# Current-induced superconducting anisotropy of ${\rm Sr_{2}RuO_{4}}$

R. Araki<EMAIL_ADDRESS>Department of Physics, Kyoto
University, Kyoto 606-8502, Japan  T. Miyoshi  Department of Physics, Kyoto
University, Kyoto 606-8502, Japan  H. Suwa  Department of Physics, Kyoto
University, Kyoto 606-8502, Japan  E. I. Paredes Aulestia  Department of
Physics, The Chinese University of Hong Kong, Shatin N.T., Hong Kong  K. Y.
Yip  Department of Physics, The Chinese University of Hong Kong, Shatin N.T.,
Hong Kong  Kwing To Lai  Department of Physics, The Chinese University of Hong
Kong, Shatin N.T., Hong Kong  S. K. Goh  Department of Physics, The Chinese
University of Hong Kong, Shatin N.T., Hong Kong  Y. Maeno  Department of
Physics, Kyoto University, Kyoto 606-8502, Japan  S. Yonezawa
<EMAIL_ADDRESS>Department of Physics, Kyoto University,
Kyoto 606-8502, Japan

###### Abstract

In the unconventional superconductor ${\rm Sr_{2}RuO_{4}}$, unusual first-
order superconducting transition has been observed in the low-temperature and
high-field region, accompanied by a four-fold anisotropy of the in-plane upper
critical magnetic field $H_{\mbox{\scriptsize{c2}}}$. The origin of such
unusual $H_{\mbox{\scriptsize{c2}}}$ behavior should be closely linked to the
debated superconducting symmetry of this oxide. Here, toward clarification of
the unusual $H_{\mbox{\scriptsize{c2}}}$ behavior, we performed the
resistivity measurements capable of switching in-plane current directions as
well as precisely controlling the field directions. Our results reveal that
resistive $H_{\mbox{\scriptsize{c2}}}$ under the in-plane current exhibits an
additional two-fold anisotropy. By systematically analyzing
$H_{\mbox{\scriptsize{c2}}}$ data taken under various current directions, we
succeeded in separating the two-fold $H_{\mbox{\scriptsize{c2}}}$ component
into the one originating from applied current and the other originating from
certain imperfection in the sample. The former component, attributable to
vortex flow effect, is weakened at low temperatures where
$H_{\mbox{\scriptsize{c2}}}$ is substantially suppressed. The latter component
is enhanced in the first order transition region, possibly reflecting a change
in the nature of the superconducting state under high magnetic field.

##  I Introduction

The layered perovskite superconductor ${\rm Sr_{2}RuO_{4}}$ with the
transition temperature $T_{\mbox{\scriptsize{c}}}$ of 1.5 K [1] has been
extensively studied due to its unconventional pairing state. Its Fermi surface
has a relatively simple topology, consisting of three cylindrical sheets
($\alpha$, $\beta$, and $\gamma$) [2, 3] with its well-characterized Fermi-
liquid behavior. Recent nuclear magnetic resonance (NMR) experiments using low
rf pulses revealed a technical problem in previous reports and clarified a
reduction of the 17O Knight shift in the superconducting states [4, 5, 6].
Recent polarized neutron scattering experiments performed under lower fields
also revealed the reduction in the magnetic susceptibility in the Ru site [7].
These results cannot be explained in the framework of the traditional spin-
triplet superconductivity scenario. Zero-field muon spin rotation ($\mu$SR) as
well as magneto-optic Kerr effect experiments showed evidence for time-
reversal-symmetry breaking (TRSB) [8, 9]. Recent $\mu$SR experiments under
uniaxial stress reveal stress-induced splitting between the onset temperatures
of superconductivity and TRSB [10] whereas such splitting does not occur as
long as the tetragonal symmetry is preserved [11]. Furthermore, ultrasound
measurements show that the superconducting order parameter of ${\rm
Sr_{2}RuO_{4}}$ consists of at least two components [12, 13, 10]. More
recently, it is revealed that the NMR spectrum near the upper critical
magnetic field $H_{\mbox{\scriptsize{c2}}}$ exhibits characteristic “double-
horn” structure, indicating the superconducting spin smecticity [14]. This
result suggests that ${\rm Sr_{2}RuO_{4}}$ is a strong candidate for the
formation of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state [15, 16].

Toward clarification of the debated superconducting order parameter of ${\rm
Sr_{2}RuO_{4}}$, properties near $H_{\mbox{\scriptsize{c2}}}$ is of primary
importance. Indeed, the superconducting transition becomes the first-order
transition (FOT) under magnetic fields aligned accurately in the $ab$ plane
and below 0.8 K [17, 18]. Considering the recent revival of the NMR data, this
first-order transition can be well interpreted as a consequence of the Pauli-
paramagnetic pair-breaking effect, which is not allowed in the traditional
chiral-$p$-wave spin-triplet scenario. More interestingly, in the same
temperature region, $H_{\mbox{\scriptsize{c2//{\it ab}}}}$ exhibits a clear
four-fold in-plane anisotropy [18, 19, 20]. Although this four-fold anisotropy
preserves the tetragonal crystalline symmetry, its origin and relation to the
first-order transition have not been clarified. The motivation of our study is
to reveal the relationship between the $H_{\mbox{\scriptsize{c2//{\it ab}}}}$
anisotropy and FOT.

We focus on the anisotropy of $H_{\mbox{\scriptsize{c2}}}$ in the $ab$ plane
under in-plane currents. First of all, $H_{\mbox{\scriptsize{c2}}}$ under
currents is expected to show the same four-fold symmetry as that under zero
current [18, 19, 20]. Secondly, since type-II superconductors under currents
and magnetic fields above the lower critical field
$H_{\mbox{\scriptsize{c1}}}$ exhibit vortex flow resistivity caused by the
Lorentz force [21], $H_{\mbox{\scriptsize{c2}}}$ in the $ab$ plane of ${\rm
Sr_{2}RuO_{4}}$ is also expected to exhibit current-induced two-fold symmetry
depending on the relative angle $\phi_{H}-\phi_{I}$ between the magnetic field
$H$ and current $I$. Here, $\phi_{H}$ and $\phi_{I}$ are the in-plane angles
of the field and current with respect to the crystal axes, respectively.
Thirdly, $H_{\mbox{\scriptsize{c2}}}$ under currents may also depend on
$\phi_{I}$. Furthermore, these anisotropies of $H_{\mbox{\scriptsize{c2}}}$
under currents are expected to show different behavior between the FOT and
second-order transition (SOT) regions due to the difference in the dominant
pair-breaking mechanisms. Thus, we consider the direction and strength of the
current as new parameters of controlling $H_{\mbox{\scriptsize{c2}}}$, and
extensively investigate the temperature dependence of the anisotropies of
$H_{\mbox{\scriptsize{c2}}}$.

(a)

(b) (c)

Figure 1: Current-direction switchable device of single-crystalline ${\rm
Sr_{2}RuO_{4}}$ for $H_{\mbox{\scriptsize{c2}}}$-anisotropy measurements. (a)
Scanning electron microscope (SEM) image of the device after the focused ion-
beam (FIB) process. FIB-cut trenches are visible as black lines with white
edges. The orange broken line indicates the original shape of the ${\rm
Sr_{2}RuO_{4}}$ crystal. To obtain low contact resistance, we used high-
temperature-cure silver paint (Dupont, 6838). (b) Schematics of the four
configurations of resistivity measurements with different current directions
under magnetic fields. The [100] direction is defined as $\phi=0^{\circ}$ and
the [010] direction as $\phi=90^{\circ}$ for both the field angle $\phi_{H}$
and current angle $\phi_{I}$. (c) Resistive transition of the device measured
on warming under zero field. The superconducting transition temperature
$T_{\mbox{\scriptsize{c}}}$ (1.45 K) remains close to the value for the
cleanest samples (1.5 K) [22] even after the FIB process, and $T_{\mathrm{c}}$
does not depend on current below 200 $\muup$A.

In order to clarify the electric current effect on the anisotropy of
$H_{\mbox{\scriptsize{c2}}}$, it is crucial to identify and eliminate the
effects of technical origins. First, because the value of
$H_{\mbox{\scriptsize{c2//{\it c}}}}$ is about one twentieth less than that of
$H_{\mbox{\scriptsize{c2//{\it ab}}}}$ [20], the presence of a small
misalignment of in-plane magnetic field itself can lead to an apparent two-
fold anisotropy of $H_{\mbox{\scriptsize{c2}}}$ in the $ab$ plane. Second,
since the distribution of currents depends on the shape and distortion of the
sample device, they can cause apparent anisotropy dependent on the direction
of the current $\phi_{I}$. To avoid detecting these extrinsic anisotropies, we
established a measurement procedure in which magnetic fields are applied in
the $ab$ plane accurately and precisely, and currents can be switched in
various crystal orientations.

In this work, we measured resistivity of a micro-structured single-crystalline
${\rm Sr_{2}RuO_{4}}$ sample under various in-plane field and current
directions. The sample was processed by focused ion-beam (FIB) as shown in
Fig. 1(a); this structured sample allows us to switch the current in the
$[100],[010],[110]$ and $[1\overline{1}0]$ directions. We find that, with a
given current direction, $H_{\mbox{\scriptsize{c2}}}$ exhibits two-fold
anisotropy as a function of the in-plane field angle, in addition to the
ordinary four-fold anisotropy. Careful analysis reveals that the two-fold
component consists of two contributions: the one caused by external current
likely due to vortex-flow effect, and the other originating from sample
inhomogeneity probably introduced by the FIB process. We revealed that the
latter is enhanced below 0.8 K, namely in the FOT regime, attributable to a
change in the superconducting order parameter.

##  II Experimental

In this study, we used ${\rm Sr_{2}RuO_{4}}$ single crystals grown with the
floating-zone method [23]. Before the micro-fabrication of the resistivity
device, $T_{\mathrm{c}}$ of a thin crystal (batch: C432; 500 $\muup$m $\times$
100 $\muup$m $\times$ 10 $\muup$m) was measured to be 1.43 K (see Supplemental
Material (SM) [24]), defined as the peak of the imaginary part of the AC
susceptibility measured using a compact susceptometer [25] that fits inside a
commerical refrigerator (Quantum Design; PPMS adiabatic demagnetization
refrigerator option). The crystalline orientation of the sample was determined
by x-ray Laue pictures. The crystal was placed on a single-crystalline ${\rm
SrTiO_{3}}$ substrate (Shinkosha), which has a thermal contraction matching
with that of ${\rm Sr_{2}RuO_{4}}$ [26, 27]. After the electrical contact
between the ${\rm Sr_{2}RuO_{4}}$ crystal and eight electrodes, made of high-
temperature-cure silver paint (Dupont, 6838), were established similarly to
previous studies [28], the surface of the crystal was protected by evaporating
a 0.5-$\mu$m layer of SiOx. We then used a focused ion beam (FIB) instrument
(JEOL, JIM-4501) with a Ga ion beam to fabricate the current-direction
switchable resistivity device. Figure 1(a) shows a scanning electron
microscope (SEM) image of the device taken from the $c$-axis direction. As
shown in the figure, the widths of the arms along the [100] and [010]
directions are both $20~{}\muup$m, and those along the [110] and
$[1\overline{1}0]$ directions are $10~{}\muup$m. The device thickness is
$10~{}\muup$m.

The ${\rm Sr_{2}RuO_{4}}$ device was cooled down to 0.12 K with a 3He-4He
dilution refrigerator (Oxford Instruments, Kelvinox 25). For controlling the
field orientation at low temperatures, we used a vector magnet system
(Cryomagnetics, VSC-3050) with orthogonally arranged SC magnets that generate
a horizontal field $H_{r}$ of up to 5 T and a vertical field $H_{z}$ of up to
3 T to control the polar field angle
$\theta_{\mbox{\scriptsize{lab}}}=\arctan(H_{z}/H_{r})$ [29]. This vector
magnet allows control of $\theta_{\mbox{\scriptsize{lab}}}$ with a typical
precision of $\Delta\theta_{\mbox{\scriptsize{lab}}}=0.005^{\circ}$ at
$\mu_{\mbox{\scriptsize{0}}}H=1$ T. Moreover, the magnet set is placed on a
horizontal rotation stage, which can control the azimuthal field angle
$\phi_{\mbox{\scriptsize{lab}}}$ with a precision of
$\Delta\phi_{\mbox{\scriptsize{lab}}}=0.001^{\circ}$. The quasi-two-
dimensional (2D) anisotropy of $H_{\mbox{\scriptsize{c2}}}$ of ${\rm
Sr_{2}RuO_{4}}$ ($H_{\mbox{\scriptsize{c2//{\it
ab}}}}/H_{\mbox{\scriptsize{c2//{\it c}}}}\sim 20$ [20]) allows us to align
the field to the $ab$ plane accurately and precisely: by rotating magnetic
field close to $H_{\mbox{\scriptsize{c2//{\it ab}}}}$, we can find a sharp
drop of resistivity when the field is exactly parallel to the $ab$ plane (see
SM [24]). The in-plane alignment was done based on the known four-fold
anisotropy of the in-plane $H_{\mbox{\scriptsize{c2}}}$ (i.e.
$H_{\mbox{\scriptsize{c2//[110]}}}>H_{\mbox{\scriptsize{c2//[100]}}}$ below
0.8 K) [19, 20]. All the data shown in this paper are plotted using the field
angles $\phi_{H}$ and $\theta_{H}$ defined in the sample coordinate.

To measure the sample resistance $R$, we employed a DC method with current
sign reversal to avoid influence of voltage offsets such as thermoelectric
voltages. We used a combination of a current source (Keithley, 6221) and a
nanovoltometer (Keithley, 2182A). We measured $R$ against various parameters:
temperature $T$, magnitude and azimuthal angle of magnetic field $H$ and
$\phi_{H}$, current amplitude and direction $I$ and $\phi_{I}$. Both
$\phi_{H}$ and $\phi_{I}$ are defined with respect to the [100] direction as
shown in Fig. 1(b). We note that for a given current value, the current
density for an orthogonal measurement ($I$//[100] or [010]) is half of that
for a diagonal measurement ($I$//[110] or $[1\overline{1}0]$) due to the
larger width of the arm along the [100] or [010] directions.

Figure 1(c) shows the temperature dependence of $R$ under zero field. The
resistivity reaches zero at $T_{\mathrm{c}}=1.45$ K. The consistent
$T_{\mathrm{c}}$ values before and after FIB assure that a possible damage or
strain due to FIB is minimal. In this study we used the current values below
200 $\muup$A for which $T_{\mathrm{c}}$ does not change under zero field.

##  III Results and Analysis

To investigate the anisotropy of the resistive $H_{\mbox{\scriptsize{c2}}}$ in
the $ab$ plane under in-plane currents, we measured $R(H)$ under various field
and current conditions. We show in Fig. 2 representative field dependence of
the resistance for $I//[100]$. Additional raw data are shown in SM [24]. The
in-plane $H_{\mbox{\scriptsize{c2}}}$, defined with the deviation from $R=0$
as indicated by the arrows, clearly depends on the in-plane field direction
and on the current strength.

Figure 2: Representative resistive transition of our ${\rm Sr_{2}RuO_{4}}$
device under various values of in-plane currents $I$. These data were taken at
0.40 K with current along the [100] direction. The curves with $I\geq
100~{}\muup$A have vertical offsets. The yellow, brown, pink and black data
points are obtained under $\phi_{H}=0^{\circ},45^{\circ},90^{\circ}$ and
$135^{\circ}$, respectively. In addition to the known four-fold anisotropy,
$H_{\mbox{\scriptsize{c2}}}$, indicated by the vertical arrows, exhibits two-
fold anisotropy as evidenced by the differences in the curves between
$\phi_{H}=0^{\circ}$ and $90^{\circ}$, or between $\phi_{H}=45^{\circ}$ and
$135^{\circ}$.

Figures 3(a) and 3(c) compare the $\phi_{H}$ dependence of the in-plane
$H_{\mbox{\scriptsize{c2}}}$ for $I//[010]$ and $I//[100]$ measured at
$200~{}\muup$A and $0.40$ K. Under in-plane currents,
$H_{\mbox{\scriptsize{c2}}}(\phi_{H})$ clearly shows not only the known four-
fold anisotropy but also an additional two-fold anisotropy. This is also
evident in the raw data in Fig. 2: For example, $H_{\mbox{\scriptsize{c2}}}$
for $\phi_{H}=45^{\circ}$ is noticeably larger than that for
$\phi_{H}=135^{\circ}$, although these two conditions should be equivalent
considering the tetragonal crystalline symmetry if the current were absent. We
have confirmed that this two-fold behavior is not caused by the misalignment
of fields with respect to the $ab$ plane (Fig. S2 in SM [24]). Moreover,
comparing Figs. 3(a) and (c), we notice that this two-fold components of
$H_{\mbox{\scriptsize{c2}}}$ are altered by changing the current direction
$\phi_{I}$. For example, the $H_{\mbox{\scriptsize{c2}}}$ value at
$\phi_{H}=0^{\circ}$ is smaller than that at $90^{\circ}$ for $I//[010]$ (Fig.
3(a)), whereas they are opposite for $I//[100]$ (Fig. 3(c)). This fact also
evidences that the external current plays an important role in the observed
two-fold behavior.

Figure 3: $H_{\mbox{\scriptsize{c2}}}$ anisotropy of ${\rm Sr_{2}RuO_{4}}$
under in-plane current. Additional data are shown in SM [24]. (a) Raw
$H_{\mbox{\scriptsize{c2}}}$ data as a function of $\phi_{H}$ under current
along the [010] direction. The data are well fitted with a combination of
four-fold and two-fold sinusoidal function (Eq. (1)) as shown with the pink
solid curve. (b) two-fold component of $H_{\mbox{\scriptsize{c2}}}$ under
$I//[010]$ (pink squares), extracted from the raw $H_{\mbox{\scriptsize{c2}}}$
data by subtracting the fitted four-fold component and the constant offset.
The pink curve present the two-fold sinusoidal fitting, whereas the solid and
broken black curves show the decomposed cosine and sine terms of the pink
curve, respectively. (c) Raw $H_{\mbox{\scriptsize{c2}}}$ data (circles) and
fitting result (solid curve) under current along the [100] direction. Compared
to (a), the data suggest that current direction $\phi_{I}$ switches the sign
of the difference between $H_{\mbox{\scriptsize{c2}}}$ at $\phi_{H}=0^{\circ}$
and that at $\phi_{H}=90^{\circ}$. (d) two-fold component of
$H_{\mbox{\scriptsize{c2}}}$ under $I//[100]$ together with the two-fold
fitting (orange curve) and the decomposed cosine (black solid curve) and sine
(black broken curve) components. (e) $I$-induced two-fold component of
$H_{\mbox{\scriptsize{c2}}}$ obtained by taking the difference between the
data in (b) and (d). This only contains the two-fold component that is
switched by the change of the current direction. (f) $I$-independent two-fold
component of $H_{\mbox{\scriptsize{c2}}}$ obtained by taking the summation of
the data in (b) and (d). This only contains the two-fold component that is not
switched by the change of the current direction.

To analyze the two-fold behavior quantitatively, we fit the
$H_{\mbox{\scriptsize{c2}}}(\phi_{H})$ data with a combination of two and
four-fold sinusoidal functions with a constant offset:

$\displaystyle\mu_{0}h_{\mbox{\scriptsize{c2}}}(\phi_{H})=a_{0}+a_{2}\cos
2\phi_{H}+b_{2}\sin 2\phi_{H}+a_{4}\cos 4\phi_{H}.$ (1)

We comment that the four-fold cosine term represents the known
$H_{\mbox{\scriptsize{c2}}}$ anisotropy under zero current
($H_{\mbox{\scriptsize{c2//[110]}}}>H_{\mbox{\scriptsize{c2//[100]}}}$) [20,
19] and we have checked that the four-fold sine term is negligible even under
currents. As exemplified by the solid curves in Figs. 3(a) and (c), the
fitting was successful for the $H_{\mbox{\scriptsize{c2}}}(\phi_{H})$ data
sets. The obtained value of $a_{4}$ is $\sim-0.011$ T and $\sim-0.010$ T for
the data in Fig. 3(a) and (c). These values are consistent with the previous
studies [19, 20]. Notice that the quantity
$\Delta(\mu_{0}H_{\mbox{\scriptsize{c2}}})=\mu_{0}H_{\mbox{\scriptsize{c2}}}[110]-\mu_{0}H_{\mbox{\scriptsize{c2}}}[100]$
in the literature corresponds to the quantity $-2a_{4}$ of our study. Then, to
extract the two-fold component $H_{\mbox{\scriptsize{c2}}}^{(2)}$, we
subtracted the fitted four-fold component $a_{4}\cos 4\phi_{H}$ and the offset
$a_{0}$ from the data. The results are shown in Figs. 3(b) and (d). Here, the
colored curves represent the two-fold terms obtained by the fitting, whereas
the black solid and broken curves are decomposed $\cos 2\phi_{H}$ and $\sin
2\phi_{H}$ components, respectively. Between $I//[100]$ and $[010]$,
$H_{\mbox{\scriptsize{c2}}}^{(2)}$ is clearly different with the opposite sign
of the cosine component.

Figure 4: Temperature dependence of $H_{\mbox{\scriptsize{c2}}}^{(2)}$
measured under current along the four directions. The pink, yellow, blue and
green points correspond to [010], [100], [110] and $[1\overline{1}0]$ current
directions. (a) Coefficient of the two-fold cosine component $a_{2}$. $a_{2}$
under $I//[100]$ and $[010]$ increases on warming and their signs are switched
by current-directions. In contrast, $a_{2}$ under $I//[110]$ and
$[1\overline{1}0]$ stays nearly zero. These results suggest that
$H_{\mbox{\scriptsize{c2}}}^{(2)}$ is mostly induced by current and thus
behaves as $\cos 2(\phi_{H}-\phi_{I})$. (b) Coefficient of the two-fold sine
component $b_{2}$. Similar to $a_{2}$, the current-direction dependence of
$b_{2}$ at high temperature is consistent with the expectation
$H_{\mbox{\scriptsize{c2}}}^{(2)}\propto\cos 2(\phi_{H}-\phi_{I})$. However,
unlike $a_{2}$, $b_{2}$ under all current-directions have an additional term
reaching nearly 0.005 T on cooling to 0 K.

We present the $T$-dependence of the obtained fitting parameters $a_{2}$ and
$b_{2}$ in eq. 1 for various current directions in Fig. 4. (Behavior of
$a_{0}$ and $a_{4}$ will be discussed later.) For the coefficient of the $\cos
2\phi_{H}$ component $a_{2}$, its magnitude under the orthogonal current
directions, namely $I//[100]$ and $I//[010]$, increases on warming while its
sign is switched by current-direction change. This is indeed expected for a
current-induced effect, where by symmetry $H_{\mbox{\scriptsize{c2}}}^{(2)}$
should behave as $\propto\cos 2(\phi_{H}-\phi_{I})=\cos 2\phi_{H}$ for
$\phi_{I}=0^{\circ}~{}(I//[100])$ and $-\cos 2\phi_{H}$ for
$\phi_{I}=90^{\circ}~{}(I//[010])$. In contrast, $a_{2}$ under the diagonal
current directions, namely $I//[110]$ and $I//[1\overline{1}0]$, are nearly
independent of temperature and stays close to zero. This is again consistent
with the $\cos 2(\phi_{H}-\phi_{I})$ behavior for the current-induced
$H_{\mbox{\scriptsize{c2}}}^{(2)}$, since $\cos 2(\phi_{H}-\phi_{I})=\pm\sin
2\phi_{H}$ for $\phi_{I}=\pm 45^{\circ}$ and thus the cosine component should
be zero. Figure 4(b) shows the temperature dependence of the coefficients
$b_{2}$. At high temperatures, the behavior of $b_{2}$ is similar to that of
$a_{2}$ except for an exchange of the roles between the orthogonal and
diagonal current directions: $b_{2}$ is finite and its sign depends on the
current direction $I$//[110] or $[1\overline{1}0]$, but is nearly zero for
$I$//[100] and [010]. Thus this high-temperature behavior in $b_{2}$ again
indicate $H_{\mbox{\scriptsize{c2}}}^{(2)}\propto\cos 2(\phi_{H}-\phi_{I})$
behavior. However, at lower temperatures, $b_{2}$ tends to converge to a
finite value $b_{2}(T\to 0)\sim 0.005$ T irrespective of the current
directions. Such convergence reveals an additional two-fold component that is
not caused by external current.

From these analyses, we revealed that the $H_{c2}^{(2)}$ data have current-
independent and current-dependent contributions, which we hereafter express as
$H_{\mbox{\scriptsize{c2}}}^{\mbox{\scriptsize{(2ind)}}}$ and
$H_{\mbox{\scriptsize{c2}}}^{\mbox{\scriptsize{(2dep)}}}$. The former
corresponds to the low-temperature current-direction independent behavior in
$b_{2}$ (Fig. 4(b)) and the latter corresponds to the $\cos
2(\phi_{H}-\phi_{I})$ term as seen in the behavior common to $a_{2}$ and
$b_{2}$ discussed in the previous paragraph. Further analyses and
interpretations of these contributions will be provided in the next section.

Before closing this section, we comment on possible heating effects due to
current. We estimate the upper limit of the actual sample temperature from the
current dependence of the coefficient $a_{0}$ by assuming that its variation
with current is solely due to Joule heating (see SM [24]). This estimation
indicates that the temperature increase is at most 0.2 K in the lowest
temperature region and less than 0.1 K above 0.4 K.

##  IV Discussion

As explained in the previous section, we extracted two-fold anisotropy of
$H_{\mbox{\scriptsize{c2}}}$ of ${\rm Sr_{2}RuO_{4}}$ under current. We reveal
systematically how this additional anisotropy depends on current-strength,
current-directions, and temperature. As a well-known effect, external current
induces vortex flows in the mixed state by the Lorentz force, leading to the
apparent reduction in $H_{\mbox{\scriptsize{c2}}}$. Thus, such an effect is
maximized under field perpendicular to current and would not occur under field
parallel to current; As a result, this effect is expressed as a two-fold
cosine-like component $\propto\cos 2(\phi_{H}-\phi_{I})$, whose sign is
negative under field perpendicular to current (i.e. $\phi_{H}-\phi_{I}=\pm
90^{\circ}$) and positive under field parallel to current (i.e.
$\phi_{H}=\phi_{I}$).

For ordinary type-II superconductors (both 2D and 3D) in which the main pair-
breaking effect is the ordinary orbital effect, the vortex-flow effect is
expected to be observed. However, in ${\rm Sr_{2}RuO_{4}}$, the FOT occurs
below 0.8 K, indicating that the dominated pair-breaking effect is not the
orbital effect. Thus, the two-fold cosine-like component can behave
differently between the FOT and the SOT regions.

In order to extract the current-dependent term including the vortex-flow
effect from the $H_{\mbox{\scriptsize{c2}}}^{(2)}$ data, we take the
difference between the
$H_{\mbox{\scriptsize{c2}}}^{(2)}(\phi_{H},\phi_{I}=90^{\circ})$ data and
$H_{\mbox{\scriptsize{c2}}}^{(2)}(\phi_{H},\phi_{I}=0^{\circ})$ data. This
process is motivated by the following expectation. We expect that the two-fold
component $H_{\mbox{\scriptsize{c2}}}^{(2)}(\phi_{H},\phi_{I})$ contain
current-independent term
$H_{\mbox{\scriptsize{c2}}}^{(2\mbox{\scriptsize{ind}})}(\phi_{H})$ and
current-dependent term
$H_{\mbox{\scriptsize{c2}}}^{(2\mbox{\scriptsize{dep}})}(\phi_{H}-\phi_{I})$,
$i.e.$

$\displaystyle
H_{\mbox{\scriptsize{c2}}}^{(2)}(\phi_{H},\phi_{I})=H_{\mbox{\scriptsize{c2}}}^{(2\mbox{\scriptsize{ind}})}(\phi_{H})+H_{\mbox{\scriptsize{c2}}}^{(2\mbox{\scriptsize{dep}})}(\phi_{H}-\phi_{I}).$
(2)

Then, due to the two-fold nature,
$H_{\mbox{\scriptsize{c2}}}^{(2\mbox{\scriptsize{dep}})}$ should change sign
upon a current-direction switching of 90∘, as

$\displaystyle
H_{\mbox{\scriptsize{c2}}}^{(2\mbox{\scriptsize{dep}})}(\phi_{H}-(\phi_{I}+90^{\circ}))=-H_{\mbox{\scriptsize{c2}}}^{(2\mbox{\scriptsize{dep}})}(\phi_{H}-\phi_{I}).$
(3)

In contrast, $H_{\mbox{\scriptsize{c2}}}^{(2\mbox{\scriptsize{ind}})}$ is, by
definition, independent of the 90∘ current direction switching. Thus, by
calculating

$\displaystyle\varDelta_{\mbox{\scriptsize{orth}}}H_{\mbox{\scriptsize{c2}}}^{\mbox{\scriptsize{(2)}}}$
$\displaystyle\equiv
H_{\mbox{\scriptsize{c2}}}^{\mbox{\scriptsize{(2)}}}(\phi_{I}=90^{\circ})-H_{\mbox{\scriptsize{c2}}}^{\mbox{\scriptsize{(2)}}}(\phi_{I}=0^{\circ}),$
(4)

$H_{\mbox{\scriptsize{c2}}}^{\mbox{\scriptsize{(2ind)}}}$ is eliminated and we
can expect that
$\varDelta_{\mbox{\scriptsize{orth}}}H_{\mbox{\scriptsize{c2}}}^{(2)}\simeq
2H_{\mbox{\scriptsize{c2}}}^{(2\mbox{\scriptsize{dep}})}(\phi_{I}=90^{\circ})$.
A representative
$\varDelta_{\mbox{\scriptsize{orth}}}H_{\mbox{\scriptsize{c2}}}^{\mbox{\scriptsize{(2)}}}$
is shown in Fig. 3(e). Similarly,

$\displaystyle\varDelta_{\mbox{\scriptsize{diag}}}H_{\mbox{\scriptsize{c2}}}^{\mbox{\scriptsize{(2)}}}$
$\displaystyle\equiv
H_{\mbox{\scriptsize{c2}}}^{\mbox{\scriptsize{(2)}}}(\phi_{I}=45^{\circ})-H_{\mbox{\scriptsize{c2}}}^{\mbox{\scriptsize{(2)}}}(\phi_{I}=-45^{\circ})$
(5)

should satisfy
$\varDelta_{\mbox{\scriptsize{diag}}}H_{\mbox{\scriptsize{c2}}}^{(2)}\simeq
2H_{\mbox{\scriptsize{c2}}}^{(2\mbox{\scriptsize{dep}})}$ (Fig. S4), providing
an independent evaluation of
$H_{\mbox{\scriptsize{c2}}}^{(2\mbox{\scriptsize{dep}})}(\phi_{I}=45^{\circ})$.
On the other hand, we can extract the current-independent term
$H_{\mbox{\scriptsize{c2}}}^{(2\mbox{\scriptsize{ind}})}(\phi_{H})$ by
evaluating sums of the datasets with 90∘ current switching:

$\displaystyle\varSigma_{\mbox{\scriptsize{orth}}}H_{\mbox{\scriptsize{c2}}}^{\mbox{\scriptsize{(2)}}}$
$\displaystyle\equiv
H_{\mbox{\scriptsize{c2}}}^{\mbox{\scriptsize{(2)}}}(\phi_{I}=90^{\circ})+H_{\mbox{\scriptsize{c2}}}^{\mbox{\scriptsize{(2)}}}(\phi_{I}=0^{\circ})$
(6)

and

$\displaystyle\varSigma_{\mbox{\scriptsize{diag}}}H_{\mbox{\scriptsize{c2}}}^{\mbox{\scriptsize{(2)}}}$
$\displaystyle\equiv
H_{\mbox{\scriptsize{c2}}}^{\mbox{\scriptsize{(2)}}}(\phi_{I}=45^{\circ})+H_{\mbox{\scriptsize{c2}}}^{\mbox{\scriptsize{(2)}}}(\phi_{I}=-45^{\circ}).$
(7)

In these sums, current-dependent terms are eliminated and we expect
$\varSigma_{\mbox{\scriptsize{orth}}}H_{\mbox{\scriptsize{c2}}}^{(2)}\simeq\varSigma_{\mbox{\scriptsize{diag}}}H_{\mbox{\scriptsize{c2}}}^{(2)}\simeq
2H_{\mbox{\scriptsize{c2}}}^{(2\mbox{\scriptsize{ind}})}$. A representative
$\varSigma_{\mbox{\scriptsize{orth}}}H_{\mbox{\scriptsize{c2}}}^{\mbox{\scriptsize{(2)}}}$
is shown in Fig. 3(f) and
$\varSigma_{\mbox{\scriptsize{diag}}}H_{\mbox{\scriptsize{c2}}}^{\mbox{\scriptsize{(2)}}}$
in Fig. S4.

In Figs. 3(e), 3(f), S4, one can clearly see that
$\varDelta_{\mbox{\scriptsize{orth}}}H_{\mbox{\scriptsize{c2}}}^{\mbox{\scriptsize{(2)}}}$
and
$\varDelta_{\mbox{\scriptsize{diag}}}H_{\mbox{\scriptsize{c2}}}^{\mbox{\scriptsize{(2)}}}$
are dominated by $\cos 2(\phi_{H}-\phi_{I})|_{\phi_{I}=90^{\circ},45^{\circ}}$
and
$\varSigma_{\mbox{\scriptsize{orth}}}H_{\mbox{\scriptsize{c2}}}^{\mbox{\scriptsize{(2)}}}$
and
$\varSigma_{\mbox{\scriptsize{diag}}}H_{\mbox{\scriptsize{c2}}}^{\mbox{\scriptsize{(2)}}}$
are by $\sin 2\phi_{H}$. To perform quantitative analysis, we fitted
$(1/2)\varSigma H_{\mbox{\scriptsize{c2}}}^{(2)}$ by $A_{2}\cos
2\phi_{H}+B_{2}\sin 2\phi_{H}$ and $(1/2)\varDelta
H_{\mbox{\scriptsize{c2}}}^{(2)}$ by $C_{2}\cos
2(\phi_{H}-\phi_{I})|_{\phi_{I}=90^{\circ},45^{\circ}}+D_{2}\sin
2(\phi_{H}-\phi_{I})|_{\phi_{I}=90^{\circ},45^{\circ}}$. The resultant
parameters are plotted in Figs. 5(c), (d), S6(a) and S6(b). We confirmed that
the coefficients evaluated from the orthogonal and diagonal combinations agree
with each other, manifesting the validity of the analyses. We then indeed find
$A_{2}$ and $D_{2}$ are small and do not exhibit temperature dependence. In
contrast, $B_{2}$ and $C_{2}$ plotted in Figs. 5(c) and (d) are much larger
than $A_{2}$ or $D_{2}$ and exhibits characteristic temperature dependence as
will be discussed later.

Figure 5:  Relation among the first-order transition (FOT) region and
$H_{\mbox{\scriptsize{c2}}}$ anisotropies under current. (a) $T$-dependence of
$a_{0}$. Since $a_{0}$ corresponds to the in-plane average value of
$H_{\mbox{\scriptsize{c2}}}$, this curve provide the
$H_{\mbox{\scriptsize{c2}}}$-$T$ phase diagram of this sample. (b)
$T$-dependence of the four-fold component $-2a_{4}$. The behavior of $-2a_{4}$
is consistent with the previous studies [19, 20]. In (a) and (b), the pink,
orange, sky-blue and green points are for the data obtained with $I$//[010],
[100], [110] and $[1\overline{1}0]$, respectively. (c) and (d); $T$-dependence
of two-fold $I$-independent component and $I$-dependent component of
$H_{\mbox{\scriptsize{c2}}}^{(2)}$. Red squares and blue diamonds are the data
obtained for orthogonal and diagonal current-directions, respectively. The
two-fold $I$-independent sine component $B_{2}$ reaches about 0.005 T in the
FOT region ($T\leq 0.8$ K) and nearly zero in the second-order transition
region ($T\geq 0.8$ K). This result indicates that the sample in a high-field
state becomes more sensitive to inhomogeneity effect, producing two-fold
anisotropy. The $\cos 2(\phi_{H}-\phi_{I})$ component $C_{2}$ in (d) increase
on warming. This observation is consistent with vortex flow effect.

In Figs. 5(a) and (b) the angle-independent component $a_{0}$ (see Eq. (1))
and four-fold component $-2a_{4}$ are compared with the coefficients $B_{2}$
and $C_{2}$. Notice that the coefficient $a_{0}$ in Eq. (1) can be considered
as the in-plane average $H_{\mbox{\scriptsize{c2}}}$ and thus provide the
$H_{\mbox{\scriptsize{c2}}}$-$T$ phase diagram of ${\rm Sr_{2}RuO_{4}}$.
Indeed, Fig. 5(a) agrees well with the phase diagram reported in literature
[20, 18]. The value $-2a_{4}$ should be equal to
$\mu_{0}H_{\mbox{\scriptsize{c2//[110]}}}-\mu_{0}H_{\mbox{\scriptsize{c2//[100]}}}$
in the absence of current, characterizing the known four-fold
$H_{\mbox{\scriptsize{c2}}}$ anisotropy. This quantity is nearly zero above
0.8 K but increase on cooling in the FOT region and reaches 0.03 T at the
lowest temperature. Such behavior is consistent with the previous studies.

As shown in Fig. 5(d), the coefficient $C_{2}$ exhibits a finite value and
decreases on cooling. The coefficient $D_{2}$ exhibits a smaller value than
$C_{2}$ and show only a weak temperature dependence (see SM [24]). These
results indicate that the current-dependent components are cosine-like and
increase on warming. It is consistent with the scenario where vortex flow
gives current-dependent two-fold anisotropy of $H_{\mbox{\scriptsize{c2}}}$.

The coefficient $B_{2}$ shown in Fig. 5(c), characterizing current-independent
two-fold behavior, is finite only below 0.8 K in the FOT region, for the
orthogonal current. A similar tendency is clear for the diagonal current.
Since other bulk measurements have not revealed such a current-independent
two-fold component of $H_{\mbox{\scriptsize{c2}}}$, the behavior of $B_{2}$ is
probably related to minute surface damage caused by the FIB processing.
However, it is not sufficient to explain the $T$-dependence of $B_{2}$ because
it does not change smoothly but appear only in the FOT region. This behavior
can be explained if the resistivity suddenly becomes more sensitive to the
inhomogeneity in the FOT region. One possible candidate for such sudden change
in the resistivity property is the formation of the FFLO state, which is
accompanied by a real-space modulation of the superconducting order parameter
and is recently revealed by an NMR experiment [30]. If there is a surface
damage, for example, the FFLO order-parameter modulation can be pinned to the
surface damage and such pinning can induce additional two-fold behavior in
$H_{\mbox{\scriptsize{c2}}}(\phi_{H})$.

##  V Summary and conclusion

Here we summarize the main results of our study.

(1) The angle-independent component $a_{0}$ in Eq. (1) can be considered as
the in-plane average $H_{\mbox{\scriptsize{c2}}}$.

(2) The quantity $-2a_{4}$ characterizes the known four-fold
$H_{\mbox{\scriptsize{c2}}}$ anisotropy. It corresponds to
$\varDelta(\mu_{0}H_{\mbox{\scriptsize{c2}}})$ in the previous studies [19,
20] and exhibits about 80$\%$ of the value of
$\varDelta(\mu_{0}H_{\mbox{\scriptsize{c2}}})$ in the first-order transition
region. In the second-order transition region, $-2a_{4}$ shows close to zero
in the same way as $\varDelta(\mu_{0}H_{\mbox{\scriptsize{c2}}})$. Thus, the
four-fold anisotropy in $H_{\mbox{\scriptsize{c2}}}$ under in-plane current
shows the same $T$-dependence as that of the four-fold anisotropy in
$H_{\mbox{\scriptsize{c2}}}$ under zero current.

(3) The coefficient $C_{2}$ shows the $\cos 2(\phi_{H}-\phi_{I})$ component of
$H_{\mbox{\scriptsize{c2}}}^{(2\mbox{\scriptsize{dep}})}$. Since it takes
positively finite values and increases with temperature, the current-induced
two-fold anisotropy in $H_{\mbox{\scriptsize{c2}}}^{(2)}$ is probably due to
the vortex-flow effect.

(4) The coefficient $B_{2}$ shows the two-fold $I$-independent sine component
of $H_{\mbox{\scriptsize{c2}}}^{(2)}$. Such a two-fold component has not been
observed in previous experiments [18, 19, 20]. This quantity revealed in our
study becomes finite only in the first-order transition region. This is
probably related to minute the crystal distortion and surface damage caused by
FIB processing. However, it is not enough to explain why $B_{2}$ becomes
finite only in the first-order transition region. As a possible scenario, we
propose that the increased sensitivity to inhomogeneity is due to the
realization of FFLO state in the first-order transition region.

In conclusion, we have established a clear approach to probe the electrical
resistivity of two-dimensional superconductors, such as ${\rm Sr_{2}RuO_{4}}$,
with a varying current and magnetic field directions. The combination of the
vector magnet and our FIB structured sample enables us to control the
azimuthal field angle $\phi_{H}$, and switch in-plane current-direction
$\phi_{I}$ along four crystal orientations. We have demonstrated that
$H_{\mbox{\scriptsize{c2}}}$ in the $ab$ plane of ${\rm Sr_{2}RuO_{4}}$
exhibits not only an ordinary four-fold anisotropy but also an additional two-
fold anisotropy. This two-fold anisotropy increases on warming and probably
expressed as $\cos 2(\phi_{H}-\phi_{I})$, suggesting that flux flow
resistivity generated by the Lorentz effect leads to underestimation of
$H_{\mbox{\scriptsize{c2}}}$ in the $ab$ plane. We find another two-fold term
behaving as $\sin 2\phi_{H}$, independent of the current direction. This
component, indicative of the sensitivity of the superconductivity to sample
inhomogenity, becomes noticible only below 0.8 K, where field-induced phases
such as the FFLO state are highly anticipated.

Our study is expected to provide hints toward clarification of the remaining
issues on ${\rm Sr_{2}RuO_{4}}$, such as the origin of the enhancement of
$T_{\mathrm{c}}$ under uniaxial strain or in eutectic crystals, and the
relationship between the superconducting order parameters and the anisotropy
of $H_{\mbox{\scriptsize{c2}}}$. In addition, our resistivity measurement
system will be helpful for investigating interesting phenomena in other
materials such as coupling between current and electronic nematicity in
nematic superconductors [31] or in nematic electron liquid systems [32].

##  VI acknowledgement

The authors thank Y. Yanase, K. Machida, K. Ishida, S. Kitagawa, G. Mattoni,
for valuable discussions. We also acknowledge technical supports from Kyoto
Univ. LTM Center. This work is supported by Japan Society for the Promotion of
Science (JSPS) Core-to-core program (No. JPJSCCA20170002) and by JSPS KAKENHI
Nos. JP15H05852, JP15K21717, JP17H06136, and JP20H05158. This work is also
supported by Research Grants Council of Hong Kong (CUHK 14301316) and CUHK
Direct Grants (No. 4053299, 4053345).

## References

  * Maeno  _et al._ [1994] Y. Maeno, H. Hashimoto, K. Yoshida, S. Nishizaki, T. Fujita, J. G. Bednorz, and F. Lichtenberg, “Superconductivity in a layered perovskite without copper” Nature 372, 532 (1994).
  * Bergemann  _et al._ [2003] C. Bergemann, A. P. Mackenzie, S. R. Julian, D. Forsythe, and E. Ohmichi, “Quasi-two-dimensional Fermi liquid properties of the unconventional superconductor Sr2RuO4” Adv. Phys. 52, 639 (2003).
  * Damascelli  _et al._ [2000] A. Damascelli, D. H. Lu, K. M. Shen, N. P. Armitage, F. Ronning, D. L. Feng, C. Kim, Z.-X. Shen, T. Kimura, Y. Tokura, Z. Q. Mao, and Y. Maeno, “Fermi Surface, Surface States, and Surface Reconstruction in Sr2RuO4” Phys. Rev. Lett. 85, 5194 (2000).
  * Pustogow  _et al._ [2019] A. Pustogow, Y. Luo, A. Chronister, Y.-S. Su, D. A. Sokolov, F. Jerzembeck, A. P. Mackenzie, C. W. Hicks, N. Kikugawa, S. Raghu, E. D. Bauer, and S. E. Brown, “Constraints on the superconducting order parameter in Sr2RuO4 from oxygen-17 nuclear magnetic resonance” Nature 574, 72 (2019).
  * Ishida  _et al._ [2020] K. Ishida, M. Manago, K. Kinjo, and Y. Maeno, “Reduction of the 17O Knight Shift in the Superconducting State and the Heat-up Effect by NMR Pulses on Sr2RuO4” Journal of the Physical Society of Japan 89, 034712 (2020).
  * Chronister  _et al._ [2021] A. Chronister, A. Pustogow, N. Kikugawa, D. A. Sokolov, F. Jerzembeck, C. W. Hicks, A. P. Mackenzie, E. D. Bauer, and S. E. Brown, “Evidence for even parity unconventional superconductivity in Sr2RuO4” Proceedings of the National Academy of Sciences of the United States of America 118, 25 (2021).
  * Petsch  _et al._ [2020] A. N. Petsch, M. Zhu, M. Enderle, Z. Q. Mao, Y. Maeno, I. I. Mazin, and S. M. Hayden, “Reduction of the Spin Susceptibility in the Superconducting State of ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$ Observed by Polarized Neutron Scattering” Phys. Rev. Lett. 125, 217004 (2020).
  * Luke  _et al._ [1998] G. M. Luke, Y. Fudamoto, K. M. Kojima, M. I. Larkin, J. Merrin, B. Nachumi, Y. J. Uemura, Y. Maeno, Z. Q. Mao, Y. Mori, H. Nakamura, and M. Sigrist, “Time-reversal symmetry-breaking superconductivity in Sr2RuO4” Nature 394, 558 (1998).
  * Xia  _et al._ [2006] J. Xia, Y. Maeno, P. T. Beyersdorf, M. M. Fejer, and A. Kapitulnik, “High Resolution Polar Kerr Effect Measurements of Sr2RuO4: Evidence for Broken Time-Reversal Symmetry in the Superconducting State” Phys. Rev. Lett. 97, 167002 (2006).
  * Grinenko  _et al._ [2021] V. Grinenko, S. Ghosh, R. Sarkar, J.-C. Orain, A. Nikitin, M. Elender, D. Das, Z. Guguchia, F. BrÃ¼ckner, M. E. Barber, J. Park, N. Kikugawa, D. A. Sokolov, J. S. Bobowski, T. Miyoshi, Y. Maeno, A. P. Mackenzie, H. Luetkens, C. W. Hicks, and H.-H. Klauss, “Split superconducting and time-reversal symmetry-breaking transitions in Sr2RuO4 under stress” Nature Phys. 17, 748 (2021a).
  * Grinenko  _et al._ [2021] V. Grinenko, D. Das, R. Gupta, B. Zinkl, N. Kikugawa, Y. Maeno, C. W. Hicks, H.-H. Klauss, M. Sigrist, and R. Khasanov, “Unsplit superconducting and time reversal symmetry breaking transitions in Sr2RuO4 under hydrostatic pressure and disorder” Nature Communications 12, 3920 (2021b).
  * Benhabib  _et al._ [2021] S. Benhabib, C. Lupien, I. Paul, L. Berges, M. Dion, M. Nardone, A. Zitouni, Z. Q. Mao, Y. Maeno, A. Georges, L. Taillefer, and C. Proust, “Ultrasound evidence for a two-component superconducting order parameter in Sr2RuO4” Nature Phys. 17, 194 (2021).
  * Ghosh  _et al._ [2021] S. Ghosh, A. Shekhter, F. Jerzembeck, N. Kikugawa, D. A. Sokolov, M. Brando, A. P. Mackenzie, C. W. Hicks, and B. J. Ramshaw, “Thermodynamic evidence for a two-component superconducting order parameter in Sr2RuO4” Nature Phys. 17, 199 (2021).
  * Ichioka  _et al._ [2007] M. Ichioka, H. Adachi, T. Mizushima, and K. Machida, “Vortex state in a Fulde-Ferrell-Larkin-Ovchinnikov superconductor based on quasiclassical theory” Phys. Rev. B 76, 014503 (2007).
  * Fulde and Ferrell [1964] P. Fulde and R. A. Ferrell, “Superconductivity in a Strong Spin-Exchange Field” Phys. Rev. 135, A550 (1964).
  * Larkin and Ovchinnikov [1964] A. I. Larkin and Y. N. Ovchinnikov, Zh. Eksp. Teor. Fiz. 47, 1136 (1964), [translation: Sov. Phys. JETP 20 (1965) 762].
  * Yonezawa  _et al._ [2013] S. Yonezawa, T. Kajikawa, and Y. Maeno, “First-Order Superconducting Transition of Sr2RuO4” Phys. Rev. Lett. 110, 077003 (2013).
  * Yonezawa  _et al._ [2014] S. Yonezawa, T. Kajikawa, and Y. Maeno, “Specific-Heat Evidence of the First-Order Superconducting Transition in Sr2RuO4” J. Phys. Soc. Jpn. 83, 083706 (2014).
  * Mao  _et al._ [2000] Z. Q. Mao, Y. Maeno, S. NishiZaki, T. Akima, and T. Ishiguro, “In-Plane Anisotropy of Upper Critical Field in Sr2RuO4” Phys. Rev. Lett. 84, 991 (2000a).
  * Kittaka  _et al._ [2009] S. Kittaka, T. Nakamura, Y. Aono, S. Yonezawa, K. Ishida, and Y. Maeno, “Angular dependence of the upper critical field of Sr2RuO4” Phys. Rev. B 80, 174514 (2009).
  * Matsushita [2014] T. Matsushita,  _Flux pinning in superconductors_ (Springer, 2014).
  * Mackenzie  _et al._ [1998] A. P. Mackenzie, R. K. W. Haselwimmer, A. W. Tyler, G. G. Lonzarich, Y. Mori, S. Nishizaki, and Y. Maeno, “Extremely Strong Dependence of Superconductivity on Disorder in Sr2RuO4” Phys. Rev. Lett. 80, 161 (1998).
  * Mao  _et al._ [2000] Z. Mao, Y. Maeno, and H. Fukazawa, “Crystal growth of Sr2RuO4” Mater. Res. Bull. 35, 1813 (2000b).
  * [24] See Supplemental Material at [URL will be inserted by publisher] for [give brief description of material].
  * Yonezawa  _et al._ [2015] S. Yonezawa, T. Higuchi, Y. Sugimoto, C. Sow, and Y. Maeno, “Compact AC susceptometer for fast sample characterization down to 0.1 K” Rev. Sci. Intrum. 86, 093903 (2015).
  * Loetzsch  _et al._ [2010] R. Loetzsch, A. LÃ¼bcke, I. Uschmann, E. FÃ¶rster, V. GroÃe, M. Thuerk, T. Koettig, F. Schmidl, and P. Seidel, “The cubic to tetragonal phase transition in SrTiO3 single crystals near its surface under internal and external strains” Appl. Phys. Lett. 96, 071901 (2010).
  * Chmaissem  _et al._ [1998] O. Chmaissem, J. D. Jorgensen, H. Shaked, S. Ikeda, and Y. Maeno, “Thermal expansion and compressibility of ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$” Phys. Rev. B 57, 5067 (1998).
  * Yasui  _et al._ [2017] Y. Yasui, K. Lahabi, M. S. Anwar, Y. Nakamura, S. Yonezawa, T. Terashima, J. Aarts, and Y. Maeno, “Little-Parks oscillations with half-quantum fluxoid features in ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$ microrings” Phys. Rev. B 96, 180507 (2017).
  * Deguchi  _et al._ [2004] K. Deguchi, T. Ishiguro, and Y. Maeno, “Field-orientation dependent heat capacity measurements at low temperatures with a vector magnet system” Rev. Sci. Instrum. 75, 1188 (2004).
  * Kinjo  _et al._ [2021] K. Kinjo, M. Manago, S. Kitagawa, Z. Q. Mao, S. Yonezawa, Y. Maeno, and K. Ishida, “Superconducting Spin Smecticity Evidencing the Fulde-Ferrell-Larkin-Ovchinnikov State in Sr2RuO4” submitted (2021).
  * Yonezawa [2019] S. Yonezawa, “Nematic Superconductivity in Doped Bi2Se3 Topological Superconductors” Condensed Matter 4 (2019), ISSN 2410-3896, doi:10.3390/condmat4010002.
  * Fernandes  _et al._ [2014] R. M. Fernandes, A. V. Chubukov, and J. Schmalian, “What drives nematic order in iron-basedÂ superconductors?” Nature Physics 10, 97 (2014), ISSN 1745-2481.

Supplemental Material for
Current-induced superconducting anisotropy of Sr2RuO4
R. Araki,1,∗ T. Miyoshi,1 H. Suwa,1 E. I. Paredes Aulestia,2 K. Y. Yip,2 K. T.
Lai,2 S. K. Goh,2 Y. Maeno,1 and S. Yonezawa1,†

1Department of Physics, Kyoto University, Kyoto 606-8502, Japan
2Department of Physics, The Chinese University of Hong Kong, Shatin N.T., Hong
Kong
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
(Dated: )

Figure S1: AC susceptibility
$\chi_{\mbox{\scriptsize{ac}}}\equiv\chi^{\prime}-i\chi^{\prime\prime}$ of the
${\rm\Large Sr_{2}RuO_{4}}$ single crystal used for our device. Before the
micro-fabrication, ac susceptibility measurement at zero DC field revealed
that this thin crystal (batch: C432; 500 $\muup$m $\times$ 100 $\muup$m
$\times$ 10 $\muup$m) exhibits superconducting transition at
$T_{\mbox{\scriptsize{c}}}=1.43~{}$K, which is defined as the peak of the
imaginary part of $\chi_{\mbox{\scriptsize{ac}}}$.  Figure S2: Check of the
alignment of magnetic field to the $ab$ plane. The figure shows the polar-
angle $\theta_{H}$ dependence of the resistance $R$ measured at 0.50 K under
1.29 T and 100-$\muup$A current along the [010] direction. The curves have
vertical offsets. The red, orange and pink curves correspond to
$\phi_{H}=135^{\circ},45^{\circ}$ and $-45^{\circ}$, respectively. For all
$\phi_{H}$, $R(\theta_{H})$ curves show sharp and symmetric dips centered at
$\theta_{H}=90^{\circ}$ due to superconductivity. These data indicate that the
misalignment of the magnetic field with respect to the $ab$ plane
$(\theta_{H}=90^{\circ})$ is at most $0.1^{\circ}$.

Figure S3: Raw magneto-resistance data. The figure represents resistive
transition of our ${\rm Sr_{2}RuO_{4}}$ device under currents. The yellow,
brown, pink and black data points are obtained under $\phi_{H}=0^{\circ}$,
$45^{\circ}$, $90^{\circ}$ and $135^{\circ}$ , respectively. (a) Data under
currents along the [010] direction at 0.4 K. Similarly to the $R(H)$ curves in
Fig. 2 of the main text, $H_{\mbox{\scriptsize{c2}}}$, indicated by the
vertical arrows, exhibits two-fold anisotropy as evidenced by the differences
between the curves of $\phi_{H}=0^{\circ}$ and $90^{\circ}$, or those of
$\phi_{H}=45^{\circ}$ and $135^{\circ}$. (b) Data under currents along the
[010] direction at 1.20 K, where superconducting transition is an ordinary
second-order transition. (c) Data under currents along the [110] direction at
1.20 K.

Figure S4: Representative $H_{\mbox{\scriptsize{c2}}}$ anisotropy of ${\rm
Sr_{2}RuO_{4}}$ under in-plane current. Similarly to Fig. 3 of the main text,
(a) and (c) represent raw $H_{\mbox{\scriptsize{c2}}}$ data (circles) and
fitting result (solid curve) under current, (b) and (d) represent two-fold
components of $H_{\mbox{\scriptsize{c2}}}$ under current, extracted from the
raw $H_{\mbox{\scriptsize{c2}}}$ data by subtracting the fitted four-fold
component and the constant offset, (e) represents $I$-induced two-fold
component of $H_{\mbox{\scriptsize{c2}}}$ obtained by taking the difference
between the data in (b) and (d), and (f) represents $I$-independent two-fold
component of $H_{\mbox{\scriptsize{c2}}}$ obtained by taking the summation of
the data in (b) and (d), respectively. The sky blue, green, pink and yellow
color indicate data under current parallel to the [110], $[1\overline{1}0]$,
[010] and [100] directions, respectively. The blue and red color indicate data
evaluated from the datasets for the orthogonal and diagonal current-
directions, respectively. Column (1) $H_{\mbox{\scriptsize{c2}}}$ under
currents along the diagonal directions at 0.30 K. Column (2)
$H_{\mbox{\scriptsize{c2}}}$ under currents along the orthogonal directions at
1.20 K. Column (3) $H_{\mbox{\scriptsize{c2}}}$ under currents along the
diagonal directions at 1.20 K.  Figure S5: Examination of heating effect from
the $T$-dependence of the coefficient $a_{0}$ considered as the in-plane
average $H_{\mbox{\scriptsize{c2}}}$. (a) Gray, sky-blue, yellow and red
squares show $a_{0}$ measured with current strength $I=50,100,150$ and
$200\muup$A respectively. Gray curves shows the fitting curve obtained from
the in-plane average $H_{\mbox{\scriptsize{c2}}}$ measured with 50$\muup$A (b)
Red squares and gray curves show the same data as those in the panel (a).
Green squares show the points with $a_{0}$ measured with $I=200\muup$A shifted
onto the fitting curve. To estimate the deference between the actual sample
temperature and the measured temperature, we assume that the change in $a_{0}$
is caused only by Joule heating and the gray curves exhibits the ideal
$H_{\mbox{\scriptsize{c2}}}$-$T$ phase diagram of the sample. This estimation
indicates the temperature increase as evaluated by the shift indicated by the
gray arrows is at most 0.2 K in the lowest temperature region and less than
0.1 K above 0.4 K. Therefore, the current-heating effect has no qualitative
effects on the analyses and conclusions.  Figure S6:  $T$-dependence of the
two-fold $I$-dependent component and $I$-independent component of
$H_{\mbox{\scriptsize{c2}}}^{(2)}$. Red squares and blue diamonds indicate
values evaluated from the datasets for the orthogonal and diagonal current-
directions, respectively. $B_{2}$ and $C_{2}$ shown in the panels (b) and (c)
respectively are the same as $B_{2}$ and $C_{2}$ in Fig. 5 of the main text.
(a) Two-fold $I$-independent cosine component. This coefficient exhibits
nearly zero at all temperatures. (b) Two-fold $I$-independent sine component.
This coefficient shows about 0.005 T in the first-order transition region
($T\leq 0.8$ K) and nearly zero in the second-order transition region ($T\geq
0.8$ K). This fact implys that the sample in a high-field state gets sensitive
to inhomogeneity effect producing two-fold anisotropy. (c) Coefficient for the
$\cos 2(\phi_{H}-\phi_{I})$ component, which increases on warming. It is
consistent with vortex flow effect. (d) Coefficient for the $\sin
2(\phi_{H}-\phi_{I})$ component, exhibiting nearly zero in the whole
temperature range.