6.730 Physics for Solid State Applications
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1 6.730 Physics for Solid State Applications Lecture 29: Electron-phonon Scattering Outline Bloch Electron Scattering Deformation Potential Scattering LCAO Estimation of Deformation Potential Matrix Element for Electron-Phonon Scattering Energy and Momentum Conservation April 21, 2004 Lundstrom, Chapter 2 General Scattering Potential final state energy is greater than initial absorption final state energy is less than initial emission Lundstrom, Sec 1.7 1
2 Matrix Elements for Bloch States Approximation for slowly varying scattering potential Overlap integral for n=n, and k=k Scattering from a Slowly Varying Potential Matrix element is just the Fourier component of the scattering potential at For more quantitative work will need to evaluate overlap integral. 2
3 Scattering Rate Calculations Example: 1-D 1 D Scattering from Traveling Wave Periodic potentials conserve total momentum.. Scattering Rate Calculations Overview Step 1: Determine Scattering Potential Step 2: Calculate Matrix Elements Step 3: Calculate State-State Transition Rates Step 4: Calculate State Lifetime Step 5: Calculate Ensemble Lifetime 3
4 Electron-Phonon Scattering Potential Beyond the Born-Oppenheimer Approximation Phonons change the electron energies by changing the bond displacement Both shear strain and local volume changes alter the electron energy change in the bandstructure due to a dilatation of solid by sound wave Relate the phonons to local changes in the volume (lattice constant). Lundstom, Electron-Phonon Scattering Potential Only LA phonons cause local changes in the volume (lattice constant). 4
5 Electron-Phonon Scattering Potential Conduction band (diamond): Valence band (diamond): Electron-Phonon Scattering Potential 5
6 Electron-Phonon Scattering Potential Silicon Example From Lecture 12: Operators for the Lattice Displacement We will now use this for electron-phonon scattering 6
7 Phonon Displacement Operator See Lecture 12 phonon displacement operator, with time dependence, relating mass for continuum solid and discrete lattice Scattering Rate Calculations Overview Step 1: Determine Scattering Potential Step 2: Calculate Matrix Elements Step 3: Calculate State-State Transition Rates Step 4: Calculate State Lifetime Step 5: Calculate Ensemble Lifetime 7
8 Electron-Phonon Matrix Element Consider the coupled electron-phonon wavefunction Phonon absorption Phonon emission Electron-Phonon Matrix Element For long wavelength phonons, can make slowly-varying approx 8
9 Scattering Rate Calculations Overview Step 1: Determine Scattering Potential Step 2: Calculate Matrix Elements Step 3: Calculate State-State Transition Rates Step 4: Calculate State Lifetime Step 5: Calculate Ensemble Lifetime Electron-Phonon Scattering Rate 9
10 Scattering Rate Calculations Overview Step 1: Determine Scattering Potential Step 2: Calculate Matrix Elements Step 3: Calculate State-State Transition Rates Step 4: Calculate State Lifetime Step 5: Calculate Ensemble Lifetime Crystal Momentum constraint Energy and Momentum Conservation Energy constraint Therefore: Lundstrom, Sec
11 Energy and Momentum Conservation For acoustic phonons and Energy and Momentum Conservation Typical acoustic phonon velocity Velocity of typical electron (300 K) final inital 11
12 Energy and Momentum Conservation Maximum momentum exchange Maximum energy exchange Acoustic phonon scattering is essentially elastic for 300K electrons Scattering Rate Calculations Overview Step 1: Determine Scattering Potential Step 2: Calculate Matrix Elements Step 3: Calculate State-State Transition Rates Step 4: Calculate State Lifetime Step 5: Calculate Ensemble Lifetime 12
13 Energy and Momentum Conservation Electron-Phonon Scattering Time Preview 13
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