Vibrational properties of Ni–Mn–Ga shape memory alloy in the martensite phases

Studying the phonon dispersion of the ferromagnetic shape memory alloy system Ni–Mn–Ga gives insight into the mechanism of the martensite transition and the forces driving the transition. Transformation of austenite single crystals under uniaxial stress results in the coexistence of two martensitic variants with perpendicular modulation vector. Here we report on inelastic neutron scattering studies of martensite crystals with off-stoichiometric compositions, varying from non-modulated (NM) to five- (5M) and seven- (7M) layer modulated martensite phases. Both the 5M and 7M crystals show fully commensurate satellite peaks along [ξξ¯?>0], corresponding to the five- and seven-layer modulation. These superstructure peaks become Γ-points of the modulated structure. Due to the coexistence of two variants within the (001) plane, both new acoustic phonons reflecting the modulation vector [ξξ¯?>0] and acoustic TA2[ξξ0] phonons corresponding to the non-modulated direction are observed. The latter display a pronounced softening around ξ = 0.2–0.4 when approaching the martensite–austenite transition from above and below, i.e. this soft mode has lowest frequency at the transition temperature. Overall the phonon dispersion of the austenite and martensite phase resemble each other very much. The coexistence of two martensitic variants after uniaxial transformation explains the particular behaviour of the low-energy excitations, in contrast to previous interpretations involving charge-density waves and associated phason modes.


Introduction
Ni-Mn-Ga magnetic shape memory materials show a magneto-structural transition, which leads to large strains of 6 and 10% under applied external magnetic field for off-stoichiometric compositions with five-layered and seven-layered martensite structures, respectively (Murray et al 2000, Sozinov et al 2002. The ferromagnetic shape memory effect covers two distinct effects: magnetic field induced structural transition (MIT) and magnetic field induced variant reorientation (MIR). For the observation of MIT, the structural transition temperature change has to be strongly coupled to the external magnetic field. Indeed, this coupling is present for all Ni-Mn-based magnetic shape memory alloys, but it is weak in Ni-Mn-Ga compared to other Ni-Mn-based alloys (Acet et al 2011). Its uniaxial magneto-crystalline anisotropy in the martensite phase is strong, however, compared to the energy barrier for twin boundary mobility. As a consequence, it is possible to move the twin boundaries just by applying an external magnetic field in the range of 0.5-1 T (Murray et al 2000, Sozinov et al 2002. This mobility of the twin boundaries allows the minimization of the Zeeman energy which is created by the external magnetic field (Acet et al 2011).
The vibrational properties of the Ni-Mn-Ga alloy system in its high temperature austenite phase have been studied extensively by inelastic neutron scattering (Zheludev et al 1995, 1996, Stuhr et al 1997, Mañosa et al 2001, Ener et al 2012. Also the structural transition properties of stoichiometric and off-stoichiometric compositions of Ni-Mn-Ga have been investigated by different methods, but are yet not fully understood (Brown et al 2008). Low temperature martensite phases of five-layer modulated (5M), seven-layer modulated (7M) and non-modulated (NM) martensite phases have been observed, depending on the specific chemical composition. The shape memory effect due to the reorientation of the variants (MIR) is restricted to the modulated martensites, because only here is the energy barrier for twin boundary mobility comparable to the Zeeman energy. The crystal structures of 7M and 5M martensites show periodic modulations of (110) planes in [110] direction with twin boundaries. A first model of the 5M martensite has been suggested by Martynov and Kokorin (1992) which defines the modulation of the (110) planes with harmonic terms. The agreement of this model with experiment has been tested with transmission electron microscopy (Pons et al 2005), x-ray (Mogylnyy et al 2003, Righi et al 2007 and neutron diffraction (Brown et al 2002, Kushida et al 2008 studies. A second approach for the modelling is to consider piecewise linear modulations instead of sinusoidal (Righi et al 2007). The expected differences in the diffraction experiments for these two models are very small, however, and experiments were not able to distinguish between these two models yet.
Up to now mainly the phonon dispersion in the austenite phase for alloys of different composition has been investigated to understand the nature of the martensite transition (Zheludev et al 1995, 1996, Stuhr et al 1997, Mañosa et al 2001, Ener et al 2012. These investigations revealed the atomistic trajectory for the transition from austenite to martensite. To comprehend the nature of MIR it is essential to know the phonon dispersion in the martensite phase. For the present work single crystals of the Ni-Mn-Ga system have been grown in the high temperature austenite structure. Note that cooling the austenite phase down into the martensite phase without precaution would fail due to the emergence of multiple martensite variants. For the work at hand, austenite crystals of different compositions have been successfully transformed to single variant non-modulated (NM) martensite and double variant 5M and 7M modulated martensite crystals by mechanical training (Rolfs et al 2012). By means of those the complete 5M martensite phonon dispersion has been measured at room temperature and compared to those previously measured in austenite (Ener et al 2012). Further temperature dependent measurements of the TA 2 [ξ ξ 0] phonon mode were carried out. A maximum in softening of the TA 2 [ξ ξ 0] mode is found at the transition temperature. In addition crystals of similar composition have been transformed to 7M and NM martensite, and their TA 2 [ξ ξ 0] phonon modes have been examined, too.

Sample characterization and single crystal diffraction
Single crystals of different composition of Ni-Mn-Ga have been grown by a modified Bridgman technique (Rolfs et al 2012) by Rolfs and Schneider from Helmholtz-Zentrum Berlin. Rectangular samples with typical dimensions of 8 × 5 × 3 mm 3 for 5M, 7 × 5 × 4 mm 3 for 7M and 6 × 4 × 3 mm 3 for NM have been cut out of the crystal bars and have then been transformed to martensite single crystals by mechanical training (Rolfs et al 2012). The transformation under uniaxial compressive stress selects the c-axis. For non-modulated martensite this leads to a single variant crystal in the strict sense. For modulated martensite there remain two perpendicular choices for the modulation vector in the (001) plane, pointing in the [ξ ξ 0] and [ξξ 0] direction (throughout this paper we use the cubic austenite notation). This is what is understood commonly in literature as a martensite single crystal (Shapiro et al 2007).
The chemical composition of each crystal has been determined by energy dispersive x-ray spectroscopy (EDX) with typical compositional changes along the sample below 1%. Table 1 summarizes the characteristics of the martensite crystals used for this paper. In accordance with the phase diagram of the Ni-Mn-Ga system non-modulated martensite is observed for  the stoichiometric composition and compositions nearby. For two samples cut out of the identical austenite single crystal rod different modulations have been found in martensite, namely 5M and 7M, indicating that the phase diagram of the Ni-Mn-Ga system depends on subtle details in composition, the differences of which are beyond the precision of EDX measurements. The crystallographic properties of the 5M single crystal have been investigated in detail by neutron diffraction at the thermal single crystal diffractometer (RESI) at the Heinz Maier-Leibnitz Zentrum (MLZ) in Garching, Germany. Room temperature diffraction measurements have been carried out on an Eulerian cradle with k i = 1.039 Å. Table 2 summarizes the refinement results of the room temperature measurements of the 5M martensite single crystal. The composition is determined directly from the occupancy factors of the corresponding elements. We settle the discrepancy in composition between EDX (Ni 49 Mn 32 Ga 19 )-which convention has been followed by Ener et al (2012)-and diffraction analysis (Ni 50 Mn 29 Ga 21 ) in favour of the diffraction results, because this can be considered as the proper average over the whole crystal. Phase transition temperatures for this composition have been measured by DSC to be M S = 361.0 K, M F = 351.2 K, A S = 357.5 K, A F = 371.2 K and T C = 371.2 K. Figure 1 shows how the peaks observed in the measured (001)-plane of the mechanically trained five-layered martensite phase follow from the existence of two variants as resulting from compressive stress along the [001]-direction. The additional variation of the peak intensities over reciprocal space is due to how neutron scattering samples the Fourier components of the structure and implies that the modulations are of predominantly transversal character.   figure 2(b). The weaker signal of the longitudinal measurement again shows the transversal character of the modulations. For the refinement of the 5M, the superspace approach has been used (Janner and Janssen 1977). A sinusoidal modulation with a commensurate modulation vector of ξ = 0.4 r.l.u. is found for 5M martensite. The observed scattering pattern and the refinement results agree well with existing literature (Righi et al (2007) and Brown et al (2002); note the differing notation conventions for tetragonal descriptions). Similar refinement for measurements at the 7M crystal failed due to overlapping peaks caused by incomplete transformation of the crystal.

Phonon dispersions and discussion
Inelastic neutron scattering experiments were carried out at the thermal three-axis spectrometer (PUMA) and cold three-axis spectrometer (PANDA) at the Heinz Maier-Leibnitz Zentrum in Garching, Germany. For room temperature phonon measurements samples were mounted on a small aluminium sample holder. Phonons are mainly measured in constant k f -mode using the PG(002) monochromator and analyser, both in focusing geometry. Depending on the slope of the dispersion some measurements were done in constant frequency-mode. The temperature dependence of the TA 2 [ξ ξ 0] phonon has been measured in a cryostat with the sample mounted in an aluminium can filled with He to ensure better thermal conductivity. Thus a temperature stability of ±0.5 K has been achieved. This qualitative behaviour will be discussed below. Similar to what has been observed in the austenite phase the TA 2 [ξ ξ 0] phonon branch exhibits a kind of dip in the interval of 0.2 < ξ < 0.4 r.l.u.

Temperature dependent phonons 5M
We first turn to the TA 2 [ξ ξ 0] phonon branch. From previous measurements in the austenite phase (Zheludev et al 1995, 1996, Stuhr et al 1997, Mañosa et al 2001, Ener et al 2012 it is known that the TA 2 [ξ ξ 0] mode shows a pronounced softening when approaching the transition, thereby indicating a low energy barrier for the modulation of the (110)  Once the (pre-)martensite phase is entered the phonons related to the transition harden again with decreasing temperature. Within a modified Ginsburg-Landau approach the square of the frequency of the soft phonon serves as order parameter for the first-order displacive transition (Krumhansl and Gooding 1989). Therefore the TA 2 [ξ ξ 0] mode around ξ = 1 3 can be interpreted as a dynamical precursor of the structural transition. Figure 4 depicts the temperature dependence of the TA 2 [ξ ξ 0]-mode in the martensite phase (coloured symbols). For comparison one measurement has been performed in the austenite phase (open black symbols). Also in the martensite phase the TA 2 [ξ ξ 0]-branch shows a softening between ξ = 0.2 and 0.4 r.l.u, albeit less pronounced than in the austenite. The situation is reversed towards the zone boundary. Here the martensite phonons are lower than the austenite ones. Most important, all of the TA 2 [ξ ξ 0] phonons in the martensite phase soften with increasing temperature towards the transition-see also figure 9(b) by Ener et al (2012)-and thereby anticipate the softened dispersion in the austenite. As these phonons do not participate in the transformation of the modulated martensite, this is all the more surprising.

Phonons reflecting 5M modulation
Now to the apparent fourth acoustic branch in the [ξ ξ 0] measurement. Due to the twovariant state of the crystal our measurement probes both phonon propagations along and perpendicular to the modulation vector. Figure 5 shows, in an iso-intensity plot, the distribution of these low lying transversal excitations as measured in high resolution constant Q-scans  figure 3) and compared to first principle calculations (Entel et al 2008) for stoichiometric tetragonal martensite (lines).
To our best knowledge first principle calculations of the phonon dispersion for the 5M martensite are not available. Such calculations are complicated by the fact that the offstoichiometric concentration and subsequent 5M modulation demands huge super-cells (Gruner and Entel 2009). However, first principle calculations of the non-modulated martensite phase are available for the stoichiometric concentration (Entel et al 2008). Figure 6 compares these calculations with the phonons measured along [ξ ξ 0] propagation. Measurement and ab initio calculation fit remarkably. The energy bands for the optical phonons are caught correctly by the calculations. Also the smoothed dip in the TA 2 [ξ ξ 0] branch is reproduced. Of course the transversal phonons related to the 5M modulation are not captured, as the modulation is not put into the calculation.

Phonons in seven-layer modulated and non-modulated martensite
The sensitivity of the Ni-Mn-Ga alloy system to subtle changes in composition is perfectly illustrated by the fact that one sample cut out of the same rod as the one for the above described 5M martensite transformed during training to 7M martensite. Figure 2(b) demonstrates the perfect commensurable structure, superstructure peaks repeat along [ξ ξ 0] with a periodicity of 2/7 r.l.u. Compared to the case for 5M ( figure 2(a)), the superstructure peaks are broadened. Single crystal diffraction shows that the transformation into 7M is only partial. The contour plot of the inelastic intensity (figure 7) shows a clearly discernible TA 2 [ξ ξ 0] branch. Phonon intensity related to the 7M modulation is evidenced only faintly (dashed line in figure 7). Magnon scattering becomes visible with its steep slope ofhω ∝ q 2 (dotted line in figure 7). Figure 8(a) collects all acoustic phonons measured for [ξ ξ 0] and [0 ξ ξ ] propagation in 7M. Also here the non-equivalence for the two propagation directions is clearly visible and can be understood by the same argument as for 5M.
For Ni 50 Mn 30 Ga 20 the austenite transforms directly to non-modulated martensite. Phonon measurements in the martensite have been performed at room temperature ( figure 8(b)). The precursor dip in the TA 2 -mode almost vanishes like in the off-stoichiometric 5M and 7M martensite. Again phonons have been measured in the two non-equivalent directions [ξ ξ 0] and

Frequency (THz)
[ξξ0] [0ξξ] [0 ξ ξ ]. The stiffening of TA 2 [ξ ξ 0] is in line with the behaviour in the modulated crystals. The optical modes seem to be decoupled from the differences in the acoustic region. Independent from concentration and 5M, 7M or NM they all lie in the same frequency range and show little dispersion with ξ .

Comparison with phonon measurements in literature
While extensive phonon studies of the Ni-Mn-Ga system exist for the austenite and premartensite phase (Zheludev et al 1995, 1996, Stuhr et al 1997, Mañosa et al 2001, Ener et al 2012, measurements of the phonon dispersion in the martensite are scarce, probably due to the difficulty of achieving single variant crystals when passing through the austenite-martensite transition. Indeed only the recent publication by Shapiro et al (2007) reports about phonon measurements in the martensite phase of stoichiometric Ni 2 MnGa. Analogous to the work reported here, they prepared their single crystal by imposing a uniaxial preferred direction, in this case via an external magnetic field of 1.5 T, during cooling into the martensite. As already reported earlier (Zheludev et al 1996) this crystal shows in the martensite an incommensurate superstructure peak along [ξ ξ 0] at ξ = 0.43. This emerging elastic diffuse scattering at incommensurate ξ = 0.43 has also been reported for the Ni 2 MnGa used in this work (see figure 8(a) by Ener et al (2012)). Shapiro et al (2007) report about a TA 2 [ξ ξ 0] branch in the martensite similar to what is shown in figure 5. Additional to this TA 2 -mode they report a kind of new transversal excitation evolving from ξ = 0.43. They argue that the peak at ξ = 0.425 r.l.u. is an elastic charge-density wave (CDW) peak. They consider the inelastic signal emerging from q CDW to be the phason associated with the CDW.
Our experimental data agree very satisfactorily with those of Shapiro et al (2007) (compare their figures 3 and 4 to figures 5 and 6 here). However, our interpretation is fundamentally different. Due to the coexistence of two martensitic variants in our sample (figure 1) we expect our data to show two TA 2 branches superimposed over each other, one corresponding to phonons with a propagation direction along the modulation and one orthogonal to it. Obviously, there is no reason for these two branches to have the same speed of sound. In light of this fact, we come to the conclusion that the observations of Shapiro et al are due to the same phenomenon: their uniaxial magnetic field a priori also favours two possible orientations equally, and if the twinning energy does not suffice in our case to ensure the whole sample to adopt a single martensitic variant, it likely also did not in their case. The different sound velocities, which are their main argument for the phason mode, are then just due to the inequivalent directions of propagation. The only difference that remains is that the nominally stoichiometric crystal used by Shapiro et al (2007) transforms to an incommensurably modulated martensite, whereas our off-stoichiometric crystals display perfect 5M or 7M modulations.

Qualitative comparison
As discussed above, the phonons of the different phases (5M, 7M, NM, austenite) do not differ noticeably for the most parts. Specifically the optical bands occupy similar energy regions. Apart from the obvious differences following from the modulations, the acoustic phonons are affected by the transformation with respect to two aspects: First, the [ξ 00] zone boundary phonons harden significantly compared to the austenite, and the [0 ξ ξ ] and [ξ ξ 0] phonons split in all the martensite phases. This latter point might be explained by a simple consideration: At the transition to any martensite phase, the cubic unit cell is deformed to a c/a-ratio of about 0.94 at nearly constant volume (Brown et al 2002). Therefore the distance between neighbouring (011)planes is shortened compared to the distance between (110)-planes. This gives rise to hardened [0 ξ ξ ] phonons and softened [ξ ξ 0] phonons at the Brillouin zone boundary, both compared to the symmetric austenite.

Summary
Single crystal diffraction of the martensite phase of off-stoichiometric Ni 50 Mn 29 Ga 21 confirm earlier high-resolution powder-and single-crystal neutron diffraction experiments of Brown et al (2002). The martensite is not exactly in tetragonal symmetry as expected but in monoclinic symmetry. The refinement results show that the Ni, Mn and Ga atoms occupy their original positions and excess Mn atoms only occupy Ga sites. The full phonon dispersion of the 5M martensite is measured for each high symmetry direction. The phonon dispersions of the austenite and martensite very much resemble each other. Optical phonons are in the same energy region, despite the structural changes. The acoustic zone boundary modes along [ξ ξ 0] and [0 ξ ξ ] split according to the symmetry breaking at the transition. In the narrow region around TA 2 [ξ ξ 0] with ξ = 0.2-0.4 r.l.u. a pronounced softening is observed when approaching the martensite-austenite transition from above and below, i.e. this soft mode has lowest frequency at the transition temperature. It can be interpreted as a dynamical precursor of the first-order phase transition within a modified Ginsburg-Landau theory (Krumhansl and Gooding 1989).
Both the 5M and 7M crystals show fully commensurate satellite peaks along [ξ ξ 0], reflecting the five-and seven-layered modulation. These superstructure peaks become -points for 'new' acoustic phonons, reflecting the reduced Brillouin zone of the modulated structures.
We explain the apparent coexistence of a seemingly unchanged TA 2 mode and the new lowenergy phonons by the two-variant state of the sample. This also provides an explanation for the earlier results by Shapiro et al (2007) without invoking phason modes associated with a charge-density wave.