Electrons in Solids
Energy Bands and Resistance in
Conductors and Semiconductors
What Have We Learned About
Optical Storage?
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Laser light is focused through a (circular) lens
onto the surface of a CD
Light striking a (smooth) land undergoes total
reflection toward the detector: 1
Light striking a (rough) pit undergoes diffuse
reflection so the detector receives less light: 0
The central max of the diffraction pattern must be
no larger than one bit if data is to be resolved
d sin q = 1.22 l
What Have We Learned About
Electrical Storage
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The electric force FE on a charge q0 can be considered due
to an electric field which is produced by other charges qi in
the area
FE = q0 E
If moving a charge between two points requires work (or
does work), the charge gains (or loses) potential energy:
DV = –  E  dx
Capacitors store charge Q in proportion to the voltage V
between the plates:
C = Q/V = C = e0 A/d
Capacitors are used in RAM
What Have We Learned About
Magnetic Storage?
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Two domains magnetized in same direction is a 0
Two domains magnetized in opposite directions is
a1
Direction of magnetization changes at start of new
bit.
Magnetic data is written by running a current
through a loop of wire near the disk
As magnetic data passes by coil of wire, changing
field induces currents according to Faraday’s Law:
e
d B
dB
 iR  
 A
dt
dt
What Have We Learned About
Magnetoresistance?
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Charges traveling through magnetic field experience
magnetic force (provided velocity and field are not
aligned):
FB = qv x B
In a current-carrying wire, this force results in more
frequent collisions and thus an increased resistance:
Magnetoresistance
Electrons traveling through magnetized material undergo
spin-dependent scattering
When magnetic field is present in magnetic superlattice,
scattering of electrons is cut dramatically, greatly
decreasing resistance: Giant magnetoresistanced
Stuff to remember about GMR
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Electrons (and other elementary “particles”) have
intrinsic magnetic fields, identified by spin
The scattering of electrons in a ferromagnetic
material depends on the spin of the electrons
Layers of ferromagnetic material with alternating
directions of magnetization exhibit maximum
resistance
In presence of magnetic field, all layers align and
resistance is minimized
What Have We Learned About
Spectra?
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ENERGY IS QUANTIZED
Different elements have different allowed energies (since
different numbers of protons and electrons provide
different structure of attraction
Light emitted when electrons move from a high energy
level to a lower energy level in an atom will have only
certain, QUANTIZED, allowed energies and wavelengths.
Those wavelengths depend solely on the element emitting
the light and compose the characteristic emission spectrum
for that element
Our Model of the Atom
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If the atom is in the “ground state” of lowest energy, electrons fill the
states in the lowest available energy levels. The first shell has two
possible states, and the second shell has eight possible states. Higher
shells have more states, but we’ll represent them with the eight states
in the first two sub-shells.
Electrons in the outermost shell are called “valence” electrons. We’ll
make them green to distinguish from e- in filled shells
E=0 (unbound)
n=4
n=3
n=2
n=1
Really eight closely spaced
energies, since no two electrons
can occupy same state
Atomic Bonding
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Electrons in an unfilled valence shell are loosely bound
Atoms will form bonds to fill valence shells, either by
sharing valence electrons, borrowing them, or loaning
them
When atoms bond in solids, sharing electrons, each
atom’s energy levels get slightly shifted
E=0 (unbound)
n=4
n=3
n=2
n=1
Electron Motion
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Electrons can only move to an open energy state
If the atom does not absorb energy, electrons can
only move to an open energy state in the same
shell
(drawing is NOT to scale)
E=0 (unbound)
n=4
n=3
n=2
n=1
Electrons in Solids
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The shifted energies in adjacent atoms combine to create a continuous
“band” of allowed energies for each original energy level; each band,
however, has a finite number of states equal to the number in original
atoms
Electrons can move from the locality of one atom to the next only if an
energy state is available within the same band
Conductors & Semiconductors
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In conductors, the valence band is only partially-full, so
electrons can easily move from being near one atom to
being near another
In semiconductors and insulators, the valence band is
completely full, so electrons must gain extra energy to
move
In semiconductors, the band gap between the full valence
band and the empty conduction band is small, so electrons
move easily with only thermal energy
In insulators, the band gap is larger, so electrons will not
easily move into the conduction band
Electrons in an Electric Field
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Conduction electrons move randomly in all directions in
the absence of a field.
If a field is applied, the electric force results in acceleration
in a particular direction:
F=ma= –eE
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 a = –eE/m
As the charges accelerate, the potential energy stored in the
electric field is converted to kinetic energy which can be
converted into heat and light as the electrons collide with
atoms in the wire
This acceleration produces a velocity
v = at = –eEt/m
Electrical current
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If an electric field points from left to right,
positive charge carriers will move toward the
right
while negative charges will move toward the
left
The result of both is a net flow of positive charge
to the right.
Current is the net change in positive charge per
time
DQ
i
Dt
A Basic Circuit
A Basic Circuit
How do we measure all this?
No, Ringo, we use the
deluxe edition of a
multimeter!
Answer the
Ohm’s Law - Before You Start
Questions in Today’s Activity,
Then Continue Through
Question 4
Electrical current
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Look at the movement of charges through a wire
with a potential difference applied - animation
The net velocity is called drift velocity vd
The charge contained in a cross-sectional volume
is Dq=Nq(Volume)=NqADx=NqAvdDt
So the charge per time crossing through A is
DQ NqAvd Dt
i

 NqAvd
Dt
Dt
How do I figure the drift velocity?
The drift velocity is the net velocity of
charge carriers after collisions
 It is not equal to the velocity caused by
acceleration of field on individual charge,
but it certainly is proportional to it.
 Since current is proportional to vd, current is
proportional to electric field
qE
v
t  vd
m

Where am I going with this?
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Electric field is proportional to potential difference or
voltage
So, . . .
i  vd  v  E  V  i  V
 Ohm
gets credit for
being the first to
notice that many
materials displayed
this proportionality.
He defined
resistance as the
ratio of V to i.
Ohm’s Law - worth remembering
V = iR
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Ohm’s law only applies to materials in which the
behavior of charge carriers can be described
statistically by the drift velocity
In short, Ohm’s law only applies to ohmic
materials.
Were both of your resistors made of ohmic
material?
A Good Analogy to Remember
On what does resistance depend?
On What Does Resistance Depend?
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If I increase the length of a wire, the current flow
decreases because of the longer path
If I increase the area of a wire, the current flow
increases because of the wider path
R = r L/A
If I change to a material with better conductivity,
the current flow
increases because charge carriers move better
If I change the temperature, the current flow
changes
Back to the water analogy
Where does the material factor enter?
Finish The Activity
Remember,
i = NqAvd, so
R a 1/N, 1/A, 1/q, 1/vd
Conductors vs.
Semiconductors
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Electrons are free to move in the conduction band
As temperatures rise, electrons collide more with vibrating
atoms; this effect reduces current
Conductors have a partially-filled valence band which
doubles as a conduction band at any temperature
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The primary effect of temperature on resistance is due to
more collisions at higher temperatures
Semiconductors have a completely-filled valence band, so
electrons have to gain energy to enter the conduction band

The primary effect of temperature on resistance is due to this
requirement: the higher the temperature, the more
conduction electrons
What have we learned today?
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When atoms bind, the energy levels in each atom get shifted
slightly (the size of the shift is very small compared with the
energy difference between different levels)
Atoms in solids form bands of closely-spaced energy levels
Electrons fill the lowest-energy bands first
The highest energy band with electrons in it is called the
valence band
If the valence band is not full, electrons can move from atom
to atom. This is the case for conductors
Electrons can not move in filled bands. Thus electrons in
semiconductors must gain energy (usually from thermal
sources) to move from atom to atom.
What else have we learned
today?
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In many, ohmic, materials, current is proportional to
voltage:
V = iR
Resistance is proportional to the length of an object and
inversely proportional to cross-sectional area:
R = rL/A
The constant of proportionality here is called the
resistivity. It is a function of material and temperature.
The resistivity of conductors increases with temperature
since atomic vibrations increase.
The resistivity of semiconductors decreases with
temperature since more conduction electrons exist, and this
effect overshadows the vibrations.