Introduction to quantum mechanics (Chap.2) Quantum theory for semiconductors (Chap. 3)

1. Basic Quantum Mechanics 18 and 20 January 2017 What Is An Energy Band And How Does It Explain The Operation Of Classical Semiconductor Devices and Quantum Well Devices? To Address These Questions, We Will Study: • Introduction to quantum mechanics (Chap.2) • Quantum theory for semiconductors (Chap. 3) • Allowed and forbidden energy bands (Chap. 3.1) • Also refer to Appendices: Table B 2 (Conversion Factors), Table B.3 (Physical Constants), and Tables B.4 and B.5 Si, Ge, and GaAs key attributes and properties. • We will expand the derivation of Schrӧdinger’s Wave Equation as summarized in Appendix E. • Will use a transmission line analogy for discussing and interpretaingSchrӧdinger’s wave equation solutions 1

2. Mechanics: the study of the behavior of physical bodies when subjected to forces or displacements Classical Mechanics: describing the motion of macroscopic objects. Macroscopic: measurable or observable by naked “eyes” or by optical microscopy Quantum Mechanics: describing behavior of systems at atomic length scales and smaller . Classical Mechanics and Quantum Mechanics 2

3. Emitted electron kinetic energy = T Incident light with frequency ν Tmax 0 ν νo Metal Plate Photoelectric Effect Experiment results The photoelectric effect ( year1887 by Hertz) • Inconsistency with classical light theory According to the classical wave theory, maximum kinetic energy of the photoelectron is only dependent on the incident intensity of the light, and independent on the light frequency; however, experimental results show that the kinetic energy of the photoelectron is dependent on the light frequency. Concept of “energy quanta” 3 3

4. Energy Quanta • Photoelectric experiment results suggest that the energy in light wave is contained in discrete energy packets, which are called energy quanta or photon • The wave behaviors like particles. The particle is photon Planck’s constant: h = 6.625×10-34 J-s Photon energy = hn (Greek lower case “nu”, Could also use “f” for frequency) Work function of the metal material = hno Maximum kinetic energy of a photoelectron: Tmax= h(n-no) 4

6. Nickel sample θ=0 Electron beam θ θ=45º Scattered beam θ=90º Davisson-Germer experiment (1927) Detector Electron’s Wave Behavior Electron as a particle has wave-like behavior 6

7. Mathematical descriptions: The momentum of a photon is: The wavelength of a particle is: Wave-Particle Duality Wave-like particle behavior (example, Davisson-Germer experiment) Particle-like wave behavior (example, photoelectric effect) Wave-particle duality λ is called the de Broglie wavelength 7

8. The Uncertainty Principle The Heisenberg Uncertainty Principle (year 1927): • It is impossible to simultaneously describe with absolute accuracy the position and momentum of a particle • It is impossible to simultaneously describe with absolute accuracy the energy of a particle and the instant of time the particle has this energy The Heisenberg uncertainty principle applies to electrons and states that we can not determine the exact position of an electron. Instead, we could determine the probability of finding an electron at a particular position. 8

9. Quantum Theory for Semiconductors How to determine the behavior of electrons )and holes) in the semiconductor? • Mathematical description of motion of electrons in quantum mechanics ─ Schrödinger’s Wave Equation • Solution of Schrödinger’s Wave Equation energy band structure and probability of finding a electron at a particular position 9

10. Schrӧdinger’s Wave EquationText Equation 2.6 which we will now derive One dimensional Schrӧdinger’s Wave Equation: , the probability to find a particle in (x, x+dx) at time t , the probability density at location x and time t Wave function Potential function Mass of the particle 10