Dynamics of Magnetoelectric Reversal of Antiferromagnetic Domain

TL;DR

This paper provides a theoretical analysis of magnetoelectric domain reversal in antiferromagnetic thin films, highlighting domain wall propagation as the dominant switching mechanism and its implications for energy-efficient memory devices.

Contribution

It introduces a comprehensive model of reversal mechanisms considering nucleation, domain wall movement, and coherent rotation, aligning theory with recent experimental observations.

Findings

Domain wall propagation dominates switching in Cr2O3 thin films.

Reversal timescales depend on applied magnetoelectric pressure.

Implications for energy-delay performance in ME memory devices.

Abstract

When electric and magnetic fields are applied together on a magnetoelectric antiferromagnet, the domain state is subject to reversal. Although the initial and final conditions are saturated single-domain states, the process of reversal may decompose into local multi-domain switching events. In thin films of Cr2O3, the magnetoelectric coercivity and the switching speed found from experiments are considerably lower than expected from magnetic anisotropy, similar to Brown's paradox in ferromagnetic materials. Multi-domain effects originate because antiferromagnetic domain walls are metastably pinned by lattice defects, not due to reduction of magnetostatic energy, which is negligible. This paper theoretically analyzes domain reversal in thin-film magnetoelectric antiferromagnets in the form of nucleation, domain wall propagation, and coherent rotation. The timescales of reversal mechanisms are modeled as a function of applied magnetoelectric pressure. The theory is assessed with reference to latest experimental works on magnetoelectric switching of thin-film Cr2O3: domain wall propagation is found to be dominant and responsible for switching in the experiments. The results bear implications in the energy-delay performance of ME memory devices utilizing antiferromagnetic insulators, which are prospective for nonvolatile technology.

Tables

Table 1. Table 1: Experiments on ME switching of thin-film Cr 2 O 3

	Ref. Toyoki et al.,2015	Ref. Kosub et al.,2017
Type	exchange bias	purely antiferromagnet
$d$	200 nm	200 nm
$V$	—	$d\times$(0.1 cm2)
$T$	268 K	292 K
$H$$\ast$	0.798 MA/m	0.5 MA/m
$E_{\text{c}}$$\ast \ast$	125 V/µm	7.50 V/µm
${\alpha }_{\parallel }$ Borisov et al. (2007)	$3.7$ ps/m	$3.3$ ps/m
${ℱ}_{\text{EB}}$$\star$	930 J/m3	$\varnothing$
${ℱ}_{\text{c}}$$†$	1.67 kJ/m3	24.8 J/m3
$\tau$	0.1 µs	—

Table 2. Table 2: Material properties of bulk Cr 2 O 3

Symbol	$𝑻=4.2$ K	$𝑻=\mathrm{𝟐𝟔𝟖}$ K	$𝑻=\mathrm{𝟐𝟗𝟐}$ K
$𝒦$ Foner (1963)	$20$ kJ/m3	$15$ kJ/m3	$7.3$ kJ/m3
$M_{\text{s}}$ Samuelsen et al. (1970); Artman et al. (1965)	$0.58$ MA/m	$0.36$ MA/m	$0.26$ MA/m
$𝒥$ Foner (1963)	$280$ MJ/m3	$\propto M_{\text{s}}$
$ℰ$ Belashchenko et al. (2016)	$0.96$ mJ/m2	$\propto \sqrt{𝒦}$
$\lambda$ Belashchenko et al. (2016)	$12$ nm	$\propto 1/\sqrt{𝒦}$
$\eta$ Wang et al. (2015)	$1.2\times {10}^{-3}$

Table 3. Table 3: Estimated domain wall propagation parameters for experiments on ME switching of thin-film Cr 2 O 3

Parameters	Ref. Toyoki et al.,2015	Ref. Kosub et al.,2017
$\ell$	$4.8$ µm	220 µm
$R$	$300$ nm
$\pi R^2/S$	0.31	$6.6\times {10}^{-3}$

Equations

$${\mathcal{F}}_{\mathrm{ME}}=-{\alpha }_{\parallel }EH\cos \theta ,$$

$$\tau =\Delta t\exp (\frac{\Delta F}{kT}),\text{if}\Delta F\gg kT,$$

$$F_{\text{N}}(r)=2\pi rd\mathcal{E}-\pi d{r}^2\mathcal{F}.$$

$$\Delta F_{\text{N}}=\frac{\pi d{\mathcal{E}}^2}{\mathcal{F}}.$$

$$\sigma =\frac{M_{\text{s}}^2}{{\gamma }^2\lambda \mathcal{J}},$$

$$v_{\text{T}}(r)=\dot{r}=\sqrt{\frac{kT}{2\pi rd\sigma }}.$$

$${\tau }_{\text{N}}=\frac{2}{3}\sqrt{\frac{2\pi d\sigma {\mathcal{E}}^3}{kT{\mathcal{F}}^3}}\exp (\frac{\pi d{\mathcal{E}}^2}{kT\mathcal{F}}).$$

$${F}_{\text{wall}}(r)=\mathcal{E}\big{[}S-\pi(R^{2}-r^{2})\big{]},$$

$$F_{\text{ME}}(r)=-\mathcal{F}Sr.$$

$$F_{\text{P}}(r)=\pi \mathcal{E}{r}^2-\mathcal{F}Sr.$$

$$R=\frac{{\mathcal{F}}_{\text{D}}S}{2\pi \mathcal{E}},\text{if}\frac{\pi R^2}{S}=\frac{{\mathcal{F}}_{\text{D}}^{2}S}{4\pi {\mathcal{E}}^2}\ll 1.$$

$$\Delta F_{\text{P}}=\pi \mathcal{E}{R}^2{\left(1-\frac{\mathcal{F}}{{\mathcal{F}}_{\text{D}}}\right)}^2=\frac{S^2}{4\pi \mathcal{E}}({\mathcal{F}}_{\text{D}}-\mathcal{F}{)}^2.$$

$$v_{\text{T}}=\sqrt{\frac{kT}{\sigma S}},$$

$${\tau }_{\text{crp}}=\sqrt{\frac{\sigma S^3}{kT}}\left(\frac{{\mathcal{F}}_{\text{D}}-\mathcal{F}}{2\pi \mathcal{E}}\right)\exp [\frac{S^2({\mathcal{F}}_{\text{D}}-\mathcal{F}{)}^2}{4\pi kT\mathcal{E}}].$$

$$v_{\text{flw}}=v_{\text{T}}+\frac{\eta \gamma \lambda }{{\eta }^2+{\zeta }^2}\left(\frac{\mathcal{F}-{\mathcal{F}}_{\text{D}}}{M_{\text{s}}}\right),\zeta =\frac{{\alpha }_{\parallel }E}{{\mu }_0{M}_{\text{s}}},$$

$$\frac{F_{\text{SW}}(\theta )}{V}=\mathcal{K}{\sin }^2\theta +{\mu }_0{M}_{\text{s}}\left(\frac{\mathcal{F}}{2{\mu }_0{M}_{\text{s}}}\right)\cos \theta ,$$

$$\Delta F_{\text{SW}}=V\mathcal{K}{\left(1-\frac{\mathcal{F}}{4\mathcal{K}}\right)}^2.$$

$${\tau }_{\text{spm}}=\frac{{\eta }^2+(\mathcal{K}/\mathcal{J}{)}^2}{2\gamma \eta \mathcal{K}/M_{\text{s}}}\sqrt{\frac{\pi kT}{\Delta F_{\text{SW}}}}\exp \left(\frac{\Delta F_{\text{SW}}}{kT}\right).$$

$${\tau }_{\text{dmp}}=\frac{{\eta }^2+(\mathcal{K}/\mathcal{J}{)}^2}{\gamma \eta }$$

$$2\mathcal{K}\ln (\frac{\mathcal{F}/2-2\mathcal{K}\cos \vartheta }{\mathcal{F}/2+2\mathcal{K}\cos \vartheta })].$$

$$S={\left(\frac{kT{d}^2{\tau }^2}{\sigma }\right)}^{1/3}.$$

$$\mathcal{M}=\frac{6\times 3{\mu }_{\text{B}}}{\sqrt{3}/2a^2}=7.83\times 10^{-4}\text{A}.$$

$${\mathcal{F}}_{\text{ms}}=\frac{1}{2}{\mu }_0{\left(\frac{\mathcal{M}}{d}\right)}^2=(3.85\times 10^{-13})\cdot \frac{1}{d^2}\text{J/mtextsuperscript3}$$

$$\mathcal{E}=4\mathcal{K}\lambda =9.6\times 10^{-4}\text{J/mtextsuperscript2},$$

Full text

I Introduction

For storing digital information on a magnetic medium, the single-domain states are of interest because they can be switched reversibly and read easily. Magnetoelectric (ME) coupling establishes a direct way to realize voltage control of the domain state in antiferromagnetic insulators which have combined space and time inversion symmetry. Fiebig (2005) This is promising for implementing compact, nonvolatile memory and logic technology with ultralow power consumption. Manipatruni et al. (2018); Dowben et al. (2018) In a recent work of interest, Kosub et al. (2017) a purely ME memory cell has been experimentally demonstrated using the ME antiferromagnet Cr2O3, which can be isothermally switched and read all-electrically at room temperature. See Appendix A for a brief timeline of the research on ME effect in Cr2O3. There is plenty of room for studies involving ME switching dynamics and its understanding particularly in thin-film systems, which are fundamental for spintronic applications.

When electric and magnetic fields are simultaneously applied on a ME antiferromagnet, the domain state is subject to reversal. Martin and Anderson (1966) Although the initial and final conditions are saturated single-domain states, the switching transient can exhibit multi-domain characteristics. Unlike in ferromagnets, the stability of multi-domain states in antiferromagnets is not apparent because the magnetostatic energy is insignificant compared to the energy of domain wall (see Appendix B). More than 60 years ago, Li Li (1956) theorized that at the Néel temperature $T_{\text{N}}$, local nucleations of the antiferromagnetic order and their subsequent growth would result in the formation of domain walls, if the wall energy is offset by the gain in entropy. Below $T_{\text{N}}$, the domain wall would owe its stability to the presence of lattice defects.

Indeed, experiments have revealed multidomain structures in Cr2O3 after zero-field cooling through $T_{\text{N}}$, both in the bulk Fiebig et al. (1995) and on the surface of thin film. Wu et al. (2011) The bulk sample shows “temperature memory effect” Brown (1969): the domain structure does not change when the sample is heated above and cooled below $T_{\text{N}}$. For the thin film sample, the surface exhibits height variations of a few nanometers over a micrometer lateral scale; the typical domain size also happens to be about a micrometer. These observations reinforce the notion that defects can metastably pin domain walls and cause multi-domain switching in antiferromagnets.

In a recent study of ME switching of exchange bias in thin-film Cr2O3/Co/Pt, Toyoki et al. (2015) the threshold ME pressure for reversal and the switching speed are found to be considerably lower than expected from magnetic anisotropy, similar to Brown’s paradox in ferromagnetic materials. Shtrikman and Treves (1960) This indicates that domain reversal is decomposing into local multi-domain switching processes, instead of undergoing uniform rotation.

In this paper, we analyze the mechanisms that govern domain reversal in thin-film ME antiferromagnets and model their timescales as a function of applied ME pressure (Sec. III). The theory primarily requires the knowledge of molecular fields of the material at a given temperature. We assess the theory with respect to recent experimental works Toyoki et al. (2015); Kosub et al. (2017) on ME switching of thin-film Cr2O3 (thickness $\leq$ 500 nm Ashida et al. (2014)) to characterize the switching dynamics (Sec. IV). The dominant reversal mechanism in the experiments is identified and the switching speed has been estimated accordingly. The results provide insights into the energy-delay prospects of ME memory devices.

II Background

II.1 Linear ME effect in Cr2O3

The linear ME effect in the presence of electric field $\mathbf{E}$ and magnetic field $\mathbf{H}$ is given by the free energy density $\mathcal{F}_{\mathrm{ME}}=-\alpha_{ij}{E}_{i}{H}_{j}$, where $\alpha_{ij}$ is the ME tensor. Fiebig (2005) The symmetry constraints in Cr2O3 allow non-zero ME effect only when the electric and magnetic fields are parallel. Dzyaloshinskii (1960) This reduces the independent components of the ME tensor to $\alpha_{\parallel}$ and $\alpha_{\perp}$, for the directions parallel and perpendicular to the trigonal $z$-axis, Mostovoy et al. (2010) respectively.

In practice, the fields are applied exclusively along the $c$-axis because $\alpha_{\parallel}\sim 1$ ps/m Borisov et al. (2007) is an order of magnitude larger than $\alpha_{\perp}$ at room temperature, Folen et al. (1961) and the uncompensated magnetization of the (0001) surface allows for easier detection. Wu et al. (2011) Hence, the ME energy density can be simplified to

[TABLE]

where $\theta$ denotes the direction of the antiferromagnetic order parameter relative to the $z$-axis. The signs of $E$ and $H$ are determined according to orientation of the $z$-axis. For a 180*∘* domain wall, the ME effect creates a pressure difference $\mathcal{F}=\absolutevalue{2\alpha_{\parallel}{EH}}$ across the wall. The ME susceptibility peaks at 266 K where $\alpha_{\parallel}^{\text{max}}=3.8$ ps/m and disappears above $T_{\mathrm{N}}=308.5$ K. Borisov et al. (2007)

The source of perturbation, which is the ME energy term (1), is of the same form as the interaction energy between magnetization and an applied magnetic field, called the Zeeman energy. For electric control of the domain state, $\alpha_{\parallel}E\gg\mu_{0}\chi_{\parallel}H$ must be satisfied, where $\mu_{0}$ is the vacuum permeability and $\chi_{\parallel}\approx 1.5\times 10^{-3}$ is the volume magnetic susceptibility parallel to the $z$-axis close to room temperature. Foner (1963) Under this assumption, ME switching of domain in antiferromagnets is subject to reversal mechanisms analogous to magnetic field-induced magnetization reversal in ferromagnets.

II.2 Mechanisms of magnetization reversal

For ferromagnetic particles, magnetization reversal occurs via coherent rotation, when the particle size is smaller than the magnetostatic exchange diameter or the macrospin limit, which is about 10-50 nm depending on the material. Abo et al. (2013) The magnetization dynamics is modeled by the Landau–Lifshitz–Gilbert equation, Landau and Lifshitz (1935); Gilbert (2004) which describes phenomenological damping of gyromagnetic precessions. When the particle size exceeds the exchange diameter, the magnetostatic interaction causes the spins to undergo incoherent modes of rotation such as curling and buckling for a prolate spheroid shape and fanning for a chain of spheres. Aharoni (2000)

In larger ferromagnetic samples, where subdivision into domains reduces the total energy, the reversal process decomposes into local nucleation and domain wall propagation events. Reversed domains tend to nucleate at centers where the energy barrier for spin rotation may be lower, like lattice defects. Cullity and Graham (2011) Once the nucleation center is stabilized, the domain wall surrounding the center can propagate across the sample to spread the reversal. Vogel et al. (2006) However, the wall motion may get interrupted by the same defects, resulting in viscous damping or even a complete halt (pinning). Ferré (2002) In imperfect crystals, nucleation centers may always be present, especially near edges, so that magnetization does not fully saturate. Magnetization reversal then becomes a problem of not nucleating new centers, but depinning domain walls that already exist. Cullity and Graham (2011) Nucleation and propagation are phenomenologically treated as thermally activated processes in Fatuzzo’s theory for switching transient in ferroelectrics, Fatuzzo (1962) which is later extended to magnetic films by Labrune et al. Labrune et al. (1989) The Fatuzzo–Labrune model neglects magnetostatic effects, which is a valid assumption for antiferromagnets.

III Theory

Multidomain behavior in antiferromagnets can originate from domain wall pinning, without any magnetostatic interaction. For an antiferromagnetic thin-film, it is sound to consider all reversal mechanisms with non-magnetostatic origin, if the in-plane dimensions of the film are larger than the domain wall width.

Each reversal mechanism has a coercive pressure $\mathcal{F}_{\text{c}}$ associated with it, which is overcome by applying ME pressure $\mathcal{F}$. If $\mathcal{F}<\mathcal{F}_{\text{c}}$, the mechanism is thermally activated from the metastable state with an escape time Hänggi et al. (1990)

[TABLE]

where $\Delta t$ is a prefactor, $\Delta F$ is the energy barrier and $kT$ is the thermal energy. For $\mathcal{F}>\mathcal{F}_{\text{c}}$, the energy barrier vanishes and the dynamics is governed by viscous damping in the system. From an observational perspective, the critical ME pressure for domain switching corresponds to the smallest coercive pressure among the mechanisms, if the measurement is performed slowly. The switching speed of a mechanism is distinguished from pulse response, if the pulse power overcomes the coercive pressure needed to trigger the mechanism. If the pulse activates multiple mechanisms in parallel, the fastest mechanism determines the switching speed.

List of symbols for magnetic properties

• $\mathcal{K}$

uniaxial anisotropy of two sublattices (J/m3)

• $\mathcal{J}$

antiferromagnetic exchange energy (J/m3)

• $\mathcal{E}$

domain wall energy (J/m2)

• $M_{\text{s}}$

sublattice magnetization (A/m)

• $\lambda$

characteristic width of domain wall (m)

• $\eta$

Gilbert damping constant

III.1 Nucleation

Consider a finite droplet of uniformly reversed order with a domain wall boundary in an otherwise single-domain state. Vogel et al. (2006) The free energy of a cylindrical droplet of radius $r$ in a film of thickness $d$ consists of the wall energy of the curved surface and the ME energy of the reversed volume, which is written as

[TABLE]

The energy is maximum when $\derivative*{F_{\text{N}}}{r}=0$, which gives $r_{\text{max}}=\mathcal{E}/\mathcal{F}$. If $r<r_{\text{max}}$ ($F_{\text{N}}(r)<0$), the droplet collapses because of wall pressure. If $r>r_{\text{max}}$ ($F_{\text{N}}(r)>0$) the droplet’s energy decreases as it expands, so that domain reversal is set in motion. The energy barrier between $r=0$ and $r=r_{\text{max}}$ that needs to be surmounted to nucleate the critical radius droplet is obtained as

[TABLE]

From field theory of antiferromagnets, Kim et al. (2014) a domain wall possesses rest mass with areal density Belashchenko et al. (2016)

[TABLE]

where $\gamma$ is the electron gyromagnetic ratio. The translational motion of the wall has average kinetic energy $kT/2$ from equipartition theorem, so the droplet wall fluctuates with thermal velocity

[TABLE]

Separate variables and integrate with the limits that correspond to $r=0$ and $r=r_{\text{max}}$ to find the attempt period. On substitution, the timescale of droplet nucleation is formulated as

[TABLE]

This equation supports the idea that nucleation is likely to occur at centers where the wall energy is locally lowered, such as dislocation lines. Aharoni (1962)

III.2 Domain wall propagation

When a domain wall intersects a non-magnetic inclusion, the wall energy is reduced by an amount proportional to the cross-sectional area of the inclusion. Suppose a spherical inclusion with radius $R$. The interaction of antiferromagnetic wall with inclusion can be modeled in the same way as for ferromagnets when $R\gg\lambda$, Krishnan (2016) but without needing the constraint on $R$ which trivializes magnetostatic energy contribution. At a general position $r\leq R$ from the center of the inclusion, the wall energy is given as

[TABLE]

where $S\propto d$ is the natural surface area of the wall. For a relatively pure crystal, $S\gg\pi R^{2}$. On applying ME pressure, the energy incurred to move the wall to a distance $r$ is given as

[TABLE]

The positive sign of $r$ is chosen along the direction of ME stress on the wall. The total free energy, ignoring the constant terms, is reduced to

[TABLE]

In equilibrium, $\derivative*{F_{\text{P}}}{r}=0$ occurs at $r_{\text{min}}=\mathcal{F}S/(2\pi\mathcal{E})$ if $r_{\text{min}}<R$. The threshold ME pressure $\mathcal{F}_{\text{D}}$, which depins the wall away from the inclusion, is obtained by setting $r_{\text{min}}=R$, such that

[TABLE]

For a uniform distribution of inclusions, the analysis differs only in the calculation of cross-sectional area, which needs to be multiplied by the number of inclusions intersected by the wall. Otherwise, the distribution of inclusions in the material should be known or accordingly, the distribution of pinning energy barriers. Ferré (2002)

For $\mathcal{F}<\mathcal{F}_{\text{D}}$, the domain wall can thermally “creep” Chauve et al. (2000) from the energy minimum at $r_{\text{min}}$ to the edge of the inclusion at $r=R$, by overcoming the pinning barrier

[TABLE]

The wall moves with uniform thermal velocity

[TABLE]

to give the attempt period $(R-r_{\text{min}})/v_{\text{T}}$. On substitution, the final expression for the timescale in the creep regime is found as

[TABLE]

For $\mathcal{F}>\mathcal{F}_{\text{D}}$, the domain wall motion is subject to viscous “flow” Chauve et al. (2000) with linear velocity Belashchenko et al. (2016)

[TABLE]

If multiple nucleation centers in the sample are activated, a wall can flow only up to a mean free path $\ell$, before it coalesces with other wall flow processes ($S\ell$ is representative of propagation activation volume in the Fatuzzo–Labrune model). The timescale in the flow regime is simply $\tau_{\text{flw}}=\ell/v_{\text{flw}}$. Since nucleation centers tend to originate around defects, the information about $\ell$ is acquired from the distribution of defects. Labrune et al. (1989)

III.3 Coherent rotation

The order parameter of antiferromagnets is represented by half the difference of oppositely aligned sublattice magnetization vectors. Baltz et al. (2018) If the exchange energy is much larger than the anisotropy, the sublattice vectors would remain antiparallel even during the reversal, so that the magnitude of the order parameter is preserved dynamically. Let $\theta$ uniformly represent the direction of the order parameter. For a ME stress applied opposite to the initial orientation along $+z$, the free energy is described using the Stoner–Wohlfarth model Tannous and Gieraltowski (2008)

[TABLE]

where $V$ is the sample volume and $\mathcal{F}/(2\mu_{0}M_{\text{s}})$ represents the staggered field experienced by sublattice magnetization. The stationary points of $F_{\text{SW}}$ are obtained by setting $\derivative*{F_{\text{SW}}}{\theta}=0$.

To reverse when $\mathcal{F}/2<2\mathcal{K}$, the order parameter must overcome the energy barrier between the minimum at $\theta=0$ and the maximum at $\theta=\arccos(\mathcal{F}/4\mathcal{K})$ , which is calculated as

[TABLE]

While the superparamagnetic limit of ferromagnetic particles is precisely formulated in the Néel–Brown theory, Brown Jr (1963) the timescale of this limit for antiferromagnets is only heuristically proposed as Atxitia et al. (2018)

[TABLE]

For small damping, the attempt period is diminished by the factor $(\mathcal{K/J})^{2}$ compared to ferromagnets, indicating faster dynamics in antiferromagnets.

For $\mathcal{F}/2>2\mathcal{K}$, the reversal occurs by damping of gyromagnetic precessions. The switching time from an initial angle $\vartheta$ to final angle $\pi-\vartheta$ is given by Mallinson (2000)

[TABLE]

The factor $(\mathcal{K}/\mathcal{J})^{2}$ replaces unity in the original expression to account for antiferromagnetic dynamics.

Notice that the expressions of $\tau_{\text{spm}}$ and $\tau_{\text{dpm}}$ are derived assuming an angle-invariant $\eta$. While the Gilbert damping phenomenology explains the linewidth of magnetic resonance, Gilbert (2004) a near-equilibrium excitation, the validity of such scheme is questionable for a $180^{\circ}$ reversal. The phenomenology of superparamagnetism as a stochastic process called the jump-noise, captures the angular dependence of damping, including a correction to the gyromagnetic ratio. Parthasarathy and Rakheja (2018a) The jump-noise representation may also be useful to calculate the timescale in the limit of small energy barrier, Parthasarathy and Rakheja (2018b) where $\tau_{\text{spm}}$ diverges otherwise. Despite the merits of the jump-noise, the classical approach is still favored due to ease of extracting $\eta$ and availability of closed-form solutions.

Also, notice that $\tau_{\text{dmp}}$ scales indefinitely as $1/\mathcal{F}$ for large $\mathcal{F}$, which is not realistic. The ultimate limit of the switching speed is a fundamental issue concerning conservation and transfer of angular momentum and energy between the heat reservoirs of the system: spins and lattice. Stöhr and Siegmann (2007) The exploration of ultrafast dynamics in antiferromagnets, however, is still in its infancy. Kirilyuk et al. (2010)

IV Application

To assess ME reversal of antiferromagnetic domain, a number sense of the energy barriers and timescales involved with reversal mechanisms is necessary. Consider the two recent works Kosub et al. (2017); Toyoki et al. (2015) on ME switching of thin-film Cr2O3, which have identical film thickness. The specifics of the experiments are summarized in Table 1. The switching in Ref. Toyoki et al., 2015 is observed for the case where the exchange bias aids in domain reversal. So, the net external pressure is the sum of applied ME and exchange bias pressures. The coercive ME pressure in Ref. Kosub et al., 2017 is significantly lower because the temperature is much closer to $T_{\text{N}}$. The material parameters of bulk Cr2O3 at relevant temperatures are listed in Table 2. The values of $\mathcal{K}$ and $M_{\text{s}}$ are extracted from Fig. 1.

Using the information in Tables 1 and 2, the timescales of nucleation and coherent rotation (assume $\vartheta=1^{\circ}$ and same $V$ for Ref. Toyoki et al., 2015) can be evaluated as shown in Fig. 2. The energy barrier of both mechanisms is orders of magnitude larger than the measured coercive ME pressure, so they are suppressed. An earlier work Brown and O’Dell (1969) indeed observed that a very large ME pressure tends to produce a fully saturated state that could not be switched further. Therefore, at moderate ME pressures, domain wall propagation is the responsible mechanism for reversal.

Estimating the timescale of domain wall propagation requires additional knowledge of the wall surface area $S$ and mean free path $\ell$. Because of symmetry of wall direction in the film plane, it is hypothesized that $S=\ell d$. At the depinning pressure $\mathcal{F}_{\text{d}}=\mathcal{F}_{\text{c}}$, $\ell=v_{\text{T}}\tau$ so that $S=dv_{\text{T}}\tau$. Using Eq. (6),

[TABLE]

Since $\tau$ is known for the experiment of Ref. Toyoki et al., 2015 (Table 1), $\ell$, $S$ and the inclusion radius $R$ (11) can be calculated. If the same $R$ is assumed for the sample of Ref. Kosub et al., 2017, $\ell$ and $S$ can be estimated by retracing the steps. The values of $\ell$, $R$ and error (11) are listed in Table 3. Note that $\pi R^{2}$ represents the total cross-sectional area of the inclusions that intersect the wall, not that of a single inclusion, so $R>d/2$ is allowed as long as $\pi R^{2}<S$ . Using the parameters in Table 3, the timescale of wall propagation is plotted in Fig. 2 (the non-monotonic behavior agrees with Eq. 15). The switching speed for Ref. Kosub et al., 2017 is estimated as 10 µs, which is different from the stated $\sim 10$ ns without measurement or theoretical validation.

Below picosecond timescale, the spin-lattice interaction in Cr2O3 comes into play, Satoh et al. (2007) and the models for nucleation and coherent rotation would become invalid. In addition to the breakdown of the models, the electric fields required to achieve ultrafast dynamics is likely to induce dielectric breakdown. In thin-films of Cr2O3, the breakdown electric field drops from 1000 V/µm of bulk to 200 V/µm. Sun et al. (2017) Even at a generous magnetic field of 1 T, the ME pressure that can be reached before breakdown is just 525 J/m3. Experimental advancements in increasing the breakdown field and reducing the value of the magnetic anisotropy Mu and Belashchenko (2018) will be needed to realize dynamics in the sub-nanosecond regime.

V Conclusion

In this paper, we theoretically analyze domain reversal in thin-film ME antiferromagnets in the form of nucleation, domain wall propagation and coherent rotation. We model the timescales of the reversal mechanisms as a function of applied ME pressure, with the knowledge of material properties and domain wall propagation parameters. We apply the theory to recent experimental works Toyoki et al. (2015); Kosub et al. (2017) on ME switching of thin-film Cr2O3, which have identical film-thickness. We find that nucleation and coherent rotation are suppressed at the ME pressures used in the experiments, and domain wall propagation is responsible for reversal.111At the time of submission of manuscript, we came across experimental works published less than a month before by Shiratsuchi et al., Shiratsuchi et al. (2018a, b); Nguyen et al. (2018) which corroborate the role of domain wall propagation in ME switching of exchange bias. We will assess our theory with their findings in a future work. We extract wall propagation parameters from the experiment of Ref. Toyoki et al., 2015 and estimate a switching speed of 10 µs for Ref. Kosub et al., 2017, which has not been measured. We remark that ultrafast dynamics of ME antiferromagnets is limited by low dielectric breakdown and high magnetic anisotropy. Experiments should focus on mitigating these limitations in order to unleash switching at terahertz speeds.

Appendix A Timeline of ME research in Cr2O3

The ME effect in Cr2O3 was predicted phenomenologically by Dzyaloshinskii in 1959, Dzyaloshinskii (1960) which was soon followed by measurement of the temperature-dependent ME susceptibility. Astrov (1961); Folen et al. (1961) A few years later, the domain switching in Cr2O3 was demonstrated by simultaneous application of electric and magnetic fields to a bulk sample. Martin and Anderson (1966); Brown and O’Dell (1969) The direction of domain was detected directly by monitoring the ME susceptibility during the switching process; the domain state could not be read in the absence of applied fields. The speed of domain reversal was measured similar to 1 ms for an applied ME pressure of about 2 J/m3 close to room temperature. With an electric-field of $10^{6}$ V/cm, the magnetization obtained from ME effect corresponds to reversal of only five of every one million spins! From the 1970s, the research in ME started declining because the direct coupling was considered too weak for ME phase control in storage elements. Fiebig (2005)

Fast-forward nearly four decades, the interest in ME research revived. The ME switching of exchange bias in a ferromagnetic layer interfacially, coupled to Cr2O3 bulk, was achieved after ME field cooling through the Néel temperature. Borisov et al. (2005) The one-to-one coupling between ferromagnet’s magnetization and antiferromagnetic order made it possible to readout the antiferromagnetic domain state independently from the switching process. Five years later, the same was accomplished isothermally at room temperature. He et al. (2010) Equilibrium magnetization at the boundary of ME antiferromagnet was proven from symmetry arguments, Belashchenko (2010) and also imaged on the surface of Cr2O3 film. Wu et al. (2011) There were theoretical Mu et al. (2013) and practical Street et al. (2014) works foussing on enhancing the Néel temperature. Research efforts from a different group confirmed isothermal switching of the exchange bias as well as the magnetization in zero magnetic field in an all-thin-film system. Ashida et al. (2015) In a parallel work, Toyoki et al. (2015) the switching speed was additionally investigated by pulse voltage measurements, and found to be around 0.1 µs. In the recent study, Kosub et al. (2017) an all-electric access to the surface magnetization was demonstrated via anomalous Hall magnetometry Kosub et al. (2015) for ferromagnet-free, thin-film ME memory. The switching speed was not examined, but prospected to be within a few tens of nanosecond.

Theoretical works have made attempts to explain the microscopic origin of the ME effect in equilibrium in Cr2O3. The earlier works Rado (1961); Date et al. (1961); Hornreich and Shtrikman (1967) were mostly phenomenolgical, while the studies during the last decade Mostovoy et al. (2010); Bousquet et al. (2011); Malashevich et al. (2012) were based on first-principles calculations. However, there is no clear consensus on the cause of ME effect yet.

Appendix B Issues with domain formation in Cr2O3

Consider the thin-film sample of Ref. Wu et al., 2011, which is 127 nm thick, and shows stripe-like surface magnetization domains, that are 1 µm wide, on the (0001) plane (see Fig. 3 for an illustration). The image of the boundary uncovers the domain structure inside the bulk because the boundary magnetization is coupled to the bulk order in a one-to-one manner. Andreev (1996); Belashchenko (2010)

Suppose that the non-trivial domain structure is formed because of the significance of the magnetostatic energy due to surface magnetization. Cr2O3 has $R\overline{3}c$ space group with lattice constant $a=0.5$ nm. Vil There are six magnetic Cr3+ ions in the unit cell in the (0001) plane, with spins aligned along the trigonal $z$-axis. Cr3+ has the electronic configuration 3d3, so each ion can at most contribute to 3$\mu_{\text{B}}$ magnetic moment. Therefore, the areal magnetic moment of the film is calculated as

[TABLE]

For a film of thickness $d$, the magnetostatic energy $\mathcal{E}_{\text{ms}}$ averaged over the sample volume is then obtained as

[TABLE]

Multidomain structures in antiferromagnets, once formed, follow the same constraints of exchange and anisotropy as their ferromagnetic counterparts and possess domain wall energy, which is calculated as Foner (1963); Belashchenko et al. (2016)

[TABLE]

where $\mathcal{K}=2\times 10^{4}$ J/m3 is the combined uniaxial anisotropy of two sublattices and $\lambda=\sqrt{\mathcal{A/K}}=12$ nm is the characteristic wall length. For an average domain width $w$, the domain wall energy density per unit volume is $\mathcal{F}_{\text{wall}}=\mathcal{E}/w$.

For the parameters $d=127$ nm and $w=1$ µm, $\mathcal{F}_{\text{ms}}=23.9$ J/m3, whereas $\mathcal{F}_{\text{wall}}=960$ J/m3. Wall formation overall costs a lot more energy because the magnetostatic contribution is negligible. For a defect-free sample, multidomain structures in antiferromagnets are not expected.

Acknowledgements.

This work was supported in part by the Semiconductor Research Corporation (SRC) and the National Science Foundation (NSF) through ECCS 1740136. S. Rakheja also acknowledges the funding support from the MRSEC Program of the National Science Foundation under Award Number DMR-1420073.