1. Introduction

Aluminum nitride (AlN) is a wide bandgap III-V semiconductor with promising optical and optoelectronic properties. It is optically uniaxial and has a large transparency window of 0.2 – 13.6 µm. Therefore, potential optical applications range from the far infrared to the deep ultraviolet spectral region, and it is a prime candidate for deep ultraviolet light sources [1,2]. Also, due to a high thermal conductivity exceeding 200 W/(m·K), it is commonly used in polycrystalline form for thermal management applications. Recent improvements in the AlN bulk and thin film crystal quality have sparked interest in nonlinear optical devices, e.g. for frequency up-conversion [3,4]. For possible down-conversion below infrared, the material optical properties will need to be characterized in more detail.

In order to study the optical properties of high quality bulk AlN at below infrared frequencies and explore the possibility of using the material for terahertz applications, we measured the ordinary and extraordinary refractive indices and the corresponding optical power loss of undoped AlN single crystals in the 1 – 8 THz region. A compilation of previous results on the dielectric function in the static to visible frequency range was published by Moore et al. [5]. However, they measured only the ordinary dielectric function. Availability of good quality bulk single crystals allowed us to determine experimentally the dielectric properties in the THz frequency range for both eigen-polarizations.

2. Experimental methods

Our optical characterization was performed on m-plane $1\overline{1}00$ single crystalline wafers, cut out from a single-crystal AlN boule grown in the -c-direction $<000\overline{1}>$. The crystal was grown by physical vapor transport from a solid AlN source and a nitrogen atmosphere. The growth temperatures exceeded 2300 °C, at a pressure of 800 Torr. The growth process is described in [6,7]. The crystals obtained from this process had dislocation densities around 10³ cm⁻² with x-ray ω-rocking curves ranging from 18 to 30 arc sec [8–10]. The samples had the optical (c-) axis <0001> parallel to the wafer surface. Two wafers were used for these measurements, with thicknesses of 610 ± 10 µm (sample I) and 1110 ± 10 µm (sample II), respectively. The optical characterization was performed with terahertz time domain spectroscopy, in which the electric fields of the broadband terahertz pulses passing through the reference (empty space) and the sample are compared [11]. We used a TeraIMAGE system (Rainbow Photonics AG) based on a pump-probe setup in transmission geometry that allows polarized measurements in the 1 – 8 THz range with low-absorption samples. A more detailed description of the apparatus is presented in [12]. Measurements were done at room temperature in nitrogen atmosphere. Each waveform consisted of 800 points, recorded over a delay of 41 ps. The crystals were mounted on an aluminum holder with a 3-mm diameter hole, which defined the optical aperture for the terahertz beam. An identical holder without the sample was used to measure the reference signals. By rotating the crystal around the probe beam axis, the optical axis of the crystal was aligned parallel or perpendicular to the polarization of the incoming THz wave, thereby probing the extraordinary or ordinary refractive index, respectively.

3. Results

The evaluation procedure is illustrated in an example of determining the refractive index and optical power loss coefficient for the ordinary beam in sample I. Measured time dependence of the amplitude of the THz pulse transmitted through the reference aperture and through the sample are shown on Fig. 1. The echoes repeating at multiples of 13 ps after the main pulse are due to reflected round-trips inside the apparatus optical filter. Additional echoes in the sample trace come from multiple reflections inside the sample. Figure 1b shows the main pulses in more detail. The peak amplitude of the pulse transmitted through this 610 µm thick sample was delayed for ~3.7 ps, giving a rough-estimate of an average ordinary refractive index of ~2.8. The pulse was attenuated by only 20%, suggesting low absorption.

 

Fig. 1 Time dependence of the amplitude of the reference (black) and signal THz pulse (red, for clarity offset vertically for −0.2). a) whole measurement interval. b) expanded time scale. The figure illustrates the example of determining the refractive index and the optical power loss coefficient for the ordinary beam in sample I.

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A Fourier transform of the measured reference and sample pulse was calculated using the whole waveform length. No waveform manipulations were applied. The corresponding amplitude spectra are shown on Fig. 2. The oscillations in the reference and sample spectra are due to the mentioned multiple reflections inside the filter and in addition due to the internal reflections inside the sample, respectively. The observed noise-floor was at $~5\times {10}^{-3}$ of the amplitude maximum. The apparent higher-than-1 transmission above 7 THz was due to signal level drift between both measurements. Variations in the measured transmission amplitude were always present, therefore limiting the useful range for fitting the amplitude to around 1.5 – 6 THz, while a range of 1 – 8 THz could still be used for fitting the phase.

 

Fig. 2 Amplitude spectra of the reference (black) and sample (red) pulse. The figure illustrates the example of determining the refractive index and the optical power loss coefficient for the ordinary beam in sample I.

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The complex spectrum of the transmitted sample pulse $E_s\left(\nu \right)$ is related to the spectrum of the reference pulse $E_r\left(\nu \right)$ through a complex transmission function of a parallel dielectric slab (Airy’s formula [13, Chap. 7, Eq. 7.1-8]):

Frequency dependence of the refractive index was obtained by fitting the phase and the absolute value of the modeled $TF(\nu )$ to the measured $E_s(\nu )/E_r(\nu )$ transmission function. AlN crystals have a simple crystal structure, which lacks optical phonons and therefore resonant absorption in the THz region or lower. Therefore, we used a 2-oscillator Sellmeier model [13, Chap. 5, Eq. 5.5-28] for $n(\nu )$:

A simultaneous fitting of the amplitude and phase of the complex transmission function proved unreliable because of the amplitude noise above 6 THz. Therefore, we fitted independently the phase in the whole frequency range and the amplitude in the reduced interval. The phase fit of our particular example (ordinary beam, sample I) is shown on Fig. 3. There was good agreement throughout the range of 1 – 6.5 THz, within 5%. Only the real part of the refractive index was fitted, since absorption negligibly influenced the phase of the $TF(\nu )$. The parameter uncertainties were estimated from the sample thickness uncertainty plus the spread of the fitting parameter that still gave a reasonable model-measurement match.

 

Fig. 3 Phase of the measured transmission function (black dots) and the fit of the modeled transmission function (red line) for the example followed in the text. The fitted parameters for the refractive index were $A_2$ = 3.72 ± 0.18 and $B_2$ = (14.2 ± 0.5) µm. Parameters of the fixed oscillator ($A_2$ = 3.131 and $B_2$ = 0.136 µm) were taken from [4].

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The absolute values of the measured and the fitted transmission functions are shown on Fig. 4. Here, the fitting parameters were $A_2$, $B_2$ and the power loss coefficient $\alpha$. The optical power loss must be due to impurities in the crystal as there are no observed phonon resonances in the THz region.

 

Fig. 4 Absolute value of the measured transmission function (black dots) and the fit of the modeled transmission function (red line). The fitted parameters for the refractive index were $A_2$ = 3.73 ± 0.26 and $B_2$ = 15 ± 0.5 µm. The optical power loss was nearly constant over the 1 – 5 THz range, with the fitted value of $\alpha$ = 2 ± 0.3 cm⁻¹. Above 5 THz the variations in amplitude from two consecutive measurements were too large to allow fitting the $\alpha$. Parameters of the fixed oscillator ($A_2$ = 3.131 and $B_2$ = 0.136 µm) were taken from [4].

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The evaluation procedures described above, namely finding the $A_2$, $B_2$ and $\alpha$ parameters by fitting the phase and the absolute value, were applied on 10 measurements for each, ordinary and extraordinary polarization, for each of the two crystals. The resulting averaged fitted parameters are given in Table 1. In the last row the weighted means over all four values for each parameter are shown. The determined ordinary refractive indices were practically equal for both samples, hence a small uncertainty in $n_o$, calculated from the variance of the weighted mean. The extraordinary refractive indices measured in the two samples were slightly shifted from one another, hence a higher uncertainty in $A_2$. One possible cause for the shift was a slight wedge in one of the samples.

Table 1. Fitted parameters of the second oscillator, for the ordinary and extraordinary refractive indices, and the associated optical power loss coefficients. Mean values over 10 measurements for samples I and II, obtained by fitting the phase and the absolute value (abs.) are shown. The averages over both samples and fitting methods give the final mean values.

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For both $n_o$ and $n_e$, the second oscillators could be ascribed to the known transverse optical phonons at 668 cm⁻¹ and 609 cm⁻¹ measured by Raman spectroscopy [15,16], as their position matched within the estimated error to our fit. As these positions have been measured with high accuracy, we thus fixed the positions of the second oscillators to the Raman-determined values and repeated the fitting procedure to find the oscillator strength only and obtained essentially the same quality of the fit, which was taken as the final result. The final set of values for the refractive indices of AlN in the 1 – 8 THz region is given in Table 2. The refractive indices, along with shaded uncertainty intervals, are shown on Fig. 5. For comparison, the estimated indices from [5] are added.

Table 2. Determined Sellmeier coefficients for the ordinary and extraordinary refractive index of AlN in the 1 – 8 THz region, as given by Eq. (2).

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Fig. 5 Ordinary and extraordinary refractive indices determined in Aluminum nitride in the 1 – 8 THz range (solid lines). The shaded regions represent the estimated uncertainties. For comparison, the indices estimated in [5] are added, (dotted lines).

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By extrapolation, we estimated the AlN static dielectric constants${\epsilon }_{DC}=n^2(\nu =0)$ to ${\epsilon }_{DC,o}=7.84\pm 0.05$ and ${\epsilon }_{DC,e}=9.22\pm 0.15$ for the polarization perpendicular and parallel to the crystal c-axis, respectively. These values agree reasonably well with the values estimated in [5].

4. Conclusion

We have measured the refractive indices and absorption of the AlN single crystals in the 1 – 8 THz region, by using time-domain terahertz spectroscopy. Indices exhibit normal dispersion with no pronounced absorption resonances, and birefringence larger than 0.2. Optical power loss in the terahertz region is approximately 2 cm⁻¹ and 4 cm⁻¹ and the estimated static dielectric constants are 7.84 and 9.22, for the ordinary and extraordinary polarization, respectively.