Terahertz time-domain spectroscopy of zone- folded acoustic phonons in 4H and 6H silicon carbide

We investigate the dielectric properties of the 4H and 6H polytypes of silicon carbide in the 0.1-19 THz range, below the fundamental transverse-optical phonons. Folding of the Brillouin zone due to the specific superlattice structure of the two polytypes leads to activation of acoustic phonon modes. We use a combination of ultrabroadband terahertz time-domain spectroscopy and simulations based on density-functional perturbation theory to observe and characterize these modes, including band splitting due to the dissimilar carbon and silicon sublattices of the structures, and an indirect measurement of the anisotropic sound velocities in the two polytypes. © 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement


Introduction
Silicon carbide (SiC) is an important material for power electronics and optoelectronics due to its uniquely high dielectric strength, high bandgap, and mechanical hardness [1,2].Recently, SiC has attracted attention as host material for solid-state single photon emitters [3] as well as dielectric substrate material for graphene growth [4,5], nanophotonics at long wavelengths with surface phonon polaritonics [6], mid-IR femtosecond pulse generation [7], and THz generation in the difficult-to-access 5-15 THz region [8].Natural SiC superlattices can emit THz electroluminescence due to Bloch oscillations [9], and it was recently demonstrated that high-quality graphene nanoribbons with ballistic electron on the µm scale at room temperature can be grown on sidewalls of SiC mesas [10,11].
It is well known that SiC naturally forms a wide range of polytypes, depending on the stacking sequence of the individual SiC layers of the material.Stacking increases the size of the crystal unit cell in the growth direction compared to the simple cubic SiC unit cell (3C SiC), and thus reduces the size of the Brillouin zone (BZ) correspondingly.This reduction can be understood as a folding of the cubic BZ, and therefore the edges of the simple BZ are projected onto the center of the BZ of the given polytype (see e.g [12,13].).The same zonefolding effect is also observed in artificial superlattice structures, such as GaAs/AlAs [14] and GaSe [15].Here we perform a detailed investigation of the infrared active zone-folded acoustic phonons below the TO frequency in the most common polytypes 4H and 6H of SiC that are commercially available in wafer dimensions.The acoustic phonons in these and other polytypes have been studied extensively by Raman spectroscopy [12,13], and their nature is well established.However, these exceedingly weak modes are not obvious in infrared spectroscopy, and have not been characterized in recent studies of the optical properties of SiC in the same spectral range [8,16].To the best of our knowledge, the modes have not been observed and documented in IR spectroscopy with the exception of the LA mode in 6H SiC [17].

Experime
Measurement femtosecond second harmo length, 800 n resulting THz central oscilla in Fig. 2).Th an intermedia (ABCD) [18, locked at f 0 /2 The sampl SiC (supplied and 502 µm, r both polytype the standard n ABCACB, re performed to All measurem path in order t

Experime
taking into account both the strong fundamental TO phonon that is responsible for the general dispersion across the frequency range explored here, and the weak polytype-specific modes in the 7-8 THz range.The resulting fit parameters are shown in Table 1.It can be noted that the zone-folded modes have intensities that are between 7 and 25 parts per million relative to the strong TO phonon.Although outside the range of the spectroscopic data, the position of the TO phonon can easily be obtained from the fits due to the strong dispersion of the real part of the permittivity at lower frequencies.Figure 3(c) shows the imaginary part of the permittivity of both polytypes, measured at 70° incidence angle, resulting in an internal propagation angle of 14.5° and giving access to the extraordinary axis of the refractive index.The high angle of incidence leads to slight shift of the THz beam path, so quantitative spectroscopic information is difficult to extract.However, we observe narrow absorption bands at 18.3 and 15.2 THz in 4H and 6H SiC, respectively.These peaks are not observed at normal incidence (Fig. 3(b)), and are due to longitudinal acoustic phonons, as will be discussed below.
where λ is the wavelength in units of µm.The deviation between these relations and our experimental data is shown in the top panel of Fig. 4. The RMS deviation is well below 10

Theoretical analysis
The nature of the observed sharp bands can be understood by considering phonons in the simplest cubic (3C) polytype.Here the phonon dispersion consist of two acoustic (TA and LA) and two optical (TO and LO) branches, where only the TO mode at wavevector q = 0 is infrared active.The phonon dispersion diagram of the higher polytypes can be qualitatively constructed by appropriate folding of the dispersion diagram along the stacking direction.The zone folding results in IR activation of several of the acoustic modes.In 4H SiC, a doubly degenerate TA mode and an LA mode is predicted, and in 6H SiC, two TA modes two LA modes are predicted, together with the strong TO phonon in both polytypes.The predicted frequencies are shown in Table 2, with a direct comparison to the experimentally determined frequencies in Fig. 3.The DFT frequencies are immediately confirmed by the experimental spectra shown in Fig. 3, with all deviations below 3% and an RMS deviation of only 0.5%.The very good agreement between the theoretical and observed frequencies may at first glance be surprising.The DFPT calculation is performed at absolute zero temperature, and thus excludes anharmonic temperature effects such as redshift of frequencies and thermal expansion of the lattice, known to have significant influence on vibrational modes in crystalline materials in the THz range [24].Figure 6 shows the calculated vibrational potential as function of the normal mode coordinates of the two motions for the 4H SiC TA mode and the upper of the 6H SiC TA modes (7.83 and 7.00 THz, respectively).The normal mode coordinates are scaled so that the curvature of the potential energy corresponds to the vibrational frequencies of the modes.The solid symbols represent the DFT potential energies, and the dashed curves represent simple harmonic fits that hardly deviates from the DFT potential within the shown range, up to energies well above 1 eV.The thermal expansion coefficients of SiC are very low, with an estimated volume expansion of 6H SiC of 9 0 / 1.5 10 V V − Δ ≈ ⋅ between 0 and 300 K [25].Pressure studies of SiC determined the shift of the TO phonon frequency to be 0 / ( / ) 19 THz , so the TO frequency shift due to thermal expansion alone between 0 and 300 K would be in the 30 kHz range, which can be safely ignored here.Thus, the harmonic vibrational potential indicates that the vibrational frequencies should be virtually independent of temperature.The precise agreement between experimental and theoretical frequencies of the IR active acoustic modes confirm the general shape of the calculated dispersion curves, and thus it is possible to accurately estimate the axial and planar sound velocities (v = ω/q) in the 4H and 6H polytypes, as shown in Table 3.The values are consistent with theoretical and experimental literature values [12,27,28], although our reported planar sound velocities are slightly lower than reported literature values.The low-frequency slope of the dispersion curves are uniquely defined by the frequencies of the IR activated acoustic modes, so we estimate the uncertainty of the extracted sound velocities to be given by the differences between optimized lattice constants in the simulation (mainly the c axis) and the experimental value at room temperature.The room temperature axial lattice parameter (c direction) for 4H and 6H SiC is 10.082 and 15.115 Å, respectively [29], and our optimized DFT unit cell parameters are 9.97 and 14.958 Å, respectively, which is 1% lower than the experimental values.This results in a similar relative uncertainty on the absolute slope of the dispersion curves in Fig. 5, and thus on the extracted sound velocities.
The detailed nature of the observed modes is detailed in Fig. 7, where we show the calculated eigenmotions of the individual ions in the unit cells, projected onto the transient dipole moment of each mode.frequencies with predictions from density functional perturbation theory, and found excellent agreement, with 0.5% RMS deviation between observed and calculated frequencies.The zone-folded acoustic modes are very weak compared to the transverse optical phonon (by a factor of approximately 10 5 ), and we believe that the present study represents the first documented observation of these modes in infrared spectroscopy.The theoretical analysis allowed us to extract good estimates of the axial and planar sound velocities in both polytypes that are consistent with previous observations and calculations, and with a precision determined by the accuracy of the calculated vibrational frequencies.
Fig. 1 crysta 400-n driven Figure 2 sho reference and insets show th has a bandwid due to strong were recorded over a time w Fig. 2 SiC a narrow measu The ampli and reference

Fig. 3 .
Fig. 3. (a) Real and (b) imaginary part of the permittivity of 4H (blue curves) and 6H (red curves) SiC.The insets show a zoom onto the resonant modes in the 6.5-8.5 THz region.Dashed curves in insets are Lorentz fits to the measurements.(c) Imaginary part of permittivity measured at 70 degrees incidence angle with longitudinal phonons at 18.4 THz (4H) and 15.2 THz (6H).The shaded areas around each curve indicate the standard deviation of the measurements.

Fig. 4 .
Fig. 4. Index of refraction of 4H (blue) and 6H (red) SiC (ordinary axis) as function of wavelength, together with Sellmeier fits in the range 15-300 µm.Shaded areas indicate standard deviation of the experimental data.Top panel shows deviation between experimental data and fits, with RMS deviations indicated in the legend.Dashed and dashed-dotted curves show the Sellmeier fits from [8,16], respectively (4H SiC).
The acoustic branches of the dispersion diagram are shown in Fig.4(a) for 4H and Fig.4(b) 6H polytypes.The dispersion diagram was calculated by ab-initio density functional perturbation theory (DFPT) as implemented in the Castep code[22,23] on optimized unit cells, using the exchange correlation functional PBEsol, a plane-wave energy cut-off of 1200 eV, 2.2⋅10 −10 eV/atom tolerance in electronic minimization, 10 −5 eV/Å 2 phonon energy tolerance, and 10 −5 Å 3 electric field convergence tolerance.A k point grid of 9x9x2 (total of 81 k points used after symmetry considerations).Figure5shows the resulting phonon dispersion curves of the transverse (red) and longitudinal (blue) acoustic phonon branches of (a) 4H and (b) 6H SiC.The IR intensities of the modes at q = 0 are indicated by gray bars (logarithmic scale).

Fig. 5 .
Fig. 5. Acoustic phonon dispersion diagrams of (a) 4H and (b) 6H SiC, calculated by DFT.Red and blue symbols represent transverse and longitudinal zone-folded acoustic branches, respectively.Grey bars indicate IR intensities (upper logarithmic scale) predicted by DFPT.Solid, black lines indicate the linear dispersion relation and speed of sound for the transverse and longitudinal directions.

Fig. 6 .
Fig. 6.Representative potential energy curves calculated by DFPT for the 4H SiC TA mode at 7.83 THz (red symbols) and the second 6H TA mode at 7.00 THz (blue symbols).Dashed lines are harmonic fits.

d
Fig. 7 modes (b + c motio Visua 6H TAThe Si and each layer bal C layers, as differences ar the polytypes the Si-C layer are, in compa motion if the layers.The 6 semi-rigid int the folded T multimedia f Visualization5.ConclusioWe have used properties of and observed

Table 1 . Lorentzian fitting parameters for 4H and 6H SiC
[16] the 15-300 µm range, excluding the narrow region of the folded-zone phonon modes that are not accounted for by the Sellmeier relations.For comparison, we show the excellent agreement with previous values for 4H SiC in the same spectral region obtained by Fischer et al.[8]and Naftaly et al.[16]