Infrared active phonons in monoclinic lutetium oxyorthosilicate

Rare-earth oxyorthosilicates offer a unique set of characteristics that make them well suited to a broad range of applications. These oxyorthosilicates include $Dy2SiO5$⁠, $Lu2SiO5$ (LSO), $Yb2SiO5$⁠, among others.¹ These materials are isostructural also with $Y2SiO5$ (YSO) and both similarities and differences are anticipated in their optical properties. Oxyorthosilicates such as LSO are attractive candidates for scintillators due to their large bandgap energy value and large mass density. Undoped LSO is an insulator with a bandgap energy near 6 eV.² Dopant states within the bandgap serve as activation centers to produce photons exploiting, for example, $4f−5d$ transitions.³ LSO has been of interest as a scintillator material when doped with cerium.⁴ In $Ce3+$ doped LSO, where Ce replaces Lu, the $4f1$ band is split by spin–orbit interactions into levels $2F5/2$ and $2F7/2$ leading to transition wavelengths of 360 nm, 300 nm, and 260 nm.⁵ 

LSO has gained substantial attraction in nuclear medicine for use as a scintillator material in positron emission tomography and combined positron emission tomography x-ray computed-tomography.^6–10 A large gamma radiation to photon conversion efficiency and a very short decay time offer increased spatial and temporal resolution in medical imaging also permitting shorter image acquisition time.^4,6 Its high efficiency also permits the development of smaller devices.⁸ Likewise, LSO has potential for use in high-energy physics. Chen et al. demonstrated that continued exposure to gamma rays did not permanently alter their optical properties.¹¹ Radiation hardness is required for use in calorimeters, which measure the energy of particles produced by collisions in accelerators. LSO was proposed for the now canceled SuperB collider project, the Mu2e collider, and in an upgrade to the Super Large Hadron Collider.^12–14 LSO is also currently investigated as an active laser material. While Ce:LSO is not a commonly used media yet, similar lanthanide doped orthosilicates have been used as effective high-powered lasers. For example, Yb:YSO and Yb:(⁠$LuxY(1−x)$⁠)SO have been shown to be highly efficient continuous-wave laser materials with emission wavelengths near or above 1000 nm.^15–17 Yb:LSO has a remarkably high saturation density of $9.2kW/cm2$ and an emission wavelength of 1083 nm.¹⁸ Neodymium doped LSO is a promising laser material with a unique emission wavelength of 1357 nm.^19,20 Nd:LSO has been proposed as a pump for strontium optical clocks, laser Doppler velocimetry, and distributed fiber sensor applications.¹⁹ Yb:LSO has been proposed for femtosecond pulsed laser applications by Thibault et al., who report optical pump pulse efficiency of 17%.²¹ 

LSO and the isostructural oxyorthosilicates crystallize as monoclinic crystals in space group 15. Density functional theory (DFT) calculations have been used to predict lattice parameters of oxyorthosilicates,^22–25 and comparison with experiment was made available by x-ray diffraction (XRD) analyses of powdered samples.^5,25–31 Key to further understanding is the availability of high quality crystals. Growth using the Czochralski method⁴ results in single crystals and deposition using high-pressure annealing techniques results in ceramics.³² While single crystal growth is slow and energy extensive, ceramics show inferior optical properties.^25,32 Presently, high quality single crystals with various doping are available commercially, while some laboratories have grown their own material.^30,33,34

Monoclinic crystals are highly anisotropic and must be described by four independent dielectric function tensor element spectra, all of which contain real and imaginary parts. Spectra of the optical properties of monoclinic crystals have only been reported very recently, and generalized spectroscopic ellipsometry (GSE) techniques for proper measurements and data analysis of arbitrarily anisotropic materials have only been developed recently.³⁵ For example, monoclinic $β-Ga2O3$⁠, which has gained attention for high power solid state transistor and switch applications, has been analyzed from the terahertz to the vacuum ultraviolet spectral regions for its complete phonon mode and free charge carrier properties,^36–39 and electronic band-to-band transitions as well as its static and high-frequency dielectric constants, indices of refraction, and extinction coefficients.^40–44 A similar analysis procedure resulted in complete sets of phonon modes for monoclinic scintillator material cadmium tungstate (⁠$CdWO4$⁠)⁴⁵ and YSO.⁴⁶ To date, the only generalized spectroscopic ellipsometry investigation of LSO was reported by Jellison et al., and the four real-valued dielectric function tensor spectra were obtained in the spectral range from 200 nm to 850 nm.³³ The spectral dependencies of the monoclinic refractive indices were calculated and the bandgap of LSO was estimated to be at 6.7 eV. Kitaura et al. reported vacuum ultraviolet reflectivity spectra to nearly 30 eV²⁹ and provided an in-depth study of the electronic band-to-band transitions in LSO. Kitaura et al. estimated the bandgap to be at 7.52 eV.

Lattice mode excitations are fundamental physical processes that influence many material properties such as thermal and electronic transport, or coupling with excitons or photon emission. The lattice thermal conductivity of LSO is important when used as an active laser material, or as an environmental barrier coating.^22,30 Since LSO is highly anisotropic, the thermal conductivity should also depend on transport direction. Gustafsson et al. reported on the thermal properties of LSO but assumed isotropic behavior.³¹ Subsequently, Cong et al. showed the effects of anisotropy in thermal conductivity and reported the behavior along each crystal axis.³⁰ While optical phonons do not directly participate in heat transfer, it is theorized that the anisotropic transport originates from coupling with anisotropic acoustic phonons.⁴⁷ Raman active modes have been well studied for a large range of oxyorthosilicates including LSO.^{26,27,34,48–50} However, in many of these reports, the crystals are approximated as isotropic materials ignoring wavevector and polarization dependencies of the Raman modes. Infrared active modes were not reported for LSO. Likewise, no DFT calculations of lattice mode properties have been reported.

In this work, we report a combined generalized spectroscopic ellipsometry and DFT analysis to obtain the complete set of long wavelength active phonon modes in cerium doped $Lu2SiO5$⁠. Ellipsometry measurements from $40to1200cm−1$ are used to determine the frequency dependent dielectric function tensor elements. Two different crystals, each with different cuts perpendicular to a crystal axis, are investigated. We closely follow the methodology recently described for YSO by Mock et al. in order to determine the complete set of long wavelength active phonon modes.⁴⁶ We expect qualitatively similar results for LSO since YSO is isostructural. As such, identical numbers of phonon pairs are expected. However, both materials differ quantitatively, which will be discussed in more detail. We find and detail 23 $Au$ symmetry and 22 $Bu$ symmetry infrared active transverse (TO) and longitudinal optical (LO) modes. We also perform DFT calculations and compare our results with the phonon modes from experiment. An excellent agreement is obtained. We also compare the phonon mode frequencies and the phonon mode order within LSO and YSO and find strong qualitative agreement. We, therefore, anticipate similar phonon mode properties among members of the isostructural group of monoclinic symmetry rare-earth oxyorthosilicates such as $Dy2SiO5$⁠, $Ho2SiO5$⁠, $Er2SiO5$⁠, $Tm2SiO5$⁠, and $Yb2SiO5$⁠.¹ We briefly describe the eigendielectric vector summation approaches for rendering the infrared optical properties of monoclinic crystals including lattice anharmonicity and the generalized spectroscopic ellipsometry measurements. We detail our DFT approach, report all results, and discuss our findings. In our ellipsometry analysis, we did not observe additional modes due to the presence of the dopant cerium. Hence, throughout this work, the effects of the dopant cerium onto the lattice modes are ignored. The results of our work will become relevant for the future understanding of optical properties such as photon–phonon coupling and lattice thermal transport processes.

$Lu2SiO5$ belongs to the space group 15 (centered monoclinic). There are 18 alternative choices for the unit cell for this space group, per the International Tables for Crystallography.⁵¹ The crystallographic standard for the unit cell choice for monoclinic crystals^52,53 promotes choosing vector b along the symmetry axis, and the lowest possible non-acute monoclinic angle in the network perpendicular to the symmetry axis. For space group 15, the $I2/c$ cell is consistent with the standard and is used throughout this work. The structural parameters for the unit cell are specified in Sec. II B. In many previous publications, the $C2/c$ cell was chosen.^{4,24,25,31,33,34} Where appropriate, we convert the literature unit cell parameters to the $I2/c$ cell used here. The $I2/c$ cell definition is shown in Fig. 1.

Four lutetium atoms form a distorted tetrahedron with isolated ionic units of $SiO4$ tetrahedrals and oxygen atoms not bonded to silicon (Fig. 1).³¹ The crystallographic axis c is distinguished by $OLu4$ tetrahedra along edge-sharing infinite chains. Two crystallographic sites, $Lu1$ and $Lu2$⁠, coordinated with either six or seven oxygen atoms, respectively, are occupied by $Lu3+$ ions.

Theoretical calculations were performed by plane wave DFT code Quantum ESPRESSO (QE).⁵⁴ We used the exchange correlation functional of Perdew, Burke, and Ernzerhof (PBE)⁵⁵ and optimized norm-conserving Vanderbilt (ONCV) scalar-relativistic pseudopotentials.^56,57 For lutetium, we used a pseudopotential with the f states frozen into the core. As $Lu2SiO5$ is isostructural with many similar rare-earth oxyorthosilicates¹ as well as with $Y2SiO5$⁠, we used the optimized crystal structure of the latter from our recent study⁴⁶ as the starting point for the calculations of $Lu2SiO5$⁠. The calculations were performed in a primitive cell $p1=a$⁠, $p2=b$⁠, $p3=(a+b+c)/2$⁠. The initial structure was first relaxed to force levels less than $10−5RyBohr−1$⁠. A regular shifted $4×4×4$ Monkhorst–Pack grid was used for sampling of the Brillouin zone.⁵⁸ A convergence threshold of $1×10−12Ry$ was used to reach self-consistency with a large electronic wavefunction cutoff of 120 Ry. The comparison of resulting optimized cell parameters with the existing literature data is listed in Table I. The relaxed cell was used for subsequent phonon calculations, which are described in Sec. IV A.

We use the eigendielectric displacement vector summation (EDVS)^36,59,60 approach and the eigendielectric displacement loss vector summation (EDLVS) approach augmented with anharmonic lattice broadening parameters for analysis of the dielectric function tensor data extracted from the ellipsometry measurements. The process for extracting the tensor data from the experiment is discussed further below. The EDVS approach permits access to the TO mode parameters. The EDVLS approach permits access to the LO mode parameters. The application of both approaches simultaneously in order to determine the amplitudes, frequencies, and broadening parameters of all TO and LO modes was shown previously by Mock et al. for YSO⁴⁶ and $NdGaO3$⁠.⁶¹ 

Both infrared frequency dependent dielectric function tensor, $ε(ω)$⁠, and dielectric loss function tensor, $ε−1(ω)$⁠, contain all information on infrared active TO and LO modes, including their directional (anisotropic) properties. Dielectric resonance for electric fields along $e^l$ with eigendielectric displacement unit vectors $e^l=e^TO,l$ define TO mode frequencies. Dielectric loss resonance for electric fields along $e^l$ with eigendielectric displacement unit vectors $e^l=e^LO,l$ define LO frequencies. Note that in the latter case, the electric field of an electromagnetic wave does interact with the medium, despite statements to the contrary often found in textbooks.^62,63 The compelling argument is that at the frequency of a LO mode, the lattice polarization (motion)—the cause of which can only be the electric field of the electromagnetic wave, is oriented so as to compensate the vacuum polarization leading to a vanishing displacement. A set of multiple TO and LO modes, where $l$ may denote a running index, can thus be determined from $ε(ω)$ and $ε−1(ω)$⁵⁹ 

where det is the determinant. We note without further proof that the total number of TO modes must always equal the total number of LO modes. In the Born and Huang approach,⁶⁴ the lattice dynamic properties in crystals with arbitrary symmetries are categorized under different electric field $E$ and dielectric displacement $D$ conditions.⁶⁵ $E=0$ and $D=0$ define the TO modes, $ωTO,l$⁠, associated with the dipole moment. $E≠0$ but $D=0$ defines the LO modes, $ωLO,l$⁠, identical with the definitions through the dielectric tensor above. $E≠0$ and $D≠0$ define the so-called limiting frequencies $ω(α)l$⁠. Here, the limiting frequencies, $ω(α)l$⁠, depend on the direction of a unit vector within the $a$–$c$ plane, $α^=cos⁡αx^+sin⁡αy^$⁠.³⁹ Frequencies $ω(α)l$ are then obtained from the subtensor within the $a$–$c$ plane as follows (T denotes the transpose):

A physical model must be selected to calculate the effect of the lattice excitation onto the optical properties. In the EDVS approach, $ε$ is obtained from a sum over all TO mode contributions, added to a high-frequency wavelength independent tensor $ε∞$⁠,

where $⊗$ is the dyadic product and $ϱTO,l$ are wavelength dependent functions. In the EDVLS approach, $ε−1$ is obtained from a sum over all LO mode contributions, added to a high-frequency wavelength independent inverse tensor $ε∞−1$⁠,

where $ϱTO,l$ are wavelength dependent functions. Note the minus sign in front of the summation in Eq. (4), which was chosen to result in real-valued LO mode amplitude parameters in Eq. (5).⁶⁶ Note that both tensors are symmetric, and six complex-valued frequency dependent functions are required to fully render $ε$ and its inverse, $ε−1$⁠. Anharmonic broadened Lorentzian oscillator functions are used to describe wavelength dependent functions in Eqs. (3) and (4),^46,61

Here, $Ak,l$⁠, $ωk,l$⁠, $γk,l$⁠, and $Γk,l$ denote the amplitude, resonance frequency, harmonic broadening, and anharmonic broadening parameter for TO (⁠$k=$ “TO”) or LO (⁠$k=$ “LO”) mode $l$⁠, respectively, and $ω$ is the frequency of the driving electromagnetic field. It can be shown that parameters $Γk,l$ vanish when there is no coupling between lattice modes, which leads to anharmonic lattice broadening.

The coordinate-invariant generalized dielectric function connects all TO and LO modes within a crystal regardless of its symmetry,⁵⁹ 

This form^59,67,68 is interpreted with TO and LO broadening parameters

The usefulness of this form during the analysis of phonon modes from ellipsometry data has been recently demonstrated for monoclinic $β-Ga2O3$⁠,^36,$CdWO4$⁠,⁶⁰ YSO,⁴⁶ and orthorhombic $NdGaO3$⁠.⁶¹ With $ω→0$⁠, Eq. (6) provides the Schubert-Lyddane–Sachs–Teller (S-LST) relationship for crystals with arbitrary symmetry, which relates the determinants of the DC and the high-frequency tensors with all TO and LO modes within a given crystal.^59,69

In materials with orthorhombic and higher symmetry, frequencies $ωTO,LO,l$ appear in associated pairs such that a TO mode is always followed by an associated LO mode, where the next mode in order must be a TO mode. This is also known as TO–LO rule.⁷⁰ It was found that this order is violated for monoclinic and triclinic symmetries. The resulting phonon order and its relationship with the reststrahlen band and the directional modes were exemplified recently for monoclinic $β-Ga2O3$⁠.³⁹ 

Coordinate systems (⁠$x^,y^,z^$⁠), (⁠$x,y,z$⁠), and (a, b, c) are needed to describe the optical properties of the monoclinic crystals within the laboratory system of the ellipsometry instrumentation. Laboratory coordinate axes $x^$⁠, $y^$⁠, and $z^$ are associated with the ellipsometer system, where $x^$ is parallel to the plane of incidence and parallel to the sample surface, $y^$ is parallel to the sample surface, and $z^$ is perpendicular to the surface pointing into the sample. The incident wave vector has a positive component in the $x^$ direction. Crystallographic axes of the monoclinic system, (a, b, c), are oriented within the Cartesian sample coordinate system, (⁠$x,y,z$⁠), such that $z$ is parallel to axis $b$⁠, and axes $a$ and $c$ are within the (⁠$x$–$y$⁠) plane. Relationships between (⁠$x,y,z$⁠), and (a, b, c) are shown in Fig. 1. Rotation transformations then connect the physical orientation of a given sample mounted onto the sample stage of the ellipsometer with the intrinsic orientations of a given phonon mode within the crystal lattice. Due to factorization according to symmetry and number of elements in the unit cell, 23 TO and 23 LO modes with $Au$ symmetry are polarized along vector b. 22 TO and 22 LO modes with $Bu$ symmetry are polarized within the monoclinic a–c plane. The orientation of a TO eigendielectric displacement vector with $Bu$ symmetry relative to $x$ within the (⁠$a$–$c$⁠) plane is denoted by $αTO,l$⁠. The orientation of a LO eigendielectric loss displacement vector with $Bu$ symmetry relative to $x$ within the (⁠$a$–$c$⁠) plane is denoted by $αLO,l$⁠.

Generalized spectroscopic ellipsometry permits measurements of the optical properties of arbitrarily anisotropic materials,^35,71,72 including crystalline materials with monoclinic^{36,40–43,46,59,60,73} and triclinic symmetry.⁷⁴ The Mueller matrix is measured and then compared with model calculations. The Mueller matrix relates the Stokes vector components before and after interaction with a sample,

with Stokes vector components defined here by $S0=Ip+Is$⁠, $S1=Ip−Is$⁠, $S2=I45−I−45$⁠, $S3=Iσ+−Iσ−$⁠. Here, $Ip$⁠, $Is$⁠, $I45$⁠, $I−45$⁠, $Iσ+$⁠, and $Iσ−$denote the intensities for the $p$-, $s$-, $+45°$⁠, $−$45$°$⁠, right handed, and left handed circularly polarized light components, respectively.⁷⁵ As discussed in detail previously,^{36,46,60,61,70,71,75–77} ellipsometry data are compared with model calculated data by best-match regression algorithms. A half-infinite, two phase model with one media being ambient air and the other monoclinic $Lu2SiO5$ separated by the planar crystal surface is applied. The angular orientations of the samples relative to their crystallographic orientations are determined together with the wavelength dependencies of the dielectric function tensor elements. Samples with different surface orientations are investigated. Data at multiple sample azimuth orientations and multiple angle of incidences are measured. All sample cuts, azimuthal rotations, and angles of incidence data are best-matched simultaneously for all wavelengths (polyfit), and complex valued functions, $εxx$⁠, $εyy$⁠, $εxy$⁠, and $εzz$ are obtained (wavelength-by-wavelength analysis). All functions are then best-match analyzed using model functions above and varying model parameters. Note that all spectra for $ε$⁠, $ε−1$⁠, $det{ε$}, and $det{ε−1}$ are evaluated simultaneously to achieve best-match and to find the best-match calculated model parameters.^46,60,61

Two single crystal samples of cerium doped $Lu2SiO5$ were purchased from MTI Corporation for this investigation. The nominal doping concentration is 0.175 mol. $%$⁠. According to Ning et al., doping with cerium causes very little change in the lattice structure of the crystal.²⁴ The sample dimensions were each $10×10×0.5mm3$⁠. Crystal orientations of our samples were (001) and (110), respectively, following the axis definition shown in Fig. 1. All model calculations were done using WVASE32™ (J. A. Woollam Co., Inc.). GSE measurements were performed with two instruments. Data within the infrared spectral range covering approximately $230$–$1200cm−1$ was acquired with a commercial IR variable angle of incidence spectroscopic ellipsometry (VASE) instrument (J. A. Woollam Co., Inc.). Data within the far-infrared (FIR) spectral range covering approximately $40$–$500cm−1$ were acquired with an in-house built FIR-VASE instrument.⁷⁸ Data were acquired in the Mueller matrix formalism. Mueller matrix data were taken for each sample at two angles of incidence, $Φa=50°$ and $70°$⁠. Five azimuthal sample orientations for each sample, with the sample rotated clockwise around its normal in $45°$ increments, were measured. Only five azimuthal angles were needed as measurements $180°$ apart are identical and no non-reciprocity effects were observed. All five azimuthal rotations are included in the analysis, but only three rotations for each surface cut are shown in our figures for brevity. Since our current instrumentation in the FIR spectral region lacks a compensator, fourth row elements of the Mueller matrix are only available from the infrared instrument, approximately $230cm−1$ and above. All samples were measured as received from the crystal manufacturer without any further surface chemical or mechanical treatment. All sample surfaces were polished and optically flat. No surface roughness or surface overlayer effects were observed in the spectral range investigated here, and no numerical surface overlayer removal procedures were performed during data analysis. The different crystallographic surfaces investigated here are stable in normal ambient conditions.

The phonon frequencies, Born effective charges, and transition dipole components were computed at the $Γ$-point of the Brillouin zone using density functional perturbation theory,⁷⁹ as implemented in the Quantum ESPRESSO package. The parameters of the TO modes were obtained from the dynamical matrix computed at the $Γ$-point. The parameters of the LO modes were obtained by setting a small displacement from the $Γ$-point in order to include the long-range Coulomb interactions of Born effective charges in the dynamical matrix. For $Au$ symmetry modes, this displacement was in the direction of the crystal vector $b$⁠. For the $Bu$ modes, the entire $a$–$c$ plane was probed with a step of $0.1°$⁠, in order to create plots of directional limiting frequencies. The extrema of the dispersion curves for each phonon mode were identified as LO modes if they did not coincide (in terms of phonon frequency and direction) with previously identified TO modes.

The results of the phonon mode calculations for all infrared active modes with $Bu$ and $Au$ symmetry (⁠$ωTO,l$⁠, $ATO,l$⁠, $αTO,l$⁠, $ωLO,l$⁠, $ALO,l$⁠, $αLO,l$⁠) are listed in Tables II and III. Note that for modes with $Au$ symmetry, all eigenvectors are oriented along direction $b$ and thus $αTO,LO,l$ are not needed. Values for $αTO,LO,$ for modes with $Bu$ symmetry are counted relative to the direction of the highest-frequency (TO, LO) mode, and the highest-frequency (TO, LO) mode is counted relative to axis $a$ within the $a$–$c$ plane. Renderings of atomic displacements for each mode were prepared using XCrysDen⁸⁰ running under Silicon Graphics Irix 6.5, and are shown in Figs. 2 and 3.

Experimental Mueller matrix data and the best-match model data are shown in Figs. 4 and 5 for both samples studied. Each unique Mueller matrix element is shown in its own pane and arranged by the corresponding matrix indices. Each pane shows three azimuthal positions denoted by P1, P2, and P3 with two angles of incidence each (⁠$50°$ and $70°$⁠). Datasets symmetric in indices are plotted together and as denoted in the corresponding panes. All Mueller matrix elements are normalized to $M11$⁠. Element $M44$ cannot be measured with our current instrumentation and is, therefore, not presented. Data obtained within the FIR range (⁠$40$–$500cm−1$⁠) and data within the IR range (⁠$500$–$1200cm−1$⁠) are shown for all elements excluding the fourth row elements. Due to limitations of our current FIR instrumentation, only IR data (⁠$250$–$1200cm−1$⁠) are shown for all fourth row elements.

In-plane anisotropy is seen in Figs. 4 and 5 in the off-block diagonal elements (⁠$M13$⁠, $M23$⁠, $M14$⁠, and $M24$⁠). For isotropic samples, these elements are zero valued across the entire spectra. Each of these elements are strongly impacted by the azimuthal rotations. To show correlation between all Mueller matrix elements and the extracted dielectric function tensor elements, the frequencies of all TO and LO phonon modes with $Au$ and $Bu$ symmetries are shown as horizontal lines in Figs. 4 and 5. Our polyfit data for each wavelength included up to 792 independent data points from the multiple cuts, azimuthal rotations, and angles of incidence. In this wavelength-by-wavelength analysis, 14 independent parameters are varied. Of these 14, eight are the real and imaginary parts of the dielectric function tensor elements (⁠$εxx$⁠, $εyy$⁠, $εxy$⁠, and $εzz$⁠). The remaining six variables are two sets of three Euler angles used to describe the sample surface and orientation, independent of wavelength. The resultant Mueller matrices of this polyfit are shown in Figs. 4 and 5 as the solid red lines. Note that experiment and best-model data lines are virtually indistinguishable due to their excellent agreement. The dielectric tensor elements found in this fit are shown in Fig. 6 as the dotted green lines. Overall, there is excellent agreement between the experimental and calculated Mueller matrix data. Longer wavelength data do become noisier as a result of lower source intensity. While these LSO samples were doped with cerium, no free charge carrier effects are detected in our data.

The real and imaginary parts of the dielectric function tensor elements found during the polyfit (⁠$εxx$⁠, $εxy$⁠, $εyy$⁠, and $εzz$⁠) are shown in Fig. 6 as green dotted lines. These tensor elements were then translated into the inverse dielectric function tensor elements (⁠$εxx−1$⁠, $εxy−1$⁠, $εyy−1$⁠, and $εzz−1$⁠) as shown in Fig. 8 as the green dotted lines again. Similarly, the determinant and inverse determinant elements (⁠$εxxεyy−εxy2$⁠) and inverse determinant [(⁠$εxxεyy−εxy2)−1$] shown in Fig. 7 are derived from the polyfit elements.

From these dielectric tensor elements, phonon modes can be observed. TO mode resonant frequencies occur at the maxima of the imaginary parts of the dielectric function tensor elements and the determinant.⁵⁹ Similarly, LO mode resonant frequencies occur at the maxima of the imaginary parts of the inverse dielectric function elements and inverse determinant.⁶⁰ In Figs. 6 and 8, panels (a), (b), and (d) have common peaks where we identify 22 TO and LO mode pairs with $Bu$ symmetry (in the a–c plane). Likewise, the peaks of panel (c) show 23 TO and LO mode pairs along the $Au$ symmetry (along the b direction).

By using a set of anharmonically broadened Lorentzian oscillators, we derive the best-match model calculations shown in Figs. 6 and 8 as the solid red lines. The best-match TO model parameters are detailed in Table IV. Parameters included in this table are amplitude (⁠$ATO,l$⁠), frequency (⁠$ωTO,l$⁠), broadening (⁠$γTO,l$⁠), anharmonic broadening (⁠$ΓTO,l$⁠), and eigenvector direction (⁠$αTO,l$⁠) for all TO modes (⁠$l=1…22$⁠) with $Bu$ symmetry.

Figure 9 is a vector representation of the amplitude and polarization direction parameters (⁠$ATO,lBu$ and $αTO,l$⁠) within the $a$–$c$ plane. Here, GSE (a), (b) model data are compared to DFT derived data (c) and (d). TO vectors are shown in panels (a) and (c) and LO vectors are in (b) and (d). Note that small amplitude modes are enlarged by factors as shown. There is generally good agreement between the GSE and DFT data, particularly for modes with large amplitudes, such as mode 1, 5, 11, or 13, for example. Some modes appear similar but are numbered differently between the two data sets. This may happen as a result of a slightly different frequency being found in GSE than DFT and hence a different number is being assigned to the mode. For instance, this happens in the TO [(a) and (c)] modes 3 and 4. There is noticeable disagreement between other modes. This lack of agreement may be attributed to a weaker mode amplitude parameter or being too close in frequency to another mode feature.

Blue solid lines in Figs. 6 and 8 indicate the resulting best-match model calculations obtained from Eq. (3) using a second independent set of anharmonically broadened Lorentzian oscillators (LO mode summation). We find excellent agreement between our wavelength-by-wavelength and model calculated $ε−1$ and $ε$⁠. All best-match LO model parameters are summarized in Table IV including amplitude (⁠$ALO,l$⁠), frequency (⁠$ωLO,l$⁠), broadening (⁠$γLO,l$⁠), anharmonic broadening (⁠$ΓLO,l$⁠), and eigenvector direction (⁠$αLO,l$⁠) for all LO modes (⁠$l=1…22$⁠) with $Bu$ symmetry. Frequencies of the LO modes are indicated by dotted vertical blue lines in Figs. 4, 5, 7, and 8 which align with the features observed in the data and the extrema seen in the imaginary part of the inverse dielectric tensor.

GSE results of mode parameters for $Au$ modes are listed in Table V. The dielectric function and inverse dielectric functions are shown in Figs. 6(c) and 8(c), respectively. Red lines show our best-match model calculation using 23 anharmonic Lorentzian oscillators. Parameter sensitivity is critical for modes which occur in close wavelength proximity and/or possess small amplitudes. Weak modes with small amplitude parameters may become subsumed by stronger modes during the regression analysis. Therefore, manual parameter adjustments and limited parameter regions for some parameters were used to reach best-match calculated model parameters. These adjustments are noted in Table V accordingly. Modes 4, 11, 13, 15, and 19–23 possess small splitting between their TO and LO frequencies. We have treated those hence as impurity-like vibrational modes as discussed recently for wurtzite-structure GaN.⁸¹ Accordingly, their TO and LO frequencies are listed equal in Table V. We note that modes 11 and 15 are observed within the TO–LO bands formed by modes 12 and 16, respectively, while all other impurity-like vibrational modes are located outside such bands. Frequencies of TO modes with $Au$ symmetry are indicated by vertical solid brown lines in Figs. 4, 5, and 6 while frequencies of LO modes with $Au$ symmetry are indicated by vertical dotted brown lines in Figs. 4, 5, and 8. Table III lists DFT calculated frequencies and amplitude parameters for $Au$ modes, and overall a good agreement is observed with our GSE results. Modes 19–23 possess small TO–LO mode splittings, with mode 20 as small exception being predicted as a band with slightly larger TO–LO splitting. Likewise, modes 4 and 11 are predicted with very small TO–LO splitting, in agreement with our observation. Note that mode 11 is located within the TO–LO band of mode 12, consistent with our GSE results. Modes 13 and 15 are predicted with somewhat larger TO–LO splitting than observed. We note that mode 15 is also observed to be located within mode 16 in GSE, while in DFT modes 15 and 16 are separated.

As pointed out by Mock et al.,⁶¹ the Schubert-BUL form [Eq. (7)] can be used to identify violations of the TO–LO order. Negative imaginary parts occur in frequency regions of TO–LO bands (inner modes, “$+$”) nested within larger TO–LO bands (outer modes, “$−$”). For example, a mode sequence of $[TO−…[TO1,+,LO1,+]…[TOn,+,LOn,+]…LO+]$ will show negative imaginary parts in frequency bands $[TO1,+,LO1,+]$⁠, $[TO2,+,LO2,+],…$ and $[TOn,+,LOn,+]$⁠. It is noted that this form does not represent a measurable dielectric function and represents an indicator of physical properties rather than representing a physical property itself. Hence, a negative imaginary part is not prohibitive. Here, we observe such occurrences between TO-1 and LO-2, in the very narrow range between TO-7 and LO-8, between TO-12 and LO-13, and between TO-15 and LO-15 and TO-16 and LO-16. These frequency regions are identical with bands of total reflection and the formation of inner and outer modes as discussed below. Note that the TO–LO rule holds true for all $Au$ modes.

In monoclinic symmetry materials with polar lattice vibrations, the order of the phonon modes is directly related to the appearance of the polarized reststrahlen bands. The existence of inner and outer modes, where inner modes are nested within the frequency range of outer modes, and their relationship with the order of the phonon modes and the reststrahlen range appearance, was discussed recently for $β-Ga2O3$ as an example.³⁹ The reststrahlen bands for frequencies within outer modes are polarization-dependent. Inner modes cause polarization-independent (totally reflective regardless of polarization) reststrahlen bands. The directional limiting frequencies within the Born–Huang approach are bound to within outer mode frequency regions not occupied by inner mode pairs. Early observations were reported by Kuzmenko for monoclinic copper monoxide and bismuth monoxide.⁸² Hence, an unusual phonon mode order can occur where both lower-frequency and upper-frequency limits for the directional modes can be both TO and/or LO modes. Figure 10 depicts the directional mode frequencies and their dispersion within the monoclinic plane for LSO, obtained by DFT and by GSE, in comparison. Symbols indicate frequency and direction of all TO and LO modes (open symbols indicate their normal directions perpendicular to the eigenvectors within the monoclinic plane). Light gray areas indicate regions of outer modes not occupied by inner modes, and within which all directional modes are confined. The dark gray areas indicate regions occupied by inner modes, within which no directional mode exists. Overall, an excellent agreement between DFT and GSE is noted. Both order of modes, their frequencies as well as direction parameters are highly consistent between theory and experiment. Small deviations can be seen for small polarity modes, which show very small dispersion only (outer or inner mode pairs with small TO–LO splitting). Also, some outer modes appear shifted and overlapping partially in the experiment, causing nested inner mode pairs and bands of total reflection (dark gray areas).

We further compare the experimentally determined phonon mode properties of LSO with those of its isostructural compound YSO, whose phonon modes we have determined recently.⁶¹ Yttrium (atomic number 39; electron configuration [Kr] 4d1 5s2; standard atomic weight 88.90584$u$⁠; covalent radius $190±7pm$⁠) has nearly half of the inertial mass of lutetium (71; [Xe] 4f14 5d1 6s2; 174.9668$u$⁠; $187±8pm$⁠) but equivalent covalent size. As can been in Table V in comparison with Table 4 in Ref. 61, the $Au$ TO modes are very similar while the LO modes are slightly different reflecting slightly different Born effective charges. Figure 10 also depicts the directional mode frequencies and their dispersion within the monoclinic plane for YSO, obtained by GSE. Comparing results from experiment for both compounds, one can see no significant differences between LSO and YSO, except for small variations in actual frequency and direction parameters. Closer inspection of Figs. 2 and 3 as well as Figs. 2 and 3 in Ref. 61 reveals that most of the phonon mode displacements in both LSO and YSO are taken up by the oxygen atoms, followed by much smaller displacement performed by the silicon atoms. The much heavier elements Y and Lu remain practically fixed within the lattice. Hence, the replacement of Lu with Y affects the phonon mode behavior only marginally. A similar behavior was reported for Raman modes measured in LSO, YSO, and $Lu1.8Y0.2SiO5$⁠.²⁷ Therefore, we expect similar phonon mode behavior in all rare-earth oxyorthosilicates $Dy2SiO5$⁠, $Ho2SiO5$⁠, $Er2SiO5$⁠, $Tm2SiO5$⁠, and $Yb2SiO5$ than observed here for LSO.

Table VI depicts results for the static and high-frequency dielectric constants obtained in this work. The static dielectric constants are obtained numerically by setting $ω=0$⁠. The high-frequency dielectric constants are determined during the best-match model GSE analysis. We note that the S-LST relation⁵⁹ is satisfied when considering all TO and LO mode frequencies and static and high-frequency dielectric constants obtained in this work.

We have determined the infrared active phonon mode parameters for the monoclinic symmetry rare-earth oxyorthosilicate $Lu2SiO5$⁠. A combined analysis method using density functional theory and spectroscopic ellipsometry was used. Our previously described approach to extract phonon mode parameters from monoclinic symmetry and hence highly anisotropic materials has been demonstrated as a versatile technique. We found all phonon modes anticipated by symmetry and in excellent agreement between theory and experiment. We determined all directional modes and established the phonon mode order with the $a$–$c$ plane. We further observe that the phonon mode properties of $Lu2SiO5$ are very similar to its isostructural compound $Y2SiO5$⁠, despite a much larger inertial mass of Y relative to Lu. This observation is explained by the large mass difference between oxygen and silicon relative to the Y and Lu atoms. We anticipate a very similar phonon mode behavior among the entire class of rare-earth monoclinic oxyorthosilicates.

This work was supported in part by the National Science Foundation (NSF) under Award No. DMR 1808715, the Air Force Office of Scientific Research under Award No. FA9550-18-1-0360, the Nebraska Materials Research Science and Engineering Center under Award No. DMR 1420645, the Swedish Research Council VR under Award No. 2016-00889, the Swedish Foundation for Strategic Research under Grant Nos. RIF14-055, and EM16-0024, the Knut and Alice Wallenbergs Foundation supported grant “Wide-bandgap semi-conductors for next generation quantum components,” the Swedish Governmental Agency for Innovation Systems (VINNOVA) under the Competence Center Program Grant No. 2016-05190, and the Swedish Government Strategic Research Area in Materials Science on Functional Materials at Linköping University, Faculty Grant SFO Mat LiU No. 2009-00971. M.S. acknowledges the University of Nebraska Foundation and the J. A. Woollam Foundation for financial support. This research was performed while author A.M. held an NRC Research Associateship award at the U.S. Naval Research Laboratory.