Carbon Nanomaterials and Technology

1. EE 6909 Carbon Nanomaterials and Technology • Dr. Md. Sherajul Islam • Assistant Professor Department of Electrical and Electronics Engineering Khulna University of Engineering & Technology Khulna, Bangladesh

2. LECTURE - 4 Band Structure: Lattice Vibrations

3. The Bloch Theorem A wave function ψ is a Bloch wave if it has the form where r is position, ψ is the Bloch wave, u is a periodic function with the same periodicity as the crystal, k is a vector of real numbers called the crystal wave vector, e is Euler's number, and i is the imaginary unit. In other words, if you multiply a plane wave by a periodic function, you get a Bloch wave.

4. A Bloch wave can be broken up into the product of a periodic function top) and a plane-wave (center). The left side and right side represent the same Bloch wave broken up in two different ways, involving the wave vector k1 (left) or k2 (right). The difference (k1−k2) is a reciprocal lattice vector. In all plots, blue is real part and red is imaginary part.

5. The Tight Binding Model In the tight binding theory, the electrons are tightly bounded to the atoms they belong to and there are only weak interactions between electrons that belong to an atom and the rest of the atoms of the crystal. We start with the single particle Schrödinger equation in the whole lattice:

8. The tight-binding model is an approach to the calculation of electronic band structure using an approximate set of wave functions based upon superposition of wave functions for isolated atoms located at each atomic site. This model describes the properties of tightly bound electrons in solids. As a result the wave function of the electron will be rather similar to the atomic orbital of the free atom where it belongs to. This method was developed by Bloch in 1928 considered only the s atomic orbital. In 1934 Jones, Mott, and Skinner considered different atomic orbitals.

12. This equation is called the secular equation, whose eigenvalues Ei(k) give the energy band structure

13. Tight Binding for Graphene

15. Calculation of π Bands Using TB We start with transfer matrix

18. The calculations of the tight binding method can be summarized in the following steps