Download Optical Properties of Metals

Advanced Materials - Lab Intermediate Physics
Ulm University
Institute of Solid State Physics
Optical Properties of Metals
Luyang Han
May 12, 2010
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
1.1
General description of optical property
Generally, the propagation of light in material can be described by Maxwell’s equations[1],
which are:
∇·D = ρ
∇·B = 0
∂B
∂t
∂D
∇×H = j+
∂t
∇×E = −
(1)
(2)
(3)
(4)
The meaning of the symbols are:
D: Electric displacement
E: Electric field
B: Magnetic induction
H: Magnetic field
ρ: Charge density
j: Electric current density
If we consider the material is not charged, non-magnetic, linear, homogeneous
and isotropic∗ , the following relations can be established:
ρ
j
D
B
=
=
=
=
0
σE
0 E
µ0 H
(5)
(6)
(7)
(8)
Equation (5) and (8) describe that the material has neutral charge and does not
have any magnetic response. Equation (6) is basically Ohm’s law and (7) describes
the dielectric response of the material. These equations describe the properties of
the materials which relates the different terms in the basic Maxwell’s equations.
Light is basically the oscillation of the electro-magnetic field. The oscillating field
∗
Non-magnetic means the magnetic permeability is 1. This assumption is valid for almost all
the material, including ferromagnetic material. One should note that the permeability discussed
here is that at optical frequencies. At low frequency the permeability of ferromagnetic material is
usually much larger than 1, but the magnetic response usually cannot follow up the fast oscillating
magnetic field at optical frequencies (1015 Hz), and the permeability at such high frequencies is
close to 1. Linear material means the relation of the response to the stimulate can be described
by a linear constant. This is usually valid when the stimulate is small. Homogeneous means the
material constant is not dependent on the position in the material. A material constant relating
two vector component is generally a tensor. For example, in single crystal material the conductivity
and permittivity depends on the direction of the field with respect to the crystal. In a isotropic
material this tensor can be simplified to a scalar.
2
1 INTRODUCTION TO THE OPTICAL PROPERTY OF MATERIAL
is generally described using complex numbers. For example, E = E0 exp(−iω)∗ . In
this case the physical electric field is taken as the real part of the complex number.
Using such convention the (3) and (4) can be simplified as:
∇ × E = iωµ0 H
∇ × H = (σ − iω0 )E
(9)
(10)
Note that if we define ˜ = + i ωσ0 , (10) can be written simply as:
∇ × H = −iω0 ˜ E
(11)
Here the ˜ is the complex permittivity of the material. For the material discussed
here, its optical properties can be completely described by this ˜ † .
Combining (11) and (9), and note that ∇ × ∇E = ∇(∇ · E) − ∇2 E, the wave
equation can be obtained:
∇2 E − ω 2 0 µ0 ˜E = 0
(12)
Here we consider the most simple case that the oscillation has the form of a planar
wave, which means the electric field can be expressed as E = E0 exp i(k · r − ωt).
Note that because the permittivity is a complex number, the wave vector k should
also be complex.
Substitute the expression into (12) the wave vector should fulfill
√
√√
√
2
|k| = ω ˜ 0 µ0 . Here c = 1/ 0 µ0 defines the speed of light in vacuum and ˜
defines the complex index of refraction in the material:
√
˜ = η + iκ
ñω
k = e
c
ñ =
(13)
(14)
Here e defines the unit vector in the direction of wave propagation. The complex
index of refraction is decomposed of two component n and κ‡ . The physical meaning
of η and κ will be clear if we substitute k back into the planar wave equation.
κω
ηω
e · r − ωt exp − e · r
(15)
E = E0 exp i
c
c
∗
Sometimes the complex exponent is also taken as exp(iω), which leads to small difference in
certain formula. However the physical phenomenon is invariant under different choice of exponent.
†
Here we assumed that σ and in (6) and (7) are real numbers. In fact one can describe both
as complex numbers, and obtain an overall ˜ parameter. However, such choice is purely mathematical. When discussing the oscillating electric field, a clear distinction between conductivity
and permittivity does not have much physical sense. Roughly speaking, the current induced by a
oscillating electric field has both an in-phase and an out-of-phase component. The in-phase component corresponds to the conductivity and since it is in phase with electric field, ohmic loss will
occur and this leads to the attenuation of the oscillation energy. The out-of-phase component is
related to the permittivity and produce no energy loss but induces the oscillating magnetic field,
i.e., the propagation of the oscillation energy.
‡
Different convention exists to express the complex index of refraction. An alternative is
ñ = n(1 + iκ).
1.2 Light scattering of small particle
3
The second exponential component describes how the amplitude of the electric
field gets attenuated along the direction of the wave propagation. The light intensity
for a planer wave is proportional to |E|2 , thus the intensity of the light will be
attenuated as:
I = I0 exp(−
2κω
e · r) = I0 exp(−αe · r)
c
(16)
The α = 2κω
is the absorption coefficient of the material, and has the dimension of
c
[L−1 ]. (16) 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
from electromagnetic radiation to some other form, scattering deviate the radiation
energy to other directions. Both absorption and scattering contribute to the attenuation 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 (16) an extinction coefficient can be defined which take into account
both scattering and absorption. If we know the concentration of the particles C .
The extinction coefficient can be normalized to each particle:
σ=
α
C
(17)
The σ here has dimension [L2 ], 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
of Maxwell’s equations considering all those parameters. Such problem is first solved
analytically by G. Mie [2] 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 significantly smaller than the wavelength of the light, the electric
field applied to the particle can be assumed homogeneous. In this case a spherical
particle behaves like a dipole and the radiation field is an oscillating dipole. Such
assumption leads to Rayleigh scattering[3], from which the absorption and scattering
cross-section are:
4
1 INTRODUCTION TO THE OPTICAL PROPERTY OF MATERIAL
00
σabs
σsca
3r
= 3kV
(2 + 0r )2 + 00r 2
0
00 2
2
k4
2 (r − 1) + r
=
(3V )
6π
(2 + 0r )2 + 00r 2
(18)
(19)
, where k is the wave vector and V is the volume of the particle. The relative
0
00
permittivity of the material is defined as ˜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 reflected, scattered or transmitted light is detected. This process
is illustrated in Fig.1.
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 1: Possible processes in an optical measurement. The reflectance, 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.
First we consider the measurement of the optical properties of a homogeneous
material. We want to determine the complex index of refraction for certain frequency of light. When the light is directed to the interface of two different media,
because of the difference in the index of refraction, part of the light is reflected.
The reflectance 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 reflectance 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
(20)
(21)
5
The reflectance and transmittance fulfill R + T = 1. 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 =
(22)
(23)
Since the reflectance and transmittance are related, measuring only R and T is
not sufficient 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 coefficient, 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 reflected. The final attenuation effect
is the sum of the absorption within the material plus the reflection on the surface.
Moreover, the light reflected from the inner surface of the material might get reflected
multiple times between the two interfaces, and the reflected 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 reflections. 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 different sample thicknesses, and
the difference of the attenuation between the two pieces is measured. This difference
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 effects 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
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 defined as τ . The free electron within
the metal is the main reason why the metal is not transparent and highly reflective.
The equation of motion for free electron in electric field is:
m
d2 x m dx
+
= −eE
dt2
τ dt
(24)
6
2 OPTICAL PROPERTIES OF METAL
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 2: The real and imaginary part of index of refraction around plasma frequency
for a metal according to (28). The 1/τ is assumed to be significantly smaller than
the plasma frequency.
The second term on the left-hand-size corresponds to an averaging effect of the
scattering to slow down the electrons. Using the same expression for oscillating
electric field as before, the electron displacement can be expressed as:
x=
eE
m(ω 2 + iω/τ )
(25)
The polarization is the dipole moment induced by the electron movement in unit
volume, thus:
ne2
P = −nex = −
E
m(ω 2 + iω/τ )
(26)
As a result the permittivity and index of refraction for free electrons are:
P (ω)
ne2
(ω) = 1 +
=1−
0 E(ω)
0 m(ω 2 + iω/τ )
s
s
ωp2
ne2
n =
1−
=
1
−
0 m(ω 2 + iω/τ )
ω 2 + iω/τ
(27)
(28)
The plasma frequency is defined as ωp2 = ne2 /0 m. 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.2. For normal metal ωp is
about 1015 − 1016 Hz and τ is around 10−14 s at room temperature. This means that
around plasma frequency the electron is oscillating much faster than the collision.
For a qualitative discussion the effect of electron collision can be neglected in (27)
and (28). If the optical frequency is lower than ωp , the permittivity is negative and
index of refraction becomes purely imaginary. This means the electric field will just
penetrate into the material, but does not form an oscillating wave. If the material
7
is thick enough, all the incoming wave will be reflected. 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
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 reflection 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 filled and the 6s is half-filled. The energy difference between 5d
and 6s level in gold is about 4 eV and this strong absorption cut out the green-blue
light from the reflection, creating the yellow color of gold. Copper has similar effect
for its 3d/4s orbit structure but with lower absorption energy [4].
3
Measurement setup
absoprtion material
UV+VIS
light source
dispersive element
light
beam
CCD detector array
Figure 3: Working principle of the spectrometer. Polychromatic light passes through
the absorption material, and then through a dispersive element, where light with
different wavelength is diverted to different directions. The light with different
wavelength is then detected with CCD detectors located at different position simultaneously.
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.3. 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 different wavelength are reflected to different directions.
Using a CCD detector array the light with different wavelength are then recorded
simultaneously. This is different from common spectrometers, where there is just
one detector and the each time just one wavelength can be recorded. In comparison,
8
4 EXPERIMENTAL PROCEDURE
the compact optical spectrometer record the different wavelength simultaneously,
thus its acquisition speed is significantly 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 different wavelength. Thanks to its much simple design, the size of
a compact spectrometer is much smaller and the cost is lower compared to common
spectrometer.
4
Experimental procedure
4.1
Preparation of Au thin film
Two transparent gold films are prepared by DC sputtering. Quartz glass is used
as the substrate due to its transparency in UV spectrum range. The substrates
are cut to 5 x 10 mm2 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 films with 10 nm and 15
nm thickness the sputtering time is 70 s and 110 s, respectively. The film 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 film might be discontinuous and
the optical properties will be different from that of the bulk. The gold film prepared
by this method might not stick very firmly on the substrate. One should take care
when handling the sample, especially scratching by tweezers shall be avoided.∗
4.2
Preparation of Au nanoparticles
The method to prepared the Au nanoparticles in this experiments is discovered by
J. Turkevich et al. in 1951 [5]. Details of the method is adapted from [6]. The
procedure is as follow:
1. Prepare 1 mM chloroauric acid (HAuCl4 · H2 O) solution and 38.8 mM sodium
citrate hydrate (Na3 C6 H5 O7 · 2H2 O) solution in water.
2. Move 5 ml 1 mM chloroauric acid solution with pipette into a beaker. Put a
magnetic stirrer in the beaker and put the beaker on the heating plate. Heat
and stir the solution simultaneously till boiling.
3. After the solution reaches the boiling point, add 0.5 ml sodium citrate solution
into the beaker using the pipette. The color of the solution will start changing
to deep red.
4. Continue to heat and stir the solution. Add water to compensate the evaporation lose of water and keep the total volume of the solution at about 5 ml.
Continue the heating till the solution becomes homogeneous and transparent.
∗
The films will be prepared by your instructor.
4.3 Measure the optical properties of the Au thin film and nanoparticles
9
5. Turn off the heating but keep stirring the solution. Wait about 20 min till it
cools down to room temperature. Dilute the solution to certain concentration
so that one can easily look through. Afterwards one can move the solution of
Au nanoparticles to some other container for storage.
The Au nanoparticles prepared by this method have an average diameter of about 20
nm. The sodium citrate acts as a reducing agent. After reduction the citrate anion
is absorbed on the surface of the particle, introducing the surface charge that repels
the particles and prevents them from aggregating. A scanning electron microscope
image of the particles is shown in Fig.4.
Figure 4: SEM image of the Au nanoparticles prepared by Turkevich’s method. A
droplet of the solution containing the Au NPs are deposited on Si substrate. After
the solvent dried the sample is investigated by SEM.
4.3
Measure the optical properties of the Au thin film and
nanoparticles
First the absorption of the Au thin film 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 difference 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 film on quartz glass
4. 15 nm Au film on quartz glass
The difference 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 difference between 2. and
3. or 2. and 4. is the total effect of absorption in Au as well as reflection at the
interface.
10
REFERENCES
To measure the optical extinction of the Au nanoparticles, first the absorption of
the same solution without the Au nanoparticles is measured as the reference. The
total extinction coefficient of Au particles can be obtained for different wavelength.
With the knowledge of chloroauric acid concentration, and assume the particles are
20 nm spheres, the particle concentration can be estimated and the extinction cross
section of the particle can be calculated.
5
Report and data treatment
Below you find some details of data analysis and questions that should be addressed
in the report. Prepare your report in accordance to the guidelines for lab reports!
1. Plot how the absorption coefficient of Au changes with different wavelength.
Try to explain the origin of the difference, and compare the result to the
literature (for example [4, 7, 8]).
2. Compare the absorption of quartz glass with and without Au. Is this difference
the same as the absorption of Au? How can one explain the possible difference?
3. Plot the extinction cross section of Au nanoparticles as a function of different
wavelength. Try to explain the extinction spectrum.
References
[1] B. Schaefer, Lehrbuch der Experimentalphysik 3: Wellen- und Teilchenoptik.
Walter de Gruyter, 2004.
[2] G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,”
Ann. Phys., vol. 330, p. 377–445, 1908.
[3] C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small
particles. Wiley, 1983.
[4] 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.
[5] J. Turkevich, P. C. Stevenson, and J. Hillier, “A study of the nucleation and
growth processes in the synthesis of colloidal gold,” Discussions of the Faraday
Society, vol. 11, pp. 55–75, 1951.
[6] A. D. McFarland, C. L. Haynes, C. A. Mirkin, R. P. V. Duyne, and H. A. Godwin,
“Color my nanoworld,” Journal of Chemical Education, vol. 81, p. 544A, 2004.
[7] S. Kupratakuln, “Relativistic electron band structure of gold,” Journal of Physics
C: Solid State Physics, vol. 3, no. 2S, p. S109, 1970.
[8] “Optical database.”
10.15.2010.
http://www.sspectra.com/sopra.html,
retrieved
on
11
A
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