Cluster-glass dynamics of the Griffiths phase in ${\mathrm{Tb}}_{5-x}{\mathrm{La}}_x{\mathrm{Si}}_2{\mathrm{Ge}}_2$

The static magnetization and dynamic susceptibility responses of the cluster system within a Griffiths phase of the magnetocaloric compound ${\mathrm{Tb}}_{5-x}{\mathrm{La}}_x{\mathrm{Si}}_2{\mathrm{Ge}}_2$ ($x=0.075$) have been investigated. A novel cluster-glass state within the Griffiths phase is formed at a characteristic freezing temperature where short-range ferromagnetic correlations set in the paramagnetic regime. Ferromagneticlike correlations are built up at around 155 K, which suddenly become frozen at a lower temperature $\sim 140\mathrm{K}$, thus in analogy with a reentrant spin glass behavior. The ac susceptibility near the freezing temperature follows a critical slowing down process characterized by ${\tau }_0={10}^{-13}s$ and dynamic exponents $\mathrm{z}\nu \sim 6$ and $\beta \sim 0.4$, similar to well-known spin glass systems. The nonlinear ac susceptibility analysis shows clearly the existence of a transition associated to the reentrant behavior. The origin of the intermediate cluster-glass phase inside the Griffiths phase is proposed to be the result of a combination of short-ranged RKKY intralayer positive exchange interactions between rare-earth ${\mathrm{Tb}}^{3+}$ ions and antiferromagnetic exchange between adjacent interlayers involving Si and Ge atoms in connection to the ${\mathrm{Tb}}^{3+}$ atoms.

Figure 1

Crystal structures of ${\mathrm{Tb}}_5({\mathrm{Si}}_2{\mathrm{Ge}}_2$) in the paramagnetic (a) and in the ferromagnetic state (b). The $R$ atoms occupying different sites are shown using green ($R_1$), light blue ($R_2$), and dark blue ($R_3$) spheres. The orange circles represent the Si/Ge atoms located at the $T_1, T_2$ positions within the slab. The red circles represent the Si/Ge atoms responsible for bonding between slabs, located at the $T_3, T_4$ positions. The thick red lines indicate the Si/Ge-Si/Ge covalently bonded pairs of atoms.

Figure 2

(a) ZFC-FC magnetization of ${\mathrm{Tb}}_{4.925}{\mathrm{La}}_{0.075}{\mathrm{Si}}_2{\mathrm{Ge}}_2$ as a function of temperature measured in an applied field of 0.02 kOe. The inset of Fig. 2 shows the Curie-Weiss fit to the inverse magnetic susceptibility in 0.02 kOe. (b) Temperature dependence of dc magnetic susceptibility of ${\mathrm{b}}_{4.925}{\mathrm{La}}_{0.075}{\mathrm{Si}}_2{\mathrm{Ge}}_2$ plotted as ${\chi }_{dc}^{-1}$ (T) in the temperature range 70–240 K, measured in a field of 0.02 kOe. Curie-Weiss fit is shown for the pure PM region (solid line) and for the temperature regimes delimited by $T_G$ and $T_{G^{*}}$ (dashed lines).

Figure 3

Temperature dependence of dc magnetic susceptibility ${\chi }_{dc}^{-1}$ as a function of magnetic field, measured on heating. The inset shows fits at 0.02 kOe in the Griffiths (GP) and PM phases, respectively.

Figure 4

Real part of the ac-susceptibility ${\chi }_1^{\prime }$ measured at different applied dc fields (0, 0.1, and 1 kOe). The inset shows the imaginary part of the ac susceptibility ${\chi }_1^{\prime \prime }$ in the temperature range 75–190 K measured at the same applied dc fields.

Figure 5

Arrott plots ($M^2\text{vs}H/M$) of the magnetization isotherms at selected temperatures in the temperature range 5–200 K.

Figure 6

Temperature dependence of the magnetic SANS intensity for ${\mathrm{Tb}}_5{\mathrm{Si}}_2{\mathrm{Ge}}_2$ collected in D16 at $q=0.1{\AA }^{-1}$ at zero magnetic field from Magén et al. [13]. The lower inset shows the temperature variation of the imaginary component of the ac susceptibility (${\chi }_1^{\prime \prime }$) in the temperature regime around 150 K, where the plateau in SANS was observed, collected under various dc bias fields (${\mathrm{H}}_{dc}$). The upper inset shows the temperature variation of the real component of the ac susceptibility (${\chi }_1^{\prime }$) in the same temperature range at different frequencies ($f=100$, 215, 463, 1000, 2150, 4630, and 10 000 Hz).

Figure 7

The real (${\chi }_1^{\prime }$) and imaginary parts (${\chi }_1^{\prime \prime }$) of the ac susceptibility as a function of temperature of ${\mathrm{Tb}}_{4.925}{\mathrm{La}}_{0.075}{\mathrm{Si}}_2{\mathrm{Ge}}_2$ measured at different frequencies from 100 Hz to 10 kHz in an applied ac field of ${\mathrm{h}}_{ac}=1$ Oe in the temperature range 115–155 K.

Figure 8

Critical slowing down analysis of the imaginary part of the ac magnetic susceptibility data ${\chi }_1^{\prime \prime }$ at $T>T_f$ [see Eqs. (3) and (4)] of the $x=0.075$ alloy. ${\chi }_1^{\prime \prime }$ at different frequencies of the ac field collapse on a master curve with $\beta =0.4\left(1\right)$. The inset shows the frequency dependence of freezing temperature $T_f$ plotted as a $ln\left(\tau \right)vsln[(T_f-T_{f_0})/T_{f_0}]$ (see text for details). The solid line represents the fit to the power-law divergence of critical slowing down process Eq. (3).

Figure 9

Temperature variation of the nonlinear susceptibility ${\chi }_2^{\prime }$ at various fixed frequencies when ${\mathrm{H}}_{dc}=0$ and ${\mathrm{h}}_{ac}=1\text{Oe}$ for (a) the temperature regime around $T_C$ and (b) temperature regime around $T_{G^{*}}$. Nonlinear component ${\chi }_3^{\prime }$ at a frequency of 10 kHz is displayed in the inset. $T_{G^{*}}$ and the freezing transitions are marked by arrows.