Interference between distortion and magnetization for the reentrant martensitic transition in Heusler-type alloys ${\mathrm{Co}}_2\mathrm{Cr}(\mathrm{Ga},\mathrm{Si})$

Heusler-type ${\mathrm{Co}}_x{\mathrm{Cr}}_{78-x}{\mathrm{Ga}}_{11}{\mathrm{Si}}_{11}$ alloys exhibit the reentrant martensitic transition, i.e., the forward martensitic transition from the paramagnetic austenite (Para-A) phase into the paramagnetic martensite (Para-M) phase and the successive reverse martensitic transition from the Para-M phase into the ferromagnetic austenite (Ferro-A) phase with decreasing temperature $T$. In the present paper, this complicated transition is analyzed by use of phenomenological free energy, which is expanded in powers of two order parameters, i.e., the tetragonal distortion $e_3$ and the magnetization $M$. It is assumed that an interference between $e_3$ and $M$ through the repulsive magnetostructural interaction, $G{e}_3^2{M}^2(>0)$, in the free energy is strong enough to suppress out the ferromagnetic martensite (Ferro-M) phase with both nonvanishing $e_3$ and $M$. With the presence of this repulsive interaction, the observed phase diagram in the $T-x$ plane is found to be successfully explained. The $T$ dependencies of $e_3$ and the entropy are also calculated to see the equilibrium state. The inverse magnetic susceptibility in the martensite phase ${\chi }^{-1}$ is derived by taking into account that the term $G{e}_3^2{M}^2(>0)$ opposes the magnetization by an external magnetic field in the martensite phase. The calculated ${\chi }^{-1}$ exhibits a negative paramagnetic Curie temperature as was observed, which verifies directly the existence of $G{e}_3^2{M}^2(>0)$. An essential role of $G{e}_3^2{M}^2(>0)$ in the phase stabilities is also found in the calculated $T$ dependence of the elastic constant $(c_{11}-c_{12})$. On the basis of the present analyses, the conditions for the appearance of the reentrant martensitic transition are described.

Figure 1

Phase diagram of ${\mathrm{Co}}_x{\mathrm{Cr}}_{78-x}{\mathrm{Ga}}_{11}{\mathrm{Si}}_{11}$ in the temperature $T$-concentration $x$ plane. The calculated transition temperatures $t_{\mathrm{c}},\mathrm{I}-\mathrm{II}\left(x^{\prime }\right)$, etc. were already transformed back into the real transition temperatures $T_{\mathrm{c}},\mathrm{I}-\mathrm{II}\left(x\right)$, etc. The phase diagram is spanned by phase I (the Para-A phase), phase II (the Para-M phase), and phase III (the Ferro-A phase). The solid lines show the phase boundaries calculated by use of the parameter values given in the text. The circles, squares, and a diamond are the transition temperatures, $T_{\mathrm{c}},\mathrm{I}-\mathrm{II}\left(x\right), T_{\mathrm{c}},\mathrm{II}-\mathrm{III}\left(x\right)$, and $T_{\mathrm{c}},\mathrm{I}-\mathrm{III}\left(x\right)(=T_{\mathrm{c}}\left(x\right))$, respectively, which were deduced experimentally [7]. (See the text.) The triangles are Curie temperatures $T_{\mathrm{c}}\left(x\right)$ observed in nonequilibrium ferromagnetic states [7].

Figure 2

Temperature $T$ dependence of the equilibrium tetragonal distortion $e_{30}$ of the alloy with $x=51.7(x^{\prime }=0.53)$. In the abscissa, temperature $t$ was already transformed back into its real temperature $T$. The solid lines are $e_{30}$ calculated by use of the parameter values given in the text. The circles are the observed values of $e_{30}$ in phase II (the Para-M phase), while the triangles are the observed values of $e_{30}$ in nonequilibrium martensite phases [27]. The broken line is the guide to the eye.

Figure 3

Calculated temperature $T$ dependencies of the minimum free energies (subtracted by the common ${\overline{F}}_0$), $\overline{F}-{\overline{F}}_0$, of the alloy with $x=51.7(x^{\prime }=0.53)$. In the abscissa, temperature $t$ was already transformed back into its real temperature $T$. The calculations were made by use of the parameter values given in the text. The bold and thin broken lines are the minimum free energy of phase III (the Ferro-A phase), ${\overline{F}}_{\mathrm{III}}(x^{\prime },t;{\overline{M}}_0)-{\overline{F}}_0(x^{\prime },t)$. The bold and thin solid lines are the minimum free energy of phase II (the Para-M phase), ${\overline{F}}_{\mathrm{II}}(x^{\prime },t;{\overline{e}}_{30})-{\overline{F}}_0(x^{\prime },t)$. The bold and thin dot-dash lines are the free energy of phase I (the Para-A phase), ${\overline{F}}_{\mathrm{I}}(x^{\prime },t;0)-{\overline{F}}_0(x^{\prime },t)=0$. All bold lines are the lowest minimum free energy among the three free energies at any $T$. The usual and unusual martensitic transitions occur at the crossing points A and B, respectively, which are at the transition temperatures $T_{\mathrm{c}},\mathrm{I}-\mathrm{II}\left(x^{\prime }\right)$ and $T_{\mathrm{c}},\mathrm{II}-\mathrm{III}\left(x^{\prime }\right)$, respectively.

Figure 4

Calculated entropies $S-S_0$ for the alloys with different values of $x$ as functions of temperature $T$. In the abscissa, temperature $t$ was already transformed back into its real temperature $T$. The solid lines were calculated by use of the parameter values in the text and an entropy change measured experimentally at a martensitic transition temperature [7]. (See the text.)

Figure 5

Inverse susceptibility ${\chi }^{-1}$ of the alloy with $x=54.2(x^{\prime }=0.89)$ as a function of temperature. In the abscissa, temperature $t$ was already transformed back into its real temperature $T$, while in the ordinate, all values of ${\chi }^{-1}$ are given in an arbitrary unit. The solid lines were calculated by use of the parameter values in the text and $g=0.42$. The broken lines were calculated by employing some different values of $g$ without changing the other parameter values. The dot-dash line was calculated in the nonequilibrium phase II (the Para-M phase). The circles show ${\chi }^{-1}$ observed in the martensite phase of this alloy, whereas the triangles show ${\chi }^{-1}$ observed in its nonequilibrium martensite phase [28].

Figure 6

Calculated temperature dependencies of the elastic constants $({\overline{c}}_{11}-{\overline{c}}_{12})$ in the alloys with $x=51.7(x^{\prime }=0.53)$ and with $x=48.5(x^{\prime }=0.071)$. In the abscissa, temperature $t$ was already transformed back into its real temperature $T$. The alloy with $x=51.7$ shows the reentrant martensitic transition, while the alloy with $x=48.5$ is always in the austenite phase. All solid lines were calculated by use of the parameter values given in the text and $g=0.42$, which was determined in Fig. 5. For comparison, the broken and dot-dash lines were calculated for $g=0.21$ and $g=0$, respectively, without changing the other parameter values.