Optical Properties of LiNbO₂ Thin Films

Lithium niobate is a member of the transition-metal oxide compounds. At normal temperature and pressure, it crystallizes in a form of three-dimensional LiNbO₃. Lithium niobate can also crystallize in the layered LiNbO₂ (LNO) modification, although it is metastable under normal conditions [1-2]. By using the epitaxial growth techniques such as pulsed laser deposition, it is now possible to grow LiNbO₂ thins films without the need for rather restricted conditions such as a reducing atmosphere [3-6]. This compound is known to be several intriguing properties such as the superconductivity [7-10] in the delithiated phase at approximately 5 K and the high ionic conductivity suitable for potential applications as a battery material [11-13].

Although stoichiometric LNO is a “mother material” of the delithiated counterpart having compatibility of transparency and the superconductivity, the little is known about the optical response of this compound. LNO is a direct gap semiconductor [14-17]. The bandgap energies were evaluated to be ca. 2.0 eV both experimentally [18] and theoretically [14]. On the other hand, the nature of these optical transitions has not been clarified yet for this material. This is partly because the experimental information is based on a diffused reflectance spectrum taken on a specimen in powdered form [18]. In this case, the conversion to the dielectric functions is very difficult. Generally, to draw electronic-structure-related knowledge from measured data, analysis based on the standard critical point (SCP) modeling [19] has been adopted so far. According to the SCP theory, the lineshape is considered to reflect that of the spectral distribution in the joint density of states of the valence and conduction bands. The lineshape of the joint density of states strongly depends on the dimensionality and the Lynch index which corresponds to the number of negative components in the effective mass vectors. Gaining insight into the nature of the optical transitions is attributed to the determination of the parameters such as the dimensionality and the Lynch index. Such a parameterization has not been, however, done so far for this material. These knowledges are very important for e.g. device applications.

In this work, we study the optical properties of LNO thin films. We present dielectric function spectra deduced from transmission and reflectance spectra at room temperature (RT) between 1.5 and 6.5 eV. A method is also described for calculating the spectroscopic distribution of the dielectric function of LNO where the relevant models are connected with the electronic energy-band structures of the compound. For comparison, we also show ab-initio calculation results with the GW level under linear-response approximations.

2.1 Electronic energy-band structure of LiNbO₂

Several groups have reported the calculated results on the electronic-energy band structures in LNO [14-17]. We also executed local-density approximation (LDA) calculations by ourselves. The reproducible results are obtained as shown in Fig. 1. To ensure computing cost-effectiveness for regression analysis, we attempted to reproduce the measured dielectric functions with critical points (CP’s), the number of which is as small as possible. From the model-dielectric-function (MDF)-based regression analysis for the measured data, the energies of the CP’s were evaluated to be ca. 2.3, 3.2, 3.9, and 5.1 eV. These transition energies are shown in Fig. 1 with vertical arrows and are labeled as E₀, E₁, E₂, and E₃. It should be noted that, for the assignments of the vertical arrows in Fig. 1, we looked for the corresponding direct transitions lower in energy if we could not find a transition having the same energy with the experiment. In the analysis, we neglect the following aspects for simplicity: (1) transitions above the E₃ gap, (2) the contributions from indirect transitions due to their extremely weaker nature in their intensities, and (3) the spin-orbit effects and the related splitting of the bands (conventionally denoted as $\Delta$ in the literature).

2.2 Model dielectric functions

We performed the line-shape analysis using the MDF approach [20]. Here, we summarize dielectric functions as a form of a complex function. To obtain $\epsilon$₁($\omega$), one can take its real parts, while for $\epsilon$₂($\omega$), imaginary parts should be taken.

The E₀, E₁, E₂ CP’s may be of the three-dimensional (3D) M₁ type. The index i in the M_i notation corresponds to earlier-mentioned Lynch index. As shown in Fig. 2, the overall feature of the $\epsilon$₁($\omega$) spectrum is characterized with monotonically decreasing behavior concerning the photon energy, accompanied with optical anomalies at the CP’s. To reproduce this behavior, the assumption of M₁ type is more appropriate. Because the M₁ CP’s longitudinal effective mass is much larger than its transverse counterpart, one can treat these 3D M₁ CP’s as a two-dimensional (2D) minimum M₀. It is known that the equation in the approximation of the 2D M₀ CP gives a series of Wannier-type excitons if taking the excitonic effects into account [21]. In other words, one can neglect the one-electron contribution to the dielectric function in the case of the 3D M₁ CP. Due to its layered structure, LNO is expected to be a material where the electron-hole correlations are rather strong. Rough calculation on the binding energy of the Wannier-type exciton yielded in ca. 70 meV for LNO by using the reported reduced effective mass (0.57 m₀) and background dielectric constant of ca. 10.3 [14]. The notation m₀ means electron’s mass in rest. Besides, the excitonic effect could not be neglected in the case of 3D M₁ (2D M₀) CP for many elemental and compound semiconductors [20,22-25]. Thus, we can justifiably take this effect into account for the analysis. For e.g. the E₀ feature, the contribution of these excitons to $\epsilon$($\omega$) can be now written with Lorentzian lineshape as:

where A₀ is a constant corresponding to the strength parameter, E₀ is the energy of the CP, G₀ is the 2D binding energy of exciton. The value of $\Gamma$₀ is the broadening parameter. Because the ground-state exciton term occupies almost 95% of the total oscillator strength, we neglected the excited-state terms ($n\geq 2$) in the current analysis.

The E₃ peak is difficult to analyze as it does not correspond to a single, well defined CP. Thus, the E₃ structure has been characterized as a damped harmonic oscillator:

with $\chi$₃ = E/E₃, where A₃ is the strength parameter and $\Gamma$₃ is a dimensionless broadening parameter.

LNO thin films were grown on MgAl₂O₄ (111) substrates using pulsed laser deposition (PLD) method with KrF excimer laser. The growth was conducted in vacuum at RT. After deposition, the films were annealed in situ at substrate temperature of 800 ^oC under a chamber pressure of 0.1 mtorr set by continuous flow of Ar/H₂ gas. Films were capped by several-nanometer-thick alumina films using PLD at RT in vacuum for avoiding reactions with air. The description of detailed growth process can be found elsewhere [6]. The film thickness is approximately 150 nm.

Optical transmittance and reflectance were measured with an ultraviolet-visible spectrometer at RT. Then, we converted these spectral intensities to those of the complex dielectric functions [26]. Here, the refractive index of the substrate is assumed to be independent of the photon energy because the bandgap of MgAl₂O₄ is significantly wider than the maximum energy of the measurement range (i.e., 6.5 eV).

Structural information of LNO has been reported by several experimental groups, leading to the assignment of space group P6₃/mmc (No. 194). We performed ab-initio calculation using the plane-wave basis set PWscf package of Quantum ESPRESSO [27-28] to evaluate the electronic-energy-band structures. For the lattice constant of LiNbO₂, a=2.938 Åand c=10.596 Åwere used. The 3D Brillouin zone was integrated using a $7\times 7\times 2$ k-point grid. We have set the energy cutoff for the plane-wave basis to 30 Ry, and have used the Perdew–Burke–Ernzerhof (PBE) functional, which belongs to the class of generalized gradient approximation (GGA) functionals. In previous works, similar calculations have been made [14-15], and qualitatively coincided results could be obtained by ourselves. For the optical spectra calculations such as dielectric functions, we used Respack ab-initio package [29-30]. This package solves Bethe-Salpeter-like equations numerically under single-excitation configuration-interaction (SECI) treatment. By doing that, the interactions between electrons and holes are taken into account.

In Fig. 2, we plot the real ($\epsilon$₁) and imaginary ($\epsilon$₂) parts of dielectric function spectra $\epsilon$($\omega$) of LNO determined at RT. As seen in the figure, the experimental data reveal clear structures at the 2.2 to 2.4-eV region. This structure originates from transition at the E₀ edge. The structures appearing at the 3.0 to 3.4, 3.6 to 4.2, and 4.7 to 5.5-eV regions are due to the E₁, E₂, and E₃ transitions, respectively.

The model dielectric function (MDF) approach given in Sec. IIB was used to fit the experimental dispersion of $\epsilon$($\omega$) over the entire range of the measurements (0 to 6.5 eV). The parameters such as A₀ and A₁ are used as adjustable constants for the calculations of $\epsilon$₁($\omega$) and $\epsilon$₂($\omega$). As has been already mentioned in Sec. IIB, the $\epsilon$₁($\omega$) for LNO is, in overall, a monotonically decreasing function with upward convex in 1.5 to 5-eV range except for several optical anomalies such as E₀. This is very peculiar compared to those for many other elemental and compound semiconductors [20,22-25]. This is somehow reminiscent of that in $\alpha$-Sn [24]. In this case, very strong absorption band is present in the $\epsilon$₂ spectrum, giving rise to consistent explanation in terms of the 3D M₀ CP with very large transition strength. This is not the case for LNO. To reproduce the $\epsilon$₁ behavior, there seems no other solution except for positive adoption of the M₁-type CP even for the lowest energy gap because the difference between the $\epsilon$₁ values at both ends is larger in the case of 3D M₁ (2D M₀). Comparison of the behavior at both ends revealed us that the lineshape of the 2D M₀ MDF is known to drop significantly, while that of the 3D M₀ MDF remains relatively flat [20]. Therefore, the 2D M₀ MDF is more suitable to reproduce the overall $\epsilon$₁ tendency. The experimental data on $\epsilon$₁($\omega$) turned out to be still somehow larger than the model fit. To improve this fit, we then introduced a phenomenological term $\epsilon_{1\infty}$ in addition to $\epsilon$₁($\omega$) to account for the contributions from higher-lying energy gaps. This term $\epsilon_{1\infty}$ is assumed to be nondispersive.

The solid and dashed lines in Fig. 2 are obtained from the sum of Eqs. (1) and (2), and $\epsilon_{1\infty}$=2.0. The vertical arrows in the figure indicate the excitonic peak energies (E₀-G₀, E₁-G₁, E₂-G₂) and the positions of the CP (E₃). The best-fit parameters are listed in Table I. Due to the earlier-mentioned assumptions, we could not determine the excitonic binding energies such as G₀ experimentally. Individual contributions to the dielectric functions of the various energy gaps for LNO are shown in Figs. 3 and 4, respectively. They are obtained from Eq. (1) for the 2D-exciton contribution in the E₀, E₁, and E₂ regions, and from Eq. (2) for the E₃ gap contribution.

The sums of Eqs. (1) and (2) fit the main features of the $\epsilon$₁ and $\epsilon$₂ data, but the agreement with the experiment is rather qualitative. The difference probably could come from several sources, including (1) the Lorentz approximation for broadening (which is known to give too much absorption below direct band edges [31]), (2) the neglection for the excited-states excitons [20], or (3) the parabolic assumption of the CP’s [19].

As understood from the table, some of the best-fit parameters from the $\epsilon$₁ analysis are different from the $\epsilon$₂ counterparts. Being compliant with SCP theory’s policy, we took the average from these values, as shown in this figure. We believe that these averaged values should be material parameters of LNO. An increase in the number of MDF’s (CP’s) is expected to lead to an improved agreement with the experiment, even with the same (common) values on the parameters. However, it might be an excessive assumption to include an additional transition to the position where no peak or anomaly is observed. Indeed, this might be justified by the fact that calculated SECI spectra include at least five CP’s in the energy range (1.5 to 6.5 eV), as can be viewed from Fig. 5.

Neglecting contributions from the optical anomalies, overall $\epsilon$₁($\omega$) in the SECI spectrum is a monotonically decreasing function with upward convex in the 1.5 to 4-eV range. This feature is in good agreement with the experiment. Besides, the agreement on the E₀ energy is also good. According to the ab-initio calculation under the LDA level, the bandgap energy is evaluated to be approximately 1.5 eV. The calculated energy is significantly lower than that of experimental value (ca. 2.3 eV). As is often the case, it is well-known that the LDA level calculation tends to underestimate the bandgap energy. On the other hand, the ab-initio SECI calculation considers the electron-hole interaction (excitonic) effects in this material, leading to the improved agreement with the experiment. The spectral blue-shift from the LDA-level result is probably due to the so-called self-energy effect. Furthermore, the energetic positions of E₁ and E₃ CP’s are in reasonable agreement with the experiment.

We notice that the calculated spectra structures are sharper and more distinct than those in the measured one. The calculated $\epsilon$₁($\omega$) amplitude as defined by the difference of the $\epsilon$₁ value at 1.5 and 6.5 eV is also larger. These looser agreements are probably related to the same origin. The broadening parameter corresponding to the width of the Green functions used in the calculation is significantly smaller than that obtained in the measurement. The doublet splitting of the E₂ is observed in the SECI spectra. Experimentally, the double seems not resolved due to the broadening effect. Thus, our results should stimulate further work for this material both experimentally and theoretically.

Finally, we mention refractive-index dispersion in the transparency region. As seen in Fig. 2, the analysis results are in agreement with the experiment ($\epsilon$₁) over the entire range of photon energies. On the other hand, the agreement is looser. An increase in the parameter $\epsilon_{1\infty}$ from 2 to 3.5 improves the agreement in the transparency region. However, this does not give a satisfactory fit to the experimental data at higher photon energies (> 3.0 eV). Several research groups have reached a similar conclusion on GaP, AlSb, and ZnSe. The looser agreement is probably related to the earlier-mentioned Lorentz approximation for broadening [20,32-33]. Indeed, Pollak’s group has introduced a phenomenological linear cutoff to eliminate this problem giving too much absorption below direct band edges [31]. We did not attempt this here because the cutoff’s adoption forced us a numerical Kramers-Kronig conversion to the $\epsilon$₁ counterpart, thus making it impossible to analyze the experimental data within the regression analysis framework based on a conventional Levenberg–Marquardt algorithm.

In summary, we have reported the dielectric functions of LNO around the fundamental band gap E₀ (ca. 2.3 eV) and up to 6.5 eV for RT. The observed spectra reveal distinct structures at energies the E₀, E₁, E₂ and E₃ CP’s. These data are analyzed based on a simplified model of the interband transitions, including these transitions as the main dispersion mechanisms. The analyzed results are in qualitative agreement with the experiment. Many important material parameters of these transitions were determined. The comparison was made with the results of ab-initio calculations, taking the electron-hole correlation effects into account.

T.M. wishes to thank the financial support of the Ministry of Education, Culture, Sports, Science and Technology Grant No. KAKENHI-19K05303. This work was partially supported from No. KAKENHI-20K15169. All of the ab-initio calculations were performed in the Supercomputer Center at the Institute of Solid-State Physics, the University of Tokyo.

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