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Abstract
Transparent Conducting Oxides (TCOs) exhibit a large and ultrafast intensity-dependent refractive index in their Epsilon-Near-Zero (ENZ) spectral region, which depends dramatically on the material properties and measurement arrangement conditions. Therefore, attempts to optimize the nonlinear response of ENZ TCOs usually involve extensive nonlinear optical measurements. In this work, we show that significant experimental work can be avoided by carrying out an analysis of the material’s linear optical response. The analysis accounts for the impact of thickness-dependent material parameters on the absorption and field intensity enhancement under different measurement conditions and estimates the incidence angle required for achieving the maximum nonlinear response for a given TCO film. We perform measurements of angle-dependent and intensity-dependent nonlinear transmittance for Indium-Zirconium Oxide (IZrO) thin films with different thicknesses and demonstrate a good agreement between the experiment and theory. Our results also indicate that the film thickness and the excitation angle of incidence can be adjusted simultaneously to optimize the nonlinear optical response, allowing a flexible design of TCO-based highly nonlinear optical devices.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Materials with vanishingly small permittivity, typically referred to as Epsilon-Near-Zero (ENZ) materials, have demonstrated great potential for controlling the properties of light at the nanoscale [1–4]. They exhibit a strong light-matter interaction within the near-zero permittivity spectral region, which leads to fascinating linear and nonlinear optical phenomena such as directional emission [5], energy squeezing [6], perfect optical absorption [7], frequency conversion [8–10], and large and ultrafast intensity-dependent refractive index [11–15].
Near-zero permittivity media can be realized using a variety of material structures, including metal-dielectric multilayered stacks [16], all-dielectric structures [17], plasmonic waveguides [18], and transparent conducting oxides (TCOs) [14,19,20]. Among these material platforms, TCOs such as Indium-Tin Oxide and Aluminum-Zinc Oxide have attracted growing attention because of their straightforward fabrication, large optical damage threshold, and tunable optical properties. TCOs are n-type semiconductors with a high concentration of free electrons occupying their nonparabolic conduction band [21,22]. The complex permittivity of these materials in the near-infrared (NIR) region of the spectrum follows the Drude model [23], enabling their ENZ properties to be tailored by controlling parameters such as the carrier concentration and effective electron mass during the fabrication process [19,20,24,25]. Furthermore, TCOs are essential for developing ENZ-plasmonic nanostructures; these engineered structures exhibit remarkable optical responses due to a strong coupling between the resonance of the plasmonic nanoantenna and the ENZ mode of the ultrathin TCO substrate [26,27].
Recent studies have reported outstanding nonlinear optical phenomena in TCO-based ENZ thin films [8,10,11]. Among them, one of the most studied is intensity-dependent refraction [11–15], which is a hot-electron-driven nonlinear process with a subpicosecond response time and potential for applications involving all-optical switching and signal processing. This nonlinear process is enabled by the power absorbed by free electrons in the material [28,29], which is proportional to the intensity of the excitation field. In turn, the field intensity in the material depends strongly on various material properties [19] (e.g. the absorption coefficient and the film thickness) and on the experimental conditions [11] (e.g. excitation field polarization and angle of incidence). Despite the substantial research on this phenomenon and its underlying physics, a study elucidating the interplay between the material properties and the experimental parameters for enabling an optimum optical nonlinear response has not been presented. Indeed, developing such an understanding is of great importance to reduce the need for intensive experimentation over a wide range of materials and experimental parameters.
In this work, we study experimentally the intensity-dependent transmittance of Indium-Zirconium Oxide (IZrO) thin films in the ENZ spectral region, taking into account the contribution of material and experimental parameters related to the optical absorption in the samples, which include the film thickness, absorption coefficient, optical intensity, and angle of incidence. We elucidate the interplay between these parameters by analyzing the angle-dependent and intensity-dependent absorption and field intensity enhancement (FIE) for the fabricated IZrO films and evaluating their combined effect on the nonlinear phenomenon using a figure of merit (FoM), which is a modified version of the FoM proposed in Ref. [28]. The results reveal that the trend of variation of the FoM with the intensity and angle of incidence for each IZrO sample closely resembles that of the measured nonlinear transmittance. Such calculations only require the film’s linear permittivity and thickness and provide a useful tool for predicting the measurement conditions corresponding to the maximum nonlinear response. The findings of this work demonstrate that the thickness and measurement parameters can be adjusted to tailor the optical response of TCOs in the ENZ region, which could be useful for developing ENZ photonics applications in the telecommunication wavelength range.
2. Linear optical properties of IZrO films
The IZrO thin films are fabricated on glass substrates (Fig. 1(a)) following the procedure described in Refs. [30,31]. The samples have a low surface roughness with an average root-mean-squared of $0.3$ nm (Fig. 1(b)), ensuring the films’ good quality. Three samples with different thicknesses, as studied in our previous work [31] are further analyzed here. For each sample, the free electron density, $N_e$, is obtained from the Hall effect measurements, and the complex permittivity spectrum is characterized by spectroscopic ellipsometry (SE) measurements (see Methods). Figure 1(c) shows the permittivity spectra of the samples as solid lines and Table 1 reports the measured values of $N_e$. Note that the films show a very similar near-zero permittivity spectral region for all the samples despite their different thicknesses.
Fig. 1. (a) schematic of the studied structures composed of an IZrO film deposited on top of a glass substrate under oblique illumination with TM-polarized laser pulses. (b) AFM image of the IZrO film showing a flat surface with an RMS surface roughness of 0.3 nm. (c) wavelength-dependent complex permittivity of the IZrO samples retrieved from SE measurements (solid lines) and the results of fitting the permittivity data to the Drude model (dashed lines).
Table 1. Material parameters and optical properties of the IZrO films evaluated at ${\rm \lambda} _{ZC}$.
The linear optical properties of the IZrO films in the NIR region of the spectrum are well described by the Drude model for the complex permittivity
where $\varepsilon _{\infty }$ is the high-frequency permittivity, $\omega$ is the angular frequency, and $\omega _p$ and $\gamma$ are respectively the plasma angular frequency and damping constant, defined as andHere, $\varepsilon _0$ is the free-space permittivity, $e$ is the elementary charge, and $N_e$, $m^*$, and $\mu _\mathrm{opt}$ are the free-electron density, effective mass, and optical mobility, respectively. As we will discuss ahead, some of these parameters play important roles in defining the strength of the intensity-dependent refraction of TCO-based ENZ materials. Thus, we first obtain their values based on the linear permittivity measurements. We obtain the Drude parameters of the material, $\varepsilon _{\infty }$, $\omega _p$, and $\gamma$, by fitting the experimental permittivity to Eq. (1). This fit is shown in Fig. 1(c) as the dashed lines. Then, we extract the $m^*$ and $\mu _\mathrm{opt}$ by using the Drude parameters and measured values of $N_e$ in Eqs. (2) and (3). These results are listed in Table 1 for the fabricated samples.
From Eqs. (1)–(3) one can obtain an expression for the wavelength at which the real part of the permittivity crosses zero
This wavelength is typically referred to as the zero-crossing wavelength and defines the central region of the ENZ spectrum. Note that $\lambda _\mathrm{ZC}$ is close to 1680 nm for all the samples, which enables us to directly compare the optical properties (linear and nonlinear) of the different samples. At $\lambda _\mathrm{ZC}$, the material’s permittivity is simply given by its imaginary part; i.e., $\varepsilon (\lambda _\mathrm{ZC})= \varepsilon _\mathrm{ZC}=\mathrm{Im}(\varepsilon _\mathrm{ZC})$. This parameter can be approximated as [19]
3. Optimizing the nonlinear optical response
The intensity-dependent refraction in TCO-based ENZ materials is a hot-electron effect, whereby free electrons undergo intraband energy transitions after being heated by an intense laser field. It is a complex process that depends on several parameters. Particularly, it depends strongly on the change of the average mass of the heated electrons, $m^*_\mathrm{avg}$, resulting from the nonparabolic conduction band of the material [13]. Furthermore, it depends critically on two other material parameters, namely $\varepsilon _{\infty }$ and $\mathrm{Im}(\varepsilon _\mathrm{ZC})$. On the one hand, the steeper dispersion relation in the ENZ spectral region caused by larger values of $\varepsilon _{\infty }$ results in larger intensity-dependent refraction because a small optically-induced change in the Drude parameters can produce a large change in the permittivity [19,28]. On the other hand, the role of $\mathrm{Im}(\varepsilon _\mathrm{ZC})$ is twofold. It is directly related to the optical absorption coefficient through $\alpha = 2 \pi \mathrm{Im}(n_\mathrm{ZC})/ \lambda _\mathrm{ZC}$, where $\mathrm{Im}(n_\mathrm{ZC}) = \sqrt {\mathrm{Im}(\varepsilon _\mathrm{ZC})/2}$ is the imaginary part of the refractive index. Thus, a large $\mathrm{Im}(\varepsilon _\mathrm{ZC})$ increases the nonlinearity of the material as electrons can be heated more efficiently due to the large optical absorption [19,28]. However, an increase in $\mathrm{Im}(\varepsilon _\mathrm{ZC})$ reduces the FIE developed in the ENZ material for oblique illumination because the electric field component normal to the interface is proportional to $1/\varepsilon _\mathrm{ZC}$ [3]. Thus, a small $\mathrm{Im}(\varepsilon _\mathrm{ZC})$ value is preferred for achieving a larger FIE.
Recently, Secondo, Khurgin, and Kinsey [28] proposed an FoM to assess the performance of the intensity-dependent refraction of this type of material. This FoM can be cast as the product of three factors: the first factor, $F_1 = (1-R) \alpha d$, characterizes the capacity of the material to absorb energy for heating the electrons; the second factor, $F_2=(1/m^*_\mathrm{avg})(\textrm {d} m^*/\textrm {d} E)$, with $E$ being the electronic energy, accounts for the degree of the nonparabolicity of the conduction band normalized to the average electron mass; and the third factor, $F_3=(1/N_e)(\textrm {d} n/\textrm {d} m^*_\mathrm{avg})$, accounts for the change in refractive index due to a given change in $m^*_\mathrm{avg}$ normalized to the carrier density and is proportional to the slope of the dispersion curve evaluated at $\lambda _\mathrm{ZC}$ [28].
The aforementioned FoM is useful for assessing the relative strength of the intensity-dependent refraction in different materials. However, it is not suitable for assessing the strength of the nonlinearity under different experimental conditions because: (i) the effect of the FIE is not included, and (ii) the parameter $F_1$ underestimates the amount of absorbed energy since it does not account for the reflections within the TCO film. Moreover, absorption of TM-polarized light in the ENZ spectral region varies significantly with the incidence angle [31], which cannot be explained by $F_1$. Therefore, we adopt a modified version of this FoM, defining it as
where $A\!=\!1\!-\!R\!-\!T$, with $T$ and $R$ being the transmittance and reflectance, respectively. The value of $A$ is calculated numerically using the transfer matrix method (TMM), thus providing a more accurate estimation of the absorption compared to $F_1$. The second term represents the averaged field intensity enhancement in the material, $\mathrm{FIE}_\mathrm{avg}$, with $\boldsymbol{E}_0$ being the incident electric field and $\boldsymbol{E}_\mathrm{avg}$ being the space-averaged electric field inside the material. Both $A$ and $\mathrm{FIE}_\mathrm{avg}$ depend on the various material and experimental parameters, such as the incidence angle $\theta _\mathrm{i}$, polarization state, the film thickness $d$, and the imaginary permittivity $\mathrm{Im}(\varepsilon _\mathrm{ZC})$, thus making this FoM representative of such experimental conditions.Among the factors composing Eq. (6), $A$ and $\mathrm{FIE}_\mathrm{avg}$ depend significantly on $\theta _\mathrm{i}$. Figures 2(a) and 2(b) show the angle-dependent values of these two factors obtained from TMM calculations using the linear permittivity data. The results show that both $A$ and $\mathrm{FIE}_\mathrm{avg}$ reach their maximum value at a specific $\theta _\mathrm{i}$, which shifts to smaller values for the thicker IZrO films. This is due to an increase in reflection at higher $\theta _\mathrm{i}$ originating from a simultaneous increase in $d$ and a decrease in $\mathrm{Im}(\varepsilon _\mathrm{ZC})$, in agreement with the analytical expression proposed in Ref. [34]. In comparison, $F_2$ and $F_3$ are mainly affected by the intrinsic properties of the TCO films. For our IZrO films, these factors are calculated using the $N_e$ and $m^*$ listed in Table 1 taking the value ${\textrm {d} m^*/\textrm {d} E=0.215\,\mathrm{(1/eV)}}$ [19], and calculating $\textrm {d} n/\textrm {d} m^*_\mathrm{avg}$ for 0.25% redshift in $\omega _p$. This particular value of redshift is chosen to approximate the FoM for low optical intensity ($\sim \!8\,\mathrm{GW/cm^2}$). The product $F_2 F_3\,\mathrm{(m^3/kgeV)}$ for the two thickest IZrO films is $1.9\times 10^4$, while that of the 36 nm sample is $\mathrm{2}.4\times 10^4$. The larger value obtained for the thinnest IZrO film originates from its smaller $m^*$ and steeper dispersion, i.e., larger $\varepsilon _{\infty }$, compared to the thicker films. Finally, Fig. 2(c) illustrates the calculated FoM, showing a dependence on $\theta _\mathrm{i}$ due to the angle-dependent $A$ and $\mathrm{FIE}_\mathrm{avg}$.
Fig. 2. The angle-dependent (a) absorption, (b) field intensity enhancement, and (c) FoM calculated for the IZrO films based on their linear optical properties. The factor $F_3$ of the FoM is calculated for a 0.25% redshift in plasma frequency ($\omega _p$), which corresponds to an intensity of $\sim \!8\,\mathrm{GW/cm^2}$.
We perform experiments to assess the suitability of Eq. (6) to describe both the experimental and the material parameters that lead to optimum intensity-dependent refraction of the fabricated IZrO samples. In particular, we carry out measurements of the intensity-dependent change in the transmittance, given by ${\Delta T = T(I)-T(I_0)}$, where $I_0$ is a sufficiently low input intensity to grant a linear optical response, and $I= I_0 + \Delta I$ with $\Delta I$ being a change in the input intensity. The intensity-dependent reflectance, $R(I)$, is also measured to obtain the nonlinear change in the absorptance as ${A(I) = 1-T(I)-R(I)}$. The range of intensities used in our experiments is $8\,\mathrm{GW/cm^2}\!<\!I\!<\!340\,\mathrm{GW/cm^2}$. Note that Eq. (6) was proposed as an FoM for the intensity-dependent refraction by considering parameters affecting the material’s refractive index, $\Delta n$ [28]. However, it has been shown that a $\Delta I\!>\!0$ leads to $\mathrm{Re}(\Delta n)\!>\!0$ and $\mathrm{Im}(\Delta n)\!<\!0$ [11], both of which lead to $\Delta T\!>\!0$. Thus, Eq. (6) is also appropriate to describe the trend of $\Delta T$. Figure 3 shows the nonlinear response of the IZrO samples as a function of $I$ and $\theta _\mathrm{i}$. It should be mentioned that the maximum $\Delta T$ obtained for each sample occurs at a wavelength shorter than the value of $\lambda _\mathrm{ZC}$ retrieved from SE analyses (see Table 1). This is attributed to the nonuniform optical properties along the thickness of the films, as we have recently reported using linear and nonlinear optical measurements [31]. The results in Figs. 3(a), 3(c), and 3(e) show the typical intensity-dependent transmission response of TCOs [11], starting with a steep linear increase in $\Delta T$ as a function of $I$ up to $30\,\mathrm{GW/cm^2}$ followed by a gradual saturation with increasing $I$, which is consistent with the saturable absorption of the IZrO films shown in Figs. 3(b), 3(d), and 3(f).
Fig. 3. Nonlinear change in the transmittance and absorptance of the fabricated IZrO films measured at different incidence angles. The measurements for the 36 nm, 59 nm, and 114 nm films are performed at wavelengths of 1630 nm, 1635 nm, and 1650 nm, respectively.
It also appears from Fig. 3 that for the thicker IZrO films at low intensities increasing $\theta _\mathrm{i}$ from $50^{\circ }$ to $60^{\circ }$ does not improve the $\Delta T$ response, indicating an optimal $\theta _\mathrm{i}$ for each IZrO film at which the nonlinear response is the largest. This can be observed in Fig. 4(a)–4(c) where $\Delta T$ is plotted for $I\!\le \!30\,\mathrm{GW/cm^2}$. According to Fig. 4(a), $\Delta T$ for the 36 nm IZrO film at $I=8\,\mathrm{GW/cm^2}$ scales up linearly with $\theta _\mathrm{i}$. However, an increase in the film’s thickness to 59 nm and 114 nm shifts the $\theta _\mathrm{i}$ corresponding to the maximum $\Delta T$ to $\sim \!50^{\circ }$ (Figs. 4(b) and 4(c)). These results are in agreement with those obtained for the angle-dependent FoM (Fig. 2(c)). Figures 4(a)–4(c) also show that the rate of change in $\Delta T\!-\!\theta _\mathrm{i}$ has a dependence on $I$ as well; increasing $I$ from $8\,\mathrm{GW/cm^2}$ to $30\,\mathrm{GW/cm^2}$ leads to a steeper slope in the $\Delta T\!-\!\theta _\mathrm{i}$ curves and shifts the $\theta _\mathrm{i}$ of maximum $\Delta T$ to larger values (see also Fig. 3). The slope increases further for $I\!>\!30\,\mathrm{GW/cm^2}$ and reaches a constant value at higher intensities due to the saturation of the nonlinear response (see Supplement 1).
Fig. 4. Angle and intensity-dependent $\Delta T$ measured for the (a) 36 nm, (b) 59 nm, and (c) 114 nm IZrO films. (d-f) show the variations in the FoM versus $\theta _\mathrm{i}$ approximated for a range of redshift in $\omega _p$ and increase in $\gamma$; the FoM values are scaled by the corresponding intensities used for the measurements in (a-c).
Finally, we calculate $A$, $\mathrm{FIE}_\mathrm{avg}$, and $\textrm {d} n/\textrm {d} m^*_\mathrm{avg}$ for different redshifts in $\omega _p$ and increments of the value of $\gamma$ corresponding to those intensities in Figs. 4(a)–4(c) to understand the angle and intensity dependence of the $\Delta T$ response. We observe an increase in $\theta _\mathrm{i}$ for the maximum $A$ and $\mathrm{FIE}_\mathrm{avg}$ as a function of the redshift in $\omega _p$. Using these values and $\textrm {d} m^*/\textrm {d} E=0.215\,\mathrm{(1/eV)}$, the FoM is calculated and scaled by the intensities used for the measurements; note that taking $I$ as the scaling factor can only be justified in the low-intensity region, where $\Delta T$ has a linear dependence on $I$. The obtained FoM values are presented in Figs. 4(d)–4(f), showing a dependence on $\theta _\mathrm{i}$ and $I$ closely resembling that of the measured $\Delta T$. These results confirm that Eq. (6) can be also used to evaluate the effects of the material parameters and the experimental conditions on the nonlinear optical response of TCOs at higher optical intensities.
4. Discussion and conclusions
As mentioned before, TCOs offer great flexibility in tailoring their optical properties in the ENZ spectral region. For instance, $\lambda _\mathrm{ZC}$ can be tuned by controlling $N_e$ [24,25], and $\mathrm{Im}(\varepsilon _\mathrm{ZC})$ can be reduced by increasing the crystal quality of the TCO films [19]. Our results show that adjusting the thickness is another approach to manipulating the optical properties of TCOs. However, using this strategy requires taking into consideration the effect of competing parameters and their variation under different measurement conditions. In this regard, we show that any attempts to improve the nonlinear response of TCOs by increasing their thickness wouldn’t yield the expected results unless the measurements are performed at the optimal angle of incidence ($\theta _\mathrm{max}$) where the maximum figure of merit ($\mathrm{FoM_{max}}$) is obtained. This can be understood by comparing the nonlinear response of the fabricated IZrO films. Table 2 shows an increase in $\Delta T$ with $d$ when the measurements are performed at the respective $\theta _\mathrm{max}$ of each IZrO film. In this scenario, the $\mathrm{FoM_{max}}$ and therefore the $\Delta T$ increases as a function of $d$ due to an increase in the factor $F_1$. On the other hand, when measured at $\theta _\mathrm{i}$ much smaller than $\theta _\mathrm{max}$ the FoM of the 114nm IZrO drops significantly, leading to a $\Delta T$ value smaller than that measured for the 36 nm film at its $\theta _\mathrm{max}$.
Table 2. Angle-dependent values of $\Delta T$ at $I\!=\!15\,{\rm GW}/{\textrm{cm}}^2$ measured for the IZrO films.
Another experimental parameter affecting the intensity-dependent refraction is the pulse duration, $\tau$. In the case of Fourier-transform-limited pulse and for $\tau \!\ll \!\tau _{el}$, with $\tau _{el}$ being the electron relaxation time, increasing $\tau$ would increase the amount of energy absorbed by free electrons and consequently lead to a larger nonlinear response [28]. Our nonlinear measurements involved pulses with a comparable duration (see Table 2); therefore, the effect of variation in $\tau$ on $\Delta T$ of the IZrO films would be negligible. It is worth mentioning that, for a given TCO, a change in $\tau$ would have an impact on the magnitude of $\Delta T$ rather than its angle and intensity dependence. Therefore, calculations of the FoM can be used to estimate the measurement conditions for an optimal nonlinear response regardless of the value of $\tau$.
In conclusion, the effects of thickness, incidence angle, and optical intensity on the nonlinear optical response of the IZrO thin films are investigated in the near-zero permittivity spectral region. The measurements reveal that an increase in thickness (intensity) decreases (increases) the incidence angle at which the maximum $\Delta T$ is achieved. These results are also supported by the calculations and originate from the variations in the amount of absorbed energy and FIE as a function of the material properties and measurement parameters. The good agreement between the experiment and calculation allows us to conclude that by estimating the FoM one can account for the impact of several important parameters on the nonlinear optical response of the TCO thin films and estimate accurately the incidence angle for inducing the strongest intraband nonlinearities. Such calculations only require the linear permittivity data and thickness of the TCO films to be known and lead to avoiding extensive experimental work when attempting to optimize the nonlinear response. Our results also show that tailoring the thickness and incidence angle simultaneously enables controlling the optical response of TCO materials, which could help to design efficient devices for ENZ-photonic applications.
5. Methods
The IZrO films were deposited on glass substrates at room temperature using a Pulsed Laser Deposition system (Twente Solid State Technologies B.V.) following the procedure of Refs. [30,31]. The complex permittivity of the films was retrieved from Spectroscopic Ellipsometry (SE) measurements using a J.A. Woollam M-2000UI ellipsometer. Surface roughness was characterized by Atomic Force Microscopy (AFM) using a Bruker Icon Dimension in Tapping mode. Electrical properties at room temperature were measured using the Hall effect technique in the van der Pauw configuration. The laser source used for the nonlinear optical experiments consisted of a pulsed Ti:Sapphire regenerative amplifier (Coherent Legend) followed by an optical parametric amplifier (Coherent Opera). The Idler part of the output (tuning range of $1600\,\mathrm{nm}\!<\!\lambda \!<\!2100\,\mathrm{nm}$; repetition rate of 5 kHz) was used for these measurements. The wavelength-dependent pulse duration was measured using an autocorrelator and varied between 63 fs - 68 fs over the studied wavelength range.
Funding
Consejo Nacional de Ciencia y Tecnología (286150, 971961); Federico Baur Endowed Chair in Nanotechnology; Optical Sciences Group, University of Twente.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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