Advanced Materials - Lab Intermediate Physics

Ulm University

Institute of Solid State Physics

Optical Properties of Metals

Luyang Han

May 12, 2010

I

Safety PrecationsMAKE SURE THAT YOU UNDERSTAND THIS

SECTION BEFORE YOU ATTEND THE EXPERIMENT!

• Always wear gloves when dealing with chemicals. Handle with care and avoidspill. Always follow the instruction of the tutor when doing the operations.

• The light source used in the spectroscopy contains strong UV radiation. DONOT 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 (1)∇ ·B = 0 (2)

∇× E = −∂B∂t

(3)

∇×H = j +∂D

∂t(4)

The meaning of the symbols are:D: Electric displacementE: Electric fieldB: Magnetic inductionH: Magnetic fieldρ: Charge densityj: Electric current densityIf we consider the material is not charged, non-magnetic, linear, homogeneous

and isotropic∗, the following relations can be established:

ρ = 0 (5)j = σE (6)

D = ε0εE (7)B = µ0H (8)

Equation (5) and (8) describe that the material has neutral charge and does nothave any magnetic response. Equation (6) is basically Ohm’s law and (7) describesthe dielectric response of the material. These equations describe the properties ofthe 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 allthe material, including ferromagnetic material. One should note that the permeability discussedhere is that at optical frequencies. At low frequency the permeability of ferromagnetic material isusually much larger than 1, but the magnetic response usually cannot follow up the fast oscillatingmagnetic field at optical frequencies (1015Hz), and the permeability at such high frequencies isclose to 1. Linear material means the relation of the response to the stimulate can be describedby a linear constant. This is usually valid when the stimulate is small. Homogeneous means thematerial constant is not dependent on the position in the material. A material constant relatingtwo vector component is generally a tensor. For example, in single crystal material the conductivityand permittivity depends on the direction of the field with respect to the crystal. In a isotropicmaterial 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)∗. Inthis 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 = iωµ0H (9)∇×H = (σ − iωε0ε)E (10)

Note that if we define ε = ε+ i σωε0

, (10) can be written simply as:

∇×H = −iωε0εE (11)

Here the ε is the complex permittivity of the material. For the material discussedhere, its optical properties can be completely described by this ε †.

Combining (11) and (9), and note that ∇ × ∇E = ∇(∇ · E) − ∇2E, the waveequation can be obtained:

∇2E− ω2ε0µ0εE = 0 (12)

Here we consider the most simple case that the oscillation has the form of a planarwave, 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 shouldalso be complex. Substitute the expression into (12) the wave vector should fulfill|k|2 = ω

√ε√ε0µ0 . Here c = 1/

√ε0µ0 defines the speed of light in vacuum and

√ε

defines the complex index of refraction in the material:

n =√ε = η + iκ (13)

k = enω

c(14)

Here e defines the unit vector in the direction of wave propagation. The complexindex of refraction is decomposed of two component n and κ‡. The physical meaningof η and κ will be clear if we substitute k back into the planar wave equation.

E = E0 exp i(ηωc

e · r− ωt)

exp(−κωc

e · r)

(15)

∗Sometimes the complex exponent is also taken as exp(iω), which leads to small difference incertain formula. However the physical phenomenon is invariant under different choice of exponent.

†Here we assumed that σ and ε in (6) and (7) are real numbers. In fact one can describe bothas complex numbers, and obtain an overall ε parameter. However, such choice is purely math-ematical. When discussing the oscillating electric field, a clear distinction between conductivityand permittivity does not have much physical sense. Roughly speaking, the current induced by aoscillating electric field has both an in-phase and an out-of-phase component. The in-phase com-ponent corresponds to the conductivity and since it is in phase with electric field, ohmic loss willoccur and this leads to the attenuation of the oscillation energy. The out-of-phase component isrelated 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 isn = n(1 + iκ).

1.2 Light scattering of small particle 3

The second exponential component describes how the amplitude of the electricfield gets attenuated along the direction of the wave propagation. The light intensityfor a planer wave is proportional to |E|2, thus the intensity of the light will beattenuated as:

I = I0 exp(−2κω

ce · r) = I0 exp(−αe · r) (16)

The α = 2κωc

is the absorption coefficient of the material, and has the dimension of[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 otherdirection in an inhomogeneous material. The scattering will also cause the atten-uation of the light intensity. Unlike absorption, where the energy is transformedfrom electromagnetic radiation to some other form, scattering deviate the radiationenergy to other directions. Both absorption and scattering contribute to the attenu-ation of light when it passes through small particles. The total attenuation is calledextinction. If the particles are dispersed homogeneously within the medium, macro-scopically the light extinction can also be described by the Lambert-Beer law. Andsimilar as in (16) an extinction coefficient can be defined which take into accountboth 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 ofthe particle. This is an intrinsic property of the particle and depends only on theoptical 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 pro-cess usually depends on the wavelength of the light, the size, shape and opticalproperties of the particles as well as the medium in which the scattering process istaking place. A complete description of the process would require the exact solutionof Maxwell’s equations considering all those parameters. Such problem is first solvedanalytically by G. Mie [2] and is usually called Mie scattering.

A formal treatment of Mie scattering is very complicated and beyond the scopeof this experiment. However, there are some simulation software packages whichsimulates the Mie scattering process, such as MiePlot (http://www.philiplaven.com/mieplot.htm). The students are encouraged to try and explore the features ofMie scattering with such software.

If the particle is significantly smaller than the wavelength of the light, the electricfield applied to the particle can be assumed homogeneous. In this case a sphericalparticle behaves like a dipole and the radiation field is an oscillating dipole. Suchassumption leads to Rayleigh scattering[3], from which the absorption and scatteringcross-section are:

4 1 INTRODUCTION TO THE OPTICAL PROPERTY OF MATERIAL

σabs = 3kV3ε

′′r

(2 + ε′r)

2 + ε′′2r

(18)

σsca =k4

6π(3V )2 (ε

′r − 1)2 + ε

′′2r

(2 + ε′r)

2 + ε′′2r

(19)

, where k is the wave vector and V is the volume of the particle. The relativepermittivity of the material is defined as εr = ε

′r + iε

′′r = εmetal/εmedium .

1.3 Optical measurement

In this part we discuss the basic formalism needed for the experimental measure-ment. The optical properties are measured by light that is directed towards thematerial and the reflected, scattered or transmitted light is detected. This processis illustrated in Fig.1.

Incoming light

Container, substrate, etc...

Homogeneous medium

particle

reflection on interface

absorption in medium

absorption in particle

scattered by particle

reflection at interfaceabsorption of substrate

Reflectance

Transmittance

Scattering

Figure 1: Possible processes in an optical measurement. The reflectance, transmit-tance and scattering can be measured. The light would interact with all the opticalelement involved in the system, not only limited to the material that we want tomeasure.

First we consider the measurement of the optical properties of a homogeneousmaterial. 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 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 theboundary conditions at the interface. The result is described by Fresnel’s equations.When the incident light is normal to the interface, the reflectance and transmittanceat the interface can be written as:

R =

∣∣∣∣ n1 − n2

n1 + n2

∣∣∣∣2 (20)

T =2(n1n

∗2 + n∗1n2)

|n1 + n2|2(21)

5

The reflectance and transmittance fulfill R+T = 1. If we consider the case thatthe light is directed from vacuum or air to certain material with n = η + iκ, theformula can be expanded as:

R =(1− η)2 + κ2

(1 + η)2 + κ2(22)

T =4η

(1 + η)2 + κ2(23)

Since the reflectance and transmittance are related, measuring only R and T isnot sufficient to determine the optical properties of material, i.e. to determine bothη and κ. It is necessary to measure also the absorbance within the material fora certain thickness. This would yield the value for the absorption coefficient, andthen κ can be extracted. This seems to be quite simple, one can just take a piece ofmaterial with known thickness and measure what is the light intensity before andafter passing though the material. However, one should notice that the materialhas at least 2 interfaces, on which the light is reflected. The final attenuation effectis 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 reflectedmultiple times between the two interfaces, and the reflected light might also interferewith the incoming light. Also in certain case, the measured material needs to bekept in certain container (liquids) or deposited on a substrate, which will cause evenmore complicated reflections. All of these make the transmission fairly complex andhard to analyze. Usually computer simulations are used to calculate the opticalcoupling in multi-layered system (such as http://thinfilm.hansteen.net/). Toovercome this problem, one may measure at two different sample thicknesses, andthe difference of the attenuation between the two pieces is measured. This differenceis 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 thetransmission is measured, it is not possible to distinguished how much is scatteredor absorbed. In this case the extinction cross section can be measured. If we havesmall particles disperse in certain medium, it is important to measure also the opticalabsorption of the medium without the particles as reference data.

2 Optical properties of metalThe optical response of metals is mainly originates from the conduction electrons.The Drude model of free electron states that the electrons in metals behave likeclassical gas molecules. There is no interaction between the electrons except scat-tering. The average scattering interval time is defined as τ . The free electron withinthe 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:

md2x

dt2+m

τ

dx

dt= −eE (24)

6 2 OPTICAL PROPERTIES OF METAL

Index of refraction

-0.5

0

0.5

1

1.5

2

ω/ωp

0.5 1 1.5 2

η

κPlasma frequency

Figure 2: The real and imaginary part of index of refraction around plasma frequencyfor a metal according to (28). The 1/τ is assumed to be significantly smaller thanthe plasma frequency.

The second term on the left-hand-size corresponds to an averaging effect of thescattering to slow down the electrons. Using the same expression for oscillatingelectric field as before, the electron displacement can be expressed as:

x =eE

m(ω2 + iω/τ)(25)

The polarization is the dipole moment induced by the electron movement in unitvolume, thus:

P = −nex = − ne2

m(ω2 + iω/τ)E (26)

As a result the permittivity and index of refraction for free electrons are:

ε(ω) = 1 +P (ω)

ε0E(ω)= 1− ne2

ε0m(ω2 + iω/τ)(27)

n =

√1− ne2

ε0m(ω2 + iω/τ)=

√1−

ω2p

ω2 + iω/τ(28)

The plasma frequency is defined as ω2p = ne2/ε0m. This property just depends

on the mass and density of the electrons. The real and imaginary part of the indexof refraction around the plasma frequency is shown in Fig.2. For normal metal ωp isabout 1015−1016 Hz and τ is around 10−14 s at room temperature. This means thataround 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 andindex of refraction becomes purely imaginary. This means the electric field will justpenetrate 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 whymetal 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. Fornormal metals this usually happens at ultra-violet frequency range. It is interestingthat at this frequency range the index of refraction is smaller than 1. This means thephase 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 UVlight will have total reflection on metal surface at large incident angle, similar asnormal light in prism.

In metals only copper, osmium and gold show certain color in visible light. Thecolor of gold and copper is related to its band structure. In the case of Au the 5dorbit is completely filled and the 6s is half-filled. The energy difference between 5dand 6s level in gold is about 4 eV and this strong absorption cut out the green-bluelight from the reflection, creating the yellow color of gold. Copper has similar effectfor its 3d/4s orbit structure but with lower absorption energy [4].

3 Measurement setup

UV+VIS

light source

absoprtion materialdispersive element

CCD detector array

light

beam

Figure 3: Working principle of the spectrometer. Polychromatic light passes throughthe absorption material, and then through a dispersive element, where light withdifferent wavelength is diverted to different directions. The light with differentwavelength is then detected with CCD detectors located at different position simul-taneously.

In this experiment a compact optical spectrometer is used to measure the ab-sorption/extinction of the sample. The working principle of the measurement isshown in Fig.3. The light source is polychromatic and contains wavelength from200 nm to 900 nm. The polychromatic light is then focused and passed throughthe absorption material directly. The transmitted light then illuminates a disper-sive element, where light of different wavelength are reflected to different directions.Using a CCD detector array the light with different wavelength are then recordedsimultaneously. This is different from common spectrometers, where there is justone 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. More-over, the compact spectrometer does not need any moving component, while thenormal spectrometer needs to move either the detector or the dispersive element to“scans” through different wavelength. Thanks to its much simple design, the size ofa compact spectrometer is much smaller and the cost is lower compared to commonspectrometer.

4 Experimental procedure

4.1 Preparation of Au thin film

Two transparent gold films are prepared by DC sputtering. Quartz glass is usedas the substrate due to its transparency in UV spectrum range. The substratesare cut to 5 x 10 mm2 size. They are then cleaned with aceton and isopropanol inultrasonic bath to get rid of the dirt. Sputtering is performed in the Balzers minisputter machine. The distance of the target to sample is set to 50 mm. Sputteringcurrent is 30 mA and Argon pressure is 0.05 mbar. Under such working conditionsthe deposition rate is about 0.14 nm/s. To prepare Au films with 10 nm and 15nm thickness the sputtering time is 70 s and 110 s, respectively. The film thicknessshould 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 andthe optical properties will be different from that of the bulk. The gold film preparedby this method might not stick very firmly on the substrate. One should take carewhen 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 byJ. Turkevich et al. in 1951 [5]. Details of the method is adapted from [6]. Theprocedure is as follow:

1. Prepare 1 mM chloroauric acid (HAuCl4 · H2O) solution and 38.8 mM sodiumcitrate hydrate (Na3C6H5O7 · 2H2O) solution in water.

2. Move 5 ml 1 mM chloroauric acid solution with pipette into a beaker. Put amagnetic stirrer in the beaker and put the beaker on the heating plate. Heatand stir the solution simultaneously till boiling.

3. After the solution reaches the boiling point, add 0.5 ml sodium citrate solutioninto the beaker using the pipette. The color of the solution will start changingto deep red.

4. Continue to heat and stir the solution. Add water to compensate the evapo-ration 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 itcools down to room temperature. Dilute the solution to certain concentrationso that one can easily look through. Afterwards one can move the solution ofAu nanoparticles to some other container for storage.

The Au nanoparticles prepared by this method have an average diameter of about 20nm. The sodium citrate acts as a reducing agent. After reduction the citrate anionis absorbed on the surface of the particle, introducing the surface charge that repelsthe particles and prevents them from aggregating. A scanning electron microscopeimage of the particles is shown in Fig.4.

Figure 4: SEM image of the Au nanoparticles prepared by Turkevich’s method. Adroplet of the solution containing the Au NPs are deposited on Si substrate. Afterthe solvent dried the sample is investigated by SEM.

4.3 Measure the optical properties of the Au thin film andnanoparticles

First the absorption of the Au thin film will be measured. The spectrum of the lightsource 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 tothe 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 candeduce the imaginary part of the index of refraction. The difference between 2. and3. or 2. and 4. is the total effect of absorption in Au as well as reflection at theinterface.

10 REFERENCES

To measure the optical extinction of the Au nanoparticles, first the absorption ofthe same solution without the Au nanoparticles is measured as the reference. Thetotal extinction coefficient of Au particles can be obtained for different wavelength.With the knowledge of chloroauric acid concentration, and assume the particles are20 nm spheres, the particle concentration can be estimated and the extinction crosssection of the particle can be calculated.

5 Report and data treatmentBelow you find some details of data analysis and questions that should be addressedin 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 theliterature (for example [4, 7, 8]).

2. Compare the absorption of quartz glass with and without Au. Is this differencethe 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 differentwavelength. 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 smallparticles. Wiley, 1983.

[4] G. P. Pells and M. Shiga, “The optical properties of copper and gold as a functionof 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 andgrowth processes in the synthesis of colloidal gold,” Discussions of the FaradaySociety, 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 PhysicsC: Solid State Physics, vol. 3, no. 2S, p. S109, 1970.

[8] “Optical database.” http://www.sspectra.com/sopra.html, retrieved on10.15.2010.

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A Optical property data of goldHere the complex index of refraction for metallic gold is plotted. The data is takenfrom [8].

Index of Refraction0 1 2 3 4 5 6

Wavelength (n

m)

200

300

400

500

600

700

800

900

η

κ