K)

MgO

Sapphire

k (

W m

-1 K

-1)

Fig. 4.46: Thermal conducitivity of Sapphire and MgO as afunction of temperature

k(T) of

non-metals



Hot Cold

Unharmonic

interaction

Direction of heat flow

1

2

3

Fig. 4.45: Phonons generated in the hot region travel towards thecold region and thereby transport heat energy. Phonon-phononunharmonic interaction generates a new phonon whose momentumis towards the hot region.

phonons not coupled

phonons bouced back

2102308 60


Ohm’s Law Fourier’s Law

k AL

TTQ

/

AL

V

R

VI

/

k

A

L

Thermal resistance ()

2102308 61


Ex. 2.19: A brass disk of electrical resistivity 50 nW-m conducts heat

from a heat source to a heat sink at a rate of 10 W. If its diameter is 20mm

and thickness is 30mm, what is the temperature drop across the disk?

CWFL = 2.44x10-8 WWK-2

Ans: 6.52 K

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