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
1
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
2
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, …
changes in:
conductivity, relaxation time,
mobility,
reflection,
transmission, …
3
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
• Assumed that positive charges are
attached to much heavier particles to
make the metal neutral
o Drude thought the electrons collided with
these heavy particles
o Electron-ion electromagnetic interaction is
neglected (careful!)
o Average time between collisions is 𝜏
o The duration of a collision is negligible
• Sommerfeld’s contribution: Electron velocity
distribution follows Fermi-Dirac statistics
http://www.pdi-berlin.de/paul-drude
4
Free carrier conductivity 𝜎(𝜔)
• Equation of motion with no restoring force
𝑑2 𝑥 1
𝑑𝑥
𝑑𝒗 1
𝑚 0 2 + 𝑚0
= 𝑚0
+ 𝑚 𝒗 = −𝑒𝑬
𝑑𝑡
𝜏
𝑑𝑡
𝑑𝑡 𝜏 0
• Seek a solution of the form:
𝑬 𝑡 = 𝑅𝑒{𝑬 𝜔 𝑒 −𝑖𝜔t }
𝒗 𝑡 = 𝑅𝑒{𝒗 𝜔 𝑒 −𝑖𝜔t }
−𝑒𝜏
1
𝒗(𝜔) =
𝑬(𝜔)
𝑚0 1 − 𝑖𝜔𝜏
−𝑛𝑒 2 𝜏
1
𝒋 𝜔 = −𝑛𝑒𝒗 =
𝑬(𝜔) = 𝜎𝑬 𝜔
𝑚0 𝑖𝜔𝜏 − 1
𝜎0
𝜎 𝜔 =
1 − 𝑖𝜔𝜏
𝑛𝑒 2 𝜏
𝜎0 =
𝑚
5
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.
24
𝑛 = 0.6022 × 10
Ag
Au
Cu
Al
𝑍𝜌𝑚
𝐴
𝜏=
𝑚𝜎0
𝑛𝑒 2
n
Drude relaxation
time 𝜏 (273 K)
𝜈𝑝
𝜆𝑝
(x 1022 cm-3)
(x 10-14 second)
(1015 Hz)
(nm)
5.86
5.9
8.47
18.1
4
3
2.7
0.8
2.17
2.18
2.61
3.82
138
138
115
79
Z is the valency
𝜌𝑚 is the mass density (g/cm3)
A is atomic mass
6
Permittivity 𝜖(𝜔)
𝑑2 𝑥 1
𝑑𝑥
𝑚0 2 + 𝑚0
+ 𝑚0 𝜔02 𝑥 = −𝑒𝑬
𝑑𝑡
𝜏
𝑑𝑡
Equation of motion
(no restoring force)
−𝑒
1
𝑥(𝜔) =
𝑬(𝜔)
𝑚0 −𝜔 2 − 𝑖𝜔/𝜏
𝑃 𝜔 = −𝑛𝑒𝑥(𝜔)
𝑫 𝜔 = 𝜖0 𝑬 𝜔 + 𝑷 𝜔
𝑛𝑒 2
1
= 𝜖0 𝑬 𝜔 + 𝜖0
𝑬 𝜔
𝑚0 𝜖0 −𝜔 2 − 𝑖𝜔/𝜏
𝜔𝑝2
𝜖𝑟 𝜔 = 1 − 2
𝜔 + 𝑖𝜔/𝜏
𝜔𝑝2
= 𝜖0 𝜖𝑟 𝑬 𝜔
𝑛𝑒 2
=
𝑚0 𝜖0
7
Linking 𝜖 (𝜔) , 𝜎 (𝜔), and 𝑛
𝑖𝜎
𝜖 𝜔 =
+ 1 = 𝜖1 + 𝑖𝜖2
𝜖0 𝜔
𝜎2
𝜖1 𝜔 = 1 −
𝜔𝜖0
𝜎1 = 𝜖2 𝜖0 𝜔
𝜎1
𝜖2 𝜔 =
𝜔𝜖0
𝜎2 = 𝜔𝜖0 (1 − 𝜖1 )
𝑛 = 𝑛 + 𝑖𝜅 = 𝜖
8
𝜔𝑝2
𝜖 =1− 2
𝜔 + 𝑖Γ𝜔
𝜔𝑝2
𝜖1 = 1 − 2
𝜔 + Γ2
For Au at 273 K:
1 Γ𝜔𝑝2
𝜖2 =
𝜔 𝜔2 + Γ2
9
𝜎0
𝜎 𝜔 =
1 − 𝑖𝜔𝜏
𝜎2 𝜔 =
𝜎0 𝜔𝜏
1 + 𝜔2𝜏 2
𝜎1 𝜔 =
𝜎0
1 + 𝜔2𝜏 2
10
𝑛=
𝜖12 + 𝜖22 + 𝜖1
2
𝜅=
𝜖12 + 𝜖22 − 𝜖1
2
11
Related parameters
• Reflectivity
(here normal incidence)
𝑛−1
𝑅=
𝑛+1
2
𝑛−1
=
𝑛+1
• Absorption coefficient
2𝜅𝜔 4𝜋𝜅
𝛼=
=
𝑐
𝜆
• Skin depth
2
𝛿=
𝛼
2
+ 𝜅2
2 + 𝜅2
12
𝑛−1
𝑅=
𝑛+1
2
13
Ep = 15.8 eV
14
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).
15
Temperature dependence
Pells and Shiga, J. Phys. C: Solid State Phys., V. 2, p. 1835 (1969).
16
Summary of the Drude-Sommerfeld model
• Allows qualitative, and often
quantitative understanding of
many optical properties of metal
• Conductivity
• Reflectivity
• Transparency if 𝜔 > 𝜔𝑝
• 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
17
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
18
Poor quantitative agreement with D.M.
10 um
1 um
19
Ordall et al. Appl. Opt. 22 1099 (1983)
Plasmon propagation length
• 1/e decay length
• Plasmon at Au/air interface
• λ = 10 μm
1
𝐿𝑖 = ′′
2𝑘𝑥
𝑘𝑥′′
𝜔
𝜖𝑣𝑎𝑐 𝜖𝐴𝑢
= 𝐼𝑚
𝑐 𝜖𝑣𝑎𝑐 + 𝜖𝐴𝑢
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
𝐿𝑖,𝑃𝑎𝑙𝑖𝑘 = 11.8 mm
𝐿𝑖,𝐵&𝐵 = 39.0 mm
20
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
Γ𝑡𝑜𝑡𝑎𝑙 = 2ℏ/𝑇𝑡𝑜𝑡𝑎𝑙
21
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).
22
Optical constants
measurement techniques
23
Kramers-Kronig method
sample
polarizer
source
monitor
• 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
detector
𝜔
𝜙𝑟 𝜔 =
𝜋
𝜎1 𝜔 = 𝜔𝜖0 𝜖2 𝜔 = 𝜔𝜖0
𝜎2 𝜔 = −𝜔𝜖0 1 − 𝜖1 𝜔
∞
0
ln 𝑅 𝜔′ − ln 𝑅(𝜔)
𝑑𝜔′
𝜔 2 − 𝜔 ′2
4 𝑅(𝜔) 1 − 𝑅 𝜔 sin 𝜙𝑟
1 + 𝑅 𝜔 − 2 𝑅(𝜔) cos 𝜙𝑟
= −𝜔𝜖0 1 −
1−𝑅 𝜔
2
[SI units]
2
− 4𝑅 𝜔 sin2 𝜙𝑟
1 + 𝑅 𝜔 − 2 𝑅 𝜔 cos 𝜙𝑟
2
Dressel & Grüner,
24
Ashcroft & Mermin, Appendix K
Kramers-Kronig relations
Handbook of Ellipsometry
Denotes that the
Cauchy principal value
must be taken
Hans Kramers
(1894-1952)
Ralph de Laer Kronig
25
(1904–1995)
Fresnel Equations
𝐸0𝑟
𝑟𝑠 =
𝐸0𝑖
𝑟𝑝 =
𝐸0𝑟
𝐸0𝑖
𝐸0𝑡
𝑡𝑠 =
𝐸0𝑖
𝐸0𝑡
𝑡𝑝 =
𝐸0𝑖
𝑠
𝑛𝑖 cos 𝜙𝑖 − 𝑛𝑡 cos(𝜙𝑡 )
=
𝑛𝑖 cos 𝜙𝑖 + 𝑛𝑡 cos 𝜙𝑡
Augustin-Jean Fresnel (1788-1827)
=
𝑝
𝑛𝑡 cos 𝜙𝑖 − 𝑛𝑖 cos(𝜙𝑡 )
𝑛𝑡 cos 𝜙𝑖 + 𝑛𝑖 cos 𝜙𝑡
𝑠
2𝑛𝑖 cos 𝜙𝑖
=
𝑛𝑖 cos 𝜙𝑖 + 𝑛𝑡 cos 𝜙𝑡
𝑝
2𝑛𝑖 cos 𝜙𝑖
=
𝑛𝑖 cos 𝜙𝑡 + 𝑛𝑡 cos 𝜙𝑖
Used for reflection-transmission measurements (like Johnson & Christy)
26
Ellipsometry
27
Ellipsometry
𝑟𝑝
𝜌 = = tan 𝜓 e𝑖Δ
𝑟𝑠
1−𝜌
𝑛 = sin 𝜙 1 +
1+𝜌
2
tan2 𝜙
28
Comparison of methods for widely
referenced optical constants for Au
Source
Author
Reference
Palik, ed.
M. L. Theye
PRB 2 3060 (1970)
Dold and
Mecke
Johnson
and Christy
energy range
measurement method
6-0.6 eV
reflectance & transmittance at
210 nm - 2070
normal incidence
nm
(requires known thickness)
Optik 22, 435 (1965)
“ellipsometric technique”;
1-0.125 eV
ERRONEOUSLY LOW K VALUES
1240 nm - 10 um
at longer wavelengths
PRB 6, 4370 (1972)
6.5 - 0.5 eV
reflectance & transmittance,
190 nm - 2000
different angles
nm
(requires significant modeling)
Optical Properties and
Bennett and Electronic Structure of 0.413 - 0.0388 eV
Ordall, ed.
Bennett
Metals and Alloys (Abeles, 3 um - 32 um
ed.)
Motulevich
Soviet Phys. JETP 20, 560
(1965)
1.24 - 0.1033 eV
1 um - 12 um
reflectance
not readily available
29
Spectroscopic Ellipsometry of bulk Au
planar surfaces
30
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
31
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.
32
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.
33
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.
34
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.
35
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.
36
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.
37