Transport Phenomena in Solid State Physics
This lecture covers topics on transport phenomena in solid state physics, focusing on key concepts like the Hall effect, magnetoresistance, and the relaxation time approximation in Boltzmann's equation. The discussion delves into the Fermi-Dirac distribution, response coefficients, and responses involving thermal and electrical gradients. Detailed explanations and theoretical calculations are provided, offering a comprehensive understanding of transport theory in the context of solid state materials.
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Presentation Transcript
PHY 752 Solid State Physics 11-11:50 AM MWF Olin 107 Plan for Lecture 23: Transport phenomena Chap. 17 & 18 in Marder Hall effect Magnetoresistance Microscopic picture Contains materials from Marder s lecture notes 3/25/2015 PHY 752 Spring 2015 -- Lecture 24 1
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3/25/2015 PHY 752 Spring 2015 -- Lecture 24 3
3/25/2015 PHY 752 Spring 2015 -- Lecture 24 4
Transport theory in the relaxation time approximation Boltzmann s equation: Relaxation time approximation to collision term Fermi-Dirac distribution 3/25/2015 PHY 752 Spring 2015 -- Lecture 24 5
Transport theory in the relaxation time approximation General solution to linear order: Estimation of resulting current: 3/25/2015 PHY 752 Spring 2015 -- Lecture 24 6
Calculation of response coefficients General form of response coeffients: where: 3/25/2015 PHY 752 Spring 2015 -- Lecture 24 7
Responses involving thermal and electrical gradients Electrochemical force: Electrochemical flux: Thermal force: Thermal flux: 3/25/2015 PHY 752 Spring 2015 -- Lecture 24 8
Responses involving thermal and electrical gradients continued Linear coefficients: Note that these can be calculated from: where 3/25/2015 PHY 752 Spring 2015 -- Lecture 24 9
Responses involving thermal and electrical gradients continued Define: Note that: ( ) E E d E ( ) F d = E E F (See Chap. 6) 3/25/2015 PHY 752 Spring 2015 -- Lecture 24 10
Example: Thermal conductivity Consider the case where there is heat flow but no current: 0 j = 3/25/2015 PHY 752 Spring 2015 -- Lecture 24 11
Example: Thermal conductivity -- continued 3/25/2015 PHY 752 Spring 2015 -- Lecture 24 12
Example: Hall effect 3/25/2015 PHY 752 Spring 2015 -- Lecture 24 13
Example: Hall effect -- continued 3/25/2015 PHY 752 Spring 2015 -- Lecture 24 14
Example: Hall effect -- continued 3/25/2015 PHY 752 Spring 2015 -- Lecture 24 15
Example: Hall effect -- continued 3/25/2015 PHY 752 Spring 2015 -- Lecture 24 16
Example: Hall effect -- continued where: 3/25/2015 PHY 752 Spring 2015 -- Lecture 24 17
Example: Hall effect -- continued Hall coefficient: 1 for electron carriers nec = R H 1 for hole carriers pec 3/25/2015 PHY 752 Spring 2015 -- Lecture 24 18
Microscopic models of conduction (Chap. 18 of Marder) Probability of electron scattering event k k distribution functions transition probability 3/25/2015 PHY 752 Spring 2015 -- Lecture 24 19
Microscopic models of conduction -- continued Estimate transition rate using Fermi Golden rule: Electron scattering potential Number of scatters Assume: 3/25/2015 PHY 752 Spring 2015 -- Lecture 24 20
Microscopic models of conduction -- continued Approximate evaluation of integral: 3/25/2015 PHY 752 Spring 2015 -- Lecture 24 21
