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

3. 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, …

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 𝜏
The duration of a collision is negligible
Sommerfeld’s contribution: Electron velocity distribution follows Fermi-Dirac statistics

5. Free carrier conductivity 𝜎(𝜔)
Equation of motion with no restoring force
𝑚 0 𝑑 2 𝑥 𝑑 𝑡 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 −𝑖𝜔𝜏 𝑬(𝜔)
𝒋 𝜔 =−𝑛𝑒𝒗= −𝑛 𝑒 2 𝜏 𝑚 𝑖𝜔𝜏−1 𝑬(𝜔)=𝜎𝑬 𝜔
𝜎 𝜔 = 𝜎 0 1−𝑖𝜔𝜏
𝜎 0 = 𝑛 𝑒 2 𝜏 𝑚

6. 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× 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 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

7. 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 𝜏 𝑚 0 𝑑𝑥 𝑑𝑡 + 𝑚 0 𝜔 0 2 𝑥=−𝑒𝑬
Equation of motion
(no restoring force)
𝑥(𝜔)= −𝑒 𝑚 − 𝜔 2 −𝑖𝜔/𝜏 𝑬(𝜔)
𝑃 𝜔 =−𝑛𝑒𝑥(𝜔)
= 𝜖 0 𝑬 𝜔 + 𝜖 0 𝑛 𝑒 2 𝑚 0 𝜖 − 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

8. 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

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

10. 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 𝜔 2 𝜏 2

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

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

13. 𝑅= 𝑛 −1 𝑛

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

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

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

17. 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

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.
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 (1983)

20. 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

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
Γ 𝑡𝑜𝑡𝑎𝑙 =2ℏ/ 𝑇 𝑡𝑜𝑡𝑎𝑙
Munechika, et al., J. Phys. Chem. C, V. 111, (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
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 𝑅(𝜔) 1−𝑅 𝜔 sin 𝜙 𝑟 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 𝑅 𝜔 −2 𝑅 𝜔 cos 𝜙 𝑟 2
Dressel & Grüner,
Ashcroft & Mermin, Appendix K

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

26. Fresnel Equations
Augustin-Jean Fresnel ( )
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)

27. Ellipsometry

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

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

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 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.

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. [
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.