Pauli Exclusion Principle
Electrons in a single atom occupy discrete levels of energy. No two
“energy levels” or “states” in an atom can have the same energy.
Each energy level can contain at most two electrons -- one with
“clockwise spin” and one with “counterclockwise spin”.
If two or more atoms are brought together, their outer (i.e., valence)
energy levels must shift slightly so they will be different from one another.
If many (e.g., N) atoms are brought together to form a solid, the Pauli
Exclusion Principle still requires that only two electrons in the entire solid
have the same energy. There will be N distinct, but only slightly different
valence energy levels, forming a valence band.
When a solid is formed, the different split energy levels of electrons
come together to form continuous bands of energies (electron energy
band).
The extent of splitting depends on the interatomic separation and
begins with the outer most electron shells as they are the first to be
perturbed as the atoms coalesce.
Band Theory: Two Approaches
• There are two approaches to finding the electron energies associated with
atoms in a periodic lattice.
• Approach #1: “Bound” Electron Approach (Solve single atom energies!)
– Isolated atoms brought close together to form a solid.
• Approach #2: “Unbound” or Free Electron Approach (E = p2/2m)
– Free electrons modified by a periodic potential (i.e. lattice ions).
• Both approaches result in grouped energy levels with allowed and forbidden
energy regions.
– Energy bands overlap for metals.
– Energy bands do not overlap (or have a “gap”) for semiconductors.
Band Theory: “Bound” Electron Approach
• For the total number N of atoms in a solid (1023 cm–3), N energy levels
split apart within a width ΔE.
– Leads to a continuous band of energies for each initial atomic
energy level (e.g. 1s energy band for 1s energy level).
Two atoms
Six atoms
Solid of N atoms
The electrical properties of a solid material are a consequence of its
electron band structure, that is, the arrangement of the outermost electron
bands and the way in which they are filled with electrons.
Four different types of band structure are possible at 0oK:
Outermost band is only partially filled with electrons (eg, Na, Cu) (Cu has a
single 4s valence electron, however for a solid comprised of N atoms, the 4s band is capable of
accommodating 2N electrons. Thus, only half of the 4s band is filled.)
Overlap of an empty band and a filled band (eg. Mg). (An isolated Mg atom has
2 electrons in its 3s level. When a solid is formed, the 3s and 3p energy levels overlap.)
A completely filled valence band is separated from the conduction band
by a relatively wide energy band gap (insulators).
A completely filled valence band is separated from the conduction band
by a relative narrow energy band gap (semiconductors)
Electron Energy
“Freedom”
Ef
Empty States
States Filled with Electrons
Distance
The Fermi Level corresponds to
the Highest Occupied Molecular
Energy Orbital (HOMO).
Band
For a Li crystal with N atoms there
are 3N electrons. The 1s band is
filled and the 2s band is half-filled.
The energy difference between adjacent states is infinitesimally small. The
electrons near the Fermi level can move from filled states to empty states
with no activation energy (metals).
Electron
Energy
“Conduction Band”
Empty
“Forbidden”
“Valence Band”
Filled with Electrons
Distance
If “valence” band is filled, no empty
space are available above the filled
states. Electrons can be promoted
Energy to the “conduction band “Lowest
Gap Unoccupied Molecular Orbital –
LUMO” with an activation energy >
energy gap (band gap). Difference
between semiconductors and
insulators is due to the size of the
bandgap.
Temperature Effect
When the temperature of the metal increases, some electrons gain energy and
are excited into the empty energy levels in a valence band. This condition
creates an equal number of empty energy levels, or holes, vacated by the
excited electrons. Only a small increase in energy is required to cause
excitation of electrons.
Both the excited electrons (free electrons) and the newly created holes
can then carry an electrical charge.
Fermi Energy is the energy corresponding to the highest filled state at 0oK.
Only electrons above the Fermi energy can be affected by an electric field
(free electrons).
Band Diagram: Fermi-Dirac “Filling” Function
The Fermi-Dirac function gives the fraction of allowed states, fFD(E), at an
energy level E, that are populated at a given temperature.
1
f FD ( E ) =
( E − EF )
e
kT
+1
where the Fermi Energy, EF, is defined as the energy where fFD(EF) = 1/2.
That is to say one half of the available states are occupied. T is the
temperature (in K) and kB is the Boltzman constant (kB = 8.62 x10-5 eV/K)
Probability of electrons (fermions) to be found at various energy levels.
1
f FD ( E ) =
( E − EF )
e
kT
+1
For E – EF = 0.05 eV ⇒ f(E) = 0.12
For E – EF = 7.5 eV ⇒ f(E) = 10 –129
Exponential dependence has HUGE effect!
Temperature dependence of Fermi-Dirac function shown as follows:
→ Step function behavior “smears” out at higher temperatures.
Fermi-Dirac Function
Metals and Semiconductors
f(E) as determined
experimentally for
Ru metal (note the
energy scale)
f(E) for a
semiconductor
Velocity of the electrons
In metals, the Fermi energy gives us information about the velocities of the
electrons which participate in ordinary electrical conduction. The amount of
energy which can be given to an electron in such conduction processes is
on the order of micro-electron volts (see copper wire example), so only
those electrons very close to the Fermi energy can participate. The Fermi
velocity of these conduction electrons can be calculated from the Fermi
energy.
Speed of light: 3x108m.s-1
Fermi Energies, Fermi Temperatures, and Fermi Velocities
Element Fermi Energy eV
Fermi Temperature x10^4 K
Li
4.74
5.51
Na
3.24
3.77
K
2.12
2.46
Rb
1.85
2.15
Cs
1.59
1.84
Cu
7.00
8.16
Ag
5.49
6.38
Au
5.53
6.42
Be
14.3
16.6
Mg
7.08
8.23
from N. W. Ashcroft and N. D. Mermin,
Fermi Velocity 10^6 m/s
1.29
1.07
0.86
0.81
0.75
1.57
1.39
1.40
2.25
1.58
The Fermi temperature is the temperature associated with the Fermi energy by
solving
EF = kB TF
for , where m is the particle mass and kB is Boltzmann's constant.
Electrical Conductivity (σ)
σ=
1
ρ
(Ω − m )
−1
Conductivity is the “ease of conduction”. Ranges
over 27 orders of magnitude!
(a)
(b)
(c)
(d)
Metals
107 (Ω.cm)-1
Semiconductors 10-6 - 104 (Ω.cm)-1
Insulators
10-10 -10-20 (Ω.cm)-1
Charge carriers, such as electrons, are deflected by atoms
or defects and take an irregular path through a conductor.
Valence electrons in the metallic bond move easily.
Covalent bonds must be broken in semiconductors and
insulators for an electron to be able to move.
Entire ions must diffuse to carry charge in many ionically
bonded materials.
•
•
Electronic conduction:
– Flow of electrons, e- and electron holes, h+
Ionic conduction
– Flow of charged ions, Ag+
Microscopic Conductivity
We can relate the conductivity of a material to microscopic parameters that
describe the motion of the electrons (or other charge carrying particles such
as holes or ions). From the equations
V = IR
and
RA
ρ=
L
⇒
V
I
= ρ
L A
If J=Current density (I/A) Ampere/m2; and ξ=electric field intensity (V/L) then
1
σ = (Ω − m )−1
J = σξ
ρ
We can also determine that the current density is
J = nqν
Where n is the number of charge carriers (carriers/cm3); q is the charge on
each carrier (1.6x10-19C); and ν is the average drift velocity (cm/s) at which
the charge carriers move.
Therefore
J = nq ν = σξ
then
ν
σ = nq
ξ
⎛ cm ⎞
ν
⎟⎟
= μ = mobility ⎜⎜
ξ
⎝ V.s ⎠
σ = nq μ
2
The charge q is a constant.
Electrons are the charge carriers in metals.
Electrons and holes are both carriers of electricity in semiconductors.
Electrons that “hop” from one defect to another or the movement of
ions are both the carriers of electricity in ceramics.
Conduction in Terms of Band
Metals
An energy band is a range of allowed electron energies.
The energy band in a metal is only partially filled with electrons.
Metals have overlapping valence and conduction bands
Drude Model of Electrical Conduction in Metals
Conduction of electrons in metals – A Classical Approach:
In the absence of an applied electric field (ξ) the electrons move in random directions colliding
with random impurities and/or lattice imperfections in the crystal arising from thermal motion of
ions about their equilibrium positions.
The frequency of electron-lattice imperfection collisions can be described by a mean free path
λ -- the average distance an electron travels between collisions.
When an electric field is applied the electron drift (on average) in the direction opposite to that
v
of the field with drift velocity
The drift velocity is much less than the effective instantaneous speed (v) of the random
motion
−2
−1
In copper v ≈ 10 cm.s while
v ≈ 10 cm.s
8
−1
where
1
3
2
mev = k BT
2
2
The drift speed can be calculated in terms of the applied electric field ξ and of v and λ
When an electric field is applied to an electron in the metal it experiences a force qξ
resulting in acceleration (a)
qξ
a=
me
Then the electron collides with a lattice imperfection and changes its direction
randomly. The mean time between collisions is
The drift velocity is
v = a ⋅τ =
τ=
q ⋅ ξ ⋅τ q ⋅ ξ ⋅ λ
=
me
me ⋅ v
If n is the number of conduction electrons per unit volume and J is the current density
Combining with the definition of resistivity gives
n ⋅q ⋅λ
σ=
me ⋅ v
2
q=1.6x10-19C
J = nq ν = σξ
q ⋅ λ q ⋅τ
μ=
=
me ⋅ v me
λ
v
Electron Energy
For an electron to become free to conduct, it must be promoted into an
empty available energy state
For metals, these empty states are adjacent to the filled states
Generally, energy supplied by an electric field is enough to stimulate
electrons into an empty state
“Freedom”
Empty States
Energy Band
States Filled with Electrons
Distance
Band Diagram: Metal
T>0
EC
Fermi “filling”
function
Conduction band
(Partially Filled)
EF
Energy band to
be “filled”
E=0
At T = 0, all levels in conduction band below the Fermi energy EF are filled
with electrons, while all levels above EF are empty.
Electrons are free to move into “empty” states of conduction band with
only a small electric field E, leading to high electrical conductivity!
At T > 0, electrons have a probability to be thermally “excited” from below
the Fermi energy to above it.
Conduction in Materials --Classical
approach
1. Drude Model
2. Temperature‐dependent conductivity 3. Matthissen’s Rule
4. Hall effect
5. Skin effect (HF resistance of a conductor)
1. Heat Capacity and thermal conductivity
Classic Model (Drude Model)
electron act as a particle
•
•
•
•
Collision: the scattering of an
electron by (and only by) an
ion core
Between collision: electrons
do not interact with each
other or with ions
An electron suffers a collision
with probability per unit time
τ-1, (τ-1 scattering rate).
Electrons achieve thermal
equilibrium with their
surrounding only through
collisions
Ex
u
Δx
Vibrating Cu + ions
(a)
V
(b)
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
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
Relaxation time approximation and Mobility
Ex
Δx
A
vdx
Velocity gained along x
vx1- ux1
Jx
Present time
Last collision
Electron 1
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.)
t1
vx1-ux1
Free time
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
t
Electron 2
t2
t
time
time
vx1-ux1
v dx = μ E x
J = en μ E x = σ E
σ = ne μ
eτ
μ =
me
(Ohm’s law)
Electron 3
t3
t
time
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
Temperature dependence of Conductivity for Metal
S = π a2
uτ
=
l
u
a
Conduction by free electrons
and scattered by lattice
vibration
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.
Resistivity (ρ) in Metals
Resistivity typically increases linearly with temperature:
ρt = ρo + αT
Phonons scatter electrons. Where ρo and α are constants for an
specific material
Impurities tend to increase resistivity: Impurities scatter electrons in
metals
Plastic Deformation tends to raise resistivity dislocations scatter
electrons
σ=
1
ρ
= nqμ
The electrical conductivity is controlled by controlling the number of
charge carriers in the material (n) and the mobility or “ease of
movement” of the charge carriers (μ)
Matthiessen’s Rule
Strained region by impurity exerts a
scattering force F = - d(PE) /dx
τI
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
Temperature Dependence, Metals
There are three contributions to ρ:
ρt due to phonons (thermal)
ρi due to impurities
ρd due to deformation
ρ = ρt + ρi+ ρd
The number of electrons in the
conduction band does not vary
with temperature.
All the observed temperature
dependence of σ in metals arise
from changes in μ
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
Scattering by Impurities and Phonons
Thermal: Phonon scattering
Proportional to temperature
ρ t = ρ o + aT
Impurity or Composition scattering
Independent of temperature
Proportional to impurity concentration
Solid Solution
Two Phase
ρ i = Aci (1 − ci )
ρ t = ραVα + ρ β Vβ
Deformation
ρ d = must be experimentally determined
100
ρ∝T
10
Resistivity (nΩ m)
1
ρ (nΩ m)
0.1
ρ∝
0.01
T5
3.5
3
2.5
2
1.5
1
0.001
ρ ∝ T5
0.5 ρ = ρR
0
0 20 40
ρ = ρR
0.0001
ρ∝T
60
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
Electron conduction in nonmetals
Insulators
Semiconductors
Conductors
Many ceramics
Superconductors
Alumina
Diamond Inorganic Glasses
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
100
Conductivity (Ωm)-1
Metals
Degenerately
Doped Si
Alloys
Te Graphite NiCr Ag
103
106
109
Figure 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
1012
Control of Electrical Conductivity
By controlling the number of charge
carriers in the material (n)
By controlling the mobility or “ease of
movement” of the charge carriers (μ)
σ = nqμ
Example: Calculation of Drift velocity of electrons in copper (valence =1)
Assuming that all of the valence electrons contribute to current flow,
(a) calculate the mobility of an electron in copper and
(b) calculate the average drift velocity for electrons in a 100cm copper wire when
10V are applied.
Data: conductivity of copper = 5.98 x 105(Ω.cm)-1 ; q = 1.6x10-19C ; lattice
parameter of copper = 0.36151x10-7cm ; copper has a FCC structure
The equation that we need to apply is:
σ = nqμ
n is the number of carriers. q is the charge . μ is the mobility
Number of carriers (n): The valence of copper is 1. Therefore, the number of
valence electrons equals the number of copper atoms in the material.
n=
4 ×1
(atoms / cell )(electrons / atoms )
22
3
=
8
466
10
electrons
cm
.
/
=
×
cell _ volume
(0.36151×10 −7 cm )3
Part (a)
σ
cm 2
cm 2
5.98 × 105 Ω −1 .cm −1
μ= =
= 44.2
= 44.2
22
3
−19
nq (8.466 × 10 electrons / cm )(1.6 ×10 C )
V .s
Ω.C
Note : 1amp = 1Coulomb / sec ond
Part (b)
The electric field intensity is:
V
10V
ξ= =
= 0.1V .cm −1
L 100cm
The equation we need to use is
cm 2
V
v = ( 44.2
)(0.1 ) = 4.42cm / s
V .s
cm
ν = μξ
What does it mean?
The drift velocity is the average velocity that a particle,
such as an electron, attains, due to an electric field.
The electron moves at the Fermi speed and it has only a tiny drift velocity
superimposed by the applied electric field.
In metals, the Fermi energy gives us information about the velocity of the electrons
which participate in ordinary electrical conduction. The amount of energy which can be
given to an electron in such conduction processes is on the order of micro-electron
volts, so only those electrons very close to the Fermi energy (EF) can participate.
2E F
vF =
m
Element
Fermi Energy
eV
Fermi Temperature
x 10^4 K
Fermi Velocity
x 10^6 m/s
Li
4.74
5.51
1.29
Na
3.24
3.77
1.07
K
2.12
2.46
0.86
Rb
1.85
2.15
0.81
Cs
1.59
1.84
0.75
Cu
7.00
8.16
1.57
Ag
5.49
6.38
1.39
Au
5.53
6.42
1.40
Be
14.3
16.6
2.25
Mg
7.08
8.23
1.58
Ca
4.69
5.44
1.28
Sr
3.93
4.57
1.18
Ba
3.64
4.23
1.13
Nb
5.32
6.18
1.37
Fe
11.1
13.0
1.98
Mn
10.9
12.7
1.96
Zn
9.47
11.0
1.83
Cd
7.47
8.68
1.62
Hg
7.13
8.29
1.58
Al
11.7
13.6
2.03
Ga
10.4
12.1
1.92
In
8.63
10.0
1.74
Tl
8.15
9.46
1.69
Sn
10.2
11.8
1.90
Pb
9.47
11.0
1.83
Bi
9.90
11.5
1.87
Sb
10 9
12 7
1 96
Insulator
The valence band and conduction band are separated by a large (> 4eV)
energy gap, which is a “forbidden” range of energies.
Electrons must be promoted across the energy gap to conduct, but the
energy gap is large. Energy gap º Eg
Electron
Energy
“Conduction Band”
Empty
“Forbidden”
“Valence Band”
Filled with Electrons
Distance
Energy
Gap
Band Diagram: Insulator
T>0
Conduction band
(Empty)
Egap
EC
EF
Valence band
(Filled)
EV
At T = 0, lower valence band is filled with electrons and upper conduction
band is empty, leading to zero conductivity.
Fermi energy EF is at midpoint of large energy gap (2-10 eV) between
conduction and valence bands.
At T > 0, electrons are usually NOT thermally “excited” from valence to
conduction band, leading to zero conductivity.
Conduction in Ionic Materials (Insulators)
Conduction by electrons (Electronic Conduction): In a ceramic, all
the outer (valence) electrons are involved in ionic or covalent bonds and thus
−Eg
they are restricted to an ambit of one or two atoms.
2 k BT
If Eg is the energy gap, the fraction of electrons in the conduction band is:
e
A good insulator will have a band gap >>5eV and 2kBT~0.025eV at room temperature
As a result of thermal excitation, the fraction of electrons in the conduction band is
~e-200 or 10-80.
There are other ways of changing the electrical conductivity in the ceramic which
have a far greater effect than temperature.
•Doping with an element whose valence is different from the atom it replaces. The
doping levels in an insulator are generally greater than the ones used in
semiconductors. Turning it around, material purity is important in making a good
insulator.
•If the valence of an ion can be variable (like iron), “hoping” of conduction can
occur, also known as “polaron” conduction. Transition elements.
•Transition elements: Empty or partially filled d or f orbitals can overlap providing a
conduction network throughout the solid.
Conduction by Ions: ionic conduction
It often occurs by movement of entire ions, since the energy gap is too large
for electrons to enter the conduction band.
Z .q.D
The mobility of the ions (charge carriers) is given by:
μ=
k B .T
Where q is the electronic charge ; D is the diffusion coefficient ; kB is
Boltzmann’s constant, T is the absolute temperature and Z is the valence of
the ion.
The mobility of the ions is many orders of magnitude lower than the mobility of
the electrons, hence the conductivity is very small:
σ = n.Z.q.μ
Example:
Suppose that the electrical conductivity of MgO is determined primarily by the
diffusion of Mg2+ ions. Estimate the mobility of Mg2+ ions and calculate the
electrical conductivity of MgO at 1800oC.
Data: Diffusion Constant of Mg in MgO = 0.0249cm2/s ; lattice parameter of
MgO a=0.396x10-7cm ; Activation Energy for the Diffusion of Mg2+ in MgO =
79,000cal/mol ; kB=1.987cal/K=k-mol; For MgO Z=2/ion; q=1.6x10-19C;
kB=1.38x10-23J/K-mol
First, we need to calculate the diffusion coefficient D
⎛ − QD
D = Do exp⎜
⎝ kT
⎛
⎞
− 79000cal / mol
cm 2
⎞
⎜
⎟⎟
=
0
.
0239
exp
⎟
⎜
s
⎠
⎝ 1.987cal / mol − Kx (1800 + 273)K ⎠
D=1.119x10-10cm2/s
Next, we need to find the mobility
2
Z .q .D ( 2carriers / ion )(1.6 × 10 −19 C )(1.1× 10 −10 )
C
.
cm
−9
=
.
μ=
=
1
12
×
10
(1.38 × 10 − 23 )(1800 + 273)
k B .T
J .s
C ~ Amp . sec ; J ~ Amp . sec .Volt
μ=1.12x10-9 cm2/V.s
MgO has the NaCl structure (with 4 Mg2+
and 4O2- per cell)
Thus, the Mg2+ ions per cubic cm is:
4Mg 2+ ions / cell
22
3
n=
=
6
.
4
×
10
ions
/
cm
(0.396 ×10 −7 cm )3
σ = nZqμ = (6.4 ×10 22 )( 2)(1.6 ×10 −19 )(1.12 ×10 −9 )
2
C
.
cm
σ = 22.94 ×10 −6 3
cm .V .s
C ~ Amp.sec ; V ~ Amp.Ω
σ = 2.294 x 10-5 (Ω.cm)-1
Example:
The soda silicate glass of composition 20%Na2O-80%SiO2 and a density of
approximately 2.4g.cm-3 has a conductivity of 8.25x10-6 (Ω-m)-1 at 150oC. If the
conduction occurs by the diffusion of Na+ ions, what is their drift mobility?
Data: Atomic masses of Na, O and Si are 23, 16 and 28.1 respectively
Solution:
We can calculate the drift mobility (μ) of the Na+ ions from the conductivity
expression
σ = ni × q × μ i
Where ni is the concentration of Na+ ions in the structure.
20%Na2O-80%SiO2 can be written as M At = 0.2 × ( 2( 23) + 1(16)) + 0.8 × (1( 281.1) + 2(16))
(Na2O)0.2-(SiO2)0.8 . Its mass can be
M At = 60.48g .mol −1
calculated as:
The number of (Na2O)0.2-(SiO2)0.8 units per unit volume can be found from the
density
n=
ρ × NA
M At
( 2.4g .cm −3 ) × (6.023x10 23 mol −1 )
=
60.48g .mol −1
n = 2.39 ×10 22 (Na 2O ) 0.2 (SiO 2 ) 0.8 units − cm −3
The concentration of Na+ ions (ni) can be obtained from the concentration of
(Na2O)0.2-(SiO2)0.8 units
⎡
⎤
0.2 × 2
22
21
−3
ni = ⎢
×
2
.
39
×
10
=
3
.
18
×
10
cm
⎥
0
.
2
×
(
2
+
1
)
+
0
.
8
×
(
1
+
2
)
⎣
⎦
And μi
σ
(8.25 × 10 −6 Ω −1m −1 )
μi =
=
q × ni (1.60 × 10 −19 C ) × (3.186 × 10 21 × 106 m −3 )
μ i = 1.62 ×10 −14 m 2V −1s −1
This is a very small mobility compared to semiconductors and metals
Conduction in ionic crystal and glasses
Mobile charges contribute to conduction
E
1×10-1
E
As3.0Te3.0Si1.2Ge1.0 glass
1×10-3
Conductivity 1/(Ωm)
Vacancy aids the diffusion of positive ion
Na+
Anion vacancy
acts as a donor
24%Na2O-76%SiO2
(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)
http://Materials.Usask.Ca
⎛ E0 ⎞
σ = σ 0 exp⎜ − ⎟
⎝ kT ⎠
1×10-5
1×10-7
12%Na2O-88%SiO2
1×10-9
1×10-11
PVAc
SiO2
PVC
1×10-13
Interstitial cation diffuses
(a)
Pyrex
1×10-15
1.2
1.6
2
2.4
2.8
103/T (1/K)
3.2
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)
http://Materials.Usask.Ca
Hopping process
Electrical Breakdown
At a certain voltage gradient (field) an insulator will break down.
There is a catastrophic flow of electrons and the insulator is fragmented.
Breakdown is microstructure controlled rather than bonding controlled.
The presence of heterogeneities in an insulator reduces its breakdown field
strength from its theoretical maximum of ~109Vm-1 to practical values of 107V.m-1
Energy Bands in Semiconductors
Electron Energy
Energy Levels and Energy Gap in a Pure Semiconductor.
The energy gap is < 2 eV.
Energy gap º Eg
“Conduction Band” (Nearly)
Empty – Free electrons
“Forbidden”
Energy Gap
“Valence Band” (Nearly) Filled with
Electrons – Bonding electrons
Semiconductors have resistivities in between those of metals and insulators.
Elemental semiconductors (Si, Ge) are perfectly covalent; by symmetry electrons
shared between two atoms are to be found with equal probability in each atom.
Compound semiconductors (GaAs, CdSe) always have some degree of ionicity. In III-V
compounds, eg. Ga+3As+5, the five-valent As atoms retains slightly more charge than is
necessary to compensate for the positive As+5 charge of the ion core, while the charge of
Ga+3 is not entirely compensated. Sharing of electrons occurs still less fairly between the
ions Cd+2 and Se+6 in the II-VI compund CdSe.
Semiconductor Materials
Semiconductor
Carbon (Diamond)
Silicon
Germanium
Tin
Gallium Arsenide
Indium Phosphide
Silicon Carbide
Cadmium Selenide
Boron Nitride
Aluminum Nitride
Gallium Nitride
Indium Nitride
Bandgap Energy EG (eV)
5.47
1.12
0.66
0.082
IIIA
1.42
10.811
1.35
5
3.00
B
Bo ro n
1.70
7.50
6.20
3.40
1.90
13
IIB
30
65.37
Zn
Zinc
48
Portion of the Periodic Table Including the Most
Important Semiconductor Elements
112.40
Cd
C a d m ium
80
200.59
Hg
M e rc ury
26.9815
Al
A lum inum
31
69.72
Ga
G a llium
49
114.82
In
IV A
6
12.01115
C
C a rb o n
14
204.37
Ti
Tha llium
Si
Silic o n
32
72.59
Ge
G e rm a nium
50
Ind ium
81
28.086
118.69
Sn
Tin
82
207.19
Pb
Le a d
VA
14.0067
7
N
N itro g e n
15
30.9738
P
V IA
15.9994
8
O
O xy g e n
16
Pho sp ho rus
33
74.922
As
A rse nic
51
121.75
Sb
A ntim o ny
83
208.980
Bi
Bism uth
32.064
S
Sulfur
34
78.96
Se
Se le nium
52
127.60
Te
Te llurium
84
(210)
Po
Po lo nium
Band Diagram: Semiconductor with No Doping
T>0
Conduction band
(Partially Filled)
EF
Valence band
(Partially Empty)
EC
EV
At T = 0, lower valence band is filled with electrons and upper conduction
band is empty, leading to zero conductivity.
Fermi energy EF is at midpoint of small energy gap (<1 eV) between
conduction and valence bands.
At T > 0, electrons thermally “excited” from valence to conduction band,
leading to measurable conductivity.
Semi-conductors (intrinsic - ideal)
Perfectly crystalline (no perturbations in the periodic lattice).
Perfectly pure – no foreign atoms and no surface effects
At higher temperatures, e.g., room temperature (T @ 300 K), some electrons are
thermally excited from the valence band into the conduction band where they
are free to move.
“Holes” are left behind in the valence band. These holes behave like mobile
positive charges.
CB electrons and VB holes can
move around (carriers).
At edges of band the kinetic energy
of the carriers is nearly zero. The
electron energy increases upwards.
The hole energy increases
downwards.
Si
Si
Si
Si
Si
Si
Si
Si
Si
Si
Si
Si
Si
Si
positive
ion core
valence
electron
Si
Si
Si
Si
Si
Si
Si
Si
Si
Si
Si
Si
Si
Si
Si
Si
Si
Si
Si
Si
Si
free
electron
free
hole
Si
Si
Si
Si
Si
Si
Si
Semiconductors in Group IV
Carbon, Silicon, Germanium, Tin
Each has 4 valence Electrons.
Covalent bond
Compound Semiconductors
(Group III and Group V)
The III-V semiconductors are prominent for applications in optoelectronics with particular
importance for areas such as wireless communications because they have a potential for
higher speed operation than silicon semiconductors.
The compound semiconductors have a crystal lattice constructed from atomic elements in
different groups of the periodic chart. Each Group III atom is bound to four Group V atoms,
and each Group
V atom is bound to four Group III atoms, giving the general arrangement shown in Figure.
The bonds are produced by sharing of electrons such that each atom has a filled (8
electron) valence band.
The bonding is largely covalent, though the shift of valence charge from the Group V atoms
to the Group III atoms induces a component of ionic bonding to the crystal (in contrast to
the elemental semiconductors which have purely covalent bonds).
Representative III-V compound semiconductors are GaP, GaAs, GaSb, InP, InAs, and InSb.
GaAs is probably the most familiar example of III-V compound semiconductors, used for both
high speed electronics and for optoelectronic devices.
Optoelectronics has taken advantage of ternary and quaternary III-V semiconductors to
establish optical wavelengths and to achieve a variety of novel device structures. The ternary
semiconductors have the general form (Ax;A’1-x)B (with two group III atoms used to filled the
group III atom positions in the lattice) or A(Bx;B’1-x) (using two group V atoms in the Group V
atomic positions in the lattice).
The quaternary semiconductors use two Group III atomic elements and two Group V atomic
elements, yielding the general form (Ax;A’1-x)(By;B’1-y). In such constructions, 0<x,y<1.
Such ternary and quaternary versions are important since the mixing factors (x and y) allow
the band gap to be adjusted to lie between the band gaps of the simple compound crystals
with only one type of Group III and one type of Group V atomic element.
The adjustment of wavelength allows the material to be tailored for particular optical
wavelengths, since the wavelength ¸ of light is related to energy (in this case the gap energy
Eg) by ¸ λ= hc/Eg, where h is Plank's constant and c is the speed of light.
Hall effect and Hall devices
q = +e
q = -e
v
B
B
F = qv×B
(a)
v
B
F = qv×B
(b)
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)
http://Materials.Usask.Ca
Hall effect in Semiconductor
Bz
Jy = 0
Jx
eEy
+
Ey
vhx
evhxBz
+
+
Ex
Jx
z
x
vex
+
+
y
eEy
evexBz
+
A
Bz
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.(E is the electric field.)
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
Thermal Conduction
HOT
• Metal
δT
dQ
dt
HEAT
HOT
COLD
A
δx
COLD
HEAT
Fig. 2.19: Heat flow in a metal rod heated at one end. Consider the
rate of heat flow, dQ/dt, across a thin section δ x of the rod. The rate
of heat flow is proportional to the temperature gradient δ T/δ x and the
cross sectional area A.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
Electron Gas
Vibrating Cu+ ions
Fig. 2.18: 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.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
The failure of Drude model
• It can explain the electric conduction, not
thermal conductivity and heat capacity
• Electronic heat capacity
– (Drude)
3
Cel = R
2
– Experimental result
CV = γT + αT
3
Need more sophisticated model (Sommerfeld or
Quantum mechanics)
Ag
400
Ag-3Cu
50000
Cu
Ag-20Cu
300
κ
= T CWFL
σ
Au
Al
200
W
Be
Mg
Mo
Brass (Cu-30Zn)
Ni
Bronze (95Cu-5Sn)
Steel (1080)
Pd-40Ag
Hg
100
0
0
10
20
30
40
50
60
Electrical conductivity, σ, 106 Ω-1 m-1
70
Fig. 2.20: Thermal conductivity, κ vs. electrical conductivity σ for
various metals (elements and alloys) at 20 °C. The solid line
represents the WFL law with CWFL ≈ 2.44×108 W Ω K-2.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
Thermal conductivity, κ (W K-1 m-1)
Thermal conductivity, κ (W K-1 m-1)
450
Copper
10000
Aluminum
1000
Brass (70Cu-30Zn)
100
Al-14%Mg
10
1
10
100
Temperature (K)
1000
Fig. 2.21: Thermal conductivity vs. temperature for two pure metals
(Cu and Al) and two alloys (brass and Al-14%Mg). Data extracted from
Thermophysical Properties of Matter, Vol. 1: Thermal Conductivity,
Metallic Elements and Alloys, Y.S. Touloukian et. al (Plenum, New
York, 1970).
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
• Non-metal
Phonon
Diamond—Good thermal conductor
Polymer- -Bad thermal conductor
Equilibrium
Hot
Cold
Energetic atomic vibrations
Fig. 2.22: Conduction of heat in insulators involves the
generation and propogation of atomic vibrations through the
bonds that couple the atoms. (An intuitive figure.)
Thermal
conductivity and
resistance
Q′ = ΔT/θ
ΔT
ΔT
Hot
Cold
Q′
A
Q′
Q′
θ
L
(b)
(a)
Fig. 2.23: Conduction of heat through a component in (a)
can be modeled as a thermal resistance θ shown in (b) where
Q′ = ΔT/θ.
Pure metal
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
Nb
Fe
Zn
W
Al
Cu
Ag
κ (W m-1 K- 52
80
113
178
250
390
420
1080
Bronze
Brass (63Cu- Dural (95Al-
Steel
(95Cu-
37Zn)
4Cu-1Mg)
125
147
1
)
Metal alloys Stainless Steel 55Cu-45Ni 70Ni-30Cu
5Sn)
κ (W m-1 K- 12 - 16
1
19.5
25
50
80
)
Ceramics
Glass-
Silica-fused S3N4
Alumina Saphire Beryllia
and glasses
borosilicate
(SiO2)
(Al2O3) (Al2O3) (BeO)
κ (W m-1 K- 0.75
1
1.5
20
30
37
260
~1000
Teflon
Polyethylene Polyethylene
)
Polymers
Polypropylene PVC
Polycarbonate Nylon
6,6
κ (W m-1 K- 0.12
1
Diamond
)
0.17
0.22
0.24
low density high density
0.25
0.3
0.5