Interplay of sample composition and anomalous Hall effect in ${\mathrm{Co}}_x\mathrm{Nb}{\mathrm{S}}_2$

Herein we present a study on the interplay of chemical composition with respect to $x$ in antiferromagnetic ${\mathrm{Co}}_x\mathrm{Nb}{\mathrm{S}}_2$ and the anomalous Hall effect (AHE) in the magnetically ordered state. The AHE diminishes virtually completely in samples with exceeding Co content compared to the idealized composition of $x=1/3$. For stoichiometric samples the effect is weak but shows a sign reversal just below the magnetic ordering temperature. With increasing deficiency, the magnitude of the AHE compared to the ordinary Hall effect increases and additionally an exchange-biaslike effect can be observed. The results reveal the chemical composition of the samples as tuning parameter for the AHE in ${\mathrm{Co}}_x\mathrm{Nb}{\mathrm{S}}_2$. Accompanying first-principles calculations demonstrate that the electronic band structure is only slightly affected by disorder and deficiency of Co, but an impact on the magnetic exchange coupling can be inferred. The results are interesting for the research on $3d$-metal inserted transition-metal dichalcogenides and in a broader context for the understanding and design of antiferromagnet-based spintronic materials.

Figure 1

Idealized structure of ${\mathrm{Co}}_{1/3}\mathrm{Nb}{\mathrm{S}}_2$ with view along [210] (a) and along [001] with additional available sites for Co [blue color, (b)] [24, 49]. The $2c$-, $2d$-, and $2b$-sites are depicted as full, half-filled, and quarter-filled spheres, respectively. Representative XRD patterns using the sample CNS6 as example. XRD carried out on oriented pieces of single crystals showing the $00l$ reflections (c) and pattern showing the $h00$ and $hk0$ reflections (d).

Figure 2

(a) Temperature-dependent resistivity data of the samples presented in this study. The inset shows an enlarged view of the low-temperature data for sample CNS6. (b) Temperature dependence of the density of charge carriers (holes, $n_{\mathrm{h}}$) extracted from the Hall effect data (vide infra) based on a single carrier model.

Figure 3

Isothermal field sweeps of ${\rho }_{xy}$ for the sample CNS1. Measurement temperatures are indicated.

Figure 4

Isothermal field sweeps of ${\rho }_{xy}$ for (a) CNS2 and (b) CNS3 (data with vertical offset). Arrows indicate the scanning direction of the magnetic field. Note the sign reversal of the anomalous Hall effect in both cases. Measurement temperatures are indicated. In (c) and (d) the anomalous component ${{\rho }^A}_{yx}$ of the data presented in (a) and (b) is presented for selected temperatures.

Figure 5

Isothermal field sweeps of ${\rho }_{xy}\left(H\right)$ for (a) CNS4 (data with vertical offset) and (b) CNS6. Measurement temperatures are indicated. In (c) ${{\rho }^A}_{yx}$ is shown extracted from the data for CNS4 shown in (a); the inset shows the small anomalous component persisting in the temperature range of 22–10 K.

Figure 6

Field dependence of ${\rho }_{xy}\left(H\right)$ after field cooling with 9 T (a) and in zero field (b) for sample CNS5. Measurement temperatures are indicated. The black vertical bars are a guide to the eye for the shift of the hysteresis loops.

Figure 7

(a) Data for ${\rho }_{xy}\left(H\right)$ collected on the samples CNS5 at $T=26\mathrm{K}$ after cooling in opposite field directions. The vanishing of the exchange-biaslike effect is demonstrated for field cooling in −9 T of applied magnetic field in a second loop. (b) Scanning of ${\rho }_{xy}\left(H\right)$ at 24 K after zfc towards ±9-T magnetic field. Hereby, the dependence of the zero field-exchange bias on magnetization directions is demonstrated.

Figure 8

(a) Exchange-biaslike effect in the Hall resistivity displayed by CNS6 at a temperature of 2 K after cooling in reversed magnetic fields. For zero field cooling no such effect is observed. (b) $M$($H$) traces recorded different temperatures after zero field cooling except indicated differently. The insets show the details of the behavior near zero field at a $T=2\mathrm{K}$ and the small hysteresis loop exemplified at $T=14\mathrm{K}$, respectively.

Figure 9

DOS of Co in the FM-ordered ${\mathrm{Co}}_{1/3}\mathrm{Nb}{\mathrm{S}}_2$ and the total DOS of ${\mathrm{Co}}_{1/3}\mathrm{Nb}{\mathrm{S}}_2$ (solid line) compared with the total DOS of $2H\text{−}\mathrm{Nb}{\mathrm{S}}_2$ (dashed line).

Figure 10

Spin-resolved Bloch spectral function A ($E$; k) calculated for stoichiometric ${\mathrm{Co}}_{1/3}\mathrm{Nb}{\mathrm{S}}_2$ for majority-spin (a) and minority-spin (b) states as well as for the majority-spin (c) and minority-spin (d) states of nonstoichiometric ${\mathrm{Co}}_{0.28}\mathrm{Nb}{\mathrm{S}}_2$.

Figure 11

Interatomic exchange coupling parameter ${J_{ij}}^{\mathrm{Co}-\mathrm{Co}}$ calculated for ${\mathrm{Co}}_{1/3}\mathrm{Nb}{\mathrm{S}}_2$ (closed circles) and for ${\mathrm{Co}}_{0.28}\mathrm{Nb}{\mathrm{S}}_2$ (open circles) with 16% vacancies on the Co sublattice. Closed squares represent the results obtained for a layer by layer AFM state used as a reference state. The inset shows the biquadratic exchange interactions parameters calculated for ${\mathrm{Co}}_{1/3}\mathrm{Nb}{\mathrm{S}}_2$ using the DLM reference state.

Figure 12

The order parameter $\xi =1/2\left(\langle M_{\mathrm{Co}1}\rangle -\langle M_{\mathrm{Co}2}\rangle \right)/\left|M_{\mathrm{Co}}\right|$ as a function of temperature obtained via MC simulations for ${\mathrm{Co}}_{1/3}\mathrm{Nb}{\mathrm{S}}_2$.