# Absence of cross-sublattice spin pumping and spin-transfer torques in
collinear antiferromagnets

Junyu Tang  Department of Physics and Astronomy, University of California,
Riverside, California 92521, USA.  Ran Cheng  Department of Electrical and
Computer Engineering, University of California, Riverside, California 92521,
USA.  Department of Physics and Astronomy, University of California,
Riverside, California 92521, USA.

###### Abstract

There have been debates over the existence of cross-sublattice (CS)
contributions to spin pumping and spin-transfer torques in an antiferromagnet
(AFM) connected to a non-magnetic metal. By studying the interfacial spin
conductance from a symmetry perspective, we first relate the CS spin pumping
to the CS spin-transfer torques in a two-sublattice AFM. Then by calculating
the interfacial spin-dependent electron scattering microscopically, we
ascertain the exact absence of the controversial CS contributions in the
collinear regime. Even in the non-collinear regime, we find that the CS
components only constitute a higher-order correction to the known theory.

††preprint: AIP/123-QED

##  I Introduction

Excitations of magnetic order parameters can generate pure spin currents of
electrons either incoherently or coherently. An incoherent generation
typically involves thermal magnons to exchange spin angular momenta with
electrons, whereas a coherent generation can be achieved by virtue of spin
pumping [1, 2]. When a collinear antiferromagnet (AFM) characterized by two
unit sublattice-magnetic vectors $\bm{m}_{A}$ and $\bm{m}_{B}$ is interfaced
with a non-magnetic metal (NM), the coherent dynamics of the Néel vector
$\bm{n}=(\bm{m}_{A}-\bm{m}_{B})/2$ and of the small magnetic moment
$\bm{m}=(\bm{m}_{A}+\bm{m}_{B})/2$ will generate a total pure spin current [3,
4, 5]

$\displaystyle\frac{e}{\hbar}\bm{I}_{s}=G^{r}(\bm{n}\times\dot{\bm{n}}+\bm{m}\times\dot{\bm{m}})-G^{i}\dot{\bm{m}},$
(1)

where $\bm{I}_{s}$ is measured in units of an electric current (unit: $A$),
$e$ is the absolute electron charge, $\hbar$ is the reduced Planck constant,
$G^{r}$ and $G^{i}$ are two independent components of the interfacial spin
conductance [6] which can be rigorously calculated by considering the
microscopic spin-dependent scattering on the AFM/NM interface [7]. Equation
(1) can be equivalently written in terms of the sublattice-magnetic vectors as

$\displaystyle\frac{e}{\hbar}\\!\bm{I}_{s}\\!=\\!G^{AA}\bm{m}_{A}\\!\times\\!\dot{\bm{m}}_{A}+G^{BB}\bm{m}_{B}\\!\times\\!\dot{\bm{m}}_{B}\\!-\\!G^{A}\dot{\bm{m}}_{A}\\!-\\!G^{B}\dot{\bm{m}}_{B},$
(2)

where $G^{AA}=G^{BB}=G^{r}/2$ and $G^{A}=G^{B}=G^{i}/2$. Recently, coherent
spin pumping in collinear AFMs described by the above equations has been
experimentally verified in a number of materials, notably in MnF2 [8], Cr2O3
[9], $\alpha-$Fe2O3 [10, 11], and synthetic AFM [12], fostering a vibrant
search of new physics in the sub-terahertz frequency range harnessing the
unique spin dynamics of AFMs.

Latest theoretical studies [13, 14, 15, 16], however, suggest that Eq. (2)
should also admit cross-sublattice (CS) terms
$G^{AB}\bm{m}_{A}\times\dot{\bm{m}}_{B}$ and
$G^{BA}\bm{m}_{B}\times\dot{\bm{m}}_{A}$, which in turns changes the Gilbert
damping constant into a matrix. Such additional terms do not contradict with
existing experimental observations, but they do modify the strength of spin
pumping predicted directly by Eqs. (1) and (2), thus affecting the numerical
extraction of the interfacial spin conductance from different materials.
Furthermore, in the non-collinear regime of a two-sublattice AFM (such as the
spin-flop phase induced by a strong magnetic field), even the conventional
form of spin pumping is questionable, especially whether
$\bm{n}\times\dot{\bm{n}}$ and $\bm{m}\times\dot{\bm{m}}$ in Eq. (1) still
share the same coefficient [17, 18]. In direct connection with the CS spin
pumping, CS spin-transfer torques are allowed by the Onsager reciprocal
relations [14, 15, 16], but their existence remains experimentally elusive.

In this Letter, we resolve the puzzle of CS contributions to spin pumping and
spin-transfer torques in collinear AFMs from a theoretical perspective. Guided
by symmetry considerations, we first clarify several essential relations
between spin pumping and spin-transfer torques in the presence of CS
contributions, where the controversial CS components are expressed in terms of
the corresponding coefficients in the spin conductance matrix. We then
quantify these coefficients by studying the microscopic spin-dependent
scattering of electrons off a fully compensated AFM/NM interface. We claim
that all CS effects vanish in the collinear regime, affirming the validity of
the established theories [viz. Eqs. (1) and (2)] and the experimental fitting
they enable. We find that even in the non-collinear regime of a two-sublattice
AFM, the CS effects only bring about higher-order corrections.

##  II Phenomenological relations

In its most general form, the coherent spin pumping by a two-sublattice AFM
into an adjacent NM can be written as

$\displaystyle\frac{e}{\hbar}\bm{I}_{s}=(\bm{m},\bm{n})\times\begin{pmatrix}G^{mm}&G^{mn}\\\
G^{nm}&G^{nn}\end{pmatrix}\begin{pmatrix}\dot{\bm{m}}\\\
\dot{\bm{n}}\end{pmatrix}-G^{m}\dot{\bm{m}},$ (3)

which differs (is generalized) from Eq. (1) by the off-diagonal terms in the
interfacial spin conductance $G^{\mu\nu}$ with $\\{\mu,\nu\\}=\\{m,n\\}$. The
dynamics of the AFM can be described by a set of coupled Landau–Lifshitz
equations as [19]

$\displaystyle\dot{\bm{m}}$
$\displaystyle=\frac{1}{\hbar}\left(\bm{f}^{n}\times\bm{n}+\bm{f}^{m}\times\bm{m}\right),$
(4a) $\displaystyle\dot{\bm{n}}$
$\displaystyle=\frac{1}{\hbar}\left(\bm{f}^{m}\times\bm{n}+\bm{f}^{n}\times\bm{m}\right),$
(4b)

where the Gilbert damping is omitted for simplicity,
$\bm{f}^{m}=-\partial\epsilon/\partial\bm{m}$ and
$\bm{f}^{n}=-\partial\epsilon/\partial\bm{n}$ are the effective fields (or
driving forces) with $\epsilon$ the magnetic free energy. In our convention,
$\bm{f}^{m}$ and $\bm{f}^{n}$ are expressed in units of energy. $\bm{I}_{s}$
can be related to these driving forces through the linear response relation

$\displaystyle
I_{s,i}=L^{sm}_{ij}f^{m}_{j}+L^{sn}_{ij}f^{n}_{j},\quad(i,j\mbox{ run over
}x,y,z)$ (5)

where the response coefficients $L^{sm}_{ij}(\bm{m},\bm{n})$ and
$L^{sn}_{ij}(\bm{m},\bm{n})$ are determined by combining Eq. (3) and Eqs. (4).
Reciprocally, the spin-transfer torques can be obtained as
$T^{m}_{i}=L^{ms}_{ij}V_{j}^{s}$ and $T^{n}_{i}=L^{ns}_{ij}V_{j}^{s}$, where
$\bm{V}^{s}=\bm{\mu}_{s}/e$ is the spin voltage with the spin chemical
potential $\bm{\mu}_{s}=(\mu_{\uparrow}-\mu_{\downarrow})\hat{\bm{s}}$
($\hat{\bm{s}}$ specifies the quantization axis). These response coefficients
satisfy the Onsager reciprocal relation

$\displaystyle L^{ms}_{ij}(\bm{m},\bm{n})=L^{sm}_{ji}(-\bm{m},-\bm{n}),$ (6)

as both $\bm{m}$ and $\bm{n}$ break the time-reversal symmetry. An identical
relation is applicable to $L^{ns}_{ij}$ and $L^{sn}_{ji}$ as well. When
$\bm{V}_{s}$ is treated as a common driving force, $L^{ms(ns)}$ and
$L^{sm(sn)}$ will share the same unit, which simplifies the following
discussions. After some straightforward algebra, we find

$\displaystyle e\bm{T}^{m}=$ $\displaystyle
G^{mm}\bm{m}\times(\bm{V}_{s}\times\bm{m})+G^{mn}\bm{n}\times(\bm{V}_{s}\times\bm{m})$
$\displaystyle+G^{nm}\bm{m}\times(\bm{V}_{s}\times\bm{n})+G^{nn}\bm{n}\times(\bm{V}_{s}\times\bm{n})$
$\displaystyle+G^{m}\bm{m}\times\bm{V}_{s},$ (7a) $\displaystyle e\bm{T}^{n}=$
$\displaystyle
G^{mm}\bm{n}\times(\bm{V}_{s}\times\bm{m})+G^{mn}\bm{m}\times(\bm{V}_{s}\times\bm{m})$
$\displaystyle+G^{nm}\bm{n}\times(\bm{V}_{s}\times\bm{n})+G^{nn}\bm{m}\times(\bm{V}_{s}\times\bm{n})$
$\displaystyle+G^{m}\bm{n}\times\bm{V}_{s},$ (7b)

where all spin-transfer torques have been scaled into the inverse-time
dimension so that $\bm{T}^{m}$ and $\bm{T}^{n}$ can be directly added to Eqs.
(4).

The spin-transfer torques exerting on the two sublattice-magnetic moments,
$\bm{m}_{A}$ and $\bm{m}_{B}$, are $\bm{T}^{A}=\bm{T}^{m}+\bm{T}^{n}$ and
$\bm{T}^{B}=\bm{T}^{m}-\bm{T}^{n}$. A simple manipulation of Eq. (7) shows
that

$\displaystyle\bm{T}^{A}=$
$\displaystyle\tau^{AA}_{D}\bm{m}^{A}\times(\bm{V}^{s}\times\bm{m}^{A})+\tau^{AB}_{CS}\bm{m}^{A}\times(\bm{V}^{s}\times\bm{m}^{B})$
$\displaystyle+\tau^{A}_{F}\bm{m}^{A}\times\bm{V}^{s},$ (8a)
$\displaystyle\bm{T}^{B}=$
$\displaystyle\tau^{BB}_{D}\bm{m}^{B}\times(\bm{V}^{s}\times\bm{m}^{B})+\tau^{BA}_{CS}\bm{m}^{B}\times(\bm{V}^{s}\times\bm{m}^{A})$
$\displaystyle+\tau^{B}_{F}\bm{m}^{B}\times\bm{V}^{s},$ (8b)

where $\tau_{D}^{AA(BB)}$, $\tau^{AB(BA)}_{CS}$ and $\tau_{F}^{A(B)}$
represent the damping-like torques, the CS torques, and the field-like
torques, respectively. To relate these torques to the matrix of interfacial
spin conductance $G^{\mu\nu}$ appearing in Eq. (3), we define
$G^{AA(BB)}=e\tau^{AA(BB)}_{D}/2$, $G^{AB(BA)}=e\tau^{AB(BA)}_{CS}/2$, and
$G^{A(B)}=e\tau^{A(B)}_{F}$, which can be expressed in terms of $G^{\mu\nu}$
as

$\displaystyle G^{AA}$
$\displaystyle=\frac{1}{4}(G^{mm}+G^{mn}+G^{nm}+G^{nn}),$ (9a) $\displaystyle
G^{BB}$ $\displaystyle=\frac{1}{4}(G^{mm}-G^{mn}-G^{nm}+G^{nn}),$ (9b)
$\displaystyle G^{AB}$
$\displaystyle=\frac{1}{4}(G^{mm}+G^{mn}-G^{nm}-G^{nn}),$ (9c) $\displaystyle
G^{BA}$ $\displaystyle=\frac{1}{4}(G^{mm}-G^{mn}+G^{nm}-G^{nn}),$ (9d)

and $G^{A}=G^{B}=G^{m}$. By invoking the Onsager reciprocal relations, we
obtain

$\displaystyle\frac{e}{\hbar}\bm{I}_{s}=(\bm{m}_{A},\bm{m}_{B})$
$\displaystyle\times\begin{pmatrix}G^{AA}&G^{AB}\\\
G^{BA}&G^{BB}\end{pmatrix}\begin{pmatrix}\dot{\bm{m}}_{A}\\\
\dot{\bm{m}}_{B}\end{pmatrix}-G^{A}\dot{\bm{m}}_{A}-G^{B}\dot{\bm{m}}_{B},$
(10)

which incorporates Eq. (2) as a special case (when the CS terms
$G^{AB}=G^{BA}=0$). Because combining Eqs. (9) and (10) can reproduce Eq. (3)
under the definitions of $\bm{n}=(\bm{m}_{A}-\bm{m}_{B})/2$ and
$\bm{m}=(\bm{m}_{A}+\bm{m}_{B})/2$, we have established consistent relations
between spin pumping and spin-transfer torques in the presence of CS
contributions, which hold in both the $(\bm{m},\bm{n})$ basis and the
$(\bm{m}_{A},\bm{m}_{B})$ basis.

Reversing the labeling of sublattices should not change the system [20], so
$\tau^{AA}_{D}=\tau^{BB}_{D}$ and $\tau_{CS}^{AB}=\tau_{CS}^{BA}$, imposing a
constraint that $G^{mn}=G^{nm}=0$. This means that $G^{\mu\nu}$ in Eq. (3)
must be diagonal. Accordingly,

$\displaystyle\tau^{AA}_{D}=\tau^{BB}_{D}=\frac{1}{2e}(G^{mm}+G^{nn}),$ (11)
$\displaystyle\tau^{AB}_{CS}=\tau^{BA}_{CS}=\frac{1}{2e}(G^{mm}-G^{nn}),$ (12)

which indicates that the CS terms exist only when $G^{mm}\neq G^{nn}$.
Therefore, to resolve the puzzle of CS contributions, we just need to quantify
and compare $G^{mm}$ and $G^{nn}$.

Figure 1: A fully compensated AFM/NM interface with cubic lattice, where $-t$
($-t_{m}$) is the hopping energy in the NM (AFM), and $a_{0}$ is the lattice
constant. On the interface plane, the magnetic unit cell is indicated by green
dashed circles, which are periodic in both the $\hat{x}$ ([1,1,0]) and $y$
([1,-1,0]) directions.

##  III Microscopic calculation

The microscopic mechanism underlying $G^{mm}$ and $G^{nn}$ is the spin-
dependent scattering of electrons off a fully compensated AFM/NM interface.
Without losing generality, let us consider a simple cubic lattice as
illustrated in Fig. 1. Following the wavefunction matching method detailed in
Chapter 4 of Ref. [7], we shall determine the scattering matrix $\mathbb{S}$
of the form

$\displaystyle\mathbb{S}=\begin{pmatrix}\mathbb{S}^{++}&\mathbb{S}^{+-}\\\
\mathbb{S}^{-+}&\mathbb{S}^{--}\end{pmatrix},$ (13)

where each block is a $2\times 2$ matrix in the spin space and $\pm$ accounts
for the sublattice (pseudo-spin) degree of freedom. Under the adiabatic
condition (i.e., the dynamics of $\bm{m}$ and $\bm{n}$ is much slower than the
electron relaxation), we have

$\displaystyle\mathbb{S}^{\pm\pm}=S$
${}_{0}^{\pm\pm}\sigma_{0}+S_{m}^{\pm\pm}\bm{m}\cdot\bm{\sigma},$ (14a)
$\displaystyle\mathbb{S}^{\pm\mp}=S$ ${}_{n}\bm{n}\cdot\bm{\sigma}\pm
S_{mn}(\bm{m}\times\bm{n})\cdot\bm{\sigma},$ (14b)

where $\bm{\sigma}=\\{\sigma_{x},\sigma_{y},\sigma_{z}\\}$ is the vectorial
Pauli spin matrices and $\sigma_{0}$ is the identity matrix. The specific
expressions of $S_{0}^{\pm\pm}$, $S_{m}^{\pm\mp}$, $S_{n}$ and $S_{mn}$ in
terms of the crystal momentum and other material parameters are listed in the
Supplementary Material. If we turn to the collinear regime,
$|\bm{n}^{2}|\approx 1$ and $|\bm{m}^{2}|\ll 1$, only $S_{0}^{\pm\pm}$ and
$S_{n}$ in Eqs. (14) will remain essential, then the spin-flip scattering will
be necessarily accompanied by the reversal of pseudo-spin [21]. When we go
beyond the collinear regime, however, the locking between spin and pseudo-spin
will be lifted. We also notice that previous studies assumed
$S_{m}^{\pm\mp}\approx S_{n}$ in the collinear regime without a rigorous
justification [3, 7], so here in a general context we treat all components in
Eq. (14) as independent quantities.

The pumped spin current polarized in the $j$ direction ($j=x,y,z$) can be
calculated by [3, 7]

$\displaystyle
I_{s,j}=-\frac{e}{2\pi}{\rm{Im}}\left\\{{\rm{Tr}}\left[\mathbb{S}^{\dagger}(\sigma_{0}\otimes\sigma_{j})\dot{\mathbb{S}}\right]\right\\},$
(15)

which, after some tedious algebra, ends up with

$\displaystyle\frac{e}{\hbar}\bm{I}_{s}=G^{nn}_{0}\bm{n}\times\dot{\bm{n}}+G^{mm}_{0}\bm{m}\times\dot{\bm{m}}-G^{m}\dot{\bm{m}}+\Delta\bm{I}_{s},$
(16)

where $\Delta\bm{I}_{s}=\Delta
G\\{[\bm{n}\cdot(\bm{m}\times\dot{\bm{m}})]\bm{n}+\bm{m}\cdot(\bm{n}\times\dot{\bm{n}})]\bm{m}\\}$
is a new term not claimed by any existing studies. Utilizing the vector
identities, we can decompose $\Delta\bm{I}_{s}$ into

$\displaystyle\Delta\bm{I}_{s}=\Delta G$
$\displaystyle\left[(\bm{n}\times\dot{\bm{n}})|\bm{m}|^{2}+(\bm{m}\times\dot{\bm{m}})|\bm{n}|^{2}\right.$
$\displaystyle\qquad\left.+2(\bm{n}\times\bm{m})(\bm{m}\cdot\dot{\bm{n}})\right],$
(17)

so Eq. (16) can be finally written as

$\displaystyle\frac{e}{\hbar}\bm{I}_{s}=$
$\displaystyle(G^{nn}_{0}+|\bm{m}|^{2}\Delta
G)\bm{n}\times\dot{\bm{n}}+(G^{mm}_{0}+|\bm{n}|^{2}\Delta
G)\bm{m}\times\dot{\bm{m}}$
$\displaystyle\quad\qquad-G^{m}\dot{\bm{m}}+2\Delta
G(\bm{n}\times\bm{m})(\bm{m}\cdot\dot{\bm{n}}),$ (18)

where we mention again that $G^{mn}$ and $G^{nm}$ are not present (they vanish
identically as required by symmetry). The generalized spin pumping formula Eq.
(18) contains four different components of the interfacial spin conductance
that are determined by the scattering matrix as

$\displaystyle
G^{nn}_{0}=\frac{e^{2}\mathscr{A}}{h\pi^{2}}\int|S_{n}|^{2}\mathrm{d}^{2}\bm{k},$
(19a) $\displaystyle
G^{mm}_{0}=\frac{e^{2}\mathscr{A}}{2h\pi^{2}}\int\left(|S_{m}^{++}|^{2}+|S_{m}^{--}|^{2}\right)\mathrm{d}^{2}\bm{k},$
(19b) $\displaystyle G^{m}=\frac{e^{2}\mathscr{A}}{2h\pi^{2}}\int{\rm
Im}\left[(S_{0}^{++})^{*}S_{m}^{++}+(S_{0}^{--})^{*}S_{m}^{--}\right]\mathrm{d}^{2}\bm{k},$
(19c) $\displaystyle\Delta
G=\frac{e^{2}\mathscr{A}}{h\pi^{2}}\int|S_{mn}|^{2}\mathrm{d}^{2}\bm{k},$
(19d)

where $\mathrm{d}^{2}\bm{k}=dk_{x}dk_{y}$ and $\mathscr{A}$ is the area of the
interface. Except the last term in Eq. (18), we can read off the effective
spin conductance $G^{nn}$ and $G^{mm}$ as

$\displaystyle G^{nn}=G^{nn}_{0}+|\bm{m}|^{2}\Delta G,$ (20a) $\displaystyle
G^{mm}=G^{mm}_{0}+|\bm{n}|^{2}\Delta G,$ (20b)

and according to Eqs. (9) we obtain the damping-like torques and the CS
torques as

$\displaystyle\tau^{AA}_{D}=\tau^{BB}_{D}$
$\displaystyle=\frac{1}{2e}(G^{mm}_{0}+G^{nn}_{0}+\Delta G),$ (21a)
$\displaystyle\tau^{AB}_{CS}=\tau^{BA}_{CS}$
$\displaystyle=\frac{1}{2e}\left[G^{mm}_{0}-G^{nn}_{0}+(|\bm{n}|^{2}-|\bm{m}|^{2})\Delta
G\right],$ (21b)

where $|\bm{n}|^{2}+|\bm{m}|^{2}=1$ is used. The above results are valid even
in the noncollinear regime.

Finally, we point out that the last term in Eq. (18), per the Onsager
relations, gives rise to an additional CS torque

$\displaystyle\Delta\bm{T}^{A}=\Delta\bm{T}^{B}=\frac{\Delta
G}{e}[\bm{V}^{s}\cdot(\bm{m}_{1}\times\bm{m}_{2})](\bm{m}_{2}\times\bm{m}_{1}),$
(22)

which is nonlinear in $\bm{m}_{1}\times\bm{m}_{2}$ thus not being captured by
the phenomenological consideration in the previous section. In the collinear
regime, this term is a negligible higher-order correction.

##  IV Numerical results

Basing on Eq. (19), we numerically plot the four components of spin
conductance in Fig. 2 as functions of the exchange coupling $J$ (between the
conduction electrons and the magnetic moments) and the ratio of kinetic
energies in the AFM and NM (i.e., hopping integrals $t_{m}$ and $t$). Here,
the spin conductance is expressed in units of $e^{2}/h$ per $a_{0}^{2}$ (area
of a magnetic unit cell on the interface), which should be multiplied by the
number of magnetic unit cells $\mathscr{N}$ on the interface to retrieve the
total spin conductance. Comparing Fig. 2(a) and (b), we find that $G^{nn}_{0}$
and $G^{mm}_{0}$ share a very similar pattern as they both culminate around
$J/t=1$ and $t_{m}/t=0.5$. They are the primary contributions to the damping-
like torques [see Eq. (21a)]. Figure 2(c) for $G^{m}$, on the other hand,
shows how the strength of field-like torques varies over $J$ and $t_{m}$. It
is clear that the damping-like (field-like) torques are dominant in the strong
(weak) exchange coupling regime, which is corroborated by a recent experiment
[22].

Figure 2: Interfacial spin conductance per unit-cell area ($e^{2}/h$ per
$a_{0}^{2}$) as a function of $J/t$ and $t_{m}/t$. Here, $J$ is the exchange
coupling between the conduction electrons and the magnetic moment, $t_{m}$
($t$) is the hopping energy in the AFM (NM).

In the collinear limit that $|\bm{n}|^{2}\rightarrow 1$ and
$|\bm{m}|^{2}\rightarrow 0$, the CS torques, according to Eq. (21b), reduce to

$\displaystyle\tau^{AB(BA)}_{CS}=\frac{1}{2e}(G^{mm}_{0}+\Delta
G-G^{nn}_{0}).$ (23)

Using the numerical results shown in Fig. 2, we find that

$\displaystyle G^{nn}_{0}=G^{mm}_{0}+\Delta G,$ (24)

which yields all CS torques exactly zero. As a matter of fact, concerning the
integrands in Eqs. (19) that are functions of the crystal momentum, one can
rigorously prove that Eq. (24) is an exact identify (see details in the
Supplementary Material). It is interesting to note that even in the highly
non-collinear regime where $|\bm{m}^{2}|$ is comparable to $|n|^{2}$, the CS
torques [proportional to
$G^{mm}_{0}-G^{nn}_{0}+(|\bm{n}|^{2}-|\bm{m}|^{2})\Delta G$] is at most a few
percents of the damping-like torques since $\Delta G$ is much smaller than
$G_{0}^{nn}$ and $G_{0}^{mm}$, as shown in Fig. 2(d).

###### Acknowledgements.

The authors acknowledge helpful discussions with Hantao Zhang. This work is
support by the Air Force Office of Scientific Research (Grant No.
FA9550-19-1-0307).

## Supplementary Material

See the supplementary materials for more mathematical details about the
interfacial spin conductance.

## Data Availability Statement

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

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  * Cogulu  _et al._ [2022] E. Cogulu, H. Zhang, N. N. Statuto, Y. Cheng, F. Yang, R. Cheng, and A. D. Kent, “Quantifying spin-orbit torques in antiferromagnet–heavy-metal heterostructures,” Phys. Rev. Lett. 128, 247204 (2022).
  * Guo, Zhang, and Cheng [2022] M. Guo, H. Zhang, and R. Cheng, “Manipulating ferrimagnets by fields and currents,” Phys. Rev. B 105, 064410 (2022).