# Challenges in extracting nonlinear current-induced phenomena in Ca2RuO4

Giordano Mattoni<EMAIL_ADDRESS>Toyota Riken – Kyoto University
Research Center (TRiKUC), Kyoto 606-8501, Japan  Department of Physics,
Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan  Kazumi
Fukushima  Department of Physics, Graduate School of Science, Kyoto
University, Kyoto 606-8502, Japan  Shingo Yonezawa  Department of Physics,
Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan
Department of Electronic Science and Engineering, Graduate School of
Engineering, Kyoto University, Kyoto 615-8510, Japan  Fumihiko Nakamura
Department of Education and Creation Engineering, Kurume Institute of
Technology, Fukuoka 830-0052, Japan  Yoshiteru Maeno  Toyota Riken – Kyoto
University Research Center (TRiKUC), Kyoto 606-8501, Japan  Department of
Physics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan

###### Abstract

An appealing direction to change the properties of strongly correlated
materials is to induce non-equilibrium steady states by the application of a
direct current. While access to these novel states is of high scientific
interest, Joule heating due to current flow often constitutes a hurdle to
identify non-thermal effects. The biggest challenge usually resides in
measuring accurately the temperature of a sample subjected to direct current,
and to use probes that give direct information of the material. In this work,
we exploit the simultaneous measurement of electrical transport and
magnetisation to probe non-equilibrium steady states in Ca2RuO4. In order to
reveal non-thermal current-induced effects, we employ a simple model of Joule
self-heating to remove the effects of heating and discuss the importance of
temperature inhomogeneity within the sample. Our approach provides a solid
basis for investigating current-induced phenomena in highly resistive
materials.

calcium ruthenate, single crystals, current-induced phenomena, nonequilibrium
steady states, nonlinear transport, Joule self-heating

##  I Introduction

Direct electric current is a powerful control parameter capable of inducing
non-equilibrium steady states in strongly correlated materials [1, 2]. Non-
equilibrium conditions can be a gateway to peculiar physics that cannot be
accessed by other means [3, 4, 5]. Despite the large scientific interest,
experiments with direct current always entail some extent of Joule heating,
which becomes particularly relevant when dealing with highly resistive
materials.

The strongly correlated oxide Ca2RuO4 is a Mott insulator at room temperature
that presents a metal–insulator transition (MIT) at about
$360\text{\,}\mathrm{K}$ and a magnetic transition towards an
antiferromagnetic state below $108\text{\,}\mathrm{K}$ [6, 7]. The MIT is
characterised by a strong coupling between Ca2RuO4 lattice and its electronic
structure [8]. Several reports showed that the MIT can be triggered by the
flow of electric current, with the suppression of its Mott gap [9, 10]. The
occurrence of this current-induced transition has been confirmed by means of
neutron scattering [11, 12], electrical transport [13, 14], and also ARPES
[15, 16]. Other reports showed the existence of a rich nanoscale structure at
the phase boundary between metallic and insulating regions, possibly induced
by the applied voltage rather than the flowing current itself [17, 18, 19].
These effects motivate a deeper investigation on the meachanism of
electrically-induced states in Ca2RuO4.

Measurements with applied current on Ca2RuO4 are particularly challenging
because they involve the application of considerable electrical currents
which, at low temperature, lead to the insurgence of a large Joule heating
because of the high resistivity of the material. Heating makes it difficult to
evaluate the sample temperature accurately, and in the case of magnetic
measurements can also introduce spurious background signals [20]. Various
efforts have been taken to measure accurately Ca2RuO4 temperature under
applied current: some groups employed thermal imaging [10, 21], Fursich et al.
looked at the shift of the Raman lines [22], Okazaki et al. used a gold
nanoparticle to locally assess the sample temperature [23], Avallone et al.
employed a nanoscale thermometer patterned right above a tiny Ca2RuO4 crystal
[24].

In this work, we employ as a “double probe” magnetic and electrical
measurements performed simultaneously to investigate the current-induced state
of Ca2RuO4. To do so, we design a special sample holder and thermometer that
maximises sample cooling and measures the sample temperature as accurately as
possible. In the presence of non-heating current-induced effects, we expect
changes in magnetisation and resistance to have a different dependence on
current. We uncover extensive sample heating that can be explained by the
coexistence of a homogeneous and inhomogeneous temperature increase. We
discuss how inhomogeneous temperature profiles may explain remaining
nonlinearities and provide a solid basis for probing current-induced effects.

##  II Experimental details

We performed measurements in a magnetic property measurement system (MPMS)
from Quantum Design. Simultaneous measurements of the sample resistance $R$
and the magnetic moment $m$ were enabled by the custom-made sample holder
described in Figs. 2a, 2b and 2c. The sample holder ensures a large sample
cooling, crucial for measurements with applied current, thanks to a large
copper strip (dimensions 200x6x0.4) that constitutes its main body. The sample
holder fits into a standard MPMS plastic straw (diameter
$6\text{\,}\mathrm{mm}$) that is used for magnetic measurements. Before sample
mounting, the in-plane crystalline axes of Ca2RuO4 were determined by separate
magnetic measurements (Fig. S1). A vertical field
$\mu_{0}H=$1\text{\,}\mathrm{T}$$ was applied along the orthorhombic $a$ axis
of Ca2RuO4, while electrical current is applied along $b$. This configuration
minimises eddy currents in the sample holder and the magnetic background.
Cooling ramps were performed by changing at a rate of
$2\text{\,}\mathrm{K}\text{\,}{\mathrm{min}}^{-1}$ the sample-space
temperature $T_{\mathrm{MPMS}}$, measured on the outer surface of the copper
jacket around the sample space (Fig. 2a). Helium exchange gas (approximately
$1\text{\,}\mathrm{mbar}$ at room temperature) ensures a thermal connection
between the sample holder and $T_{\mathrm{MPMS}}$.

Electrical contact to the sample was provided by Au wires (diameter
$50\text{\,}\mathrm{\SIUnitSymbolMicro m}$) and Ag paint (DuPont 4929N with
diethyl succinate, cured at room temperature), which were then linked to
thicker copper wires (diameter $0.2\text{\,}\mathrm{mm}$). This configuration
leads to a low contact resistance, typically much below
$100\text{\,}\mathrm{\SIUnitSymbolOhm}$ at room temperature (Fig. S2). A thin
sheet of cigarette paper was used to eletrically insulate the sample from the
copper holder, and GE Varnish 7031 was used to glue the elements together and
ensure a good thermal contact. Electrical measurements were performed by
sourcing a current to the sample with a Keysight B2912A (voltage compliance
$210\text{\,}\mathrm{V}$) and measuring the 2- or 4-probe voltage with a
Keithley Electrometer 6514 (input impedance
$>$200\text{\,}\mathrm{T\SIUnitSymbolOhm}$$). The experiments were performed
on several Ca2RuO4 crystals and we here report two representative examples.
Sample #1 (CRO16-4, size 2.8x1x0.6, mass $7.98\text{\,}\mathrm{mg}$), measured
in two-probe configuration, is presented in Fig. 1. Sample #2 (CR19-17, size
2x1x0.1, mass $2.51\text{\,}\mathrm{mg}$), measured in four-probe
configuration with an additional thermometer connected directly to its top-
surface, is presented in Fig. 2. Further technical details are given in the
following sections.

##  III Results and discussion

###  III.1 Simultaneous electrical-transport and magnetic measurements

(a)

(b)

(c)

(d)

(e)

(f)

Figure 1: Simultaneous magnetic and electrical transport measurements with
constant current. LABEL:sub@fig:Res Experimental sample resistance as a
function of temperature for different applied currents. The resistance is
measured in a two-probe configuration on Sample #1. LABEL:sub@fig:IVs
Voltage–current characteristics for selected temperatures as indicated by the
vertical linecuts in LABEL:sub@fig:Res. The voltage curves are normalised to
their maximum value $V_{\mathrm{max}}$. LABEL:sub@fig:Mag Sample magnetic
moment measured simultaneously with the resistance. For magnetic measurements,
the data for $I=0$ is also included. LABEL:sub@fig:ResSim Simulated data of
sample resistance, LABEL:sub@fig:IVsSim voltage–current characteristics, and
LABEL:sub@fig:MagSim magnetic moment calculated by using the self-heating
model of Eq. 1 with
$\alpha=$180\text{\,}\mathrm{K}\text{\,}{\mathrm{W}}^{-1}$$.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

Figure 2: Sample self-heating evaluated with a top thermometer.
LABEL:sub@fig:Setup_MPMS Schematics of the sample and holder inside the MPMS
and location of the system thermometer $T_{\mathrm{MPMS}}$.
LABEL:sub@fig:Setup_copper Schematic drawing and corresponding photograph of
the sample holder with kapton wire spacer and LABEL:sub@fig:Setup_sample
detail of the sample itself. The spatial directions are indicated with Ca2RuO4
orthorhombic crystalline axes $a$, $b$, and $c$. LABEL:sub@fig:Ttop Top-
thermometer temperature as a function of system temperature for several
currents and LABEL:sub@fig:dTtop corresponding increase of sample temperature
due to current heating. LABEL:sub@fig:P2p Electrical power supplied to the
sample by the flowing current and LABEL:sub@fig:alpha experimental values of
$\alpha(T)$ if the temperature increase is described by the Joule self-heating
model of Eq. 1.

In Fig. 1a, we show the resistance of Sample #1 in a 2-probe configuration
$R_{\mathrm{2p}}$. For the smallest current
$I=$1\text{\,}\mathrm{\SIUnitSymbolMicro A}$$, the resistance increases
several orders of magnitude upon lowering temperature and it goes beyond the
measurement limit at about $150\text{\,}\mathrm{K}$, consistent with previous
reports of high-quality Ca2RuO4 crystals [6, 25]. For larger values of current
up to $I=$10\text{\,}\mathrm{mA}$$, the resistance curves gradually become
lower, in accordance with other reports [10, 13, 14, 26]. By taking vertical
linecuts in Fig. 1a (dotted lines), we extract voltage–current characteristics
at three fixed values of $T_{\mathrm{MPMS}}$. The resulting curves in Fig. 1b
show a non-linear behaviour that becomes more pronounced at lower
temperatures. Similar trends have been previously reported for voltage–current
characteristics of Ca2RuO4 [27, 24]. We note that the location of the region
of negative differential resistance, after the voltage peak, is strongly
dependent on the thermal couplings and the sample temperature, and it has been
suggested to be the fingerprint of Ca2RuO4 metal–insulator transition [10, 28,
22, 17].

As an important additional probe for Ca2RuO4 properties under applied current,
we also measure the sample magnetic moment $m$ simultaneously with the
resistance. We present in Fig. 1c the magnetic moment for $I=0$ which, upon
decreasing temperature, shows a gradual increase, a peak at about
$110\text{\,}\mathrm{K}$ indicating Ca2RuO4 antiferromagnetic transition, and
a final saturating trend, consistent with literature [29]. Note that we
intentionally report the measured magnetic moment $m$ instead of the sample
magnetisation $M=m_{\mathrm{sample}}/V_{\mathrm{sample}}$ because $m$ may
contain additional background signals as discussed in the following section.
With applied current $I>0$, the antiferromagnetic transition disappears and
the magnetic moment decreases. As for the resistance, the measurements are
interrupted whenever it becomes impossible to source the chosen current to the
sample.

In order to identify non-thermal current-induced effects, we attempt to
estimate and subtract the Joule self-heating. For this purpose, we consider a
simple model in which the sample is at an effective temperature
$T_{\mathrm{MPMS}}+\Delta T$, and the $\Delta T$ is determined solely by the
electrical power $P=I^{2}R$ as

$\Delta T(T_{\mathrm{MPMS}})=\alpha P(T_{\mathrm{MPMS}}+\Delta T)=\alpha
I^{2}R_{0}(T_{\mathrm{MPMS}}+\Delta T),$ (1)

where $\alpha$ is a constant expressing thermal resistance between the sample
and the cryostat and $R_{0}$ is the sample resistance with close-to-zero
current. By using as only input
$R_{0}(T)=R_{\mathrm{2p}}(T,I{=}$1\text{\,}\mathrm{\SIUnitSymbolMicro A}$)$,
we simulate the data in Figs. 1d and 1e by solving Eq. 1. We also simulate the
magnetisation data in Fig. 1f using $m_{0}=m(I{=}0)$. The data is produced by
adjusting the value of the phenomenological constant
$\alpha=$180\text{\,}\mathrm{K}\text{\,}{\mathrm{W}}^{-1}$$, for which we find
a striking qualitative agreement between experiment and simulation. The model
correctly captures the current-induced reduction of both $R_{\mathrm{2p}}$ and
$m$, and also the non-linear trend of the current–voltage characteristics.
This indicates that a significant portion of the observed behaviour can be
explained by Joule self-heating of a temperature-homogeneous insulating phase,
underlying the importance of developing a special technique to accurately
measure the sample temperature.

###  III.2 Joule self-heating of Ca2RuO4

In order to accurately assess the sample temperature, we perform another set
of measurements on a similar Ca2RuO4 crystal (Sample #2) connected in a
4-probe configuration (full data in Fig. S3). Direct comparison of the two-
and four-probe resistance indicates that the contact resistance is negligible
(Fig. S2). A thermometer glued by GE Varnish 7031 directly on the sample top
surface is used to measure $T_{\mathrm{top}}$ as shown in Fig. 2a. For this
purpose, we chose a platinum resistive sensor (Heraeus Pt1000, SMD0603) whose
substrate was thinned down to about $80\text{\,}\mathrm{\SIUnitSymbolMicro m}$
thickness by mechanical polishing in order to enhance its proximity to the
sample. We also mechanically removed the sensor contact pads, which contain
magnetic materials such as nickel, in order to bring its magnetic signal to a
negligible value (Fig. S4). We provided electrical contact to the sensor (Fig.
2c) by using Ag paint and two thin Au wires (diameter
$18\text{\,}\mathrm{\SIUnitSymbolMicro m}$, length $0.5\text{\,}\mathrm{cm}$)
which are then connected in a four-probe configuration to a set of phosphor
bronze wires that have low thermal conductivity. As shown in Fig. 2b, we
minimise heat escape from the temperature sensor by using kapton spacers that
keep its wires physically separated from the highly conductive sample holder.
Because the magnetic moment of Sample #2 is rather small, we perform magnetic
measurements with both a positive and negative applied current in order to
identify the magnetic signal generated by the electrical leads (detailed
description in Fig. S5). This background signal is subtracted to extract the
value of $m$ for this sample shown in Fig. 3.

In Fig. 2d, we observe that $T_{\mathrm{top}}$ significantly deviates from
$T_{\mathrm{MPMS}}$ (dotted line), indicating that the sample is substantially
heated by the flowing current, especially at lower temperatures. We quantify
such sample heating in Fig. 2e as $\Delta
T_{\mathrm{top}}=T_{\mathrm{top}}-T_{\mathrm{MPMS}}$ and also calculate in
Fig. 2f the electrical power dissipated by the flowing current as
$P_{\mathrm{2p}}=I^{2}R_{\mathrm{2p}}$. Since the power dissipation through
the low-resistance copper leads is negligible, most of $P_{\mathrm{2p}}$ is
dissipated through the sample and at its electrical contacts. To test whether
the dissipated power determines a Joule self-heating in accordance with the
model of Eq. 1, we calculate the experimental $\alpha_{\mathrm{top}}(T)=\Delta
T_{\mathrm{top}}/P_{\mathrm{2p}}$ in Fig. 2g. We find values
$\alpha_{\mathrm{top}}=$150250\text{\,}\mathrm{K}\text{\,}{\mathrm{W}}^{-1}$$
which are consistent with the value
$\alpha=$180\text{\,}\mathrm{K}\text{\,}{\mathrm{W}}^{-1}$$ used in the
simulation of Fig. 1, thus supporting our model choice. The increase of
$\alpha_{\mathrm{top}}$ at lower temperatures indicates a worse sample
cooling, possibly due to a decreased thermal conductivity of the components or
to a lower pressure of the He exchange gas.

###  III.3 Universal relationship between magnetic moment and resistance

(a)

(b)

(c)

(d)

Figure 3: Magnetic moment as internal thermometer. LABEL:sub@fig:Rmu
Relationship between the resistance and magnetic moment of Sample #2 for
different temperatures and currents. The colour indicates different applied
currents, while the temperature dependence is implicit. LABEL:sub@fig:dTm
Sample heating estimated from the magnetic moment $T_{m}$, the top thermometer
$T_{\mathrm{top}}$, and LABEL:sub@fig:dTm_ratio their ratio.
LABEL:sub@fig:IVs_different Voltage–current characteristics at a fixed sample
temperature evaluated with different probes. As the probe of sample
temperature becomes more accurate, the curves tend to a more ohmic behaviour.

To reveal possible non-thermal current-induced effects that would induce
different changes of resistance and magnetic moment, we plot in Fig. 3a the
data of the 4-probe resistance $R_{\mathrm{4p}}$ vs $m$ for Sample #2. The
data mostly collapses on the same curve, suggesting a universal correlation
between $R_{\mathrm{4p}}$ and $m$, irrespective of the applied current value.
This is a surprising result because if the band structure of Ca2RuO4 is
changed by the flowing current, there is no expectation that both
$R_{\mathrm{4p}}$ and $m$ change in the same manner. Despite the extensive
overlap of the curves, some deviation is observed in the high-resistance high-
magnetisation region, which corresponds to lower temperatures.

To investigate these deviations, we tentatively assume that $m$ does not
depend on current but only on temperature and we use the experimental data of
$m(T_{\mathrm{MPMS}})$ to calculate the sample “magnetic temperature” $T_{m}$.
Under our assumption, $T_{m}$ provides an internal probe of sample
temperature, which we use to estimate the sample heating as $\Delta
T_{m}=T_{m}-T_{\mathrm{MPMS}}$. This heating is systematically larger than
what is measured by $T_{\mathrm{top}}$, and their ratio in Fig. 3c shows that
$\Delta T_{m}$ is up to $40\text{\,}\mathrm{\char 37\relax}$ larger than
$\Delta T_{\mathrm{top}}$, implying that the top thermometer measures a value
which is significantly lower than the sample average temperature.

We show in Fig. 3d the voltage–current characteristics for Sample #2 extracted
at a constant sample temperature of $270\text{\,}\mathrm{K}$ estimated by
different temperature probes. Changing the temperature probe from
$T_{\mathrm{MPMS}}$, to $T_{\mathrm{top}}$, to $T_{m}$, the non-linear curves
become more and more straight, and approach the ohmic behaviour. This
indicates that a large component of the observed non-linearity can be ascribed
to an underestimation of the average sample temperature caused by the Joule
self heating. Non-thermal current-induced effects, if present, should be
investigated after removing this large heating component. We note that some
deviation from the ohmic behaviour persists even when using $T_{m}$, which may
indicate the presence of non-thermal current-induced effects that will be
further investigated in the following section.

###  III.4 Possible sample temperature inhomogeneity

(a)

(b)

(c)

(d)

(e)

(f)

Figure 4: Possible spatial temperature inhomogeneity. LABEL:sub@fig:Series
Schematic of a simplified sample temperature inhomogeneity with vertical
(i.e., in series) and LABEL:sub@fig:Parallel horizontal (i.e., in parallel)
boundaries. In both cases, we set the temperature of the colder regions to
$T_{\mathrm{cold}}=T_{\mathrm{top}}$. LABEL:sub@fig:perc_Series Simulated hot
volume fraction and LABEL:sub@fig:T_Series temperature of the hotter region
for the series configuration, and LABEL:sub@fig:perc_Parallel,
LABEL:sub@fig:T_Parallel for the parallel configuration. Also here, all the
temperature increases $\Delta T$ are referred to $T_{\mathrm{MPMS}}$.

We now discuss whether residual deviations of the $R$ vs $m$ curves can be
described by possible inhomogeneities of the sample temperature. We note that
the magnetic moment is a bulk measurement averaged over the entire sample
volume, while the resistance is dominated by the most-conductive electrical
channel. To account for possible inhomogeneities, we consider two simplified
model scenarios in which the sample temperature presents hotter regions with
vertical boundaries, which we call in series (Fig. 4a), or horizontal, which
we call in parallel (Fig. 4d). The first scenario can be related to excess
sample heating in proximity of the current leads, possibly due to contact
resistance. The second scenario can be related to an excess heating in the
internal part of the sample, that is further away from the colder bottom and
top surfaces that are in contact with the sample holder and exchange gas,
respectively. The formulation of the following analysis allows the location
and extent of the hotter and colder regions to be different from the one in
the schematic drawings, as long as the directionality is respected (for
example, the hotter regions could be multiple or spatially asymmetric).

In this simplified model, we consider two sharply defined regions at a hotter
($T_{\mathrm{hot}}$) and colder ($T_{\mathrm{cold}}$) temperature whose extent
is defined by the volume fraction $x_{\mathrm{hot}}$. For both scenarios, the
sample magnetisation is given by the volume average

$m(T)=x_{\mathrm{hot}}m_{0}(T_{\mathrm{hot}})+(1-x_{\mathrm{hot}})m_{0}(T_{\mathrm{cold}}),$
(2)

where $m_{0}$ is the magnetic moment at zero current. The sample resistance,
instead, is calculated differently in the two scenarios as

$\displaystyle R^{\mathrm{series}}(T)=$ $\displaystyle
x_{\mathrm{hot}}R_{0}(T_{\mathrm{hot}})+(1-x_{\mathrm{hot}})R_{0}(T_{\mathrm{cold}}),$
(3) $\displaystyle\frac{1}{R^{\mathrm{parallel}}(T)}=$
$\displaystyle\frac{x_{\mathrm{hot}}}{R_{0}(T_{\mathrm{hot}})}+\frac{(1-x_{\mathrm{hot}})}{R_{0}(T_{\mathrm{cold}})}.$

Following the discussion of the previous section, we expect the sample average
temperature to be on average larger than what is measured by
$T_{\mathrm{top}}$. We thus tentatively set
$T_{\mathrm{cold}}=T_{\mathrm{top}}$ and use the experimental data of $m(T)$
and $R(T)$ as inputs to solve the coupled set of equations Eqs. 2 and 3 to
find numerical solutions for $x_{\mathrm{hot}}$ and $T_{\mathrm{hot}}$.

In the series scenario of Figs. 4b and 4c, the values of $x_{\mathrm{hot}}$
are not well defined in the higher-temperature region because the sample self-
heating is small (i.e., $T_{\mathrm{hot}}\sim T_{\mathrm{cold}}$). In this
region, the universality of resistance vs magnetisation is satisfied (Fig.
3a). At lower temperatures, $x_{\mathrm{hot}}$ shoots up to
$100\text{\,}\mathrm{\char 37\relax}$, indicating that a large portion of the
sample is at $T_{\mathrm{hot}}$. For larger applied currents, no numerical
solution is found below about $200\text{\,}\mathrm{K}$, indicating that the
series temperature inhomogeneity cannot explain the experimental behaviour.
The formation of a hotter regions with vertical boundary is thus unlikely,
indicating that sample self-heating at the current leads is negligible,
consistent with our estimate of a low contact resistance (Fig. S2).

In the parallel scenario of Fig. 4e, a numerical solution is found for all
temperatures and currents. At low temperature, $x_{\mathrm{hot}}$ approaches a
value of about $80\text{\,}\mathrm{\char 37\relax}$ that is consistent for all
experimental currents, indicating that the coexistence of a broad hotter
region and a thin colder region is a possible description of the experimental
behaviour. From Fig. 4f, we note that the temperature difference between the
hotter and colder regions $\Delta\widetilde{T}=\Delta T_{\mathrm{hot}}-\Delta
T_{\mathrm{cold}}$ is of a few Kelvin at room temperature (electrical power
$P_{\mathrm{2p}}\sim$50\text{\,}\mathrm{mW}$$ from Fig. 2f), while it grows to
about $\Delta\widetilde{T}\sim$30\text{\,}\mathrm{K}$$
($P_{\mathrm{2p}}\sim$500\text{\,}\mathrm{mW}$$) at low temperature.
Considering that thermal conduction within the sample is given by
$P=\kappa\dfrac{A}{t}\Delta\widetilde{T}$, where the room-temperature
conductivity is
$\kappa_{c,\ce{Ca2RuO4}}=$1.8\text{\,}\mathrm{W}\text{\,}{\mathrm{m}}^{-1}\text{\,}{\mathrm{K}}^{-1}$$
[30] and $A/t$ is the sample cross-sectional area over its thickness, we
estimate that at room temperature
$\Delta\widetilde{T}\sim$1.4\text{\,}\mathrm{K}$$ along the $c$ direction of
Ca2RuO4. At lower temperatures, this vertical temperature inhomogeneity grows
up to a factor 10 due to the increasing electrical power, and may be further
enhanced by the decreasing thermal conductivity of Ca2RuO4. The presence of
hot and cold regions in parallel is thus a reasonable possibility, and their
extent may depend on sample size, thermal couplings, and cooling conditions.

##  IV Conclusions

We have investigated current-induced phenomena in Ca2RuO4 through a wide
temperature range by means of simultaneous magnetic and electrical
measurements. Despite the purpose-made setup, the sample experienced a large
Joule self-heating that we quantified by means of a simple model and a
thermometer in direct contact with the sample. While most deviations from
ohmic behaviour can be explained by homogeneous sample heating, additional
effects are present. Temperature inhomogeneity is intrinsic to a current-
induced steady state where the continuous heat input is balanced by the heat
escape. Therefore, we introduced a model of inhomogeneous sample heating which
explained most of these additional non-linearities as due to a temperature
gradient in the direction perpendicular to the current flow. This analysis
allowed us to identify that a combination of homogeneous and inhomogeneous
current-induced heating are responsible for the observed behaviour. Non-
thermal current-induced effects in Ca2RuO4, if present, are below the
detection limit of our experiment. Our results pose a solid basis for
investigating current-induced phenomena in insulators, where large current
heating is unavoidable.

## Acknowledgements

The authors thank N. Manca for valuable comments on the manuscript and H.
Michishita for machining the copper components. This work was supported by
JSPS Grant-in-Aids KAKENHI Nos. JP26247060, JP15H05852, JP15K21717,
JP17H06136, and JP18K04715, as well as by JSPS Core-to-Core program. G.M.
acknowledges support from the Dutch Research Council (NWO) through a Rubicon
grant number 019.183EN.031, and support from the Kyoto University Foundation.

## References

  * Kumai  _et al._ [1999] R. Kumai, Y. Okimoto, and Y. Tokura, Current-induced insulator-metal transition and pattern formation in an organic charge-transfer complex, Science 284, 1645 (1999).
  * Myers  _et al._ [1999] E. Myers, D. Ralph, J. Katine, R. Louie, and R. Buhrman, Current-induced switching of domains in magnetic multilayer devices, Science 285, 867 (1999).
  * Fausti  _et al._ [2011] D. Fausti, R. Tobey, N. Dean, S. Kaiser, A. Dienst, M. C. Hoffmann, S. Pyon, T. Takayama, H. Takagi, and A. Cavalleri, Light-induced superconductivity in a stripe-ordered cuprate, Science 331, 189 (2011).
  * Stojchevska  _et al._ [2014] L. Stojchevska, I. Vaskivskyi, T. Mertelj, P. Kusar, D. Svetin, S. Brazovskii, and D. Mihailovic, Ultrafast switching to a stable hidden quantum state in an electronic crystal, Science 344, 177 (2014).
  * Mattoni  _et al._ [2018] G. Mattoni, N. Manca, M. Hadjimichael, P. Zubko, A. van der Torren, C. Yin, S. Catalano, M. Gibert, F. Maccherozzi, Y. Liu, S. S. Dhesi, and A. D. Caviglia, Light control of the nanoscale phase separation in heteroepitaxial nickelates, Physical Review Materials 2, 085002 (2018).
  * Nakatsuji  _et al._ [1997] S. Nakatsuji, S.-i. Ikeda, and Y. Maeno, Ca2RuO4: New Mott insulators of layered ruthenate, Journal of the Physical Society of Japan 66, 1868 (1997).
  * Cao  _et al._ [1999] G. Cao, C. Alexander, S. McCall, J. Crow, and R. Guertin, From antiferromagnetic insulator to ferromagnetic metal: A brief review of the layered ruthenates, Materials Science and Engineering: B 63, 76 (1999).
  * Han and Millis [2018] Q. Han and A. Millis, Lattice energetics and correlation-driven metal-insulator transitions: The case of Ca2RuO4, Physical Review Letters 121, 067601 (2018).
  * Nakamura  _et al._ [2013] F. Nakamura, M. Sakaki, Y. Yamanaka, S. Tamaru, T. Suzuki, and Y. Maeno, Electric-field-induced metal maintained by current of the Mott insulator Ca2RuO4, Scientific Reports 3, 2536 (2013).
  * Okazaki  _et al._ [2013] R. Okazaki, Y. Nishina, Y. Yasui, F. Nakamura, T. Suzuki, and I. Terasaki, Current-induced gap suppression in the Mott insulator Ca2RuO4, Journal of the Physical Society of Japan 82, 103702 (2013).
  * Bertinshaw  _et al._ [2019] J. Bertinshaw, N. Gurung, P. Jorba, H. Liu, M. Schmid, D. Mantadakis, M. Daghofer, M. Krautloher, A. Jain, G. Ryu, _et al._ , Unique crystal structure of Ca2RuO4 in the current stabilized semimetallic state, Physical Review Letters 123, 137204 (2019).
  * Jenni  _et al._ [2020] K. Jenni, F. Wirth, K. Dietrich, L. Berger, Y. Sidis, S. Kunkemöller, C. P. Grams, D. I. Khomskii, J. Hemberger, and M. Braden, Evidence for current-induced phase coexistence in Ca2RuO4 and its influence on magnetic order, Phys. Rev. Mater. 4, 085001 (2020).
  * Cirillo  _et al._ [2019] C. Cirillo, V. Granata, G. Avallone, R. Fittipaldi, C. Attanasio, A. Avella, and A. Vecchione, Emergence of a metallic metastable phase induced by electrical current in Ca2RuO4, Physical Review B 100, 235142 (2019).
  * Zhao  _et al._ [2019] H. Zhao, B. Hu, F. Ye, C. Hoffmann, I. Kimchi, and G. Cao, Nonequilibrium orbital transitions via applied electrical current in calcium ruthenates, Physical Review B 100, 241104 (2019).
  * Ootsuki  _et al._ [2022] D. Ootsuki, A. Hishikawa, T. Ishida, D. Shibata, Y. Takasuka, M. Kitamura, K. Horiba, Y. Takagi, A. Yasui, C. Sow, _et al._ , Metallic surface state in the bulk insulating phase of Ca_2-xSr_xRuO4 ($x=0.06$) studied by photoemission spectroscopy, Journal of the Physical Society of Japan 91, 114704 (2022).
  * Curcio  _et al._ [2023] D. Curcio, C. E. Sanders, A. Chikina, H. E. Lund, M. Bianchi, V. Granata, M. Cannavacciuolo, G. Cuono, C. Autieri, F. Forte, G. Avallone, A. Romano, M. Cuoco, P. Dudin, J. Avila, C. Polley, T. Balasubramanian, R. Fittipaldi, A. Vecchione, and P. Hofmann, Current-driven insulator-to-metal transition without Mott breakdown in Ca2RuO4, Phys. Rev. B 108, L161105 (2023).
  * Zhang  _et al._ [2019] J. Zhang, A. S. McLeod, Q. Han, X. Chen, H. A. Bechtel, Z. Yao, S. G. Corder, T. Ciavatti, T. H. Tao, M. Aronson, _et al._ , Nano-resolved current-induced insulator-metal transition in the Mott insulator Ca2RuO4, Physical Review X 9, 011032 (2019).
  * Vitalone  _et al._ [2022] R. A. Vitalone, A. J. Sternbach, B. A. Foutty, A. S. McLeod, C. Sow, D. Golez, F. Nakamura, Y. Maeno, A. N. Pasupathy, A. Georges, _et al._ , Nanoscale femtosecond dynamics of Mott insulator (Ca_0. 99Sr_0. 01)2RuO4, Nano Letters 22, 5689 (2022).
  * Gauquelin  _et al._ [2023] N. Gauquelin, F. Forte, D. Jannis, R. Fittipaldi, C. Autieri, G. Cuono, V. Granata, M. Lettieri, C. Noce, F. Miletto-Granozio, A. Vecchione, J. Verbeeck, and M. Cuoco, Pattern formation by electric-field quench in a Mott crystal, Nano Letters 23, 7782 (2023), pMID: 37200109\.
  * Mattoni  _et al._ [2020a] G. Mattoni, S. Yonezawa, and Y. Maeno, Diamagnetic-like response from localized heating of a paramagnetic material, Applied Physics Letters 116, 172405 (2020a).
  * Mattoni  _et al._ [2020b] G. Mattoni, S. Yonezawa, F. Nakamura, and Y. Maeno, Role of local temperature in the current-driven metal–insulator transition of Ca2RuO4, Physical Review Materials 4, 114414 (2020b).
  * Fürsich  _et al._ [2019] K. Fürsich, J. Bertinshaw, P. Butler, M. Krautloher, M. Minola, and B. Keimer, Raman scattering from current-stabilized nonequilibrium phases in Ca2RuO4, Physical Review B 100, 081101 (2019).
  * Okazaki  _et al._ [2020] R. Okazaki, K. Kobayashi, R. Kumai, H. Nakao, Y. Murakami, F. Nakamura, H. Taniguchi, and I. Terasaki, Current-induced giant lattice deformation in the Mott insulator Ca2RuO4, Journal of the Physical Society of Japan 89, 044710 (2020).
  * Avallone  _et al._ [2021] G. Avallone, R. Fermin, K. Lahabi, V. Granata, R. Fittipaldi, C. Cirillo, C. Attanasio, A. Vecchione, and J. Aarts, Universal size-dependent nonlinear charge transport in single crystals of the Mott insulator Ca2RuO4, npj Quantum Materials 6, 91 (2021).
  * Nakatsuji and Maeno [2000] S. Nakatsuji and Y. Maeno, Quasi-two-dimensional Mott transition system Ca_2-xSr_xRuO4, Physical Review Letters 84, 2666 (2000).
  * Terasaki  _et al._ [2020] I. Terasaki, I. Sano, K. Toda, S. Kawasaki, A. Nakano, H. Taniguchi, H. J. Cho, H. Ohta, and F. Nakamura, Non-equilibrium steady state in the Mott insulator Ca2RuO4, Journal of the Physical Society of Japan 89, 093707 (2020).
  * Tsurumaki-Fukuchi  _et al._ [2020] A. Tsurumaki-Fukuchi, K. Tsubaki, T. Katase, T. Kamiya, M. Arita, and Y. Takahashi, Stable and tunable current-induced phase transition in epitaxial thin films of Ca2RuO4, ACS applied materials & interfaces 12, 28368 (2020).
  * Sakaki  _et al._ [2013] M. Sakaki, N. Nakajima, F. Nakamura, Y. Tezuka, and T. Suzuki, Electric-field-induced insulator–metal transition in Ca2RuO4 probed by X-ray absorption and emission spectroscopy, Journal of the Physical Society of Japan 82, 093707 (2013).
  * Braden  _et al._ [1998] M. Braden, G. André, S. Nakatsuji, and Y. Maeno, Crystal and magnetic structure of Ca2RuO4: Magnetoelastic coupling and the metal-insulator transition, Physical Review B 58, 847 (1998).
  * Kawasaki  _et al._ [2021] S. Kawasaki, A. Nakano, H. Taniguchi, H. J. Cho, H. Ohta, F. Nakamura, and I. Terasaki, Thermal diffusivity of the Mott insulator Ca2RuO4 in a non-equilibrium steady state, Journal of the Physical Society of Japan 90, 063601 (2021).

Supplementary Material

Figure S1:  Determination of in-plane crystalline axes. Before mounting on the
copper sample holder, the magnetic signal of Sample #2 alone is measured in a
standard MPMS straw in two different sample orientations. The absence of the
peak at $110\text{\,}\mathrm{K}$ in the orange curve allows us to identify the
crystalline orthorhombic $b$ axis which is the easy axis for antiferromagnetic
ordering in Ca2RuO4.

(a)

(b)

Figure S2:  Comparison of two- and four-probe resistance for Sample #2.
LABEL:sub@fig:Resistance24p_curves Resistance vs temperature curves measured
with a current of $1\text{\,}\mathrm{\SIUnitSymbolMicro A}$ and
LABEL:sub@fig:Resistance24p_ratio their ratio. At higher temperature, the
ratio $R_{\mathrm{2p}}/R_{\mathrm{4p}}$ is compatible with the geometrical
ratio of the sample width $3.8\text{\,}\mathrm{mm}$ and the inner distance
between the voltage probes $2.8\text{\,}\mathrm{mm}$. This indicates that
$R_{\mathrm{2p}}$ is larger merely due to geometry (i.e., due to a longer
electrical channel), thus indicating that contact resistance is negligible.
The data indicates that the contact resistance is smaller than
$100\text{\,}\mathrm{\SIUnitSymbolOhm}$ at room temperature and it does not
increase significantly at lower temperatures. At lower temperature, instead,
the ratio significantly deviates from the geometrical one. This devtiation
cannot be explained as to be due to contact resistance, which would determine
an increase of $R_{\mathrm{2p}}$, but it is possibly related to the formation
of temperature inhomogeneity within the sample as discussed in Fig. 4 of the
main text.

(a)

(b)

(c)

(d)

Figure S3:  Resistance and magnetic curves as a function of temperature for
Sample #2. LABEL:sub@fig:S2_RvsT Four-probe resistance and
LABEL:sub@fig:S2_mvsT magnetic moment measured for different values of current
similar to what presented for Sample #1 in Fig. 1 of the main text. The data
is plotted as a function of the system thermometer $T_{\mathrm{MPMS}}$.
LABEL:sub@fig:S2_RvsTtop Resistance and LABEL:sub@fig:S2_mvsTtop magnetic
moment replotted as a function of the top thermometer $T_{\mathrm{top}}$. Even
if the effects of current look smaller in the data with $T_{\mathrm{top}}$,
the current-induced reduction of $R_{\mathrm{4p}}$ and $m$ is qualitatively
the same. This data is used for the analysis of Figs. 2, 3 and 4 in the main
text.  Figure S4:  Comparison of magnetic backgrounds. Magnetic signals of
isolated materials measured in the MPMS. (1) Ca2RuO4 Sample #2; (2) Pt1000
sensor after mechanical removal of the electrode coating and (3) same Pt1000
sensor after also thinning the substrate down to about
$80\text{\,}\mathrm{\SIUnitSymbolMicro m}$ by mechanical polishing; (4) A
small cubic copper piece of about $80\text{\,}\mathrm{mg}$. When compared to
Ca2RuO4, the magnetic signal of the polished Pt1000 temperature sensor is
negligible. The copper material has a considerable magnetic moment, but the
signal of the long copper stripe used as sample holder does not appear in the
magnetic measurements performed in this work thanks to its homogeneous shape.
This is because the MPMS employs a SQUID magnetometer that is only sensitive
to spatial variations of magnetic moment. Fits in the range
$130300\text{\,}\mathrm{K}$ show that the magnetic temperature coefficient of
Ca2RuO4 is more than one order of magnitude larger than the other materials.
This indicates that the insurgence of magnetic signals due to local heating of
the background materials, which is proportional to their temperature
coefficient, is minimal in our setup [G. Mattoni et al., APL 116 (2020)].

(a)

(b)

(c)

(d)

Figure S5:  Removal of magnetic signal generated by the current leads. In
order to identify and subtract the magnetic signal coming from the current
leads, each magnetic measurement is repeated with opposite values of current.
LABEL:sub@fig:VSQUID_pm SQUID response measured with a positive and negative
current of $10\text{\,}\mathrm{mA}$. LABEL:sub@fig:VSQUID_symm Symmetric
[$(V_{\mathrm{SQUID}}(I^{+})+V_{\mathrm{SQUID}}(I^{-}))/2$] and asymmetric
[$(V_{\mathrm{SQUID}}(I^{+})-V_{\mathrm{SQUID}}(I^{-}))/2$] components of the
SQUID signal. The latter is amplified by a factor 10 for clarity. Both data
are fitted with the SQUID response function to extract the corresponding
magnetic moment (for details see [G. Mattoni et al., APL 116 (2020)]). The fit
for the symmetric component has good agreement with the data and provides a
magnetic signal which is centred at position $x=$2.2\text{\,}\mathrm{cm}$$,
consistent wtih the sample spatial location. The fit for the asymmetric
component, instead, reveals a magnetic signal located at a shifted position
$x=$1.8\text{\,}\mathrm{cm}$$ which is not produced by the sample but from the
current flowing through the current leads. LABEL:sub@fig:mSymm Magnetic signal
of Sample #2 that is used in the manuscript and LABEL:sub@fig:mAsymm spurious
magnetic signal of the current leads that is correctly discarded thanks to the
described measurement technique.


# A NOTE ON TRANSCENDENTAL ANALYTIC FUNCTIONS WITH RATIONAL COEFFICIENTS
MAPPING $\mathbb{Q}$ INTO ITSELF

JEAN LELIS  FACULDADE DE MATEMÁTICA, ICEN, UNIVERSIDADE FEDERAL DO PARÁ,
BELÉM, PA, BRAZIL<EMAIL_ADDRESS>,  DIEGO MARQUES  DEPARTAMENTO DE
MATEMÁTICA, UNIVERSIDADE DE BRASÍLIA, BRASÍLIA, DF, BRAZIL<EMAIL_ADDRESS>,
CARLOS GUSTAVO MOREIRA  INSTITUTO DE MATEMÁTICA PURA E APLICADA, RIO DE
JANEIRO, RJ, BRAZIL<EMAIL_ADDRESS>and  PAVEL TROJOVSKÝ  FACULTY OF SCIENCE,
UNIVERSITY OF HRADEC KRÁLOVÉ, CZECH REPUBLIC<EMAIL_ADDRESS>
###### Abstract.

In this note, the main focus is on a question about transcendental entire
functions mapping $\mathbb{Q}$ into $\mathbb{Q}$ (which is related to a
Mahler’s problem). In particular, we prove that, for any $t>0$, there is no a
transcendental entire function $f\in\mathbb{Q}[[z]]$ such that
$f(\mathbb{Q})\subseteq\mathbb{Q}$ and whose denominator of $f(p/q)$ is
$O(q^{t})$, for all rational numbers $p/q$, with $q$ sufficiently large.

###### Key words and phrases:

Transcendental functions, rational functions, Mahler’s question, Liouville
numbers

###### 2020 Mathematics Subject Classification:

Primary 11J82, Secondary 30Dxx

##  1\. Introduction

Transcendental number theory began in 1844, when Liouville [1] proved the
existence of transcendental numbers. In fact, he was able to explicit an
infinite class of such numbers. These numbers are the well-known Liouville
numbers: a real number $\xi$ is called a Liouville number, if there exists a
sequence of distinct rational numbers $(p_{k}/q_{k})_{k}$, with $q_{k}>1$,
such that

$0<\left|\xi-\frac{p_{k}}{q_{k}}\right|<\frac{1}{q_{k}^{\omega_{k}}},$

where $\omega_{k}$ tends to infinite as $k\to\infty$. The set of all Liouville
numbers is denoted by $\mathbb{L}$.

In his pioneering book, Maillet [2], in 1906, proved that
$f(\mathbb{L})\subseteq\mathbb{L}$, for any non-constant rational function
$f\in\mathbb{Q}(z)$. In light of this fact, in 1984, Mahler [4] raised the
following question:

######  Question 1.

Are there transcendental entire functions $f(z)$ such that if $\xi$ is any
Liouville number, then so is $f(\xi)$?

In 2015, Marques and Moreira [5] showed the existence of uncountably many
transcendental entire functions $f$ such that
$f(\mathbb{Q})\subseteq\mathbb{Q}$ and for which $\mathop{\rm
den}(p/q)<q^{8q^{2}}$, for all rational number $p/q$, with $q>1$ (here, and in
what follows, $\mathop{\rm den}(z)$ denotes the denominator of the irreducible
rational number $z$). It follows from their argument that Question 1 has a
positive answer if the following question also has:

######  Question 2.

Are there transcendental entire functions $f(z)$ such that
$f(\mathbb{Q})\subseteq\mathbb{Q}$ and

$\mathop{\rm den}(f\left(p/q\right))=O(q^{t}),$

for all rational numbers $p/q$, with $q$ sufficiently large, where $t\geq 0$
is a given positive integer?

There are some progress in this question. For instance, in 2016, Marques,
Ramirez and Silva [6] showed that there is ‘no’ transcendental entire function
$f(z)\in\mathbb{Q}[[z]]$ such that $f(\mathbb{Q})\subseteq\mathbb{Q}$ and
$\mathop{\rm den}f(p/q)=o(q)$ (this is the case $t\in[0,1)$ in Question 2). By
using Whittaker’s theory [7] of polynomial expansions of analytic functions,
in 2020, Lelis and Marques [3] proved that the answer is also ‘no’ for the
case in which $f(z)\in\mathbb{C}[[z]]$ and $t=1$.

The goal of this note is to prove, in particular, the non-existence of
functions as in Question 2 with rational coefficients. More precisely,

######  Theorem 1.

Let $t$ be a positive integer and $\Omega\subseteq\mathbb{C}$ be a
neighborhood of origin. Then, there is no a transcendental function
$f(z)=\sum_{k\geq 0}a_{k}z^{k}\in\mathbb{C}[[z]]$, analytic in $\Omega$, such
that $a_{k}\in\mathbb{Q}$, for all $k\in[0,t]$,
$f(\mathbb{Q}\cap\Omega)\subseteq\mathbb{Q}$ and

$\mathop{\rm den}f(p/q)=O(q^{t/2}),$

for all rational numbers $p/q\in\Omega$, with $q$ sufficiently large.

As an immediate consequence, we infer that

######  Corollary 1.

Let $t$ be a positive integer. Then, there is no a transcendental entire
function $f(z)\in\mathbb{Q}[[z]]$ such that $f(\mathbb{Q})\subseteq\mathbb{Q}$
and

$\mathop{\rm den}f(p/q)=O(q^{t}),$

for all rational numbers $p/q$, with $q$ sufficiently large.

##  2\. The Proof of Theorem 1

In order to simplify the argument, and causing no loss of generality, we shall
prove the theorem for $2t$ (instead of $t$). Aiming for a contradiction, let
$t\geq 1$ be an integer and suppose that $f(z)\in\mathbb{C}[[z]]$ is a
transcendental function which is analytic in the neighborhood
$\Omega\subseteq\mathbb{C}$ of the origin given by

$f(z)=\sum_{k=0}^{\infty}a_{k}z^{k}$

such that $a_{k}\in\mathbb{Q}$, for all $k\in[0,2t]$,
$f(\mathbb{Q}\cap\Omega)\subseteq\mathbb{Q}$ and $\mathop{\rm
den}(f(p/q))=O(q^{t})$, for all $p/q\in\Omega\cap\mathbb{Q}$ with $q$
sufficiently large. Note that, without loss of generality, we may assume that
$f(0)=0$, that is, $a_{0}=0$.

Let $F(z)\in\mathbb{Q}[z]$ be the polynomial given by

$F(z)=\sum_{n=1}^{2t}a_{n}z^{n}.$

Then, there exist polynomials $P(z)$ and $Q(z)$ in $\mathbb{Q}[z]$ of degrees
at most $t$ such that

$R(z):=\frac{P(z)}{Q(z)}=F(z)+z^{2t+1}K(z),$

where $K(z)\in\mathbb{C}[[z]]$ is analytic in a neighborhood of origin.
Indeed, suppose that

$Q(z)=q_{0}+q_{1}z+\cdots+q_{t}z^{t},$

and consider the product $Q(z)F(z)$ given by

$Q(z)F(z)=b_{1}z+\cdots+b_{3t}z^{3t}.$

Therefore, we need to determine $q_{0},q_{1},\ldots,q_{t}$ in $\mathbb{Q}$,
not all zero, such that $b_{j}=0$, for all $j\in[t+1,2t]$. In other words, we
want a non-trivial rational solution for the $t\times(t+1)$ homogeneous linear
system

$\displaystyle\left\\{\begin{array}[]{ccccccccc}a_{2t}q_{0}&+&a_{2t-1}q_{1}&+&\cdots&+&a_{t}q_{t}&=&0\\\
a_{2t-1}q_{0}&+&a_{2t-2}q_{1}&+&\cdots&+&a_{t-1}q_{t}&=&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
a_{t+1}q_{0}&+&a_{t}q_{1}&+&\cdots&+&a_{1}q_{t}&=&0\end{array}\right.$

in the variables $q_{0},q_{1},\ldots,q_{t}$. Since the number of variables is
larger than the number of equations, then a basic Linear Algebra result
ensures the existence of a solution
$(q_{0},\ldots,q_{t})\in\mathbb{Q}^{t+1}\backslash\\{(0,\ldots,0)\\}$. Now,
let $j\geq 0$ be the smallest non-negative integer such that $q_{j}\neq 0$.
Thus, we may suppose that $q_{j}=1$ (because
$(0,\ldots,0,1,q_{j+1}/q_{j},\ldots,q_{t}/q_{j})$ is also a solution of the
previous linear system). So, we define

$P(z):=b_{j+1}z^{j+1}+b_{j+2}z^{j+2}+\cdots+b_{t}z^{t}.$

Therefore

$P(z)=Q(z)F(z)+z^{2t+1}S(z),$

with $S(z)\in\mathbb{Q}[z]$ and so

(2.2) $R(t)=\frac{P(z)}{Q(z)}=F(z)+z^{2t+1-j}K(z),$

where $K(z)=S(z)/(1+q_{j+1}z+\cdots+q_{t}z^{t-j})$ is analytic in a
neighborhood $\Omega^{\prime}\subseteq\mathbb{C}$ of origin.

On the other hand, we have that

(2.3) $f(z)=F(z)+z^{2t+1}\tilde{K}(z),$

where $\tilde{K}(z)$ is analytic in $\Omega$. Let $M^{\prime}$ be the smallest
positive integer such that $1/M^{\prime}\in\Omega\cap\Omega^{\prime}$. Thus,
combining (2.2) together with (2.3) and for all positive integers
$M>M^{\prime}$, we get

(2.4)
$\left|f\left(\frac{1}{M}\right)-R\left(\frac{1}{M}\right)\right|=\frac{1}{M^{2t+1-j}}\Theta(M),$

where

$\Theta(M)=\left|\frac{1}{M^{j}}\tilde{K}\left(\frac{1}{M}\right)-K\left(\frac{1}{M}\right)\right|.$

Now, by hypothesis, $f(1/M)=A_{M}/B_{M}\in\mathbb{Q}$, with
$\gcd(A_{M},B_{M})=1$ and $B_{M}\leq C_{1}M^{t}$ (for some positive constant
$C_{1}$). Moreover, we have that

$R\left(\frac{1}{M}\right)=\frac{b_{j+1}M^{t-j-1}+b_{j+2}M^{t-j-2}\cdots+b_{t}}{M^{t-j}+q_{j+1}M^{t-j-1}+\cdots+q_{t}}$

is a rational number. Note that, if we suppose that $f(1/M)\neq R_{t}(1/M)$
for infinitely many integers $M>M^{\prime}$, then

(2.5)
$\left|f\left(\frac{1}{M}\right)-R\left(\frac{1}{M}\right)\right|\geq\frac{1}{C_{1}M^{t}(M^{t-j}+q_{j+1}M^{t-j-1}+\cdots+q_{t})}\geq\frac{1}{C_{2}M^{2t-j}},$

for all integer $M$ sufficiently large (say $M>M_{0}>M^{\prime}$), where
$C_{2}=2C_{1}$. By (2.4) and (2.5), we obtain that

$\frac{1}{M^{2t-j}}\ll\left|f\left(\frac{1}{M}\right)-R\left(\frac{1}{M}\right)\right|=\frac{1}{M^{2t+1-j}}\Theta(M)\ll\frac{1}{M^{2t+1-j}}$

yielding to the absurdity that $M^{2t+1-j}\ll M^{2t-j}$, for infinitely many
positive integers $M$ for which $f(1/M)\neq R(1/M)$.

Therefore, $f(1/M)=R(1/M)$ for all large enough integer $M$, say $M\geq
M_{1}$. Thus, $f(z)$ and $R(z)$ coincide in the set
$\\{1/M:M>\max\\{M_{0},M_{1}\\}\\}\subseteq\Omega\cap\Omega^{\prime}$ which
has a limit point. By the Identity Theorem for Analytic Functions, we infer
that $f(z)=R(z)$, for all $z\in\Omega\cap\Omega^{\prime}$ with contradicts the
transcendence of $f(z)$. The proof is then complete. ∎

## Acknowledgement

D.M was supported by CNPq-Brazil. Part of this work was done during a visit of
D.M and J.L to IMPA (Rio de Janeiro) which provided them excellent working
conditions. P.T. was supported by the Project of Excellence, Faculty of
Science, University of Hradec Králové, No. 2210/2023-2024.

## References

  * [1] J. Liouville, Sur des classes très-étendues de quantités dont la Valeur n’est ni algébrique ni même réductible à des irrationnelles algébriques, C. R. Acad. Sci. Paris, 18 (1844), 883–885.
  * [2] E. Maillet, Introduction à la Théorie des Nombres Transcendants et des Propriétés Arithmétiques des Fonctions. Gauthier–Villars, Paris (1906).
  * [3] J. Lelis and D. Marques, On transcendental entire functions mapping $\mathbb{Q}$ into itself. J. Number Theory 206 (2020), 310–319.
  * [4] K. Mahler, Some suggestions for further research, Bull. Austral. Math. Soc. 29 (1984), no. 1, 101–108.
  * [5] D. Marques and C. G. Moreira, On a variant of a question proposed by K. Mahler concerning Liouville numbers. Bull. Austral. Math. Soc. 91 (2015), 29–33.
  * [6] D. Marques, J. Ramirez and E. Silva, A note on lacunary power series with rational coefficients. Bull. Austral. Math. Soc., 93 (2016), 372–374.
  * [7] J. M. Whittaker, On series of polynomials. The Quarterly J. Math., 1 (1934), 224–239.