Semiconductor Electronics - Chapter 1 Electromagnetic Wave Interaction with Solids PDF

Chapter 1: Electromagnetic Wave Interaction with Solids Semiconductor Electronics / Dr. -Ing Andreas Bablich 11. April 2023 uni-siegen.de www.uni-siegen.de Chapter 1: Electromagnetic Wave Interaction with Solids Conservation of Mechanical Quantities...

Chapter 1: Electromagnetic Wave Interaction with Solids Semiconductor Electronics / Dr. -Ing Andreas Bablich 11. April 2023 uni-siegen.de www.uni-siegen.de Chapter 1: Electromagnetic Wave Interaction with Solids Conservation of Mechanical Quantities Semiconductor Electronics 11. April 2023 2 Chapter 1: Electromagnetic Wave Interaction with Solids Harmonic Waves Semiconductor Electronics 11. April 2023 3 Chapter 1: Electromagnetic Wave Interaction with Solids Light Waves Planck: Light is not a continuous wave, but a stream of photons. Photon: 𝑊𝑊𝑝𝑝ℎ𝑜𝑜𝑜𝑜 = ћ𝜔𝜔 "A photon is an energy package, a so-called wave packet, propagating with the speed of light. " Semiconductor Electronics 11. April 2023 4 Chapter 1: Electromagnetic Wave Interaction with Solids Spectrum of Electromagnetic Radiation Source: Wikipedia Semiconductor Electronics 11. April 2023 5 Chapter 1: Electromagnetic Wave Interaction with Solids Energy and Momentum of the Photon Semiconductor Electronics 11. April 2023 6 Chapter 1: Electromagnetic Wave Interaction with Solids Wave-Particle Duality 𝑝𝑝 = 𝑚𝑚𝑚𝑚 ℎ 2𝜋𝜋 ℎ De Broglie: 𝑝𝑝 = ћ𝑘𝑘 𝑚𝑚𝑚𝑚 = ћ𝑘𝑘 = = Speed [cm/s] Mass [g] Wavelenght λ [nm] 2𝜋𝜋 𝜆𝜆 𝜆𝜆 Handgun 28000 9,6 2,5 E-26 Assault Rifle 71500 16,5 5,6 E-27 Cannon 175000 21400 1,8 E-26 Housefly 180 0,06 6,2 E-21 ℎ Automobile 2917 1,5 E+06 1,5 E-29 𝜆𝜆 = Matter Wave 𝑚𝑚𝑚𝑚 Airplane 27777 400 E+06 5,9 E-30 Fighter aircraft 51250 9,4 E+06 1,4 E-28 𝑞𝑞 2 For comparison: rElectron (classic): 2 𝐸𝐸 = 𝑚𝑚𝑒𝑒 𝑐𝑐 ≡ 𝐸𝐸 = 4𝜋𝜋𝜀𝜀0 𝑟𝑟 𝑞𝑞 2 ⇒ 𝑟𝑟 = 2 ≈ 2,8 ∗ 10−15 𝑚𝑚 = 2,8 ∗ 10−6 𝑛𝑛𝑛𝑛 4𝜋𝜋𝜀𝜀0 𝑚𝑚𝑒𝑒 𝑐𝑐 Semiconductor Electronics 11. April 2023 7 Chapter 1: Electromagnetic Wave Interaction with Solids Electron Wave 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 (1𝑒𝑒𝑒𝑒) 𝜆𝜆 ≈ 10−7 𝑐𝑐𝑐𝑐 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑐𝑐𝑐𝑐 𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 (𝑆𝑆𝑆𝑆 ) 𝑎𝑎 ≈ 5 ∗ 10−8 𝑐𝑐𝑐𝑐 Semiconductor Electronics 11. April 2023 8 Chapter 1: Electromagnetic Wave Interaction with Solids Source: Wikipedia Semiconductor Electronics 11. April 2023 9 Chapter 1: Electromagnetic Wave Interaction with Solids Interference of Electron Waves in the Crystal Structure Due to the interference of electron waves with the crystal lattice, electrons in the solid cannot take any arbitrary energy values, but only those that lead to wavelengths that do not cancel each other out through interference. Consequently, there must be "allowed" and "forbidden" values of the electron energy in a solid. Semiconductor Electronics 11. April 2023 10 Chapter 1: Electromagnetic Wave Interaction with Solids How to calculate the allowed values of electron energy in a crystal? One needs to derive a wave equation for the electron waves, adjust its general solution to boundary conditions, and determine which values of electron energy allow for solutions. The wave equation for electron waves is the Schrödinger equation. The boundary conditions are determined by the potential profile of the crystal lattice. In the Kronig/Penney model, the potential profile of an idealized, one- dimensional periodic crystal is approximated using a rectangular function. Semiconductor Electronics 11. April 2023 11 Chapter 1: Electromagnetic Wave Interaction with Solids Schrödinger Equation (Time-dependent) Harmonic Wave : 𝛹𝛹(𝑥𝑥, 𝑡𝑡) = 𝛹𝛹0 exp[𝑗𝑗𝑗𝑗(𝑥𝑥, 𝑡𝑡)] Quantum Terms: 𝑘𝑘 = 𝑝𝑝⁄ћ ; 𝜔𝜔 = 𝑊𝑊 ⁄ћ 𝑗𝑗 Electron Wave: 𝛹𝛹(𝑥𝑥, 𝑡𝑡) = 𝛹𝛹0 exp (𝑝𝑝𝑝𝑝 − 𝑊𝑊𝑊𝑊) ћ 𝜕𝜕𝜕𝜕 𝑗𝑗 ћ 𝜕𝜕𝜕𝜕 = − 𝑊𝑊𝑊𝑊 → 𝑊𝑊𝑊𝑊 = − 𝜕𝜕𝜕𝜕 ћ 𝑗𝑗 𝜕𝜕𝜕𝜕 𝜕𝜕 2 𝛹𝛹 𝑗𝑗 2 2 𝑝𝑝2 2 𝜕𝜕 2 𝛹𝛹 2 = 𝑝𝑝 𝛹𝛹 = − 2 𝛹𝛹 → 𝑝𝑝 𝛹𝛹 = −ћ 𝜕𝜕𝑥𝑥 2 ћ ћ 𝜕𝜕𝑥𝑥 2 Energy Conservation Law: 1 2 𝑝𝑝2 𝑊𝑊 = 𝑊𝑊𝑘𝑘𝑘𝑘𝑘𝑘 + 𝑊𝑊𝑝𝑝𝑝𝑝𝑝𝑝 = 𝑚𝑚𝑣𝑣 + 𝑊𝑊𝑝𝑝𝑝𝑝𝑝𝑝 = +𝑊𝑊𝑝𝑝𝑝𝑝𝑝𝑝 ∗ 𝛹𝛹 2 2𝑚𝑚 𝑝𝑝2 𝑊𝑊𝑊𝑊 = 𝛹𝛹+𝑊𝑊𝑝𝑝𝑝𝑝𝑝𝑝 𝛹𝛹 2𝑚𝑚 Schrödinger Equation (Time-dependent): ћ 𝜕𝜕𝜕𝜕 ћ2 𝜕𝜕 2 𝛹𝛹 − =− + 𝑊𝑊𝑝𝑝𝑝𝑝𝑝𝑝 𝛹𝛹 𝑗𝑗 𝜕𝜕𝜕𝜕 2𝑚𝑚 𝜕𝜕𝑥𝑥 2 Semiconductor Electronics 11. April 2023 12 Chapter 1: Electromagnetic Wave Interaction with Solids Schrödinger Equation (Time-independent) ћ 𝜕𝜕𝜕𝜕 ћ2 𝜕𝜕2 𝛹𝛹 Schrödinger Equation (Time-independent):− =− + 𝑊𝑊𝑝𝑝𝑝𝑝𝑝𝑝 𝛹𝛹 𝑗𝑗 𝜕𝜕𝜕𝜕 2𝑚𝑚 𝜕𝜕𝑥𝑥 2 Separation of variables (Bernoulli): 𝛹𝛹 𝑥𝑥, 𝑡𝑡 = 𝛹𝛹 𝑥𝑥 ∗ 𝑓𝑓 𝑡𝑡 ; 𝑊𝑊𝑝𝑝𝑝𝑝𝑝𝑝 = 𝑊𝑊𝑝𝑝𝑝𝑝𝑝𝑝 (𝑥𝑥) ≠ 𝑓𝑓(𝑡𝑡) Electron Wave: 𝑗𝑗 𝑗𝑗 𝑗𝑗 𝑗𝑗 𝛹𝛹 𝑥𝑥, 𝑡𝑡 = 𝛹𝛹0 exp 𝑝𝑝𝑝𝑝 − 𝑊𝑊𝑊𝑊 = 𝛹𝛹0 exp 𝑝𝑝𝑝𝑝 ∗ exp 𝑊𝑊𝑊𝑊 = 𝛹𝛹(𝑥𝑥)exp − 𝑊𝑊𝑊𝑊 ћ ћ ћ ћ 𝜕𝜕𝜕𝜕 𝑗𝑗 𝑗𝑗 = 𝛹𝛹(𝑥𝑥) − 𝑊𝑊 exp − 𝑊𝑊𝑊𝑊 𝜕𝜕𝜕𝜕 ћ ћ 𝜕𝜕 2 𝛹𝛹 𝜕𝜕 2 𝛹𝛹(𝑥𝑥) 𝑗𝑗 = exp − 𝑊𝑊𝑊𝑊 𝜕𝜕𝑥𝑥 2 𝜕𝜕𝑥𝑥 2 ћ Schrödinger Equation (Time-independent): 𝑗𝑗 𝑗𝑗 ћ2 𝜕𝜕 2 𝛹𝛹(𝑥𝑥) − 𝛹𝛹 𝑥𝑥 − 𝑊𝑊 = − + 𝑊𝑊𝑝𝑝𝑝𝑝𝑝𝑝 𝛹𝛹(𝑥𝑥) ћ ћ 2𝑚𝑚 𝜕𝜕𝑥𝑥 2 𝜕𝜕 2 𝛹𝛹(𝑥𝑥) 2𝑚𝑚 + 2 𝑊𝑊 − 𝑊𝑊𝑝𝑝𝑝𝑝𝑝𝑝 (𝑥𝑥) 𝛹𝛹(𝑥𝑥) = 0 𝜕𝜕𝑥𝑥 2 ћ Semiconductor Electronics 11. April 2023 13 Chapter 1: Electromagnetic Wave Interaction with Solids Potential Energy in Crystals Point Charge Q: 𝑄𝑄 1 𝑉𝑉 (𝑥𝑥) = Potential 4𝜋𝜋 𝜀𝜀 0 𝜀𝜀 𝑟𝑟 𝑥𝑥 𝑑𝑑𝑑𝑑 𝑄𝑄 1 𝐸𝐸 (𝑥𝑥) = − = Field 𝑑𝑑𝑑𝑑 4𝜋𝜋𝜀𝜀 0 𝜀𝜀 𝑟𝑟 𝑥𝑥 2 𝑄𝑄 1 𝐹𝐹 (𝑥𝑥) = −𝑞𝑞𝑞𝑞(𝑥𝑥) = −𝑞𝑞 4𝜋𝜋𝜀𝜀 𝑥𝑥 2 Force (Coulomb) 0 𝜀𝜀 𝑟𝑟 𝑥𝑥 𝑥𝑥 𝑄𝑄 1 𝑄𝑄 1 𝑥𝑥 𝑊𝑊𝑝𝑝𝑝𝑝𝑝𝑝 (𝑥𝑥) = − 𝐹𝐹 (𝑥𝑥)𝑑𝑑𝑑𝑑 = +𝑞𝑞 𝑑𝑑𝑑𝑑 = 𝑞𝑞 − = −𝑞𝑞𝑞𝑞(𝑥𝑥) 4𝜋𝜋𝜀𝜀0 𝜀𝜀𝑟𝑟 𝑥𝑥 2 4𝜋𝜋𝜀𝜀0 𝜀𝜀𝑟𝑟 𝑥𝑥 ∞ ∞ ∞ 𝑊𝑊𝑝𝑝𝑝𝑝𝑝𝑝 (𝑥𝑥) = −𝑞𝑞𝑞𝑞(𝑥𝑥) Potential Energy Semiconductor Electronics 11. April 2023 14 Chapter 1: Electromagnetic Wave Interaction with Solids One-dimensional Periodic Crystal April 11, 2023 Semiconductor Electronics 15 Chapter 1: Electromagnetic Wave Interaction with Solids Rectangular Potential Approximation, Kronig/Penney-Model Semiconductor Electronics 11. April 2023 16 Chapter 1: Electromagnetic Wave Interaction with Solids Kronig/Penney-Model... one-dimensional periodic Crystal Region 1 (0 < 𝑥𝑥 < 𝑎𝑎): Region 2 (−𝑏𝑏 < 𝑥𝑥 < 0): 𝑊𝑊𝑝𝑝𝑝𝑝𝑝𝑝 = 0 𝑊𝑊𝑝𝑝𝑝𝑝𝑝𝑝 = 𝑊𝑊0 Schrödinger Equation: 𝜕𝜕 2 𝛹𝛹1 (𝑥𝑥) 2𝑚𝑚 𝜕𝜕 2 𝛹𝛹2 (𝑥𝑥) 2𝑚𝑚 + 2 𝑊𝑊 𝛹𝛹1 (𝑥𝑥) = 0 + 2 (𝑊𝑊 − 𝑊𝑊0 ) 𝛹𝛹2 (𝑥𝑥) = 0 𝜕𝜕𝑥𝑥 2 ћ 𝜕𝜕𝑥𝑥 2 ћ =𝛼𝛼 2 =−𝛽𝛽 2 𝜕𝜕 2 𝛹𝛹1 (𝑥𝑥) 𝜕𝜕 2 𝛹𝛹2 (𝑥𝑥) + 𝛼𝛼 2 𝛹𝛹1 (𝑥𝑥) = 0 − 𝛽𝛽 2 𝛹𝛹2 (𝑥𝑥) = 0 𝜕𝜕𝑥𝑥 2 𝜕𝜕𝑥𝑥 2 Solution: invoice in the script 𝛹𝛹1 (𝑥𝑥) = 𝛾𝛾1 (𝑥𝑥)𝑒𝑒𝑒𝑒𝑒𝑒(𝑗𝑗 𝑘𝑘𝑘𝑘) 𝛹𝛹2 (𝑥𝑥) = 𝛾𝛾2 (𝑥𝑥)𝑒𝑒𝑒𝑒𝑒𝑒(𝑗𝑗 𝑘𝑘𝑘𝑘) Chapter 2.3.1, 𝛾𝛾1 (𝑥𝑥), 𝛾𝛾2 (𝑥𝑥) Bloch Function equations (17.a), (17.b) 𝛾𝛾1′ ′ + 2𝑗𝑗𝑗𝑗𝛾𝛾1′ + (𝛼𝛼 2 − 𝑘𝑘 2 )𝛾𝛾1 = 0 𝛾𝛾2′ ′ + 2𝑗𝑗𝑗𝑗𝛾𝛾2′ − (𝛽𝛽 2 − 𝑘𝑘 2 )𝛾𝛾2 = 0 Two linear homogeneous differential equations arise for the Bloch functions. Semiconductor Electronics 11. April 2023 17 Chapter 1: Electromagnetic Wave Interaction with Solids Linear Homogeneous Second-order Differential Equation With Constant Coefficients Differential Equation: 𝑦𝑦 ′ ′ + 𝑝𝑝𝑦𝑦 ′ + 𝑞𝑞𝑞𝑞 = 0 Solution approach: 𝑦𝑦 = 𝑒𝑒 𝑟𝑟𝑟𝑟 → 𝑦𝑦 ′ = 𝑟𝑟𝑟𝑟 𝑟𝑟𝑟𝑟 , 𝑦𝑦 ′ ′ = 𝑟𝑟 2 𝑒𝑒 𝑟𝑟𝑟𝑟 → 𝑟𝑟 2 𝑒𝑒 𝑟𝑟𝑟𝑟 + 𝑝𝑝𝑝𝑝𝑝𝑝 𝑟𝑟𝑟𝑟 + 𝑞𝑞𝑞𝑞 𝑟𝑟𝑟𝑟 = 0 | ∗ 𝑒𝑒 −𝑟𝑟𝑟𝑟 Characteristic Equation: 𝑟𝑟 2 + 𝑝𝑝𝑝𝑝 + 𝑞𝑞 = 0 1⁄2 𝑝𝑝 𝑝𝑝 2 𝑟𝑟1,2 = − ± − 𝑞𝑞 2 2 Solution: 𝑦𝑦 = 𝐴𝐴 ∗ 𝑒𝑒 𝑟𝑟1 𝑥𝑥 + 𝐵𝐵 ∗ 𝑒𝑒 𝑟𝑟2 𝑥𝑥 Semiconductor Electronics 11. April 2023 18 Chapter 1: Electromagnetic Wave Interaction with Solids Solutions of the Equations for Bloch functions Region 1 (0 < 𝑥𝑥 < 𝑎𝑎): Region 2 (−𝑏𝑏 < 𝑥𝑥 < 0): General solution: 𝛾𝛾1 (𝑥𝑥) = 𝐴𝐴1 𝑒𝑒 𝑗𝑗 (𝛼𝛼−𝑘𝑘)𝑥𝑥 + 𝐵𝐵1 𝑒𝑒 −𝑗𝑗 (𝛼𝛼 −𝑘𝑘)𝑥𝑥 𝛾𝛾2 (𝑥𝑥) = 𝐴𝐴2 𝑒𝑒 (𝛽𝛽−𝑗𝑗𝑗𝑗 )𝑥𝑥 + 𝐵𝐵2 𝑒𝑒 −(𝛽𝛽+𝑗𝑗𝑗𝑗 )𝑥𝑥 Undamped oscillation Damped oscillation Calculation in the script chapter 2.3.1 Boundary conditions: Equations (21.a)-(21.d) 𝑥𝑥 = 0: 𝛾𝛾1 (0) = 𝛾𝛾2 (0); 𝛾𝛾1′ (0) = 𝛾𝛾2′ (0) 𝑥𝑥 = 𝑎𝑎: 𝛾𝛾1 (𝑎𝑎) = 𝛾𝛾2 (𝑎𝑎); 𝛾𝛾1′ (𝑎𝑎) = 𝛾𝛾2′ (𝑎𝑎) Determination of the constants: A linear system of equations with 4 equations and 4 unknowns arises Semiconductor Electronics 11. April 2023 19 Chapter 1: Electromagnetic Wave Interaction with Solids Linear System of Equations System of Equations: 𝑘𝑘11 𝐶𝐶1 + 𝑘𝑘12 𝐶𝐶2 + ⋯ 𝑘𝑘1𝑛𝑛 𝐶𝐶𝑛𝑛 = 0 𝑘𝑘21 𝐶𝐶1 + 𝑘𝑘22 𝐶𝐶2 + ⋯ 𝑘𝑘2𝑛𝑛 𝐶𝐶𝑛𝑛 = 0 ⋮ ⋮ ⋮ 𝑘𝑘𝑛𝑛1 𝐶𝐶1 + 𝑘𝑘𝑛𝑛2 𝐶𝐶2 + ⋯ 𝑘𝑘𝑛𝑛𝑛𝑛 𝐶𝐶𝑛𝑛 = 0 The system of equations only has non-zero solutions 𝐶𝐶1 … 𝐶𝐶𝑛𝑛 ≠ 0 if the coefficient determinant 𝛥𝛥𝑘𝑘 is zero. 𝛥𝛥𝑘𝑘 = 0 Coefficient Determinant: 𝑘𝑘11 … 𝑘𝑘1𝑛𝑛 𝛥𝛥𝑘𝑘 = ⋮ ⋮ =0 𝑘𝑘𝑛𝑛1 … 𝑘𝑘𝑛𝑛𝑛𝑛 The solutions of the differential equations for the Bloch functions with the boundary conditions given by Bloch's theorem lead to an eigenvalue problem. The eigenvalues are the solutions of the coefficient determinant; for these eigenvalues, there exist eigenfunctions. However, we are not interested in the eigenfunctions themselves, but only in the eigenvalues for which solutions exist. That is, we seek to determine the values of electron energy for which the Schrödinger equation has solutions. Semiconductor Electronics 11. April 2023 20 Chapter 1: Electromagnetic Wave Interaction with Solids Eigenvalue Equation for Square-Wave Energies After calculation (chapter 2.3.1), the eigenvalue equation can be determined by setting the coefficient determinant to zero. 𝛽𝛽 2 − 𝛼𝛼 2 sinh(𝛽𝛽𝛽𝛽) sin(𝛼𝛼𝛼𝛼) + cosh(𝛽𝛽𝛽𝛽) cos(𝛼𝛼𝛼𝛼) = cos[𝑘𝑘(𝑎𝑎 + 𝑏𝑏)] 2𝛽𝛽𝛽𝛽 𝑒𝑒 𝑥𝑥 −𝑒𝑒 −𝑥𝑥 𝑒𝑒 𝑥𝑥 +𝑒𝑒 −𝑥𝑥 A further simplification of the eigenvalue equation results from the transition from rectangular 𝑠𝑠𝑠𝑠𝑠𝑠ℎ = ; 𝑐𝑐𝑐𝑐𝑐𝑐ℎ = 2 2 energies to δ energies. Boundry Conditions: 𝑏𝑏 → 0, 𝑊𝑊0 → ∞, 𝑊𝑊0 𝑏𝑏 = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐. 1 1 vlim[𝛽𝛽𝛽𝛽] = lim ћ 2𝑚𝑚(𝑊𝑊0 − 𝑊𝑊)𝑏𝑏 = lim ћ 2𝑚𝑚(𝑊𝑊0 𝑏𝑏 − 𝑊𝑊𝑊𝑊)√𝑏𝑏 = 0 𝐾𝐾𝐾𝐾 𝐾𝐾𝐾𝐾 𝐾𝐾𝐾𝐾 2𝑚𝑚 2𝑚𝑚 2𝑚𝑚 2𝑚𝑚 lim[𝛽𝛽 2 𝑏𝑏] = lim 2 (𝑊𝑊0 − 𝑊𝑊)𝑏𝑏 = lim 2 𝑊𝑊0 𝑏𝑏 − lim 2 𝑊𝑊𝑊𝑊 = 2 𝑊𝑊0 𝑏𝑏 = 𝛽𝛽 2 𝑏𝑏 𝐾𝐾𝐾𝐾 𝐾𝐾𝐾𝐾 ћ 𝐾𝐾𝐾𝐾 ћ 𝐾𝐾𝐾𝐾 ћ ћ 𝑊𝑊0 ≫ 𝑊𝑊 2𝑚𝑚 Eigenvalue equations for δ energies (Kronig/Penney model): lim[𝛼𝛼 2 𝑏𝑏] = lim 𝑊𝑊𝑊𝑊 = 0 𝐾𝐾𝐾𝐾 𝐾𝐾𝐾𝐾 ћ2 𝐬𝐬𝐬𝐬𝐬𝐬(𝜶𝜶𝜶𝜶) ↪ lim[cosh(𝛽𝛽𝛽𝛽)] = 1 𝑷𝑷 + 𝐜𝐜𝐜𝐜𝐜𝐜(𝜶𝜶𝜶𝜶) = 𝐜𝐜𝐜𝐜𝐜𝐜(𝒌𝒌𝒌𝒌) 𝐾𝐾𝐾𝐾 𝜶𝜶𝜶𝜶 𝛽𝛽 2 − 𝛼𝛼 2 𝛽𝛽 2 𝑏𝑏 − 𝛼𝛼 2 𝑏𝑏 sinh(𝛽𝛽𝛽𝛽) 𝟏𝟏 ↪ lim sinh(𝛽𝛽𝛽𝛽) = lim = 𝜶𝜶 = √𝟐𝟐𝟐𝟐𝟐𝟐 ~ √𝑾𝑾 𝐾𝐾𝐾𝐾 2𝛽𝛽𝛽𝛽 𝐾𝐾𝐾𝐾 2𝛼𝛼 𝛽𝛽𝛽𝛽 ћ 𝛽𝛽 2 𝑏𝑏 − 𝛼𝛼 2 𝑏𝑏 sinh(𝛽𝛽𝛽𝛽) 𝛽𝛽 2 𝑏𝑏 𝛽𝛽 2 𝑏𝑏 𝜷𝜷𝟐𝟐 𝒎𝒎 = lim 2𝛼𝛼 lim 𝛽𝛽𝛽𝛽 = 2𝛼𝛼 ∗1 = 2𝛼𝛼 𝑷𝑷 = 𝒂𝒂𝒂𝒂 ≈ 𝒂𝒂𝒂𝒂 𝟐𝟐 𝑾𝑾𝟎𝟎 ~ 𝑾𝑾𝟎𝟎 𝐾𝐾𝐾𝐾 𝐾𝐾𝐾𝐾 𝟐𝟐 ћ ↪ lim{cos[𝑘𝑘(𝑎𝑎 + 𝑏𝑏)]} = cos(𝑘𝑘𝑘𝑘) 𝐾𝐾𝐾𝐾 𝛽𝛽 2 𝑏𝑏 ⇒ sin(𝛼𝛼𝛼𝛼) + cos(𝛼𝛼𝛼𝛼) = cos(𝑘𝑘𝑘𝑘) 2𝛼𝛼 Semiconductor Electronics 11. April 2023 21 Chapter 1: Electromagnetic Wave Interaction with Solids Semiconductor Electronics 11. April 2023 22 Chapter 1: Electromagnetic Wave Interaction with Solids Semiconductor Electronics 11. April 2023 23 Chapter 1: Electromagnetic Wave Interaction with Solids Width of the allowed energy intervals = f(αa), P = parameter Semiconductor Electronics 11. April 2023 24 Chapter 1: Electromagnetic Wave Interaction with Solids f(ka), P = const. W(k) diagram Simultaneous representation of energy and momentum of the electron in the solid! Semiconductor Electronics 11. April 2023 25 Chapter 1: Electromagnetic Wave Interaction with Solids Free Electron and Crystal Electron 1 1 𝑚𝑚2 𝑣𝑣 2 𝑝𝑝2 ћ2 𝑘𝑘𝑘 (𝑘𝑘𝑘𝑘)2 ћ 2 1 𝑊𝑊𝑘𝑘𝑘𝑘𝑘𝑘 = 𝑚𝑚𝑣𝑣 2 = = = = ~ (𝑘𝑘𝑘𝑘)2 Parabola 2 2 𝑚𝑚 2𝑚𝑚 2𝑚𝑚 2𝑚𝑚 𝑎𝑎 𝑚𝑚 Free Electron Crystal Electron Semiconductor Electronics 11. April 2023 26 Chapter 1: Electromagnetic Wave Interaction with Solids Effective Mass Concept 1 𝑊𝑊𝑘𝑘𝑘𝑘𝑘𝑘 ~ (𝑘𝑘𝑘𝑘)2 𝑚𝑚𝑒𝑒𝑒𝑒𝑒𝑒 An electron in a crystal can be approximated as a free electron in the vicinity of the extrema of the W(k) diagram, with the only parameter that needs to be changed from that of a free electron being its mass. Therefore, an 'effective mass' (𝑚𝑚𝑒𝑒𝑒𝑒𝑒𝑒 ) is introduced to describe the impact of the solid state on the electron. This effective mass summarizes the effects of the crystal on the electron. In result an electron in a crystal is often referred to as a 'quasi-free' electron. This concept is crucial for understanding the electronic properties of materials and has numerous applications in the fields of solid-state physics and materials science. Semiconductor Electronics 11. April 2023 27 Chapter 1: Electromagnetic Wave Interaction with Solids What is a Hole? 1 𝑊𝑊3,𝑘𝑘𝑘𝑘𝑘𝑘 = 𝑚𝑚3,𝑒𝑒𝑒𝑒𝑒𝑒 𝑣𝑣 2 = 𝑊𝑊3 − 𝑊𝑊3 𝑘𝑘𝑘𝑘 = 0 = 𝛥𝛥𝑊𝑊3 = −𝑞𝑞𝛥𝛥𝑉𝑉3 2 1 𝑊𝑊2,𝑘𝑘𝑘𝑘𝑘𝑘 = − 𝑚𝑚3,𝑒𝑒𝑒𝑒𝑒𝑒 𝑣𝑣 2 = 𝑊𝑊2 − 𝑊𝑊2 𝑘𝑘𝑘𝑘 = 0 = −𝛥𝛥𝑊𝑊3 = − −𝑞𝑞𝑞𝑞𝑉𝑉2 = +𝑞𝑞𝛥𝛥𝑉𝑉2 2 Due to the interference of electron waves in a solid, the relationship between the energy and the momentum of the electron can be distorted in such a way that it appears to describe not a negative electron but a positive charge carrier, a "hole." Although "holes" are a theoretical construct, treating them as equally mobile charge carriers alongside electrons facilitates the understanding of electrical processes in semiconductors. Holes can possess other properties that differentiate them from electrons, such as a different effective mass, in addition to having opposite charge. Semiconductor Electronics 11. April 2023 28 Chapter 1: Electromagnetic Wave Interaction with Solids W(k)-Diagram Si and W(k)-Diagram GaAs Semiconductor Electronics 11. April 2023 29 Chapter 1: Electromagnetic Wave Interaction with Solids Effective Mass Approximation for quasi-free electron: 2 cos 𝑥𝑥 ≈ 1 − 𝑥𝑥 2 2 2 1 − cos 𝑥𝑥 ≈ 1 − 1 − 𝑥𝑥 2 = 𝑥𝑥 2 W(k)-Diagram: 𝑥𝑥 ↪ 𝑘𝑘𝑘𝑘 1 1 − cos(𝑘𝑘𝑘𝑘) ≈ (𝑘𝑘𝑘𝑘)2 |∗ 𝛥𝛥𝑊𝑊𝑛𝑛 2 𝑘𝑘𝑘𝑘 𝑞𝑞𝑞𝑞𝑞𝑞𝑞𝑞𝑞𝑞𝑞𝑞𝑞𝑞𝑞𝑞𝑞𝑞 1 𝑝𝑝 2 (𝑘𝑘𝑘𝑘 ) 2 ћ 2 𝛥𝛥𝑊𝑊𝑛𝑛 [1 − cos(𝑘𝑘𝑘𝑘)] ≈ 𝛥𝛥𝑊𝑊 (𝑘𝑘𝑘𝑘)2 ≈ = 𝑛𝑛 2 2𝑚𝑚 𝑛𝑛 ,𝑒𝑒𝑒𝑒𝑒𝑒 2𝑚𝑚 𝑛𝑛 ,𝑒𝑒𝑒𝑒𝑒𝑒 𝑎𝑎 Semiconductor Electronics 11. April 2023 30 Chapter 1: Electromagnetic Wave Interaction with Solids General Expression for Effective Mass The effective mass is inversely proportional to the curvature of the W(k) diagram 1 1 𝑊𝑊 = 𝑚𝑚𝑒𝑒𝑒𝑒𝑒𝑒 𝑣𝑣 2 ↪ 𝑚𝑚𝑒𝑒𝑒𝑒𝑒𝑒 = 2𝑊𝑊 2 2 𝑣𝑣 𝛹𝛹(𝑥𝑥, 𝑡𝑡) = 𝛹𝛹0 exp(𝑗𝑗𝑗𝑗) Matarial Wave 𝜑𝜑 = 𝑘𝑘𝑘𝑘 − 𝜔𝜔𝜔𝜔 Phase 𝜕𝜕𝜕𝜕 𝜔𝜔 𝜕𝜕𝜕𝜕 = 𝑘𝑘𝑘𝑘𝑘𝑘 − 𝜔𝜔𝜔𝜔𝜔𝜔 ≝ 0 ↪ 𝜕𝜕𝜕𝜕 = 𝑘𝑘 𝜔𝜔 1 𝑊𝑊 𝑣𝑣𝑝𝑝ℎ = = 𝑘𝑘 ћ 𝑘𝑘 Phase Velocity 𝑊𝑊 ≠ 𝑓𝑓(𝑘𝑘) 𝜕𝜕𝜕𝜕 1 𝜕𝜕𝜕𝜕 𝑣𝑣𝑔𝑔𝑔𝑔 = = ≈ 𝑣𝑣 Group Velocity 𝜕𝜕𝜕𝜕 ћ 𝜕𝜕𝜕𝜕 𝑊𝑊 = 𝑓𝑓(𝑘𝑘) 1 1 𝑚𝑚𝑒𝑒𝑒𝑒𝑒𝑒 = 2𝑊𝑊 1 𝜕𝜕𝜕𝜕 2 = ћ2 1 𝜕𝜕𝜕𝜕 2 ћ2 𝜕𝜕𝜕𝜕 𝜕𝜕 2 𝑊𝑊 2𝑊𝑊 𝜕𝜕𝜕𝜕 = 𝜕𝜕𝑘𝑘 2 ћ2 𝑚𝑚𝑒𝑒𝑒𝑒𝑒𝑒 = 𝜕𝜕 2 𝑊𝑊 Effective Mass (generally) 𝜕𝜕𝑘𝑘 2 Semiconductor Electronics 11. April 2023 31 Chapter 1: Electromagnetic Wave Interaction with Solids 1 𝜕𝜕𝜕𝜕 2 𝜕𝜕 2 𝑊𝑊 Proof 2𝑊𝑊 𝜕𝜕𝜕𝜕 = 𝜕𝜕𝑘𝑘 2 𝜕𝜕 1 𝜕𝜕𝜕𝜕 2 1 𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 = = 𝜕𝜕𝜕𝜕 2 𝜕𝜕𝜕𝜕 2 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 1 𝜕𝜕 2 𝑊𝑊 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕 2 𝑊𝑊 𝜕𝜕𝜕𝜕 𝜕𝜕 2 𝑊𝑊 = 2 ∗ + ∗ = ∗ 2 𝜕𝜕𝑘𝑘 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝑘𝑘 2 𝜕𝜕𝜕𝜕 𝜕𝜕𝑘𝑘 2 𝜕𝜕 1 𝜕𝜕𝜕𝜕 2 𝜕𝜕𝜕𝜕 𝜕𝜕 2 𝑊𝑊 = ∗ ∗ 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 2 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝑘𝑘 2 1 𝜕𝜕𝜕𝜕 2 𝜕𝜕 2 𝑊𝑊 𝜕𝜕 = 𝜕𝜕𝜕𝜕 2 𝜕𝜕𝜕𝜕 𝜕𝜕𝑘𝑘 2 1 𝜕𝜕𝜕𝜕 2 𝜕𝜕 2 𝑊𝑊 = 𝑊𝑊 2 𝜕𝜕𝜕𝜕 𝜕𝜕𝑘𝑘 2 1 𝜕𝜕𝜕𝜕 2 𝜕𝜕 2 𝑊𝑊 = 2𝑊𝑊 𝜕𝜕𝜕𝜕 𝜕𝜕𝑘𝑘 2 Semiconductor Electronics 11. April 2023 32 Chapter 1: Electromagnetic Wave Interaction with Solids Bandgap 𝑇𝑇 2 𝑇𝑇1 Si: 𝑇𝑇1 = 3.120𝐾𝐾 𝑇𝑇2 = 636𝐾𝐾 𝛥𝛥𝛥𝛥(𝑇𝑇) = 𝛥𝛥𝛥𝛥(𝑇𝑇 = 0) − 𝑇𝑇+𝑇𝑇2 eV GaAS: 𝑇𝑇1 = 1.850𝐾𝐾 𝑇𝑇2 = 204𝐾𝐾 Semiconductor Electronics 11. April 2023 33 Chapter 1: Electromagnetic Wave Interaction with Solids Silicon 0 Electron in Conduction Band 3 Electron in Conduction Band 0 Holes in Valence Band 3 Holes in Valence Band Semiconductor Electronics 11. April 2023 34 Chapter 1: Electromagnetic Wave Interaction with Solids Max Planck Max Karl Ernst Ludwig Planck (born April 23, 1858 in Kiel – died October 4, 1947 in Göttingen) was an important German physicist and Nobel Prize winner for physics. The German physicist was one of the most important natural scientists of the 20th century. He was the founder of quantum physics, which transitioned from classical to modern physics in 1900. Before that, Max Planck, based on thermodynamics (1894), dealt with the thermal radiation of "black bodies", which absorb all frequencies of electromagnetic radiation and emit them again when heated. In 1900 he developed the exact law of black thermal radiation, the "Planckian radiation network" for the entire frequency range by interpolating between the radiation laws. In 1905, Albert Einstein applied quantum theory for the first time to the photoelectric effect, which had not been explained by classical physics until then. After the Second World War, the Kaiser Wilhelm Society was renamed the Max Planck Society for the Advancement of Science in his honour. Semiconductor Electronics 11. April 2023 35 Chapter 1: Electromagnetic Wave Interaction with Solids Erwin Schrödinger Erwin Rudolf Josef Alexander Schrödinger (born August 12, 1887 in Vienna-Erdberg; † January 4, 1961 in Vienna-Erdberg) was an Austrian physicist and philosopher of science. He is considered one of the founders of quantum mechanics and received the Nobel Prize in Physics together with Paul Dirac in 1933 for the discovery of new productive forms of atomic theory. The Austrian physicist is one of the fathers of quantum physics. In 1926, Erwin Schrödinger formulated the "Schrödinger equation" named after him. It forms one of the axiomatic foundations of quantum mechanics and describes the temporal development of quantum mechanical systems. Schrödinger was awarded the Nobel Prize in Physics. In his treatise "What is Life?" he introduces the concept of negentropy and thus made a lasting contribution to the development of molecular biology by explaining biological topics in physical terms... Semiconductor Electronics 11. April 2023 36 Chapter 1: Electromagnetic Wave Interaction with Solids Albert Einstein Albert Einstein (March 14, 1879 in Ulm, Germany – April 18, 1955 in Princeton, USA) was a theoretical physicist. His contributions significantly changed the physical world view; In 1999, 100 leading physicists voted him the greatest physicist of all time. During his life he held partially overlapping citizenships of the following nations: Germany (1879-1896 and 1914-1933), Switzerland (1901-55), Austria (1911-1912), USA (1940-1955).Einstein's main work is the theory of relativity, which revolutionized the understanding of space and time. In 1905 his work entitled On the electrodynamics of moving bodies was published, the content of which is now referred to as the special theory of relativity. In 1916 Einstein published the general theory of relativity. He also made significant contributions to quantum physics: for his explanation of the Albert Einstein Formula photoelectric effect, which he also published in 1905, he was awarded the 1921 Nobel E=mc2 Prize in Physics in November 1922. Contrary to popular belief, his theoretical work played only an indirect role in the construction of the atomic bomb and the development of nuclear energy. Albert Einstein is considered the epitome of the researcher and genius. However, he also used his extraordinary reputation outside of the scientific community in his commitment to international understanding and peace. In this context he saw himself as a pacifist, socialist and Zionist. Semiconductor Electronics 11. April 2023 37 Chapter 1: Electromagnetic Wave Interaction with Solids Louis-Victor de Broglie Louis-Victor Pierre Raymond de Broglie (born August 15, 1892 in Dieppe, Normandy – died March 19, 1987 in Louveciennes, Yvelines department) was a French physicist. He belonged to the French noble family of the Broglies and was the younger brother of the experimental physicist Maurice de Broglie. De Broglie is considered one of the most important physicists of the 20th century, for his discovery of the wave nature of the electron (wave-particle duality) in his dissertation Recherches sur la theorie des Quanta and the resulting theory of matter waves, he received the 1929 Nobel Prize in Physics. Semiconductor Electronics 11. April 2023 38 Chapter 1: Electromagnetic Wave Interaction with Solids Ralph Kronig, William Penney Ralph de Laer Kronig, born Ralph Kronig, (March 10, 1904 in Dresden – November 16, 1995) was a German-American theoretical physicist. He discovered particle spin before Uhlenbeck and Goudsmit, but did not publish this. Kronig was involved in the early development of quantum mechanics and its application, e.g. in atomic physics and molecular physics. The Kronig-Penney model and the Kramers-Kronig relation go back to him. Sir William George Penney, Baron Penney (June 24, 1909 – March 3, 1991) was a British physicist. Penney is considered the father of the British atomic bomb. In 1946 he was elected a member ("Fellow") of the Royal Society, which awarded him the Rumford Medal "in recognition of his distinguished and paramount personal contribution to the establishment of economic nuclear energy in Great Britain" in 1966. The Kronig-Penney model goes back to him. Semiconductor Electronics 11. April 2023 39 Chapter 1: Electromagnetic Wave Interaction with Solids Felix Bloch Felix Bloch (October 23, 1905 in Zurich – September 10, 1983 in Zurich) was a Swiss-American physicist. He received the Nobel Prize in Physics in 1952. Between 1924 and 1927 Bloch studied mathematics and physics at the ETH Zurich. He then continued his studies in Leipzig with Werner Heisenberg. The subject of his diploma thesis was the Schrödinger equation. The doctoral thesis dealt with the behavior of electrons in crystal lattices and was the starting point for his life's work: the quantum mechanical treatment of solid state physics, to the foundations of which he made many contributions, such as the band model of electrons in solid bodies and the Bloch function. Semiconductor Electronics 11. April 2023 40 Chapter 1: Electromagnetic Wave Interaction with Solids Werner Heisenberg Werner Karl Heisenberg (* December 5, 1901 in Würzburg; † February 1, 1976 in Munich) was one of the most important physicists of the 20th century and Nobel Prize winner. In 1927 he formulated the Heisenberg uncertainty principle, which was named after him and which meets one of the fundamental statements of quantum mechanics - namely that certain parameters of a particle (such as its position and momentum) cannot be determined with any degree of accuracy at the same time. The German physicist published the most important discovery since the concept of quantum theory: the "uncertainty principle". For example, the "Heisenberg uncertainty notation" states that the location and momentum (i.e. the product of mass and velocity) of an atomic particle can never be determined with absolute accuracy. Accordingly, the more precise the measurement of the spatial coordinates, the less precise the determination of the momentum components and vice versa. Awarded the Nobel Prize in Physics in 1932, he made the search for the "world formula" the goal of his work from 1953, while Werner Heisenberg also dealt with the social and political difficulties of physics in a philosophical and popular scientific manner. On June 16, 2000, a memorial stone for Werner Heisenberg was unveiled on the Heligoland Oberland, who had the idea for formulating quantum mechanics there in June 1925. Semiconductor Electronics 11. April 2023 41