Broadband plasmon resonance enhanced third-order optical

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Broadband plasmon resonance enhanced third-order optical nonlinearity in refractory titanium nitride nanostructures Rodrigo Sato, Satoshi Ishii, Tadaaki Nagao, Masanobu Naito, and Yoshihiko Takeda ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.8b00357 • Publication Date (Web): 13 Aug 2018 Downloaded from http://pubs.acs.org on August 14, 2018

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Broadband plasmon resonance enhanced thirdorder optical nonlinearity in refractory titanium nitride nanostructures Rodrigo Sato,1,* Satoshi Ishii,2 Tadaaki Nagao,2,3 Masanobu Naito4,5 and Yoshihiko Takeda1,6, †

1

Research Center for Advanced Measurement and Characterization, National Institute for Materials Science (NIMS), Tsukuba, Ibaraki 305-0003, Japan

2

International Center for Materials Nanoarchitectonics (WPI-MANA), National Institute for Materials Science (NIMS), Tsukuba, Ibaraki 305-0044, Japan 3

Department of Condensed Matter Physics, Graduate School of Science, Hokkaido University, Sapporo, Hokkaido 060-0810, Japan

4

Research Center for Structural Materials, National Institute for Materials Science (NIMS), Tsukuba, Ibaraki 305-0047, Japan

5

Graduate School of Frontier Sciences, The University of Tokyo, Kashiwa, Chiba 277-8561, Japan 6

School of Pure and Applied Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8577, Japan

Keywords: nonlinear optics, third-order susceptibility, nonlinear plasmonics, titanium nitride.

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Abstract Ultrafast control of light by light at the nanoscale may enable numerous long-awaited applications in nanophotonics. However, traditional plasmonic materials suffer from low optical thresholds that limit their usages. Titanium nitride has shown superior properties such as thermal stability, low cost and CMOS-compatible fabrication process. Even though titanium nitride is prominent alternative plasmonic material, little is known about its optical nonlinearities and underlying mechanisms. Specifically, the third-order nonlinearity results in modifications of the refractive index, allowing all-optical modulation and switching functionalities. Here, we experimentally obtained the third-order optical susceptibility of the titanium nitride nanoparticles in an unprecedented wide bandwidth range and compared to those of traditional materials. The experiments show a much broader nonlinear enhancement compared to gold and silver nanoparticles. This work demonstrates that titanium nitride is a valid alternative plasmonic material for efficient active nanophotonics devices in the near infrared region without the need for complex nanostructures.

In the field of plasmonics and metamaterials, metals such as gold, silver and copper have been extensively studied.1,2 These metals support the light-driven collective oscillation of free electrons, known as surface plasmon (SP) resonance.3 As a consequence, light can be concentrated and manipulated at the nanoscale. The unprecedented properties of plasmonic resonances opened up a wide range of applications, including bio-sensing,4 SP-assisted thermal cancer therapy,5 SP-assisted magnetic recording,6 and SP-enhanced photodetectors.7 Although noble metals enabled important applications as passive plasmonic devices, they have critical limitations for active plasmonic applications. These metals substantially suffer


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from thermal instability under intense electromagnetic field that greatly restrict them as efficient and active plasmonic building blocks.8,9 Thus, alternative plasmonic materials are needed to address these technical challenges and significantly improve the design and optimization of active devices. Recently, transition metal nitrides were suggested as alternative plasmonic materials. 9,10

Titanium nitride (TiN) nanostructures have been the subject of growing interest in

nanophotonics due to their superior properties compared to noble metals, such as high melting point (>2900 oC), chemical stability, hardness and compatibility with complementary metal–oxide–semiconductor (CMOS) fabrication processes.9 With these advantages, TiN nanostructures offer several promising applications, including active nanophotonic functionalities such as all-optical ultrafast switches, and second- and third-harmonic generation.11 The suitability of a metamaterial for those applications also essentially requires large nonlinear optical coefficients and tunability. Owing to an interplay between the interband transitions and SP resonances, linear and nonlinear optical properties are strongly wavelength-dependent. Consequently, the complex third-order nonlinear susceptibility, which is a crucial parameter for the control of light at the nanoscale, rapidly changes in this wavelength region and could result in induced transparency and optical limiting.12,13 Nevertheless, to the best of our knowledge, the ultrafast wavelength-dependent nonlinearities of TiN nanostructures have yet to be experimentally investigated. Using a single wavelength measurement at 532 nm, S. Divya et al.14 investigated TiN nanoparticles (NPs) embedded in polyvinyl alcohol (PVA). However, their results on the observed nonlinearities primarily reflect cumulative thermo-optic contributions due to nanosecond pulses.15 Furthermore, microscopic theories and Miller’s rule fail to accurately predict the relationship between linear and nonlinear optical coefficients.16


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In the present work, we obtained the wavelength-dependent nonlinearities of both TiN thin film and NPs by spectroscopic ellipsometry and ultrafast pump-probe spectroscopy. Combining these two techniques enabled us to extract the TiN/PVA composite (effective) and inherent TiN particle (intrinsic) nonlinear complex third-order susceptibilities χ(3).17 These results allowed us to identify and disentangle the underlying physical mechanisms involved. Our results indicate that the wavelength-dependent χ(3) of the TiN/PVA composite reflects an interplay between the SP resonances and interband transitions. Also, we will discuss the intrinsic nonlinearity of the TiN NP. From these results, we validated the TiN as a major alternative plasmonic material at the near infrared region without the need for complex nanostructures. Also, we clarified the discrepancies in the results at a few wavelengths available in the literature. Ultimately, these results can boost the exploitation of transition metal nitrides as building blocks for ultrafast light manipulation at the nanoscale where lowpower, robust and broadband metamaterials are expected. The preparation method of TiN NPs has been previously reported.18 Using radiofrequency thermal plasma processing, plasma was generated from Ti powder in argonnitrogen and then quenched to form single crystalline NPs with an average size of 30 nm. Xray diffraction (XRD) patterns showing a single phase of TiN is provided in Figure S1 in the supporting information. The surface of these NPs is protected by a native oxide layer of a few nanometers. Energy-dispersive X-ray spectroscopy (EDS) element mapping showing the oxide layer is provided in Figure S2 in the supporting information. The following procedure describes the fabrication process of TiN NPs embedded in PVA composite. The PVA solution was prepared by dissolving 1.0 g of PVA in 50 ml of deionized water under stirring and heated (75 oC) for 5 h and then cooled at room temperature. Afterward, 1.0 mg of the TiN NPs was dissolved in 100 µl of deionized water and sonicated. The resulting TiN NPs solution was added to 40 µl of PVA solution. Finally, the TiN NPs dispersed PVA was spin-


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coated on a silica glass substrate to form a film. For comparison, a 25-nm TiN thin film was fabricated on a silica glass substrate using DC magnetron sputtering similar to the method described in ref 19. The linear optical properties were investigated by spectroscopic Mueller-matrix ellipsometry (J. A. Woollam, VASE).20 The Mueller matrices were obtained from the spectroscopic ellipsometry measurements at angles of incidence from 50o to 60o by 1o step and from 50o to 70o by 10o step for TiN/PVA composite and TiN thin film, respectively (see Figures S3 and S4 for more details in the supporting information). Assuming a homogeneous composite layer with thickness of 573 nm, the TiN/PVA composite was described by an effective medium approximation (EMA). The intrinsic TiN NP with filling factor of 0.085 was described by Drude term, Lorentz and Tauc-Lorentz oscillators. Drude term models the free electrons. Lorentz and Tauc-Lorentz model the interband electrons and the native oxide layer. It was demonstrated by Blanckenhagen et al.21 that the Tauc-Lorentz model is a great tool to parameterize the bound electrons of oxide thin film materials. The quality of the fitting is described by the mean square error (MSE),20 the MSE of the TiN/PVA composite was 17.0. For the TiN thin film, TiN was described by Drude term and two Lorentz oscillators. The oxide layer on the surface of the thin film was disregarded. Considering the MSE of 3.07, the present model is adequate to describe the 25-nm TiN thin film. The oscillators’ parameters of these samples are shown in Table S1 and S2 in the supporting information. Note that these samples were measured in different facilities using a similar spectroscopic ellipsometry (J. A. Woollam, VASE) but with an extended wavelength region for the TiN nanoparticles. Owing to a broad plasmon resonance centered at 730 nm, it was measured from 350 to 1200 nm at J. A. Woollam Corp., Tokyo. For the thin film, the wavelength range was measured up to 1000 nm due to the limitation of the ellipsometry system at our institute. The aim of the thin film analysis was to clarify the contribution of


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interband electrons to the optical nonlinearity of the NPs. Since the contribution of the interband loss to the linear optical properties takes place in the vicinity of the interband electrons (see Figures S3 for more details in the supporting information), the range from 350 to 1000 nm is adequate.

Figure 1: (a) Extinction spectra of TiN NPs dispersed in PVA matrix and 25-nm thin film. Real and imaginary components of the dielectric function of intrinsic TiN (b) NPs and (c) 25-nm thin film. Wavelength region from 350 to 1200 nm and 350 to 1000 nm for composite and thin film, respectively. The extinction spectra ⁄ , where T is the transmittance of the substrate SiO2 with TiN thin film or TiN/PVA composite and T0 is the transmittance of the substrate, were measured by a spectrometer (Jasco, V670). Figure 1(a) shows the extinction spectrum of the TiN/PVA composite. The spectrum is composed by two well-defined features whose physical


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origin reflects the interband transitions and SP resonance, respectively, at shorter wavelength and centered at 730 nm. The broad SP resonance peak agrees well with the previous work,18 verifying that in our sample the TiN NPs are well dispersed into the PVA matrix. The complex dielectric function (or relative permittivity) ε of the intrinsic TiN NPs is shown in Figure 1(b). The wavelength at which Re[ε] = 0, known as screened plasma wavelength, phenomenologically indicates the stoichiometry of TiNx, and it is frequently reported in the literature.22-24 Values vary from 2.00 to 2.95 eV; value of 2.65 eV indicates x=1.25 The extracted Re[ε] shows the screened plasma wavelength of 573 nm (2.16 eV). For the sake of later discussion, the extinction spectrum of the 25-nm TiN thin film is plotted in Figure 1(a). In contrast to NPs, the extinction spectrum of the thin film does not present the SP resonance peak. The absorption at shorter and longer wavelength is due to the interaction of light with interband and free electrons, respectively. The complex dielectric function is shown in Figure 1(c). Note that higher intensities of the negative Re[ε] component compared to NPs indicate higher carrier concentration and mobility.22 The extracted Re[ε] shows the screened plasma wavelength of 565 nm (2.19 eV). The screened plasma values of composite and thin film are within the range of previous reports.22-24 Agreement of the measured linear optical properties of the TiN NPs and thin film samples to the literature22-25 verifies our ellipsometric results (see Figures S3 and S4 and Tables S1 and S2 for more details in the supporting information). Thus, the obtained dielectric functions can be used to extract the nonlinear quantity χ(3). Having described the static properties of the TiN/PVA composite and thin film, we now turn to the spectral and relaxation dynamics of the optical nonlinearity. The nonlinear transmission changes ∆T/T, defined as the difference between the transmitted light with and without light excitation, were measured by a custom-made femtosecond pump-and-probe spectroscopy.17 The fundamental laser beam at 800 nm was generated by a regenerative amplifier (Spectra Physics, Spitfire) using Ti:sapphire (Spectra Physics, Mai Tai) and


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Nd:YLF (Spectra Physics, Empower) lasers. Schematic representation of the pump-probe spectroscopy and sample along with pump and probe beams are shown in Figure 2(a) and 2(b). The samples were excited by a fraction of the fundamental laser beam with a maximum peak intensity of 41 GW/cm2 at 800 nm. To avoid laser-induced damage, repetition rate and pulse width were kept at 0.5 kHz and 130 fs, respectively. The supercontinuum probe pulses were generated by focusing a fraction of the fundamental laser beam into a CaF2 and YAG crystals for visible and near-IR, respectively. The chirping effect was corrected by using the Kerr gate technique.26

Figure 2: (a) Schematic representation of the pump-and-probe spectroscopy. (b) Schematic representation of the TiN NPs dispersed in PVA matrix on a silica glass. (c) Nonlinear transmission changes of the TiN/PVA composite and the TiN thin film. The composite and the thin film were excited at a pump energy per pulse of 8.2 and 6.0 µJ,


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respectively. Pumping wavelength was 800 nm. Wavelength region from 350 to 1200 nm and 350 to 1000 nm for composite and thin film, respectively.

The ∆T/T of the TiN/PVA composite is plotted in Figure 2(c) and compared to the 25nm TiN thin film. Note that the wavelength range and intensity of the white-light continuum of the probe light restrict the measurements. Within the low excitation regime, the TiN/PVA composite and thin film were excited with a peak intensity of 41 and 30 GW/cm2 (8.2 and 6.0 µJ energy per pulse), respectively. In order to qualify the influence of the PVA matrix to the nonlinearities of the TiN/PVA composite, ∆T/T of pure PVA film was measured and no significant nonlinearity was obtained. Hence, the contribution of the matrix to the composite nonlinearities was neglected. Our measurement technique allows us to reveal the spectral information of the transient response. Photoinduced absorption and transparency (negative and positive ∆T/T) can be observed at shorter and longer wavelength, respectively, for the TiN/PVA composite. This photoinduced features are due to the changes of the dielectric function of TiN NPs ∆εTiN modulated by the local-field enhancement fl at the SP resonance. Such photoinduced features stem from interband electrons and SP resonance and were extensively reported for traditional plasmonic NPs.27,28 Thus, to clarify the contribution of the SP resonance, the TiN thin film was analyzed. For the thin film, ∆T/T shows induced absorption (negative region) up to 730 nm. Since the thin film has no local-field enhancement, the induced absorption observed is a consequence of the interband electrons. This phenomenon will be later explained, when discussing the intrinsic nonlinear

susceptibility of the TiN NPs .

Next, we discuss the effective of the TiN/PVA composite and intrinsic of the TiN NPs obtained by combining ∆T/T with the dielectric function. This analysis allowed us


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to clarify the physical origin of the optical nonlinearities. Following the procedure described by Sato et al.,17 the χ(3)s were evaluated from the nonlinear dielectric function changes ∆ε and were expressed as

Δ = ,

(1) $

Δ = ! "# ,

(2)

where ∆εeff and ∆εTiN are the dielectric function changes of the TiN/PVA composite and intrinsic TiN NP, respectively.

!

is the local-field enhancement and I is the pump peak

irradiance. and "# are the applied wave circular frequency of the probe and pump, respectively. ∆ε was evaluated by minimizing the MSE of the ellipsometric model using ∆T+T (∆T+T along with the steady state transmission is shown in Figure S5 in the supporting information). For that, we used the same ellipsometric model for linear optical properties and fitted only the oscillators and Drude term parameters of the TiN. Here it should be noted that the PVA and SiO2 do not have a significant contribution to the nonlinearity. For the composite and thin film, the MSE was 5.79 and 4.03, respectively. The oscillators’ parameters at the excited state are shown in Table S1 and S2 in the supporting information. The effective and intrinsic χ(3)’s spectra are shown in Figure 3. The complex

of the TiN/PVA composite shows strong wavelength dependence as can be observed in

Figure 3(a). In the vicinity of the SP resonance and interband transitions, the Im[ ] component varies from – 0.80 x 10-19 (at 880 nm) to 0.19 x 10-19 m2/V2 (at 460 nm) and the

Re[ ] component varies from – 1.05 x 10-19 (at 1200 nm) to 0.22 x 10-19 m2/V2 (at 580 nm). These results emphasize the importance of taking into consideration the wavelength dispersion of the third-order nonlinearity. It is worthwhile to note that the dispersion of the

of composites reflect the local electric-field enhancement (SP resonance) and the


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intrinsic nonlinearity .29 Therefore, to understand the macroscopic nonlinearities, we

studied the of the TiN NPs.

Figure 3: Effective third-order susceptibility of the TiN/PVA composite and (b) intrinsic third-order susceptibility of the TiN nanoparticle. Wavelength region from 350 to 1200 nm.

Figure 3(b) shows the complex of the TiN NPs. Note that, the wavelength dependence is non-trivial and can be either positive or negative; thus, modulated by optical nonlinearities, the dielectric function of the TiN NPs can increase or decrease depending on the wavelength region. The intrinsic nonlinearity of the TiN NPs is a direct consequence of the interband and intraband transitions.30 In order to identify and disentangle these

contributions, the of the 25-nm TiN thin film was evaluated (Fig. 4). The observed spectral feature of the thin film is a result of the changes of the dielectric function of the TiN. This was attained by the modifications of the Lorentz oscillators in the ellipsometric model. Note that the Drude term modification between the linear and nonlinear states is negligible (see Table S1 for more details in the supporting information). As the Lorentz oscillators describe the interband electrons, these results indicate that the observed nonlinearity of the TiN thin film solely reflects the interband electrons. Indeed, the optical nonlinearity of the


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traditional plasmonic thin films originate from the interband electrons as suggested in ref 31 and 32. Comparison was made with traditional plasmonic metals because, to the best of our knowledge, the wavelength-dependent nonlinearities of TiN thin film have yet to be theoretically or experimentally investigated. For TiN, the interband transitions mainly occur from hybridized Ti p with N p electrons to Ti d electrons.26 Although these transitions are located at 3.5 eV (354 nm) and 5.0 eV (248 nm) below the Fermi level, the cut-off energy is much lower than 3.5 eV. Cut-off energy is defined as the maximum energy below the Fermi level to reach non-zero values in the electronic density of states.25 Non-zero values of the interband absorption at 1.55 eV (800 nm), described by two Lorentz oscillators, can be observed in the dielectric function of the TiN thin film provided in Figure S3 in the supporting information. Similarly, by characterizing TiN thin films with spectroscopic ellipsometry, Patsalas et al.33 shows non-zero values of the interband absorption at 800 nm. Thus, by exciting the TiN thin film at 800 nm, the excitation energy is absorbed by the interband electrons. Subsequently, these electrons are excited to the conduction band.34 This process ends up heating the conduction electrons and leads to a modification in the FermiDirac distribution (Fermi-smearing contribution). Since the population of energies increase (decrease) above (below) the Fermi level, the dispersion of the dielectric function of the TiN is modulated. This mechanism is also referred as hot-electron contribution. To support this attribution, we analyzed the electron dynamics of the TiN. For Au thin film, Fermi-smearing contribution has a slower relaxation time of several picoseconds compared to interband contribution due to the energy transfer from the thin film to the lattice as electron-phonon interactions.31 The observed relaxation time of the TiN NPs and thin film is on the order of hundreds of picoseconds (Fig. 5). This slower relaxation time compared to noble metals may be due to lower carrier concentration and more complex electronic band structures near the Fermi level, however, the exact reason remains unknown.


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Figure 4: Third-order susceptibility of the 25-nm TiN thin film. Wavelength region from 350 to 1000 nm.

After elucidating the influence of the interband transitions to the of bulk TiN, we can clarify the interaction region between the interband transitions and SP resonance to the

of the NPs (Fig. 3b). With electron confinement in nanostructures, intraband absorption creates a strongly athermal electron distribution. Subsequently, the energy is released as electron-electron and electron-phonon interactions.35 This contribution leads to the strong macroscopic optical nonlinearity enhanced by the SP resonance (Fig. 2c and 3). However, due to the partial spectral overlap with the interband transitions, the SP resonance contribution to the optical nonlinearity is altered in the vicinity of the interband transitions. By comparing the ∆/ of the TiN NPs with the thin film (Fig. 2c), one can observe that the SP resonance contribution is spectrally coupled to the interband contribution. This detrimental effect is more pronounced in Au NPs where complex structures have been employed to shift the SP resonance away from the interband electrons.36


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Figure 5: Electron dynamics of the TiN/PVA composite and 25-nm TiN thin film at 470 and 700 nm. Time delay range from -10 to 500 ps.

As already noted by several studies and reviews,28,37

data available in the literature

of TiN and traditional metals have huge variations. This is mainly caused by the simplistic single wavelength measurements performed by Z-scan method. These reports have conflicting results regarding the underlying physical mechanisms. As seen in Figures 3 and 4,

is strongly wavelength-dependent. Also, quantitative comparison is difficult due to

measurements performed at different wavelengths, pulse width and film thickness.28 Long pulse duration has an additional thermo-optical contribution to the optical nonlinearities and, for thin films, thin thickness of a few nm’s has an additional size quantization contribution.35 Using Z-scan technique at 532 nm, S. Divya et al.14 measured TiN NPs with average size of 55 nm dispersed in water. They observed an induced absorption and attributed the values of

to SP resonance and thermal effects. However, with no SP resonance, ∆T/T of the TiN thin film (Fig. 2c) shows induced absorption at 532 nm. Therefore, we posit that the induced absorption observed in the TiN NPs at 532 nm in ref 14 reflects the TiN interband electrons. Using dual-arm Z-scan at 780 and 1550 nm, N. Kinsey et al.37 measured 52-nm TiN thin


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film. They concluded that the observed optical nonlinearities values of TiN thin film were

similar to traditional noble metals and TiN NPs. Conversely, we emphasize that the is strongly wavelength-dependent and strongly varies among materials. Note that these

materials have positive and negative , and therefore induced absorption and transparency, at different wavelength regions. One example of the wavelength-dependent

of gold thin film can be found in a recent paper by A. Marini et al.33 Thus, dispersions of

the '" of Au and of TiN thin films are remarkably different. Also, as seen in Figure 3(a) and 4, TiN/PVA composite have great difference in its optical nonlinearities compared to the thin film due to the influence of the SP resonances combined with the interband transitions that can be complex to interpret in single wavelength measurements. After clarifying the nonlinear plasmonic properties of the TiN NPs, we compared to those of noble metals in the literature.11,35 As in traditional plasmonic NPs, effective optical nonlinearities of TiN/PVA composite are enhanced by the SP resonance. Significantly, the

contribution of the SP resonance to the of the TiN/PVA composite is much broader, from visible to near infrared region (Fig. 3a). Interestingly, for TiN NPs, the SP resonance only causes induced transparency in the vicinity of the resonance. Furthermore, the decay time is nearly 100-fold slower than noble metals.35 However, its time decay is faster than semiconductors NPs.38 Thus, with these unique nonlinear optical properties, TiN NPs can benefit in nanophotonic applications where broadband materials from visible to near infrared are highly expected. To summarize, we have experimentally investigated the nonlinear optical properties of the TiN/PVA composite in a broad wavelength range, from 350 to 1200 nm. These NPs have a 1-2 nm native oxide layer. The nonlinear optical modulation of the TiN NPs leads to induced transparency or absorption depending on the wavelength. As a result, single


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wavelength measurements have led to conflicting outcomes. The combination of pump-probe spectroscopy with spectroscopic ellipsometry presents numerous opportunities for exploiting the optical nonlinearities of plasmonic platform materials. From these results, we have succeeded in identifying the Fermi-smearing and SP resonance contributions and provided the spectra of the effective (macroscopic) and intrinsic (microscopic) third-order optical nonlinearities. Understanding these contributions pave the way towards optimization of nonlinear responses because one can engineer the local-field enhancement to control the optical nonlinearities of the nonlinear plasmonic building blocks. Furthermore, the optical nonlinearity of TiN NPs due to the SP resonance is much broader and slower than traditional noble metals. We believe that the present results may provide new insights into the exploitation of TiN as a novel nanoplasmonic platform material where broadband, robust and CMOS-compatibility are required, i.e., for integrated ultrafast frequency conversion and modulation of light at the nanoscale.

ASSOCIATED CONTENT Supporting information See supporting information for additional information on characterization of TiN NPs, including TEM-EDS and XRD pattern; ellipsometric fitting results of TiN NPs and thin film; transient transmission changes of TiN NPs and thin film along with steady state spectra.

AUTHOR INFORMATION


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Corresponding authors * E-mail: [email protected]

† E-mail: [email protected]

NOTES The authors declare no competing financial interest.

Acknowledgments This work was partially supported by the Japan Society for the Promotion of Science (JSPS) Kakenhi grant numbers 16H06364, 17H04801, and 17K19045, the Core Research for Evolutional Science and Technology (CREST) “Phase Interface Science for Highly Efficient Energy Utilization” grant number JPMJCR13C3 from Japan Science and Technology Agency, JFE 21st Century Foundation, and the Kao Foundation for Arts and Sciences.

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For Table of Contents Use Only Broadband plasmon resonance enhanced third-order optical nonlinearity in refractory titanium nitride nanostructures

Rodrigo Sato, Satoshi Ishii, Tadaaki Nagao, Masanobu Naito and Yoshihiko Takeda

Schematic representation of the TiN NPs dispersed in PVA matrix on a silica glass substrate and the pump-probe spectroscopy setup. Intrinsic third-order optical susceptibility of TiN nanoparticles.


Broadband plasmon resonance enhanced third-order optical

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