Tunable exchange bias in the magnetic Weyl semimetal ${\mathrm{Co}}_3{\mathrm{Sn}}_2{\mathrm{S}}_2$

Exchange bias is a phenomenon critical to solid-state technologies that require spin valves or nonvolatile magnetic memory. The phenomenon is usually studied in the context of magnetic interfaces between antiferromagnets and ferromagnets, where the exchange field of the former acts as a means to pin the polarization of the latter. In the present study, we report an unusual instance of this phenomenon in the topological Weyl semimetal ${\mathrm{Co}}_3{\mathrm{Sn}}_2{\mathrm{S}}_2$, where the magnetic interfaces associated with domain walls suffice to bias the entire ferromagnetic bulk. Remarkably, our data suggest the presence of a hidden order parameter whose behavior can be independently tuned by applied magnetic fields. For micron-size samples, the domain walls are absent, and the exchange bias vanishes, suggesting the boundaries are a source of pinned uncompensated moment arising from the hidden order. This mechanism suggests that exciting opportunities lie ahead for the application of topological materials in spintronic technologies.

Figure 1

Controlling the exchange bias of ${\mathrm{Co}}_3{\mathrm{Sn}}_2{\mathrm{S}}_2$ single crystal sample by applying out-of-plane and in-plane field protocols at 4.2 K. (a) Optical image of the sample and a schematic depicting paths for current $I_x$, voltage probe $V_y$, and the crystal axes. (b) Hall resistance $R_{xy}=V_y/I_x$ as a function of the out-of-plane field ${\mu }_0{H}_z$. After ZFC the field is swept between $\pm 1$ T resulting in a rectangular $R_{xy}$ with $H_{EB}=50\mathrm{mT}$ (black). Controlling the positive coercive field $H_c^{+}$ by performing a larger field excursion in the negative direction ${\mu }_0{H}_{\max }^{-}=-2$, −3, −6 T while keeping ${\mu }_0{H}_{\max }^{+}=1$ T resulting in $H_{\mathrm{EB}}=-170$, −200, $-300\mathrm{mT}$, respectively (dark blue to light blue). (c) $H_c^{+}$ as a function of ${\mu }_0{H}_{\max }^{-}$, with colored dots corresponding to the measurements shown in (b). (d) Each measurement is performed at $H_{\mathrm{IP}}=0$ after ZFC (black), after an excursion to $H_{\mathrm{IP}}^{\mathrm{ex}}=2$ T resulting in $H_c^{+}=$ 139 mT and $H_c^{-}=-275$ mT, $H_{\mathrm{EB}}=68$ mT (blue), and to ${\mu }_0{H}_{\mathrm{IP}}^{\mathrm{ex}}=-2$ T yielding $H_c^{+}=$ 244 mT, $H_c^{-}=-128$ mT, and $H_{EB}=-58$ mT (red). Three loops performed for each measurement gave similar results, and one representative result is shown for clarity. In both (b) and (d) the curves for different protocols are shifted vertically for clarity.

Figure 2

Demagnetization and magnetization. (a) Demagnetization protocol, the field swept from −1 to 1 T to initialize the sample and from +1 until $H_c^{-}$ is reached (blue). Sweeping the field between the distinct $H_c^{\pm }$ after three repetitions produces a vanishingly small $R_{xy}$ (red). (b) Magnetization of the sample. The field is first swept to −1 T (red) and then $\mathrm{a}+/-1$-T loop is performed (blue).

Figure 3

Scanning SOT microscopy images of ${\mathrm{Co}}_3{\mathrm{Sn}}_2{\mathrm{S}}_2$ bulk at 4.2 K. (a) Optical image of the SOT pointing down, the sample, and the reflection of the SOT on the sample. (b) $R_{xy}$ as a function of applied magnetic field for different phases of the sample. ZFC to saturated field (blue), demagnetized state to saturated field (green), and magnetization reversal with transitory state (red). $R_{xy}$ corresponding to a loop with ${\mu }_0\left|H_{\max }^{\pm }\right|=1\mathrm{T}$ is shown for comparison (black dashed). The fields at which the images were taken are marked with dots. The curves for different protocols are shifted vertically for clarity. (c)–(n) Sequence of magnetic images of different states of the sample at distinct values of applied out-of-plane field ${\mu }_0{H}_z$. (c)–(f). Evolution from ZFC to the saturation field. ${\mu }_0{H}_z=45$ mT (c), ${\mu }_0{H}_z=65$ mT (d), ${\mu }_0{H}_z=85$ mT (e), ${\mu }_0{H}_z=95$ mT (f). (g)–(j) Evolution of demagnetization to the saturation field. ${\mu }_0{H}_z=30$ mT (g), ${\mu }_0{H}_z=50$ mT (h), ${\mu }_0{H}_z=75$ mT (i), ${\mu }_0{H}_z=95$ mT (j). (k)–(n) Magnetization reversal with transitory state. ${\mu }_0{H}_z=35$ mT (k), ${\mu }_0{H}_z=45$ mT (l), ${\mu }_0{H}_z=75$ mT (m), ${\mu }_0{H}_z=95$ mT (n). All images are $45\times 45\mu {\mathrm{m}}^2$, pixel size 480 nm (c)–(f) and 980 nm (g)–(n), acquisition time 8 min/image (c)–(f) and 5.8 min/image (g)–(n). The bright to dark color scale represents 50 mT and is the same for all images. See Supplemental Material Movies 1–3 corresponding to images (c)–(f), (g)–(j), (k)–(n), respectively.

Figure 4

$R_{xy}$ measurements of ${\mathrm{Co}}_3{\mathrm{Sn}}_2{\mathrm{S}}_2$ FIB samples with distinct thicknesses ranging 6–42 μm at 10 K. (a) SEM image of four different ${\mathrm{Co}}_3{\mathrm{Sn}}_2{\mathrm{S}}_2$ single crystals cut by FIB with an area of $60\times 30\mu {\mathrm{m}}^2$ and thicknesses ranging 6–42 μm. (b) SEM image at higher magnification of the 6-μm-thick sample including a schematic of the measurement and the crystal axes. (c)–(f) $R_{xy}$ measurements under different out-of-plane protocols. (c),(d) Sample thickness 6 μm. (e),(f) Sample thickness 42 μm. (c),(e) Blue: After ZFC, the field is swept three times between $\pm 1$ T. Red: Finding the positive and negative coercive fields. Sweeping the field six times between the positive and negative coercive fields does not change the coercive field, therefore no demagnetization process is possible in the FIB sample. Only two curves are shown for clarity. (d),(f) Applying a positive 6-T field and then sweeping the field until the negative coercive field does not change the coercive field of the 6-μm sample. The negative coercive field of the 42-μm sample grows by 200 mT resulting in $H_{\mathrm{EB}}=100$ mT. The $R_{xy}$ measurements of samples with other thickness are presented in Supplemental Material Movie 4 [38]. (c) Inset: SOT imaging of the 6-μm sample before and after $H_c^{+}$ presenting a single domain structure which flips at $H_c^{+}$. All images are $45\times 45\mu {\mathrm{m}}^2$, pixel size 480 nm, acquisition time 9 min/image. The bright to dark color scale represents 10 mT and is the same for all images. See Supplemental Material Movie 8 [38].