Abstract
Strong coupling between discrete phonon and continuous electron–hole pair excitations can induce a pronounced asymmetry in the phonon line shape, known as the Fano resonance. This effect has been observed in various systems. Here we reveal explicit evidence for strong coupling between an infraredactive phonon and electronic transitions near the Weyl points through the observation of a Fano resonance in the Weyl semimetal TaAs. The resulting asymmetry in the phonon line shape, conspicuous at low temperatures, diminishes continuously with increasing temperature. This behaviour originates from the suppression of electronic transitions near the Weyl points due to the decreasing occupation of electronic states below the Fermi level (E_{F}) with increasing temperature, as well as Pauli blocking caused by thermally excited electrons above E_{F}. Our findings not only elucidate the mechanism governing the tunable Fano resonance but also open a route for exploring exotic physical phenomena through phonon properties in Weyl semimetals.
Introduction
The Weyl semimetal (WSM) phase, a novel topological state of quantum matter, has been proposed to exist in materials with two nondegenerate bands crossing at E_{F} in threedimensional momentum space^{1}. At the band crossing points (Weyl points), the electronic dispersion is linear in all three directions, resembling a threedimensional version of graphene, and the lowenergy excitations can be described by Weyl equations, producing a condensedmatter realization of Weyl fermions^{2}. Recently, such a WSM state has been discovered in noncentrosymmetric transitionmetal monoarsenides and monophosphides (TaAs, TaP, NbAs and NbP)^{3,4,5,6,7,8}, where 12 pairs of Weyl points have been found. Since the Weyl points are located in close proximity to E_{F} in these materials, interband electronic transitions near the Weyl points occur at a very low energy, 2μ, where μ represents the chemical potential with respect to the Weyl points^{9,10}. This energy scale overlaps optical phonon frequencies^{11}. Consequently, strong coupling between electronic transitions near the Weyl points (Weyl fermions) and phonons may arise, manifested through a temperaturetunable Fano resonance.
This is actually a rare occurrence. Fano resonances generally do not occur in conventional metals or semiconductors, since the energy scale of interband electronic transitions in these materials is usually much higher than phonon excitations. However, this interesting phenomenon has been reported in bilayer or fewlayer graphene^{12,13,14}, topological insulators^{15,16} and some strongly correlated electron systems, such as stripephase nickelates^{17} and highT_{c} superconductors^{18,19}. The observed Fano resonance in strongly correlated materials is very weak, with 1/q^{2}, a parameter that describes the asymmetry of the Fano line shape, only reaching ∼0.04 (refs 17, 18, 19). In topological insulators, the effect is also weak and cannot be observed without manipulating extrinsic parameters, such as magnetic field^{15}, chemical doping^{16} or microfabrication on the surface^{20}. In contrast, bilayer or fewlayer graphene exhibits a strong Fano resonance, with 1/q^{2}≥≥1, but the resonance energy is ∼1,600 cm^{−1} (200 meV), much greater than E_{F}, making it impossible to tune the resonance via temperature^{12,13,14}. Moreover, the Fano physics revealed in all the above materials is associated with or has considerable contributions from coupling between phonons and conventional massive fermions.
Here we observe an intrinsic, strong and temperaturetunable Fano resonance, which arises purely from quantum interference between phonons and massless Weyl fermions, in the recently discovered Weyl semimetal TaAs. We further demonstrate that the Fano line shape can be tuned by changing the occupation of the electronic states near the Weyl points. These observations not only open a novel avenue for exploring exotic quantum phenomena in WSMs, such as the chiral anomaly^{21,22,23,24}, but also set the stage for a variety of potential applications that take advantage of the ability to tune the Fano resonance using different parameters (for example, temperature, light or magnetic/electric fields).
Results
Sample growth and characterization
Highquality single crystals of TaAs were synthesized through a chemical vapour transport method^{23}. The asgrown crystals are polyhedrons with shiny facets up to 1.5 mm in size. Xray diffraction measurements reveal that the asgrown facets are the (001), (107) and (112) surfaces (Supplementary Fig. 1). Systematic optical measurements were carried out on all three surfaces.
Reflectivity and optical conductivity
Figure 1a shows the farinfrared reflectivity R(ω) measured on the (107) surface of TaAs at 11 different temperatures from 5 to 300 K (ref. 10). The relatively high R(ω) that approaches unity at zero frequency is consistent with the metallic nature of TaAs. A welldefined plasma edge in the farinfrared region suggests very low carrier density, in agreement with the tiny volumes enclosed by the Fermi surfaces in this material^{3,4,5,6}. In addition to the broad features in R(ω), a sharp feature can be clearly identified at ∼253 cm^{−1} (31 meV), as indicated by the arrow, which is associated with the infraredactive A_{1} phonon mode (Supplementary Note 1).
In order to gain direct information about this mode, we calculated the optical conductivity σ_{1}(ω) from R(ω) using a Kramers–Kronig analysis^{10} (more details in Methods section). Figure 1b displays σ_{1}(ω) of TaAs on the (107) surface in the farinfrared region at different temperatures. The lowfrequency σ_{1}(ω) is dominated by a narrow Drude response alongside prominent linear features, whose origin and temperature dependence have been previously discussed in detail^{10}. The A_{1} mode manifests itself as a sharp peak in the σ_{1}(ω) spectrum, as indicated by the arrow. Figure 1c shows an enlarged view of σ_{1}(ω) in the frequency region of 240–270 cm^{−1}, where the A_{1} mode can be seen more clearly. It is well known that, in the absence of strong electron–phonon coupling, the phonon exhibits a symmetric line shape in σ_{1}(ω) that can be described by a Lorentz oscillator, as schematically illustrated in Fig. 1d. In contrast, strong electron–phonon coupling gives rise to an asymmetric phonon profile in σ_{1}(ω), known as the Fano line shape^{13,14,25} (Fig. 1e). As shown in Fig. 1c, the A_{1} mode in TaAs exhibits a striking asymmetric line shape at low temperatures, which is an unequivocal signature of strong electron–phonon coupling. More interestingly, the asymmetry of the phonon line shape diminishes as the temperature rises, suggesting that the couplinginduced Fano resonance in TaAs can be tuned by temperature.
Mechanism governing the temperaturetunable Fano resonance
To quantify the temperature dependence of the A_{1} mode, we extract the phonon line shape by subtracting a linear electronic background in a narrow frequency range at all measured temperatures, as shown in Fig. 2a. At each temperature, the phonon is fit with the Fano line shape^{25},
where Z_{0} is the vacuum impedance; ω_{0}, γ and Ω correspond to the resonance frequency, linewidth and strength of the phonon, respectively; q is a dimensionless parameter that describes the asymmetry of the Fano profile. A larger 1/q^{2} indicates more conspicuous asymmetry in the phonon line shape, while for 1/q^{2}=0, the symmetric Lorentz line shape is fully recovered. The solid lines in Fig. 2a represent the fitting curves, which describe the measured phonon line shapes reasonably well at all temperatures. This procedure also returns the temperature dependence of the fitting parameters.
Figure 2b depicts 1/q^{2} as a function of temperature. While 1/q^{2} adopts a large value of 1.1 at 5 K (refs 12, 13, 14, 18), it decreases dramatically with increasing temperature. This suggests that the A_{1} mode is strongly coupled to a continuum of electron–hole excitations, and the resulting Fano resonance varies significantly with temperature. We first trace the origin of this coupling by examining the band structure of TaAs. Both firstprinciple calculations and angleresolved photoemission spectroscopic measurements have revealed 12 pairs of Weyl points in TaAs^{3,4,5,6}. These Weyl points are categorized into two types^{3}. Four pairs in the k_{z}=0 plane, about 2 meV above E_{F}, are defined as W1, while another eight pairs off the k_{z}=0 plane, lying about 21 meV below E_{F}, are named W2. Interband electronic transitions in the vicinity of a Weyl point start at ω=2μ (refs 9, 10), making it easy to calculate that electronic transitions near W2 turn on at ω=42 meV (∼336 cm^{−1}). Thus electronic transitions at the frequency of the A_{1} mode (253 cm^{−1}) do not occur near W2. This implies that the A_{1} mode is unlikely to be coupled to the electronic transitions near W2. However, electronic transitions near W1 set in at ω>4 meV (∼32 cm^{−1}) and can therefore overlap with the frequency of the A_{1} mode, suggesting that this mode is coupled to the electronic transitions near W1.
Having attributed the asymmetric line shape of the A_{1} mode to its strong coupling with the electronic transitions near W1, we proceed to understand the temperature dependence of this mode. In WSMs, since the Weyl points are in close proximity to E_{F}, electronic transitions near the Weyl points can be dramatically affected by thermal excitations, thus changing the line shape of the phonon that is coupled to these transitions. Figure 3 shows the band structure along three momentum directions near W1 in TaAs. The occupation probabilities of the electronic states in these bands (colour maps) are calculated using the Fermi–Dirac distribution function at three different temperatures (5, 150 and 300 K), from which we see that the occupation of the electronic states near W1 depends strongly on the temperature. At 5 K (Fig. 3a–c), the electronic states below E_{F} are fully occupied (blue), while the states above E_{F} are empty (white). In this case, interband transitions at the energy of the A_{1} mode ħω_{0} are strong, as indicated by the thick arrows. As the temperature increases from 5 to 300 K, thermal excitations cause vacant states to appear below E_{F} and electronic states above E_{F} to be partially occupied. Electronic transitions at ħω_{0} are significantly suppressed (illustrated by the thin arrows in Fig. 3), because the available initial states for these transitions decrease, and many of the final states are Pauli blocked by thermally excited electrons. This suppression of the electronic transitions near W1 is directly responsible for the change in the A_{1} phonon line shape.
For a more quantitative analysis, the dimensionless parameter q in the Fano theory is given by^{13,25}
where V_{e−ph} is the electron–phonon coupling strength; D_{e−h}(ω_{0}, T) is the joint electron–hole pair density of states at the frequency of the A_{1} mode ω_{0} for a given temperature T; and μ_{ph} and μ_{e−h} represent the optical matrix elements for phonon and electron–hole pair excitations, respectively. In this equation, we note that raising T mainly modifies D_{e−h}(ω_{0}, T) by thermally exciting electrons to the electronic states above E_{F} and creating holes below E_{F}. Near the Weyl points, the finitetemperature joint electron–hole pair density of states at ħω_{0} takes the form
where is the Fermi function, with representing the energy of the singleparticle state with respect to the Weyl points and being the zerotemperature joint electron–hole pair density of states at ħω_{0}. The red solid curve in Fig. 2b is the leastsquares fit to the experimental temperature dependence of 1/q^{2} (blue solid circles) using equation (2). The excellent agreement between our experimental data and the model further underlines the intimate link between the line shape of the A_{1} mode and the electronic transitions near W1, which are continuously suppressed with increasing temperature due to the reduced occupation of the electronic states below E_{F} and Pauli blocking from thermally excited electrons above E_{F}.
A careful examination of the temperature dependence of the A_{1} phonon linewidth γ (Fig. 2c) leads us to the same conclusion. In the case of weak electron–phonon coupling, phonon decay is dominated by the anharmonic effect: a zonecentre phonon decays into two acoustic modes with the same frequencies and opposite momenta^{26,27}. The temperature dependence of the phonon linewidth γ^{ph−ph}(T) for this process follows
where γ_{0}^{ph−ph} is the residual linewidth at zero temperature. Apparently, this model does not account for the behaviour of the A_{1} mode in TaAs, since it gives an increasing γ as the temperature is raised, which is opposite to our experimental observation. Instead, strong electron–phonon coupling must be taken into account to understand the temperature dependence of γ in TaAs. In a system with strong electron–phonon coupling, a phonon can also decay by creating an electron–hole pair^{28}. This process is sensitive to D_{e−h}(ω_{0}, T) and is thus suppressed with increasing temperature due to thermal excitations, resulting in a temperaturedependent phonon linewidth γ^{e−ph}(T)
where γ_{0}^{e−ph} represents a residual linewidth. Consequently, the temperaturedependent linewidth of a phonon mode that is strongly coupled to electronic excitations is given by γ(T)=γ^{ph−ph}(T)+γ^{e−ph}(T). This equation gives an excellent description to the measured temperature dependence of the linewidth for the A_{1} mode in TaAs, as shown by the red solid curve in Fig. 2c. The above observations explicitly demonstrate that the A_{1} mode is strongly coupled to the electronic transitions near W1, and both the line shape and linewidth of this phonon are closely tied to these transitions, which can be continuously tuned by temperature through varying the occupation of the electronic states near W1.
To lend further credence to our experimental results and analysis, we performed optical measurements at 12 different temperatures on the (112) surface. Essentially identical behaviour was revealed for the A_{1} mode (Supplementary Fig. 3). We then utilized the same methods and models to analyse the line shape and linewidth of this mode observed on the (112) surface (Supplementary Fig. 4), reaching the same conclusions.
Finally, we note that although electronic transitions at ħω_{0} are absent near W2 at low temperatures, Fermi smearing at high temperatures may relax the Pauli blocking, allowing these transitions to occur. From equations (2 and 3), we can easily calculate the asymmetry of the A_{1} mode arising from coupling to W2 at 300 K: . This small value of 1/, which vanishes quickly with decreasing temperature, suggests that electronic transitions near W2 do not give rise to a noticeable Fano resonance in the A_{1} mode of TaAs over the measured temperature range (5–300 K). However, the enhancement of these transitions with increasing temperature may produce a strong Fano resonance at high enough temperatures. It is also worth pointing out that changing the occupation of the electronic states near W1 via other methods, such as electrical gating or femtosecond optical excitation, should induce a similar change in the Fano line shape of the A_{1} mode in TaAs.
Discussion
The strong coupling between Weyl fermions and phonons enables the study of exotic quantum phenomena in WSMs by tracking the properties of phonons. One interesting proposal derived from our findings is to provide experimental evidence for the chiral anomaly^{21,22,23,24}, in which the application of parallel electric (E) and magnetic (B) fields pumps electrons from one Weyl point to the other with opposite chirality at a rate proportional to E·B, leading to a shift of E_{F} in opposite directions at different Weyl points. This E_{F} shift caused by the chiral anomaly can significantly affect the electronic transitions near the Weyl points, which accordingly changes the line shape of the phonon that is coupled to these transitions.
For more insight, we can calculate the chiral anomalyinduced E_{F} shift, given by^{9,29}
where v_{F} is the Fermi velocity and τ is the scattering time between different Weyl points. To obtain a realistic estimate, we take v_{F}=5 × 10^{5} m s^{−1} (refs 6, 23, 30), τ=5 × 10^{−11} s (refs 9, 24, 29), B=5 T and E=1 V mm^{−1}, giving ΔE_{F}=31.34 meV. Such an E_{F} shift would completely block the electronic transitions at ħω_{0} near W1 (Fig. 3), resulting in a full recovery of the symmetric Lorentz line shape. This theoretical estimate thus suggests that, at low temperatures, the striking asymmetry of the A_{1} phonon line shape in TaAs should decrease with increasing E·B, due to the chiral anomalyinduced E_{F} shift, until a symmetric Lorentz line shape is fully recovered at high E·B fields. Experimental observation of this effect would be a strong evidence for the chiral anomaly in WSMs.
Furthermore, our observations suggest a variety of potential device applications. The Fano resonance in optical spectra arises from strong quantum interference between topologically nontrivial Weyl fermions and phonons, and the properties of Weyl fermions can be manipulated by a variety of different parameters, such as temperature, light^{31,32} and electric/magnetic fields^{23,24}. This could thus lead to a new class of devices operating at the Fano resonance that could be controlled by any one of these parameters, making them unusually flexible and thus potentially useful in a wide range of applications.
In conclusion, we have found compelling evidence for strong coupling between the infraredactive A_{1} phonon mode and electronic transitions near W1 (Weyl fermions) in TaAs through the observation of a Fano resonance. Varying the temperature, which changes the occupation of the electronic states in proximity to E_{F}, can continuously tune the amplitude of the electronic transitions near W1, thus manipulating the line shape of the A_{1} mode. Our findings not only suggest a new approach for investigating exotic quantum phenomena in WSMs, such as the chiral anomaly, through the behaviour of phonons, but also imply a broad range of potential applications, such as in thermo, magneto and electrooptical devices. Please note that, after our manuscript was submitted, we noticed two preprints (refs 33, 34) that theoretically propose detecting the chiral anomaly in WSMs through phonon dynamics.
Methods
Sample synthesis
Highquality TaAs single crystals were grown through a chemical vapour transport method^{23}. A previously reacted polycrystalline TaAs was filled in a quartz ampoule using iodine (2 mg cm^{−3}) as the transporting agent. After evacuating and sealing, the ampoule was kept at the growth temperature for 3 weeks. Large polyhedral crystals with dimensions up to 1.5 mm are obtained in a temperature field of ΔT=1,150–1,000 °C.
Optical measurements
Near normal incident reflectivity R(ω) was measured over a broad frequency range on a Bruker VERTEX 80v FTIR spectrometer. In order to accurately measure the absolute R(ω) of the samples, an in situ gold overcoating technique^{35} was used. R(ω) from 40 to 15,000 cm^{−1} were measured at 11 different temperatures from 300 down to 5 K on three different surfaces of asgrown single crystals in an ARSHelitran cryostat. Since a broad spectral range is required for a KramersKronig analysis, we extended our data to the ultraviolet range (up to 50,000 cm^{−1}) at room temperature using an AvaSpec2048 × 14 fiber optic spectrometer.
Kramers–Kronig analysis
The real part of the optical conductivity σ_{1}(ω) has been determined via a Kramers–Kronig analysis of R(ω)^{36}. Below the lowest measured frequency, we used a Hagen–Rubens (R=1−A) form for the lowfrequency extrapolation. Above the highest measured frequency, we assumed a constant reflectivity up to 12.5 eV, followed by a freeelectron (ω^{−4}) response.
Data availability
All data that support the findings of this study are available from the corresponding authors upon request.
Additional information
How to cite this article: Xu, B. et al. Temperaturetunable Fano resonance induced by strong coupling between Weyl fermions and phonons in TaAs. Nat. Commun. 8, 14933 doi: 10.1038/ncomms14933 (2017).
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Acknowledgements
We thank Ricardo Lobo, Yongkang Luo, Simin Nie and Hongming Weng for illuminating discussions. Work at IOP CAS was supported by MOST (973 Project Nos. 2015CB921303 and 2015CB921102) and NSFC (Grant Nos. 91121004, 91421304 and 11374345). Work at LANL was performed at the Center for Integrated Nanotechnologies, a US Department of Energy, Office of Basic Energy Sciences user facility, and funded by the LANL LDRD program and by the UC Office of the President under the UC Lab Fees Research Program, Grant ID No. 237789. H.X. is supported by NSFC, Grant No. U1530402.
Author information
Author notes
 B. Xu
 & Y. M. Dai
These authors contributed equally to this work.
Affiliations
Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, P.O. Box 603, Beijing 100190, China
 B. Xu
 , L. X. Zhao
 , K. Wang
 , R. Yang
 , W. Zhang
 , J. Y. Liu
 , G. F. Chen
 & X. G. Qiu
Center for High Pressure Science and Technology Advanced Research, Beijing 100094, China
 B. Xu
 & H. Xiao
Center for Integrated Nanotechnologies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
 Y. M. Dai
 , S. A. Trugman
 , JX Zhu
 , D. A. Yarotski
 & R. P. Prasankumar
Collaborative Innovation Center of Quantum Matter, Beijing 100190, China
 G. F. Chen
 & X. G. Qiu
Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
 S. A. Trugman
 & JX Zhu
Associate Directorate for Chemistry, Life and Earth Sciences, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
 A. J. Taylor
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Contributions
B.X. carried out the optical measurements with the assistance of K.W., R.Y., W.Z. and J.Y.L.; L.X.Z. and G.F.C. synthesized the single crystals; B.X., Y.M.D., H.X., A.J.T., D.A.Y., R.P.P. and X.G.Q. analysed the data; S.A.T. and J.X.Z. contributed to theoretical models; Y.M.D. wrote the manuscript; all authors made comments on the manuscript; R.P.P. and X.G.Q. supervised the project.
Competing interests
The authors declare no competing financial interests.
Corresponding authors
Correspondence to Y. M. Dai or R. P. Prasankumar or X. G. Qiu.
Supplementary information
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Supplementary Information
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