Band theory of solids

1
 11.1 Band Theory of Solids
 11.2 Semiconductor Theory
 11.3 Semiconductor Devices
 11.4 Nanotechnology
CHAPTER 11
Semiconductor Theory and DevicesSemiconductor Theory and Devices
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

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11.1: Band Theory of Solids
 In Chapter 10 you learned about structural, thermal,
and magnetic properties of solids.
In this chapter we concentrate on electrical
conduction.

3
Categories of Solids
 There are three categories of solids, based on their
conducting properties:
 conductors
 semiconductors
 insulators

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Electrical Resistivity
and Conductivity of
Selected Materials
at 293 K

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Reviewing the previous table reveals that:
 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

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Semiconductor Conduction
 The free-electron model from Chapter 9 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.

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Resistivity vs. Temperature
Figure 11.1: (a) Resistivity versus temperature for a typical conductor. Notice the linear rise in
resistivity with increasing temperature at all but very low temperatures. (b) Resistivity versus
temperature for a typical conductor at very low temperatures. Notice that the curve flattens and
approaches a nonzero resistance as T → 0. (c) Resistivity versus temperature for a typical
semiconductor. The resistivity increases dramatically as T → 0.

8
 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.
Band Theory of Solids

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 Consider initially the known wave functions of two
hydrogen atoms far enough apart so that they do
not interact.

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 Interaction of the wave functions occurs as the atoms get closer:
 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.
Symmetric Antisymmetric

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 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.

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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.
 R. de L. Kronig and W. G. Penney developed a
useful one-dimensional model of the electron lattice
interaction in 1931.

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Kronig-Penney Model
 Kronig and Penney assumed that an electron
experiences an infinite one-dimensional array of finite
potential wells.
 Each potential well models attraction to an atom in the
lattice, so the size of the wells must correspond roughly
to the lattice spacing.

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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:

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Tunneling
 In the region between a < x < a + b the electron can
tunnel through and the wave function loses its
oscillatory solution and becomes exponential:

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Kronig-Penney Model
 Matching solutions at the boundary, Kronig and
Penney find
Here K is another wave number.

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 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.
Kronig-Penney Model

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Figure 11.5 (a) Plot of the left side of
Equation (11.3) versus ka for κ2
ba / 2 =
3π / 2. Allowed energy values must
correspond to the values of k for
for which the plotted
function lies between -1 and +1.
Forbidden values are shaded in light
blue. (b) The corresponding plot of
energy versus ka for κ2
ba / 2 = 3π / 2,
showing the forbidden energy zones
(gaps).
The Forbidden Zones

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Important differences between the Kronig-
Penney model and the single potential well
1) 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.
2) In a real three-dimensional crystal it is appropriate to
speak of a wave vector . The allowed ranges for
constitute what are referred to in solid state theory as
Brillouin zones.
k
r
k
r

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And…
3) 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.

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Band Theory and Conductivity
 Band theory helps us understand what makes a
conductor, insulator, or semiconductor.
1) Good conductors like copper can be understood using the free
electron
2) 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.
3) 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.

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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.

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 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.
Smaller energy gaps create semiconductors

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11.2: Semiconductor Theory
 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 (9.34):
β = (kT)−1
and EF is the Fermi energy.

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Temperature and Resistivity
 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.

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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:
1) 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.
2) 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.

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Clement-Quinnell Equation
 A useful empirical expression developed by Clement
and Quinnell for the temperature variation of standard
carbon resistors is given by
where A, B, and K are constants.

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Test of the Clement-Quinnell Equation
Figure 11.7: (a) An experimental test of the Clement-Quinnell equation, using resistance versus
temperature data for four standard carbon resistors. The fit is quite good up to 1 / T ≈ 0.6,
corresponding to T ≈ 1.6 K. (b) Resistance versus temperature curves for some thermometers used in
research. A-B is an Allen-Bradley carbon resistor of the type used to produce the curves in (a). Speer
is a carbon resistor, and CG is a carbon-glass resistor. Ge 100 and 1000 are germanium resistors.
From G. White, Experimental Techniques in Low Temperature Physics, Oxford: Oxford University
Press (1979).

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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.

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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.

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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.

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Acceptor Levels
 Consider what happens when indium is added to silicon.
 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).
 acceptor levels p-Type semiconductors
 In addition to intrinsic and impurity semiconductors, there are many
compound semiconductors, which consist of equal numbers of
two kinds of atoms.

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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 p-
type carriers dominate in a complex system.

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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.
 This thermoelectric effect is sometimes called the Seebeck
effect.

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The Thomson and Peltier Effects
 In a normal conductor, heat is generated at the rate of I2
R.
But a temperature gradient across the conductor causes
additional heat to be generated.
This is the Thomson Effect.
Here heat is generated if current flows toward the higher
temperature and absorbed if toward the lower.
 The Peltier effect occurs when heat is generated at a
junction between two conductors as current passes
through the junction.

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The 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.

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11.3: Semiconductor Devices
pn-junction Diodes
 Here 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.

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Operation of a pn-junction Diode
Figure 11.12: The operation of a pn-junction diode. (a) This is the no-bias case. The small
thermal electron current (It) is offset by the electron recombination current (Ir). The net
positive current (Inet) is zero. (b) With a DC voltage applied as shown, the diode is in reverse
bias. Now Ir is slightly less than It. Thus there is a small net flow of electrons from p to n and
positive current from n to p. (c) Here the diode is in forward bias. Because current can
readily flow from p to n, Ir can be much greater than It. [Note: In each case, It and Ir are
electron (negative) currents, but Inet indicates positive current.]

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 The diode is an important tool in many kinds of electrical circuits. As an example, consider
the bridge rectifier circuit shown in Figure 11.14. The bridge rectifier is set up so that it
allows current to flow in only one direction through the resistor R when an alternating
current supply is placed across the bridge. The current through the resistor is then a
rectified sine wave of the form
 This is the first step in changing alternating current to direct current. The design of a power
supply can be completed by adding capacitors and resistors in appropriate proportions.
This is an important application, because direct current is needed in many devices and the
current that we get from our wall sockets is alternating current.
Figure 11.14: Circuit diagram for a diode bridge rectifier.
Bridge Rectifiers
(11.10)

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 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.
Zener Diodes
Figure 11.15: A typical I-V curve for a
Zener diode.
Figure 11.16: A Zener diode reference
circuit.

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Light Emitting Diodes
 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.

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Photovoltaic Cells
 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).

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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 pn-junction diodes back to back.
Transistors
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 two-diode model of the npn transistor. (c) The npn transistor symbol used in circuit
diagrams. (d) The pnp transistor symbol used in circuit diagrams.

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 Consider now the npn junction in the circuit shown in Figure 11.23a. If the
emitter is more heavily doped than the base, then there is a heavy flow of
electrons from left to right into the base. The base is made thin enough so that
virtually all of those electrons can pass through the collector and into the output
portion of the circuit. As a result the output current is a very high fraction of the
input current. The key now is to look at the input and output voltages. Because
the base-collector combination is essentially a diode connected in reverse bias,
the voltage on the output side can be made higher than the voltage on the input
side. Recall that the output and input currents are comparable, so the resulting
output power (current × voltage) is much higher than the input power.
Transistors
Figure 11.23: (a) The npn transistor in a voltage amplifier circuit. (b) The circuit has been modified to
put the input between base and ground, thus making a current amplifier. (c) The same circuit as in (b)
using the transistor circuit symbol.

45
Field Effect Transistors (FET)
 The three terminals of the FET are known as the drain,
source, and gate, and these correspond to the
collector, emitter, and base, respectively, of a bipolar
transistor.
Figure 11.25: (a) A schematic of a FET. The two gate regions are connected internally. (b)
The circuit symbol for the FET, assuming the source-to-drain channel is of n-type material
and the gate is p-type. If the channel is p-type and the gate n-type, then the arrow is
reversed. (c) An amplifier circuit containing a FET.

46
Schottky Barriers
 Here a direct contact is made between a metal and a
semiconductor. If the semiconductor is n-type, electrons from it tend
to migrate into the metal, leaving a depleted region within the
semiconductor.
This will happen as long as the work function of the metal is higher
(or lower, in the case of a p-type semiconductor) than that of the
semiconductor.
 The width of the depleted region depends on the properties of the
particular metal and semiconductor being used, but it is typically on
the order of microns. The I-V characteristics of the Schottky barrier
are similar to those of the pn-junction diode. When a p-type
semiconductor is used, the behavior is similar but the depletion
region has a deficit of holes.

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Schottky Barriers
Figure 11.26: (a) Schematic drawing of a typical Schottky-barrier FET. (b) Gain versus
frequency for two different substrate materials, Si and GaAs. From D. A. Fraser,
Physics of Semiconductor Devices, Oxford: Clarendon Press (1979).

48
Semiconductor Lasers
 Like the gas lasers described in Section 10.2, 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.

49
Semiconductor Lasers
 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.

50
Integrated Circuits
 The most important use of all these semiconductor
devices today is not in discrete components, but rather in
integrated circuits called chips.
 Some integrated circuits contain a million or more
components such as resistors, capacitors, and
transistors.
 Two benefits: miniaturization and processing speed.

51
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.

52
11.4: Nanotechnology
 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 .1 nm up to 1 micron (1000 nm).
 This technology has applications in engineering,
chemistry, and the life sciences and, as such, is
interdisciplinary.

53
Carbon Nanotubes
 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.

54
Structure of a Carbon Nanotube
Figure 11.30: Model of a carbon
nanotube, illustrating the hexagonal
carbon pattern superimposed on a
tubelike structure. There is virtually no
limit to the length of the tube. From
http://www.hpc.susx.ac.uk/
~ewels/img/science/nanotubes/.

55
Carbon Nanotubes
 The basic structure shown in Figure 11.30 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.

56
Applications of Nanotubes
 By 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.

57
Nanoscale Electronics
 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.

58
Nanoscale Electronics
 These nanowires have been shown useful in
connecting nanoscale transistors and
memory circuits.
These are referred to as nanotransistors.

59
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.

60
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.
 Possible quantum effects as devices become smaller,
specifically the superposition of quantum states
possibly leading to quantum computing.

Band theory of solids

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