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Riccardo Tomasello 1,2 , Roman Verba 3 , Victor Lopez-Dominguez 1 , Francesca Garescì 4 , Mario Carpentieri 2 , Massimiliano Di Ventra 5 , Pedram Khalili Amiri 1,* , Giovanni Finocchio 6,* 1 Department of Electrical and Computer Engineering, Northwestern University, Evanston, Illinois 60208, USA 2 Department of Electrical and Information Engineering, Politecnico of Bari, 70125 Bari, Italy 3 Institute of Magnetism, Kyiv 03142, Ukraine 4 Department of Engineering, University of Messina, I-98166, Messina, Italy 5 Department of Physics, University of California, San Diego, La Jolla, CA 92093, USA 6 Department of Mathematical and Computer Sciences, Physical Sciences and Earth Sciences, University of Messina, I-
98166, Messina, Italy * corresponding authors: [email protected], [email protected] Abstract
Voltage controlled magnetic anisotropy (VCMA) is a low-energy alternative to manipulate the
ferromagnetic state, which has been recently considered also in antiferromagnets (AFMs). Here, we
theoretically demonstrate that VCMA can be used to excite linear and parametric resonant modes in
easy-axis AFMs with perpendicular anisotropy, thus opening the way for an efficient electrical
control of the Néel vector, and for detection of high-frequency dynamics. Our work leads to two key results: (i) VCMA parametric pumping experiences the so-called “exchange enhancement” of the
coupling efficiency and, thus, is 1-2 orders of magnitude more efficient than microwave magnetic
fields or spin-orbit-torques, and (ii) it also allows for zero-field parametric resonance, which cannot
be achieved by other parametric pumping mechanisms in AFMs with out-of-plane easy axis.
Therefore, we demonstrate that the VCMA parametric pumping is the most promising method for
coherent excitation and manipulation of AFM order in perpendicular easy-axis AFMs.
1
Parametric resonance, discovered first in mechanical systems, occurs in various harmonic
oscillators in nature, when at least one of their parameters varies periodically in time with an
amplitude overcoming a threshold value. In Magnetism, the parametric resonance and other
parametric phenomena are very rich, including excitation and amplification of spin waves (SWs),
wave front reversal and reversal of momentum relaxation, magnetic soliton compression, and
condensation of magnons ([1–5] and references therein). Parametric pumping is usually achieved by
external microwave magnetic fields[1,6]. However, in ferromagnets, more energetically-efficient
methods involve parametric pumping created by acoustic waves[7,8] or by microwave electric fields
via the voltage-controlled magnetic anisotropy (VCMA)[9]. The latter is especially efficient at
nanoscale size due to vanishing Ohmic losses[10], and offers a uniquely efficient mechanism of
parametric coupling to short (tens of nanometers wavelength) exchange SWs[11].
VCMA has been extensively studied in ferromagnetic metal/dielectric interfaces, typically Fe/MgO or CoFeB/MgO [11–15]. Recently, a very large linear voltage control coefficient 0.3 pJ/Vm [16] has been achieved, which can be increased to 1.2 pJ/Vm [17] in interfaces where ion
migration controls the anisotropy. VCMA-driven magnetization dynamics have shown beneficial
effects in different applications, such as memory [18,19], magnonic devices [20,21], low-energy
motion of skyrmions [22,23], as well as ferromagnetic resonance excitation [14].
Lately, the VCMA effect has been theoretically predicted in antiferromagnets (AFMs) and
proposed as an efficient mechanism for the excitation of AFM dynamics[24–28]. In particular, magnetoelectric coefficients as large as 1.5 pJ/Vm have been predicted in MgO-capped MnPt
films[28]. In addition, indirect evidence of VCMA in AFM thin films was experimentally found in
Ref. [29]. AFMs are more abundant in nature than ferromagnets. Recently, they have been receiving
a renewed interest thanks to their outstanding properties such as low susceptibility, lack of stray fields,
and THz dynamics [30,31]. However, manipulation of AFM order is a more complex task than in
ferromagnets due to the high exchange interaction, making it hard to manipulate them by magnetic
fields. Recent experiments have demonstrated that the Néel vector dynamics can be driven by
electrical currents via spin-orbit torque (SOT) effects [32–38]. This is especially important for the
implementation of AFMs in hybrid CMOS/spintronic circuits. VCMA constitutes another viable
alternative for electrically driving AFM dynamics, which promises to be more energy-efficient due
to reduced Ohmic losses[14,22].
In this work, we demonstrate that VCMA can be successfully applied for the excitation of
AFM resonance modes, both in the linear and parametric resonance regimes. Our results point out
that parametric resonance in AFMs is very efficient for the excitation of large-amplitude precession
2 of the Néel vector. Although parametric resonance in AFMs driven by microwave magnetic fields
has been known for fifty years [39–42], here we show that the mechanism of parametric coupling of
VCMA drive to AFM resonance modes is completely different from the one of microwave magnetic
field. Therefore, VCMA parametric pumping is not only 1-2 orders of magnitude more efficient than
microwave magnetic field pumping, but also allows for parametric excitation in zero bias magnetic
field, which is impossible to achieve with magnetic field pumping. Our results open the possibility of
realization of power-efficient high frequency AFM-based devices, such as tunable electrical detectors
[43,44].
The paper is organized as follows. In Sec. II, theoretical analysis of VCMA-driven AFM
dynamics is performed. Section III presents results of micromagnetic simulations of AFM dynamics
under VCMA drive and comparison with theoretical prediction. In Sec. IV, we consider the efficiency
of VCMA drive by comparing it to alternative excitation mechanisms, i.e. microwave magnetic field,
and SOT. Finally, conclusions are made in Sec. V.
II.
A.
THEORY
Model and basic equations
We study the magnetization dynamics of a thin AFM nanoelement with out-of-plane (OOP)
uniaxial magnetic anisotropy, as AFMs with OOP anisotropy were predicted to demonstrate VCMA.
In both analytical calculations and micromagnetic simulations, we use a continuous two-sublattice
model of an AFM, which rigorously applies for many AFMs, and is one of the most comprehensive
model of AFM dynamics[45]. In particular, this model has been already used to describe AFM-based
THz oscillators [46], detectors [44], and soliton dynamics [47–49]. The model describes the
antiferromagnetic order by considering two sublattices characterized by a normalized magnetization vectors m j = M j / M s , j = 1,2 ( M s is the saturation magnetization of the two sublattices M s 1 = M s 2 = M s ), which dynamics is governed by two coupled Landau-Lifshitz-Gilbert (LLG)
equations[46,47]
where 0 is the gyromagnetic ratio, G is the Gilbert damping parameter, and B eff , j is the effective
field acting on j-th sublattice, which consists of the exchange (homogeneous intersublattice AFM
interaction and nonuniform intrasublattice exchange), uniaxial perpendicular anisotropy, VCMA,
magneto-dipolar B dip , as well as external field contributions.
3 Here, we are concentrating on the VCMA contribution. When an electric field E ac is applied
to the AFM surface - using a gate separated by a dielectric layer, see Sec. III below - at a microwave
frequency f ac , the perpendicular magnetic anisotropy is modulated with the same frequency. The
resulting effective anisotropy field acting on j-th sublattice, thus, becomes
where B a,0 = 2 K u ,0 / M s , K u,0 is the uniaxial anisotropy constant at zero applied voltage, B VCMA = 2 E ac / ( t AFM M s ) is the anisotropy field modulation due to VCMA, with being the magnetoelectric coefficient (VCMA efficiency), and t AFM being the AFM film thickness, and VCMA
is the phase of the applied microwave voltage signal. It is worth noting that the VCMA-related
effective field is proportional both to the external voltage, and to the OOP (z) component of the
sublattice magnetization (in general, time-dependent). This aspect makes the effect of the VCMA
completely different from the microwave magnetic field effect, as shown in the following.
The expression of the VCMA-related effective field in Eq. (2) is used for the analysis of the
VCMA action on AFM dynamic modes. Here, we restrict the analysis to the case of spatially-uniform
AFM dynamics, i.e. we consider the case of AFM resonance, which is typically achieved in
submicron thin AFM elements, as confirmed below by micromagnetic simulations. The VCMA
interaction with propagating SWs or spatially-nonuniform AFM modes can be carried out within the
same formalism.
We assume that the AFM can be biased by an in-plane magnetic field B x , applied, for definiteness, in the x direction. Under such a field, the static magnetization μ j = m j ,0 of the
sublattices sin tilts, B x / B a,0 so that μ 1 sin , 0, cos , μ 2 sin , 0, cos , where
2 B ex . Here, B ex is the effective fields of the homogeneous AFM exchange, which
is the strongest effective field in an AFM, B ex B a , B dip . Excitations of an AFM with uniaxial
anisotropy are well-known. The excitation spectrum consists of 2 linear modes, lower frequency
Mode1 and higher frequency Mode 2. Their frequencies, when accounting for the demagnetization
fields of thin AFM nanoelement, are[1,4,50]:
4 (3)
(4)
where a = 0 B a , ex = 0 B ex , H = 0 B x , M = 0 0 M s . At low and moderate fields ( B x 2 B ex
), the frequencies of linear modes can be approximated to [1,51]:
2 (1) 2 a ex − H 2 a 2 ;
(2) 2 a ex + H
2 2 ex (5)
The frequency of Mode 1 is slightly dependent (decreasing) on the field, while the frequency of Mode
2 increases with B x (examples are shown below in Sec. III). At zero field, the modes are, naturally,
degenerated. Comparing to ferromagnets, where the linear mode has a frequency of the order of FM ~ a (in an unbiased case), the frequencies of the AFM modes are 2 ex / a 1 times larger,
which constitutes the so-called effect of “exchange enhancement” of AFM dynamic characteristics,
caused by huge effective fields of homogeneous AFM exchange[45,51]. Many, but not all AFM
characteristics experiences this enhancement.
The structure of the AFM modes are fully characterized by a net dynamic magnetization m ( ) = ( m 1,( ) + m 2,( ) ) / 2 , and a dynamic Neel vector l ( ) = ( m 1,( ) − m 2,( ) ) / 2 , where ν = 1,2 is the
mode index (in order not to mix mode index and sublattice index, here and in the following the mode
index is in the brackets). The net magnetic moment and dynamic AFM vector of the lower mode
(Mode 1) are:
m 1 1
(1), one finds, in a general case, that the linear coupling to the VCMA is proportional to the OOP
dynamic component of the AFM vector l z . Therefore, there is no coupling to the Mode 2, and the coupling to the Mode 1 appears only at a nonzero bias field and is proportional to sin . The fact that
the linear coupling is defined by l z and not by m z - as the coupling with the external field -, underlines
a crucial difference in the action of the VCMA on the AFM dynamics compared to the magnetic field
action, in which it is also the key element to excite parametric dynamics (see below). Indeed, the
linear components (at the excitation frequency) of the VCMA effective field are opposite in the two
sublattices B eff , j B VCMA z , j e z because of the opposite orientations of the OOP static
magnetization components, hence it could excite modes with anti-phase OOP dynamic magnetization.
In contrast, an external magnetic field, obviously, acts in phase on both sublattices.
The effective driving field acting on the Mode 1 is equal to
B VCMA sin cos b
2 / 1 , (8)
, (9)
a
and corresponding excited mode amplitude is
b c
1
i ac 1 1 2 which is related with the amplitude of Neel vector oscillations as l max 2 c l 1 c /4 .
In Eq. (9), Γ is the damping rate, which is given by ( ) = G ( ) ( ) with 2 1 1 2 2 a 1 , 2 ex a 2 a 2 2 a 2 2
2 2 ex . (10)
being the coefficients dependent on the precession ellipticity of the modes[50]. In the range of small
bias fields 1 2 ex / 2 a , we can approximate the damping rate to 1 2 G ex . It is
clear that the damping rate also exhibits the effect of exchange enhancement, leading to wide AFM
resonance curves.
The enhanced damping rate results in a weak efficiency of the linear excitation of the AFM
modes. The coupling efficiency in Eq. (8) shows no exchange enhancement, and it is also proportional to small values of sin , and a / 1 1 / 4 2 ex / a . Therefore, the amplitude of the excited
mode, given by the coupling efficiency and damping, is small. We can conclude that the VCMA is a
good mechanism for the linear excitation of AFM resonance modes. Nevertheless, in the case of
sufficient bias magnetic field, the VCMA could be the most efficient mechanism for the excitation of
the Mode 1. This aspect is promising from an experimental point of view because the VCMA can
6 provide a larger dynamical contribution to the effective field and low parasitic losses simultaneously.
The same problem of small excitation rate stands true for other linear excitation mechanisms,
especially for microwave field, as discussed in Sec. IV.
Now, we consider the parametric coupling of the VCMA to the AFM modes. The calculation
of the parametric interaction efficiency due to the VCMA pumping is done within the same SW
perturbation formalism[50]. In a general case, we obtain V 1 V 2 i A 1
iA 2 2 ex cos 2 sin 2 cos 2 sin 2 2 2 , a 1 . 2 (2) 2 ex ) 2 ( a cos sin (11)
The parametric coupling coefficients V are defined as usual, so that the parametric term in the equation of motion of the mode amplitude is derived as dc ( ) / dt + ... = V ( ) b p c ( * ) e − i p t , where b p = ΔB VCMA is the pumping amplitude, and ω p is the VCMA pumping frequency. In the range of small
bias fields, expressions (11) are reduced to:
V 1 i cos
3
4 2 ex , V
2 a i cos 2
4 2 ex ,
(12) a
These expressions clearly demonstrate that the parametric coupling to VCMA pumping in small bias
fields also exhibit the exchange enhancement, as it follows from the multiplier 2 ex / a . This
exchange enhancement is related to strong ellipticity of the magnetization precession, which creates
a large longitudinal component at the double oscillation frequency, responsible for the parametric
coupling.
The exchange enhancement of the parametric coupling to VCMA is a key result of our
theoretical calculations. This enhancement retains the parametric excitation mechanism efficient
despite the exchange enhanced damping rates. Indeed, the threshold of the parametric excitation is
determined by the ratio b th = V/Γ, and the exchange enhancement of the coupling efficiency “compensates” the one of the damping rate, resulting in accessible values of the excitation threshold.
Below, by means of micromagnetic simulations, we also show that, above the threshold, the
amplitudes of parametrically-excited modes grows fast, and can reach large values up to c ~ 1.
With the increase of the bias field, the efficiency of ellipticity-caused parametric coupling decreases, as it is usual for the so-called “tilted” parametric pumping[1,2], and vanishes at /2
(this contribution is described by the first term in Eq. (11)). Another contribution to the parametric coupling efficiency ( sin 2 term in Eq. (11)) is specific for anisotropy-driven parametric 7 pumping[8,20], and therefore for VCMA, and it increases with the static magnetization tilt. However,
this contribution does not exhibit the exchange enhancement, and the VCMA driven parametric
excitation becomes much less efficient at
III.
/2 .
MICROMAGNETIC SIMULATIONS
In order to verify our theoretical predictions, we performed systematic micromagnetic
simulations of the AFM dynamics using an in-house numerical framework[46,47]. In addition,
micromagnetic simulations allow us to study overthreshold dynamics, find stationary amplitudes of
parametrically excited modes, and compare excitation efficiency by VCMA with other possible
driving forces. Last but not least aim is to validate the above used approximation of spatially-uniform
dynamics, and excitation of linear eigenmodes of the AFM sample. Indeed, AFMs are characterized
by nonlinear dynamics even in small samples, as proved by excitations of magnetic solitons and
domain walls [45,51]. Therefore, it is not straightforward that the parametric excitation gives rise to
uniform AFM modes.
The magnetization dynamics of an AFM nanoelement with OOP uniaxial magnetic
anisotropy, which can be also biased by an in-plane (IP) magnetic field B x , is studied, and different driving forces – magnetic fields, SOT [32,34,35,52], as well as VCMA – are considered and
compared. A sketch of the studied structure is shown in Fig. 1. It is a 5-terminal device where a squared 200x200 nm 2 AFM voltage gate [53,54] is placed on top of a Heavy Metal (HM) cross bar.
The four terminals of the cross bar allow for injecting an ac electrical current j HM ( t ) = J HM sin ( 2 f ac t + HM ) into the Platinum HM, thus generating the SOT with spin-polarization
p y (p x ) along the y-direction (x-direction) when the current flows between the terminals A-A’ (B-B’). The fifth terminal is used to apply an ac voltage V ( t ) = V sin ( 2 f ac t + VCMA ) (Fig. 1(b)), which modulates the OOP anisotropy K u ( t ) = K u ,0 + K u sin ( 2 f ac t + VCMA ) of the AFM because of the VCMA
effect. Also, a microwave magnetic field b ( t ) with different polarizations could be applied to the
structure. The proposed structure can be used for experimental investigation of the VCMA and SOT-
driven AFM dynamics in the same sample, as well as charting an experimentally-viable pathway
towards practical implementation of the discussed phenomena.
8
The micromagnetic simulations are based on the above mentioned two-sublattice continuous model of AFM in Eq. (1). The exchange effective field acting on the 1 st sublattice include three terms:
(13)
and the expression for B 2,ex is achieved by the index permutation 1 2 . Here, a is the lattice constant, A 11 0 and A 22 0 are the inhomogeneous intra-lattice contributions (assumed to be equal, A 11 = A 22 ), A 12 0 is the inhomogeneous inter-sublattice contribution, which is neglected in this study, and A 0 0 , is the homogeneous inter-sublattice contribution to the exchange energy. The effect of
SOT from the spin-Hall effect (SHE) is described by addition of the following torque [35] to Eq. (1):
j ( t ) Τ j = d J i − DLT HM ( m j m j p ) t AFM
with d J being the torque coefficient given by d J = (14)
g B , where g is the Landè factor, B is the 2 eM
SBohr magneton, and e is the electron charge. The coefficient i − DLT takes into account the efficiency of the charge/spin current conversion of the current j HM ( t ) = J HM sin ( 2 f ac t + HM ) flowing in the
HM. The vector p is the direction of the spin-polarization. The physical parameters in the simulations are similar to previous studies on metallic AFMs, such as PtMn, FeMn and FeRh, namely M s = 566 kA/m, A 11 = A 22 = 5.0 pJ/m, A 0 = − 0.248 pJ/m, K u = 28.3 kJ/m 3 , a=0.5 nm, i − DLT = 0.15 , and = 0.02 [35]. The thickness of AFM layer is t AFM 1 nm . The cell size in the simulation was set to 4 x 4 x 1 nm 3 .
First, we performed characterization of linear eigenmodes by exciting them with different
drives (details on which mode is excited by microwave field or SOT with a given polarization are 9 discussed in Sec. IV). The frequencies of the simulated linear modes (which, obviously, does not depend on the type of linear drive) coincide with analytical calculations – in the range of small and
moderate bias fields, the frequency of Mode 1 is weakly field-dependent, while the frequency of
Mode 2 increases with the field (Fig. 2). In the following, if other is not stated, the external filed is
fixed to B x = 1800 mT in order to easily distinguish between the modes.
Now, we focus on the VCMA-driven excitation of the AFM resonance modes. The simulated
resonance curves under the VCMA drive of different amplitude are shown in Fig. 3. At small
amplitudes of VCMA drive (B VCMA < 45 mT), the resonance curve exhibits a broad resonance peak
at the frequency of 33 GHz (Fig. 3(a)), which is the Mode 1 eigenfrequency. In contrast, we find no
linear response of the Mode 2 to the VCMA drive. This behaviour is in full accordance with our
theoretical calculations, which shows that the linear VCMA couples only to the Mode 1. The
amplitude of the linearly-excited Mode 1 nicely follows the calculated linear dependence in Eq. (9). However, even applying a large VCMA drive of ΔB VCMA = 80 mT leads to very small oscillations amplitude of the linearly-excited mode l y 0.08 .
At larger VCMA drive amplitudes, B VCMA ≥ 45 mT, the simulations show the appearance of
a second peak, which is located at about the double frequency of Mode 1. The amplitude of this peak abruptly increases with VCMA beyond a threshold value (see Figs. 3(a) – (c)). These characteristics
are a clear evidence of the parametric resonance. Importantly, the parametrically-driven dynamics of
the AFM nanoelement remains almost perfectly spatially-uniform (in all the studied range of VCMA
drives, as well as at different bias magnetic fields,), thus validating our theoretical approach.
The theoretically-calculated parametric excitation threshold for B x = 1800 mT is B VCMA,th = 48.6 mT , which agrees well with the simulation result B VCMA,th 48 mT . If we consider
10 the predicted VCMA magnetoelectric coefficient of 1.5 pJ/Vm [28], and an MgO thickness of 2
nm in the structure, the required voltage to achieve the parametric resonance is about 17 mV, which
should be not hard to access in experiments and utilize for various applications. At higher excitation
frequency and higher VCMA drive (central frequency 124.4 GHz and the threshold of 96.5 mT), we
also observe parametric resonance of the Mode 2 in the simulations (not shown), which also correlates
well with theoretical predictions.
Just above the parametric excitation threshold, the resonance peak is narrow, especially in
comparison with linear resonance (Fig. 3(a)). With the increase of the parametric pumping, the peak becomes wider and acquires a characteristic antisymmetric “triangle-like” shape (Fig. 3(b)). To
describe the peak shape as well as finding the amplitudes of the parametrically-excited modes, we
need to consider nonlinear effects, which limit the growth of the parametric instability. In a confined
geometry with discrete SW spectrum, typically, the most important nonlinear effect is the nonlinear
11 2 frequency shift of the ν-th mode frequency ( ) = ( ) ,0 + T ( ) c ( ) , accounting for which, the
amplitude c (ν) of the parametrically-excited mode is calculated as[9]:
2
V ( ) B VCMA − ( 2 )
, (15) T ( )
where = p 2 − ( ) ,0 is the detuning from the exact parametric resonance. Although the
calculation of the nonlinear frequency shift is a complex task, one can use the following trick in the
case of the low-frequency mode. The relation between both static and dynamic magnetization
components of the sublattices is the same (m x,1 = m x,2 , m y,1 = -m y,2 , m z,1 = -m z,2 ), which allows for
reducing the two coupled LLG equations for the sublattices into one effective equation, and apply
ready-to-use equations based on the Hamiltonian formalism [55,56]. In this way, we find
T 1 ex / 2 (for B x 2 B ex ). In fact, the excited mode amplitude given by Eq. (15) not always
can be reached. In the case of low thermal noise, the SW mode is excited within the frequency range
th = 2 V ( ) B VCMA − ( 2 ) , in which small-amplitude SWs become unstable due to the
parametric pumping. In this case, the parametric resonance curve has a characteristic “triangular”
shape (see green curve in the inset of Fig. 3(b)) with the maximum at the left (right) edge for negative (positive) nonlinear frequency shift equal to c ( ) ,max
reproduces well the micromagnetic outcomes (Fig. 3(c)). Some discrepancy at large VCMA drive is
common and related with the utilization of Taylor expansion in the Hamiltonian formalism, which
becomes less accurate at large precession amplitudes (typically, for l y 0.5 ). However, if thermal
fluctuations are large (overcoming the dashed curve in the inset of Fig. 3(b)), or the excitation frequency is continuously swept from the right, one can access the part of the curve beyond − th .
The maximal frequency detuning and peak amplitude in this more complex case are determined by
other nonlinear mechanisms (nonlinearity of parametric interaction efficiency or/and nonlinearity of
damping), consideration of which lies beyond the scope of this work.
Above, we found that, by means of VCMA parametric pumping, it is possible to excite large-
amplitude SW modes in an AFM nanoelement. Let us briefly consider alternative mechanism and the
efficiency of spin wave excitation by their means, starting in this subsection from the linear excitation.
12 The oldest and well-established mechanism involves the use of microwave magnetic fields.
Depending on the polarization of the magnetic field, it can excite either Mode1 or Mode 2, or both
modes, which is determined by the net dynamic magnetization of the modes m ( ) . Thus, according to
Eqs. (6) and (7), the microwave field b x excites only Mode 1, while fields b y and b z excite Mode 2
(the last one only in the presence of static bias field B x ), in full accordance with micromagnetic data.
The coupling efficiency, however, shows the same problem as the linear coupling with VCMA drive – they are inversely proportional to 4 2 ex / a and, accounting for the enhanced damping, the
excited SW amplitudes at the experimentally-achievable microwave drive are very low. The exception is OOP microwave field. The coupling to it exhibits a “partial exchange enhancement” b ~ 4 2 ex / a , being, however, proportional to another weak value sin . Thus, microwave
magnetic field is not a choice for the linear excitation of OOP easy-axis AFM independently of the
polarization.
Much recent approach to excite AFM dynamics is the application of SOT, e.g., by means of SHE.
In the simulations, Mode 1 (2) was observed when a SOT with a spin-polarization parallel
(perpendicular) to the external field is applied and the dynamic AFM vector has y (x)-component of
the Néel vector l y (x) at the excitation frequency. Exemplary resonance curves of SOT-driven AFM
dynamics are shown in the inset of Fig. 4, additionally underlining large width of the resonance curves
in the linear excitation regime. The mode selectivity is also easy to find theoretically considering the
mode structure and SOT effective field.
Figure 4 shows the Néel vector oscillation amplitude excited by SOT. Here, the ac electrical current is chosen large but achievable in experiment – it is sufficiently small to prevent non-magnetic
phenomena, such as electromigration [38], but is not far from this limit. The amplitude of Mode 1 is
larger than the one of Mode 2 and is weakly dependent on the field in the studied range. Contrarily,
the amplitude of Mode 2 decreases with the field. Overall, the maximum achievable oscillation
amplitude is reasonable l max ~ 0.075, and significantly larger than ones achievable with microwave
magnetic field excitation. Nevertheless, such oscillation level can be insufficient for some
applications, e.g., for application of these devices as detectors, considering that the magnetization
precession is at not large amplitude and the electrical readout mechanisms for AFM order, developed
so far, are not very efficient. Also, large precession amplitudes are indispensable for devices based
on nonlinear SW interactions.
13
We have already mentioned above that the coupling efficiency of SOT to AFM modes experiences “partial exchange enhancement”, being proportional to b ~ 4 2 ex / a . It is much better
than the microwave field excitation, but not enough to fully compensate the exchange enhancement
of the damping. Thus, SOT in AFM is less efficient for linear SW excitation than in ferromagnets. In
a general case, field-like SOT can be also present in AFMs, but it does not lead to any significant
differences in the excitation efficiency, as is discussed in the Appendix A.
Now, let us look on the parametric excitation. First, it is worth noting that microwave field-driven
parametric resonance in AFMs is generally well-known. However, almost all the previous works
considered easy-plane AFMs [40–42], which demonstrate good parametric coupling with magnetic
field pumping. Easy-axis AFMs, considered in this work, show a completely different behavior. The
parametric coupling with OOP magnetic field b z is identically zero since it affects the sublattices with opposite phases. The parametric resonance is possible only in a biased state (B x >0) under “parallel pumping” b p =b x e x . The efficiency of the parametric coupling to this pumping is easy to calculate
within the same approach, as for VCMA, which yields
sin V where ellipticity-related coefficient demonstrates exchange enhancement ( i
2 , (16)
is given by Eq. (10). Although this coupling efficiency also ~ 2 ex / a ), it is also proportional to sin , and,
consequently, at low and moderate bias fields, is small. In particular, it is much smaller than the
parametric coupling for the VCMA pumping, as shown in Fig. 5. In addition, this figure clearly shows
that the VCMA-driven parametric excitation is achievable in zero bias magnetic field, as also 14 confirmed by the micromagnetic simulations (see Fig. 3(d)). This is important both from a
fundamental and practical point of view, since this behavior cannot be obtained by any other here-
studied means, and underlines one more time the crucial difference between the VCMA action on
AFM materials compared to the field action.
Finally, let us consider the possibility of parametric resonance excitation by SOT. The
effective field of
( SOT
is given by ) b SOT, j = b SOT ( m j p ) = b SOT c m j , ( ) p + c * m * j , ( ) p + ( μ j p ) . Within the framework of
perturbation theory[50], the coefficients, which describe parametric coupling, are b ( ) for the part of
perturbation field, linear in c * , and S , in calculation of which only the static part of perturbation
field is accounted for (see Eqs. (2.4-2.5) in[50]). Both of them are identically zero independently of the spin current polarization, since ( ) b ( ) ~ j m * j , ( ) m * j , ( ) p j 0 and S ( ) , ( ) ~ j μ j ( μ j p j ) 0 . Thus, degenerate parametric process cannot be driven by SOT from SHE. Nondegenerate process (splitting of pumping into 2 different modes, P → ( 1 ) + ( 2 ) ) is also ( ) impossible – its efficiency is given by j m * j , ( 1 ) m * j , ( 2 ) p j 0 and one can easily check that
contributions from different sublattices compensate each other.
To summarize, we put all the considered cases in Table I, that shows the correspondence
between drive and excited AFM mode, as well as how the coupling relates to the exchange enhancement rate = 2 ex / a . One can see that, among all the considered mechanisms, SOT is
the most suitable for linear excitation of AFM eigenmodes. The most efficient mechanism is, 15 however, the VCMA-driven parametric resonance, as it is the only mechanism exhibiting a “full exchange enhancement” V ~ 2 ex / a both for unbiased and weakly biased AFMs. It is worth
noting that the exchange enhancement rate is, indeed, large and is of a principal importance for the AFM dynamics. For the studied AFM, this ratio is 2 ex / a = 11.84 , and could reach even higher
values for other AFMs with stronger homogeneous AFM exchange interaction. In addition to high “magnetic efficiency”, the VCMA drive has a perfect electric efficiency in terms of low Joule heating
losses and other parasitic losses. Overall, the VCMA parametric pumping results to be the most
promising method for coherent excitation and manipulation of AFM order in easy-axis AFMs with
OOP anisotropy.
| Drive Type | Coupling efficiency | ||
|---|---|---|---|
| Drive Type | Linear coupling, b | Parametric coupling, V | |
| Drive Type | Mode 1 | Mode 2 | Both modes |
| SOT with spin-polarization perpendicular to Bx | 0 | ~ | 0 |
| SOT with spin-polarization parallel to Bx | ~ | 0 | 0 |
| Microwave field bx | ~ 1/ | 0 | ~ sin |
| Microwave field by | 0 | ~ 1/ | 0 |
| Microwave field bz | 0 | ~ sin | 0 |
| VCMA | ~ sin / | 0 | ~ |
Microwave field b x ~ 1/ 0 ~ sin Microwave field b y 0 ~ 1/ 0 Microwave field b z 0 ~ sin 0 Drive Type Linear coupling, b ~ ~ sin / VCMA 0 TABLE I. Summary of the excited modes of an AFM with OOP easy axis for different excitation sources. The order of the coupling efficiency is also indicated, where = 2 ex / a 1 is the “exchange enhancement ratio”. The
proportionality to sin underlines that the excitation mechanism can only work for an AFM biased by an external
magnetic field.
In summary, we have analyzed, by means of micromagnetic simulations and analytical theory,
the excitation of resonant modes in a uniaxial perpendicular AFM comparing different excitation
source: magnetic fields, SOT, and VCMA. The linear excitation can be achieved by all the sources,
where the particular excited mode depends on combination of field/SOT polarization, and bias
magnetic field. However, amplitudes of SW modes, which could be reached in an experiment, are not
large and does not exceed l i ~ 0.05-0.1, because of the exchange enhancement of the damping rate,
which cannot be completely balanced by any linear drive source.
In contrast, VCMA parametric pumping demonstrates exchange enhancement of the coupling
rate, allowing, thus, for the excitation of SW modes with unprecedentedly large precession amplitude.
The parametric resonance could in principle be excited by microwave magnetic fields, however, the 16 parametric interaction efficiency for the field pumping is much lower than the one for VCMA
pumping. In addition, the VCMA advantageously allows for parametric excitation even at zero
magnetic field.
Compared to SOT, which can only excite linear modes, the VCMA stands as an unique electrical
pumping source for efficient excitation of large-amplitude coherent dynamics in easy-axis AFMs.
This is a key and promising result for AFMs device applications, which should not be based on linear
modes but on parametric excitation.
ACKNOWLEDGEMENT
This work was supported under the Grant 2019-1-U.0. ("Diodi spintronici rad-hard ad elevata sensitività - DIOSPIN") funded by the Italian Space Agency (ASI) within the call “Nuove idee per la componentistica spaziale del futuro”, and the project PRIN 2020LWPKH7 funded by the Italian
Ministry of University and Research. R. V. acknowledges support by the Ministry of Education and
Science of Ukraine (project # 0121U110090). This work was also supported by a grant from the U.S.
National Science Foundation, Division of Electrical, Communications and Cyber Systems (NSF
ECCS-1853879), and by the National Science Foundation Materials Research Science and
Engineering Center at Northwestern University (NSF DMR-1720319). MD acknowledges support
from the DOE under Grant No. DESC0020892. The work of R.T. at the Northwestern University was
sponsored by the Petaspin Association (www.petaspin.com).
APPENDIX A: EFFECT OF THE FIELD-LIKE TORQUE
We simulated the effect of the SO-field-like torque (FLT) B FLT . We performed systematic
micromagnetic simulations to study how B FLT affects the Néel vector dynamics acting simultaneously
with the damping-like torque-related field (DLT) B DLT d J = 2 0 d j i DLT j HM t t 0 AFM , where g B , with g being the Landè factor, B the Bohr magneton, e the electron charge, and M S 2 eM
Sthe saturation magnetization. 0 is the vacuum permeability, i − DLT takes into account the efficiency of the charge/spin current conversion of the current j HM t flowing in the heavy metal, 0 is the
gyromagnetic ratio, and t AFM is the AFM thickness.
In Fig. 6, we summarize the results (the time evolution of the spatially-averaged Néel vector
components) achieved at the resonance frequency, where the larger effect is observed, for spin-
polarization p x and p y and three different bias fields B x . We consider four values of the ratio 17 B FLT B DLT respectively equal to 0.0, 0.2, 0.4, and 0.6 (which we also refer to as FLT0.0, FLT0.2,
FLT0.4, and FLT0.6). The FLT is considered along the x- (y-) direction according to the direction of
the spin-polarization p y (p x ), and the two torques are perpendicular in the sample plane.
18 For spin-polarization p x , the y- and z-components do not exhibit significant changes under the FLT.
On the other hand, the x-component
strength for each B x , but such an increase is smaller as B x gets larger (see also Fig. 7(a)).
For spin-polarization p y , the x- and z-components do not exhibit significant changes under the FLT.
Whereas, the y-component
an oscillation at twice the input frequency, while a non-zero FLT leads the oscillation frequency to
be at the same frequency as the input one. Therefore, we observe a trade-off between these two effects,
which yields the existence of a threshold value of the FLT for each B x . Below such a threshold,
is characterized by those two modes simultaneously, and beyond the threshold,
input frequency. For instance, we can compare the blue curve in Fig. 6(d) for FLT<0.3DLT (double
mode) with the grey curve at zero FLT (single mode at twice the input frequency), and with the green
and red curves at higher FLT (single mode at the input frequency).
The different behavior for different spin-polarizations is ascribed to the relative orientation between
the ac field responsible for the FLT and the bias field B x . In the first case (p x ), they are perpendicular
to each other and the only effect is the increase of the oscillation amplitude. In the second case (p y ),
they are parallel to each other and both amplitude and frequency change.
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24