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

2. Band Structure: Lattice Vibrations
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.
The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).
In other words, if you multiply a plane wave by a periodic function, you get a Bloch wave.

4. The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).
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:
The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).

6. The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).

7. The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).

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.
The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).

9. The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).

10. The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).

11. The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).

12. The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).
This equation is called the secular equation, whose eigenvalues Ei(k) give the energy band structure

13. Tight Binding for Graphene
The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).

14. The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).

15. Calculation of π Bands Using TB
We start with transfer matrix
The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).

16. The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).

17. The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).

18. The calculations of the tight binding method can be summarized in the following steps
The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).

19. The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).

20. The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).

21. The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).

22. The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).

23. The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).

24. The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).