Optical Constants of metals (Au), the Drude model, and Ellipsometry

1. Optical Constants of metals (Au), the Drude model, and Ellipsometry Robert L. Olmon, Andrew C. Jones, Tim Johnson, David Shelton, Brian Slovick, Glenn D. Boreman, Sang-Hyun Oh, Markus B. Raschke

2. Physical phenomena sensitive to optical constants in metal • Plasmon propagation length • Polarizability of a metal cluster • Impedance of nanoparticles (e.g. for impedance matching optical antennas) • Optical/IR antenna resonance frequency • Skin depth • Casimir force • Radiative lifetime of plasmonic particles

3. Intrinsic vs. Extrinsic size effects • Optical material parameters can be divided between intrinsic and extrinsic

4. Drude-Sommerfeld Model • Negatively charged particles behave like in a gas • Particles of mass m move in straight lines between collisions (assuming no external applied field) • Electron-electron electromagnetic interactions are neglected • Assumed that positive charges are attached to much heavier particles to make the metal neutral • Drude thought the electrons collided with these heavy particles • Electron-ion electromagnetic interaction is neglected (careful!) • Average time between collisions is • The duration of a collision is negligible • Sommerfeld’s contribution: Electron velocity distribution follows Fermi-Dirac statistics http://www.pdi-berlin.de/paul-drude

5. Free carrier conductivity • Equation of motion with no restoring force • Seek a solution of the form:

6. Drude parameters • Number of conduction electrons is equal to the valency • Measuring the conductivity (or resistivity) of a metal gives a way to find . Z is the valency is the mass density (g/cm3) A is atomic mass

7. Permittivity Equation of motion (no restoring force)

8. Linking , , and

9. For Au at 273 K: For Au at 273 K:

12. Related parameters • Skin depth • Reflectivity (here normal incidence) • Absorption coefficient

14. Ep = 15.8 eV Ehrenreich, H, Philipp, H.R. and Segall, B. Phys. Rev. 132 1918 (1962).

15. Interband transitions Energy band diagram for Au Ramchandani, J. Phys. C: Solid State Phys.,V. 3,P. S1 (1970).

16. Temperature dependence Pells and Shiga, J. Phys. C: Solid State Phys., V. 2, p. 1835 (1969).

17. Summary of the Drude-Sommerfeldmodel • Allows qualitative, and often quantitative understanding of many optical properties of metal • Conductivity • Reflectivity • Transparency if • Relaxation time • Plasma frequency • Links refractive index to conductivity • Predicts mean-free path, Fermi Energy, Fermi velocity • Does not take into account absorption due to interband transitions • Fails to predict non-metallic behavior of elements like boron (an insulator), which has the same valency as Al, or different conductive behavior of allotropese.g. of carbon • Interpreting Drude collisions purely as electron-ion collisions does not allow prediction of • The role of the ions in physical phenomena (e.g. specific heat or thermal conductivity) is ignored • The role of sub-valence electrons is ignored • EXTRINSIC effects are not considered

18. Available data • “the infrared data are very limited and agreement in the n spectra is not good.” – Lynch and Hunter (in Palik) • “Agreement at the junctions of the data sets is rare” (ibid.) • Sometimes unspecified yet critical parameters: • Sample quality • Temperature • Sample preparation methods • Measurement methods

19. Poor quantitative agreement with D.M. 10 um 1 um Ordallet al. Appl. Opt. 22 1099 (1983)

20. Plasmon propagation length • 1/e decay length • Plasmon at Au/air interface • λ = 10 μm Optical constants at 10 um

21. Homogeneous line widths of silver nanoprisms • Single particle localized surface plasmon resonance sensing: sensitivity is inversely proportional to resonance line width. • Require high local field enhancement and low damping FDTD Munechika, et al., J. Phys. Chem. C, V. 111, 18906 (2007).

22. Modeling metal clusters Ag clusters Sonnichsen et al., New J. Phys. V. 4, 93 (2002).

23. Optical constants measurement techniques

24. Kramers-Kronig method • Measure reflected power at the sample, R (or transmitted, T) • Compare to a known sample • Use K-K relation to obtain lost phase information • Requires broad spectral range [SI units] Dressel & Grüner, Ashcroft & Mermin, Appendix K

25. Kramers-Kronig relations Denotes that the Cauchy principal value must be taken Hans Kramers (1894-1952) Ralph de LaerKronig (1904–1995) Handbook of Ellipsometry

26. Fresnel Equations Augustin-Jean Fresnel (1788-1827) Used for reflection-transmission measurements (like Johnson & Christy)

27. Ellipsometry

28. Ellipsometry

29. Comparison of methods for widely referenced optical constants for Au

30. Spectroscopic Ellipsometry of bulk Au planar surfaces

31. Broadband SE of bulk Au • Available optical constants data = largely unreliable • Require source for • Continuous • Broadband (200 nm – 20 um) • High spectral resolution • Three samples: • Single-crystal (SC) gold, 1mm thick • Thermally evaporated gold, 200 nm thick • Evaporated, template stripped gold, 200 nm thick • VASE and VASE-IR measurements

32. SE measurements on bulk Au • All three samples agree well with respect to the real permittivity in the visible, and they are in good agreement with JC at 500 nm and longer. • In the region of interband sp-d band transitions, JC deviates significantly. • Anomaly in Palik, centered at about 650 nm.

33. SE measurements on bulk Au • Good agreement at short wavelengths • Deviation begins at about 600 nm, with JC and Palik systematically too high toward longer wavelengths, and not really in agreement.

34. SE measurements on bulk Au • Measured values are within the large range given by previous measurements. • The evaporated and smooth template-stripped samples show nearly identical behavior, while the SC has a lower negative permittivity, indicating a dependence on crystallinity, but not surface roughness.

35. SE measurements on bulk Au • The three samples show good agreement with each other, particularly at long wavelengths, •  indicates that loss in the IR has a low dependence on sample preparation. • Their trend is steeper than Palik’s, crossing to higher permittivity at about 5 μm.

36. Conclusion • The Drude model gives a way to predict someoptical properties of metals. • However, the Drude model does not provide a full understanding of what is happening in the metal. • For accurate prediction of optical phenomena: Direct measurement of the sample under study is preferable to looking in a data table. • We give a high resolution, continuous data set for a broad frequency range, suitable for plasmonic studies.

37. References • Handbook of Optical Constants of Solids, 3rd. Ed., Palik, ed. Academic Press (1998). • M. Dressel and G. Grüner, Electrodynamics of Solids, Cambridge University Press, 2002. • N. W. Ashcroft and N. D. Mermin, Solid State Physics, Brooks/Cole, 1976. • H. G. Tompkins and E. A. Irene, Handbook of Ellipsometry, William Andrew Publishing, 2005. • J. A. Woolum Co. [www.jawoollam.com] • Johnson and Christy, “Optical Constants of the Noble Metals,” PRB V. 6, 4370 (1972). • D. Fleisch, A student’s guide to Maxwell’s equations, Cambridge University Press, 2008. • M. Fox, Optical Properties of Solids, Oxford University Press, 2001. • Ordalet al., Appl. Optics V. 22, 1099 (1983) • Born and Wolf, Principles of Optics, Pergamon, New York, 1964. • H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, Springer-Verlag.