Download Drude theory. Conductivity of metals. Scattering. Matthiessen`s and

Drude model of electrical conduction
Ex
Δx
A
vdx
Jx
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.)
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
J x (t ) = envdx (t )
Drift velocity = average velocity
in x-direction
1
v dx = [v x1 + v x 2 + vx 3 + ⋅⋅⋅ + v xN ]
N
Current density
Jx =
Δq enAvdx Δt
I
=
=
= envdx
A AΔt
AΔt
Jx = current density in the x direction,
e = electronic charge,
n = electron concentration,
vdx = drift velocity in x direction,
N = number of conduction electrons
Drude model of electrical conduction
Ex
Δx
A
vdx
Jx
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.)
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
Drift velocity = average velocity
in x-direction
1
v dx = [v x1 + v x 2 + vx 3 + ⋅⋅⋅ + v xN ]
N
Current density
Jx =
Δq enAvdx Δt
I
=
=
= envdx
A AΔt
AΔt
?
J x (t ) = envdx (t )
Jx = current density in the x direction,
e = electronic charge,
n = electron concentration,
vdx = drift velocity in x direction,
N = number of conduction electrons
Electrical resistance is caused by scattering on “perturbations” of ideal crystal
Ex
Velocity gained along x
vx1°ux1
Present time
Last collision
Electron 1
time
u
Δx
Vibrating Cu+ ions
t1
vx1°ux1
t
V
Electron 2
(b)
(a)
Free time
t2
t
time
vx1°ux1
Fig. 2.2 (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 absence
of 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, Δx, from its initial position toward the
positive terminal
Fig. 2.3: Velocity gained in the x-direction at time t from the
electric field (Ex) for three electrons. There will be N electrons to
consider in the metal.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
Electron 3
t3
eE x
(t − ti )
me
t
time
where uxi=thermal velocity,
Full velocity of i-th electron
v xi = u xi +
Average drift velocity vdx =
eE
eE
1
[v x1 + v x 2 + v x 3 + ⋅ ⋅ ⋅ + v xN ] = x (t − ti ) = x τ
N
me
me
Ex=electric field e=electron charge,
me=mass of electron
τ = average time between collisions = scattering time = relaxation time
Definition of Drift Mobility
vdx =
eE x
τ
me
vdx = μdEx
vdx = drift velocity, μd = drift mobility, Ex = applied field
Drift Mobility and Mean Free Time
eτ
μd =
me
μd = drift mobility, e = electronic charge, τ = mean scattering time
(mean time between collisions) = relaxation time, me = mass of an
electron in free space.
Drift Mobility, Concentration and Conductivity
vdx = μdEx
J x = envdx = enμ d E x
J x = σE x
σ = enμ d
σ = conductivity
Electron drift mobility in metals (Example 2.2)
Conductivity of copper (Cu) σ = 5.9×105 Ω-1cm-1
Density of copper
Atomic mass of copper
ρ = 8.96 g cm-3
Μat = 63.5 g mol-1
μd = ?
τ =?
σ = enμ d
n ≈ N at
ρN A
N at =
NA
μd = 43.4 cm2V-1s-1
M at
= 6.0220 × 1023
- an Avogadro's number
eτ
μd =
me
τ = 2.5×10-14 s
Subshell
n
Shell
l=0
s
1
p
2
d
1
K
2
2
L
2
6
3
M
2
6
10
4
N
2
6
10
3
f
Table 1.1. Maximum possible number
of electrons in the shells and subshells
of an atom
Quantum numbers:
14
Principal
Orbital
Magnetic
Spin
n = 1, 2, 3...
l = 0,1,2 ... n-1 (l < n)
ml = -l , -l +1, ...0... l -1, l or (2l +1)
ms = + ½ , - ½
[Ar] 3d104s1
Scattering cross-section
S = π a2
l=uτ
a
u
Mean Free Time Between Collisions
1
τ=
SuNs
τ = mean free time,
u = mean speed (thermal + drift),
Ns = concentration of scatterers,
S = cross-sectional area
Electron
Fig. 2.4: Scattering of an electron from the thermal vibrations of the
atoms. The electron travels a mean distance l = u τ between
collisions. Since the scattering cross sectional area is S, in the volume
Sl there must be at least one scatterer, Ns(Suτ) = 1.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
Scattering on Atomic Vibrations
1
τ∝
T
or
eτ
e C
1
= en × ⇒ σ ∝
σ = enμ d = en
me
me T
T
C
τ=
T
Resistivity Due to Thermal
Vibrations of the Crystal
ρT = AT
ρT = resistivity of the metal,
A = temperature independent constant,
T = temperature
Scattering from impurities
Strained region by impurity exerts a
scattering force
Scattering time (τΙ)
does NOT
depend on
temperature
τI
1
τΤ
Fig. 2.5: Two different types of scattering processes involving scattering
from impurities alone and thermal vibrations alone.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
τ
1
μ
=
=
1
τT
1
μT
+
1
τI
+
1
μI
ρ = ρT + ρ I
eτ
μd =
me
1
ρ=
nμ
Matthiessen’s Rule
ρ = ρT + ρI
ρ = effective resistivity, ρT = resistivity due to scattering by thermal
vibrations, ρI = resistivity due to scattering of electrons from
impurities.
ρ = ρT + ρ R
ρ = ΑΤ + Β
ρ = overall resistivity, ρT = resistivity due to scattering from thermal
vibrations, ρR = residual resistivity
Definition of Temperature
Coefficient of Resistivity
1 ⎡ δρ ⎤
⎢ ⎥
αo =
ρo ⎣ δT ⎦ T =To
αo = TCR (temperature coefficient of resistivity), δρ = change in
resistivity, ρo = resistivity at reference temperature To , δT = small
increase in temperature, To = reference temperature
Temperature Dependence of Resistivity
ρ = ρο [1 + α0(T−T0)]
ρ = resistivity, ρo = resistivity at reference temperature, α0 = TCR
(temperature coefficient of resistivity), T = new temperature, T0 =
reference temperature
Temperature dependence of resistivity of some metals and alloys
2000
Inconel-825
NiCr Heating Wire
1000
Iron
Tungsten
Resistivity (nΩ m)
Monel-400
ρ∝T
Tin
100 Platinum
Copper
Nickel
Silver
10
100
1000
10000
Temperature (K)
Fig. 2.6: The resistivity of various metals as a function of temperature
above 0 °C. Tin melts at 505 K whereas nickel and iron go through a
magnetic to non-magnetic (Curie) transformations at about 627 K and
1043 K respectively. The theoretical behavior (ρ ~ T) is shown for
reference.
[Data selectively extracted from various sources including sections in
Metals Handbook, 10th Edition, Volumes 2 and 3 (ASM, Metals
Park, Ohio, 1991)]
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
Temperature dependence of resistivity of copper
100
ρ∝T
10
Resistivity (nΩ m)
1
ρ (nΩ m)
3.5
0.1
ρ∝
0.01
3
2.5
ρ∝T
2
1.5
ρ ∝ T5
1
0.5 ρ = ρR
0
0 20 40 60
0.001
ρ = ρR
0.0001
T5
80 100
T (K)
0.00001
1
10
100
1000
10000
Temperature (K)
Fig.2.7: The resistivity of copper from lowest to highest temperatures
(near melting temperature, 1358 K) on a log-log plot. Above about
100 K, ρ ∝ T, whereas at low temperatures, ρ ∝ T 5 and at the
lowest temperatures ρ approaches the residual resistivity ρR . The
inset shows the ρ vs. T behavior below 100 K on a linear plot ( ρR
is too small on this scale).
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
Definition of Temperature Coefficient of
Resistivity
1 ⎡ δρ ⎤
⎢ ⎥
αo =
ρo ⎣ δT ⎦ T =To
αo = TCR (temperature coefficient of resistivity), δρ = change in
resistivity, ρo = resistivity at reference temperature To , δT = small
increase in temperature, To = reference temperature
Temperature Dependence of Resistivity
ρ = ρο [1 + α0(T−T0)]
ρ = resistivity, ρo = resistivity at reference temperature, α0 = TCR
(temperature coefficient of resistivity), T = new temperature, T0 =
reference temperature
2000
Inconel-825
NiCr Heating Wire
1000
Iron
Tungsten
Monel-400
Resistivity (nΩ m)
Temperature dependence
of resistivity of some metals
and alloys
ρ∝T
Tin
100 Platinum
Copper
Nickel
Silver
10
100
1000
10000
Temperature (K)
Fig. 2.6: The resistivity of various metals as a function of temperature
above 0 °C. Tin melts at 505 K whereas nickel and iron go through a
magnetic to non-magnetic (Curie) transformations at about 627 K and
1043 K respectively. The theoretical behavior (ρ ~ T) is shown for
reference.
[Data selectively extracted from various sources including sections in
Metals Handbook, 10th Edition, Volumes 2 and 3 (ASM, Metals
Park, Ohio, 1991)]
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
What is the temperature of the filament?
40 W
0.333 A
120 V
Power radiated from a light bulb at 2408 °C is equal to the electrical
power dissipated in the filament.
P = 40 W
V = 120 V
L = 38.1 cm
D = 33 μm
ρ (273K) = 5.51×10-8 Ωm
ρ (T)~ T1.2
Resistivity of Cu alloyed with small amount of Ni
60
Cu-3.32%Ni
Cu-2.16%Ni
40
Cu-1.12%Ni (Deformed)
Cu-1.12%Ni
ρCW
Resistivity
(nΩ m)
20
100%Cu (Deformed)
100%Cu (Annealed)
ρI
ρT
0
0
100
200
Temperature (K)
300
Fig. 2.8: Typical temperature dependence of the resistivity
of annealed and cold worked (deformed) copper containing
various amount of Ni in atomic percentage (data adapted
from J.O. Linde, Ann. Pkysik, 5, 219 (1932)).
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
ρ = ΑΤ + ρ r
Scattering on grain boundaries
Grain 1
Grain 2
Grain
Boundary
(a)
(b)
Fig. 2.31: Grain boundaries cause scattering of the electron and
therefore add to the resistivity by Matthiessen's rule. For a very
grainy solid, the electron is scattered from grain boundary to grain
boundary and the mean free path is approximately equal to the
mean grain diameter.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
Scattering on surface
Jx
D
Fig. 2.32: Conduction in thin films may be controlled by scattering
from the surfaces.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
Scattering on grain boundaries
Scattering on surface
300
35
(b)
(a)
30
As deposited
Annealed at 100 C
Annealed at 150 C
100
25
50
20
ρbulk = 16.7 n m
10
15
0 0.05 0.01 0.015 0.02 0.025
1/d (1/micron)
5
10
50
100
Film thickness (nm)
(a) ρfilm of the Cu polycrystalline films vs. reciprocal mean grain size (diameter), 1/d. Film
thickness D = 250 nm- 900 nmdoes not affect the resistivity. The straight line is ρfilm = 17.8
n m+ (595 n mnm)(1/d),
(b) ρfilm of the Cu thin polycrystalline films vs. filmthickness D. In this case, annealing (heat
treating) the films to reduce the polycrystallinity does not significantly affect the resistivity
because ρfilm 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)
500
300
35
(b)
(a)
30
As deposited
Annealed at 100 C
Annealed at 150 C
100
25
50
20
ρbulk = 16.7 n m
10
15
0
0.05 0.01 0.015 0.02 0.025
5
10
ρ
ρ crystal
≈ 1 + 1.33β
β=
λ⎛ R ⎞
⎜
⎟
d ⎝1− R ⎠
R = probability of reflection on the grain boundary
100
500
Film thickness (nm)
1/d (1/micron)
Mayadas and Shatzkes formula
50
ρ
ρ bulk
3λ
= 1+
(1 − p)
8D
D
λ
> 0.3
p = fraction of elastic collisions
Resistivities of Cu – Ni and Ni – Cr alloys. Nordheim rule
Temperature (°C)
1500
1400
LIQUID PHASE
1300
S
L+
1200
1100
1000
ρ(200C)
Ni
69
Cr
129
Nichrome 1120
US
UID
Q
I
L
S
IDU
L
SO
SOLID SOLUTION
0
20
100% Cu
40
60
at.% Ni
80
α×103(200C)
6
3
0.3
100
100% Ni
(a)
Resistivity (nΩ m)
600
500
Cu-Ni Alloys
400
Nordheim’s Rule for Solid Solutions
300
200
100
0
0
20
100% Cu
40
60
at.% Ni
80
100
100% Ni
(b)
Fig. 2.10(a) Phase diagram of the Cu-Ni alloy system. Above the liquidus
line only 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 (a solid solution) exists. (b) The resistivity of the Cu-Ni alloy as a
function of Ni content (at.%) at room temperature. [Data extracted from
Metals Handbook-10th Edition, Vols 2 and 3, ASM, Metals Park, Ohio,
1991 and Constitution of Binary Alloys, M. Hansen and K. Anderko,
McGraw-Hill, New York, 1958]
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
ρI = CX(1 − X)
ρI = resistivity due to scattering of
electrons from impurities,
C = Nordheim coefficient,
X = atomic fraction of solute atoms in a
solid solution
ρI = CX(1 − X)
Au
Mn
Cu
Zn
Violation of Nordheim’s Rule due to formation of chemical compounds
Resistivity (nΩ m)
160
140
Quenched
ρI = CX(1 − X)
120
100
80
60
Annealed
40
20
0
Cu 3 Au
CuAu
0 10 20 30 40 50 60 70 80 90 100
Composition (at.% Au)
Fig. 2.11: Electrical resistivity vs. composition at room temperature in
Cu-Au alloys. The quenched sample (dashed curve) is obtained by
quenching the liquid and has the Cu and Au atoms randomly mixed.
The resistivity obeys the Nordheim rule. On the other hand, when the
quenched 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 an
ordered fashion in the crystal and the scattering effect is reduced.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
Combined Matthiessen and Nordheim Rules
ρ = ρmatrix + CX(1 − X)
ρ = resistivity, ρmatrix = resistivity of the matrix due to scattering from
thermal vibrations and other defects, C = Nordheim coefficient, X =
atomic fraction of solute atoms in a solid solution
Predictions using Nordheim's rule
Alloy
Ag-Au
Au-Ag
Cu-Pd
Ag-Pd
Au-Pd
Pd-Pt
Pt-Pd
Cu-Ni
X (at.%)
8.8%
Au
8.77%
Ag
6.2%
Pd
10.1%
Pd
8.88%
Pd
7.66%
Pt
7.1%
Pd
2.16%
Ni
ρ0 (nΩ m)
16.2
22.7
17
16.2
22.7
108
105.8
17
ρ at X
(nΩ m)
44.2
54.1
70.8
59.8
54.1
188.2
146.8
50
X′
15.4%
Au
24.4%
Ag
13% Pd
15.2%
Pd
17.1%
Pd
15.5%
Pt
13.8%
Pd
23.4%
Ni
ρ′ at X′
(nΩ m)
??
??
??
??
??
??
??
??
Ceff
ρ′ at X′
(nΩ m)
Experimental
Percentage
Difference
ρ alloy = ρ 0 + Ceff X (1 − X )
Ceff =
ρ alloy − ρ 0
X (1 − X )
ρ ( X ' ) alloy = ρ 0 + Ceff X ' (1 − X ' )
Predictions using Nordheim's rule
Alloy
Ag-Au
Au-Ag
Cu-Pd
Ag-Pd
Au-Pd
Pd-Pt
Pt-Pd
Cu-Ni
X (at.%)
8.8%
Au
8.77%
Ag
6.2%
Pd
10.1%
Pd
8.88%
Pd
7.66%
Pt
7.1%
Pd
2.16%
Ni
ρ0 (nΩ m)
16.2
22.7
17
16.2
22.7
108
105.8
17
ρ at X
(nΩ m)
44.2
54.1
70.8
59.8
54.1
188.2
146.8
50
Ceff
348.88
392.46
925.10
480.18
388.06
1133.85
621.60
1561.51
X′
15.4%
Au
24.4%
Ag
13% Pd
15.2%
Pd
17.1%
Pd
15.5%
Pt
13.8%
Pd
23.4%
Ni
ρ′ at X′
(nΩ m)
??
??
??
??
??
??
??
??
ρ′ at X′
(nΩ m)
Experimental
Percentage
Difference
ρ alloy = ρ 0 + Ceff X (1 − X )
ρ ( X ) alloy = ρ 0 +
'
Ceff =
ρ ( X ) alloy − ρ 0
X (1 − X )
ρ alloy − ρ 0
X (1 − X )
X ' (1 − X ' )
Predictions using Nordheim's rule
Alloy
Ag-Au
Au-Ag
Cu-Pd
Ag-Pd
Au-Pd
Pd-Pt
Pt-Pd
Cu-Ni
X (at.%)
8.8%
Au
8.77%
Ag
6.2%
Pd
10.1%
Pd
8.88%
Pd
7.66%
Pt
7.1%
Pd
2.16%
Ni
ρ0 (nΩ m)
16.2
22.7
17
16.2
22.7
108
105.8
17
ρ at X
(nΩ m)
44.2
54.1
70.8
59.8
54.1
188.2
146.8
50
Ceff
348.88
392.46
925.10
480.18
388.06
1133.85
621.60
1561.51
X′
15.4%
Au
24.4%
Ag
13% Pd
15.2%
Pd
17.1%
Pd
15.5%
Pt
13.8%
Pd
23.4%
Ni
ρ′ at X′
(nΩ m)
61.65
95.09
121.63
78.09
77.71
256.51
179.74
296.89
ρ′ at X′
(nΩ m)
66.3
107.2
121.6
83.8
82.2
244
181
300
7.01%
less
11.29%
less
0.02%
more
6.81%
less
5.46%
less
4.88%
more
0.69%
less
1.04%
less
Experimental
Percentage
Difference
ρ alloy = ρ 0 + Ceff X (1 − X )
ρ ( X ' ) alloy = ρ 0 +
Ceff =
ρ ( X ) alloy − ρ 0
X (1 − X )
ρ alloy − ρ 0
X (1 − X )
X ' (1 − X ' )
Phase diagram of 60% Pb – 40% Sn alloy
400
L
L
200
α+L
C
US
M
N
O
VUS
P
100
E
183°C
F
Q
L (61.9%Sn)
Eutectic
US B
UID
Q
I
L
SOLIDUS
L+β D
61.9%
β
α+β
SO L
Primary α
Eutectic
α
LIQUID
UI
D
SO LVUS
Eutectic
L (61.9%Sn)
α (19%Sn)
LI Q
S
IDU
α
L
SOL
Temperature (oC)
A
300
L (61.9%Sn)
L
Eutectic
α (light)
and β (dark)
G
0
Pb
20
40
60
80
Sn
Composition in wt.% Sn
T
235°C
183°C
Cooling of a 60%Pb40%Sn alloy
L
L+α
M
N O L+β+α
P
β +α
Q
T
Cooling of 38.1%Pb61.9%Sn alloy
L
183 °C
t
L + Eutectic solid (β+α)
E
F
Eutectic solid (β+α)
G
t
Fig. 1.69: The alloy with the eutectic composition cools like a pure element exhibiting a
single solidification temperature at 183°C. The solid has the special eutectic structure.
The alloy with the composition 60%Pb-40%Sn when solidified is a mixture of primary
α and eutectic solid.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
The simplest models of non-isomorphous alloys
Continuous phase
Dispersed phase y
Jy
L
A
A
x
Jx
Jx
L
α β
(a)
A
(b)
L
(c)
Fig. 2.12: The effective resistivity of a material having a layered
structure. (a) Along a direction perpendicular to the layers. (b) Along
a direction parallel to the plane of the layers. (c) Material with a
dispersed phase in a continuous matrix.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
Resistivity-Mixture Rule
Reff
Lα ρα Lβ ρ β
=
+
A
A
Reff =
ρeff = χαρα + χβρβ
ρeff = effective resistivity,
χα = volume fraction of the α-phase,
ρα = resistivity of the α-phase,
χβ = volume fraction of the β-phase,
ρβ = resistivity of the β-phase
L ρ eff
A
Conductivity-Mixture Rule
σeff = χασα + χβσβ
σeff = effective conductivity,
χα = volume fraction of the α-phase,
σα = conductivity of the α-phase,
χβ = volume fraction of the β-phase,
σβ = conductivity of the β-phase
Resistivity-Mixture Rule (ρα ≈ ρβ)
L
A
(a)
α
β
α
β
A/N
(b)
L
Fig. 2.13: (a) A two phase solid. (b) A thin fiber cut out from the
solid.
ρeff = χαρα + χβρβ
ρeff = effective resistivity,
χα = volume fraction of the α-phase,
ρα = resistivity of the α-phase,
χβ = volume fraction of the β-phase,
ρβ = resistivity of the β-phase
Mixture Rule (ρd > 10ρc )
ρeff
1
(1 + χ d )
2
= ρc
(1− χ d )
Mixture Rule (ρd < 0.1ρc )
ρeff
ρeff = effective resistivity,
ρc = resistivity of the continuous phase,
χd = volume fraction of the dispersed phase,
ρd = resistivity of the dispersed phase
(1 − χ d )
= ρc
(1 + 2 χ d )
General case: Reynolds and Hough rule
σ eff − σ c
σd −σc
= χd
σ eff + 2σ c
σ d + 2σ c
σeff = effective resistivity,
σc = resistivity of the continuous phase,
χd = volume fraction of the dispersed phase,
σd = resistivity of the dispersed phase
Resisitivity of bronze (95% Cu + 5% Sn) containing 15 vol. % of pores
Mixture Rule (ρd > 0.1ρc )
Bronze
Air
ρeff
1
(1 + χ d )
2
= ρc
(1− χ d )
ρc = 1 ×10-7 Ωm
ρeff = 1.27 ×10-7 Ωm
Resistivity of Ag-Ni contact alloys for switches
Ni% in Ag-Ni
0
10
15
20
30
100
d, g cm-3
10.5
10.3
9.76
9.4
9.47
8.9
Hardness, VHN
30
50
55
60
65
80
χNi
0
0.12
0.16
0.21
0.32
100
Res Mix Rule
23.21
25.86
28.41
34.30
Cond Mix Rule
18.54
19.33
20.15
22.34
Reynolds&Hough
19.07
20.11
21.17
23.96
20.90
23.60
25.00
31.10
(volume fraction)
ρexp, nΩ m
16.9
wNi d
χ Ni =
d Ni
71.4
Electrical resistivity and structure of non-isomorphous alloy
400
L
L
A
L (61.9%Sn)
L
TB
LIQUID
(a)
UI Liquid, L
DU
One phase L (61.9%Sn)
S
β
+L
β
Eutectic
αα
+L+L M
TE αα
DUS Bregion: β
I
U
Q
TwoN
regionLI L+β D only
SOLIDUS
200
Ophase E
C
α 183°C
+β
61.9%
β
T1
P
F
Temperature (oC)
Temperature
Primary α
Eutectic
VU
S
α+β
SO L
100
100%A
SOLVUS
Eutectic
L (61.9%Sn)
α (19%Sn)
LIQ
S
IDU
α
L
SOL
300TA
X2 100%B
X (% B)
X1
Q
G
Eutectic
α (light)
and β (dark)
0
T
235°C
183°C
Resistivity
Pb
Cooling of a 60%Pb40%Sn alloy
L
L+α
M
N O L+β+ρα
PA
β+α
0 X1
Q
t
20
40
(b) 60
80
Sn
Composition in wt.% Sn
Mixture Rule
T
Nordheim's Rule
183 °C
Composition X (% B)
Cooling of 38.1%Pb61.9%Sn alloy
L
ρB
L + Eutectic solid (β+α)
E X F100%B
Eutectic solid (β+α)
2
G
t
Fig. 2.14 (a) The phase diagram for a binary, eutectic forming alloy.
Fig. 1.69: The(b)
alloy
the eutectic
composition for
cools
a pure
element exhibiting a
Thewith
resistivity
vs composition
thelike
binary
alloy.
single solidification
temperature at 183°C. The solid has the special eutectic structure.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
The alloy with
the composition 60%Pb-40%Sn when solidified is a mixture of primary
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α and eutectic solid.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
Temperature
Electrical resistivity of Ag – Ni alloy
(a)
Liquid, L
TA
TE α
T1
100%A
X1
α+L
TB
β+L
Two phase region
α+β
X (% B)
β
One phase
region: β
only
X2 100%B
Resistivity
(b)
Mixture Rule
Nordheim's Rule
ρA
0 X1
Composition X (% B)
ρB
X2 100%B
Fig. 2.14 (a) The phase diagram for a binary, eutectic forming alloy.
(b) The resistivity vs composition for the binary alloy.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
Resistivity of Ag-Ni contact alloys for switches
0
10
15
20
30
100
d, g cm-3
10.5
10.3
9.76
9.4
9.47
8.9
Hardness, VHN
30
50
55
60
65
80
χNi
0
0.12
0.16
0.21
0.32
100
Res Mix Rule
23.21
25.86
28.41
34.30
Cond Mix Rule
18.54
19.33
20.15
22.34
Reynolds&Hough
19.07
20.11
21.17
23.96
20.90
23.60
25.00
31.10
wNi d
χ Ni =
d Ni
16.9
(a)
Liquid, L
TA
TE α
T1
100%A
X1
α+L
71.4
TB
β+L
Two phase region
α+β
X (% B)
β
One phase
region: β
only
X2 100%B
(b)
Resistivity
ρexp, nΩ m
Temperature
Ni% in Ag-Ni
Mixture Rule
Nordheim's Rule
ρA
0 X1
Composition X (% B)
ρB
X2 100%B
Fig. 2.14 (a) The phase diagram for a binary, eutectic forming alloy.
(b) The resistivity vs composition for the binary alloy.
Classification of conductors
Insulators
Semiconductors
Conductors
Many ceramics
Superconductors
Alumina
Diamond Inorganic Glasses
Metals
Mica
Polypropylene
PVDF Soda silica glass
Borosilicate Pure SnO2
PET
Intrinsic Si
Amorphous
SiO2
Intrinsic GaAs
As2Se3
10-18
10-15
10-12
10-9
10-6
10-3
Degenerately
Doped Si
Alloys
Te Graphite NiCr Ag
100
103
106
Conductivity (Ωm)-1
Fig. 2.24 Range of conductivites exhibited by various materials
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
109
1012
Conductivity of ceramic and glass insulators
⎛ Eσ ⎞
σ = σ o exp⎜ − ⎟
⎝ kT ⎠
E
E
Vacancy aids the diffusion of positive ion
O2–
Si4+
Na+
Anion vacancy
acts as a donor
Interstitial cation diffuses
(a)
(b)
Fig. 2.27: Possible contributions to the conductivity of ceramic and
glass insulators (a) Possible mobile charges in a ceramic (b) A Na+
ion in the glass structure diffuses and therefore drifts in the
direction of the field. (E is the electric field.)
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
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Temperature Dependence of Conductivity
1×10-1
Conductivity 1/(Ωm)
1×10-3
24%Na2O-76%SiO2
As3.0Te3.0Si1.2Ge1.0 glass
Pyrex
⎛ Eσ ⎞
σ = σ o exp⎜ − ⎟
⎝ kT ⎠
1×10-5
1×10-7
1×10-9
1×10-11
PVAc
SiO2
conductivity,
k = Boltzmann constant,
T = temperature
PVC
1×10-13
1×10-15
1.2
12%Na2O-88%SiO2
1.6
2
2.4
2.8
103/T (1/K)
3.2
σ = conductivity, σο = constant,
Εσ = activation energy for
3.6
4
Fig. 2.28: Conductivity vs reciprocal temperature for various low
conductivity solids. (PVC = Polyvinyl chloride; PVAc = Polyvinyl
acetate.) Data selectively combined from numerous sources.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
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Thermally activated diffusion
A
A*
B
Arrenius type behavior ∼exp(EA/kT)
U = PE(x)
U A*
ϑ= Avο exp(−EA/kT)
A*
EA
UA= UB
B
A
Displacement
X
Fig. 1.28: Diffusion of an interstitial impurity atom in a crystal from
one void to a neighboring void. The impurity atom at position A
must posses an energy EA to push the host atoms away and move
into the neighboring void at B.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
Unstable (Activated State)
U(X) = PE = mgh
Metastable
Stable
E
A
ΔU
UA*
UA
UB
A*
A
B
X
XA
XA*
XB
System Coordinate, X = Position of Center of Mass
Fig. 1.27: Tilting a filing cabinet from state A to its edge in state A*
requires an energy EA. After reaching A*, the cabinet
spontaneously drops to the stable position B. PE of state B is lower
than A and therefore state B is more stable than A.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
where
EA = UA* − UA
ϑ = frequency of jumps,
A = a dimensionless constant that has only a
weak temperature dependence,
vo = vibrational frequency,
EA = activation energy,
k = Boltzmann constant,
T = temperature,
UA* = potential energy at the activated state A*,
UA = potential energy at state A.
Classification of conductors
Insulators
Semiconductors
Conductors
Many ceramics
Superconductors
Alumina
Diamond Inorganic Glasses
Metals
Mica
Polypropylene
PVDF Soda silica glass
Borosilicate Pure SnO2
PET
Intrinsic Si
Amorphous
SiO2
Intrinsic GaAs
As2Se3
10-18
10-15
10-12
10-9
10-6
10-3
Degenerately
Doped Si
Alloys
Te Graphite NiCr Ag
100
103
106
Conductivity (Ωm)-1
Fig. 2.24 Range of conductivites exhibited by various materials
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
109
1012
Silicon unit cell
C
a
a
a
Fig. 1.33: The diamond unit cell is cubic. The cell has eight atoms.
Grey Sn (α-Sn) and the elemental semiconductors Ge and Si have
this crystal structure.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
Each atom has
four neighbors
and all bonds are
saturated
Conductivity of Semiconductor
E
C
a
a
a
Fig. 1.33: The diamond unit cell is cubic. The cell has eight atoms.
Grey Sn (α-Sn) and the elemental semiconductors Ge and Si have
this crystal structure.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
Conductivity of Semiconductor
E
C
a
a
a
eFig. 1.33: The diamond unit cell is cubic. The cell has eight atoms.
Grey Sn (α-Sn) and the elemental semiconductors Ge and Si have
this crystal structure.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
σ = enμe + epμh
n = concentration of electrons
p = concentration of holes
μe = drift mobility of the electrons
μh = drift mobility of the electrons
e = electronic charge
Conductivity of Semiconductor
E
C
a
a
a
eFig. 1.33: The diamond unit cell is cubic. The cell has eight atoms.
Grey Sn (α-Sn) and the elemental semiconductors Ge and Si have
this crystal structure.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
σ = enμe + epμh
n = concentration of electrons
p = concentration of holes
μe = drift mobility of the electrons
μh = drift mobility of the electrons
e = electronic charge
Conductivity of Semiconductor
E
C
a
a
a
eFig. 1.33: The diamond unit cell is cubic. The cell has eight atoms.
Grey Sn (α-Sn) and the elemental semiconductors Ge and Si have
this crystal structure.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
σ = enμe + epμh
n = concentration of electrons
p = concentration of holes
μe = drift mobility of the electrons
μh = drift mobility of the electrons
e = electronic charge
Conductivity of Semiconductor
E
C
a
a
a
eFig. 1.33: The diamond unit cell is cubic. The cell has eight atoms.
Grey Sn (α-Sn) and the elemental semiconductors Ge and Si have
this crystal structure.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
σ = enμe + epμh
n = concentration of electrons
p = concentration of holes
μe = drift mobility of the electrons
μh = drift mobility of the electrons
e = electronic charge
Conductivity of Semiconductor
E
C
a
a
h+
a
eFig. 1.33: The diamond unit cell is cubic. The cell has eight atoms.
Grey Sn (α-Sn) and the elemental semiconductors Ge and Si have
this crystal structure.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
σ = enμe + epμh
n = concentration of electrons
p = concentration of holes
μe = drift mobility of the electrons
μh = drift mobility of the electrons
e = electronic charge
Temperature Dependence of Conductivity
⎛ Eσ ⎞
σ = σ o exp⎜ − ⎟
⎝ kT ⎠
Conductivity of semiconductor
∞
σ = conductivity,
σο = constant,
Εσ = activation energy for conductivity,
k = Boltzmann constant,
T = temperature
σ = enμe + epμh
EA
)
probability (E>EA) = ∫ nE dE = A exp(−
kT
EA
General Conductivity
σ = Σqi ni μi
σ = conductivity,
qi = charge carried by the charge carrier species i (for electrons and holes qi = e),
ni = concentration of the charge carrier,
μi = drift mobility of the charge carrier of species i
Hall effect. Monopolar conduction
Lorentz Force F = qv × B
Jy = 0
VH
Bz
V
+
+
+
+
eEH
Jx
EH
+
Ex
vdx
y
evdxBz
Jx
z
x
A
Bz
V
+
Fig. 2.15: Illustration of the Hall effect. The z-direction is out from the
plane of paper. The externally applied magnetic field is along the zdirection.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
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F = force, q = charge,
v = velocity of charged particle,
B = magnetic field
eEH = evdxBz
Jx=envdx
⎛ 1⎞
E H = ⎜ ⎟ J x Bz
⎝ en ⎠
Hall coefficient
EH
⎛ 1⎞
RH =
= −⎜ ⎟
J x Bz
⎝ en ⎠
⎛ 1⎞
RH = −⎜ ⎟
⎝ en ⎠
σ = enμ
μ H = − RH σ
Face Centered Cubic (FCC)
FCC Unit Cell
(a)
2R
a
a
a
(b)
a
(c)
Fig. 1.30: (a) The crystal structure of copper is Face Centered Cubic
(FCC). The atoms are positioned at well defined sites arranged
periodically and there is a long range order in the crystal. (b) An FCC unit
cell with closed packed spheres. (c) Reduced sphere representation of the
FCC unit cell. Examples: Ag, Al, Au, Ca, Cu, γ-Fe (>912 C), Ni, Pd, Pt,
Rh
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
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Simple wattmeter using Hall effect
IL
Wattmeter
IL
Load
Source
VL
RL
Bz ∝ I L
VL
IL
VH ∝ E H ∝ RH J x Bz ∝ I L2 ∝ W
IL
C
C
V
VH
Bz
w
R
VL
Ix = VL/R
Fig. 2.17: Wattmeter based on the Hall effect. Load voltage and load
current have L as subscript. C denotes the current coils. for setting
up a magnetic field through the Hall effect sample (semiconductor)
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
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Hall effect. Ambipolar conduction
Jy = 0
+
Ex
Jx
vhx
+
eEy
+
+
Jx
z
x
B
v
B
B
F = qv×B
F = qv×B
(a)
vex
evhxBz
q = –e
v
y
+
Ey
eEy
q = +e
Bz
evexBz
+
A
Bz
Fig. 2.16 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.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
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pμ h − nμ e
p − nb 2
RH =
=
2
e ( pμ h + nμ e )
e( p + nb) 2
2
V
Fig. 2.26: 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.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
(b)
2
RH = Hall coefficient,
p = concentration of the holes,
μh = hole drift mobility,
n = concentration of the electrons,
μe = electron drift mobility,
b = μe/μh where μe is the electron drift
mobility and μh is the hole drift mobility,
e = electronic charge
Skin Depth for Conduction
ωL
R
δ=
δ = Skin depth
2a
At high frequencies, the core region exhibits more inductive
impedance than the surface region, and the current flows in the
surface region of a conductor defined approximately by the skin
depth, δ.
1
1
ωσμ
2
δ = skin depth, ω = angular frequency
of current, σ = conductivity, μ =
magnetic permeability of the medium
HF Resistance per Unit Length Due to Skin Effect
ρ
ρ
rac = ≈
A 2πaδ
rac = ac resistance , ρ = resistivity, A = cross-sectional area, a =
radius, δ = skin depth
Skin effect
ωL
R
δ = Skin depth
2a
At high frequencies, the core region exhibits more inductive
impedance than the surface region, and the current flows in the
surface region of a conductor defined approximately by the skin
depth, δ.
B2
(a) Total current into
paper is I
B1
(b) Current in hollow outer
cylinder is I/2
B2
B1
B2
B1
(c) Current in inner
cylinder is I/2
Illustration of the skin effect. A hypothetical cut produces a hollow outer
cylinder and a solid inner cylinder. Cut is placed where it would give equal
current in each section. The two sections are in parallel so that the currents
in (b) and (c) sum to that in (a).
Skin effect
rf
Copper wire with radius1 mm
σdc = 5.9×107 Ω-1cm-1
ρ dc
rdc = 2
πa
ρ dc
rf =
2πaδ
rdc
=?
rf
a
=
rdc 2δ
δ=
1
1
ωσμ
2
r10 MHz
≈ 24
rdc
r1GHz
≈ 240
rdc
Interconnects in microelectronics
M7
M6
Low
permittivity
dielectric
M5
M3
Cu
interconnects
M2
M4
M3
M2
M1
M1
Silicon
Metal interconnects wiring devices on a silicon
crystal. Three different metallization levels M1,
M2, and M3 are used. The dielectric between the
interconnects has been etched away to expose the
interconnect structure.
Cross section of a chip with 7 levels of
metallization, M1 to M7. The image is
obtained with a scanning electron microscope (SEM).
|SOURCE: Courtesy of IBM
|SOURCE: Courtesy of Mark Bohr, Intel.
Effective multilevel interconnect capacitance
C H = ε 0ε r
TL
X
WL
CV = ε 0ε r
H
ρL
R=
Ceff
⎛T W⎞
= 2(CV + C H ) = 2ε 0ε r L⎜ + ⎟
⎝X H⎠
T
Ar ≡
= 1÷ 2 = aspect ratio
W
L = 10 mm
X ≈W ≈ P
T = 1 μm
ρ = 17 nΩ m (Cu)
RCeff
2
P = 1 μm
εr = 3.6 (FSG)
TW
L2
= 2ε 0ε r ρ
TW
RCeff
⎛T W⎞
⎜ + ⎟
⎝X H⎠
1 ⎞
⎛ 4
= 2ε 0ε r ρL ⎜ 2 + 2 ⎟
⎝P T ⎠
RCeff = 0.43 ns
2
Electromigration
(a)
(b)
Void
and
failure
Hillock
Grain boundary Void
Electron
Hillock
Current
Interconnect
In
a ce
terf
Cold
(c)
Hot
Hillocks
Cold
Grain
boundary
Current
(a) Electrons bombard the metal ions and force themto slowly migrate
(b) Formation of voids and hillocks in a polycrystalline metal interconnect by the
electromigration of metal ions along grain boundaries and interfaces. (c) Accelerated tests on 3 mmCVD(chemical vapor deposited) Cu line. T = 200 oC, J = 6
MAcm-2: void formation and fatal failure (break), and hillock formation.
|SOURCE: Courtesy of L. Arnaud et al, Microelectronics Reliability, 40, 86, 2000.
Black’s equation
t MTF = A B J
−n
⎛ EA ⎞
exp⎜
⎟
⎝ kT ⎠
tMTF = mean time to 50% failure
AB = constant
J = current density
n = integer
EA = activation energy