A Large Effective Phonon Magnetic Moment in a Dirac Semimetal
We have investigated the low frequency terahertz response of the Dirac semimetal CdAs concentrating on its phonon dynamics and their response to magnetic field. Due to the very small effective mass in these materials, we observed a very prominent and sensitive cyclotron resonance as field is tuned as well as an optical phonon mode. As the cyclotron resonance is tuned with field to pass through the phonon frequency, we observed a notable splitting in the energies of right- and left-hand polarized phonons. For positive fields, the left-handed phonon shows notable Fano asymmetry, while the right-handed phonon shows little asymmetry. The splitting between the left- and right-hand phonons can be expressed as an effective phonon magnetic moment approximately 2.7 times the Bohr magneton, which is almost four orders of magnitude larger than the prediction of ab initio calculations of the effect in nonmagnetic insulators. We ascribe this exceedingly large value to the resonant coupling to the cyclotron motion. This circular-polarization selective coupling that can be manipulated by magnetic field provides new insight in understanding the complicated magneto transport in CdAs and new functionality for nonlinear optics to utilize exotic light-induced topological phases in Dirac semimetals.
A number of linear and nonlinear magneto-optical effects from relativistic fermions and Berry curvatures are anticipated in 3D topological semimetals (TSMs) Armitage et al. (2018). They are of great current interest due to their relation to the nontrivial nature of many of these materials. Besides their appealing electronic features, the interplay between electronic states and other degrees of freedom, such as lattice vibrations and magnons, have also begun to attract attention Song et al. (2016); Rinkel et al. (2017); Liu and Shi (2017); Kuroda and et al. (2015). For instance, it has been predicted that in a Weyl semimetal, the “chiral current” which corresponds to the transfer of charge between Weyl nodes, can interact with Raman-active phonon modes and make them infrared-active Song et al. (2016); Rinkel et al. (2017). Such phonon modes may also hybridize with the plasmon mode of Weyl fermions in a similar fashion to Kondo hybridization in heavy-fermion systems Barnes et al. (1991a); Wang et al. (1997).
In this letter, we report the observation of another interesting charge-phonon coupling effect in a TSM. We have investigated the magnetoterahertz properties of high-quality epitaxial CdAs films and found a novel interplay between intraband inter-LL excitations and circularly polarized phonons in this Dirac semimetal. The temperature-dependent terahertz (THz) conductivity exhibits coherent metallic transport and a sharp optical phonon mode in THz region. Due to the low cyclotron mass ( 0.03), a very sharp cyclotron resonance mode develops and moves rapidly to higher frequency with increasing field. As the cyclotron resonance is tuned to pass through the optical phonon, we note a large splitting in the energies of right- (R-) and left- (L-hand) polarized phonons. This is accompanied by a notable Fano asymmetry of the L-hand branch. The splitting is almost four orders of magnitude larger than the prediction of first-principle calculations of the phonon Zeeman effect in undoped semimetalsJuraschek and Spaldin (2018) and leads to an effective phonon magnetic moment of almost 2.7 Bohr magnetons – an unprecedentedly large value in a nonmagnetic system. We attribute this large enhancement to the resonant interaction between the circularly polarized cyclotron motion and lattice degrees of freedom. This field controllable and circular polarization selective coupling provides new opportunity for nonlinear optical methods to induce and study exotic light-induced topological phases.
CdAs is a tetragonal Dirac semimetal system with lattice structure Ali et al. (2014). It has a pair of four-fold degenerate Dirac nodes located along the axis. Both Dirac nodes are protected by a symmetry around the axis and cannot be removed except by breaking this symmetry. Recently, high quality epitaxial CdAs films grown by molecular-beam epitaxy are gradually availableSchumann et al. (2016); Nakazawa et al. (2018) and the films in this work were grown on (111)B GaAs substrates to a thickness of 280 nm. Further details of the film growth can be found elsewhere Schumann et al. (2016). Magnetoterahertz measurement was performed by time-domain THz spectroscopy Cheng et al. (2019).
The real part of the zero-field optical conductivity is displayed in Fig. 1. At 6 K, shows a zero frequency Drude-like peak with a well-defined phonon mode at 0.67 THz. With increasing temperature, the Drude part of becomes larger and sharper and the phonon mode becomes broader. A Drude-Lorentz fit for real and imaginary parts of conductivity at 6 K are shown in the inset of Fig. 1. This fit mainly includes one Drude term and one finite frequency Lorentz term. One can see the optical conductivity at 6 K is well reproduced by this fitting. The plasma frequency (/2) and scattering rate (1/2) of the Drude oscillator are 22 THz and 0.9 THz, respectively at 6 K. The oscillator strength and linewidth of the phonon are 4.5 THz and 0.12 THz, respectively at 6 K. The weak features above 1.6 THz probably indicate other phonon modes with large damping. By carefully inspection of the lineshape of the phonon at 0.67 THz, we can see that the phonon mode exhibits a weak asymmetry which indicates it has a detectable coupling to the continuum of electronic states. This will be discussed in more detail below.
We measure the optical conductivity in the circular basis by using the fast rotating polarizer method  to extract both the diagonal () and off-diagonal () complex transmissions and then generating the transmission for R- and L- polarized light by the expression . The conductivity in the circular basis can be extracted by the usual correspondence between transmission and conductivity in thin films after recognizing that in Faraday geometry circular polarization is the eigenpolarization for transmission Cheng et al. (2019).
In Figs. 2(a) and 2(b), we show the real and imaginary parts of the optical conductivity in the circular polarization basis (). It is quite illustrative to display the response to the R- and L-hand polarized light as positive and negative frequencies respectively and hence the conductivity in the circular basis becomes a single continuous function of frequency that smoothly extends through zero frequency. At zero field, the real part of is a function peaked at zero frequency, which is typical metallic response. With increasing positive field, the peak moves quickly to finite negative frequency, while the conductivity is suppressed on the positive frequency side. This large shift of the peak with relatively small magnetic field can be identified as the cyclotron resonance (CR) mode of the -type carriers [Inset of Fig. 2(d)] with a small cyclotron mass. One of the most interesting aspects of is the field evolution of the 0.7 THz phonon. One can see that in the R-hand channel the phonon’s peak position and lineshape evolves only slightly with increasing field. In contrast, the L-hand phonon shows a large responses to field. As shown in Fig. 2(a), the low frequency side of the L-hand phonon develops a weak dip around 0.6 THz. This weak feature cannot be interpreted in terms of electronic excitations alone and comes from a “Fano” asymmetry induced by magnetic field-enhanced electron-phonon coupling. Fano resonance is a very general phenomenon that arises from a interference between a sharp mode and a continuum background that it couples to Homes et al. (2018).
To separate electronic and phonon components, we used a Drude/Drude-Lorentz model to fit the complex R- and L-hand optical conductivities simultaneously. The total THz conductivity is = + plus a very weak electron-like oscillator that allows the spectrum to be fit on the positive frequency side at high fields. Its incorporation has no impact on the conclusions of the paper. The contribution of the Drude response in magnetic field is the conventional Drude model but including the effect of cyclotron resonance:
In the above expression, runs from positive to negative frequency and is the CR frequency. For hole (electron) carriers, is positive (negative). As goes to zero, this formula automatically recovers the usual Drude form. The expression for the phonon conductivity with the Fano asymmetry is Homes et al. (2018):
Here, is the phonon’s oscillator strength, is the phonon’s central frequency, is the phonon linewidth, and is the Fano coupling/asymmetry parameter. As approaches zero, the asymmetry vanishes and the phonon recovers the usual symmetric Lorentzian lineshape. In Fig. 2(c), we show a Drude-Lorentz fit for the conductivity at 8 kG. One can see that the fit well reproduces the conductivity over the whole frequency region. We also show the pure Drude simulation without the phonon (green) in Fig. 2(c). By comparing with , we can see the L-hand phonon exhibits a clear Fano shape but the phonon in R hand is more symmetric.
The CR as a function of magnetic field shows a linear field dependence (Fig. 2(d)). Although CdAs is a 3D Dirac semimetal and its carriers are massless Dirac fermions, under these weak fields the system’s response is semi-classical and its CR dispersion can feature a linear field evolution: /, where is the cyclotron mass of electron carriers ( in a linear dispersion approximation). By fitting the field dependence of the CR, is found to be 0.03 free electron masses, which is in agreement with mass from T-dependent Shubnikov-de Haas oscillations Goyal et al. (2018). The small value of arises in the very low chemical potential of these TSMs. Combining with the extracted zero field scattering rates, the electronic mobility is then estimated to be cmVs, consistent with previous dc transport measurement Schumann et al. (2017). In Fig. 2(e) we plot the scattering rate of the Drude oscillator as a function of field. With magnetic field, the scattering rate initially decreases before increasing above 8 kG.
To exhibit the field evolution of the phonons clearly, we subtract the electronic signal and plot the optical conductivity of L- and R- phonons in Fig. 3(a) with offsets. The L- phonon develops a clear asymmetry with increasing field, while the R- phonon shows smaller changes. In Fig. 3(b), we show a Fano fit (Eq. 2) to the L- and R- phonon at 8 kG. We can see that the fit captures all features of both channels. Besides the Fano asymmetry, the oscillator strength of the L- phonon becomes larger than of the R- channel. In Fig. 3(c), we show the field-dependent oscillator strength for L and R- phonons. One can see that of the L- phonon shows an enhancement around 6 kG, before decreasing with increasing field. In contrast, in the R- channel an enhancement is not observed. The central frequencies show distinct field dependencies. As shown in Fig. 3(d), in the L- channel increases quickly from 0 to 8 kG and then stays constant. In contrast, in the R- channel shows a small initial decrease before saturating. Near 6 kG, the splitting of phonons is 0.04 THz.
The field dependence of the Fano parameter is shown in Fig. 3(e). At zero field, of both channels is small. With increasing field, the R-hand phonon still shows a small . However, the L-hand channel, increases its magnitude and shows a strong resonance feature near 8 kG. With further increasing field, decreases but is still larger than of R-hand. The resonance features presented in L-hand phonon at 6 kG strongly indicate the CR mode plays an important role in the asymmetry between L- and R-hand phonon because the CR energy is around 0.7 THz and crosses the phonon’s central frequency when the field is 6 - 8 kG.
For this case of a coupling of a phonon to an electronic continuum, the Fano parameter is determined by the expression Tang et al. (2009). Here D is the electronic joint density of states that arises from the electron-hole pair intraband inter-LL transitions near the phonon frequency . is the electron-phonon coupling strength, and and are the optical matrix elements of phonons and electron-hole pairs respectively. The electron-phonon coupling strength is not expected to have a strong field dependence, but obviously D will. As shown in Fig. 2(a), optical conductivity on the negative frequency side is gradually enhanced but on the positive frequency side it is suppressed as the CR moves in the negative frequency direction. D would reach its maximum when the CR resonates with the phonon in the L-hand channel. This is presumably why the phonon oscillator strength and the Fano parameter in the L-hand channel show a resonance structure near . It is also straightforward to understand the mechanism for the increasing field dependence of Drude scattering rate [Fig. 2(e)] as a field-enhanced coupling between the optical phonon and massless Dirac fermions. Above 8 kG, the magnetic field enhanced electron-phonon scattering surpasses the decreasing trend of LL broadening from impurity potentials, and modifies the scattering rate to be an increasing function of field. This coupling of the cyclotron resonance to phonons has much in common with previous observations of anomalous broadening of the cyclotron resonance when tuned resonantly to a phonon in graphene, BiSe, and GaAs-(Ga,Al)As heterojunctions Faugeras et al. (2009); Yan et al. (2010); Barnes et al. (1991b); Wu et al. (2015). In those cases however, the phonon was Raman-active and its coupling could be only inferred from its influence on the linewidth of the CR mode; its effect was not measured directly on the phonon itself. Here, the increasing linewidth of the CR mode above 8 kG indicates the field-enhanced interaction between CR and the phonon. The crucial observation in the present case is that since the phonon is IR active, the breaking of degeneracy between R- and L- branches can be seen directly in the THz experiments.
We ascribe the large splitting in phonon frequencies to the resonant enhancement of the cyclotron resonance circulating with or against the circular motion of the phonons. Typically, degenerate phonons can be represented in terms of either Cartesian or circular coordinates. However, in high symmetry lattices that nevertheless have no mirror planes or break time-reversal symmetry, the degeneracy between L- and R-hand phonons can be broken and the circular basis becomes the unique and proper one. There has been recent (and earlier) interest in systems where phonons can be imbued with characteristics found in other lattice excitations such as angular momentum and Berry phase structures Qin et al. (2012); Zhang and Niu (2015). Extensive experiments in the 1970s showed in insulating magnetic crystals a splitting between and polarized phonon branches could be triggered by a magnetic field. For instance, Schaack showed in strongly paramagnetic CeF a two-fold degenerate optical phonon at 49.6 meV can be split by as much 0.8 meV at 1 Tesla Schaack (1976). Much more recently, Juraschek and Spaldin used density functional theory to study the field-induced phonon splitting in even nonmagnetic compounds and found the relative splitting () in most nonmagnetic compounds would be to Juraschek and Spaldin (2018). Zhang and Niu showed that in inversion symmetry broken 2D chalcogenides that valley phonons could possess intrinsic angular moment in the finite (but opposite) angular momentum in the two and valleys Zhang and Niu (2015). This has recently been confirmed in optical pump-probe spectroscopy experiments Zhu et al. (2018). In the present case, the relative energy splitting in weak magnetic field between the L- and R-hand phonons is 0.06 and equivalent to a magnetic moment of 2.5 mA at 6 kG, which is 2.7 Bohr magnetons. This is approximately 3 to 4 orders of magnitude larger than predicted in nonmagnetic insulatorsJuraschek and Spaldin (2018).
Our work features the first comprehensive observation for an ultralow magnetic field ( 0.7 T) enhanced electron-phonon coupling effect in the non-magnetic Dirac semimetal CdAs. However, we believe our findings are not limited to this particular case; the general idea should be more widely applicable in other TSMs. By tuning their charge densities via gating TSM films, one also may study the gating- and field-tunable CR-phonon resonance in these TSMs. Aside from being interesting in their own right, the observation of these rich charge and lattice dynamics and their response to magnetic field in CdAs may provide new pathways to study novel light-induced phases in TSMs by ultrafast manipulation of lattice degrees of freedom Hubener et al. (2017). Among other aspects, one may use a narrow-spectrum multicycle intense THz pump pulse resonating with the phonon to excite a CdAs film. The strong stimulus of the phonon will create notable strains and lattice deformations that break lattice symmetries and drive the Dirac semimetal into a light-induced topological insulator phase. Moreover, one may use intense circularly polarized pump pulse to excite the sample and drive it into a light-induced Floquet-Weyl semimetal phase where the fourfold degenerate Dirac node is split into two separate Weyl nodes.
Experiments at JHU were supported by the Army Research Office Grant W911NF-15-1-0560. Work at UCSB was supported by the Vannevar Bush Faculty Fellowship program by the U.S. Department of Defense (grant no. N00014-16-1-2814).
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