# Study of magnetoelastic interaction in MnF2 by the acoustoelectric
transformation method

I. V. Bilych1, K. R. Zhekov1, T. N. Haidamak1∗∗, G. A. Zvyagina1, D. V.
Fil2,3, V. D. Fil1<EMAIL_ADDRESS><EMAIL_ADDRESS>1B. Verkin
Institute for Low Temperature Physics and Engineering, National Academy of
Sciences of Ukraine, Nauky Avenue, 47 Kharkiv 61103, Ukraine
2Institute for Single Crystals, National Academy of Sciences of Ukraine, 60
Nauky Avenue, Kharkiv 61072, Ukraine
3V.N. Karazin Kharkiv National University, 4 Svobody Square, Kharkiv 61022,
Ukraine

###### Abstract

The mechanisms of magnetoelastic interaction in MnF2 are studied using the
method of acoustoelectric transformation. The temperature dependence of the
piezomagnetic coupling coefficient and its anisotropy are determined in the
antiferromagnetic phase. We observe a new effect consisting in the appearance
of a non-diagonal component of the magnetic susceptibility tensor, which is
proportional to the square of the order parameter, under the action of the
shear wave. A phenomenological interpretation of the effect, which takes into
account a small-angle rotation of the crystal lattice, is presented. In the
paramagnetic state, the effect of acoustic deformation is reduced to the
modulation of the diagonal component of the susceptibility tensor.

##  I Introduction

In this paper, the influence of elastic strains on the state of the magnetic
subsystem of MnF2 single crystals is investigated by acoustic methods. The
motivation for our reference to this rather well-studied compound is due to
the following circumstances. Manganese fluoride together with cobalt fluoride
were the first antiferromagnets in which piezomagnetism was detected Ref. [1].
The measurements [1] were performed using the standard balance method of the
measurement of magnetic moments. Unfortunately, in MnF2, due to a very small
value of the corresponding piezomagnetic modulus, it was only possible to
establish the existence of the effect and to estimate its intensity, but no
measurements of the temperature dependence and anisotropy were made. Later the
presence of piezomagnetism in MnF2 was confirmed [2] using the SQUID
magnetometer, but in that experiment the characteristics mentioned above were
not studied either. The reason for a relative failure in both cases was the
need to detect a weak piezomagnetic response at a significant background level
unrelated to the strains.

In the present work, the piezomagnetic response in MnF2 is studied using the
acoustoelectric transformation (AET) method. Previously the AET method was
successfully used to study piezomagnetism in CoF2 [3]. Significant advantages
of the method are the practical absence of a background signal and a
flexibility in implementing various experimental geometries. As a result, the
temperature dependence of the effect and its anisotropy were reliably
established, and a new phenomenon, the appearance of nondiagonal components of
the magnetic susceptibility tensor in antiferromagnets in the field of a
displacement gradient, was observed.

The impact of acoustic deformations on the characteristics of the magnetic
subsystem in the paramagnetic state is also investigated.

Some disadvantage of the AET method is the impossibility to provide direct
quantitative measurements of the intensity of the studied effects. The
measurement algorithm gives only relative quantities. But this shortage can be
easily overcome by comparing the measured parameters with the response of a
calibrated sample.

##  II Experimental details

The basis of the AET method for studying the piezomagnetic effect in
magnetodielectrics and the description of its experimental implementation were
given in detail in Ref. [3]. Here we briefly outline the features of the
measurement procedure that are essential for further discussion and relevant
to the specific experiment.

A sound wave of a given polarization, introduced into the sample through an
acoustic delay line (DL), propagates along one of the main directions of the
elastic modules tensor. In certain situations, due to the magnetoelastic
interaction, magnetization oscillations, synchronous with acoustic ones, are
excited in the sample. The magnetization oscillations are transformed at the
output interface into an electromagnetic field, which is registered by a
polarized antenna. To study the influence of the magnetic field $\bm{H}$ on
the AET process in the described experiments the geometry
$\bm{H}\parallel\bm{q}$ ($\bm{q}$ is the sound wave vector) was used.

The waist of the sound beam is assumed to be much larger than the sound wave
length. Then the problem is reduced to the one-dimensional one. Two scenarios
of the AET response are possible.

In the first scenario, the vector of magnetization $\bm{m}$ (excited by the
sound) is orthogonal to the wave vector $\bm{q}$. This configuration arises
due to the piezomagnetic interaction, whether an external magnetic field is
present or not. The magnetization induced by the external field $\bm{H}$ can
also have a component which is acoustically modulated and orthogonal to
$\bm{q}$. In the discussed geometry ($\bm{H}\parallel\bm{q}$), this is
possible only in the presence of non-diagonal components of the magnetic
susceptibility. As we will see below, this is exactly the situation in the
antiferromagnetic phase of MnF2. For such a scenario (let us call it a ”wave”
scenario) in the harmonic approximations ($\bm{E},\bm{m}\propto\exp(i\omega
t)$ ) in the sound beam one can write the following wave equation

$\Delta\bm{E}=-k_{c}^{2}\varepsilon\mu\bm{E}+4\pi ik_{c}\nabla\times\bm{m}$
(1)

(in Gauss system of units), where $k_{c}=\omega/c$. In the one-dimensional
case under discussion, the spatial dependence is proportional to
$\exp(-i\bm{q}\cdot\bm{r})$, and in our experiments $\bm{q}$ is orthogonal to
the radiating interface.

Since $|\bm{q}|\gg k_{c}$, the first term in the right-hand side of Eq. (1)
can be neglected. The solution of Eq. (1) yields the electric field $\bm{E}$
tangential to the radiating surface and orthogonal to $\bm{m}$. The
electromagnetic radiation arises due to the continuity of $\bm{E}$ at the
boundary. In the near zone it can be approximated by a linearly polarized
plane wave. The magnetic component $\bm{h}$ of this wave is collinear to
$\bm{m}$. The field $\bm{h}$ is registered by a frame antenna, which can be
rotated to determine the polarization of the wave. Since $\bm{h}$ and $\bm{m}$
are collinear and proportional to each other, to simplify the following
discussion we identify the electromagnetic field registered by the antenna
with the acoustically induced magnetization in the sample.

At $\bm{H}\neq 0$, another scenario (magneto-dipole) is also possible. Sound
perturbation modulates the diagonal component of the magnetic susceptibility,
so that a collinear to $\bm{H}$ and sign-alternating component of the
magnetization appears in the sample. Averaged over the sample, it creates the
magnetic field outside the sample and affects the antenna. The characteristics
of this field can be calculated by methods of magnetostatics. Its structure is
similar to the dipole field, i.e., it has both longitudinal and transverse
components. But still the amplitude of the response registered by the antenna
is related by a linear dependence to the magnetization contribution induced by
the sound. However, the identification of the measured response with a
particular component of $\bm{m}$ is no longer possible.

All measurements were performed in the pulse mode. The DL allows us to
separate the signals under study from the parasitic leakage of the radio
pulse, which excites the piezoelectric transducer. A Ge monocrystal oriented
along the second-order axis was used as the DL. The latter ensured the
possibility of working with the elastic strains of a given polarization.

The apparatus described in Ref. [4] was used, which allows one to register
simultaneously the amplitude and the phase of the electromagnetic signals at
fixed frequency ($\approx 55$ MHz).

Monocrystalline MnF2 samples were in the form of plane-parallel plates with
the thickness $\approx 0.3$ mm, the transverse dimensions $\approx 3$ mm, and
the C4 axis orthogonal to the plane of the plates. All experiments were
performed in the $\bm{q}\parallel$ C4 geometry. Two samples were investigated,
which showed identical results. The results stated below refer to the
temperature range up to 200 K, including the range of the acoustic grease
melting ($\approx 120$ K). Since the acoustic path ”piezoelectric transducer –
DL – sample” includes two transition grease layers, a temperature-dependent
correction was introduced into the high-temperature part of the amplitude
data, correcting variations in the transition damping. To obtain it, in the
same experiment we measured the temperature variations in the amplitude of the
signal rereflected in the DL, which also passed twice through the transient
layer of the acoustic putty. At our frequencies the temperature variations of
the attenuation in the germanium DL due to its smallness can be neglected [5].

##  III Results and discussion

###  III.1 The $u_{xz}$ deformation

Let us first recall the information necessary for further discussion on some
physical characteristics of MnF2. The crystal structure of MnF2 belongs to the
centrosymmetric tetragonal syngony, spatial group P42/mnm. The main symmetry
elements are the 4th order helical axis – C4 axis (4z), the sliding symmetry
plane $\bm{n}$, parallel to C4, and the second order axis 2d, orthogonal to C4
and directed along the diagonal of the basic square. Naturally, the last two
elements are duplicated by the C4 axis. The generally accepted canonical
choice of coordinate axes is that the $z$-axis is parallel to C4, the $x$-axis
is orthogonal to one of the $n_{x}$ sliding planes.

At $T_{N}\approx 68$ K there is a transition to a magnetically ordered phase –
a two-sublattice uniaxial fully compensated antiferromagnet. The order
parameter is the antiferromagnetism vector $\bm{L}$ oriented along the
$z$-axis.

Within the framework of crystallochemical symmetry the piezomagnetic moment
$\bm{m}^{0}$, which appears due to the partial decompensation of the
antiferromagnet under the strain, is described by the tensor relation [6]:

$m_{i}^{0}=P_{ijkl}L_{j}u_{kl}.$ (2)

Here and hereafter, the index 0 indicates the response observed in the absence
of an external magnetic field. The invariance of relations (2) with respect to
the operations of crystallochemical symmetry determines the structure of the
tensor $P_{ijkl}$. It is also necessary to take into account the parity of a
particular symmetry element [6]. The latter is considered odd if it connects
atoms in different sublattices. An odd element changes the sign of the
antiferromagnetism vector in addition to the rules for converting the
components of the axial vector. In the P42/mnm group the elements $n_{x}$ and
4z are odd. It means, for example, that when reflected in the plane $n_{x}$
the $L_{z}$ component does not change its sign.

The tensor structure of $P_{ijkl}$ for different spatial groups is presented
in many publications (see, for example, Ref. [6]), but it is given, as a rule,
only for the single choice of the canonical variant of the coordinate system.
The transition to another system by the tensor algebra algorithms is rather
cumbersome, but one can use more simple and physically more transparent
procedure.

In the particular situation under consideration
($\bm{q}\parallel\bm{L}\parallel\bm{z}$), the relation (2) reduces to

$m_{i}^{0}=P_{i3k3}L_{z}u_{kz}.$ (3)

Let the canonical coordinate system with the $x$-axis orthogonal to the
sliding plane be chosen. Applying the $n_{x}$ operation to Eq. (3), we can see
that only the $P_{1323}$ and $P_{2313}$ components remain nonzero. The
presense of 2d operation, which swaps the $x$\- and $y$-axes, allows to
conclude that these components of the piezomagnetism tensor are equal each
other. Thus, with this choice of the coordinate axes, the piezomagnetic moment
is orthogonal to the vector of displacement inducing this moment.

The sign of these components remains undefined. Let us assume that it is
positive. The scheme of the mutual arrangement of the displacement vectors and
excited by them piezomagnetic moments is shown in Fig. 1 by solid arrows. Note
that the non-equivalence of the x- and y-axes is due to the disappearance of
the C4 axis in the MnF2 magnetic group.

Figure 1: Scheme of the piezomagnetic response at different polarizations of
the elastic field.

If the 2d axis is present in the symmetry group, this axis can be chosen as
the $x^{\prime}$-axis. In such a systems of coordinates applying C2 and then
4z operation to Eq. (3), one can easily see that only two components equal in
absolute value and opposite in sign,
$P^{\prime}_{1^{\prime}31^{\prime}3}=-P^{\prime}_{2^{\prime}32^{\prime}3}$,
remain nonzero. Hence, if we introduce a deformation into the sample with the
polarization along [110] (in the canonical system), then the excited
piezomagnetic moment will be collinear to the displacement vector. A simple
illustration of this statement can easily be seen from the scheme shown in
Fig. 1. The elastic displacement along [110] (the dotted line in Fig. 1)
emerges as a result of interference of two in-phase and equal in amplitude
normal modes ($u_{xz}$ and $u_{yz}$). Each mode generates its own
piezomagnetic moment orthogonal to it. Obviously, the resulting moment is
collinear to the resulting displacement.

Experiments have shown that in MnF2 below the Neel temperature a transverse
acoustic wave propagating along the C4 axis always generates a piezomagnetic
response. Its orientation is determined by the polarization of the elastic
field. Figure 2 shows typical examples of amplitude-phase rotation diagrams
observed in various experimental situations. The coordinate of the abscissa
axis is the angle $\psi$ between the normal to the plane of the loop antenna
and the displacement vector of the sound. Accuracy of its tuning to an initial
reference level is $\pm 5^{\circ}$. The ordinates show the amplitude (Fig. 2
(a), (c)) and the phase (Fig. 2 (b), (d)) of the recorded piezomagnetic
response.

Figure 2: Polarization diagrams at different polarizations of the elastic
field measured in the zero magnetic field and in the presence of the magnetic
field $\bm{H}$ ($\bm{H}\parallel z$): (a), (b) – $\bm{u}\parallel[100]$, black
curves corresponds to $T=53$ K, $H=0$, measurements without preliminary
switching on the magnetic field; red curves corresponds to $T=53$ K, $H=0$,
measurements after preliminary switching on the magnetic field $H=3$ T; green
curves corresponds to $T=53$ K, $H=3$ T; blue curves corresponds to $T=72$ K,
$H=3$ T; (c), (d) – $u\parallel[110]$, red curves correspond to $T=4.2$ K,
$H=0$; green curves correspond to $T=4.2$ K, $H=2$ T.

We emphasize that in Fig. 2 the coordinate corresponding to the maximum of the
amplitude response determines the angle between the displacement $\bm{u}$ in
the elastic deformation and the magnetic moment $\bm{m}$ excited by it. The
amplitude dependences are given in semi-logarithmic scale for the sake of
better visibility. The maximum signal level in the absence of a magnetic field
is chosen as the reference level. The shape of phase characteristics makes it
possible to evaluate the degree of ellipticity of the registered radiation. In
particular, in Fig. 2 (b), (d) the step-like phase change at the angle
$\varphi\approx 180^{\circ}$ indicates that in these cases the polarization is
practically linear.

In Fig. 2, in addition to the dependences taken in the zero fields, the
rotation diagrams measured in the presence of the magnetic field $\bm{H}$
($\bm{H}\parallel\bm{z}$) are shown. We will turn to their interpretation
later.

It follows from Fig. 2 that at $\bm{u}\parallel[100]$ (Fig. 2(a)) $\bm{m}^{0}$
is orthogonal to $\bm{u}$, while at the displacement along [110] (Fig. 2(c))
the corresponding vectors are collinear. These results are in complete
agreement with the above analysis.

It follows from Eqs. (2) and (3) that in antiphase domains the moments
generated by the deformation are reversed, so that the resulting response can
be greatly underestimated. In this connection Fig. 2 (a) illustrates a
”successful” experiment of monodomainization of the sample by the short-term
switching on the external field $H_{z}=3$ T before measurements. The result is
not obvious at all, since it is necessary to specify the reason why the order
parameter entered to the expansion of thermodynamic potentials in an odd
degree, providing the realization of energy minimization at a given direction
of $\bm{L}$. In our opinion, the piezomagnetic moment allowed by symmetry,
$m_{z}^{0}\sim L_{z}\sigma_{xy}$, emerges in the sample rigidly glued to the
DL due to the appearance of thermoelastic $\sigma_{xy}$ static stresses. Its
Zeeman interaction with the external magnetic field $H_{z}$ leads to the
monodomainization of the antiferromagnetic structure.

The temperature dependence of the piezomagnetic response in the absence of an
external magnetic field, measured when the antenna is tuned to the maximum
signal, is shown in Fig. 3.

Figure 3: Temperature dependence of $m^{0}$. Symbols are the normalized
experimental values of $m^{0}$, red (dash) curve is the function
$\sqrt{1-(T/68.7)^{2}}$, blue (dash dot) curve is the function
$1.12(1-T/68.7)^{0.382}$. (a), the dependencies in whole temperature range;
(b), the dependences in the vicinity of $T_{N}$.

On average, it is satisfactorily described by the dependence
$m^{0}(T)/m^{0}(0)\approx\sqrt{1-(T/T_{N})^{2}}$ (typical to the type II phase
transitions), although it is not good enough near $T_{N}$. We will return to
the discussion of the behavior in the region of critical fluctuations later.
The absolute value of the piezomagnetic modulus
$\lambda_{123}=P_{1z23}L_{z0}=4\pm 1$ G ($L_{z0}$ is the upper limit value of
the order parameter) is determined from a comparison of the response from MnF2
with the previously measured response from CoF2 ($1000$ G [3]). It is close to
one found from the estimate presented in [1] ($6.4$ G).

Let us now discuss the nature of the AET response $\bm{m}^{H}$ observed in the
magnetic field, which in the case of $\bm{u}\parallel[100]$ has a polarization
diagram different from that of $\bm{m}^{0}$. It follows from the rotation
diagrams presented in Fig. 2 that its orientation is always almost collinear
to the elastic displacement. In particular, at $\bm{u}\parallel[100]$, when
$\bm{m}^{0}$ is orthogonal to the displacement, one can fix the angular
coordinates at which only the $\bm{m}^{H}$ signal is mostly registered. The
temperature behavior of the amplitude and phase, measured in such geometry, is
shown in Fig. 4. The amplitude is given in the semi-logarithmic scale, which
allows to demonstrate more clearly the existence of the response even at
$T>T_{N}$. The phase at $T_{N}$ changes almost in a step-like manner.

Figure 4: Temperature dependence of the amplitude (red line) and the phase
(blue line) of $m^{H}$ ($H=3$ T).

Our interpretation of the dependences presented in Fig. 4 is as follows. The
response $\bm{m}^{H}$ is the result of the interference of two contributions
whose phases are different and are almost independent of the temperature. The
first one, rather weak, apparently exists at temperatures both above and below
$T_{N}$, while the second one, with rapidly increasing intensity, appears only
in the antiferromagnetic phase.

Figure 5 shows the dependence of $m_{x}^{H}$ on the magnetic field, measured
at the orientation of the receiving antenna, which provides the minimum
piezomagnetic response $\bf{m}^{0}$. It is practically linear.

Figure 5: Magnetic field dependence of $m^{H}$ ($T=47$ K).

Within the scenarios described in the previous subsection, the following
variants are possible to explain the origin of $\bm{m}^{H}$. The ”wave”
scenario. The deformation initiates the emergence of a nondiagonal component
in the magnetic susceptibility tensor, i.e., a magnetization component
orthogonal to $\bm{H}$ appears. From the formal point of view, this effect is
quite possible because the field of shear displacement gradients changes the
symmetry of the lattice, making it more anisotropic. The ”magneto-dipole”
scenario. The deformation modulates the longitudinal relative to $\bm{H}$
magnetization, and the antenna only registers the transverse component of the
magnetic field induced by magnetization. In our opinion, this scenario is
unlikely. It should be expected that in this case the recorded response should
be correlated with the behavior of the longitudinal magnetic susceptibility.
In antiferromagnets MnF2, the longitudinal magnetic susceptibility decreases
with decreasing temperature, while $m_{x}^{H}$ only increases below $T_{N}$
(Fig. 4).

According to the general theory of type II phase transitions [7], the
temperature behavior of thermodynamic characteristics in the region of
critical fluctuations is described by power law dependences on the quantity
$(1-T/T_{c})$, and the exponents are called critical indices. Their numerical
values are universal: they depend mostly on the system dimensionality. In
particular, a critical index $\beta\approx 0.33$ is expected for the order
parameter in the three-dimensional case of the antiferromagnet.
Experimentally, such behavior is carried out only approximately, and the
deviation of values of critical indices from theoretical ones is about
$10-20$%.

Figure 6: Approximations of the temperature dependences by critical exponents.
Curve 1 is $m^{0}$, curve 2 is $m^{H}$ ($H=3$ T).

Figure 6 shows the temperature dependence of the measured signals in double
logarithmic coordinates. In these coordinates the piezomagnetic response
$m^{0}(T)$ is almost a straight line (curve 1) in the whole temperature range.
A small deviation from the straight line in the left-hand part of the figure
may be caused by some smearing of the transition due to the heterogeneity of
the sample. The slope of the linear dependence determines the parameter
$\beta=0.38\pm 0.01$. It practically coincides with the critical index value
found in Ref. [8] ($0.36\pm 0.02$), characterizing the temperature behavior of
the order parameter. This conclusion is in the complete agreement with the
relation (3). The accuracy of the description of the behavior $m^{0}(T)$ by
this critical exponent is illustrated in Fig. 3. Concidering the maximum
length linear segment of the curve 1 in Fig. 6, we refine the value of
$T_{N}=68.7$ K. This value is used in all evaluations in this work. Curve 2 in
Fig. 6 illustrates the temperature dependence of the response $m_{x}^{H}$. The
left-hand part of this curve deviates significantly from the straight line
already at $T>66$ K due to the existence of the previously mentioned small
response of a different nature at all temperatures near $T_{N}$. Nevertheless,
the slope of the straight line section ($0.78\pm 0.01$) is twice as large as
the slope of line 1. It is natural to assume that the temperature dependence
of the ampliture of the response is determined by the square of the order
parameter in a certain temperature range. The accuracy of the approximation by
the assumed dependence is illustrated in Fig. 7. Surprisingly, the critical
exponent describes the behavior over the amlost entire temperature range below
$T_{N}$. Thus, the response $\bm{m}^{H}$ can be presented as

$m_{i}^{H}=\left(B_{i3k3}L_{z}^{2}\frac{\partial u_{k}}{\partial
z}\right)H_{z}.$ (4)

Figure 7: Approximation of the normalized experimental temperature dependence
of $m_{x}^{H}$ (symbols) at $H=3$ T by the function $1.11(1-T/68.7)^{0.78}$
(blue line).

From the symmetry considerations similar to ones used above, it is easy to see
that only two components of the tensor in Eq. (4) can be nonzero:
$B_{1313}=B_{2323}$. This conclusion is valid at any choice of the coordinate
system, i.e., the response is always collinear to the displacement that
initiated it, which fully corresponds to the results presented in Fig. 2. It
is natural to interpret the combination in parentheses in Eq. (4) as a non-
diagonal component of the susceptibility tensor. The presence of $L_{z}^{2}$
in Eq. (4) means that the discussed response appears only in the magnetically
ordered phase.

Let us give a simple phenomenological interpretation of the experimentally
derived relation (4). One considers that the shear wave is accompanied by a
rotation of the crystal lattice by a small angle
$\delta\varphi=0.5\nabla\bm{u}$ [9]. In the framework of the Landau’s approach
to the expansion of the thermodynamic potential of antiferromagnets by
magnetic variables, in the main approximation the magnetization is given by
equation [9]

$\bm{M}=\chi_{p}\bm{H}-2D(\bm{L}\cdot\bm{H})\bm{L},$ (5)

where $\chi_{p}\approx 10^{-3}$ is the susceptibility in the paramagnetic
phase at $T=T_{N}$, and $D$ is a dimensional (1/G2) numerical coefficient.

Along with the rotation of the anisotropy axis, the antiferromagnetism vector
also rotates. As a result, a component
$L_{x}=L_{z}\delta\varphi=0.5L_{z}(\partial u_{x}/\partial z)$ appears, and in
the main approximation we have the same as Eq. (4) equation for the
magnetization induced by joint action of the magnetic field and sound,

$m_{x}^{H}=\left(DL_{z}^{2}\frac{\partial u_{x}}{\partial z}\right)H_{z}.$ (6)

Let us also estimate the value of the effect predicted by Eq. (6). Since in
MnF2 at $T\to 0$ the susceptibility $\chi$ goes to zero, from Eq. (5) we
obtain $DL_{z0}^{2}\sim 0.5\chi_{p}\sim 0.5\cdot 10^{-3}$. From Fig. 6, in
which both dependences are measured at the same value of $\partial
u_{x}/\partial z$, it can be found that at low temperature the $m^{H}$ signal
(at $H=3$ T) is in 4 times larger than the $m^{0}$ sygnal. Taking into account
the linear dependence of $m^{H}$ on magnetic field (Fig. 5) we estimate that
$m^{H}$ compares with $m^{0}$ at $H\approx 0.7$ T. Consequently, the estimate
of the piezomagnetic modulus from Eq. (6) gives $\lambda_{xxz}\sim 3.5$ G that
agrees with the value found above ($4$ G). Note that the realistic value of
$\delta\varphi$ is $10^{-5}$, and apparently for this reason, as far as the
authors know, no such effect has been observed in other experiments. We
believe that this result nicely illustrates the capabilities of the AET
technique.

Let us say a few words about possible nature of the response registered at
$T>T_{N}$. The multiplier $L_{z}^{2}$ does not introduce any symmetry
restrictions (or simplifications), and in the paramagnetic phase one would
expect a realization of the relation similar to (4), but without $L_{z}^{2}$.
However, in this case, the amplitude-phase diagrams at $T>T_{N}$ would repeat
those for the antiferromagnetic phase in the position of characteristic points
and shape, which in fact does not occur (Fig. 2 (a), (b)). We believe that at
$T>T_{N}$ the second ”magneto-dipole” scenario is the case, namely, the sound
deformation modulates the longitudinal susceptibility, and the antenna
registers the transverse component of the dipole magnetic field. The
polarization diagram in this case simply reflects the asymmetry of the
measuring cell together with the sample relative to the antenna rotation axis.

###  III.2 The $u_{zz}$ deformation.

In this subsection we consider the effect of $u_{zz}$ deformation on the
magnetic subsystem. Such a deformation emerges under propagating of the
longitudinal wave along the C4 axis. This deformation does not violate the
original symmetry, so the relations of the type (4) with $j=k=z$ are only
allowed. In this case the coefficient at $H_{z}$ is simply the first term of
the expansion of the diagonal component of the magnetic susceptibility tensor
with respect to $u_{zz}$, and hence, the response registered by the antenna is
completely within the framework of the ”magneto-dipole” scenario.

Figure 8 shows the temperature dependence of the AET response to the
deformation in MnF2 and of the magnetic susceptibility measured for the same
sample. Both dependences were taken in the $\bm{H}\parallel z$ geometry and
are normalized for convenience of comparison. In the antiferromagnetic state,
both dependences demonstrate almost the same change, close to the linear one.
The susceptibility is obviously quite well described by the Landau theory,
which predicts a linear decrease in $\chi$ with decreasing temperature [9].
The AET response, which is proportional to the derivative of $\chi$ with
respect to strain, should naturally exhibit a similar behavior.

Now let’s turn to the paramagnetic phase. MnF2 belongs to the family of
classical antiferromagnets, and in the molecular field approximation its
susceptibility at $T>150$ K is well described by the Curie-Weiss law
$\chi(T)=C/(T+\Theta)$. The value of the paramagnetic Curie temperature
$\Theta=86$ K was evaluated from processing of the curve 1 in Fig. 8. This
temperature is determined by the exchange interaction of given magnetic atom
with the average field created by the entire ensemble of magnetic atoms. Small
deviations from this law at $T_{N}<T<150$ K are associated with the
development of pair correlations between the nearest neighbors in the magnetic
subsystem, which are not taken into account accurately by the mean-field
model.

Figure 8:  Temperature dependence of the AET response to $u_{zz}$ deformation
and of the susceptibility. Symbols are the AET experiment, curve 1 is
$\chi(T)$, curve 2 is $\chi^{2}(T)$ , curve 3 is the function
$4.91/(1+T/86)^{3.5}$

Acoustic deformation, modulating the mutual arrangement of magnetic atoms,
thereby modulates the exchange field, changing the value of $\Theta$, and with
it, changing $\chi(T)$. As a result, the sample placed in a constant magnetic
field acquires an additional magnetization oscillating in time and space,
producing also the corresponding dipole magnetic field $h(T)\sim
d\chi/du_{zz}\sim(T+\Theta)^{2}(d\Theta/du_{zz})$. If we assume that the
derivative $d\Theta/du_{zz}$ does not depend on temperature, then the field
$h(T)$ should change as $\chi^{2}(T)$. In Fig. 8 for $T>T_{N}$ this dependence
is also shown for comparison. As one can see, the behavior of $h(T)$ is
noticeably different from the expected one.

The behavior of the MnF2 susceptibility under the elastic deformation was
studied by a direct method in experiments on hydrostatic compression [10]. It
was also estimated from the results of the study of the magnetostriction [11].
Comparison of the results of this work with the literature data is shown in
Fig. 9. In this figure, in semi-logarithmic coordinates, the behavior at
$T>T_{N}$ of the dependences $\chi(T)$, $\chi^{2}(T)$, $d\chi/du$ [10],
$d\chi/du$ [11], and our data for $h(T)$ (from Fig.8) are shown.

Figure 9: Comparative analysis of temperature dependences of various
thermodynamic functions. Curve 1 is $\chi(T)$ , curve 2 is $\chi^{2}(T)$ ,
curve 3 is $d\chi(T)/du$ [11], curve 4 is $d\chi(T)/du$ [10], curve 5 is the
$h(T)$ response to $u_{zz}$ (Fig.8), curve 6 is $m_{H}$ response to $u_{xz}$
(Fig. 4).

We emphasize that the results presented in Fig. 9 display relative temperature
changes in the corresponding dependences. Their mutual relations are chosen to
provide convenience of comparison. It can be seen that the results of [10, 11]
reach the expected proportionality $\chi^{2}(T)$ only at $T\geq 150$ K. It
seems that the dependence $h(T)$ also tends to reach the expected behavior,
although for larger certainty the temperature interval should be extended.
Below 150 K, deviations from the expected dependence correspond to a more
pronounced effect of deformation on pair correlations. It is especially
significant in our measurements. Perhaps this is due to the frequency
dependence of the correlation corrections.

It is interesting to note that the temperature change $h(T)$ in the case under
discussion is almost perfectly approximated by the power law dependence
$h(T)\propto(T+\Theta)^{-3.5}$ (see Fig. 8). The deviation of the
approximation from the experiment in the interval 70 K $\leq T\leq 200$ K does
not exceed 5%. It is not clear whether there is any hidden physical meaning of
this behavior, or that coincidence is accidental.

In Fig. 9, the dependence of $h(T)$ measured in the paramagnetic phase under a
shear deformation is also displayed (see the data of Fig. 4). Its behavior
seems to be proportional to $\chi^{2}(T)$ once $T$ exceeds $T_{N}$. One can
think that the shear deformation, which does not change, in the first
approximation, interatomic distances, has a little effect on pair
correlations.

##  IV Conclusion

In summary, the main result is that the acoustic-electrical transformation can
be considered as highly informative method of the study of various mechanisms
of magneto-elastic interaction in magnetoactive media. In particular, the
manifestations of piezomagnetism in MnF2 were studied in detail, the
anisotropy of the effect and its temperature behavior were determined, and a
phenomenological interpretation of the observed dependences was given.

The novel for an antiferromagnet phenomenon, namely, the occurrence of new
(off-diagonal) components of the magnetic susceptibility tensor in the field
of displacement gradients, has been discovered. The form of the term of the
thermodynamic potential responsible for the appearance of such an effect,
which is proportional to the square of the antiferromagnetic order parameter,
is experimentally established. It is shown that the effect is fully
(qualitatively and quantitatively) explainable by taking into account the low-
angle rotation of the lattice in the transverse acoustic wave.

In the paramagnetic phase, AET measurements made it possible to determine the
deformation contribution to the temperature behavior of the diagonal component
of the magnetic susceptibility tensor, which turned out to be significant in
the region of pair correlations.

## ACKNOWLEDGMENT

G.A.Z. acknowledges the support from Pauli Ukraine Project, funded in the
Wolfgang Pauli Institut Thematic Program ”Mathematics-Magnetism-Materials
(2021/2022)”.

∗∗Present affiliation is Charles University, Prague, Czech Republic

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