Download Lecture_33 - Ch.42- Crystals, Band gaps, Semiconductors

Question on Van der Waals Interactions
Consulted with Professor Lam and Marfatia, who
are more expert.
Indeed, by definition, Van der Waals binding arise
from dipole-dipole interactions.
However, they can occur from induced dipoles in
neutral atoms or molecules. In particular, a QM
dipole fluctuation in one atom can induce a dipole
in another atom and then produce a Van der Waals
force/potential (also mentioned on p.1407 of the
text).
So permanent dipole moments are not required at
all. Also note Van der Waals binding is weak.
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Visual Demonstration
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Rotation and vibrational levels combined
• Figure 42.8 (right) shows an
energy-level diagram for
rotational and vibrational energy
levels of a diatomic molecule.
Figure 42.9 (below) shows a
typical molecular band spectrum.
Question: Do the n values label
rotational or vibrational modes ?
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Clicker question on rotational energy levels
When a diatomic molecule rotates very rapidly, the atoms
that comprise the molecule move slightly farther apart (the
molecule “stretches”). What effect does this have on the
rotational energy levels of the molecule?
A. The low-energy levels are spaced closer together
than if the molecule did not stretch.
B. The low-energy levels are spaced farther apart than if
the molecule did not stretch.
C. The high-energy levels are spaced closer together
than if the molecule did not stretch.
D. The high-energy levels are spaced farther apart than
if the molecule did not stretch.
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Clicker question on rotational modes
When a diatomic molecule rotates very rapidly, the atoms
that comprise the molecule move slightly farther apart (the
molecule “stretches”). What effect does this have on the
rotational energy levels of the molecule?
A. The low-energy levels are spaced closer together
than if the molecule did not stretch.
B. The low-energy levels are spaced farther apart than if
the molecule did not stretch.
C. The high-energy levels are spaced closer together
than if the molecule did not stretch.
I=mr2,; E=L2/2I
D. The high-energy levels are spaced farther apart than
if the molecule did not stretch.
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Clicker question on energy levels of a molecule
This diagram shows the
vibrational and rotational energy
levels of a diatomic molecule.
Consider two possible transitions
for this molecule:
A. n = 2, l = 5 to n = 1, l = 4
B. n = 2, l = 1 to n = 1, l = 0
The energy change is
A. greater for transition A.
B. greater for transition B.
C. the same for both transitions.
D. any of the above, depending on circumstances.
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Clicker question on molecular energy levels
This diagram shows the
vibrational and rotational energy
levels of a diatomic molecule.
Consider two possible transitions
for this molecule:
A. n = 2, l = 5 to n = 1, l = 4
B. n = 2, l = 1 to n = 1, l = 0
The energy change is
A. greater for transition A.
B. greater for transition B.
A. l(l+1)h2; [5(5+1)4(4+1)]h2=14h2; B.
C. the same for both transitions.
[2(2+1)-1(2)]h2=4h2
D. any of the above, depending on circumstances.
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Crystal lattices for crystalline solids
• A crystal lattice is a repeating pattern of
mathematical points. The figure (below)
shows some common types of lattices.
Two common types of binding in crystals:
ionic and covalent.
Also “metallic crystal”, electrons are shared
among many atoms.
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Crystal lattices and structures
Important to note, the crystal
lattice extends to infinity
(repeating each element)
• The figure (below) shows the
diamond structure, covalent
bonding
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Types of crystals
• Figure 42.15 (left) shows a metallic solid, and Figure
42.16 (right) shows an edge dislocation in two
dimensions.
Also note the contrast between crystals and amorphous
materials (e.g. glass), no long range order
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Energy bands
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Energy bands
• Consider a solid with N
identical atoms, push the atoms
closer together. The
wavefunctions of the valence
electrons distort. By Heisenberg
the wave functions are less
localized and extend large
distances if the energy is fixed.
• This leads to formation of
continuous energy bands. There
can be discrete gaps between
the energy bands (band gap)
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According to Pauli,
when all states in a
band are “full”
cannot add more
electrons.
Energy bands
Band gap of 5
eV or more
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Small band gap
e.g. 1.12 eV for
Si, 0.67 for
germanium
Metallic sodium is
an example, gap
is 2.1 eV but even
at T=0, conduction
electrons.
Semiconductors
• A semiconductor has an electrical resistivity that
is intermediate between those of good conductors
and good insulators.
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Resistivity
• One type of thermometer works by measuring the
temperature dependent electrical resistivity of a
sample.
• Question: Which of the following types of
material displays the greatest change in resistivity
for a given temperature change ?
(i) insulator (ii) semiconductor (iii) resistor or
(iv) superconductor.
Ans: semiconductor. The resistivity of
conductors and insulators changes more
gradually with temperature
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Holes
• A hole is a vacancy in a
semiconductor.
• A hole in the valence band
behaves like a positively
charged particle.
• The figure on the right
shows the motions of
electrons in the conduction
band and holes in the
valence band with an
applied electric field.
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Question: What is
the force on a
charged particle in
a uniform E field
Impurities
• Doping is the deliberate addition of impurity
elements.
• In an n-type semiconductor, the conductivity is
due mostly to negative charge (electron) motion.
• In a p-type semiconductor, the conductivity is due
mostly to positive charge (hole) motion.
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n-type and p-type semiconductors
• Figure 42.26 (left) shows an n-type semiconductor, and
Figure 42.27 (right) shows a p-type semiconductor.
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Clicker question on band gaps and materials
At absolute zero (T = 0 K), what is the difference
between a semiconductor and an insulator?
A. The conduction band is empty in a semiconductor
but partially filled in an insulator.
B. The conduction band is partially filled in a
semiconductor but empty in an insulator.
C. The energy gap between the valence and
conduction bands is large in a semiconductor but
small in an insulator.
D. The energy gap between the valence and
conduction bands is small in a semiconductor but
large in an insulator.
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A42.4
At absolute zero (T = 0 K), what is the difference
between a semiconductor and an insulator?
A. The conduction band is empty in a semiconductor
but partially filled in an insulator.
B. The conduction band is partially filled in a
semiconductor but empty in an insulator.
C. The energy gap between the valence and
conduction bands is large in a semiconductor but
small in an insulator.
D. The energy gap between the valence and
conduction bands is small in a semiconductor but
large in an insulator.
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Fermi-Dirac distribution
• The Fermi-Dirac
distribution f(E) is the
probability that a state
with energy E is occupied
by an electron (identical
spin ½ particles)
f (E) =
1
e
(E-E f )/kBT
+1
There is a different distribution
for spin 1 (photons) and spin 0
particles [Bose-Einstein]
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Look at the
change with
temperature
Clicker question on semiconductors
How would you expect the electric conductivity of a
semiconductor to vary with increasing temperature?
A. It should increase, because more electrons are
thermally excited from the valence band into the
conduction band.
B. It should increase, because more electrons are
removed from their parent atoms and added to the
valence band.
C. It should decrease, because the added thermal
energy breaks apart correlated electron pairs.
D. It should decrease, because the atoms in the
crystal will vibrate more and thus block the flow of
electrons.
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A42.6
How would you expect the electric conductivity of a
semiconductor to vary with increasing temperature?
A. It should increase, because more electrons are
thermally excited from the valence band into the
conduction band.
B. It should increase, because more electrons are
removed from their parent atoms and added to the
valence band.
C. It should decrease, because the added thermal
energy breaks apart correlated electron pairs.
D. It should decrease, because the atoms in the
crystal will vibrate more and thus block the flow of
electrons.
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