Flat Optical Conductivity in ZrSiS due to Two-Dimensional Dirac Bands

M. B. Schilling; L. M. Schoop; B. V. Lotsch; M. Dressel; A. V. Pronin

doi:10.1103/PhysRevLett.119.187401

Flat Optical Conductivity in ZrSiS due to Two-Dimensional Dirac Bands

M. B. Schilling, L. M. Schoop, B. V. Lotsch, M. Dressel, A. V. Pronin

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Abstract

ZrSiS exhibits a frequency-independent interband conductivity σ(ω)=const(ω)≡σflat in a broad range from 250 to 2500 cm-1 (30-300 meV). This makes ZrSiS similar to (quasi-)two-dimensional Dirac electron systems, such as graphite and graphene. We assign the flat optical conductivity to the transitions between quasi-two-dimensional Dirac bands near the Fermi level. In contrast to graphene, σflat is not universal but related to the length of the nodal line in the reciprocal space, k0. Because of spin-orbit coupling, the discussed Dirac bands in ZrSiS possess a small gap Δ, for which we determine an upper bound max(Δ)=30 meV from our optical measurements. At low temperatures the momentum-relaxation rate collapses, and the characteristic length scale of momentum relaxation is of the order of microns below 50 K.

Original language	English (US)
Article number	187401
Journal	Physical review letters
Volume	119
Issue number	18
DOIs	https://doi.org/10.1103/PhysRevLett.119.187401
State	Published - Nov 1 2017
Externally published	Yes

All Science Journal Classification (ASJC) codes

Physics and Astronomy(all)

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10.1103/PhysRevLett.119.187401

Cite this

@article{5b8d774f10ac4ef9a55b2cad807863a7,

title = "Flat Optical Conductivity in ZrSiS due to Two-Dimensional Dirac Bands",

abstract = "ZrSiS exhibits a frequency-independent interband conductivity σ(ω)=const(ω)≡σflat in a broad range from 250 to 2500 cm-1 (30-300 meV). This makes ZrSiS similar to (quasi-)two-dimensional Dirac electron systems, such as graphite and graphene. We assign the flat optical conductivity to the transitions between quasi-two-dimensional Dirac bands near the Fermi level. In contrast to graphene, σflat is not universal but related to the length of the nodal line in the reciprocal space, k0. Because of spin-orbit coupling, the discussed Dirac bands in ZrSiS possess a small gap Δ, for which we determine an upper bound max(Δ)=30 meV from our optical measurements. At low temperatures the momentum-relaxation rate collapses, and the characteristic length scale of momentum relaxation is of the order of microns below 50 K.",

author = "Schilling, {M. B.} and Schoop, {L. M.} and Lotsch, {B. V.} and M. Dressel and Pronin, {A. V.}",

note = "Funding Information: Schilling M. B. 1 Schoop L. M. 2 Lotsch B. V. 2 Dressel M. 1 Pronin A. V. 1 Physikalisches Institut, 1 Universit{\"a}t Stuttgart , Pfaffenwaldring 57, 70569 Stuttgart, Germany 2 Max Planck Institute for Solid State Research , Heisenbergstr. 1, 70569 Stuttgart, Germany 1 November 2017 3 November 2017 119 18 187401 30 July 2017 {\textcopyright} 2017 American Physical Society 2017 American Physical Society ZrSiS exhibits a frequency-independent interband conductivity σ ( ω ) = const ( ω ) ≡ σ flat in a broad range from 250 to 2500 cm - 1 (30–300 meV). This makes ZrSiS similar to (quasi-)two-dimensional Dirac electron systems, such as graphite and graphene. We assign the flat optical conductivity to the transitions between quasi-two-dimensional Dirac bands near the Fermi level. In contrast to graphene, σ flat is not universal but related to the length of the nodal line in the reciprocal space, k 0 . Because of spin-orbit coupling, the discussed Dirac bands in ZrSiS possess a small gap Δ , for which we determine an upper bound max ( Δ ) = 30 meV from our optical measurements. At low temperatures the momentum-relaxation rate collapses, and the characteristic length scale of momentum relaxation is of the order of microns below 50 K. Deutsche Forschungsgemeinschaft http://dx.doi.org/10.13039/501100001659 German Research Foundation DFG http://sws.geonames.org/2921044/ DFG DR228/51-1 Max-Planck-Gesellschaft http://dx.doi.org/10.13039/501100004189 Max Planck Society MPG http://sws.geonames.org/2921044/ Succeeding the intense investigations of the linearly dispersing energy bands in two-dimensional (2D) graphene, line-node semimetals (LNSMs) were predicted in 2011 [1] . Here, 2D electronic bands with linear dispersion (Dirac bands) cross each other along continuous lines (loops) in reciprocal space [2–15] . In general, the topology of the nodal lines within a Brillouin zone (BZ) may be very complex; e.g., nodal lines may be linked and knotted in different ways [16–18] . Important is that 2D Dirac electrons exist in the three-dimensional (3D) bulk of a LNSM. Such 3D materials with 2D Dirac electrons (i.e., the 3D analogues of graphene) are supposed to demonstrate a number of unusual electronic properties that can potentially be useful for applications [1–3] . Many theoretical propositions are around for materials realizing the LNSM phase, such as Cu 3 PdN [7] , SrIrO 3 [8] , CaAgP, and CaAgAs [9] , as well as a new crystallographic form of Ca 3 P 2 [10] . Evidence for a possible realization of a LNSM state has been recently obtained via angle-resolved photoemission spectroscopy (ARPES) in PbTaSe 2 [5] . A state similar to LNSM, but with Dirac arcs instead of closed loops, is reported in PtSn 4 [6] . Further, much attention is currently paid to ZrSiS [11–13,15] and its structural analogues, such as HfSiS [15,19] , ZrSiTe [14,20] , and CeSbTe [21] . The presence of effectively 2D Dirac bands in ZrSiS and its family is well established by several methods, including ARPES [12–15,22] , Hall measurements [23,24] , and quantum oscillations [20,22–28] , as well as by electronic-structure calculations [12–15,22] . These studies demonstrate that ZrSiS possesses two types of line nodes. The line nodes of the first type are situated far away ( ∼ 0.7 eV ) from the Fermi level; we dub them as high-energy nodes. Spin-orbit coupling (SOC) opens a gap along certain portions of this nodal line [12,15] . The second-type line nodes appear close to the Fermi level (low-energy nodes), but are believed to be fully gapped due to SOC, similarly to such LNSM candidates as Cu 3 PdN [7] and SrIrO 3 [8] . The gap, however, is calculated to be very small, of the order of 10 meV [12] . Such a small value has indeed been confirmed by recent ARPES measurements [15] , although the resolution was not sufficient to accurately determine the gap size. At higher energies (up to a few hundreds meV), the linearity of the line-node Dirac bands in ZrSiS remains uncompromised [12,15] . These low-energy line nodes are the focus of the present study. Dirac electrons in solids are known to manifest themselves in peculiar ways in different experiments [29–33] . One such manifestation is in their optical response (i.e., the ac transport), usually expressed in terms of the complex optical conductivity, σ ( ω ) = σ 1 ( ω ) + i σ 2 ( ω ) . For example, in the simplest case of electron-hole symmetric d -dimensional ungapped Dirac (Weyl) bands, σ 1 ( ω ) is supposed to follow a power law, σ 1 ( ω ) ∝ ω d - 2 [34,35] . This optical-conductivity behavior—unusual for conventional materials—has indeed been confirmed for quasi-2D electrons in graphite and graphene, where σ 1 ( ω ) ≈ const ( ω ) [36,37] , and for 3D Dirac electrons in such point-node Dirac or Weyl semimetals as ZrTe 5 [38] , Cd 3 As 2 [39] , and TaAs [40] , where σ 1 ( ω ) ∝ ω was reported. As mentioned above, the Dirac electrons in a (3D) LNSM live effectively in two dimensions. Thus, the optical conductivity of a LNSM should be similar to the one of graphene, i.e., frequency independent. Such flat optical conductivity in LNSMs has indeed been recently predicted by theory [41–43] . Here, we report observation of frequency-independent optical conductivity in ZrSiS. This evidences the existence of quasi-2D Dirac states and a quasi-2D electronic ac transport in this material. From our optical measurements, we extract the length of the nodal line (the node is understood here as the Dirac point of the gapped Dirac band) and estimate the size of the gap in this band. The investigated single crystals were grown by loading equimolar amounts of Zr, Si, and S together with a small amount of iodine in a sealed quartz tube, which was kept at 1100 ° C for 1 week. A temperature gradient of 100 ° C was applied and the crystals were collected at the cold end of the tube. The crystal structure (tetragonal, space group P 4 / mmm ) was confirmed with x-ray and electron diffraction similarly to Ref. [12] . The optical reflectivity R ( ν ) was measured at 10 to 300 K over a broad frequency range from ν = ω / 2 π ≈ 50 to 25 000 cm - 1 using commercial Fourier-transform infrared spectrometers. All measurements were performed on freshly cleaved (001) surfaces. In accordance with the tetragonal structure, no in-plane optical anisotropy was detected. At low frequencies, an in situ gold evaporation technique was utilized for reference measurements, while above 1000 cm - 1 gold and protected silver mirrors served as references. The high-frequency range was extended by room-temperature ellipsometry measurements up to 45 000 cm - 1 in order to obtain more accurate results for the Kramers-Kronig analysis [44] . The Kramers-Kronig analysis was made involving the x-ray atomic scattering functions for high-frequency extrapolations [45] and dc-conductivity values, σ dc ( T ) , and the reflectivity-fitting procedure [46] for zero-frequency extrapolations. Important to note is that our optical measurements reflect the bulk material properties, since the skin depth is above 40 nm for any measurement frequency. The measured frequency-dependent reflectivity R ( ν ) is shown in Fig. 1 for selected temperatures. Above 1000 cm - 1 , the temperature has only a minor influence on the spectra. In the low-frequency range, the reflectivity is rather high (above 99%), in agreement with the very low dc resistivity [23,25] . The results of the Kramers-Kronig analysis are shown in Figs. 2–4 in terms of the real and imaginary parts of optical conductivity, as well as the real part of permittivity, ϵ 1 ( ω ) = 1 – 4 π σ 2 ( ω ) / ω . 1 10.1103/PhysRevLett.119.187401.f1 FIG. 1. Frequency-dependent reflectivity of ZrSiS at 10 and 300 K. For intermediate temperatures the reflectivity curves lie in between, and are not shown for clarity. No temperature dependence is seen above ∼ 8000 cm - 1 . The sketches show the position of the low-energy nodal line (red dashed line) in BZ and the Dirac bands near the Fermi level. 2 10.1103/PhysRevLett.119.187401.f2 FIG. 2. Real part of the optical conductivity of ZrSiS as a function of frequency. An important result of this work is presented in Fig. 2 : the real part of the optical conductivity is almost frequency independent, σ 1 ( ω ) = σ flat ≈ 6600 Ω - 1 cm - 1 , in the range from 250 to 2500 cm - 1 [30–300 meV] basically at all temperatures investigated (at T ≥ 100 K , the flat region starts at a bit higher frequencies because of a rather broad free-electron contribution). Such frequency-independent behavior of σ 1 ( ω ) is similar to what has been predicted [47] and observed [37] in graphene and matches the theory for the optical response due to transitions between the 2D Dirac states in LNSMs [41–43] . In contrast to graphene, no universal sheet conductance is expected for LNSMs; instead, σ flat is related to the length of the nodal line k 0 in a BZ. For a circular nodal line, one has σ 1 ( ω ) = σ flat = e 2 k 0 16 ℏ . (1) It is assumed here that the plane of the nodal circle is perpendicular to the electric-field component of the probing radiation and that there is no particle-hole asymmetry [41–43] . In ZrSiS, the low-energy nodal line is not circular and not even flat; instead, the BZ contains a 3D “cage” of nodal lines [12] . Thus, a straightforward application of this formula is not rigourously validated. Nevertheless, having no better model at hand, we use Eq. (1) for a rough estimate of k 0 = 4.3 {\AA} - 1 . This value seems to be reasonable: according to the band-structure calculations, the total lengths of the nodal line projections on the [100] and [001] directions are about 3.5 and 6 {\AA} - 1 per BZ, respectively. At ν > 3000 cm - 1 , σ 1 ( ω ) is not frequency independent anymore; it decreases with frequency. This is likely because the 2D Dirac band loses its linearity at such large energies [12] . At even higher frequencies, σ 1 ( ω ) starts to rise as ω increases. It would be of interest to detect the transitions between the high-energy Dirac bands above approximately 1.3 eV = 10 500 cm - 1 , but we do not see any clear signatures of such transitions: as discussed above, these Dirac bands are partly gapped by SOC and for bands of such complex shapes the optical conductivity is not expected to be flat or linear in frequency. Additionally, non-Dirac bands above the Fermi level also contribute to the absorption processes at these frequencies, as one can see from the band structure calculations of Refs. [12,15] . At the lowest frequencies measured, ν < 500 cm - 1 , σ 1 ( ω ) also deviates from the flat behavior, as can be seen in Figs. 2 and 3 , and exhibits several features. The contribution of free carriers (so-called Drude response) is present at all temperatures, but best seen at 100 and 300 K. A free-electron component is expected in the optical response because the Fermi level in ZrSiS is slightly above the nodal line [12,15] . As the temperature drops, the Drude band narrows, revealing two distinct modes at around 100 and 150 cm - 1 . A brief discussion on the possible origins of these modes is given below. The narrowing of the Drude band reflects the strong suppression of the dc resistivity [23,25] and of the carrier scattering rate (or, more accurately, the momentum-relaxation rate) as T → 0 . 3 10.1103/PhysRevLett.119.187401.f3 FIG. 3. Examples of the optical conductivity fits for ZrSiS at low frequencies. Thick solid lines are experimental data; thin solid lines are total fits; dashed lines represent fit contributions of the Drude (red), Pauli-edge (black), and Lorentzian (olive, shown only for T = 10 K ) terms. Thin dotted (blue) lines represent attempts to fit the 300 K data without the Lorentzians. At the lowest temperatures, σ 1 ( ω ) develops a minimum around 200 cm - 1 (bottom panel of Fig. 3 ). We relate this dip and the corresponding feature in σ 2 ( ω ) , see Fig. 4 , to the Pauli blocking of the transitions in the 2D band. Such features, related to the position of the Fermi level, are well known in semiconductors [48] and have recently been discussed in relation to graphene and Dirac or Weyl semimetals [49–51] . As already mentioned, band-structure calculations and ARPES measurements locate the Fermi level in ZrSiS in the upper (conduction) Dirac band [12,15] ; see the sketch in Fig. 1 . Thus, the Pauli edge must be seen in the interband portion of optical conductivity. An onset of the interband transitions commonly shows up at the frequency equal to max { Δ , 2 μ } [39,42,43,49–51] , with Δ being the band gap and μ the position of the Fermi level relative to the Dirac point. Thus, Eq. (1) can be modified to σ 1 ( ω ) = e 2 k 0 16 ℏ × θ ( ℏ ω - max { Δ , 2 μ } ) , (2) where θ ( x ) is the Heaviside step function. From Eq. (2) and the bottom panel of Fig. 3 , one can conclude that max { Δ , 2 μ } must be smaller than approximately 250 cm - 1 or some 30 meV and, hence Δ < 30 meV . 4 10.1103/PhysRevLett.119.187401.f4 FIG. 4. Main frame: imaginary part of the optical conductivity in ZrSiS. Arrow indicates the position of the Pauli edge at 10 K. Inset: real part of the dielectric constant near the plasma frequency. This estimate of the upper limit of Δ from optical data is in good agreement with the value obtained from band-structure calculations (15 meV, [12] ). The ARPES value of 60 meV [15] likely overestimates the gap due to the limited resolution. The fact that the relative position of the Dirac points and the Fermi level is slightly k -dependent [12,15] might lead to a broadening of the optical Pauli-edge feature observed. Importantly, even if the Fermi level appears within the gap for some values of k , our conclusion on the upper limit of Δ still holds. Let us now turn to the modes at 100 and 150 cm - 1 and consider possible scenarios for their origin. (i) Phonons. It is extremely unlikely that the observed modes originate from phonons as such, because the frequencies of the modes do not agree with calculations of the infrared-active phonons [52] and because the modes are unrealistically strong for any phonon except soft modes in ferroelectrics. However, no ferroelectricity is reported or expected in ZrSiS and our modes do not shift with temperature, as would be the case for ferroelectric soft modes. (ii) Bulk electron localization. A disorder-induced localization of bulk electrons seems also rather unlikely, since the material is very clean [28] . Its dc resistivity monotonically decreases with temperature and is extremely low at T → 0 [23,25] . (iii) Surface contributions. Although our skin depth is above 40 nm for any measurement frequency, some surface contributions cannot be fully excluded. Very recently, extremely strong surface states in ZrSiS have been theoretically predicted and observed by ARPES measurements [53] . However, no surface contributions are seen in dc-conductivity measurements on microstructure samples with different thicknesses [54] . Appearance of inhomogeneities in the electron surface density, possibly with electron localization, can reconcile the results of Refs. [53] and [54] , as well as provide an explanation for our low-frequency modes. (iv) Excitons. At first glance, it does not seem plausible that excitons are the reason for the observed features, because of the strong screening effects from free carriers. Nevertheless, if we calculate the exciton binding energy E b within the simplest hydrogenic exciton model [55] using the effective mass of the carriers in the low-energy 2D Dirac band, m * = 0.025 m e [28] , and an estimate of the static lattice dielectric constant, ϵ 0 ∼ 7.5 [56] , we obtain E b ∼ 100 cm - 1 . Excitons with such E b are supposed to be seen in the spectra at Δ - E b ≤ 150 cm - 1 ( Δ ≤ 250 cm - 1 ), i.e., right in the frequency range of our modes. One can propose that the excitons may manifest themselves on or near the surface, where free-electron screening is weaker. Furthermore, large strengths of the observed modes in this case can be quantitatively explained by bounding between excitons and surface inhomogeneities [57] . Further studies are necessary to fully clarify the nature of the low-energy absorption peaks in ZrSiS. To get some more quantitative estimates of the parameters determining the optical response, we fit the optical conductivity with a model consisting of a Drude term, two Lorentzians, and a term describing the Pauli edge. Scattering and other processes, leading to broadening of the sharp step in Eq. (2) , may be taken into account by replacing the Heaviside function with 1 2 + 1 π arctan ω - max { Δ , 2 μ } / ℏ Γ , (3) for example, where Γ represents a broadening parameter due to the k -dependent gap, impurity scattering, or temperature. At T = 10 K , reasonable fits can be obtained with a very sharp Pauli edge, i.e., with Γ of a few cm - 1 . We set Γ = k B T / ℏ for all temperatures, since smaller values seem not to be physical. This yields Γ = 7 , 35, and 210 cm - 1 for 10, 50, and 300 K, respectively; see Fig. 3 . In all our fits we keep the zero-frequency limit of the Drude term equal to σ dc at all temperatures. Owed to the broad Drude tail, the description of the 300 K data is straightforward. It provides the momentum-relaxation rate of free carriers, γ = 1 / ( 2 π τ ) = ( 120 ± 10 ) cm - 1 ( τ is the corresponding relaxation time), and a plasma frequency, ω pl / 2 π = ( 24 000 ± 1000 ) cm - 1 [58] . On the other hand, the screened plasma frequency, ω pl scr = ω p l / ϵ ∞ ( ϵ ∞ is the contribution of the higher-frequency optical transitions to ϵ 1 ), can be directly determined from optical measurements as the zero-crossing point of ϵ 1 ( ν ) [44] . We find ω pl scr to be temperature independent and situated at 8900 cm - 1 , cf. the inset of Fig. 4 . Hence, ϵ ∞ = ( ω pl / ω pl scr ) 2 ≈ 7 , which is in good agrement with the optical measurements, presented in same figure. As one can see from Figs. 2–4 , the Drude term becomes narrower as T → 0 . At low temperatures, γ is below our measurement window and our fits thus might become ambiguous. To avoid this, we first tried to keep the plasma frequency of the Drude term constant as a function of T , but this turned out to be unsatisfactory. Some spectral weight had to be redistributed between the Drude term and the Lorentzians. Nevertheless, we tried to have this spectral weight transfer as small as possible and the total plasma frequency of the three terms (Drude plus two Lorentzians) to be temperature independent in accordance with the temperature-independent ω pl . Examples of the fits obtained in this way are shown in Fig. 3 . At T ≤ 50 K , we found that γ ≈ 2 to 2.5 cm - 1 and the momentum-relaxing τ is 2.1–2.7 ps. Interestingly, at low temperatures the momentum-relaxation length, ℓ mr = v F τ , obtained from our estimate of τ , becomes macroscopically large. Using v F = 5 × 10 5 m / s as an average Fermi velocity in the low-energy Dirac bands [23] , we obtain ℓ mr ≥ 1 μ m for T ≤ 50 K . This implies that the hydrodynamic behavior of electrons, reported recently in clean samples of graphene [59,60] and the Weyl semimetal WP 2 [61] , might also be realized in ZrSiS. This proposition seems reasonable, because only linear bands with highly mobile carriers (typical mobilities are 1 0 3 – 1 0 4 cm 2 / V s [23,24,28] ) cross the Fermi level in ZrSiS. Summarizing, the real part of the optical conductivity of ZrSiS was found to be independent on frequency in a rather broad range from 250 to 2500 cm - 1 (30–300 meV). Our observations are supported by recent theoretical predictions for the optical response of LNSMs, and constitute an independent confirmation of 2D Dirac bands in ZrSiS near the Fermi level. The characteristic features of the Pauli edge, appearing in the low-frequency spectra, provide the upper limit ( 250 cm - 1 , 30 meV) for the gap between the 2D Dirac bands. The momentum-relaxation length is at the micrometer scale at T ≤ 50 K . 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year = "2017",

month = nov,

day = "1",

doi = "10.1103/PhysRevLett.119.187401",

language = "English (US)",

volume = "119",

journal = "Physical Review Letters",

issn = "0031-9007",

publisher = "American Physical Society",

number = "18",

}

TY - JOUR

T1 - Flat Optical Conductivity in ZrSiS due to Two-Dimensional Dirac Bands

AU - Schilling, M. B.

AU - Schoop, L. M.

AU - Lotsch, B. V.

AU - Dressel, M.

AU - Pronin, A. V.

N1 - Funding Information: Schilling M. B. 1 Schoop L. M. 2 Lotsch B. V. 2 Dressel M. 1 Pronin A. V. 1 Physikalisches Institut, 1 Universität Stuttgart , Pfaffenwaldring 57, 70569 Stuttgart, Germany 2 Max Planck Institute for Solid State Research , Heisenbergstr. 1, 70569 Stuttgart, Germany 1 November 2017 3 November 2017 119 18 187401 30 July 2017 © 2017 American Physical Society 2017 American Physical Society ZrSiS exhibits a frequency-independent interband conductivity σ ( ω ) = const ( ω ) ≡ σ flat in a broad range from 250 to 2500 cm - 1 (30–300 meV). This makes ZrSiS similar to (quasi-)two-dimensional Dirac electron systems, such as graphite and graphene. We assign the flat optical conductivity to the transitions between quasi-two-dimensional Dirac bands near the Fermi level. In contrast to graphene, σ flat is not universal but related to the length of the nodal line in the reciprocal space, k 0 . Because of spin-orbit coupling, the discussed Dirac bands in ZrSiS possess a small gap Δ , for which we determine an upper bound max ( Δ ) = 30 meV from our optical measurements. At low temperatures the momentum-relaxation rate collapses, and the characteristic length scale of momentum relaxation is of the order of microns below 50 K. Deutsche Forschungsgemeinschaft http://dx.doi.org/10.13039/501100001659 German Research Foundation DFG http://sws.geonames.org/2921044/ DFG DR228/51-1 Max-Planck-Gesellschaft http://dx.doi.org/10.13039/501100004189 Max Planck Society MPG http://sws.geonames.org/2921044/ Succeeding the intense investigations of the linearly dispersing energy bands in two-dimensional (2D) graphene, line-node semimetals (LNSMs) were predicted in 2011 [1] . Here, 2D electronic bands with linear dispersion (Dirac bands) cross each other along continuous lines (loops) in reciprocal space [2–15] . In general, the topology of the nodal lines within a Brillouin zone (BZ) may be very complex; e.g., nodal lines may be linked and knotted in different ways [16–18] . Important is that 2D Dirac electrons exist in the three-dimensional (3D) bulk of a LNSM. Such 3D materials with 2D Dirac electrons (i.e., the 3D analogues of graphene) are supposed to demonstrate a number of unusual electronic properties that can potentially be useful for applications [1–3] . Many theoretical propositions are around for materials realizing the LNSM phase, such as Cu 3 PdN [7] , SrIrO 3 [8] , CaAgP, and CaAgAs [9] , as well as a new crystallographic form of Ca 3 P 2 [10] . Evidence for a possible realization of a LNSM state has been recently obtained via angle-resolved photoemission spectroscopy (ARPES) in PbTaSe 2 [5] . A state similar to LNSM, but with Dirac arcs instead of closed loops, is reported in PtSn 4 [6] . Further, much attention is currently paid to ZrSiS [11–13,15] and its structural analogues, such as HfSiS [15,19] , ZrSiTe [14,20] , and CeSbTe [21] . The presence of effectively 2D Dirac bands in ZrSiS and its family is well established by several methods, including ARPES [12–15,22] , Hall measurements [23,24] , and quantum oscillations [20,22–28] , as well as by electronic-structure calculations [12–15,22] . These studies demonstrate that ZrSiS possesses two types of line nodes. The line nodes of the first type are situated far away ( ∼ 0.7 eV ) from the Fermi level; we dub them as high-energy nodes. Spin-orbit coupling (SOC) opens a gap along certain portions of this nodal line [12,15] . The second-type line nodes appear close to the Fermi level (low-energy nodes), but are believed to be fully gapped due to SOC, similarly to such LNSM candidates as Cu 3 PdN [7] and SrIrO 3 [8] . The gap, however, is calculated to be very small, of the order of 10 meV [12] . Such a small value has indeed been confirmed by recent ARPES measurements [15] , although the resolution was not sufficient to accurately determine the gap size. At higher energies (up to a few hundreds meV), the linearity of the line-node Dirac bands in ZrSiS remains uncompromised [12,15] . These low-energy line nodes are the focus of the present study. Dirac electrons in solids are known to manifest themselves in peculiar ways in different experiments [29–33] . One such manifestation is in their optical response (i.e., the ac transport), usually expressed in terms of the complex optical conductivity, σ ( ω ) = σ 1 ( ω ) + i σ 2 ( ω ) . For example, in the simplest case of electron-hole symmetric d -dimensional ungapped Dirac (Weyl) bands, σ 1 ( ω ) is supposed to follow a power law, σ 1 ( ω ) ∝ ω d - 2 [34,35] . This optical-conductivity behavior—unusual for conventional materials—has indeed been confirmed for quasi-2D electrons in graphite and graphene, where σ 1 ( ω ) ≈ const ( ω ) [36,37] , and for 3D Dirac electrons in such point-node Dirac or Weyl semimetals as ZrTe 5 [38] , Cd 3 As 2 [39] , and TaAs [40] , where σ 1 ( ω ) ∝ ω was reported. As mentioned above, the Dirac electrons in a (3D) LNSM live effectively in two dimensions. Thus, the optical conductivity of a LNSM should be similar to the one of graphene, i.e., frequency independent. Such flat optical conductivity in LNSMs has indeed been recently predicted by theory [41–43] . Here, we report observation of frequency-independent optical conductivity in ZrSiS. This evidences the existence of quasi-2D Dirac states and a quasi-2D electronic ac transport in this material. From our optical measurements, we extract the length of the nodal line (the node is understood here as the Dirac point of the gapped Dirac band) and estimate the size of the gap in this band. The investigated single crystals were grown by loading equimolar amounts of Zr, Si, and S together with a small amount of iodine in a sealed quartz tube, which was kept at 1100 ° C for 1 week. A temperature gradient of 100 ° C was applied and the crystals were collected at the cold end of the tube. The crystal structure (tetragonal, space group P 4 / mmm ) was confirmed with x-ray and electron diffraction similarly to Ref. [12] . The optical reflectivity R ( ν ) was measured at 10 to 300 K over a broad frequency range from ν = ω / 2 π ≈ 50 to 25 000 cm - 1 using commercial Fourier-transform infrared spectrometers. All measurements were performed on freshly cleaved (001) surfaces. In accordance with the tetragonal structure, no in-plane optical anisotropy was detected. At low frequencies, an in situ gold evaporation technique was utilized for reference measurements, while above 1000 cm - 1 gold and protected silver mirrors served as references. The high-frequency range was extended by room-temperature ellipsometry measurements up to 45 000 cm - 1 in order to obtain more accurate results for the Kramers-Kronig analysis [44] . The Kramers-Kronig analysis was made involving the x-ray atomic scattering functions for high-frequency extrapolations [45] and dc-conductivity values, σ dc ( T ) , and the reflectivity-fitting procedure [46] for zero-frequency extrapolations. Important to note is that our optical measurements reflect the bulk material properties, since the skin depth is above 40 nm for any measurement frequency. The measured frequency-dependent reflectivity R ( ν ) is shown in Fig. 1 for selected temperatures. Above 1000 cm - 1 , the temperature has only a minor influence on the spectra. In the low-frequency range, the reflectivity is rather high (above 99%), in agreement with the very low dc resistivity [23,25] . The results of the Kramers-Kronig analysis are shown in Figs. 2–4 in terms of the real and imaginary parts of optical conductivity, as well as the real part of permittivity, ϵ 1 ( ω ) = 1 – 4 π σ 2 ( ω ) / ω . 1 10.1103/PhysRevLett.119.187401.f1 FIG. 1. Frequency-dependent reflectivity of ZrSiS at 10 and 300 K. For intermediate temperatures the reflectivity curves lie in between, and are not shown for clarity. No temperature dependence is seen above ∼ 8000 cm - 1 . The sketches show the position of the low-energy nodal line (red dashed line) in BZ and the Dirac bands near the Fermi level. 2 10.1103/PhysRevLett.119.187401.f2 FIG. 2. Real part of the optical conductivity of ZrSiS as a function of frequency. An important result of this work is presented in Fig. 2 : the real part of the optical conductivity is almost frequency independent, σ 1 ( ω ) = σ flat ≈ 6600 Ω - 1 cm - 1 , in the range from 250 to 2500 cm - 1 [30–300 meV] basically at all temperatures investigated (at T ≥ 100 K , the flat region starts at a bit higher frequencies because of a rather broad free-electron contribution). Such frequency-independent behavior of σ 1 ( ω ) is similar to what has been predicted [47] and observed [37] in graphene and matches the theory for the optical response due to transitions between the 2D Dirac states in LNSMs [41–43] . In contrast to graphene, no universal sheet conductance is expected for LNSMs; instead, σ flat is related to the length of the nodal line k 0 in a BZ. For a circular nodal line, one has σ 1 ( ω ) = σ flat = e 2 k 0 16 ℏ . (1) It is assumed here that the plane of the nodal circle is perpendicular to the electric-field component of the probing radiation and that there is no particle-hole asymmetry [41–43] . In ZrSiS, the low-energy nodal line is not circular and not even flat; instead, the BZ contains a 3D “cage” of nodal lines [12] . Thus, a straightforward application of this formula is not rigourously validated. Nevertheless, having no better model at hand, we use Eq. (1) for a rough estimate of k 0 = 4.3 Å - 1 . This value seems to be reasonable: according to the band-structure calculations, the total lengths of the nodal line projections on the [100] and [001] directions are about 3.5 and 6 Å - 1 per BZ, respectively. At ν > 3000 cm - 1 , σ 1 ( ω ) is not frequency independent anymore; it decreases with frequency. This is likely because the 2D Dirac band loses its linearity at such large energies [12] . At even higher frequencies, σ 1 ( ω ) starts to rise as ω increases. It would be of interest to detect the transitions between the high-energy Dirac bands above approximately 1.3 eV = 10 500 cm - 1 , but we do not see any clear signatures of such transitions: as discussed above, these Dirac bands are partly gapped by SOC and for bands of such complex shapes the optical conductivity is not expected to be flat or linear in frequency. Additionally, non-Dirac bands above the Fermi level also contribute to the absorption processes at these frequencies, as one can see from the band structure calculations of Refs. [12,15] . At the lowest frequencies measured, ν < 500 cm - 1 , σ 1 ( ω ) also deviates from the flat behavior, as can be seen in Figs. 2 and 3 , and exhibits several features. The contribution of free carriers (so-called Drude response) is present at all temperatures, but best seen at 100 and 300 K. A free-electron component is expected in the optical response because the Fermi level in ZrSiS is slightly above the nodal line [12,15] . As the temperature drops, the Drude band narrows, revealing two distinct modes at around 100 and 150 cm - 1 . A brief discussion on the possible origins of these modes is given below. The narrowing of the Drude band reflects the strong suppression of the dc resistivity [23,25] and of the carrier scattering rate (or, more accurately, the momentum-relaxation rate) as T → 0 . 3 10.1103/PhysRevLett.119.187401.f3 FIG. 3. Examples of the optical conductivity fits for ZrSiS at low frequencies. Thick solid lines are experimental data; thin solid lines are total fits; dashed lines represent fit contributions of the Drude (red), Pauli-edge (black), and Lorentzian (olive, shown only for T = 10 K ) terms. Thin dotted (blue) lines represent attempts to fit the 300 K data without the Lorentzians. At the lowest temperatures, σ 1 ( ω ) develops a minimum around 200 cm - 1 (bottom panel of Fig. 3 ). We relate this dip and the corresponding feature in σ 2 ( ω ) , see Fig. 4 , to the Pauli blocking of the transitions in the 2D band. Such features, related to the position of the Fermi level, are well known in semiconductors [48] and have recently been discussed in relation to graphene and Dirac or Weyl semimetals [49–51] . As already mentioned, band-structure calculations and ARPES measurements locate the Fermi level in ZrSiS in the upper (conduction) Dirac band [12,15] ; see the sketch in Fig. 1 . Thus, the Pauli edge must be seen in the interband portion of optical conductivity. An onset of the interband transitions commonly shows up at the frequency equal to max { Δ , 2 μ } [39,42,43,49–51] , with Δ being the band gap and μ the position of the Fermi level relative to the Dirac point. Thus, Eq. (1) can be modified to σ 1 ( ω ) = e 2 k 0 16 ℏ × θ ( ℏ ω - max { Δ , 2 μ } ) , (2) where θ ( x ) is the Heaviside step function. From Eq. (2) and the bottom panel of Fig. 3 , one can conclude that max { Δ , 2 μ } must be smaller than approximately 250 cm - 1 or some 30 meV and, hence Δ < 30 meV . 4 10.1103/PhysRevLett.119.187401.f4 FIG. 4. Main frame: imaginary part of the optical conductivity in ZrSiS. Arrow indicates the position of the Pauli edge at 10 K. Inset: real part of the dielectric constant near the plasma frequency. This estimate of the upper limit of Δ from optical data is in good agreement with the value obtained from band-structure calculations (15 meV, [12] ). The ARPES value of 60 meV [15] likely overestimates the gap due to the limited resolution. The fact that the relative position of the Dirac points and the Fermi level is slightly k -dependent [12,15] might lead to a broadening of the optical Pauli-edge feature observed. Importantly, even if the Fermi level appears within the gap for some values of k , our conclusion on the upper limit of Δ still holds. Let us now turn to the modes at 100 and 150 cm - 1 and consider possible scenarios for their origin. (i) Phonons. It is extremely unlikely that the observed modes originate from phonons as such, because the frequencies of the modes do not agree with calculations of the infrared-active phonons [52] and because the modes are unrealistically strong for any phonon except soft modes in ferroelectrics. However, no ferroelectricity is reported or expected in ZrSiS and our modes do not shift with temperature, as would be the case for ferroelectric soft modes. (ii) Bulk electron localization. A disorder-induced localization of bulk electrons seems also rather unlikely, since the material is very clean [28] . Its dc resistivity monotonically decreases with temperature and is extremely low at T → 0 [23,25] . (iii) Surface contributions. Although our skin depth is above 40 nm for any measurement frequency, some surface contributions cannot be fully excluded. Very recently, extremely strong surface states in ZrSiS have been theoretically predicted and observed by ARPES measurements [53] . However, no surface contributions are seen in dc-conductivity measurements on microstructure samples with different thicknesses [54] . Appearance of inhomogeneities in the electron surface density, possibly with electron localization, can reconcile the results of Refs. [53] and [54] , as well as provide an explanation for our low-frequency modes. (iv) Excitons. At first glance, it does not seem plausible that excitons are the reason for the observed features, because of the strong screening effects from free carriers. Nevertheless, if we calculate the exciton binding energy E b within the simplest hydrogenic exciton model [55] using the effective mass of the carriers in the low-energy 2D Dirac band, m * = 0.025 m e [28] , and an estimate of the static lattice dielectric constant, ϵ 0 ∼ 7.5 [56] , we obtain E b ∼ 100 cm - 1 . Excitons with such E b are supposed to be seen in the spectra at Δ - E b ≤ 150 cm - 1 ( Δ ≤ 250 cm - 1 ), i.e., right in the frequency range of our modes. One can propose that the excitons may manifest themselves on or near the surface, where free-electron screening is weaker. Furthermore, large strengths of the observed modes in this case can be quantitatively explained by bounding between excitons and surface inhomogeneities [57] . Further studies are necessary to fully clarify the nature of the low-energy absorption peaks in ZrSiS. To get some more quantitative estimates of the parameters determining the optical response, we fit the optical conductivity with a model consisting of a Drude term, two Lorentzians, and a term describing the Pauli edge. Scattering and other processes, leading to broadening of the sharp step in Eq. (2) , may be taken into account by replacing the Heaviside function with 1 2 + 1 π arctan ω - max { Δ , 2 μ } / ℏ Γ , (3) for example, where Γ represents a broadening parameter due to the k -dependent gap, impurity scattering, or temperature. At T = 10 K , reasonable fits can be obtained with a very sharp Pauli edge, i.e., with Γ of a few cm - 1 . We set Γ = k B T / ℏ for all temperatures, since smaller values seem not to be physical. This yields Γ = 7 , 35, and 210 cm - 1 for 10, 50, and 300 K, respectively; see Fig. 3 . In all our fits we keep the zero-frequency limit of the Drude term equal to σ dc at all temperatures. Owed to the broad Drude tail, the description of the 300 K data is straightforward. It provides the momentum-relaxation rate of free carriers, γ = 1 / ( 2 π τ ) = ( 120 ± 10 ) cm - 1 ( τ is the corresponding relaxation time), and a plasma frequency, ω pl / 2 π = ( 24 000 ± 1000 ) cm - 1 [58] . On the other hand, the screened plasma frequency, ω pl scr = ω p l / ϵ ∞ ( ϵ ∞ is the contribution of the higher-frequency optical transitions to ϵ 1 ), can be directly determined from optical measurements as the zero-crossing point of ϵ 1 ( ν ) [44] . We find ω pl scr to be temperature independent and situated at 8900 cm - 1 , cf. the inset of Fig. 4 . Hence, ϵ ∞ = ( ω pl / ω pl scr ) 2 ≈ 7 , which is in good agrement with the optical measurements, presented in same figure. As one can see from Figs. 2–4 , the Drude term becomes narrower as T → 0 . At low temperatures, γ is below our measurement window and our fits thus might become ambiguous. To avoid this, we first tried to keep the plasma frequency of the Drude term constant as a function of T , but this turned out to be unsatisfactory. Some spectral weight had to be redistributed between the Drude term and the Lorentzians. Nevertheless, we tried to have this spectral weight transfer as small as possible and the total plasma frequency of the three terms (Drude plus two Lorentzians) to be temperature independent in accordance with the temperature-independent ω pl . Examples of the fits obtained in this way are shown in Fig. 3 . At T ≤ 50 K , we found that γ ≈ 2 to 2.5 cm - 1 and the momentum-relaxing τ is 2.1–2.7 ps. Interestingly, at low temperatures the momentum-relaxation length, ℓ mr = v F τ , obtained from our estimate of τ , becomes macroscopically large. Using v F = 5 × 10 5 m / s as an average Fermi velocity in the low-energy Dirac bands [23] , we obtain ℓ mr ≥ 1 μ m for T ≤ 50 K . This implies that the hydrodynamic behavior of electrons, reported recently in clean samples of graphene [59,60] and the Weyl semimetal WP 2 [61] , might also be realized in ZrSiS. This proposition seems reasonable, because only linear bands with highly mobile carriers (typical mobilities are 1 0 3 – 1 0 4 cm 2 / V s [23,24,28] ) cross the Fermi level in ZrSiS. Summarizing, the real part of the optical conductivity of ZrSiS was found to be independent on frequency in a rather broad range from 250 to 2500 cm - 1 (30–300 meV). Our observations are supported by recent theoretical predictions for the optical response of LNSMs, and constitute an independent confirmation of 2D Dirac bands in ZrSiS near the Fermi level. The characteristic features of the Pauli edge, appearing in the low-frequency spectra, provide the upper limit ( 250 cm - 1 , 30 meV) for the gap between the 2D Dirac bands. The momentum-relaxation length is at the micrometer scale at T ≤ 50 K . 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PY - 2017/11/1

Y1 - 2017/11/1

N2 - ZrSiS exhibits a frequency-independent interband conductivity σ(ω)=const(ω)≡σflat in a broad range from 250 to 2500 cm-1 (30-300 meV). This makes ZrSiS similar to (quasi-)two-dimensional Dirac electron systems, such as graphite and graphene. We assign the flat optical conductivity to the transitions between quasi-two-dimensional Dirac bands near the Fermi level. In contrast to graphene, σflat is not universal but related to the length of the nodal line in the reciprocal space, k0. Because of spin-orbit coupling, the discussed Dirac bands in ZrSiS possess a small gap Δ, for which we determine an upper bound max(Δ)=30 meV from our optical measurements. At low temperatures the momentum-relaxation rate collapses, and the characteristic length scale of momentum relaxation is of the order of microns below 50 K.

AB - ZrSiS exhibits a frequency-independent interband conductivity σ(ω)=const(ω)≡σflat in a broad range from 250 to 2500 cm-1 (30-300 meV). This makes ZrSiS similar to (quasi-)two-dimensional Dirac electron systems, such as graphite and graphene. We assign the flat optical conductivity to the transitions between quasi-two-dimensional Dirac bands near the Fermi level. In contrast to graphene, σflat is not universal but related to the length of the nodal line in the reciprocal space, k0. Because of spin-orbit coupling, the discussed Dirac bands in ZrSiS possess a small gap Δ, for which we determine an upper bound max(Δ)=30 meV from our optical measurements. At low temperatures the momentum-relaxation rate collapses, and the characteristic length scale of momentum relaxation is of the order of microns below 50 K.

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U2 - 10.1103/PhysRevLett.119.187401

DO - 10.1103/PhysRevLett.119.187401

M3 - Article

C2 - 29219545

AN - SCOPUS:85032814978

VL - 119

JO - Physical Review Letters

JF - Physical Review Letters

SN - 0031-9007

IS - 18

M1 - 187401

ER -

Flat Optical Conductivity in ZrSiS due to Two-Dimensional Dirac Bands

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