Optical Properties of Semiconductors and Insulators in Solid State Physics
Explore the exciting world of optical properties in semiconductors and insulators with a focus on excitons, interband transitions, matrix elements, and real spectra analysis. Learn about the frequency dependence of optical properties and a simple treatment of exciton effects in a two-band model. Delve into electronic Hamiltonians, ground state wavefunctions, and solving the Schrödinger equation in this illuminating lecture session.
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Presentation Transcript
PHY 752 Solid State Physics 11-11:50 AM MWF Olin 103 Plan for Lecture 27: Optical properties of semiconductors and insulators (Chap. 7 & 12 in GGGPP) Excitons 10/30/2015 PHY 752 Fall 2015 -- Lecture 26 1
10/30/2015 PHY 752 Fall 2015 -- Lecture 26 2
Interband transitions 10/30/2015 PHY 752 Fall 2015 -- Lecture 26 3
In general the matrix element Mcv(k) is a smooth function of k and the joint density of states often determines the frequency dependence of the optical properties: 10/30/2015 PHY 752 Fall 2015 -- Lecture 26 4
Real spectra and more complete analysis From Michael Rohlfing and Steven Louie, PRB 62 4927 (2000) 10/30/2015 PHY 752 Fall 2015 -- Lecture 26 5
Real spectra and more complete analysis From Michael Rohlfing and Steven Louie, PRB 62 4927 (2000) 10/30/2015 PHY 752 Fall 2015 -- Lecture 26 6
Real spectra and more complete analysis From Michael Rohlfing and Steven Louie, PRB 62 4927 (2000) 10/30/2015 PHY 752 Fall 2015 -- Lecture 26 7
Real spectra and more complete analysis From Michael Rohlfing and Steven Louie, PRB 62 4927 (2000) 10/30/2015 PHY 752 Fall 2015 -- Lecture 26 8
Simple treatment of exciton effects in a two-band model 10/30/2015 PHY 752 Fall 2015 -- Lecture 26 9
Electronic Hamiltonian Ground state wavefunction Excited state from two-band model summing over wavevectors k Solving Schroedinger equation in this basis: 10/30/2015 PHY 752 Fall 2015 -- Lecture 26 10
Exciton equations -- continued where: After several steps: Ignoring U2for the moment -- 10/30/2015 PHY 752 Fall 2015 -- Lecture 26 11
1 1 1 = + Reduced mass: m m ex v c Equation for kex=0: Define an envelope function 10/30/2015 PHY 752 Fall 2015 -- Lecture 26 12
Introduced electron-hole screening Hydrogen-like eigenstates: Exciton Eigenstates E n = ex E E n G 2 10/30/2015 PHY 752 Fall 2015 -- Lecture 26 13
Some details Considered a closed shell system with N electrons We can write the effective Hamiltonian: with 10/30/2015 PHY 752 Fall 2015 -- Lecture 26 14
Consider an excitation where an occupied state is moved to an excited stat e : m 10/30/2015 PHY 752 Fall 2015 -- Lecture 26 15
Define: Eigenstates: 10/30/2015 PHY 752 Fall 2015 -- Lecture 26 16
More detailed treatment of U2 (J) term: Effective dipole moment: 10/30/2015 PHY 752 Fall 2015 -- Lecture 26 17
Resulting equation for envelope function: Relationships of envelope function: 10/30/2015 PHY 752 Fall 2015 -- Lecture 26 18
Summary of Wannier exciton analysis Hydrogen-like eigenstates: Exciton Eigenstates E n = ex E E n G 2 Wannier analysis is reliable for loosely bound excitons found in semiconductors; for excitons in insulators (such as LiF) Frenkel exciton analysis applies. 10/30/2015 PHY 752 Fall 2015 -- Lecture 26 19
Optical absorption due to excitons (Chap. 12) Transition probability from ground state with 10/30/2015 PHY 752 Fall 2015 -- Lecture 26 20
For spherically symmetric excitons (first class transitions) For p-like excitons ( second class transitions) 10/30/2015 PHY 752 Fall 2015 -- Lecture 26 21
Example of Cu2O: 10/30/2015 PHY 752 Fall 2015 -- Lecture 26 22
