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Abstract
Transparent conducting oxides (TCOs) have recently attracted much attention in the fields of optics and photonics because of their outstanding linear and nonlinear optical response in the near-zero permittivity spectral region. The optical response of these materials can be further enhanced by optimizing the material properties through fabrication. In particular, two important TCO parameters affecting the strength of the optical interactions are the optical mobility and effective mass of free electrons. In this work, we fabricate epitaxial, highly-textured, and polycrystalline tin-doped indium oxide (ITO) films to experimentally study the effect of the crystal quality on the optical mobility and effective electron mass, and on the optical properties of the material in the near-zero permittivity spectral region. Compared to polycrystalline ITO, we report an increase in the optical mobility from 38 to 67 cm2/Vs and a reduction in the effective mass from 0.3 m0 to 0.24 m0 in oxygen-deficient epitaxially grown ITO films. The improved material parameters reduces the imaginary part of the permittivity (from 0.56 to 0.42) and results in a steeper material dispersion for the high-crystal-quality ITO films. Based on these results, an analysis of the figure of merit for nonlinear refraction reveals that epi- and tex-ITO films can achieve a stronger nonlinear response than poly-ITO samples. Our results show that controlling the free-electron optical mobility and effective mass through crystal quality along with tuning the free-electron density allows for tailoring simultaneously the near-zero-permittivity wavelength and the optical losses at that wavelength, which is of utmost importance for the ENZ photonics applications.
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1. Introduction
Materials with vanishingly small permittivity, typically known as Epsilon-Near-Zero (ENZ) materials, have gained an increased attention in the optics community over the past decade because they enable fascinating and technologically important optical effects, such as super-coupling, radiation-emission control, and giant ultrafast nonlinear response [1–4]. A class of materials with great potential for ENZ applications is transparent conducting oxides (TCOs), which includes materials such as tin-doped indium oxide (ITO) or aluminum-doped zinc oxide (AZO). TCOs offer several key advantages in the near infrared (NIR) region of the spectrum. One of them is the capacity to reliably and accurately adjust the near-zero permittivity spectral region through free-electron doping. Previous reports have demonstrated the possibility of adjusting the zero-crossing wavelength ($\lambda _{\textrm{ZC}}$) – i.e., the wavelength at which the real part of the permittivity crosses zero – over spectral ranges as broad as an octave [5–11]. Another advantage is the availability of well-established fabrication techniques to produce ultra-thin layers of these materials [12,13]. In particular, this is important for the development of ENZ-photonic structures based on thin-film and multilayer systems [3,14–16]. In addition, TCOs feature a high damage threshold, which is important for nonlinear optical applications [17–23].
Several optical phenomena have been explored recently in TCO-based ENZ materials offering promise for potential applications in optics and photonics in the NIR region of spectrum. These phenomena include perfect absorption of TM-polarized light [24], electric field enhancement [25], ultrafast nonlinear refraction [17–20], and optical frequency conversion [21–23]. Although these studies reported remarkable outcomes, the results can be further enhanced by tailoring the TCO material properties. For example, increasing the electron optical mobility results in a reduction of the imaginary part of the material permittivity in the near-zero permittivity region, which leads to reduced optical losses. A small optical loss is desirable for several applications in ENZ photonics, such as beam shaping [26], wave mixing [16], and harmonic generation [27], as well as applications based on near-zero-index photonics [3]. Moreover, a reduction in the electron effective mass is expected to enhance the nonlinear optical response based on the intensity-dependent refractive index [28]. Therefore, it is of great importance to generate knowledge and develop methods for the optimization of the optical properties of TCO-based ENZ media by controlling material parameters through fabrication.
Significant efforts have been made on improving the optoelectrical properties of these materials [5,6,29,30], and recent studies have started to optimize the optical properties in the near-zero permittivity region as well [7–11]. Several studies have shown control of the free-electron density through deposition or annealing of TCO films under different pressure, temperature or doping content [5–13]. Also, investigations on the electron mobility ($\mu _e$) in samples of different crystal quality have reported electrical mobilities as large as 100 $\textrm {cm}^2/\textrm{Vs}$ for epitaxially grown ITO films [29,31–34], which is more than three times larger than the values typically observed in polycrystalline samples [5,35,36]. Finally, studies on the electron effective mass have shed light on the relationship between this parameter and the free-electron density [5,6] as well as the potential impact of the crystal quality [29]. Despite these considerable progress in material optimization, the effect of crystal quality on the optical properties of TCOs relevant to the applications in ENZ photonics has not been studied, leaving this material parameter open to be explored.
In this work, we study the optical properties of highly-crystalline ITO films and assess two relevant material parameters: the free-electron optical mobility and effective mass. For this purpose, epitaxial and textured ITO films are grown on Yttria-stabilized Zirconia (YSZ) and Magnesium Oxide (MgO) single crystal substrates, respectively. The epitaxial ITO (epi-ITO) is single-crystalline, whereas the textured film (tex-ITO) is polycrystalline but has a preferential crystallographic orientation. A polycrystalline (poly-ITO) film, with random crystallographic orientation, is also fabricated on a glass substrate to evaluate the variation of the material properties with crystallinity. We report a considerable increase in the optical mobility from $\sim \!38\,\textrm{cm}^2/Vs$ for poly-ITO to $\sim \!59\,\textrm{cm}^2/Vs$ and $\sim \!67\,\textrm{cm}^2/Vs$ for tex-ITO and epi-ITO, respectively. Furthermore, small values for the electron effective mass ranging between $0.19\,m_0$ to $0.3\,m_0$, where $m_0$ is the electron rest mass, are obtained. We perform spectroscopic ellipsometry measurements and show that the high-crystal-quality samples exhibit smaller optical losses and a steeper dispersion around $\lambda _{\textrm{ZC}}$. More importantly, controlling the crystal quality of ITO enables us to tailor its optical properties at a particular $\lambda _{\textrm{ZC}}$, which is of significance for applications in ENZ photonics. Finally, using the obtained results we asses the figure of merit for intraband nonlinearities to be 116% (51%) larger for epi-ITO (tex-ITO) than for the poly-ITO sample, which implies an improved nonlinear optical performance with crystal quality. Our findings could contribute to developing TCOs with improved linear and nonlinear optical response in the near-zero permittivity spectral region.
2. ITO material properties
TCOs are degenerately doped wide-bandgap semiconductors. As such, they are highly transparent in the visible and NIR spectral range, while exhibiting metallic-like properties due to the high free-electron density (typically in the order of $10^{20}\,\textrm{cm}^{-3}$). These materials have been widely used as transparent electrodes in optoelectronic devices such as solar cells, flat-panel displays, and light emitting diodes [5,6,37]. Recently, they have also been explored for potential applications in linear and nonlinear optics in the near-zero permittivity spectral region [2,17,24,38]. The linear optical properties of TCOs are dictated by their complex permittivity function, which can be well approximated by the Drude model [39]
Clearly, the parameters $N$, $\mu _{\textrm{opt}}$, $m^*$, and $\epsilon _{\infty }$ play important roles in defining the optical properties of TCOs. Within the context of ENZ photonics, the two most important optical properties are the zero-crossing wavelength, $\lambda _{\textrm{ZC}}$, and the imaginary part of the permittivity evaluated at that wavelength. Each application would require a specific $\lambda _{\textrm{ZC}}$, whereas a small $\textrm{Im}(\epsilon )$, i.e. a low optical loss, is beneficial for many applications. Using Eqs. (2b) and (3), evaluating at $\lambda _{\textrm{ZC}}$, i.e. $\omega =\omega _p/\sqrt {\epsilon _{\mathrm{\infty}}}$, and using the approximation $\gamma \ll \omega _p$, the imaginary part of the permittivity in terms of material parameters can be expressed as
Applications based on ENZ materials, require controlling the material parameters to achieve a particular $\lambda _{\textrm{ZC}}$, which implies that N needs to be kept fixed. This, further, implies constant values for $\epsilon _{\infty }$ and $m^*$ [5,6]. Therefore, at a given $\lambda _{\textrm{ZC}}$, the only approach to reducing $\textrm{Im}(\epsilon )$ is to increase $\mu _{\textrm{opt}}$. Consequently, fabrication strategies that significantly increase the optical mobility would be another way of reducing the optical loss. This can be realized through fabricating highly-crystalline TCO films with a low defect concentration. It is also possible to obtain a smaller effective mass in TCO materials with a high crystal quality [29].
Fig. 1. XRD patterns of ITO films deposited in oxygen atmosphere and annealed under a flow of $\textrm{Ar}\!+\!{\textrm H}_2$. (a) and (b) represent, respectively, the $2\theta -\theta$ and $\phi$ scans of epitaxial ITO on YSZ (001) annealed at $400\,\rm ^{\circ }C$. (c) $2\theta -\theta$ scan of textured ITO on MgO (001) annealed at $400\,\rm ^{\circ }C$. (d) $2\theta -\theta$ scan of polycrystalline ITO on glass annealed at $450\,\rm ^{\circ }C$.
3. ITO film fabrication and characterization
ITO films were deposited by pulsed laser deposition in $\textrm{O}_2$ (0.2 mbar) and $\textrm{O}_2+Ar$ (0.004 mbar, $\textrm{1}\%\,\textrm{O}_2$) atmospheres using a KrF excimer laser ($\lambda =248\,\textrm{nm}$) at a repetition rate of 2 Hz and a fluence of 1.2 $\textrm{J}/cm^2$. The target used for depositions had a low Sn doping (1.6 wt%), which could help to further improve the material properties of interest by decreasing the number of ionized impurity scattering centers. The epi-ITO (tex-ITO) and poly-ITO samples were deposited at 700 $\rm ^{\circ }C$ and 500 $\rm ^{\circ }C$, respectively. The films fabricated in oxygen atmosphere were annealed at 400 $\rm ^{\circ }C$ and 450 $\rm ^{\circ }C$ for 30 min under a flow of $\textrm{Ar}\!+\!\textrm{H}_2$ ($\textrm{4}\,\%\textrm{H}_2$) to increase the free-electron density by creating oxygen vacancies. X-ray diffraction (XRD) patterns were collected using an X’Pert-PANalytical analyzer to evaluate the crystal quality. Hall effect measurements in the van der Pauw configuration were performed to evaluate N and $\mu _e$ at room temperature. Surface roughness and grain morphology were characterized by Atomic Force Microscopy (AFM) using a Bruker Icon Dimension in Tapping mode. The linear permittivity was measured by a J.A.Woollam M-2000UI spectroscopic ellipsometer.
Figure 1(a) shows the $2\theta - \theta$ XRD scan of epi-ITO film including the characteristic peaks of ITO (200), (400), (600), and (800) along with the (200) and (400) of YSZ. The absence of diffraction peaks belonging to the crystallographic orientations other than [001] confirms the highly-oriented nature of the deposited film. Figure 1(b) illustrates a 360$^\circ$ scan ($\phi$-scan) around (222) ITO on YSZ (001) in which a four-fold rotational symmetry confirms the heteroepitaxial growth [31]. The diffraction pattern of tex-ITO, shown in Fig. 1(c), is composed of peaks originating from the (222) and (444) planes of ITO and (200) and (400) of MgO. Clearly, there is no epitaxial bonding between ITO and MgO because of a large unit cell mismatch ($\sim \!16.7\%$). However, appearance of the ITO peaks only with [111] orientation indicates a highly-textured growth. Moreover, the full width at half maximum values of out-of-plane rocking curves for (400) ITO on YSZ and (222) ITO on MgO are, respectively, 0.225$^\circ$ and 0.255$^\circ$, indicating a small crystal tilting in those samples. The XRD pattern of ITO deposited on glass shown in Fig. 1(d) is composed of peaks with different crystallographic orientations, revealing the polycrystalline nature of the film.
Fig. 2. AFM images of the samples deposited in oxygen atmosphere and annealed under a flow of $\textrm{Ar}\!+\!\textrm{H}_2$. (a) epitaxial ITO on YSZ (001) annealed at $400\,\rm ^{\circ }C$, (b) textured ITO on MgO (001) annealed at $400\,\rm ^{\circ }C$, and (c) polycrystalline ITO on glass annealed at $450\,\rm ^{\circ }C$.
The AFM images of the annealed ITO films are presented in Fig. 2. A columnar growth with a root-mean-squared (RMS) surface roughness value of $7.6\rm \,nm$ is observed for epi-ITO sample (Fig. 2(a)), in agreement with the literature [31]. As can be seen from Fig. 2(b), the majority of the grains in tex-ITO film have a triangular morphology consistent with [111]-oriented growth [32,40]. In addition, the textured film is flat with $\textrm{RMS}=0.9\,nm$. In contrast, the poly-ITO sample has a RMS value of 39 nm caused by the overgrown grains because of annealing at a higher temperature.
Figure 3 illustrates the linear permittivity (solid lines) of the ITO films retrieved from spectroscopic ellipsometry measurements, applying a combination of the Gaussian (or Tauc-Lorentz) and Drude oscillators. We obtained an excellent fit to the ellipsometry data characterized by a mean-squared-error (MSE) ranging from 0.9 to 6.9 (see the Supplement 1). The permittivity data in the NIR region are fitted to the Drude model (dashed lines) using a nonlinear least-squares minimization method to extract $\omega _p$, $\gamma$, and $\epsilon _{\infty }$. An extrapolation using the fit parameters is also performed to find the zero-crossing wavelengths. The optical parameters from the fit and the electrical properties collected from the Hall effect measurement are summarized in Tables 1 and 2. Table 1 also includes $m^*$ for the fabricated ITO films, which are obtained from Eq. (3a) using the value of N obtained through the Hall effect measurements and the extracted values of $\omega _p$. These values of $m^*$ along with the extracted values of $\gamma$ are then used to find $\mu _{\textrm{opt}}$ from Eq. (3b).
Fig. 3. Wavelength-dependent permittivity of the films deposited in oxygen atmosphere (as-deposited and annealed under a flow of $\textrm{Ar}+\textrm{H}_2$). (a) epitaxial ITO on YSZ (001), (b) textured ITO on MgO (001), and (c) polycrystalline on glass. (d) variation in the imaginary part of the permittivity with thickness in annealed textured ITO films.
Table 1. Optical parameters of epitaxial, textured, and polycrystalline ITO films deposited in oxygen atmosphere. $^\circ$ As deposited. $^\dagger$ Annealed under a flow of $\textrm{Ar}+\textrm{H}_2$. The maximum error bars for $m^*$ and ${\mu} _{opt}$ are $\pm 1.6\%$ and $\pm 0.6\%$, respectively. Th error bars for the other parameters are negligible.
Table 2. The optical loss obtained for textured ITO fabricated in different atmospheres; $(\dagger )$ deposited in oxygen atmosphere and annealed under a flow of $\rm {Ar+\textrm{H}_2}$, $(\circ )$ as-deposited in $\rm {O_2+Ar}$ atmosphere. Results are compared to the reported loss values for ITO films with different levels of crystallinity. The maximum error bar for ${\mu} _{opt}$ is $\pm 1\%$. Th error bars for the other parameters are negligible.
4. Effect of crystallinity on optical properties
Optical mobility: Depending on N and the crystal quality, values in the range of $38-87\rm \, cm^2/Vs$ are obtained for $\mu _{\textrm{opt}}$ of the ITO films. According to Table 1, annealing the as-deposited films leads to a reduction in $\mu _{\textrm{opt}}$ and an increase in $\mu _{\textrm{e}}$. It is important to mention that $\mu _{\textrm{opt}}$ is typically attributed to the in-grain mobility and is primarily limited by ionized impurities scattering (for films with such N) [5]. Therefore, the increase of N, as potential scattering centers, is leading to the drop of $\mu _{\textrm{opt}}$ after the annealing. Conversely, $\mu _{\textrm{e}}$ extracted from the Hall effect measurements includes the effect of the grain boundaries [41], which may lead to the difference between $\mu _{\textrm{opt}}$ and $\mu _{\textrm{e}}$. The post-deposition annealing is likely to improve the crystal quality and/or grain size of ITO films leading to the suppressed grain boundary scattering and the improved $\mu _{\textrm{e}}$. On the other hand, the effect of crystal quality on free-electron mobility becomes clear by comparing this parameter for the three different annealed samples. For similar values of N, both $\mu _{\textrm{e}}$ and $\mu _{\textrm{opt}}$ of the annealed epi- and tex-ITO films are notably higher than those measured for poly-ITO (see Table 1). The higher $\mu _{\textrm{opt}}$ of epi- and tex-ITO, in turn, leads to an $\textrm{Im}(\epsilon )$ of 0.42, which is 25% smaller than 0.56 obtained for poly-ITO. According to Eq. (4), a simultaneous increase in $\mu _{\textrm{opt}}$ and $m^*$ favors the reduction in the optical loss. However, $m^*$ of epi- and tex-ITO films is smaller than that of poly-ITO, allowing us to conclude that the reduced loss of the high-crystal-quality films results only from the increased $\mu _{\textrm{opt}}$ [42,43]. The crystallinity can also be improved by increasing the film thickness, which is attributed to an increased crystallite size [43,44]. This, in turn, could further reduce $\textrm{Im}(\epsilon )$. We investigated the effect of thickness on the optical loss by fabricating tex-ITO films with different thicknesses. As summarized in Table 2, increasing the thickness of tex-ITO, deposited in oxygen atmosphere, from 324 nm to 488 nm reduces the imaginary part of the permittivity from 0.4 to 0.365 ($\sim \!10\%$). This is caused by the higher $\mu _{\textrm{opt}}$ for the thicker sample, since $\lambda _{\textrm{ZC}}$ is similar for both samples. Compared to the 100 nm-thick sample, however, the reduced $\textrm{Im}(\epsilon )$ of the thicker films results from their higher N (5.1$\times 10^{20}\,\rm {cm^{-3}}$). We also fabricated tex-ITO films in $\textrm{O}_2+Ar$ atmospheric gas, which exhibit a reduction in the optical loss with thickness as well. Noteworthy, $\mu _{\textrm{opt}}$ ($\textrm{Im}(\epsilon )$) for those films is higher (smaller) than that obtained for samples deposited in oxygen atmosphere, suggesting more control over the optical properties of our high-crystal-quality films through deposition atmosphere.
We note that $\textrm{Im}(\epsilon )$ values comparable to those of our highly-crystalline films have already been reported for polycrystalline ITO [5,17,22] (see Table 2). However, it is important to mention that those ITO films had N values in the order of $10^{21}\,\textrm{cm}^{-3}$, resulting in a $\lambda _{\textrm{ZC}}$ much shorter than that of the ITO samples fabricated in this work. According to Eq. (5), $\textrm{Im}(\epsilon )$ linearly scales up with $\lambda _{\textrm{ZC}}$. Therefore, we calculate the optical loss coefficient $\alpha =2\pi \textrm {Im}(\sqrt {\epsilon })/\lambda$, where $\lambda$ is the free-space wavelength, to take into account the impact of $\lambda _{\textrm{ZC}}$. As included in Table 2, smaller $\alpha$ values are obtained for tex-ITO films. In principle, the loss coefficient of our highly-crystalline ITO can be further reduced by fabricating films with higher N.
Free-electron effective mass: It is well-known that the nonparabolic conduction band of TCOs results in a dependence of $m^*$ on N. Values of $m^*$ in the range of $0.19-0.45\,m_0$ corresponding to N in the range of $10^{17}-10^{21}\,\textrm{cm}^{-3}$ have been reported for undoped and Sn-doped In2O3 [5,29,34,45,46]. We also observe such dependence of $m^*$ on N. Compared to the case of as-deposited films, the higher N of the annealed samples leads to their larger $m^*$. However, for similar N values, $m^*$ of epi-ITO (tex-ITO) is $\sim \!20\%$ ($\sim \!10\%$) smaller than that of poly-ITO (see Table 1). This could arise from different levels of crystallinity of the investigated films. The smaller $m^*$ of our highly-crystalline films, in turn, results in their shorter $\lambda _{\textrm{ZC}}$ compared to poly-ITO. Using the values in Table 1, it is found that the term $\sqrt {\epsilon _\infty /N}$ in Eq. (5) is practically the same for all ITO films (maximum deviation of 10%). Therefore, the difference in $\lambda _{\textrm{ZC}}$ of the annealed ITO films is mainly caused by the variations in $m^*$ with crystallinity.
Interestingly, the $m^*$ values obtained in this work are smaller than those reported for ITO at similar $\lambda _{\textrm{ZC}}$ [5,29,47,48] (see Table 3), which could be caused by the relatively low Sn content (1.6 wt%) of our ITO films. Density Functional Theory calculations of ITO band structure revealed an increased flattening of the conduction band at higher Sn dopings [49]. As a consequence of this flattening, the nonparabolicity of the conduction band increases, leading to an increase in $m^*$ [50]. This implies that both N and the extrinsic dopant source influence $m^*$. Here, we employed annealing to generate the excess electrons required for a blue-shift in $\lambda _{\textrm{ZC}}$ by creating oxygen vacancies, rather than increasing the extrinsic doping level. Consequently, the low Sn doping and the high crystal quality of epi- and tex-ITO samples lead to their smaller $m^*$ in contrast to the values reported for ITO films with different levels of crystal quality [5,29,47,48].
Table 3. The free-electron effective mass for ITO films with different levels of doping and crystal quality.
Material dispersion: As listed in Table 1, the $\mathrm{\epsilon} _{\infty }$ values for the annealed epi- and tex-ITO samples are larger than for poly-ITO, which suggests an enhancement of the interband resonance strength in the UV spectral region as a result of the improved crystal quality. This results in a steeper material dispersion around $\lambda _{\textrm{ZC}}$ and is beneficial for intensity-dependent refractive index [28], as explained in the discussion section.
5. Discussion
Reducing the optical loss of TCOs can offer more promise for both linear and nonlinear optical applications. For instance, it is shown that the thickness required for perfect absorption of TM-polarized light is directly related to $\textrm{Im}(\epsilon )$ [24]. Therefore, low-loss TCOs can achieve perfect absorption for deep-subwavelength thicknesses, enabling more compact devices. Low-loss ENZ media also offer an approach to realizing near-zero index (NZI) media [3]. Recent studies on the optical response of NZI materials have led to demonstration of interesting optical phenomena such as controlled emission [51], super-coupling [52,53], and strong nonlinear interactions [16,54]. For example, a small refractive index enhances the conversion efficiency of nonlinear processes such as wave mixing and harmonic generation through a phase-mismatch free interaction [16,54]. In addition, the conversion efficiency would further increase in low-loss ENZ materials due to a larger field enhancement [27]. The field intensity enhancement factor (FIE) relates the longitudinal component of the fields inside ($E_{\textrm{ENZ}}$) and outside ($E_i$) the ENZ medium by FIE=$|E_{\textrm{ENZ}}/E_i |^2=|\epsilon _i/\epsilon _{\textrm{ENZ}}|^2$, where $\epsilon _{\textrm{ENZ}}$ and $\epsilon _i$ are the permittivities of the ENZ and outside media, respectively. From the results obtained here, around two-fold enhancement of the FIE factor is expected for epi-ITO compared to the poly-ITO sample. Although modest, this improvement could be very significant in films patterned with resonant plasmonic structures, increasing further the large FIE factors (in the order of 50) typically obtained for such structures [55].
The variations in material properties with crystal quality would also influence the intensity-dependent refractive index. This phenomenon is particularly strong in TCO-based ENZ media [17,18] and originates from an intraband excitation of the free electrons in the nonparabolic conduction band [56]. Excited electrons redistribute and occupy higher energy states associated with larger effective masses, leading to an increase in the average effective mass, $m^*_{\textrm{avg}}$. In turn, this causes a red-shift of $\omega _p$ (see Eq. (3a)) and increases the refractive index. To quantify the efficiency of this complex nonlinear mechanism, a figure of merit (FOM) has been recently suggested [28], being equal to the product of three factors: $F_1=(1/m^*_{\textrm{avg}})(dm^*/dE)$, which represents the normalized change in $m^*_{\textrm{avg}}$ for a fixed amount of absorbed energy and depends on the level of nonparabolicity of the conduction band; $F_2=(1/N)(dn/dm^*_{\textrm{avg}})$, which quantifies the variation in the refractive index with changes in $m^*_{\textrm{avg}}$ normalized to N; and $F_3=(1-R)\alpha d$, which is proportional to the energy absorbed by the medium. In order to calculate the FOM for our samples, we take the $m^*_{\textrm{avg}}$ and N values for the annealed samples in Table 1, approximate $dm^*/dE$ through the procedure proposed in Ref. [28], and use the ITO band structure in Ref. [57]. The quantity $dn/dm^*_{\textrm{avg}}$ is calculated using the Drude model and assuming a 1% increase in $m^*_{\textrm{avg}}$ as a result of the nonlinear process. As shown in Figs. 4(a) and (b), the factors $F_1$ ad $F_2$ are larger for epi- and tex-ITO than for poly-ITO, which despite the smaller $F_3$ factor (Figs. 4(c)), lead to significantly larger values of FOM for those samples (Figs. 4(d)). This is caused by the smaller $m^*_{\textrm{avg}}$ and larger $\epsilon _{\mathrm{\infty}}$ of the high-crystal-quality samples. A reduction in $m^*_{\textrm{avg}}$ leads to a larger modification to the free-electron distribution due to an intraband excitation, whereas an increase in $\epsilon _{\mathrm{\infty}}$ enhances sensitivity of the refractive index to the changes of $m^*_{\textrm{avg}}$ as a result of steeper dispersion around $\lambda _{\textrm{ZC}}$. Note that the FOM obtained in this work is at least $\sim \! 25$ times larger than the value reported for ITO in Ref. [28]. Such a difference is expected because we calculated $F_3$ at $\lambda _{\textrm{ZC}}$, where the absorption is higher, as opposed to Ref. [28] where this factor is taken for a wavelength shorter than $\lambda _{\textrm{ZC}}$. For a fair comparison with respect to the results in Ref. [28], we calculated the FOM at a wavelength of 785 nm, which is much shorter than $\lambda _{\textrm{ZC}}$ and the results for the poly-ITO, tex-ITO, and epi-ITO are 92, 113, and 182, respectively. These values are at least 58% larger than the FOM reported for ITO in Ref. [28].
Fig. 4. Factors (a) $F_1$, (b) $F_2$, and (c) $F_3$ along with (d) the FOM calculated for ITO samples deposited in oxygen atmosphere an annealed under a flow of $\textrm{Ar}+\textrm{H}_2$.
6. Conclusion
ITO films with low doping content and different levels of crystallinity were fabricated and evaluated in terms of material parameters and linear optical properties at the zero-crossing wavelength. Our measurements revealed a significant increase in the optical mobility up to 76% in the epitaxial ITO film with respect to that of the polycrystalline sample. This increased optical mobility, in turn, resulted in a reduction of the imaginary part of the permittivity by 25% while maintaining the zero-crossing wavelength relatively constant in the range of 1740-1774, which represents a variation of $\sim$1.9% with respect to poly-ITO. Moreover, we observed further reduction in the optical loss as the film thickness is increased. A decrease in the effective mass with crystallinity was also obtained. Compared to polycrystalline ITO, the effective mass of the annealed epitaxial ITO sample was 20% smaller, which could be attributed to the higher crystal quality. We also discussed how the improved material properties of the highly-crystalline ITO films can help to enhance the optical interactions and quantitatively showed the figure of merit for intraband nonlinearities to be 116% larger for the epitaxial than for the polycrystalline films reported here.
Funding
CONACyT (Ciencia Básica) grant number 286150.
Acknowledgments
I.D.L. acknowledges the support of the Federico Baur Endowed Chair in Nanotechnology. We are also thankful to Prof. Nathaniel Kinsey for helpful discussion about the FOM for intraband nonlinearities.
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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