Transcript CHAPTER 11: Semiconductor Theory and Devices

CHAPTER 11
Semiconductor Theory and Devices
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11.1 Band Theory of Solids
11.4 Nanotechnology
Categories of Solids
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There are three categories of solids, based on their
conducting properties:
 conductors
 semiconductors
 insulators
Electrical Resistivity
and Conductivity of
Selected Materials
at 293 K
Reviewing the previous table reveals that:
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The electrical conductivity at room temperature is
quite different for each of these three kinds of solids
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Metals and alloys have the highest conductivities
followed by semiconductors
and then by insulators
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.
Band Theory of Solids
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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.
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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.
Band Theory of Solids
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Interaction of the wave functions occurs as the atoms get closer:
Symmetric
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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.
Band Theory of Solids
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In the symmetric case the binding energy is slightly
stronger resulting in a lower energy state.
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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
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An effective way to understand the energy gap in
semiconductors is to model the interaction between
the electrons and the lattice of atoms.
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R. de L. Kronig and W. G. Penney developed a
useful one-dimensional model of the electron lattice
interaction in 1931.
Kronig-Penney Model
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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.
Kronig-Penney Model
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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:
Tunneling
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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
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Matching solutions at the boundary, Kronig and
Penney find
Here K is another wave number.
Kronig-Penney Model
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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.
The Forbidden Zones
Figure 11.5 (a) Plot of the left side of
Equation (11.3) versus ka. 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 κ2ba / 2 = 3π / 2,
showing the forbidden energy zones
(gaps).
Important differences between the KronigPenney 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.
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.
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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.
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.
Valence and Conduction Bands
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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.
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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.
Smaller energy gaps create semiconductors
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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.
11.4: Nanotechnology
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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.
Carbon Nanotubes
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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.
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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
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
Chris Ewels/www.ewels.info
Carbon Nanotubes
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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.
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In a multi-walled nanotube, multiple layers are
nested like the rings in a tree trunk.
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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
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Because of their strength they are used as structural
reinforcements in the manufacture of composite
materials
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(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.
Nanoscale Electronics
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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.
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In addition to the nanotubes already described,
semiconductor wires (for example indium phosphide)
have been fabricated with diameters as small as 5 nm.
Nanoscale Electronics
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These nanowires have been shown useful in
connecting nanoscale transistors and
memory circuits.
These are referred to as nanotransistors.
Graphene
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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
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Quantum dots are nanostructures made of
semiconductor materials.
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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.
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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
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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
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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.
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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.
4-rod Segmented Linear Paul Trap
Trapped Ions
Magnification ×6
1mm
Magnification ×6
×8
100µm