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ELECTRONIC
MATERIALS
Electronic Materials
Superconductors
Conductors
Semi-Conductors
Dielectrics
Type I
Hg, MgB2
Linear
Ag, Au, Cu, Hg, W
Elemental
Si, Ge, a-Sn, C
Type II
YBCO,BSCCO
Non-Linear
ZnO, diodes
Compound
GaAs, AlAs
Organic
Inorganic
ITO,YSZ, BaTiO3
Dielectrics
Insulators
Porcelain, glasses, Al2O3
Linear
Al2O3, SiO2, ZrO2
NonLinear
BaTiO3, PZT
Electrical Properties
Objectives
To understand:
Electronic Conduction in materials
Band Structure
Conductivity
Metals
Semiconductors
Ionic conduction in ceramics
Dielectric Behavior
Polarization
Important Terms
Acceptor: element introduced to semiconductor to generate free hole (by “accepting” electron
from semiconductor atom and “releasing” hole at the same time); acceptor atom must have one
valence electron less than semiconductor; boron (B) is the most common acceptor in silicon
technology; alternatives include indium and gallium.
Donor: element introduced to semiconductor to generate free electron (by “donating” a electron
to semiconductor); must have one more valence electron than semiconductor; common donors in
Si: P, As, Sb and in GaAs: S, Se, Sn.
Dopant: element introduced to semiconductor to establish either p-type (acceptor) or n-type
(donor) conductivity
Electron: negatively charged particle in an atom; carrier of smallest (elemental) charge of 1.6x1019C;
carrier of negative charge in semiconductors
Hole: positive charge carrier in a semiconductors which materially does not exits; lack of electron
moving in the direction opposite to that of electron and carrying a positive charge; features higher
effective mass than electron, hence, lower mobility.
Fermi level: energy level in solids at which the Fermi-Dirac distribution function is equal to 0.5.
Fermi-Dirac distribution function: formula describing the probability of a state being
occupied by an electron depending on the state's energy level.
Semiconductor: solid-state material in which, in contrast to metals and insulators: (i) electrical
conductivity can be controlled by orders of magnitude by adding very small amounts of alien
elements known as dopants, (ii) electrical conductivity can be controlled not only by negatively
charged electrons, but also by positively charged holes and (iii) electrical conductivity is
sensitive to temperature, illumination, and magnetic fields; these properties result from the fact
that, due to the nature of interatomic bonds (mostly covalent), in semiconductors, in contrast to
metals but similarly to insulators, the valence band and the conduction band are separated by the
energy gap; in semiconductors, however, width of the energy gap rarely exceeds about 3.5 eV
while in insulators energy gap is typically wider that about 5 eV; semiconductor properties are
displayed by the elements from the IVth group of periodic table: carbon, (C) in the form of
diamond, silicon (Si), germanium (Ge) and tin (Sn); numerous compound semiconductors can be
formed by combining elements from IVth group, e.g. SiGe and SiC, as well as from groups III
and V,e.g. GaAs, InP, or GaN, and groups II and Vi, e.g. CdTe, ZnS, etc. Furthermore, select
organic materials display semiconductor properties.
Semiconductor Diode: two-terminal device in which current is a very strong function of the
direction of applied voltage; large current is allowed to flow under forward voltage bias while
almost no current will flow under the reverse voltage bias, hence, displays rectifying
characteristics; typically implemented by p-n junction.
Intrinsic semiconductor: semiconductor materials in which free electrons and holes result only
from band-to band generation, and hence, feature the same concentration; undoped; does not display
neither n- or p-type conductivity.
Extrinsic semiconductor: doped semiconductor featuring either p- or n-type conductivity.
Scattering: process responsible for electron in a certain state defined by its crystal momentum
suddenly moving into a different state; fundamental effect defining electron transport in
semiconductor.
Drift velocity: velocity of carriers under electric field; as opposed to carriers in free space carriers
in a semiconductor are not "infinitely" accelerated by the electric field due to scattering - thus they
reach a finite velocity regardless of the period of time over which the field is acting; at a given
electric field drift velocity is determined the carrier mobility.
Electron mobility: measure of electron scattering in semiconductor; proportionality factor
between electron drift velocity and electric field as well as carrier concentration and conductivity of
semiconductor; unit cm2/V s; the same way as effective mass of an electron, electron mobility is
different for different semiconductors; electron mobility at 300 K for three key semiconductors: Si 1500 cm2/V s, GaAs -7500 cm2/V s, 6H-SiC -400 cm2/V s; higher mobility makes semiconductor
better suited for high speed applications.
Hall Effect: magnetic field applied to semiconductor through which current is flowing causes
generation of electric field perpendicular to the direction of current flow and the magnetic field; used
to derive some electrical properties of semiconductor.
Band gap: energy band separating conduction and valence bands in the solid; no electron energy
levels are allowed in the forbidden band; no energy gap in metals in which case conduction and valence
bands overlap; solids featuring energy gap are defined as either semiconductors or insulators based on
the width of energy gap; values of Eg (at 300K) for common semiconductors: InSb - 0.17 eV, Ge - 0.67
eV, Si - 1.12 eV, GaAs - 1.43 eV GaP - 2.26 eV, 6H-SiC - 2.9 eV, GaN - 3.5 eV, and insulators Ta2O5 4.2 eV , TiO2 -5 eV, Si3N4 - 5.1, Al2O3 ~5 eV, SiO2 - 8.0.
Dielectric: a solid displaying insulating properties (energy gap wider than 4 eV); upper most energy
band is completely empty, hence, extremely low conductivity; characteristics of material are
independent of the applied voltage.
Lithography: process used to transfer pattern from the mask/reticle to the layer of resist deposited on
the surface of the wafer; kind of lithography depends on the wavelength of radiation used to expose
resist: photolithography (or optical lithography) uses UV radiation, X-ray lithography uses X-ray, ebeam lithography uses electron bean, ion beam lithography uses ion beam.
Electron beam (e-beam) lithography, EBL: lithography technique using focused beam of
electrons to expose the resist; no mask is used as pattern is "written" directly into the resist by very fast
scanning of electron beam; pattern transfer resolution below 0.1 µm is possible.
Transistor: three-terminal semiconductor device in which input signal (voltage or current depending on
the type of transistor) controls output current, the most important semiconductor device; performs
switching and amplifying functions; early on replaced bulky and inefficient vacuum triode in electronic
circuits; invention of transistor (Schockley, Brattain and Bardeen) triggered electronic revolution after the
World Word II; can operate as a discrete device or a building cell of integrated circuits; numerous kinds
of transistors can be distinguished based on their design and principles of operation; two major types:
field-effect (unipolar) transistor and bipolar transistor.
Definitions
Ohm’s Law
V = IR
V - Voltage, I - current, R -Resistance
Units
V - Volts (or W/A (Watts/amp) or J/C Joules/Coulomb))
I – amperes (or C/s - Coulombs/second)
R - ohms (Ω)
The resistance is influenced by specimen configuration
Consider current moving through a conductor
R = V/I
with cross sectional area, A and a length, L
I
Area (A)
Length (L)
Resistivity (ρ) and Conductivity (σ)
Geometrically independent forms of
Ohm’s Law
Probe Conductivity Measurements
L
R = Rsample + Rcontact
R = V/I
RA Ω − m 2
=
=Ω−m
ρ=
L
m
A
V
It can give erroneous values if
contact resistance, Rcontact, is not
negligible with respect to Rsample
I = V1/R1
Rsample = V2/I
Rsample = (V2R1)/V1
ρ = Rsample (A/L)
Eliminates contact resistance
I
Voltmeter
L
A
I
V2
R1
Current
Source
V1
Voltmeter
Conductivity (σ)
σ=
Conductivity is the “ease of conduction”
Ranges over 27 orders of magnitude!
Metals
Semiconductors
Insulators
1
ρ
(Ω − m )
−1
107 (Ω.cm)-1
10-6 - 104 (Ω.cm)-1
10-10 -10-20 (Ω.cm)-1
Resistivities of Real Materials
Compound
Compound
Ca
Resistivity
(Ω-cm)
3.9 × 10-6
Si
Resistivity
(Ω-cm)
~ 0.1
Ti
Mn
Zn
Cu
42 × 10-6
185 × 10-6
5.9 × 10-6
1.7 × 10-6
Ge
ReO3
Fe3O4
TiO2
~ 0.05
36 × 10-6
52 × 10-6
9 × 104
Ag
Pb
1.6 × 10-6
21 × 10-6
ZrO2
Al2O3
1 × 109
1 × 1019
(a) Charge carriers, such as electrons, are deflected by
atoms or defects and take an irregular path through a
conductor. The average rate at which the carriers move
is the drift velocity (v).
(b) Valence electrons in the metallic bond move easily.
(c) Covalent bonds must be broken in semiconductors and
insulators for an electron to be able to move.
(d) 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+
Electrical Conductivity in Metals
If the nuclei are arranged in a perfectly ordered lattice then there should be no
resistance to oppose the flow of current.
Also, an increase in temperature causes an increase in the thermal population of the
higher levels and it should increase the electrical conductivity.
BUT, thermal vibrations of the nuclei increase electrical resistance so conductivity
actually decreases with temperature.
Movement of an electron through (a) a perfect crystal,
(b) a crystal heated to a high temperature, and (c) a
crystal containing atomic level defects. Scattering of
the electrons reduces the mobility and conductivity.
The effect of temperature on the electrical
resistivity of a metal with a perfect crystal
structure. The slope of the curve is the
temperature resistivity coefficient.
For most metals, resistivity increases approx.
linearly with temperature.
ρ = ρ o [1 + α (T − T o )]
ρ is the resistivity at temperature T
(measured in Celsius).
ρο is the reference resistivity at
the reference temperature To
(usually taken to be 20 oC).
α is a parameter called
temperature coefficient of
resistivity.
Platinum Resistance Thermometer: A resistance thermometer (conductor with a fixed
cross section), which measures temperature by measuring the change in the resistance of a
conductor, is made of platinum and has a resistance of 50.0Ω at 20oC. When the device is
immersed in a vessel containing melting indium, its resistance increases to 76.8Ω. Find the
melting point of Indium. Use α=3.92x10-3(oC)-1 for platinum.
Solution:
Ro=50.0 Ω.
To=20oC.
R=76.8 Ω.
R − Ro
76.8Ω − 50.0Ω
o
=
T − To =
=
137
C
−
1
α × R o 3.92 × 10 −3 o C × [50.0Ω ]
[
T = 157 o C
( )
]
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.
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
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)
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.