Session 3 Band Model

3.1 Introduction to Quantum Mechanics

The motion of large objects，such as planets and satellites，can be predicted to a high degree of accuracy using classical theoretical physics based on Newton’s laws of motion. But certain experimental results，involving electrons and high-frequency electromagnetic waves，appear to be inconsistent with classical physics. However，these experimental results can be predicted by the principles of quantum mechanics. The quantum mechanical wave theory is the basis for the theory of semiconductor physics. We are ultimately interested in semiconductor materials whose electrical properties are directly related to the behavior of electrons in the crystal lattice. The behavior and characteristics of these electrons can be described by the formulation of quantum mechanics called wave mechanics. There are three principles we need to consider：the principle of energy quanta，the wave-particle duality principle，and the uncertainty principle .

3.1.1 The Principle of Energy Quanta

Planck postulated in 1900 that thermal radiation is emitted from a heated surface in discrete packets of energy called quanta. The energy of these quanta is given by E=hν，where ν is the frequency of the radiation and h is a constant now known as Planck’s constant. Then in 1905，Einstein interpreted the photoelectric results by suggesting that the energy in a light wave is also contained in discrete packets. The particle-like packet of energy is called a photon，whose energy is also given by E=hν. A photon with sufficient energy，then，can knock an electron from the surface of the material. The minimum energy required to remove an electron is called the work function of the material and any excess photon energy goes into the kinetic energy of the photoelectron.

In 1924，de Broglie postulated the existence of matter waves. He suggested that since waves exhibit particle-like behavior，then panicles should be expected to show wave-like properties. The hypothesis of de Broglie was the existence of a wave-particle duality principle. In some cases electromagnetic waves behave as if they are particles（photons）and sometimes particles behave as if they are waves. This wave-particle duality principle of quantum mechanics applies primarily to small particles such as electrons，but it has also been shown to apply to protons and neutrons. The wave-particle duality principle is the basis on which we will use wave theory to describe the motion and behavior of electrons in a crystal.

The Heisenberg uncertainty principle，given in 1927，also applies primarily to very small particles，and states that we cannot describe with absolute accuracy the behavior of these subatomic particles. The uncertainty principle describes a fundamental relationship between conjugate variables，including position and momentum and also energy and time. The first statement of the uncertainty principle is that it is impossible to simultaneously describe with absolute accuracy the position and momentum of a particle. The second statement of the uncertainty principle is that it is impossible to simultaneously describe with absolute accuracy the energy of a particle and the instant of time the particle has this energy.

3.1.2 Schrodinger’s Wave Equation

The various experimental results involving electromagnetic waves and particles，which could not be explained by classical laws of physics，showed that a revised formulation of mechanics was required. Schrodinger，in 1926，provided a formulation called wave mechanics，which incorporated the principles of quanta introduced by Planck，and the wave-particle duality principle introduced by de Broglie. Based on the wave-particle duality principle，we will describe the motion of electrons in a crystal by wave theory. This wave theory is described by Schrodinger’s wave equation .

The one-dimensional Schrodinger’s wave equation is given by

Where Ψ（x，t）is the wave function，V（x）is the potential function assumed to be independent of time，m is the mass of the particle，and j is the imaginary constant . There are theoretical arguments that justify the form of Schrodinger’s wave equation，but the equation is a basic postulate of quantum mechanics. The wave function Ψ（x，t）will be used to describe the behavior of the system and，mathematically，Ψ（x，t）can be a complex quantity.

We are ultimately trying to use the wave function Ψ（x，t）to describe the behavior of an electron in a crystal. The function Ψ（x，t）is a wave function，so it is reasonable to ask what the relation is between the function and the electron. The total wave function is the product of the position-dependent，or time-independent，function and the time-dependent function. Max Born postulated in 1926 that the function |Ψ（x，t）|2dx is the probability of finding the particle between x and x+dx at a given time，or that |Ψ（x，t）|2 is a probability density function. We have that

｜Ψ（x，t）｜2=ψ（x）ψ*（x）=｜ψ（x）｜2 （3.2）

|Ψ（x，t）|2 is the probability density function and is independent of time. One major difference between classical and quantum mechanics is that in classical mechanics，the position of a particle or body can be determined precisely，whereas in quantum mechanics，the position of a particle is found in terms of a probability. We will determine the probability density function for several examples，and，since this property is independent of time，we will，in general，only be concerned with the time-independent wave function.