Ch2 3rdEd Extended

8/3/2019 Ch2 3rdEd Extended

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From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)

Chapter 2

Electrical Engineering

Materials EET3066


2/81


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.

|SOURCE: Dr. Don Scansen, Semiconductor Insights, Kanata, Ontario, Canada


3/81

Fig 2.1


Drift of electrons in a conductor in the presence of an applied

electric field. Electrons drift with an average velocity vdx in the x-

direction.(Ex is the electric field.)


4/81

Definition of Drift Velocity

vdx ! 1N

[vx1 vx2 vx3 vxN]

vdx = drift velocity inx direction,N= number of conduction electrons,

vxi =x direction velocity ofith electron

Current Density and Drift Velocity

Jx = current density in thex direction, e = electronic charge, n =

electron concentration, vdx = drift velocity

Jx (t) =envdx(t)


5/81

Fig 2.2


Vibrating Cu+ ions

u(x

Ex

V

(a) (b)

(a) A conduction electron in the electron gas moves about randomly in a metal (with a mean speed

u) being frequently and randomly scattered by by thermal vibrations of the atoms. In the absenceof an applied field there is no net drift in any direction.

(b) In the presence of an applied field, Ex, there is a net drift along the x-direction. This net drift

along the force of the field is superimposed on the random motion of the electron. After many

scattering events the electron has been displaced by a net distance, (x, from its initial position

toward the positive terminal


6/81

Fig 2.3


timet

2t

Electron 2

vx2

-ux2

time

vx1

-ux1

t1

t

Last collision

Free time

Electron 1

Present timeVelocity gained along x

timet3

t

Electron3

vx3

-ux3

Velocity gained in the x-direction at time t from the electric field(Ex) for three electrons. There will be Nelectrons to consider in

the metal.


7/81


Definition of Drift Mobility

vdx = drift velocity, Qd= drift mobility,Ex = applied field

vdx =QdEx

Qd= drift mobility, e = electronic charge, X = mean scattering time

(mean time between collisions) = relaxation time, me = mass of an

electron in free space.

Drift Mobility and Mean Free Time

Qd!

eX

me


8/81


Unipolar Conductivity

W = conductivity, e = electronic charge, n = number of electrons per

unit volume, Qd= drift velocity, X = mean scattering (collision) time =

relaxation time, me = mass of an electron in free space.

W ! enQd! e

2

nX

me

Drift Velocity

(x = net displacement parallel to the field, (t= time interval,

vdx = drift velocity

(x(t !

vdx


9/81

Fig 2.4


t1

s = (x

Collision

Start

Finish

Electric field

E

t2

x

Collision

0

2

1

p

s1

ux1

uy1

3

4

t3

Distance drifted in total time (t

The motion of a single electron in the presence of an electric field E. During a time

interval ti, the electron traverses a distance si along x. Afterp collisions, it has drifted

a distance s = (x.


10/81

Fig 2.5


a

u

Electron

S= T a2

N=uX

Scattering of an electron from the thermal vibrations of the atoms. The

electron travels a mean distanceN= u X between collisions. Since the

scattering cross sectional area is S, in the volume SNthere must be at

least one scatterer, Ns(SuX) = 1.


11/81


Mean Free Time Between Collisions

X = mean free time, u= mean speed of the electron,Ns =

concentration of scatterers, S= cross-sectional area of the scatterer

X !1

SuNs

Resistivity Due to Thermal Vibrations of the

Crystal

VT

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

= temperature

VT=AT


12/81

Fig 2.6


XI

X8

Strained region by impurity exerts a

scattering force F= - d(PE) /dx

Two different types of scattering processes involving scattering fromimpurities alone and thermal vibrations alone.


13/81



14/81


Matthiessens Rule

V = effective resistivity, VT

= resistivity due to scattering by thermal

vibrations only, VI = resistivity due to scattering of electrons fromimpurities only.

V =VT+V

I

V = overall resistivity, VT

= resistivity due to scattering from thermal

vibrations,VR = residual resistivity

V =VT

+ VR


15/81


Definition of Temperature Coefficient ofResistivity

Eo= TCR(temperature coefficient of resistivity),

HV= change in the

resistivity, Vo = resistivity at reference temperature To, HT= small

increase in temperature, To = reference temperature

oTTo

oT !

-

!HHV

VE 1

V = resistivity, Vo = resistivity at reference temperature, E0 = TCR

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

reference temperature

Temperature Dependence ofResistivity

V!VS[1 + Eo(TTo)]


16/81

Fig 2.7


The resistivity of various metals as a function of temperature above 0C. Tin melts at 505 K whereas nickel and iron go through a magneticto non-magnetic (Curie) transformations at about 627 K and 1043 K

respectively. The theoretical behavior(V ~ T) is shown for reference.[Data selectively extracted from various sources including sections inMetals Handbook, 10th Edition, Volumes 2 and 3 (ASM, MetalsPark, Ohio, 1991)]

V w T

Tungsten

Silver

Copper

Iron

Nickel

Platinum

NiCr Heating Wire

Tin

Monel-400

Inconel-825

10

100

1000

2000

100 1000 10000

Temperature(K)

Res

istivit

y(

nm

)


17/81

Fig 2.8


0.00001

0.0001

0.001

0.01

0.1

10

100

10 100 1000 10000

Temperature(K)

0

0.51

1.5

2

2.5

3

3.5

0 20 40 60 80 100T(K)

V w T

V w T5

V = VR

V ! VR

V w T5

V w T

Res

isti

vit

y(

n

m)

V (n; m)

The resistivity of copper from lowest to highest temperatures (nearmelting temperature, 1358 K) on a log-log plot. Above about 100 K,

V w T, whereas at low temperatures, V w T5 and at the lowest

temperatures V approaches the residual resistivity VR . The inset

shows the V vs. Tbehavior below 100 K on a linear plot ( VRis too

small on this scale).


18/81

Fig 2.9


Temperature (K)

0

20

40

60

0 100 200 300

VT

VI

VCW

100%Cu (Annealed)

100%Cu (Deformed)

Cu-1.12%Ni

Cu-1.12%Ni (Deformed)

Cu-2.16%Ni

Cu-3.32%Ni

R

esisti

vity

(n

;

m)

Typical temperature dependence of the resistivity of annealed andcold worked (deformed) copper containing various amount of Ni inatomic percentage (data adapted from J.O. Linde, Ann. Pkysik, 5,219 (1932)).


19/81

Fig 2.10


120 V

40 W

0.333 A

Power radiated from a light bulb is approximately equal to theelectrical power dissipated in the filament.


20/81



21/81

Fig 2.11


1000

1100

1200

1300

1400

1500

20 40 60 80 100

LIQUID PHASE

SOLID SOLUTION

0

100% Cu 100% Niat.% Ni

Temperature

(C)

(a)

LIQUID

US

SOLID

US

L+S

100% Cu at.% Ni

Cu-Ni Alloys

0

100

200

300

400

500

600

0 20 40 60 80 100

100% Ni

Res

isti

vity

(n

;

m)

(b)

(a) Phase diagram of the Cu-Ni alloy system. Above the liquidus lineonly the liquid phase exists. In the L + S region, the liquid (L) and solid(S) phases coexist whereas below the solidus line, only the solid phase (asolid solution) exists. (b) The resistivity of the Cu-Ni alloy as a functionof Ni content (at.%) at room temperature. [Data extracted from MetalsHandbook-10th Edition, Vols 2 and 3, ASM, Metals Park, Ohio, 1991 andConstitution of Binary Alloys, M. Hansen and K. Anderko, McGraw-Hill,

New York, 1958]


22/81



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Nordheims Rule for Solid Solutions

VI

= resisitivity due to scattering of electrons from

impurities

C= Nordheim coefficient

X= atomic fraction of solute atoms in a solid solution

VI

=CX(1X)


24/81

Fig 2.12


0

20

40

60

80

100

120

140

160

0 10 20 30 40 50 60 70 80 90 100

Composition (at.% Au)

Annealed

Quenched

Cu3Au CuAu

Res

isti

vity(

n;

m)

Electrical resistivity vs. composition at room temperature in Cu-Aualloys. The quenched sample (dashed curve) is obtained by quenchingthe liquid and has the Cu and Au atoms randomly mixed. The

resistivity obeys the Nordheim rule. On the other hand, when thequenched sample is annealed or the liquid slowly cooled (solid curve),certain compositions (Cu3Au and CuAu) result in an ordered

crystalline structure in which Cu and Au atoms are positioned in anordered fashion in the crystal and the scattering effect is reduced.


25/81


Combined

Matthiessen and Nordheim Rules

V = resistivity of the alloy (solid solution)

Vmatrix = resistivity of the matrix due to scattering from

thermal vibrations and other defects

C= Nordheim coefficientX= atomic fraction of solute atoms in a solid solution

V =Vmatrix + CX(1X)


26/81

Fig 2.13


L

A

Jy

(b)

Jx

E F

A

L

(a)

L

A

Dispersed phaseContinuous phase y

x

Jx

(c)

The effective resistivity of a material having a layered structure. (a)

Along a direction perpendicular to the layers. (b) Along a directionparallel to the plane of the layers. (c) Material with a dispersed phasein a continuous matrix.


27/81


Effective Resistance of Mixtures

Reff= effective resistance

LE = total length (thickness) of the E-phase layers

VE= resistivity of the E-phase layers

A = cross-sectional area

LF = total length (thickness) of theF-phase layersVF = resistivity of the F-phase layers

Reff !LEVEA

LFVF

A


28/81


Resistivity-Mixture Rule

Veff= effective resistivity of mixture, GE= volume fraction of the E-

phase,VE = resistivity of the E-phase, GF= volume fraction of the F-

phase,VF= resistivity of the F-phase

Veff=GEVEGFVF

Conductivity-MixtureRule

Weff= effective conductivity of mixture, GE= volume fraction of the

E-phase, WE = conductivity of the E-phase, GF= volume fraction of

theF-phase, WF= conductivity of theF-phase

Weff=G

EW

EG

FW

F


29/81


Mixture Rule (Vd> 10V

c)

Veff= effective resistivity, Vc = resistivity of continuous phase, Gd=

volume fraction of dispersed phase,Vd= resistivity of dispersed

phase

Veff ! Vc

(1 1

2

Gd)

(1 G d)

MixtureRule (Vd< 0.1V

c)

Veff= effective resistivity, Vc = resistivity of the continuous phase, Gd= volume fraction of the dispersed phase,Vd= resistivity of the

dispersed phase

Veff

! Vc

(1 Gd)

(1 2Gd)


30/81

Fig 2.14


L

A

F

E(a)

FA/N

E

L(b)

(a) A two phase solid. (b) A thin fiber cut out from the solid.


31/81

Fig 2.15


X1

X2

100%B100%A X (% B)

Liquid, L

E+L F+LE

F

TA

TB

E FT1

Two phase region

One phase

region: F

onlyT

E

(a)

Tempera

ture

VB

VA

Composition X (% B)0 X1 X2 100%B

Nordheim's Rule

Mixture Rule

(b)

Res

isti

vit

y

(a) The phase diagram for a binary, eutectic forming alloy. (b) Theresistivity vs composition for the binary alloy.


32/81

Fig 2.16


Illustration of the Hall effect. The z-direction is out from the plane ofpaper. The externally applied magnetic field is along the z-direction.

Jxvdx

VVHeE

H

Jy

= 0

xz

y

evdxBz

Bz

V

Bz

A

Jx

EH

Ex

+ + ++ +

+


33/81

Fig 2.17


B

F = qvvB

v

B

F = qvvB

v

B

q=+e q= -e

(a) (b)

A moving charge experiences a Lorentz force in a magnetic field. (a) A

positive charge moving in the x direction experiences a force downwards. (b)

A negative charge moving in the -x direction also experiences a force

downwards.


34/81


Lorentz Force

F = force, q = charge, v = velocity of charged particle, B = magnetic

field

F =qv v B

RH= Hall coefficient,Ey = electric field in the y-direction,Jx =

current density in thex-direction,Bz = magnetic field in the z-

direction

Definition of Hall Coefficient

zx

y

H

BJ

ER !


35/81



36/81



37/81

Fig 2.18


V

IL

VL

VL

Bz

Ix =VL/RR

VH

w

IL

C C

Wattmeter

Load

RLSource

ILIL

VL

Wattmeter based on the Hall effect.Load voltage and load currenthave L as subscript. C denotes the current coils. for setting up amagnetic field through the Hall effect sample (semiconductor)


38/81

Fig 2.19


Vibrating Cu+

ionsElectron Gas

HOT COLDHEAT

Thermal conduction in a metal involves transferring energy from the hot region to the cold

region by conduction electrons. More energetic electrons (shown with longer velocity vectors)

from the hotter regions arrive at cooler regions and collide there with lattice vibrations and

transfer their energy. Lengths of arrowed lines on atoms represent the magnitudes of atomic

vibrations.


39/81

Fig 2.20


HT

Hx

HOT COLD

dQ

dtHEAT A

Heat flow in a metal rod heated at one end. Consider the rate of heat

flow, dQ/dt, across a thin section H x of the rod. The rate of heat flow

is proportional to the temperature gradient H T/H x and the crosssectional area A.


40/81


Fouriers Law of Thermal Conduction

Qd = rate of heat flow, Q = heat, t= time, O = thermal conductivity,

A = area through which heat flows, dT/dx = temperature gradient

dQ !dQ

dt!OA

HT

Hx

I= electric current, A = cross-sectional area, W = electrical

conductivity, dV/dx = potential gradient (represents an electric field),

HV= change in voltage across Hx, Hx = thickness of a thin layer atx

Ohms Law ofElectrical Conduction

I ! AWHV

Hx


41/81

Fig 2.21


0

100

200

300

400

450

0 10 20 30 40 50 60 70

Electrical conductivity, W 106 ;-1 m-1

Al

Ag

Au

Cu

Brass (Cu-30Zn)

Be

Pd-40Ag

Bronze (95Cu-5Sn)

Ag-3Cu

Ag-20Cu

W

Mo

Hg

Mg

Steel (1080)

Ni

OW

= T CWFL

The

rmalconduc

tivi

ty,O

(W

K-1m-1

)

Thermal conductivity, O vs. electrical conductivity W for variousmetals (elements and alloys) at 20 C. The solid line represents the

WFL law with CWFL } 2.44v108 W ; K-2.


42/81


Wiedemann-Franz-Lorenz Law

O = thermal conductivity

W = electrical conductivity

T= temperature in Kelvins

CWFL =Lorenz number

O

WT!CWFL ! 2.45v10

8W ; K

2


43/81

Fig 2.22


Copper

Brass (70Cu-30Zn)

Al-14%Mg

Aluminum

10

100

1000

10000

50000

1 10 100 1000

The

rma

lcon

duct

ivit

y,O

(W

K-1

m-1

)

Temperature (K)

Thermal conductivity vs. temperature for two pure metals (Cu and Al)and two alloys (brass and Al-14%Mg). Data extracted fromThermophysical Properties ofMatter, Vol. 1: Thermal Conductivity,Metallic Elements and Alloys, Y.S. Touloukian et. al (Plenum, NewYork, 1970).


44/81



45/81

Fig 2.23


Energetic atomic vibrations

Hot Cold

Equilibrium

Conduction of heat in insulators involves the generation andpropogation of atomic vibrations through the bonds that couple theatoms. (An intuitive figure.)


46/81


Fouriers Law

dQ ! AO(T

L!

(T

(L /OA)

Qd = rate of heat flow or the heat current, A = cross-sectional area,

O = thermal conductivity (material-dependent constant), (T=

temperature difference between ends of component,L = length of

component

Ohms Law

I= electric current, (V= voltage difference across the conductor,R =

resistance,L = length, W = conductivity, A = cross-sectional area

)/( AL

V

R

VI

W

(

!

(

!


47/81


Definition of Thermal Resistance

Qd = rate of heat flow, (T= temperature difference,U = thermal

resistance

dQ !

(T

U

Thermal Resistance

U = thermal resistance,L = length, A = cross-sectional area, O =

thermal conductivity

U !L

AO


48/81

Fig 2.24


Conduction of heat through a component in (a) can be modeled as a

thermal resistance U shown in (b) where Qd= (T/U.

Hot

L

(T

(a)

Qd

Cold

A

Qd

(T

Qd

(b)

U

Qd= (T/U


49/81

Fig 2.25


Range of conductivites exhibited by various materials

10610310010-310-610-910-1210-1510-18 109

Semiconductors Conductors

1012

Conductivity ( m)-1

AgGraphite NiCrTe

DegeneratelyDoped Si

Insulators

Diamond

SiO2

Superconductors

PET

PVDF

AmorphousAs

2Se

3

Mica

Alumina

Borosilicate

Inorganic Glasses

Alloys

Soda silica glass

Many ceramics

Metals

Polypropylene

Pure SnO2

Intrinsic GaAs

Intrinsic Si


50/81

Fig 2.26


(a) Thermal vibrations of the atoms rupture a bond and release a free electron into the crystal. A hole

is left in the broken bond which has an effective positive charge.(b)An electron in a neighbouring bond can jump and repair this bond and thereby create a hole in its

original site; the hole has been displaced.

(c) When a field is applied both holes and electrons contribute to electrical conduction.

e-hole

E

a b c


51/81


Conductivity of a Semiconductor

W = conductivity, e = electronic charge, n = electron concentration, Qe= electron drift mobility,p = hole concentration, Q

h= hole drift

mobility

W =enQe + epQh

ve = drift velocity of the electrons, Qe = drift mobility of the electrons,

e = electronic charge,Fnet = net force

Drift Velocity and Net Force

ve !Qe

eFnet


52/81

Fig 2.27


Jx

vex

eEy

Jy

= 0

x

z

y

evexBz

Bz

Bz

Jx

Ey

Ex

vhx

evhxBz+

++

+ +

eEy

V

+

Hall effect for ambipolar conduction as in a semiconductor where there

are both electrons and holes. The magnetic field Bz is out from the plane of the

paper. Both electrons and holes are deflected toward the bottom surface of the

conductor and consequently the Hall voltage depends on the relative mobilities and

concentrations of electrons and holes.


53/81


Hall Effect for Ambipolar Conduction

RH= Hall coefficient,p = concentration of the holes, Qh = hole drift

mobility, n = concentration of the electrons, Qe = electron drift

mobility, e = electronic charge

RH !pQ

h

2 nQe

2

e(pQh

nQe )2

OR

b =Qe,/Qh

RH

!p nb

2

e(p nb)2


54/81

Fig 2.28


(a)

Anion vacancy

acts as a donor

E

Interstitial cation diffuses

Vacancy aids the diffusion of positive ion

Na+

O2-

Si4+

(b)

E

Possible contributions to the conductivity of ceramic and glass

insulators (a) Possible mobile charges in a ceramic (b) A Na+ ionin the glass structure diffuses and therefore drifts in the direction ofthe field. (E is the electric field.)


55/81


General Conductivity

W = conductivity

qi = charge carried by the charge carrier species i (for electrons and

holes qi =e)

ni

= concentration of the charge carrier

Qi = drift mobility of the charge carrier of species i

W ! 7qi niQi


56/81


From Principles ofElectronic Materials and Devices, ThirdEdition, S.O. Kasap ( McGraw-Hill, 2005)


57/81

Fig 2.29


1v10-15

1v10-13

1v10-11

1v10-7

1v10-5

1v10-3

1v10-1

1.2 1.6 2 2.4 2.8 3.2 3.6 4

103/T (1/K)

SiO2

Pyrex

12%Na2O-88%SiO2

24%Na2O-76%SiO2

PVAc

PVC

As3.0Te3.0Si1.2Ge1.0 glass

1v10-9

Con

duct

ivit

y1

/(

m

)

Conductivity vs reciprocal temperature for various low conductivitysolids. (PVC = Polyvinyl chloride; PVAc = Polyvinyl acetate.) Dataselectively combined from numerous sources.


58/81


Temperature Dependence ofConductivity

W = conductivity

WS!constant

)W= activation energy for conductivityk= Boltzmann constant, T= temperature

W ! W o exp EW

kT


59/81

Fig 2.30


B1

B2

B2

B1

B1

B2

(a) Total current into

paper is I (b)Current inhollow outercylinder isI/2

(c) Current in inner

cylinder is I/2

Illustration of the skin effect. A hypothetical cut produces a hollow outercylinder and a solid inner cylinder. Cut is placed where it would give equalcurrent in each section. The two sections are in parallel so that the currentsin (b) and (c) sum to that in (a).


60/81

Fig 2.31


2a

H = Skin depth

R

[L

At high frequencies, the core region exhibits more inductiveimpedance than the surface region, and the current flows in thesurface region of a conductor defined approximately by the skin

depth, H.


61/81


Skin Depth for Conduction

H = skin depth, [ = angular frequency of current, W = conductivity, Q

= magnetic permeability of the medium

H !1

12

[WQ

rac = ac resistance ,V = resistivity, A = cross-sectional area, a =

radius, H = skin depth

rac !V

A

}V

2TaH

HFResistance per Unit Length Due to Skin Effect


62/81

Fig 2.32


(b)

(a) Grain boundaries cause scattering of the electron and

therefore add to the resistivity by Matthiessen's rule .(b) For a very grainy solid, the electron is scattered from grainboundary to grain boundary and the mea n free path isapproximate ly equal to the mea n grain diameter.

Grain1

Grain2

Grain

Boundary

(a)


63/81

Fig 2.33


D

Jx

Conduction in thin films may be controlled by scattering from thesurfaces.


64/81

Fig 2.34


Scattering 1

Scattering 2

U

N= D

+x

x

N= D/cosU

+y

y

The mean free path of the electron depends on the angle U after scattering.

From Principles ofElectronic Materials and Devices, ThirdEdition, S.O. Kasap ( McGraw-Hill, 2005)


65/81



66/81


67/81

Fig 2.36


M3

Silicon

M2

M1

Cuinterconnects

Metal interconnects wiring devices on a siliconcrystal. Three different metallization levels M1,

M2, and M3 are used. The dielectric between theinterconnects has been etched away to expose the

interconnect structure.

|SOURCE: Courtesy of IBM

Lowpermittivity

dielectric

M1M2

M3

M4

M5

M6

M7

Cross section of a chip with 7 levels of

metallization, M1 to M7. The image isobtained with a scanning electron micro-

scope (SEM).

|SOURCE: Courtesy of Mark Bohr, Intel.


68/81


Three levels of interconnects in a flash memory chip. Different levels are connected

through vias.

|SOURCE: Courtesy of Dr. Don Scansen, Semiconductor Insights, Kanata, Ontario, Canada


69/81

Fig 2.37



70/81

Fig 2.38


Hillocks

Void

and

failure

Current

Grain boundary Void

Hillock

Cold

Cold

Hot

Current

Hillock

Grainboundary

(a) (b)

(c)

Interconnect

Interf

aceElectron

(a) Electrons bombard the metal ions and force them to slowly migrate(b) Formation of voids and hillocks in a polycrystalline metal interconnect by theelectromigration of metal ions along grain boundaries and interfaces. (c) Acceler-

ated tests on 3 mm CVD (chemical vapor deposited) Cu line. T= 200 oC, J= 6

MA cm-2: void formation and fatal failure (break), and hillock formation.

|SOURCE: Courtesy ofL. Arnaud et al, Microelectronics Reliability, 40, 86, 2000.


71/81



72/81

Fig 2.39


Resistivity versus temperature for pure iron and 4% C steel.


73/81



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75/81


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78/81

Fig 2.40


Hall effect in a rectangular material with length L, widthW, and thicknessD.

The voltmeter is across the width W.


79/81

Fig 2.41


The strain gauge consists of a long, thin wire folded several times along its length to

form a grid as shown and embedded in a self-adhesive tape. The ends of the wire are

attached to terminals (solder pads) for external connections. The tape is stuck on the

component for which the strain is to be measured.


80/81


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Ch2 3rdEd Extended

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