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

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 Skin depth = c/(omega*k) = lambda / (2pi*k)

Intrinsic vs. Extrinsic size effects Optical material parameters can be divided between intrinsic and extrinsic Intrinsic Extrinsic Related to atomic-scale properties: bond strength, bond length, crystallography, composition (doping etc.) geometry: crystal size, surface roughness, layer thickness, finite size effects Manipulated by Surrounding environment frequency of light propagation direction sample preparation aspect ratio annealing applied external fields Results in: changes in: conductivity, relaxation time, mobility, reflection, transmission, …

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 𝜏 The duration of a collision is negligible Sommerfeld’s contribution: Electron velocity distribution follows Fermi-Dirac statistics http://www.pdi-berlin.de/paul-drude

Free carrier conductivity 𝜎(𝜔) Equation of motion with no restoring force 𝑚 0 𝑑 2 𝑥 𝑑 𝑡 2 + 1 𝜏 𝑚 0 𝑑𝑥 𝑑𝑡 = 𝑚 0 𝑑𝒗 𝑑𝑡 + 1 𝜏 𝑚 0 𝒗=−𝑒𝑬 𝑬 𝑡 =𝑅𝑒{𝑬 𝜔 𝑒 −𝑖𝜔t } 𝒗 𝑡 =𝑅𝑒{𝒗 𝜔 𝑒 −𝑖𝜔t } Seek a solution of the form: 𝒗(𝜔)= −𝑒𝜏 𝑚 0 1 1−𝑖𝜔𝜏 𝑬(𝜔) 𝒋 𝜔 =−𝑛𝑒𝒗= −𝑛 𝑒 2 𝜏 𝑚 0 1 𝑖𝜔𝜏−1 𝑬(𝜔)=𝜎𝑬 𝜔 𝜎 𝜔 = 𝜎 0 1−𝑖𝜔𝜏 𝜎 0 = 𝑛 𝑒 2 𝜏 𝑚

Drude relaxation time 𝜏 (273 K) Drude parameters Number of conduction electrons is equal to the valency Measuring the conductivity (or resistivity) of a metal gives a way to find 𝜏. 𝑛=0.6022× 10 24 𝑍 𝜌 𝑚 𝐴 𝜏= 𝑚 𝜎 0 𝑛 𝑒 2   n Drude relaxation time 𝜏 (273 K) 𝜈 𝑝 (1015 Hz) 𝜆 𝑝 (nm) (x 1022 cm-3) (x 10-14 second) Ag 5.86 4 2.17 138 Au 5.9 3 2.18 Cu 8.47 2.7 2.61 115 Al 18.1 0.8 3.82 79 Z is the valency 𝜌 𝑚 is the mass density (g/cm3) A is atomic mass

Permittivity 𝜖(𝜔) 𝑚 0 𝑑 2 𝑥 𝑑 𝑡 2 + 1 𝜏 𝑚 0 𝑑𝑥 𝑑𝑡 + 𝑚 0 𝜔 0 2 𝑥=−𝑒𝑬 𝑚 0 𝑑 2 𝑥 𝑑 𝑡 2 + 1 𝜏 𝑚 0 𝑑𝑥 𝑑𝑡 + 𝑚 0 𝜔 0 2 𝑥=−𝑒𝑬 Equation of motion (no restoring force) 𝑥(𝜔)= −𝑒 𝑚 0 1 − 𝜔 2 −𝑖𝜔/𝜏 𝑬(𝜔) 𝑃 𝜔 =−𝑛𝑒𝑥(𝜔) = 𝜖 0 𝑬 𝜔 + 𝜖 0 𝑛 𝑒 2 𝑚 0 𝜖 0 1 − 𝜔 2 −𝑖𝜔/𝜏 𝑬 𝜔 𝑫 𝜔 = 𝜖 0 𝑬 𝜔 +𝑷 𝜔 =𝜖 0 𝜖 𝑟 𝑬 𝜔 Ep1 is polarization Ep2 is energy dissipation 𝜖 𝑟 𝜔 =1− 𝜔 𝑝 2 𝜔 2 +𝑖𝜔/𝜏 𝜔 𝑝 2 = 𝑛 𝑒 2 𝑚 0 𝜖 0

Linking 𝜖 (𝜔) , 𝜎 (𝜔), and 𝑛 𝜖 𝜔 = 𝑖 𝜎 𝜖 0 𝜔 +1= 𝜖 1 +𝑖 𝜖 2 𝜖 𝜔 = 𝑖 𝜎 𝜖 0 𝜔 +1= 𝜖 1 +𝑖 𝜖 2 𝜖 1 𝜔 =1− 𝜎 2 𝜔 𝜖 0 𝜎 1 = 𝜖 2 𝜖 0 𝜔 𝜖 2 𝜔 = 𝜎 1 𝜔 𝜖 0 𝜎 2 =𝜔 𝜖 0 (1− 𝜖 1 ) 𝑛 =𝑛+𝑖𝜅= 𝜖

𝜖=1− 𝜔 𝑝 2 𝜔 2 +𝑖Γ𝜔 𝜖 1 =1− 𝜔 𝑝 2 𝜔 2 + Γ 2 𝜔 𝑝 =1.37× 10 16 𝑠 −1 Γ=3.33× 10 13 𝑠 −1 For Au at 273 K: For Au at 273 K: 𝜖 2 = 1 𝜔 Γ 𝜔 𝑝 2 𝜔 2 + Γ 2

𝜎 𝜔 = 𝜎 0 1−𝑖𝜔𝜏 𝜎 2 𝜔 = 𝜎 0 𝜔𝜏 1+ 𝜔 2 𝜏 2 𝜎 1 𝜔 = 𝜎 0 1+ 𝜔 2 𝜏 2 𝜎 𝜔 = 𝜎 0 1−𝑖𝜔𝜏 𝜎 2 𝜔 = 𝜎 0 𝜔𝜏 1+ 𝜔 2 𝜏 2 Intersection is omega = 1/tau 𝜎 1 𝜔 = 𝜎 0 1+ 𝜔 2 𝜏 2

𝑛= 𝜖 1 2 + 𝜖 2 2 + 𝜖 1 2 𝜅= 𝜖 1 2 + 𝜖 2 2 − 𝜖 1 2 At very low frequencies, n = k Jump at omega_p: becomes transparent (absorption drops, n -> 1)

Related parameters Reflectivity (here normal incidence) 𝑅= 𝑛 −1 𝑛 +1 2 = 𝑛−1 2 + 𝜅 2 𝑛+1 2 + 𝜅 2 Reflectivity (here normal incidence) 𝛼= 2𝜅𝜔 𝑐 = 4𝜋𝜅 𝜆 Absorption coefficient 𝛿= 2 𝛼 Skin depth

𝑅= 𝑛 −1 𝑛 +1 2

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

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

Temperature dependence More scattering at higher temperatures; Pells and Shiga, J. Phys. C: Solid State Phys., V. 2, p. 1835 (1969).

Summary of the Drude-Sommerfeld model Allows qualitative, and often quantitative understanding of many optical properties of metal Conductivity Reflectivity Transparency if 𝜔> 𝜔 𝑝 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 allotropes e.g. of carbon Interpreting Drude collisions purely as electron-ion collisions does not allow prediction of 𝜏 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 Transition out: The Drude model gives a good qualitative – and often quantitative -- understanding in the free electron regime, but to understand

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

Poor quantitative agreement with D.M. Drude model gives good qualitative trend, but neglects EXTRINSIC effects How should we predict the behavior of our system? 10 um 1 um Ordall et al. Appl. Opt. 22 1099 (1983)

Plasmon propagation length 1/e decay length Plasmon at Au/air interface λ = 10 μm 𝑘 𝑥 ′′ =𝐼𝑚 𝜔 𝑐 𝜖 𝑣𝑎𝑐 𝜖 𝐴𝑢 𝜖 𝑣𝑎𝑐 + 𝜖 𝐴𝑢 𝐿 𝑖 = 1 2 𝑘 𝑥 ′′ Optical constants at 10 um n k Palik 12.4 55.0 Bennett & Bennett 7.62 71.5 Motulevich 11.5 67.5 Padalka 7.41 53.4 … and if you’re trying to connect the resonant wavelength scaling of antennas with their optical properties, good luck with n varying by 60% depending on what source you choose 𝐿 𝑖,𝑃𝑎𝑙𝑖𝑘 =11.8 mm 𝐿 𝑖,𝐵&𝐵 =39.0 mm

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 Γ 𝑡𝑜𝑡𝑎𝑙 =2ℏ/ 𝑇 𝑡𝑜𝑡𝑎𝑙 Munechika, et al., J. Phys. Chem. C, V. 111, 18906 (2007).

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

Optical constants measurement techniques

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 𝜙 𝑟 𝜔 = 𝜔 𝜋 0 ∞ ln 𝑅 𝜔 ′ − ln 𝑅(𝜔) 𝜔 2 − 𝜔 ′2 𝑑𝜔′ 𝜎 1 𝜔 =𝜔 𝜖 0 𝜖 2 𝜔 =𝜔 𝜖 0 4 𝑅(𝜔) 1−𝑅 𝜔 sin 𝜙 𝑟 1+𝑅 𝜔 −2 𝑅(𝜔) cos 𝜙 𝑟 2 [SI units] 𝜎 2 𝜔 =−𝜔 𝜖 0 1− 𝜖 1 𝜔 =−𝜔 𝜖 0 1− 1−𝑅 𝜔 2 −4𝑅 𝜔 sin 2 𝜙 𝑟 1+𝑅 𝜔 −2 𝑅 𝜔 cos 𝜙 𝑟 2 Dressel & Grüner, Ashcroft & Mermin, Appendix K

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

Fresnel Equations Augustin-Jean Fresnel (1788-1827) 𝑟 𝑠 = 𝐸 0𝑟 𝐸 0𝑖 𝑠 = 𝑛 𝑖 cos 𝜙 𝑖 − 𝑛 𝑡 cos⁡( 𝜙 𝑡 ) 𝑛 𝑖 cos 𝜙 𝑖 + 𝑛 𝑡 cos 𝜙 𝑡 𝑟 𝑝 = 𝐸 0𝑟 𝐸 0𝑖 𝑝 = 𝑛 𝑡 cos 𝜙 𝑖 − 𝑛 𝑖 cos⁡( 𝜙 𝑡 ) 𝑛 𝑡 cos 𝜙 𝑖 + 𝑛 𝑖 cos 𝜙 𝑡 𝑡 𝑠 = 𝐸 0𝑡 𝐸 0𝑖 𝑠 = 2𝑛 𝑖 cos 𝜙 𝑖 𝑛 𝑖 cos 𝜙 𝑖 + 𝑛 𝑡 cos 𝜙 𝑡 𝑡 𝑝 = 𝐸 0𝑡 𝐸 0𝑖 𝑝 = 2𝑛 𝑖 cos 𝜙 𝑖 𝑛 𝑖 cos 𝜙 𝑡 + 𝑛 𝑡 cos 𝜙 𝑖 -Cannot get R(n,k) and T(n,k) analytically, -so you have to model n,k curves depending on R and T, and narrow in on the correct values of n and k. This is cumbersome, obviously, especially with high spectral resolution; -requires a known thickness Used for reflection-transmission measurements (like Johnson & Christy)

Ellipsometry

Ellipsometry 𝜌= 𝑟 𝑝 𝑟 𝑠 = tan 𝜓 e 𝑖Δ 𝑛 = sin 𝜙 1+ 1−𝜌 1+𝜌 2 tan 2 𝜙

Comparison of methods for widely referenced optical constants for Au Source Author Reference energy range measurement method Palik, ed. M. L. Theye PRB 2 3060 (1970) 6-0.6 eV 210 nm - 2070 nm reflectance & transmittance at normal incidence (requires known thickness)   Dold and Mecke Optik 22, 435 (1965) 1-0.125 eV 1240 nm - 10 um “ellipsometric technique”; ERRONEOUSLY LOW K VALUES at longer wavelengths Johnson and Christy PRB 6, 4370 (1972) 6.5 - 0.5 eV 190 nm - 2000 nm reflectance & transmittance, different angles (requires significant modeling) Ordall, ed. Bennett and Bennett Optical Properties and Electronic Structure of Metals and Alloys (Abeles, ed.) 0.413 - 0.0388 eV 3 um - 32 um reflectance Motulevich Soviet Phys. JETP 20, 560 (1965) 1.24 - 0.1033 eV 1 um - 12 um not readily available

Spectroscopic Ellipsometry of bulk Au planar surfaces

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

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.

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.

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.

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.

Conclusion The Drude model gives a way to predict some optical 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.

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. Ordal et 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.