A Raman probe of phonons and electron–phonon interactions in the Weyl semimetal NbIrTe4

There is tremendous interest in measuring the strong electron–phonon interactions seen in topological Weyl semimetals. The semimetal NbIrTe4 has been proposed to be a Type-II Weyl semimetal with 8 pairs of opposite Chirality Weyl nodes which are very close to the Fermi energy. We show using polarized angular-resolved micro-Raman scattering at two excitation energies that we can extract the phonon mode dependence of the Raman tensor elements from the shape of the scattering efficiency versus angle. This van der Waals semimetal with broken inversion symmetry and 24 atoms per unit cell has 69 possible phonon modes of which we measure 19 modes with frequencies and symmetries consistent with Density Functional Theory calculations. We show that these tensor elements vary substantially in a small energy range which reflects a strong variation of the electron–phonon coupling for these modes.

www.nature.com/scientificreports/ the known Type-II Weyl semimetal WTe 2 , so it is expected to break inversion symmetry. Several ab initio calculations including spin-orbit coupling indicate that NbIrTe 4 is a type-II Weyl semimetal with 16 Weyl points which contains 8 pairs of opposite chirality located at energies very close to the Fermi energy 39 . With 24 atoms per unit cell, 69 phonon modes are possible in this structure 40 . While several magneto-transport measurements in late 2019 have shown non saturating magneto-resistance at low temperature, and quantum oscillations 41,42 which are consistent with a complex Fermi surface, there are no angle-resolved photoemission spectroscopy (ARPES) measurements to investigate the expected surface Fermi arcs or band crossings expected in this material.
In the first section, we describe angle-resolved polarized micro-Raman scattering measurements on a nanoflake of NbIrTe 4 to investigate the symmetry and frequency of the Stokes scattered Raman modes. We show that the observed modes and symmetries are consistent with ab-initio density functional theoretical (DFT) calculations and extract the normal modes of atomic vibrations for these modes.
Then we consider whether there is evidence of variable electron phonon interactions for different laser excitations. We compare in detail the symmetries and intensities of the observed Raman active modes in NbIrTe 4 , which are excited at 633 nm (red) and at 514 nm (green) wavelengths. We observe enhanced intensity for most Raman modes for 633 nm excitation, and also a notable variation between the two excitation wavelengths for different rotational symmetries of the modes which identify changes in the electron-phonon interaction. These changes with excitation wavelength vary substantially from mode to mode. We conclude with a discussion which shows that we can extract detailed information about each of the Raman tensor elements (electron-phonon coupling constants) for 13 phonon modes ranging from 45 to 260 cm −1 (5-32 meV).

Sample morphology and experimental setup
A single crystal ternary NbIrTe 4 alloy was grown by the flux method (see Methods). The orthorhombic primitive cell of this structure (Fig. 1a) includes 24 atoms with four formula units 43 . Chains of Nb and Ir atoms alternatively follow a zigzag pattern along the 'a' axis, and hybridize with each other long the 'b' axis to form a conducting plane. Te atoms bond on top and bottom of the 'ab' planes to form layers which are van der Waals bonded along the 'c' direction 43 .
A transmission electron microscopy (TEM) image and electron diffraction pattern of NbIrTe 4 sample taken along the 'c' direction is illustrated at Fig. 1b. The TEM diffraction pattern is taken with the electron beam aligned with the central part of a NbIrTe 4 exfoliated flake dispersed onto a holey carbon grid. The TEM image exhibits the uniformity of the exfoliated flake, while the diffraction pattern confirms the symmetry of the orthorhombic structure of this material with the a and b directions of the diffraction pattern corresponding to the 'a' and 'b' axes of the flake.
X-ray diffraction measurements (XRD) from a NbIrTe 4 single crystal grown in the same way determined lattice constants of 3.79(03), 12.52(07) and 13.14(35) Angstroms for the 'a' , 'b' and 'c' lattice constants, respectively. These lattice constants are very close to the calculated values in the literature 43 and are used as inputs to our DFT calculation (see Methods) in order to obtain appropriate interatomic potentials for this crystal. Details of the DFT calculations are in the supplementary document (S1).
The nanoflakes were exfoliated from a NbIrTe 4 single crystal by using NITTO blue tape, and then mechanically transferred on a Si/SiO 2 substrate for angle-resolved polarized micro-Raman spectroscopy (see optical image in Fig. 1c).  www.nature.com/scientificreports/ Figure 1d illustrates a schematic diagram of our polarized micro-Raman experiment system (see Methods). A linearly polarized 632.9 nm or 514.5 nm laser is focused to a ~ 1.5 µm spot onto the nanoflake with the optic axis aligned parallel to the 'c' axis of the nanoflake. The substrate with the nanoflake is rotated keeping the laser spot fixed so that the polarization is at an angle θ relative to the 'a' axis (Fig. 1c). The double-sided black arrow in the image shows the polarization direction of the incident light relative to the 'a' and 'b' axes of the nanoflake marked by white arrows. A polarization analyzer is set so that the polarization of the scattered light (e s ) is either parallel (e i || e s ) or perpendicular ( e i ⊥ e s ) to the incident laser polarization (e i ). Data for both configurations is collected with a 10 minute exposure time with the sample rotated through 360 degrees with 10-degree increments.

Experimental and theoretical results
The orthorhombic primitive cell of NbIrTe 4 structure belongs to the space group Pmn2 1 or point group C 2v including 4 irreducible representations, A 1,2 and B 1,2 . From the character table, the NbIrTe 4 unit cell includes 24 atoms so that 72 phonon modes are possible with 69 Raman-active modes: 23 A 1 + 12 A 2 + 11 B 1 + 23 B 2 . Detecting modes with different irreducible representations can be achieved by selecting the laser excitation direction along with the polarization configuration of the experiment.
The incoming and backscattered beams are along the 'c' axis of NbIrTe 4 lattice (see Fig. 1d), and perpendicular to the nanoflake. We align the 'x' , 'y' and 'z' directions of lab coordinate system with the 'a' , 'b' and 'c' crystallographic axes of the structure, respectively. The angle θ is thus between the polarization direction of the incident beam and the 'x' axis. The Raman experiments are taken either in the parallel configuration (e i || e s ) or in the perpendicular configuration ( e i ⊥ e s ).
We recall that the scattered intensity in Raman experiments is proportional to the excitation frequency and polarization direction of the incident and scattered light 44 : where A provides the scaling with incident laser power, ω i and ω s are the frequencies of the incident and scattered photons, R is the Raman tensor reflecting selection rules and containing the electron-phonon coupling parameters, e i and e s are the incident and scattered polarization vectors. E L is the incident laser energy, E α and E β are the energies of the intermediate electronic states |α > and |β >, ω v is the phonon frequency, E i is the energy of the initial electronic state. H eo and H ep are the electron-photon and electron-phonon operators. Thus the denominator describes resonance conditions for the incoming and outgoing photons 44 . After removing the ω s 4 dependence, the normalized scattered intensities at two different excitation wavelengths provide a view into the Raman tensor dependence on crystal orientation and excitation energy and geometry for each mode and thus connect that to the electron-phonon interactions in this material. The Raman tensors corresponding to the A 1 , A 2 , B 1 and B 2 irreducible representations are given by 45,46 where d, f, g, h, k and l are complex tensor elements of R which depend on the derivatives of the complex dielectric response and are connected to the electron-phonon coupling between electronic and lattice states 47 . Each complex element includes a phase term in the form of the x = |x|e iϕ x .
According to Porto's notation, we see from Eq. (2) that the Z(XX)Z and Z(YY )Z measurements (θ = 0° and θ = 90° when e i || e s ) reveal A 1 modes and, Z(XY )Z and Z(YX)Z measurements (θ = 0° and θ = 90° when e i ⊥ e s ) reveal A 2 modes. Also, because the incident and scattered light propagates along 'c' (z) axis direction (the electric fields are parallel to the xy plane for any θ), Raman experiments do not probe the B 1 or B 2 phonon modes.
For the incident laser polarization at an angle of θ relative to the a axis, the elements of e i and e s are (cos(θ), sin(θ), 0) for the e i || e s a measurement, while for the e i ┴ e s measurement e s has (− sin(θ), cos(θ), 0) elements. Thus, at a fixed excitation frequency, by using e i and e s in Eq. (1), the intensity of Raman modes as a function of rotation angle will take the form: where ϕ df = ϕ d − ϕ f is the relative complex phase factor between the d and f tensor elements. From this result we see that the angular dependence of the A 1 intensity when measured in the parallel configuration depends   can be found in the in the supplementary documents (S5). The details of the DFT calculations which provide both the frequency and symmetry of active Raman modes are provided in the supplementary documents (S1). Figure 2c, d show spectra and fits to a representative A 1 and A 2 mode at both red (633 nm) and green (514 nm) excitations. The frequencies extracted from these fits are 88.4 cm −1 for the A 1 mode and 126.7 cm −1 for the A 2 mode. For comparison, the frequency of the DFT calculated mode is shown as symbols: 85.42 cm −1 for the A 1 mode (3% error) and 125.17 cm −1 for the A 2 mode (1% error). The atomic vibrations for these two modes are shown with two different perspectives, namely viewed along either the 'a' or 'b' axis. Note that the motion of atoms for the A 1 modes is restricted to the 'cb' crystal plane, while the atoms for the A 2 modes can only vibrate parallel Table 1. Calculated A 1 and A 2 Raman modes of the NbIrTe 4 nano-flake by using density functional theory (DFT) and experimentally extracted values in parallel and cross configurations using 633 nm and 514 nm excitation. The last four columns express the parameters d/f and ϕ df determined from the angular dependence for each experimentally detected A 1 modes. Comparing the Raman spectra in Fig. 2 obtained for green (514 nm) or red (633 nm) laser excitation, one can immediately see that generally the scattering efficiencies for most modes increase strongly for red excitation. From Eq. (1) the scattering efficiency usually is dominated by the ω 4 term which thus would increase the intensities for green excitation. In order to see clearly the change in the matrix elements, the spectra in this paper are normalized to the measured integrated intensity of the well-known 520 cm −1 silicon (optical phonon) Raman line. This removes both the ω 4 term and any dependence of the measurement from the Raman instrument and detector. By comparison with reflectivity measurements in the closely related materials TaIrTe 4 48 and WTe 2 49 we do not expect any large change in the dielectric response (real or imaginary parts) with laser excitation energy which might also impact the Raman measurements. The penetration depth is also substantially less than the nanoflake thickness. The normalized intensity ratios (I 633 /I 514 ) for each mode are recorded in Table 1. While two modes at 85.42 cm −1 and 102.16 cm −1 do not show significant change, the two modes at 119.86 cm −1 and 135.31 cm −1 are not detected for 514 nm excitation at all. There are no modes which show a weaker response for red laser excitation, and most show an enhancement of the matrix element by a factor of 2-8. As we shall now discuss, there are also strong changes in the angular dependence of A 1 modes measured for e i || e s for different excitation energies.

The Raman tensor, Raman intensities and variation of electron phonon interaction
As captured by Eqs. 1-7, the intensity of the Raman scattering for different modes depends on both the values of |d|, |f|, and |h| as well as the geometry as reflected in the angle θ. We will see that presentation of the data in polar plots reflects this variation and note that, as well, the response can be significantly different for the two different excitation wavelengths. Figure 3a illustrates a segment of our Z(XX)Z and Z(YY )Z Raman measurements where two 121.7 and 135.9 cm −1 modes are only observed with 633 nm excitation (red dots and line); no response of these modes is observed for 514 nm excitation measurements (blue dots and line). Thus, these two modes clearly demonstrate a strong dependency on excitation energy suggesting that the coupling between electronic states and lattice states is very weak for 514 nm excitation. Fitting spectra of our angle-resolved measurements with the e i || e s configuration for these two modes is obtained by fixing the frequency and varying only the linewidth and intensity for all fits. The results of the intensity versus angle are fit to Eq. (3), as shown with solid lines in Fig. 3b, c. As detailed in the supplementary information (S2), these plots allow us to extract the relative value of '|d|' and '|f|' for each mode. For instance, for the mode at 121.7 cm −1 , |d| and |f| are nearly equal. For the mode at 135.9 cm −1 |d| is about 40% larger than |f|.
The other observed 10 A 1 modes can be classified into two categories. In one category, regardless the effect of φ df on rotational symmetry of the modes (S3), the |d|/|f| ratio stays either larger than or smaller than one as the excitation energy is changed. Figure 4 shows polar plots of 6 modes at two 633 nm (red curves) and 514 nm (blue curves) excitations for a e i || e s configuration. This figure displays 5 modes at 46.4, 75.4, 172.4, 194.4 and 261.2 cm −1 where the maximum for both excitation wavelengths is along 'x' direction with the ratio of |d|/|f| larger than 1 (S2) for both excitations. We see that the maximum of plot at 152.5 cm −1 is along 'y' direction which means |d|/|f| is smaller than 1 for both excitations.
For each of these 10 modes, we note that the integrated intensity with 633 nm excitation is larger than for the comparable 514 nm measurements. The blue numbers within each plot show the factor by which the of 514 nm measurements are multiplied to have equal intensity with 633 nm measurements. As noted in Table 1, this factor is different for each mode, with the largest value of 8.3 for the mode at 46.4 cm −1 and the smallest value of 1.3 for 102.2 cm −1 .
In the second category, which includes 4 (different) modes, the |d|/|f| ratio equals one for one excitation and is distinctly different from one for another excitation. Figure 5 shows polar plots of 4 modes at both excitation  Clearly coupling between electron and lattice states strongly depends on excitation energy for these modes.

Discussion
In the experiments described above, we have explored Raman scattering from 13 A 1 modes and 6 A 2 modes which are excited using both green (514 nm) and red (633 nm) laser excitation. We have shown that we can extract |d|/|f| and ϕ df for every A 1 mode by fitting the shape of the angular dependence of the Raman spectra for (e i ||e s ) using Eq. (3). It is straightforward to show that by combining this data with the intensity ratios, I 633 /I 514 , one can extract directly the excitation energy dependence of |d| and |f| directly (S7). Specifically, for A 1 modes which exhibit a maximum intensity for θ = 0°, we find: If the maximum intensity for the mode is at θ = 90° then: From this we can extract the ratio for the d Raman tensor element: For the A 2 modes where we can measure the intensity ratios I 633 /I 514 which is directly related to the h Raman tensor element: In Table 2 we show all of the results from these experiments. While all the A 1 phonon modes essentially have the same symmetries, the modes show a wide range of responses in both intensity and changes in the relative electron-phonon coupling parameters, |d| and |f|. Most modes show a substantial increase in Raman scattering efficiency in the red which range from factors of 2-8 (see Tables 1, 2 or Fig. 6), while two A 1 modes are not seen at all in the green, and two modes do not show any intensity enhancement. Such differing behavior from mode to mode is not what is usually seen in resonant Raman experiments. In most cases, all Raman modes are seen to increase in intensity substantially when a virtual intermediate electronic state is created and destroyed in the Raman scattering process because the energy denominator in the Raman tensor element tends towards zero for all modes (see denominator of Eq. 1). One concludes that the variation in the intensity and also in |d/|f| must result from differences in the matrix elements in the numerator. An incoming photon creates a real or virtual electronic state which creates a phonon through electron-phonon coupling by scattering to an intermediate electronic state before the electron and hole recombine emitting a photon at a lower energy. To show the wide variation in the energy variation of the Raman tensor elements, we use Eqs. 8-11 to extract |d 633 |/|d 514 |, |f 633 |/|f 514 | and |h 633 |/|h 514 |. These values are recorded in Table 2. In Fig. 6 we display I 633 /I 514 , |d 633 |/|d 514 |, and |f 633 |/|f 514 | as a function of phonon energy for the A 1 phonon modes. One can clearly see strong increases in the Raman scattering efficiencies for red (633 nm) excitation in many of the Raman modes. However, the behavior of |d| and |f| are different. For example, maximum intensity enhancements are seen (8)   4 cm −1 mode shows that the enhancement for the |d| tensor element is larger than for |f|. However, the 172.4 cm −1 mode shows that the enhancement for the |f| tensor element is larger than for |d|, while the 261.2 cm −1 phonon shows the same enhancement for |d| and for |f|. More interestingly, there is a peak enhancement in the |f| tensor element where the intensity enhancement is a minimum. It is the variation of the electron-phonon coupling which results in the creation and scattering of phonons which results in such a wide variation in intensities and angular polarization response moving from green to red excitation. A number of theoretical works have shown that the coupling of phonons to the chiral Fermions in these topological Weyl semimetals is quite complex 29,30,32,33 Substantial changes (even Fano resonances) have been observed as a function of temperature for both Raman scattering and also infrared absorption, and also most recently in a magnetic field [34][35][36][37][38]40 . The results here shows that substantial changes can also occur in the electron-phonon coupling during the creation of virtual intermediate states in resonant Raman scattering.

Conclusion
A total of 19 Raman active modes, 13 A 1 and 6 A 2 from NbIrTe 4 have been detected through angle-resolved polarized micro-Raman scattering for 633 nm and 514 nm excitations. The frequency and symmetry of the detected modes is consistent with expectations from DFT calculations. The DFT calculations also provide the normal modes for each mode in which A 1 modes have out-of-plane motions while A 2 modes have in-plane motions. Utilizing the symmetry of the A 1 modes, we are able to explore the relative change of the scattering efficiency and |d|/|f| electron-phonon constants as a function of excitation energy for each phonon mode. We observe efficiency increases for the majority of modes for red excitation rather than green, but these increases vary substantially from mode to mode. For example, some modes are unexpectedly absent for green excitation while others are seen to be unchanged for both red and green excitation. Thus, some modes clearly show increases or decreases of the |d|/|f| electron-phonon coupling parameters, while others do not change. Through analysis of these results, we are able to extract the quantitative enhancements of the electron-phonon tensor elements |d|, |f| and |h| for 10 A 1 phonons and 2 A 2 phonons ranging from 5 to 32 meV (45 to 260 cm −1 ). We conclude that these changes cannot be related to common excitations of the same intermediate electronic states, but rather result from the complex coupling of the phonons with the complex electronic states (possibly the Chiral Fermion states) which have been seen in a number of other systems [34][35][36][37][38] . Sample characterization. Lattice parameters of NbIrTe 4 were obtained at 300 K using single crystal X-ray diffraction (SCXRD). Data was collected using a Bruker Kappa Apex II single-crystal diffractometer with Mo Ka radiation and a TRIUMPH monochromator. www.nature.com/scientificreports/ Angular-dependent Raman scattering measurements. Figure 1d illustrates a schematic diagram of our polarized micro-Raman experiment system. For excitation we have used a polarized 632.8 nm wavelength of a He/Ne laser and a polarized 514.5 nm wavelength of an Argon-ion laser. The laser beam is focused to a ~ 1.5 µm spot size onto the nanoflake by using a 100 × objective. The laser power at 633 and 514 nm measurements is limited to less than 200 µW on the sample. The incoming and backscattered laser beams are along the 'c' axis of the flake and the linear polarization of the incident beam is fixed during experiment. By rotating the measurement stage which carries NbIrTe 4 flake on a Si/SiO 2 substrate, it is possible to align the incident laser polarization along the 'a' or 'b' axes of the crystal, or any arbitrary angle in between while keeping the laser spot constant on the nanoflake. The backscattered light passes through a polarizer which is fixed to provide the maximum efficiency through a Dilor triple-spectrometer with two 1800 mm −1 gratings used in the subtractive mode to remove the scattered laser light followed by a dispersive spectrograph with a 1800 mm −1 grating. The signal is detected by a LN2 cooled CCD camera. A half-wave plate before the polarizer is allows us to select the polarization of the scattered light (e s ) to be either parallel (e i || e s ) or perpendicular ( e i ⊥ e s ) to the incident laser polarization (e i ).

Methods
DFT calculations of phonon modes. Our calculations are performed using density functional theory (DFT) as implemented in the Vienna ab initio simulation package (VASP) code [50][51][52] . The Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional and the projector-augmented-wave (PAW) approach are used. Throughout the work, the cutoff energy is set to be 550 eV for expanding the wave functions into plane-wave basis, and the number of k points was set to 4 × 4 × 4 for a 3 × 1 × 1 supercell. The real-space force constants of the supercells were calculated in the density-functional perturbation theory (DFPT) 53 and the phonon frequencies were calculated from the force constants using the PHONOPY code 54 . In our calculations, we adopt the experimental structural parameters (a = 3.7903 Å, b = 12.5207 Å, and c = 13.1435 Å).