Anomalous depinning of magnetic domain walls within the ferromagnetic phase of the Weyl semimetal Co₃Sn₂S₂

Single crystals of Co₃Sn₂S₂ were synthesized by solution growth [8, 23, 24]. A ternary mixture with initial composition Co₁₂S₈Sn₈₀ was first heated to $400$°C over two hours and held there for another two hours. It was then heated to $1050$°C over six hours and held there for $10$ hours, followed by a slow, $90$ hour cool down to $740$°C. The remaining molten flux was removed by centrifugation. Shiny hexagonal crystals were obtained. The powder x-ray diffraction data refines to a hexagonal unit cell with $a=5.3641(8)$ Å and $c=13.1724$ Å, which are consistent with previously reported values [25].

Magnetization measurements of Co₃Sn₂S₂ single crystals were performed in a Quantum Design MPMS SQUID up to $7$ T with the magnetic field parallel to the $c$ axis. Magnetization as a function of temperature was measured under various applied fields in both field-cooled-cooling (FCC) and field-cooled-warming (FCW) conditions. Hysteresis loops were measured by first cooling down to the desired temperature in zero applied field and then sweeping the field. Before each measurement, we followed a demagnetization procedure to minimize the remanent field in the magnet and in the sample chamber by systematically oscillating the magnet to zero field at room temperature.

Polar MOKE images are acquired from as-grown Co₃Sn₂S₂(0001) single crystal samples, using a normal-incidence Sagnac interferometric scanning microscope  [26]. The microscope measures Kerr rotation as opposed to Kerr ellipticity [9]. It has a spatial resolution of 0.85 $\mu$m and a sensitivity of $4.4$ $\mu$rad. The single crystal sample is mounted on a cold finger inside a cryostat. The sample is optically accessible from outside through an optical window. The temperature of the sample is variable from $300$ K to $5$ K. We use a permanent magnet placed outside of the cryostat to produce a variable magnetic field of up to 100 Oe (0.01 T) along the $c$ axis (the surface normal) of the sample. For this experiment, the field is held constant at 94 Oe, measured by a Hall effect sensor.

The FCC curve with a magnetic field of $1000$ Oe applied parallel to the $c$ axis is shown in Fig. 1, and illustrates the ferromagnetic-paramagnetic transition. The inverse susceptibility is fit to the Curie-Weiss law and we obtain an effective moment of $1.14$ $\mu_{B}$ per Co and a Curie-Weiss temperature of $\theta_{\mathrm{CW}}=175$ K, which are consistent with previously published work [23, 25, 27, 4]. There is no difference in the FCC and FCW magnetization at fields above $300$ Oe. If the applied field is below $300$ Oe, however, two anomalies emerge. First, the FCC and FCW curves start to deviate near $T_{\mathrm{C}}$. Second, a local maximum arises at $T_{\mathrm{A}}$ $\approx$ 130 K in the FCC data, which is also reported in other studies [18, 10, 11, 12, 13, 14, 15, 16]. The local maximum disappears above 300 Oe and below 50 Oe, while a small hysteresis between FCC and FCW still remains at 50 Oe. The field dependence of $T_{\mathrm{A}}$ is relatively small, as shown in the inset of Fig. 1(b). The fact that the anomaly disappears above $300$ Oe, where the magnetization is saturated, suggests its origin is related to magnetic domains. A local minimum also emerges in the FCW magnetization albeit at higher temperature. The hysteresis between FCC and FCW magnetizations indicates it is unlikely that the local maximum and minimum in the FCC and FCW magnetization are due to a second order phase transition, which should not depend on magnetic history.

We will compare the fraction of magnetic domains assuming no pinning with the fraction extracted from our experimental magnetization where pinning effects are inevitable. In the absence of domain wall pinning, the demagnetising field $H_{d}=N_{c}M$ perfectly counteracts the applied field, $H_{\mathrm{applied}}$, as long as $H_{\mathrm{applied}}<N_{c}M_{\mathrm{s}}$. $N_{c}$ is the demagnetization factor along the $c$-direction. Because the $c$ axis is the easy magnetization axis, $N_{c}$ can be measured experimentally from a plot of $M^{2}$ versus $H_{\mathrm{applied}}/M$ for the $M(H)$ data along the easy axis of magnetization as shown in Fig. 2(c). The x-intercept gives the value of the demagnetization factor along the easy magnetization axis [28, 29]. Due to the existence of domain pinning, the curves at low temperatures tend to shift to the right. Therefore, we used the x-intercept at $170$ K, where domain pinning is negligible, as the demagnetization factor and obtained $N_{c}=0.608$. $M_{\mathrm{s}}$ is the spontaneous magnetization, which is obtained from the extrapolation to zero field of $M(H)$ curves, as shown in Fig. 4 (c) and (d). The magnetization of the sample can be related to the fraction of up domains $n_{\uparrow}$, where $M=(2n_{\uparrow}-1)M_{\mathrm{s}}$. From $H_{\mathrm{applied}}=N_{c}M$, we can determine $n_{\uparrow}$ as follows:

The applied field, $H_{\mathrm{applied}}$, and the demagnetization factor, $N_{c}$, are temperature independent. Equation 1 gives us the result that, without pinning, the fraction of up domains should decrease as temperature decreases because $M_{\mathrm{s}}$ increases.

In Fig. 2(a) we compare the calculated fraction of magnetically aligned (up) domains with the value deduced from the experimental magnetization. The blue and red curves are the temperature dependence of the up-domain fraction extracted from the FCC and FCW magnetization respectively at $100$ Oe, with the following formula:

The black curve is the fraction of up-domain calculated from a demagnetization theory [30] in Eq. 1. We can see that domain wall pinning keeps the up-domain fraction above the theoretical value for all but the temperatures near $T_{C}$. While cooling down below $T_{C}$, the system encounters a domain-wall pinning effect where the blue curve begins to deviate from the black curve. This domain-wall pinning causes the magnetization to increase nearly proportionally to $M_{s}$ as shown by the blue curve in Fig. 2(b). Below around $132$ K, the system enters a partial depinning region where the up-domain fraction decreases towards its theoretical no pinning value. The magnetization consequently decreases, which causes a local maximum near $132$ K. Upon further cooling, there is a competition between this anomalous depinning effect and the reduced thermal energy which tends to favor pinning. The reduced thermal energy eventually dominates around $120$ K and domain walls tend to pin again. Therefore, the magnetization tends to be proportional to $M_{\mathrm{s}}$, i.e. $M=\delta M_{\mathrm{s}}$ with $\delta=0.31$. The repinning is consistent with the fact that domain walls in Co₃Sn₂S₂ rarely move below 119 K as reported by a Lorentz microscopy study [31]. Upon warming up, as shown by the red curve in Fig. 2, since domain walls are pinned, the domain fraction remains unchanged until around 155 K, where the increased thermal energy allows more up domains to form. Due to the presence of domain-wall pinning, the total magnetization of the sample when cooled down and warmed up follows two different pathways, resulting in the hysteresis between the FCC and FCW magnetization curves.

For evidence of domain-wall depinning effects, we measured hysteresis loops with the applied field only up to $100$ Oe at various temperatures. An example is shown in Fig. 3a. We note that domain structures depend on defects and the shape of a sample [32, 33]. For this reason, we use the same sample for all the bulk magnetization measurements presented in this article. The virgin curve at $130$ K remains flat at first, then increases with a constant slope, which is the hallmark of a domain-wall pinning-type virgin curve[30]. The magnetization of the field-increasing part of the hysteresis loop is smaller than that of the virgin curve, indicating that the domains encounter some pinning effect. Asymmetry between positive and negative coercive field emerges up to $160$ K. The asymmetry can be attributed to the Exchange Bias (EB), which has been observed by other groups [19, 34], or the remanent field during the measurement. At $130$ K, the negative coercive field is $H_{\mathrm{c}}^{-}=-34.3(5)$ Oe, while the positive coercive field is $H_{\mathrm{c}}^{+}=42.1(3)$ Oe, and the actual coercive field, $H_{\mathrm{c}}$, is taken to be the average of the absolute value of the two. The nucleation field $H_{\mathrm{nuc}}^{\downarrow}$, below which the down domains start to form, was obtained as shown in Fig. 3(a). If there were no domain-wall pinning, the $M(H)$ curve should be a straight line. Therefore, the coercive field should always be zero and the nucleation field $H_{\mathrm{nuc}}^{\downarrow}$ should be the highest field of the hysteresis loop, i.e., $100$ Oe. However, Fig. 3(b) reveals a local maximum of $H_{\mathrm{c}}$ and a local minimum of $H_{\mathrm{nuc}}^{\downarrow}$ at $132$ K, below which $H_{\mathrm{c}}$ starts to decrease and $H_{\mathrm{nuc}}^{\downarrow}$ starts to increase, indicating a partial domain-wall depinning. The domain wall depinning around 130 K is consistent with the high mobility of domain walls observed near 130 K with a magnetic force microscope [22]. Further below $120$ K, domain-wall pinning strength increases again. $H_{\mathrm{c}}$ consequently increases and $H_{\mathrm{nuc}}^{\downarrow}$ decreases. We note that the local maximum disappears at fields above $300$ Oe. As shown in Fig 5, $300$ Oe is the largest value of the nucleation field when this sample is fully magnetized. Therefore, down domains cannot form above $300$ Oe and the local maximum disappears. The actual value of the nucleation field depends on the shape of the sample and defects [32, 33]. Although the field above which the local maximum in the FCC curve disappears varies among different samples, the depinning around $130$ K is a general observation in this compound [18, 10, 11, 12, 13, 14, 15, 16]. While domain-wall pinning upon cooling is very common in many systems, depinning upon cooling is anomalous. The origin of the depinning, however, is still under investigation. It is possible that the depinning is enabled by the recently observed local structure distortion [20] which can reduce the local anisotropy energy and thus allows domains to flip, causing the depinning around $130$ K.

Since the sample is single domain at high fields, one can wonder if there is still evidence for the magnetic anomaly in the hysteresis loops up to 7 T. As shown in Fig. 4, the spontaneous magnetization, $M_{\mathrm{s}}$, is about 0.3 $\mu_{B}$/Co at $5$ K [6, 25]. We also found that increasing temperature causes the shape of the hysteresis loop to change from a rectangular shape (5 K), to a triangular-tailed rectangle (100 K), to a bi-triangular shape (132 K), until the hysteresis disappears completely (160 K). The change from triangular-tailed-rectangle to rectangular shape of hysteresis loops can be explained with a domain-wall motion model. As shown in Fig. 5, the magnitude of the nucleation field increases as cooling down, until, around $85$ K, it exceeds the magnitude of the demagnetization field, causing the hysteresis loop changes from a triangular-tailed shape to a rectangular shape [35]. While there is no minimum in the nucleation field, we find that the nucleation field changes its sign around $130$ K, causing the hysteresis loop changes from a bi-triangular shape to a triangular-tailed shape.

Except for a recent MOKE study reported by Lee and coworkers  [21], domain structures of Co₃Sn₂S₂(0001) single crystal samples are largely unknown. By measuring spatially resolved magnetization through the Kerr effect as a function of temperature, we can examine whether the magnetic anomaly is intrinsic to the physics in a single domain, or, as we suggested previously, is due to the presence of multiple domains and domain walls. Such detailed information is elusive from bulk measurements averaged over the entire sample.

To examine the domain structures near $T=130$ K, we acquired a set of MOKE images at $140$ K, $136$ K, $132$ K, $128$ K, $124$ K, and $120$ K after the sample is field cooled in a magnetic field of H${}_{\parallel}=+94$ Oe. These images are shown in Fig. 6. Other than the disappearance of a small embedded red domain in the upper left corner (between $136$ K and $132$ K) and that of an embedded blue domain to its immediate right (between $128$ K and $124$ K), the domain structure remains essentially unchanged between $140$ K and $120$ K. Limited by the spatial resolution of the scanning microscope ($0.85\,\mu$m), from these MOKE images alone, we cannot rule out transitions that involve domain walls, such as those proposed in [21].

We extract the fraction of up domains, down domains, and domain walls from these images as follows. First, we measure the average absolute Kerr rotation of up and down domains. Then, we count points above $75\%$ of the average value as up or down domains, and points below that value as domain walls. Fig. 7 shows the evolution of these fractions as a function of temperature. As the sample is cooled down, the fraction of up domains shrinks while that of down domains grows due to the demagnetization effect. The fact that the temperature variation of the domain fractions follows the same trend as the evolution expected from the case with no pinning confirms that the magnetic domain walls are indeed able to move between 120K and 140K.