Abstract
The spintorque driven dynamics of antiferromagnets with DzyaloshinskiiMoriya interaction (DMI) were investigated based on the LandauLifshitzGilbertSlonczewski equation with antiferromagnetic and ferromagnetic order parameters (l and m, respectively). We demonstrate that antiferromagnets including DMI can be described by a 2dimensional pendulum model of l. Because m is coupled with l, together with DMI and exchange energy, close examination of m provides fundamental understanding of its dynamics in linear and nonlinear regimes. Furthermore, we discuss magnetization reversal as a function of DMI and anisotropy energy induced by a spin current pulse.
Introduction
Since the first discovery of subpicosecond demagnetization of ferromagnetic nickel film using femtosecond infrared lasers, ultrafast manipulation of magnetization has raised much interest in terms of both condensed matter physics and applications in information storage devices^{1}. Together with the development of femtosecond lasers, a considerable number of research studies have been conducted to explore the microscopic dynamics experimentally as well as theoretically for various magnetic systems^{2,3,4,5,6,7,8,9,10,11,12,13}.
The antiferromagnet (AF) system is a promising structure for ultrafast processes because it has a relatively strong exchange interaction that shifts the precession frequency into the terahertz range. The AF system can be excited or switched at picosecond timescales (significantly faster than ferromagnetic precession^{14,15,16,17}), and AF switching through currentinduced spin transfer torque has been recently measured electrically^{18}.
AF systems with weak ferromagnetism (AWF) might be useful for memory devices because of their weak ferromagnetism and selectively controllable excitation modes^{19}. The weak ferromagnetism is associated with broken inversion symmetry in the material and is independent of any ferromagnetic impurities^{20}. This type of magnetism has been studied experimentally in the rare earth orthoferrites^{21,22,23,24,25,26} and rhombohedral antiferromagnet FeBO_{3}^{27,28} by many research groups. However, analytical approaches to describe AWF dynamics are rare except for AF cases^{15,29}.
This paper shows that AWF dynamics is governed by the classical pendulum equations on the antiferromagnetic order parameter (l), similar to the simple AF case^{14}. We demonstrate quantitatively that the occurrence of the second harmonic of the ferromagnetic order parameter (m) is direct evidence for a nonlinear regime, including resonant frequency softening^{30}, and that the ellipticity of the precessional motion of m determines the DzyaloshinskiiMoriya interaction (DMI) energy. Additionally, we propose that sublattice dynamics (s_{1}, s_{2}) can be revealed experimentally. We discuss switching efficiency as a function of anisotropy, DMI energy and damping constant (α) using spin current pulse with various durations and densities.
Theory
AWF dynamics
Figure 1 shows AWF static and dynamic configurations based on two sublattice models below the Néel temperature^{31}. Antiferromagnetically coupled spins lie along the xaxis because the anisotropy of the spins occurs along the uniaxial direction, with the magnetic easy axis parallel to xaxis, and the spins are tilted along the zaxis due to the DMI vector, , as shown in Fig. 1(a). The DMI produces two resonant modes, called the Sigma mode and the Gamma mode (S and Gmode, respectively)^{19}.
LandauLifshitzGilbertSlonczewski equation
To better understand the kinetics of AWF, the total energy based on two sublattices with i = 1, 2 is expressed as
where the normalized magnetization, s_{i} = S_{i}/S_{0} with S_{0} = S_{i} is dimensionless, and ħ is the reduced Plank constant. The first term is related to the exchange energy, where J is the nearestneighbor symmetric exchange constant, with the positive sign accounting for AF coupling. The second term describes DzyaloshinskiiMoriya (DM) energy, where the DM vector, D, is , D_{y} > 0, and its magnitude is relatively weak. The third and fourth terms are anisotropy energies where anisotropy constants are K_{x} > 0 and K_{z} < 0, indicating magnetic inplane and outofplane anisotropy, respectively. These energy combinations cause the antiparallel spins to be tilted slightly along the zaxis. The dynamics can be described by the coupled LandauLifshitzGilbertSlonczewski equation of motion:
The fifth term is phenomenological damping, which is characterized by the Gilbert damping constant (α). The final term is the Slonczewskitype spin transfer torque (STT), where p is the unit vector of spin polarization, and Ω is the STT strength with angular frequency unit, defined as εħγJ_{s}/(2VS_{0}e), which is proportional to the spin current density, J_{s}, where ε and V are the scattering efficiency and volume of AWF region, respectively^{14,29,32}.
We use staggered magnetization, l = (s_{1} − s_{2})/2, and weak magnetization, m = (s_{1} + s_{2})/2, so that Eq. (2) becomes
where Damping and STT are α(m × l + l × m) and Ω[m × (l × p) + l × (m × p)], respectively. Equations (3) and (4) are constrained by
Consider the Smode excited by spin current with polarization, p. When the STT turns on, s_{1} and s_{2} are dragged slightly toward the yaxis by the exchange coupling between the conduction electrons and the magnetic moments, as shown in Fig. 1(b). Consequently, as m_{y} increases in magnitude, l is moved away from its equilibrium position. After the STT turns off, l and m are subject to an internal magnetic field torque, and m precesses on the xyplane and fluctuates along the zaxis, as shown in Fig. 1(d). (In a simple AF, only m_{y} fluctuation is shown^{14,17,29}.) This is ascribed to the DMI, coupled with m_{x} and m_{z}, which causes elliptical polarization of precessional motion ofm, as shown in Fig. 1(d). The details are discussed below, in conjunction with the second harmonic oscillation of m_{z}.
Because the l_{y} component is much smaller than l_{x} and l_{z}, the dynamics of lcan be regarded as approximately 2dimensional (2D) pendulum motion oscillating with angle φ_{1} on the xzplane (see Supplementary information). Therefore, we expand Eqs (3) and (4) by using the effective vectors (l_{x}, 0, l_{z}) and (0, m_{y}, 0), and take the cross product of l on Eq. (4) to extract only m,
where the equations are simplified by employing Eq. (5) and ignoring terms coupled with anisotropy energy because K_{x} and K_{z} ≪ D_{y} and J. Substituting Eq. (7) into Eq. (3), we have the 2D pendulum equations,
where .
In Gmode, p; thus, s_{1} and s_{2} are dragged slightly toward the zaxis, as shown in Fig. 1(c). As m_{z} increases in magnitude, with m_{x} and m_{y} remaining zero, l is moved away from the equilibrium position (see Supplementary information). After STT turns off, m_{z} shows only fluctuation motion and loscillates on the xyplane, as shown in Fig. 1(e). The methodology in Smode is used again, so that
Substituting Eq. (9) into Eq. (3), the 2D pendulum equation becomes
where , with spin canting angle β in equilibrium (l and l_{z} = 0) because tan[β] = m_{z}/l_{x} = D_{y}/(2J). These outcomes confirm White et al.^{19}.
Results and Discussion
Ultrafast dynamics in the terahertz regime
We introduce the DMI or antisymmetric superexchange interaction from the triangle spanned among three ions (magnetic ions and oxygen ion). Such an interaction was discovered in the interface between AF and ferromagnet^{33}, and between AF and ferrimagnet superlattices^{34}, as well as bulk crystals^{20,21,22,23,24,25,26}. Here, we suppose a twolayer system consisting of two antiferromagnetic oxides, where the interaction between two magnetic ions arranged along the xaxis, and the oxygen ion, shifted slightly to the zaxis, gives rise to DM vector,, as shown in the inset of Fig. 1(a). Because the magnetic easy axis is the xaxis, sublattice spins are canted toward the zaxis. Additionally, we assume that the magnetic unit cell exhibits weak ferromagnetism, as in the case of rare earth ferrite, ReFeO_{3} single crystal (Re: Er, Tm, and Y, etc). The parameters chosen were J = 113.5 meV, K_{x} = 4.14 μeV, K_{z} = 0, and D_{y} = 0.01J so that spin precession motion is in the terahertz frequency range. To inject spin current into AWF, we exploit the spin hall effect in Pt with strong spinorbit coupling. Figure 2(a,b) show Smode and Gmode in AWF (see Supplementary Movie 1 & 2). Moreover, we checked that our analytical results are validated by numerical calculations based on Eq. (1) (see Supplementary Figure 2 and 3). However, for stronger DM energy, we found that the analytical solution deviates from the numerical one because the approximation () is no longer valid. (see Supplementary information).
Second harmonic oscillation of M_{z} as a nonlinear effect
S and Gmode dynamics have common characteristic motion: second harmonic oscillations along the zaxis. According to Eqs (6) and (9), and are both responsible for the nonlinearity of l_{x}, together with the resonant frequency softening^{30}. For example, φ_{1}(t) is sinusoidal in Ω = 0.8 GHz; as a results, m_{z} = l_{x} · D_{y}/ħ ∼ cos[φ_{1}(t)] = cos[Asin[ω_{Sigma}t]] is replaced with ~(1 − A^{2}/4) − A^{2}cos[2ω_{Sigma}t]/4 by its first order Taylor expansion. Likewise, and are shown in Fig. 2(a,b), fourth row, right.
Determination of DMI strength
The DMI strength can be obtained by examining the first harmonic precession on the xyplane in the Smode, . Using Eqs (6) and (7), the ellipticity, ε is calculated as m_{y}(t)/m_{x}(t) = [−2J/(D_{y}l_{z})] ħ()/(2J)] = Aħω_{Sigma}cos[ω_{Sigma}t]/(D_{y}sin[Asin[ω_{Sigma}t]]). If we assume A is small enough, ε. For example, is constant within a few percent with , as shown in Fig. 2(a), fourth row, left. Experimentally, the precessional polarization in Smode can be measured using optical tools: terahertz time domain spectroscopy^{23,24,25,26,27,28,30}, or time resolved magneto optical Kerr/Faraday rotation^{16,21,22}.
s_{1} and s_{2} deduced from m and l
Once the DMI strength is determined, J is easily estimated by using wellknown antisymmetric exchange model, M_{s} ∼ M_{0}D_{y}/(2J)^{35}, where saturation magnetization, M_{s} can be measured by using a conventional sample vibrating magnetometer. M_{0} is the number of magnetic ions per volume or mole. Conventional time domain terahertz spectroscopy (or time resolved magneto optical Kerr/Faraday rotation technique) can be used to observe m(t). From the Fourier transform of m_{x}(t) (or m_{y}(t)) and m_{z}(t), the resonant frequencies (and thereby K_{x} and K_{z}) are obtained. As m, D, and J are determined, l could be estimated using Eqs (6) and (9). For example, in Smode, l_{z} and l_{x} can be deduced from m_{x}and m_{z} using Eqs (5) and (6). In Gmode, l_{x} and l_{z} can be estimated using Eqs (8) and (9) and the spectral amplitude and phase information. The resulting s_{1} and s_{2} are shown in Fig. 1(b,c). In contrast, l_{x} is not extractable in simple AF because of the lack of DMI^{14}.
Switching mechanism and efficiency
Figure 3(a,b) show the switching process for Smode and Gmode, respectively (see Supplementary Movie 3 and 4). In Smode, m, defined as , (or m_{z} in Gmode), increases in magnitude less than 1% for l_{max} ~ 1 with the spin current pulse. Additional canting is converted into kinetic energy, and if the kinetic energy exceeds the maximum potential energy, spin reversal occurs after the pulse has been turned off. This is inertiadriven switching^{17}, and the switching is identically applied in Gmode.
Because either ω_{Sigma} or ω_{Gamma} could be manipulated by K_{z} or D_{y}, switching efficiency should be considered for these parameters. Figure 4 shows the periodic patterns for the terminal phase of l_{x} for various values of K_{x} and D_{y} after excitation by spin current pulses for several values of τ, Ω and p. From left down to right up, terminal phases of l_{x} are indicated by nπ, n = 0, 1, 2.
When the potential barrier increases from K_{z} = 0 in Fig. 4(d) to K_{z} = −0.5K_{x} in Fig. 4(e), l in Smode must overcome the higher potential barrier on the xzplane. Thus, phases of l_{x} are shifted upward in Fig. 4(b), compared to Fig. 4(a). Another factor to modify the switching efficiency is the DMI strength in Gmode, where K_{z} does not play a role in the control of the energy barrier on the xyplane because the energy barrier on the xzplane is controllable by K_{z}. When D_{y} = 0.01 J in Fig. 4(c), the pendulum system energy is higher than that of D_{y} = 0.005 J in Fig. 4(f), and the first switching demands higher STT strength. Experimentally, magnetic materials have temperature dependence on anisotropy energies or thickness dependence on D_{y}^{33,36}. Additionally, the interface engineering is used to change DMI strength^{37}. Applying these properties, one can expect to switch magnetization under optimal conditions. In particular, AWF systems, which have anisotropy with two easyaxes (K_{x} and K_{z} > 0), would undergo switching at a lower critical STT strength (Ω_{c}) in Smode, because K_{z} lowers the switching barrier. Finally, we checked the α dependence. When the pulse duration (τ) is short, the damping effect obviously lowers Ω_{c}: For example, Ω_{c} = 4.4 GHz, τ = 5 ps for α = 0.007 and Ω_{c} = 3.8 GHz, τ = 5 ps for α = 0.005 in Fig. 5(a,b). In general, as α becomes smaller, the periodic patterns become narrower^{14}. In particular, in short τ, the slope of the phase boundary is steep and dependent on α; thus, one might doubt its stable functionality as a device. In long τ, Ω_{c} is much reduced and finally become minimized, but its magnitude is not further reduced for variation of α.
Conclusion
In summary, we investigated the process of precession motion in antiferromagnets with weak ferromagnetism through spin transfer torque. Although DMI splits the AF resonant mode into S and Gmodes, the modes are also be interpreted as pendulum models on l. Because l and DMI energy are coupled and independently extractable through measurement of m, dynamic analysis of m provides fundamental understanding of sublattice dynamics, as shown in Fig. 1(b,c). Adjustment of appropriate parameters, such as the anisotropy barrier and DMI strength, provide more efficient magnetization reversal.
Additional Information
How to cite this article: Kim, T. H. et al. Ultrafast spin dynamics and switching via spin transfer torque in antiferromagnets with weak ferromagnetism. Sci. Rep. 6, 35077; doi: 10.1038/srep35077 (2016).
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Acknowledgements
This work was supported by the National Research Foundation of Korea (NRF) grants funded by the Ministry of Science, ICT and Future Planning (MSIP) of Korea (Bank for Quantum Electronic Materials, No. 20110028736 and 2013K000315).
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B.C. and T.H.K. conceived the project idea and planned the analytical and numerical calculations. T.H.K. performed the analytical and numerical calculations. T.H.K., P.G., S.H.H. and B.C. analyzed the data. B.C. led the work and wrote the manuscript with T.H.K. The results of the theoretical and numerical findings were discussed by all coauthors.
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Kim, T., Grünberg, P., Han, S. et al. Ultrafast spin dynamics and switching via spin transfer torque in antiferromagnets with weak ferromagnetism. Sci Rep 6, 35077 (2016). https://doi.org/10.1038/srep35077
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