Chiral domain walls of Mn$_3$Sn and their memory

TL;DR

This paper investigates the unique chiral domain walls in Mn$_3$Sn, revealing their contribution to Hall effects and magnetization, and demonstrates control over their chirality through magnetic field history.

Contribution

It uncovers the chiral nature of domain walls in Mn$_3$Sn and their impact on Hall conductivity and magnetization, introducing new insights into antiferromagnetic domain wall behavior.

Findings

Domain walls generate additional Hall conductivity contributions.

In-plane spin rotation explains observed effects.

Chirality of domain walls depends on magnetic history.

Abstract

Magnetic domain walls are topological solitons whose internal structure is set by competing energies which sculpt them. In common ferromagnets, domain walls are known to be of either Bloch or N\'eel types. Little is established in the case of Mn$_3$Sn, a triangular antiferromagnet with a large room-temperature anomalous Hall effect, where domain nucleation is triggered by a well-defined threshold magnetic field. Here, we show that the domain walls of this system generate an additional contribution to the Hall conductivity tensor and a transverse magnetization. The former is an electric field lying in the same plane with the magnetic field and electric current and therefore a planar Hall effect. We demonstrate that in-plane rotation of spins inside the domain wall would explain both observations and the clockwise or anticlockwise chirality of the walls depends on the history of the field orientation and can be controlled.

Tables

Table 1. Supplementary table 1: Size and the aspect ratio l y / l x subscript 𝑙 𝑦 subscript 𝑙 𝑥 l_{y}/l_{x} of six different samples used in this work.

	$l_x$(mm)	$l_y$(mm)	$l_z$(mm)	$l_y/l_x$
#5	0.5	0.6	2	1.2
#13-1	0.5	0.54	0.66	1.08
#13-2	0.28	0.54	0.66	1.93
#13-3	0.16	0.54	0.66	3.38
#15	1	0.2	1.8	0.2
#triangle	0.63	0.73	1.8	1.16

Full text

I Introduction

A domain wall is the topological defect of a discrete symmetry. In ferromagnetic materials, these are narrow boundaries separating magnetic domains with different polarities. Their width and structure are set by the competition between the exchange energy and the magneto-crystalline anisotropy energy Getzlaff . They are either of Bloch type, where the the magnetization rotates in a plane parallel to the wall plane, or of Néel type, whose magnetization vector rotates in a plane perpendicular to the wall. Thanks to high-resolution scanning probes of local magnetization, they can be visualized Tetienne . Theoretical proposals for other more sophisticated spin textures have recently emergedCheng . In addition to their fundamental interest, the attention to domain walls is driven by the quest for new spintronic devices Stamps . Much less is known about antiferromagnetic domain walls.

A large anomalous Hall effect (AHE) was recently discovered Nakatsuji ; Nayak ; Kiyohara in the Mn3X(X=Sn,Ge) family of non-collinear antiferromagnets Zimmer ; Tomiyoshi1982 ; Tomiyoshi1982b ; Yang2017 . The discovery followed theoretical predictions Chen2014 ; Kubler2014 and preceded the observation of a variety of other anomalous transverse responses by thermal and optical probes Ikhlas2017 ; Li2017 ; Higo2018 ; Xu2018 ; Balk2019 ; Wuttke2019 ; Sugii2019 . These materials constitute new platforms for antiferromagnetic spintronics Zhang2017 ; Smejkal2018 . The structure of domain walls have been a subject of theoretical Liu2017 and experimental studies Li2018 . Evidence and arguments for a non-trivial spin texture in domain walls are available, but no direct image of their magnetic structure, yet.

Here, we report on three distinct experimental observations leading us to identify the in-plane structure of the domain walls in Mn3Sn. The first observation is that in the narrow magnetic field window of multiple domains, there is a planar Hall effect (PHE) which consists in an electric field oriented parallel (and not perpendicular) to the applied magnetic field. The thermoelectric counterpart of this effect, namely a planar Nernst effect (PNE) was also detected. The second observation is the existence of a transverse magnetic response in the same narrow field window. Employing micron-size Hall sensors in close proximity with the sampleBehnia2000 ; Collignon2017 , we monitored the local magnetic field at the surface and found in the same field window a finite off-diagonal magnetization: a finite magnetization oriented perpendicular to the orientation of the applied magnetic field. We will argue below that a satisfactory explanation of both these observations is provided by a specific spin texture inside the domain walls. The third result is that the sign of the emergent electric field (set by the clock-wise or anti-clockwise rotation of the spins inside walls) depends on the history of the magnetic field orientation. We will show that this is caused by residual minority domains promoting a specific chirality. This last observation constitutes a new case of memory formation in condensed matter recording a direction Keim2018 .

II Results

II.1 Planar Hall effect and Planar Nernst effect

Fig. 1 shows an additional hitherto unreported component in the Hall and the Nernst responses of Mn3Sn, which we call planar Hall effect (PHE) and planar Nernst effect (PNE). The experimental configuration is sketched in Fig. 1(a). Charge current was applied along the $z$-axis ($J//z$) and the magnetic field was oriented along the y-axis ($H//y$). Electric field was measured simultaneously along both $x$- and $y$-axes. $E_{x}$, which represents an electric field vector perpendicular the magnetic field and the charge current, is the Hall response. As seen in Fig. 1(b), it displays a hysteretic jump as reported previously Nakatsuji ; Li2017 ; Li2018 . As the magnetic field is swept, three different regimes succeed each other Li2018 . In regime I, the system hosts one single domain. When the applied magnetic field (opposite to the magnetization of the dominant domain) exceeds a threshold, new domains nucleate and regime II starts. At sufficiently large magnetic field, the system becomes single-domain again (regime III). As seen in (Fig. 1c), in regime II, $E_{y}$, the component of the electric field parallel to the magnetic field, becomes finite. The result was reproduced in several other samples and was also detected when the applied magnetic field was along the $x$-axis, see Supplementary Figure 2. In other words,in the presence of multiple domains, when $J//z$ and $H//y(//x)$, there is a non-vanishing $E_{y}$ ($E_{x}$). This is a planar Hall effect, with an electric field, which is parallel and not perpendicular to the magnetic field. Note that this signal only emerges in the presence of domain walls. Its amplitude is comparable to the amplitude of the topological Hall effect (THE) Fig. 1(d) extracted by subtracting Hall and magnetization hysteresis loops Li2018 , see Supplementary Note 4. Interestingly, the THE is present in the same field interval as the PHE, but shows different signs for the two sweeping orientations.

The experimental configuration for probing the Nernst response is shown in Fig. 1e. The thermal gradient is applied along the $z$-axis. When the magnetic field is oriented along the $y$-axis, there is a finite $E_{x}$. It represents the anomalous Nernst effect, which also displays a hysteretic jump (Fig. 1f), as reported previously Ikhlas2017 ; Li2017 . In addition to this, however, when the magnetic field is along the $x$-axis, there is a finite $E_{x}$ in regime II (Fig. 1g). This is the planar Nernst effect (PNE). Like its Hall counterpart, it becomes non-zero in a narrow field window when there are multiple domains and its amplitude is comparable to the amplitude of the topological Nernst effect (TNE) (Fig. 1h) extracted by subtracting Nernst and magnetization hysteresis loops, see Supplementary Note 4.

We carried out an extensive set of temperature-dependent measurements, see Supplementary Figure 4. In the whole temperature window of the triangular order in Mn3Sn ($50K<T<300K$), the magnitude of PHE (PNE) remain a sizable fraction ($\approx 0.3-0.4$) of the total AHE (ANE) and there is no significant evolution with temperature. We will show below how the PHE, the PNE and their odd parity in field, are set by the internal structure of domain walls Liu2017 in this system.

II.2 Magnetization (bulk vs. surface; longitudinal vs. transverse)

Fig. 2 presents the data magnetization obtained in two different ways. In addition to measuring bulk magnetization with a conventional vibrating sample magnetometer (VSM), we used two-dimensional electron gas (2DEG) Hall sensors, attached to one edge of the sample to monitor the local magnetic field at its surface (See method). By choosing the mutual orientation of the sensor and the applied magnetic field, we could extract both diagonal and off-diagonal magnetization at the surface of the sample.

As seen in Fig. 2a and Fig. 2b, the hysteresis loop of bulk magnetization depends on the aspect ratio $l_{y}/l_{x}$, where $l_{y}(l_{x}$) is the length of the sample along the $y$-axis($x$-axis). When $l_{y}/l_{x}\approx 1$ [inset.(a)], bulk magnetization for the field along two orientations are almost coincident. But when $l_{y}/l_{x}\approx 3$, the hysteresis loop is wider when the field is oriented along the longer axis, in agreement with that was reported previously Li2017 . As seen in the Fig. 2a, domain nucleation occurs at the same magnetic field for the two orientations, but the loop closes later when the field is oriented along the longer axis. A straightforward interpretation of this observation is that the new domain(s) occupy the whole sample more efficiently when the magnetic field is oriented along a shorter axis.

Additional insight is brought by surface magnetization data obtained with Hall sensors. As shown in Fig. 2c and Fig. 2d, no matter the sample’s aspect ratio, the hysteresis loop of surface magnetization is always narrow. The surface magnetization shows a sharp jump at the threshold field of bulk magnetization. We conclude that when the field is oriented along $y$- ($x$-) axis, the new domains nucleate at the $xz(yz)$- surface and immediately occupy the area ($5\times 5\mu m^{2}$) probed by a Hall sensor. The wide hysteresis loop of the bulk magnetization monitors the gradual enhancement produced by the smooth occupation of the center of the sample.

We used the Hall sensors to look for an off-diagonal magnetic response, namely a finite magnetic field perpendicular to the applied field. The mutual configuration of the sample, the magnetic field and the Hall sensors for quantifying longitudinal and transverse magnetization are shown in [inset.(e)] and [inset.(f)]. The obtained data at room temperature is shown in Fig. 2e and Fig. 2f. The transverse response is restricted to regime II and has symmetric and asymmetric components.

II.3 Chiral domain walls

A spin texture for domain walls (see supplemental material in Liu2017 ), which would explain our results, is sketched in Fig. 3. One domain (oriented along $\theta=\rm{\pi}$) is located at the center and another domain with opposite polarity ($\theta=0$) at the periphery. [In the convention used here Liu2017 , $\theta$ is the angle between the $x$-axis and a pair of parallel spins of the unit cell.] In the (more or less thick) wall separating these two domains, spins rotate smoothly and concomitantly in the $x-y$ plane. The texture along $x$-axis is such that at the center of the domain wall, the adopted configuration has an orientation perpendicular to the two domains. Fig. 3b shows different versions of the same structure with a narrower wall. One can see that the two possible configurations are $+\rm{\pi}/2$ and $-\rm{\pi}/2$. This would correspond to an either clockwise or anticlockwise rotation of spins depending on the specific domain configuration at the center and the periphery. Note that domain walls of this type, with in-plane rotation of two possible signs, follow directly from the hierarchy of scales discussed in Liu2017 , in which the Dzyaloshinskii-Moriya (DM) interaction is much stronger than an in-plane two-fold anisotropy. The origin of the two-fold anisotropy will be discussed in future work.

We note that a study using Magneto-Optical Kerr Effect (MOKE) microscopy Higo2018 detected oppositely aligned domains in the multidoamin regime at small magnetic fields. The domains were found to extend over tens of microns. However, the fine structure of the walls separating these domains Liu2017 could not be resolved in this study.

Such a texture would provide a natural explanation for the transverse magnetization (TM) and the planar Hall effect observed in regime II. The in-plane tilt of spins (and the magnetic octupole Suzuki2017 ) would generate a magnetic field perpendicular to and electric field parallel to the orientation of the applied magnetic field. This is the origin of the transverse magnetization and planar Hall effect. The angle-dependent study of the AHE Li2018 has established that the orientation of the electric field associated with anomalous Hall effect is set by the orientation of spins (and not the crystal axes). Therefore, the $\rm{\pi}/2$ spin configuration in the center of domain wall would naturally gives rise to an electric field perpendicular to those generated by the $\theta=0$ and $\theta=\rm{\pi}$ domains.

In this picture, the sign of the signals reflects the chirality of the domain wall. Consider a hysteresis loop with the magnetic field swept from a $\theta=\rm{\pi}$ single-domain to another $\theta=0$ single domain regime and then back to the original $\theta=\rm{\pi}$ single-domain (Fig. 3c). If during this sequence, for both sweeping orientations, the spin configuration inside the domain walls remains the same (either +$\rm{\pi}/2$ or -$\rm{\pi}/2$), then the PHE and the TM signals will be even (symmetric) in field. On the other hand, if what remains fixed is the sense of the rotation (clockwise or anticlockwise), then the signals will be odd (or asymmetric) in field, because the spin configuration inside the domain wall will be opposite during the two sweeps.

II.4 Domain walls have a memory

Keeping this in mind, let us turn our attention to another outcome of this study, a memory effect. The experimental protocol is defined in (fig. 4a). We performed the measurement twice for identical configurations, but with different prior histories. The measurement consisted in sweeping the magnetic field oriented along $x$-axis from 0.5 T to -0.5 T and back. This corresponds to switching domains from $\theta=0$ to $\theta=\rm{\pi}$ configurations and back to the starting point. The measurement was preceded in the first case by a field rotation from $+\rm{\pi}/2$ to 0 and in the second case, by a rotation from $-\rm{\pi}/2$ to 0. As one can see in fig. 4b), the results are strikingly different. In the first case the PHE signals are positive, in the second are negative. We note that this is a phenomenon belonging to the category dubbed memory of direction Keim2018 . By subtracting the two sets of data or adding them, one can extract the symmetric (fig. 4c) and the asymmetric (fig. 4d) components of the PHE signal. The symmetric part is seven times larger than the asymmetrical part.

Note the small gap seen between the two sets of $\rho^{\rm{PHE}}_{xz}$ data obtained with two different prior histories in Fig. 4b. It arises because we have assumed an identical offset for both sets of data. This offset is caused by an unavoidable misalignment between lateral contacts. The difference between the two sets of data obtained with different prior histories implies that history affects the offset too.

The transverse magnetization displays also a similar memory (See the protocol defined in Fig. 4e and the data shown in Fig. 4f-h). One can see, however, that in this case, the main component is asymmetric, which is three times larger than the symmetric one.

III Discussion

Recalling that PHE is a bulk effect, we conclude that the orientation of the spins inside the walls is mainly set by the past history. On the other hand, in the case of transverse magnetization at the surface, the spin orientation mainly depends on the sign of the magnetic field and the rotation orientation is less affected by the prior history. This raises an obvious question: Where does the system stock the information regarding the previous orientation of the magnetic field?

A plausible answer to this question is provided by the scenario sketched above. When the magnetic field is oriented along $\theta=+\rm{\pi}/2$, at the end of a ($-y\leftrightarrow+y$) hysteretic loop, the sample is practically single-domain with $\rm{\pi}/2$ spin configuration. In principle rotating from $\rm{\pi}/2$ to [math] before the measurement would change the spin configuration of the whole sample from $\rm{\pi}/2$ to [math]. However, if residual domains remain stuck in the $\rm{\pi}/2$ configuration, they will set the spin configuration of the domain walls along $\rm{\pi}/2$. If this is the case, then one would expect to see a dependence of the memory effect on the strength of the prior magnetic field. The larger the magnetic field at which the ($\rm{\pi}/2$ to [math]) rotation takes place before the measurement, the smaller the fraction of the domains which had stayed in place and the smaller their role in setting the chirality.

As seen in Fig. 5, this is indeed the case. We measured PHE after cycling and rotating the magnetic field at B= 0.5 T, 1 T, 2 T. One can see that the magnitude of the PHE and in its symmetric component steadily decreases. This implies that the symmetric component of the PHE set by the chirality of the wall is promoted by the presence of minority domains, whose population decrease with increasing magnetic field. The asymmetric component, on other hand, does not show significant evolution with magnetic field.

If domain walls with opposite chiralities were evenly distributed in the sample no PHE or transverse magnetization signal would have been observed. This is not the case. The dominance of a symmetric and history-dependent component in the PHE signal implies that deep inside the sample, minority domains set the chirality of the domain wall. The dominance of the asymmetric and history-independent component in surface transverse magnetization indicates that wall spin orientation at surface is principally set by the orientation of the magnetic field with only a minor role proposed by the minority domains. We note that the domain wall spin texture proposed here can also generate a topological Hall response as reported previously Li2018 , provided that we assume an additional off-plane tilt of spins residing inside the domain walls. Indeed, if the unit vector of magnetization has a finite z dependence ($\frac{\partial\overrightarrow{n}}{\partial z}\neq 0$), then combined with the finite $\frac{\partial\overrightarrow{n}}{\partial r}$, it generates an axially oriented emergent magnetic field (B${}_{\theta}\neq 0$ in cylindrical coordinates) Everschor2014 and the skyrmionic number will be finite, producing real-space Berry curvature. Such an assumption would not alter the conclusions drawn above. Yet, it is not necessary for explaining the observations reported in the present study.

Heating the sample above $T_{\rm{N}}=$ 420 K would presumably erase all history dependence. It would be interesting to compare field-cooled and zero-field-cooled behaviors across the transition temperature in future experiments combining a furnace and a magnet.

In summary we put under scrutiny a narrow field window in which there are multiple magnetic domains in Mn3Sn and found that in this regime, one can observe a planar Hall and planer Nernst effect as well as transverse magnetization. These observations can be explained by a specific spin texture for domain walls where spins rotate in the pseudo-Kagomé plane. The choice of clockwise or anti-clockwise rotation can be controlled by the prior magnetic history of the sample, providing a new platform for memory formation.

IV Methods

Sample preparation and Transport measurements: Mn3Sn single crystals with a typical size in the range of centimeter were grown by the vertical Bridgman technique [24]. They were cut to desired dimensions by a wire saw. All transport experiments were performed in a commercial measurement system (Quantum Design PPMS), using the Horizontal Rotator Option. Hall resistivity was measured by a standard four-probe method using a current source and a DC-nanovoltmeter. Two Chromel- Constantan (type E) thermocouples were employed to measure the temperature difference in the case of Nernst measurements.

magnetization: Bulk magnetization was measured using a vibrating sample magnetometer (VSM) method. For surface magnetization measurements, we employed an array of Hall sensors based on high-mobility AlGaAs/GaAs heterostructure; The density of the two-dimensional electron gas (2DEG) was n = 2.5 $\times$ 10${}^{1}1$cm*-2* (300 K) and it was located 160 nm below the surface. The device was fabricated using electron beam lithography and 250 V argon ions to define the mesa. Supplementary Figure 1 shows an array of ten sensors each $5\times 5\mu\rm{m}^{2}$ square with a 100 $\mu$m interval between two neighboring sensors Collignon2017 . Attaching the device to the surface of the sample, the local magnetic field was determined by measuring the Hall resistivity of the sensor using an AC current source and a lock-in amplifier.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

• zengwei.zhu@hust.edu.cn

• kamran.behnia@espci.fr

Supplementary Note 1 Experimental methods

For surface magnetization measurements, we employed an array of Hall sensors based on high-mobility AlGaAs/GaAs heterostructure; The density of the two-dimensional electron gas (2DEG) was $n=2.5\times 10^{11}$cm*-2* (300K) and it was located 160 nm below the surface. The device was fabricated using electron beam lithography and 250 V argon ions to define the mesa.  Supplementary Figure 1 shows an array of ten sensors each $5\times 5\mu$m2 square with a 100 $\mu$m interval between two neighboring sensors Collignon2017 . Attaching the device to the surface of the sample, the local magnetic field was determined by measuring the Hall resistivity of the sensor using an AC current source and a lock-in amplifier.

Supplementary Note 2 Sample details

For different experiments, six samples were used in this work with different aspect ratios ($l_{y}/l_{x}$) and cross-section shapes. The list of samples is given in Supplementary table 1. Samples #5 and #13-1 have square cross-section samples and aspect ratio ($l_{y}/l_{x}$) is close to unity. Samples #13-2, #13-3 and #5 have rectangular cross sections with aspect ratio ($l_{y}/l_{x}$) deviating from unity; The sample dubbed #triangle, has a equilateral triangle as cross section, with three sides along y axis.

Supplementary Note 3 Planar Hall effect (PHE) in different samples

The existence of the planar Hall effect was reproduced in more than three samples and with different set-ups and cross-sections ( Supplementary Figure 2).  Supplementary Figure 2a and  Supplementary Figure 2b represent anomalous and planar Hall resistivity in sample #5 with the magnetic field along x axis, and the electric field measured simultaneously along both $x$- and $y$-axes.  Supplementary Figure 2c and  Supplementary Figure 2d show anomalous and planar Hall resistivity in sample #15, the one in which the Nernst data was shown in the main text. The electric field $E_{x}$ was measured for two different orientations of the magnetic field.  Supplementary Figure 2e and  Supplementary Figure 2f show the Hall data in a sample with triangle cross-section. The planar Hall effect is present in all these three samples and the ratio ($\rho^{\rm{PHE}}$/$\rho^{\rm{AHE}}$) is always between 0.3 to 0.4. It’s worth noting that the width of the regime II is different and depended on the aspect ratio $l_{y}/l_{x}$ Li2018 .

Supplementary Note 4 Extraction of topological Hall effect (THE) and topological Nernst effect (TNE)

Supplementary Figure 3a and  Supplementary Figure 3d compares the hysteretic loops of the Hall and the Nernst response with the magnetization.  Supplementary Figure 3b and  Supplementary Figure 3e show the same comparison between normalized signals (after subtracting the high-field slope). One can see that the two responses do not scale with each other in regime II.  Supplementary Figure 3c and  Supplementary Figure 3f show $\rho^{\rm{THE}}_{xz}(\mu_{0}H)=\rho^{\rm{A}}_{xz}(\mu_{0}H)-C(M(\mu_{0}H)-\mu_{0}H\chi)$ and $S^{\rm{TNE}}_{xz}(\mu_{0}H)=S^{\rm{A}}_{xz}(\mu_{0}H)-C^{*}(M(\mu_{0}H)-\mu_{0}H\chi)$ , where $\chi$ is the high-field susceptibility (the slope of the magnetization outside the hysteresis loop), $C=\rho^{\rm{A}}_{xz}(\mu_{0}H=0T)/M(\mu_{0}H=0T)$ and $C^{*}=S^{\rm{A}}_{xz}(\mu_{0}H=0T)/M(\mu_{0}H=0T)$ are constant two fitting constants.

Supplementary Note 5 Temperature dependence of PHE and PNE

Supplementary Figure 4, shows the evolution of PHE and PNE as the temperature changes from 300 K to 50 K. In the whole temperature range studied, the ratios of both PHE(THE) and PNE(TNE) to AHE and ANE remain constant.

Supplementary Note 6 Spin texture with rectangular boundaries

Supplementary Figure 5, shows two domains separated by a domain wall in a rectangular configurationLiu2017 .