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DEE4521 Semiconductor Device Physics Lecture 2: Band Structure Prof. Ming-Jer Chen Department of Electronics Engineering National Chiao-Tung University 09/24/2013 1 Electron Distribution Function f(x, y, z, kx, ky, kz, t) According to Heisenberg’s Uncertainty Principle, We have a 6 dimensionality space at a time for a Semiconductor in a Real x-y-z Space; and at each point (x,y.z), Electrons, Holes, Phonons, and Photons are all better dealt with in another space: kx-ky-kz Space or Wavevector Space or Momentum Space 2 by Analogy • De Broglie’s Wave and Particle Duality • Degree of Freedom (DOF) – Kinetic Energy • Potential Energy and its Reference 3 Electrons in Solid A ball in the air Ball’s Mass m in x direction Electron Effective Mass mx* in x direction Ball’s Momentum mvx Crystal Momentum ħkx (kx: wave vector in x direction) Effective Mass m* Electron Momentum ħ(kx-kxo) Ball’s Kinetic Energy mvx 2/2 Crystal momentum Electron Kinetic Energy Ek = 2 2 E = ħ kx /2m* ħ2(kkx-kxo)2/2m x* 1. kxo: a point in k space around which electrons are likely found. 2. Crystal momentum (global) must be conserved in k space, not Electron 4 Momentum (local). Si Conduction-Band Structure in wave vector k-space (Constant-Energy Surfaces in k-space)Effective mass approximation: Kinetic energy m* (to reflect electron confinement in solid) Ek = ħ2(ky – kcy)2/2m* + ħ2kx2/2m* + ħ2kz2/2m* Ellipsoidal energy surface (silicon) E = Ek + Ec 6-fold valleys Potential energy total electron energy Kcy 0.85 (2/a); Longitudinal Effective Mass m* (or ml*)= 0.92 mo Transverse Effective Mass m* (or mt*)= 0.197 mo a: Lattice Constant 5 Effective Masses of Commonly Used Materials (You may then find that these effective masses are far from the rest mass. This is just one of the quantum effects.) Electron and hole effective mass are anisotropic, depending on the orientation direction. Electron (not hole) effective mass is isotropic, regardless of orientation. Rest mass of electron mo (by Prof. Robert F. Pierret) = 0.9110-30 kg Ge Si GaAs ml*/mo 1.588 0.916 mt*/mo 0.081 0.190 me*/mo 0.067 mhh*/mo 0.347 0.537 0.51 mlh*/mo 0.0423 0.153 0.082 mso*/mo 0.077 0.234 0.154 6 Electron Energy E-k Relation in a Crystal Zinc blende a = 5.6533 Å Diamond a = 5.43095 Å Quasi-Classical Approximation Diamond a = 5.64613 Å ( 3/2 )2/a 1 d 2 E 2 m* dK 2 K 0 Bottom of valley 7 k-Space Definition <001> 3-D View (out-of-plane) The zone center (Gamma at k = 0) The zone end along <100> On (001) Wafer <100> (in-plane) Length = 2/a (Gamma to X) <010> (in-plane) Length =( 3/2 )2/a (Gamma to L) (001) The zone end along <111> a: Lattice Constant (Principal-axis x, y, and z coordinate system usually aligned to match the k coordinate system) 8 Electron E-k Diagram Indirect gap Direct gap EG: Energy Gap 9 Comparisons between Different Materials Conduction Band (Constant-Energy Surface) 8-fold valleys along <111> (half-ellipsoid in Brillouin) one valley at the zone center (sphere) 6-fold valleys along <100> (ellipsoid) 10 Valence-Band Structure 11 Conduction-Band Electrons and Valence-Band Holes and Electrons Hole: Vacancy of Valence-Band Electron 12 No Electrons in Conduction Bands All Valence Bands are filled up. 13 14 Work Function E (Electron Affinity) (= 4.05 eV for Si) Ec x 15