1. 1 Prof. Ming-Jer Chen Department of Electronics Engineering National Chiao-Tung University 09/20/2012 DEE4521 Semiconductor Device Physics Lecture 2: Lecture 2: Band Structure in Semiconductors Band Structure in Semiconductors

2. 2 ….Standing on the Shoulders of Giants… ….Standing on the Shoulders of Giants… -- Isaac Newton -- Isaac Newton With this in mind, then device physics can be With this in mind, then device physics can be interesting and useful! interesting and useful!

3. 3 We see Semiconductors in x-y-z Space but but Electrons and Holes and Phonons and Photons also Live in another Space: k x -k y -k z Space or Wavevector Space or Wavevector Space or Momentum Space or Momentum Space Remember: Our EE’s Terminologies like V and I want us to see Semiconductors in this additional space as well.

4. SOP for Band Structure 4 1.Think a Photoelectric Effect Experiment by Einstein  Potential Energy of Interest 2. Degree of Freedom (DOF) and Kinetic Energy 3. Combine Newton’s Mechanisms and De Broglie’s Hypothesis Then, You have Conduction-Band Structure Again think a Photoelectric Experiment with High Energy Photons Or A Photon Transparent Experiment through a Very Thin Sample Finally, You Get Energy Gap and hence a Valence-Band Structure

5. by Analogy 5 1.De Broglie’s Wave and Particle Duality 2.Degree of Freedom (DOF) – Kinetic Energy 3.Potential Energy and its Reference

6. Effective Mass m* Crystal momentum E k = ħ 2 k x 2 /2m* Ball’s Mass m in x direction Ball’s Momentum mv x Ball’s Kinetic Energy mv x 2 /2 Electron Effective Mass m x * in x direction Crystal Momentum ħk x (k x : wave vector in x direction) Electron Momentum ħ(k x -k xo ) Electron Kinetic Energy E k = ħ 2 (k x -k xo ) 2 /2m x * 1. k xo : a point in k space around which electrons are likely found. 2. Crystal momentum (global) must be conserved in k space, not Electron Momentum (local). A ball in the air Electrons in Solid 6

7. 7 + ħ 2 k z 2 /2m  * + ħ 2 k x 2 /2m  * Effective mass approximation: m* (to reflect electron confinement in solid) E k = ħ 2 (k y – k cy ) 2 /2m  * Si Conduction-Band Structure in wave vector k-space (silicon) K cy  0.85 (2  /a); Longitudinal Effective Mass m  * (or m l *)= 0.92 m o Transverse Effective Mass m  * (or m t *)= 0.197 m o a: Lattice Constant 6-fold valleys Ellipsoidal energy surface (Constant-Energy Surfaces in k-space) E = E k + E c total electron energy Potential energy Kinetic energy

8. 8 (by Prof. Robert F. Pierret) Effective Masses of Commonly Used Materials Ge Si GaAs m l */m o 1.588 0.916 m t */m o 0.081 0.190 m e */m o 0.067 m hh */m o 0.347 0.537 0.51 m lh */m o 0.0423 0.153 0.082 m so */m o 0.077 0.234 0.154 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 m o = 0.91  10 -30 kg (You may then find that these effective masses are far from the rest mass. This is just one of the quantum effects.)

9. 9 Electron Energy E-k Relation in a Crystal Diamond a = 5.43095 Å Diamond a = 5.64613 Å Zinc blende a = 5.6533 Å ( )2  /a Quasi-Classical Approximation Bottom of valley

10. 10 k-Space Definition The zone center (Gamma at k = 0) The zone end along (in-plane) (out-of-plane) (in-plane) (001) Length = 2  /a (Gamma to X) Length =( )2  /a (Gamma to L) 3-D View (Principal-axis x, y, and z coordinate system usually aligned to match the k coordinate system) On (001) Wafer a: Lattice Constant

11. 11 Electron E-k Diagram Indirect gap Direct gap E G : Energy Gap

12. 12 Conduction Band 8-fold valleys along (half-ellipsoid in Brillouin) 6-fold valleys along (ellipsoid) one valley at the zone center (sphere) (Constant-Energy Surface) Comparisons between Different Materials

13. 13 Valence Band Structure

14. 14 Conduction-Band Electrons and Valence-Band Holes Hole: Vacancy of Valence-Band Electron

15. 15 No Electrons in any Conduction Bands All Valence Bands are filled up.

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17. 17  (Electron Affinity) (= 4.05 eV for Si) Work Function Ec E x