Damping and Anti-Damping Phenomena in Metallic Antiferromagnets: An ab-initio Study

TL;DR

This study uses first-principles calculations to analyze damping phenomena in metallic antiferromagnets, revealing the roles of relativistic and exchange components in AFMR linewidth and the effects of current-induced torques.

Contribution

It provides a detailed ab-initio analysis of AFMR phenomena, distinguishing relativistic and exchange contributions and exploring current-induced torques in metallic antiferromagnets.

Findings

Exchange component dominates at low temperatures.

Relativistic component linked to spin orbit coupling.

Current-induced torques influence AFMR linewidth and exchange coupling.

Abstract

We report on a first principles study of anti-ferromagnetic resonance (AFMR) phenomena in metallic systems [MnX (X=Ir,Pt,Pd,Rh) and FeRh] under an external electric field. We demonstrate that the AFMR linewidth can be separated into a relativistic component originating from the angular momentum transfer between the collinear AFM subsystem and the crystal through the spin orbit coupling (SOC), and an exchange component that originates from the spin exchange between the two sublattices. The calculations reveal that the latter component becomes significant in the low temperature regime. Furthermore, we present results for the current-induced intersublattice torque which can be separated into the Field-Like (FL) and Damping-Like (DL) components, affecting the intersublattice exchange coupling and AFMR linewidth, respectively.

Tables

Table 1. Table 1: Calculated sublattice magnetic moment ( M s subscript 𝑀 𝑠 M_{s} ), magnetocrystalline anisotropy energy per unit cell, ( K 2 ⟂ superscript subscript 𝐾 2 perpendicular-to K_{2}^{\perp} ), intersublattice exchange coupling per unit cell ( λ 𝜆 \lambda ), ratio of the resistivity ( ρ x ​ x subscript 𝜌 𝑥 𝑥 \rho_{xx} ) to the broadening parameter, η 𝜂 \eta , and the experimental values of ρ x ​ x subscript 𝜌 𝑥 𝑥 \rho_{xx} . We also list values of α d subscript 𝛼 𝑑 \alpha_{d} , α 0 m → s ∥ a → ​ ( c → ) superscript subscript 𝛼 0 conditional subscript → 𝑚 𝑠 → 𝑎 → 𝑐 \alpha_{0}^{\vec{m}_{s}\parallel\vec{a}(\vec{c})} for sublattice magnetization parallel to the a → → 𝑎 \vec{a} ( c → → 𝑐 \vec{c} ) axis, the relativistic ( α ⟂ r subscript superscript 𝛼 𝑟 perpendicular-to \alpha^{r}_{\perp} ) and exchange ( α ⟂ e ​ x subscript superscript 𝛼 𝑒 𝑥 perpendicular-to \alpha^{ex}_{\perp} ) damping parameter for out of plane oscillation mode, for η 𝜂 \eta and η 𝜂 \eta /10 corresponding to the high- and low-temperature regimes, respectively. Finally, we list values of the sublattice current-induced field-like ( τ F ​ L 0 , E → ∥ a → ​ ( c → ) ) superscript subscript 𝜏 𝐹 𝐿 0 conditional → 𝐸 → 𝑎 → 𝑐 \big{(}{\tau}_{FL}^{0,{\vec{E}\parallel\vec{a}(\vec{c})}}\big{)} and antidamping-like ( τ D ​ L 0 , E → ∥ a → ​ ( c → ) superscript subscript 𝜏 𝐷 𝐿 0 conditional → 𝐸 → 𝑎 → 𝑐 {\tau}_{DL}^{0,{\vec{E}\parallel\vec{a}(\vec{c})}} ) components of the spin-orbit torques under an external electric field along the a → → 𝑎 \vec{a} ( c → → 𝑐 \vec{c} )-axis for room-temperature broadening.

	$\left|M_s\right|$	$c/a$	$K_2^{⟂}$	$\lambda$	${\rho }_{xx}/\eta$	${\rho }_{xx}^{exp}$	$\eta$	${\alpha }_d$	${\alpha }_0^{{\overrightarrow{m}}_s\parallel \overrightarrow{c}}$	${\alpha }_0^{{\overrightarrow{m}}_s\parallel \overrightarrow{a}}$	${\alpha }_{⟂}^r$	${\alpha }_{⟂}^{ex}$	${\tau }_{FL}^{0,\overrightarrow{E}\parallel \overrightarrow{a}\left(\overrightarrow{c}\right)}$	${\tau }_{DL}^{0,\overrightarrow{E}\parallel \overrightarrow{a}\left(\overrightarrow{c}\right)}$
	(${\mu }_B$)		(meV)	(eV)	$\left(\frac{\mu \Omega cm}{meV}\right)$	$\left(\mu \Omega cm\right)$	(meV)		(${10}^{-3}$)	(${10}^{-3}$)	(${10}^{-3}$)	(${10}^{-3}$)	(${10}^{-3}$Å)	(${10}^{-3}$Å)
FeRh	3.1	1+$x$	-1.2$x$	0.44	3.4	$\approx$ 100 111Ref.Mankovsky2017 ;   bRef.KOUVEL1963 ;   cRef.Zhang2014 ;   dRef.Ou2016 ;  eRef.Kim2016 ; Moriyama2014 ; Tshitoyan2015 ; Zhang2014	29	0.25	0.8	0.8	$1.7/\sqrt{\left|x\right|}$	1.5$\sqrt{\left|x\right|}$	33 (33)	-14 (-14)
							2.9	2.5	0.27	0.27	$0.6/\sqrt{\left|x\right|}$	15$\sqrt{\left|x\right|}$
MnRh	3.1	0.94	-0.7	0.42	0.57	9522footnotemark: 2	166	0.13	3.3	3.9	10	0.6	10 (7)	6 (-3)
							16.6	0.45	1.5	1.7	5	2
MnPd	3.9	0.93	-0.6	0.5	2.6	223 33footnotemark: 3	103	0.3	0.5	0.6	1.6	0.9	-2 (-5)	93 (4)
							10.3	1.8	0.1	0.5	1.3	5.6
MnPt	3.8	0.93	0.45	0.48	2.7	119,44footnotemark: 416455footnotemark: 5	48	0.43	2.2	7.1	6.7	1.2	-15 (17)	1 (11)
							4.8	3.5	1.5	21	4.6	10
MnIr	2.6	0.97	-5.9	0.4	0.5	176-26966footnotemark: 6	350	0.22	36	35	39	3.6	7 (13)	18 (-7)
							35	0.36	14	11	12	6

Equations

$$\frac{d{\vec{m}}_s(t)}{dt}=$$

$${\vec{H}}_s^{eff}={\vec{B}}_{ext}+\sum _{i=xyz}\left(K_{i;s}^{(2)}+K_s^{(4)}(1-m_{i;s}^2(t))\right)\frac{m_{i;s}(t)}{M_s}{\hat{e}}_i$$

$$\displaystyle+e\vec{\tau}_{DL}^{0}\cdot\vec{E}_{ext}\vec{m}_{s}(t)\times\vec{m}_{\bar{s}}(t)+\Big{(}\frac{\lambda}{M_{s}}+e\vec{\tau}_{FL}^{0}\cdot\vec{E}_{ext}\Big{)}\vec{m}_{\bar{s}}(t).$$

$$(\frac{{\omega }_j}{\gamma }-i{\vec{\tau }}_{DL}^0\cdot {\vec{E}}_{ext}{)}^2=({\omega }_j^0{)}^2+2i{\Gamma }_j\frac{{\omega }_j}{\gamma },j=x,y$$

$${\omega }_j^0=\frac{\sqrt{{\mathcal{K}}_x{\mathcal{K}}_y+2{\lambda }^{\prime }{\mathcal{K}}_j}}{M}$$

$$\Gamma_{j}\equiv\Gamma^{r}+\Gamma_{j}^{ex}=\frac{1}{2M}\big{(}\lambda^{\prime}\alpha_{0}+\mathcal{K}_{j}\alpha_{d}\big{)},$$

$${\alpha }_j^{eff}\equiv \frac{\delta Im({\omega }_j)}{\delta Re({\omega }_j)}=\frac{\lambda {\alpha }_0+{\mathcal{K}}_j{\alpha }_d}{2M\sqrt{{\mathcal{K}}_x{\mathcal{K}}_y+2\lambda {\mathcal{K}}_j}},j=x,y.$$

$${\alpha }_0=\frac{\hslash }{\pi N_{k}M}\sum _{\vec{k}}\mathrm{Tr}({\hat{A}}_{\vec{k}}[{\hat{H}}_{SOC},{\sigma }^{+}]{\hat{A}}_{\vec{k}}[{\hat{H}}_{SOC},{\sigma }^{-}]).$$

$${\alpha }_d$$

$${\vec{\tau }}_{s;i}^S$$

$${\vec{\tau }}_{s;i}^{AS}$$

$${\hat{H}}_{\vec{k}}$$

$${\sigma }_{xx}$$

$${\alpha }_{s{s}^{\prime }}$$

$$i\omega m_s^x-\gamma {\vec{\tau }}_{DL}\cdot {\vec{E}}_{ext}(m_s^x+m_{\bar{s}}^x)=\sum _{s^{\prime }}{\Omega }_{s{s}^{\prime }}^y{m}_{s^{\prime }}^y$$

$$i\omega m_s^y-\gamma {\vec{\tau }}_{DL}\cdot {\vec{E}}_{ext}(m_s^y+m_{\bar{s}}^y)=-\sum _{s^{\prime }}{\Omega }_{s{s}^{\prime }}^x{m}_{s^{\prime }}^x,$$

$${\hat{\Omega }}^j$$

$$(\omega /\gamma -i{\vec{\tau }}_{DL}\cdot {\vec{E}}_{ext}{)}^2={\Omega }_{\parallel }^2+(B_{ext}^z{)}^2-\frac{{\mathcal{K}}_{\perp }^2}{M^2}-({\vec{\tau }}_{DL}\cdot {\vec{E}}_{ext}{)}^2\pm 2{\omega }_0,$$

$${\Omega }_{\parallel }^2=\frac{({\mathcal{K}}_{\parallel }+{\beta }_d-{\beta }_{\bar{d}})({\mathcal{K}}_{\parallel }+{\beta }_d+{\beta }_{\bar{d}}+2\lambda )}{M^2},$$

$${\omega }_0^2=(B_{ext}^z{)}^2\left({\Omega }_{\parallel }^2-({\vec{\tau }}_{DL}\cdot {\vec{E}}_{ext}{)}^2\right)+\frac{{\mathcal{K}}_{\perp }^2(\lambda +{\beta }_{\bar{d}}{)}^2}{M^4},$$

$$(\omega /\gamma -i{\vec{\tau }}_{DL}\cdot {\vec{E}}_{ext}{)}^2=\frac{({\mathcal{K}}_{\parallel }\mp {\mathcal{K}}_{\perp }+{\beta }_d-{\beta }_{\bar{d}})(2\lambda +{\mathcal{K}}_{\parallel }\pm {\mathcal{K}}_{\perp }+{\beta }_d+{\beta }_{\bar{d}})}{M^2}$$

Full text

Spintronics is a field of research exploiting the mutual influence between the electrical field/current and the magnetic ordering. Todate the realization of conventional spintronic devices has relied primarily on the ferromagnetic (FM) based heterostructuresSlonczewski1996 ; Berger1996 ; Manchon2008 ; Miron2010 ; Miron2011 ; Liu2012 . On the other hand, antiferromagnetic (AFM) materials, have been recently revisited as potential alternative candidates for active elements in spintronic devicesBaltz2018 ; Gomonay2014 . In contrast to their FM counterparts, AFM systems have weak sensitivity to magnetic field perturbations, produce no perturbing stray fields, and can offer ultra-fast writing schemes in terahertz (THz) frequency range. The THz spin dynamics due to AFM ordering has been experimentally demonstrated using all-opticalKirilyuk2010 ; Wienholdt2012 , and Néel SOTWadley2016 ; Bhattacharjee2018 mechanisms.

One of the most important parameters in describing the dynamics of the magnetic materials is the Gilbert damping constant, $\alpha$. Intrinsic damping in metallic bulk FMsKambersky2007 ; mahfouziPRB2017_GD is associated with the coupling between the conduction electrons and the time-dependent magnetization, $\vec{m}(t)$, where the latter in the presence of spin-orbit coupling (SOC) leads to a modulation (breathing) of the Fermi surfaceKambersky2007 and hence excitation of electrons near the Fermi energy. The excited conduction electrons in turn relax to the ground state through interactions with the environment (e.g. phonons, photons, etc), leading to a net loss of the energy/angular momentum in the system. While the damping in FMs has been extensively studied both experimentally and theoretically, the damping in metallic AFM has not received much attention thus far.

Manipulation of the damping constant in magnetic devices is one of the prime focuses in the field of spintronics. Conventional approaches to manipulate the damping rate of a FM rely on the injection of a spin polarized current into the FM. The spin current is often generated either through the Spin Hall Effect (SHE)Dykanov1971 ; Sinova2015 by a charge current passing through a heavy metal (HM) adjacent to the FM in a lateral structure, or spin filtering in a magnetic tunnel junction (MTJ) in a vertical heterostructureRalph2008 . Similar mechanisms have also been proposedNunez2006 ; Gomonay2014 ; Gulyaev2014 ; Cheng2016 ; Gulbrandsen2018 ; Khymyn2017 for AFM materials, where the goal is often to cause spontaneous THz-frequency oscillations or reorientationWadley2016 ; Kriegner2016 ; Chen2018 ; Moriyama12018 of the AFM Néel ordering, $\vec{n}(t)=(\vec{m}_{1}-\vec{m}_{2})/2$. Here, $\vec{m}_{s}$ is a unit vector along the magnetization orientation of the sublattice $s$. In contrast to the aforementioned studies that require breaking of inversion symmetry to induce Néel ordering switching, in this work we focus on the current-induced excitation of the sublattice spin dynamics of bulk metallic AFM materials with inversion symmetry intact, and hence no Néel SOTWadley2016 ; Bhattacharjee2018 ; Zelezny2014 ; Zelezny2017 .

The objective of this work is to, (1) provide a general analytical expression for the AFMRKittel1951 frequency and linewidth in the presence of current-induced sublattice torque, and (2) employ the Kubo-like formalism with first principles calculations to calculate the Gilbert damping tensor, $\alpha_{ss^{\prime}}$ ($s,s^{\prime}=\uparrow,\downarrow$), and the field-, $\vec{\tau}_{FL}$, and damping-like, $\vec{\tau}_{DL}$, components of the sublattice torque for a family of metallic AFM materials including MnX (X=Ir,Pt,Pd,Rh) and FeRh, shown in Fig. 1. We demonstrate that the zero-bias AFMR linewidth can be separated into the relativistic, $\Gamma^{r}=\lambda\alpha_{0}/2M$, and exchange, $\Gamma^{ex}=\mathcal{K}\alpha_{d}/2M$ componentsMentink2012 , where $\alpha_{d}\equiv\sum_{s}\alpha_{ss}$, $\alpha_{0}\equiv\alpha_{d}-\sum_{s}\alpha_{s\bar{s}}$, $M$ is the magnetic moment of each sublattice, $\lambda$ is the intersublattice exchange interaction, and $\mathcal{K}$ is the magnetocrystalline anisotropy energy. In agreement with recent first principles calculationsLiu2017 , we find that $\alpha_{d}$ is about 3 orders of magnitude larger than $\alpha_{0}$, indicating the crucial role of the exchange component to the AMFR linewidth. Our calculations reveal that at high temperatures the interband contribution to the relativistic component is the dominant term in the AFMR linewidth, while at low temperatures both exchange and relativistic components contribute to the AFMR linewidth on an equal footing. We further demonstrate that the current-induced antidamping- (field-) like torque changes the AFMR linewidth (intersublattice exchange interaction), thereby allowing the manipulation of the damping constant (Néel temperature) in bulk AFM materials.

Precessional magnetization dynamics of AFMs is often described by a system of coupled equations for each spin sublattice,Cheng2016 ; Khymyn2017 ; Checinski2017 where a local damping constant $\alpha$ is assigned to each of the two sublattices ignoring the effects of the rapid (atomic scale) spatial variation of the magnetization on the damping constant due to the AFM ordering. Taking into account the Gilbert damping tensor, $\alpha_{ss^{\prime}}$, the coupled LLG equations of motion for the two sublattices can be written as,

[TABLE]

where the local effective field in the presence of the external electric ($\vec{E}_{ext}$) and magnetic ($\vec{B}_{ext}$) fields, is given by,

[TABLE]

Here, $\lambda$ is the exchange coupling between the two sublattices, $\vec{\tau}_{DL}^{0}$($\vec{\tau}_{FL}^{0}$) is the current-induced intersublattice damping-like (field-like) torque component and $K^{(2)}_{i;s}$ ($K^{(4)}_{s}$) is the second (fourth) order magneto-crystalline anisotropy energy (MCAE). Eq. (Damping and Anti-Damping Phenomena in Metallic Antiferromagnets: An ab-initio Study) shows that the effect of $\vec{\tau}_{FL}^{0}$ is to renormalize the intersublattice exchange coupling, $\lambda^{\prime}=\lambda+M_{s}e\vec{\tau}_{FL}^{0}\cdot\vec{E}_{ext}$.

In the following, without the loss of generality, we assume $K^{z}_{2}=0$ and $K^{x,y}_{2}\geq 0$, where in the absence of an external magnetic field the magnetization relaxes towards the $\hat{e}_{z}$-axis which can be either in- or out-of-plane. Consequently, we consider $\vec{m}_{s}(t)=m^{z}_{s}\hat{e}_{z}+\delta\vec{m}_{s}(t)$, where, $m^{z}_{s}=\pm 1$ and $\delta\vec{m}_{s}(t)$ is small deviation of the magnetic moment normal to the easy ($\hat{e}_{z}$) axis. Solving the resulting linearized LLG equations of motions, the poles of the transverse dynamical susceptibility yield two oscillating modes with resonance frequencies, $\omega_{j}$, given by

[TABLE]

where, $M=|M_{s}|$, $\mathcal{K}_{j}=K_{j}^{(2)}+K^{(4)}$ and the AFMR linewidth

[TABLE]

can be separated into a relativistic component originating from the angular momentum transfer between the collinear AFM orientation and the crystal through the SOC, and an exchange component that originates from the spin current exchange between the two AFM sublattices. For a system with uniaxial MCAE, Eq.(3a) can be used in both cases of out-of- and in-plane precessions with $K_{x,y}^{(2)}=|K_{\perp}^{(2)}|$ and $K_{y}^{(2)}=0,K_{x}^{(2)}=|K_{\perp}^{(2)}|$, respectively, where $|K_{\perp}^{(2)}|$ is the amplitude of the out of plane MCAE. Eq. (3a) is the central result of this paper which is used to calculate the AFMR frequency and linewidth and their corresponding current-induced effects. A more general form of Eq. (3a) in the presence of an external magnetic field along the precession axis is presented in the Appendix. C.

Eq. (3a) also yields the effective Gilbert damping

[TABLE]

Similarly to the linewidth, $\alpha^{eff}_{j}$ can be separated into the relativistic, $\alpha^{r}_{j}=\Gamma^{r}_{j}/\omega_{j}^{0}$ and exchange, $\alpha^{ex}_{j}=\Gamma^{ex}_{j}/\omega_{j}^{0}$, contributions. To understand the origin of the relativistic component of the AFMR linewidth, one can use a unitary transformation into the rotating frame of the AFM direction, where $\alpha_{0}$ can be written in terms of the matrix elements of $\hat{H}_{SOC}$ using the spin-orbital torque correlation (SOTC) expression,mahfouziPRB2017_GD also often referred to as Kambersky’s formulaKambersky2007 ,

[TABLE]

Here, $\hat{A}_{\vec{k}}=Im(G_{\vec{k}}^{r})$ is the spectral function, $\hat{G}^{r}_{\vec{k}}$ is the retarded Green function calculated at the Fermi energy, $2\sigma^{\pm}=\sigma_{x}\pm i\sigma_{y}$ are the spin ladder operators, and $N_{k}$ is the number of k-point sampling in the first Brillouin zone.

A similar approach applied to the intersublattice elements of the damping tensor leads to a relationship between different elements of $\alpha_{ss^{\prime}}$, rather than an explicit expression for each element. This is due to the fact that in the rotating frame of one sublattice, the other sublattices precesses. Therefore, to calculate $\alpha_{d}$ we employ the original torque correlation expressionmahfouziPRB2017_GD ,

[TABLE]

where $\hat{\Delta}_{\vec{k}}^{s}$ is the exchange spitting of the conduction electrons for sublattice $s$.

Since, for AFMs with Néel temperature above room temperature $\lambda\gg\mathcal{K}_{j}$, one might conclude that $\alpha^{r}\gg\alpha^{ex}$ and the effects of the intersublattice spin exchange on the AFMR line-width becomes negligible. However, since $|\alpha_{ss}|$ is proportional to the intersublattice hopping strength[see Appendix. B] one can expect to have $\|\alpha_{ss^{\prime}}\|\gg\alpha_{0}$. Therefore, the interplay between the relativistic and exchange terms is material dependent, where, for systems with $\lambda\gg\mathcal{K}_{j}$, the effect of the intersublattice spin exchange on the AFMR linewidth may dominate.

The crystal structure, conventional and primitive cell, and the AFM ordering of the MnX (X=Pt,Pd,Ir,Rh) family of metallic bulk AFMs and the biaxially strained AFM bulk FeRh is shown in Fig. 1. The details of the electronic structure calculations of the various damping and antidamping properties are described in detail in the Appendix. A. Table 1 lists the ab initio results of the sublattice magnetic moment, $M_{s}$, $c/a$ ratio, magnetocrystalline anisotropy energy, $K_{\perp}^{(2)}$ , intersublattice exchange interaction, $\lambda$, and ratio of the longitudinal conductivity to the broadening parameter, $\rho_{xx}/\eta$, for the FeRh and MnX systems, respectively. We also list experimental values of the room-temperature $\rho_{xx}$ which were used to determine the broadening parameter. For FeRh we provide the linear dependence of $K_{\perp}^{(2)}$ as a function of biaxial strain, $x\equiv c/a-1$, which shows that under compressive (tensile) biaxial strain the magnetization is along the $c$ ($a$) axisBordel2012 . For the MnX family the magnetization is along the $a$ axis except for MnPt. The MCA values for both MnX and FeRh are in good agreement with previous ab-nitio calculations Umetsu2006 ; Bordel2012 ; Shick2010 ; Chang2018 .

We also list in Table 1 values of $\alpha_{d}$ and $\alpha_{0}^{\vec{m}_{s}\parallel\vec{a}(\vec{c})}$ for sublattice magnetization parallel to the $\vec{a}$($\vec{c}$) axis, and the relativistic ($\alpha^{r}_{\perp}$) and exchange ($\alpha^{ex}_{\perp}$) damping components of the effective Gilbert damping for $\eta$ at room temperature and $\eta/10$ corresponding to low temperature. The decrease (increase) of the damping constants with decreasing temperature is associated with the conductivity (resistivity)-like regime where the inter-(intra-) band scattering contribution is dominant. We find that for the $\eta$ value corresponding to room temperature the AFMR linewidth is mostly dominated by the relativistic component, while at low temperatures the two components are comparable in magnitude. For FeRh a relatively large strain (i.e. $x\approx 0.1$) is required to render the exchange component have a significant contribution to the AFMR linewidth at low temperature.

In Fig. 2(a) we show the variation of $\alpha_{d}$ and $\alpha_{0}^{\vec{m}_{s}\parallel\vec{a}}$ with $\eta$ for cubic FeRh as a representative example. We find that in the experimentally relevant range of $\eta$ ($\approx$ 10 - 100 meV) $\alpha_{0}$ is in the resistivity regime where the interband component is dominant. On the other hand, $\alpha_{d}$ decreases monotonically with $\eta$, suggesting that the intraband component is dominant. Unlike $\alpha_{0}$ which may depend on the orientation of the Néel ordering, $\alpha_{d}$ is relatively isotropic.

Finally, Table 1 lists the values for the current-induced FL- and DL- intersublattice torque coefficients, $\tau^{0,i}_{FL/DL}$, under an external electric field along the $i$ ($a$ or $c$) direction. The sublattice torques are determined by fixing the orientation of the $\bar{s}$ sublattice magnetization and calculating the torque for different magnetization orientations of the $s$-sublattice, using the symmetric and antisymmetric correlation expressionsmahfouziPRB2018_SOT ,

[TABLE]

Here, $f_{n\vec{k}}$ is the Fermi-Dirac distribution function and $\varepsilon_{m\vec{k}}$ are the eigenvalues of the Hamiltonian $\hat{H}_{\vec{k}}$. Having determined the torques, we fit the results to the expected $\tau_{FL}^{0,i}\vec{m}_{s}\times\vec{m}_{\bar{s}}$ and $\tau_{DL}^{0,i}\vec{m}_{s}\times(\vec{m}_{s}\times\vec{m}_{\bar{s}})$ expressions and find the values for the FL and DL torque coefficients. The calculations reveal that the symmetric (anti-symmetric) torque expression leads to the DL (FL) component, in contrast to the SOT results in HM/FM bilayersmahfouziPRB2018_SOT .

Fig. 2(b) displays the current-induced FL and DL intersublattice torques under an external electric field along the $a$ direction for FeRh, as a representative example, versus the broadening parameter $\eta$. Note, the FL component that originates from the antisymmetric torque term [Eq. 8b] is relatively insensitive to $\eta$ (or temperature). On the other hand, the DL intersublattice torque varies almost linearly with $\eta$ (for $\eta$ <0.1 eV) and is of extrinsic origin. Thus, in the ballistic regime where the electronic spin diffusion length is infinite, there is no current-induced transfer of angular momentum between the two sublattices, as it would violate the conservation law of total angular momentum. In the extreme opposite limit, where the spin diffusion length is much smaller than the lattice constant, each sublattice can be viewed as a magnetic lead in a spin valve system where the intersublattice DL torque is analogous to the DL-spin transfer torque.

Im summary, we have employed ab-initio based calculations to investigate the AFMR phenomena in MnX (X=Ir,Pt,Pd,Rh) and biaxially strained FeRh metallic AFMs in the presence or absence of an external electric field. We demonstrate that both the AFMR linewidth and effective Gilbert damping parameter can be separated into a relativistic and exchange contributions, where the former dominates at room temperature while the latter becomes significant at low temperatures. We find that both the AFMR linewidth and the intersublattice exchange interaction (and hence the AFMR frequency and Néel temperature) can be tuned by the external electric field. For example for AFM FeRh an external electric field of 1 V/$\mu m$ (current density of $\approx$ 1012 A/m2) yields an intersublattice FL torque of 3.3 meV ($\approx 0.01\lambda$) and DL torque of 1.4 meV $\equiv$ 2.1 THz change of AFMR linewidth.

The work is supported by NSF ERC-Translational Applications of Nanoscale Multiferroic Systems (TANMS)- Grant No. 1160504 and by NSF-Partnership in Research and Education in Materials (PREM) Grant Nos. DMR-1205734 and DMR-1828019.

Appendix A Density Functional Theory Calculations

We have carried out density functional theory (DFT) calculations for the MnX (X=Pt,Pd,Ir,Rh) family of metallic bulk AFMs (L10 structure) and the biaxially strain G-AFM FeRh (bcc B2 structure) shown in Fig. 1 of the main text. The DFT calculations employed the Vienna ab initio simulation package (VASP) Kresse96a ; Kresse96b . The pseudopotential and wave functions are treated within the projector-augmented wave (PAW) method Blochl94 ; KressePAW . Structural relaxations were carried using the generalized gradient approximation as parameterized by Perdew et al. PBE where the largest atomic force is smaller than 0.01 eV/Å. The plane wave cutoff energy was 500 eV and a 15 $\times$ 15 $\times$ 15 $k$ points mesh was used in the 3D Brillouin Zone (BZ) sampling for the self consistent charge relaxation. A $k$-point mesh of 8 $\times$ 8 $\times$ 8 $k$ was used to obtain the tight-binding Hamiltonian in Wannier basis set using the VASP-Wannier90 calculations Mostofi . The time-dependent electronic Hamiltonian of the system is given by,

[TABLE]

where, $\hat{H}_{\vec{k}}^{0}$ is the spin-independent term of the Hamiltonian where for simplicity we have dropped the Kronecker matrix product symbol between matrices in the orbital and spin Hilbert spaces and $\hat{H}_{SOC}=\sum_{I,l}\xi_{I,l}\vec{\hat{L}}_{I,l}\cdot\vec{\hat{\sigma}}$ is the spin orbit term of the Hamiltonian. The $\vec{\hat{L}}$ and $\xi_{I,l}$ are the angular momentum operator and spin-orbit coupling strength for orbital $l$ of the $I$th atom, respectively. In the following subsections we present the details of the methods that were used to calculate the various physical quantities in Table I.

A.1 Magneto-Crystalline Anisotropy Energy

The uniaxial magneto-crystalline anisotropy energy, $K^{\perp}_{2}$ was determined from the total energy difference between in-plane, $\vec{m}\parallel a$, and out of plane, $\vec{m}\parallel c$ magnetization orientations, $K^{\perp}_{2}=E^{tot}_{\vec{m}\parallel a}-E^{tot}_{\vec{m}\parallel c}$. For the FeRh where $c/a$= 1.0 the MCAE is small. Thus, we have calculated the variation of the MCAE with biaxial strain $x=c/a-1$ which is shown in Fig. 3. The MCAW can be fitted by $K_{2}^{\perp}=-1.2(c/a-1)meV/u.c.$, indicating that under tensile (compressive) biaxial strain the magnetization orientation is along the $a$ ($c$ axis), in agreement with previous ab initio calculations. Bordel2012

A.2 Calculation of Intersublattice Exchange Coupling

The intersublattice exchange coupling is determined from total energy calculations with SOC where one varies the the angle between the sublattice magnetic moments (inset in Fig. 4). The total energy is calculated by imposing an orientation constrain on the magnetic moment configuration using the constrained moment method implemented in VASP where a penalty functional is added to the total energy to align the magnetic moment along a preferred direction. In Fig. 4 we show the variation of $E(\theta)-E(180)$ with the angle $\theta$ (filled circles) for AFM FeRh. The energy difference was fitted to the $E(\theta)-E(180^{0})=\lambda_{1}(1+\cos(\theta))+\lambda_{2}\sin^{2}(\theta)$ expression (blue curve). The exchange coupling is in turn determined from $\lambda=\partial^{2}E(\theta)/\partial\theta^{2}|_{\theta=180^{0}}$, which yields $\lambda=\lambda_{1}+2\lambda_{2}=445\ meV/u.c.$. It is worth mentioning that in our calculations, $\lambda_{2}$, also referred to as the biquadratic exchange termPapanicolaou1988 ; Ivanov2003 ; Chubukov1990 , is significant only in the case of FeRh which undergoes an AFM to FM transition at about 350 K. The biquadratic exchange interaction provides the energy barrier for the AFM to FM transition.

A.3 Calculation of Conductivity

The longitudinal conductivity is determined from Kubo’s expression,

[TABLE]

where $V$ is the volume of the unit cell and the resistivity is in turn given by $\rho_{xx}=1/\sigma_{xx}$. Since the relaxation time approximation is unreliable in the limit of large broadening parameter, $\eta$, we consider only the small $\eta$ limit, were the resistivity is dominated by the intraband component and is proportional to $\eta$. In this case $\rho_{xx}/\eta$ is independent of the broadening parameter that can be used to deduce an estimate of the broadening parameter by replacing the theoretical values of the resistivity with the experimental values.

A.4 Calculation of Damping Parameters

For circular dynamics of the magnetization close to the easy axis, $\vec{m}^{0}_{s}=m^{z}_{s}\vec{e}_{z}$, ($m^{z}_{s}=\pm 1$), the intersublattice damping constant tensor within torque correlation (TC) method is given by,

[TABLE]

Here, $N_{k}$ is the number of $k$-points in the summation, $\hat{G}^{r}_{\vec{k}}=1/(E_{F}-i\eta-\hat{H}_{\vec{k}})$ is the retarded Green’s function, $Im()$ is the imaginary part, $\hat{\sigma}^{\pm}=\hat{\sigma}_{x}\pm i\hat{\sigma}_{y}$ with $\hat{\sigma}_{i}$ being the Pauli matrices, and $\hat{\Delta}_{\vec{k}}^{s}=\frac{\hat{\Delta}_{\vec{k}}\hat{1}_{s}+\hat{1}_{s}\hat{\Delta}_{\vec{k}}}{2}$, is the sublattice exchange splitting where, $\hat{1}_{s}$ is the diagonal matrix with identity elements for orbitals corresponding to sublattice $s$ and zero elsewhere.

Appendix B Toy Model for Gilbert damping tensor

It is instructive to apply the approach of the Gilbert damping constant tensor to a toy model and calculate the matrix elements, analytically. The simplest AFM toy model consists of a four band model Hamiltonian without SOC, $\hat{H}(\vec{k})=\varepsilon(\vec{k})\hat{1}+\Delta\hat{\tau}_{z}\vec{m}\cdot\vec{\hat{\sigma}}+T\hat{\tau}_{x}$, where, $\hat{\sigma}_{i}$s are the Pauli matrices, $\hat{\tau}_{i}$s are Pauli matrices in sublattice space and $T$ is the intersublattice hopping parameter.

For the intraband component of the intrasublattice damping parameter tensor elements we obtain, $\alpha_{ss}=\frac{1}{4MN_{k}}\sum_{n\vec{k}}\frac{T^{2}\Delta^{2}}{\Delta^{2}+T^{2}}\delta(E_{F}-\varepsilon_{n})^{2}$, where $\varepsilon_{1,2}(\vec{k})=\varepsilon(\vec{k})\pm\sqrt{T^{2}+\Delta^{2}}$. Within the relaxation time ($\tau_{el}$) approximation and introducing the parameter $\eta=\hbar/2\tau_{el}$ to broaden the Dirac delta function we find, $\alpha_{ss}\approx\frac{1}{8M\eta\pi}\frac{T^{2}\Delta^{2}}{\Delta^{2}+T^{2}}g_{0}$, where $g_{0}=\frac{2}{N_{k}}\sum_{n\vec{k}}\delta(E_{F}-\varepsilon_{n})$ is the density of states per unit cell at the Fermi energy. Similarly, for the intersublattice elements we obtain, $\alpha_{s\bar{s}}=\alpha_{ss}$, where, as expected due to the absence of the SOC $\alpha_{0}=0$. This suggests that while the microscopic origin of the intrinsic damping, $\alpha_{0}$, is rooted in the transfer of the angular momentum from local spin moments to the crystal mediated by the SOC, the individual sublattice Gilbert damping tensor elements, $\alpha_{ss^{\prime}}$ are governed by the hopping strength of the electrons between different sublattices.

Appendix C Derivation of AFMR Frequency and Linewidth

Since, experimental measurements of the magnetic resonance phenomena is often performed by sweeping the amplitude of the time independent external magnetic field and fixed frequency for the microwave, we define $\beta_{ss^{\prime}}=i\alpha_{ss^{\prime}}\omega$, where $\omega$ is the microwave frequency. Eq.1 in the main text for an AFM with, $m_{1}^{z}=-m_{2}^{z}$, $M=M_{s}$ and $K_{2;s}^{x,y}=K_{2}^{x,y}$, can be rewritten as,

[TABLE]

where,

[TABLE]

Here, we assumed $K^{z}_{2}=0$, $\beta_{d}=\beta_{ss}$ and $\beta_{\bar{d}}=\beta_{s\bar{s}}$. The eigen-frequencies of the system are given by

[TABLE]

where,

[TABLE]

and, we define, $\mathcal{K}_{\parallel}=(\mathcal{K}_{x}+\mathcal{K}_{y})/2$ and $\mathcal{K}_{\perp}=(\mathcal{K}_{x}-\mathcal{K}_{y})/2$. In the absence of an external magnetic field and in linear response regime to the external electric field we obtain,

[TABLE]