Order parameter dynamics in Mn₃Sn driven by DC and pulsed spin–orbit torques

Antiferromagnets (AFMs) are a class of magnetically ordered materials that exhibit negligible net magnetization owing to the unique arrangement of strongly exchange-coupled spins on the atoms of their unit cells. As a result, AFMs produce negligible stray fields and are robust to external magnetic field perturbations, and their precession frequency, set by the geometric mean of exchange and anisotropy energies, is in the terahertz (THz) regime.^1–3 The past decade has witnessed a rapid rise in theoretical and experimental research focused on the fundamental understanding and applications of AFM materials as active spintronic device elements.^3–8 There exists a broad range of AFM materials, including insulators, metals, and semiconductors with unique properties that could be exploited to realize magnonic devices,⁹ high-frequency signal generators and detectors,^5,6,10–15 and non-volatile memory.^16–18 For example, insulators such as NiO^19,20 and MnF₂²¹ are well studied and have the potential to carry charge-less spin waves or magnons. Insulating AFM Cr₂O₃ shows magnetoelectricity below its Néel temperature of 307 K, which was exploited to demonstrate voltage-controlled exchange-bias memory and fully electrically controlled memory devices.²² On the other hand, metallic AFMs have been mostly used as sources of exchange bias in spin valves and tunnel junction-based devices.^23,24 More recently, there has been significant research activity in non-collinear, chiral metallic AFMs of the form Mn₃X with a triangular spin structure and several intriguing magneto-transport characteristics, such as a large spin Hall effect (SHE),²⁵ anomalous Hall effect (AHE),^26–28 and ferromagnet-like spin-polarized currents.²⁹ 

Negative chirality materials, such as Mn₃Sn, Mn₃Ge, and Mn₃Ga, perhaps best represent the promise of non-collinear metallic AFMs with a potential for ferromagnet-like spintronic devices in which the order parameter could be fully electrically controlled and read-out.^30–32 In a recent experiment,³³ conducted in a bilayer of heavy metal and Mn₃Sn, a characteristic fluctuation of the Hall resistance was measured in response to a DC current in the heavy metal. This observation could be explained in terms of the rotation of the chiral spin structure of Mn₃Sn driven by spin–orbit torque (SOT). Pal et al. and Krishnaswamy et al. independently showed that Mn₃Sn layers thicker than the spin diffusion length could be switched by seeded SOTs.^34,35 Here, the SOT sets the spin texture of the AFM in a thin layer at the interface, which acts as the seed for the subsequent setting of the domain configuration of the entire layer. The seeded SOT also requires bringing the temperature of the AFM above its ordering temperature and then cooling it in the presence of the SOT generated in a proximal heavy metal layer. Very recently, a tunneling magnetoresistance (TMR) of ∼2% at room temperature in an all antiferromagnetic tunnel junction consisting of Mn₃Sn/MgO/Mn₃Sn was experimentally measured.³⁶ The TMR in Mn₃Sn originates from the time reversal symmetry breaking and the momentum-dependent spin splitting of bands in the crystal. These recent works highlight the tremendous potential of Mn₃Sn and other negative chirality AFMs to explore and develop spintronic device concepts.

In this paper, we discuss the energy landscape of a thin film of Mn₃Sn in the mono-domain limit and deduce the weak six-fold magnetic anisotropy of the film via perturbation and numerical solutions (Sec. III). Consequences of the sixfold anisotropy on the equilibrium states, the origin of the weak ferromagnetic moment, and SOT-induced non-equilibrium dynamics are carefully modeled in Secs. III and IV. The analytic model of the threshold spin current to drive the system into steady-state oscillations is validated against numerical simulations of the equation of motion. Because of the weak in-plane magnetic anisotropy, on the order of 100 J/m³, we find that the critical spin current to induce dynamic instability of the order parameter in Mn₃Sn could be orders of magnitude lower than that in other AFMs, such as NiO^5,6,12 and Cr₂O₃.¹² Additionally, the oscillation frequency of the order parameter in Mn₃Sn could, in principle, be tuned from the gigahertz (GHz) to the terahertz (THz) scale.¹⁴ We also examine the response of the system when subject to pulsed spin current. Our results show that by carefully tuning the pulse width of the current (t_pw) and the spin charge density injected into the system (J_st_pw, where J_s is the spin current density), the evolution of the order parameter across different energy basins, separated by 60° in the phase space, can be controlled. Our results of the SOT-driven dynamics in single-domain Mn₃Sn are closely related to the experimental results of Ref. 33. Previous theoretical works on the current-driven dynamics in collinear as well as non-collinear AFMs have only addressed systems with zero net magnetization and those with two-fold anisotropy.^14,37,38 Another work focusing on NiO with sixfold anisotropy had only presented numerical aspects of pulsed SOT switching.³⁹ On the other hand, here, we address both the numerical and analytical aspect of current-driven dynamics in a sixfold anisotropic system with a small non-zero net magnetization. We also discuss the AHE and TMR detection schemes and present the magnitude of output voltages for typical material parameters in both the cases (Sec. V). Finally, we highlight the impact of thermal effects on the dynamics of the order parameter and the temperature rise of the sample when DC spin currents are acting on Mn₃Sn to generate high-frequency oscillations (Sec. VI). Our models highlight the limits, potential, and opportunities of negative chirality metallic AFMs, prominently Mn₃Sn, for developing functional magnetic devices that are fully electrically controlled.

Mn₃Sn has a Néel temperature of ∼420 K and can only be stabilized in excess of Mn atoms. Below this temperature, it crystallizes into a layered hexagonal D0₁₉ structure, where the Mn atoms form a Kagome-type lattice in basal planes that are stacked along the c axis ([0001] direction), as shown in Fig. 1(a). In each plane, the Mn atoms are located at the corners of the hexagons, whereas the Sn atoms are located at their respective centers. The magnetic moments on the Mn atoms, on the other hand, have been shown by neutron diffraction experiments⁴⁰ to form a noncollinear triangular spin structure with negative chirality, as shown in Fig. 1(b). The nearest neighbor Mn moments (m₁, m₂, and m₃) are aligned at an angle of ∼120°, with respect to each other, resulting in a small net magnetic moment m. The weak ferromagnetism in Mn₃Sn, and similar antiferromagnets, such as Mn₃Ge and Mn₃GaN, has been attributed to the geometrical frustration of the triangular antiferromagnetic structure, which leads to slight canting of Mn spins toward in-plane easy axes.⁴¹ The chirality of the spin structure breaks time reversal symmetry and leads to momentum-dependent Berry curvature and modifies the magneto-transport signals in Mn₃Sn.^26,41 Switching from one chirality to another flips the sign of the Berry curvature and thus the sign of magneto-transport signals that are odd with respect to Berry curvature.

Previous theoretical works^42,43 posit the aforementioned crystal structure to a hierarchy of interactions typical for 3d transition metal ions—strong Heisenberg exchange, followed by weaker Dzyaloshinskii–Moriya (DM) interaction, with single-ion anisotropy being the weakest. Their model of the magnetic free energy interaction in a unit cell of bulk Mn₃Sn crystal, based on this hierarchy of interactions, predicts the existence of a small in-plane net magnetization, sixfold anisotropy degeneracy, and the deformation of the magnetic texture by an external magnetic field. The basic spin structure of the free energy model in the case of Mn₃Sn comprises two triangles, with three spins each, one on each basal plane.^42,43 While the free energy model of Ref. 42 explicitly considers an interplane exchange interaction in addition to the intraplane exchange interaction, the free energy model of Ref. 43 considers an effective exchange parameter, which includes both these effects. The latter approach simplifies the system of six spins to that of three spins. Therefore, in our work, we assume a free energy model based on the latter approach, viz., a system of three magnetic moments.^14,44

Compared to previous works discussing micromagnetic modeling of AFMs,^14,45 monodomain modeling is applicable for AFM particles that are smaller than the domain size, which is typically 200–250 nm for Mn₃Sn.^33,46 For such length scales, spatial variation in exchange and DM interactions can be safely ignored.⁴⁷ The details of the micromagnetic simulation framework, including the boundary conditions, can be consulted from our previous work.¹⁴ Since in this work we focus on the mono-domain limit, the cell dimensions are the same as the physical dimensions of the AFM layer. The thickness of the AFM film plays a role in establishing the spin angular momentum acting on the AFM from the proximal NM. In contrast to the continuum modeling approach presented in this work and in Ref. 33, some previous works have employed a three spin atomistic model.^34,46,48 Although the atomistic modeling approach is more accurate, it becomes computationally expensive for large systems of size 10’s of nm and above since the number of atoms and the associated spins increase. On the other hand, the continuum modeling framework is more suitable for large systems and has been shown to agree well with atomistic simulations in the case of another manganese-based AFM, Mn₂Au.⁴⁹ Therein, the material parameters used in the continuum framework were well calibrated against those in the atomistic model. The material parameters used in our work are also calibrated against an atomistic model as mentioned in Ref. 44.

In exchange energy dominant AFMs, such as Mn₃Sn, the hierarchy of energy interactions leads to J_E ≫ D ≫ K_e. As a result, the exchange energy is minimized if the three sublattice vectors are at an angle of $2π3$ with respect to each other. The minimization of the DM interaction energy confines the sublattice vectors to the x–y plane with a zero z-component and enforces a clockwise ordering between the vectors m₁, m₂, and m₃. The magnetocrystalline anisotropy energy would be minimized if each m_i coincides with its respective u_e,i. However, counterclockwise ordering of u_e,1, u_e,2, and u_e,3 along with the clockwise ordering of m₁, m₂, and m₃ implies that all the magnetization vectors cannot coincide with their respective easy axis simultaneously. This would lead to six equilibrium or minimum energy states wherein only of the sublattice vectors is coincident with its easy axis.

Minimizing Eq. (1) with respect to m_i leads to six equilibrium states, as shown in Fig. 2. They are given as $φmeq=nπ/3$⁠, where n = {0, 1, 2, 3, 4, 5}. In each case, only one of the three magnetization vectors coincides with its easy axis. A small in-plane average magnetization, $m=m1+m2+m33$⁠, is also obtained, depicted by the red arrows in Fig. 2. In each case, m coincides with the sublattice vector that is aligned along its easy axis. Our numerical calculations reveal the norm of the average magnetization to be equal but very small for all the six ground states, approximately $m≈3.66×10−3$⁠. Therefore, in Fig. 2, m is zoomed-in by 100× for the sake of clear presentation. The small non-zero value of $m$ suggests that the angle between the sublattice vectors is not exactly 120°, but deviates slightly. Defining this deviation as $ηij=cos−1(mi⋅mj)−2π3$⁠, we find that η_ij ≈ −0.36° if either of m_i or m_j is coincident with its easy axis, while η_ij ≈ 0.72° if neither m_i nor m_j coincides with its easy axes. This small canting of sublattice vectors is well known, as mentioned in previous theoretical and experimental works.^40,41,50 Our numerical results show that η₁₂ + η₂₃ + η₃₁ = 0.

The setup used for analyzing the dynamics of the antiferromagnet when subject to SOT is shown in Fig. 3. The spin current generated due to charge-to-spin conversion in the heavy metal is polarized along n_p, which coincides with the z axis, per our convention. The Kagome lattice of Mn₃Sn is formed in the x–y plane, while the z axis coincides with [0001] direction, as shown in Fig. 1(b). This setup resembles the experimental setup from Refs. 33 and 46, where an MgO (110)[001] substrate leads to the selective growth of Mn₃Sn in the [0001] direction. The effect of SOT on the AFM order is maximum in this setup since the sublattice vectors predominantly lie in the x–y plane with a small z-component, while the spin polarization is perpendicular to the x–y plane.³³ Other methods of spin injection, such as a spin-polarized electric current from a proximal ferromagnet, could also be used. As mentioned previously, our analysis of the SOT dynamics will be carried out in the single-domain limit.

For currents above the threshold current $(Js>Jsth)$⁠, the net energy pumped into the system overcomes the intrinsic barrier imposed by the effective anisotropy of the system. The z-component of the order parameter increases due to the SOT, as shown in Fig. 4(b). For the case of the spin polarization perpendicular to the Kagome plane, as considered in this work, an equal spin torque acts on all the sublattice vectors along the z-direction.^46,51 As a result, their z-components increase in magnitude, which, in turn, increases the z-component of the average magnetization since $mz=m1,z+m2,z+m3,z/3$⁠. Consequently, an effective exchange field (∝J_Em_z) arises in the z-direction, which leads to oscillation of the order parameters around the z axis with frequencies in the GHz–THz range.^14,44

If the input spin current is turned off during the course of the oscillation, the order parameter loses energy to the intrinsic damping of the AFM. As a result, m_z reduces to zero, while φ_m settles into the nearest equilibrium state, as shown in Fig. 4(c). Here, π/2 < φ_m < 5π/6 when the current is turned off; therefore, the order parameter settles into the $φmeq=2π/3$ equilibrium state. Comparing Figs. 4(b) and 4(c), one can conclude that it is possible to switch to all six states if the current pulse is switched off at an appropriate time. Indeed, the same is shown in Fig. 7, where different final states are achieved by changing the duration of the current pulse of magnitude J_s = 0.6 MA/cm². Another way to switch between stable states is by varying the magnitude of current pulse for a fixed duration. Figure 8 shows switching of the order parameter from $φmeq=0$ to $φmeq=π/3,2π/3$ and π as the magnitude of the input current is increased, while its duration is fixed at 100 ps. In Figs. 7 and 8, we only present switching in one direction, that is, from $φmeq=0$ through $φmeq=π$ since $φmeq=4π/3$ and $φmeq=5π/3$ could be achieved by changing the direction of input current. Finally, $mz(∝−φ̇m)$ increases in both magnitude and frequency with J_s [Fig. 8(b)] since $φ̇m$ increases with current [Eq. (11)].

This equation suggests that for large currents $(Js≫Jsth)$⁠, the final equilibrium state only depends on the input spin charge density (J_st_pw) as $Jstpw∼α2eℏMsdaγ(2n−1)π6$⁠. Such a large current density above the threshold current density corresponds to the small pulse width regime. For current magnitude near the threshold current $(Js≳Jsth)$ and large t_pw, the final state scales almost linearly with t_pw as $JsthtpwJsJsth2−1∼α2eℏMsdaγnπ3$⁠. To verify these claims, we solved Eq. (7) for t_pw in the range [1, 100] ps and spin charge density in the range [0, 1) C/m² and evaluated the final states. It can be observed from Fig. 9 that for small t_pw, the final states $(φmeq)$ are, indeed, only dependent on J_st_pw, more evidently for $φmeq=2π/3$ and π. This is because the input current is large for these cases. On the other hand, the final states for large pulse width clearly depend on t_pw. This behavior is more noticeable for $φmeq=π/3$ and $φmeq=2π/3$⁠, whereas for $φmeq=π$⁠, it would be evident for longer pulse widths. The overlaid dashed lines correspond to Eq. (15) for n = 1, 2, 3 and represent the minimum spin charge density required to switch from $φmeq=0$ to $φmeq=nπ/3$ as a function of the pulse width. The fluctuations of the Hall resistance under the effect of SOT in Ref. 33 were credited to the aforementioned pulsed dynamics. However, an expression describing the interdependence of J_s and t_pw required to switch from one known state to another known state, as a function of the material parameters, was missing.

To calculate the Hall voltage in our device setup, we consider ρ_HM = 43.8 µΩ cm and d_HM = 7 nm, corresponding to that of W (listed in Table I), while the other parameters are taken as $ρMn3Sn=330μΩ cm$⁠,^33, $dMn3Sn=4nm$⁠, and $|ρHMn3Sn|=3μΩcm$⁠.⁵⁴ If a read current of I_R = 50 µA is applied to the bilayer, a Hall voltage of V_H = 1.86 µV is generated in the HM layer. As indicated in Eq. (16), a larger read current would increase⁵³ the detected Hall voltage. Alternatively, a larger Hall resistivity of Mn₃Sn would be desired to generate larger read voltages. Since the read and write terminals in this device setup are the same, this setup is more suitable for detecting the switching dynamics.

An all antiferromagnetic tunnel junction built using two Mn₃Sn electrodes has recently been shown to exhibit a finite TMR at room temperature.³⁶ It was experimentally demonstrated that a TMR value of about 2% at room temperature was possible for thicker tunnel barrier (MgO) and large resistance-area (RA) product. The high (low) resistance state corresponds to an angle of π(0) between the magnetic octupoles of the two Mn₃Sn layers. The electronic bands in Mn₃Sn show momentum-dependent spin splitting of the Fermi surface due to the broken time reversal symmetry.^36,55 As a result, a spin-polarized current is generated. The relative orientation of the magnetic octupoles in the two AFM layers controls the magnitude of the spin-polarized current and, hence, the magnetoresistance.⁵⁵ A simple representation of an all-Mn₃Sn tunnel junction is shown in Fig. 11. Here, a thick layer of Mn₃Sn, with fixed orientation m_R, is used as the reference layer, while the order parameter m of the thinner layer is modulated using the device setup of Fig. 3. The different orientations of m with respect to m_R are measured as the read voltage, V_L, across a resistive lead, R_L, resulting in a finite read voltage, V_R.

Theoretical calculations have revealed that for vacuum as a tunnel barrier, RA product increased from ∼0.05 Ω μm² in the parallel configuration to 0.2 Ω μm² in the antiparallel configuration.⁵⁵ For the states corresponding to $φmeq=π/3$ and $φmeq=2π/3$⁠, the RA product is evaluated to be ∼0.07 and 0.11 Ω μm², respectively.⁵⁵ Assuming the same value of RA product in our device setup, we estimate the voltage drop across R_L = 50 Ω as V_L = 9.1 mV for the parallel configuration, while V_L = 7.1 mV for the antiparallel configuration of the free and fixed layers of the tunnel junction. On the other hand, V_L = 8.8 mV and V_L = 8.2 mV for $φmeq=π/3$ and $φmeq=2π/3$⁠, respectively. Here, we considered V_R = 10 mV, while the area of the tunnel barrier was assumed to be $A=100×100nm2$⁠. This setup can be used for detecting both the oscillation and switching dynamics.

When Mn₃Sn is driven by large DC SOT to excite oscillatory dynamics in the 100’s of GHz range, it is likely that Joule heating in the device structure would impose practical limits on the operating conditions. For example, our simulations show that we need ∼17 MA/cm² spin current to generate oscillations of the order parameter at 100 GHz in a 4 nm thick film of Mn₃Sn. Due to the spin Hall angle of the heavy metal being less than unity, the required charge current could be 2-20× higher than the spin current. The large charge current requirement will increase the temperature of the device structure, which, if significant, could alter the properties of the magnetic material or in extreme cases change the magnetic order completely. In addition to the magnitude of the input current pulse, the Joule heating in the device also depends on the electrical properties and the size of the heavy metal that the charge current passes through. The temperature rise of the device due to the generated heat depends on the ambient temperature and the thermal properties of the conducting (magnetic and non-magnetic) media and the substrate. Finally, in the pulsed response case, the time-domain temperature profile is expected to be sensitive to the duration of the applied current, and thus, the peak temperature in this case could be markedly different from that achieved when DC charge currents are used in the device.

A recent experimental investigation of field-assisted current-driven switching dynamics in Mn₃Sn attributes the switching process to Joule heating.^34,35 They show that despite the spin diffusion length in Mn₃Sn being extremely small (roughly 1 nm), the measured AHE signals showed switching behavior for films of thicknesses ranging from 30 to 100 nm. Subsequently, this switching dynamics was explained as a demagnetization of the AFM order in Mn₃Sn due to Joule heating over a time scale of 10’s of nanosecond (ns), followed by a remagnetization of the AFM order due to cooling in the presence of SOT. We must, however, point out that the SOT device setups in these experiments were different from our setup of Fig. 3. Unlike our case, the spin polarization associated with the SOT was in the Kagome plane, which increases the current requirement.^33,46 Nevertheless, an accurate investigation of the current-driven magnetization dynamics in our device setup requires a careful consideration of the temperature rise due to Joule heating.

Figure 12 shows the maximum rise in the junction temperature for different substrate thicknesses and DC input charge current densities. As expected, ΔT_M increases with both d_sub and J_c. An increase in J_c increases the heat generated in the metallic layers, whereas an increase in d_sub increases the thermal resistance in the direction of heat flow. Assuming the heat sink temperature to be 300 K, ΔT_M ≳ 120 K would change the antiferromagnetic order since the Nèel temperature for Mn₃Sn is ∼420 K. If we consider d_sub = 2 µm, then ΔT_M < 120 K for J_c < 9.4 × 10⁷ A/cm². This limits the use of a 4 nm thick Mn₃Sn film as a signal generator to about 33 GHz. On the other hand, for d_sub = 5 µm, the charge current density is limited to J_c < 6 × 10⁷ A/cm², which limits the frequency of the signal source to about 21 GHz. The white dotted line overlaid in Fig. 12 represents this upper limit for the charge current for different substrate thicknesses. A thinner substrate, or the metallic heat sources with a smaller area of cross section, could help lower the maximum temperature rise.

The time to reach ΔT_M, on the other hand, depends on τ. For a constant input current, ΔT_j(t) reaches ∼95% of ΔT_M in t = 3τ. If we consider d_sub = 1 µm, then τ = 33.7 ns, and the time to reach 95% of the steady-state temperature is ∼100 ns. Consequently, input currents larger than J_c = 1.3 × 10⁸ A/cm² could be safely applied if the duration of the pulse is of the order of few 10’s of ps. Figure 13 shows the non-quasi-static response of the rise in junction temperature, ΔT_j(t), as a function of time for current pulses of duration t_pw = 100 ps. It can be observed that even for charge current density as large as J_c = 3 × 10⁸ A/cm², the temperature rise is less than 24 K. Therefore, such large currents could be used during the switching operation, if required. Similar behavior is expected for other values of t_pw ≪ τ. A drawback of the high value of τ, due to large d_sub, is the slow removal of heat from the system, resulting in a slow decrease of temperature, as shown in Fig. 13. A thinner substrate or a substrate with lower thermal capacitance is preferred to reduce the time for heat to flow from the source to the sink, decreasing τ.

The aforementioned analysis does not consider the temperature dependence of the material parameters (electrical and thermal) of the AFM, the HM, or the substrate. In general, the material properties of the AFM, such as the uniaxial anisotropy constant, could decrease with an increase in temperature due to Joule heating.¹⁸ Therefore, if the input current is slowly increased from zero to a higher value, the stationary steady-state solutions could be different from those given in Eq. (12), while the onset of dynamics could occur at an input current lower than that predicted in Eq. (13). The oscillation frequency at an input current [Eq. (14)], on the other hand, could increase owing to the lower threshold current as well as Joule heating induced lower saturation magnetization. Consequently, the order parameter could be switched at a faster rate using a current pulse of smaller duration since J_st_pw [Eq. (15)] would decrease. However, an exact value of the threshold current, stationary state, oscillation frequency, or minimum pulse width required to switch between two states as a function of the input current would depend on the relative change in the antiferromagnetic material parameters with temperature.

Non-zero temperature leads to random fluctuation of the AFM spins, and as a result, the order parameter would exhibit a distribution around the equilibrium states in the six degenerate energy wells. When acted upon by external spin current, the order parameter would exhibit stochastic dynamics—each stationary steady-state below the threshold current would show a distribution around a mean value, the oscillation frequency would exhibit a non-zero linewidth, and J_st_pw would show thermal noise-induced randomness. The effect of thermal fluctuations would be more prominent below and near threshold current, while its effect on the dynamics would be weak for large values of input currents. Similar to ferromagnets, thermal noise in AFMs could be modeled as three random Gaussian fields, one for each sublattice, with zero mean and standard deviation equal to one.⁵⁸ In our modeling approach, this field would be added to Eq. (8), and the total field would consist of those due to exchange interaction, DM interaction, uniaxial anisotropy, external field, and thermal noise. The goal of our work, however, was to understand the deterministic dynamics in monodomain Mn₃Sn, and therefore, the effect of thermal noise is not considered here.

In this work, we investigated the SOT-driven oscillation and switching dynamics in sixfold magnetic anisotropy degenerate single-domain Mn₃Sn thin films. The spin polarization associated with the SOT was assumed perpendicular to the Kagome plane. Numerical simulations of the SOT-driven dynamics in the DC regime reveal that the frequency of oscillation of the order parameter in Mn₃Sn could be tuned from 100’s of MHz to 100’s of GHz by increasing the input spin current density. Analytic models of frequency vs input spin current density presented here for the first time show excellent agreement with the numerical results, obtained by solving the classical coupled LLG equations of motion for the sublattice vectors of Mn₃Sn. In addition to the oscillatory dynamics, switching of the order parameter between the six energy minima could also be accomplished by utilizing pulsed input spin current. In this case, the switching of the order parameter can be controlled by tweaking the amplitude and the pulse width of the input spin current. To provide a physical insight into the physics and functionality of Mn₃Sn in the pulsed-SOT regime, we developed analytic models of the final state that the order parameter settles into as a function of the spin charge density and revealed excellent match with numerical results. Motivated by recent experimental advances in electrical detection of the magnetic state of Mn₃Sn, we discussed AHE and TMR as possible schemes to detect the state of the AFM in a microelectronics-compatible manner. We showed that AHE voltage signals could be in the range of ∼μV, while the TMR voltage signals could be around 7–9 mV with a further possibility of increasing this voltage range for higher RA product values. Finally, we evaluated the temperature rise as a function of the input current pulse. Our results showed that the maximum temperature rise for MgO substrate of thickness d_sub = 1 µm is less than 24 K for current pulses with t_pw ≤ 100 ps and J_c ≤ 3 × 10⁸ A/cm². On the other hand, in the DC mode, the maximum temperature rise is found to be 120 K for J_c = 1.3 × 10⁸ A/cm², thereby limiting the application of Mn₃Sn as a coherent signal generator for frequencies of up to 45 GHz. In this work, we considered thin films with thicknesses less than 10 nm. Typically, such films show twofold degeneracy, instead of the sixfold, owing to epitaxial strain. This will be addressed in our future work.

See the supplementary material for the perturbative analysis of the ground state.

This research was primarily supported by the NSF through the University of Illinois at Urbana-Champaign Materials Research Science and Engineering Center under Grant No. DMR-1720633.

The authors have no conflicts to disclose.

Ankit Shukla: Conceptualization (equal); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Software (lead); Validation (lead); Writing – original draft (lead); Writing – review & editing (lead). Siyuan Qian: Data curation (supporting); Formal analysis (supporting); Software (supporting). Shaloo Rakheja: Conceptualization (lead); Funding acquisition (lead); Project administration (lead); Resources (lead); Supervision (lead); Writing – original draft (supporting); Writing – review & editing (supporting).

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