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8/14/2019 PC Chapter 43
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Chapter 43
Molecules and Solids
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Molecular Bonds
Introduction The bonding mechanisms in a molecule
are fundamentally due to electric forces
The forces are related to a potential
energy function
A stable molecule would be expected at
a configuration for which the potential
energy function has its minimum value
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Molecular Bonds Feature 1 The force between atoms is repulsive at
very small separation distances This repulsion is partially electrostatic and
partially due to the exclusion principle Due to the exclusion principle, some
electrons in overlapping shells are forced
into higher energy states The energy of the system increases as if a
repulsive force existed between the atoms
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Molecular Bonds Feature 2 The force between the atoms is
attractive at larger distances The attractive force (for many molecules) is
due to the dipole-dipole interactionbetween charge distributions within theatoms of the molecules
The electric fields of two dipoles willinteract, resulting in a force between thedipoles
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Potential Energy Function The potential energy for a system of two
atoms can be expressed in the form
ris the internuclear separation distance
m and n are small integers A is associated with the attractive force B is associated with the repulsive force
( )n m
A BU rr r
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Potential Energy Function,
Graph At large separations,
the slope of the curve ispositive Corresponds to a net
attractive force
At the equilibriumseparation distance, theattractive and repulsive
forces just balance At this point the potential
energy is a minimum The slope is zero
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Molecular Bonds Types Simplified models of molecular bonding
include Ionic
Covalent
van der Waals
Hydrogen
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Ionic Bonding Ionic bonding occurs when two atoms
combine in such a way that one or more
outer electrons are transferred from oneatom to the other
Ionic bonds are fundamentally caused
by the Coulomb attraction betweenoppositely charged ions
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Ionic Bonding, cont. When an electron makes a transition
from the E= 0 to a negative energy
state, energy is released The amount of this energy is called the
electron affinity of the atom
The dissociation energy is the amountof energy needed to break themolecular bonds and produce neutralatoms
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Ionic Bonding, NaCl Example
The graph shows the total energy of the molecule
vs the internuclear distance
The minimum energy is at the equilibrium
separation distance
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Ionic Bonding,final The energy of the molecule is lower
than the energy of the system of two
neutral atoms It is said that it is energetically
favorable for the molecule to form The system of two atoms can reduce its
energy by transferring energy out of the
system and forming a molecule
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Covalent Bonding A covalent bond between two atoms is
one in which electrons supplied by
either one or both atoms are shared bythe two atoms
Covalent bonds can be described interms of atomic wave functions
The example will be two hydrogenatoms forming H2
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Wave Function Two Atoms
Far Apart Each atom has a
wave function
There is little overlap
between the wavefunctions of the two
atoms
1 31( ) or as
o
r ea
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Wave Function Molecule The two atoms are
brought close together
The wave functionsoverlap and form the
compound wave shown
The probability
amplitude is larger
between the atoms than
on either side
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Active Figure 43.3
(SLIDESHOW MODE ONLY)
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Covalent Bonding, Final The probability is higher that the electrons
associated with the atoms will be located
between them This can be modeled as if there were a fixed
negative charge between the atoms, exerting
attractive Coulomb forces on both nuclei
The result is an overall attractive force
between the atoms, resulting in the covalent
bond
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Van der Waals Bonding Two neutral molecules are attracted to each
other by weak electrostatic forces called vander Waals forces Atoms that do not form ionic or covalent bonds are
also attracted to each other by van der Waalsforces
The van der Waals force is due to the fact
that the molecule has a charge distributionwith positive and negative centers at differentpositions in the molecule
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Van der Waals Bonding, cont. As a result of this charge distribution,
the molecule may act as an electric
dipole Because of the dipole electric fields, two
molecules can interact such that there
is an attractive force between them Remember, this occurs even though the
molecules are electrically neutral
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Types of Van der Waals
Forces Dipole-dipole force
An interaction between two molecules each
having a permanent electric dipole moment Dipole-induced dipole force
A polar molecule having a permanent
dipole moment induces a dipole moment ina nonpolar molecule
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Types of Van der Waals
Forces, cont. Dispersion force
An attractive force occurs between two nonpolarmolecules
The interaction results from the fact that, althoughthe average dipole moment of a nonpolarmolecule is zero, the average of the square of thedipole moment is nonzero because of charge
fluctuations The two nonpolar molecules tend to have dipole
moments that are correlated in time so as toproduce van der Waals forces
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Hydrogen Bonding In addition to covalent bonds, a
hydrogen atom in a molecule can also
form a hydrogen bond Using water (H2O) as an example
There are two covalent bonds in themolecule
The electrons from the hydrogen atoms aremore likely to be found near the oxygenatom than the hydrogen atoms
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Hydrogen Bonding
H2O Example, cont.
This leaves essentially bare protons at
the positions of the hydrogen atoms
The negative end of another molecule
can come very close to the proton
This bond is strong enough to form a
solid crystalline structure
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Hydrogen Bonding, Final The hydrogen bond
is relatively weakcompared with otherelectrical bonds
Hydrogen bonding isa critical mechanismfor the linking ofbiological moleculesand polymers
DNA is an example
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Energy States of Molecules The energy of a molecule (assume one
in a gaseous phase) can be divided into
four categories Electronic energy
Due to the interactions between the moleculeselectrons and nuclei
Translational energy Due to the motion of the molecules center of
mass through space
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Energy States of Molecules, 2 Categories, cont.
Rotational energy
Due to the rotation of the molecule about itscenter of mass
Vibrational energy Due to the vibration of the molecules constituent
atoms The total energy of the molecule is the
sum of the energies in these categories: E= Eel + Etrans + Erot + Evib
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Spectra of Molecules The translational energy is unrelated to
internal structure and therefore
unimportant to the interpretation of themolecules spectrum
By analyzing its rotational and
vibrational energy states, significantinformation about molecular spectra
can be found
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Rotational Motion of
Molecules A diatomic model will be
used, but the same
ideas can be extended
to polyatomic molecules A diatomic molecule
aligned along anxaxis
has only two rotational
degrees of freedom Corresponding to
rotations about the yand
xaxes
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Rotational Motion of
Molecules, Energy The rotational energy is given by
I is the moment of inertia of the molecule
is called the reduced mass of the molecule
2
rot
1
2
IE
2 21 2
1 2
Im m
rr
m m
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Rotational Motion of Molecules,
Angular Momentum Classically, the value of the molecules
angular momentum can have any value
L = I Quantum mechanics restricts the values
of the angular momentum to
J is an integer called the rotationalquantum number
1 0 1 2 , , ,L J J J h K
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Rotational Kinetic Energy of
Molecules, Allowed Levels The allowed values are
The rotational kinetic energy isquantized and depends on its moment
of inertia As Jincreases, the states become
farther apart
2
rot 1 0 1 22 , , ,IE J J J
hK
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Allowed Levels, cont. For most molecules,
transitions result in radiationthat is in the microwave
region Allowed transitions are given
by the condition
Jis the number of the higher
state
2 2
rot 24
1 2 3
I I
, , ,
h
E J J
J
h
K
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Active Figure 43.5
(SLIDESHOW MODE ONLY)
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Sample Transitions
CO Example
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Vibrational Motion of
Molecules A molecule can be
considered to be aflexible structurewhere the atoms arebonded by effectivesprings
Therefore, themolecule can bemodeled as a simpleharmonic oscillator
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Vibrational Motion of
Molecules, Potential Energy A plot of the
potential energyfunction
ro is the equilibrium
atomic separation For separations
close to ro, theshape closelyresembles aparabola
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Vibrational Energy Classical mechanics describes the frequency
of vibration of a simple harmonic oscillator
Quantum mechanics predicts that a moleculewill vibrate in quantized states
The vibrational and quantized vibrational
energy can be altered if the molecule
acquires energy of the proper value to cause
a transition between quantized states
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Vibrational Energy, cont. The allowed vibrational energies are
vis an integer called the vibrational quantum
number
When v= 0, the molecules ground state
energy is h The accompanying vibration is always present,
even if the molecule is not excited
vib
10 1 2
2 , , ,E v h v
K
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Vibrational Energy, Final The allowed vibrational
energies can be expressedas
Allowed transitions arev = 1
The energy betweenstates is Evib = h
vib
1
2 2
0 1 2 , , ,
h kE v
v
K
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Some Values for Diatomic
Molecules
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Molecular Spectra In general, a molecule vibrates and
rotates simultaneously
To a first approximation, these motionsare independent of each other
The total energy is the sum of the
energies for these two motions:
21
12 2
I
E v h J J
h
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Molecular Energy-Level
Diagram For each allowed state ofv,
there is a complete set of
levels corresponding to the
allowed values ofJ The energy separation
between successive
rotational levels is much
smaller than betweensuccessive vibrational levels
Most molecules at ordinary
temperatures vibrate at v= 0
level
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Molecular Absorption
Spectrum
The spectrum consists of two groups of lines
One group to the right of center satisfying the selection rulesJ= +1 and v= +1
The other group to the left of center satisfying the selection
rules J= -1 and v= +1
Adjacent lines are separated by /2I
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Active Figure 43.8
(SLIDESHOW MODE ONLY)
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Absorption Spectrum of HCl
It fits the predicted pattern very well
A peculiarity shows, each line is split into a doublet Two chlorine isotopes were present in the same
sample
Because of their different masses, different Is are
present in the sample
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Intensity of Spectral Lines The intensity is determined by the
product of two functions ofJ The first function is the number of available
states for a given value ofJ There are 2J+ 1 states available
The second function is the Boltzmannfactor
2B( 1)/(2 )IJ J k T
on n e
h
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Intensity of Spectral Lines,
cont Taking into account both factors by
multiplying them,
The 2J+ 1 term increases with J
The exponential term decreases
This is in good agreement with theobserved envelope of the spectral lines
2
B( 1)/(2 )2 1 II J J k T J e h
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Bonding in Solids Bonds in solids can be of the following
types Ionic Covalent
Metallic
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Ionic Bonds in Solids The dominant interaction between ions
is through the Coulomb force
Many crystals are formed by ionicbonding
Multiple interactions occur among
nearest-neighbor atoms
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Ionic Bonds in Solids, 2 The net effect of all the interactions is a
negative electric potential energy
is a dimensionless number known as the
Madelung constant The value of depends only on the
crystalline structure of the solid
2
attractive e
eUk
r
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Ionic Bonds, NaCl Example
The crystalline structure is shown (a) Each positive sodium ion is surrounded by six negative
chlorine ions (b) Each chlorine ion is surrounded by six sodium ions (c) = 1.747 6 for the NaCl structure
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Total Energy in a
Crystalline Solid As the constituent ions of a crystal are
brought close together, a repulsive
force exists The potential energy term B/rm accounts
for this repulsive force
This repulsive force is a result ofelectrostatic forces and the exclusion
principle
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Total Energy in a Crystalline
Solid, cont The total potential energy
of the crystal is
The minimum value, Uo, is
called the ionic cohesive
energy of the solid It represents the energy
needed to separate the solid
into a collection of isolated
positive and negative ions
m
2
etotalr
B
r
ekU +=
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Properties of Ionic Crystals They form relatively stable, hard
crystals
They are poor electrical conductors They contain no free electrons
Each electron is bound tightly to one of the
ions They have high melting points
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More Properties of Ionic
Crystals They are transparent to visible radiation, but
absorb strongly in the infrared region
The shells formed by the electrons are so tightlybound that visible light does not possess sufficient
energy to promote electrons to the next allowed
shell
Infrared is absorbed strongly because the
vibrations of the ions have natural resonant
frequencies in the low-energy infrared region
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Final Properties of Ionic
Crystals Many are quite soluble in polar liquids
Water is an example of a polar liquid
The polar solvent molecules exert anattractive electric force on the charged ions
This breaks the ionic bonds and dissolves
the solid
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Properties of Solids with
Covalent Bonds Properties include
Usually very hard
Due to the large atomic cohesive energies High bond energies
High melting points
Good electrical conductors
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Cohesive Energies for Some
Covalent Solids
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Covalent Bond Example
Diamond
Each carbon atom in a diamond crystal iscovalently bonded to four other carbon atoms
This forms a tetrahedral structure
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Another Carbon Example --
Buckyballs
Carbon can form
many different
structures The large hollow
structure is called
buckminsterfullerene Also known as a
buckyball
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Metallic Solids
Metallic bonds are generally weaker
than ionic or covalent bonds
The outer electrons in the atoms of ametal are relatively free to move
through the material
The number of such mobile electrons ina metal is large
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Metallic Solids, cont.
The metallic structure can
be viewed as a sea or
gas of nearly free
electrons surrounding alattice of positive ions
The bonding mechanism
is the attractive force
between the entirecollection of positive ions
and the electron gas
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Properties of Metallic Solids
Light interacts strongly with the free
electrons in metals
Visible light is absorbed and re-emittedquite close to the surface
This accounts for the shiny nature of metal
surfaces
High electrical conductivity
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More Properties of Metallic
Solids
The metallic bond is nondirectional This allows many different types of metal
atoms to be dissolved in a host metal invarying amounts The resulting solid solutions, oralloys, may
be designed to have particular properties
Metals tend to bend when stretched Due to the bonding being between all of
the electrons and all of the positive ions
f
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Free-Electron Theory of
Metals
The quantum-based free-electron theory of
electrical conduction in metals takes into
account the wave nature of the electrons The model is that the outer-shell electrons
are free to move through the metal, but are
trapped within a three-dimensional box
formed by the metal surfaces Each electron can be represented as a
particle in a box
F i Di Di ib i
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Fermi-Dirac Distribution
Function
Applying statistical physics to a
collection of particles can relate
microscopic properties to macroscopicproperties
For electrons, quantum statistics
requires that each state of the systemcan be occupied by only two electrons
F i Di Di t ib ti
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Fermi-Dirac Distribution
Function, cont.
The probability that a particular statehaving energy Eis occupied by one of
the electrons in a solid is given by
(E) is called the Fermi-Diracdistribution function EF is called the Fermi energy
B( )
1( )
1
FE E k T E
e
F i Di Di t ib ti
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Fermi-Dirac Distribution
Function at T= 0
At T= 0, all states
having energies less
than the Fermienergy are occupied
All states having
energies greater
than the Fermienergy are vacant
F i Di Di t ib ti
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Fermi-Dirac Distribution
Function at T> 0
As Tincreases, the
distribution rounds
off slightly States near and
below EF lose
population
States near and
above EF gain
population
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Active Figure 43.15
(SLIDESHOW MODE ONLY)
El t P ti l i
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Electrons as a Particle in a
Three-Dimensional Box
The energy levels for the electrons are
very close together
The density-of-states function givesthe number of allowed states per unit
volume that have energies between E
and dE:
F B
3 2 1 2
( )3
8 2( )
1e
E E k T
m E dE g E dE
h e
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Fermi Energy at T= 0 K
The Fermi energy at T= 0 K is
The order of magnitude of the Fermi
energy for metals is about 5 eV
The average energy of a free electron in
a metal at 0 K is Eav = (3/5) EF
2 32
F
3
(0) 2 8e
e
h n
E m
F i E i f S
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Fermi Energies for Some
Metals
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Wave Functions of Solids
To make the model of a metal more
complete, the contributions of the
parent atoms that form the crystal mustbe incorporated
Two wave functions are valid for an
atom with atomic numberA and asingle s electron outside a closed shell:
( ) ( ) ( ) ( ) o oZr na Zr nas s r A r e r A r e
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Combined Wave Functions
The wave functionscan combine in thevarious ways shown
s- + s
- is equivalent to
s+ + s
+
These two possiblecombinations of wavefunctions representtwo possible states ofthe two-atom system
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Splitting of Energy Levels The states are split into
two energy levels due tothe two ways of combiningthe wave functions
The energy difference isrelatively small, so the twostates are close togetheron an energy scale
For large values ofr, theelectron clouds do notoverlap and there is nosplitting of the energylevel
S litti f E
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Splitting of Energy
Levels, cont.
As the number ofatoms increases,the number of
combinations inwhich the wavefunctions combineincreases
Each combinationcorresponds to adifferent energylevel
Splitting of Energy
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Splitting of Energy
Levels, final
When this splitting is
extended to the large
number of atoms
present in a solid, thereis a large number of
levels of varying energy
These levels are so
closely spaced they canbe thought of as a band
of energy levels
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Energy Bands in a Crystal
In general, a crystalline solidwill have a large number ofallowed energy bands
The white areas representenergy gaps, correspondingto forbidden energies
Some bands exhibit anoverlap
Blue represents filled bandsand gold represents emptybands in this example ofsodium
Electrical Conduction
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Electrical Conduction
Classes of Materials
Good electrical conductors contain a highdensity of free charge carriers
The density of charge carriers in an insulatoris nearly zero
Semiconductors are materials with a chargedensity between those of insulators andconductors
These classes can be discussed in terms of amodel based on energy bands
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Metals
To be a good conductor, the charge carriersin a material must be free to move inresponse to an electric field We will consider electrons as the charge carriers
The motion of electrons in response to anelectric field represents an increase in theenergy of the system
When an electric field is applied to aconductor, the electrons move up to anavailable higher energy state
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Metals Energy Bands
At T= 0, the Fermi energylies in the middle of the band All levels below EF are filled
and those above are empty If a potential difference is
applied to the metal,electrons having energiesnearEF require only a small
amount of additional energyfrom the applied field toreach nearby higher energylevels
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Metals As Good Conductors
The electrons in a metal experiencing
only a small applied electric field are
free to move because there are manyempty levels available close to the
occupied energy level
This shows that metals are goodconductors
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Insulators
There are no available states that lie close inenergy into which electrons can move upwardin response to an electric field
Although an insulator has many vacant statesin the conduction band, these states areseparated from the filled band by a largeenergy gap
Only a few electrons can occupy the higherstates, so the overall electrical conductivity isvery small
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Insulator Energy Bands
The valence band isfilled and the conductionband is empty at T= 0
The Fermi energy liessomewhere in theenergy gap
At room temperature,very few electrons
would be thermallyexcited into theconduction band
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Semiconductors
The band structure
of a semiconductor
is like that of an
insulator with a
smaller energy gap
Typical energy gap
values are shown inthe table
Semiconductors
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Semiconductors
Energy Bands
Appreciable numbers
of electrons are
thermally excited into
the conduction band
A small applied
potential difference can
easily raise the energyof the electrons into the
conduction band
Semiconductors
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Semiconductors
Movement of Charges
Charge carriers in asemiconductor canbe positive,
negative, or both When an electron
moves into theconduction band, it
leaves behind avacant site, called ahole
Semiconductors
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Semiconductors
Movement of Charges, cont.
The holes act as charge carriers Electrons can transfer into a hole, leaving
another hole at its original site
The net effect can be viewed as theholes migrating through the material inthe direction opposite the direction of
the electrons The hole behaves as if it were a particle
with charge +e
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Intrinsic Semiconductors
A pure semiconductor material
containing only one element is called an
intrinsic semiconductor It will have equal numbers of conduction
electrons and holes
Such combinations of charges are calledelectron-hole pairs
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Doped Semiconductors
Impurities can be added to asemiconductor
This process is called doping Doping
Modifies the band structure of thesemiconductor
Modifies its resistivity Can be used to control the conductivity of
the semiconductor
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n-Type Semiconductors
An impurity can addan electron to thestructure
This impurity would bereferred to as a donoratom
Semiconductorsdoped with donoratoms are called n-typesemiconductors
n-Type Semiconductors
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n-Type Semiconductors,
Energy Levels
The energy level ofthe extra electron is
just below the
conduction band The electron of the
donor atom canmove into the
conduction band asa result of a smallamount of energy
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p-Type Semiconductors
An impurity can add a holeto the structure This is an electron
deficiency
This impurity would bereferred to as a acceptoratom
Semiconductors doped
with acceptor atoms arecalled p-typesemiconductors
p-Type Semiconductors
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p-Type Semiconductors,
Energy Levels
The energy level of thehole is just above thevalence band
An electron from thevalence band can fill thehole with an addition of asmall amount of energy
A hole is left behind in
the valance band This hole can carry
current in the presence ofan electric field
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Extrinsic Semiconductors
When conduction in a
semiconductor is the result of
acceptor or donor impurities, thematerial is called an extrinsic
semiconductor
Doping densities range from 1013
to1019 cm-3
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Semiconductor Devices
Many electronic devices are based on
semiconductors
These devices include Junction diode
Light-emitting and light-absorbing diodes
Transistor Integrated Circuit
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The Junction Diode
Ap-type semiconductor is joined to an
n-type
This forms a p-n junction Ajunction diode is a device based on
a singlep-n junction
The role of the diode is to pass currentin one direction, but not the other
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The Junction Diode, 2
The junction has three
distinct regions ap region
an n region a depletion region
The depletion region is
caused by the diffusion of
electrons to fill holes This can be modeled as if
the holes being filled were
diffusing to the n region
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Because the two sides of the depletionregion each carry a net charge, aninternal electric field exists in thedepletion region
This internal field creates an internalpotential difference that prevents further
diffusion and ensures zero current inthe junction when no potentialdifference is applied
The Junction Diode, 3
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Junction Diode, Biasing
A diode is forward biasedwhen thep side isconnected to the positive terminal of a battery This decreases the internal potential difference
which results in a current that increasesexponentially
A diode is reverse biasedwhen the n side isconnected to the positive terminal of a battery
This increases the internal potential difference andresults in a very small current that quickly reachesa saturation value
Junction Diode:
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Junction Diode:
I-VCharacteristics
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LEDs and Light Absorption
Light emission and absorption in semiconductors issimilar to that in gaseous atoms, with the energy bands ofthe semiconductor taken into account
An electron in the conduction band can recombine with ahole in the valance band and emit a photon
An electron in the valance band can absorb a photon andbe promoted to the conduction band, leaving behind a
hole
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Transistors
A transistor is formed from twop-n
junctions
A narrow n region sandwiched betweentwop regions or a narrowp region between
two n regions
The transistor can be used as An amplifier
A switch
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Integrated Circuits
An integrated circuit is a collection ofinterconnected transistors, diodes,resistors and capacitors fabricated on asingle piece of silicon known as a chip
Integrated circuits Solved the interconnectedness problem
posed by transistors Possess the advantages of miniaturization
and fast response
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Superconductivity
A superconductor expels magnetic
fields from its interior by forming surface
currents Surface currents induced on the
superconductors surface produce a
magnetic field that exactly cancels theexternally applied field
Superconductivity and
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Superconductivity and
Cooper Pairs
Two electrons are bound into a Cooper pair
when they interact via distortions in the array
of lattice atoms so that there is a net
attractive force between them
Cooper pairs act like bosons and do not obey
the exclusion principle
The entire collection of Cooper pairs in ametal can be described by a single wave
function
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Superconductivity, cont.
Under the action of an applied electric field,
the Cooper pairs experience an electric force
and move through the metal
There is no resistance to the movement of
the Cooper pairs They are in the lowest possible energy state
There are no energy states above that of theCooper pairs because of the energy gap
Superconductivity -
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Superconductivity
Critical Temperatures
A new family ofcompounds was found thatwas superconducting at
high temperatures The critical temperature is
the temperature at whichthe electrical resistance of
the material decreases tovirtually zero
Critical temperatures for