Download 2nd Set of Notes - Idaho State University

REVIEW FOR MIDTERM
Overall we have covered a great deal of basic terminology this first part of the
semester that will hopefully allow you read papers for your projects. The midterm
will be multiple choice, matching, and short answer. The short answer questions
will be something like comparing and contrasting the two band theories we
discussed, indicating how band gaps occur, or describing how the junction potential
forms and how it helps create the voltage in a solar cell. Terminology that you
should know or at least be familiar with for the multiple choice and matching is:
Terminology:
crystalline
amorphous
polycrystalline
molecular orbitals
Solar Cells
Photovoltaics
Semiconductor
p,n junction
n-type doping
p-type doping
crystalline
polycrystalline
amorphous
bandgap
direct
indirect
conductor
insulator
ionic solid
molecular solid
network covalent
metallic
unit cell
primitive cell
Bravais Lattice
Miller Indicies
Coordination number
Giant Molecules
Band Theory
Tight Binding Approximation
Nearly Free Electron Approx.
Molecular Orbitals
LCAO-MO
Bonding Orbital
Antibonding Orbital
Variation Theory
Nonbonding MO
Huckel Molecular Orbital Theory
Secular Determinant
s-band
p-band
HOMO
LUMO
Fermi Level
Fermi Dirac Distribution
Conduction Band
Brillouin Zones
Intrinsic Conductivity
Indirect Semiconductor
Direct Semiconductor
Hall Effect
Conversion Efficiency
Junction Potential Barrier
Inversion Layer Solar Cell
Schottky Barrier Solar Cell
Fill Factor
Voc
Isc
What are the four different types of solids? Each of the four different types may form
crystalline solids. The regular arrangement of the components of a crystalline solid at the
microscopic level produces the beautiful, characteristic shapes of crystals.
There are seven types of unit cells for crystal systems. They differ from one another in
terms of the relationships between the lengths of the sides and the angles.
Cubic
Tetragonal
Orthorhombic
Rhombohedral Monoclinic Triclinic Hexagonal
How may atoms be arranged in the unit cell? It depends on which of the four
arrangements the crystal is set.
Placement
Arrangement
Symbol Atoms/unit cell
Placed at the vertices of the parallelepiped Primitive P
1
Additional atom in center of cell
Innerzeintrein
Additional on opposing faces of the cell
body-centered
I
2
c-centered
C
2
F
4
Atoms on six faces of the unit cell
face-centered
(centering on four faces is forbidden by symmetry)
When these four arrangements are considered together with the seven crystal systems
considered earlier, not all combinations are possible. Some are forbidden by symmetry
and some may be expressible in terms of a smaller unit cell.
The different specifications on a, b, c and and and P, I, C, and F give 14 different
three dimensional space lattices called Bravais lattices. There are 1 triclinic, 2
monoclinic, 4 rhombic, 2 tetragonal, 1 hexagonal, 1 trigonal, and 3 cubic lattices. Can
there be a c-centered cubic lattice? Why or Why not?
Packing plays a roll in the type of unit cell that results, how? Remember that the higher
the coordination number – the number of atoms (or ions) surrounding an atom (or ion) in
a crystal lattice the less free space there is in the unit cell.
The Miller Indicies, an essential method of characterizing a particular plane that cuts
through a crystal is to define the reciprocal of the intersection of the plane with each of
the unit cell axes. For instance if the plane cuts the a axis at ½ a, and the b axis at ½ b,
and the c axis at c, the reciprocal would be 2, 2, 1 You should be able to give me the
Miller indicies corresponding to a particular figure. How can you calculate the spacing
between the planes?
We discussed two models related to band theory. You should be able to compare and
contrast these models.
In the first model we discussed, the Tight Binding Approximation, we discussed how a
crystal could be described in terms of the Molecular Orbitals, MOs, corresponding to a
Giant Molecule. We discussed how it is impossible to solve Schrodinger’s equation
analytically and exactly for any molecule other than H2+ . So we looked at
approximations where MOs could be made from linear combinations of Atomic Orbitals,
and the fact that the total number of molecular orbitals was equal to the total number of
AOs that we used to construct them. So for instance for a hydrogen molecule, combining
two !s atomic orbitals we made a one bonding and one antibonding or a total of two MOs.
MOs can be bonding antibonding or nonbonding.
We looked at the Variation Theory which can be used to combine many AO by
modifying the coefficients to give the best possible Molecular Orbitals. The basic tenant
of this was that the best ground state MO could be constructed by minimizing the
coefficients with respect to the energy. That is the minimum energy solution supplied the
best approximation. This boiled down to solving the secular determinant. Be able to
write out the secular determinant in terms of the and terms for a system with
conjugated pi electrons. Remember that Huckel Theory applies. What are , and S?
Remember we found that as there are more and more atoms in the conjugated system the
energy separation between the Molecular Energy states became smaller and smaller.
The Tight Binding Approximation is essentially an extension of the Huckel molecular
theory to solid crystalline systems such as alkali metals.
First atom supplies an s orbital at a certain energy, the 2nd atom brought up overlaps with
1st atom and forms bonding and antibonding orbitals, the 3rd atom added overlaps it
nearest neighbor (and only slightly its next nearest neighbor and three molecular orbitals
result, one bonding, one antibonding and one in between that is nonbonding, the 4th atom
added leads to the formation of a fourth MO and so on and so forth.
How does the band gap arise? One band is from overlap of s, the other overlap of p
electrons, etc. Where is the Fermi level in regard to the s band for an alkali metal>
What is the shape of the Fermi-Dirac distribution, and how does it explain
semiconductors or insulators.
As the temp. is increased, the tail of the Fermi-Dirac distribution extends across the band
gap and the electrons leave the valence band and populate the empty orbitals of the
upper band This upper band is called, the conduction band and holes are left
behind. The holes and promoted electrons are mobile, and the solid becomes
conducting. So this material is a semiconductor.
What is missing in the tight binding approximation is a accurate description of the
electron movement.
In the Nearly Free Electron Approximation has had success in accounting for many
properties of metals, Electrical conductivity, Thermal conductivity, Light absorption of
metals, Wiedemann-Franz law (universal proportionality). However, until quantum
theory was incorporated there were problems with predicting specific heat.
Quantum Treatment
We assumed that the electrons in a lattice act as if they were particles in a box of length
L. This is just the particle in a box model used to model translational motion in
elementary quantum mechanics. The wavefunction exp (ikx) is the wavefunction of a
free (unbound particle), that is one moving without a potential present, but k is not
quantized as it is in the particle in a box. The edges of the entire crystal at x=0 and x=L
quantizes k. L is very large compared to the lattice spacing and this means that there will
be many many quantum states that are very closely spaced in energy: En = n2 h 2 /
(8mL2) or letting k = n/L k =2  p/h and E = k2 h2/ 82me as the electron is
essentially a free electron. Also there are many many electrons in a macroscopic crystal,
but there are a vastly greater number of k states. k is related to the momentum of the
electons moving in the lattice.
In the Nearly Free Electron Approximation, the band gaps are a result of the fact that the
electrons do not move completely freely, but they are periodically disrupted by an
attractive potential of the atomic nuclei at the lattice points. Then electrons of a certain
wavelength are scattered  = 2k when meets the Bragg condition. 2a sin  = n  =
2n/k. So at 90o the Bragg reflections occur when k = n  / a. Again, electrons of a
wavelength that satisfies the Bragg condition will be scattered and cannot pass through
the lattice. That means that the energies corresponding to the values of k where k = n  /
a will not be allowed. This gives discontinuities in the energy.
Can a force be impressed on the electrons to cause them to drift in a particular direction?
Below the Fermi level all states are occupied, so there cannot be a net change into any
particular direction or particular set of them, but if they can reach an empty or partially
empty band, then the electons can have a net momentum in one direction.
Definitions of Metal, Semiconductor, and Insulator based on resistance are given
generally as follows:
Good Conductors 10- 6
ohm-cm
Semiconductors
10- 2 – 109
Insulators
101 4 – 102 2
We found that there are two general classifications of semiconductors:
Direct Band Gap Semiconductor
Lowest point in the conduction band is at the same k value as the high point in the
conduction band. Here the threshold frequency vg determines the bandgap directly.
Indirect Band Gap Semiconductor
The lowest point in the conduction band is at a different value than the highest point in
the valence band. They have different k values and a phonon is necessary to conserve
momentum.
Solar Cell operation can be thought of in terms of the most basic description of
photovoltaic effect in silicon.
Photons of light interact with the electrons in the p doped silicon and impart
enough energy in some of these electrons to free them. Then at the electrons move to the
pn interface and the junction potential barrier present at the interface acts on these
electrons such that the electrons are segregated to one side and the holes on the other
giving a “light induced potential” that can be used to drive a current through an external
circuit.
Low energy infrared light can be absorbed by the material causing it to vibrate or twist
but the vibrating atoms do not break loose. Light of higher frequency and higher energy
can strike an electron on an atom in the crystalline lattice and excite it to a higher energy
level or if the light is energetic enough, the bound electron can be torn from its place in
the crystal, leaving behind a silicon bond missing an electron. This electron is free to
move about the crystal and is said to be in the crystal’s conduction band, because free
electrons are the means by which electricity flows. There is also a bond missing an
electron and this is often called a hole.
At the interface between the p an n-doped silicon a junction potential is set up such that
there are opposite electric charges facing each other, on either side of a dividing line.
This potential barrier is able to separate electrons and holes generated by light by sending
the electrons to one side of the barrier and the holes to the opposite side of the barrier
where they are less likely to rejoin each other. This separation creates a voltage
difference between one side of the cell and the other side of the solar cell.
Acceptor-doped crystals are known as p-type because it has freely moving holes which
act like positive charges. Donor-doped crystals are known as n-type because it has freely
moving electrons.
Major phenomena limiting cell efficiency are:
1. Reflection from the cells surface, 2. Light that is not energetic enough
3. Light having extra energy beyond needed to put the electron in the conduction band.
4. Light-generated electrons and holes (empty bonds) that randomly encounter each other
and recombine before they can contribute to the cell performance 5. Light generated
electrons and holes that are brought together by surface and material defects in the cells.
6. Resistance to current flow. 7. Self shading resulting from top-surface electrical
contacts. 8. Performance degradation at non-optimal (high or low) operating temps.
Recombination of Electron-Hole Pairs
This is inadvertent recombination of electrons and holes before they can contribute to an
electric current. Usually small losses because solar cells are designed to minimize this.
a) Direct Recombination – happens when an electron and a hole randomly meet heat is
made. It is a problem only before the carriers are separated by light.
b) Indirect Recombination – occurs in many ways don’t just run into each other. This is
the predominant mechanism for recombination. Electrons or holes can run into dangling
bonds, impurities, of fractures (defects).
Resistance is ever-present where the flow of current is accompanied by collisions
between charge carriers and the material the charges are flowing through.
Resistance losses occur in: a) the bulk of the base material. b) the narrow top surface, c)
the interface between the cell and the electrical contacts with the outside. Usually it is
better to highly dope silicon to reduce resistance, but lattice damage and eventually there
can be so many free carriers that the junction no longer exists.
Two predominant causes for efficiency losses at high temp.1) Lattice vibrations interfere
with the free passage of charge carriers. 2) Junction begins to lose its ability to separate
charges. Number 1 occurs even at room temp. Number 2 does not occur until about 300C.
Low Temp Losses
Two effects play a role. As temp falls, thermal energy is less able to free charge carriers
from either dopant atoms or intrinsic silicon. Light generated charge carrier mobility
drops. At very low temp, so little energy that even dopants behave as if they were normal
silicon atoms.
n-type retains it extra electron, p-type its extra holes so junction disappears.
A Schottky Barrier Cell is a solar cell whose junction is induced when a metal contact is
applied to the surface of silicon that has been suitably doped. Due to the electronic
properties of the Si and the metal, the charge carriers are distributed in side the surface
similar to that of a p-n junction. Light striking the Schottky barrier cell, made with n-
type silicon, causes electron-hole pairs to be generated in the silicon. Holes then migrate
into the metal contact and electrons from the metal migrate into the silicon. This causes
a current to flow.
In an Inversion Layer Solar Cell, a layer of SiO deposited on SiO2 coated with p-type
silicon inducing a junction near the top of the p-type silicon. Vapor SiO loses electrons
as it solidifies, so SiO has a positive charge. The positive charge pulls the few free
electrons in the p-type silicon to the interface between the Si and SiO2, causing that
region to behave as if it were n-type Si. The value of the insulating SiO2 layer is to
prevent electrons from entering and neutralizing the SiO.
What is the basic definition of conversion efficiency in a solar cell, and how can the
efficiency be measured? The Power is the current times the voltage. What is the short
circuit current Isc and the open circuit voltage Voc. The definition of the fill factor is:
(measured maximum power output from the cell) / (Isc * Voc)