Advanced Materials - Lab Intermediate Physics Ulm University Institute of Solid State Physics Optical Properties of Metals Burcin Özdemir Luyang Han April 28, 2014
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. I
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1 Introduction to the optical property of material Light is a form of electromagnetic radiation which can be dened by dierent wavelengths ranging between long radio waves (10 10 m) and gamma rays (10 12 m) as depicted in Fig. 1. 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 = E 0 exp i(k r ωt). (1) The wave vector k is a complex number which can be dened as k 2 = ω ɛ ɛ 0 µ 0 (ɛ 0 and µ 0 are permittivity and permeability of free space respectively). Here c = 1/ ɛ 0 µ 0 denes the speed of light in vacuum and ɛ denes the complex index of refraction in the material:
2 1 INTRODUCTION TO THE OPTICAL PROPERTY OF MATERIAL ñ = ɛ = η + iκ, (2) k = eñω c. (3) Here e denes 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. ( ηω ) E = E 0 exp i c e r ωt exp ( κω ) c e r. (4) The second exponential component describes how the amplitude of the electric eld gets attenuated along the direction of the wave propagation. The light intensity for a planar wave is proportional to E 2, thus the intensity of the light will be attenuated as: I = I 0 exp( 2κω c e r) = I 0 exp( αe r). (5) The α = 2κω is the absorption coecient of the material, and has the dimension of c [L 1 ]. (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 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 (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 = α C. (6) The σ here has dimension [L 2 ], 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. ñ = n(1 + iκ). An alternative is
1.3 Optical measurement 3 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: σ abs = 3ɛ r 3kV (2 + ɛ r) 2 + ɛ 2 r σ sca = k4 6π (3V (ɛ )2 r 1) 2 + ɛ 2 r (2 + ɛ r) 2 + ɛ 2 r (7) (8), where k is the wave vector and V is the volume of the particle. The relative permittivity of the material is dened as ɛ r = ɛ r + iɛ r = ɛ metal / ɛ 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. Homogeneous medium Container, substrate, etc... Incoming light reflection on interface absorption in medium reflection at interface particle absorption in particle absorption of substrate 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 frequency 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: R = ñ 1 ñ 2 ñ 1 + ñ 2 2, (9) T = 2(ñ 1ñ 2 + ñ 1ñ 2 ) ñ 1 + ñ 2 2. (10) The reectance and transmittance fulll 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: R = (1 η)2 + κ 2 (1 + η) 2 + κ, (11) 2 4η T = (1 + η) 2 + κ. (12) 2 Since the reectance and transmittance are related, measuring only 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. The equation of motion for free electron in electric eld is: m d2 x dt + m dx = ee. (13) 2 τ dt 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 = ee m(ω 2 + iω/τ). (14) The polarization is the dipole moment induced by the electron movement in unit volume, thus: ne 2 P = nex = E. (15) m(ω 2 + iω/τ) As a result the permittivity and index of refraction for free electrons are: ɛ(ω) = 1 + P (ω) ɛ 0 E(ω) = 1 ne 2 ɛ 0 m(ω 2 + iω/τ), (16) ne n = 1 2 ɛ 0 m(ω 2 + iω/τ) = ωp 2 1 ω 2 + iω/τ. (17) The plasma frequency is dened as ω 2 p = ne 2 /ɛ 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.3. For normal metal ω p is about 10 15 10 16 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 eect of electron collision can be neglected in (16) and (17). If the optical frequency is lower than ω p, 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 ω p, 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 1 0.5 0 Plasma frequency η κ -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 dispersive element UV+VIS light source 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 are cut to 5 x 10 mm 2 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 Ag NP suspension in citrate buer 10 nm, C: 6E 12 particles/ml 10 nm, C: 3.6E 12 particles/ml 50 nm, C: 3E 10 particles/ml 40 nm, C: 5.7E 10 particles/ml 80 nm, C: 7.8E 9 particles/ml 60 nm, C: 1.7E 10 particles/ml 100 nm, C: 3.8E 9 particles/ml 100 nm, C: 3.6E 9 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. Please bring a 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.
REFERENCES 9 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 OPTICAL PROPERTY DATA OF GOLD 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 η κ 200 300 400 500 600 700 800 900 Wavelength (nm)