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Abstract
Pure YAG single crystal was optically characterized in the wide spectral range (from far IR to vacuum UV) by applying the universal dispersion model. Data obtained from a broad range of characterization instruments and methods was simultaneously processed using least-square method and the result were compared with literature findings. The universal dispersion model describes individual elementary electron and phonon excitations in materials as separate contributions. For the first time recorded, an asymmetric Voigt peak approximation was used for modeling the contribution of one-phonon absorption in crystalline material. The optical constants are presented both graphically and in detailed dispersion parameters sets.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Single crystal yttrium aluminum garnet (YAG, Y3Al5O12) is the one of the most important host medium used for laser system, which together with the emerging solid-state pumping sources, made possible construction of high-power and high-efficiency laser systems. When doped with rare-earth ions (Nd, Er, Yb, Ho, Tm), it forms efficient active medium used in a wide range of systems, from low-power laser oscillators to multi-kW industrial or research systems [1]. Beside thin-disks, slabs or rods, the single crystal YAG fibers, both pure and doped, showed great potential in a variety of application [2]. The combination of its very low internal stress, high hardness and high heat and chemical resistance, makes the YAG single crystal an interesting alternative to sapphire for a variety of optics [3,4]. In high-power/high-energy laser system, the YAG can be even more attractive due to lack of birefringence, while its laser-induced damage threshold value scores high [5–8]. Due to its high non-linearity, the generation of supercontinuum in this solid-state material was also investigated [9].
Fabrication of the large size YAG substrates with low absorption for the specific laser wavelengths is still a technological challenge. The mechanism of the optical absorption is incomplete and depends on a number of manufacturing and post-processing. So understanding the optical properties of the YAG single crystal over a wide spectral range is crucial for using this material up to its very limits.
The broad range optical constants of crystalline materials can be precisely determined using different optical instruments combined with advanced modeling. Following the above mentioned, the universal dispersion model (UDM) [10–12] is crucial for response function modeling, based on simultaneous experimental data processing, while a single physically realistic dispersion model is used. The UDM is a collection of elementary excitation models, fulfilling all tree fundamental conditions: time-reversal symmetry, Kramers-Kronig consistency and conformity with the sum rules.
2. Experimental data and data processing
The optical characterization presented in this paper is based on simultaneous processing of our experimental data (spectrophotometry and spectroscopic ellipsometry) and refractive index published by Zelmon et al. [13]. Our measurements were performed on four spectrophotometers: Bruker Vertex 70v, Bruker Vertex 80v, Perkin Elmer Lambda 1050 and McPherson VUVAS 1000 and two ellipsometers: Woollam IR-VASE and Horiba Jobin Yvon UVISEL (see Table 1). The experimental data were measured at room temperature (296–300 K) in the wide spectral range from far IR to vacuum UV. The experimental data enable us to precisely determine the spectral distribution of the phonon absorption in the IR and distribution of electronic excitations in UV regions. However, due to the systematic errors of these methods, it is not possible to determine the response function in the region of transparency with absolute accuracy as it is possible by minimal deviation method (MDM). Zelmon et al. measured a YAG prism by the MDM and tabulated the refractive index $n$ in his work. We assumed that YAG material of their prism is the same quality as our 2.555 mm thick (determined by a micrometer screw gauge) YAG plate – Czochralski grown in reducing atmosphere (Crytur [4]).
The data processing presented in this work is based on simultaneous fit of all measurements using equipments detailed in Table 1. For this purpose was used our internally developed optical characterization software newAD2 [15], enabling to perform heterogeneous data processing using least-square method (LSM) with Levenberg–Marquardt algorithm (LMA) [16] in the modified version, which automatically equalize the contributions of individual experimental data sets to the residual sum of squares [10,11]. This equalization allows the combination of individual data from different spectral ranges, obtained using different instruments, having distinct systematic errors at different spectral and angular densities of data points, while keeping sensitivity of the method to features in individual data sets. This modified LMA does not allow to define exactly the global quantity $\chi$ characterizing the quality of the fit for all the experimental data. For this reason why only the $\chi _i$ quantities defined for individual data sets are presented in Table 1. Note that the modified LMA provides all information about statistical uncertainties of the fit parameters and their correlations. For further details see Supplement 1 and out-final in Dataset 1 [17].
Table 1. List of the experimental and tabulated data with spectral ranges (including resolution) and angle of incidence (AOI). The fitted experimental data $R$, $T$ and $I_\textrm {s},I_\textrm {c},I_\textrm {n}$ represent the reflectance, transmittance and associated ellipsometric quantities [14], respectively. The $\chi _i$ represents the quality of fits for corresponding data.
3. Results
3.1 Region of transparency
The refractive index of the undoped YAG single crystal, according to Zelmon et al. [13], was measured with the absolute accuracy better than $2 \times 10^{-4}$. The authors did not provide more precise estimation of errors, thus, we assumed for fitting a uniform accuracy $1 \times 10^{-4}$ for all the spectral points plotted in Fig. 1. Fit of YAG’s refractive index data by the UDM is also presented in the same figure. The root mean square (RMS) value of the difference between the UDM and the experimental data was found to be $1.26\times 10^{-4}$. It is known that the refractive index data in the region of transparency can be modeled by empirical Sellmeier formula [18]:
Fig. 1. Spectral dependence of the refractive index $n$ of YAG crystal measured by Zelmon et al. [13] using the MDM. The curves represents fit of these data by universal dispersion model (UDM) and by harmonic oscillator model (HOM).
The empirical Sellmeier formula is equivalent to real part of harmonic oscillator model (HOM) response function. Thus, the two-term Sellmeier formula can be understood as two independent harmonic oscillators with two resonant energies. The HOM is derived on the basis of classical equations of motion for bound charged particles. The complex response function can be written using generalized functions, as follows
Fig. 2. Spectral dependencies of the complex dielectric function $\hat \varepsilon$, calculated by universal dispersion model (UDM) and harmonic oscillator model (HOM) fitting Zelmon et al. [13] experimental data.
Table 2. Dispersion parameters of HOM by fitting Zelmon et al. [13] experimental data, compared with the transition strengths determined from UDM.
Four dispersion parameters of HOM are introduced in Table 2, while the corresponding spectral dependence is plotted in Fig. 2. Since HOM approximates the spectral distribution of electron and phonon excitations using only two discrete excitations, it is clear that the dielectric response in the region of transparency is a worse approximation than the one calculated by UDM.
In our opinion, it is a coincidence that systematic errors better correspond to the less precise model. Within UDM the distribution of electron and phonon excitations, i. e. transition strength function $F(E)$, are known relatively well (details are published in the following sections).
Fig. 3. Spectral dependencies of ellipsometric quantities displayed by the complex pseudodielectric function $\langle \hat \varepsilon \rangle$ and the degree of polarization $P$.
The strength of all electron and phonon excitations calculated from UDM model as the sum of corresponding $N_x$ parameters can be compared to the transition strengths, predicted by HOM, from the fit of refractive index data (see Table 2). The differences between transition strengths are due to the asymmetric distribution of excitations described by $F(E)=E\varepsilon _{\mathrm {i}}(E)$ around the mean excitation energies $E_\textrm {el}$ and $E_\textrm {ph}$ (see Fig. 2). In addition, the actual electron density (transition strength) is much higher than determined from UDM, where the excitations of the core electrons have been neglected.
As mentioned earlier, the spectroscopic methods are not very sensitive to the absolute value of the refractive index or the extinction coefficient in the region of transparency. In the case of spectroscopic ellipsometry, this is mainly due to the uncertainty of the response from the sample surface. Figure 3 shows the ellipsometric data displayed using a pseudodielectric function $\langle \hat \varepsilon \rangle$ and the degree of polarization $P$. Because the measurement was performed on a relatively thick plate, the depolarization effects caused by the sample back side reflections had to be taken into account. A special algorithm truncating back side reflections implemented in newAD2 software was applied [14,15]. From Fig. 3 can be observed that the pseudodielectric function strongly depends on the angle of incidence and, in addition, contains jumps in the spectral points where the monochromators or photomultipliers are switched (for details see Supplement 1). Furthermore, the imaginary part of the pseudodielectric function, which in ideal case corresponds to the energy absorbed in the sample, is relatively high (0.2–0.3 at 6 eV). Later on, we will show that the real value of the imaginary part of the dielectric function is approximately $10^{-5}$. The high value of $\langle \varepsilon _{\mathrm {i}}\rangle$ can be explained by the non-ideal sample surface. Several surface effects can be accounted: surface roughness, adsorbed surface layer, polish-induced surface damage or naturally occurring surface effects associated with crystal termination (the surface dielectric response differs from bulk response, even for an ideal crystal surface, due to the broken symmetry).
Three approximations (theories) were tested for the description of surface and near-surface effects: Drude approximation (DA) [22,23], effective medium approximation (EMA) [24] and the Rayleigh-Rice theory (RRT) [25–28]. The fits of experimental data using all models (including combinations) were practically equivalent (see Supplement 1). Although the choice of the surface model does not affect the fit of the experimental data, its choice may be important to minimize systematic errors in dispersion model parameter determination. The resulting model, combining RRT and EMA, is believed to realistically describe the surface. The RRT was used in this combination because this theory truthfully describes the bonding conditions at rough interfaces. For this model, the roughness is described using power spectral density function (PSDF). Then, if the Gaussian PSDF is assumed, the roughness is parameterized by the RMS of heights $\sigma$ and the autocorrelation length (lateral dimension of roughness) $\tau$. The EMA together with the Bruggeman formula [29] is a suitable choice for the description of the surface layer, because it simply allows to parameterize the optical response of this layer using thickness $d_\textrm {o}$ and packing density parameter $p_\textrm{m}$. The parameters obtained using this model were as follows: $\sigma =1.49(14)$ nm, $\tau =10.5(12)$ nm, $d_\textrm {o}=3.6(1.7)$ nm and $p_{\textrm{m}}=0.91(6)$.
From AFM data it is visible a very smooth surface, while the Fourier analysis reveals spatial frequencies corresponding to $\sigma =0.100(5)$ nm and $\tau =8.5(1.0)$ nm (see Supplement 1). Discrepancy between $\sigma$ parameter determined by optical method and by AFM can be explained by tip-surface convolution effects which underestimate $\sigma$ (tip radius is about 50 nm), therefore, the agreement can be considered as good as possible.
3.2 Absorption on localized states and Urbach tail
The existence of imperfections in the crystal network causes weak scattering (absorption). From the structure point of view, this scattering can be interpreted as excitations of the localized electrons with energies lying inside the forbidden energy band. Distribution of the localized states depends on purity and grow technology of the specific crystal. Response function can be effectively described as the sum of Gaussian broadened discrete transitions. Four discrete transitions were sufficient to describe the YAG sample. Thus, the contribution to the dielectric function can be written as follows [12,30–33]
The proof data to demonstrate absorption at localized states is the UV-visible transmittance data shown in Fig. 4, where the Gaussian absorption bands are evident. The values of the dispersion parameters describing these absorption bands are introduced in Table 3.
Fig. 4. Spectral dependencies of the transmittance $T$ measured in the visible and UV spectral range and corresponding modeled imaginary part of dielectric function $\varepsilon _{\mathrm {i}}$. The separated UDM contributions of the absorption on localized states and Urbach tail are plotted by dash and dash–dot lines.
Fig. 5. Spectral dependence of modeled imaginary part of dielectric function $\varepsilon _{\mathrm {i}}$ in the region of the electron excitations. The separated UDM contributions of the absorption on localized states and Urbach tail are plotted by dash and dash–dot lines.
Figures 4 and 5 show that for energies higher than $6.3$ eV, the absorption in the YAG crystal increases exponentially. This exponential increase is associated with the well-known Urbach tail, which corresponds to the weak absorption below the band gap energy $E_\textrm {g}$. The Urbach tail can be explained as the transitions of the electrons from the localized valence states to the extended unoccupied conduction states and the transitions from the extended valence states to the localized unoccupied states [35]. In other worlds, the Urbach tail is the transition region between the absorption on localized states and the interband transitions regions. The Fermi energy is assumed to be in center of forbidden energy band. Thus, the minimum excitation energy is half of the band gap energy, i. e. excitation from the occupied Fermi level to the bottom of the unoccupied conduction band or from the occupied maximum of the valence band to the unoccupied Fermi level. In this case, at photon energies $E \in (0; E_\textrm {g}/2)$ no absorption is assumed. Then, up to the band gap energy, i. e. for photon energies $E \in (E_\textrm {g}/2; E_\textrm {g})$, the absorption corresponds to exponential Urbach tail. Finally, the absorption above the band energy is modeled by a rational function in which its behavior corresponds to the classical asymptotic behavior of bound states, i. e. $\varepsilon _\textrm {i,ut}(E) \approx 1/E^{3}$ for large energies. Thus, the exponential and rational part have the following forms
Table 3. Dispersion parameters describing electronic excitations.
3.3 Interband electron excitations
The interband transitions correspond to the excitations of the electrons from the extended valence states to the extended conduction states. The interband contribution is modeled as the sum of eight contributions. Each contribution represent an absorption structure above the band gap energy (see reflectance in Fig. 6). These structures (excitons) are modeled by Campi–Coriasso model [12,36–38], where the imaginary part is zero for energies below the $E_\textrm {g}$, otherwise
Fig. 6. Spectral dependencies of the reflectance $R$ measured in the UV and VUV spectral range and corresponding modeled complex dielectric function $\hat \varepsilon$. The separated UDM Urbach tail contribution is plotted by dashed line. The exciton position are shown by arrows.
The first-principles calculations [41] predict direct band gap for energy 4.71 eV while the presented UDM assumes an indirect band gap at energy 6.652(8) eV. The typical direct square-root band gap with the discrete excitons was not observed in the optical data, thus, a well known fact must be stated: today’s ab-initio calculations do not allow to correctly predict either the energy or the nature of the band gap [42] for such complicate system as the YAG crystal is.
3.4 One-phonon absorption
Primitive cell of YAG crystal contains 80 atoms, thus 240 normal modes of vibration exist in this crystal. However, due to the crystal symmetry only 17 modes are IR-active [43–46]. The UDM incorporates these one-phonon absorption structures using Voigt peaks [12,47]. This model represents generalization of coupled harmonic oscillator model replacing Lorentzian distribution function by a more general Voigt profile. Moreover, asymmetric peak approximation is used for description of the coupling effects compared to the classical model, where the 17 vibration modes generates 136 coupling parameters (one parameter for each pair of peaks). Using this approximation, the coupling parameters are reduced to 16 independent parameters. As a result of this reduction, the asymmetric Voigt peak approximation (AVPA) yields better fit compared to the exact solution for classical equations of motion or by other approximations tested (for details see Supplement 1). The AVPA contribution to the dielectric function of one-phonon absorption peaks can be written using complex Faddeeva function $\hat {\mathrm {W}}(\cdot )$ in the following closed form
The parameter $L_{\textrm {ph},j}$ takes values between 0 (fully Gaussian model) and 1 (fully Lorentzian model). Finally, the spectral position and width of phonon excitations is usually introduced in wavenumber unit, thus, instead $E_{\textrm {ph},j}$ and $B_{\textrm {ph},j}$ in eV unit are used the parameters $\nu _{\textrm {ph},j}$ and $\beta _{\textrm {ph},j}$ in $\textrm{cm}^{-1}$ unit ($E_{\textrm {ph},j}= h \nu _{\textrm {ph},j}$ and $B_{\textrm {ph},j}= h \beta _{\textrm {ph},j}$, where $h$ is Planck’s constant).
For fully Lorentzian model the (7) must be calculated as an asymmetric Lorentzian peak approximation (ALPA)
The previous ALPA form can be rewritten in the factorized form of damped harmonic oscillator model [12,48]. The parameter $M_{\textrm {ph},j}$ expresses the coupling between the $j$-th vibration mode and the rest of the system and it can have both positive and negative values. The factor $1/\mathcal {C}_{\mathrm {N}}$ is a normalization constant. Each absorption peak depends on five parameters: $N_{\textrm {ph},j}$ transition strength; $M_{\textrm {ph},j}$ asymmetry of the peak; $\nu _{\textrm {ph},j}$ wavenumber of vibration mode; $\beta _{\textrm {ph},j}$ FWHM of peak; $L_{\textrm {ph},j}$ strength of Lorentzian part. The condition for $M_{\textrm {ph},j}$ parameters together with the normalization constant
Table 4. Dispersion parameters of the phonon excitations. $^{\dagger}$ Internal parameter $M_\textrm {ph17}$ is calculated according to condition (10).
All 84 dispersion parameters together with the internal parameter $M_\textrm {ph17}$ are introduced in Table 4. In the one-phonon absorption spectral range, the sample is non-transparent, so the response function can be measured by ellipsometry or by reflectivity in the FIR and MIR regions (see Figs. 7 and 8). From these figures it is seen that only 16 absorption peaks can be clearly identified. These 16 absorption peaks corresponds to vibration modes $j=1,\dots,11,13,\dots,17$. In the Hofmeister and Campbell work [45] a very weak 12-th vibration mode is assumed at $\textrm{460}\,cm^{-1}$. In the presented work, this vibration mode was found at similar frequency (see parameters in Table 4). The parameters of this peak was not possible to fit because due to the remaining systematic errors the fitting procedure is unstable and it have tendency to use the parameters for correction of the shape of peak 11 or 13. For details see Supplement 1. Assignment of all 17 vibration modes was done in Hofmeister and Campbell work [45] (see also [46]).
Fig. 7. Spectral dependencies of ellipsometric quantities displayed by the complex pseudodielectric function $\langle \hat \varepsilon \rangle$ in the one-phonon absorption spectral range.
Fig. 8. Spectral dependencies of reflectance $R$ and transmittance $T$ measured in the spectral range of the one-phonon excitations and corresponding modeled complex dielectric function $\hat \varepsilon$. For the 12-th vibrational mode concern to Table 4 and details in text.
Fig. 9. Spectral dependencies of reflectance $R$ and transmittance $T$ measured in the one- and multi-phonon regions and corresponding modeled imaginary part of dielectric function $\varepsilon _{\mathrm {i}}$.
3.5 Multi-phonon absorption
In the case of multi-phonon absorption, all the phonons from all the branches and the entire Brillouin zone participate on absorption process. Thus, the two-phonon processes cover spectral range from zero frequency up to the double of maximum phonon frequency, three-phonon processes cover spectral range up to the triple of maximum phonon frequency and so on (see Figs. 9 and 10). The maximum phonon frequency ($818.2\,\textrm {cm}^{-1}$) was estimated as a longitudinal frequency of 17th vibrational mode. Multi-phonon absorption can be modeled by Gaussian broadened bands described by piecewise continuous functions, corresponding to the additive and subtractive joint density of states for phonon branches [12,49]. In this study, the modeling was simplified by model of Gaussian broadened discrete spectrum, where each excitation is described by 3 parameters (see Table 4). Because the probability of absorption processes decreases with increasing number of phonons participating on these processes, then, also the transition strength of phonon absorption processes decreases with the number of phonons. This is the reason why it is very difficult to separate them. For separation of individual absorption processes, it is necessary to perform multi-temperature measurements and to use the different temperature dependencies of individual absorption processes, as it was performed for crystalline silicon [50]. Therefore, it is impossible to say that wide structures 18 and 19 are multi-phonon excitations. These broad peaks can be interpreted as some corrections of the one-phonon excitations.
Fig. 10. Spectral dependencies of ellipsometric quantities displayed by pseudodielectric function $\langle \hat \varepsilon \rangle$ and the degree of polarization $P$ in the multi-phonon region.
4. Conclusion
The goal of this work was to study the optical properties of YAG single crystal in wide spectral range by applying a universal dispersion model. The YAG single crystal, which is a popular laser host material, was studied from VUV to FIR range, using reflectance, transmittance and ellipsometric measurements, all performed at room temperature. The universal dispersion model described individual elementary electron and phonon excitations in material, presented in a unified formalism, which is parameterized using normalized transition strength functions, respecting conformity with the sum rules [10–12]. Absorption phenomena (interband electronic excitations, absorption on localized states, Urbach tail and phonon absorptions) contributing to the dielectric response were discussed in detail. For the one-phonon contribution, the model used an asymmetric Voigt peak approximation (AVPA), which represented the best fit of the experimental data. The optical constants modeled in this study were determined with high absolute precision (in order of 10$^{-4}$). The results presented are an important starting point for a more complex research, where various functional dopants (Nd, Yb, Er, Ce, Sm, Tu, Ho, Cr, V, etc.) in the YAG’s matrix can be comprehensively studied. The work can be extended to a temperature-dependent dispersion model or it can be extrapolated to similar crystalline host material such as YAP and LuAG. Therefore, we showed that the UDM can be successfully used for determination of optical constants with absolute precision and for dissemination of optical constants data instead of tabulated values.
Funding
Ministerstvo Školství, Mládeže a Tělovýchovy (CZ.02.1.01/0.0/0.0/15_006/0000674, LM2018097, LM2018110); Horizon 2020 Framework Programme (NO 739573).
Acknowledgments
The authors thank Dr. Petr Klapetek for AFM measurement of YAG sample. The optical measurements were performed on the CEPLANT and CEITEC Nano Research Infrastructure.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are available in Dataset 1 [17].
Supplemental document
See Supplement 1 for supporting content.
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