Electronic and dynamical properties of cobalt monogermanide CoGe phases under pressure

We present the pressure dependence of the electronic and dynamical properties of six different CoGe phases with orthorhombic Cmmm, hexagonal P6/mmm and P ̄ 6 2m, monoclinic C2/m, cubic P2 1 3, and orthorhombic Pnma symmetries. Using first-principles DFT calculations and the direct force-constants method, we study the dynamical stability of individual phases under external pressure. We show that the orthorhombic (Cmmm) and hexagonal (P6/mmm) structures are unstable over a broad pressure range and most pronounced imaginary phonon soft mode in both cases leads to a stable hexagonal (P ̄ 6 2m) structure of the lowest ground-state energy of all studied phases at ambient and low (below ∼ 3 GPa ) external pressure. Under these conditions, the cubic structure has the highest energy, however, together with monoclinic and orthorhombic phases, it is dynamically stable and all these three structures can potentially coexist as meta-stable phases. Above ∼ 3 GPa , the cubic phase becomes the most energetically favorable. Fitting the Birch–Murnaghan equation of state, we derive bulk modulus for all mentioned phases. The results indicate relatively high resistance of CoGe to compression. Such conclusions are confirmed by band structure calculations. Additionally, we show that electronic bands of the hexagonal (P ̄ 6 2m) phase reveal characteristic features of the kagome-like structure, while in the cubic phase the electronic bands contain spin-1 and double Weyl fermions. In both cases, the external pressure induces the Lifshitz transition, related to the modification of the Fermi surface topology.


I. INTRODUCTION
Exploring the structure of materials is a fundamental first research step in physics, chemistry, and material science.The crystal structure as well as its other properties are inherently determined by bonds between atoms, molecules, or ions.However, a structure exposed to changing external conditions, such as temperature or pressure, may undergo a structural phase transition between different arrangements of atoms, causing the crystal symmetry change.Inspired by the recent discovery of several distinct structures of cobalt monogermanide, we performed an extensive study of stability and pressure dependence of different CoGe crystalline phases.
Typically, the single crystal of CoGe is grown using a chemical vapor transport method [14].CoGe can also be synthesized in the cubic FeSi-type structure (B20) at high pressures and temperatures [1,2].The cubic phase is a simple, low carrier density, metal, similar to CoSi [5].Furthermore, the B20 phase was investigated by measuring the specific heat, resistivity, and 59 Co nuclear magnetic resonance, which uncovered a phase transition at 13.7 K [11].FeGe also crystallizes with the B20 structure and monoclinic phase.At 893 K, the cubic B20 phase transforms into the CoSn-type structure, which in turn undergoes a transition at 1013 K to the hightemperature monoclinic polymorph, isostructural with CoGe [14].Furthermore, FeGe exhibits a hexagonal hightemperature P6/mmm structure, while decreasing temperature leads to the cubic P2 1 3 structure [8,26].Highpressure conditions should favor the cubic B20 structure which has the highest density of all monogermanides polymorphs [2,14].Nevertheless, the question of CoGe structure under pressure still remains open.In this work we discuss the crystal stability of CoGe under pressure and its structural, electronic, and dynamical properties.
The paper is organized as follows.Our results are presented and discussed in Sec.II: First, we analyze the investigated crystal structures (Sec.II A).Next, we describe lattice dynamics and system stability at zero pressure (Sec.II B) and under external hydrostatic pressure (Sec.II C).Afterwards, the electronic band structures of the most favorable crystal structures are presented (Sec.III).Finally, we summarize and conclude our findings in Sec.IV.Details of the numerical calculation can  As we mentioned in the Introduction, TM monogermanides [1][2][3][4][5][6][7][8][9][10][11] and TM monosilicides [16][17][18][19][20][21] crystallize within several structures.Early stage study of CoGe suggests existence of monoclinic C2/m [13][14][15] and cubic P2 1 3 (B20) structures [1].Recently, theoretical studies have also predicted the hexagonal P 62m structure [27].Additionally, one should expect a strong impact of external conditions, such as temperature or pressure, on structure and stability of the material.For example, the crystal structure of FeGe undergoes transformation from the hexagonal P6/mmm to cubic P2 1 3 phase with decreasing temperature, around 625 K [8,26].Motivated by this, we take up here the topic of stability under pressure of a few plausible structures of CoGe.
The last two symmetries were taken into account due to the chemical affinity of CoGe with other similar systems [36].

B. Zero pressure
To check the system's stability, we calculate the phonon dispersion relations for the symmetries mentioned above (Fig. 2).Since the number of degrees of freedom of the primitive unit cell determines an amount of dispersion relations, the phonon spectrum of C2/m [Fig.2(a)] is the most complex.Similar crystal structures [containing the kagome-like net, cf.Fig. 1(a)-(c)] exhibit comparable phonon dispersion curves [cf.Fig. 2(c)-(e)].The phonon frequency ranges for all presented structures are analogous.
Such similarities are also visible in the volume dependence of the ground state energy calculated for systems of different symmetries (Fig. 3).All structures have a comparable volume and nearly the same (within 0.5%) energy per one formula unit.Fitting the Birch-Murnaghan Volume dependence of the ground-state energy calculated at zero pressure for different structures of CoGe.equation of state [55]: to energy versus volume data, we found a bulk modulus B 0 and its pressure derivative B ′ 0 at the equilibrium volume V 0 (Tab.I).All symmetries are characterized by relatively large bulk modulus, which indicates a weak impact of external pressure on the system's mechanical properties.
Some of the structures discussed above can be eliminated at zero pressure due to instability of harmonic phonons.This applies especially to Cmmm and P6/mmm structures, which show imaginary soft modes at the Γ point.Interestingly, in both structures, this soft mode is associated with the same deformation of the kagome-net, i.e. mutually opposite rotation of the Co triangles forming this sublattice [cf.Figs 1(a) and 1(b) with Fig. 1(c)] around the c axis [27].After optimization, the distorted kagome lattice in P 62m is stable [Fig.1(c)] without imaginary frequencies in the phonon spectrum [Fig.2(d)].In the final structure, with the P 62m symmetry, the Co-triangles within the kagome-like structure are rotated by 4 • , which is close to the rotation angle observed experimentally in RhPb [56], i.e. ∼ 4.5 • .The phonon dispersion curves of the cubic P2 1 3 phase [Fig.2(f)] are similar to those reported for RhGe [12].Under zero pressure, the cubic P2 1 3 (B20) phase has the highest energy among the reported structures (Fig. 3).However, as mentioned earlier, the transition from the hexagonal (P6/mmm) to cubic (P2 1 3) symmetry occurs in FeGe also due to temperature increase [8,26].

C. Role of external pressure
The stability of the system depends on external conditions.Below, we discuss the effect of the external pressure on the system's stability.The comparison of enthalpy (per formula unit) for the discussed symmetries is presented in Fig. 4(a).As the reference level of energy, we choose the energy of the C2/m structure (red line).At low pressures, the P 62m structure is the most favorable energetically.Then, above ∼ 3 GPa, the cubic phase has the lowest energy and should be preferred, which is in agreement with the previous predictions [2,14].However, regardless of the mutual ground-state energy relations, under specific conditions, crystal can grow in some metastable structures mentioned earlier (i.e.C2/m, Pnma, or P2 1 3 structure).Based on this, we expect that the experimentally reported monoclinic C2/m phase [14] can come from the cubic P2 1 3 structure at low temperatures.
The unit cell of the unstable Cmmm phase can be constructed by doubling P6/mmm or P 62m unit cells.For the Cmmm and P6/mmm symmetries, the ground-state energies and equilibrium volumes are mostly the same over the entire pressure range (cf.green and orange lines in Figs 3 and 4(a)).However, the imaginary soft mode in the phonon spectra indicates existence of a structure with lower energy.Indeed, our group-theoretical analysis of both soft modes points out at the dynamically stable structure of P 62m symmetry with energy systematically lower than those of the Cmmm and P6/mmm phases.Even though all these structures have the same volume under pressure [cf.green, orange, and blue lines on Fig. 4(b)] only the P 62m structure is stable over the entire pressure range [cf.blue line with green and orange lines on Fig. 4(a)].
As expected, due to the relatively large value of bulk modulus, volume (per formula unit) does not strongly depend on pressure [see Fig. 4(b)].Therefore, independently of the structure symmetry (i.e.arrangement of atoms), the atomic density of systems is approximately the same and inversely proportional to volume.Similarly, all structures exhibit similar compressibility, which is reflected in the relatively weak pressure dependence of volume (independently of the symmetry of the system).For example, the hexagonal P 62m structure under external pressure of 10 GPa changes the lattice constants from a = b = 5.009 Å and c = 3.857 Å to a = b = 4.914 Å and c = 3.790 Å.Similarly, the cubic P2 1 3 phase lattice constant shrinks from 4.640 Å to 4.556 Å.In both cases, the relative modification of the lattice constants induced by such pressure is around ∼ 2 %.
Due to small compressibility and volume modification, the orbital overlap does not change much.Consequently, electronic band structures are unaffected by external pressure (see top and bottom panels in Fig. 5).Also, there is only small variation in corresponding phonon dispersion relations (not shown).For example, in the cubic P2 1 3 structure only the frequency of the highest phonon mode at the Γ point is apparently changing from 8.54 THz to 9.29 THz, while the main features of the other phonon dispersion curves remain unchanged.
From the above analysis we can conclude that, under some "critical" pressure (estimated from theoretical calculations as ∼ 3 GPa), the studied system transforms from the P 62m to P2 1 3 symmetry -similarly to FeGe, which undergoes the phase transition from the hexagonal P6/mmm to cubic P2 1 3 symmetry when temperature decreases [8,26].During the described transition, the volume of CoGe increases by about 2.75 Å3 per formula unit.

III. ELECTRONIC PROPERTIES
For phases with the lowest energies on the enthalpy vs. pressure graph (Fig. 4), i.e.P 62m and P2 1 3, we investigate the electronic band structure.Their dispersion relations (Fig. 5) are characteristic for these symmetries and, generally, a whole class of similar materials.In the case of the hexagonal P 62m phase, the electronic band structure exhibits the unique features of a system con-  taining "kagome" net.For an ideal kagome lattice, the band structure contains a Dirac crossing at the K point, a strong van Hove singularity at the M point, and an almost flat band [37][38][39][40][41][42][43][44][45].We should notice that, contrary to the ideal 2D kagome lattice, where a perfectly flat band is realized, in the three-dimensional (3D) multi-orbital systems the kagome-related "flat" band has a finite bandwidth.The nearly-flat band is mostly associated with the d xz/yz and d xy/x 2 −y 2 orbitals of Co atoms [38,39,42] forming the distorted kagome net [bands marked with the green background line in Fig. 5(c)].Analogously to CoSn-like compounds, the flat bands are located around −1 eV [38,45,56].In case of the cubic P2 1 3 phase, the electronic band structure exhibits characteristic features of TM monosilicide compounds [19,20,[57][58][59][60][61][62][63][64], such as features of spin-1 fermions at the Γ point, and double degenerate Weyl points at the R point [marked by green and cyan background lines in Fig. 5(d)].The spin-1 fermions are related to the crossing of three doublydegenerate bands (in the absence of spin-orbit coupling).Similarly, one can observe the double Weyl point built by two Dirac-like cones centered at the same point.Both band structures of spin-1 fermions and double Weyl point are presented in Fig. 6.As a consequence, a large Fermi arc is observed in the surface spectral function of CoGe with the cubic structure [53].
The main features of the electronic band structure remain mostly unchanged under pressure.Nevertheless, in both cases, external hydrostatic pressure leads to the Lifshitz transition [65], i.e. change in the Fermi surface topology (see Fig. 7).In both structures, the compression shifts electronic bands and modifies the Fermi surface.In the case of hexagonal P 62m structure, new Fermi pockets emerge around the A point [cf.

IV. SUMMARY
In summary, we investigated the stability of several structures of cobalt monogermanide CoGe: monoclinic C2/m, orthorhombic Cmmm and Pnma, hexagonal P6/mmm and P 62m, and cubic P2 1 3. From the study of lattice dynamics, we found that the monoclinic Cmmm and hexagonal P6/mmm structures are unstable and have the imaginary soft modes in the phonon spectra.Based on group-theoretical analysis, we reveal that both soft modes lead to the same stable P 62m structure, containing the distorted kagome lattice of Co atoms.Surprisingly, under ambient pressure, the P 62m structure has the lowest energy among studied phases.The cubic P2 1 3 structure is energetically favored under pressure above ∼ 3 GPa.
We also discussed the electronic band structure of the most stable hexagonal P 62m and cubic P2 1 3 phases.We demonstrated that the former one shows characteristic features of the compounds containing the kagome net, while the latter exhibits traits of the chiral cubic structure, such as spin-1 fermions and double Weyl fermions.In fact, the P 62m structure contains the distorted kagome net of Co atoms, with two triangles forming the kagome-like net rotated in the opposite directions about 4 • around the c axis.Furthermore, we show that external pressure weakly affects the main features of the electronic band structure.Nevertheless, the external hydrostatic pressure leads to the Lifshitz transition in both cases.
ACKNOWLEDGMENTS Some figures in this work were rendered using Vesta [66] and XCrySDen [67]  The first-principles density functional theory (DFT) calculations were performed using the Vienna Ab initio Simulation Package (Vasp) code [68][69][70] with the projector augmented-wave (PAW) potentials [71].For the exchange-correlation energy, the generalized gradient approximation (GGA) in the Perdew, Burke, and Ernzerhof for solids (PBEsol) parametrization was used [72].The energy cutoff for the plane-wave expansion was set to 350 eV.
The optimization of the lattice constants and atom positions, including the spin-orbit coupling, was performed in the conventional unit cells.As a convergence condition of the optimization loop, we took the energy change below 10 −6 eV and 10 −8 eV for the ionic and electronic degrees of freedom, respectively.The following k-point grids within the Monkhorst-Pack [73] scheme were used for particular symmetries: 4 × 12 × 10 for monoclinic C2/m, 6 × 5 × 3 for orhoromic Pnma, 10 × 6 × 12 for orhorombic Cmmm, 10 × 10 × 6 for hexagonal P6/mmm and P 62m, and 10 × 10 × 10 for cubic P2 1 3.The symmetries of the system were analyzed using FindSym [74] and Spglib [75], while momentum space analysis was performed with SeeK-path [76].

FIG. 2 .
FIG. 2. The phonon dispersion curves of CoGe along the high symmetry directions of considered structures at zero pressure.Corresponding symmetry groups are indicated in the graphs.

FIG. 7 .
FIG. 7. Modification of the Fermi surface by the external pressure for the hexagonal P 62m and cubic P213 phases (as labeled).
Fig 7(a) and 7(c)].On the other hand, in the cubic P2 1 3 structure the small Fermi pocket at the M point disappears under pressure [cf.Fig 7(b) and 7(d)].Such modifications of the Fermi surface under pressure are related to a relatively small modification of the electronic band structure under pressure.In the hexagonal structure, the bottom of the electron-like band at the A point is shifted to lower energies [cf. the electronic band structure at the A point, marked by a red background in Fig 5(a) and 5(c)].Similarly, in the case of the cubic structure, the top of the hole-like band at the M point is also shifted to lower energies [cf. the electronic band structure at the M point, marked by the red background in Fig 5(b) and 5(d)].

TABLE I .
The ground-state energy and equilibrium volume calculated per formula unit, as well as bulk modulus B0 and its pressure derivative B ′ 0 fitted with the Birch-Murnaghan equation of state for different structures of CoGe.