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Adaptive modulation in Ni2Mn1.4In0.6 magnetic shape memory Heusler alloy

The origin of incommensurate structural modulation in Ni-Mn based Heusler type magnetic shape memory alloys (MSMAs) is still an unresolved issue inspite of intense focus on this due to its role in the magnetic field induced ultra-high strains. In the archetypal MSMA Ni2MnGa, the observation of non-uniform displacement of atoms from their mean positions in the modulated martensite phase, premartensite phase and charge density wave as well as the presence of phason broadening of satellite peaks have been taken in support of the electronic instability model linked with a soft acoustic phonon. We present here results of a combined high resolution synchrotron x-ray powder diffraction (SXRPD) and neutron powder diffraction (NPD) study on Ni2Mn1.4In0.6 using (3+1)D superspace group approach, which reveal not only uniform atomic displacements in the modulated structure of the martensite phase with physically acceptable ordered magnetic moments in the antiferromagnetic phase at low temperatures but also the absence of any premartensite phase and phason broadening of the satellite peaks. Our HRTEM studies and first principles calculations of the ground state also support uniform atomic displacements predicted by powder diffraction studies. All these observations suggest that the structural modulation in the martensite phase of Ni2Mn1.4In0.6 MSMA can be explained in terms of the adaptive phase model. The present study underlines the importance of superspace group analysis using complimentary SXRPD and NPD in understanding the physics of the origin of modulation as well as the magnetic and the modulated ground states of the Heusler type MSMAs. Our work also highlights the fact that the mechanism responsible for the origin of modulated structure in different Ni-Mn based MSMAs may not be universal and it must be investigated thoroughly in different alloy compositions.

moments in the antiferromagnetic phase at low temperatures but also the absence of any premartensite phase and phason broadening of the satellite peaks. Our HRTEM studies and first principles calculations of the ground state also support uniform atomic displacements predicted by powder diffraction studies. All these observations suggest that the structural modulation in the martensite phase of Ni 2 Mn 1. 4 In 0.6 MSMA can be explained in terms of the adaptive phase model.
The present study underlines the importance of superspace group analysis using complimentary SXRPD and NPD in understanding the physics of the origin of modulation as well as the magnetic and the modulated ground states of the Heusler type MSMAs. Our work also highlights the fact that the mechanism responsible for the origin of modulated structure in different Ni-Mn based MSMAs may not be universal and it must be investigated thoroughly in different alloy compositions.
Magnetic shape memory Heusler alloys (MSMAs) in the Ni-Mn-X (X= Ga, In, Sn) system have enormous potential for technological applications due to a rich variety of properties ranging from generation of extremely large magnetic field induced strain (~10%) to pronounced magnetocaloric and barocaloric effects, large magnetoresistance, anomalous Hall effect and large exchange bias [1][2][3][4][5][6] . The technologically significant physical properties of these alloys are intimately linked with the coupling between structural and magnetic degrees of freedom below the magnetostructural (martensite) phase transition. Since the modulated crystal structure of martensite phase due to its low detwinning stress is known to play an important role in deciding the response of these alloys to the external magnetic field, there is currently a lot of interest in understanding the origin of the modulated structure of the martensite phase itself [7][8][9][10][11][12][13][14][15][16][17] . Two different models have been proposed in the literature for the origin of modulation in the MSMAs. The first one is the 3 adaptive phase model in which the modulated structure is considered as a nanotwinned state of the Bain distorted phase, which maintains the invariance of the habit plane between the high temperature austenite and the low temperature martensite phases 14,15 . Commensurate modulated structure, uniform atomic displacements and absence of any premartensite phase transition as well as phason strains are the key manifestations of this model. The second model, based on charge density wave (CDW) coupled to a soft transverse acoustic (TA2) mode 18-20 , can, on the other hand, explain non-uniform atomic displacements, incommensurate nature of modulation and existence of premartensite phase as well as phasons 10-12, 21, 22 . It has been proposed that the formation of discommensurations in the form of stacking faults and antiphase boundaries can in principle result into an average incommensurate modulated structure even for the adaptive phase model [14][15][16] . However, one of the key features for distinguishing between the two models for the origin of modulation is the identification of the nature (uniform versus non-uniform) of the atomic displacements.
Recently we have shown that the origin of modulation in Ni 2 MnGa shape memory Heusler alloy cannot be explained within the framework of adaptive phase model as the modulated structure has non-uniform atomic displacements 12 . Also the fact that the incommensurate martensite phase results from an incommensurate premartensite phase and not directly from the austenite phase does not support the adaptive phase model 11 . The presence of phasons 23 and the broadening of the superlattice peaks due to phason strains 12 in Ni 2 MnGa also goes against the concept of adaptivity. On the other hand, the Ni-Mn-In MSMAs do not exhibit the premartensite (precursor) phase formation as the austenite phase transforms directly to the martensite phase. This suggests that these alloys may be model systems for investigating the applicability of the adaptive phase model for structural modulation through a careful analysis of the structure of the martensite 4 phase. Here, we present results of Rietveld analysis of high resolution synchrotron x-ray powder diffraction (SXRPD) and neutron powder diffraction (NPD) data on a Ni-Mn-In alloy, Ni 2 Mn 1.4 In 0.6 , using (3+1) D superspace group approach. Our analysis reveals that the modulated structure of the martensite phase of Ni 2 Mn 1.4 In 0.6 involves uniform displacements of atoms with respect to their positions in the Bain distorted basic cell. This conclusion is well supported by high-resolution transmission electron microscopic studies as well as first principles calculations of the ground state of these alloys.. All these observations comprehensively rule out the applicability of the CDW based soft mode model and support the adaptive modulation model for  Fig.1a. It reveals a sharp jump in magnetization at the Curie temperature T c~ 315 K due to a ferromagnetic (FM) transition, followed by a decrease in magnetization at the first order martensite transition temperature T M~2 95 K. The bifurcation of the ZFC and FCC curves below T~145 K is in agreement with earlier reports 3 where it has been attributed to the coexistence of antiferromagnetic and ferromagnetic exchange interactions 5 in the martensite phase 37 . Our isothermal magnetization M(H) plot at 2 K (see Fig. 1b) indicates typical antiferromagnetic ground state with a spin flop transition occurring at a very low magnetic field of ±0.05 T leading to opening up of double hysteresis loop above this field. The fact that the spin flop transition occurs at such a low field suggests that both the FM and AFM states are nearly degenerate in the martensite phase, even though the ground state is dominated by AFM interactions.
We now turn towards the structure of the austenite and martensite phases using SXRPD patterns recorded at 350 K (austenite phase) and 235 K (martensite phase), respectively.
In the first step of the structure analysis, we performed indexing of the powder diffraction patterns by LeBail technique, which refines the unit cell parameters and profile broadening functions to obtain the best fit between the observed and calculated profiles in the least squares sense for a given space group. At 350 K, all the observed Bragg peaks could be indexed well with the cubic austenite structure (space group Fm-3m) and the refined lattice parameter is found to be 6.00483(4) Å. The presence of the superstructure peaks like (111) and (200) in the SXRPD pattern (inset of Fig.2a) confirms that the structure corresponds to the ordered L2 1 type 38 . At 235 K, many more reflections appear and the cubic austenite peaks split into two or more peaks clearly indicating a non-cubic structure. A careful analysis of all the observed low intensity peaks revealed that the martensite structure at 235 K cannot be explained in terms of a simple Bain distorted unit cell and requires consideration of modulation of the Bain distorted unit cell as reported in other MSMAs 7,8,11,39,40 . Superspace (3+1) D formalism 41-44 is a powerful tool to investigate such complex modulated structures and we employed this formalism to investigate the structure of the modulated martensite phase in Ni 2 Mn 1.4 In 0.6 . Following the superspace group formalism, the SXRPD pattern was divided into two sets of reflections: (1) main reflections 6 corresponding to the Bain distorted basic structure and (2) satellite reflections due to the modulation whose intensity is in general much less than the intensity of the main reflections. All the main reflections corresponding to the basic structure could be indexed with a monoclinic cell with space group I2/m and Le-Bail refinement gave us lattice parameters as a=4.3983(1) Å, b= 5.6453(2) Å, c= 4.3379(1) Å and β= 92.572(2) 0 . After obtaining the cell parameters for the basic structure, the full SXRPD pattern including both the main and the satellite reflections was considered for Le-Bail refinement using the superspace group formalism. The satellite reflections were indexed using a modulation wave vector q= (0, 0, 1/3) and superspace group I2/m(α0γ)00.
Although this commensurate wave vector could index many of the satellite reflections some of the calculated satellite reflections were found to be shifted away from the observed reflection positions, as can be clearly seen in the inset of Fig.2b. Therefore the wave vector q was allowed to be refined and an excellent match between the observed and calculated profiles was obtained for an incommensurate modulation wave vector q= 0.35987(8) c*= (1/3+δ) c* (where δ=0.02653 is the degree of incommensuration), including those which could not be accounted for using the commensurate wave vector q=1/3 (see Fig.2c). This indicates that the martensite phase of Further, the deviation of atoms ̅ ) from the average structure due to the modulation was modelled using a harmonic atomic modulation function: where and are the Fourier amplitudes of the displacement modulation of the j th atom while "n"is the order of the Fourier series, which is taken as equivalent to the highest order of the satellite reflections observed 47 which is n=2 in the present case. In the Rietveld refinement, the amplitudes of the atomic modulation function were refined without any constraints for different atomic sites as per the non-uniform displacement model used in the refinements of Ni 2 MnGa 12 system. While this refinement yields reasonable fit between the observed and the calculated peak profiles (see Fig.3a), the calculated interatomic distances are physically unrealistic for the Ni- To summarise, we have critically evaluated the applicability of two existing models (electronic instability and the nano-twinning based adaptivity models) for the origin of modulation in       To obtain the low temperature martensitic phase, a N2-cooled sample holder was used. Highresolution TEM data were analyzed by using the Digital Micrograph (DM) software (Gatancompany, USA). The high-resolution TEM images were filtered by the ASBF filter method using a script in DM created by Dave Mitchell based on the work of R. Kilaas [3] where the noisy background is subtracted. Visualization of the models was performed with the VESTA 3 software [4] Electron diffraction patterns were simulated with the program JEMS (version:3.5930U2010) [5].

(I) Structural parameters obtained from the Rietveld refinement of SXRPD data considering non-uniform displacement (electronic instability model):
The refined structural parameters (atomic positions and amplitude of modulation) obtained from the Rietveld refinements of SXRPD data using non-uniform atomic displacement model are shown in Table S1. As in this model the amplitudes of the atomic modulation function were refined without any constraints for different atomic sites therefore the amplitudes (A 1 , B 1 , A 2 , B 2 ) are shown separately for each atom. Table S1: Atomic positions (x, y, z), amplitudes (A 1 ,B 1 , A 2 ,B 2 ) of the modulation function, and atomic displacement parameter (U iso ) of the modulated martensite phase of Ni 2 Mn 1.4 In0.6 obtained from the Rietveld refinement using SXRPD data considering non-uniform displacement model.

(II)
Structural parameters obtained from the Rietveld refinement of SXRPD data considering distance constraint:   After confirming the absence of any descernible anti-site disorder from the analysis of the RT neutron powder diffraction, we proceed to discuss the refinement of the structure of the martensite phase. The neutron diffraction data for the martensite phase was collected at the lowest possible temperature (3K). To investigate the modulated structure, we employed superspace (3+1) D formalism as for the SXRPD data. In the first step of Rietveld refinement, the refinement was carried out without any constraints on the amplitude or direction of atomic displacements for the atomic modulation functions of the different atoms, similar to that used for the analysis of the SXRPD pattern. The observed and calculated peak profiles are shown in Fig.   S2 and corresponding refined structural parameters are given in Table S3.
It is interesting to note that while the refinement converged for a non-uniform atomic displacement model, but it led to unreasonable interatomic distances, as shown in Fig.S4a for some selected atomic pairs obtained using powder neutron diffraction. The interatomic distances given in Fig.S4a obtained from a non-uniform displacement model clearly indicates that the use of powder neutron diffraction also cannot resolve the issue of implausible interatomic distances and the problem lies with the modulation model itself, in agreement with SXRPD analysis. In the next step, we therefore considered a uniform displacement model for Rietveld refinement (Fig.S3) in which the amplitude of modulation for all the atomic sites was constrained to be identical (Table S4). The derived interatomic distances from the uniform atomic displacement model shown in the Fig.S4b clearly reveals that this model gives physically realistic interatomic distances that are acceptable for the shape memory Heusler compounds/alloys. Thus, our results based on both PND and SXRPD data reveal that the modulation in the martensite phase of Ni 2 Mn 1.4 In 0.6 involves uniform displacement of atoms and is, therefore, consistent with the predictions of the adaptive modulation model.    values that are expected for these kind of intermetallic compounds/alloys.

Rational approximant 3D superstructures obtained from the (3+1) D analysis of neutron diffraction data:
It is possible to obtain the 3D superstructure from the analysis of diffraction data using (3+1) D superspace group. The refined structural parameters for the 3D rational approximant superstructures, which are obtained from the (3+1) D superspace group analysis of neutron powder diffraction data at 3K are shown in Table S5 for non-uniform displacement (electronic instability) model and in Table S6 for uniform displacement (adaptive phase) model.

(V) High resolution Transmission electron microscopy (HRTEM):
To further support our results obtained from the diffraction studies about the uniform displacement model as origin of modulation in Ni 2 Mn 1.4 In 0.6, we performed HRTEM studies.   In order to verify the results of Rietveld refinements and HRTEM studies, we also performed first-principles calculations for martensite phase for both adaptive phase model and non-uniform displacement modulation model using the fully relativistic Green function SPR-KKR package [6] and employing the CPA alloy theory to account for the chemical Mn-In disorder. The exchange-correlation potential was treated within the conventional local density approximation [7]. For first-principles calculations 3D rational approximant superstructures for both non-uniform displacement (Table S5) and adaptive phase (Table S6) models were used.
For non-uniform displacement model, many quantities (such as the Fermi energy, local magnetic moments, etc.) appeared to be unrealistic. In particular, the total energy appears to be