Low-temperature infrared dielectric function of hyperbolic $\alpha$-quartz

TL;DR

This study measures and models the infrared dielectric properties of $ extalpha$-quartz from 1.5 K to 200 K, revealing temperature-dependent phonon behavior and predicting high-quality polaritons in its hyperbolic spectral region.

Contribution

The paper provides a comprehensive temperature-dependent dielectric function model for $ extalpha$-quartz, extending previous high-temperature data down to 1.5 K and predicting low-temperature polariton quality factors.

Findings

Dielectric function characterized from 1.5 K to 200 K.

Phonon parameters fitted with a multioscillator model.

Predicted high-Q polaritons in the hyperbolic spectral region.

Abstract

We report the infrared dielectric properties of $\alpha$-quartz in the temperature range from $1.5\ \mathrm{K}$ to $200\ \mathrm{K}$. Using an infrared free-electron laser, far-infrared reflectivity spectra of a single crystal $y$-cut were acquired along both principal axes, under two different incidence angles, in S- and P-polarization. These experimental data have been fitted globally for each temperature with a multioscillator model, allowing to extract frequencies and damping rates of the ordinary and extraordinary, transverse and longitudinal optic phonon modes, and hence the temperature-dependent dispersion of the infrared dielectric function. The results are in line with previous high-temperature studies, allowing for a parametrized description of all temperature-dependent phonon parameters and the resulting dielectric function from $1.5\ \mathrm{K}$ up to the $\alpha$-$\beta$-phase transition temperature, $T_C = 846\ \mathrm{K}$. Using these data, we predict remarkably high quality factors for polaritons in $\alpha$-quartz's hyperbolic spectral region at low temperatures.

Tables

Table 1. Table 1: Results for E 𝐸 E -type phonon mode frequencies and damping rates used as free parameters to fit the experimental reflectivity spectra. Values for ε ∞ ⟂ subscript 𝜀 subscript perpendicular-to \varepsilon_{\infty_{\perp}} have been averaged over all temperatures.

$j$	$T$	${\Omega }_{{\mathrm{TO}}_j}$	${\gamma }_{{\mathrm{TO}}_j}$	${\Omega }_{{\mathrm{LO}}_j}$	${\gamma }_{{\mathrm{LO}}_j}$
	($K$)	(${\mathrm{cm}}^{-1}$)	(${\mathrm{cm}}^{-1}$)	(${\mathrm{cm}}^{-1}$)	(${\mathrm{cm}}^{-1}$)
3	1.5	$391.5$		$403.0$
	20	$392.1$		$402.8$
	100	$391.9$		$403.0$
	200	$392.4$		$402.8$	$2.6$
4	1.5	$454.0$	$2.6$	$510.5$
	20	$452.4$	$3.4$	$510.4$	$2.3$
	100	$453.6$	$2.2$	$510.0$	$2.1$
	200	$451.2$	$4.0$	$508.8$	$2.8$
5	1.5	$695.9$	$4.9$	$698.4$	$4.0$
	20	$695.8$	$4.9$	$698.5$	$4.5$
	100	$696.0$	$4.2$	$698.1$	$3.9$
	200	$695.3$	$5.6$	$697.7$	$5.0$
6	1.5	$797.2$	$4.8$	$810.0$	$4.3$
	20	$797.2$	$4.4$	$810.2$	$5.0$
	100	$796.9$	$5.1$	$809.9$	$4.1$
	200	$796.5$	$6.0$	$809.1$	$5.2$
7	1.5	$1063.7$	$6.1$	$1230.7$	$8.2$
	20	$1062.9$	$6.3$	$1231.9$	$10.9$
	100	$1063.9$	$6.8$	$1231.2$	$12.0$
	200	$1063.0$	$7.1$	$1230.0$	$12.1$
8	1.5	$1157.2$	$6.2$	$1154.9$	$6.1$
	20	$1156.9$	$6.9$	$1154.4$	$6.2$
	100	$1157.0$	$6.2$	$1154.9$	$6.0$
	200	$1157.0$	$7.2$	$1154.8$	$6.3$
		${\epsilon }_{{\infty }_{⟂}}=2.296$

Table 2. Table 2: Results for A 2 subscript 𝐴 2 A_{2} -type phonon mode frequencies and damping rates used as free parameters to fit the experimental reflectivity spectra. Values for ε ∞ ∥ subscript 𝜀 subscript parallel-to \varepsilon_{\infty_{\parallel}} have been averaged over all temperatures.

$j$	$T$	${\Omega }_{{\mathrm{TO}}_j}$	${\gamma }_{{\mathrm{TO}}_j}$	${\Omega }_{{\mathrm{LO}}_j}$	${\gamma }_{{\mathrm{LO}}_j}$
	($K$)	(${\mathrm{cm}}^{-1}$)	(${\mathrm{cm}}^{-1}$)	(${\mathrm{cm}}^{-1}$)	(${\mathrm{cm}}^{-1}$)
1	1.5	$360.7$		$384.8$
	20	$360.1$		$384.8$
	100	$360.9$		$384.3$
	200	$361.2$	$2.1$	$385.2$
2	1.5	$497.9$	$3.1$	$553.6$	$2.8$
	20	$498.4$	$3.6$	$554.4$	$3.3$
	100	$498.0$	$3.2$	$553.1$	$3.1$
	200	$496.8$	$4.7$	$552.2$	$4.0$
3	1.5	$773.7$	$5.4$	$789.9$	$6.3$
	20	$774.4$	$5.6$	$789.9$	$6.5$
	100	$774.1$	$5.8$	$789.9$	$7.1$
	200	$775.3$	$5.9$	$788.8$	$6.8$
4	1.5	$1073.0$	$6.2$	$1238.7$	$12.4$
	20	$1072.7$	$3.5$	$1241.2$	$11.2$
	100	$1072.8$	$4.9$	$1239.2$	$11.1$
	200	$1070.9$	$5.3$	$1239.6$	$11.5$
		${\epsilon }_{{\infty }_{\parallel }}=2.334$

Table 3. Table 3: Comparison of α 𝛼 \alpha -quartz’s hyperbolic regions, H1 and H2, at 20   K times 20 kelvin 20\text{\,}\mathrm{K} and 300   K times 300 kelvin 300\text{\,}\mathrm{K} , with the naturally hyperbolic hexagonal boron nitride, comprising the parametrized TO and LO phonon frequencies, Q 𝑄 Q -factors at Re ⁡ ( ε ) = − 2 Re 𝜀 2 \operatorname{Re}(\varepsilon)=-2 , and Re ⁡ ( ε ⟂ ( ∥ ) ) \operatorname{Re}(\varepsilon_{\perp(\parallel)}) where Re ⁡ ( ε ∥ ( ⟂ ) ) = − 2 \operatorname{Re}(\varepsilon_{\parallel(\perp)})=-2 for both principal crystal axes, based on the power law fits.

Material	$T$ ($K$)	${\Omega }_{⟂}$ (${\mathrm{cm}}^{-1}$)		${\Omega }_{\parallel }$ (${\mathrm{cm}}^{-1}$)		${Q|}_{\mathrm{Re}\left(\epsilon \right)=-2}$		${\mathrm{Re}\left({\epsilon }_{⟂(\parallel )}\right)|}_{\mathrm{Re}\left({\epsilon }_{\parallel (⟂)}\right)=-2}$
		TO	LO	TO	LO	$Q_{⟂}$	$Q_{\parallel }$	$\mathrm{Re}\left({\epsilon }_{⟂}\right)$	$\mathrm{Re}\left({\epsilon }_{\parallel }\right)$
$\alpha$-SiO2 H1	$20$	$391.6$	$402.7$	$358.9$	$384.4$	$464$	$681$	$11.0$	$2.2$
[Gervais and Piriou, 1975]	$300$	$393.1$	$402.7$	$364.3$	$387.2$	$87$	$69$	$11.9$	$2.2$
$\alpha$-SiO2 H2	$20$	$454.8$	$511.3$	$503.4$	$555.3$	$214$	$249$	$1.1$	$15.0$
[Gervais and Piriou, 1975]	$300$	$449.0$	$508.2$	$492.2$	$550.6$	$91$	$76$	$0.9$	$23.9$
h-BN [Caldwell et al., 2014]	$300$	$1360$	$1614$	$760$	$825$	$221$	$399$	$8.0$	$2.8$

Equations

$$R_{xx}$$

$$R_{yy}$$

$$R_{zz}$$

$$k_{{\mathrm{i}}_z}$$

$$k_{{\mathrm{t}}_z}^{\mathrm{o}}$$

$$k_{{\mathrm{t}}_z}^{\mathrm{e}}$$

$$\epsilon (\omega )={\epsilon }_{\infty }\prod _j\frac{{\Omega }_{{\mathrm{LO}}_j}^2-{\omega }^2-\mathrm{i}{\gamma }_{{\mathrm{LO}}_j}\omega }{{\Omega }_{{\mathrm{TO}}_j}^2-{\omega }^2-\mathrm{i}{\gamma }_{{\mathrm{TO}}_j}\omega },$$

$$I_{\mathrm{P}}$$

$$I_{\mathrm{S}}$$

$$y_{j}(T)=\left|y_{j}($0\text{\,}\mathrm{K}$)+k(T_{C}-T)^{1/2}\right|,$$

$$Q=\frac{\omega\frac{\mathrm{d}\operatorname{Re}(\varepsilon)}{\mathrm{d}\omega}}{2\operatorname{Im}(\varepsilon)}\Biggm{\lvert}_{\operatorname{Re}(\varepsilon)=-2.}$$

Full text

Infrared spectroscopy, anharmonic lattice dynamics, optical phonons, dielectric properties, hyperbolicity

I Introduction

To date, quartz is of great technological importance, mainly thanks to its pronounced piezoelectricity and abundance in nature.Bosak et al. (2012) Despite being extremely well-studied, quartz and, in particular, its lattice dynamics have been subject to recent research. For instance, its broken inversion symmetry makes it an excellent model system for nonlinear optical techniques, including SFG and SHG phonon spectroscopy.Liu and Shen (2008); Winta et al. (2018) Often, however, these advanced techniques require good knowledge of the linear optical behavior of the sample which is fully described by its frequency $\omega$-dependent dielectric tensor, \overset{\text{\tiny\bm{\leftrightarrow}}}{\varepsilon}(\omega).Liu and Shen (2008); Paarmann et al. (2015, 2016); Winta et al. (2018)

The dielectric properties of $\alpha$-quartz are of particular interest.Spitzer and Kleinman (1961); Merten (1968); Falge and Otto (1973); Gervais and Piriou (1975); Humlíček (1994) The uniaxial crystal naturally features hyperbolicity in the far-infrared (far-IR), i.e., its diagonal permittivity tensor has both, positive and negative principal components, such that $\operatorname{Re}(\varepsilon_{\parallel})\cdot\operatorname{Re}(\varepsilon_{\perp})<0$.da Silva et al. (2012); Passler and Paarmann (2017) This results in the momentum $k$-space’s isofrequency surfaces being open hyperboloids (as opposed to, e.g., closed spheres for isotropic media), supporting high-$k$ waves.Guo et al. (2012); Caldwell et al. (2014); Dai et al. (2015a); Li et al. (2018); Ma et al. (2018) These states, thanks to their large spatial frequency, can be utilized in nanophotonic devices for, e.g., subdiffractional imaging or nanolithography using hyperlenses.Liu et al. (2007); Xiong et al. (2009) Moreover, $\alpha$-quartz exhibits both, type I ($\mathrm{Re}(\varepsilon_{\parallel})<0$, $\mathrm{Re}(\varepsilon_{\perp})>0$) and type II ($\operatorname{Re}(\varepsilon_{\parallel})>0$, $\operatorname{Re}(\varepsilon_{\perp})<0$) hyperbolicity in close spectral proximity, further adding to its design flexibility.

Despite its prevailing relevance, low-temperature studies of $\alpha$-quartz’s dielectric function have only been reported in the terahertz range.Davies et al. (2018) In 1975, Gervais and Piriou performed extensive temperature-dependent IR reflectivity measurements on $\alpha$-quartz at high temperatures ranging from $295\text{\,}\mathrm{K}975\text{\,}\mathrm{K}$.Gervais and Piriou (1975) Here, we determine $\alpha$-quartz’s IR dielectric function at low temperatures, ranging from $1.5\text{\,}\mathrm{K}200\text{\,}\mathrm{K}$, using an IR free-electron laser (FEL) to acquire reflectivity spectra for various polarization and orientation combinations. These we then fit globally with a multi-oscillator model taking independently into account TO and LO phonon frequencies and damping rates along both principal axes.

II Experimental setup

The experimental setup features a noncollinear autocorrelator geometry in reflection with two focused FEL beams impinging on the sample incident at [math] and [math]. The intensity of the reflected beams is then detected by two home-built pyroelectric photodetectors.

The sample is placed inside a helium bath cryostat (CryoVac GmbH & Co KG), facilitating the setup’s low-temperature operation, ranging from $1.5\text{\,}\mathrm{K}325\text{\,}\mathrm{K}$ with <$$0.1\text{\,}\mathrm{K} precision. Optical KRS-5 and diamond windows grant IR accessibility to the sample chamber while the built-in motorized sample stage allows for alignment of the vertical position as well as the polar and azimuthal angle of the sample.

Details on the FEL are reported elsewhere.Schöllkopf et al. (2015) For these measurements, the electron energy was set to $23\text{\,}\mathrm{MeV}$ and $32\text{\,}\mathrm{MeV}$, allowing to tune the output wavelength, $\lambda$, from $12\text{\,}\mathrm{\SIUnitSymbolMicro m}28\text{\,}\mathrm{\SIUnitSymbolMicro m}$ ($350\text{\,}{\mathrm{cm}}^{-1}850\text{\,}{\mathrm{cm}}^{-1}$) and from $7\text{\,}\mathrm{\SIUnitSymbolMicro m}18\text{\,}\mathrm{\SIUnitSymbolMicro m}$ ($550\text{\,}{\mathrm{cm}}^{-1}1400\text{\,}{\mathrm{cm}}^{-1}$), respectively, through variation of the motorized undulator gap. Polarization rotation of the linearly polarized FEL beam is achieved through two subsequent wire-grid polarizers (Thorlabs, Inc.) set to [math] and either [math] or [math] with respect to the incoming P-polarization, for P- and S-polarization, respectively.

The sample studied here is an optically polished $\alpha$-quartz $y$-cut single crystal (MaTecK GmbH) with the [0001] crystal axis, i.e., the optic $c$-axis, parallel to the surface plane. In this arrangement, both, the ordinary ($E$-type) and extraordinary ($A_{2}$-type) IR-active vibrational modes can be probed either exclusively or simultaneously, depending on the $c$-axis orientation, adjustable via the sample’s azimuthal angle, and the FEL beam polarization.

III Dispersion model

In order to model the IR reflectivity of the $\alpha$-quartz $y$-cut for arbitrary angles of incidence, $\alpha_{\mathrm{i}}$, as well as horizontal ($c\parallel x$) and vertical ($c\parallel y$) $c$-axis orientations, we evaluate the elements of the Fresnel reflection tensor. These are straightforwardly derived from Maxwell’s equations under the boundary condition of in-plane field conservation at the interface with air and read:

[TABLE]

Here, $k_{\mathrm{i}_{z}}$, $k_{\mathrm{t}_{z}}^{\mathrm{o}}$, and $k_{\mathrm{t}_{z}}^{\mathrm{e}}$ denote, respectively, the normal-to-surface ($z$) components of the complex wave vectors of the incoming, and ordinary and extraordinary transmitted waves inside the sample. They read:Mosteller, Jr. and Wooten (1968)

[TABLE]

All of these dispersive quantities are ultimately dependent on the elements of the diagonal dielectric tensor, $\varepsilon_{\perp}$ and $\varepsilon_{\parallel}$, perpendicular and parallel to the optic $c$-axis, respectively. The highly dispersive dielectric functions, $\varepsilon_{\perp}(\omega)$ and $\varepsilon_{\parallel}(\omega)$, can be described using a four-parameter semi-quantum (FPSQ) model:Adachi (1999); Gervais and Piriou (1974, 1975)

[TABLE]

where $\varepsilon_{\infty}$ is the high-frequency contribution to the dielectric function. $\Omega_{\mathrm{TO(LO)}_{j}}$ denotes the TO (LO) phonon frequency of the $j$th vibrational mode, and $\gamma_{\mathrm{TO(LO)}_{j}}$ its respective damping rate. We note that the FPSQ model in Eq. 7 can result in the imaginary part of the dielectric function, $\operatorname{Im}(\varepsilon)$, taking on negative values for large TO-LO splittings of the damping rates, $\Delta\gamma_{j}=\gamma_{\mathrm{LO}_{j}}-\gamma_{\mathrm{TO}_{j}}$.Gervais and Piriou (1974) To avoid this unphysical regime of the multi-oscillator model, we later apply a penalty to negative values of $\operatorname{Im}(\varepsilon)$ during the least-squares fitting routine.

Finally, the reflected light intensities for P- and S-polarization, $I_{\mathrm{P}}$ and $I_{\mathrm{S}}$, respectively, are then straightforwardly given by:

[TABLE]

where $E_{\mathrm{i}}$ denotes the incident electric field. This leaves the TO and LO phonon frequencies, $\Omega_{\mathrm{TO}_{j}}$ and $\Omega_{\mathrm{LO}_{j}}$, their respective damping rates, $\gamma_{\mathrm{TO}_{j}}$ and $\gamma_{\mathrm{LO}_{j}}$, as well as the high-frequency contributions, $\varepsilon_{\infty_{\perp}}$ and $\varepsilon_{\infty_{\parallel}}$, as the only independent variables in the model.

IV Results

We have measured all possible combinations of the sample’s $c$-axis orientation (vertical and horizontal), the FEL beam polarization (P and S), as well as the angle of incidence ([math] and [math]) for four temperatures: $1.5\text{\,}\mathrm{K}$, $20\text{\,}\mathrm{K}$, $100\text{\,}\mathrm{K}$, and $200\text{\,}\mathrm{K}$. As an example, we show the experimental data for all geometries and both incidence angles at the base temperature, $T=$1.5\text{,}\mathrm{K}$$, in Fig. 1.

Spectral regions of near-perfect reflectivity emerge between corresponding TO and LO phonon resonances where $\operatorname{Re}(\varepsilon)$ takes on negative values. These so-called Reststrahlen bands are particularly pronounced for modes with large TO-LO splittings, i.e., high oscillator strengths.Adachi (1999) Each measurement configuration, i.e., P- or S-polarization with horizontal or vertical $c$-axis orientation, results in a specific direction of the IR electric field with respect to the principal crystal axes, and thus probes one of the two unique elements of the dielectric tensor, $\varepsilon_{\perp}$ and $\varepsilon_{\parallel}$, exclusively, or both simultaneously. For instance, S-polarized reflectivity with $c\parallel x$ as well as P-polarized reflectivity with $c\parallel y$ (blue and orange curves in Fig. 1, respectively) are exclusively sensitive to the ordinary $E$-type modes and therefore solely probe $\varepsilon_{\perp}$. Similarly, S-polarized reflectivity with $c\parallel y$ (red curve), probes the extraordinary $A_{2}$-type modes, thus $\varepsilon_{\parallel}$. Comparing these spectra of exclusive sensitivity to $\varepsilon_{\perp}$ and $\varepsilon_{\parallel}$, e.g. blue vs. red curves in Fig. 1, respectively, reveals $\alpha$-quartz’s pronounced uniaxial anisotropy, being the result of different numbers of IR-active modes and significant frequency shifts between its principal axes. P-polarized reflectivity with $c\parallel x$ (green curve), on the other hand, is sensitive to both principal components. This becomes apparent in the reflectivity spectra (green curves) as features attributable to both, $E$- and $A_{2}$-type modes, are present, the former being more pronounced for the more grazing [math] incidence angle, i.e., for a larger incoming out-of-plane field component.

The experimental spectra have been fitted with a nonlinear least squares regression procedure applied globally to the entire data set for each temperature using Eqs. 1–9, yielding the frequencies and damping rates of both, $E$- and $A_{2}$-type phonon modes, as well as the high-frequency contributions, $\varepsilon_{\infty_{\perp}}$ and $\varepsilon_{\infty_{\parallel}}$.

The fits are in good agreement with the experimental data across the whole data set. Noticeable deviations from the experimental data can be observed in the lower frequency region from $450\text{\,}{\mathrm{cm}}^{-1}550\text{\,}{\mathrm{cm}}^{-1}$ which we suspect are caused by two parasitic spectral contributions: (i) The cryostat windows—although being wedged—cause a Fabry-Perot-like spectral modulation on top of the measured raw data. This modulation happens to be particularly pronounced at the lower Reststrahlen region at $\alpha_{\mathrm{i}}=$$$ and becomes more prominent where signal levels are constantly high, i.e., in Restrahlen bands. (ii) CO2 bending modes in air cause strong IR absorption in this spectral region which we minimize by flooding the optical setup with N2 gas. Both of these unwanted effects are being corrected for by measuring reference spectra in S- and P-polarization using a gold mirror as a nominally perfect reflector in place of the sample and dividing the raw reflectivity spectra by their corresponding reference. However, a certain spectral modulation remains and is observable in the corrected spectra. Nevertheless, these modulations do not have a strong influence on the fit procedure as positions and widths of Reststrahlen edges remain largely unaffected.

The resulting fit parameters from all measurements for $E$- and $A_{2}$-type phonon modes are summarized in Tables 1 and 2, respectively. Here, damping rates that fall below the FEL linewidth of $\Delta\omega\gtrsim$2\text{,}{\mathrm{cm}}^{-1}$$ are indicated as <2.0. Values for $\varepsilon_{\infty_{\perp}}$ and $\varepsilon_{\infty_{\parallel}}$ are averaged over all four measured temperatures as the values do not show any significant temperature dependence.

V Discussion

In Figs. 2 and 3, we plot the temperature dependence of our fit results for TO and LO phonon frequencies and damping rates together with the values determined by Gervais and Piriou who studied the dielectric properties of $\alpha$-quartz at high temperatures.Gervais and Piriou (1975) Overall, our results are largely consistent with Gervais and Piriou’s previous work as the extension of our low-temperature results to the high-temperature values taken from Ref. 9 is rather gradual. Notably, the majority of modes still experience a significant decrease in damping rates below $295\text{\,}\mathrm{K}$. In particular, the spectrally lower $E$-type $j=3,4$ and $A_{2}$-type $j=1,2$ modes which support $\alpha$-quartz’s pronounced hyperbolicity, experience a reduced damping rate by nearly a factor of 2 compared to room temperature.

To describe the temperature dependence of both, the phonon frequencies and damping rates, we apply a power law fit with the vertex at $T_{C}$ to the entire temperature range including Gervais and Piriou’s high-temperature data:Polyakova (2014)

[TABLE]

where $y_{j}$ denotes either the phonon frequency, $\Omega_{j}$, or the damping rate, $\gamma_{j}$, of the $j$th mode. The fitted curves are also shown in Figs. 2 and 3 as solid lines which describe the temperature-dependent behavior of $\Omega_{j}(T)$ and $\gamma_{j}(T)$ with good accuracy.

From the phonon frequencies and damping rates, we can now easily compute the dielectric functions, $\varepsilon_{\perp}(\omega)$ and $\varepsilon_{\parallel}(\omega)$, using Eq. 7. A fully parametrized and temperature-dependent dielectric tensor based on the power law fits to our as well as Gervais and Piriou’s high-temperature data can be calculated using the Python Jupyter Notebook or MATLAB script supplied in Ref. Winta et al., 2019 which give physical results for $\varepsilon_{\perp}(\omega)$ and $\varepsilon_{\parallel}(\omega)$ over the full wavelength range studied ($350\text{\,}{\mathrm{cm}}^{-1}1380\text{\,}{\mathrm{cm}}^{-1}$). We note, that we restrict the validity of the resulting dielectric function to quartz’s $\alpha$-phase, where the power law fits describe the data with high accuracy. However, our simple model fails to describe the resonant behavior of damping rates at $T_{C}$ and leads to instabilities of the FPSQ model for $T>T_{C}$.

We exemplarily plot the resulting real and imaginary parts of $\varepsilon_{\perp}(\omega)$ and $\varepsilon_{\parallel}(\omega)$ at $T=$20\text{,}\mathrm{K} (Fig. [4](#S5.F4)), a typical temperature where numerous current polariton studies are being performed, e.g. using cryogenic scanning near-field optical microscopy.Ni *et al.* ([2018](#bib.bib27)) Light grey shades mark type I hyperbolic regions where $\operatorname{Re}(\varepsilon_{\parallel})<0$ and $\operatorname{Re}(\varepsilon_{\perp})>0$, while darker shades indicate type II hyperbolic bands where $\operatorname{Re}(\varepsilon_{\parallel})>0$ and $\operatorname{Re}(\varepsilon_{\perp})<0$. Especially in the lower spectral region, between $\sim360\text{,}{\mathrm{cm}}^{-1}$and$\sim$$550\text{,}{\mathrm{cm}}^{-1}$, pronounced hyperbolic bands of both types, I and II, emerge.

To assess whether these hyperbolic bands are suitable for potential nanophotonic applications, we evaluate the quality factor $Q$ as a figure of merit (FOM) which reads:Caldwell et al. (2015)

[TABLE]

For comparability, we determine $Q$ in all hyperbolic bands in the lower spectral region between $361\text{\,}{\mathrm{cm}}^{-1}$ and $554\text{\,}{\mathrm{cm}}^{-1}$ at $\operatorname{Re}(\varepsilon)=-2$. Thus, we here follow the common practice in the field of plasmonics where $\operatorname{Re}(\varepsilon)=-2$ marks the peak of the absorption cross-section of a spherical nanoparticle in air due to a localized surface plasmon resonance.Caldwell et al. (2015) Another important property of hyperbolic materials is the ratio $\varepsilon_{\perp}/\varepsilon_{\parallel}$ as it defines the rigid propagation direction of hyperbolic polaritons in the given material and plays a crucial role in the design of, e.g., hyperlenses.Caldwell et al. (2014) For this reason, we also evaluate $\operatorname{Re}(\varepsilon_{\perp(\parallel)})$ at $\operatorname{Re}(\varepsilon_{\parallel(\perp)})=-2$. Both FOMs are given in Table 3 for $\alpha$-quartz (SiO2) at $20\text{\,}\mathrm{K}$ and $300\text{\,}\mathrm{K}$ as well as hexagonal boron nitride (h-BN) for comparison. Hexagonal boron nitride has recently been subject to various studies, making use of its natural hyperbolicity and excellent $Q$-factors.Caldwell et al. (2014, 2015); Giles et al. (2018); Dai et al. (2014, 2015b); Li et al. (2018) Here, we refer to h-BN as a benchmark system to evaluate $\alpha$-quartz’s potential for nanophotonic device applications.

The comparison shows that $\alpha$-quartz at low temperatures offers very good $Q$-factors, even surpassing those reported for h-BN at room temperature. We note, that reflectivity-based methods can only offer limited sensitivity to the small off-resonance imaginary part of the dielectric function. Therefore, $Q$-factors should be understood as estimates and a precise determination of the polariton performance requires a more direct measurement. The analysis also shows that $\alpha$-quartz enables the high-$k$ states characteristic for hyperbolic materials in a distinctly different spectral range, i.e., between $361\text{\,}{\mathrm{cm}}^{-1}$ and $554\text{\,}{\mathrm{cm}}^{-1}$, as opposed to h-BN which exhibits type I hyperbolicity from $760\text{\,}{\mathrm{cm}}^{-1}$ to $825\text{\,}{\mathrm{cm}}^{-1}$ and type II hyperbolicity from $1360\text{\,}{\mathrm{cm}}^{-1}$ to $1614\text{\,}{\mathrm{cm}}^{-1}$. Furthermore, $\alpha$-quartz possesses both, type I and type II hyperbolic bands within the same spectral region, opening up additional design opportunities for nanophotonic devices. Also in terms of propagation direction of hyperbolic polaritons which is defined by the ratio $\varepsilon_{\perp}/\varepsilon_{\parallel}$, $\alpha$-quartz offers additional flexibility as its lower pair of hyperbolic bands (indicated as “H1” in Fig. 4 and Table 3) exhibits similar properties as h-BN whereas band “H2” provides significantly different values while being in very close spectral proximity to H1. While the latter is also true at room temperature, $Q$-factors of hyperbolic polaritons experience a substantial improvement by at least a factor of 2 as compared to room temperature, promising superior performance of nanophotonic devices utilizing $\alpha$-quartz’s hyperbolicity at cryogenic temperatures.

VI Conclusion

We report the dielectric properties of $\alpha$-quartz in the temperature range from $1.5\text{\,}\mathrm{K}200\text{\,}\mathrm{K}$ using IR reflectivity measurements and a global fit procedure to extract the temperature-dependent phonon frequencies and damping rates. From the results, we calculate the in-plane and out-of-plane dielectric function, $\varepsilon_{\perp}$ and $\varepsilon_{\parallel}$, respectively, which indicate remarkably high $Q$-factors for polaritons in $\alpha$-quartz’s naturally hyperbolic spectral region while offering additional design opportunities over established hyperbolic materials like h-BN.

Acknowledgment

The authors thank Wieland Schöllkopf and Sandy Gewinner for their assistance with operating the free-electron laser.