The Kronig-Penney Model
In the previous section, we discussed qualitatively the spitting of allowed electron
energies as atoms are brought together to form a crystal. The concept of allowed and
forbidden energy bands can be developed more rigorously by considering quantum
mechanics and Schrodinger's wave equation. It may be easy for the reader to "get
lost" in the following derivation, but the result forms the basis for the energy-band
theory of semiconductors.
The potential function of a single, noninteracting, one-electron atom is shown in
Figure 3.5a. Also indicated on the figure are the discrete energy levels allowed for
the electron. Figure 3.5b shows the same type of potential function for the case when
several atoms are in close proximity arranged in a one-dimensional array. The potential
functions of adjacent atoms overlap, and the net potential function for this
case is shown in Figure 3 . 5 c I.t is this potential function we would need to use in
Schrodinger's wave equation to model a one-dimensional single-crystal material.
The solution to Schrodinger's wave equation, for this one-dimensional single crystal
lattice, is made more tractable by considering a simpler potential function.
Figure 3.6 is the one-dimensional Kronig-Penney model of the periodic potential
function, which is used to represent a one-dimensional single-crystal lattice. We need
to solve Schrodinger's wave equation in each region. As with previous quantum mechanical
problems, the more interesting solution occurs for the case when E <>o,
which corresponds to a particle being bound within the crystal. The electrons are
contained in the potential wells, but we have the possibility of tunneling between
wells. The Kronig-Penney model is an idealized periodic potential representing a
one-dimensional single crystal. but the results will illustrate many of the important
features of the quantum behavior of electrons in a periodic lattice.
To obtain the solution to Schrodinger's wave equation, we make use of a mathematical
theorem by Bloch. The theorem states that all one-electron wave functions,
Figure 3.5 (a) Potential function of a single isolated
atom. (b) Overlapping potential functions of adjacent
atoms. (c) Net potential function of a one-dimensional
single crystal.
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energies as atoms are brought together to form a crystal. The concept of allowed and
forbidden energy bands can be developed more rigorously by considering quantum
mechanics and Schrodinger's wave equation. It may be easy for the reader to "get
lost" in the following derivation, but the result forms the basis for the energy-band
theory of semiconductors.
The potential function of a single, noninteracting, one-electron atom is shown in
Figure 3.5a. Also indicated on the figure are the discrete energy levels allowed for
the electron. Figure 3.5b shows the same type of potential function for the case when
several atoms are in close proximity arranged in a one-dimensional array. The potential
functions of adjacent atoms overlap, and the net potential function for this
case is shown in Figure 3 . 5 c I.t is this potential function we would need to use in
Schrodinger's wave equation to model a one-dimensional single-crystal material.
The solution to Schrodinger's wave equation, for this one-dimensional single crystal
lattice, is made more tractable by considering a simpler potential function.
Figure 3.6 is the one-dimensional Kronig-Penney model of the periodic potential
function, which is used to represent a one-dimensional single-crystal lattice. We need
to solve Schrodinger's wave equation in each region. As with previous quantum mechanical
problems, the more interesting solution occurs for the case when E <>o,
which corresponds to a particle being bound within the crystal. The electrons are
contained in the potential wells, but we have the possibility of tunneling between
wells. The Kronig-Penney model is an idealized periodic potential representing a
one-dimensional single crystal. but the results will illustrate many of the important
features of the quantum behavior of electrons in a periodic lattice.
To obtain the solution to Schrodinger's wave equation, we make use of a mathematical
theorem by Bloch. The theorem states that all one-electron wave functions,
Figure 3.5 (a) Potential function of a single isolated
atom. (b) Overlapping potential functions of adjacent
atoms. (c) Net potential function of a one-dimensional
single crystal.
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