Neutron diffraction and symmetry analysis of the martensitic transformation in Co-doped ${\mathrm{Ni}}_2\mathrm{MnGa}$

Martensitic transformations are strain driven displacive transitions governing the mechanical and physical properties in intermetallic materials. This is the case in ${\mathrm{Ni}}_2\mathrm{MnGa}$, where the martensite transition is at the heart of the striking magnetic shape memory and magnetocaloric properties. Interestingly, the martensitic transformation is preceded by a premartensite phase, and the role of this precursor and its influence on the martensitic transition and properties is still a matter of debate. In this work we report on the influence of Co doping (${\mathrm{Ni}}_{50-x}{\mathrm{Co}}_x{\mathrm{Mn}}_{25}{\mathrm{Ga}}_{25}$ with $x=3$ and 5) on the martensitic transformation path in stoichiometric ${\mathrm{Ni}}_2\mathrm{MnGa}$ by neutron diffraction. The use of the superspace formalism to describe the crystal structure of the modulated martensitic phases, joined with a group theoretical analysis, allows unfolding the different distortions featuring the structural transitions. Finally, a general Landau thermodynamic potential of the martensitic transformation, based on the symmetry analysis, is outlined. The combined use of phenomenological and crystallographic studies highlights the close relationship between the lattice distortions at the core of the ${\mathrm{Ni}}_2\mathrm{MnGa}$ physical properties and, more in general, on the properties of the martensitic transformations in the Ni-Mn based Heusler systems.

Figure 1

(a) Crystal structure of the austenite phase in the Heusler alloys ${\mathrm{X}}_2\mathrm{YZ}$ (X blue, Y purple, Z gray). (b) Example of the tetragonal strain effect on the coordination environment the Z (or X) sites that resemble the Jahn-Teller effect in perovskites. (c) Example of shuffle that resemble the octahedral tilting in perovskite oxides.

Figure 2

Magnetization measurements of (a) ${\mathrm{Ni}}_{47}{\mathrm{Co}}_3{\mathrm{Mn}}_{25}{\mathrm{Ga}}_{25}$ and (b) ${\mathrm{Ni}}_{45}{\mathrm{Co}}_5{\mathrm{Mn}}_{25}{\mathrm{Ga}}_{25}$ measured in an applied field of 5 mT following a zero-field-cooled (ZFC), field-cooled (FC), and field-cooled-warming (FCW) procedure.

Figure 3

Rietveld plot of the austenitic phase at 600 K in the $Fm\overline{3}m{1}^{\prime }$ space group for the ${\mathrm{Ni}}_{47}{\mathrm{Co}}_3{\mathrm{Mn}}_{25}{\mathrm{Ga}}_{25}$ compound. Observed (x, black), calculated (red line), and difference (blue line) patterns are shown. The tick marks indicate the Bragg reflection positions. The agreement factors are $R_p=3.8\%$ and $R_{\text{wp}}=3.9\%$. The second and third tick-marks rows indicate the reflections from MnO and V, respectively.

Figure 4

Rietveld plot of the ferromagnetic austenitic phase at 295 K in the $I4/m{m}^{\prime }{m}^{\prime }$ magnetic space group for the (a) ${\mathrm{Ni}}_{47}{\mathrm{Co}}_3{\mathrm{Mn}}_{25}{\mathrm{Ga}}_{25}$ and (b) ${\mathrm{Ni}}_{45}{\mathrm{Co}}_5{\mathrm{Mn}}_{25}{\mathrm{Ga}}_{25}$ composition. Observed (x, black), calculated (red line), and difference (blue line) patterns are shown. The tick marks indicate the Bragg reflection positions of the main phase (first row), MnO, and V (second and third row, respectively). The agreement factors are $R_p=4.8\%$ and $R_{\text{wp}}=3.9\%$ and $R_p=3.7\%$ and $R_{\text{wp}}=3.8\%$. The inset shows the evolution of the refined magnetic moment as a function of temperature pointing out the continuous character of the transition, the red line shows the best fit of the data with a power law returning $T_C^A$ of 431(3) K with critical exponent $\beta =0.24\left(3\right)$ for ${\mathrm{Ni}}_{47}{\mathrm{Co}}_3{\mathrm{Mn}}_{25}{\mathrm{Ga}}_{25}$ and $T_C^A$ of 449(7) K and $\beta =0.25\left(4\right)$ for ${\mathrm{Ni}}_{45}{\mathrm{Co}}_5{\mathrm{Mn}}_{25}{\mathrm{Ga}}_{25}$.

Figure 5

Top: Rietveld plot of the first martensitic phase at 130 K in the $I{m}^{\prime }m{m}^{\prime }\left(00\gamma \right)s00$ magnetic superspace space group for the ${\mathrm{Ni}}_{47}{\mathrm{Co}}_3{\mathrm{Mn}}_{25}{\mathrm{Ga}}_{25}$ composition. The agreement factors are $R_p=5.4\%$ and $R_{\text{wp}}=6.8\%$. The inset shows a zoom of the diffraction pattern highlighting some satellite reflections. Bottom: Rietveld plot of the second martensitic phase at 5 K in the $I{m}^{\prime }m{m}^{\prime }\left(00\gamma \right)s00$ magnetic superspace space group for the ${\mathrm{Ni}}_{47}{\mathrm{Co}}_3{\mathrm{Mn}}_{25}{\mathrm{Ga}}_{25}$ composition, The agreement factors are $R_p=4.04\%$ and $R_{\text{wp}}=5.05\%$. The inset shows a zoom of the diffraction pattern highlighting some satellite reflections. In both panels observed (x, black), calculated (line, red), and difference (line, blue) patterns are reported. The first tick-marks row indicates the Bragg reflections position of the martensite phase with the main reflections in black and the first order satellite reflections in green. The second and third tick-marks rows indicate the reflections from the MnO magnetic impurity and V, respectively.

Figure 6

(a) Average nuclear and magnetic structure for the 3% Co sample in the martensitic phase at 5 K. (b) Modulated structure at 5 K in the martensite structure, the solid line shows the average unit cell and the dashed lines are a guide for the eye to appreciate the sinusoidal modulation along the $a$ direction. (c) Plot of the sinusoidal displacement along $x$ as a function of the internal coordinate $x_4$.

Figure 7

Temperature evolution of the cell parameter (top) and $\gamma$ component of the modulation vector across the two martensitic transformations of the ${\mathrm{Ni}}_{47}{\mathrm{Co}}_3{\mathrm{Mn}}_{25}{\mathrm{Ga}}_{25}$ compound. The dashed lines indicate the two transition temperatures.

Figure 8

Rietveld plot of the martensitic phase at 5 K in the $I{m}^{\prime }m{m}^{\prime }\left(00\gamma \right)s00$ magnetic superspace space group for the ${\mathrm{Ni}}_{45}{\mathrm{Co}}_5{\mathrm{Mn}}_{25}{\mathrm{Ga}}_{25}$ composition. Observed (x, black), calculated (line, red), and difference (line, blue) patterns are reported. The first tick-marks row indicates the Bragg reflections position of the martensite phase with the main reflection in black and the satellite reflection in green. The second and third tick-marks rows indicate the reflection from the MnO magnetic impurity and V, respectively. The agreement factors are $R_p=3.7\%$ and $R_{\text{wp}}=6.5\%$. The inset shows a zoom of the diffraction patter highlighting some satellite reflections around the 111 reflection.

Figure 9

Temperature evolution of the cell parameters (top) and $\gamma$ component of the modulation vector across the martensitic transformation of the ${\mathrm{Ni}}_{45}{\mathrm{Co}}_5{\mathrm{Mn}}_{25}{\mathrm{Ga}}_{25}$ compound. The dashed line indicates the $T_{\text{II}}$ transition temperature.

Figure 10

Mode decomposition analysis of the ${\mathrm{Ni}}_{50-x}{\mathrm{Co}}_x{\mathrm{Mn}}_{25}{\mathrm{Ga}}_{25}$ samples. In the top panels is shown the total amplitude of the ${\mathrm{\Sigma }}_2$ displacive mode representative of the sinusoidal displacement of the atoms in the martensite states, the red line represents the best fit with a critical law $\propto {(T-T_C)}^{\beta }$ see text for details. The bottom panel shows the evolution across the martensitic transition of the tetragonal strain ${\mathrm{\Gamma }}_3^{+}$ (left axis) and of the orthorhombic strain ${\mathrm{\Gamma }}_5^{+}$ (right axis), the red line represent a linear fit of the ${\mathrm{\Gamma }}_3^{+}$ strain close to the $T_{\text{I}}$ transition temperature. The dashed lines indicate the critical temperatures $T_{\text{I}}$ and $T_{\mathrm{II}}$.

Figure 11

Sketch of the magnetic structures derived from the parent $Fm\overline{3}m{1}^{\prime }$ symmetry, corresponding to the action of the $m{\mathrm{\Gamma }}_4^{+}$ irreps having different order parameter direction and magnetic anisotropy (see Table 1). The purple planes in the monoclinic magnetic space groups indicate the anisotropy plane.