Interplay of phase sequence and electronic structure in the modulated martensites of ${\mathrm{Mn}}_2\mathrm{NiGa}$ from first-principles calculations

We investigate the relative stability, structural properties, and electronic structure of various modulated martensites of the magnetic shape memory alloy ${\mathrm{Mn}}_2\mathrm{NiGa}$ by means of density functional theory. We observe that the instability in the high-temperature cubic structure first drives the system to a structure where modulation shuffles with a period of six atomic planes are taken into account. The driving mechanism for this instability is found to be the nesting of the minority band Fermi surface, in a similar way to that established for the prototype system ${\mathrm{Ni}}_2\mathrm{MnGa}$. In agreement with experiments, we find 14M modulated structures with orthorhombic and monoclinic symmetries having energies lower than other modulated phases with the same symmetry. In addition, we also find energetically favorable 10M modulated structures which have not been observed experimentally for this system yet. The relative stability of various martensites is explained in terms of changes in the electronic structures near the Fermi level, affected mostly by the hybridization of Ni and Mn states. Our results indicate that the maximum achievable magnetic field-induced strain in ${\mathrm{Mn}}_2\mathrm{NiGa}$ would be larger than in ${\mathrm{Ni}}_2\mathrm{MnGa}$. However, the energy costs for creating nanoscale adaptive twin boundaries are found to be one order of magnitude higher than that in ${\mathrm{Ni}}_2\mathrm{MnGa}$.

Figure 1

(a) Cubic ${\mathrm{Hg}}_2\mathrm{CuTi}$ structure of ${\mathrm{Mn}}_2\mathrm{NiGa}$. (b) Nonmodulated tetragonal $\left(\mathrm{L}{1}_0\right)$ structure. (c) 6M structure. (d) 10M structure. (e) $10\mathrm{M}{\left(3\overline{2}\right)}_2$ structure. (f) $14{\mathrm{M}}_{3/7}$ structure. (g) $14{\mathrm{M}}_{2/7}$ structure. (h) $14\mathrm{M}{\left(5\overline{2}\right)}_2$ structure for the martensitic phase.

Figure 2

The total energies of pseudotetragonal 10M structure (with orthorhombic lattice parameters) relative to the cubic ${\mathrm{Hg}}_2\mathrm{CuTi}$ structure, as a function of $c/a$ ratio for five different volumes: 490 $\AA {}^3$, 495 $\AA {}^3$, 497.94 $\AA {}^3$(equilibrium volume of cubic structure), 500 $\AA {}^3$, and 505 $\AA {}^3$.

Figure 3

The variation of the total energy in NM tetragonal $\left(\mathrm{L}{1}_0\right)$ and in the different modulated structures of ${\mathrm{Mn}}_2\mathrm{NiGa}$ as a function of the $c/a$ ratio. The zero energy is taken to be the energy of the cubic ${\mathrm{Hg}}_2\mathrm{CuTi}$-type inverse Heusler structure. The energies of the pseudocubic structures (modulated structures inscribed in an orthorhombic cell with $c/a=1)$ are shown in the inset.

Figure 4

Total density of states of different pseudocubic modulated phases. The total density of states in the cubic structure is shown for comparison.

Figure 5

Total and atom-projected density of states of ${\mathrm{Mn}}_2\mathrm{NiGa}$ in the pseudocubic 6M modulated structure. The total density of states in the cubic ${\mathrm{Hg}}_2\mathrm{CuTi}$-type structure is shown for comparison.

Figure 6

Total density of states of fully relaxed modulated structures. The density of states in the NM tetragonal $\left(\mathrm{L}{1}_0\right)$ structure is also plotted.

Figure 7

Full phonon dispersion curve for ${\mathrm{Mn}}_2\mathrm{NiGa}$ (solid line) and ${\mathrm{Ni}}_2\mathrm{MnGa}$ (dashed line) in the cubic structure. The calculated dispersion for ${\mathrm{Ni}}_2\mathrm{MnGa}$ was taken from Ref. [81].

Figure 8

Minority spin (top row) and majority spin (bottom row) Fermi surfaces of ${\mathrm{Mn}}_2\mathrm{NiGa}$ (left column) and ${\mathrm{Ni}}_2\mathrm{MnGa}$ (right column) in the extended zone scheme, represented by a cubic box with the edge length $2\pi /a$. The center and the corners of the box refer to the $\mathrm{\Gamma }$ point. Each band crossing the Fermi energy (three in the case of the majority spin channel and two for the minority spin channel) is indicated by an individual color (green: $17\mathrm{th}$ spin-up and the $13\mathrm{th}$ spin-down band of ${\mathrm{Ni}}_2\mathrm{MnGa}$ and $14\mathrm{th}$ spin-up and $13\mathrm{th}$ spin-down band of ${\mathrm{Mn}}_2\mathrm{NiGa}$; orange: $18\mathrm{th}$ spin-up and $14\mathrm{th}$ spin-down band of ${\mathrm{Ni}}_2\mathrm{MnGa}$ and $15\mathrm{th}$ spin-up and $14\mathrm{th}$ spin-down band of ${\mathrm{Mn}}_2\mathrm{NiGa}$; blue: $19\mathrm{th}$ spin-up band of ${\mathrm{Ni}}_2\mathrm{MnGa})$. The Fermi surfaces agree excellently with previous calculations for ${\mathrm{Mn}}_2\mathrm{NiGa}$ [77, 82] and ${\mathrm{Ni}}_2\mathrm{MnGa}$ [23, 32, 48, 49, 83, 84, 85]. The red arrows indicate possible nesting vectors according to the respective first maxima of ${\chi }_0^{\prime }(\underline{q},0)$ at $\underline{q}={\xi }_1\left[110\right]$ as calculated in this work.

Figure 9

Real part of the generalized susceptibility in the low-frequency limit ${\chi }_0^{\prime }(\underline{q},0)$ of ${\mathrm{Ni}}_2\mathrm{MnGa}$ and ${\mathrm{Mn}}_2\mathrm{NiGa}$ for wave vectors $\underline{q}$ in the $\left[100\right]–\left[010\right]$ plane: (a) and (b) show the minority spin contribution, (c) and (d) the majority spin channel. Panel (e) shows the cross sections along the $\left[110\right]$ direction indicated by the dashed lines in (a)–(d). In all cases, a moderate electronic temperature of $T=50$ K was assumed to smoothen the features. The long-wavelength limit ${\chi }_0^{\prime }(0,0)$ has been subtracted, such that the absolute magnitude of the features represented by the color coding can be compared. The difference between two neighboring contour lines is 0.1 (arbitrary units) in (a)–(d).