Refractive index and dielectric constant evolution of ultra-thin gold from clusters to films

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1 Physics Physics Research Publications Purdue University Year 010 Refractive index and dielectric constant evolution of ultra-thin old from clusters to films X. F. Wan K. P. Chen M. Zhao D. D. Nolte This paper is posted at Purdue e-pubs. articles/1338

2 Refractive index and dielectric constant evolution of ultra-thin old from clusters to films Xuefen Wan, 1 Kuo-pin Chen, Min Zhao, 1 and David D. Nolte 1 1 Department of Physics, Purdue University, West Lafayette, Indiana 47907, USA Department of Electrical and Computer Enineerin, Purdue University, West Lafayette, Indiana 47907, USA *nolte@physics.purdue.edu Abstract: Usin hih-speed picometroloy, the complete cluster-to-film dielectric trajectories of ultra-thin old films on silica are measured at 488 nm and 53 nm wavelenths for increasin mass-equivalent thickness from 0. nm to 10 nm. The trajectories are parametric curves on the complex dielectric plane that consist of three distinct reimes with two turnin points. The thinnest reime (0. nm 0.6 nm) exhibits increasin dipole density up to the turnin point for the real part of the dielectric function at which the clusters bein to acquire metallic character. The mid-thickness reime (0.6 nm ~ nm) shows a linear trajectory approachin the turnin point for the imainary part of the dielectric function. The third reime, from nm to 10 nm, clearly displays the Drude circle, with no observable feature at the eometric percolation transition. 010 Optical Society of America OCIS codes: ( ) Instrument, measurement and metroloy; (40.040) Optics at surfaces; ( ) Thin film, optical properties; ( ) metal optics. References and links 1. E. Ozbay, Plasmonics: merin photonics and electronics at nanoscale dimensions, Science 311(5758), (006).. R. B. Laibowitz, and Y. Gefen, Dynamic Scalin near the Percolation-Threshold in Thin Au Films, Phys. Rev. Lett. 53(4), (1984). 3. J. J. Tu, C. C. Homes, and M. Stronin, Optical properties of ultrathin films: evidence for a dielectric anomaly at the insulator-to-metal transition, Phys. Rev. Lett. 90(1), (003). 4. A. Gray, M. Balooch, S. Alleret, S. De Gendt, and W. E. Wan, Optical detection and characterization of raphene by broadband spectrophotornetry, J. Appl. Phys. 104(5), (008). 5. P. Grosse, and V. Offermann, Analysis of Reflectance Data Usin the Kramers-Kroni Relations, Appl. Phys., A Mater. Sci. Process. 5(), (1991). 6. D. A. Crandles, F. Eftekhari, R. Faust, G. S. Rao, M. Reedyk, and F. S. Razavi, Kramers-Kroni-constrained variational dielectric fittin and the reflectance of a thin film on a substrate, Appl. Opt. 47(3), (008). 7. M. Hövel, B. Gompf, and M. Dressel, Dielectric properties of ultrathin metal films around the percolation threshold, Phys. Rev. B 81(3), (010). 8. F. L. McCrackin, E. Passalia, R. R. Stromber, and H. Steinber, Measurememt of Thickness and Refractive Index of Very Thin Films and Optical Properties of Surfaces by Ellipsometry, J. Res. Natl. Bur. Stand. A 67, (1963). 9. R. J. Archer, Determination of Properties of Films on Silicon by Method of Ellipsometry, J. Opt. Soc. Am. 5(9), (196). 10. M. Yamamoto, and T. Namioka, In situ ellipsometric study of optical properties of ultrathin films, Appl. Opt. 31(10), (199). 11. X. Wan, M. Zhao, and D. D. Nolte, Common-path interferometric detection of protein monolayer on the BioCD, Appl. Opt. 46(3), (007). 1. X. F. Wan, Y. P. Chen, and D. D. Nolte, Stron anomalous optical dispersion of raphene: complex refractive index measured by Picometroloy, Opt. Express 16(6), (008). 13. O. S. Heavens, Optical Properties of thin solid films (Academic Press Inc., New York, 1955), p. 66~ V. M. Shalaev, Nonlinear Optics of Random Media: Fractal Composites and Metal-Dielectric Films, Spriner Tracts in Modern Physics (Berlin Heidelber, 000). 15. H. Y. Li, S. M. Zhou, J. Li, Y. L. Chen, S. Y. Wan, Z. C. Shen, L. Y. Chen, H. Liu, and X. X. Zhan, Analysis of the drude model in metallic films, Appl. Opt. 40(34), (001). 16. F. Abeles, Optical Properties of Solids (North-Holland, Amsterdam, 197), p M. Walther, D. G. Cooke, C. Sherstan, M. Hajar, M. R. Freeman, and F. A. Hemann, Terahertz conductivity of thin old films at the metal-insulator percolation transition, Phys. Rev. B 76(1), (007). (C) 010 OSA November 010 / Vol. 18, No. 4 / OPTICS EXPRESS 4859

3 18. U. Kreibi, and C. V. Frastein, Limitation of Electron Mean Free Path in Small Silver Particles, Z. Naturforsch. B 4, (1969). 19. M. G. Blaber, M. D. Arnold, and M. J. Ford, Search for the Ideal Plasmonic Nanoshell: The Effects of Surface Scatterin and Alternatives to Gold and Silver, J. Phys. Chem. C 113(8), (009). 0. C. Nouez, and C. E. Roman-Velazquez, Dispersive force between dissimilar materials: Geometrical effects, Phys. Rev. B 70(19), (004). 1. M. Kreiter, S. Mittler, W. Knoll, and J. R. Sambles, Surface plasmon-related resonances on deep and asymmetric old ratins, Phys. Rev. B 65(1), (00).. T. Zychowicz, J. Krupka, and J. Mazierska, Measurements of Conductivity of Thin Gold Films at Microwave Frequencies Employin Resonant Techniques, Proceedins of Asia-Pacific Microwave Conference (006). 1. Introduction The noble metals lie at the interface between optical and electronic physics [1]. As the scale of metal nanophotonic and plasmonic elements shrink, metallic behavior becomes size dependent, and thin layers frament into clusters that have different optical properties than the bulk [,3]. Despite the importance of old as an archetypical metal and its importance for photonic applications, there has been no study revealin the continuous cluster-to-film-to-bulk transition of the complex refractive index n and dielectric constant, especially when the metal film averae thickness is below nm. The ultra-thin reime is inaccessible to current techniques for thin film optical studies. Because n is complex with two variables (real part and imainary part), a one-parameter measurement cannot acquire the full information needed to calculate n. Traditional methods select two or more photonic states of the probe liht, such as wavelenth, polarization or incidence anle [4] to execute multiple-parameter measurements. For example, spectral reflectometry measures the reflectance chane caused by the thin film in a broad wavelenth rane and uses the Kramers-Kroni relation to find n [5 7]. Ellipsometry measures n by monitorin reflectances at two different polarization states of oblique-incident liht [8,9]. However, chanin the photonic state of the probe liht potentially limits the measurement of n because of optical dispersion or anisotropy which occurs for sub-nanometer metal films. These consequences limit the validity of traditional methods in the ultra-thin reime. For example, ellipsometric measurements become erratic when the old film thickness is below 6 nm [7,10]. We have developed interferometric picometroloy which measures n usin a sinle photonic state (sinle wavelenth, sinle polarization and normal incidence) [11]. Picometroloy monitors the complex chane of the normal-incidence reflection coefficient r instead of reflectance R to calculate n of the film by analyzin the far-field diffraction of a patterned film in a sinle measurement which efficiently excludes errors introduced by multiple measurements. The chane of r is obtained by monitorin the chane of the far-field diffraction pattern as the focal spot scans over the ede of a thin film. The phase chane of () r is obtained by monitorin the spatial asymmetry of the diffraction pattern, and the amplitude is obtained by monitorin the overall intensity. The scannin system shown in Fi. 1a is employed to perform the picometroloy measurement. Both the modulus and the phase chane of r caused by the thin film are acquired usin a split detector that simultaneously monitors intensities and asymmetries of the reflected beam. The split detector is located on the Fourier plane and consists of two semi-circular halves that contribute sinals A and B. The intensity (I) sinal of the diffraction pattern is acquired as I = A + B, and the spatial asymmetry sinal (phase contrast, PC) is acquired as PC = A-B. Typical I and PC channel sinals are shown in Fi. 1b. The complex index n is calculated usin the picometroloy equation [1] (1 r) d PC I ( n 1) Ai ( x) Ai ( x) j, r where r is the reflection coefficient of the bare substrate, is the free space wavelenth, I PC I i ( x ) and i ( x ) are the normalized sinals of the I and PC channels, and A i ( x) and (1) (C) 010 OSA November 010 / Vol. 18, No. 4 / OPTICS EXPRESS 4860

4 PC I PC A i ( x) represent the amplitudes of i ( x ) and i ( x ). The mass-equivalent thickness is defined as d m / m where m is the surface mass density measured by a quartz crystal monitor, and m is the molecular mass density of old. The platform consists of a spinner and a linear stae to produce a two-dimensional picometroloy scan. We previously applied this system to measure anomalous dispersion of raphene [1].. Experiments To study n and of old at arbitrary thicknesses in the rane of 0~10 nm, we created old samples with averae old film thicknesses that increased continuously and linearly from 0 to 10 nm alon the y-direction (Fi. 1b) on a sinle substrate. This was achieved by a metal evaporator modified by attachin a stepper-motor-controlled shutter between the evaporation source and the silicon chip. The shutter moves with constant speed and exposes the chip surface continuously and incrementally durin evaporation. The old deposition rate in these experiments was 0. nm/s, and the silicon chip was at room temperature (300 K). The optical study of old with mass-equivalent thicknesses from 0 to 10 nm is performed on a sinle chip to eliminate the error due to variations amon different chips. Because picometroloy is based on Gaussian beam diffraction from the ede of a film, the old film is deposited in a stripe pattern (Fi. 1b) fabricated usin photolithoraphy. The periodic pattern alon the x-direction provides multiple edes and improves the detection sensitivity. The substrate is a silicon chip with a 134 nm SiO layer (measured by a spectroscopic analysis of the reflectance) that is chosen to ive nominally equal I and PC sinals in Eq. (1). Prior to old evaporation, the substrate was cleaned by acetone in a sonic bath for 15 min and rinsed with methanol and deionized water and dried with a nitroen stream. Fi. 1. Schematics of picometroloy and the old sample. (a) Picometroloy measures the complex refractive index of ultra-thin films at sinle wavelenths at normal incidence. The reflected Gaussian beam forms an asymmetric diffraction pattern on the Fourier plane when scannin across film edes. By analyzin the intensity variation (I sinal) and spatial asymmetry (PC sinal) of the diffraction pattern with a split detector, full information is acquired of the complex refractive index calculation. (b) The old sample is fabricated for the optical study for thicknesses in the rane of 0~10 nm on a sinle chip. The old effective thickness (measured mass per area divided by molecular mass volume density) increases from 0 to 10 nm alon the Y direction, and the old film has a stripe pattern alon the X direction to provide multiple film edes for the picometroloy analysis. Two-dimensional picometroloy scannin was performed on the old sample at wavelenths of 488 nm and 53 nm. Imaes of I and PC scans are shown in Fi. b and c. Normalization is performed by dividin the I and PC sinal by the intensity of the reflected beam on the adjacent land of the substrate. As a demonstration, in Fi. 3a, the I and PC sinals are extracted for 4 nm old, and n for old is calculated at this thickness. Althouh the approximation in Eq. (1) is suitable to calculate n when the thickness is much less than (C) 010 OSA November 010 / Vol. 18, No. 4 / OPTICS EXPRESS 4861

the probe wavelenth, to improve calculation precision, we used a riorous computational transfer matrix method [13] to extract the precise n from the data. The r of the substrate ( r 0.38 0.

5 the probe wavelenth, to improve calculation precision, we used a riorous computational transfer matrix method [13] to extract the precise n from the data. The r of the substrate ( r i at 488 nm wavelenth and r i at 53 nm wavelenth), the film thickness d, and the I and PC responses are measured, hence we plot n n ik on the complex plane to find a value of n that satisfies both the I and PC responses. For each thickness there is a sinle curve on which all n ive a constant I response (which explains why reflectometry alone cannot determine the n ). All n satisfyin the PC response form a separate curve. The intersection of the I and PC curves uniquely determines the taret n (Fi. 3b) at this thickness. By these means, n of old is calculated for any thickness less than 10 nm. The curves of n n ik are shown in Fi. 3d. Gold with effective thickness below 10 nm is usually not a continuous medium but has the topoloy of heteroeneous clusters with a connectivity that oes throuh a percolation transition [14]. Fi.. Gold film pattern and I, PC response imaes. (a) Stripe-patterned old film with thickness varyin continuously from 0 to 10 nm on thermal oxide on silicon (134 nm SiO ). The sample is prepared usin a thermal metal evaporator in which the evaporation time is controlled by a stepper motor. The imae is captured by microscope under white liht. The color of the old shows a rich transition within 10 nm (Real color). (b) and (c) show the imaes of the sample in the I and PC channels scanned by the picometroloy system at 53 nm (Pseudo color). (C) 010 OSA November 010 / Vol. 18, No. 4 / OPTICS EXPRESS 486

6 Fi. 3. Data processin to calculate refractive indices of ultra-thin old. (a) One example of I and PC sinals from 4 nm thick old (one track taken from imaes Fi. 1b and 1c). The normalized amplitudes of both channels are acquired, and the refractive index of old at this thickness can be calculated by Eq. (1). (b) The I response is satisfied by multiple n which form a curve on the complex plane, and similarly for the PC response. The intersection point uniquely determines n of the old film. (c). Normalized amplitudes of I and PC responses of old from 0 to 10 nm at 488 nm and 53 nm wavelenths. (d) The n is calculated for old films with 0~10 nm thicknesses at both 488 nm and 53 nm wavelenths. The trajectory of the dielectric constant ' i " n across the complex plane, parameterized by the film thickness, is shown in Fi. 4 for the wavelenths 488 nm and 53 nm for which the collective plasmon contributions to the Drude response are small [7]. Three distinct reimes are observed in the trajectories as film thickness decreases from 10 nm to 0. nm. In the first reime (old thickness decreases from 10 nm to nm), evolves in a circular trajectory. In the second reime (old thickness from nm to 1 nm), evolves in a linear trajectory. In the third reime (old thickness from 1 nm to 0. nm), evolves with the imainary part approachin zero as the discontinuous film framents completely into finely scattered clusters. (C) 010 OSA November 010 / Vol. 18, No. 4 / OPTICS EXPRESS 4863

7 3. Discussion Fi. 4. The dielectric trajectories of old film as the film thickness chanes from 0. to 10 nm. The dielectric trajectory has three distinct reimes that oriinate from three types of old topoloy: 1) Sparse reime (0. nm to 1 nm). In this reime, coverae of old clusters oes from zero to 0%, and the imainary part of vanishes at ultra-low coverae. ) Intermediate reime (1 nm to nm) connectin the first and third reime. 3) Drude circle reime ( nm to 10nm). In this reime, the effective dielectric constant of old film evolves alon a circular trajectory (Drude circle). This pattern is predicted by the Drude equation as the size of the metal clusters increases above the electron mean free path. To interpret the trajectory of ultra-thin old, the microscopic topoloy of the film was analyzed by a Field-Emission scannin electron microscopy (SEM, Hitachi S-4800 Field Emission SEM). For instance, a old film with a mass-equivalent thickness of a nanometer is a collection of isolated old clusters. The size, shape (aspect ratio) and surface coverae of the old clusters determines the effective dielectric constant. SEM imain was performed to measure the surface coverae, aspect ratio and size of the old clusters at thicknesses of 1,, 5 and 10 nm. The data and topoloy analysis are shown in Fi. 5, enablin us to interpret the evolution of. (C) 010 OSA November 010 / Vol. 18, No. 4 / OPTICS EXPRESS 4864

8 Fi. 5. SEM imaes of the old samples and old topoloy analysis. (a), (b), (c), (d) show the imaes of old clusters at different mass-equivalent film thicknesses. (e) and (f) show Gold topoloy in terms of coverae, cluster lateral size and film thicknesses. The cluster heiht was calculated by dividin the averae old thickness by the coverae, and the cluster lateral size was calculated by auto-correlation analysis of the SEM imaes. Cluster coverae rapidly and almost linearly increases from 0 to 37% as the averae thickness of old film chanes from 0 to nm. The coverae increases at a much slower rate when film thickness rows beyond 3 nm. As a result, coverae increases from 54% to 60% as the old thickness chanes from 5 to 10 nm. In the thinnest reime (mass-equivalent thickness nm), the dipole density on the surface increases causin an increase of the real part of the dielectric constant as the old clusters row in lateral size up to 10 nm (4 nm in heiht) up to a surface coverae of 15%. Approximately at this size, the clusters bein to take on metallic character, and the dielectric trajectory reaches the turnin point in the real part of the dielectric constant as the imainary part increases rapidly. In the mid-thickness reime (mass-equivalent thickness 0.6 to nm) the coverae increases from 15% up to 40%, as the lateral size of the clusters increases to 16 nm (6 nm heiht). The trajectory is approximately linear in this reime, with the real part decreasin and the imainary part increasin. In the thickest reime of the trajectory of (Drude circle reime where old thickness increases from nm to 10 nm), the aspect ratio of old clusters chanes little as the coverae varies slowly (coverae above 50%), but the averae lateral size of the old clusters chanes (C) 010 OSA November 010 / Vol. 18, No. 4 / OPTICS EXPRESS 4865

9 from 16 nm to 4 nm (6 nm to 16 nm in heiht). The cluster size is the major factor determinin the effective of the old film in this reime causin to evolve in a circular trajectory in Fi. 4 as the film thickness increases. This phenomenon is caused by radual deconfinement of the free electrons in old clusters caused by the increasin cluster size. Accordin to the Drude theory [15,16], the of nanoscale old structures is described as bound i p, d () where p is the bulk plasmon frequency, is the probe liht anular frequency, bound is the dielectric constant contributed by bound electrons, and d is the collision frequency of free electrons. is a constant (bulk value), but when the cluster size is on the order of the electron mean free path (5 nm in Au [17]), d increases due to surface collisions. In Ref [18], the approximate behavior was parametrized by ( bulk ) vf / s where v F is the Fermi velocity of free electrons and s is the radius of a metal nano-particle. Therefore, evolves in a circular trajectory, called the Drude Circle, as constant. This is a direct consequence of the equation d varies and is held p 1 i / p p ( bound ) 1 i / (3) that describes a parametric circular trajectory for,. p The radius of this circle on the complex plane is p / centered at bound. The Drude circle describes the parametric trajectory of for metal clusters as varies. By fittin the data in the first reime on Fi. 4, the radii are found to be 3.8 at 488 nm and 5. at 53 nm, ivin a plasmon enery of 7.3 ev and bound 6 1.6i. The plasmon frequency of bulk Au is between ev in [19 1]. Therefore, the Drude equation is consistent with our experimental results. The turnin point in the imainary part of the dielectric constant (the apex of the trajectory) occurs when the free electron scatterin rate equals the optical anular frequency. To study Drude behavior, previous approaches tuned the frequency while treatin as a constant. Here, we observe the Drude circle for the first time by usin the raded-thickness old sample combined with picometroloy in a sinle photonic state, allowin d to vary parametrically and to observe at different old thicknesses, while keepin constant (constant radius on the complex plane). Our experiment tracks the metal-to-insulator (I/M) transition of old film from an optical perspective. The eometric percolation threshold is reached when a sinle iant cluster spans the probe scale, which occurs around a mass-equivalent thickness of 7 nm in our samples. Therefore, althouh the eometric percolation thickness is well within the Drude circle, there is no measurable feature or deviation from the circular Drude trajectory in that thickness rane. For optical frequencies, continuous percolation across lon-rane old clusters has little or no influence on the optical properties of a old film. There is evidence [] in the microwave reion (4.9 GHz, 60 mm wavelenth) that the conductivity of a old film decreases abruptly below a thickness of 8 nm, which is consistent with the percolation threshold. Hiher-frequency infrared liht ( 4 m ) displayed a transmission anomaly at the thinner value of nm [3], which is where, in our results, the Drude circle commences Bulk (C) 010 OSA November 010 / Vol. 18, No. 4 / OPTICS EXPRESS 4866

10 (metallic optical behavior) as a continuation of the mid-thickness reime of the dielectric trajectory. 4. Conclusion In conclusion, we have measured the dielectric trajectory on the complex plane of thermallyevaporated old on silica at the wavelenths 53 nm and 488 nm with mass-equivalent old film thickness as the parametric variable. Picometroloy is able to extract the full complex dielectric trajectory usin a sinle-wavelenth measurement without polarization control and at normal incidence. We extracted of old films with arbitrary thickness in the rane of 0. nm to 10 nm, and demonstrated that the trajectory traces out the Drude circle for massequivalent thicknesses reater than nm. Acknowledement This project is sponsored by a rant from the 1st Century Fund of the Indiana Economic Development Corporation. We sincerely thank Dr. Vladimir Shalaev for fruitful discussions. We also thank Jacob Millspaw and Chris Petrovitch for help processin the old samples. (C) 010 OSA November 010 / Vol. 18, No. 4 / OPTICS EXPRESS 4867

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Interactions with Matter Manetic Lenses Manetic fields can displace electrons Manetic field can be produced by passin an electrical current throuh coils of wire Manetic field strenth can be increased by usin a soft ferromanetic