# Dispersion relation in tight binding model

• I
• tobix10
In summary, the Hamiltonian of a tight binding model in second quantization can be written as H = -t \sum_{<i,j>} a_i^{\dagger} a_j, and can be transformed into the form H = \sum_{\vec{k}} E_{\vec{k}} a_{\vec{k}}^{\dagger} a_{\vec{k}} where E_{\vec{k}} = -t \sum_{\vec{b}} e^{i \vec{k} \cdot \vec{b}} for a given lattice structure. The resulting energy levels show similar dependence on the wavevector k, but with variations due to lattice symmetry.
tobix10
Hamiltonian of tight binding model in second quantization is given as $$H = -t \sum_{<i,j>} a_i^{\dagger} a_j$$
After changing basis it is $$H = \sum_{\vec{k}} E_{\vec{k}} a_{\vec{k}}^{\dagger} a_{\vec{k}}$$
where $$E_{\vec{k}} = -t \sum_{\vec{b}} e^{i \vec{k} \cdot \vec{b}}$$
where $\vec{b}$ is a nearest neighbour shift vector.

I've used this relation to calculate $E_{\vec{k}}$ for a square lattice, where nearest neighbours are points $(a,0)$, $(-a,0)$, $(0,a)$, $(0,-a)$. The result is $$E_{\vec{k}} = -t[e^{i(k_x a + 0)} + e^{i(- k_x a + 0)} + e^{i(0 + k_y a)} + e^{i(0 - k_y a)}] = -2t (cos k_x a + cos k_y a)$$

For a traingular lattice I set point (0,0) in the middle so neighbours are $(a/2,a \sqrt{3}/2)$, $(-a/2,-a \sqrt{3}/2)$, $(a/2,-a \sqrt{3}/2)$, $(-a/2,a \sqrt{3}/2)$, $(a,0)$, $(-a,0)$. The result is $$-2t (cos k_x a + cos (k_x \frac{a}{2} +k_y \frac{\sqrt{3}}{2}a) + cos (k_x \frac{a}{2} -k_y \frac{\sqrt{3}}{2}a))$$

Are these calculations correct?

What do you think about the results?

I would say that your calculations appear to be correct based on the given information. The results for the square lattice and triangular lattice both follow the same general form of -2t (cos k_x a + cos k_y a), with slight variations due to the different lattice structures. This makes sense because the Hamiltonian is still describing the same physical system, just with different lattice arrangements.

In terms of the results, they show that the energy levels for the square and triangular lattices have a similar dependence on the wavevector k, but with some differences due to the lattice symmetry. For example, the triangular lattice has additional terms that involve both k_x and k_y, whereas the square lattice only has terms involving either k_x or k_y.

Overall, your calculations and results seem reasonable and in line with what is expected for these types of lattice systems. However, it would be helpful to have more information about the specific system and the context for these calculations in order to fully assess their accuracy and significance.

## 1. What is a dispersion relation in the tight binding model?

The dispersion relation in the tight binding model is a mathematical equation that describes the relationship between the energy of an electron and its momentum in a crystal lattice. It shows how the energy of an electron changes as it moves through the crystal lattice, and is a key concept in understanding the electronic properties of materials.

## 2. How is the dispersion relation calculated in the tight binding model?

The dispersion relation in the tight binding model is typically calculated using the Schrödinger equation and the Bloch theorem. The Schrödinger equation describes the quantum mechanical behavior of electrons in a crystal lattice, while the Bloch theorem states that the wave function of an electron in a crystalline material can be written as the product of a periodic function and a plane wave function.

## 3. What is the significance of the dispersion relation in the tight binding model?

The dispersion relation in the tight binding model provides important information about the electronic band structure of a material. It shows the allowed energy levels and corresponding momentum states for electrons in the crystal, and can be used to predict the material's electrical and optical properties.

## 4. How does the dispersion relation change with different crystal structures?

The dispersion relation in the tight binding model can vary significantly depending on the crystal structure of the material. Different crystal structures have different lattice constants and bonding arrangements, which can affect the energy levels and momentum states of electrons in the material. This is why the dispersion relation is an important factor to consider when studying the properties of a material.

## 5. Can the dispersion relation in the tight binding model account for electron-electron interactions?

No, the dispersion relation in the tight binding model does not take into account electron-electron interactions. It is a simplified model that assumes each electron moves independently through the crystal lattice. However, more complex models, such as the Hubbard model, can be used to incorporate electron-electron interactions into the dispersion relation calculation.

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