Chapter 2: Electrical and Thermal Conduction in Solids

2.1 Classical Theory: The Drude Model

2.2 Temperature Dependence of Resistivity
2.3 Matthiessen’s rule
2.5 Hall Effect and Hall Devices
2.7 Electrical Conductivity of Nonmetals
2.9 Thin Metal Films
2.10 Interconnections in Microelectronics
2.11 Electromigration

“Free mobile charge carriers have average velocity, called the drift velocity, that
depends on the field.
Applying Newton’s second law to electron motion and using such concepts as
mean free time between electron collisions with lattice vibrations, crystal defects,
impurities, etc., we will derive the fundamental equations that govern electrical
conduction in solids. A key concept will be the drift mobility, which is a measure of
the ease with which charge carriers in the solid drift under the influence of an
external electrical field.”

Physical Electronics, Spring 2017 1

 2.1 Classical Theory: The Drude Model

Derivation of Current Equation J x = enµd Ex

∆q
J= where ∆q is the net quantity of charge flowing through an area A in time ∆t
A∆t

Drift velocity is the average velocity of the

electrons in the x direction at time t :

1
υdx
= [υ x1 + υ x 2 + υ x 3 + ⋅⋅⋅ + υ xN ]
N
where vxi is the x direction velocity of the ith electron, and N is the number of conduction
electrons in the metal.
The current density in the x direction is then

∆q enAυdx ∆t n = the number of electrons per unit

=
Jx = = enυdx volume in the conductor (n=N/V)
A∆t A∆t
The average velocity at one time may not be the same as at another time,
because the applied field may be changing: Ex = Ex(t)

Therefore, the time-dependent current density is J x (t ) = enυdx (t )

Physical Electronics, Spring 2017 2
 2.1 Classical Theory: The Drude Model

Calculation of Drift Velocity

Ex

u
∆x

Vibrating Cu+ ions 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
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, ∆x, from its initial position
toward the positive terminal

Physical Electronics, Spring 2017 3

 2.1 Classical Theory: The Drude Model

Derivation of Current Equation Velocity gained along x

Present time
eEx vx1-ux1
υ xi =
u xi + ( t − t1 ) Last collision
me
uxi, initial velocity : the velocity of electron Electron1
time
i in the x direction just after the collision. t1 Free time t
eEx vx2-ux2
F= me a= eEx → a=
me
1
υdx
= [υ x1 + υ x 2 + υ x 3 + ⋅⋅⋅ + υ xN ] Electron 2
N t2 t
time
eEx
= ( t − ti ) vx3-ux3
me
( t − ti ) =
τ : the average free time for N electrons Electron3
between collisions (mean scattering time
time or relaxation time) t3 t

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.
Physical Electronics, Spring 2017 4
 2.1 Classical Theory: The Drude Model

eτ
Therefore, the drift velocity υdx is υdx = Ex
me
eτ eτ
The proportional constant is defined as the drift mobility µd µd =
me me
υdx = µd Ex The drift velocity is dependent on the applied electric field.

J x = enµd Ex =σ Ex : Ohm's law A large drift mobility does not necessarily

imply high conductivity, because
conductivity also depends on the
concentration of conduction electrons
u : mean speed of conduction electron 
 → l = u ⋅τ
l : mean free path 

υdx is a number average at one instant

Physical Electronics, Spring 2017 5

 2.1 Classical Theory: The Drude Model

∆x
υdx
= ( ∆t >> τ ) : an average velocity for one electron over many collisions
∆t
Electric field
E
s1 x
1 Finish
Collision
uy1 p
t1 t2
Start
ux1 2
0 t3
3
Collision
4

s = ∆x
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. After p collisions, it has drifted
a distance s = ∆x.

Physical Electronics, Spring 2017 6

 2.1 Classical Theory: The Drude Model

Example 2.2 : Electron Drift mobility in Metals

Cu : σ =5.9 ×105 Ω-1cm −1 , d =
8.96 g / cm3 , 63.5 g / mol
Calculate µd , τ ?
N A 6.02 ×1023 cm −3 , 63.5 g
One mole of Cu: =

Number of Cu atom =number of conduction electron:

d ⋅ N A 8.96 g / cm3 × 6.02 ×1023 atoms / mol
=
n = = 8.5 ×1022 electrons / cm3
M at 63.5 g / mol

σ 5.9 ×105 Ω-1cm −1

µd =
= −19
=
−3
43.4 cm 2
⋅ V −1 −1
s
e ⋅ n 1.6 ×10 C × 8.5 ×10 cm22

µd me 43.4 ×10−4 m 2 ⋅ V −1s −1 × 9.1×10−31 kg

τ
= = −19
= 2.5 × 10 −14
s
e 1.6 ×10 C

If u = 1.5 ×106 m / s, mean free path, l = u ⋅τ = 37 nm

Physical Electronics, Spring 2017 7

 2.1 Classical Theory: The Drude Model

Example 2.3 : Drift velocity and Mean speed

What is the applied electric field that will impose a drift velocity equal to 0.1% of the
mean speed u(~106 m/s) of conduction electrons in copper? What is the current
density?

υd 0.001×106 m / s
=
E = = 2.3 × 10 5
V / m : unattainably large E-field
µd 43.4 ×10 m /(V ⋅ s )
−4 2

J = σ ⋅ E = 5.9 ×107 Ω −1m −1 × 2.3 ×105 V / m

= 1.4 ×107 A / m 2 : unattainably large current density

The drift velocity is much smaller than the mean speed of the electrons.
The mean speed is not affected by electric field.

Physical Electronics, Spring 2017 8

 2.2 Temperature Dependence of Resistivity: Ideal Pure Metal

Thermal / Electrical Conductance HOT COLD

HEAT

“Good electrical conductors

(e.g., metals) are also known
to be good thermal
conductors. The conduction
of thermal energy from
higher to lower temperature
regions in a metal involves
the conduction electrons
carrying the energy.
Consequently, there is an
innate relationship between Electron Gas Vibrating Cu+ ions
the electrical and thermal
conductivities.” 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
How about temperature velocity vectors) from the hotter regions arrive at cooler
dependence of ‘mobility’ regions and collide there with lattice vibrations and transfer
and ‘resistivity’ ? their energy. Lengths of arrowed lines on atoms represent
the magnitudes of atomic vibrations.

Physical Electronics, Spring 2017 9

 2.2 Temperature Dependence of Resistivity: Ideal Pure Metal

Temperature Dependence of Mean Free Time

1
One scatter in the volume S ⋅ l → ( S ⋅ l ) N s = ( S ⋅ u ⋅τ ) N s = 1 → τ =
S ⋅ u ⋅ Ns
τ = mean free time
S = π a2
u = mean speed of the electron
Ns = concentration of scatters τ
=u a
(vibrating atoms, impurity, vacancy, etc)
S = cross-sectional area of the scatterer u

“Temperature dependence
of τ arises essentially from
that of the cross-sectional
Electron
area S.”

“u : slightly temp. Scattering of an electron from the thermal vibrations of the atoms. The
dependent, and electron travels a mean distance  = u τ between collisions. Since the
Ns ≅ Nat.” scattering cross sectional area is S, in the volume S there must be at
least one scatterer, Ns(Suτ) = 1.

Physical Electronics, Spring 2017 10

 2.2 Temperature Dependence of Resistivity: Ideal Pure Metal

The thermal vibrations of the atom (i.e., lattice vibration) can be considered to be
simple harmonic motion, much the same way as that of a mass M attached to a
spring.

1 1
Average kinetic energy = Ma ω ≈ kT
2 2

4 2
1 1 1 C
τ∝ = ∝ → τ= : C is temp.-independent costant
S πa 2
T T

eτ eC 1 1 mT
µ= = ρ= = = 2 e = AT
d
me meT
T
σT enµd e n C

ρT , σ T : Lattice-scattering-limited resistivity/conductivity

Physical Electronics, Spring 2017 11

 2.2 Temperature Dependence of Resistivity: Ideal Pure Metal

Example 2.6 : Drift mobility and Resistivity due to Lattice Vibrations

1.5 ×106 m / s, Vibration frequency ω =
u of Cu = 4 ×1012 s −1
The density of copper is 8.96 g / cm3 , atomic mass is 63.56 g / mol

d ⋅ N A 8.96 ×103 kg / m3 × 6.02 ×1023 mol −1 −3

N= N= = = 8.5 × 10 28
m
63.56 ×10−3 kg / mol
s at
M at

2π kT 2π ×1.38 ×10−23 J / K × 300 K

S π a=
= 2
= = 3.9 × 10 −22
m 2

M ω 2  63.56 ×10−3 kg / mol 

 −1  × (2π × 4 × 1012
rad / s ) 2

 6.02 × 10 23
mol 
1 1
τ
= = = 2.0 × 10 −14
s
S ⋅ u ⋅ Ns ( 3.9 ×10 m ) × (1.5 ×10 m / s ) × (8.5 ×10 m )
−22 2 6 28 −3

eτ
µd = = 35 cm 2V −1 ⋅ s −1 σ = enµd = 4.8 ×105 Ω −1cm −1
me

Physical Electronics, Spring 2017 12

 2.3 Matthiessen’s Rules

Lattice Scattering and Impurity Scattering

The theory of conduction that considers scattering from lattice vibrations only works
well with pure metals; fails for metallic alloys.
As long as the impurity atom results in a local distortion of the crystal lattice, it will
be effective in scattering. What actually scatters the electron is a local, unexpected
change in the potential energy.
Strained region by impurity exerts a
scattering force F = - d(PE) /dx
Since in unit time, 1/τ is the net
probability of scattering, 1/τT is the
probability of scattering from lattice
τI
vibrations alone, and 1/τI is the
probability of scattering from
impurities alone. If each scattering
mechanisms are independent;
1 1 1 τΤ
= +
τ τT τI
:sum of the numbers
of each collisions Two different types of scattering processes involving scattering from
impurities alone and thermal vibrations alone.

Physical Electronics, Spring 2017 13

 2.3 Matthiessen’s Rules

Matthiessen’s Rule
eτ 1 1 1 µ L : lattice-scattering-limited drift mobility
µd = → = +
me µd µ L µ I µ I : impurity-scattering-limited drift mobility
1 1 1
ρ= = + =ρT + ρ I : Matthiessen's rule
enµd enµ L enµ I
For those near-perfect pure metal crystals, ρT=1/enμT, is the dominating contribution,
As soon as we add impurities, however, there is an additional resistivity, ρI=1/enμI,
which arises from the scattering of the electrons from the impurities.

τT ∝ T −1
( ρT ∝ T ) τ I ⋅ u :temperature-independent (lI ∝ N I−3 )
lI =
where lI is the mean separation between the impurities
ρ ρT + ρ R
= ρR is called the residual resistivity and is due to the scattering
≈ AT + B of electrons by impurities, dislocations, interstitial atoms,
vacancies, grain boundaries, etc. (which means that ρR also
( ρT = AT ) includes ρI ). ρR shows very little temperature dependence.
ρ = resistivity, ρo = resistivity at reference temperature, α0 = TCR
ρ = ρ0 1 + α 0 (T − T0 )  (temperature coefficient of resistivity), T = new temperature, T0 = reference
temperature

Physical Electronics, Spring 2017 14

 2.5 The Hall Effect and Hall Devices

Hall Effect (Metal)

  
= qυ × B
F F = force, q = charge, v = velocity of charged particle, B = magnetic field

 1 
eEH= eυdx Bz → EH=   J x Bz
 en 
EH 1 Ey 1 Jy =0
→− =− → = RH = − Bz
J x Bz en J x Bz en y
VH V
+ + + +eE +
RH: Hall coefficient: measures the Hall H x
Jx Jx z
field per unit transverse applied current Ex
and magnetic filed vdx
EH
q=+e q =-e evdxBz
v v A
Bz
B B V
B
F = qv×B F = qv×B +
A moving charge experiences a Illustration of the Hall effect. The z-direction is out from the plane of
Lorentz force in a magnetic field. paper. The externally applied magnetic field is along the z-direction.

Physical Electronics, Spring 2017 15

 2.5 The Hall Effect and Hall Devices
Wattmeter
Example 2.16: Hall Effect Wattmeter IL IL

VH = wEH = wRH J x Bz ∝ I x Bz ∝ VL I L Load

VL
( Bz ∝ I L , I x = Source RL
VL / R)
→ Power dissipated in road resistor
VL
Example 2.17: Hall Mobility IL IL
C C
V VH
1
− ,σ =
RH = enµd → µ H =
− RH σ Bz
en
w
R Ix = VL/R
VL

Example 2.18: Conduction

Electron Concentration
1 1
n=
− =
−
eRH (1.6 ×10−19 C )(−5.5 ×10−11 m3 A−1s −1 )
1.14 ×1029 m −3 (cf . N at =
= 8.5 ×1028 m −3 )

Physical Electronics, Spring 2017 16