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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
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From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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
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Fig 2.1
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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.)
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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)
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Fig 2.2
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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
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Fig 2.3
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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.
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From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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
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From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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
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Fig 2.4
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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.
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Fig 2.5
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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.
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From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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
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Fig 2.6
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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.
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From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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
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From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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)]
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Fig 2.7
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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
)
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Fig 2.8
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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).
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Fig 2.9
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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)).
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Fig 2.10
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
120 V
40 W
0.333 A
Power radiated from a light bulb is approximately equal to theelectrical power dissipated in the filament.
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From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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Fig 2.11
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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]
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From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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)
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Fig 2.12
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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.
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From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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)
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Fig 2.13
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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.
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From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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
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From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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
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From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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)
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Fig 2.14
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
L
A
F
E(a)
FA/N
E
L(b)
(a) A two phase solid. (b) A thin fiber cut out from the solid.
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Fig 2.15
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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.
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Fig 2.16
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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
+ + ++ +
+
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Fig 2.17
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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.
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From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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 !
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From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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Fig 2.18
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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)
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Fig 2.19
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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.
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Fig 2.20
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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.
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From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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
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Fig 2.21
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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.
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From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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
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Fig 2.22
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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).
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From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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Fig 2.23
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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.)
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From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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
(
!
(
!
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From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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
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Fig 2.24
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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
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Fig 2.25
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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
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Fig 2.26
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
(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
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From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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
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Fig 2.27
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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.
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From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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
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Fig 2.28
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
(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.)
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From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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
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From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
From Principles ofElectronic Materials and Devices, ThirdEdition, S.O. Kasap ( McGraw-Hill, 2005)
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Fig 2.29
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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.
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From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
Temperature Dependence ofConductivity
W = conductivity
WS!constant
)W= activation energy for conductivityk= Boltzmann constant, T= temperature
W ! W o exp EW
kT
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Fig 2.30
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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).
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Fig 2.31
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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.
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From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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
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Fig 2.32
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
(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)
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Fig 2.33
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
D
Jx
Conduction in thin films may be controlled by scattering from thesurfaces.
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Fig 2.34
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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)
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From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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Fig 2.36
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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.
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From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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
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Fig 2.37
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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Fig 2.38
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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.
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From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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Fig 2.39
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
Resistivity versus temperature for pure iron and 4% C steel.
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From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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Fig 2.40
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
Hall effect in a rectangular material with length L, widthW, and thicknessD.
The voltmeter is across the width W.
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Fig 2.41
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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.
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