Effect of temperature and compositional changes on the phonon properties of Ni-Mn-Ga shape memory alloys

We report on the vibrational properties of the ferromagnetic shape memory alloy system Ni-Mn-Ga in its stoichiometric Ni${}_2$MnGa and off-stoichiometric Ni${}_{49}$Mn${}_{32}$Ga${}_{19}$ compositions. Elastic and inelastic neutron scattering measurements at different temperatures are presented with a focus on the austenite phase and compared to first-principles calculations. The overall behavior of the full phonon dispersion is similar for both compositions with remarkable exceptions for the TA${}_2$[$\xi \xi 0$] acoustic branch and optical phonon branches. Less dispersion is found in the optical phonons for Ni${}_{49}$Mn${}_{32}$Ga${}_{19}$ in the whole reciprocal space when compared to Ni${}_2$MnGa and is explained by the occupation of regular Ga sites by excess Mn atoms. A pronounced softening in the TA${}_2$[$\xi \xi 0$] phonon branch within the austenite phase is observed in both samples when approaching the martensitic transition. Its location in reciprocal space reveals the martensitic transition mechanism. The austenite $L$2${}_1$ structure transforms to the tetragonal modulated martensite structure by shuffling (110) planes in the [1$\overline{1}$0] direction, similarly to what has been observed at the martensitic transitions of the $d^1$ and $d^2$ transition metals. Whereas the temperature dependence of the softening of the TA${}_2$[$\xi \xi 0$] phonons in the stoichiometric sample coincides perfectly with the magnetic and structural transitions, this is not the case for the off-stoichiometric sample. Here the relation between the magnetic ordering and the vibrational properties is still an open question.

Figure 1

Magnetization of (a) Ni${}_2$MnGa and (b) Ni${}_{49}$Mn${}_{32}$Ga${}_{19}$ as a function of temperature. ZFC indicates the zero-field-cooled, FC, the field-cooled, and FH the field-heated measurements. The insets show the corresponding DSC measurements.

Figure 2

The phonon dispersion calculations of the stoichiometric Ni${}_2$MnGa sample for the high-symmetry directions of the $L$2${}_1$ structure. The color coding represents the elemental contribution to the vibrational modes.

Figure 3

The phonon dispersion of the stoichiometric Ni${}_2$MnGa sample for the high-symmetry directions of the $L$2${}_1$ structure. The symbols indicate the experimental results taken at room temperature. The lines represent the first-principles calculations from Fig. 2, here shown without color coding.

Figure 4

Phonon dispersion of stoichiometric Ni${}_2$MnGa. Symbols indicate the experimental results taken at room temperature (as in Fig. 3). Solid lines indicate the phonon dispersion obtained by fitting BvK force constants up to the ninth nearest-neighbor shell to the experimental results.

Figure 5

Total and partial vibrational densities of states in Ni${}_2$MnGa calculated from (a) first principles and (b) the BvK model.

Figure 6

The phonon dispersion of the Ni${}_{49}$Mn${}_{32}$Ga${}_{19}$ sample in the austenite state. Lines indicate the first-principles calculations for the stoichiometic Ni${}_2$MnGa at absolute zero temperature. The inelastic neutron scattering measurement were done at 373 K.

Figure 7

The TA${}_2$[$\xi \xi$0] phonon modes of (a) Ni${}_2$MnGa and (b) Ni${}_{49}$Mn${}_{32}$Ga${}_{19}$ for different temperatures. Dashed lines in (a) show BvK fits to the corresponding temperatures of the austenite phase. In (b) the dashed and solid lines are guides for the eye in the austenite phase and martensite phase, respectively.

Figure 8

(a) Intensity plot of the occurrence of the 3$M$ phase on cooling stoichiometric Ni${}_2$MnGa. The color code represents the neutron intensity in a logarithmic scale. (b) Elastic measurement of 5$M$ martensite at 298 K. Superstructure Bragg peaks of the martensite phase show commensurate positions.

Figure 9

Frequency squared as a function of temperature for different $\xi$ values around the softening minimum of (a) the stoichiometric Ni${}_2$MnGa alloy and (b) the off-stoichiometric Ni${}_{49}$Mn${}_{32}$Ga${}_{19}$ alloy. $T_{A-3M}$ defines the premartensite transition temperature. Above this temperature the alloy has $L{2}_1$ Heusler structure and below this temperature the structure is three-layered modulated premartensite. $T_{A-5M}$ defines the transition temperature from austenite to five-layered martensite of off-stoichiometric Ni${}_{49}$Mn${}_{32}$Ga${}_{19}$. The solid lines are a guide to the eye. $T_C$ represents the Curie temperatures for both stoichiometric and off-stoichiometric compositions.

Figure 10

Vibrational entropy calculations of Ni${}_2$MnGa in the quasiharmonic (open dots) and in the harmonic (dashed line) approximations. The quasiharmonic approximation entropies are calculated from the DOS at the corresponding temperature. Harmonic entropies are extrapolated from the room-temperature DOS. The solid line depicts the first-principles calculations obtained using the harmonic approximation.