Defect-induced ultimately fast volume phonon-polaritons in the wurtzite Zn0.74Mg0.26Se mixed crystal

Volume-phonon-polaritons (VPP’s) propagating at a light-in-vacuum-like speed are identified in the wurtzite-type Zn0.74Mg0.26Se mixed crystal by near-forward Raman scattering. Their detection is selective to both the laser energy and the laser polarization, depending on whether the ordinary (n0) or extraordinary (ne) refractive index is addressed. Yet, no significant linear birefringence (n0 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\simeq }}$$\end{document}≃ ne) is observed by ellipsometry. The current access to ultrafast VPP’s is attributed to the quasi-resonant Raman probing of an anomalous dispersion of n0 due to impurity levels created deep in the optical band gap by oriented structural defects. The resonance conditions are evidenced by a dramatic enhancement of the Raman signals due to the polar modes. Hence, this work reveals a capacity for the lattice defects’ engineering to “accelerate” the VPP’s of a mixed crystal up to light-in-vacuum-like speeds. This is attractive for ultrafast signal processing in the terahertz range. On the fundamental side we provide an insight into the VPP’s created by alloying ultimately close to the center of the Brillouin zone.

Replacement of electrons by photons as the carrier of information, leading to the development of photonics besides electronics, is highly promising in view to ultimately accelerate the signal processing in matter. On this way one resorts to elementary crystal excitations able to couple with light. Plasmon-polaritons spanning large to small plasma pulsations offer an option for data processing in the visible to terahertz spectral ranges [1][2][3] . In the latter range, the phonon-polaritons (PP) resulting from the coupling between optical lattice vibrations (phonons) and a photon-like (i.e., transverse) electric field → E ( ) in polar dielectrics are the other option 4 . The existence of PP's in common high-purity and high-structural-quality materials for electronics, e.g., the III-V and II-VI binary semiconductor compounds with zincblende (cubic) structure, has been known for decades [5][6][7] . The pioneering study of the PP coupling in GaP (III-V) compound done by near-forward ('transmission'-like) Raman scattering goes back to the sixties 8,9 . The interest is revived at each novel generation of semiconductors; nowadays the focus has shifted onto the emerging class of N-based (III-V) compounds 10,11 . The physics behind the PP's in pure compounds is well understood. Basically the PP's can be classified into surface (SPP, ε ω < ( ) 0 r ) or volume (VPP, ε ω > ( ) 0 r ) ones depending on the sign of the relative dielectric function ε ω ( ) r , which changes on crossing the Reststrahlen ω ω − ( ) TO LO band spanning the transverse (TO) and longitudinal (LO) pulsations of an optical phonon 12 .
In view of applications, the SPP's are preferred over the VPP's, because the propagation of the electromagnetic energy at the surface is less prone to dissipative effects due to defects and impurities 13,14 . Moreover, the confinement of light at the surface offers a possibility to achieve light localization into volumes down to a few nanometers, i.e. at a length scale much below the diffraction limit 1,15 . This is promising in view of optical signal processing in miniaturized devices at a level of integration comparable to that achieved in modern electronics. However, the SPP is bound to a fixed pulsation inside the Reststrahlen band, given by ε ω = − ( ) 1 selectivity. In contrast, the VPP is naturally dispersive. However, a major problem is that the VPP is hardly supported by a pure crystal since phonon decoupling occurs as soon as the VPP enters the highly-dispersive/ high-velocity regime. The main features of the SPP's and VPP's related to a zincblende-type compound, corresponding to an unique optical phonon, are summarized in the Supplementary Section S1 for reference purpose, using ZnSe (II-VI) as a case study. Elegant 'extrinsic' means to circumvent the absence of dispersion of the SPP are nanoarchitectural design or hybridization with a plasmon at metal/semiconductor-like interfaces 16,17 . An alternative 'intrinsic' option likely to reconcile the advantages of the SPP (signal processing at light-like speeds) and VPP (dispersive character) might be to (re)consider the PP coupling in volume, albeit in mixed crystals (multi-phonon systems) rather than in pristine (sole-phonon) ones.
Such option, so far unexplored, may seem doomed to fail as the chemical and positional disorders inherent to a mixed crystal are expected to aggravate the dissipative effects that are nearly prohibitive already for pure crystals. In fact, we are not aware of any mixed crystal being considered so far for applications in photonics. Half a century separates the original near-forward Raman study of the PP coupling in GaP 9 (still arising interest 18 ) from the pioneering experimental works on the mixed crystals, engaged over the past decade. Nevertheless, several attractive features already emerge, summarized below, that might stimulate an interest in photonics for using the VPP created by alloying. The ω-dependent VPP Raman cross section shown in Fig. 1a for the particular mixed crystal studied in this work can be considered as generic, and thus used as a visual support to fix ideas in the following. (b) Polarized phonon-polariton Raman spectra in the near-forward (θ ~ 0°) X(ZZ)X and X(YZ)X scattering geometries. In panel (a), asymptotic regimes are defined by the dispersions of light well-beneath and wellbeyond the phonon resonances (dashed-dotted curves, dictated by ε s and ε ∞ , respectively) and by the LO's and TO's close to Γ and away from Γ, respectively. I c refers to the photon-like extinction of the PP int mode. The dispersion within the darkened area close to Γ has no physical meaning. The scan lines achievable experimentally by using the effective refractive index (n eff ) in the perfect forward scattering geometry (θ = 0°) with the red (R, 632.8 nm), green (G, 514.5 nm) and blue (B, 488.0 nm) laser lines are indicated (dashed curves), for reference purpose. The scan line corresponding to the experimental A 1 -like X(ZZ)X Raman spectrum displayed in panel (b) is shown (θ = 0.55°). The curved arrow indicates an apparent upward-tilt of the corresponding scan line when shifting to the E 1 -like X(YZ)X geometry.
www.nature.com/scientificreports www.nature.com/scientificreports/ Calculations of the VPP dispersion via the Maxwell's equations 8,19,20 have shown that AB 1-x C x alloying generates an intermediary VPP with (A-B, A-C)-mixed character, i.e. PP int , between the PP − and PP + parent-like ones mostly related to the soft/heavy (say A-B) and stiff/light (A-C) bonds, respectively. PP int differs in nature from PP − and PP + in that it exhibits a characteristic dispersion with a S-like shape between the Reststrahlen bands of the two bonds governed by two horizontal phonon asymptotes, and not by one phonon asymptote and one photon asymptote as is the case for the PP − and PP + dispersions. These two phonon asymptotes are positioned, away from the center Γ of the Brillouin zone = q ( 0), at the pulsation of the upper TO mode (TO A-C ), and, near Γ, at the pulsation of the lower LO mode (LO A-B ). The S-like dispersion can thus be adjusted to suit the user's needs depending on the choice of parent compounds and on the crystal composition.
On the experimental side, pioneering far-infrared reflectivity studies of the PP coupling in thin films of wurtzite-type GaN-based mixed crystals were concerned with the SPP only 20,21 . Regarding the VPP, we are only aware of our own near-forward Raman studies on two zincblende-type ZnSe-based mixed crystals, namely Zn 1-x Be x Se and ZnSe 1-x S x (refs [22][23][24][25][26]. The Reststrahlen bands of the parent compounds are well separated in these systems, so that the PP int mode exhibits a large S-like dispersion, covering as much as ~200 cm −1 (~6 THz) in the case of Zn 1-x Be x Se (see, e.g., ref. 26 ), and shows up as a distinct feature in the Raman spectra. As such, the PP int mode could be studied experimentally in detail.
The S-dispersion was to a certain extent probed by slightly varying the scattering angle θ between the incident → k ( ) i and scattered → k ( ) s light wavevectors inside the crystal near the perfect forward scattering geometry (θ = 0°), and also by changing the laser pulsation ω ( ) i . This latter option implies the dispersion of the refractive index ω n( ) around ω i that is decisive to how far (with respect to the ω min value achieved at θ = 0°) a given laser line would 'penetrate' downward the S-shaped PP int dispersion towards Γ (ref. 8 ). The probing of the S-dispersion has revealed several attractive features of the PP int mode: c c near the S-inflexion, where the PP int mode becomes photon-like. It is revived on both sides of I c , as the PP int mode recovers a dominant phonon character. This conforms to intuition that only matter-like (phonon-like) excitations scatter light efficiently [25][26][27] .
(ii) On both sides of I c , the PP int mode obeys the same Raman selection rules as its native purely-mechanical TO phonons away from Γ (ref. 26 ). This means that the PP int mode preserves its transverse character throughout its entire S-like dispersion, even in the LO-like asymptotic regime near Γ. As such, it keeps an ability to couple strongly with light at any stage. (iii) The agreement between the experimental and theoretical PP int selection rules is excellent ante ω ω > ( ) c and post ω ω < ( ) c photon-like extinction 26 . Apparently the long wavelength in the PP regime ∼ q ( 0) operates a natural average on the alloy disorder, so that the PP int mode does 'see' neither the chemical disorder nor the local lattice distortions inherent to a mixed crystal, and propagates like in a perfect medium. (iv) While the PP int mode ante-extinction exhibits a large Raman linewidth comparable to that of its native TO phonon away from Γ, it becomes sharp post-extinction while approaching Γ (refs 25,26 ) -an explanation is given in the Supplementary Section S4, with concomitant increase on the PP int lifetime 28 . This opposes to the experimental trend in a pure crystal (a direct insight in the case of ZnSe is given in the Supplementary  Fig. S2) 8 .
Summarizing, the naturally dispersive PP int mode created by alloying is interesting in many respects: (i) it remains supported by the crystal (phonon-like) throughout most of its S-like dispersion, (ii) it is likely to couple with light throughout its whole dispersion, (iii) it 'feels' like propagating in a pure crystal, and, beyond all expectation, (iv) it gains lifetime while acquiring high velocities ( → q 0). Of upmost interest in view of applications is the latter 'high-velocity/long-lifetime' regime of the PP int dispersion situated beneath I c . So far, only its early stage near I c could be probed experimentally, leaving the lower part of the S-like dispersion totally unexplored across all studied mixed crystals [22][23][24][25][26] . An access near Γ is needed to complete the PP int picture in view of applications, but also for the sake of fundamental understanding. As already mentioned, the limiting factor is the finite dispersion of the refractive index n ω ( ) around ω i that so far hindered to approach sufficiently small q values. In this work, the lacking insight into the PP int mode near Γ is searched for by trying near-forward Raman scattering on a wurtzite-type ZnSe-based mixed crystal, using Zn 0.74 Mg 0.26 Se as a case study. The ZnSe-based crystals are transparent in the visible (the optical band gap of ZnSe is 2.7 eV at room temperature 29 ) and thus well-suited for a Raman study in 'transmission' . Moreover, owing to a large difference in the Zn (~65) and Mg (~24) atomic masses, the Reststrahlen bands of the wurtzite-type ZnSe 30 and MgSe 31 compounds are well separated, i.e. by ~20 cm −1 , which is needed for the PP int mode to exhibit a pronounced S-like dispersion. This separation is preserved with alloying 32-34 . Last but not least, Zn 0.74 Mg 0.26 Se is interesting for its wurtzite structure -in part because the PP int mode remained unexplored in anisotropic crystals so far, but moreover in a hope that the vibrational and optical anisotropies behind the structural anisotropy would help to diversify the access to PP int . Indeed, for each phonon that is likely to support the VPP coupling, in reference to the A 1 and E 1 polar vibrations along and perpendicular to the → c -crystal axis, respectively 10,11 , the PP int insight can be optimized by playing with the dispersion of either the ordinary (n 0 ) or extraordinary (n e ) refractive index. Altogether the access to the PP int mode by near-forward Raman scattering is thus potentially quadruple (2 phonons × 2 refractive indices) with a wurtzite-type system such as Zn 0.74 Mg 0.26 Se, and not only unique (1 phonon × 1 refractive index) as with the zincblende-type ZnSe-based mixed crystals studied so far. The opportunities to capture novel information near Γ are correspondingly enriched.

Results and Discussion
No significant linear birefringence could be detected by ellipsometry in the visible where operates the Raman scattering. An effective refractive index ω n ( ) eff is thus considered from now on for Zn 0.74 Mg 0.26 Se. Besides, in the Zn 0.74 Mg 0.26 Se near-forward Raman spectra taken with the red laser line (632.8 nm, Fig. 1b), the native Zn-Se (~200 cm −1 ) and Mg-Se (~280 cm −1 ) purely-mechanical TO's of the VPP's, and also the related LO's, both theoretically forbidden but activated due to multi-reflection of the laser beam between faces of the crystal 9 , appear to be quasi degenerate in the A 1 and E 1 symmetries (within ~5 cm −1 ). Altogether, the quasi negligible linear birefringence  n n ( ) e 0 and the apparent A 1 − E 1 degeneracy ruin all hopes of achieving a diversified PP int insight by playing with the optical and vibrational anisotropies behind the wurtzite structure of Zn 0.74 Mg 0.26 Se, independently evidenced by X-ray diffraction. Details concerning the ellipsometry and X-ray diffraction measurements are given in the Supplementary Section S2.
Based on the observed pulsations of the Zn-Se and Mg-Se purely-mechanical TO's, an overview of the VPP dispersion is achieved by solving numerically that captures the resonances behind the dispersion of a TO mode, using a classical two-oscillator . A more refined expression in which the resonance-term is weighted by a pre-factor involving the Faust-Henry coefficients of ZnSe and MgSe, derived in ref. 22 , provides the actual VPP Raman cross section in its ω-dependence, shown in Fig. 1a. Note the punctual extinction (at I c ) of the PP int Raman signal in the photon-like regime. Detail is given in the Supplementary Section S4.
The near-forward Raman spectrum taken in the X(ZZ)X pure-A 1 geometry (Fig. 1b), using the Porto's notation 35 , reveals a sharp PP int mode in the shallow reinforcement regime (ω ~ 260 cm −1 < ω c ~ 265 cm −1 ). The corresponding Raman 'scan' line (θ ~ 0.55°) in Fig. 1a, as derived from the wavevector conservation law that governs the Raman scattering using the effective refractive index measured by ellipsometry (see the Supplementary Section S1) to express the magnitudes of → k i and → k s , crosses the S-like PP int dispersion exactly between ω c (θ ~ 0.80°) and ω min (~258 cm −1 , θ = 0°). As the PP int mode emerges near its native -TO Mg Se A1 mode, it is referred to as PP TO int . As soon as θ increases, the PP TO int mode vanishes. This is consistent with its entry into the photon-like extinction regime ω ω ∼ ( ) c on its way back to the collapse regime ω ω . The tilt suggests an anomalous dispersion of n 0 near ω i , presumably due to impurity levels created by structural defects. These give rise to a local absorption reflected by a maximum in ε ω Im{ ( )} r (as sketched out in Fig. 2a -dotted curve) going with a sigmoidal distortion of ε ω Re{ ( )} r , with concomitant impact on the dispersion of n 0 , being inverted locally (thick curve) -basic relations are given, e.g., in ref. 36 . In Fig. 2a, the dispersion of the refractive index for the zincblende-type Zn 0.74 Mg 0.26 Se single-crystalline epitaxial film 37 is added for comparison.
In fact, the activation of the PP LO int and + PP LO modes in the X(YZ)X geometry is selective to ω i . It is strong with the red (632.8 nm) laser line and also, to a less extent, with the near-infrared (785.0 nm) one, but remains weak with the green (514.5 nm) and blue (488.0 nm) ones (examples are given in the Supplementary Fig. S10). This suggests a resonance mechanism involving impurity levels mostly situated deep in the optical band gap. The resonance is searched for by extending the Raman analysis away from ω i using the red laser line (Fig. 2b). A typical two-feature set, combining a dramatic enhancement of the Raman signal preceding its sudden extinction at ~840 cm −1 beneath ω i (considering the used Stokes scattering), reveals a resonance with an impurity level positioned right at the extinction (i.e., at ω ω ∼ − − 840 cm ). The enhanced signal is of the PP type since it vanishes along with the PP LO int and + PP LO modes when θ increases (Fig. 2b). It is assigned as a resonance-induced PP + mode (Res − PP + ) away from the LO-like limit of the PP + dispersion. At large scattering angle (θ ≥ 2°) the PP's disappear. The only left polar ( → E -equipped) modes likely to be resonantly enhanced (via the → E -mediated Fröhlich mechanism 38 ) are then the LO ones, notably the sharp and intense LO Zn-Se peak, giving rise, in fact, to resonance-activated harmonics up to the third order.
Earlier photoluminescence measurements done with Zn 1-x Mg x Se (0 < x < 0.6) monocrystals from the same source as the current one (x = 0.26) have revealed a strong red emission, providing a direct evidence for a (2019) 9:7817 | https://doi.org/10.1038/s41598-019-44273-5 www.nature.com/scientificreports www.nature.com/scientificreports/ continuum of impurity levels deep in the optical band gap 39 . Based on positron annihilation measurements, the red emission has been related to Zn vacancies (V Zn ), possibly arranged as dimers 39 . As the activation of the PP LO int and + PP LO Raman features occurs only when → → e c i and in a given phonon (E 1 ) symmetry, we conclude that the V Zn -related complexes have a preferential orientation with respect to the → c -axis of the crystal. The concentration of such complexes is small, though, since no singularity is detected in the red spectral range of the linear dichroism measured by transmission ellipsometry (see the Supplementary Fig. S4). Their massive impact on the X(YZ)X Raman spectra is all due to the resonance conditions.
Technically, a crude contour modeling of the bimodal (PP LO int , + PP LO ) Raman signal can be achieved (see the Supplementary Fig. S9) by considering that the distorted dispersion of the refractive index in the studied ω ω ω ≤ ≤ Res s i spectral domain (emphasized in Fig. 2a) is opposite (negative) and roughly twice as large in magnitude as that (positive) measured for ω n ( ) eff by ellipsometry. To fix ideas, the amount of oscillator strength (taken as the only adjustable parameter -see details in the Supplementary Section S4) needed to achieve such inversion (shown in Fig. 2a) by introducing one unique damped Lorentz oscillator at ω Res was estimated at ~1% of the oscillator strength carried by the optical phonon of ZnSe.
Summarizing, the Zn 0.74 Mg 0.26 Se mixed crystal with wurtzite structure is studied by near-forward Raman scattering. Despite the almost negligible linear birefringence ∼ n n ( ) e 0 observed by ellipsometry, an unexpectedly large variety of VPP's is revealed, contrasting with the quasi-indiscernibility of the native A 1 and E 1 phonons. This is due to an artificial linear birefringence created by deep impurity levels associated with oriented crystal defects. www.nature.com/scientificreports www.nature.com/scientificreports/ The artificial linear birefringence grants access to the PP int mode both in its shallow reinforcement regime just beneath the photon-like extinction I c ( → → e c i ) and far from I c deep in its reinforcement regime ultimately close to Γ ( → ⊥ → e c i ), where it acquires a light-in-vacuum-like speed. An added bonus in this case is the access to PP + . On the practical side, the easy-to-handle shift from the shallow PP ( )

Methods
This Sec. introduces the experimental methods needed for interpretation and replication of the reported data in the main part of the manuscript. Additional experimental and theoretical aspects, covering X-ray diffraction, ellipsometry measurements done in transmission, conventional backward Raman scattering, ab initio calculations on the native phonons behind the phonon-polaritons, are given in the course of the discussion of the corresponding data in the Supplementary Sections S2 and S3.
Sample growth and preparation. The used sample consists of a large high-quality (Zn,Mg)Se single crystal (cylinder, 3 mm in height and 8 mm in diameter) with 'pure' wurtzite structure, grown by the Bridgman method. The composition estimates from the a and c lattice constants measured by X-ray diffraction at the PSICHÉ beamline of synchrotron SOLEIL are 26.2 and 25.9 at.%Mg, respectively. Detail is given in the Supplementary Section S2. A fragment (~2.5 mm in length) of the crystal was oriented by conoscopy, using a cross-polarized microscope, so as to dispose of one pair of parallel (within 1°) faces (~0.5 mm 2 , giving rise to hyperbolic fringes by conoscopy) with in-plane → c -axis (within 5°), plus one face perpendicular to the → c -axis (~2 mm 2 , circular fringes). The three faces were polished to optical quality in view of polarized Raman measurements.
Ellipsometry measurements. The dispersion of the refractive index of the main non-oriented crystal measured by conventional ellipsometry (using a HORIBA UVISEL phase modulated spectroscopic ellipsometer) at a near-Brewster incidence in the visible spectral range, where operates the Raman scattering, does not reveal any sign of linear birefringence (LB). The measured dispersion of the refractive index, hence an effective one, noted n eff , was found to obey the Cauchy's formula .  X 2 3524 0 0021, = .
.  Z n m 8 9110 0 1772 4 (with λ in nm). More refined ellipsometry measurements done in transmission through the small → c -containing faces of the oriented fragment at normal incidence (see the Supplementary Fig. S4), treated within the approach of the Mueller's matrix 36 26 Se crystal the linear birefringence is constant and presumably smaller than 0.020 (detail is given in the Supplementary Section S2). Whether considering that n e ω ( ) is step-increased/decreased from ω n ( ) eff by ±0.02 throughout the visible we have checked that the 'scan' lines achieved by Raman scattering at near-normal incidence (θ = 0.55°) with the used (R) laser line polarized perpendicular to → c (the relevant refractive index is n 0 , then) superimpose exactly, i.e., the variation remains within the thickness of the 'scan' line (Fig. 1a).
Near-forward Raman measurements. Near-forward Stokes (ω ω > i s ) Raman spectra are taken by focusing the exciting laser beam at normal incidence onto the rear crystal face containing the → c -crystal axis. The scattered light originates from the focus plane of the incident lens, and is detected in the same direction as the incident laser beam, i.e., normally to the front crystal face. The limiting factor to achieve the perfect forward scattering geometry (θ = 0°) is the numerical aperture of the microscope objective used to collect the scattered light, corresponding to a moderate focal length (~3.00 cm, slightly depending on the used laser line) and a significant diameter (0.40 cm). In contrast the incident laser beam has a small diameter (0.20 cm) and moreover the focal length of the incident lens is large (~15.00 cm). With this, the scattered light detected outside the sample fits into a pencil-like 3.80° solid cone, to compare with a narrower 0.35° solid cone for the exciting laser beam.
Polarized near-forward Raman spectra are taken by placing half-wave plates on each side of the crystal with their neutral axis disposed either parallel or rotated by 45° to each other and with respect to the → c -crystal axis. The scattered light is analyzed parallel to the entrance slit of the spectrometer corresponding to maximum spectrometer throughput. The as-obtained scattering geometries with parallel and crossed polarizations of the incident laser ( → e i ) and scattered light ( → e s ) write X(ZZ)X, X(ZY)X, X(YZ)X and X(YY)X using Porto's notation → → → → k e e k ( , ) i i s s (ref. 35 ). The X and Z axis of the laboratory coordinate system coincide with the directions of the incident laser ( → k i ) and of the scattered light ( → k s ) and with the → c -crystal axis, respectively. Both X(ZZ)X and X(YY)X consist of 'pure' A 1 -like geometries for the TO modes (the VPP's as well as their native TO's, depending on whether the analysis is placed near Γ using near-forward Raman scattering or away from it as in a backscattering Raman experiment, respectively). Nevertheless the A 1 -TO Raman signal shows up clearly only in the first geometry. It is negligible in the second one (hence not shown). In contrast the X(ZY)X and X(YZ)X geometries are of the pure E 1 type. The only difference is that in the former geometry the incident laser beam propagates as an extraordinary wave ( → → e c i ) inside the crystal whereas as an ordinary one ( → ⊥ → e c i ) in the latter geometry, the outgoing scattered light being ordinary ( → ⊥ → e c s ) in both cases. When needed (Fig. 2b), the departure from the nominally perfect forward scattering geometry (θ = 0°) was operated by finely adjusting the incidence of the laser beam in the (x, y) plane, while detecting the scattered light along the fixed X direction.
The theoretical dependence of the crossed-and parallel-polarized A 1 and E 1 TO Raman intensities on the azimuth angle (α) between the incident polarization and the → c -crystal axis for a complete α-revolution at the www.nature.com/scientificreports www.nature.com/scientificreports/ sample surface, calculated by using the relevant wurtzite-type Raman tensors, is reported in the Supplementary  Fig. S5, for reference purpose.

Data Availability
All data regarding the work presented here including the Matlab routine for contour modeling of the Raman cross section of the VPP modes in their θ ω y ( , , )-dependence is available upon request to the corresponding author.