1 Introduction

Quartz has been extensively utilised as a substrate for optical characterisation of non-free standing samples, both in the near-infrared to visible region [1,2,3,4], and the terahertz region [5,6,7,8,9,10] of the electromagnetic spectrum, owing to quartz having low absorption and high transmission in these regions [11, 12]. Samples supported by a substrate form a heterostructure and optical characterisation of these samples will often require optical modelling of the sample–substrate heterostructure [10, 13]. For precise optical modelling of heterostructures with quartz, knowledge of the complex refractive index, \(\tilde n = n + i\kappa \), of quartz is required where n is the refractive index and κ is the extinction coefficient. While there have been reports previously on the complex refractive index of quartz at room temperature and low temperature [12], little is known on the temperature-dependence of the complex refractive index of quartz in the terahertz region.

Quartz is an earth-abundant mineral form of tetrahedrally bonded silica (i.e. silicon dioxide, SiO2). At temperatures below 846 K, the stable crystalline form of quartz is the α-phase which is uniaxial and has a chiral structure that may be right or left handed [14]. Thus light propagating along the optic axis (also called c-axis or z-axis) suffers no birefringence but experiences optical activity, while light propagating along other directions exhibits birefringence. At terahertz frequencies, x-cut quartz has been shown to be highly birefringent over the range 100 GHz to 3.5 THz [15], and stacks of birefringent quartz plates have been used to create an achromatic quarter waveplate operating at terahertz frequencies [16]. When used as substrates for optical and THz spectroscopy, wafers of z-cut quartz are typically used in order to avoid birefringence affecting light propagating normal to the surface of the wafer, which is the usual measurement geometry.

α-quartz has a 3-formula unit (i.e. 9 atom) basis in its trigonal primitive unit cell (space group P3221) and thus possesses 24 optical and 3 acoustic phonon branches. The lowest energy pair of optical phonon modes is both infrared and Raman active. These modes have been calculated to have an energy of 133.3 cm− 1 for the TO mode and 133.4 cm− 1 for the LO mode at the Brillouin zone centre assuming a temperature of 0 K [17]. The small splitting of the energy degeneracy of these E-symmetry TO and LO modes is expected as a consequence of anharmonicity, but is difficult to observe experimentally owing to phonon-lifetime broadening exceeding the splitting [18]. The next highest infrared-active mode is also expected to be a (near) doubly degenerate E-mode at 262 cm− 1 [17, 19]. Bréhat and Wyncke used infrared spectroscopy to measure phonon-modes at 128 cm− 1 (\(\sim \) 3.8 THz) and 265 cm− 1 (\(\sim \) 8 THz) in close agreement with theory [12]. There is also an A1-symmetry phonon at an energy of \(\sim \) 220 cm− 1 [17, 19]; however, this mode is not infrared active. Since the lowest energy E-mode (128 cm− 1) lies close to the region generally of interest in terahertz time-domain spectroscopy (TDS) and optical pump terahertz probe spectroscopy (OPTPS) measurements, it is important to understand the temperature-dependence of this phonon mode and how it could affect TDS and OPTPS measurements.

Previous measurements of the complex refractive index at terahertz frequencies were performed using either Fourier spectroscopy [20, 21] or infrared grating spectroscopy [22]. Interestingly, one of the earliest demonstrations of THz-TDS was by Grischkowsky et al. [23] who extracted the refractive index and absorption coefficient of quartz at room temperature over a range from 0.2–2.0 THz. Over the past decade the bandwidth available to terahertz TDS and OPTPS has increased owing to improvements of emitters and detectors [24]. These improvements now allow the complex refractive index of materials to be measured with high precision over a large frequency range.

Here, we measure the complex refractive index of α-quartz (SiO2) at 10, 40, 80, 120, 160, 200 and 300 K, between 0.5 THz and 5.5 THz using terahertz TDS. The electric field of the terahertz probe was aligned perpendicular to the c-axis of the α-quartz sample. We present the temperature-dependence of the \(\sim \) 3.9 THz phonon mode (E-symmetry mode [25]), showing that the phonon mode red shifts with increasing temperature from 3.96 THz at 10 K to 3.86 THz at 300 K. The oscillator strength of the 3.9 THz phonon mode also decreases with increasing temperature. We also show that THz-TDS allows the dispersion relation for the upper and lower branches of quartz’s phonon-polariton dispersion to be determined experimentally.

2 Experimental Methods

2.1 Terahertz Time-domain Spectroscopy (TDS)

A terahertz time-domain spectrometer built at Oxford was used for this study. The system has been described previously [26]. Briefly, the output of an amplified Ti:Sapphire laser system with 30 fs pulses at a repetition rate of 5 kHz with a central wavelength of 800 nm and an average power of 4 W was split into two beam paths: a probe beam; and a gate beam. Terahertz pulses were generated using a spintronic emitter [27, 28]. Terahertz pulses were detected by electro-optic sampling using a 0.2 mm thick GaP(110) crystal, Wollaston prism and a pair of balanced photodiodes. Temperature-dependent TDS measurements of quartz were performed using a cold-finger cryostat (Oxford Instruments). By measuring the signal in the presence and absence of the sample the terahertz transmission in the frequency domain was obtained by Fourier transforming the signal and dividing the sample signal by the reference.

2.2 Fourier Transform Infrared Spectroscopy (FTIR)

The reflection (R) and transmission (T) spectra were measured from 100 to 350 cm− 1 using a Bruker Vertex 80v Fourier transform infrared (FTIR) spectrometer with a globar source. A liquid helium cooled silicon bolometer (Infrared Laboratories Inc.) and Mylar beamsplitter were used for the measurements. Reflection and transmission spectra were converted into extinction coefficient spectra, κ by

$$ \kappa(\bar\nu) = \frac{-1}{4\pi \bar\nu d} \ln\left( \frac{T}{1-R}\right), $$
(1)

where d is the thickness of the sample and \(\bar \nu \) is the wavenumber.

3 Results

3.1 Temperature-dependent Refractive Index of α-Quartz

Single-crystal α-quartz wafers were purchased from Bright Crystals Technology Inc. The wafers were z-cut to an accuracy of ± 15′, had a specified thickness of 2 ± 0.05 mm and diameter of approximately 13 mm. The wafers were double-sided polished with a scratch/dig specification of 40/20. Time-domain spectroscopy, detailed in Section 2, was used to measure the amplitude and phase of the transmission through α-quartz of a broadband terahertz pulse in the time-domain. The thickness of the particular wafer used in the data presented here was measured using a micrometer screw gauge at the centre and edge of the wafer and was determined to be 2.05 ± 0.03 mm. The custom-built TDS system at Oxford is well-suited for determining the dielectric function at different temperatures as the cryostat and THz optics are enclosed in the same vacuum space, avoiding the need for cryostat windows. The absence of windows greatly simplifies the extraction of the dielectric function. The terahertz transmission in the frequency-domain was obtained by measuring the signal response of quartz and vacuum in the time-domain and dividing the Fourier transformed signal of the quartz by that of vacuum. The method for obtaining the complex refractive index from TDS is described later.

Figure 1a shows the extinction coefficient measured by TDS and FTIR spectroscopy at 10 K demonstrating the absorption of the first two E-symmetry phonon modes of α-quartz at \(\sim \) 3.95 THz (132 cm− 1) and \(\sim \) 7.94 THz (265 cm− 1). Figure 1b-e shows the refractive index (b and c) and extinction coefficient (d and e) of quartz measured by TDS between 10 and 300 K. Table 1 also tabulates the refractive index of quartz as a function of both frequency and temperature. The lowest energy E-mode is measured to have a resonant frequency of \(\sim \) 3.9 THz (132 cm− 1) as is demonstrated by the derivative-like feature in the refractive index (Fig. 1c) and peak in the extinction coefficient (Fig. 1e). A more detailed discussion of the 3.9 THz E-mode is provided in Section 3.2. If this phonon mode is ignored, the (real) refractive index only has a slight temperature dependence, thus a reasonable phenomenological expression for the refractive index of quartz along its optical axis is,

$$ n(\nu) = \left\{\begin{array}{lllllll} 2.0928 & \text{if } 0.5 \text{THz} < \nu < 1 \text{THz}\\ 2.0959 - 0.0066\nu + 0.0035 \nu^{2} & \text{if } 1 \text{THz} < \nu < 5.5 \text{THz} \end{array}\right., $$
(2)

where ω = 2πν.

Fig. 1
figure 1

a Extinction coefficient measured by TDS (ν < 4.8 THz) and FTIR (ν > 4.8 THz) spectroscopy at 10 K showing the absorption of the first two E-symmetry phonon modes of α-quartz at 3.96 THz (132 cm− 1) and 7.94 THz (265 cm− 1). b, c Refractive index and d and e extinction coefficient of α-quartz as a function of frequency at 10, 40, 80, 120, 160, 200 and 300 K. Refractive indices and extinction coefficients were measured using terahertz time-domain spectroscopy

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Table 1 Refractive index of α-quartz
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Now we describe the conversion from the terahertz transmission function (with phase information) of the sample to the complex refractive index. Consider an incident propagating monochromatic wave, Ei = E0ei(kz−ωt) with a dispersion relation \(k = \tilde {n}\omega /c\) where \(\tilde {n}\) is the complex refractive index of a material. The transmitted electric fields Er and Es for (a) vacuum and (b) a thick sample of thickness d, respectively (Fig. 2), are defined in the frequency domain by:

$$ E_{\text{r}} = e^{i(\tilde{n}_{\text{r}}d)\omega /c} E_{\text{i}} $$
(3)
$$ E_{\text{s}} = t_{\text{r,s}} t_{\text{s,r}} e^{i(\tilde{n}_{\text{s}}d)\omega /c} E_{\text{i}} $$
(4)

where the subscripts r and s denote the vacuum reference and sample, \(t_{i,j}= 2\tilde {n}_{i}/(\tilde {n}_{i}+\tilde {n}_{j})\) is the Fresnel transmission at the boundary between materials i and j. The terahertz sample transmission is given by:

$$ T(\omega) = \frac{E_{\text{s}}}{E_{\text{r}}} = \frac{4\tilde{n}_{\text{s}}}{(1+\tilde{n}_{\text{s}})^{2}} e^{i((\tilde{n}_{\text{s}}-1)d)\omega /c} $$
(5)
Fig. 2
figure 2

Transmission of a terahertz pulse through: a vacuum and b a thick sample of thickness d. The reflections from the back surface of the sample are time windowed out during the experiment

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There are no analytic solutions for the complex refractive index, \(\tilde {n}_{\text {s}}\), for the above equation so \(\tilde {n}_{\text {s}}\) will need to be found numerically. Owing to the complex exponent in Eq. 5, T(ω) is oscillatory in n-space which would lead to convergence issues if a standard least squares fitting method is applied. To overcome this potential fitting difficulty, an alternate error function is used in the fitting algorithm [29, 30], given by:

$$ \xi(n,\kappa) = \xi_{\text{mod}}^{2}(n,\kappa) + \xi_{\arg}^{2}(n,\kappa) $$
(6)

where n and κ are the real and imaginary parts of the complex refractive index. The functions ξmod and \(\xi _{\arg }\) are defined by:

$$ \xi_{\text{mod}}(\omega;n,\kappa) = \log(|T(\omega;n,\kappa)|) - \log(|T_{\text{meas}}(\omega)|) $$
(7)
$$ \xi_{\arg}(\omega;n,\kappa) = \arg(T(\omega;n,\kappa)) - \arg(T_{\text{meas}}(\omega)) $$
(8)

where Tmeas is the measured value and T is the fitted value. The objective of the fitting algorithm is to minimise the error function, ξ.

3.2 Terahertz Frequency Phonon Mode in α-Quartz

The spectral response of the 3.9 THz E-phonon mode in α-quartz (Fig. 3) was modelled by a Lorentz oscillator with temperature-dependent fitting parameters. The refractive index was fitted using two Lorentz oscillators to account for the \(\sim \)3.9 THz E-mode and the presence of a second broad E-mode at \(\sim \)8 THz (= 265 cm− 1) [12, 17]. The A1 mode at 220 cm− 1 is not infrared active and thus does not contribute [12]. The refractive index was fitted with the following function [31],

$$ n(\omega) = \sqrt{\frac{|\tilde\epsilon_{L}(\omega)| + \text{Re}\{\tilde\epsilon_{L}(\omega)\}}{2}} $$
(9)

where 𝜖L is the relative dielectric function of two Lorentz oscillators given by:

$$ \tilde\epsilon_{L}(\omega) = \epsilon_{\infty} - \sum\limits_{j = 1}^{2}\frac{A_{j}{\omega_{j}^{2}}}{\omega^{2} - {\omega_{j}^{2}} - i{\Gamma}_{j}\omega} $$
(10)

where \(\epsilon _{\infty }\) is the high frequency dielectric constant, Aj is the oscillator strength, ωj is the resonant phonon frequency with ωj= 2 fixed at 8 THz [12], and Γj is the damping parameter.

Fig. 3
figure 3

Refractive index of α-quartz near the 3.9 THz phonon mode with the data shown as red circles and fits shown as black lines. The fitted data are modelled by a Lorentz oscillator as described in the text (Eq. 9)

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Figure 4 shows (a) the oscillator strength, (b) the phonon resonant frequency and (c) the damping parameter as a function of temperature that were extracted by fitting Eq. 9. The damping parameter, Γ increases with increasing temperature as expected. Even at 10 K, splitting of the energy degeneracy of the 3.9 THz LO and TO modes could not be resolved indicating that the splitting is less than 0.8 cm− 1, which is in agreement with variational density-functional perturbation theory predictions [32]. The electric dipole oscillator strength decreases with increasing temperature and the phonon resonance redshifts with increasing temperature, in agreement with theoretical predictions based on a model of cubic anharmonicity [18]. This trend is expected as a harmonic (parabolic) potential is only a good approximation for low vibrational energy levels. At higher temperatures, higher energy vibrational energy levels also become occupied according to a Bose-Einstein distribution and the widening of the potential from a parabola at the higher energies (i.e. the weakening of interatomic forces for large amplitude displacement) means that the energy levels are spaced closer together. Thus, phonon modes generally redshift and broaden at higher temperatures.

Fig. 4
figure 4

Extracted parameters from fitting an oscillator model (Eq. 9) to the refractive index near the 3.9 THz phonon mode (Fig. 3) showing a the oscillator strength, b the phonon resonant frequency, and c the damping parameter as a function of temperature

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3.3 Polariton Dispersion in α-Quartz

The polariton dispersion relation demonstrates the coupling between a photon and the longitudinal (LO) and transverse (TO) optical phonons. Figure 5 shows the polariton dispersion relation of α-quartz with the lowest energy E mode near the zone centre. The dispersion relation is a plot of frequency against wave vector, where the wave vector is calculated using the measured refractive index by the following equation [31],

$$ k(\omega) = \frac{\omega |n(\omega)|}{c} $$
(11)
Fig. 5
figure 5

Dispersion relation of the polariton with the lowest energy doubly degenerate E mode of α-quartz measured at 10 K and 300 K. The inset shows is an enlarged section of the dispersion relation near the E mode. The solid black line is a fit to the dispersion relation at 10 K using Eq. 12 with the damping parameter set to Γ = 5.3 cm− 1 from the previous fitting of the refractive index. The dotted grey line is the calculated dispersion relation in the absence of damping, Γ = 0 cm− 1. The horizontal dashed black line is the energy of the E mode (132 cm− 1 ≈ 3.954 THz)

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Here, the photon couples with the doubly degenerate E mode and no energy gap between the LO and TO phonons can be resolved. The polariton dispersion relation at 10 K was fitted with the following equation,

$$ k^{2} = \frac{\omega^{2}}{c^{2}}\epsilon_{\infty} \left( \frac{\omega_{\text{LO}}^{2} - \omega^{2} -i{\Gamma}\omega}{\omega_{\text{TO}}^{2} - \omega^{2} -i{\Gamma}\omega}\right), $$
(12)

where ωLO and ωTO are the frequencies of the LO and TO phonon modes and Γ is the damping parameter. Γ and average of ωLO and ωTO were fixed at 5.3 cm− 1 and 132 cm− 1 respectively, as determined from the previous fitting of the refractive index. \(\epsilon _{\infty }\) is the high frequency (i.e. frequencies much greater than ωLO) dielectric constant and was found to be 4.52. The splitting of the LO and TO phonon modes was determined to be less than 0.03 cm− 1 and is significantly less than the damping parameter.

4 Conclusion

Quartz is a cheap, inert and durable substrate for supporting samples during optical spectroscopy measurements at terahertz frequencies. Here, we have presented the complex refractive index of α-quartz at terahertz frequencies (0.5 to 5.5 THz) at temperatures between 10 and 300 K. α-quartz shows excellent transmission properties for spectroscopy over this frequency range apart from a band of 0.2 THz width centred on a phonon absorption feature at 3.9 THz. We have also used terahertz TDS to study the temperature dependence of this phonon feature. It originates from the lowest energy TO and LO phonon modes of quartz, for which we extracted the phonon-polariton dispersion and found the splitting of the degeneracy of the two modes to be less than 0.03 cm− 1 at 10 K. The phonon lifetime, energy and dipole oscillator strength were measured as a function of temperature with experimental trends agreeing with previously published theoretical models of anharmonicity. Our results will assist with future optical characterisation of non-free standing samples in the terahertz region, which relies on accurate knowledge of the refractive index and extinction coefficient of the substrate.