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6.5: The Tight Binding Approximation

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    50039
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    Each atom in a conductor typically possesses many electrons. We can simplify molecular orbital calculations significantly by neglecting all but a few of the electrons. The basis for discriminating between the electrons is energy. The electrons occupy different atomic orbitals: some electrons require a lot of energy to be pulled out the atom, and others are more weakly bound.

    Our first assumption is that electrons in the deep atomic orbitals do not participate in charge transport. Recall that charge conduction only occurs though states close to the Fermi level. Thus, we are concerned with only the most weakly bound electrons occupying so-called frontier atomic orbitals.

    In this class, we will exclusively consider carbon-based materials. Furthermore, we will only consider carbon in the triangular geometry that yields \(sp^{2}\) hybridized atomic orbitals; see Appendix 4 for a full discussion. In these materials, each carbon atom has one electron in an unhybridized \(p_{z}\) orbital. The unhybridized \(p_{z}\) atomic orbital is the frontier orbital. It is the most weakly bound and also contributes to \(\pi\) molecular orbitals that provide a convenient conduction path for electrons along the molecule. We will assume that the molecular orbitals of the conductor relevant to charge transport are linear combinations of frontier atomic orbitals.

    For example, let's consider the central carbon atom in Figure 6.6.1. Assume that the atom is part of a triangular network and that consequently it contains one electron in a frontier \(p_{z}\) atomic orbital. Let‟s consider the effect of the neighboring carbon atom to the right of the central atom.

    Screenshot 2021-05-24 at 17.06.32.png
    Figure \(\PageIndex{1}\): One carbon atom with a single frontier electron and its neighboring nucleus. The Hamiltonian of the system contains potential terms for each of the two nuclei.

    Assuming the positions of the atoms are fixed, the Hamiltonian of the system consists of a kinetic energy operator, and two Coulombic potential terms: one for the central atom and one for its neighbor:

    \[ H = T + V_{1} + V_{2} \nonumber \]

    Now, consider an integral of the form:

    \[ \left< \phi_{r} | H | \psi \right> \left< \phi_{r} | E | \psi \right> \nonumber \]

    Following Equation (6.4.1), the wavefunction in this two atom system can be written as

    \[ \psi = c_{1}\phi_{1}+c_{2}\phi_{2} \nonumber \]

    We can expand the LHS of Eq. (6.6.2) as follows:

    \[ \left< \phi_{r}|H|\psi \right> = c_{1}\left< \phi_{r}|T+V_{1}|\phi_{1} \right> + c_{1}\left< \phi_{r}|V_{2}|\phi_{1} \right> + c_{2}\left< \phi_{r}|T+V_{2}|\phi_{2} \right> + c_{2}\left< \phi_{r}|V_{1}|\phi_{2} \right> \nonumber \]

    The RHS expands as

    \[ \left< \phi_{r}|E|\psi \right> = c_{1}E\left< \phi_{r}|\phi_{1} \right> + c_{2}E\left< \phi_{r}|\phi_{2} \right> \nonumber \]

    The terms in these expansions are not equally important. We can considerably simplify the calculation by categorizing the various interactions and ignoring the least important.

    Overlap integrals

    First of all, let's define the overlap integral between frontier orbitals on atomic sites s and r:

    \[ S_{sr} = \left< \phi_{s}|\phi_{r} \right> \nonumber \]

    These integrals yield the overlap between atomic orbitals at different sites in the solid. Spatial separation usually ensures that \(S_{sr} \ll 1\) for \(s \neq r\). Of course, for normalized atomic orbitals \(S_{sr} = 1\) for \(s = r\).

    Screenshot 2021-05-24 at 17.19.15.png
    Figure \(\PageIndex{2}\): The overlap between two adjacent atomic orbitals is shaded in yellow. In the tight binding approximation we will assume that the overlap between frontier atomic orbitals on different sites is zero.

    The self-energy

    Next, let's define the self-energy. At a particular atomic site, we have

    \[ T+V_{r}|\phi_{r}\big \rangle = \alpha_{r}|\phi_{r}\big \rangle \nonumber \]

    where \(\alpha_{r}\) is the self energy, i.e.:

    \[ \alpha_{r}= \left< \phi_{r}|T+V_{r}|\phi_{r}\right> \nonumber \]

    The self energy, \(\alpha\), is defined to be negative for an electron in a positively charge nuclear potential. Note that if the interaction between the atoms is weak then the self energy is similar to the energy, E, of the combined system.

    Screenshot 2021-05-24 at 17.24.10.png
    Figure \(\PageIndex{3}\): The interaction between a nucleus and its frontier atomic orbital is known as the self energy.

    Hopping interactions

    Let's define the hopping interaction between different sites s and r:

    \[ \beta_{sr} = \left< \phi_{s}|V_{s}|\phi_{r} \right> \nonumber \]

    The hopping interaction, \(\beta\), is defined to be negative for an electron in a positively charge nuclear potential.

    Screenshot 2021-05-24 at 17.26.07.png
    Figure \(\PageIndex{4}\): The interaction between a nucleus and the neighboring frontier atomic orbital is known as the hopping interaction.

    The remaining interactions

    The remaining interaction considers the interaction of a frontier orbital on one site with the potential on another site. It has the form

    \[ \left< \phi_{r}|V_{s}|\phi_{r} \right> \nonumber \]

    where \(s \neq r\). It may not be immediately evident that this interaction is usually much weaker than the hopping interaction of Equation (6.6.9). But if the individual frontier orbitals decay exponentially with distance as exp[-ka] where a is the spacing between the atoms, then this terms behaves as exp[-2ka] whereas the hopping term and overlap integral \(S_{sr}\) for \(s \neq r\) both follow exp[-ka].

    Consequently, we will neglect this interaction.

    Thus, Equation (6.6.2) can be re-written for r = 1 and r = 2 as

    \[ c_{1}\alpha_{1}+c_{2}\beta_{12}+c_{2}\alpha_{2}S_{12} = c_{1}E+c_{2}ES_{12} \nonumber \]

    \( c_{2}\beta_{21}+c_{1}\alpha_{1}S_{21}+c_{2}\alpha_{2} = c_{1}ES_{21}+c_{2}E \)

    Terms containing only the self energy or energy, E, of the combined system are large. The small terms are highlighted below in red:

    \[ c_{1}\alpha_{1}+\textcolor{red}{c_{2}\beta_{12}}+\textcolor{red}{c_{2}(\alpha_{2}-E)S_{12}} = c_{1}E \nonumber \]

    \( \textcolor{red}{c_{1}\beta_{21}}+\textcolor{red}{c_{1}(\alpha_{1}-E)S_{21}} + c_{2}\alpha_{2} = c_{2}E \)

    Next, we note that the difference between the self energies, \(\alpha_{1}\) and \(\alpha_{2}\), and the energy, E, of the combined system may be small. Under this limit, we can reduce the equations further to

    \[ c_{1}\alpha_{1} + c_{2}\beta_{12} = c_{1}E \\
    c_{1}\beta_{21}+c_{2}\alpha_{2} = c_{2}E \nonumber \]

    Written as a matrix, we get

    \[ \left(\begin{array}{ll}
    \alpha_{1} & \beta_{12} \\
    \beta_{21} & \alpha_{2}
    \end{array}\right)\left(\begin{array}{l}
    c_{1} \\
    c_{2}
    \end{array}\right)=E\left(\begin{array}{l}
    c_{1} \\
    c_{2}
    \end{array}\right) \nonumber \]

    Thus, we can ignore the overlap integrals of separated atoms.

    In summary, tight binding theory makes the following approximations:

    1. Consider only frontier atomic orbitals
    2. Consider only interactions between the frontier atomic orbitals of nearest neighbors. This is the tight binding approximation.
    3. Ignore the overlap integrals of separated atoms, i.e. \(S_{sr} = \delta_{sr}\). This is valid only when \(\alpha_{1} \approx \alpha_{2} \approx E\). We will assume \(S_{sr} = \delta_{sr}\) generally to simplify the mathematics.

    The self energy, \(\alpha\), and the hopping interaction, \(\beta\), could be calculated numerically given the potential and the frontier atomic orbital. But, in this class, we will not actually determine \(\alpha\) and \(\beta\). Rather we are interested in the form of the molecular wavefunctions and the dispersion relations for their energies. With this information we can determine whether the conductor is a metal or an insulator, and its density of states.


    This page titled 6.5: The Tight Binding Approximation is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Marc Baldo (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.