Double-Weyl phonons in transition-metal monosilicides

TL;DR

This paper predicts topological phonons with double Weyl points in certain transition-metal monosilicides, revealing novel surface states and providing new avenues for experimental study of topological phononics at THz frequencies.

Contribution

It identifies a new class of topological phonons in noncentrosymmetric materials with specific Weyl points and complex surface dispersions, expanding the understanding of topological states in atomic vibrations.

Findings

Presence of double Weyl points in phonon spectra of $M$Si materials.

Surface phonon dispersions form double helicoidal sheets.

Analytical expression of surface bands as Weierstrass elliptic functions.

Abstract

Topological states of electrons and photons have attracted significant interest recently. Topological mechanical states also being actively explored, have been limited to macroscopic systems of kHz frequency. The discovery of topological phonons of atomic vibrations at THz frequency can provide a new venue for studying heat transfer, phonon scattering and electron-phonon interaction. Here, we employed ab initio calculations to identify a class of noncentrosymmetric materials of $M$Si ($M$=Fe,Co,Mn,Re,Ru) having double Weyl points in both their acoustic and optical phonon spectra. They exhibit novel topological points termed "spin-1 Weyl point" at the Brillouin zone~(BZ) center and "charge-2 Dirac point" at the zone corner. The corresponding gapless surface phonon dispersions are double helicoidal sheets whose isofrequency contours form a single non-contractible loop in the surface BZ. In addition, the global structure of the surface bands can be analytically expressed as double-periodic Weierstrass elliptic functions. Our prediction of topological bulk and surface phonons can be experimentally verified by neutron scattering and electron energy loss spectroscopy, opening brand new directions for topological phononics.

Figures

Equations

$$\omega (k_x,k_y)\sim \wp (z;2\pi ,2\pi )=$$

$$\text{Im}\{\log [\frac{1}{z^2}+\sum _{n,m\ne 0}(\frac{1}{(z+2m\pi +2n\pi i{)}^2}-\frac{1}{(2m\pi +2n\pi i{)}^2})]\},$$

Full text

Double-Weyl phonons in transition-metal monosilicides

Tiantian Zhang

Institute of Physics, Chinese Academy of Sciences/Beijing National Laboratory for Condensed Matter Physics, Beijing 100190, China

  

Zhida Song

Institute of Physics, Chinese Academy of Sciences/Beijing National Laboratory for Condensed Matter Physics, Beijing 100190, China

  

A. Alexandradinata

Department of Physics, Yale University, New Haven, Connecticut 06520, USA

  

Hongming Weng

Institute of Physics, Chinese Academy of Sciences/Beijing National Laboratory for Condensed Matter Physics, Beijing 100190, China

Collaborative Innovation Center of Quantum Matter, Beijing, 100084, China

  

Chen Fang

cfang@iphy.ac.cn

Institute of Physics, Chinese Academy of Sciences/Beijing National Laboratory for Condensed Matter Physics, Beijing 100190, China

  

Ling Lu

linglu@iphy.ac.cn

Institute of Physics, Chinese Academy of Sciences/Beijing National Laboratory for Condensed Matter Physics, Beijing 100190, China

  

Zhong Fang

Institute of Physics, Chinese Academy of Sciences/Beijing National Laboratory for Condensed Matter Physics, Beijing 100190, China

Collaborative Innovation Center of Quantum Matter, Beijing, 100084, China

**Topological states of electrons Hasan and Kane (2010); Qi and Zhang (2011); Chiu et al. (2016); Bansil et al. (2016) and photons Lu et al. (2014) have attracted significant interest recently. Topological mechanical states Huber (2016); Prodan and Prodan (2009); Kane and Lubensky (2014); Chen et al. (2014); Yang et al. (2015); Wang et al. (2015); Xiao et al. (2015); Nash et al. (2015); Süsstrunk and Huber (2015); Mousavi et al. (2015); Fleury et al. (2016); Rocklin et al. (2016); He et al. (2016); Lu et al. (2016); Süsstrunk and Huber (2016), also being actively explored, have been limited to macroscopic systems of kHz frequency. The discovery of topological phonons of atomic vibrations at THz frequency can provide a new venue for studying heat transfer, phonon scattering and electron-phonon interaction. Here, we employed ab initio calculations to identify a class of noncentrosymmetric materials of $M$Si ($M$=Fe,Co,Mn,Re,Ru) having double Weyl points in both their acoustic and optical phonon spectra. They exhibit novel topological points termed “spin-1 Weyl point” at the Brillouin zone (BZ) center and “charge-2 Dirac point” at the zone corner. The corresponding gapless surface phonon dispersions are double helicoidal sheets whose isofrequency contours form a single non-contractible loop in the surface BZ. In addition, the global structure of the surface bands can be analytically expressed as double-periodic Weierstrass elliptic functions. Our prediction of topological bulk and surface phonons can be experimentally verified by neutron scattering and electron energy loss spectroscopy, opening brand new directions for topological phononicsLiu et al. (2016); Ji and Shi (2016); Zhang et al. (2010). **

Double Weyl points

The two-band Hamiltonian near a Weyl point Murakami (2007); Wan et al. (2011); Xu et al. (2015); Lv et al. (2015); Lu et al. (2015); Soluyanov et al. (2015); Burkov (2016); Deng et al. (2016), $H_{2}(\mathbf{k})\propto\mathbf{k}\cdot\mathbf{S}=\frac{\hbar}{2}\mathbf{k}\cdot\mathbf{\sigma}$, is the simplest possible Lorentz invariant theory for three-dimensional (3D) massless fermions, where $\mathbf{S}$ is the rotation generator for spin-1/2 particles and $\sigma_{i}$’s are the Pauli matrices. The two bands have spin components $S_{k}=\mathbf{S}\cdot\hat{\mathbf{k}}=+\hbar/2$ and $-\hbar/2$ and form inward or outward hedgehog configuration on each equal-energy surface. The Chern number of $+1$ or $-1$ is the topological invariant that distinguishes between these two cases. For spin-1 bosons like phonon, photon and magnons, the natural extension of a Weyl Hamiltonian is the three-band Hamiltonian $H_{3}(\mathbf{k})\propto\mathbf{k}\cdot\mathbf{L}$, where $L_{i}$ are the spin-1 matrix representations of the rotation generators. The Chern numbers of the resultant three bands, corresponding to $L_{k}=\mathbf{L}\cdot\mathbf{\hat{k}}=\hbar,0,-\hbar$ are $+2$, [math] and $-2$, doubling those of the spin-1/2 Weyl point. We hence refer to such a band crossing point as the “spin-1 Weyl point” Bradlyn et al. (2016) in this paper. Another possibility of band crossing having Chern number of 2 is the direct sum of two identical spin-1/2 Weyl points, referred as “charge-2 Dirac point” in this paper, whose four-band Hamiltonian is H_{4}(k)\sim\left(\begin{array}[]{cc}\mathbf{k}\cdot\mathbf{\sigma}&0\\ 0&\mathbf{k}\cdot\mathbf{\sigma}\end{array}\right) Geilhufe et al. (2016), in contrast to a regular 3D Dirac point consists of two Weyl points of opposite Chern numbers Liu et al. (2014) and quadratic double-Weyl points between two bands Xu et al. (2011); Fang et al. (2012); Chen et al. (2015); Huang et al. (2016). In this Letter, we predict the presence of the spin-1 Weyl in the acoustic and optical branches, as well as charge-2 Dirac points in the optical branches of the phonon spectra in existing materials. The topological surface state are explicitly calculated and we analytically described the surface state dispersions with the Weierstrass elliptic function.

Phonon calculations

Phonons are quantized excited vibrational states of interacting atoms. Solids with more than one atom in the primitive cell have both acoustic and optical branches in their phonon band structure, which we computed for the $M$Si family ($M$=Fe,Co,Mn,Re,Ru) in Fig. 2. $M$Si belongs to the simple cubic crystal structure with space group $P2_{1}3$ (No.198). Each primitive cell contains 8 atoms, both $M$ and Si atoms occupy Wyckoff positions (4a). The crystal structure of $M$Si Pauling and Soldate (1948); Zhao et al. (2009) is shown in Fig.2(a) and the corresponding BZ is shown in Fig. 2(b). In this paper, the vibrational modes of $M$Si family are calculated by using Vienna $Ab$ $initio$ Simulation Package (VASP)Kresse and Furthmüller (1996), based on density functional perturbation theoryGonze and Lee (1997). The exchange-correlation functional within a generalized gradient approximation was used Perdew et al. (1996). The wave functions are expanded in plane waves up to a kinetic energy cutoff of 420 eV and a sum on a Monkhorst-pack grid of $5\times 5\times 5$ is used for integrals over the BZ. Wilson loop methodSoluyanov and Vanderbilt (2011); Yu et al. (2011) is used to find the Chern numbers of the double-Weyl points.

Spin-1 Weyl acoustic phonons

Shown in Fig. 2(c), the three branches of acoustic phonons form a spin-1 Weyl point. The longitudinal branch has a Chern number of 0 and the two transverse branches have Chern numbers of $\pm$ 2, reflecting the fact that phonons are spin-1 particles. We note that the individual Chern numbers can be defined only when the three acoustic dispersions are well separated from each other. This only happens when $\mathcal{PT}$ symmetry is broken, since, otherwise, Berry curvature strictly vanishes and the Chern number of any band is zero.. Here $\mathcal{P}$ is parity inversion and $\mathcal{T}$ is time-reversal. $M$Si is non-centrosymmetric, so $\mathcal{P}$ is broken in the lattice. We emphasize that these spin-1 Weyl dispersions is a general property for both phonons and photons at zero frequency; the difference lies in the vanishing longitudinal mode for photons Gao et al. (2015). We note that the topological surface states related to this zero-frequency spin-1 Weyl point may not be observed, since the two transverse Chern bands both have positive group velocity in all directions, not leaving a bulk gap when projected on any open surface.

Double-Weyl optical phonons

Both the spin-1 Weyl points at $\Gamma$ and charge-2 Dirac points at R were found in the optical phonon spectrum of FeSi in Fig. 2(c). They are stabilized by the lattice symmetries and $\mathcal{T}$. The No. 198 ($P2_{1}3$) space group has twofold screw rotations along the each $\langle 100\rangle$ axes ($\{C_{2x}|(\frac{1}{2},\frac{1}{2},0)\}$ along x axis) and three-fold rotations $C_{3}$ along the $\langle 111\rangle$ axes. We performed a $k\cdot{p}$ analysis at the two $\mathcal{T}$-invariant momenta ($\Gamma$ and R) in the Supplementary Information.

At $\Gamma$ point, the irreducible representations for the optical branches are $\Gamma=2A+2E+5T$, in which $A$, $E$ and $T$ represent the singly, doubly and triply degenerate multiplets respectively. All five three-fold band degeneracies are spin-1 Weyl points enforced by the little group symmetry at $\Gamma$, which is the point group $T(23)$.

At R point of the BZ corner, all bands form charge-2 Dirac points. In other words, in $M$Si materials, every phonon band connects to the four-fold degenerate double-Weyl points at R. This is due to both $\mathcal{T}$ and the non-symmorphic nature of the No. 198 space group containing three screw axes.

As examples, we plotted the zoom-in dispersions of the two types of double-Weyl points of the highest phonon bands ($\sim$14.5 THz) in Fig. 2(d) and (e). These four bands are frequency-isolated from the rest. Since a topological point carrying non-zero Chern number cannot exist alone in the BZ, the spin-1 Weyl and charge-2 Dirac points come in pairs and their Chern numbers cancel exactly.

Here we offer an intuitive perspective to understand the charge-2 Dirac point at R. The three screw axes, $C_{2x,2y,2z}$, at this point anti-commute with each other and satisfy $C_{2i}^{2}=-1$, just like half-integer spin rotations. From this we know that all irreducible representations have even dimensions, with the smallest representation being two-dimensional and the rotations represented by $\pm i\sigma_{x,y,z}$. But since R is also time-reversal invariant, these screw axes must commute with $\mathcal{T}$, so that all matrice representations of the three screws must be real. This is impossible for two dimensional representations of SU(2), but can be met by four-dimensional representations. Finally, since time-reversal preserves the Chern number of a Weyl point, the four-dimensional representation has charge $\pm 2$.

Single non-contractible surface loop

An iso-energy contour of any 2D (non-interacting) system or trivial surface state is a closed loop, which can be continuously deformed into an even number of non-contractible loops which wraps around the BZ torus. The unique feature of a Weyl crystal is its open surface arcs whose ends are pinned by the projection of Weyl points in the bulk. In contrast, the $M$Si surface contour is a single non-contractible loop.

The surface local density of states (LDOS) is plotted along the surface momentum lines in Fig. 3(a) and the iso-frequency surface contours are plotted in Fig. 3(b), (c) and (d) for different frequencies. For computing the surface LDOS, we first calculated the second rank tensor of force constant in Cartesian coordinates from density functional perturbation theory, from which we can get the tight-binding parameters for the bulk and surface atoms.Togo and Tanaka (2015). Then we obtain the surface Green’s function iteratively and take its imaginary part as the LDOSLopez Sancho et al. (1985); Sancho et al. (1984); WU (2015).

On the (001) surface, the symmetries of the $M$Si lattice are broken on the surface and $\mathcal{T}$ is the only symmetry left invariant other than the in-plane translations. The (001) surface BZ is a square. The spin-1 Weyl point at $\Gamma$ is projected to $\bar{\Gamma}$, and charge-2 Dirac point at R is projected to $\bar{\mathrm{M}}$. Note that the four BZ corners are the same $\bar{\mathrm{M}}$ point. Since surface arcs always connect two Weyl points with opposite Chern numbers, there should be two arcs connecting the double-Weyl points at the BZ center ($\bar{\Gamma}$) and BZ corner ($\bar{\mathrm{M}}$). Constrained by $\mathcal{T}$, the two arcs must be related by a $\pi$ rotation about Gamma; when viewed together, the two arcs stretch diagonally across the BZ. This forms a single non-contractible loop connected by two surface arcs where the connection points are projections of the bulk double-Weyl points. These novel non-contractible surface loops are shown at three different frequencies in Fig. 3(b), (c) and (d), where the bulk pockets are connected by the two arcs.

Double-helicoid surface states

Iso-energy surface arcs are only the local description (in energy) of a topological surface state in the momentum space, which are gapless under arbitrary surface conditions, in the presence of the protecting symmetries. They are sheets noncompact (infinitely connecting) in the energy axis. Fang et al. Fang et al. (2016) pointed out that the Weyl surface states are equivalent to a helicoid, one of the common noncompact Riemann sheets. Here, it is natural to expect that the surface state of a double-Weyl crystal is a double helicoid: two surface sheets wind around the double Weyl point. This is indeed confirmed by the results plotted in Fig. 3 and illustrated in Fig. 4.

Analytical description by Weierstrass elliptic function

We show that the double-helicoid surface of $M$Si phonons is topologically equivalent to the Weierstrass elliptic function, that is double-periodic and analytical in the whole BZ. In Ref. Fang et al. (2016), Fang et al. showed that the surface state dispersion near the projection of topological band crossings can be mapped to the Riemann surfaces of analytic functions with surface momentum $k\equiv{k_{x}}+ik_{y}$ as a complex variable. Near the projection of a Weyl point having Chern number ($C$), the dispersion is topologically equivalent to the winding phase of an analytic function having an order-$C$ zero ($C\textgreater 0$) or pole ($C\textless 0$), or symbolically $\omega(z)\sim\text{Im}[\log{(z^{C})}]$, where $z$ is the planar momentum relative to the projection of the Weyl point.

However, this analytic functions $z^{C}$ is defined in a noncompact momentum-space and hence cannot provide a global description for surface dispersions in the whole surface BZ, which is a compact torus. To establish the global picture, we notice that analytic functions having two or more zero-pole-pairs and periodic in both directions are elliptic functions. Therefore, we use the Weierstrass elliptic function ($\wp$), having one second-order pole at $\mathbf{k_{+}}=(\pi,\pi)$ and one second-order zero at $\mathbf{k_{-}}=(0,0)$, to reveal the global structure of the double-Weyl surface states. The explicit mapping is given by

[TABLE]

where $z\equiv{}k-(1+i)\pi$.

We plot, in Fig. 4, the surface state in the whole BZ with the illustrated bulk double-Weyl points at the pole and zero. Two surface arcs are plotted in green that match the calculation results in Fig. 3(b) and (c).

Conclusion

We predicted topological double-Weyl phonons in both acoustic and optical branches of a family of existing crystalline materials: the transition-metal monosilicides. The corresponding surface phonon dispersions can be globally expressed by Weierstrass elliptic functions in the whole BZ. Our findings reveal the existence of topological phonons in this material system that was previously studied as thermal-electric materialsWolfe et al. (1965); Sales et al. (2011) and superconductors Frigeri et al. (2004). This work also paves the way for studying topological phonons at the atomic level with quantum-mechanical treatment.

Acknowledgments

We thank Jiawei Zhou, Yuanfeng Xu, Gregory Moore, Shiyan Li, Xuetao Zhu for useful discussion. This work was supported by the National Key Research and Development Program under grant No. 2016YFA0302400 (C.F, L.L.) and 2016YFA0300600 (C.F., H.W.), NSFC under grant number 11674370 (C.F.) and 11421092 (T. Z., H.W., Z.F.), the National 973 program of China No. 2013CB921700 (Z.F.), the Yale Postdoctoral Prize Fellowship (A.A.) and the National Thousand-Young Talents Program of China (C.F., L.L.).