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Chapter23_24_Bandstructure

# Chapter23_24_Bandstructure - 2.3 Semiconductor Models 2.3.1...

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2.3 – Semiconductor Models 2.3.1 Electron States in Atoms Quantum Mechanics Hydrogen Atom Multi-Electron Atoms 2.3.2 Semiconductor Bond Model Covalent Bonds 2.3.3 Semiconductor Band Model Energy Bands Band Gap Electron and Holes Band Structure & Effective Mass Simplified Semiconductor Band Model Literature: Pierret, Chapter 2.1-2.3, page 23-40 Pierret, Appendix A, page 733-748 Slides courtesy of Prof. Oliver Brand

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2 2.3.1 Electron States in Atoms – Formalism of Quantum Mechanics – Systems with atomic dimensions , such as the electrons in a semiconductor atom, are described by the quantum mechanics and not the classical “Newtonian” mechanics The quantum mechanics leads to the concept of quantized energy values for the electrons of an atom, necessary to explain e.g. the discrete spectral lines emitted by heated gases The (time-dependent) Schrödinger equation describes the dynamic behavior of a single-particle system, e.g. the behavior of an electron in the potential of the hydrogen H + nucleus m is the particle mass, U the system’s potential energy and The complex wavefunction Ψ = Ψ (x,y,z,t) describes the dynamic behavior of the particle in the potential U 2 2m 2 Ψ + U(x,y,z) Ψ = i ∂Ψ t i = 1
3 Formalism of Quantum Mechanics (cont.) The product Ψ * Ψ dV of the wavefunction Ψ and its complex conjugate Ψ * gives the probability to find the particle in a certain volume dV The expectation values for the particles position and momentum can be calculated from Assuming a single-particle system with fixed total energy E, we can simplify the Schrödinger equation by separating the wave function as follows yielding Ψ * Ψ dV = 1 V ∫∫∫ x = Ψ * x Ψ dV ∫∫∫ p x = Ψ * i ∂Ψ x dV ∫∫∫ Ψ (x,y,z,t) = ψ (x,y,z) e iEt/ 2 2m 2 ψ + E U(x,y,z) [ ] ψ = 0

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4 The Hydrogen Atom Solving the Schrödinger equation for an electron in the electrostatic potential U of the hydrogen nucleus yields the wavefunctions Ψ nlm of the different electron states of the hydrogen atom and their energies The electron state is described by three quantum numbers n,l and m (and spin s) Even mathematically simpler models, such as an electron in a 1-D potential barrier, result in a quantized energy U = q 2 4 πε 0 r E n = 13.6 n 2 eV Note: 1eV = 1.6 · 10 -19 Joules Pierret, Fig. 2.1
5 Multi-Electron Atoms Similar to the hydrogen atom, the electron states in multi- electron atoms are uniquely characterized by 4 quantum numbers n, l, m and s Pauli Exclusion Principle: No two electrons in a system can be characterized by the same set of quantum numbers Thus, at T = 0K, the electron states are filled from the one

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Chapter23_24_Bandstructure - 2.3 Semiconductor Models 2.3.1...

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