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

Consider a one-dimensional crystal of atoms with one valence orbital $\phi$. The tight binding wavefunction for this case is,

\[ \begin{equation}
 \psi_{k}\left(x\right)=\frac{1}{\sqrt{L}}\sum\limits_{n}e^{inka} \phi\left(x-na\right). 
\end{equation} \]

Here $n$ is an integer. Multiply the Schrödinger equation from the left by $\phi^*\left(x\right)$ and integrate over all space.

\[ \begin{equation}
 \langle\phi\left(x\right)|\hat{H}|\psi_{k}\left(x\right)\rangle = E\langle\phi\left(x\right)|\psi_{k}\left(x\right)\rangle.
\end{equation} \]

In the tight binding approximation, only the on-site and nearest neighbor matrix elements are retained on the left side and only the on-site term is retained on the right side. It is assumed that all the other terms are small enough that they can be ignored.

\[ \begin{equation}
 \langle\phi\left(x\right)|\hat{H}|\phi\left(x-a\right)\rangle e^{-ika} +\langle\phi\left(x\right)|\hat{H}|\phi\left(x\right)\rangle+\langle\phi\left(x\right)|\hat{H}|\phi\left(x+a\right)\rangle e^{ika} + \text{small terms}= E + \text{small terms} .
\end{equation} \]

Let $\epsilon = \langle\phi\left(x\right)|\hat{H}|\phi\left(x\right)\rangle$ and $t = - \langle\phi\left(x\right)|\hat{H}|\phi\left(x-a\right)\rangle$. The dispersion relation can then be written as,

\[ \begin{equation}
 E= \epsilon -t\left(e^{-ika} + e^{-ika}\right).
\end{equation} \]

The exponential terms can be combined into a cosine term.

\[ \begin{equation}
 E= \epsilon -2t\cos\left(ka\right) .
\end{equation} \]

One-dimensional crystal with two atoms in the basis

Consider a one-dimensional crystal with two atoms in the basis. The tight binding wavefunction is written in terms of two atomic orbitals, one centered on each atom in the basis.

\[ \begin{equation}
 \psi_{k}\left(x\right)=\frac{1}{\sqrt{L}}\sum\limits_{n}e^{inka} \left( c_1\phi_1\left(x-na\right) +c_2\phi_2\left(x-na\right)\right) .
\end{equation} \]

To determine the dispersion relation, multiply the time independent Schrödinger equation from the left by each of the atomic orbitals and integrate over all space. This results in the following two equations,

\[ \begin{equation}
 \begin{array}{a}
\langle\phi_1|\hat{H}|\psi_{k}\rangle = E\langle\phi_1|\psi_{k}\rangle , \\ \langle\phi_2|\hat{H}|\psi_{k}\rangle = E\langle\phi_2|\psi_{k}\rangle . \end{array} 
\end{equation} \]

In the tight binding approximation, only the on-site and nearest neighbor matrix elements are retained on the left side of these equations and only the on-site term is retained on the right side. It is assumed that all the other terms are small enough that they can be ignored.

\[ \begin{equation}
 \begin{array}{a}
\epsilon_1 c_1 -tc_2(1+e^{-ika}) = Ec_1 ,\\
\epsilon_2 c_2 -tc_1(1+e^{ika}) = Ec_2. 
\end{array} 
\end{equation} \]

Where $\epsilon_1 = \langle\phi_1(x)|\hat{H}|\phi_1(x)\rangle$, $\epsilon_2 = \langle\phi_2(x)|\hat{H}|\phi_2(x)\rangle$, and $t = - \langle\phi_1(x)|\hat{H}|\phi_2(x)\rangle= - \langle\phi_2(x-a)|\hat{H}|\phi_1(x)\rangle$. Due to symmetry, the matrix element $t$ is the same for both nearest neighbors. 
This can be written in matrix form.

\[ \begin{equation}
 \begin{bmatrix} 
\epsilon_1 -E & -t\left(1+e^{-ika}\right) \\
 -t\left(1+e^{ika}\right) & \epsilon_2 -E\end{bmatrix}\left[ \begin{array}{c} c_1 \\ c_2 \end{array} \right] =0.
 \end{equation} \]

This matrix equation will have solutions when the determinant equals zero.

\[ \begin{equation}
 E^2-(\epsilon_1+\epsilon_2)E + \epsilon_1\epsilon_2 -2t^2(1+\cos(ka))=0.
\end{equation} \]

Solving for the energy, the dispersion relation is,

\[ \begin{equation} 
 E=\frac{(\epsilon_1+\epsilon_2)\pm \sqrt{(\epsilon_1-\epsilon_2)^2+8t^2(1+\cos(ka))}}{2}.
\end{equation} \]

Lithium (bcc)

The metal lithium forms crystals with a body centered cubic Bravais lattice and one atom per primitive unit cell. The primitive lattice vectors for a bcc lattice with a lattice constant of $a$ are,

\[ \begin{equation}
 \begin{array}{a}
\vec{a}_1 = \frac{a}{2} \left( \hat{x}+\hat{y}-\hat{z} \right), \\
\vec{a}_2 = \frac{a}{2} \left( -\hat{x}+\hat{y}+\hat{z} \right), \\
\vec{a}_3 = \frac{a}{2} \left( \hat{x}-\hat{y}+\hat{z} \right). \end{array}
\end{equation} \]

Lithium has one valence electron in a 2s orbital, $\phi_{\text{2s}}$. The tight binding wavefunction for lithium is,

\[ \begin{equation}
 \psi_{\vec{k}}\left(\vec{r}\right)=\frac{1}{\sqrt{N}}\sum\limits_{h,j,l}e^{i\left(h\vec{k}\cdot\vec{a}_1 + j\vec{k}\cdot\vec{a}_2 + l\vec{k}\cdot\vec{a}_3\right)} \phi_{\text{2s}}\left(\vec{r}-h\vec{a}_1-j\vec{a}_2-l\vec{a}_3\right) .
\end{equation} \]

Multiply the time-independent Schrödinger equation from the left by $\phi^*_{\text{2s}}\left(\vec{r}\right)$ and integrate over all space.

\[ \begin{equation}
 \langle\phi_{\text{2s}}\left(x\right)|\hat{H}|\psi_{k}\left(x\right)\rangle = E\langle\phi_{\text{2s}}\left(x\right)|\psi_{k}\left(x\right)\rangle .
\end{equation} \]

In the tight binding approximation, only the on-site and nearest neighbor matrix elements are retained on the left side and only the on-site term is retained on the right side. It is assumed that all the other terms are small enough that they can be ignored. There are eight nearest neighbor matrix elements for a bcc lattice.

\[ \begin{equation}
 \epsilon - t e^{i\vec{k}\cdot\vec{a_1}} - t e^{i\vec{k}\cdot\vec{a_2}} - t e^{i\vec{k}\cdot\vec{a_3}} - t e^{-i\vec{k}\cdot\vec{a_1}} - t e^{-i\vec{k}\cdot\vec{a_2}} - t e^{-i\vec{k}\cdot\vec{a_3}} - t e^{i\vec{k}\cdot\left(\vec{a_1} +\vec{a}_2 +\vec{a}_3\right)} - t e^{-i\vec{k}\cdot\left(\vec{a_1} +\vec{a}_2 +\vec{a}_3\right)} + \text{small terms}= E + \text{small terms}. 
\end{equation} \]

Where $\epsilon = \langle\phi_{\text{2s}}\left(\vec{r}\right)|\hat{H}|\phi_{\text{2s}}\left(\vec{r}\right)\rangle$ and $t = - \langle\phi_{\text{2s}}\left(\vec{r}\right)|\hat{H}|\phi_{\text{2s}}\left(\vec{r}-\vec{a}_1\right)\rangle$. Due to symmetry, the matrix element $t$ is the same for all nearest neighbors. Using the definition of the primitive lattice vectors, this can be rewritten as,

\[ \begin{equation}
 \begin{array}{b}
 E = \epsilon - te^{ik_xa/2}e^{ik_ya/2}e^{-ik_za/2} -te^{-ik_xa/2}e^{ik_ya/2}e^{ik_za/2} -te^{ik_xa/2}e^{-ik_ya/2}e^{ik_za/2} 
-te^{-ik_xa/2}e^{-ik_ya/2}e^{ik_za/2} \\
-te^{ik_xa/2}e^{-ik_ya/2}e^{-ik_za/2} -te^{-ik_xa/2}e^{ik_ya/2}e^{-ik_za/2} 
 -te^{ik_xa/2}e^{ik_ya/2}e^{ik_za/2} -te^{-ik_xa/2}e^{-ik_ya/2}e^{-ik_za/2} 
\end{array} .
\end{equation} \]

Using the definition of relation $2\cos x = e^{ix} + e^{-ix}$, this can be written as,

\[ \begin{equation} 
 E = \epsilon - 8t\cos\left(k_xa/2\right)\cos\left(k_ya/2\right)\cos\left(k_za/2\right) .
\end{equation} \]

The dispersion relation for the 2s band for lithium in the tight binding approximation is drawn below.

Hexagonal - one atomic orbital

Consider a crystal with a hexagonal Bravais lattice and one atom in the basis. If a single spherically symmetric atomic orbital is used and all atoms are equidistant from each other, the band can be calculated relatively simply. The primitive lattice vectors for a hexagonal lattice are,

$$\vec{a}_1=a\hat{x},\qquad\vec{a}_2=\frac{a}{2}\hat{x}+\frac{\sqrt{3}a}{2}\hat{y},\qquad\vec{a}_3=c\hat{z}.$$

For equidistant atoms, $c=a$. With one valence orbital, $\phi$, the tight binding wavefunction is,

$$ \psi_{\vec{k}}\left(\vec{r}\right)=\frac{1}{\sqrt{N}}\sum\limits_{h,j,l}e^{i\left(h\vec{k}\cdot\vec{a}_1 + j\vec{k}\cdot\vec{a}_2 + l\vec{k}\cdot\vec{a}_3\right)} \phi\left(\vec{r}-h\vec{a}_1-j\vec{a}_2-l\vec{a}_3\right).$$

Multiply the time-independent Schrödinger equation from the left by $\phi^*\left(\vec{r}\right)$ and integrate over all space.

$$\langle\phi\left(x\right)|\hat{H}|\psi_{k}\left(x\right)\rangle = E\langle\phi\left(x\right)|\psi_{k}\left(x\right)\rangle .$$

In the tight binding approximation, only the on-site and nearest neighbor matrix elements are retained on the left side and only the on-site term is retained on the right side. It is assumed that all the other terms are small enough that they can be ignored. There are eight nearest neighbor matrix elements for a hexagonal lattice.

\[ \begin{equation}
 \epsilon - t e^{i\vec{k}\cdot\vec{a_1}} - t e^{i\vec{k}\cdot\vec{a_2}} - t e^{i\vec{k}\cdot\vec{a_3}} - t e^{-i\vec{k}\cdot\vec{a_1}} - t e^{-i\vec{k}\cdot\vec{a_2}} - t e^{-i\vec{k}\cdot\vec{a_3}} - t e^{i\vec{k}\cdot\left(-\vec{a_1} +\vec{a}_2\right)} - t e^{-i\vec{k}\cdot\left(-\vec{a_1} +\vec{a}_2 \right)} + \text{small terms}= E + \text{small terms}. 
\end{equation} \]

Where $\epsilon = \langle\phi\left(\vec{r}\right)|\hat{H}|\phi\left(\vec{r}\right)\rangle$ and $t = - \langle\phi\left(\vec{r}\right)|\hat{H}|\phi\left(\vec{r}-\vec{a}_1\right)\rangle$. Since the distance to all nearest neighbors is the same, we assume that the matrix element $t$ is the same for all nearest neighbors. Using the definition of the primitive lattice vectors, this can be rewritten as,

$$ E = \epsilon - te^{ik_xa} -te^{ik_xa/2}e^{ik_y\sqrt{3}a/2} -te^{ik_za} 
-te^{-ik_xa/2}e^{ik_y\sqrt{3}a/2} \\
- te^{-ik_xa} -te^{-ik_xa/2}e^{-ik_y\sqrt{3}a/2} -te^{-ik_za} 
-te^{ik_xa/2}e^{-ik_y\sqrt{3}a/2} .$$

Using the definition of relation $2\cos x = e^{ix} + e^{-ix}$, this can be written as,

$$ E = \epsilon - 2t\cos(k_xa) -4t\cos(k_xa/2)\cos(k_y\sqrt{3}a/2) -2t\cos(k_za).$$

The dispersion relation in the tight binding approximation is drawn below.

Graphene

Graphene is a two-dimensional material consisting of carbon atoms in a hexagonal lattice.

There are two carbon atoms in the primitve unit cell. Each carbon atom has one 2pz valence orbital. The $\phi_{\text{2p}_{z1}}$ orbital is centered at the site of the C1 atoms and the $\phi_{\text{2p}_{z2}}$ orbital is centered at the site of the C2 atoms. The primitive lattice vectors are,

\[ \begin{equation}
 \begin{array}{a}
\vec{a}_1 = \frac{\sqrt{3}a}{2} \hat{x}+ \frac{a}{2}\hat{y}, \\
\vec{a}_2 = \frac{\sqrt{3}a}{2} \hat{x}- \frac{a}{2}\hat{y}. \end{array}
 \end{equation} \]

The tight binding wavefunction for graphene is,

\[ \begin{equation}
 \psi_{\vec{k}}\left(\vec{r}\right)=\frac{1}{\sqrt{N}}\sum\limits_{h,j}e^{i\left(h\vec{k}\cdot\vec{a}_1 + j\vec{k}\cdot\vec{a}_2\right)} \left( c_1\phi_{\text{2p}_{z1}}\left(\vec{r}-h\vec{a}_1-j\vec{a}_2\right) + c_2 \phi_{\text{2p}_{z2}}\left(\vec{r}-h\vec{a}_1-j\vec{a}_2\right) \right) .
\end{equation} \]

To determine the dispersion relation, multiply the time-independent Schrödinger equation from the left by each of the pz orbitals and integrate over all space. This results in the following two equations,

\[ \begin{equation}
 \begin{array}{a}
\langle\phi_{\text{2p}_{z1}}|\hat{H}|\psi_{k}\rangle = E\langle\phi_{\text{2p}_{z1}}|\psi_{k}\rangle , \\ \langle\phi_{\text{2p}_{z2}}|\hat{H}|\psi_{k}\rangle = E\langle\phi_{\text{2p}_{z2}}|\psi_{k}\rangle . \end{array} 
\end{equation} \]

\[ \begin{equation}
 \begin{array}{a}
\epsilon c_1 -tc_2\left(1+e^{-i\vec{k}\cdot\vec{a_1}} +  e^{-i\vec{k}\cdot\vec{a_2}}\right) = Ec_1 ,\\
\epsilon c_2 -tc_1\left(1+e^{i\vec{k}\cdot\vec{a_1}} +  e^{i\vec{k}\cdot\vec{a_2}}\right) = Ec_2. 
\end{array} 
\end{equation} \]

Where $\epsilon = \langle\phi_{\text{2p}_{z1}}\left(\vec{r}\right)|\hat{H}|\phi_{\text{2p}_{z1}}\left(\vec{r}\right)\rangle$ and $t = - \langle\phi_{\text{2p}_{z1}}\left(\vec{r}\right)|\hat{H}|\phi_{\text{2p}_{z2}}\left(\vec{r}\right)\rangle= - \langle\phi_{\text{2p}_{z1}}\left(\vec{r}\right)|\hat{H}|\phi_{\text{2p}_{z1}}\left(\vec{r}-\frac{a}{\sqrt{3}}\hat{x}\right)\rangle$. Due to symmetry, the matrix element $t$ is the same for all nearest neighbors. Using the definition of the primitive lattice vectors, this can be written in matrix form.

\[ \begin{equation}
 \begin{bmatrix} 
\epsilon -E & -t\left(1+e^{-i\sqrt{3}k_xa/2}e^{-ik_ya/2} +  e^{-i\sqrt{3}k_xa/2}e^{+ik_ya/2}\right) \\
 -t\left(1+e^{i\sqrt{3}k_xa/2}e^{ik_ya/2} +  e^{i\sqrt{3}k_xa/2}e^{-ik_ya/2}\right) & \epsilon -E\end{bmatrix}\left[ \begin{array}{c} c_1 \\ c_2 \end{array} \right] =0.
\end{equation} \]

This matrix equation will have solutions when the determinant equals zero.

\[ \begin{equation}
 \left(\epsilon-E\right)^2 -t^2\left(1+e^{-i\sqrt{3}k_xa/2}e^{-ik_ya/2} +  e^{-i\sqrt{3}k_xa/2}e^{+ik_ya/2}\right)\left(1+e^{i\sqrt{3}k_xa/2}e^{ik_ya/2} +  e^{i\sqrt{3}k_xa/2}e^{-ik_ya/2}\right)=0.
\end{equation} \]

Using the definition of relation $2\cos x = e^{ix} + e^{-ix}$, this can be written as,

\[ \begin{equation}
 \left(\epsilon-E\right)^2 -t^2\left(3+2\cos\left(\frac{\sqrt{3}k_xa}{2}-\frac{k_ya}{2}\right)+2\cos\left(\frac{\sqrt{3}k_xa}{2}+\frac{k_ya}{2}\right)+2\cos\left(k_ya\right)\right)=0.
 \end{equation} \]

Using the trigonometric identities, 
\[ \begin{equation}
 \begin{array}{a}
\cos\left(a+b\right)=\cos\left(a\right)\cos\left(a+b\right)-\sin\left(a\right)\sin\left(a\right), \\\cos\left(a-b\right)=\cos\left(a\right)\cos\left(a+b\right)+\sin\left(a\right)\sin\left(a\right)\text{, and} \\
\cos\left(2a\right)=2\cos\left(a\right)-1, 
\end{array} 
\end{equation} \]

this can be written as,

\[ \begin{equation}
 \left(\epsilon-E\right)^2 -t^2\left(1+4\cos\left(\frac{\sqrt{3}k_xa}{2}\right)\cos\left(\frac{k_ya}{2}\right)+4\cos^2\left(\frac{k_ya}{2}\right)\right)=0.
\end{equation} \]

Solving for the energy, the dispersion relation for graphene is,

\[ \begin{equation}
E=\epsilon \pm t\sqrt{1+4\cos\left(\frac{\sqrt{3}k_xa}{2}\right)\cos\left(\frac{k_ya}{2}\right)+4\cos^2\left(\frac{k_ya}{2}\right)}.
\end{equation} \]