Hagen-Rubens: from the solution of Maxwell‘s equations (  = n ) for small frequencies

1. 100% Reflectivity of metals Hagen-Rubens: from the solution of Maxwell‘s equations ( = n) for small frequencies Drude: free damped electrons (classical electron theory), determine the color of materials Lorentz: strongly bound electrons (classical electron theory for dielectric materials)

2. Hagen-Rubens Relation Relationship between the optical reflection and the electrical conductivity In the IR* range ( < 1013 s-1): /   Metals with good electrical conductivity have a large reflectivity in the IR range (small ) * Infrared

3. 100% Reflectivity of metals Hagen-Rubens: from the solution of Maxwell‘s equations ( = n) for small frequencies Drude: free damped electrons (classical electron theory), determine the color of materials Lorentz: strongly bound electrons (classical electron theory for dielectric materials)

4. Free Electrons (Classical Drude Theory of Electrical Conductivity) Electron gas within the material Number of atoms/electrons in alkali metals per m3 … Avogadro constant … density … molar mass … drift velocity … electron mass … electric field … damping Free electrons … Interaction with the crystal lattice …

5. v vF t Free Electrons (Classical Drude Theory of Electrical Conductivity) … equation of motion … maximum drift velocity … solution of the equation of motion … time between two collisions … Fermi velocity

6. Free Electrons without Damping (Classical Theory) Excitation of electrons via electromagnetic wave (light): Equation of motion: This equation can be solved by the substitution: Dipole moment of an electron: Total polarization: N … number of free electrons (number of electron at the Fermi surface)

7. Free Electrons without Damping(Classical Theory) Permittivity: Free electrons without damping Frequency Frequency

8. Free Electrons without Damping (Classical Theory) Free electrons without damping Reflection: For high frequencies becomes real. Therefore imag() = 0 Reflective Transparent Refelctivity (%) For low frequencies becomes imaginary. Therefore real() = 0 f … number of free electrons per cm³ Frequency

9. The Plasma Frequency Good compliance with the experiment for alkali metals

10. Free Electrons without Damping 100%

11. v vF t Free Electrons with Damping (Classical Drude Theory) Excitation of electrons via an electromagnetic wave (Light): Equation of motion: Constant velocity of electrons: Equation of motion with maximal drift velocity: Drift velocity: Ohm’s law: Damping:

12. Free Electrons with Damping (Classical Drude Theory) Equation of motion: This equation can be solved by the substitution: Complex amplitude of oscillations Dipole moment of an electron: Total polarization:

13. Free Electrons with Damping (Classical Drude Theory) Total polarization: Permittivity:

14. Free Electrons with Damping (Classical Drude Theory) Permittivity (dielectric function, dielectric constant):

15. Free Electrons with Damping (Classical Drude Theory) Free electrons with damping Frequency 1 … Plasma frequency 2 … Damping frequency Frequency

16. Free Electrons with Damping (Classical Drude Theory) 1

17. Free Electrons with Damping (Classical Drude Theory) Free electrons with damping Absorption Reflectivity: Reflective Transparent Reflectivity (%) Frequency

18. Free Electrons with Damping (Classical Drude Theory) 100% Small spectrum for the absorption of light (absorption band), investigated for metals and nonmetals

19. Strongly Bound Electrons(Electron Theory for Dielectric Materials) Bonding between electron and atom is quasi-elastic  harmonic oscillator with natural frequency and damping

20. Strongly Bound Electrons(Electron Theory for Dielectric Materials) Equation of motion: m … electron mass, ´ … damping, k … spring constant (bond to the core) This equation can be solved by the substitution: : Drude theory

21. Strongly Bound Electrons(Electron Theory for Dielectric Materials) Total polarization: … Eigenfrequency of electrons … damping (electrical conductivity, emission of photons) Permittivity: Index of refraction:

22. Model of Strongly Bound ElectronsPermittivity Bound electrons with damping and eigenfrequency eigenfrequency Frequency Frequency

23. Model of Strongly Bound ElectronsIndex of Refraction Bound electrons with damping and eigenfrequency eigenfrequency Frequency Frequency

24. Model of Strongly Bound ElectronsReflectivity Bound electrons with damping and eigenfrequency eigenfrequency Reflectivity (%) Frequency

25. Free Electrons with Damping and Bound Electrons with Damping and Natural Frequency Free and bound electrons with damping eigenfrequency Frequency Frequency

26. Free Electrons with Damping and Bound Electrons with Damping and Natural Frequency Free and bound electrons with damping IR absorption (reflection) Reflectivity (%) Absorption of visible light Frequency

27. Free Electrons with Damping and Bound Electrons with Damping and Eigenfrequency 100%

28. Dispersion Curve Polarizability (proportional to permittivity) as function of frequency (wavelength) Slow permanent dipoles Interaction between ions Interaction between electrons and atomic nuclei Polarizability Dipole relaxation Ion resonance Validity of Maxwell’s equations Electron resonance Frequency Microwave radiation Infrared radiation Ultraviolet radiation and x-rays

29. Optical Absorption • Lattice vibrations • Absorption in IR range – small natural frequencies of lattice vibrations • IR and Raman spectroscopy – investigation of lattice dynamics • Conducting electrons • Especially in metals • Ionic crystals and insulators are normally transparent • Core electrons • Interaction between electron and atomic nucleus • High natural frequency • Absorption and emission of radiation in the x-ray range (selective filters, fluorescence spectroscopy) hard soft Visible light X-rays Conducting electrons Absorption coefficient Core electrons Vibrations Wavelength

30. Overview of scattering processes Electron spectroscopy with x-rays XPS Raman process IR absorption with two phonons Photon ´, k´ Photon ´, k´ Photon , k Photon , k X-ray photon Phonon , K Phonon , K Photoelectron Photon – light quantum Phonon – quasiparticle to describe lattice vibrations

31. Overview of scattering processes Emission of characteristic x-rays + absorption Thomson process Compton process Photon ´, k´ Photon ´, k´ Photon , k Photon , k X-ray photon Phonon (for neutrons) Electron (for x-rays) X-ray photon … Increase of electron energy Elastic scattering – x-ray diffraction, neutron diffraction, electron diffraction Inelastic scattering – x-rays, neutrons

32. Special CasesHigh Frequencies Real (n) < 1, Real (n)  1, Imag (n)  0 Low reflectivity, high absorption X-ray radiation Example: gold (CuKa)  = 1.5418 10-10 m d = 4.2558 10-5 b = 4.5875 10-6

33. Special CasesWeak Damping

34. Multiple Oscillators Multiple electrons per atom with damping and natural frequency 0 0i,   i Weak damping

35. Free Electrons with Damping and Bound Electrons with Damping and Eigenfrequency

36. Free Electrons with Damping and Bound Electrons with Damping and Eigenfrequency Free and bound electrons with damping Frequency Frequency

37. Quantum Mechanical Description of Optical Properties Quantum jump (band transition) Direct Indirect Phonon = lattice vibration

38. Polarizability Polarizability of molecules: … electric susceptibility … relative permittivity … vacuum permittivity … number density of molecules … polarizability … Boltzmann constant … temperature Simplified dispersion curve: „slow“ permanent dipoles can’t change their polarization easily – decrease of permittivity Fig. 6.32. Dielectric orientation polarization: Top: orientation of molecular dipoles for three different field frequencies. Center: orientation distribution of dipoles (length of arrows is equal to the probability of a polarization in direction of the same arrow). Bottom: resulting relaxation curve of the permittivity

39. Piezoelectricity and Pyroelectricity Polarization without an external electric field Change in length of the crystal  Polarization of dipole moments  Generation of a surface charge Fig. 6.37. Orientation of a piezoelectric quartz plate to the parent crystal Crystal with external voltage  Polarization of dipole moments  Change in length of the crystal … generated surface charge … material constant … length of crystal … thickness of crystal … force Temperature change of crystal  Change in length of the crystal (thermal expansion)  Polarization of dipole moments  Generation of a surface charge Fig. 6.38. Evidence of the transversal piezoelectric effect

40. Piezoelectricity Mechanical stress Mechanical stress Mechanical stress Mechanical stress

41. Ferroelectricity Spontaneous polarization (arrangement) of dipole moments without an external electric field Dielectric material Spontaneous polarization Ferroelectric material

42. Ferroelectric Crystals Perovskite structure Wyckoffpositions: Ca: 1a (0,0,0) Ti: 1b (½,½,½) O: 3c (0,½,½) Ferroelectric materials with perovskite structure: SrTiO3, BaTiO3, PbTiO3, KNbO3, LiTaO3, LiNbO3 Ferroelectricity is connected to the crystal structure

43. Ferroelectric Domains The whole polarization of a crystal with ferroelectric domains is smaller than the polarization of a crystal without domains – the microstructure plays an important role

44. Ferroelectric Domains in a BaTiO3 Single Crystal Total polarization of crystal increases with external voltage