Optical Response of Chiral Multifold Semimetal PdGa

Abstract

We present a theoretical study of the band structure and optical conductivity for the chiral multifold semimetal PdGa. We identify several characteristic features in the optical conductivity and provide their origins within the band structure. As experimental optical studies for the mentioned compound have not been reported, we contrast our results with the related compounds, RhSi and CoSi. We believe that the presented hallmarks will provide guidance to future experimental works.

1. Introduction

First hints of topological characters of materials were found in crystals, where avoided band crossings caused a separation of band energies accompanied by a mixture of band characters. Brought into contact with a different material, that hosts a regular order of bands, without band mixing, this enforces surface states at the interface, which would host linear, intersecting bands with opposite spin characters. Later, it was found that this can be extended to be a bulk property, when the Weyl semimetal was born [1], hosting a three-dimensional bulk realization of linear, spin non-degenerate bands. Their equivalency to the surface states of topological insulators is immediatelly obvious, but can further be quantized by the Chern number $\chi$, for which the Weyl semimetal is the lowest integer realization with $\chi =\pm 1$ [2,3,4]. Weyl semimetals were promising for applications in spintronics [5,6,7], in opto-electronics [8,9,10,11], or even in chemistry [12,13,14,15,16]. Of essence, here, was either the breaking of the time-reversal symmetry, or the inversion symmetry in combination with sufficiently strong spin-orbit coupling (SOC). The first confirmed Weyl-semimetal is TaAs [17,18,19], which lacks inversion symmetry.

This can be extended to crystals, which host, besides broken inversion symmetry, also a lack of mirror symmetries. Among the 230 space groups, the 65 Sohncke groups fall under this condition and can provide a chiral crystal structure. Note that the 65 Sohnke groups are not necessarily chiral space groups, but can provide chiral crystal structures. In total, there are only 22 chiral space groups (11 enantiomorphic pairs). These 22 chiral space groups are contained within the 65 Sohnke groups. The group of PdGa, $P{2}_{1}3$ is not a chiral space group, but a Sohnke group [20,21].

It has recently been suggested, that in chiral crystals a new type of fermionic state can be realized, extending the pool of topological quasiparticles by the so called multifold fermion [22]. They differ from the Weyl fermion in that they are hosted by conical band intersection with Chern numbers $\left|\chi \right|>1$. To this end, chiral crystal structures belonging to the RhSi family, space group $P{2}_{1}3$, number 198, have garnered attention as angle resolved photoemission spectroscopy (ARPES) measurements provided strong indications that these materials host surface states, which are maximally extended in k-space [23,24,25,26,27,28]. The lack of inversion and mirror symmetries, in combination with SOC, leads to the splitting of bands around the high symmetry points $\Gamma$ and R, giving rise to a non-collinear spin arrangement with Chern numbers $\chi =\pm 4$ [29] with maximally extended surface states, ranging from the center to the edge of the Brillouin zone [25,27,30]. In such semimetals, the quantized circular photogalvanic effect has been predicted, which, by effect of topological states, constitutes a photocurrent, quantized in units of material-independent fundamental constants [31,32,33,34].

In this paper, we theoretically investigated the material PdGa, which belongs to the same family. So far, PdGa has not been as extensively investigated as its silicon-based counterparts. In contrast to CoSi and RhSi, the Fermi surface of PdGa is considerably larger and is expected to host a larger number of free carriers, perhaps making the experimental observation of the low energy dynamics challenging, as in the case of RhSi [35]. Hence, the theoretical approach would be beneficiary for comparison and provide some guide for the interpretation of the high energy part of the measured spectra.

Furthermore, the optical conductivity provides a useful tool to identify fingerprints of Dirac fermions [36,37,38,39,40], which should express themselves as linear signatures $\sigma {\left(\omega \right)}_1\propto \omega$ [41,42]. In the silicides, they are thought to occur as low energy excitations near the $\Gamma$-point, as the Fermi level almost coincides with the multifold chiral intersection. Even though the Fermi level in PdGa is much higher, we nonetheless find a nearly linear ridge in the optical spectrum and identify its origin, as discussed in the results.

The combination with external stimuli, such as magnetic fields [43,44,45] or external pressure [46,47], which provide clean, non-invasive tuning mechanisms, makes the optical conductivity especially desirable in the search for topological materials.

We provide theoretical estimates for intra- and interband optical conductivities, which we contrast to and interpret along with the published experimental results of PdGa’s sister compounds CoSi [48] and RhSi [35], as no experimental data for PdGa are available.

2. Results and Discussion

We performed DFT calculations using Wien2k’s full-potential linearized augmented plane wave (LAPW) methods with the Perdew–Burke–Ernzerhof (PBE) exchange correlation, accounting for the semimetallic nature of the chiral multifold compound PdGa [49,50]. The lattice parameters were adopted from Ref. [29]. Figure 1 shows the chiral atomic structure and the Fermi surface of PdGa. The Fermi surface is comparatively large, with reference to RhSi and CoSi, indicating a much stronger intraband response. In Figure 1d the Fermi surfaces of the separate contributing bands are shown, sorted by energy from left to right. Within the cube-like structure, a set of eight droplet-shaped Fermi surfaces are hidden, positioned within the corners of the cube.

The band structures of PdGa were calculated on an $18\times 18\times 18$ k-mesh with and without SOC, as shown in Figure 2a–c. The calculations were converged within 14 cycles down to the charge ${10}^{-5}$e, with e being the electron charge. Core leakage was well within acceptable levels: the core charges for Pd and Ga integrated to 29.998 e and 17.999 e, respectively. These core electrons arise from choosing a generous energy interval of $-9.0$ Ry for outer electrons. We chose the parameter $R{K}_{\mathrm{Max}}=7.0$ and excluded relativistic linear orbitals (RLOs). Due to SOC in combination with the lack of inversion symmetry, bands with opposite spin split throughout the Brillouin zone (BZ) under the action of the Dresselhaus effect. Exempt from the splitting is the $\Gamma$-point, at the BZ center, and points or paths on the BZ boundaries. The $\Gamma$-point hosts a four-fold degenerate spin 3/2 fermion, also known as the Rarita–Schwinger fermion [51], and the R-point two three-fold degenerate spin 1 fermions, each with the maximum possible Chern number of $\left|\chi \right|=4$ [29,30]. A detailed view of bandstructures around the $\Gamma$-point is given in Figure 2b,c without and with SOC, respectively.

The calculations summarized in Figure 3 and Figure 4 were performed on a denser $32\times 32\times 32$ k-mesh with SOC, using the optic module [52]. Figure 3 shows the density of states (DOS) of PdGa, as well as the atomic densities of Pd and Ga separately. It can be clearly seen that the majority of carriers at the Fermi surface, as well as electrons involved in optical transitions far beyond the visible spectrum, are entirely contributed by Pd. Significant contributions to the DOS from Ga arise at energies only as low as $-15\mathrm{eV}$, and beyond.

The interband contribution to the optical conductivity of PdGa, without scattering, is shown in Figure 4a. It can be obtained from the imaginary part of the relative permittivity ${\epsilon }_2$, which is the standard output of Wien2k, via the convention (CGS units)

$${\sigma }_1=\frac{\omega {\epsilon }_2}{4\pi }.$$

(1)

Several distinct features can be seen. Two features of special interest have been marked ①, at around 150–300 meV, and ②, which shows a distinct peak at around 600 meV. The former originates from bands near the $\Gamma$-point, see again Figure 2c, corresponding to transitions from the Fermi surface droplets in Figure 1d to the corners of the cube-shaped Fermi surface. The latter is of particular interest as it seems to reproduce a known feature in CoSi around 560 meV [48] very well and is also in the energy range of a broader feature, likely of similar origin, in RhSi, around 750 meV [35], both measured at $T=10$ K. The sharp peak at 600 meV can be attributed to transitions between the parallel bands dispersing from $\Gamma$ to R and similar transitions between M and R. Since for CoSi and RhSi the corresponding bands between $\Gamma$ and R are not as parallel, we predict that the optical transition for PdGa will be distinctly sharper. Note that the interband conductivities were not broadened in our calculations. In the case of RhSi, the mentioned feature merges with a rising edge the in optical conductivity around 1 eV [35], which may indicate what could be identified with the lower edge of feature ③ in our calculations, which arises from a combination of transitions $A\to C,D$ and $B\to D$.

As mentioned earlier, another interesting aspect of the the calculated ${\sigma }_1\left(\omega \right)$, is the nearly linear ridge extending over a rather large energy range, from roughly 0.9 eV to 1.5 eV. Linear features in the optical conductivity are readily identified as signatures of Dirac fermions [36,37,38,39,40]. We can assign it to the transition $B\to C$. In an experimental setting, this feature may mistakenly be interpreted as transitions between bands extending from the lower lying multifold point, intersecting $\Gamma$ at around $-1.0$ eV, to bands above, which we can safely exclude. Note further that the contributing bands contain accidental near-degeneracies, akin to gapped Dirac points, between $\Gamma -X$ and $\Gamma -R$. Away from the high-symmetry lines, these features may develop into real conical intersections, implying that the discussed transition may very well contain character of Dirac fermions. This warrants closer inspection in future works and may motivate experimental investigations.

In Figure 4b the intra- and interband contributions are plotted alongside the total optical conductivity, ${\sigma }_1\left(\omega \right)$. For the intraband contribution we make the reasonable assumption for the dc conductivity, ${\sigma }_0=1·{10}^6{\Omega }^{-1}{\mathrm{cm}}^{-1}$ supported by on-site dc measurements, and use the theoretically obtained value for the plasma frequency, ${\omega }_p=4.43\mathrm{eV}$ (note that this is the unscreened plasma frequency). Both values are significantly larger than the experimentally obtained parameters for CoSi and RhSi [35,48], which is consistent with the much larger Fermi surface of PdGa. These values, under assumption of a single Drude contribution for the intraband transitions, yield a scattering rate of $\Gamma =2.5\mathrm{meV}$, which is expressed by a very sharp Drude peak, seen in Figure 4b. All discussed interband features remain clearly visible after inclusion of the intraband contribution. Whether such a sharp Drude peak is experimentally reproducible remains to be seen. Accounting for interband broadening, the feature ② might be brought to closer coincidence with the CoSi peak, as well as the much broader RhSi feature. Note that the low energy feature ① is inherent only to PdGa. This is, again, due to fact that bands, originating from the $\Gamma$-point in PdGa, disperse in parallel towards R, while in CoSi and RhSi these bands take the form of a double flat band overlaid with a Dirac cone, contributing an energetically much broader joint DOS, smearing out the transition.

3. Conclusions

In summary, we performed DFT calculations on one of the chiral multifold semimetals, PdGa, and estimated the optical conductivity in a broad frequency range, providing a detailed picture the band structure and a guide for interpretation of future optical experiments on this currently popular compound. A plethora of tuning mechanisms, such as application of pressure, magnetic fields, gating or doping may be applied to effectively alter the band structure, which our work should provide a helpful reference to. Optical transitions were assigned to the specific bands based on the band structure of the compound. Several common features were found in the related compounds, CoSi and RhSi, based on the experimental reports: A sharp, prominent mid-infrared absorption, which originates from the parallel bands between the $\Gamma$ and R-points and the successive linear-in-frequency increase. In addition, PdGa seems to possess a low lying optical transition at around 150–300 meV that is predicted to be absent in the sister compounds. Indeed, it has not been reported for either CoSi or for RhSi in previous experimental studies [35,48].

Furthermore, a linear-in-frequency section of the optical conductivity, often taken as a hint towards the presence of Dirac or Weyl fermions, around 0.9 eV to 1.5 eV has been identified, excluding the origin as transitions between the multifold chiral points intersecting $\Gamma$ at around 0.5 eV and 1.0 eV. Nearly degenerate bands between $\Gamma -X$ and $\Gamma -M$ may hint at accidental Dirac-like degeneracies near the high-symmetry lines, warranting further investigation.