PHY.K02UF Molecular and Solid State Physics

Tight binding

Tight binding is a method to calculate the electronic band structure of a crystal. It is similar to the method of Linear Combination of Atomic Orbitals (LCAO) used to construct molecular orbitals. Although this approximation neglects the electron-electron interactions, it often produces qualitatively correct results and is sometimes used as the starting point for more sophisticated approaches.

A wave function is constructed from the valence orbitals of all of the atoms in a primitive unit cell of the crystal.

\[ \begin{equation} \psi_{\text{unit cell}}\left(\vec{r}\right)= \sum\limits_{a} \sum\limits_{ao} c_{ao,a}\phi^{Z_a}_{ao}\left(\vec{r}-\vec{r}_a\right). \end{equation} \]

Where $a$ sums over the atoms in the basis and $ao$ sums over the atomic orbitals. It is conventional to relabel the atomic orbitals with an index $i$ that sums over the atomic orbitals.

\begin{equation} \psi_{\text{unit cell}}\left(\vec{r}\right)=\sum\limits_{i}c_{i}\phi_{i}\left(\vec{r}-\vec{r}_i\right). \end{equation}

The coefficients $c_{i}$ are determined by substituting the wavefunction into the Schrödinger equation. For instance, for calcium carbonate CaCO3, the valence orbitals would be the 4s orbital for calcium, the 2s and 3 × 2p orbitals for carbon and the 2s and 3 × 2p orbitals for oxygen. In this case there would be 15 terms in the wavefunction for the unit cell, $(i=1,\cdot\cdot\cdot,15)$.

The wave function for the one unit cell is then repeated at every unit cell in the crystal with a complex prefactor.

\[ \begin{equation} \psi_{\vec{k}}\left(\vec{r}\right)=\frac{1}{\sqrt{N}}\sum\limits_{h,j,l}e^{i\left(h\vec{k}\cdot\vec{a}_1 + j\vec{k}\cdot\vec{a}_2 + l\vec{k}\cdot\vec{a}_3\right)} \psi_{\text{unit cell}}\left(\vec{r}-h\vec{a}_1-j\vec{a}_2-l\vec{a}_3\right). \end{equation} \]

Here $N$ is the number of unit cells in the crystal; $h$, $j$, and $l$ are integers that are used to label all the unit cells in the crystal; $\vec{k}$ is a wave vector; and $\vec{a}_1$, $\vec{a}_2$, and $\vec{a}_3$ are primitive lattice vectors in real space. Because of the translational symmetry of the crystal, the energy eigen functions must also be eigen functions of the translation operator. The complex prefactor ensures that the wavefunction is an eigen function of the translation operator $\hat{T}_{pqs}$.

\[ \begin{equation} \hat{T}_{pqs}\psi_{\vec{k}}\left(\vec{r}\right)=\psi_{\vec{k}}\left(\vec{r}+p\vec{a}_1+q\vec{a}_2+s\vec{a}_3\right)=e^{i\left(p\vec{k}\cdot\vec{a}_1 + q\vec{k}\cdot\vec{a}_2 + s\vec{k}\cdot\vec{a}_3\right)}\psi_{\vec{k}}\left(\vec{r}\right). \end{equation} \]

The operator $T_{pqs}$ translates the function by $p\vec{a}_1+q\vec{a}_2+s\vec{a}_3$ where $p$, $q$, and $s$ are integers.

The energy of the tight binding wave function can be evaluated by substituting $\psi_{\vec{k}}$ into the time independent Schrödinger equation.

\[ \begin{equation} \hat{H}\psi_{\vec{k}}=E\psi_{\vec{k}} . \end{equation} \]

Multiply the Schrödinger equation from the left by one of the valence orbitals $\phi_{n}^*\left(\vec{r}\right)$ and integrate over all space.

\[ \begin{equation} \langle\phi_{n}|\hat{H}|\psi_{\vec{k}}\rangle = E\langle\phi_{n}|\psi_{\vec{k}}\rangle. \end{equation} \]

In the tight binding wavefunction, the $h$, $j$, and $l$ indices sum over all of the unit cells in the crystal so $\psi_{\vec{k}}$ is finite everywhere. However, the valence orbital $\phi_n$ goes to zero rapidly as the distance from the center of this orbital increases. On the left side of the equation above, the only significant terms are $\langle\phi_n|\hat{H}|\phi_n\rangle$ and the matrix elements involving $\phi_n$ and and its nearest neighbor orbitals. On the right side of the equation typically only the largest term is included.

\[ \begin{equation} c_n\langle\phi_{n}|\hat{H}|\phi_{n}\rangle + \sum\limits_{m=\text{nearest neighbors}}c_m\langle\phi_{n}|\hat{H}|\phi_{m}\rangle e^{i\left(h\vec{k}\cdot\vec{a}_1 + j\vec{k}\cdot\vec{a}_2 + l\vec{k}\cdot\vec{a}_3\right)} + \text{small terms} = Ec_n\langle\phi_{n}|\phi_{n}\rangle + \text{small terms} . \end{equation} \]

There is one equation like this for each of the valence orbitals $\phi_n$. These equations can be solved to determine the coefficients $c_i$ and energy $E$.

One-dimensional crystal with one atom in the basis

One-dimensional crystal with two atoms in the basis

Lithium (bcc)

Hexagonal - one atomic orbital