Terahertz permittivity parameters of monoclinic single crystal lutetium oxyorthosilicate Open Access

Lutetium oxyorthosilicate, Lu₂SiO₅ (LSO), has proven to be an important wide bandgap material for a number of applications.¹ Cerium doped LSO (LSO:Ce) can be used as a scintillating crystal for medical imaging^2–4 and for particle detection in high-energy physics.⁵ LSO:Ce has been grown in monocrystalline,⁶ as well as polycrystalline form.^7,8 LSO doped with other rare-earth elements has been studied as laser materials.^9,10 To improve its performance in devices, the defect characteristics of Ce-doped LSO (LSO:Ce) have been researched extensively.^11–14 For example, microwave resonant cavity measurements were used to obtain photoconductivity spectra, which provide insight into scintillation efficiency and defect trapping.^15,16 Electron paramagnetic resonance (EPR) spectroscopy is another microwave-based technique employed to study Ce³⁺ sites in single crystal LSO:Ce.^13,17,18 Despite the importance of these microwave-based characterizations, not much research has been done to precisely determine the optical properties of LSO near gigahertz (GHz) or terahertz (THz) frequencies. Due to its monoclinic lattice structure, LSO is optically anisotropic, and therefore, its crystal axes directions should be considered when performing EPR or microwave cavity resonance measurements. Two previous studies on LSO characterize its anisotropic optical response: Jellison et al.¹⁹ in the visible spectral range (200–850 nm) and Stokey et al.²⁰ in the infrared (IR) spectral range (30–1200 cm^–1). Stokey et al. also reported anisotropic static permittivity values, which come from an extrapolation of the IR permittivity tensor model along certain crystallographic directions. Since no significant contributions from phonon modes are expected below 30 cm^–1, these values should be similar to those found at GHz or THz frequencies.²⁰ Dominiak-Dzik et al.²¹ reported one real permittivity value of 14.8 for single crystal dysprosium-doped LSO determined by electrical measurements in a frequency range below 1 MHz. However, no crystal axes assignment was given. To facilitate more advanced studies at frequencies below the IR spectral range, it is important to investigate the static/quasi-static permittivity and its anisotropy in LSO single crystals.

In this work, we report the use of THz generalized spectroscopic ellipsometry (THz-GSE) to determine the anisotropic permittivity parameters of LSO. This THz-GSE study expands on a previous IR-GSE study²⁰ and provides a comparison to the DC permittivity parameters from the IR optical model's extrapolation. Knowledge on low symmetry materials properties and measurement approaches is not exhaustive. Given the complexity of monoclinic crystals, it is not trivial that the THz optical properties can be derived by extrapolation from previously investigated infrared optical properties. Our work provides an independent experimental verification that extrapolation provides a good approximation. Our work also demonstrates in necessary yet brief details how such analyses can be performed, and which are not trivial either. X-ray diffraction (XRD) measurements were also used to establish the crystal axes orientations for the investigated samples. The combined XRD/THz-GSE analysis allows a full characterization of LSO's THz optical response relative to its lattice vectors.

LSO has a monoclinic crystal structure and belongs to space group No. 15. For this space group, there are 18 possible choices for the unit cell definition. Here, we chose to employ the $I 2 / c$ cell as was done in Stokey et al.,²⁰ for which the lattice constants are a = 12.362, b = 6.640, c = 10.247 Å, and the monoclinic angle is $β = 102.3 °$⁠. Note that a comparison to the commonly implemented $C 2 / c$ cell^17,22 is also provided in the supplementary material. Shown in Fig. 1 is the LSO $I 2 / c$ unit cell definition used throughout this work. The LSO lattice vectors are initially aligned so that $x ∥ a , y ∥ c ⋆$⁠, and $z ∥ − b$⁠. The Cartesian coordinates of the XRD instrument and THz ellipsometer (x, y, and z) are fixed throughout the experiments, whereas the crystal axes are rotated by the appropriate Euler angles to match the sample surface cut and azimuth orientation for the corresponding measurement.

In order to generate the correct optical response for a given crystal surface and azimuth orientation, the permittivity tensor must be rotated by the corresponding Euler angles.^24,25 For the Euler rotations, we use the z–x′–z″ convention, in which the crystal axes are first rotated about z by the angle $ϕ$⁠, then about x′ by θ, and then about z″ by ψ. The first rotation by $ϕ$ can be thought of as the azimuth orientation for a sample with a given surface orientation. The remaining Euler angles θ and ψ determine a sample's surface orientation.

The samples studied in this work are single crystal LSO:Ce with (110) and approximate $( 9 ¯ 1 4 ¯ )$ surface orientations purchased from MTI Corporation. The dimensions of both samples are (10 × 10) mm² with a nominal thickness of 0.5 mm. The cerium doping concentration is nominally 0.175 mol. %. According to Ref. 26, Ce impurities should not introduce much change in the LSO lattice structure. Therefore, we assume the THz optical properties of LSO and LSO:Ce to be similar when considering relatively low dopant concentrations.

X-ray diffraction measurements in the Bragg–Brentano geometry were employed to gain information on the crystallographic orientation of the LSO samples using a PANalytical X'Pert Pro MRD diffractometer, where data were collected in the 5°–85° 2θ-range. Pole figure measurements were performed using Cu Kα radiation in a PANalytical Empyrean X-ray diffractometer. A polycapillary x-ray lens was used at the incident beam, while a parallel plate collimator was used on the detector side.

THz-GSE was employed to determine the LSO permittivity tensor values in the spectral ranges of 350–510 and 700–970 GHz. All data were recorded using the Mueller matrix formalism, in which the sample's optical response is represented by the 4 × 4 Mueller matrix.²⁷ A home-built THz ellipsometer capable of acquiring the upper-left 3 × 3 block of the complete Mueller matrix was used for all THz-GSE measurements.²⁸ For each wavelength, angle of incidence, and sample azimuth orientation, a single Mueller matrix is recorded. The (110) and approximate $( 9 ¯ 1 4 ¯ )$ LSO samples were measured at two angles of incidence (⁠ $Φ a = 40 °$ and $60 °$⁠), and four different azimuth orientations. All Mueller matrix elements have been normalized to $M 11$ (total reflected intensity). Note that all THz-GSE measurements were performed at room temperature and in the reflection mode. THz-GSE measurements and model analysis were performed with WVASE (J.A. Woollam Co., Inc.). The optical model is comprised of three layers (air–substrate–air), where the layer interfaces are assumed to be plane-parallel and infinitely wide. Model-calculated data can then be fit to the THz-GSE experimental data, where the dielectric tensor elements are unknown variables in the fitting process.

In order to verify the crystal axes directions for our (110) and approximate $( 9 ¯ 1 4 ¯ )$ oriented LSO samples, the XRD pole figure data were acquired and analyzed (see the supplementary material). This analysis provides a set of Euler angles for each sample, which describe the rotations needed to properly align the crystal axes with respect to the XRD diffractometer's x, y, and z directions. By choosing a similar alignment of the Cartesian coordinate frame fixed to the THz ellipsometer, the XRD-determined Euler angles can be used as input parameters for the THz-GSE analysis. The Euler angles determined by XRD for the (110) LSO sample are $ϕ = 0 ° , θ = 29 °$⁠, and $ψ = − 102 °$⁠. These are nearly identical to the expected values of $θ = 28.8 ° = tan − 1 ( b a sec ( β − 90 ) ) °$ and $ψ = − 102.3 ° = − β °$ for a (110) surface oriented crystal, which comes from the geometry of the $I 2 / c$ unit cell. Further details regarding all pole figure measurements, simulations, and Euler angles for the approximate $( 9 ¯ 1 4 ¯ )$ oriented sample can be found in the supplementary material.

All THz-GSE data for the (110) sample are shown in Fig. 2, whereas THz-GSE data for the approximate $( 9 ¯ 1 4 ¯ )$ LSO sample are included in the supplementary material. The oscillating features in the Mueller matrix spectra are due to Fabry–Pérot interferences caused by multiple reflections off of the front- and backside interfaces of the LSO substrates (Fig. 2, upper-left diagram). These additional reflections are advantageous since they constructively interfere with one another resulting in increased sensitivity to the dielectric tensor elements.^29–32 The THz-GSE spectra are also sensitive to other model parameters, such as the substrate thickness and azimuth orientation angles. Therefore, the unknown parameters included in the model fit were: complex-valued permittivity tensor elements (⁠ $ε a , ε b , ε c ⋆ , ε a c ⋆$⁠), (110) and $( 9 ¯ 1 4 ¯ )$ sample thicknesses, and an azimuth angle offset $ϕ off$ for each sample. The offset angles account for the small difference in $ϕ$ when remounting the samples on the XRD and THz-GSE instruments. All model parameters are varied simultaneously to produce the final best-match fit (solid red lines, Fig. 2). The resulting substrate thickness is $( 498 ± 1 ) μ m$ for the (110) sample, and $( 541 ± 1 ) μ m$ for the $( 9 ¯ 1 4 ¯ )$ sample. This agrees well with thicknesses measured by a vernier caliper of 0.51 and 0.54 mm for the (110) and $( 9 ¯ 1 4 ¯ )$ samples, respectively. Only small azimuth offset parameters were produced by the model fit: $ϕ off = ( − 1 ± 0.3 ) °$ for the (110) sample and $ϕ off = ( − 0.3 ± 0.1 ) °$ for the $( 9 ¯ 1 4 ¯ )$ sample.

Although only minimal dispersion is expected within our spectral range, frequency-dependent permittivities were allowed in the optical model by assigning separate permittivity tensors to 350–510 and 700–970 GHz. The values for each tensor were varied independently during the fitting process. This approach was sufficient to achieve a good match to the experimental THz-GSE data. The resulting best-match real-valued permittivity parameters are given in Table I. Only minimal dispersion can be seen when comparing the parameters for 350–510 and 700–970 GHz. Since the strongest optical resonances occur at frequencies higher than 1 THz, the THz permittivities should be similar to the static values.³³ In Table I, the static dielectric constants reported by Stokey et al. are also included.²⁰ Values from the density functional theory (DFT) were calculated using Quantum Espresso³⁴ according to the formula in Ref. 35. Static permittivities from IR-GSE were obtained numerically by extrapolating the IR optical model to zero frequency. We note here that the permittivities reported in Stokey et al.²⁰ were accidentally calculated using an incorrect assignment of the a and c lattice vectors. However, the corrected parameters are included throughout this work (further details on the corrected DFT/IR-GSE analysis are discussed in the supplementary material). Both DFT and IR-GSE DC parameters listed in Table I agree well those found in this work by THz-GSE. Note that the IR-GSE model predicts real permittivity values at 430 GHz (835 GHz) to be $≈ 0.03$ (⁠ $≈ 0.1$⁠) larger than those at zero frequency, demonstrating the existence of slight dispersion inside our spectral range. In addition to the real permittivities, the THz-GSE analysis also reveals small imaginary contributions to LSO's permittivity tensor. The best-fit results for these values are shown in Table II. The imaginary components are almost negligible when compared to the real values (⁠ $< 1 %$⁠). According to the IR-GSE model, the imaginary parts should be $≈ 0.01$ (⁠ $≈ 0.02$⁠) at 430 GHz (835 GHz), which is primarily due to absorption from the lowest-lying IR-active phonon modes. Figure 3 provides a visual comparison between the THz-GSE, IR-GSE, and DFT calculated principle axes parameters as a function of frequency.

Shown in Fig. 4 are the principal dielectric axes overlaid onto the lattice vectors for LSO. As is well known for monoclinic crystals, the principal axes in the low-symmetry crystallographic plane may not necessarily share the same direction as the lattice vectors.³⁶ This is observed for the $I 2 / c$ unit cell definition in the static/quasi-static case, where the principal directions fall near the midpoint between a and $c ⋆$⁠. However, for the more common $C 2 / c$ cell definition, the principal directions do fall close to a and $c ⋆$ (see the supplementary material).

In summary, we have determined the permittivity tensor values for monoclinic single crystal LSO in the spectral ranges of 350–510 and 700–970 GHz. Our permittivity results at THz frequencies agree well with static values reported previously. However, slight dispersion and absorption are still observed within our spectral range. This is in accordance with the eigen dielectric vector summation model implemented by Stokey et al., which includes permittivity contributions from IR-active phonon modes and other high-frequency dielectric resonances. The combined XRD/THz-GSE analysis allows LSO's tensor diagonalization angle to be calculated. We find that the principal dielectric axes fall farther away from the a and $c ⋆$ lattice vectors for the $I 2 / c$ unit cell definition, but fall rather close for the commonly implemented $C 2 / c$ cell definition. We believe that our work provides a timely and valuable contribution to the rapidly expanding field of THz materials and applications, and readers may appreciate technical details for their own investigations. This information is important to consider when performing advanced microwave/THz characterizations involving LSO single crystals.

See the supplementary material for additional information regarding the LSO XRD pole figure analysis, THz-GSE characterization of the ( 9 ¯ 1 4 ¯ ) sample, and revised IR-GSE optical model.

This work is performed in the framework of the Knut and Alice Wallenbergs Foundation funded grant “Wide-bandgap semiconductors for next generation quantum components” (Grant No. 2018.0071), by the Swedish Research Council VR Award Nos. 2016-00889 and 2022-04812, the Swedish Foundation for Strategic Research Grant Nos. RIF14-055 and EM16-0024, by the Swedish Governmental Agency for Innovation Systems VINNOVA under the Competence Center Program Grant No. 2022-03139, by the Swedish Government Strategic Research Area in Materials Science on Functional Materials at Linköping University, Faculty Grant SFO Mat LiU No. 2009-00971, and NanoLund, Lund University. We acknowledge support by the National Science Foundation Award Nos. DMR 1808715 and OIA-2044049, Air Force Office of Scientific Research Award Nos. FA9550-19-S-0003 and FA9550-21-1-0259, the University of Nebraska Foundation, and the J. A. Woollam Foundation.

The authors have no conflicts to disclose.

Sean Knight: Formal analysis (equal); Investigation (equal); Writing – original draft (lead). Steffen Richter: Formal analysis (equal); Writing – review & editing (equal). Alexis Papamichail: Formal analysis (supporting); Investigation (equal). Megan Stokey: Formal analysis (supporting). Rafal Korlacki: Formal analysis (supporting). Vallery Stanishev: Formal analysis (supporting). Philipp Kühne: Formal analysis (supporting). Mathias Schubert: Conceptualization (equal); Supervision (supporting); Writing – review & editing (equal). Vanya Darakchieva: Conceptualization (equal); Funding acquisition (lead); Resources (lead); Supervision (lead); Writing – review & editing (supporting).

The data that support the findings of this study are available from the corresponding author upon reasonable request.