Cluster spin-glass behavior and memory effect in ${\mathrm{Cr}}_{0.5}{\mathrm{Fe}}_{0.5}\mathrm{Ga}$

We report the structural, static, and dynamic properties of ${\mathrm{Cr}}_{0.5}{\mathrm{Fe}}_{0.5}\mathrm{Ga}$ by means of powder x-ray diffraction, dc magnetization, heat capacity, ac susceptibility, magnetic relaxation, and magnetic memory effect measurements. The dc magnetization and ac susceptibility studies reveal a spin-glass transition at around $T_{\mathrm{f}}\simeq 22$ K. An intermediate value of the relative shift in freezing temperature $\delta T_{\mathrm{f}}\simeq 0.017$, obtained from the ac susceptibility data reflects the formation of cluster spin-glass states. The frequency dependence of $T_{\mathrm{f}}$ is also analyzed within the framework of dynamic scaling laws such as power law and Vogel-Fulcher law. The analysis using power law yields a characteristic time constant for a single spin flip $\tau *\simeq 1.1\times {10}^{-10}$ s and critical exponent $z{\nu }^{\prime }=4.2\pm 0.2$. On the other hand, the Vogel-Fulcher law yields the characteristic time constant for a single spin flip ${\tau }_0\simeq 6.6\times {10}^{-9}$ s, Vogel-Fulcher temperature $T_0=21.1\pm 0.1$ K, and an activation energy $E_{\mathrm{a}}/k_{\mathrm{B}}\simeq 16$ K. The value of $\tau *$ and ${\tau }_0$ along with a nonzero value of $T_0$ provide further evidence for the cluster spin-glass behavior. The magnetic field dependent $T_{\mathrm{f}}$ follows the de Almeida-Thouless (AT) line with a non-mean-field type instability, reflecting either a different universality class or strong anisotropy in the spin system. A detailed nonequilibrium dynamics study via relaxation and memory effect experiments demonstrates the evolution of the system through a number of intermediate metastable states and striking memory effects. All the above observations render a cluster spin-glass behavior in ${\mathrm{Cr}}_{0.5}{\mathrm{Fe}}_{0.5}\mathrm{Ga},$ which is triggered by magnetic frustration due to competing antiferromagnetic and ferromagnetic interactions and magnetic site disorder. Moreover, the asymmetric response of magnetic relaxation with respect to the change in temperature, below the freezing temperature can be explained by the hierarchical model.

Figure 1

Rietveld refinement of the x-ray diffraction pattern of ${\mathrm{Cr}}_{0.5}{\mathrm{Fe}}_{0.5}\mathrm{Ga}$ at $T=300$ (top) and 15 K (bottom), respectively. The open circles and solid lines are the observed and calculated patterns, respectively. The Bragg positions are indicated by ticks. Solid line at the bottom represents the difference between the observed and calculated intensities.

Figure 2

Variation of lattice constants ($a$ and $c$) and unit cell volume ($V_{\mathrm{cell}}$) with temperature for ${\mathrm{Cr}}_{0.5}{\mathrm{Fe}}_{0.5}\mathrm{Ga}$. The solid line represents the fit of $V_{\mathrm{cell}}\left(T\right)$ using Eq. (1).

Figure 3

Temperature dependent dc susceptibility $\chi \left(T\right)$ measured under different applied fields for ZFC and FC protocols. The arrows point to $T_{\mathrm{f}}$. Inset: The variation of $T_{\mathrm{f}}$ with $H$. The solid line represents the fit using Eq. (4).

Figure 4

$1/\chi$ vs $T$ measured at $H=0.02$ T and the solid line is the CW fit for $T>50$ K. Inset: Isothermal magnetization $M\left(H\right)$ curves at $T=2$ K and 100 K.

Figure 5

Temperature dependent heat capacity $C_{\mathrm{p}}$ in the absence of magnetic field between 2 and 250 K. Lower inset: $C_{\mathrm{p}}/T$ vs $T^2$. Upper inset: $C_{\mathrm{p}}$ vs $T$ in the low temperatures regime and solid line is the fit in the temperature range 2 to 10 K, as described in the text.

Figure 6

Upper panel: Real part of the ac susceptibility $\left[{\chi }^{\prime }\left(T\right)\right]$ measured at different frequencies ($\nu$) and at an ac field $H_{\mathrm{ac}}=5$ Oe. The solid downward arrow points to the peak shift. Lower panel: ${\chi }^{\prime }\left(T\right)$ measured at different dc applied fields, fixing $\nu =200$ Hz and $H_{\mathrm{ac}}=5$ Oe. The dotted downward arrow guides the peak position. Inset: The variation of $T_{\mathrm{f}}$ with $H$. The solid line represents the fit using Eq. (4).

Figure 7

Upper panel: ${\chi }^{\prime }\left(T\right)$ measured at different frequencies ($\nu$) at a fixed dc field $H_{\mathrm{dc}}=0.25$ T and at a fixed ac field $H_{\mathrm{ac}}=5$ Oe. The downward arrow points to the peak shift. Lower left panel: ${log}_{10}\left(\tau \right)$ vs ${log}_{10}(T_f/T_g-1)$ for $H_{\mathrm{dc}}=0$ and 0.25 T. The solid lines represent the fits using Eq. (7). Lower right panel: $ln\left(\tau \right)$ vs $1/(T_f-T_0)$ for $H_{\mathrm{dc}}=0$ and 0.25 T. The solid lines represent the fits using Eq. (10).

Figure 8

Relaxation of the zero-field-cooled (ZFC) magnetization measured at different temperatures $T=5$, 10, and 15 K for a waiting time $t_{\mathrm{w}}=60$ s, as discussed in the text. The solid lines represent the fit using stretched exponential function given in Eq. (12).

Figure 9

Memory effect as a function of temperature in (a) FC and (b) ZFC protocols in $H=200$ Oe, as discussed in the text. The measurements were interrupted at $T{}_1^{\mathrm{int}}=12$ K and $T{}_2^{\mathrm{int}}=5$ K for 2 hours each. Inset: Difference in magnetization $\mathrm{\Delta }M=(M{}_{\mathrm{ZFCW}}^{\mathrm{mem}}-M{}_{\mathrm{ZFCW}}^{\mathrm{ref}})$ vs $T$ for the ZFC condition.

Figure 10

Magnetic relaxation measurements in the negative $T$ cycle in an applied field of $H=200$ Oe for (a) ZFC and (b) FC methods. Insets: $M\left(t\right)$ data at 12 K for negative FC and ZFC $T$ cycles along with the fit by Eq. (12). For the positive $T$ cycle, ZFC and FC data are shown in (c) and (d), respectively.