Allowed and Forbidden energy bands
we treated the one-electron, or hydrogen, atom. That analysis showed that the energy of the bound electron is quantized: Only discrete values of electron energy are allowed. The radial probability density for the electron was also determined. This function gives the probability of finding the electron at a particular distance from the nucleus and shows that the electron is not localized at a given radius. We can extrapolate these single-atom results to a crystal and qualitatively derive the concepts of allowed and forbidden energy bands. We can then apply quantum mechanics and Schrodinger's wave equation to the problem of an electron in a single crystal. We find that the electronic energy states occur in hands of allowed Elates that are separated by forbidden energy bands.
Formation of Energy Bands
Figure 3.la shows the radial probability density function for the lowest electron
energy state of the single, non interacting hydrogen atom, and Figure 3.lb shows the
same probability curves for two atoms that are in close proximity to each other. The
wave functions of the two atom electrons overlap, which means that the two electrons
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Formation of Energy Bands
Figure 3.la shows the radial probability density function for the lowest electron
energy state of the single, non interacting hydrogen atom, and Figure 3.lb shows the
same probability curves for two atoms that are in close proximity to each other. The
wave functions of the two atom electrons overlap, which means that the two electrons
will interact. This interaction or perturbation results in the discrete quantized energy
level splitting into two discrete energy levels, schematically shown in Figure 3.1~.
The splitting of the discrete state into two states is consistent with the Pauli exclusion
principle.
A simple analogy of the splitting of energy levels by interacting particles is the
following. Two identical race cars and drivers are far apart on a race track. There is
no interaction between the cars, so they both must provide the same power to
achieve a given speed. However, if one car pulls up close behind the other car, there
is an interaction called draft. The second car will be pulled to an extent by the lead
car. The lead car will therefore require more power to achieve the same speed, since
it is pulling the second car and the second car will require less power since it is
being pulled by the lead car. So there is a "splitting" of power (energy) of the two
interacting race cars. (Keep in mind not to take analogies too literally.)
Now, if we somehow start with a regular periodic arrangement of hydrogen type
atoms that are initially very far apart, and begin pushing the atoms together, the
initial quantized energy level will split into a band of discrete energy levels. This effect
is shown schematically in Figure 3.2, where the parameter ro represents the
equilibrium inter atomic distance in the crystal. At the equilibrium inter atomic distance,
there is a band of allowed energies, but within the allowed band, the energies
are at discrete levels. The Pauli exclusion principle states that the joining of atoms
to form a system (crystal) does not alter the total number of quantum states regardless
of size. However, since no two electrons can have the same quantum number.
the discrete energy must split into a band of energies in order that each electron can
occupy a distinct quantum state.
We have seen previously that, at any energy level, the number of allowed quantum
states is relatively small. In order to accommodate all of the electrons in a crystal,
then, we must have many energy levels within the allowed band. As an example,
suppose that we have a system with 1019 one-electron atoms and also suppose that,
at the equilibrium inter atomic distance, the width of the allowed energy band is I eV.
For simplicity, we assume that each electron in the system occupies a different energy
level and, if the discrete energy states are equidistant, then the energy levels ate
separated by 10-19 eV. This energy difference is extremely small, so that for all practical
purposes, we have a quasi-continuous energy distribution through the allowed
level splitting into two discrete energy levels, schematically shown in Figure 3.1~.
The splitting of the discrete state into two states is consistent with the Pauli exclusion
principle.
A simple analogy of the splitting of energy levels by interacting particles is the
following. Two identical race cars and drivers are far apart on a race track. There is
no interaction between the cars, so they both must provide the same power to
achieve a given speed. However, if one car pulls up close behind the other car, there
is an interaction called draft. The second car will be pulled to an extent by the lead
car. The lead car will therefore require more power to achieve the same speed, since
it is pulling the second car and the second car will require less power since it is
being pulled by the lead car. So there is a "splitting" of power (energy) of the two
interacting race cars. (Keep in mind not to take analogies too literally.)
Now, if we somehow start with a regular periodic arrangement of hydrogen type
atoms that are initially very far apart, and begin pushing the atoms together, the
initial quantized energy level will split into a band of discrete energy levels. This effect
is shown schematically in Figure 3.2, where the parameter ro represents the
equilibrium inter atomic distance in the crystal. At the equilibrium inter atomic distance,
there is a band of allowed energies, but within the allowed band, the energies
are at discrete levels. The Pauli exclusion principle states that the joining of atoms
to form a system (crystal) does not alter the total number of quantum states regardless
of size. However, since no two electrons can have the same quantum number.
the discrete energy must split into a band of energies in order that each electron can
occupy a distinct quantum state.
We have seen previously that, at any energy level, the number of allowed quantum
states is relatively small. In order to accommodate all of the electrons in a crystal,
then, we must have many energy levels within the allowed band. As an example,
suppose that we have a system with 1019 one-electron atoms and also suppose that,
at the equilibrium inter atomic distance, the width of the allowed energy band is I eV.
For simplicity, we assume that each electron in the system occupies a different energy
level and, if the discrete energy states are equidistant, then the energy levels ate
separated by 10-19 eV. This energy difference is extremely small, so that for all practical
purposes, we have a quasi-continuous energy distribution through the allowed
energy band. The fact that 10-19 eV is a very small difference between two energy
states can be seen from the following example.
To calculate the change in kinetic energy of an electron when the velocity changes by a small value
Consider an electron traveling at a velocity of 107 cm/s. Assume the velocity increases by
a value of 1 cm/s. The increase in kinetic energy is given by
states can be seen from the following example.
To calculate the change in kinetic energy of an electron when the velocity changes by a small value
Consider an electron traveling at a velocity of 107 cm/s. Assume the velocity increases by
a value of 1 cm/s. The increase in kinetic energy is given by
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