Anomalous vibrational effects in nonmagnetic and magnetic Heusler alloys

First-principles calculations are used in order to investigate phonon anomalies in nonmagnetic and magnetic Heusler alloys. Phonon dispersions for several systems in their cubic $L{2}_1$ structure were obtained along the [110] direction. We consider compounds that exhibit phonon instabilities and compare them with their stable counterparts. The analysis of the electronic structure allows us to identify the characteristic features leading to structural instabilities. The phonon dispersions of the unstable compounds show that, while the acoustic modes tend to soften, the optical modes disperse in a way which is significantly different from that of the stable structures. The optical modes that appear to disperse at anomalously low frequencies are Raman active, which is considered an indication of a stronger polarizability of the unstable systems. We show that phonon instability of the ${\mathrm{TA}}_2$ mode in Heusler alloys is driven by interaction (repulsion) with the low-energy optical vibrations. The optical modes show their unusual behavior due to covalent interactions which are additional bonding features incommensurate with the dominating metallicity in Heusler compounds.

Figure 1

(Color online) (a) The $L{2}_1$ Heusler structure with unit cell of space group $Fm\overline{3}m\left(O_5^h\right)$ of ${\mathrm{Ni}}_2\mathrm{MnGa}$ (Ref. 3) and (b) conventional tetragonal cell used in the electronic structure calculations. All compounds considered in this work have the same structure and formula unit $X_{2}YZ$.

Figure 2

Phonon dispersion curves of (a) FM ${\mathrm{Ni}}_2\mathrm{MnGa}$, (b) FM ${\mathrm{Ni}}_2\mathrm{MnAl}$, (c) FM ${\mathrm{Ni}}_2\mathrm{MnGe}$, (d) FM ${\mathrm{Ni}}_2\mathrm{MnIn}$, (e) FM ${\mathrm{Co}}_2\mathrm{MnGa}$, (f) FM ${\mathrm{Co}}_2\mathrm{MnGe}$, (g) NM ${\mathrm{Ni}}_2\mathrm{TiGa}$, and (h) FerriM ${\mathrm{Fe}}_2\mathrm{MnGa}$ in the $L{2}_1$ structure. Here, the reduced wave vector coordinate $\zeta$ spans the fcc Brillouin zone from $\Gamma$ to $X$. Imaginary frequencies of the unstable modes are shown in the real negative frequency range. The frequency of the optical modes $T_{2g}$ at $\Gamma$ is marked by a black dot; note that it appears at lower values as compared to the stable systems.

Figure 3

Site-projected vibrational density of states (VDOS) showing the distribution of amplitudes of the atoms over the frequency range. This information shows which atom dominates in the vibrations of a given frequency range. In particular, the anomalous behavior of optical vibrations of Ni lying at frequencies below those of the the heavier Ga atom in unstable ${\mathrm{Ni}}_2\mathrm{MnGa}$ is clearly visible (see also Fig. 2). In the case of the stable system ${\mathrm{Co}}_2\mathrm{MnGe}$, the sequence of the optical modes is regular, i.e., increasing as the mass of the atom decreases.

Figure 4

(Color online) The $\Gamma$ point $T_{2g}$ optical mode which vibrates at anomalously low energy. Note that this is the only phonon mode with $T_{2g}$ symmetry in this structure and it is Raman active. Moreover, this mode is pure Ni vibrations, or Co, i.e., the atoms sitting in the $8c$ Wyckoff positions. Other atoms are not involved in this mode. Other optical modes involve vibrations of all atoms together and have symmetry $T_{1u}$ at $\Gamma$, being IR active.

Figure 5

(Color online) Site-projected electronic density of states of ${\mathrm{Ni}}_2\mathrm{MnGa}$ and ${\mathrm{Co}}_2\mathrm{MnGe}$ in the $L{2}_1$ structure. There is a difference in the density of spin-down electronic states of both compounds close to the Fermi level. In ${\mathrm{Ni}}_2\mathrm{Mn}X$ ($X=\mathrm{Ga}$, Ge, Al, In) compounds, the peak in the DOS marked by the arrow arises from $p$ states of the $X$ atom. The double structure of this peak is related to the hybridization splitting cause by the interaction of $4p_{\downarrow }$ and Ni $3d_{\downarrow }$ states (Ref. 32).

Figure 6

(Color online) Site-projected electronic density of states of ${\mathrm{Ni}}_2\mathrm{TiGa}$. The double-peak structure at the Fermi level is caused by the hybridization of Ga $4p$ and Ni $3d$ states showing that nonmagnetic Heusler systems can also exhibit the same kind of characteristic peak structure as the FM systems.

Figure 7

(Color online) Nearest-neighbor Ni-Ni and Co-Co force constants ($X$ atoms in $X_{2}YZ$ Heusler compounds). The force constants strongly depend on the valence electron number per atom. Possibly, there is a critical value of $e∕a$ allowing the separation of structurally unstable Heusler systems with negative force constants from stable ones with positive force constants, but this has to be verified by additional calculations. Note that these Ni-Ni negative force constants do not lead directly to the unstable acoustic mode ${\mathrm{TA}}_2$, but lead to the inversion of the optical modes.

Figure 8

(Color online) Schematic representation of the deformation of the outer $3d$ electron cloud around the Ni atoms in the unstable Heusler alloys from present ab initio calculations. There are two effects which contribute to the deformation. One is due to covalent bonding of Ga $4p$ and Ni $3d$ states; the other (stronger) effect is due electrostatic repulsion of Mn and Ni atoms resulting from their nearly spherical charge distribution. The deformation of the $3d$ orbitals of Ni is of tetrahedral nature and strengthens the covalent Ni-Ga bond.