In Situ Observation of a Phase Transition in a Thin Molecular Film by

Molecular Film by Optical Second Harmonic Generation. Theodore Sjodin,† Thomas Troxler, and Hai-Lung Dai*. Laboratory for the Research on the Struct...
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In Situ Observation of a Phase Transition in a Thin Molecular Film by Optical Second Harmonic Generation Theodore Sjodin,† Thomas Troxler, and Hai-Lung Dai* Laboratory for the Research on the Structure of Matter, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6202 Received June 28, 1999. In Final Form: November 19, 1999 By use of optical second harmonic generation (SHG), an irreversible phase transition from a disordered to an ordered structure is observed in thin pyridine (C5H5N) films dosed on a Ag(111) surface under ultrahigh vacuum. This phase transition occurs when the surface temperature is raised above 125 K. An analytical model of SHG from the annealed pyridine film including optical interference is presented. It is most likely that the annealed films are composed of crystallites randomly oriented on the surface plane with an average length scale comparable to or larger than half a micrometer.

I. Introduction Recently, there has been considerable interest in thin molecular films on solid substrates. A notable example is the amorphous water film,1-3 which is an important system in many areas ranging from atmospheric to interstellar chemistry. In the study of thin molecular films, determining the structures and the associated phase transitions has always been an experimental challenge. In this paper, we demonstrate the ability of optical second harmonic generation (SHG) to determine the structural properties of such thin films. A preliminary report4 from this laboratory has shown an irreversible phase transition in thin pyridine (C5H5N) films on a Ag(110) surface detected by using optical second harmonic generation in an unique optical configuration. In this paper, we present measurements in a more generalized experimental configuration on the (111) surface and an analytical model describing SHG from a thin molecular film on a metal substrate. Analysis based on this model allows us to assign this phase transition to an amorphous to polycrystalline state transition. Optical second harmonic generation has been used extensively to characterize properties of surface and interfacial regions.5,6 Second harmonic generation,7 and the more general form, sum-frequency generation,8 have been used to characterize, in situ, the growth and thickness of molecular thin films on metal substrates. The interference between the second harmonic generated from a thin film and its substrate has been studied in detail for a semiconductor heterojunction.9 These studies have shown that optical interference manifests dominant effects in the observed SHG signal. When the film’s thickness is on * To whom all correspondence should be addressed at the Department of Chemistry, University of Pennsylvania, Philadelphia, PA 19104-6323. E-mail: [email protected]. † Present address: Department of Physics, Northeastern University, Boston, MA 02115. (1) Angell, C. A. Annu. Rev. Phys. Chem. 1983, 34, 593. (2) Angell, C. A. Science 1995, 267, 1924. (3) Smith, R. S.; Kay, B. D. Surf. Rev. Lett. 1997, 4, 781. (4) Sjodin, T.; Li, C.-M.; Dai, H.-L. Proc. SPIE 1995, 2547, 419. (5) Shen, Y. R. Solid State Commun. 1997, 102, 221. (6) Richmond, G. L.; Robinson, J. M.; Shannon, V. L. Prog. Surf. Sci. 1988, 28, 1. (7) Li, C. M.; Ying, Z. C.; Dai, H.-L. J. Chem. Phys. 1994, 101, 7058. (8) Hirose, C.; Ishida, H.; Iwatsu, K.; Watanabe, N.; Kubota, J.; Wada, A.; Domen, K. J. Chem. Phys. 1998, 108, 5948. (9) Yeganeh, M. S.; Qi, J.; Culver, J. P.; Yodh, A. G. Phys. Rev. B 1992, 46, 1603.

the order of optical wavelengths, strong oscillations in SH intensity as a function of film thickness have been observed in all cases.7-9 These strong oscillations can be used to determine optical properties of the thin film. In general, for a system consisting of a molecular film and a metal substrate, there are multiple sources contributing to the SHG signal. SHG originates from the metal surface in the dipole approximation because of the breakdown of symmetry of the centrosymmetric crystal at the surface. It may arise, to a lesser degree, from the bulk of the metal due to quadrupole contributions. SHG from the molecular film on the other hand depends on the alignment of the molecules in the film. For a molecular solid, the nonlinear response of the medium can be considered to arise from the sum of the hyperpolarizability of each individual molecule within the coherence length of the fundamental light. The directionality of the molecular hyperpolarizability is well defined in the molecular frame according to the molecular symmetry. Therefore, the total second harmonic response is highly dependent on the local order of the molecules in the film. Furthermore, the film’s structure, defined in the laboratory frame, can be explored by the dependence of SHG from the thin film on the light polarization. A pyridine molecular film on the order of 103 Å thickness on a Ag(111) surface will be tested as a prototypical system for the effectiveness of this SHG probe. The pyridine molecule with C2v symmetry has a directional hyperpolarizability well defined in the molecular frame. When the molecules are randomly oriented, the hyperpolarizability of many molecules will sum up to zero. With such an amorphous structure, the ensemble of pyridine molecules produces no SHG. But when the pyridine molecules are ordered in such a way that the vector sum of the individual hyperpolarizabilities is not zero, SHG will result. Experimental results presented here will show that observations based on SHG clearly indicate an irreversible phase transition when pyridine films are annealed above 125 K. From the observed SH intensity’s azimuthal angle and film thickness dependence, information about the film’s structure is obtained. By comparison of the experimental results to theoretical calculations, which are based on an analytical model of the SHG generated in a thin film and include interference, a model of a polycrystalline annealed film with large domains is proposed.

10.1021/la990837b CCC: $19.00 © 2000 American Chemical Society Published on Web 02/05/2000

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Figure 2. SH intensity as a function of azimuthal angle of (a) clean Ag(111) and (b) Ag(111) exposed to 500 langmuirs of pyridine at 100 K. The solid lines are fits to eq 1. Figure 1. Schematic diagram of the experimental setup: P, polarizer; F1 and F2, optical filters; MC, monochromator; PMT, photomultiplier tube; BS, beam splitter.

II. Experimental Section All experiments were performed in an ultrahigh vacuum chamber with a base pressure of 1 × 10-10 Torr. The Ag(111) crystal was mounted on a sample manipulator which allowed the sample to translate in all three directions and rotate about the surface normal (over a range of 140°). The sample temperature could be varied from 100 to 730 K using liquid nitrogen cooling and heating from electron bombardment. Before each experiment the sample was cleaned by sputtering with Ne+ and then annealed at 730 K. All pyridine adsorption on to the Ag surface was performed at 100 K. Pyridine was admitted into the ultrahigh vacuum (UHV) chamber through a variable leak valve. The dosing pressures used varied from 1 × 10-7 to 8 × 10-7 Torr with the majority of experiments dosed at 5 × 10-7 Torr. The ion gauge pressure and, for some experiments, the pyridine partial pressure from a mass spectrometer were recorded in real time using a personal computer. The pyridine pressures and exposures reported here are not corrected for ion-gauge sensitivity. A schematic diagram of the optical setup is shown in Figure 1. The 532 nm output of a 20 Hz pulsed Nd:YAG laser (Quantel 581C) was directed toward the Ag(111) sample at an angle of 60°. The incident laser pulse had a pulse duration of 8 ns (full width at half-maximum), a beam diameter of 0.4 cm, and a maximum peak intensity of 20 MW cm-2. The estimated peak surface temperature rise per pulse was 10 K.10 The reflected SH signal was collected by a photomultiplier and integrated by a gated CAMAC board interfaced with a personal computer. Three short pass optical filters (Corning 7-54) and a monochromator (Thermo Jarrell Ash MonoSpec 18) were used to ensure that only the SH light at 266 nm was detected. Glan-laser polarizers were used before the sample and after to select the input and output polarization. For all experiments, the input fundamental light was s-polarized, and only the s-polarized output SH light was detected. This will be referred to as ss-polarized. To correct for laser intensity fluctuations, about 4% of the incident beam was split off and directed onto a quartz plate. For most experiments, SH generated from the quartz surface in a reflection configuration was collected and recorded. For a few experiments, the SH generated from the quartz in a transmission configuration was used. The quartz SH signal was used to normalize the SH signal from the Ag(111) sample.

III. Results A. SHG from Amorphous Pyridine Films on Ag(111). Pyridine is weakly chemisorbed to silver with a saturated monolayer desorption temperature of 190 K and forms multilayers with a desorption temperature of

160-165 K.11 In a previous paper,7 the SHG from thin pyridine films grown on Ag(111) at 90 K was studied in detail. The interference between the reflected fundamental and the second harmonic from the Ag surface was used to monitor the film growth and thickness. The azimuthal angle dependence of the SH intensity generated from the clean Ag(111) crystal and the pyridinecovered Ag(111) crystal with an exposure of 500 langmuirs (1 langmuir ) 10-6 Torr s) at 100 K is shown in Figure 2. The azimuthal angle dependence of the ss-polarized SH intensity, Iss, from the (111) surface can be theoretically described as7,12

Iss) I0|bss sin(3Φ)|2

(1)

where the azimuthal angle, Φ, is the angle between the light’s incident plane and the [21 h1 h ] direction of the crystal and bss is the anisotropic parameter for the ss-polarized light. This equation suffices for a pyridine-covered surface if the origin of the SH arises from the metal surface alone. In this case, bss is a function of pyridine film thickness. This is exactly what we have observed in Figure 2. In Figure 3 the pyridine coverage dependence of the ss-polarized SHG at Φ ) 30° is shown at a temperature of 100 K. The strong oscillatory pattern mainly originates from interference between multiple reflections of the fundamental light within the film, as shown in Figure 4. The observed interference pattern, shown as a function of exposure in Figure 3, depends on the wavelength of the fundamental and SH light. As described in detail in ref 7, the pyridine film thickness dependence of SHG for a film deposited at 100 K can be modeled assuming the SH light is generated only from the silver substrate. At liquid nitrogen temperatures, the adsorbed pyridine film is presumably disordered,11 reflecting the random distribution of the molecules in the gas phase upon adsorption. Water films adsorbed on a variety of metal substrates below 140 K are also observed to be amorphous.4 No bulk SHG is generated from the amorphous pyridine film since a random distribution of molecules with nonzero hyperpolarizability results in a zero sum of nonlinear polarization. The ratio of the SHG from the amorphous pyridine-covered Ag(111), IamorP/Ag, (10) Hicks, J. M.; Urbach, L. E.; Plummer, E. W.; Dai, H. L. Phys. Rev. Lett. 1988, 61, 2588. (11) Wu, Q.; Hanely, L. J. Phys. Chem. 1993, 97, 2677. (12) Sipe, J. E.; Moss, D. J.; van Driel, H. M. Phys. Rev. B 1987, 35, 1129.

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Sjodin et al. Table 1. Optical Constants Used in the Calculation constants n1(ω)a n1(2ω)a k1(2ω)a n2(ω)b

values

constants

values

1.51 1.70 0.095 0.037

(ω)b

3.54 1.40 1.46

k2 n2(2ω)c k2(2ω)c

a For liquid pyridine from ref 13. b For silver from: Otter, M. Z. Phys. 1961, 161, 539. c For silver from: Winsemius, P. Thesis, University of Leiden, 1973.

Figure 3. SH intensity versus pyridine exposure for unannealed films at 100 K. The dots are experimental data. The solid line is a fit using eq 2.

Figure 5. SH intensity from a pyridine film as a function of temperature. The solid line is the SH intensity during heating and the dashed line is during cooling. The film thickness was 140 ( 10 nm. The arrows represent the heating and cooling directions. The heating rate was 3 K/min, and the cooling rate was uncontrolled.

Figure 4. Diagram depicting the propagation of the fundamental (solid line) and the second harmonic (dashed line) in the three media.

to that from clean Ag(111), IAg, is7

F(x) )

IamorP/Ag 4 2 ) |t(ω) 02 | |T20| IAg

(2)

where t(ω) 02 is the transmission coefficient from the vacuum (medium 0), through the pyridine film (medium 1), into the silver (medium 2) of the fundamental light and T20 is from the silver to the vacuum for the SH light. (Lower case constants will refer to values at the fundamental wavelength, and upper case constants will refer to values at the SH wavelength. Also, any constant without an explicit frequency dependence is assumed to be evaluated 7 at 2ω.) t(ω) 02 is given by, (ω) (ω) (ω) (ω) t(ω) 02 ) g02 t01 t12 exp(ik1,z d)

(3)

-1 (ω) (ω) (ω) g(ω) 02 ) [1 - r01 t12 exp(2ik1,z d)]

(4)

where

(ω) is the and is the result of multiple reflections, k1,z z-component of the fundamental wave vector in medium (ω) 1, and r(ω) 01 (t01 ) is the Fresnel reflection (transmission) coefficient from medium 0 to medium 1. T20 is defined the same as t(ω) 20 except all quantities are evaluated at 2ω. The thickness dependence of the unannealed pyridine film can be fit to eq 2 as shown in Figure 3. There are no reported values for the refractive index of solid amorphous pyridine; therefore the pyridine optical constants used

and shown in Table 1 are for liquid pyridine.13 The exposure scale is linear to film thickness, and the growth rate of the film can be obtained from the theoretical fits to eq 2. From the fitted calculation shown in Figure 3, a growth rate of 0.11 ( 0.02 nm/langmuir is obtained. All film thicknesses presented in this paper are obtained from fitting the exposure data to eq 2. We have shown that the pyridine films grown at 100 K on Ag(111) produce no measurable SHG. The observed SHG is only from the Ag(111) substrate. The azimuthal angle dependence of the SHG from pyridine-covered Ag(111) shows a 3-fold symmetry pattern identical to the Ag(111) substrate. Also, the thickness dependence of the SHG can be modeled using eq 2, which assumes SH is generated only from the silver. The oscillations in the SHG observed as a function of pyridine film thickness originate from interference mainly due to multiple reflections of the fundamental light within the film. B. The Temperature Dependence of SHG and the Phase Transition. The temperature dependence of SHG from a 140 ( 10 nm thick (1270 ( 90 langmuir exposure) amorphous pyridine film is shown in Figure 5. The sample has been rotated to Φ ) 0°, so no SHG from Ag(111) can be observed. Initially, at 100 K no SHG signal is detected. As the sample is heated to above 125 K, there is a large increase in the SHG observed. This signal persists up to the multilayer desorption temperature of 165 K. If the annealed film is allowed to cool before the desorption temperature is reached, as in Figure 5, the SHG is observed to persist below 125 K and is roughly temperature independent. We propose the additional SHG observed when the film is annealed above 125 ( 5 K is due to a transition of the film structure from an amorphous to a crystalline state. (13) MacRae, R. A.; Williams, M. W.; Arakawa, E. T. J. Phys. Chem. 1974, 61, 861.

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Figure 6. SH intensity from annealed pyridine films as a function of azimuthal angle at film thicknesses of (a) 120 ( 10 nm and (b) 350 ( 40 nm. The solid lines are fits to eq 5.

The crystalline structure of solid pyridine at liquid nitrogen temperatures has been solved by X-ray crystallography.14 The crystals are orthorhombic and belong to space group Pna21, which is noncentrosymmetric and allows for the generation of SHG from the bulk of the film. The crystalline phase is more thermodynamically stable than the amorphous film. Therefore, when the annealed film is cooled below the transition temperature, it remains in the crystalline state. C. SHG from Annealed Films on Ag(111). The azimuthal angle dependence of the SH intensity of the annealed films is shown in Figure 6 for two different film thickness. The data show a 3-fold anisotropic component and an isotropic component which is absent from both the clean Ag(111) and the amorphous pyridine-covered Ag(111) as shown in Figure 2. The azimuthal angle dependence of the annealed pyridine-Ag(111) is given by

Iss,ann ) |aann + bann sin(3Φ)|2

(5)

where aann represents the isotropic component and bann the anisotropic component. Both aann and bann are complex, and phase differences may exist between them allowing for the unequal ratios of the maxima and minima as a function of Φ. Films of various thickness were annealed, and the magnitudes of the isotropic, |aann|2, and anisotropic, |bann|2, components were obtained by fitting eq 5 to the observed SHG from the annealed films. |aann|2 and |bann|2 are shown as a function of amorphous film thickness in Figure 7. Both components show oscillatory behavior that can be related to the film structure and thickness. In the next section, the origin of the anisotropic and isotropic components will be discussed in an attempt to elucidate the structure of the annealed films. IV. Discussion: Structure of the Annealed Films The SHG from the annealed pyridine-covered Ag(111) consists of an isotropic component and an anisotropic component with a 6-fold symmetry pattern. The point group symmetry of crystalline pyridine is C2ν.14 The azimuthal angle dependence of the anisotropic component, shown in Figure 6, clearly shows a 6-fold symmetry pattern which is not possible for a crystal with C2ν symmetry, but is possible for Ag(111) which has C3ν symmetry. Therefore, we first consider the possibility that this anisotropic component originates from the Ag(111) substrate. (14) Mootz, D.; Wussow, H.-G. J. Chem. Phys. 1981, 75, 1517.

Figure 7. (a) The anisotropic component, |bann|2, of SH intensity of annealed pyridine films as a function of unannealed film thickness. The solid line is a fit to eq 2. (b) The isotropic component, |aann|2, of SH intensity of annealed pyridine films as a function of unannealed film thickness. The solid line is a fit to eq A17.

The thickness dependence of the anisotropic component of the annealed samples displays very similar oscillation patterns to that of the amorphous film. If the origin of the anisotropic component is from the Ag(111) substrate, and the film thickness and optical constants were approximately unchanged by the annealing, the anisotropic SHG as a function of film thickness should be similar. The thickness dependence of the anisotropic component was calculated using eq 2, the same one used for the amorphous film. The fit of eq 2 to |bann|2 is shown in Figure 3 using the values for the optical constants shown in Table 1. A very good fit can be obtained by adjusting only the overall magnitude of the calculated SH intensity. The annealed anisotropic SH intensity is reduced by ∼40% compared to the amorphous films. This decrease is attributed to increased scattering in the pyridine films and is confirmed by visual inspection. The decrease could not be modeled by adding an imaginary component to the index of refraction at the fundamental wavelength. This indicates that the excess scattering occurs at the interfaces and boundaries of crystallites (see below) and not in the bulk of the film. From the symmetry consideration and film thickness dependence we conclude that the anisotropic component originates from the Ag(111) substrate. The observation of the isotropic SHG in an ss-polarization configuration presents an interesting dilemma. From the analysis of the nonzero second-order susceptibility tensor elements for a single crystal of C2ν symmetry or a surface of C3ν symmetry, it can be shown that the SHG is completely anisotropic for the ss-polarization configuration. This symmetry restriction is in contradiction of the experimental observation. A simple explanation for this observation is that if the pyridine film has optical anisotropy, both an s and p polarization component would be present at the pyridine-silver interface. This would

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produce an isotropic contribution but it would also produce a component with a cos(3Φ) dependence, which was not observed. Therefore, we do not attribute the isotropic component to optical anisotropy within the pyridine film. From these observations, we believe that the isotropic component does not arise from a single crystal. To understand this observation, we propose that the film is polycrystalline. The unannealed film is in an amorphous or glasslike state. The phase transition to a crystalline state could occur via a classic nucleation and growth model.15 This has been observed in the crystallization of amorphous water films.4 In a nucleation and growth model, nucleation sites of the second phase occur from statistical fluctuations, and growth of the second phase occurs at the interfaces between the domains with the second phase structure. This type of phase transition would lead to a polycrystalline sample with randomly oriented crystallites. This has been observed in water films.3 The SHG from an ensemble of crystallites randomly oriented is isotropic. We can further infer from the coherence length of the SHG process about the length scale of the crystallized domain size. SHG is a coherent process. The coherence length for SHG, λc, along the axis of light propagation, is16

λc )

λ 4[n(2ω) - n(ω)]

(6)

Using the values of the refractive index from Table 1, λc ∼ 700 nm for pyridine. The SHGs from crystallites with a characteristic length much smaller than λc are phase correlated. If the crystallite domain size is much smaller than λc, the total hyperpolarizability summed over a random distribution of crystallites within λc would be zero. No SHG should be observed in this case. For crystallites that are larger than or comparable to λc, on the other hand, the SHG from nearby crystallites is not phase correlated and does not interfere with one another. Considering that the light incident angle is 60°, we should really use λc cos 30° (∼600 nm) for comparison of dimensions on the surface plane. The SHG from a random distribution sums up linearly to an isotropic constant. Therefore, we propose that the SHG from the annealed films is from a random distribution of crystallites with an average length comparable to or larger than half a micrometer. The hypothesis that the isotropic SHG originates from the polycrystalline pyridine film can be confirmed by its thickness dependence. The model we use to calculate the SHG thickness dependence for the polycrystalline film consists of one layer of crystallites which are disklike, or platelike with a diameter, l, and a height, h, equal to the film thickness, d. This is shown schematically in Figure 8. The diagram, showing cylindrical crystallites, is for illustrative proposes only. The actual shape and size distribution of the crystallites are unknown and are not important for this analysis. What is important is that the film thickness is approximately equal to the average heights of the crystallites and that the thickness of the film is less than the coherence length. We also require that the lateral dimensions of the crystallites, defined in this model by l, are sufficiently small that we have a large number of crystallites within the beam diameter of 0.4 cm to provide a random distribution of crystallites. In addition, to calculate the thickness dependence of the SHG, (15) Doremus, R. H. Rates of Phase Transformations; Academic Press: Orlando, FL, 1985. (16) Kurtz, S. K.; Perry, T. T. J. Appl. Phys. 1968, 39, 3798.

Figure 8. A representation of polycrystalline pyridine films from (a) top and (b) side views. l is the characteristic diameter of a crystallite, and h is the height of a crystallite which is equal to film thickness, d.

the SHG is modeled to originate along the z-axis in the film and interference at the two interfaces is included. On the basis of the reasonable assumption that the average crystallite length is larger than the film thickness, the SHG can be modeled as an average over an isotropic distribution on the surface plane of crystallites with the same thickness as the film. Various techniques have been developed to model the SHG from thin films.8,9,17 This calculation will be done parallel to ref 9, where a solution to the nonlinear wave equation is obtained by applying appropriate boundary conditions at the vacuum-pyridine and pyridine-silver interfaces. Since the isotropic component SHG is only from the pyridine film, no SHG from the silver substrate is included. A calculation for the SHG from the pyridine film using the model described in the Appendix is shown in Figure 7b using the optical constants in Table 1. The agreement between the calculation and the experimental data is good. The oscillations are dominated by the g(ω) 02 factor in Y (see eq A16). This term originates from the multiple reflections of the fundamental light in the pyridine film. It dominates over the other terms due to the very high reflectivity of the silver substrate at the fundamental wavelength. This term also dominates in the unannealed and the anisotropic SHG explaining the very similar interference pattern between them. This calculation confirms that the origin of the isotropic components of the SHG is from the annealed molecular film rather than the substrate. V. Conclusion An irreversible amorphous to polycrystalline phase transition was observed and characterized in a thin pyridine film on Ag(111) at 125 K using optical SHG. The SHGs from the polycrystalline films were observed to have both an isotropic and anisotropic component, whereas SHGs from the amorphous films were only observed to have an anisotropic component. By comparing the film thickness dependence of the polycrystalline anisotropic SHG to a previously derived model for the amorphous film, the anisotropic component was determined to originate from the Ag(111) substrate. For the isotropic component, an analytical expression was derived to model the SHG from the polycrystalline film with crystallites larger that the SHG coherence length. This model included interference effects from multiple reflection of the fundamental and SH light within the film. The calculated film thickness dependence of the SHG from the film was in good agreement with the experimental results of the isotropic component. This observation, that the crystalline pyridine film generates isotropic SHG, supports the theory that the annealed pyridine molecular film consists of polycrystallites with an average length scale comparable (17) Bethune, D. S. J. Opt. Soc. Am., B 1989, 6, 910.

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P(2ω) ) yˆ (Ptei(kt1b‚r-2ωt) + Prei(kr1b‚r-2ωt) + Pcei(kc1b‚r-2ωt)) (A3)

Figure 9. Schematic diagram that defines the boundaries and propagation directions of (a) the fundamental, (b) the SH free (homogeneous), and (c) the SH bound (inhomogeneous) electric fields.

to or larger than half a micrometer. This study demonstrates that this nonlinear optical technique can by used on a variety of thin film materials to obtain structural information, particularly the kinetic aspect of phase transitions. Acknowledgment. The authors thank Drs. C. M. Li and K. J. Song for work on preliminary experiments and Professor E. Burstein and Dr. J. Dvorak for stimulating discussions. This work was supported in part by the National Science Foundation, MRSEC Program, under Grant No. DMR96-32598, and a grant from the New Energy and Industrial Technology Development Organization of Japan. Appendix: Calculation of Isotropic SHGs from a Thin Nonlinear Dielectric Film on a Metal Substrate In this calculation, a monochromatic plane wave with frequency ω is incident on the pyridine film at an angle θ0 with respect to the surface normal in a vacuum. The incident wave is refracted into the pyridine film and interacts with the medium to produce a second-harmonic nonlinear polarization P(2ω). The nonlinear polarization radiates a wave at frequency 2ω. The propagation of the second harmonic field is governed by the nonlinear wave equation. In this film, the equation is18

∇×∇× E(2ω) + 1

1(2ω) ∂2 (2ω) 4π ∂2 (2ω) E1 ) 2 2 P c2 ∂t2 c ∂t

(A1)

(ω) (ω) where Pt ) |yˆ 5 χ(2):yˆ yˆ |E(ω) ˆ5 χ(2):yˆ yˆ |E(ω) 1t E1t , Pr ) |y 1r E1r , Pc ) (ω) (ω) (ω) (ω) (2):y 2|yˆ 5 χ ˆ yˆ |E1t E1r , ktlb ) 2kr1 , krlb ) 2kr1 , and kclb ) k(ω) r1 + k(ω) r1 , and the polarization is assumed to be ss, which corresponds to a y-polarization for the incident fundamental field and P(2ω) using the coordinate system defined in Figure 9. The solution of eq A1 will be a superposition of the inhomogeneous, or particular, solution which will be referred to as a bound solution with subscript, b, and the homogeneous solution which will be referred to as free with subscript f. The free waves in media (1), (2), and (3), respectively, are

Ef0 ) yˆ Ef0rei(kfor‚r-2ωt)

(A4)

Ef1 ) yˆ (Ef1tei(kf1t‚r-2ωt) + Ef1rei(kf1r‚r-2ωt))

(A5)

Ef2 ) yˆ Ef2tei(kf2t‚r-2ωt)

(A6)

and are depicted graphically in Figure 9b. The bound waves only exist in medium 1 and are depicted graphically in Figure 9c. The bound field is

Eb1 )

4π yˆ (Ptei(kt1b‚r-2ωt) + Prei(kr1b‚r-2ωt) + b - f Pcei(kc1b‚r-2ωt)) (A7)

where b ) (ω) and f ) (2ω). The magnitudes of the wave vectors in medium (1) are |kb1| ) n1(ω)(2ω/c) and |kf1| ) n1(2ω)(2ω/c). The directions of propagation of the free and bound waves are determined by applying Snell’s law at the various interfaces. In general, the free and bound waves propagate in different directions with different phase velocities. The interference of the free and bound waves has been clearly demonstrated in the observation of Maker fringes in thin nonlinear slabs.19,20 To solve for the observed SHG, eqs A4-A7 need to be expressed in terms of the incident fundamental field. First, the fundamental electric field inside medium (1) must be determined in terms of the incident electric field, Ei0. The fundamental electric field amplitude in medium (1) is determined by satisfying the boundary conditions; i.e., the tangential components of E and H must be continuous at the pyridine-silver interface at z ) -d and at the pyridine-vacuum interface at z ) 0. The solutions for the fundamental fields are

where 1(ω) is the linear dielectric constant at frequency ω and the nonlinear polarization is

(ω) (ω) (ω) E(ω) t1 ) g02 t01 Ei0

(A8)

(ω) P(2ω) ) 5 χ (2):E(ω) 1 E1

(ω) (ω) (ω) -2ikt1,z(ω)d (ω) Ei0 E(ω) r1 ) g02 r12 t01 e

(A9)

(A2)

where is 5 χ(2) the second-order nonlinear susceptibility tensor of the film material and E(ω) 1 is the fundamental electric field in medium 1. As shown in Figure 9a, there are two components of E(ω) 1 , a transmitted component, (ω) E(ω) t1 , propagating with wave vector kt1 , and a reflected (ω) (ω) one, Er1 , with wave vector kr1 . P(2ω) can be expressed in (ω) terms of E(ω) t1 and Er1 and is

(ω) where g(ω) 02 is given by eq 4, kt1,z is the z-component of the wave vector of the transmitted fundamental field, and d is the thickness of medium (1). The SH field can be solved by satisfying the boundary conditions that tangential components of E and H must be continuous at z ) -d and z ) 0. The boundary conditions at z ) -d of the SH fields for Etan and Htan, respectively, are

(18) Shen, Y. R. The Principles of Nonlinear Optics; John Wiley: New York, 1984. (19) Maker, P. D.; Terhune, R. W.; Nisenoff, M.; Savage, C. M. Phys. Rev. Lett. 1962, 8, 21. (20) Jerphagnon, J.; Kurtz, S. K. J. Appl. Phys. 1970, 41, 1667.

kf1,z(Ef1te-ikf1,zd - Ef1reikf1,zd) +

4π k (P e-ikb1,zd + b - f b1,z t

Preikb1,zd + Pc) ) kf2,zEf2,te-ikf2,zd (A10)

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Ef1te-ikf1,zd + Ef1reikf1,zd +

Sjodin et al.

4π (P e-ikb1,zd + Preikb1,zd) ) b - f t Ef2te-ikf2,zd (A11)

Using eqs A10-A11, Ef1r is determined in terms of Ef1t:

Ef1r ) R12Ef1te-2ikf1,zd -

4π e-i(kb1,z+kf1,z)d × (b - f) (kf1z + kf2z)

[(kb1,z + kf2,z)e2ikb1,zd Pr - (kb1,z - kf1,z)Pt + kf2,zeikb1,zdPc] (A12) The boundary conditions at z ) 0 for the SH fields for Etan and Htan, respectively, are

Ef1t + Ef1r +

4π (P + Pr + Pc) ) Ef0r (b - f) t

kf1,z(Ef1t - Ef1r) +

(A13)

4π k (P - Pr) ) kf0,zEf0r (b - f) b1,z t (A14)

Using eqs A12-A14, the solution for Ef0r in terms of is

E(2ω) f0r where

(ω) (ω) :yˆ yˆ |Ei0 Ei0

(2)

) Y|yχ 5

Y)

4π g(ω)2t(ω)2G T × (b - f) 02 01 02 10 kf2,z 2 -i(kb1,z+kf1,z)d (1 + r(ω) + 12 ) e kf1,z + kf2,z 1 -ikb1,zd 2 ) (1 - R12e-2ikf1,zd) + (1 + r(ω) 12 e 2 kb1,z -i(kb1,z+kf1,z)d (1 - r(ω)2 )12 e kf1,z + kf2,z kb1,z -2ikb1,zd (1 - r(ω)2 )(1 + R12e-2ikf1,zd) (A16) 12 e 2kf1,z

[

]

Y depends on film thickness, d, incidence angle, and the linear optical constants of the medium. Equation A15 is valid only for a single crystal. The solution for the polycrystalline film is obtained by averaging over a random distribution of the orientation of the crystallites. The orientation of the crystallite only affects the susceptibility tensor. The average SHG intensity, I(2ω) poly is

I(2ω) poly )

(ω) E0i

(A15)

c (ω) (ω) 2 |Y〈yˆ 5 χ (2):yˆ yˆ 〉Ei0 Ei0 | 8π

(A17)

where 〈yˆ 5 χ(2):yˆ yˆ 〉 represents an angle averaged value. I(2ω) poly is the isotropic component of the SHG from the annealed films. LA990837B