Transcript phys3313-fall12-112612

PHYS 3313 – Section 001
Lecture #21
Monday, Nov. 26, 2012
Dr. Jaehoon Yu
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Isotope Effect
Cooper Pair
Application of Superconductivity
Semi-Conductor
Nano-technology
Monday, Nov. 26, 2012
PHYS 3313-001, Fall 2012
Dr. Jaehoon Yu
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Announcements
• Your presentations are in classes on Dec. 3 and
Dec. 5
– All presentation ppt files must be sent to me by 8pm
Sunday, Dec. 2
• Final exam is 11am – 1:30pm, Monday, Dec. 10
• Reading assignments
– CH10.1, 10.3 and 10.4
• Colloquium this week
Monday, Nov. 26, 2012
PHYS 3313-001, Fall 2012
Dr. Jaehoon Yu
2
Superconductivity
0.5
Isotope effect:
M Tc = constant
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M is the mass of the particular superconducting isotope. Tc is a
bit higher for lighter isotopes.
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Mercury with M=119.5u, Tc=4.815K
Mercury isotope with M=203.4u, Tc=4.146K
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It indicates that the lattice ions are important in the
superconducting state unlike classical conduction where zero
resistivity can only come from non-interaction between electrons
and the lattice
Bardeen-Cooper-Schrieffer theory (electron-phonon interaction):
1) Electrons form Cooper pairs which propagate throughout the
lattice.
2) Propagation is without resistance because the electrons move in
resonance with the lattice vibrations (phonons).
Monday, Nov. 26, 2012
PHYS 3313-001, Fall 2012
Dr. Jaehoon Yu
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Superconductivity
• How is it possible for two electrons to form a coherent pair?
• Consider the crude model.
• Each of the two electrons experiences a net attraction
toward the nearest positive ion.
• Relatively stable electron pairs can be formed. The two
fermions combine to form a boson. Then the collection of
these bosons condense to form the superconducting state.
Monday, Nov. 26, 2012
PHYS 3313-001, Fall 2012
Dr. Jaehoon Yu
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Superconductivity
• Considering just one of the two electrons, the propagation wave that is
created by lattice deformation due to the Coulomb attraction between
the electron and ions is associated with phonon transmission, and the
electron-phonon resonance allows the electron to move without
resistance.
• The complete BCS theory predicts other observed phenomena.
1)
2)
3)
An isotope effect with an exponent very close to 0.5.
é æ T ö2ù
It gives a critical field.
Bc (T ) = Bc ( 0 ) ê1- ç ÷ ú
êë è Tc ø úû
Predicts that metals with higher resistivity in room temperature are better
superconductors
Monday, Nov. 26, 2012
PHYS 3313-001, Fall 2012
Dr. Jaehoon Yu
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Superconductivity
Magnetic flux through a superconducting ring
nh
F = nF0 =
Quantized!!
2e
h
F0 =
» 2.068 ´10 -15 Wb Quantum Fluxoid
2e
3) A conducting electron in the first
excited state with the energy gap Eg
wrt the ground state no longer moves
with 0 resistance. This means that Eg
is the energy needed to break a
Cooper pair apart Eg(0) ≈ 3.54kTc at
T = 0.
Eg (T ) » 1.74Eg ( 0 )
Monday, Nov. 26, 2012
PHYS 3313-001, Fall 2012
Dr. Jaehoon Yu
T
1Tc
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The Search for a Higher Tc
• Keeping materials at extremely low temperatures is very
expensive and requires cumbersome insulation techniques.
• Making liquid He is very hard
Monday, Nov. 26, 2012
PHYS 3313-001, Fall 2012
Dr. Jaehoon Yu
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The Search for a Higher Tc
• The copper oxide superconductors fall into a category of ceramics.
• Most ceramic materials are not easy to mold into convenient shapes.
• There is a regular variation of Tc with n.
Tc of thallium-copper oxide with n = 3
Monday, Nov. 26, 2012
PHYS 3313-001, Fall 2012
Dr. Jaehoon Yu
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The Search for a Higher Tc
• Higher values of n correspond to more stacked layers of
copper and oxygen.
thallium-based superconductor
Monday, Nov. 26, 2012
PHYS 3313-001, Fall 2012
Dr. Jaehoon Yu
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Superconducting Fullerenes
• Another class of exotic superconductors is based on the
organic molecule C60.
• Although pure C60 is not superconducting, the addition of
certain other elements can make it so.
Monday, Nov. 26, 2012
PHYS 3313-001, Fall 2012
Dr. Jaehoon Yu
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Applications of Superconductivity
Josephson junctions:
• The superconductor / insulator / superconductor layer
constitutions.
• In the absence of any applied magnetic or electric field, a DC
current will flow across the junction (DC Josephson effect).
• Junction oscillates with frequency when a voltage is applied
(AC Josephson effect).
2eV
fj =
h
• They are used in devices known as SQUIDs. SQUIDs are
useful in measuring very small amounts of magnetic flux.
Monday, Nov. 26, 2012
PHYS 3313-001, Fall 2012
Dr. Jaehoon Yu
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Applications of Superconductivity
Maglev: Magnetic levitation of trains
 In an electrodynamic (EDS) system,
magnets on the guideway repel the
car to lift it.

Monday, Nov. 26, 2012
In an electromagnetic (EMS)
system, magnets attached to the
bottom of the car lie below the
guideway and are attracted upward
toward the guideway to lift the car.
PHYS 3313-001, Fall 2012
Dr. Jaehoon Yu
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Generation and Transmission of Electricity
• Significant energy savings if the heavy iron cores used
today could be replaced by lighter superconducting magnets.
• Expensive transformers would no longer have to be used to
step up voltage for transmission and down again for use.
• Energy loss rate for transformers is
P = I R=P
2
RV
2
2
lostclear pictures of thetrans
• MRI obtains
body’s soft tissues,
allowing them to detect tumors and other disorders of the
brain, muscles, organs, and connective tissues.
Monday, Nov. 26, 2012
PHYS 3313-001, Fall 2012
Dr. Jaehoon Yu
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CHAPTER 11
Semiconductor Theory and Devices
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11.1
11.2
11.3
11.4
Band Theory of Solids
Semiconductor Theory
Semiconductor Devices
Nanotechnology
It is evident that many years of research by a great many people, both
before and after the discovery of the transistor effect, has been required
to bring our knowledge of semiconductors to its present development.
We were fortunate to be involved at a particularly opportune time and to
add another small step in the control of Nature for the benefit of mankind.
- John Bardeen, 1956 Nobel lecture
Categories of Solids
• There are three categories of solids, based on their
conducting properties:
– conductors
– semiconductors
– insulators
• The electrical conductivity at room temperature is quite
different for each of these three kinds of solids
– Metals and alloys have the highest conductivities
– followed by semiconductors
– and then by insulators
Semiconductor Conduction
• The free-electron model does not apply to semiconductors and
insulators, since these materials simply lack enough free
electrons to conduct in a free-electron mode.
• There is a different conduction mechanism for semiconductors
than for normal conductors.
Band Theory of Solids
• In order to account for decreasing resistivity with increasing
temperature as well as other properties of semiconductors, a new
theory known as the band theory is introduced.
• The essential feature of the band theory is that the allowed energy
states for electrons are nearly continuous over certain ranges, called
energy bands, with forbidden energy gaps between the bands.
• Consider initially the known wave functions of two hydrogen atoms
far enough apart so that they do not interact.
Band Theory of Solids
• Interaction of the wave functions occurs as the atoms get closer:
Symmetric
Antisymmetric
• An atom in the symmetric state has a nonzero probability of being
halfway between the two atoms, while an electron in the antisymmetric
state has a zero probability of being at that location.
• In the symmetric case the binding energy is slightly stronger resulting in
a lower energy state.
– Thus there is a splitting of all possible energy levels (1s, 2s, and so on)
• When more atoms are added (as in a real solid), there is a further
splitting of energy levels. With a large number of atoms, the levels are
split into nearly continuous energy bands, with each band consisting of
a number of closely spaced energy levels.
Kronig-Penney Model
• An effective way to understand the energy gap in semiconductors is to
model the interaction between the electrons and the lattice of atoms.
• Kronig and Penney assumed that an electron experiences an infinite onedimensional array of finite potential wells in 1931
• Each potential well models attraction to an atom in the lattice, so the size of
the wells must correspond roughly to the lattice spacing.
Kronig-Penney Model
• Since the electrons are not free their energies are less than
the height V0 of each of the potentials, but the electron is
essentially free in the gap 0 < x < a, where it has a wave
function of the form
where the wave number k is given by the usual relation:
In the region between a < x < a + b the electron can tunnel
through and the wave function loses its oscillatory solution
and becomes exponential:
Kronig-Penney Model
• Matching solutions at the boundary, Kronig and Penney find
Here K is another wave number.
• The left-hand side is limited to values between +1 and −1
for all values of K.
• Plotting this it is observed there exist restricted (shaded)
forbidden zones for solutions.
Important differences between the Kronig-Penney model
and the single potential well
• For an infinite lattice the allowed energies within each band are
continuous rather than discrete. In a real crystal the lattice is not
infinite, but even if chains are thousands of atoms long, the allowed
energies are nearly continuous.
• In a real three-dimensional crystal it is appropriate to speak of a wave
vector k . The allowed ranges for k constitute what are referred to in
solid state theory as Brillouin zones.
• In a real crystal the potential function is more complicated than the
Kronig-Penney squares. Thus, the energy gaps are by no means
uniform in size. The gap sizes may be changed by the introduction of
impurities or imperfections of the lattice.
• These facts concerning the energy gaps are of paramount importance
in understanding the electronic behavior of semiconductors.
Band Theory and Conductivity
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Band theory helps us understand what makes a
conductor, insulator, or semiconductor.
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Good conductors like copper can be understood using the free
electron.
It is also possible to make a conductor using a material with its
highest band filled, in which case no electron in that band can
be considered free.
If this filled band overlaps with the next higher band, however
(so that effectively there is no gap between these two bands)
then an applied electric field can make an electron from the
filled band jump to the higher level.
This allows conduction to take place, although typically
with slightly higher resistance than in normal metals. Such
materials are known as semimetals.
Valence and Conduction Bands
• The band structures of insulators and semiconductors resemble each
other qualitatively. Normally there exists in both insulators and
semiconductors a filled energy band (referred to as the valence
band) separated from the next higher band (referred to as the
conduction band) by an energy gap.
• If this gap is at least several electron volts, the material is an
insulator. It is too difficult for an applied field to overcome that large
an energy gap, and thermal excitations lack the energy to promote
sufficient numbers of electrons to the conduction band.
• For energy gaps smaller than about 1 electron volt, it is possible for
enough electrons to be excited thermally into the conduction band,
so that an applied electric field can produce a modest current.
The result is a semiconductor.
Semiconductor Theory
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At T = 0 we expect all of the atoms in a solid to be in the ground state. The
distribution of electrons (fermions) at the various energy levels is governed by the
Fermi-Dirac distribution of Equation:
1
FFD =
exp éë b ( E - EF ) ùû +1
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β = (kT)−1 and EF is the Fermi energy.
When the temperature is increased from T = 0, more and more atoms are found in
excited states.
The increased number of electrons in excited states explains the temperature
dependence of the resistivity of semiconductors.
Only those electrons that have jumped from the valence band to the conduction
band are available to participate in the conduction process in a semiconductor.
More and more electrons are found in the conduction band as the temperature is
increased, and the resistivity of the semiconductor therefore decreases.
Some Observations
• Although it is not possible to use the Fermi-Dirac factor to derive an
exact expression for the resistivity of a semiconductor as a function of
temperature, some observations follow:
• The energy E in the exponential factor makes it clear why the band gap
is so crucial. An increase in the band gap by a factor of 10 (say from 1
eV to 10 eV) will, for a given temperature, increase the value of
exp(βE) by a factor of exp(9βE).
• This generally makes the factor FFD so small
that the material has to be an insulator.
• Based on this analysis, the resistance of a semiconductor is expected
to decrease exponentially with increasing temperature.
• This is approximately true—although not exactly, because the
function FFD is not a simple exponential, and because the band
gap does vary somewhat with temperature.
Holes and Intrinsic Semiconductors
• When electrons move into the conduction band, they leave behind
vacancies in the valence band. These vacancies are called holes.
Because holes represent the absence of negative charges, it is useful
to think of them as positive charges.
• Whereas the electrons move in a direction opposite to the applied
electric field, the holes move in the direction of the electric field.
• A semiconductor in which there is a balance between the number of
electrons in the conduction band and the number of holes in the
valence band is called an intrinsic semiconductor.
• Examples of intrinsic semiconductors include pure carbon and
germanium.
Impurity Semiconductor
• It is possible to fine-tune a semiconductor’s properties by adding a
small amount of another material, called a dopant, to the
semiconductor creating what is a called an impurity semiconductor.
• As an example, silicon has four electrons in its outermost shell (this
corresponds to the valence band) and arsenic has five.
• Thus while four of arsenic’s outer-shell electrons participate in
covalent bonding with its nearest neighbors (just as another silicon
atom would), the fifth electron is very weakly bound.
• It takes only about 0.05 eV to move this extra electron into the
conduction band.
• The effect is that adding only a small amount of arsenic to silicon
greatly increases the electrical conductivity.
n-type Semiconductor
• The addition of arsenic to silicon creates what is
known as an n-type semiconductor (n for negative),
because it is the electrons close to the conduction
band that will eventually carry electrical current.
• The new arsenic energy levels just below the
conduction band are called donor levels because
an electron there is easily donated to the
conduction band.
Acceptor Levels
• Consider what happens when indium is added to silicon.
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Indium has one less electron in its outer shell than silicon. The result is one extra
hole per indium atom. The existence of these holes creates extra energy levels
just above the valence band, because it takes relatively little energy to move
another electron into a hole
Those new indium levels are called acceptor levels because they can easily
accept an electron from the valence band. Again, the result is an increased flow
of current (or, equivalently, lower electrical resistance) as the electrons move to
fill holes under an applied electric field
• It is always easier to think in terms of the flow of positive charges
(holes) in the direction of the applied field, so we call this a p-type
semiconductor (p for positive).
• In addition to intrinsic and impurity semiconductors, there are many
compound semiconductors, which consist of equal numbers of two
kinds of atoms.
Thermoelectric Effect
• When there is a temperature gradient in a thermoelectric
material, an electric field appears.
– This happens in a pure metal since we can assume the system
acts as a gas of free electrons.
– As in an ideal gas, the density of free electrons is greater at the
colder end of the wire, and therefore the electrical potential
should be higher at the warmer end and lower at the colder end.
– The free-electron model is not valid for semiconductors;
nevertheless, the conducting properties of a semiconductor are
temperature dependent, as we have seen, and therefore it is
reasonable to believe that semiconductors should exhibit a
thermoelectric effect.
Thermoelectric Effect
• In one dimension the induced electric field E in a
semiconductor is proportional to the temperature gradient,
so that
where Q is called the thermoelectric power.
• The direction of the induced field depends on whether the
semiconductor is p-type or n-type, so the thermoelectric
effect can be used to determine the extent to which n- or ptype carriers dominate in a complex system.
• This thermoelectric effect is sometimes called the
Seebeck effect.
The Thomson, Peltier Effects &
Thermocouple
• Thompson Effect: In a normal conductor, heat is generated at the
rate of I2R. But a temperature gradient across the conductor causes
additional heat to be generated. Here heat is generated if current
flows toward the higher temperature and absorbed if toward the
lower.
• Peltier effect occurs when heat is generated at a junction between
two conductors as current passes through the junction.
• Thermocouple: An important application of the Seebeck
thermoelectric effect is in thermometry. The thermoelectric power of
a given conductor varies as a function of temperature, and the
variation can be quite different for two different conductors. This
difference makes possible the operation of a thermocouple.
Semiconductor Devices
pn-junction Diodes: p-type and n-type semiconductors are
joined together
The principal characteristic of a pn-junction diode is that it
allows current to flow easily in one direction but hardly at all
in the other direction.
We call these situations forward bias and reverse bias,
respectively.
Zener Diodes
• The Zener diode is made to operate under reverse bias once a
sufficiently high voltage has been reached. The I-V curve of a Zener
diode is shown in Figure 11.15. Notice that under reverse bias and
low voltage the current assumes a low negative value, just as in a
normal pn-junction diode. But when a sufficiently large reverse bias
voltage is reached, the current increases at a very high rate.
A Zener diode reference circuit.
A typical I-V curve for a Zener diode.
Light Emitting Diodes (LED)
• Another important kind of diode is the light-emitting diode (LED).
Whenever an electron makes a transition from the conduction band
to the valence band (effectively recombining the electron and hole)
there is a release of energy in the form of a photon (Figure 11.17). In
some materials the energy levels are spaced so that the photon is in
the visible part of the spectrum. In that case, the continuous flow of
current through the LED results in a continuous stream of nearly
monochromatic light.
Figure 11.17: Schematic of an
LED. A photon is released as an
electron falls from the conduction
band to the valence band. The band
gap may be large enough that the
photon will be in the visible portion
of the spectrum.
Photovoltaic Cells
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An exciting application closely related to the LED is the solar cell, also known as the photovoltaic cell.
Simply put, a solar cell takes incoming light energy and turns it into electrical energy. A good way to think
of the solar cell is to consider the LED in reverse (Figure 11.18). A pn-junction diode can absorb a
photon of solar radiation by having an electron make a transition from the valence band to the
conduction band. In doing so, both a conducting electron and a hole have been created. If a circuit is
connected to the pn junction, the holes and electrons will move so as to create an electric current, with
positive current flowing from the p side to the n side. Even though the efficiency of most solar cells is
low, their widespread use could potentially generate significant amounts of electricity. Remember that
the “solar constant” (the energy per unit area of solar radiation reaching the Earth) is over 1400 W/m2,
and more than half of this makes it through the atmosphere to the Earth’s surface. There has been
tremendous progress in recent years toward making solar cells more efficient.
Figure 11.18: (a) Schematic of a photovoltaic cell. Note the similarity to Figure 11.17. (b) A
schematic showing more of the working parts of a real photovoltaic cell. From H. M. Hubbard,
Science 244, 297-303 (21 April 1989).
Transistors
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Another use of semiconductor technology is in the fabrication of transistors,
devices that amplify voltages or currents in many kinds of circuits. The first
transistor was developed in 1948 by John Bardeen, William Shockley, and Walter
Brattain (Nobel Prize, 1956). As an example we consider an npn-junction
transistor, which consists of a thin layer of p-type semiconductor sandwiched
between two n-type semiconductors. The three terminals (one on each
semiconducting material) are known as the collector, emitter, and base. A good
way of thinking of the operation of the npn-junction transistor is to think of two pnjunction diodes back to back.
Figure 11.22: (a) In the npn transistor, the base is a p-type material, and the emitter and collector are n-type. (b) The twodiode model of the npn transistor. (c) The npn transistor symbol used in circuit diagrams. (d) The pnp transistor symbol
used in circuit diagrams.
Semiconductor Lasers
• Like the gas lasers described, semiconductor lasers operate using
population inversion—an artificially high number of electrons in excited
states
• In a semiconductor laser, the band gap determines the energy difference
between the excited state and the ground state
• Semiconductor lasers use injection pumping, where a large forward
current is passed through a diode creating electron-hole pairs, with
electrons in the conduction band and holes in the valence band.
• A photon is emitted when an electron falls back to the valence band
to recombine with the hole.
• Since their development, semiconductor lasers have been used in a
number of applications, most notably in fiber-optics communication.
• One advantage of using semiconductor lasers in this application is their
small size with dimensions typically on the order of 10−4 m.
• Being solid-state devices, they are more robust than gas-filled tubes.
Integrated Circuits
• The most important use of all these semiconductor
devices today is not in discrete components, but
rather in integrated circuits commonly called
chips.
• Some integrated circuits contain a million or more
components such as resistors, capacitors,
transistors, and logic switches.
• Two benefits: miniaturization and processing
speed
Moore’s Law and Computing Power
Figure 11.29: Moore’s law, showing the progress in computing power over a 30-year span,
illustrated here with Intel chip names. The Pentium 4 contains over 50 million transistors. Courtesy
of Intel Corporation. Graph from http://www.intel.com/research/silicon/mooreslaw.htm.
Nanotechnology & Carbon Nanotubes
• Nanotechnology is generally defined as the scientific study
and manufacture of materials on a submicron scale.
• These scales range from single atoms on the order of 0.1
nm up to 1 micron (1000 nm).
• This technology has applications in engineering, chemistry,
and the life sciences and, as such, is interdisciplinary.
• In 1991, following the discovery of C60
buckminsterfullerenes, or “buckyballs,” Japanese physicist
Sumio Iijima discovered a new geometric arrangement of
pure carbon into large molecules.
• In this arrangement, known as a carbon nanotube,
hexagonal arrays of carbon atoms lie along a cylindrical tube
instead of a spherical ball.
Structure of a Carbon Nanotube
There is virtually no limit to the
length of the tube. From Chris
Ewels/www.ewels.info
• leads to two types of nanotubes. A
single-walled nanotube has just the
single shell of hexagons as shown.
• In a multi-walled nanotube, multiple
layers are nested like the rings in a
tree trunk.
• Single-walled nanotubes tend to
have fewer defects, and they are
therefore stronger structurally but
they are also more expensive and
difficult to make.
•
Applications of Nanotubes
• Because of their strength they are used as structural reinforcements in
the manufacture of composite materials
– (batteries in cell-phones use nanotubes in this way)
• Nanotubes have very high electrical and thermal conductivities, and as
such lead to high current densities in high-temperature
superconductors.
• One problem in the development of truly small-scale electronic devices
is that the connecting wires in any circuit need to be as small as
possible, so that they do not overwhelm the nanoscale components
they connect.
• In addition to the nanotubes already described, semiconductor wires
(for example indium phosphide) have been fabricated with diameters
as small as 5 nm.
• These nanowires have been shown useful in connecting nanoscale
transistors and memory circuits. These are referred to as
nanotransistors
Graphene
• A new material called graphene was first isolated in
2004. Graphene is a single layer of hexagonal
carbon, essentially the way a single plane of atoms
appears in common graphite.
• A. Geim and K. Novoselov received the 2010 Nobel
Prize in Physics for “ground-breaking experiments.”
Pure graphene conducts electrons much faster than
other materials at room temperature.
• Graphene transistors may one day result in faster
computing.
Graphene
Figure 11.33 Schematic diagram of graphene-based transistor developed at the University
of Manchester. The passage of a single electron from source to drain registers 1 bit of
information—a 0 or 1 in binary code.
Quantum Dots
• Quantum dots are nanostructures made of semiconductor
materials.
– They are typically only a few nm across, containing up to 1000
atoms.
– Each contains an electron-hole pair ]confined within the dot’s
boundaries, (somewhat analogous to a particle confined to a
potential well discussed in Chapter 6.
• Properties result from the fact that the band gap varies over
a wide range and can be controlled precisely by
manipulating the quantum dot’s size and shape.
– They can be made with band gaps that are nearly continuous
throughout the visible light range (1.8 to 3.1 eV) and beyond.
Nanotechnology and the Life Sciences
• The complex molecules needed for the variety of
life on Earth are themselves examples of nanoscale
design.
• Examples of unusual materials designed for specific
purposes include the molecules that make up
claws, feathers, and even tooth enamel.
Information Science
• It’s possible that current photolithographic techniques for
making computer chips could be extended into the hard-UV
or soft x-ray range, with wavelengths on the order of 1 nm,
to fabricate silicon-based chips on that scale
• In the 1990s physicists learned that it is possible to take
advantage of quantum effects to store and process
information more efficiently than a traditional computer. To
date, such quantum computers have been built in prototype
but not mass-produced.