Tight Binding Calculation

In carbon materials except for diamond, the π electrons are valence electrons which are relevant for the transport. A tight binding calculation provides insights for understanding the electronic structure of the π energy levels or bands for graphite.
Because of the translational simmetry of the unit cells in the direction of the lattice vectors $\vec{a}_i$ (i = 1, 2, 3), any wave function of the lattice $\Psi$ should satisfy Bloch theorem $T_{\vec{a}_i}\Psi=e^{i\vec{k}\cdot\vec{a}_i}\Psi$ , where $T_{\vec{a}_i}$ is a translational operation along the lattice vector $\vec{a}_i$ and $\vec{k}$ is the wave vector.

A form which satisfies the equation is based on the j-th atomic orbital in the unit cell (or atom). A tight binding, Bloch function $\Phi_j(\vec{k},\vec{r})$ is given by $\Phi_j(\vec{k},\vec{r})=\frac{1}{N}\sum_R^N{e^{i\vec{k}\cdot\vec{R}}\varphi_j(\vec{r}-\vec{R}), j=1,...,N)}$ . Here $\vec{R}$ is the position of the atom and $\varphi_j$ denotes the atomic wavefunction in state j. The boundary condition of the Bloch theorem gives the wave number $k=\frac{2p\pi}{Ma_i}, (p=0,1,...,M-1), (i=1,...,3)$ .

The eigenfunctions in the solid $\Psi_j(\vec{k},\vec{r}), (j=1,...,n)$ , where n is the number of Bloch functions, are expressed by a linear combination of Bloch functions $\Phi_{j^{'}}(\vec{k},\vec{r}), (j=1,...,n)$ as follows: $\Psi_j(\vec{k},\vec{r})=\sum_{j^{'}=1}^n{C_{jj^{'}}(\vec{k})\Phi_j^{'}(\vec{k},\vec{r})}$ where $C_{jj^{'}}(\vec{k})$ are coefficients to be determined.

The j-th eigenvalue $E_j(\vec{k}) (j=1,...,n)$ as a function of $\vec{k}$ is given by $E_j(\vec{k})=\frac{\langle \Psi_j|\mathcal{H}|\Psi_j \rangle}{\langle \Psi_j|\Psi_j \rangle}=\frac{\int \Psi_j^{*}\mathcal{H}\Psi_j d\textbf{r}}{\int \Psi_j^{*}\Psi_j d\textbf{r}}$ where $\mathcal{H}$ is the Hamiltonian of the solid. Sobstituting the equations we obtain $E_i(\vec{k})=\frac{\sum_{j,j^'=1}^n{C_{ij^{*}}C_{ij^{'}}\langle\Phi_j|\mathcal{H}|\Phi_{j^{'}}}\rangle} {\sum_{j,j^'=1}^n{C_{ij^{*}}C_{ij^{'}}\langle\Phi_j|\Phi_{j^{'}}}\rangle}=\frac{\sum_{j,j^'=1}^n{\mathcal{H}_{jj^{'}}C_{ij^{*}}C_{ij^{'}}}} {\sum_{j,j^'=1}^n{\mathcal{S}_{jj^{'}}C_{ij^{*}}C_{ij^{'}}}}$ where the integrals over the Bloch orbitals, $\mathcal{H}_{j,j^{'}}(\vec{k})$ and $\mathcal{S}_{j,j^{'}}(\vec{k})$ are called transfer integral matrices and overlap integral matrices, respectively, which are defined by $\mathcal{H}_{jj^{'}}(\vec{k})=\langle\Phi_j|\mathcal{H}|\Phi_{j^{'}}\rangle$ and $\mathcal{S}_{jj^{'}}(\vec{k})=\langle\Phi_j|\Phi_{j^{'}}\rangle$ .

Two-Dimensional Graphite

Graphite is a three-dimensional (3D) layered hexagonal lattice of carbon atoms. A single layer of graphite, forms a two-dimensional (2D) material, called 2D graphite or graphene layer. Graphite surface lattice is hexagonal and is defined by the unit vector
$\vec{a}=\left(\frac{\sqrt{3}}{2}a,\frac{a}{2}\right)$
as shown in the image below.

(a)The unit cell and (b) Brillouin zone of two dimensional graphite are shown as the dotted rombus and the shaded hexagon, respectively. $\vec{a}_i$ and $\vec{b}_i$ (i = 1, 2) are unit vectors and reciprocal lattice vectors, respectively. Energy dispersion relation are obtained along the perimeter of the dotted triangle connecting the high simmetry point $\Gamma$ , K and M.

π bands for two-dimensional graphite

Two Bloch functions, $T_{\vec{a}_i}\Psi=e^{i\vec{k}\cdot\vec{a}_i}\Psi$ , constructed from atomic orbitals for the two inequivalent carbon atoms at A and B in figure above, provide the basis functions for 2D graphite. When we consider only nearest-neighbor interactions, then there is only an integration over a single atom in $\mathcal{H}_{AA}$ and $\mathcal{H}_{BB}$ as shown in $\mathcal{H}_{AA}=\frac{1}{N}\sum_{(R,R^{'})}{e^{ik(R-R^{'})}\langle \varphi_A(r-R^{'})|\mathcal{H}|\varphi_A(r-R)\rangle}=\frac{1}{N}\sum_{(R=R^{'})}{\epsilon_{2p}} \frac{1}{N}\sum_{(R=R^{'} \pm a)}{e^{(\pm ika)}\langle \varphi_A(r-R^{'})|\mathcal{H}|\varphi_A(r-R)\rangle}$ and thus $\mathcal{H}_{AA}=\mathcal{H}_{BB}=\epsilon_{2p}$ . For the off-diagonal matrix element $\mathcal{H}_{AB}$ , we must consider the three nearest-neighbor B atoms relative to an A atom, which are denoted by the vectors $\vec{R}_1$ , $\vec{R}_2$ and $\vec{R}_3$ . We then consider the contribution to $\mathcal{H}_{AB}(r)=\frac{1}{N}\sum_R\{e^{ika/2}\langle \varphi_A(r-R)|\mathcal{H}|\varphi_B(r-R-a/2) \rangle e^{-ika/2} \langle \varphi_A(r-R)|\mathcal{H}|\varphi_B(r-R+a/2) \rangle \}=2t\cos(ka/2)$ from $\vec{R}_1$ , $\vec{R}_2$ and $\vec{R}_3$ as follows $\mathcal{H}_AB=t(e^{i\vec{k}\cdot\vec{R}_1}+e^{i\vec{k}\cdot\vec{R}_2}+e^{i\vec{k}\cdot\vec{R}_3})=tf(k)$ where t is $t=\langle\varphi_A(r-R)|\mathcal{H}|\varphi_B(r+R\pm a/2\rangle$ .

Solving the secular equation $\det(\mathcal{H}-E\mathcal{S})=0$ , the eigenvalue $E(\vec{k})$
are obtained as a function $\omega(\vec{k}), k_x$
and $k_y$ : $E_{g2D}(\vec{k})=\frac{\epsilon_{2p}\pm t\omega(\vec{k})}{1 \pm s\omega(\vec{k})}$
where the + signs in the numerator and denominator go together giving the bonding $\pi$ energy band, and likewise for the - signs, which give the antibonding
$\pi^*$ band.

The function $\omega(\vec{k})$ is given by: $\omega(\vec{k})=\sqrt{|f(\vec{k}|^2}=\sqrt{1+4\cos{\frac{\sqrt{3}k_xa}{2}}+\cos{\frac{k_ya}{2}}+4\cos^2{\frac{k_ya}{2}}}$
t is the transfer integral $t=\langle\varphi_A(r-R)|\mathcal{H}|\varphi_B(r+R\pm a/2\rangle$
s is the overlap integral between the nearest A and B atoms, $s=\langle\varphi_A(r-R)|\varphi_B(r-R\pm a/2\rangle$ .

Application preview

The following image shows the application preview. You can modify the parameters of equation
$E_{g2D}(\vec{k})=\frac{\epsilon_{2p}\pm t\omega(\vec{k})}{1 \pm s\omega(\vec{k})}$ where t is the transfer integral and s is the overlap integral between the nearest A and B atoms.

The energy dispersion relations in the case of $s=0$ are commonly used as a simple approximation for the electronic structure of a graphene layer:
$E_{g2D}(k_x,k_y)=\pm t{\sqrt{1+4\cos{\frac{\sqrt{3}k_xa}{2}}+\cos{\frac{k_ya}{2}}+4\cos^2{\frac{k_ya}{2}}}}$

As you can see, in the image are shown three points $\Gamma$ , M and K that are high simmetry points shown in the triangle $\Gamma M K$ in the little hexagon.

MATLAB Graphic Application

You can download the installer for the standalone application (without Matlab) here using “MyAppInstaller_web.exe”.

How to intall

Open “MyAppInstaller_web.exe”
You need to install the MATLAB Runtime to run this app.
Install the application EnergyDispersionRelation2DGraphite
Run the app

Bibliography

R. Saito, G. Dresselhaus - Physical properties of carbon nanotubes