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Polarization-resolved Raman study of bulk-like and Davydovinduced vibrational modes of exfoliated Black Phosphorus Anne-Laurence Phaneuf-L'Heureux, Alexandre Favron, Jean-Francis Germain, Patrick Lavoie, Patrick Desjardins, Richard Leonelli, Richard Martel, and Sébastien Francoeur Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.6b03907 • Publication Date (Web): 09 Nov 2016 Downloaded from http://pubs.acs.org on November 10, 2016

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Polarization-resolved Raman study of bulk-like and Davydov-induced vibrational modes of exfoliated Black Phosphorus Anne-Laurence Phaneuf-L’Heureux,† Alexandre Favron,‡ Jean-Francis Germain,† Patrick Lavoie,† Patrick Desjardins,† Richard Leonelli,‡ Richard Martel,¶ and Sebastien Francoeur∗,† †Department of Engineering Physics, Polytechnique Montréal, Montréal, Québec, H3C 3A7, Canada ‡Department of Physics, Université de Montréal, Montréal, Québec, H3C 3J7, Canada ¶Department of Chemistry, Université de Montréal, Montréal, Québec, H3C 3J7, Canada E-mail: [email protected]

Abstract Owing to its crystallographic structure, black phosphorus is one of the few 2D materials expressing strongly anisotropic optical, transport, and mechanical properties. We report on the anisotropy of electron-phonon interactions through a polarizationresolved Raman study of the four vibrational modes of atomically thin black phosphorus (2D-phosphane): the three bulk-like modes A1g , B2g , A2g and the Davydov-induced mode labeled Ag (B2u ). The complex Raman tensor elements reveal that the relative variation in permittivity of all Ag modes is lowest along the zigzag direction, irrespective of the atomic motion involved, the basal anisotropy of these variations is most pronounced for A2g and Ag (B2u ), and interlayer interactions in multi-layer samples lead to reduced

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anisotropy. The bulk-forbidden Ag (B2u ) mode appears for n ≥ 2 and quickly subsides in thicker layers. It is assigned to a Davydov-induced IR to Raman conversion of the bulk IR mode B2u and exhibits characteristics similar to A2g . Although this mode is expected to be weak, an electronic resonance significantly enhances its Raman efficiency such that it becomes a dominant mode in the spectrum of bilayer 2D-phosphane.

Keywords Black phosphorus, Vibrational modes, Angle-resolved Raman spectroscopy, Anisotropic properties, Davydov effects Black phosphorus (P-black) is an elemental semiconductor with a direct gap of 0.34 eV 1 and relatively high mobilities (350 cm2 V−1 s−1 for intrinsic P-black 2 and up to 3000 cm2 V−1 s−1 18 5 for p-doped 3,4 ). Its orthorhombic structure (D2h ) leads to anisotropic optical, electrical,

and mechanical characteristics. 1 Like graphene, boron nitride, and metal dichalcogenides, Pblack exhibits a lamellar structure bonded together through van der Walls forces 6 and hence can be exfoliated in samples of thicknesses down to the monolayer. 7–10 Often called phosphorene, the resulting 2D material has the structure of a phosphane. 7,11 In contrast to other known 2D materials, additional interlayer interactions 12,13 lead to important variations of the band gap with thickness, reaching a value of 1.73 eV for the monolayer. 14 Along with its strong anisotropic properties, this important confinement-induced band gap variation makes 2D-phosphane one of the most interesting 2D materials for electronic and optoelectronic applications. By providing important structural and electronic information, Raman spectroscopy has played an outstanding role in the development of 2D materials. 15–17 A number of studies have described in some details the Raman signature of P-black as a function of thickness close to 18,19 or down 7–10 to the monolayer limit, oxidation or degradation, 7,18,20 and polarization. 21–25 This last aspect is particularly interesting, as the polarization response allows determining tensor elements of Raman-allowed vibrational modes and their dependence 2


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on sample thickness, interlayer coupling, external strain, doping, structural imperfections and chemical modifications, thereby shedding light on many characteristics of P-black. In this work, we present a comprehensive analysis of the Raman-allowed vibrational modes and their polarization as a function of the number of atomic layers, n, in the atomically thin regime, n = 2, 3, 4, and 5. X-ray diffraction is used to identify crystal directions and all Raman signals are corrected for the linear birefringence and dichroism of P-black. This allows establishing a reliable dependence of the Raman tensor elements as a function of sample thickness. Finally, two significant Davydov-induced effects are discussed: the Davydovbroadening of all bulk-allowed modes and the presence of a bulk-forbidden Davydov-induced additional Raman mode. Figure 1(a) presents the Raman spectra of 2D-phosphane as a function of the number of atomic layers n, from the monolayer (n = 1) to an optically thick bulk sample. The three vibrational modes (A1g , B2g , and A2g ) of bulk 26,27 and exfoliated P-black 7–9,22,23 are clearly observed. From bulk to atomically thin samples, several changes are observed: slight frequency variations, non-monotonic broadenings, and the appearance of two new Ramanactive vibrational modes. 7 The first mode appears slightly above A1g and is most obvious in the monolayer spectrum. This mode is associated to a second-order Raman process and will be described elsewhere. The second mode is located on the high-frequency shoulder of A2g and it is most obvious from the bilayer spectrum. It is labeled Ag (B2u ) for its connection with the IR-allowed B2u mode to be described herein. As discussed below, the analysis of the polarization response of these Raman modes requires identifying crystal axes and applying corrections for the optical anisotropy of black-phosphorus. Bulk P-black is represented by a base-centered orthorhombic Bravais lattice and a two18 atom motif. 28 Its D2h space group symmetry is non-symmorphic and, in accordance with the

International Tables for Crystallography, crystallographic directions are defined as follows: the b axis is perpendicular to the atomic planes, c is along to the armchair (puckering) direction, and a is along the zigzag direction. In this work, crystal directions and properties

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Figure 1: (a) Raman spectra of 2D-phosphane as function thickness n. In the back scattering configuration, three modes are expected for the bulk: A1g , B2g , and A2g . In thin layers, a new vibrational mode labeled Ag (B2u ) is also present. (b) Corresponding atomic motions of the three bulk modes. (c) Full width at half maximum (FWHM) intensity of A1g , B2g and A2g as a function of n, measured at 300 K. Error bars were determined from the results from several samples. For several data points, errors are smaller than the marker size. The dotted lines are guides to the eye. are defined through a, b, and c instead of arbitrarily defining a second set of directions (e.g. x, y, z). Taking advantage of the anisotropy of optical, electrical or mechanical properties of Pblack requires a reliable method for the identification of crystallographic directions. Through the anisotropy of the Raman tensor elements, polarization-resolved measurements of Ag modes can in principle provide a rigorous identification. However, conflicting assignments have been reported. The angle at which A2g (or sometimes A1g ) is maximum has been assigned to the c (armchair) direction in Ref. 8,21,22,24,29,30 and to the a (zigzag) direction in. 23,31 Three important aspects are worth noting in regards to these results. First, reference directions were obtained through various means or considerations (transmission electron microscopy (TEM), 23,24 optical effects, 21,31 theoretical calculations, 29 transport measurements and theoretical considerations, 22 or simple arguments relating atomic motions and changes in polarizability 8,30,32 ). Although TEM allows for a direct determination of crystal directions, conflicting assignments have nonetheless been reported in Ref. 23,24 Second, assignments were 4


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done on samples with various degrees of degradation and with widely varying thicknesses. Both of these factors may have an effect on the quality of the data and the nature of the results. Finally, it has been shown that the anisotropic optical properties 24 and polarization selection rules 25 of P-black can singularly affect the Raman scattering efficiency and renormalize its angular response. In this work, the angular dependence of both A1g and A2g is defined with respect to the a and c directions determined from single crystal X-ray diffraction. Figure 2(a) shows the logarithm of the diffraction intensity measured from (021) and (111) crystallographic planes as a function of θ, a rotation in the ac basal plane with the origin oriented along a (zigzag). In a skew-symmetric configuration, two diffraction maxima are obtained at a diffraction angle 2θXRD = 26.57◦ and a tilt angle ψXRD = 50.38◦ for (021) and four diffraction maxima are obtained at 2θXRD = 35.05◦ and ψXRD = 75.88◦ for (111). Either of these two diffraction patterns unambiguously identify the crystal axis of this P-black sample. Fig.2(c) shows the (021) diffraction peak maxima over a narrower range of angles. The observed multiplicity indicates the presence of several domains slightly rotated around the b axis, but all within ±4◦ of the (021) direction. The crystal axes are indicated on the optical microscopy image shown in Fig 2(d). Although cleavage always occurred along crystal directions, the shortest dimension could not be reliably correlated to either main crystallographic directions. Fig. 2(b) presents the polarization-resolved Raman intensities of A1g and A2g measured in a parallel excitation and detection polarization configuration (θ = θexc. = θdet. ). For this optically thick bulk sample, absorption in the black phosphorus layer suppresses interference effects in either the P-black or the silicon oxide layer. Indeed, in this regime, identical angular dependencies were obtained from several samples irrespective of their actual thicknesses: the aspect ratio I(a)/I(c) is 2.37 ± 0.11 for A1g and 0.98 ± 0.04 for A2g . The polarization-resolved Raman shown in Fig. 2(b) is therefore associated with the intrinsic response of bulk P-black. The angular variation of A1g exhibits a clear maximum along a (zigzag), while A2g exhibits a relatively symmetric quatrefoil shape. These results indicate that A1g is the preferred

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Figure 2: (a) Diffracted intensity (log scale) from (021) and (111) planes and (b) polarizationresolved Raman intensity (linear scale) from Ag vibrational modes as a function of θ, a rotation in the sample basal plane (ac plane). The Raman intensity is collected in a parallel polarization configuration. The P-black sample used is an optically thick bulk sample. (c) Diffracted intensity of (021) as a function of θ in a narrow range of angles. (d) Crystal directions superposed on an optical image of the P-black sample.

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candidate for identifying crystal directions from the raw Raman response of optically thick P-black probed with a 532 nm excitation. 24,25 The Raman response shown in Fig. 2 is, however, significantly altered by the linear birefringence and dichroism. 24 This optical anisotropy leads to polarization-dependent transmission and absorption, and a transformation from linear to elliptical polarization. Therefore, polarization-resolved Raman scattering of bulk and exfoliated P-black does not directly express the relative values of the Raman tensor as it has been assumed in Ref. 21,22 Their determination requires taking into account optical effects as described in Supporting Information. Figure 3 shows the optically corrected Raman response of all three main Raman modes for 2D-phosphane with n =2, 3, 4, 5 and an optically thick bulk as a function of the excitation polarization. Fig. S1 in Supporting Information presents the raw data. Two measurement configurations shown correspond to parallel (θexc = θdet ) and orthogonal (θdet = θexc + 90◦ ) excitation and detection polarizations. As can be seen from the comparison of the raw (Fig. 2(b)) and corrected (Fig. 3) responses of bulk P-black, optical effects are very important: they reverse the direction of A1g and reshapes the A2g quatrefoil, such that both intensity maxima are in reality aligned with the c (armchair) direction. This important correction for optically thick samples is dominated by the linear dichroism of P-black and is given by fa /fc , where fa and fc are factors taking into account the optical effects due to birefringence, dichroism, and interference in both the P-black and SiO2 layers. Their mathematical expressions and all related optical parameters are given in Supporting information. The scattering volume is ni,c /ni,a ≈ 7.3 times larger for light polarized along a, ni,j being the imaginary part of the refractive index along direction j. For n-layer 2D-phosphane, however, interference effects are relatively unimportant and the scattering volume is determined by the sample thickness. Enhancement factors are fa /fc =1.08, 1.12, 1.14, 1.16 and 1.19 for n =1, 2, 3, 4 and 5, respectively. In the intermediate regime between atomically thin and optically thick P-black, which is not investigated here, interference effects within the P-black layers enhance

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the natural dichroism and can lead to large correction factors. 24,25 The polarization responses shown in Fig. 3 as a function of n match those expected from these modes, but a number of significant aspects can be noted: the A1g and A2g parallel responses differ significantly and evolve with sample thickness, whereas orthogonal responses are qualitatively very similar for all modes and all sample thicknesses. These behaviors are quantitatively analyzed using Eqs. (S1) and (S2) presented in Supporting Information by simultaneously optimizing the fit of both parallel (dark curves) and perpendicular polarization measurements (lighter curves). As revealed in Fig. 3, this model captures all the important features of the data and allows the determination of the norms of the Raman tensor elements, Raa and Rcc , and their relative, Ψac . Fig. 4 shows the optimal values of the relative magnitude R = Raa /Rcc and the relative phase Ψac of Raman tensor elements. Error bars were determined from the residuals of the regression analysis. Although the scattering efficiency of the monolayer proved to be too weak to be reliably analyzed as a function of polarization, this data reveal the evolution of both R and Ψac as a function of thickness, the atomic motion involved, and temperature. R and Ψac express the anisotropy of the variation of the real and imaginary parts of the permittivity (ǫ′ = dǫ/dQAg ) associated with the vibrational motion Q of a mode of Ag representation,

ǫ′ 2r,aa + ǫ′ 2i,aa ǫ′ 2r,cc + ǫ′ 2i,cc ′ ′ ǫi,aa ǫi,cc = arctan ′ − arctan ′ , ǫr,aa ǫr,cc

R= Ψac

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where the permittivity tensor elements are separated in real and imaginary parts. The data of Fig. 4(a) shows that R < 1. It implies that the variation in permittivity is the lowest along the stiffer a (zigzag) direction regardless of the atomic motion (A1g or A2g ) involved. R(A1g ) is consistently larger than R(A2g ), indicating that anisotropy is more important for A2g . This 8


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Figure 3: Polarization-resolved Raman scattering for n = 2, 3, 4, and 5 2D-phosphane and bulk P-black as a function of the polarization of the excitation. For each plot, the results from two polarization configurations are shown: dark and solid symbols correspond to a parallel polarization configuration and light and hollow symbols correspond to an orthogonal polarization configuration. Dark and light curves show the fitted responses for the parallel and orthogonal configurations, respectively.

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is consistent with the fact that the atomic motion for A2g is principally oriented along the c (armchair) direction. The atomic motion for A1g is predominantly along b and its anisotropy is consequently weaker, but nonetheless important. R(300 K) appears slightly larger than R(78 K) for both modes and for all thicknesses. This suggests that the permittivity variation along a increases more rapidly than that along c with temperature and that the Raman anisotropy is maximum at low temperatures. Figure 4(a) also suggests that this anisotropy evolves with sample thickness: R increases towards unity with n and reaches a maximum value at the bulk, indicating that interlayer interactions reduce the anisotropy of monolayer 2D-phosphane. Fig. 4(b) shows the value of Ψac as a function thickness for both Ag modes at two temperatures. Ψac (A2g ) is generally larger than Ψac (A1g ), indicating again that anisotropy is more important for an atomic motion along c than along b. The variation as a function of n is opposite for the two vibration modes. Ψac (A1g ) decreases from 50◦ to 10◦ in thick layers, indicating that the ratio r = ǫ′i /ǫ′r becomes very similar for both a and c directions, raa ≈ rcc . In contrast, Ψac (A2g ) increases due to an increasing difference between raa and rcc , which represents yet another form of anisotropy expressed by P-black. Although a deeper analysis of R and Ψac calls for a comprehensive model taking into the account the significant evolution of the band structure with thickness and the anisotropy of electronphoton interactions, 14,25 this results provides important clues on the effect of the vibrational modes on the permittivity of P-black and phonon-interactions. Fig. 1(a) indicates the presence of an additional mode in vicinity of A2g . First reported in Ref., 7 this bulk-forbidden mode labeled Ag (B2u ) is located at 3.1 cm−1 above A2g . Its width is very well defined (3.0 cm−1 ) and is narrower than that of A2g (4.5 cm−1 ). Ag (B2u ) only appears as a distinct and well-resolved Raman mode for the bilayer under specific excitation conditions. It is systematically absent from all monolayers studied and it appears as a weak shoulder in three- and 4-layer 2D-phosphane. Fig. 5(a) shows the polarized Raman response of Ag (B2u ) for a 2D-phosphane bilayer at 300 K. It unambiguously corresponds to a vibrational mode exhibiting the transformation properties and selection rules of Ag

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Figure 4: Relative magnitude R = Raa /Rcc (a) and phase Ψac (b) of the Raman tensor elements for both Ag modes as a function of thickness at 300 and 78 K. symmetry. The relative magnitude and phase of the tensor elements are R = 0.12 ± 0.04 and Ψac = 50 ± 15◦ at 300 K and R = 0.20 ± 0.03 and Ψac = 63 ± 13◦ at 78 K. A comparison with the data of Fig. 4 reveals that values for R are quite similar to that of A2g , but that Ψac is somewhat smaller. Their similarity results from their similar atomic motion as will be discussed below. This new bulk-forbidden Raman mode is a assigned to a Davydov-induced conversion of a bulk-like IR mode. 0 315

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modes. For n-layer 2D-phosphane, the lack of periodicity along b imposes, however, a larger primitive cell containing 4n atoms. Table S1 in Supporting Information lists important symmetry characteristics of atomically thin samples. Since the space groups of all atomically thin samples share the same factor group, the D2h point group, the bulk nomenclature for the assignment of Raman modes remains valid irrespective of n. The 12n vibrational modes in n-layer 2D-phosphane can be expressed as in-phase or outof-phase combinations of monolayer vibrational modes in symmetrically opposed layers, such that a given monolayer mode X, where X = Ap , B1p , B2p , or B3p and p = {g, u}, generates f1 (n)(X ⊗ Ag ) modes and f2 (n)(X ⊗ B2u ) modes of opposite parity. The multiplicities of these in-phase (f1 ) and out-of-phase (f2 ) combinations of atomic motions are given by,

f1 (n) =

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Using the trilayer as example, the monolayer mode Ag generates f1 (n = 3) = 2 two Ramanallowed Ag and f2 (n = 3) = 1 IR-allowed B2u modes. Similarly, B2u generates two B2u and one Raman-allowed Ag modes. Depending on the strength of interlayer interactions and their effects on a given vibrational mode, these f1 modes can appear as spectrally resolved features, referred to as Davydov splitting 15 or as a single broadened feature referred to as Davydov broadening. 33 The f2 modes of opposite parity are referred to as a Davydov mode conversion. 15,17 These effects are expected to be important towards their onset (low n) and to subside quickly as n eventually exceeds the interaction range of a monolayer vibrational mode with that of its neighbors. In thin 2D-phosphane, both Davydov broadening and Davydov mode conversion are observed.

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A Davydov broadening is observed for all three bulk-allowed Raman modes. As shown in Fig. 1(c), the full-width at half-maximum averaged over all samples of identical thickness follows a non-monotonic variation: the maximum broadening observed at n = 3 matches the onset of this effect and exceeds that of both the monolayer and bilayer. As n increases beyond this maximum, the widths evolve towards bulk values. A1g and B2g recover their bulk values at n ≈ 5, but A2g remains higher: interlayer coupling, as measured from this Davydov broadening, appears longer in range for A2g . The presence of Ag (B2u ) for n = 2 cannot be associated to a Davydov splitting of A2g , since this effect only occurs for n ≥ 3, as f1 (n = 1) = f1 (n = 2) = 1. It is instead associated with a Davydov mode conversion of the IR-active mode B2u : the out-of-phase combination of two B2u monolayer modes generates f2 (n = 2) = 1 Raman-allowed Ag mode, as shown in Fig. S2 of Supporting Information. Two strong arguments support this assignment. First, Ag (B2u ) is systematically absent from monolayers, as the minimal volume of the unit cell is incompatible with any Davydov-related effects (f1 (n = 1) = 1 and f2 (n = 1) = 0). It appears, as expected, for n = 2 and then it quickly subsides as n increases, in a fashion similar to the Davydov broadening described above. Second, the frequency of Ag (B2u ) is very 1 close to that of IR mode B2u . For bulk P-black, the transverse and longitudinal B2u modes

have been reported at 468 and at 470 cm−1 , 27 very close to A2g at 470 cm−1 . The vibrational motions of A2g and B2u are both along the a (zigzag) direction and they only differ by the phase of the motion between weakly coupled zigzag chains (see Fig. S2), explaining their near degeneracy in bulk P-black. Assuming that interlayer interactions are relatively weak, which is the case in lamellar structures and in P-black, the frequency of Ag (B2u ) will be close to that of the parent B2u vibrational frequency. As discussed in Ref. 34 and obvious from Fig. S2(b), the main difference is that atomic motions have opposite parity with respect to inversion and their frequency difference is proportional to the strength of the interlayer coupling. Therefore, we expect Ag (B2u ) to appear within a few wavenumbers from either A2g or B2u . Indeed, several theoretical estimates have reported B2u at frequencies only slightly

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superior to that of A2g , 13,35,36 indicating a reversal of B2u and A2g in atomically thin layers. This corresponds very well to our experimental observations. As can been noted in Fig. 1(a), the intensity of Ag (B2u ) is almost equal to that of A2g for the bilayer. The Raman efficiency of this Davydov-converted mode is somewhat unexpected, since it becomes allowed only through relatively weak interlayer interactions. Fig.1(a) also reveals that the scattering efficiency of both modes is about 10 times that measured from other thin layers, suggesting the occurrence of a resonant enhancement of the Raman efficiency. This is confirmed with Raman spectra measured at several different excitation wavelengths. With respect to excitation at 633 nm, the intensity of both A2g and Ag (B2u ) at 532 nm is significantly enhanced, up by a factor of 50 for Ag (B2u ), by an electronic resonance with the second conduction band of the bilayer. 14 In contrast, the enhancement of Ag1 and B2g is considerably less. The simultaneous resonance of A2g and Ag (B2u ) and their similar Raman tensor elements can be explained again by their similar atomic motion (see Fig. S2). Several other modes may become Raman-allowed in 2D-phosphane with n ≥ 2: B1u , B2u , B3u generate B3g , Ag , B1g , respectively. The IR-mode B1u observed in absorption at 137 cm−1 should become observable in Raman in a a backscattering configuration. However, as the theoretical analysis of Ref. 37 suggests, the IR to Raman conversion may be experimentally observable only when the IR mode is nearly degenerate with a Raman-mode, thereby providing the means for these two modes to couple and mix their characters. The polarization-resolved Raman response of the first-order Raman modes of P-black reveals a surprisingly rich behavior in the atomically thin regime: a significant evolution of its Raman tensor elements with sample thickness and the appearance of a new Ramanallowed vibrational mode. These aspects shed light on 2D-phosphane, its electron-phonon interactions, its interlayer interactions, and its anisotropic properties. For the engineering of new device functionalities based on anisotropic properties, it appears that the thinnest layers are more promising.

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Experimental methods Sample preparation. The preparation of atomically thin samples free from oxidation and other forms of degradation requires minimal exposure to oxygen and humidity. 7 Exfoliation, optical contrast and AFM measurements were carried out in a nitrogen-filled glove box. Pblack samples were exfoliated using PDMS stamps onto Si substrates with a 290 nm SiO2 layer. Prior to exfoliation, substrates were cleaned with a piranha solution and then heated to 300 ◦C for several hours. Optical contrast measurements were used to screen for atomically thin samples and thicknesses were measured by contact-mode AFM. XRD X-ray measurements were carried out on a thick bulk sample exposed to ambient air. A 0.3 mm collimator with 5 mm slits and a point detector were employed to reduce the size of the measured spot. The (020) plane was first probed to ensure the sample was properly aligned within the X-ray beam and to identify b as the out-of-plane axis. θ-scans were then carried out for both (021) and (111) planes in a skew-symmetric configuration. Raman measurements. P-black and 2D-phosphane samples were excited at 532 nm or 633 nm in a back-scattering configuration (b crystal direction) with a fluence varying between 200 and 500 µW µm−2 in a vacuum chamber (P ≤ 10−5 mbar). Most measurements were done at 300 K, except for some data sets presented in Fig. 4. Near-diffraction limited resolution was achieved using an objective with a numerical aperture of 0.5. This relatively low numerical aperture minimized the electric field polarized along the optical axis. The spectral resolution was 0.2 cm−1 . The polarization of the excitation and scattered beams were controlled independently using a rotating half-wave plate and fixed polarizers. Calibration measurements confirmed the polarization isotropy of the optical system in both parallel and orthogonal polarization configurations.

Supporting Information Available Uncorrected polarization-resolved Raman scattering; Raman scattering response of an anisotropic 15


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crystal; Symmetry and Raman modes of bulk black phosphorus and thin 2D-phosphane; Illustration of the Davydov-induced vibrational modes in bilayer 2D-phosphane. This material is available free of charge via the Internet at http://pubs.acs.org/.

Acknowledgement ´ The authors would like to acknowledge the financial support of the Institut de l’Energie Trottier. This work was made possible by funding from Natural Sciences and Engineering Research Council of Canada (NSERC) and the Fonds de Recherche du Québec-Nature et Technologie (FRQNT). A.-L. P.-L. acknowledges financial support from NSERC and FRQNT.

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