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Energy band structure and electrical conductivity mechanisms of metals and semiconductors
- 1. 12/17/2019 1 Energy Band Structure And Electrical Conductivity Mechanisms of Metals And Semiconductors 1 Krishna Jangid PhD Student Department of Engineering Physics
- 2. Energy band structure of Metals and Semiconductors Formation of band (Metal) Categorization of materials 2
- 3. 𝑓 𝐸 = 1 ⅇ 𝐸−𝐸 𝐹 𝑘 𝐵 𝑇+1 Energy band structure of Metals and Semiconductors 3
- 4. Energy band structure of Metals and Semiconductors Metals Semiconductors/Insulators 4 Overlapping of bands in Mg Bands in Silicon
- 5. Electrical conductivity of metals 𝐹 = 𝑚 ⅆ𝑣 ⅆ𝑡 = ℏ ⅆ𝑘 ⅆ𝑡 = −ⅇ𝐸 𝛿𝑘 = − ⅇ𝐸𝑡 ℏ 𝑗 = 𝑛𝑞𝑣 = 𝑛ⅇ2 𝜏 𝐸 𝑚 𝑗 = 𝜎𝐸 𝜌 = 𝑚 𝑛ⅇ2 𝜏𝜎 = 𝑛ⅇ2 𝜏 𝑚 𝑣 = ℏ𝑘 𝑚 𝜏 is the collision time 5 Fermi sphere under the influence of Force F
- 6. Factors affecting conductivity in metals Scattering of electrons by Phonons Normal scattering Umklapp scattering Electrons Role of Pauli exclusion principle Imperfections Temperature Phonon concentration Scattering 𝝈 Scattering 𝝆 6
- 7. Electrical conductivity of semiconductors Intrinsic semiconductors where, 𝑁𝐶 = 2 2𝛱𝑚 𝑛 ∗ 𝑘𝑇 ℏ2 3/2 Concentration of electrons in the conduction band 𝒏 = 𝐸 𝐶 ∞ 𝐷𝑒 𝐸 𝑓𝑒 𝐸 ⅆ𝐸 𝒏 = 𝑁𝑐 exp − 𝐸𝑐 − 𝐸 𝐹 𝑘𝑇 𝒏𝒊 = 2 2𝜋𝑘𝑇 ℎ2 3 2 𝑚 𝑛 ∗ 𝑚 𝑝 ∗ 3 4 exp −𝐸𝑔 2𝑘𝑇 where, mn* and mp* are the effective mass of electrons and holes 7 Electron concentration dependence on temperature: (a) Ge; (b) Si
- 8. Electrical conductivity: ln 𝜎 = − 𝐸 𝑔 2𝑘 1 𝑇 + 3 2 ln 𝑇 + constant𝝈 = 𝜎 𝑛 + 𝜎 𝑝 = ⅇ𝑛𝑖 𝜇 𝑛 + 𝜇 𝑝 = ⅇ 𝜇 𝑛 + 𝜇 𝑝 2 2𝜋𝑘𝑇 3/2 ℎ3 𝑚 𝑛 ∗ 𝑚 𝑝 ∗ 3 4 exp −𝐸𝑔 2𝑘𝑇 Intrinsic semiconductors (cont.) Mobilities were assumed to be independent of temperature As scattering of electrons affects conductivity 8 Electrical conductivity vs Temperature
- 9. Extrinsic semiconductors Electrical conductivity: 𝜎 = ⅇ𝑛𝜇 𝑛 • Mobilities were assumed to be independent of temperature. • Nd is assumed to be constant. 𝜎 = ⅇ𝜇 𝑛 𝑁𝑑 𝑁𝑐 1 2 exp −𝐸𝑔 2𝑘𝑇 𝒍𝒏 𝝈 = −𝐸𝑔 2𝐾 1 𝑻 + 1 2 ln 𝑵 𝒅 𝑁𝑐 + constant Concentration of electrons in the conduction band Carrier concentration: 𝒏 = 𝑵 𝒅 𝑁𝐶 1 2 exp −𝐸𝑔 2𝐾𝑻 9 Carrier concentration dependence over temperature Conductivity dependence over temperature
- 10. References • Solid state physics by Puri and Babbar. • Introduction to solid state physics by Charles kittel. • Nanotechnology: Understanding Small Systems by Ben Rogers. • NPTEL videos.
- 11. Questions.. 10