Statistical Inference for Stochastic Processes,

25, no. , pp. , 2022, DOI: 10.1007/s11203-022-09278-4.

M. V. BURNASHEV 111Burnashev M. V. is with Institute for Information
Transmission Problems, Russian Academy of Sciences, Moscow; email:
<EMAIL_ADDRESS>
ON STEIN’S LEMMA IN HYPOTHESES TESTING IN GENERAL NON-ASYMPTOTIC CASE 222This
work was supported by the Russian Foundation for Basic Research under Grant
19-01-00364.

Abstract –  The problem of testing two simple hypotheses in a general
probability space is considered. For a fixed type-I error probability, the
best exponential decay rate of the type-II error probability is investigated.
In regular asymptotic cases (i.e., when the length of the observation interval
grows without limit) the best decay rate is given by Stein’s exponent. In the
paper, for a general probability space, some non-asymptotic lower and upper
bounds for the best rate are derived. These bounds represent pure analytic
relations without any limiting operations. In some natural cases, these bounds
also give the convergence rate for Stein’s exponent. Some illustrating
examples are also provided.

Index Terms –  Testing of hypotheses, type-I and type-II error probabilities,
Stein’s exponent.

##  1 Introduction and Main result

We assume that there are given probability measures $\mathbf{P}$ and
$\mathbf{Q}$ on a measurable space $(\cal X,{\cal B})$. We consider the
problem of testing the simple hypothesis $\mathcal{H}_{0}$ against the simple
alternative $\mathcal{H}_{1}$, based on observation ${\mathbf{x}}$:

$\begin{gathered}\mathcal{H}_{0}:{\mathbf{x}}\sim{\mathbf{P}},\\\
\mathcal{H}_{1}:{\mathbf{x}}\sim{\mathbf{Q}}.\end{gathered}$ (1)

If for the testing of hypotheses $\mathcal{H}_{0}$ and $\mathcal{H}_{1}$ a
decision (acceptance) region ${\mathcal{D}}\in{\cal X}$ is chosen, such that

$\begin{gathered}{\mathbf{x}}\in\mathcal{D}\Rightarrow\mathcal{H}_{0},\qquad{\mathbf{x}}\not\in\mathcal{D}\Rightarrow\mathcal{H}_{1},\end{gathered}$
(2)

then the type-I error probability $\alpha({\mathcal{D}})$ and the type-II
error probability $\beta({\mathcal{D}})$ are defined by formulas,
respectively,

$\alpha({\mathcal{D}})=\mathbf{P}({\mathbf{x}}\not\in\mathcal{D}|\mathcal{H}_{0})$
(3)

and

$\begin{gathered}\beta({\mathcal{D}})=\mathbf{P}({\mathbf{x}}\in\mathcal{D}|\mathcal{H}_{1}).\end{gathered}$
(4)

In the paper, we consider the case when the type-I error probability $\alpha$,
$0<\alpha<1$, is fixed, and we are interested in the minimal possible type-II
error probability $\beta(\alpha)$

$\beta(\alpha)=\inf\limits_{{\mathcal{D}}:\alpha({\mathcal{D}})\leq\alpha}\beta({\mathcal{D}}).$
(5)

The corresponding optimal decision region ${\mathcal{D}}(\alpha)$ to this
problem is given by Neyman-Pearson lemma [1, 2].

We will need the following notion [3].

Definition 1.  For probability distributions ${\mathbf{P}}$ and ${\mathbf{Q}}$
on a measurable space $(\cal X,{\cal B})$ the function
$D({\mathbf{P}}||{\mathbf{Q}})$ (Kullback–Leibler distance or divergence for
measures ${\mathbf{P}}$ and ${\mathbf{Q}}$) is defined as

$\begin{gathered}D({\mathbf{P}}||{\mathbf{Q}})={\mathbf{E}}_{\mathbf{P}}\ln\frac{d{\mathbf{P}}}{d{\mathbf{Q}}}({\mathbf{x}})\geq
0,\end{gathered}$ (6)

where the expectation is taken over the measure ${\mathbf{P}}$.

In order to describe Stein’s lemma and its relation to the value
$\beta(\alpha)$ via the function $D({\mathbf{P}}||{\mathbf{Q}})$, we consider
first the case ${\cal X}={\mathbf{R}}^{n}$, when an observation ${\mathbf{x}}$
from (1) has the form
${\mathbf{x}}={\mathbf{x}}_{n}=(x_{1},\ldots,x_{n})\in{\mathbf{R}}^{n}$. We
assume that the sample ${\mathbf{x}}_{n}$ consists of independent and
identically distributed (i.i.d.) random variables $x_{i}$, $i=1,\ldots,n$ with
a given distribution ${\mathbf{P}}$ (in the case of $\mathcal{H}_{0}$), or
with a given distribution ${\mathbf{Q}}$ (in the case of $\mathcal{H}_{1}$):

$\begin{gathered}\mathcal{H}_{0}:x_{i}\sim{\mathbf{P}},\qquad i=1,\ldots,n;\\\
\mathcal{H}_{1}:x_{i}\sim{\mathbf{Q}},\qquad i=1,\ldots,n.\end{gathered}$ (7)

If, in model (7) ${\mathbf{P}}\not\equiv{\mathbf{Q}}$, then
$D({\mathbf{P}}||{\mathbf{Q}})>0$ and the value $\beta(\alpha)$ decreases
exponentially as $n\to\infty$. Moreover, we have [3, Theorem 3.3]

$\begin{gathered}\lim_{\alpha\to
0}\lim_{n\to\infty}\frac{1}{n}\ln\beta(\alpha)=-D({\mathbf{P}}||{\mathbf{Q}}).\end{gathered}$
(8)

Relation (8) is called Stein’s lemma [4], [5]. 111The reference [4] is due to
remarks from [5, p. 18] and [3, Theorem 3.3] with attribute to the unpublished
paper of C. Stein. In the case of i.i.d. random variables, its proof can be
found in [3, Theorem 3.3], [6, Theorem 12.8.1]. It is natural to expect that
the relation (8) holds not only for i.i.d. random variables as in model (7),
but in much more general cases. Following [6, Theorem 12.8.1], some particular
analogs of formula (8) have already appeared for the cases of stationary
Gaussian [7] and Poisson [8] random processes (with observation time $T$
instead of $n$).

Note that the relation (8) is essentially oriented to the case when the type-I
and the type-II errors imply very different losses for us, and we are mainly
interested in minimization of the type-II error probability
$\beta={\mathbf{P}}\\{H_{0}|H_{1}\\}$. The case is quite popular in many
applications (see, e.g., [7] and references therein). Also, Chernov [5, p. 17]
wrote on such case: “It occasionally happens in practice that it is important
to obtain $\beta$ very small, whereas a relatively large value of $\alpha$,
like $.05$ or $.10$, is not disastrous”

In this paper, for the general model (1) some non-asymptotic generalization
and strengthening of the relation (8) is derived. Such non-asymptotic results
allowed to generalize the relation (8) for testing a simple hypothesis versus
a composite one [8]. Also, such non-asymptotic results would allow essentially
simplify similar results for Gaussian processes [7, 9].

###  1.1 Assumption

In order to simplify formulation of the results and to avoid pathological
cases, we assume that the following assumption is satisfied:

I. Probability measures $\mathbf{P}$ and $\mathbf{Q}$ on a measurable space
$(\cal X,{\cal B})$ are equivalent and Radon-Nikodim derivative
$d{\mathbf{P}}/d{\mathbf{Q}}({\mathbf{x}})$ for them is defined, positive and
finite for ${\mathbf{x}}\in{\cal X}$.

###  1.2 Main result

The main result of this paper is the following.

Theorem. If the assumption ${\bf I}$ holds, then the minimal possible
$\beta(\alpha)$, $0<\alpha<1$, satisfies the bounds

$\begin{gathered}\ln\beta(\alpha)\geq-\frac{D({\mathbf{P}}||{\mathbf{Q}})+h(\alpha)}{1-\alpha},\\\
h(\alpha)=-\alpha\ln\alpha-(1-\alpha)\ln(1-\alpha),\end{gathered}$ (9)

and

$\begin{gathered}\ln\beta(\alpha)\leq-D({\mathbf{P}}||{\mathbf{Q}})+\mu_{0}(\alpha),\end{gathered}$
(10)

where $\mu_{0}(\alpha)$ is the minimal value $\mu_{0}$, satisfying the
relation

$\begin{gathered}{\mathbf{P}}_{{\mathbf{P}}}\left\\{\ln\frac{d{\mathbf{P}}}{d{\mathbf{Q}}}({\mathbf{x}})\leq
D({\mathbf{P}}||{\mathbf{Q}})-\mu_{0}\right\\}\leq\alpha.\end{gathered}$ (11)

With some loss of accuracy, we can simplify the upper bound (10), using some
upper bound for the value $\mu_{0}(\alpha)$ (see (24) below). For example, we
get

Corollary. The minimal possible $\beta(\alpha)$, $0<\alpha<1$, satisfies also
the upper bound

$\begin{gathered}\ln\beta(\alpha)\leq-D({\mathbf{P}}||{\mathbf{Q}})+\alpha^{-1/2}r_{1}({\mathbf{P}},{\mathbf{Q}}),\end{gathered}$
(12)

where

$\begin{gathered}r_{1}({\mathbf{P}},{\mathbf{Q}})=\left\\{{\mathbf{E}}_{{\mathbf{P}}}\left[\ln\frac{d{\mathbf{P}}}{d{\mathbf{Q}}}({\mathbf{x}})\right]^{2}-D^{2}({\mathbf{P}}||{\mathbf{Q}})\right\\}^{1/2}.\end{gathered}$
(13)

Note that all bounds (9), (10) and (12) are pure analytical relations without
any limiting operations. Also, for a fixed $\alpha$, both lower bound (9) and
upper bound (12) are close to each other, if the value
$r_{1}({\mathbf{P}},{\mathbf{Q}})$ is much smaller than
$D({\mathbf{P}}||{\mathbf{Q}})$.

Remark 1. For the particular case of Poisson processes Theorem was proved in
[8] (as Theorem 3).

Remark 2. Note that if in some asymptotic setting the Central Limit Theorem is
applicable to the random variable
$D({\mathbf{P}}||{\mathbf{Q}})-\ln(d{\mathbf{P}}/d{\mathbf{Q}})({\mathbf{x}})$,
then the remaining term $r_{1}({\mathbf{P}},{\mathbf{Q}})$ in (12) has the
optimal order (in $D({\mathbf{P}}||{\mathbf{Q}})$) for fixed $\alpha$ and
large $D({\mathbf{P}}||{\mathbf{Q}})$ (see Example 3 below).

##  2 Proofs

###  2.1 Proof of Theorem

Derive first the lower bound (9). Without loss of generality and in order to
avoid bulky formulas, note that using a natural randomization on the “bound”
of a decision region ${\mathcal{D}}\in{\cal X}$, we have with ${\cal
D}^{c}={\cal X}\setminus{\cal D}$ and $\beta=\beta({\cal D})$,
$\alpha=\alpha({\cal D})$

$\begin{gathered}\beta={\mathbf{Q}}({\cal D})=\int\limits_{{\cal
D}}\frac{d{\mathbf{Q}}}{d{\mathbf{P}}}({\mathbf{x}})d{\mathbf{P}}({\mathbf{x}}),\qquad\alpha={\mathbf{P}}({\cal
D}^{c}).\end{gathered}$ (14)

Since ${\mathbf{P}}({\cal D})=1-{\mathbf{P}}({\cal D}^{c})=1-\alpha$, then
considering $d{\mathbf{P}}/(1-\alpha)$ as the probability distribution on
${\cal D}$, and using the inequality
$\ln{\mathbf{E}}\xi\geq{\mathbf{E}}\ln\xi$, we have

$\begin{gathered}\ln\frac{\beta}{1-\alpha}=\ln\left[\frac{1}{(1-\alpha)}\int\limits_{{\cal
D}}\frac{d{\mathbf{Q}}}{d{\mathbf{P}}}({\mathbf{x}})d{\mathbf{P}}({\mathbf{x}})\right]\geq\\\
\geq\frac{1}{(1-\alpha)}\int\limits_{{\cal
D}}\ln\frac{d{\mathbf{Q}}}{d{\mathbf{P}}}({\mathbf{x}})d{\mathbf{P}}({\mathbf{x}})=\\\
=-\frac{D({\mathbf{P}}||{\mathbf{Q}})}{1-\alpha}-\frac{1}{(1-\alpha)}\int\limits_{{\cal
D}^{c}}\ln\frac{d{\mathbf{Q}}}{d{\mathbf{P}}}({\mathbf{x}})d{\mathbf{P}}({\mathbf{x}}).\end{gathered}$
(15)

Since ${\mathbf{P}}({\cal D}^{c})=\alpha$, similarly to (15), the last term in
the right-hand side of (15) gives

$\begin{gathered}\frac{1}{(1-\alpha)}\int\limits_{{\cal
D}^{c}}\ln\frac{d{\mathbf{Q}}}{d{\mathbf{P}}}({\mathbf{x}})d{\mathbf{P}}({\mathbf{x}})\leq\\\
\leq\frac{\alpha}{(1-\alpha)}\ln\left[\frac{1}{\alpha}\int\limits_{{\cal
D}^{c}}d{\mathbf{Q}}({\mathbf{x}})\right]=\frac{\alpha}{(1-\alpha)}\ln\frac{1-\beta}{\alpha}\leq\frac{\alpha}{(1-\alpha)}\ln\frac{1}{\alpha}.\end{gathered}$
(16)

Therefore, from (15) and (16) we have

$\begin{gathered}\ln\frac{\beta}{1-\alpha}\geq-\frac{D({\mathbf{P}}||{\mathbf{Q}})}{1-\alpha}-\frac{\alpha}{(1-\alpha)}\ln\frac{1}{\alpha},\end{gathered}$
(17)

from where the lower bound (9) follows.

In order to prove the upper bound (10), we set a value $\mu>0$, and define the
acceptance region in favor of ${\mathbf{P}}$

$\begin{gathered}{\cal A}_{\mu}=\left\\{{\mathbf{x}}\in{\cal
X}:\ln\frac{d{\mathbf{P}}}{d{\mathbf{Q}}}({\mathbf{x}})\geq
D({\mathbf{P}}||{\mathbf{Q}})-\mu\right\\}.\end{gathered}$ (18)

Denote by $\alpha_{\mu}$ and $\beta_{\mu}$ the type-I and the type-II error
probabilities for the acceptance region ${\cal A}_{\mu}$, respectively. Then
by (18)

$\begin{gathered}\beta_{\mu}=\int\limits_{{\cal
A}_{\mu}}d{\mathbf{Q}}=\int\limits_{{\cal
A}_{\mu}}\frac{d{\mathbf{Q}}}{d{\mathbf{P}}}({\mathbf{x}})d{\mathbf{P}}\leq
e^{-D({\mathbf{P}}||{\mathbf{Q}})+\mu}.\end{gathered}$ (19)

Also

$\begin{gathered}\alpha_{\mu}={\mathbf{P}}_{{\mathbf{P}}}\left\\{\ln\frac{d{\mathbf{P}}}{d{\mathbf{Q}}}({\mathbf{x}})\leq
D({\mathbf{P}}||{\mathbf{Q}})-\mu\right\\}={\mathbf{P}}(\eta\geq\mu),\end{gathered}$
(20)

where

$\eta=D({\mathbf{P}}||{\mathbf{Q}})-\ln\frac{d{\mathbf{P}}}{d{\mathbf{Q}}}({\mathbf{x}}),\qquad{\mathbf{E}}\eta=0.$
(21)

The best solution is to set $\mu$ such that $\alpha_{\mu}=\alpha$. Therefore,
we define $\mu_{0}=\mu_{0}(\alpha)$ by the formula (11). Then we have

$\begin{gathered}\alpha_{\mu_{0}(\alpha)}=\alpha,\end{gathered}$ (22)

and by (19)

$\begin{gathered}\ln\beta_{\mu_{0}(\alpha)}\leq-D({\mathbf{P}}||{\mathbf{Q}})+\mu_{0}(\alpha),\end{gathered}$
(23)

which proves the upper bound (10).

###  2.2 Proof of Corollary

In order to prove the upper bound (12), we simplify the relation (11), using
the classic Chebyshev inequality

$\begin{gathered}{\mathbf{P}}(\eta\geq\mu)\leq\mu^{-2}{\mathbf{E}}(\eta_{+}^{2}),\end{gathered}$
(24)

where $z_{+}=\max\\{0,z\\}$. Then from (20) and (24)

$\begin{gathered}\alpha_{\mu}\leq\mu^{-2}{\mathbf{E}}_{\mathbf{P}}\left[D({\mathbf{P}}||{\mathbf{Q}})-\ln\frac{d{\mathbf{P}}}{d{\mathbf{Q}}}({\mathbf{x}})\right]_{+}^{2}.\end{gathered}$
(25)

We set $\mu=\mu_{1}>0$ as follows (see (13))

$\begin{gathered}\mu_{1}=\alpha^{-1/2}\left\\{{\mathbf{E}}_{\mathbf{P}}\left[D({\mathbf{P}}||{\mathbf{Q}})-\ln\frac{d{\mathbf{P}}}{d{\mathbf{Q}}}({\mathbf{x}})\right]_{+}^{2}\right\\}^{1/2}=\\\
=\alpha^{-1/2}r_{1}({\mathbf{P}},{\mathbf{Q}}).\end{gathered}$ (26)

Then we get

$\begin{gathered}\alpha_{\mu_{1}}\leq\alpha,\end{gathered}$ (27)

and by (19) and (26)

$\begin{gathered}\ln\beta_{\mu_{1}}\leq-D({\mathbf{P}}||{\mathbf{Q}})+\mu_{1}=\\\
=-D({\mathbf{P}}||{\mathbf{Q}})+\alpha^{-1/2}r_{1}({\mathbf{P}},{\mathbf{Q}}),\end{gathered}$
(28)

which proves the upper bound (12). $\Box$

##  3 Examples

Example 1 (Poisson processes). Denote by
${\mathbf{x}}_{T}=x_{{\mathbf{u}}_{T}}=\\{x_{{\mathbf{u}}_{T}}(t),0\leq t\leq
T\\}$, the Poisson process with the intensity
${\mathbf{u}}_{T}=\\{u_{T}(t),0\leq t\leq T\\}$ Kutoyants (1984) and let
${\cal X}_{T}$ denote the observation space (i.e. ${\mathbf{x}}_{T}\in{\cal
X}_{T}$). Consider testing a simple hypothesis $\mathcal{H}_{0}$ versus a
simple alternative $\mathcal{H}_{1}$, based on observations
${\mathbf{x}}_{T}=\\{{\mathbf{x}}_{T}(t),0\leq t\leq T\\}$:

$\begin{split}&\mathcal{H}_{0}:{\mathbf{x}}_{T}=x_{{\mathbf{p}}_{T}},\\\
&\mathcal{H}_{1}:{\mathbf{x}}_{T}=x_{{\mathbf{q}}_{T}},\end{split}$ (29)

where intensities ${\mathbf{p}}_{T}$ and ${\mathbf{q}}_{T}$ are given.

Poisson processes on $[0,T]$ with intensities ${\mathbf{p}}_{T}$ and
${\mathbf{q}}_{T}$ generate on $({\cal X}_{T},{\cal B}_{T})$ measures
${\mathbf{P}}_{{\mathbf{p}}_{T}}$ and ${\mathbf{P}}_{{\mathbf{q}}_{T}}$,
respectively. The following formula for the Radon–Nikodim derivative holds
(see, e.g., Kutoyants (1984))

$\begin{gathered}\frac{d{\mathbf{P}}_{{\mathbf{p}}_{T}}}{d{\mathbf{P}}_{{\mathbf{q}}_{T}}}({\mathbf{x}})=\exp\left\\{(\ln({\mathbf{p}}_{T}/{\mathbf{q}}_{T}),dx)-({\mathbf{p}}_{T}-{\mathbf{q}}_{T},{\mathbf{1}})\right\\},\end{gathered}$
(30)

where for functions ${\mathbf{f}}=\\{f(t),0\leq t\leq T\\}$,
${\mathbf{s}}=\\{s(t),0\leq t\leq T\\}$ and Poisson process $x(t)$ we used the
notation

$({\mathbf{f}},{\mathbf{s}})=\int\limits_{0}^{T}f(t)s(t)dt,\qquad({\mathbf{f}},dx)=\int\limits_{0}^{T}f(t)dx(t).$
(31)

It is supposed that the right-hand side of (30) is well defined and therefore
the measures ${\mathbf{P}}_{{\mathbf{p}}_{T}}$ and
${\mathbf{P}}_{{\mathbf{q}}_{T}}$ are equivalent (see, e.g., Kutoyants
(1984)).

From (30) for any integrable function ${\mathbf{f}}_{T}=\\{f_{T}(t),0\leq
t\leq T\\}$ the ”expectation” formula follows (see, e.g., Kutoyants (1984))

$\ln{\mathbf{E}}_{{\mathbf{p}}_{T}}e^{(\ln{\mathbf{f}}_{T},dx)}=({\mathbf{f}}_{T}-{\mathbf{1}},{\mathbf{p}}_{T}),$
(32)

where ${\mathbf{E}}_{{\mathbf{p}}_{T}}$ means expectation with respect to the
measure ${\mathbf{P}}_{{\mathbf{p}}_{T}}$. In particular, from (32) we get

$\begin{gathered}{\mathbf{E}}_{{\mathbf{p}}_{T}}({\mathbf{f}}_{T},dx)=({\mathbf{f}}_{T},{\mathbf{p}}_{T}),\\\
{\mathbf{E}}_{{\mathbf{p}}_{T}}({\mathbf{f}}_{T},dx)^{2}=({\mathbf{f}}_{T},{\mathbf{p}}_{T})^{2}+({\mathbf{f}}_{T}^{2},{\mathbf{p}}_{T}).\end{gathered}$
(33)

Then, from (6), (30) and (33) we have

$\begin{gathered}D({\mathbf{P}}_{{\mathbf{p}}_{T}}||{\mathbf{P}}_{{\mathbf{q}}_{T}})=D({\mathbf{p}}_{T}||{\mathbf{q}}_{T})=(\ln({\mathbf{p}}_{T}/{\mathbf{q}}_{T}),{\mathbf{p}}_{T})-({\mathbf{p}}_{T}-{\mathbf{q}}_{T},{\mathbf{1}}).\end{gathered}$
(34)

For the remaining term $r_{1}({\mathbf{P}},{\mathbf{Q}})$ from (12)-(13) we
get by (33) and (34)

$\begin{gathered}r_{1}({\mathbf{P}},{\mathbf{Q}})=\left\\{(\ln^{2}({\mathbf{p}}_{T}/{\mathbf{q}}_{T}),{\mathbf{p}}_{T})\right\\}^{1/2},\end{gathered}$
(35)

which gives an estimate to the convergence rate in (8) (provided the right-
hand side of (35) is finite).

Example 2 (Gaussian processes). Note that by the Karhunen-Loéve expansion most
of detection problems on continuous-time Gaussian processes can be reduced to
the discrete-time case with Gaussian random vectors [11, Ch. VI]. Formally,
such representation consists of infinite number of expansion terms, but in
practice such infinite series can be well approximated by a finite summation
(without essential loss for performance characteristics).

For that reason we consider testing of simple hypotheses $\mathcal{H}_{0}$ and
$\mathcal{H}_{1}$, based on observation of Gaussian random vectors
${\mathbf{x}}_{n}\in{\mathbf{R}}^{n}$:

$\begin{gathered}\mathcal{H}_{0}:{\mathbf{x}}_{n}=\boldsymbol{\xi}_{n},\qquad\boldsymbol{\xi}_{n}\sim{\mathcal{N}}(\boldsymbol{0},\mathbf{I}_{n}),\\\
\mathcal{H}_{1}:{\mathbf{x}}_{n}=\boldsymbol{\eta}_{n},\qquad\boldsymbol{\eta}_{n}\sim{\mathcal{N}}(\boldsymbol{0},\mathbf{M}_{n}),\end{gathered}$
(36)

where the sample $\boldsymbol{\xi}_{n}=(\xi_{1},\ldots,\xi_{n})$ represents
“noise” and consists of independent and identically distributed Gaussian
random variables with zero means and variances $1$. The stochastic “signal”
$\boldsymbol{\eta}_{n}=(\eta_{1},\ldots,\eta_{n})$ is a Gaussian random vector
with zero mean and covariance matrix $\mathbf{M}_{n}$ (which is symmetric and
nonnegative-definite).

Denote by ${\mathbf{P}}_{n}$ and ${\mathbf{Q}}_{n}$ distributions of
$\boldsymbol{\xi}_{n}$ and $\boldsymbol{\eta}_{n}$, respectively. Note that if
$\det\mathbf{M}_{n}=0$, then distributions ${\mathbf{P}}_{n}$ and
${\mathbf{Q}}_{n}$ are orthogonal, and therefore, hypotheses $\mathcal{H}_{0}$
and $\mathcal{H}_{1}$ can be tested with $\alpha=\beta=0$.

Hence, we assume that $\det\mathbf{M}_{n}>0$, and then $\mathbf{M}_{n}$ is a
symmetric and positive-definite matrix. Denote by
$\lambda_{1},\ldots,\lambda_{n}$ the eigenvalues of $\mathbf{M}_{n}$ (all
positive).

In order to simplify calculations, we reduce model (36) to the equivalent one
with the diagonal matrix $\mathbf{M}_{n}$. Indeed, for the covariance matrix
$\mathbf{M}_{n}$ there is the orthogonal matrix $\mathbf{T}_{n}$ and the
diagonal matrix $\mathbf{\Lambda}_{n}$ such that
$\mathbf{M}_{n}=\mathbf{T}_{n}\mathbf{\Lambda}_{n}\mathbf{T}_{n}^{\prime}$.
Moreover, the diagonal matrix
$\mathbf{\Lambda}_{n}=\mathbf{T}_{n}^{\prime}\mathbf{M}_{n}\mathbf{T}_{n}$
consists of the eigenvalues $\\{\lambda_{i}\\}$ of $\mathbf{M}_{n}$ [12, Ch.
4.7-9].

Also, for any orthogonal matrix $\mathbf{T}_{n}$, the distribution of
$\mathbf{T}_{n}^{\prime}\boldsymbol{\xi}_{n}$ (for simple hypothesis
$\mathcal{H}_{0}$ from (36)) remains the same as of $\boldsymbol{\xi}_{n}$.
Therefore, multiplying both sides of (36) by $\mathbf{T}_{n}^{\prime}$, we
reduce model (36) to the equivalent model with the diagonal matrix
$\mathbf{\Lambda}_{n}$. In other words, model (36) takes the form

$\begin{gathered}\mathcal{H}_{0}:x_{i}=\xi_{i},\qquad\xi_{i}\sim{\mathcal{N}}(0,1),\\\
\mathcal{H}_{1}:x_{i}=\eta_{i}=\lambda_{i}^{1/2}\xi_{i},\quad
i=1,\ldots,n.\end{gathered}$ (37)

Then we have

$\begin{gathered}\ln\frac{d{\mathbf{P}}_{n}}{d{\mathbf{Q}}_{n}}({\mathbf{x}})=\frac{1}{2}\sum\limits_{i=1}^{n}\left[\ln\lambda_{i}+\left(\frac{1}{\lambda_{i}}-1\right)x_{i}^{2}\right]\end{gathered}$
(38)

and

$\begin{gathered}D(\mathbf{P}_{n}||\mathbf{Q}_{n})=\frac{1}{2}\sum_{i=1}^{n}\left(\ln\lambda_{i}+\frac{1}{\lambda_{i}}-1\right).\end{gathered}$
(39)

Also

$\begin{gathered}r_{1}^{2}({\mathbf{P}}_{1}^{n},{\mathbf{Q}}_{1}^{n})={\mathbf{E}}_{{\mathbf{P}}_{n}}\left[D({\mathbf{P}}_{n}||{\mathbf{Q}}_{n})-\ln\frac{d{\mathbf{P}}_{n}}{d{\mathbf{Q}}_{n}}({\mathbf{x}})\right]^{2}=\\\
=\frac{1}{4}\mathbf{E}\left[\sum_{i=1}^{n}\left(\frac{1}{\lambda_{i}}-1\right)(\xi_{i}^{2}-1)\right]^{2}=\\\
=\frac{1}{4}\mathbf{E}\sum_{i=1}^{n}\left(\frac{1}{\lambda_{i}}-1\right)^{2}(\xi_{i}^{2}-1)^{2}=\frac{1}{2}\sum_{i=1}^{n}\left(\frac{1}{\lambda_{i}}-1\right)^{2}.\end{gathered}$
(40)

For simplicity, we assume that for a constant $C>0$ and large $n$ the
following condition is satisfied

$\begin{gathered}\sum_{i=1}^{n}\left(\frac{1}{\lambda_{i}}-1\right)^{2}\leq
C^{2}\sum_{i=1}^{n}\left(\ln\lambda_{i}+\frac{1}{\lambda_{i}}-1\right).\end{gathered}$
(41)

Then, by (9), (12) and (39)-(41) we have

$\begin{gathered}-\frac{D({\mathbf{P}}_{n}||{\mathbf{Q}}_{n})+1}{1-\alpha}\leq\ln\beta(\alpha)\leq\\\
\leq-D({\mathbf{P}}_{n}||{\mathbf{Q}}_{n})+C\alpha^{-1/2}\sqrt{D({\mathbf{P}}_{n}||{\mathbf{Q}}_{n})},\end{gathered}$
(42)

which yields an estimate to the convergence rate in (8).

Example 3 (independent random variables). Consider a natural generalization of
model (7), when observations
${\mathbf{x}}_{1}^{n}=(x_{1},\ldots,x_{n})\in{\mathbf{R}}^{n}$ are independent
random variables with different distributions $\\{{\mathbf{P}}_{i}\\}$ or
$\\{{\mathbf{Q}}_{i}\\}$, $i=1,\ldots,n$:

$\begin{gathered}\mathcal{H}_{0}:{\mathbf{x}}_{1}^{n}\sim{\mathbf{P}}_{1}^{n}={\mathbf{P}}_{1}\times\ldots{\mathbf{P}}_{n},\\\
\mathcal{H}_{1}:{\mathbf{x}}_{n}\sim{\mathbf{Q}}_{1}^{n}={\mathbf{Q}}_{1}\times\ldots{\mathbf{Q}}_{n}.\end{gathered}$
(43)

Since

$\begin{gathered}D({\mathbf{P}}_{1}^{n}||{\mathbf{Q}}_{1}^{n})=\sum\limits_{i=1}^{n}D({\mathbf{P}}_{i}||{\mathbf{Q}}_{i}),\end{gathered}$
(44)

the lower bound (9) takes the form

$\begin{gathered}\ln\beta\geq-\frac{1}{(1-\alpha)}\left[D({\mathbf{P}}_{1}^{n}||{\mathbf{Q}}_{1}^{n})+1\right].\end{gathered}$
(45)

We also have for the remaining term
$r_{1}({\mathbf{P}}_{1}^{n},{\mathbf{Q}}_{1}^{n})$ from (13)

$\begin{gathered}r_{1}^{2}({\mathbf{P}}_{1}^{n},{\mathbf{Q}}_{1}^{n})\leq{\mathbf{E}}_{{\mathbf{P}}_{1}^{n}}\left[D({\mathbf{P}}_{1}^{n}||{\mathbf{Q}}_{1}^{n})-\ln\frac{d{\mathbf{P}}_{1}^{n}}{d{\mathbf{Q}}_{1}^{n}}({\mathbf{x}})\right]^{2}=\\\
=\sum\limits_{i=1}^{n}{\mathbf{E}}_{{\mathbf{P}}_{i}}\left[D({\mathbf{P}}_{i}||{\mathbf{Q}}_{i})-\ln\frac{d{\mathbf{P}}_{i}}{d{\mathbf{Q}}_{i}}(x_{i})\right]^{2}=\\\
=\sum\limits_{i=1}^{n}\left[{\mathbf{E}}_{{\mathbf{P}}_{i}}\left(\ln\frac{d{\mathbf{P}}_{i}}{d{\mathbf{Q}}_{i}}(x_{i})\right)^{2}-D^{2}({\mathbf{P}}_{i}||{\mathbf{Q}}_{i})\right].\end{gathered}$
(46)

For simplicity, we assume that for a constant $C>0$ and all sufficiently large
$n$ the following condition is satisfied

$\begin{gathered}\sum\limits_{i=1}^{n}\left[{\mathbf{E}}_{{\mathbf{P}}_{i}}\left(\ln\frac{d{\mathbf{P}}_{i}}{d{\mathbf{Q}}_{i}}(x_{i})\right)^{2}-D^{2}({\mathbf{P}}_{i}||{\mathbf{Q}}_{i})\right]\leq\\\
\leq
C^{2}\sum\limits_{i=1}^{n}D({\mathbf{P}}_{i}||{\mathbf{Q}}_{i})=C^{2}D({\mathbf{P}}_{1}^{n}||{\mathbf{Q}}_{1}^{n}).\end{gathered}$
(47)

Then, by (46)-(47)

$\begin{gathered}{\mathbf{E}}_{{\mathbf{P}}_{1}^{n}}\left[D({\mathbf{P}}_{1}^{n}||{\mathbf{Q}}_{1}^{n})-\ln\frac{d{\mathbf{P}}_{1}^{n}}{d{\mathbf{Q}}_{1}^{n}}({\mathbf{x}})\right]^{2}\leq
C^{2}D({\mathbf{P}}_{1}^{n}||{\mathbf{Q}}_{1}^{n}).\end{gathered}$ (48)

Therefore, by (9) and (12) we have for $\ln\beta(\alpha)$

$\begin{gathered}-\frac{D({\mathbf{P}}_{1}^{n}||{\mathbf{Q}}_{1}^{n})}{1-\alpha}\leq\ln\beta(\alpha)\leq\\\
\leq-D({\mathbf{P}}_{1}^{n}||{\mathbf{Q}}_{1}^{n})+C\alpha^{-1/2}\sqrt{D({\mathbf{P}}_{1}^{n}||{\mathbf{Q}}_{1}^{n})},\end{gathered}$
(49)

which estimates the convergence rate in (8).

The author would like to thank Kutoyants Yu. A. and anonymous reviewers for
useful discussions and constructive critical remarks, which improved the
paper.

## References

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††thanks: Authors to whom correspondence should be addressed:
<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Microwave Spin-Pumping from an Antiferromagnet FeBO3

D.A.Gabrielyan  Kotel’nikov Institute of Radioengineering and Electronics,
Russian Academy of Sciences, Moscow 125009, Russia  Moscow Power Engineering
Institute, Moscow 111250, Russia  D.A. Volkov  Kotel’nikov Institute of
Radioengineering and Electronics, Russian Academy of Sciences, Moscow 125009,
Russia  Moscow Power Engineering Institute, Moscow 111250, Russia  E.E.
Kozlova  Kotel’nikov Institute of Radioengineering and Electronics, Russian
Academy of Sciences, Moscow 125009, Russia  Moscow Institute of Physics and
Technology, Dolgoprudny 141701, Russia  A.R. Safin  Kotel’nikov Institute of
Radioengineering and Electronics, Russian Academy of Sciences, Moscow 125009,
Russia  Moscow Power Engineering Institute, Moscow 111250, Russia  D.V.
Kalyabin  Kotel’nikov Institute of Radioengineering and Electronics, Russian
Academy of Sciences, Moscow 125009, Russia  HSE University, Moscow, 101000,
Russia  A.A. Klimov  Kotel’nikov Institute of Radioengineering and
Electronics, Russian Academy of Sciences, Moscow 125009, Russia  MIREA -
Russian Technological University, Moscow 119454, Russia
V.L. Preobrazhensky  Prokhorov General Physics Institute RAS, Moscow 119991,
Russia  M.B. Strugatsky  V.I. Vernadsky Crimean Federal University, Simferopol
295007, Russia  S.V. Yagupov  V.I. Vernadsky Crimean Federal University,
Simferopol 295007, Russia  I.E. Moskal  Kotel’nikov Institute of
Radioengineering and Electronics, Russian Academy of Sciences, Moscow 125009,
Russia  G.A. Ovsyannikov  Kotel’nikov Institute of Radioengineering and
Electronics, Russian Academy of Sciences, Moscow 125009, Russia  S.A. Nikitov
Kotel’nikov Institute of Radioengineering and Electronics, Russian Academy of
Sciences, Moscow 125009, Russia  Moscow Institute of Physics and Technology,
Dolgoprudny 141701, Russia  Saratov State University, Saratov 410012, Russia

###### Abstract

Recently, canted antiferromagnets offer great potential for fundamental
research and applications due to their unique properties. The presence of the
Dzyaloshinskii-Moriya interaction leads to the existence of a weak
ferromagnetic moment at room temperature. We study both theoretically and
experimentally microwave spin pumping by the quasi-ferromagnetic mode from a
canted easy plane antiferromagnet with weak ferromagnetism FeBO3. The
conversion of a microwave signal into the constant voltage is realized using
the inverse spin Hall effect in an iron borate/heavy metal heterostructure. We
use an additional bias magnetic field to selectively tune the resonance
frequency of such a microwave detector over a wide range up to 43.5 GHz with
potential sensitivity near 2.5 $\mu$V/W. We confirm the pure spin current
nature by changing polarity of the detected via inverse spin Hall effect
voltage by switching the direction of the bias magnetic field. We believe that
our results will be useful for the development of highly tunable, portable and
sensitive microwave antiferromagnet-based functional devices.

††preprint: APS/123-QED

Spin pumping in magnetic materials is one of the most powerful methods for
detecting pure spin currents. Spin currents in ferromagnetic metals were the
primary focus of spin pumping research [1, 2], but these studies were
gradually extended to magnetic semiconductors and isolators [3, 4].
Researchers in modern spintronics are focused on multisublattice magnetic
materials such as antiferromagnets (AFM), because due to exchange enhancement,
they exhibit a number of unique properties [5, 6, 7]: subterahertz and
teraherz (THz) resonance frequencies, high speed of spin waves and domain
walls, amplification of spin current due to excitation of evanescent modes,
etc. These materials are the most promising candidates for applications in the
field of ultrafast magnonics and spintronics [8, 9] such as the new generation
of 5G and 6G telecommunications. This is also supported by low radiation
losses to AFM, meaning that they create virtually no electromagnetic microwave
scattering fields due to zero total magnetic moment. Various AFM-based
functional devices have been described in detail previously: oscillators [10,
11, 12, 13, 14], detectors [15, 16, 17, 18, 19], emitters and amplifiers [20,
21], neuromorphic processors [22, 23, 24], memory elements [25, 26], spectrum
analyzers [27, 28] etc.

Spin pumping experiments have been conducted with various AFM materials,
namely Cr2O3 [29], MnF2 [30, 31] and NiO [32]. However, the above-mentioned
materials have resonance frequencies from hundreds of gigahertz (GHz) to units
of terahertz, which significantly complicates the experimental measurement of
resonance characteristics by detecting spin currents. But there are AFM
materials with weak ferromagnetism caused mostly by the Dzyaloshinskii-Moriya
interaction (DMI), having resonance frequencies as in several hundred GHz
(”optical” antiferromagnetic modes), and in units and tens of GHz (”acoustic”
quasi-ferromagnetic modes). It opens up new opportunities for experimental
research of spin currents in AFM. The generation of spin currents in various
canted multisublattices magnets is a crucial and weakly explored field.
Hematite, $\alpha$-Fe2O3, is an AFM with weak ferromagnetism, it is an
excellent candidate for these investigations. Recently [33, 34, 35], it has
been published the spin-pumping from the hematite on the lower frequency mode
across the Morin phase transition.

Figure 1:  (a) Theoretical and experimental dependences measured by the VNA-
FMR method magnetic field of the resonant frequency of the FeBO3. The insert
shows the precession of the magnetic sublattices $\mathbf{M}_{1,2}$ in the
antiferromagnet and of the magnetization vector $\mathbf{m}$. (b) FMR-spectra
of the FeBO3 at various frequencies measured by the VNA-FMR method. Under the
action of the external magnetic field, the precession of the total
magnetization vector of the sublattices in the antiferromagnet is occurred.

In this work, we study both theoretically and experimentally microwave spin
pumping from the iron borate/platinum heterostructure. Iron borate (FeBO3) is
an antiferromagnet with weak ferromagnetism and easy plane magnetic
anisotropy. It has a trigonal (rhombohedral) crystal structure, a high Neel
temperature (348 K) and narrow lines of magnetic resonances [36, 37, 38, 39,
40, 41, 42, 43]. Iron borate crystals have two magnetic sublattices. The
magnetic moments of their ions are located at a small angle of 55′ [44] caused
by the Dzyaloshinskii-Moriya interaction [43, 45]. Along with the existence of
a non-zero antiferromagnetic moment, there is also a weak ferromagnetic moment
$4\pi M$ = 115-120 G at room temperature [46]. In this work we use a highly
advanced single-crystalline iron borate samples synthesized by a flux growth
technique [47]. FeBO3 has two modes of its eigen-oscillations - quasi-
ferromagnetic (tens of GHz) [40, 41] and quasi-antiferromagnetic (hundreds of
GHz) [48]. All the above makes iron borate a promising material for
fundamental research, as well as for creating various magnetoelectronics
devices on its basis.

We measure the spin current using the FeBO3/Pt heterostructure for the first
time, and also compare the results with a theoretical model, taking into
account the Dzyaloshinskii-Moriya interaction and the magnetoelastic coupling.
The analysis demonstrates the important role of these components in describing
antiferromagnetic processes with weak ferromagnetism.

We start with the theoretical and experimental investigation of the low-
frequency mode and S21-parameter, which gives information about the absorbed
power of the material. The insert in Fig. 1(a) demonstrates that a homogeneous
precession of the magnetic sublattices $\mathbf{M}_{1,2}$ in antiferromagnet
is initiated by the action of an external variable microwave radiation with
linear polarization. The dynamics of the magnetization vector
$\mathbf{m}=\left(\mathbf{M}_{1}+\mathbf{M}_{2}\right)/2\textrm{M}_{\tiny{\textrm{s}}}$
and the Néel vector
$\mathbf{l}=\left(\mathbf{M}_{1}-\mathbf{M}_{2}\right)/2\textrm{M}_{\tiny{\textrm{s}}}$
($\textrm{M}_{\tiny{\textrm{s}}}$ is the saturation magnetization of the AFM)
are governed by the equations, respectively [49, 50, 51, 52]:

$\begin{array}[]{c}\mathbf{m}=\frac{1}{\omega_{\tiny{\textrm{ex}}}}\left(\left[\frac{d\mathbf{l}}{dt}\times\mathbf{l}\right]+\left[\mathbf{l}\times\gamma\left(\mathbf{H}_{0}+\mathbf{h}_{\tiny{\textrm{RF}}}\right)\right]\times\mathbf{l}+\left[\mathbf{l}\times\gamma\mathbf{H}_{\tiny{\textrm{DMI}}}\right]\right),\end{array}$
(1)
$\begin{array}[]{c}\mathbf{l}\times\left[\frac{d^{2}\mathbf{l}}{dt^{2}}+\gamma_{\tiny{\textrm{eff}}}\frac{d\mathbf{l}}{dt}+\frac{\partial{W}}{\partial{\mathbf{l}}}-2\gamma\left[\frac{d\mathbf{l}}{dt}\times\mathbf{H}\right]\right]\\\
=\gamma\left[\frac{d\mathbf{H}}{dt}\times\mathbf{l}\right],\end{array}$ (2)

where
$\gamma_{\tiny{\textrm{eff}}}=\alpha_{\tiny{\textrm{eff}}}\omega_{\tiny{\textrm{ex}}}$
is the spectral linewidth of the AFM resonance, $\alpha_{\tiny{\textrm{eff}}}$
is the effective damping including Gilbert constant and spin-pumping terms
(see for more details [6, 53]), $\omega_{\tiny{\textrm{ex}}}$ is the exchange
frequency, $\gamma=2\pi\cdot 28$ GHz/T is the gyromagnetic ratio,
$\mathbf{H}=\mathbf{H_{0}}+\mathbf{H_{\tiny{\textrm{DMI}}}}+\mathbf{h_{\tiny{\textrm{RF}}}}$,
here $\mathbf{H_{0}}=\textrm{H}_{0}\mathbf{x}$ is the external constant
magnetic field,
$\mathbf{H_{\tiny{\textrm{DMI}}}}=\textrm{H}_{\tiny{\textrm{DMI}}}\mathbf{z}$
is the DMI field and the magnetic field component of a linearly polarized
microwave field
$\mathbf{h_{\tiny{\textrm{RF}}}}=\textrm{h}_{\tiny{\textrm{RF}}}\cos{\omega\textrm{t}}\cdot\mathbf{y}$,
$W(\mathbf{l,H_{0}})$ is the potential function [54, 18]:

$\begin{array}[]{c}W(\mathbf{l,H_{0}})=\frac{\omega_{\tiny{\textrm{ex}}}\omega_{\tiny{\textrm{A}}}}{2}(\mathbf{l}\cdot\mathbf{z})^{2}+\frac{\gamma^{2}}{2}(\mathbf{H_{\tiny{\textrm{DMI}}}}\cdot\mathbf{l})^{2}\\\
+\frac{\omega_{\tiny{\textrm{ex}}}\omega_{\tiny{\textrm{me}}}}{2}(\mathbf{l}\cdot\mathbf{x})^{2}+\frac{\gamma^{2}}{2}(\mathbf{H_{0}}\cdot\mathbf{l})^{2}.\end{array}$
(3)

Here characteristic frequencies are
$\omega_{\tiny{\textrm{ex}}}=2\gamma\textrm{H}_{\tiny{\textrm{ex}}}$,
$\omega_{\tiny{\textrm{A}}}=\gamma\textrm{H}_{\tiny{\textrm{A}}}$,
$\omega_{\tiny{\textrm{me}}}=\gamma\textrm{H}_{\tiny{\textrm{me}}}$, and
$\textrm{H}_{\tiny{\textrm{ex}}}$ is the AFM internal exchange magnetic field,
$\mathbf{H_{\tiny{\textrm{A}}}}=\textrm{H}_{\tiny{\textrm{A}}}\mathbf{y}$ is
the AFM anisotropy field,
$\mathbf{H_{\tiny{\textrm{me}}}}=\textrm{H}_{\tiny{\textrm{me}}}\mathbf{x}$ is
the effective magnetostriction field [55]. For the weak ferromagnets, such as
hematite, iron borate, manganese carbonate, etc., having the space group of
symmetry $D_{3d}^{6}$, the field of spontaneous magnetostriction, which
determines the gap in the spectrum of magnetic excitations at zero field,
should be chosen as $\textrm{H}_{\tiny{\textrm{me}}}$. Magnetic moment is
always directed along the external magnetic field and the direction of
spontaneous striction “follows” the direction of the external magnetic field.

The low-frequency resonant mode of the AFM, leading from the solving of the
equation (2) (see for more details Suppl. Inf.) taking into account the
applied constant magnetic field and DMI field, is equal to [36]

$\begin{array}[]{c}\omega_{\tiny{\textrm{QFMR}}}=\sqrt{\omega_{\tiny{\textrm{H}}}^{2}+\omega_{\tiny{\textrm{H}}}\omega_{\tiny{\textrm{DMI}}}+\omega_{\tiny{\textrm{ex}}}\omega_{\tiny{\textrm{me}}}}.\end{array}$
(4)

Figure 2:  (a) Experimental setup for measuring the ISHE voltage by phase-
locking method. A RF signal generator up to 43.5 GHz is used as a RF signal
source. The amplitude-modulated signal is applied to the coplanar waveguide.
The output of the amplitude modulation signal of the generator is connected to
the lock-in amplifier (LIA) reference input thus LIA is phase-locked at the
generator modulation frequency. Under the action of the amplitude-modulated
signal, a homogeneous precession of the magnetic sublattices of FeBO3 is
initiated, which leads to spin-pumping at the boundary of FeBO3/Pt and the
emergence of a spin current $\mathbf{j}_{\tiny{\textrm{sp}}}$ in the platinum
layer. Due to the inverse spin Hall effect, the appearance of spin current in
Pt leads to the emergence of ISHE voltage. (b) Experimentally obtained ISHE
voltage of FeBO3/Pt at different frequencies normalized by actual RF power
with offset voltage.

It should be noted that the frequency of quasi-ferromagnetic resonance is
significantly affected by the DMI field, which is mostly present in the canted
antiferromagnets, as well as the magnetoelastic coupling. The iron borate has
a large value of the latter. The application of external mechanical strain
allows a significant increase in the resonant frequency.

In order to compare the obtained theoretical expressions with the experimantal
results, we measure the ferromagnetic resonance (FMR) using the VNA-FMR method
(see Suppl. Inf. for more details). We use two port Vector Network Analyser
(VNA) up to 20 GHz for radio frequency (RF) signal generation and detection.
VNA’s ports are connected to the coplanar waveguide designed for frequencies
up to 50 GHz, which in turn is located between the electromagnet’s poles. We
measure real and imaginary parts of S21-parameter at the fix frequency with
magnetic field sweeping (see more details in Supplemental material). Also, the
influence of irregularities in the amplitude-frequency response of the
microwave path is taken into account. Fig. 1(a) shows the theoretical and
experimental dependence of the resonant frequency of FeBO3 for the low-
frequency mode on the applied external magnetic field. These experimental
points are derived from the FMR spectra (Fig. 1(b)) of FeBO3, measured by the
VNA-FMR method. As it is shown in Fig. 1(a) theoretical and experimental
results are consistent with high accuracy. From FMR spectra we can determine
the linewidth. In our case, the linewidth is 50 Oe at 10 GHz, which is
significantly smaller than in a similar experiment with hematite [34, 33, 35].
It is also worth noting a slight increase in the linewidth with a rising
frequency of input signal.

Let us calculate the inverse spin Hall voltage caused by the spin pumping from
the AFM. As mentioned earlier, an external variable microwave wave with linear
polarization initiates a homogeneous precession of the magnetic sublattices
$\mathbf{M}_{1,2}$ in the AFM. This precession causes spin pumping at the
$\textrm{FeBO}_{3}/\textrm{Pt}$ boundary and the excitation of the spin
current $\mathbf{j}_{\textrm{sp}}$ in the Pt layer [6]:

$\begin{array}[]{c}\mathbf{j}_{\tiny{\textrm{sp}}}=\frac{hg_{r}}{4\pi^{2}}\left[\frac{d\mathbf{l}}{dt}\times\mathbf{l}+\frac{d\mathbf{m}}{dt}\times\mathbf{m}\right],\end{array}$
(5)

where $g_{r}$ is the real part of the spin-mixing conductance and $h$ is the
Planck constant. Due to the inverse spin Hall effect (ISHE), the appearance of
spin current in Pt leads to a non-zero potential difference at the output
electrodes $V_{\text{out}}$ placed on the distance $L$. Note that the main
contribution to the spin current will be made by the second term on the right
side of expression (5), that is, the influence due to a varying of the
magnetic moment $\mathbf{m}(t)$, in contrast to most experiments with spin
pumping from antiferromagnets that do not have weak ferromagnetism [30, 31,
32]. As can be seen, expression (5) determines the quadratic nature of the
spin current dependence on the magnitude of the resonant amplitude of the
magnetic moment. In (5) we do not consider the imaginary part of the spin
mixing conductance, the corresponding term is proportional to the first
derivative of magnetization and does not affect the DC component of the
rectified voltage [6].

We experimentally demonstrate the phenomenon described above using an
installation illustrated in Fig. 2(a). For this measurement we utilize the
lock-in technique. A RF signal generator up to 43.5 GHz is used as the RF
signal source. The amplitude-modulated signal is applied to the coplanar
waveguide, the board of which has two contacts for ISHE voltage measurement.
Output of the amplitude modulation signal of the generator is connected to the
lock-in amplifier (LIA) reference input thus LIA is phase-locked on the
modulation frequency of the generator. The sample was separated from the
coplanar waveguide by adhesive tape due to the conductivity of platinum layer.

Figure 3:  (a) Comparison of the theoretical curve obtained by Eq. (8) with
the experimental points of ISHE voltage at 14.257 GHz, normalized by actual RF
power. (b) Experimentally measured dependence of ISHE voltage on the applied
external magnetic field at various RF power at 23.514 GHz. (c) Theoretical and
experimentally measured power dependence of the ISHE voltage at 23.514 GHz.

We obtain a set of experimentally received dependencies of ISHE voltage on
different frequencies normalized by actual RF power with offset voltage from
magnetic field (Fig. 2(b)). One can see, as the frequency increases, the
resonant peaks drift apart, and their amplitude decreases. An important
feature is that at positive magnetic field values, the resonant peak is also
positive, and at negative magnetic fields, it is negative.

For a more detailed study of this behavior let us derive an expression for the
inverse spin Hall voltage $V_{\tiny{\textrm{out}}}$ induced by spin pumping
from the AFM into the HM layer [18, 53, 10]:

$\begin{array}[]{c}V_{\tiny{\textrm{out}}}=\left.\kappa\left<\frac{d\mathbf{l}}{dt}\times\mathbf{l}+\frac{d\mathbf{m}}{dt}\times\mathbf{m}\right>\right|_{\mathbf{x}},\end{array}$
(6)

where $\left<...\right>$ is average of time during the period
$2\pi/\omega_{\tiny{\textrm{QFMR}}}$, $\kappa$ is the proportionality
coefficient [15]:

$\begin{array}[]{c}\kappa=\frac{Lg_{r}\theta_{\tiny{\textrm{SH}}}e\lambda_{\tiny{\textrm{Pt}}}\rho}{2\pi
d_{\tiny{\textrm{Pt}}}}\tanh\left(\frac{d_{\tiny{\textrm{Pt}}}}{2\lambda_{\tiny{\textrm{Pt}}}}\right),\end{array}$
(7)

and $L$ is the distance between output electrodes,
$\theta_{\tiny{\textrm{SH}}}$ is the spin-Hall angle, $e$ is the electron
charge, $\lambda_{\tiny{\textrm{Pt}}}$ is the spin-diffusion length, while
$\rho$ and $d_{\tiny{\textrm{Pt}}}$ are the electrical resistivity and
thickness of the Pt layer, respectively.

For the input alternating power of the RF signal
$P_{\tiny{\textrm{RF}}}=\frac{c}{2\mu_{0}}S\cdot\left(h_{\tiny{\textrm{RF}}}\right)^{2}$,
where $c$ is the speed of light, $\mu_{0}$ is the vacuum magnetic
permeability, $S$ is the AFM layer cross-section, one can find the detector
sensitivity defined as (see Suppl. Inf.):

$\begin{array}[]{c}R\left(\omega\right)=\frac{\left|V_{\tiny{\textrm{out}}}\left(\omega\right)\right|}{P_{\tiny{\textrm{RF}}}}=\frac{2\mu_{0}\gamma^{2}\kappa\cdot\kappa^{{}^{\prime}}}{cS\omega_{\tiny{\textrm{ex}}}^{2}}\cdot\frac{\left(\omega_{\tiny{\textrm{H}}}+\omega_{\tiny{\textrm{DMI}}}\right)\omega_{\tiny{\textrm{DMI}}}^{2}}{\left(\omega^{2}-\omega_{\tiny{\textrm{QFMR}}}^{2}\right)^{2}+\left(\gamma_{\tiny{\textrm{eff}}}\omega\right)^{2}}\omega^{2}.\end{array}$
(8)

where $\kappa^{{}^{\prime}}$ is phenomenological fit parameter that has been
obtained by comparing experimental data with theoretical model. By analyzing
the expression, we can determine that the sensitivity’s dependence on the
magnetic field is a linear function. All numerical values of constants used
for constructing theoretical curves are given in Table I [40, 56]. Some
parameters, i.e., the spin Hall $\theta_{SH}$ angle and the spin mixing
constant $g_{r}$, were taken to coincide with similar parameters for hematite
[33, 34]. Fig. 3(a) shows a comparison of the theoretical curve obtained by
Eq. (8) with the experimental dependence of ISHE voltage on the applied
external magnetic field at a frequency of 14.257 GHz. A number of curves is
measured (Fig. 3(b)) at different RF power to determine the nature of the
change in the ISHE voltage level from the applied RF power (Fig. 3(c)) at
23.514 GHz.

TABLE I. Numerical values of the parameters used to calculate the
characteristics for the FeBO3 [40, 56, 33, 34].

Parameter Value Physical meaning $\theta_{\textrm{SH}}$ 0.1 Spin-Hall angle
$\textrm{g}_{\textrm{r}}$ 6 $\cdot 10^{18}$ m-2 Real part of the spin-mixing
conductance $\alpha_{\textrm{eff}}$ $5.5\cdot 10^{-3}$ Effective damping
including Gilbert constant and spin-pumping terms $\textrm{H}_{\textrm{ex}}$
6500 kOe Exchange magnetic field $\textrm{H}_{\textrm{me}}$ 0.2 Oe Effective
magnetostriction field $\textrm{H}_{\textrm{DMI}}$ 100 kOe DMI field
$\lambda_{\textrm{Pt}}$ 1.2 nm Spin-diffusion length
$\textrm{d}_{\textrm{Pt}}$ 2 nm Thickness of the Pt layer $\rho$ $2.5\cdot
10^{-7}\Omega\cdot$ m Electrical resistivity of the Pt layer $L$ 3 mm Distance
between output electrodes S 5 $\times$ 5 mm2 AFM layer cross-section

Comparison of theoretical and experimental results presented in Fig. 3(a)
shows a good confirmation of the theoretical model presented in current
article. Small differences are due to the parasitic effect of the output
contacts in the experimental measurement, which is not taken into account in
the theoretical description. Eq. (8) shows that when the input frequency
increases, a larger magnetic field is required to achieve resonance, which in
turn leads to a decrease in the resonance peak amplitude. Fig. 3(b)
demonstrates the experimentally measured dependence of ISHE voltage on the
applied external magnetic field at various RF power. This dependence has a
resonant nature. Also, it can be noticed that the ISHE voltage amplitude and
the linewidth are grow by increasing the power of the RF signal. The
theoretical and experimental dependence of ISHE voltage on RF signal power can
be seen in Fig. 3(c). The slope of this characteristic determines the volt-
watt sensitivity of the structure. ISHE voltage at a frequency of 23.514 GHz
is 13.2 nV, which at an input power of 5.403 mW corresponds to the sensitivity
of 2.5 $\mu$V/W.

Finally, we have demonstrated both theoretically and experimentally that an
antiferromagnet with weak ferromagnetism FeBO3 can be used as a sensitive
element to detect the resonance of a linearly polarized RF wave in tens of
GHz. We studied FMR spectra of this structure and showed that the theoretical
and experimental dependence of resonance frequency from magnetic field were
consistent. In addition, we measured ISHE voltage from the structure and
compared it to the theoretical model. It was noted that an increase in the
frequency of input microwave field causes an increase in the distance between
the resonance peaks as well as a decrease in their amplitude. A study of the
ISHE voltage dependence on the magnetic field showed that an increase in the
power of the applied RF signal increases the amplitude of the resonance peak
as well as the width of the line. We obtained a value of the sensitivity of
2.5 $\mu$V/W, typical of similar structures (for example, in [33] a value of
the sensitivity is about 0.4 $\mu$V/W). Comparing our results with the results
of similar works, we can conclude that they are consistent with each other,
but we demonstrated a narrower linewidth. In our opinion, the study of
structures based on antiferromagnets with weak ferromagnetism allows to expand
the understanding of spin dynamics, as well as to implement various microwave
and terahertz applications, developments of antiferromagnetic spintronics,
magnonics, and data storage.

The work was carried out within the framework of the state assignment of the
Kotel’nikov Institute of Radio Engineering and Electronics of the Russian
Academy of Sciences

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