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Chapter 2.
Electrical and Thermal Conduction
in Solids
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An Overview
In this chapter,
we will treat conduction ‘e’ in metal as
“free charges” that can be accelerated
by an applied electric field, to explain
the electrical and thermal conduction in
a solid.
Electrical conduction involves the motion of charges in a material under the
Influence of an applied electric field. By applying Newton’s 2nd law to ‘e’ motion
& using a concept of “mean free time” between ‘e’ collisions with lattice vibrations,
crystal defects, impurities, etc., we will derive the fundamental equations that
govern electrical conduction in solids.
Thermal conduction,i.e., the conduction of thermal E from higher to lower
temperature regions in a metal, involves the conduction ‘e’ carrying the energy.
Therefore, the relationship between the electrical conductivity and thermal
conductivity will be reviewed in this textbook.
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CONTENTS
Electrical Conductivity of Metals:
2.1 Classical theory : The DRUDE model
2.2 Temperature dependence of resistivity
2.3 MATTHIESSEN’s and NORDHEIM’s Rules.
2.4 Resistivity of mixtures and porous materials
2.5 The Hall Effect and Hall Devices
Thermal Conductivity:
2.6 Thermal conduction
Electrical Conductivity of Nonmetals:
2.7 Electrical conductivity of nonmetals
Additional Issues:
2.9 Thin metal films
2.10 Interconnects in microelectronics
2.11 Electromigration and Black’s equations
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2.1 Classical theory : The DRUDE model
Goal: To find out the relation between the conductivity (or resistivity) and drift velocity
, and thereby its relation to mean free time and drift mobility, from the description of the
current density
 In a conductor where ‘e’ drift in the presence of an electric field,
current density is defined as the net amount of charge flowing across a unit area per
unit time
q
J
At
J : current density
q : net quantity of charge
flowing through an area A at Ex
In this system, electrons drift with an average velocity vdx in the x-direction, called the drift velocity.
(Here Ex is the electric field.)
 Drift velocity is defined as
the average velocity of electrons in the x direction at time t, denote by vdx(t)
vdx 
1
[v x1  v x 2  v x 3  ...  v xN ]
N
vxi : x direction velocity of the ith electrons
N : # of conduction electrons in the metal
[2.1]
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2.1.1 Metals and conduction by electrons
 Current density in the x direction can be rewritten as a function of the drift velocity
Jx 
q enAv dx t

At
At
J x (t )  envdx (t )
[2.2]
: In time Δt, the total charge Δq crossing the area A is enAΔx, where
Δx=vdxΔt and n is assumed to be the # of ‘e’ per unit volume in the
conductor (n=N/V).
: time dependent current density is useful since the average velocity at
one time is not the same as at another time, due to the change of Ex
 Think of motions of a conduction ‘e’ in metals before calculating Vdx.
(a)
A conduction ‘e’ in the electron gas moves
about randomly in a metal (with a mean speed
u) being frequently and randomly scattered by
thermal vibrations of the atoms.
In the absence of an applied field there is no
net drift in any direction.
Ex  Ex (t )
(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
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2.1.1 Metals and conduction by electrons
To calculate the drift velocity vdx of the ‘e’ due to applied field Ex, we first consider the
eEx
velocity vxi of the ith ‘e’ in the x direction at t.
Since
is the acceleration a of the ‘e’ [F=qE=ma],
Let uxi be the initial velocity of ‘e’ i in the x direction just after the
collision. Vxi is written as the sum of uxi and the acceleration of
the ‘e’ after the collision. Here, we suppose that its last collision
was at time ti; therefore, for time (t-ti), it accelerated free of
collisions, as shown in Fig.2.3.
me vxi in the x direction at t is given by
uxi is velocity of ith‘e’ in the x direction after the collision
However, this is only for the ith electron. We
need the average velocity vdx for all such
electrons along x as the following eqn.
vdx 
1
eEx
[vx1  vx 2  vx3  ...  vxN ] 
(t  ti )
N
me
(t-ti) : average free time for N electrons between
collision (~ τ = mean free time or mean
scattering time)
Fig 2.3 Velocity gained in the x direction at time t from the electric field ( E x) for three different electrons.
There will be N electrons to consider in the metal.
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2.1.1 Metals and conduction by electrons
 Drift mobility (vs. mean free time)
: widely used electronic parameter in semiconductor device physics.
Suppose that τ is the mean free time or mean scattering time. Then, for some electrons, (tti) will be greater than ,and for others, it will be shorter, as shown in Fig 2.3. Averaging (tti) for N electrons will be the same as . Thus we can substitute for (t-ti) in the previous
expression to obtain



vdx 
e
Ex
me
[2.3]
Equation 2.3 shows that the drift velocity increases linearly with the applied field. The
constant of proportionality e / me has been given a special name and symbol, called drift
mobility  d , which is defined as
vdx  d Ex
where d 
e
me
[2.4]
[2.5]
 , which is often called the relaxation time, is directly related to the microscopic
processes that cause the scattering of the electrons in the metal; that is, lattice vibration,
crystal imperfections, and impurities.
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2.1.1 Metals and conduction by electrons
From the expression for the drift velocity vdx the current density Jx follows immediately…
by substituting Equation 2.4 into 2.2, that is,
J x  en d Ex
 Ex [2.6]
Therefore, the current density is proportional to the electric field and the conductivity
term is given by
  en d

[2.7]
Then, let’s find out temperature dependence of conductivity (or resistivity) of a metal
by considering the mean time .


The mean time between collisions has further significance. Its 1/ represents the mean
frequency of collisions or scattering events; that is 1/ is the mean probability per unit
time that the electron will be scattered. Therefore, during a small time interval t , the
probability of scattering will be t /  .
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2.2 Temperature dependence of resistivity
To find the temperature dependence of  , let’s consider the temperature dependence
of the mean free time  , since this determines the drift velocity.
Fig 2.5 scattering of an electron from the
thermal vibration of the atoms.
The electron

travels a mean distance l  u between
collisions.
Since the scattering cross-sectional area is S,
in the volume Sl there must be at least one
scatterer as
NsSu   1
volume
  a1
2

1
SuNs
[2.11]
Ns : concentration of scattering centers
When the conduction electrons are only scattered
by thermal vibrations of the metal ion, then  in the
mobility expression d  e m refers to the mean
e
time between scattering events by this process.
S : cross-sectional area
u : mean speed
a : amplitude of the vibrations
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2.2 Temperature dependence of resistivity
 Lattice-scattering-limited conductivity
: the resistivity of a pure metal wire increase linearly with the temperature, due to the
scattering of conduction electrons by thermal vibrations of the atoms.  feature of a metal
(cf. semiconductors)
The thermal vibrations of the atom can be considered to be simple harmonic motion, much
the same way as that of a mass M attached to a spring. From the kinetic theory of matter,
1
1
Ma 2 w2 (average kinetic energy of the oscillatio ns )  kT
4
2
So a 2  T . This makes sense because raising the T increases atomic vibrations. Thus
1
1
C Since the mean time between scattering events τ is inversely

or


2
a 2 T
T proportional to the area a that scatters the ‘e’,
eC
(to show a relation with T)
substituting for  in d  e / me results in  d 
meT
meT
1
1
So, the resistivity of a metal T 


[2.12]
T end e 2 nC T  AT

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2.3 MATTHIESSEN’s and NORDHEIM’s Rules.
2.3.1 Matthiessen’s rule and the temperature coefficient of resistivity
: The theory of conduction that considers scattering from lattice vibrations only works well
with pure metal and it fails for metallic alloys. Their resistivities are weakly T-dependent,
and so, different type of scattering mechanism is required for metallic alloys.
Let’s consider a metal alloy that has randomly distributed impurity atoms.
Strained region by impurity exerts a
scattering force F = - d (PE) /dx
I
We have two mean free times between collision.
 T : scattering from thermalvibrationonly
 i : scattering from impurityonly
In unit time,1 a net probability of
scattering, is given by

1 1 1
 
[2.13]


Two different types of scattering processes involving scattering from
impurities alone and thermal vibrations alone.
T
i
Then, since drift mobility depends
on effective scattering time,
effective drift mobility is given by
1
1 1
 
ud u L u I
[2.14]
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2.3.1 Matthiessen’s rule and the temperature coefficient
of resistivity
where u L is the lattice-scattering-limited drift mobility,
uI is the impurity-scattering-limited drift mobility.
Since effective resistivity  of the material is simply

1
1
1


enud enuL enuI
1 / enud
which can be written   T   I
[2.15]
This summation rule of resistivities from different scattering mechanisms
is called Matthiessen’s rule.
Furthermore, in a general from, effective resistivity can be given by
  T   R (  R : residual resistivity)
: scattering E of im purities, dislocations, in ternal atom,
vacancies, gain boundaries, etc
Since residual resistivity shows very little T-dependence
whereas ρT = AT .
  AT  B
[2.17]
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2.3.1 Matthiessen’s rule and the temperature coefficient
of resistivity
 Temperature coefficient of resistivity (TCR)
Eqn. 2.17 indicates that the resistivity of a metal varies with T, with A and B depending on
the material. Instead of listing A and B in resistivity tables, we prefer a temperature coefficient
that refers to small, normalized changes around a reference temperature.
0 
1   
0 T T T0
[2.18]
- temp sensitivity of the resistivity of metals
If the resistivity follows the behavior like
in Eqn. 2.17, then Eqn. 2.18 leads to
  0 1  0 (T  T0 )
[2.19]
where a0 is constant over a
temperature range T0 to T,
    o
&
T  T  To
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Resistivity of various metals vs. T
However   AT  B is only an approximation for some metals and not true for all metals.
This is because the origin of the scattering may be different depending on the temperature.
100
T
2000
Inconel-825
NiCr Heating Wire
1000
10
Scattering from vibration
Iron
Resistivity (n m)
Resistivity(n m)
Tungsten
Monel-400
T
Tin
100 Platinum
 (n m)
3.5
0.1
  T5
0.01
T
3
2.5
2
Copper
Nickel
0.001
Silver
0.0001
  T5
1.5
1
0.5  = R
  R
0
0
Scattering from impurity
0.00001
10
100
1000
10000
10
100
20
40
60
80
100
T (K)
1000
10000
Temperature(K)
Temperature (K)
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 ( ~ 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)]
The resistivity of copper from lowest to highest temperatures (near
melting temperature, 1358 K) on a log-log plot. Above about 100 K,
  T, whereas at low temperatures,   T 5 and at the lowest
temperatures  approaches the residual resistivity R . The inset
shows the  vs. T behavior below 100 K on a linear plot ( R is too
small on this scale).
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2.3.2 Solid solution and Nordheim’s rule
How does the resistivity of solid solutions change with alloy composition ?
In an isomorphous alloy of two metals, that is, a binary alloy that forms a solid solution
(Ni-Cr alloy), we would expect Eqn 2.15 to apply, with the temperature-independent
impurity contribution  I increasing with the concentration of solute atoms.
  T   I
[2.15]
This means that as alloy concentration
increases, resistivity increases and
becomes less temperature dependent as
ρI, overwhelms ρT, leading to αo << 1/273.
This (temperature independency) is the advantage of alloys in resistive components.
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2.3.2 Solid solution and nordheim’s rule
 I  CX (1  X )
[2.21]
C (Nordheim’s coefficient):
represents effectiveness of the solute atom in
increasing the resistivity.
Nordheim rule is useful for predicting the
resistivities of dilute alloys, particularly in
the low-concentration region.
Temperature (°C)
 Nordheim’s rule for solid solutions: an
important semiempirical Eqn. that can be used to
predict the resistivity of an alloy, which relates the
impurity resistivity to the atomic fraction X of
solute atoms in a solid solution, as follows:
1500
US
UID
Q
I
L
US
LID
O
S
1400
LIQUID PHASE
1300
L+
1200
1100
1000
S
SOLID SOLUTION
20
0
40
100% Cu
60
80
at.% Ni
100
100% Ni
(a)
600
Resistivity (n m)
How does the concentration of solute atoms
affect on ρI ?
500
Cu-Ni Alloys
400
300
200
100
consistent
0
0
20
100% Cu
40
60
at.% Ni
80
100
100% Ni
(b)
(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
%Nordheim’s rule assumes that the solid solution has the solid solution) exists. (b) The resistivity of the Cu-Ni alloy as a function
solute atoms randomly distributed in the lattice, and these of Ni content (at.%) at room temperature. [Data extracted from Metals
random distributions of impurities cause the ‘e’ to become Handbook-10th Edition, Vols 2 and 3, ASM, Metals Park, Ohio, 1991 and
Constitution of Binary Alloys, M. Hansen and K. Anderko, McGraw-Hill,
scattered as they whiz around the crystal.
New York, 1958]
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2.3.2 Solid solution and Nordheim’s rule
 Combination of Matthiessen and Nordheim rules leads to a general
expression for ρ of the solid solution:
  matrix  CX (1  X )
[2.22]
where matrix  T   R is the resistivity of the matrix due to scattering
from thermalvibrationsand from other defect, absenceof alloyingelements.
160
Quenched
140
Resistivity (n m)
Exception: at some concentrations of certain binary alloys,
Cu and Au atoms are not randomly mixed but occupy
regular sites, which decrease the resistivity. ------------
120
100
80
Annealed
60
40
20
Cu3Au
CuAu
0
0
10 20 30 40 50 60 70 80 90 100
Composition (at.% Au)
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.