Strain behavior and lattice dynamics in Ni₅₀Mn₃₅In₁₅

In combination, the magnetization and RUS data allow a straightforward comparison of the strength of strain coupling associated with each transition. Clearly, the dominant changes in the shear modulus are connected with the large shear strains associated with the martensitic transition. The very small anomaly in f ² at $T_{\text{C}}^{\text{A}}$ implies only weak coupling of the ferromagnetic order parameter with shear strain, and, similarly, for the magnetic ordering at $T_{\text{C}}^{\text{M}}$ there is little or no deflection in the trend of f ² which might indicate any significant magnetoelastic coupling below the second magnetic ordering transition.

The martensitic phase below ${{T}_{\text{M}}}/{{T}_{\text{A}}}$ in a sample with a close composition (Ni₅₀Mn_50−xIn_x at x  =  15.2) is known to consist of a mixture of 10 M and 14 M structures [22], each of which has substantial shear strains with respect to the parent cubic structure. The magnitudes of these at room temperature can be estimated using lattice parameter data for the 10 M structure given by Khovaylo et al [22]: a  =  4.377, b  =  5.564, c  =  21.594 Å, $\beta ={{91.93}^{{}^\circ}}$. If orthogonal reference axes, X, Y and Z, are chosen as being parallel to the crystallographic a, b and ${{c}^{*}}$, respectively, the non-zero strain components with respect to a cubic structure with lattice parameter a_o are given by

$$\begin{eqnarray}&&{{e}_{1}}=\frac{a-{{a}_{o}}}{{{a}_{o}}}\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}\,\,\,\,\,\,\,\,\,\begin{array}{*{35}{l}} {{e}_{2}} & =\frac{b/\sqrt{2}-{{a}_{o}}}{{{a}_{o}}} \\ \end{array}\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}\,\,\,\,\begin{array}{*{35}{l}} {{e}_{3}} & \approx \frac{c/5-{{a}_{o}}}{{{a}_{o}}} \\ \end{array}\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{{e}_{5}}\approx \cos \beta \end{eqnarray} \tag{ 4 }$$

where an approximation for the value of a_o is

$$\begin{eqnarray}&&{{a}_{o}}\approx {{\left(\frac{a\,\cdot\, b\,\cdot\, c}{5\sqrt{2}}\right)}^{1/3}}.\end{eqnarray} \tag{ 5 }$$

In symmetry-adapted form, tetragonal and orthorhombic strains are given by

$$\begin{eqnarray}&&{{e}_{t}}=\frac{1}{\sqrt{3}}\left(2{{e}_{3}}-{{e}_{1}}-{{e}_{2}}\right)\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{{e}_{o}}=\left({{e}_{1}}-{{e}_{2}}\right).\end{eqnarray} \tag{ 7 }$$

On this basis, values of the room temperature shear strains are e_t  =  0.036, e_o  =  0.091, e₅  =  −0.034. As is well known for martensitic transitions these are large in comparison with values in the range 1–3% for more typical ferroelastics, and would be expected to give rise to very substantial changes in the shear elastic constants according to the classical model of strain/order parameter coupling described originally by Slonczewski and Thomas [23].

The overall behavior requires an understanding of both the order parameters involved in the phase transitions and the nature of their coupling with strain, as set out here.

There are three order parameters to consider but, as already noted, coupling between the magnetic order parameter(s) and shear strain is evidently weak. This leaves two order parameters driven by changes in electronic structure, the first of which, Q₁, would give rise to the symmetry change Fm$\bar{3}$m–I4/mmm and the second, Q₂, is responsible for the multiple repeat of the 10 M structure. Q₁ has the symmetry properties of the irreducible representation $\Gamma\text{ }_{3}^{+}$ and, if the complication of incommensurate ordering is ignored, Q₂ would have symmetry properties corresponding to a point away from the Brillouin zone center along the $k=\left[\xi ,\xi ,0\right]$ line. The essential point is that Q₁ would couple bilinearly with a tetragonal strain ($\lambda eQ$) to give pseudoproper ferroelastic behaviour and Q₂ would have linear-quadratic coupling ($\lambda e{{Q}^{2}}$) to give an improper ferroelastic transition. With regard to the elastic constants, the former would give characteristic softening of (${{C}_{11}}-{{C}_{12}}$) as T approaches the martensitic transition point from above and below while the latter would give a step-like softening below the transition temperature [24]. The shear modulus of a cubic crystal comprises of both (${{C}_{11}}-{{C}_{12}}$) and C₄₄, but there is no real indication in the data for f ² in figure 3 of significant pseudoproper ferroelastic softening with cooling towards the martensitic transition point from the stability field of the cubic phase. It is therefore concluded that the martensitic transition is driven primarily by Q₂ and is essentially improper ferroelastic. Rather than there being an abrupt softening below the transition point, however, there is an apparently continuous and large stiffening. From this it is clear that any contributions from relaxations due to $\lambda e{{Q}^{2}}$ coupling must be small, where e could be any of the three shear strains, and the overall elastic properties are determined by the next higher order (biquadratic) terms, $\lambda {{e}^{2}}{{Q}^{2}}$. The amount of elastic stiffening with respect to the cubic parent structure will then scale with $Q_{2}^{2}$. The transition is first order in character, so the apparently continuous variation of the effective shear modulus is most likely to be a consequence of averaging the elastic constants of the two (or more) coexisting structures in the close vicinity of the transition point.

Elastic softening due to coupling of the form ($\lambda e{{Q}^{2}}$) requires that at the time scale of some applied stress there is a relaxation of the strain and, consequently, of the order parameter. It is inevitable that there must be some frequency above which an approach to equilibrium elastic properties is not observed because there is insufficient time for these relaxations to occur. In this case the contributions of higher order coupling of the form $\lambda {{e}^{2}}{{Q}^{2}}$ might still be detected, but such an effect might normally be expected to be small. The data in figure 3 show stiffening of the shear modulus by over 100%, however, indicating that the biquadratic coupling effect is dominant. Similar stiffening has been observed in association with the martensitic transition in single crystal of Cu_74.08Al_23.13Be_2.79 both at 0.1 MHz and 0.25–8 Hz [25], implying that the issue is not simply a matter of dispersion with respect to frequency, but this is not a group/subgroup transition and the elastic stiffening was attributed to pinning of twin walls by dislocations. Substantial stiffening in an entirely unrelated material, SrAl₂O₄, also has a pattern that appears to be due to $\lambda {{e}^{2}}{{Q}^{2}}$ coupling [26], and is most unlikely to have been affected by dislocations. The common feature is that the hexagonal-monoclinic transition in SrAl₂O₄ also involves a combination of two order parameters, one of which belongs to a zone center irreducible representation and is responsable for pseudoproper ferroelastic character, while the other belongs to a zone boundary irreducible representation [24, 27]. The two non-zero shear strains have values up to about 1% and 6%. We therefore speculate that the strains or the combination of twin walls from the two order parameters with different symmetry lock together in both Ni₅₀Mn₃₅In₁₅ and SrAl₂O₄ in such a way that strain/order parameter relaxation is suppressed. The outcome could then be a substantial increase in rigidity, i.e. the structures become elastically stiffer rather than softer.

With respect to elastic relaxation behavior at the ferromagnetic transition, the onset of elastic softening occurs more or less at $T_{\text{C}}^{\text{A}}\approx 313$ K. The effect is too subtle to be seen in relatively noisy ultrasonic data obtained from a single crystal of Ni₅₀Mn₃₄In₁₆ by Moya et al [9], but a similar pattern of softening occurs in (${{C}_{11}}-{{C}_{12}}$) below the equivalent ferromagnetic transition and ahead of the premartensite transition in a single crystal with composition close to stoichiometric Ni₂MnGa [28, 29]. Seiner et al reported that the lattice geometry remains cubic in this temperature interval, again signifying that direct magnetoelastic coupling is weak. The form of the expected elastic anomaly should follow from coupling of the form $\lambda e{{m}^{2}}$ and would be a stepwise softening at $T=T_{\text{C}}^{\text{A}}$. Instead, there is continuously increasing softening with respect the trend extrapolated from above $T=T_{\text{C}}^{\text{A}}$. Again in the absence of contributions from relaxations due to $\lambda e{{m}^{2}}$, the next high order terms have the form $\lambda {{e}^{2}}{{m}^{2}}$ and softening would scale with m², as perhaps is the case. Softening rather than stiffening implies that the coupling coefficient has opposite sign (and is much smaller) in comparison with stiffening below the martensitic transition temperature.

Although Moya et al [9] obtained data over a relatively small temperature interval, 200–360 K, above the martensitic transition, there is close agreement with the trend of only slight softening of (${{C}_{11}}-{{C}_{12}}$) at $T\to {{T}_{\text{M}}}$. The Voigt-Reuss average, ${{G}_{\text{RV}}}$, value of the shear modulus obtained from their single crystal measurements at room temperature is 42 GPa. Using this value to calibrate the f ² data in figure 3, allows absolute values of the shear modulus to be estimated over the complete temperature range of the RUS data, as depicted on the right axis of figure 3, emphasizing the large change in shear stiffness which arises as a consequence of the martensitic transition.

The variation of Q⁻¹ in figure 5 has a pattern which is typical of acoustic loss accompanying a ferroelastic phase transition [30]. The conventional explanation would be of a steep increase at the transition point associated with the appearance of twin walls which are mobile under application of an external stress. In the austenite phase there is no twinning or any other significant damping mechanism present, in contrast with the martensite phase (e.g. [31]). The plateau of Q⁻¹ below ${{T}_{\text{M}}}/{{T}_{\text{A}}}$ is due to thermally activated mobility of the twin walls in an effectively viscous medium. At some lower temperature the twin walls become pinned by defects in a freezing interval which is readily identifiable by the development of a Debye peak in the loss and an increase in stiffness with respect to shear, as appears to occur at ${{T}_{\text{B}}}$. The loss peak can be fit according to [32]

$$\begin{eqnarray}&&{{Q}^{-1}}(T)=Q_{\text{B}}^{-1}{{\left[\cosh \left\{\frac{{{E}_{a}}}{{{r}_{2}}(\beta )}\left(\frac{1}{T}-\frac{1}{{{T}_{\text{B}}}}\right)\right\}\right]}^{-1}}\end{eqnarray} \tag{ 8 }$$

where $Q_{\text{B}}^{-1}$ is the maximum of the peak at ${{T}_{\text{B}}}$, E_a is the activation energy and ${{r}_{2}}(\beta )$ is a width parameter, which arises from any spread in relaxation times for the dissipation processes. The inset of figure 5 shows the results of a fit to Q⁻¹(T) from the peak in the RUS spectra near 0.43 MHz. After subtracting a baseline, the fitting gives $Q_{\text{B}}^{-1}=0.009$, ${{E}_{a}}/{{r}_{2}}(\beta )=31$ kJ mol⁻¹, and ${{T}_{\text{B}}}=177.8$ K. If the loss mechanism is assumed to be due to a single relaxation process [${{r}_{2}}(\beta )=1$] related to displacements of domain walls, the Debye peak in Q⁻¹ signifies the freezing of the domain-wall movement with a thermal activation energy of about 31 kJ mol⁻¹ (0.3 eV). The peak in Q⁻¹ occurs at $\omega \tau =1$, where ω is the angular frequency ($=2\pi f$) and

$$\begin{eqnarray}&&\tau ={{\tau}_{o}}\exp \left(\frac{{{E}_{a}}}{RT}\right).\end{eqnarray} \tag{ 9 }$$

Using the fit values of ${{T}_{\text{B}}}$ and E_a and $f=5.65\times {{10}^{5}}$ Hz then gives an estimate for the inverse attempt frequency as ${{\tau}_{o}}=2.2\times {{10}^{-16}}$ s. Variations of Q⁻¹ below the Debye loss peak are noisy but decrease steeply with falling temperature.

The RUS data are consistent with a previous report of pinning of twin walls below  ∼170 K in Ni–Mn–Ga martensite [33], which is comparable with the freezing behavior found here at ${{T}_{\text{B}}}\approx 178$ K. Discussions of twin wall mobility in the 10 M phase of Ni–Mn–Ga alloys have focused on the stress felt by two different types of twin walls under the influence of an externally applied magnetic field [34–36]. Of these, one has a thermally activated mechanism for sideways displacements in response to the field, and the mechanism is understood to involve nucleation and subsequent migration of ledges along the walls. An activation energy barrier of 0.15–0.30 eV has been reported to give good agreement with observations [34], and the present result is consistent with this. Internal friction measurements at 2 Hz of Ni–Mn–Ga alloys with unspecified superstructure types gave activation energy values in the range 0.02–0.04 eV, however [37]. The pinning mechanisms are not understood but the higher value could imply a mechanism involving interaction with impurity atoms [37].

A freezing process described by equation (8) must be accompanied by elastic stiffening according to the Debye equations in the usual way. The relationships required in this context are

$$\begin{eqnarray}&&\tan \delta =\Delta \text{ }\frac{\omega \tau}{1+{{\omega}^{2}}{{\tau}^{2}}}\end{eqnarray} \tag{ 10 }$$

where δ is the phase angle, and, in the case of a standard linear solid

$$\begin{eqnarray}&&\Delta \text{ }=\frac{{{C}_{\text{U}}}-{{C}_{\text{R}}}}{{{C}_{\text{R}}}}~\text{for}~\left({{C}_{\text{U}}}-{{C}_{\text{R}}}\right)\ll {{C}_{\text{R}}}.\end{eqnarray} \tag{ 11 }$$

${{C}_{\text{U}}}$ is the relevant elastic modulus for the unrelaxed state, excluding strain due to movement of the wall, and ${{C}_{\text{R}}}$ the modulus of the relaxed state, including the strain due to movement of the wall to its new equilibrium position [38]. The relationship between $\tan \delta$ and Q⁻¹ is [39–41]

$$\begin{eqnarray}&&\tan \delta \approx \frac{1}{\sqrt{3}}\frac{\Delta \text{ }f}{f}=\frac{1}{\sqrt{3}}{{Q}^{-1}}.\end{eqnarray} \tag{ 12 }$$

On this basis, and using $Q_{\text{B}}^{-1}=0.009$ at $T={{T}_{\text{B}}}$ ($\omega \tau =1$), the expected change in f ² at T_B in figure 3 is $0.033\times {{10}^{11}}$ Hz² which is sufficiently close to the observed change, $0.06\times {{10}^{11}}$ Hz², to confirm that both anomalies could be due simply to freezing of the twin wall motion. The nature of the pinning mechanism is not known. It may only be coincidence that it occurs just below $T_{\text{C}}^{\text{M}}\approx 200$ K, but by comparing Q⁻¹(T) with the ZFC magnetization curve, we see that the decrease in the acoustic loss goes along with a decrease in the ZFC magnetization. This behavior might be related to competing antiferromagnetic (AFM) and FM interactions which are blocked in a glassy magnetic state [42]. The strong frequency dependence of the peak below ${{T}_{\text{B}}}$ can be taken as a further hint for the glassy character of the strain distribution in the sample. The existence of a strain-glass phase has been proposed in the shape-memory alloys Ti₅₀Ni_50−xFe_x [43] or, recently, in the Heusler material Ni–Co–Mn–Ga [44]. It has been further predicted for Ni–Co–Mn– Z compounds with $Z=\text{In}$, Sn, and Sb [45]. Because of the magnetoelastic coupling, supermagnetoelastic behavior (existence of strong FM and AFM interactions in Mn-rich Heusler alloys, which allows elastic softening of the magnetic sublattices, in particular, near the magnetostructural transition), magnetic-cluster-spin glass and strain glass should mutually interact in FM shape memory Heusler alloys. The kinetic arrest phenomenon as a relict of the magnetostructural transformation also supports the glassy behavior and may even lead to strain-glass formation interacting with the magnetic glasses [45]. The kinetic arrest phenomenon has also been observed in Ni₅₀Mn₃₅In₁₅ [15, 46]. Thus, the present finding in RUS data might be an indication of the presence of a strain-glass phase in Ni₅₀Mn₃₅In₁₅.