Influence of the type of intercalation on spin-glass formation in the Fe-doped ${\mathrm{TaS}}_2\left({\mathrm{Se}}_2\right)$ polytype family

We suggest an explanation based on the Blume-Capel model of why some layered compounds of the iron-intercalated transition metal dichalcogenides ${\mathrm{TaS}}_2\left({\mathrm{Se}}_2\right)$ exhibit spin-glass behavior, while another group of this family demonstrates low-temperature paramagnetism. In these materials, the doped Fe atoms either substitute the Ta atoms with losing their magnetic moments or sit between the ${\mathrm{TaS}}_2\left({\mathrm{Se}}_2\right)$ layers keeping their spin states. The Blume-Capel model allows us to introduce a chemical potential to control a balance of the intercalated elements of both types. The Ghatak-Sherrington theory of spin-glass behavior of this model predicts an existence of a tricritical point that means that there is a concentration threshold of Fe ions retaining their magnetic moments, above which spin-glass ordering occurs. Below the threshold, Fe ions behave as independent paramagnetic centers. We build temperature dependencies of magnetic susceptibility and field dependencies of magnetization to highlight specific features of the model related with a variable content of Fe ions in the high-spin state. A specific crystal structure of the layered transition metal dichalcogenides gives an opportunity to increase the concentration of ions with nonzero magnetic moments by co-intercalating non-Kramers $3d$ ions into the van der Waals gaps. This process may trigger spin-glass ordering in the initially paramagnetic Fe-doped ${\mathrm{TaS}}_2\left({\mathrm{Se}}_2\right)$ polytype complexes.

Figure 1

Types of Fe intercalation into the $\mathrm{Ta}{X}_2$ ($X$ = S, Se) layered system: (a) insertion; (b) substitution.

Figure 2

Phase diagram as a function of temperature and the parameter ${\mathrm{\Delta }}_0$. The boundary of instability is shown by the solid lines in the replicon sector, and by the broken lines in the anomalous/longitudinal sector. The cases of the ratio ${\sigma }_{\mathrm{\Delta }}/J$ are taken as (1) 0.0; (2) 0.1; (3) 0.2; (4) 0.3; (5) 0.4; and (6) 0.5.

Figure 3

The zero-field critical frontier $T_f/J$ separating the spin-glass and the singlet paramagnetic phases at $J_0=0$ as a function of the average concentration $\mu$ of intercalated ions retaining their magnetic moments. The cases of ${\sigma }_{\mathrm{\Delta }}/J$ are taken: (1) 0.0; (2) 0.3; (3) 0.5. The black dots mark tricritical points.

Figure 4

Plots of the order parameters (the spin glass $q$, the magnetization $\mu$, and the concentration of magnetically active ions $\mu$) in the absence of random Gaussian fields (${\sigma }_h=0$): (a)–(c) ${\mathrm{\Delta }}_0/J=10.0$; (d)–(f) ${\mathrm{\Delta }}_0/J=-0.3$; (g)–(i) ${\mathrm{\Delta }}_0/J=-0.9$. Everywhere, ${\sigma }_{\mathrm{\Delta }}=0.5$ is taken.

Figure 5

The zero-field ($h_0=0$) phase diagrams in the absence of random Gaussian fields (${\sigma }_h=0$): (a) ${\mathrm{\Delta }}_0/J=10.0, {\sigma }_{\mathrm{\Delta }}=0.5$; (b) ${\mathrm{\Delta }}_0/J=-0.3, {\sigma }_{\mathrm{\Delta }}=0.5$. The colors of phases: violet (SG), blue (PM), and green (FM).

Figure 6

The phase diagram with the stability line. The PM phase is stable, the SG is entirely unstable, while the FM is only stable over the AT line (dashed). The cases ${\mathrm{\Delta }}_0/J=10.0$ (blue lines) and ${\mathrm{\Delta }}_0/J=-0.3$ (red lines) are taken. Another parameters are $h_0=0, {\sigma }_h=0$, and ${\sigma }_{\mathrm{\Delta }}=0$.

Figure 7

Zero-field phase diagrams in the presence of random Gaussian fields (${\sigma }_h=0.1$): (a) ${\mathrm{\Delta }}_0/J=10.0, {\sigma }_{\mathrm{\Delta }}=0.0$; (b) ${\mathrm{\Delta }}_0/J=10.0, {\sigma }_{\mathrm{\Delta }}=0.5$; (c) ${\mathrm{\Delta }}_0/J=-0.3, {\sigma }_{\mathrm{\Delta }}=0.5$. The I/FM phase is shown by violet/green color. (d)–(f) Plots of the order parameters at ${\mathrm{\Delta }}_0/J=-0.3, {\sigma }_{\mathrm{\Delta }}=0.5$.

Figure 8

Magnetic susceptibility without external field for $J_0=0.0$ and different values of ${\mathrm{\Delta }}_0/J$ shown by figures in the inset. Here, ${\sigma }_{\mathrm{\Delta }}=0.0$ and ${\sigma }_h=0.0$.

Figure 9

Evolution of the Almeida-Thouless lines as ${\mathrm{\Delta }}_0/J$ varies. The instability region (under the line) is reduced as ${\mathrm{\Delta }}_0/J$ decreases: ${\mathrm{\Delta }}_0/J=10.0$ (black), ${\mathrm{\Delta }}_0/J=0.0$ (magenta), ${\mathrm{\Delta }}_0/J=-0.3$ (blue), ${\mathrm{\Delta }}_0/J=-0.6$ (red), ${\mathrm{\Delta }}_0/J=-0.9$ (brown).

Figure 10

Evolution of temperature dependence of the susceptibility with an external magnetic field $h_0/J$ marked by colors in the inset of the plot (f): (a), (b) ${\mathrm{\Delta }}_0/J=10.0$; (c), (d) ${\mathrm{\Delta }}_0/J=-0.3$; (e), (f) ${\mathrm{\Delta }}_0/J=-0.9$. Here, ${\sigma }_{\mathrm{\Delta }}=0.5$ and ${\sigma }_h=0.1$.

Figure 11

Magnetization curves for ${\mathrm{\Delta }}_0/J=10.0$ (a) and ${\mathrm{\Delta }}_0/J=-0.3$ (b). The inset: The plot colors are marked by figures of temperature $T/J$. The parameters $J_0/J=0, {\sigma }_{\mathrm{\Delta }}=0.5$, and ${\sigma }_h=0.1$ are taken in all cases.