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

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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

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

extrinsicIntrinsic 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 bySurrounding environment

frequency of lightpropagation direction

sample preparationaspect ratioannealing

applied external fields

Results in:

changes in:conductivity, relaxation time,

mobility,reflection,

transmission, …

changes in:conductivity, relaxation time,

mobility,reflection,

transmission, …

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Drude-Sommerfeld Model• Negatively charged particles behave like in a gas

o Particles of mass m move in straight lines between collisions (assuming no external applied field)

o Electron-electron electromagnetic interactions are neglected

http://www.pdi-berlin.de/paul-drude

• Assumed that positive charges are attached to much heavier particles to make the metal neutralo Drude thought the electrons collided with

these heavy particleso Electron-ion electromagnetic interaction is

neglected (careful!)o Average time between collisions is o The duration of a collision is negligible

• Sommerfeld’s contribution: Electron velocity distribution follows Fermi-Dirac statistics

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Free carrier conductivity

𝜎 (𝜔 )=𝜎0

1−𝑖 𝜔𝜏𝜎 0=

𝑛𝑒2𝜏𝑚

• Equation of motion with no restoring force

𝒗 (𝜔 )=−𝑒𝜏𝑚0

11−𝑖𝜔𝜏

𝑬 (𝜔)

𝒗 (𝑡 )=𝑅𝑒 {𝒗 (𝜔 )𝑒−𝑖 𝜔 t }

𝑬 (𝑡 )=𝑅𝑒 {𝑬 (𝜔 )𝑒−𝑖𝜔 t }

𝒋 (𝜔 )=−𝑛𝑒𝒗=−𝑛𝑒2𝜏

𝑚0

1𝑖𝜔𝜏−1

𝑬 (𝜔)=𝜎 𝑬 (𝜔 )

𝑚0𝑑2𝑥𝑑𝑡2

+ 1𝜏𝑚0

𝑑𝑥𝑑𝑡

=𝑚0𝑑𝒗𝑑𝑡

+ 1𝜏𝑚0𝒗=−𝑒𝑬

• Seek a solution of the form:

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Drude parameters

𝜏=𝑚𝜎 0

𝑛𝑒2

• Number of conduction electrons is equal to the valency• Measuring the conductivity (or resistivity) of a metal gives a way to find .

n Drude relaxation time (273 K)

(1015 Hz) (nm)(x 1022 cm-3) (x 10-14 second)

Ag 5.86 4 2.17 138Au 5.9 3 2.18 138Cu 8.47 2.7 2.61 115Al 18.1 0.8 3.82 79

𝑛=0.6022×1024𝑍 𝜌𝑚𝐴

Z is the valency is the mass density (g/cm3)A is atomic mass

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Permittivity

𝑚0𝑑2𝑥𝑑𝑡2

+ 1𝜏𝑚0

𝑑𝑥𝑑𝑡

+𝑚0𝜔02𝑥=−𝑒𝑬

𝑥 (𝜔)=−𝑒𝑚0

1

−𝜔2−𝑖 𝜔/𝜏𝑬 (𝜔 )

𝑃 (𝜔 )=−𝑛𝑒𝑥 (𝜔 )

𝑫 (𝜔 )=𝜖0𝑬 (𝜔 )+𝑷 (𝜔 ) ¿𝜖0𝑬 (𝜔 )+𝜖0𝑛𝑒2

𝑚0𝜖01

−𝜔2− 𝑖𝜔 /𝜏𝑬 (𝜔 )¿𝜖0𝜖𝑟 𝑬 (𝜔 )

𝜖𝑟 (𝜔 )=1−𝜔𝑝2

𝜔2+𝑖𝜔 /𝜏𝜔𝑝2= 𝑛𝑒2

𝑚0𝜖0

Equation of motion (no restoring force)

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Linking , , and

�̂� (𝜔 )= 𝑖 �̂�𝜖0𝜔

+1=𝜖1+𝑖𝜖2

𝜖1 (𝜔 )=1−𝜎2𝜔𝜖0

𝜖2 (𝜔 )=𝜎 1

𝜔𝜖0

𝜎 1=𝜖2𝜖0𝜔

𝜎 2=𝜔𝜖0(1−𝜖1)

�̂�=𝑛+𝑖 𝜅=√ �̂�

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𝜖=1−𝜔𝑝2

𝜔2+𝑖 Γ𝜔

𝜖1=1−𝜔𝑝2

𝜔2+Γ2

𝜖2=1𝜔

Γ 𝜔𝑝2

𝜔2+Γ2For Au at 273 K:

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𝜎 (𝜔 )=𝜎0

1−𝑖 𝜔𝜏

𝜎 1 (𝜔 )=𝜎0

1+𝜔2𝜏2

𝜎 2 (𝜔 )=𝜎0𝜔𝜏

1+𝜔2𝜏2

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𝑛=√ √𝜖12+𝜖22+𝜖12

𝜅=√ √𝜖12+𝜖22−𝜖12

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Related parameters

𝛼=2𝜅𝜔𝑐

=4𝜋𝜅𝜆

𝑅=|~𝑛−1~𝑛+1 |2

=(𝑛−1 )2+𝜅 2

(𝑛+1 )2+𝜅2

𝛿=2𝛼

• Skin depth

• Absorption coefficient

• Reflectivity (here normal incidence)

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𝑅=|~𝑛−1~𝑛+1 |2

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

Ep = 15.8 eV

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Interband transitions

Ramchandani, J. Phys. C: Solid State Phys., V. 3, P. S1 (1970).

Energy band diagram for Au

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

Temperature dependence

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Summary of the Drude-Sommerfeld model

• 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

• 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

• 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

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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

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1 um10 um

Ordall et al. Appl. Opt. 22 1099 (1983)

Poor quantitative agreement with D.M.

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Plasmon propagation length

• 1/e decay length• Plasmon at Au/air interface• λ = 10 μm

𝐿𝑖=1

2𝑘𝑥′ ′ 𝑘𝑥

′ ′=𝐼𝑚{𝜔𝑐 √ 𝜖𝑣𝑎𝑐𝜖𝐴𝑢𝜖𝑣𝑎𝑐+𝜖𝐴𝑢 }n k

Palik 12.4 55.0

Bennett & Bennett 7.62 71.5

Motulevich 11.5 67.5

Padalka 7.41 53.4

𝐿𝑖 ,𝑃𝑎𝑙𝑖𝑘=11.8mm

𝐿𝑖 ,𝐵∧𝐵=39.0mm

Optical constants at 10 um

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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).

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Modeling metal clusters

Sonnichsen et al., New J. Phys. V. 4, 93 (2002).

Ag clusters

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Optical constants measurement techniques

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Kramers-Kronig method

Dressel & Grüner,Ashcroft & Mermin, Appendix K

monitor

source

sample

polarizer

detector

• 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

𝜙𝑟 (𝜔 )=𝜔𝜋∫

0

∞ln 𝑅 (𝜔 ′ )−ln𝑅 (𝜔)

𝜔2−𝜔 ′ 2 𝑑𝜔 ′

𝜎 1 (𝜔 )=𝜔𝜖0𝜖2 (𝜔 )=𝜔𝜖04√𝑅 (𝜔) [1−𝑅 (𝜔 ) ] sin𝜙𝑟

[1+𝑅 (𝜔 )−2√𝑅 (𝜔 )cos𝜙𝑟 ]2

𝜎 2 (𝜔 )=−𝜔𝜖0 [1−𝜖1 (𝜔 ) ]=−𝜔𝜖0(1− [1−𝑅 (𝜔 ) ]2−4 𝑅 (𝜔 )sin2𝜙𝑟

[1+𝑅 (𝜔 )−2√𝑅 (𝜔 ) cos𝜙𝑟 ]2 )

[SI units]

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Kramers-Kronig relations

Hans Kramers (1894-1952)∫❑ Denotes that the

Cauchy principal value must be takenHandbook of Ellipsometry

Ralph de Laer Kronig (1904–1995)

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Fresnel Equations

𝑟 𝑠=(𝐸0𝑟𝐸0 𝑖)𝑠

=𝑛𝑖cos (𝜙𝑖 )−𝑛𝑡cos (𝜙𝑡)

𝑛𝑖cos (𝜙𝑖 )+𝑛𝑡 cos (𝜙𝑡 )

𝑟𝑝=(𝐸0𝑟𝐸0 𝑖)𝑝

=𝑛𝑡 cos (𝜙𝑖 )−𝑛𝑖cos (𝜙𝑡)

𝑛𝑡cos (𝜙 𝑖 )+𝑛𝑖cos (𝜙𝑡 )

𝑡 𝑠=(𝐸0𝑡

𝐸0 𝑖 )𝑠=2𝑛𝑖cos (𝜙𝑖 )

𝑛𝑖cos (𝜙𝑖 )+𝑛𝑡 cos (𝜙𝑡 )

𝑡𝑝=( 𝐸0𝑡

𝐸0 𝑖 )𝑝=2𝑛𝑖cos (𝜙𝑖 )

𝑛𝑖cos (𝜙𝑡 )+𝑛𝑡 cos (𝜙 𝑖 )

Augustin-Jean Fresnel (1788-1827)

Used for reflection-transmission measurements (like Johnson & Christy)

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Ellipsometry

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Ellipsometry

𝜌=𝑟 𝑝𝑟 𝑠

= tan𝜓 e𝑖 Δ

~𝑛=sin 𝜙√1+( 1−𝜌1+𝜌 )2

tan 2𝜙

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Comparison of methods for widely referenced optical constants for AuSource 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 eV190 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 eV3 um - 32 um reflectance

Motulevich Soviet Phys. JETP 20, 560 (1965)

1.24 - 0.1033 eV1 um - 12 um not readily available

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Spectroscopic Ellipsometry of bulk Au planar surfaces

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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

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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.

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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.

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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.

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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.

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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.

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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.