16 Bandstructure 1 - Fundamentals of Nanoelectronics...

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Fundamentals of Fundamentals of Nanoelectronics Nanoelectronics Prof. Supriyo Datta ECE 453 Purdue University Network for Computational Nanotechnology 10.01.2004 Lecture 16: Bandstructure 1 Ref. Chapter 5.1
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Basis Functions 00:03 • What we are trying to learn are the energy levels of a given solid. The concept of bandstructure is very useful in understanding the energy levels of periodic solids. Note that the atoms in solids we are interested in are arranged periodically. • The starting point is the Schrödinger equation: • The direct war of solving this equation is to set up a lattice of points and turn the equation into a matrix equation. But this direct approach gets intractable because the matrices get huge for problems dealing with more than 1 dimension. • Instead what is widely used is the method of basis functions: one uses his/her knowledge of the material to write the function Φ in terms of a set of basis functions. • Example: suppose you are interested in the lowest energy levels of a hydrogen molecule. There, one knows the wavefunctions of 1s levels of each H atom. When the two atoms are brought close to each other the wavefunctions mix up somewhat but the overall wavefunction is composed of the two individual wave functions: • Instead of looking for a function we only need to find the proper coefficients which in the case of H2 molecule there are only 2 of them. Φ
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16 Bandstructure 1 - Fundamentals of Nanoelectronics...

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