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Abstract
The formation and relaxation of plasmon-induced hot electrons in Au nanodisks (AuNDs) with relatively large particle size was studied by femtosecond pump-probe spectroscopy. We temporally resolved the plasmon-induced formation of nonthermal electrons through intraband transitions and subsequent redistribution of nonthermal electrons to thermal electrons. The transient extinction spectra were found to exhibit clear dichroism for the pump polarization on a time scale of a few hundred femtoseconds. The electric field distribution was calculated using the Finite-difference time-domain (FDTD) method to be inhomogeneous inside the AuNDs under irradiation by linearly polarized light. The formation and relaxation of hot electrons occur locally inside the AuNDs, which results in the observed ultrafast dichroic response.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Substantial attention has been given to the unique and exotic optical behavior of metal nanoparticles due to localized surface plasmons (LSP) [1,2]. These findings have led to the core technology in the fields of the modern optics and optoelectronics. The intense and confined optical fields due to LSPs are useful for substantially exciting materials. Enhanced Raman cross sections [3,4], amplified fluorescence quantum yields [5,6], and increased efficiencies for nonlinear wave conversions [7] are some of the most important applications of LSPs. Researchers have turned their attentions towards generating plasmon-induced hot electrons. Meaningful discussions have taken place regarding the use of plasmon-induced hot electrons in the fields of energy conversion [8,9], photocatalysis [10–12] and photodetection [13].
Numerous experimental and theoretical studies have been performed to understand the formation and relaxation of plasmon-induced hot electrons. Ultrafast spectroscopy has demonstrated the formation of hot electrons during plasmon decay and relaxation due to electron-electron and electron-photon scatterings [14–17]. Ultrafast dynamics at the single particle levels has also been reported [18].
The semiclassical two or three temperature models are the most popular models among reported theoretical approaches [18–21]. These models consist of coupled equations for the populations of nonthermal electrons, the electron temperature and the lattice temperature. These models can explain the interplay of energy between the exciting light and electron gas, the redistributions of nonthermal electrons with nonequilibrium distributions to thermal electrons with Fermi-Dirac distributions at higher temperature, the cooling of the electron temperature and the increase in the lattice temperature. More detailed models incorporating the electronic structures of the metals have also been recently proposed [22–25].
Most previous works have dealt with the metal nanoparticles of which sizes were much smaller than the light wavelength. A quasi-static model has been used for explaining the ultrafast optical responses [2]. In the model, the electric fields inside the metal nanoparticles was assumed to be homogeneous. Hence, the discussion was made under the assumption of spatially homogeneous distributions of plasmon-induced hot electrons inside metal nanoparticles. Research on the particles with relatively larger sizes has also dealt with a model in the limit of small-particles [18]. However, in reality, LSP-enhanced optical fields are inhomogeneously distributed inside particles, and intense fields are localized mostly in the vicinity of the edges of the particles. Thermal diffusion produces a spatial redistribution of excess energy within a few hundred nm-region on a time-scale of subpicosecond [26]. It is not obvious to what extent the model for the small-particle limit is applicable to ultrafast dynamics for plasmon-induced hot electrons in relatively large metal nanoparticles.
The present paper reports femtosecond pump-probe spectroscopy for Au cylindrical nanodisks (AuNDs) with a diameter of a few hundred nanometers. The size of the AuNDs were comparable to the LSP resonance wavelength, and the model for the small-particle limit was not applicable for the systems in the strict sense. The geometry of the AuNDs is invariant against the rotation operation around the center of the disk. Hence, the linear optical properties are isotropic within the disk plane. We examined the polarization dependency of the transient spectrum so as to address the influence of the inhomogeneous field distributions inside the AuNDs on the ultrafast responses of the plasmon-induced hot electrons.
Polarization-resolved femtosecond spectroscopy has already been performed for small Au nanospheres [27]. The authors of that paper claimed that polarization-resolved spectroscopy was useful for probing particle asphericity. In the present study, the polarization-resolved technique was used to address the dynamics related to the spatially inhomogeneous generation of hot electrons inside AuNDs. Our experimental results demonstrated that AuNDs irradiated by linearly polarized exciting light exhibit dichroism within a femtosecond time scale. The origin of the transient dichroism was explained by using the numerically calculated field distribution inside the AuNDs.
2. Experiment
The AuNDs were fabricated onto SiO2 substrates coated with ITO thin films using electron beam lithography. The thickness of the ITO films was 150 nm. The conductivity of the ITO layer prevented the charge-up phenomenon on the substrate during the irradiation by the electron beam. A lift-off procedure was used for patterning the Au thin films on the substrates. The diameter and the height of the disks was 2R=350 nm and H=30 nm, respectively. AuNDs with the same dimensions were arrayed onto a two-dimensional square lattice. The edge-to-edge separation of the AuNDs was Λ=500 nm. Figure 1(a) shows an SEM image of the AuNDs. As will be seen later in Fig. 2(a), the linear extinction spectrum $Ext.(\lambda )$ shows a peak at${\lambda _{LSP}}$=1080 nm due to an LSP resonance.
Fig. 1. (a) SEM image of AuNDs. The scale bar is 1 µm. (b) A schematics of femtosecond pump-probe spectroscopy system.
Fig. 2. (a) Extinction spectrum for the AuNDs. The spectrum for the pump beams is also shown as a red curve. (b) Transient differential extinction spectra for the AuNDs at several delay times. Solid and dashed curves correspond to the data in the pump//probe and pump⊥probe configurations, respectively. (c) Calculated spectra for $\partial Ext.(\lambda )/\partial {\varepsilon _1}$ (black solid curve) along with transient differential extinction spectra at 50 fs (red dashed curve).
A schematics of the pump-probe spectroscopy system is shown in Fig. 1(b) [28,29]. Femtosecond optical pulses from a Ti: sapphire regenerative amplifier (center wavelength: 800 nm, pulse width: 100 fs, and pulse energy: 0.5 mJ at a 1 kHz repetition rate.) were used as the light source for the pump-probe spectroscopy. The optical pulses were divided in two by using a beam splitter. The wavelength of one portion was converted into the second harmonic at 400 nm. Concurrently, the wavelength was down-converted by an optical parametric amplifier and used for the pump pulses. The center wavelength was tuned to ${\lambda _{pump}}$ = 1100 nm, which was close to the peak of $Ext.(\lambda )$ [Fig. 2(a)]. The pulse width, compressed with a prism compressor, was 40 fs. The infrared optical pulse can be down-converted directly from the fundamental optical pulses at 800nm. However, an infrared optical pulse with a broader spectral width, which is more suitable for the shorter pulse width, was obtained with the present two-step approach. The other portion of the optical pulses was converted into a white light continuum by the self-phase modulation effect in a 3 mm-fused silica substrate and then used as probe pulses. The chirp of the optical pulses was compensated by using a grating compressor before irradiation of the samples.
The pump and probe pulses were loosely focused with an off-axis parabolic mirror and spatially overlapped onto the sample surfaces. The incidence of the probe beams was normal to the surface of the disks. The pump and probe beams were noncollinearly propagated, and the angle between the pump and probe beams was approximately 5°. The diameters of the pump and probe beams on the sample surfaces were 100 and 40 µm, respectively. The fluence of the pump beam on the samples was Fpump∼1.2 mJ/cm2.
A retroreflector was set in the optical path of the pump beams. The position of the retroreflector was controlled to tune the delay time, t, between the pump and probe lights. The polarization direction for the pump beams was rotated with a half-wave plate. The probe beams for the extinction spectra were spectrally resolved and detected by using a multichannel InGaAs photodiodes with fast electronics. These beams were sequentially recorded shot by shot by with and without irradiation of the pump beams. The time-resolved differential extinction spectra $\Delta Ext.(\lambda )$ at the probe wavelength $\lambda$ were obtained by subtracting the former from the latter. The measurements were performed in two optical configurations: either with the polarization of the pump beams parallel and perpendicular to that of the probe beams, or with pump//probe and pump⊥probe configurations.
The transient change due to cross-phase modulation (XPM) for the substrates was contaminated in the measured spectra. The intense pump light induces a change in the refractive index of the substrates, which resulted in the phase modulations in the probe optical fields [30]. The LSP-enhanced optical fields might also induce an XPM effect. However, the phase modulation due to the latter was found to be negligibly small as stated below. The transient extinction signal due to the AuNDs and the signal due to the XPM were regarded as being independent of each other. The time-resolved spectra were measured for the samples, and successively for the bare substrate. The transient spectra were later displayed by subtracting the latter from the former.
When the light-matter interaction length is much shorter than the light wavelength, the pump-induced phase change $\Delta \phi $ can be approximated to be as,
where ${n_2}$, $\lambda $, $l$, and ${I_{pump}}$ are the nonlinear refractive index of the substrate, probe light wavelength, light-matter interaction length inside the substrate and the intensity of the pump beams, respectively. In the present case related to the LSP, the field enhancement factor, ${\eta _{LSP}}$, must be multiplied by the intensity of the pump light. According to a numerical simulation carried out via the Finite difference time-domain (FDTD) method, as discussed later, the maximum LSP enhancement factor was determined to be ${\eta _{LSP}}$∼20. Hence, the pump light intensity in the vicinity of the Au nanorods was calculated to be ${I_{pump}}\sim {\eta _{LSP}} \cdot {F_{pump}}/\Delta {T_{pump}}$=20×1.2 mJ/cm2/40 fs=600 GW/cm2. On the other hand, the LSP-enhanced fields spread out over a distance of l∼10 nm, and this is an appropriate measure of the light-matter interaction length. The phase change induced by the LSP enhanced optical field was estimated to be as small as $\Delta {\phi _{LSP}}$ ∼0.056 rad at $\lambda $ ∼1100 nm by using the parameter ${n_2}$ = 0.82×10−16 m2/W reported in the literature [31].3. Results and discussion
Figure 2(a) shows the linear extinction spectrum for the AuNDs. Figure 2(b) shows $\Delta Ext.(\lambda )$ for the AuNDs at several different delay times. The solid and dashed curves show data measured in the pump//probe and pump⊥probe configurations, respectively. In both optical geometries, $\Delta Ext.(\lambda )$ consists of a positive change in the wavelength range of $\lambda > {\lambda _{LSP}}$ and a negative change for $\lambda < {\lambda _{LSP}}$. The spectral shape of $\Delta Ext.(\lambda )$ did not substantially change as t increased. The spectral shape is explained from the viewpoint of the redshift of the linear extinction spectrum as reported in the previous studies [17,18].
According to these works, the transient spectra are associated with optically induced changes in the dielectric function $\varepsilon (\lambda )$, as shown by Eq. (2).
Figure 2(c) presents $\partial Ext.(\lambda )/\partial {\varepsilon _1}(\lambda )$ calculated using Eq. (2) along with $\Delta Ext.(\lambda )$ at a delay time of 50 fs. In this calculation, $\partial Ext.(\lambda )/\partial \lambda $ was obtained by numerically differentiating the data in Fig. 2(a). $\partial {\varepsilon _1}(\lambda )/\partial \lambda $ was calculated by using data reported in the literature [33]. The calculated spectra matched well with the experimental curves. It was concluded that the observed transient differential extinction spectrum can be mostly attributed to optically induced changes in $\Delta {\varepsilon _1}({\lambda ,t} )$.
Figure 3 shows the intensity of $\Delta Ext.(\lambda )$ against the delay time t at $\lambda $=1040 and 1130 nm. The former and latter probe wavelengths correspond to the peak and valley for $\Delta Ext.(\lambda )$ in Fig. 2(b). The data for the pump//probe and pump⊥probe configurations are shown in Figs. 3(a) and 3(b), respectively. The temporal profiles are reproduced by summation of an exponential decay and constant component, as shown by Eq. (4).
Fig. 3. (a) Temporal profiles for the transient differential extinction of the AuNDs at 1030 and 1140 nm in the pump//probe configuration. Red solid curves are fit to Eq. (3) and the blue and green dashed curves are the contributions of the exponential and constant components in Eq. (3), respectively. (b) The same data as (a) but for the pump⊥probe configuration. (c) The peak and valley position for the transient differential extinction spectra against delay time. Open circles and filled triangles show the data for the pump//probe and pump⊥probe configurations, respectively.
According to previous studies, LSP-enhanced fields can promote intraband transitions, in which the electrons below the Fermi level EF are excited to the unoccupied levels above EF [14,17,19,21]. The excited electrons form nonequilibrium distributions immediately after the intraband transitions due to the LSP-enhanced fields. These nonthermal electrons are redistributed into Fermi-Dirac distributions at higher temperatures through electron-electron scattering within a time scale of 100 fs. The thermal electrons further exchanged energy with the lattice, and the electron temperature is cooled within a time scale of few picoseconds. The change in the distribution of the conducting electrons results in a change in the joint density of states after the interband transition and subsequent changes in the dielectric function of Au. In reference to previous studies, the exponential and constant components in Eq. (4) were attributed to the distributions of the nonthermal and thermal electrons.
Figure 3(c) shows the positions of the peak and valley in $\Delta Ext.(\lambda )$ against t. Both of the datasets obtained for the pump//probe and pump⊥probe configurations are displayed. The peak position was redshifted, and the valley position was blueshifted at $t$< 40 fs. The positions did not substantially change with longer delay times. The separation between the peak and valley gradually increased on the time scale of the temporal resolution of the present spectrometer. No substantial difference between the data in the two configurations was observed. The plasmon decay time was estimated from the inverse of the width of $\Delta Ext.(\lambda )$ to be ${\tau _{LSP}}$∼7 fs. The 40 fs-temporal resolution for the present spectrometer may be insufficient to resolve the formation of nonthermal electron distributions. However, the present results show that the intraband transition did not occur instantaneously upon photoexcitation, rather, the intraband transitions occurred during the plasmon decay.
To summarize the experimental results discussed so far, the amplitude of $\Delta Ext.(\lambda )$ was found to be dependent on the configuration of the pump and probe light polarization, while the temporal profile was found to be almost independent of these factors. These observations probably indicate that the formation and the redistribution of nonthermal electrons did not occur over the entire AuNDs, but that this rather occurred locally inside the AuNDs themselves.
The field distributions inside the AuNDs were calculated by using the FDTD method in order to address the dichroism in $\Delta Ext.(\lambda )$ (Fig. 4). Here, the xy-plane is the plane of the disk, and the z-axis is set to the direction of the height. The azimuthal angle ϕ is defined with respect to the x-axis. The x-polarized pump light at 1100 nm propagates along the z-axis. The top and bottom faces of the disk correspond to z=0 and 30 nm, respectively. The cross-sectional view of the electric field density $1/2\varepsilon {E^2}$ is displayed at z=15 nm in Fig. 4(a). The line profiles along the x-axis at y=0 nm and the y-axis at x=0 nm are also shown in Figs. 4(b) and 4(c), respectively.
Fig. 4. (a) Calculated distribution of (a) the electric field density $1/2\varepsilon {E^2}$ against the x-polarized 1100 nm-pump light. The line profiles along the x-axis at y=0 and along the y-axis at x=0 nm are also shown in (b) and (c), respectively. (d) Calculated scattering spectrum for the AuNDs with a homogeneous distribution of the dielectric function. (e) Calculated differential scattering spectra for x- (red) and y-polarizations (black curves). Inset shows a schematic of the distribution of the dielectric function for the AuNDs used for calculating the differential scattering spectrum. The dielectric function in the shaded area is as$\varepsilon + \Delta {\varepsilon _1}$, and $\varepsilon $ in the other area.
The electric field is densely distributed at the dielectric/metal boundary regions outside of the AuND. The electric field density is the highest in the vicinities of the points (±R, 0), and then it becomes lower as ϕ becomes larger in the region 0< ϕ <π/2. On the other hand, the field density at the inner surface of the AuND is higher in the edge region on the vicinities of the points (0, ±R). The electric field density near (0, ±R) just inside the Au side is approximately 60 times higher than that near (±R, 0). The electric field near (0, ±R) can excite hot electrons more intensely than in other regions.
This feature is slightly different from the case of metal nanospheres, where the electric field is found to be uniformly distributed and the intensity is independent of the positions inside the nanospheres [34]. An inhomogeneous electric field distribution is observed due to the y-dependent oscillation distance of the conducting electrons inside the nanodisks.
The electric field oscillates nearly along the polarization direction of the pump light. In the present case, the electric field oscillates along the x-axis. In the central region around y∼0, the electric field oscillated almost across the central region of the disk. The oscillation of the electric field suffers from absorption losses inside the metal. The longer oscillation distance results in higher absorption losses. As |y| becomes larger and approaches |y|=R, the electric field oscillates at shorter distances. The electric field avoids the absorption due to the metal, and a stronger field remains in the edge regions on the vicinities of the points (0, ±R).
Next, the differential spectrum was calculated using the following procedure. First, the distribution of the dielectric function was set in reference to the optical field distribution in Fig. 4(a). This was set to $\varepsilon ^{\prime}(\lambda )= \varepsilon (\lambda )+ \Delta {\varepsilon _1}$ in the circular segmental areas shaded in Fig. 4(e), while it was $\varepsilon (\lambda )$ in the other regions. Here, $\varepsilon (\lambda )$ is the dielectric function of pristine Au, and $\Delta {\varepsilon _1}$ is the optically induced change in the real part of it. The shaded areas correspond to the regions in which the intense optical fields are distributed.
Second, the scattering spectra ${\sigma _x}(\lambda )$ and ${\sigma _y}(\lambda )$ were calculated for the x- and y-polarized light, respectively. The former and latter correspond to the transient spectra in the pump//probe and pump⊥probe configurations, respectively. The scattering spectrum for the AuNDs ${\sigma _0}(\lambda )$ was also calculated, for which the dielectric function $\varepsilon (\lambda )$ was uniform in the whole area [Fig. 4(d)]. Finally, the spectral derivative $\partial {\sigma _x}(\lambda )/\partial {\varepsilon _1} = [{{\sigma_x}(\lambda )- {\sigma_0}(\lambda )} ]/\Delta {\varepsilon _1}$ and$\partial {\sigma _y}(\lambda )/\partial {\varepsilon _1} = [{{\sigma_y}(\lambda )- {\sigma_0}(\lambda )} ]/\Delta {\varepsilon _1}$ were calculated [Fig. 4(e].
We reproduced the spectral peak ${\lambda _{LSP}}$ due to the LSP in the calculated spectrum ${\sigma _0}(\lambda )$. Both $\partial {\sigma _x}(\lambda )/\partial {\varepsilon _1}$ and $\partial {\sigma _y}(\lambda )/\partial {\varepsilon _1}$ exhibit a positive change in $\lambda > {\lambda _{LSP}}$ and a negative change in $\lambda < {\lambda _{LSP}}$. Additionally, the magnitude of the change is higher in $\partial {\sigma _x}(\lambda )/\partial {\varepsilon _1}$ than in $\partial {\sigma _y}(\lambda )/\partial {\varepsilon _1}$. These observations for the numerical data qualitatively reproduced the spectroscopic behaviors of the polarization dependence in $\Delta Ext.({\lambda ,t} )$.
Finally, we discuss the persistent time for the inhomogeneous distribution of the hot electrons. The thermal diffusion time for the inhomogeneous hot electrons is expressed using an electron heat capacity constant ${\gamma _e}$, the electron temperature ${T_e}$, and the electron energy diffusion constant ${\kappa _e}$ as shown Eq. (5) [26].
Here, d is the typical length scale for the spatial distribution of the hot electrons. From the spatial distribution of the electric field in Fig. 4(a), the diameter of the disk 2R=350 nm is a good estimate for d in the present case. According to the literature, the electron heat capacity constant and the electron energy diffusion constant are ${\gamma _e}$=68 mJ/m3·K2 and ${\kappa _e}$=318 W/K·m, respectively [26,35]. A ${\tau _{ETD}}$ of ∼800 fs was obtained by using these parameters. Here, room temperature, ${T_e}$ = 300 K, was used for the electron temperature. Part of the nonthermal electrons are probably converted into thermal electrons after electron-electron scattering, resulting in the electrons being heated. In practice, the electron temperature is higher than room temperature, and ${\tau _{ETD}}$ is longer than the approximation of ∼800 fs.The thermal diffusion might also occur in the thickness direction of the AuNDs. The thermal diffusion time along the direction can be estimated by using d∼H=30 nm in place of 2R=350nm, and it is as short as ∼6 fs, which is much faster than the temporal resolution of the present spectrometer. The contribution of the thermal diffusion along the direction is negligibly small.
The present experimental results demonstrated that the dichroism in $\Delta Ext.$ is persistent over ${\tau _d}$ ∼200 fs, which is the decay time for the nonthermal electrons. This time is much faster than ${\tau _{ETD}}$. The nonthermal electron distribution disappeared before the electron distribution was spread out entirely over the nanodisks. As a consequence, a significant depolarization of $\Delta Ext.({\lambda ,t} )$ was not observed during ${\tau _d}$ due to the redistribution of the nonthermal electrons.
4. Conclusions
This paper presents femtosecond pump-probe spectroscopy for AuNDs at the LSP resonance. The size of the AuNDs is comparable to the LSP resonance wavelength, and it is much larger than those in the most of the previous studies. The measured time-resolved differential extinction spectra were attributed to changes in the dielectric function of Au due to plasmon-induced hot electrons. The temporal profiles obtained for the differential extinction spectra were decomposed into three components, namely, plasmon-induced formations of nonthermal electrons, redistribution of the nonthermal electrons to thermal electrons through electro-electron scattering, and long-lived thermal electrons. The temporal behaviors obeyed the mechanism led by the quasi-static model for the small-particle limit in the previous study.
On the other hand, the polarization dependencies of the transient spectra were newly observed in the present study, and they were not explained in the framework of the previous model. Although the linear optical response of the AuNDs was found to be isotropic in the plane of the disk surface, the transient spectra exhibited dichroism with respect to the pump light polarization within the femtosecond time scale. This is because of an inhomogeneous electric field distribution inside the AuNDs, and the local formation and relaxation of hot electrons inside the disks.
Funding
Japan Society for the Promotion of Science (26286059); Adaptable and Seamless Technology Transfer Program through Target-Driven R and D (AS2720000X).
Disclosures
The authors declare no conflicts of interest.
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