Optical Properties of Metals

```Advanced Materials - Lab Intermediate Physics
Ulm University
Institute of Solid State Physics
Optical Properties of Metals
Burcin Özdemir
Luyang Han
April 28, 2014
I
Safety Precations
MAKE SURE THAT YOU UNDERSTAND THIS
SECTION BEFORE YOU ATTEND THE EXPERIMENT!
•
Always wear gloves when dealing with chemicals. Handle with care and avoid
spill. Always follow the instruction of the tutor when doing the operations.
•
The light source used in the spectroscopy contains strong UV radiation. DO
NOT look into the light source.
II
1
1
Introduction to the optical property of material
Light is a form of electromagnetic radiation which can be dened by dierent wave10
−12
lengths ranging between long radio waves (10 m) and gamma rays (10
m) as
depicted in Fig. 1.
Figure 1: The electromagnetic spectrum with a focus on the visible light region [1].
A human eye can respond to the wavelengths between 400-700 nm (<800 nm
gender dependent) which is visible as the colors of violet to red respectively. Light
interacts with the material by transmission, refraction, absorption, scattering or
reection. What we see as the color of the material results from all these interactions
and transmitted to the human eye.
1.1 General description of optical property
Generally, the propagation of light in material can be described by Maxwell's equations [2] which can be solved to understand the relation of magnetic and electrical
properties of the material. Light is basically the oscillation of the electro-magnetic
eld. The oscillating eld is generally described using complex numbers.
Here we consider the most simple case that the oscillation has the form of a
planar wave, which means the electric eld can be expressed as:
E = E0 exp i(k · r − ωt).
The wave vector
k
(1)
√√
|k|2 = ω ˜ 0 µ0
respectively). Here c =
is a complex number which can be dened as
(0 and
µ0 are permittivity and permeability of free
√ space
√
1/ 0 µ0 denes the speed of light in vacuum and ˜ denes
refraction in the material:
the complex index of
2
1
INTRODUCTION TO THE OPTICAL PROPERTY OF MATERIAL
ñ =
√
˜ = η + iκ,
ñω
k = e .
c
(2)
(3)
e denes the unit vector in the direction of wave propagation. The complex
n and κ∗ . The physical meaning
η and κ will be clear if we substitute k back into the planar wave equation.
κω
ηω
e · r − ωt exp − e · r .
(4)
E = E0 exp i
c
c
Here
index of refraction is decomposed of two component
of
The second exponential component describes how the amplitude of the electric
eld gets attenuated along the direction of the wave propagation. The light intensity
2
for a planar wave is proportional to |E| , thus the intensity of the light will be
attenuated as:
I = I0 exp(−
2κω
e · r) = I0 exp(−αe · r).
c
(5)
2κω
The α =
is the absorption coecient of the material, and has the dimension of
c
−1
[L ]. (5) proves the Lambert-Beer law in optics.
1.2 Light scattering of small particle
Scattering is the process by which the intensity of the light get directed to other
direction in an inhomogeneous material. The scattering will also cause the attenuation of the light intensity.
Unlike absorption, where the energy is transformed
energy to other directions.
Both absorption and scattering contribute to the at-
tenuation of light when it passes through small particles. The total attenuation is
called extinction. If the particles are dispersed homogeneously within the medium,
macroscopically the light extinction can also be described by the Lambert-Beer law.
And similar as in (5) an extinction coecient can be dened which take into account
both scattering and absorption.
If we know the concentration of the particles
C,
the extinction coecient can be normalized to each particle:
σ=
The
σ
α
.
C
(6)
2
here has dimension [L ], thus it is called the extinction cross-section of
the particle. This is an intrinsic property of the particle and depends only on the
optical properties of the particle and of its surrounding medium.
The interaction of light with small particles is a fairly complicated phenomenon.
The scattered light usually has certain angular distribution, and the scattering process usually depends on the wavelength of the light, the size, shape and optical
properties of the particles as well as the medium in which the scattering process is
taking place. A complete description of the process would require the exact solution
∗
Dierent convention exists to express the complex index of refraction. An alternative is
ñ = n(1 + iκ).
1.3
3
Optical measurement
of Maxwell's equations considering all those parameters. Such problem is rst solved
analytically by G. Mie [3] and is usually called Mie scattering.
A formal treatment of Mie scattering is very complicated and beyond the scope
of this experiment.
However, there are some simulation software packages which
simulates the Mie scattering process, such as MiePlot (http://www.philiplaven.
com/mieplot.htm).
The students are encouraged to try and explore the features of
Mie scattering with such software.
If the particle is signicantly smaller than the wavelength of the light, the electric
eld applied to the particle can be assumed homogeneous. In this case a spherical
particle behaves like a dipole and the radiation eld is an oscillating dipole. Such assumption leads to Rayleigh scattering [4], from which the absorption and scattering
cross-section are:
00
, where
k
3r
(2 + 0r )2 + 00r 2
0
00 2
2
k4
2 (r − 1) + r
(3V )
=
6π
(2 + 0r )2 + 00r 2
σabs = 3kV
(7)
σsca
(8)
is the wave vector and
V
is the volume of the particle. The relative
permittivity of the material is dened as
0
00
˜r = r + ir = ˜metal /˜medium
.
1.3 Optical measurement
In this part we discuss the basic formalism needed for the experimental measurement.
The optical properties are measured by light that is directed towards the
material and the reected, scattered or transmitted light is detected. This process
is illustrated in Fig.2.
Container, substrate, etc...
Homogeneous medium
absorption in medium
Incoming light
reflection at interface
absorption of substrate
particle
reflection on interface
absorption in particle
Transmittance
Reflectance
scattered by particle
Scattering
Figure 2: Possible processes in an optical measurement. The reectance, transmittance and scattering can be measured. The light would interact with all the optical
element involved in the system, not only limited to the material that we want to
measure.
4
1
INTRODUCTION TO THE OPTICAL PROPERTY OF MATERIAL
First we consider the measurement of the optical properties of a homogeneous
material.
We want to determine the complex index of refraction for certain fre-
quency of light. When the light is directed to the interface of two dierent media,
because of the dierence in the index of refraction, part of the light is reected.
The reectance can be derived by solving the Maxwell's equations considering the
boundary conditions at the interface. The result is described by Fresnel's equations.
When the incident light is normal to the interface, the reectance and transmittance
at the interface can be written as:
ñ1 − ñ2 2
,
R = ñ1 + ñ2 2(ñ1 ñ∗2 + ñ∗1 ñ2 )
T =
.
|ñ1 + ñ2 |2
The reectance and transmittance fulll
R + T = 1.
(9)
(10)
If we consider the case that
the light is directed from vacuum or air to certain material with
ñ = η + iκ,
the
formula can be expanded as:
(1 − η)2 + κ2
,
(1 + η)2 + κ2
4η
T =
.
(1 + η)2 + κ2
R =
Since the reectance and transmittance are related, measuring only
(11)
(12)
R
and
T
is
not sucient to determine the optical properties of material, i.e. to determine both
η
and
κ.
It is necessary to measure also the absorbance within the material for
a certain thickness. This would yield the value for the absorption coecient, and
then
κ can
be extracted. This seems to be quite simple, one can just take a piece of
material with known thickness and measure what is the light intensity before and
after passing though the material.
However, one should notice that the material
has at least 2 interfaces, on which the light is reected. The nal attenuation eect
is the sum of the absorption within the material plus the reection on the surface.
Moreover, the light reected from the inner surface of the material might get reected
multiple times between the two interfaces, and the reected light might also interfere
with the incoming light. Also in certain case, the measured material needs to be
kept in certain container (liquids) or deposited on a substrate, which will cause even
more complicated reections. All of these make the transmission fairly complex and
hard to analyze.
Usually computer simulations are used to calculate the optical
coupling in multi-layered system (such as
http://thinfilm.hansteen.net/).
To
overcome this problem, one may measure at two dierent sample thicknesses, and
the dierence of the attenuation between the two pieces is measured. This dierence
is usually only due to the absorption within the material.
If the material is inhomogeneous, both the scattering and absorption can occur.
The total attenuation is the sum of both eects and it is called extinction. If just the
transmission is measured, it is not possible to distinguished how much is scattered
or absorbed. In this case the extinction cross section can be measured. If we have
5
small particles disperse in certain medium, it is important to measure also the optical
absorption of the medium without the particles as reference data.
2
Optical properties of metal
The optical response of metals is mainly originates from the conduction electrons.
The Drude model of free electron states that the electrons in metals behave like
classical gas molecules. There is no interaction between the electrons except scattering. The average scattering interval time is dened as
τ.
The free electron within
the metal is the main reason why the metal is not transparent and highly reective.
m
d2 x m dx
= −eE.
+
dt2
τ dt
(13)
The second term on the left-hand-size corresponds to an averaging eect of the
scattering to slow down the electrons.
Using the same expression for oscillating
electric eld as before, the electron displacement can be expressed as:
x=
m(ω 2
eE
.
+ iω/τ )
(14)
The polarization is the dipole moment induced by the electron movement in unit
volume, thus:
ne2
P = −nex = −
E.
m(ω 2 + iω/τ )
(15)
As a result the permittivity and index of refraction for free electrons are:
ne2
P (ω)
=1−
,
(ω) = 1 +
0 E(ω)
0 m(ω 2 + iω/τ )
s
s
2
ωp2
ne
n =
1−
= 1− 2
.
0 m(ω 2 + iω/τ )
ω + iω/τ
The plasma frequency is dened as
ωp2 = ne2 /0 m.
(16)
(17)
This property just depends
on the mass and density of the electrons. The real and imaginary part of the index
of refraction around the plasma frequency is shown in Fig.3. For normal metal ωp is
15
16
−14
Hz and τ is around 10
s at room temperature. This means that
around plasma frequency the electron is oscillating much faster than the collision.
For a qualitative discussion the eect of electron collision can be neglected in (16)
and (17). If the optical frequency is lower than
ωp ,
the permittivity is negative and
index of refraction becomes purely imaginary. This means the electric eld will just
penetrate into the material, but does not form an oscillating wave. If the material
is thick enough, all the incoming wave will be reected.
This is the reason why
metal surface looks colorless and shiny. If the optical frequency is higher than
ωp ,
the index of refraction is real, which means the material becomes transparent. For
normal metals this usually happens at ultra-violet frequency range. It is interesting
6
3
MEASUREMENT SETUP
2
Index of refraction
1.5
η
κ
Plasma frequency
1
0.5
0
-0.5
0.5
1
1.5
2
ω/ω p
Figure 3: The real and imaginary part of index of refraction around plasma frequency
for a metal according to (17). The
1/τ
is assumed to be signicantly smaller than
the plasma frequency.
that at this frequency range the index of refraction is smaller than 1. This means the
phase velocity of light within the material is larger than the speed of light in vacuum.
This leads to many interesting phenomenon and applications. For example, the UV
light will have total reection on metal surface at large incident angle, similar as
normal light in prism.
In metals only copper, osmium and gold show certain color in visible light. The
color of gold and copper is related to its band structure. In the case of Au the 5d
orbit is completely lled and the 6s is half-lled. The energy dierence between 5d
and 6s level in gold is about 4 eV and this strong absorption cut out the green-blue
light from the reection, creating the yellow color of gold. Copper has similar eect
for its 3d/4s orbit structure but with lower absorption energy [5].
3
Measurement setup
In this experiment a compact optical spectrometer is used to measure the absorption/extinction of the sample. The working principle of the measurement is shown
in Fig.4. The light source is polychromatic and contains wavelength from 200 nm
to 900 nm. The polychromatic light is then focused and passed through the absorption material directly. The transmitted light then illuminates a dispersive element,
where light of dierent wavelength are reected to dierent directions.
Using a
CCD detector array the light with dierent wavelength are then recorded simultaneously.
This is dierent from common spectrometers, where there is just one
detector and the each time just one wavelength can be recorded.
In comparison,
the compact optical spectrometer record the dierent wavelength simultaneously,
thus its acquisition speed is signicantly faster than standard spectrometers. Moreover, the compact spectrometer does not need any moving component, while the
normal spectrometer needs to move either the detector or the dispersive element to
scans through dierent wavelength. Thanks to its much simple design, the size of
7
absoprtion material
UV+VIS
light source
dispersive element
light
beam
CCD detector array
Figure 4: Working principle of the spectrometer. Polychromatic light passes through
the absorption material, and then through a dispersive element, where light with
dierent wavelength is diverted to dierent directions.
The light with dierent
wavelength is then detected with CCD detectors located at dierent position simultaneously.
a compact spectrometer is much smaller and the cost is lower compared to common
spectrometer.
4
Experimental procedure
4.1 Preparation of Au thin lm
Two transparent gold lms are prepared by DC sputtering.
Quartz glass is used
as the substrate due to its transparency in UV spectrum range. The substrates
2
are cut to 5 x 10 mm size. They are then cleaned with aceton and isopropanol in
ultrasonic bath to get rid of the dirt. Sputtering is performed in the Balzers mini
sputter machine. The distance of the target to sample is set to 50 mm. Sputtering
current is 30 mA and Argon pressure is 0.05 mbar. Under such working conditions
the deposition rate is about 0.14 nm/s.
To prepare Au lms with 10 nm and 15
nm thickness the sputtering time is 70 s and 110 s, respectively. The lm thickness
should not exceed 25 nm, as then there would hardly be any light passing through.
If the nominal thickness is smaller than 8 nm the lm might be discontinuous and
the optical properties will be dierent from that of the bulk. The gold lm prepared
by this method might not stick very rmly on the substrate. One should take care
∗
when handling the sample, especially scratching by tweezers shall be avoided.
4.2 The Au/Ag nanoparticle solutions
Commercially available 4 Au and 4 Ag nanoparticle (NP) solutions will be used
to compare the size dependent optical properties [6,7]. Related information on the
∗
The lms will be prepared by your instructor.
8
5
REPORT AND DATA TREATMENT
particle diameters as well as the particle concentration in the solutions are depicted
in Table 1.
Au NP suspension in citrate buer
12
10 nm, C: 6E particles/mL
50 nm, C: 3E 10 particles/mL
80 nm, C: 7.8E 9 particles/mL
100 nm, C: 3.8E 9 particles/mL
Ag NP suspension in citrate buer
10 nm, C: 3.6E 12
40 nm, C: 5.7E 10
60 nm, C: 1.7E 10
100 nm, C: 3.6E 9
particles/mL
particles/mL
particles/mL
particles/mL
Table 1: Particle concentrations of commercially available Ag/Au nanoparticle solutions [6,7].
4.3 Measure the optical properties of the Au thin lm and
Au/Ag nanoparticles
First the absorption of the Au thin lm will be measured. The spectrum of the light
source must be recorded as reference. To measure the attenuation of certain material,
record the spectrum of the light passing through the material, and the dierence to
the reference is the attenuation. The following spectra shall be measured:
1. The original light source
2. The quartz glass substrate
3. 10 nm Au lm on quartz glass
4. 15 nm Au lm on quartz glass
The dierence of 2. and 3. should be the absorption of the Au, from which one can
deduce the imaginary part of the index of refraction. The dierence between 2. and
3. or 2. and 4. is the total eect of absorption in Au as well as reection at the
interface.
To measure the optical extinction of the Au/Ag nanoparticle solutions, rst
the absorption of the same solution without the nanoparticles is measured as the
reference. The total extinction coecient of Au and Ag particles can be obtained
for dierent wavelengths and particle sizes.
With the knowledge of the particle
concentration the extinction cross section of the particle can be calculated.
5
Report and data treatment
Below you nd some details of data analysis and questions that should be addressed
in the report.
The data evaluation will be done during the lab.
laptop with an Excel or compatible program. Prepare your report in accordance to
the guidelines for lab reports!
1. Plot the transmission spectra of the light source, quartz glass and related Au
lms.
9
REFERENCES
2. Plot the trasmittance and optical density of the Au lms with dierent thicknesses. Try to explain the origin of the dierence.
3. Plot how the absorption coecient of Au changes with dierent wavelength.
Try to explain the origin of the dierence, and compare the result to the
literature (8,9).
4. Calculate the imaginary part of the index of refraction and compare the result
to the literature [8].
5. Plot the transmission spectra of the light source, cuvette and related Au/Ag
NP solutions.
6. Plot the trasmittance and optical density of the Au/Ag NP solutions with
dierent diameters. Try to explain the origin of the dierence.
7. Plot the extinction cross section of Au/Ag NPs as a function of dierent wavelength and size. Try to explain the extinction spectrum.
References
[1] http://zebu.uoregon.edu/~imamura/122/lecture-2/em.html
[2] B. Schaefer, Lehrbuch der Experimentalphysik 3:
Wellen- und Teilchenoptik.
Walter de Gruyter, 2004.
[3] G. Mie, Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen
Ann. Phys., vol. 330, p. 377-445, 1908.
[4] C. F. Bohren and D. R. Human, Absorption and scattering of light by small
particles. Wiley, 1983.
[5] G. P. Pells and M. Shiga, The optical properties of copper and gold as a function
of temperature Journal of Physics C: Solid State Physics, vol. 2, no. 10, p. 1835,
1969.
[6] http://www.sigmaaldrich.com/materials-science/nanomaterials/goldnanoparticles.html
[7] http://www.sigmaaldrich.com/materials-science/nanomaterials/silvernanoparticles.html
[8] http://www.sspectra.com/sopra.html
[9] S. Kupratakuln, Relativistic electron band structure of gold Journal of Physics
C: Solid State Physics, vol.3, no. 2S, p. S109, 1970.
10
A
A
OPTICAL PROPERTY DATA OF GOLD
Optical property data of gold
Here the complex index of refraction for metallic gold is plotted. The data is taken
from [8].
Index of Refraction
6
5
4
3
2
1
0
200
η
κ
300
400
500
600
700
Wavelength (nm)
800
900
```