# Anomalous dissipation in KTaO3 (111) interface superconductor in the absence
of external magnetic field

Shashank Kumar Ojha  Contributed equally<EMAIL_ADDRESS>Department of
Physics, Indian Institute of Science, Bengaluru 560012, India  Prithwijit
Mandal  Contributed equally  Department of Physics, Indian Institute of
Science, Bengaluru 560012, India  Siddharth Kumar  Department of Physics,
Indian Institute of Science, Bengaluru 560012, India  Srimanta Middey
<EMAIL_ADDRESS>Department of Physics, Indian Institute of Science,
Bengaluru 560012, India

###### Abstract

Understanding the nature of dissipation mechanisms in two-dimensional (2D)
superconductors under applied perpendicular magnetic field ($B$⟂) using very
small current ($I$) excitations has been a subject of huge interest in the
last two decades. However, dissipation mechanisms at large $I$ drive remain
largely unexplored. We investigate this fundamental question for a newly
fabricated KTaO3 (111)-based interfacial superconductor, which lies in the
intermediate disorder regime. We demonstrate two distinct regimes of
dissipation across the Berezinskii Kosterlitz Thouless phase transition
temperature ($T_{\mathrm{BKT}}$). Below $T_{\mathrm{BKT}}$, $I$ driven
breaking of vortex-antivortex pairs and superconducting weak links are found
to be the major sources of dissipation. Most importantly, we uncover a new
source of dissipation arising from large electric field driven electronic
instability in the temperature range
$T_{\mathrm{BKT}}<T<T$${}_{\textnormal{C}}$ (where $T$${}_{\textnormal{C}}$ is
superconducting transition temperature), leading to a rare observation of
clockwise hysteresis loop in $I$-$V$ curve. While such behavior had been
reported earlier in type II superconductors in the presence of $B$⟂,
experimental demonstration in the absence of external $B$ remains elusive so
far. Our results not only reveal the microscopic structure of 2D
superconductors close to BKT transition, but also deepens the understanding of
how a BKT system is ultimately transferred to a normal state by increasing
$I$.

Answering how robust is the superconducting condensate against external
perturbation requires an intense knowledge about the role of disorder and
dimensionality of the system tinkham:2004p . With the recent advent of high
quality sample fabrication techniques, there has been a surge of activity in
the field of two dimensional (2D) superconductors in recent times
Saito:2016p16094 ; Qiu:2021p2006124 . These systems undergo Berezinskii
Kosterlitz Thouless (BKT) type of phase transition berezinskii:1971P493 ;
Kosterlitz:1973P1181 . In a pure BKT system, current ($I$) driven unbinding of
thermally generated bound vortex-antivortex pairs is the major cause of
dissipation under zero magnetic field ($B$) below $T$${}_{\textnormal{BKT}}$
Kosterlitz:1973P1181 ; Beasley:1979p1165 ; Epstein:1981p534 . This phenomenon
happens over small $I$ drive and has been thoroughly investigated. Beyond this
limit, there exists a large region in the $I$-$T$ phase space below
$T$${}_{\textnormal{BKT}}$, which is characterized by a dilute plasma of
vortices and antivortices. This region is microscopically similar to the
regime of proliferated free vortices in temperature range
$T$${}_{\textnormal{BKT}}$ $\leq$ $T$ $\leq$$T$${}_{\textnormal{C}}$
Resnik:1981p1542 . Unfortunately, studying dissipation in these two regimes of
free vortices has been highly challenging due to non-trivial dynamics of
massively generated non-equilibrium quasi particles from the collapse of ultra
fast moving vortices under large $I$ drive larkin1975:960 ; samoilov:1995p4118
.

The presence of disorder in samples, which is inevitable in reality, further
complicates this problem by turning BKT system inhomogeneous. Such
inhomogeneities might range from atomic level point defects to macroscopically
phase separated regions Caprara:2012p196401 ; Caprara:2014p014002 ;
Caprara:2013p020504 ; Ariando:2011p188 . While the former determines the
vortex pinning strength, the latter very often leads to a network of
superconducting puddles joined by weak superconducting links. Such weak links
are very fragile under large electric field and are another competing source
of dissipation under large $I$ in the absence of $B$ Likharev:1979p101 . In
the past, much of the attention has been paid to understanding the dissipation
in either very clean or dirty system. Notably, all of these measurements have
been primarily performed in the presence of $B$ (under very small $I$) and
very little is known about the nature of dissipation under large $I$
Benyamini:2019p947 ; Saito:2020p074003 . Further, what happens in the
intermediate disordered regime also remains an open question. Understanding
dissipation in such unexplored regimes is not only critical to answering some
of the fundamental questions about BKT phase transition in inhomogeneous
system but will also be pivotal in realizing next generation applications such
as superconducting digital memory, cavities for particle accelerators and THz
radiation sources Embon:2017p85 ; Devoret:2013p1169 ; Gurevich:2008p104501 ;
Welp:2013p9 .

Figure 1: Device geometry and transport behavior of 2DEG at AlOx/KTO (111)
interface a. In a pure ionic picture, (111) oriented KTaO3 can be considered
as a sequence of alternating [KO3]5- and Ta5+ planes. Arrangement of
Ta${}^{\text{+5}}$ ions in two adjacent (111) planes (labelled by Ta-I and Ta-
II) has been shown in the upper panel Xiao:2011p596 . (Lower panel) Schematics
of two Hall bars made on a AlOx/KTO (111) heterostructure. b. Temperature-
dependent $R_{S}$ of 2DEG for the two Hall bars for a 7 nm AlOx/KTO (111)
sample. Inset shows a magnified view around the superconducting transition
temperature. Interestingly, the normal state $R_{S}$ ($T$) shows a non Fermi
liquid behavior ($R_{S}$ $\propto$ $T$ α where $\alpha$ $<$ 2) in a broad
range of temperatures from 75 K to 300 K with $\alpha$ $=$ 1.5 and 1.3 for
current along [11$\bar{2}$] and [1$\bar{1}$0] respectively. This behavior is
in sharp contrast with the $T$ 3 behavior observed in bulk electron doped KTO,
where no superconductivity has been observed (see supplementary Fig. 2). Low
temperature variation of $R_{S}$ under $B_{\perp}$ has been shown in c. (from
0 T to 0.1 T) and d. (from 0.2 T to 9 T) for the Hall bar along [11$\bar{2}$].
Dotted lines in (d) show logarithmic dependence of $R_{S}$ with the
temperature near the avoided SIT transition.

Oxide heterostructure based interfacial superconductor turns out to be a
promising platform for such an investigation as they intrinsically lie near
the intermediate disorder regime and hence would allow for simultaneous
investigation of dissipation pertaining to a pure BKT system and also arising
from the inhomogeneous electronic structure using a single sample
Devarakonda:2020p231 . Recently, superconductivity (SC) has been discovered at
the interface and surface of (111) oriented KTaO3 (KTO) with
$T$${}_{\textnormal{C}}\sim$ 1.5-2.2 K Changjiang:2021p716 ; Zheng:2021p721 ;
Ren:2022peabn4273 ; mallik:2022 . This $T$${}_{\textnormal{C}}$ is one order
of magnitude higher than any STO based heterostructure Reyren:2007p1196  and
hence has generated tremendous excitement in the field of interfacial SC. Most
importantly, these samples are a bit cleaner compared to STO based
superconducting interfaces. As a result, KTO (111) based superconductors lie
exactly at the boundary between clean and dirty limits, making it a unique
system for investigating dissipation in 2D superconductors. Interestingly, SC
was also found to be strongly influenced by the choice of over-layer grown on
KTO (111) substrate. For example, the presence of a magnetic element in the
overlayer could lead to a stripe order near superconducting transition
Changjiang:2021p716 . Such a situation would lead to additional directional
anisotropy apart from the anisotropy arising from the two inequivalent
crystallographic directions [11$\bar{2}$] and [1$\bar{1}$0] for (111) oriented
KTO (Fig. 1a).

In order to avoid the additional influence of overlayer on nature of
dissipation, we have fabricated a new superconducting interface by ablating
non-magnetic Al2O3 on KTO (111) substrate [dimension 5 mm $\crossproduct$ 5 mm
$\crossproduct$ 0.5 mm] by pulsed laser deposition technique (see Methods and
Supplementary Fig. 1). The resultant film is amorphous. For electrical
transport measurements, two Hall bars were patterned along two in-equivalent
crystallographic directions : [11$\bar{2}$] and [1$\bar{1}$0] (Fig. 1a) by
selective scratching of film deep into the substrate CuiZu:2013p167 . Fig. 1b
shows the sheet resistance ($R_{S}$) vs. temperature plot of a 7 nm AlOx/KTO
(111) sample. As evident, the interface exhibits metallic behavior up to low
temperature confirming the formation of two dimensional electron gas (2DEG).
The origin of the 2DEG is connected to the creation of oxygen vacancies (OVs)
Ojha:2021p054008  within the KTO substrate. Further, a clear superconducting
transition is observed with little anisotropy e.g. $T$${}_{\textnormal{C}}$ =
1.55 K and 1.51 K for current along [11$\bar{2}$] and [1$\bar{1}$0]
respectively (inset of Fig. 1b) ($T$${}_{\textnormal{C}}$ is estimated from
the condition $R_{S}$($T$${}_{\textnormal{C}}$) = 0.5$\crossproduct$$R_{S}$(5
K)). While the value of $T$${}_{\textnormal{C}}$ is very similar to the
previous reports Changjiang:2021p716 ; Zheng:2021p721 , the observation of
little anisotropy is in sharp contrast with the observation of large in-plane
anisotropy in EuO/KTO (111) near the superconducting $T$${}_{\textnormal{C}}$
Changjiang:2021p716 .

Figure 2: Critical field, weak anti-localization, and the extent of disorder
a. Temperature dependence of upper critical field ($B$${}_{\text{C}}$⟂) under
$B$⟂ for $I$ along [11$\bar{2}$] and [1$\bar{1}$0]. The solid line denotes
fitting with Ginzburg-Landau theory. b. Temperature dependence of upper
critical field ($B$${}_{\text{C}}$∥) under $B\parallel$ for $I$ along
[11$\bar{2}$] and [1$\bar{1}$0]. Further, $B$ is parallel to the current
direction. The solid line denotes fitting with Tinkham’s model. c. Sheet
conductance difference ($\Delta$$\sigma$=$\sigma$($B$)-$\sigma$($B$=0),
$\sigma$=1/$R_{S}$ ($B$)) in the units of $e$2/$\pi$$h$ for the Hall bar with
$I$ along [11$\bar{2}$]. The black solid curves show the fitting with ILP
(Iordanskii, Lyanda-Geller, and Pikus) theory iordanskii:1994p199 ;
Ojha2020:p2000021  (without considering linear Rashba term) including a
classical $B^{2}$ term (also see Supplementary note S7 for fitting details).
d. Phase diagram of several superconducting compounds categorized by their
extent of 2D character and cleanliness. 2D character is resembled by the
anisotropy of critical field defined by
($B$${}_{\text{C}}$⟂/$B$${}_{\text{C}}$∥) and extent of disorder is quantified
by the ratio between phase coherence length and electronic mean free path
($\xi_{0}$/$l$). Assuming a single isotropic band in 2D, $l$${}_{\text{mfp}}$
is given by $l$${}_{\text{mfp}}$=$h$/($e$2$k_{\text{F}}$$R_{S}$), where
$k_{\text{F}}$=(2$\pi$$n_{s}$)1/2 is the Fermi wave vector and $n_{s}$ is the
sheet carrier density. From the measured $n_{s}$ (at 5 K) and $R_{S}$ (at 5
K), the $l$${}_{\text{mfp}}$ is estimated to be $\sim$12 nm for the present
case. The value of all the parameters for other compounds have been largely
taken from the reference Devarakonda:2020p231  except for the LaTiO3/STO
interface which has been taken from Biscaras:2010p89 . AlOx/KTO (111) is
located very near to the boundary between clean and dirty limits, denoted by a
horizontal solid line.

To further probe the nature of this new superconducting system, temperature
dependent measurements of $R_{S}$ ($T$) under perpendicular ($B_{\perp}$) and
parallel ($B_{\parallel}$) magnetic fields have been carried out. Fig. 1c
shows one representative set of data for $I$ $\parallel$ [11$\bar{2}$] under
low $B_{\perp}$ [$R_{S}$ ($T$) under $B_{\perp}$ for $I$ along [1$\bar{1}$0]
has been shown in Supplementary Fig. 3]. Clearly, the SC is disrupted at very
low $B_{\perp}$, which can be attributed to the low pinning of vortices in 2D
superconductors. Upon increasing the $B_{\perp}$, the sample undergoes an
avoided superconductor to insulator transition (SIT) around $R_{S}$ $\sim$ 1
k$\Omega$/sq. (see Fig. 1d). This result is markedly distinct from the
conventional theoretical framework that predicts a direct transition to an
insulating state when the normal state $R_{S}$ approaches the quantum of
resistance $h$/4$e^{2}$ $=$ 6.4 k$\Omega$/sq. in the limit $T$ $\rightarrow$ 0
($h$ is the Planck’s constant and -$e$ is the electron’s charge)
Goldman:1998p39 ; Haviland:1989p2180 . Such avoided SIT is generally observed
in 2D superconductors with relatively less disorder and has proven monumental
in studying phases beyond the Landau Fermi liquid theory
Kapitulnik:2019p011002 . Interestingly, at higher $B_{\perp}$ and lower $T$,
our sample exhibits a logarithmic dependence of $R_{S}$ on $T$. This
logarithmic divergence is incompatible with the prediction of weak
localization correction in 2D or Kondo effect Kapitulnik:2019p011002  and is
ultimately connected with the emergent granular nature of our conducting
interface Beloborodov:2007p469 ; Zhang:2021p30 . In the later part of this
article, we will establish the microscopic origin behind such a granular
structure and further study about its impact on dissipation under large $I$.

Fig. 2a shows the temperature dependence of upper critical field
($B_{C_{\perp}}$), obtained by tracking the evolution of
$T$${}_{\textnormal{C}}$ with $B_{\perp}$ in $R_{S}$ vs. $T$ plot. An
appreciable difference in magnitude of $B_{C_{\perp}}$ is observed for $I$
along [11$\bar{2}$] and [1$\bar{1}$0]. Higher value of $B_{C_{\perp}}$ along
[11$\bar{2}$] is consistent with the observation of higher
$T$${}_{\textnormal{C}}$ for $I$ $\parallel$ to [11$\bar{2}$]. The solid line
shows fitting with the Ginzburg-Landau (G-L) theory which predicts linear $T$
behavior of $B_{C_{\perp}}$ given by

$B_{\textnormal{C}_{\perp}}=\frac{\Phi_{0}(1-T/T_{\textnormal{C}})}{2\pi(\xi_{0})^{2}}$
(1)

where $\Phi_{0}$ is the magnetic flux quantum and $\xi_{0}$ is the G-L
coherence length at $T$ = 0 K. $\xi_{0}$ from fitting is found to be $\sim$
23.4 nm and 21.4 nm for $I$ along [11$\bar{2}$] and [1$\bar{1}$0],
respectively.

To determine the thickness of the superconducting region, upper critical field
($B$${}_{\text{C}}$∥) has also been determined under $B_{\parallel}$ with two
different configurations: $I$ $\parallel$ $B$ and $I$ $\perp$ $B$. Fig. 2b
shows the temperature dependence of $B$${}_{\text{C}}$∥ for the case with
$I$$\parallel$$B$ (see Supplementary Fig. 4 to Fig. 6 for raw $R_{S}$ vs. $T$
plot at different fixed $B$ and $B$${}_{\text{C}}$∥ vs. $T$ plot for $I$
$\perp$ $B$). Similar to the out of plane measurement, the magnitude of
$B$${}_{\text{C}}$∥ is found to be larger for the Hall bar with $I$ along
[11$\bar{2}$]. The temperature dependence of $B$${}_{\text{C}}$∥ shows a
characteristic square-root dependence (shown by the solid lines in Fig. 2b).
This is consistent with the Tinkham’s model Tinkham:1963p2413  with

$B_{\textnormal{C}_{\parallel}}=\frac{\Phi_{0}[12(1-T/T_{\textnormal{C}})]^{1/2}}{2\pi\textit{d}\xi_{0}}$
(2)

where $d$ is the effective thickness of the superconducting region. The
estimated $d$ ($\sim$ 5 nm) is much less than $\xi$0, signifying 2D nature of
the SC at the AlOx/KTO (111) interface. Interestingly, the extrapolated value
of $B$${}_{\text{C}}$∥ to 0 K ($\sim$10 T) is much larger than Clogston
Chandrasekhar (CC) limit Chandrasekhar:1962p7 ; Clogston:1962p266 . Such a
large value of $B$${}_{\text{C}}$∥ is generally expected in systems with a
strong spin orbit coupling (SOC) Werthamer:1966p295  and the observation of
weak antilocalization (WAL) characteristics in longitudinal magnetoconductance
data within the normal phase (see Fig. 2c) demonstrates the importance of SOC
in the present case.

Figure 3: Current voltage ($I$-$V$) characteristics and determination of
$T$${}_{\textnormal{BKT}}$ a. Temperature dependent $I$-$V$ curves measured in
current bias mode for the Hall bar along [11$\bar{2}$]. Solid and dotted
curves denote forward and backward sweeps, respectively. Curves have been
shifted upward for visual clarity. b. $I$-$V$ curves in logarithmic scale
during the forward sweep. The solid black line shows the fit with the power
law given by $V$ $\propto$ $I$α. A dotted gray line corresponds to $\alpha$=3
where the BKT transition takes place. c. Temperature dependence of $\alpha$
for $I$ along [11$\bar{2}$] and [1$\bar{1}$0]. A dotted green line shows a
constant line for $\alpha$=3. d. To estimate $T_{\textnormal{BKT}}$ using the
Halperin-Nelson model Halperin:1979p599 ; Minnhagen:1987p1001 ,
[dln($R_{S}$)/d$T$]-2/3 has been plotted as a function of $T$, near the
superconducting transition temperature. By finding the $x$ axis intercept of
this plot, we find $T$${}_{\textnormal{BKT}}$ $\sim$ 1.51 K and 1.43 K for the
Hall bar along [11$\bar{2}$] and [1$\bar{1}$0], respectively. These values are
very close to the $T_{\textnormal{BKT}}$, obtained in (c). e. Temperature
dependent $I$-$V$ curves measured in current bias mode for another sample with
14 nm AlOx thickness. Solid and dotted curves denote forward and backward
sweeps, respectively. Curves have been shifted upward for visual clarity.

In order to examine the extent of disorder in our system, we have estimated
the ratio of $\xi_{0}$ and the electronic mean free path $l$${}_{\text{mfp}}$.
The ratio is close to 2, emphasizing that the SC at AlOx/KTO (111) interface
falls in the intermediate disorder regime (see Fig. 2d, which summarizes
several superconductors on the basis of disorder and 2D character). The
presence of OVs within KTO in the present heterostructure are natural source
of inhomogeneities. Clustering of OVs can also lead to a very local
inhomogeneous electronic structure in the real space Ojha:2021p085120 . Apart
from such local inhomogeneities, there is another source of inhomogeneity,
which happens at a much larger scale, known as electronic phase separation
(EPS) Caprara:2012p196401 ; Caprara:2014p014002 ; Caprara:2013p020504 ;
Ariando:2011p188 . EPS has been routinely observed in STO based 2DEGs and is
very often associated with the presence of multi carriers at the interface.
The observation of two types of electrons with densities $n_{1}$ and $n_{2}$
with mobility $\mu_{1}$, and $\mu_{2}$, respectively ($n_{1}>>n_{2}$ and
$\mu_{1}<\mu_{2}$ ) in our Hall effect measurements (see Supplementary note
S8-S10) strongly suggests that a similar scenario might also be applicable in
our samples as well. As a general consequence of this, superconducting puddles
(joined by weak links) would emerge naturally in real space Chen:2018p4008 ,
making the SC strongly inhomogeneous. We believe this is the dominant cause
for the observed granular nature of our system. We also note that EPS could
also arise due to the Rashba SOC Caprara:2012p196401  which is also quite
generic to our system.

Having established the nature of inhomogeneities in our 2D superconducting
system, we explore the nature of dissipation under current bias in the absence
of $B$. For this, comprehensive $I$-$V$ measurements have been performed. Fig.
3a shows the $I$-$V$ curves (in the positive quadrant) taken in forward and
backward sweeps at several fixed temperatures from 1.26 K to 10 K for $I$
along [11$\bar{2}$] (see supplementary Fig. 12 for data of $I$ along
[1$\bar{1}$0]). All data (except the curve of 1.26 K) has been shifted
vertically upwards for visual clarity. Broadly four distinct regimes can be
identified in the $I$-$V$ curve at the lowest temperature (1.26 K) of our
measurements: (1) At small currents (up to 60-70 $\mu$A), voltage drop is
almost independent of $I$ signifying a dissipation-less regime. (2) Above
this, a non-linear behavior appears in a very short window from $\sim$80
$\mu$A-110 $\mu$A. (3) This regime then translates into a region (from 110
$\mu$A to 175 $\mu$A), where the majority of the dissipation happens as
observed by a large change in the voltage drop. (4) Above 175 $\mu$A, the
magnitude of $V$ grows almost in proportion to the applied $I$ and finally
enters into the regime of ohmic dissipation. All these different regions in
$I$-$V$ characteristics are strongly $T$ dependent. The first and fourth
regimes are well understood tinkham:2004p  and are skipped from further
discussions for the sake of brevity.

We first discuss the origin of non-linear $I$-$V$, observed just above the
dissipation less regime. Notably, this regime corresponds to the intrinsic
dissipation of a BKT system which is characterized by power law behavior
($V\propto I^{\alpha}$) arising from $I$ driven unbinding of thermally
generated vortex-antivortex pairs near the BKT transition Beasley:1979p1165 ;
Epstein:1981p534 . This behavior becomes much more evident in the logarithmic
plot (Fig. 3b), where power law translates into a linear behavior. The value
of $\alpha$ becomes exactly 3 at the $T_{\textnormal{BKT}}$ (shown by a dotted
gray line ($V$ $\propto$ $I$3) in Fig. 3b) and is routinely used to trace out
BKT phase transition in 2D superconductors. Fig. 3c shows the temperature
dependence of $\alpha$ obtained from the linear fit in Fig. 3b. From the
crossover of $\alpha$ around 3, $T$${}_{\textnormal{BKT}}$ is found out to be
1.39 K and 1.30 K for the Hall bar along [11$\bar{2}$] and [1$\bar{1}$0],
respectively. These values of $T$${}_{\textnormal{BKT}}$ are also very close
to the estimated values (see Fig. 3d) using the Halperin-Nelson model
($R_{S}$=$R_{0}$exp[-$b$/($T$-$T$${}_{\textnormal{BKT}}$)1/2] where $b$ is the
vortex-antivortex interaction strength) Halperin:1979p599 ;
Minnhagen:1987p1001 .

We next focus on the nature of dissipation beyond power law regime. At the
lowest temperature of our measurement 1.26 K, which is below
$T$${}_{\textnormal{BKT}}$, dissipation happens via several discrete jumps in
voltage during the forward sweep. This is much more evident from the $dV$/$dI$
plot shown in supplementary Fig. 11. Such discrete behavior reminds us of
phase slip events, which are generally observed in one dimensional (1D)
superconducting wire tinkham:2004p . We attribute discrete jumps in our sample
to the breaking of quasi 1D superconducting weak links joining superconducting
puddles Likharev:1979p101 . Weak links are very fragile under electric field
and break at a much lesser current density than the required amount to break
the global SC. However, such discrete voltage jumps could also arise from the
movement of free vortices across the sample boundary. Interestingly, an
anticlockwise hysteresis is observed upon backward $I$ sweep. This could be
understood from the thermal effect argument wherein the local temperature of
the sample goes higher (due to Joule heating) than the thermal bath, once it’s
driven into the normal state Gurevich:1987p941 ; Likharev:1979p101 . Since the
critical current ($I_{c}$) of superconductors decreases with an increase in
$T$ (supplementary Fig. 11), sweeping back $I$ immediately would lead to a
shift of $I$-$V$ curve towards left, leading to anticlockwise hysteresis. With
the increase in $T$, the position of the hysteresis is also found to shift
towards lower $I$. Surprisingly, the nature of hysteresis changes completely
from anticlockwise to clockwise above a certain temperature (highlighted by
arrows in Fig. 3a). Fig. 3e shows the $I$-$V$ curves (in the positive
quadrant) at several fixed temperatures for another sample with 14 nm AlOx
thickness (for $I$ along [11$\bar{2}$]) exhibiting similar clockwise
hysteresis above a certain temperature. Such clockwise hysteresis is extremely
rare samoilov:1995p4118  and has never been observed in any interfacial
superconductors to the best of our knowledge.

To visualize this drastic change in $I$-$V$ hysteresis, we further plot the
maximum width of hysteresis, defined as
$\Delta$$I$c=($I$c)${}_{\text{forward}}$-($I$c)${}_{\text{backward}}$
[($I$c)${}_{\text{forward}}$ and ($I$c)${}_{\text{backward}}$ are the values
of critical current in the middle of hysteresis in the forward and backward
sweep, respectively] as a function of $T$ (Fig. 4a and Fig. 4b). The upper and
lower panels in Fig. 4a correspond to $\Delta$$I$c for $I$ along [11$\bar{2}$]
and [1$\bar{1}$0], respectively. Fig. 4b contains a similar set of data for
another sample with AlOx thickness $\sim$14 nm (other characteristics of this
sample have been shown in Supplementary note S14-S16). As clearly evident,
hysteresis always changes its sign around the $T$${}_{\textnormal{BKT}}$ and
vanishes around $T$${}_{\textnormal{C}}$ in all the four Hall bars, that we
have investigated in this work. Such a peculiar behavior goes beyond the Joule
heating argument and strongly emphasizes presence of an additional dissipation
mechanism in the temperature range $T$${}_{\textnormal{BKT}}$ $\leq$ $T$
$\leq$$T$${}_{\textnormal{C}}$ at large $I$ drive.

Figure 4: Sign change of hysteresis across BKT transition a. Maximum width of
hysteresis
($\Delta$$I$c=($I$c)${}_{\text{forward}}$-($I$c)${}_{\text{backward}}$) for 7
nm AlOx film on KTO (111). The upper and lower panels correspond to $I$ along
[11$\bar{2}$] and [1$\bar{1}$0]. b. Similar data for another sample with
double the thickness (14 nm) of AlOx.

In order to understand such rare clockwise hystersis, we consider Larkin and
Ovchinnikov’s (LO) model of non-linear $I$-$V$ larkin1975:960 , which was
originally formulated to deal with a highly non-equilibrium problem of ultra-
fast moving vortices under large $I$ drive in the mixed state of type II
superconductors. It was predicted that, at large vortex velocities,
quasiparticles at the core of the vortex can reach energies above the
superconducting energy gap and ultimately diffuse away from the core,
resulting in an electronic instability. Such a situation would lead to a non-
linear $I$-$V$ characteristics samoilov:1995p4118 ; Doettinger:1994p1691 ,
which would be the strongest at temperatures close to the
$T$${}_{\textnormal{C}}$ in the presence of very weak $B$
($B$$\ll$$B$${}_{\text{C}}$⟂). It is well known that even in the absence of
$B$, 2D superconductors in the temperature range $T$${}_{\textnormal{BKT}}$
$\leq$ $T$ $\leq$$T$${}_{\textnormal{C}}$ are known to be microscopically
similar to a type II superconductor in the mixed state owing to the presence
of thermally generated free vortices above $T$${}_{\textnormal{BKT}}$
Resnik:1981p1542 . This immediately suggests an exciting possibility of
observing LO type of electronic instability in 2D superconductors close to
$T$${}_{\textnormal{C}}$ even in the absence of externally applied $B$. We
also emphasize that while all of our $I$-$V$ measurements have been performed
in the absence of $B$, the self field from the applied $I$ in the sample
should be sufficient to meet the criteria for the weak $B$ limit of LO model.
Under these circumstances, functional form for non-linear $I$-$V$ is given by

$I-I_{c}=\bigg{[}\frac{V}{1+(V/V^{*})^{2}}+V\bigg{(}1-\frac{T}{T_{c}}\bigg{)}^{1/2}\bigg{]}\frac{1}{R_{f}}$
(3)

where $R_{f}$ is the flux flow resistance and $V$∗ is the critical voltage
given by

$V^{*}=\frac{D^{1/2}[14\xi(3)]^{1/4}(1-T/T_{\textnormal{C}})^{1/4}BL}{(\pi\tau_{e})^{1/2}}$
(4)

Here $D$ is the diffusion coefficient, $\tau$e is the inelastic electron
scattering time, $\xi$ is the Riemann zeta function and $L$ is the length
between voltage probes. It was further proposed that, once the system is
driven into the normal resistive state in the forward sweep, the electron-
electron (inelastic) scattering rate is higher (smaller $\tau$e)
samoilov:1995p4118 . This would mean that during the backward sweep the value
of $V$∗ will be higher ($V$∗ $\sim$ $\tau$e-1/2) than that of the forward
sweep. This would automatically pull the $I$-$V$ curve towards the right
leading to a clockwise hysteresis. We believe such a hysteretic $\tau$e is the
dominant cause behind the observation of clockwise $I$-$V$ in our samples.
What remains unclear at this stage is the in-plane anisotropy observed in the
onset of clockwise hysteresis e.g., there is a delay in the onset of clockwise
hysteresis for $I$ along [1$\bar{1}$0]. This could arise from the in-plane
anisotropy of critical vortex velocity for the onset of electronic
instability. Such an observation is beyond the LO theory and calls for further
investigations in the future. Since the vortex structure in BKT system is
strongly influenced by the presence of strong SOC Devreese:2014p165304 , an
extention of LO theory in presence of SOC and finite heating effects will be
essential to understand such non-trivial feature. In Fig. 5 we show a detailed
$I$-$T$ phase diagram describing various active dissipation mechanisms in the
absence of the $B$.

Figure 5: Current temperature ($I$-$T$) phase diagram at $B$ = 0 $I$-$T$ phase
diagram constructed using the parameters for Hall bar along [11$\bar{2}$] for
7 nm AlOx/KTO(111) sample. Below the BKT transition, the sample is
characterized by randomly distributed superconducting puddles joined by weak
superconducting links. Increasing $I$ first lead to unbinding of vortex-
antivortex pairs. This transition is shown by the square symbol. Further
increase in $I$ leads to breaking of weak links. Diamond symbol marks the
locus of current just before the closure of hysteresis in the forward sweep
and triangle symbol defines the onset of clockwise hysteresis above
$T$${}_{\textnormal{BKT}}$.

In conclusion, we have demonstrated a new route to achieve 2D
superconductivity in (111)-oriented KTO substrate by ablating Al2O3 using
pulsed laser deposition. The underlying interface is found to be intrinsically
inhomogeneous in nature, which is attributed to the presence of two types of
carriers. Interestingly, one of the bands is extremely dilute ($n_{2}$ $\sim$
109cm-2) in nature. While, it remains unclear at this stage whether electrons
in such a dilute band take part in SC or not, such a possibility would point
towards an unusual pairing mechanism beyond BCS theory. Scanning tunnelling
spectroscopy measurements will be monumental in understanding the exact nature
of inhomogeneous SC.

Further, we uncover a rich variety of dissipation mechanisms close to the BKT
transition. Observation of electronic instability induced dissipation (between
$T$${}_{\textnormal{BKT}}$ $\leq$ $T$ $\leq$$T$${}_{\textnormal{C}}$) could be
ultimately traced back to the presence of proliferated free vortices above
$T$${}_{\textnormal{BKT}}$, which is quite generic to the BKT phase transition
Hadzibabic:2006P1118 . Future studies will focus on measurements beyond the
intermediate disorder regime under simultaneous top and bottom gate, which
will provide an independent investigation of the role of disorder and carrier
density in determining the nature of dissipation under large $I$ drive. We
believe that the observation of electronic instability near the
$T$${}_{\textnormal{C}}$ is not particularly limited to the oxide interfaces
and it would be interesting to explore this highly non-equilibrium phenomenon
in other 2D superconducting platforms, e.g. transition metal dichalcogenides,
twisted bilayer graphene etc.

## References

  * (1) Tinkham, M.  _Introduction to superconductivity_ (Courier Corporation, 2004).
  * (2) Saito, Y., Nojima, T. & Iwasa, Y.  Highly crystalline 2d superconductors.  _Nature Reviews Materials_ 2, 16094 (2016).  https://doi.org/10.1038/natrevmats.2016.94.
  * (3) Qiu, D. _et al._ Recent advances in 2d superconductors.  _Advanced Materials_ 33, 2006124 (2021).  https://doi.org/10.1002/adma.202006124.
  * (4) Berezinskii, V.  Destruction of long-range order in one-dimensional and two-dimensional systems having a continuous symmetry group i. classical systems.  _Sov. Phys. JETP_ 32, 493–500 (1971).
  * (5) Kosterlitz, J. M. & Thouless, D. J.  Ordering, metastability and phase transitions in two-dimensional systems.  _Journal of Physics C: Solid State Physics_ 6, 1181–1203 (1973).  https://doi.org/10.1088/0022-3719/6/7/010.
  * (6) Beasley, M. R., Mooij, J. E. & Orlando, T. P.  Possibility of vortex-antivortex pair dissociation in two-dimensional superconductors.  _Phys. Rev. Lett._ 42, 1165–1168 (1979).  https://link.aps.org/doi/10.1103/PhysRevLett.42.1165.
  * (7) Epstein, K., Goldman, A. M. & Kadin, A. M.  Vortex-antivortex pair dissociation in two-dimensional superconductors.  _Phys. Rev. Lett._ 47, 534–537 (1981).  https://link.aps.org/doi/10.1103/PhysRevLett.47.534.
  * (8) Resnick, D. J., Garland, J. C., Boyd, J. T., Shoemaker, S. & Newrock, R. S.  Kosterlitz-thouless transition in proximity-coupled superconducting arrays.  _Phys. Rev. Lett._ 47, 1542–1545 (1981).  https://link.aps.org/doi/10.1103/PhysRevLett.47.1542.
  * (9) Larkin, A. & Ovchinnikov, Y.  Nonlinear conductivity of superconductors in the mixed state.  _Sov. Phys. JETP_ 41, 960–965 (1975).
  * (10) Samoilov, A., Konczykowski, M., Yeh, N.-C., Berry, S. & Tsuei, C.  Electric-field-induced electronic instability in amorphous mo 3 si superconducting films.  _Physical review letters_ 75, 4118 (1995).
  * (11) Caprara, S., Peronaci, F. & Grilli, M.  Intrinsic instability of electronic interfaces with strong rashba coupling.  _Phys. Rev. Lett._ 109, 196401 (2012).  https://link.aps.org/doi/10.1103/PhysRevLett.109.196401.
  * (12) Caprara, S. _et al._ Inhomogeneous multi carrier superconductivity at laxo¡sub¿3¡/sub¿/srtio¡sub¿3¡/sub¿(x = al or ti) oxide interfaces.  _Superconductor Science and Technology_ 28, 014002 (2014).  https://doi.org/10.1088/0953-2048/28/1/014002.
  * (13) Caprara, S. _et al._ Multiband superconductivity and nanoscale inhomogeneity at oxide interfaces.  _Phys. Rev. B_ 88, 020504 (2013).  https://link.aps.org/doi/10.1103/PhysRevB.88.020504.
  * (14) Ariando _et al._ Electronic phase separation at the laalo3/srtio3 interface.  _Nature Communications_ 2, 188 (2011).  https://doi.org/10.1038/ncomms1192.
  * (15) Likharev, K. K.  Superconducting weak links.  _Rev. Mod. Phys._ 51, 101–159 (1979).  https://link.aps.org/doi/10.1103/RevModPhys.51.101.
  * (16) Benyamini, A. _et al._ Fragility of the dissipationless state in clean two-dimensional superconductors.  _Nature Physics_ 15, 947–953 (2019).  https://doi.org/10.1038/s41567-019-0571-z.
  * (17) Saito, Y., Itahashi, Y. M., Nojima, T. & Iwasa, Y.  Dynamical vortex phase diagram of two-dimensional superconductivity in gated $\mathrm{Mo}{\mathrm{s}}_{2}$.  _Phys. Rev. Materials_ 4, 074003 (2020).  https://link.aps.org/doi/10.1103/PhysRevMaterials.4.074003.
  * (18) Embon, L. _et al._ Imaging of super-fast dynamics and flow instabilities of superconducting vortices.  _Nature Communications_ 8, 85 (2017).  https://doi.org/10.1038/s41467-017-00089-3.
  * (19) Devoret, M. H. & Schoelkopf, R. J.  Superconducting circuits for quantum information: An outlook.  _Science_ 339, 1169–1174 (2013).  https://www.science.org/doi/abs/10.1126/science.1231930.  eprint https://www.science.org/doi/pdf/10.1126/science.1231930.
  * (20) Gurevich, A. & Ciovati, G.  Dynamics of vortex penetration, jumpwise instabilities, and nonlinear surface resistance of type-ii superconductors in strong rf fields.  _Phys. Rev. B_ 77, 104501 (2008).  https://link.aps.org/doi/10.1103/PhysRevB.77.104501.
  * (21) Welp, U., Kadowaki, K. & Kleiner, R.  Superconducting emitters of thz radiation.  _Nature Photonics_ 7, 702–710 (2013).  https://doi.org/10.1038/nphoton.2013.216.
  * (22) Xiao, D., Zhu, W., Ran, Y., Nagaosa, N. & Okamoto, S.  Interface engineering of quantum hall effects in digital transition metal oxide heterostructures.  _Nature Communications_ 2, 596 (2011).  https://doi.org/10.1038/ncomms1602.
  * (23) A., D. _et al._ Clean 2d superconductivity in a bulk van der waals superlattice.  _Science_ 370, 231–236 (2020).  https://doi.org/10.1126/science.aaz6643.
  * (24) Changjiang, L. _et al._ Two-dimensional superconductivity and anisotropic transport at ktao3 (111) interfaces.  _Science_ 371, 716–721 (2021).  https://doi.org/10.1126/science.aba5511.
  * (25) Zheng, C. _et al._ Electric field control of superconductivity at the laalo3/ktao3(111) interface.  _Science_ 372, 721–724 (2021).  https://doi.org/10.1126/science.abb3848.
  * (26) Ren, T. _et al._ Two-dimensional superconductivity at the surfaces of ktao¡sub¿3¡/sub¿ gated with ionic liquid.  _Science Advances_ 8, eabn4273 (2022).  https://www.science.org/doi/abs/10.1126/sciadv.abn4273.  eprint https://www.science.org/doi/pdf/10.1126/sciadv.abn4273.
  * (27) Mallik, S. _et al._ Superfluid stiffness of a ktao3-based two-dimensional electron gas.  _arXiv preprint arXiv:2204.09094_ (2022).
  * (28) Reyren, N. _et al._ Superconducting interfaces between insulating oxides.  _Science_ 317, 1196–1199 (2007).
  * (29) Iordanskii, S., Lyanda-Geller, Y. B. & Pikus, G.  Weak localization in quantum wells with spin-orbit interaction.  _ZhETF Pisma Redaktsiiu_ 60, 199 (1994).
  * (30) Ojha, S. K. _et al._ Oxygen vacancy-induced topological hall effect in a nonmagnetic band insulator.  _Advanced Quantum Technologies_ 3, 2000021 (2020).  https://doi.org/10.1002/qute.202000021.
  * (31) Biscaras, J. _et al._ Two-dimensional superconductivity at a mott insulator/band insulator interface latio3/srtio3.  _Nature Communications_ 1, 89 (2010).  https://doi.org/10.1038/ncomms1084.
  * (32) Cui-Zu, C. _et al._ Experimental observation of the quantum alous hall effect in a magnetic topological insulator.  _Science_ 340, 167–170 (2013).  https://doi.org/10.1126/science.1234414.
  * (33) Ojha, S. K. _et al._ Electron trapping and detrapping in an oxide two-dimensional electron gas: The role of ferroelastic twin walls.  _Phys. Rev. Applied_ 15, 054008 (2021).  https://link.aps.org/doi/10.1103/PhysRevApplied.15.054008.
  * (34) Goldman, A. M. & Marković, N.  Superconductor‐insulator transitions in the two‐dimensional limit.  _Physics Today_ 51, 39–44 (1998).  https://doi.org/10.1063/1.882069.
  * (35) Haviland, D. B., Liu, Y. & Goldman, A. M.  Onset of superconductivity in the two-dimensional limit.  _Phys. Rev. Lett._ 62, 2180–2183 (1989).  https://link.aps.org/doi/10.1103/PhysRevLett.62.2180.
  * (36) Kapitulnik, A., Kivelson, S. A. & Spivak, B.  Colloquium: Anomalous metals: Failed superconductors.  _Rev. Mod. Phys._ 91, 011002 (2019).  https://link.aps.org/doi/10.1103/RevModPhys.91.011002.
  * (37) Beloborodov, I. S., Lopatin, A. V., Vinokur, V. M. & Efetov, K. B.  Granular electronic systems.  _Rev. Mod. Phys._ 79, 469–518 (2007).  https://link.aps.org/doi/10.1103/RevModPhys.79.469.
  * (38) Zhang, X., Hen, B., Palevski, A. & Kapitulnik, A.  Robust anomalous metallic states and vestiges of self-duality in two-dimensional granular in-inox composites.  _npj Quantum Materials_ 6, 30 (2021).  https://doi.org/10.1038/s41535-021-00329-2.
  * (39) Halperin, B. I. & Nelson, D. R.  Resistive transition in superconducting films.  _Journal of Low Temperature Physics_ 36, 599–616 (1979).  https://doi.org/10.1007/BF00116988.
  * (40) Minnhagen, P.  The two-dimensional coulomb gas, vortex unbinding, and superfluid-superconducting films.  _Rev. Mod. Phys._ 59, 1001–1066 (1987).  https://link.aps.org/doi/10.1103/RevModPhys.59.1001.
  * (41) Tinkham, M.  Effect of fluxoid quantization on transitions of superconducting films.  _Phys. Rev._ 129, 2413–2422 (1963).  https://link.aps.org/doi/10.1103/PhysRev.129.2413.
  * (42) Chandrasekhar, B. S.  A note on the maximum critical field of high‐field superconductors.  _Applied Physics Letters_ 1, 7–8 (1962).  https://doi.org/10.1063/1.1777362.
  * (43) Clogston, A. M.  Upper limit for the critical field in hard superconductors.  _Phys. Rev. Lett._ 9, 266–267 (1962).  https://link.aps.org/doi/10.1103/PhysRevLett.9.266.
  * (44) Werthamer, N. R., Helfand, E. & Hohenberg, P. C.  Temperature and purity dependence of the superconducting critical field, ${H}_{c2}$. iii. electron spin and spin-orbit effects.  _Phys. Rev._ 147, 295–302 (1966).  https://link.aps.org/doi/10.1103/PhysRev.147.295.
  * (45) Ojha, S. K. _et al._ Oxygen vacancy induced electronic structure modification of ${\mathrm{ktao}}_{3}$.  _Phys. Rev. B_ 103, 085120 (2021).  https://link.aps.org/doi/10.1103/PhysRevB.103.085120.
  * (46) Chen, Z. _et al._ Carrier density and disorder tuned superconductor-metal transition in a two-dimensional electron system.  _Nature Communications_ 9, 4008 (2018).  https://doi.org/10.1038/s41467-018-06444-2.
  * (47) Gurevich, A. V. & Mints, R. G.  Self-heating in normal metals and superconductors.  _Rev. Mod. Phys._ 59, 941–999 (1987).  https://link.aps.org/doi/10.1103/RevModPhys.59.941.
  * (48) Doettinger, S. G. _et al._ Electronic instability at high flux-flow velocities in high-${T}_{c}$ superconducting films.  _Phys. Rev. Lett._ 73, 1691–1694 (1994).  https://link.aps.org/doi/10.1103/PhysRevLett.73.1691.
  * (49) Devreese, J. P. A., Tempere, J. & Sá de Melo, C. A. R.  Effects of spin-orbit coupling on the berezinskii-kosterlitz-thouless transition and the vortex-antivortex structure in two-dimensional fermi gases.  _Phys. Rev. Lett._ 113, 165304 (2014).  https://link.aps.org/doi/10.1103/PhysRevLett.113.165304.
  * (50) Hadzibabic, Z., Krüger, P., Cheneau, M., Battelier, B. & Dalibard, J.  Berezinskii–kosterlitz–thouless crossover in a trapped atomic gas.  _Nature_ 441, 1118–1121 (2006).  https://doi.org/10.1038/nature04851.

METHODS

Sample growth and characterization: AlOx/KTO(111) samples were fabricated by
ablating a single crystalline Al2O3 target on (111) oriented KTO substrate
using a pulsed laser deposition system (Neocera LLC, USA) equipped with a high
pressure reflection high energy electron diffraction setup (Staib instruments,
Germany). A KrF excimer laser (Coherent, Germany) operated at a repetition
rate of 1 Hz ($\lambda$=248 nm) and an energy density$\sim$ 1J/cm2 (on the
target) was used for ablating the target. Target to substrate distance was
fixed at 5.6 cm. The substrate was heated using a resistive heater whose
temperature was maintained at 560 ∘C during the growth. The growth chamber
pressure was 5 $\crossproduct$ 10-6 Torr during the deposition. Immediately
after the ablation, the sample was cooled to room temperature at a rate of
15∘C/min under the vacuum. The surface morphology of the as received substrate
and the film was monitored by performing atomic force microscopy (AFM) in non-
contact mode using a Park AFM system. The thickness of the films was
determined from X-ray reflectivity measurement performed in a lab based Rigaku
Smartlab diffractometer. For more details, see supplementary note S1.

Transport Measurements: All the transport measurements were performed in an
Oxford Integra LLD system using the standard four probe method in the Hall bar
geometry. Ohmic contacts were made by ultrasonically bonding Al wire.
Electrical resistance was measured using a dc delta mode with a Keithley 6221
current source and a Keithley 2182A nanovoltmeter and also using standard low-
frequency lock-in technique. For $I$-$V$ measurements a Keithley 2450 source
meter was used in current bias mode with a sweep rate of 10 $\mu$A/sec.

ACKNOWLEDGEMENTS

Authors are thankful to Prof. Sumilan Banerjee, Prof. Manish Jain, Prof.
Vibhor Singh , Prof. Jak Chakhalian, and Sanat Kumar Gogoi for fruitful
discussions and valuable comments about the manuscript. SM acknowledges a
Department of Science and Technology (DST) Nanomission grant (DST/NM/NS/20
18/246), SERB, India (Early Career Research Award:ECR/2018/001512, I.R.H.P.A
Grant No. IPA/2020/000034), MHRD, Government of India under STARS research
funding (STARS/APR20 19/PS/156/FS) for financial support. The authors
acknowledge the AFM, XRD, and wire bonding facility at the Department of
Physics, IISc Bangalore. We are also thankful to Ranjan Kumar Patel for
proofreading.

COMPETING INTERESTS

The authors declare no competing interests.

Supplementary Material

List of contents

S1. Growth and characterization of AlOx/KTO (111) sample.

S2. $R_{S}$ vs $T$ data of the bulk oxygen deficient KTaO3 (001) single
crystal.

S3. $R_{S}$ vs. $T$ data under perpendicular $B$ for Hall bar along
[1$\bar{1}$0].

S4. $R_{S}$ vs. $T$ data under in-plane $B$ for Hall bar along [11$\bar{2}$]
for two configurations ($B$ $\parallel$ $I$ and $B$ $\perp$ $I$).

S5. $R_{S}$ vs. $T$ data under in-plane $B$ for Hall bar along [1$\bar{1}$0]
for two configurations ($B$ $\parallel$ $I$ and $B$ $\perp$ $I$).

S6. Temperature dependent upper critical field under in-plane $B$ for both the
Hall bars for two configurations ($B$ $\parallel$ $I$ and $B$ $\perp$ $I$).

S7. Fitting of weak-anti localization (WAL) data with Iordanskii, Lyanda-
Geller, and Pikus (ILP) theory.

S8. Transverse resistance ($R_{xy}$) as a function of the out-of-plane
magnetic field at different temperatures.

S9. Non-linear Hall effect and evidence for two-band transport.

S10. Temperature dependence sheet carrier density ($n_{S}$), mobility ($\mu$)
and evidence for carrier freezing effect.

S11. Temperature dependent $I$-$V$ characteristics and critical current
($I_{C}$) for Hall bar along [11$\bar{2}$].

S12. Temperature dependent $I$-$V$ characteristics and observation of
clockwise hysteresis for Hall bar along [1$\bar{1}$0].

S13. $R_{S}$ vs. $T$ data of another sample with 14 nm AlOx thickness.

S14. Temperature dependent $I$-$V$ characteristics of another sample with 14
nm AlOx thickness for $I$ along [11$\bar{2}$].

S15. Temperature dependent $I$-$V$ characteristics and observation of
clockwise hysteresis on another sample with 14 nm AlOx thickness for $I$ along
[1$\bar{1}$0].

S1. Growth and characterization of AlOx/KTO (111) sample

Figure S1: a. RHEED image of KTO (111) substrate. b. The intensity of the
specular RHEED spot during the growth of the film. c. RHEED image just after
the deposition of the film. d. AFM image of the substrate. e. AFM image of the
film. f. X-ray reflectivity pattern of the heterostructure along with the
simulation.

Fig. S1a shows the reflection high energy electron diffraction (RHEED) image
of as received KTO (111) substrate. Observation of intense diffraction spots
along with Kikuchi lines establishes the flat and single crystalline nature of
the surface and further excludes the possibility of faceting ichimiya:2004p ;
Middey:2012p261602 . This is further evident from the atomic force microscopy
(AFM) image of the substrate, which exhibits a very smooth surface morphology
with mean roughness ($R_{q}$)$\sim$ 100 pm (Fig. S1d). Fig. S1b shows the
temporal evolution of the intensity of the specular RHEED spot during the film
deposition. As evident, intensity decreases gradually during the growth. This
is due to the amorphous nature of the film which is evident from the absence
of any diffraction spots in the RHEED image after the deposition (Fig. S1c).
The resultant film has very flat surface morphology (Fig. S1e) with
$R_{q}$$\sim$ 175 pm. In order to determine the thickness of the films, X-ray
reflectivity measurements have been performed. Fig. S1f shows the measured
data along with the simulation (using GenX bjorck:2007p1174 ) for two
representative samples with thicknesses $\sim$7 nm and 14 nm.

S2. $R_{S}$ vs. $T$ data of the bulk oxygen deficient KTaO3 (001) single
crystal.

Figure S2: Sheet resistance vs. temperature plot for three bulk oxygen
deficient (001) oriented KTO single crystal with sheet carrier densities 2.2
$\crossproduct$ 1015cm-2, 1.6 $\crossproduct$ 1015cm-2 and 8.4 $\crossproduct$
1014cm-2 measured at room temperature. The solid black line denotes fitting
with $T$3. Data have been reproduced with permission from references 30 and 45
of the main text.

S3. $R_{S}$ vs. $T$ data under perpendicular $B$ for Hall bar along
[1$\bar{\textbf{1}}$0]

Figure S3:  Low temperature variation of $R_{S}$ under $B_{\perp}$ for the
Hall bar along [1$\bar{1}$0] has been shown in a. (from 0 T to 0.1 T) and b.
(from 0.2 T to 9 T) for 7 nm AlOx/KTO (111) sample. Similar to the Hall bar
along [11$\bar{2}$], an avoided SIT is observed around $R_{S}$ $\sim$ 0.9
k$\Omega$/sq. A logarithmic dependence is also observed at higher $B$ and low
$T$.

S4. $R_{S}$ vs. $T$ data under in-plane $B$ for Hall bar along
[11$\bar{\textbf{2}}$] for two configurations ($B$ $\parallel$ $I$ and $B$
$\perp$ $I$)

Figure S4: Low temperature variation of $R_{S}$ under $B_{\parallel}$ (for 7
nm AlOx/KTO (111) sample) when $I$ $\parallel$ [11$\bar{2}$] and $B$
$\parallel$ $I$ has been shown in a. (from 0 T to 4 T) and b. (from 5 T to 9
T). Plots for the case when $I$ $\parallel$ [11$\bar{2}$] and $B$ $\perp$ $I$
has been shown in c. (from 0 T to 3.25 T) and d. (from 5 T to 9 T).

S5. $R_{S}$ vs. $T$ data under in-plane $B$ for Hall bar along
[1$\bar{\textbf{1}}$0] for two configurations ($B$ $\parallel$ $I$ and $B$
$\perp$ $I$)

Figure S5: (a) Low temperature variation of $R_{S}$ under $B_{\parallel}$ (for
7 nm AlOx/KTO (111) sample) when $I$ $\parallel$ [1$\bar{1}$0] and $B$
$\parallel$ $I$ have been shown in a. (from 0 T to 3.25 T) and b. (from 5 T to
9 T). Plots for the case when $I$ $\parallel$ [11$\bar{2}$] and $B$ $\perp$
$I$ has been shown in c. (from 0 T to 4 T) and d. (from 5 T to 9 T).

S6. Temperature dependent upper critical field under in-plane $B$ for both the
Hall bars for two configurations ($B$ $\parallel$ $I$ and $B$ $\perp$ $I$)

Figure S6:  Temperature dependent upper critical field ($B_{C_{\parallel}}$),
obtained by tracking the evolution of $T$${}_{\textnormal{C}}$ with $B$∥ in
$R_{S}$ vs. $T$ plot for all the four configurations discussed in sections S4
and S5. For the case of Hall bar along [11$\bar{2}$], $B_{C_{\parallel}}$ is
found to be lower for the case when $B$ $\parallel$ $I$ (configuration 1) than
$B$ $\perp$ $I$ case (configuration 2). Interestingly, this trend is
completely opposite for the Hall bar along [1$\bar{1}$0] where
$B_{C_{\parallel}}$ for the case when $B$ $\parallel$ $I$ (configuration 3) is
found to be higher than $B$ $\perp$ $I$ case (configuration 4). Such a
behavior could arise from a small $p$-wave component in the superconducting
order parameter as proposed recently zhang:2021p .

S7. Fitting of weak-anti localization (WAL) data with Iordanskii, Lyanda-
Geller, and Pikus (ILP) theory

Figure S7: a. Temperature dependence of the phase coherence time ($\tau$ϕ) and
spin precession time ($\tau$${}_{\text{SO}}$) obtained from the fitting for
both the Hall bars. b. Temperature dependence of the number of independent
channels contributing to WAL.

As shown in the main text, ILP theory (with only cubic Rashba term) along with
a small Kohler $B^{2}$ term provides an excellent fit at all temperatures. In
the absence of linear Rashba term, the correction to the sheet conductance
($\Delta$$\sigma$) is given by

$\displaystyle\Delta\sigma(B)$ $\displaystyle=$ $\displaystyle
N\frac{e^{2}}{\pi
h}\Bigg{[}\Psi(\frac{1}{2}+\frac{B_{\phi}}{B}+\frac{B_{SO}}{B})-\frac{1}{2}\Psi(\frac{1}{2}+\frac{B_{\phi}}{B})$
(5)
$\displaystyle+\frac{1}{2}\Psi(\frac{1}{2}+\frac{B_{\phi}}{B}+\text{2}\frac{B_{SO}}{B})-\text{ln}\frac{B_{\phi}+B_{SO}}{B}$
$\displaystyle-\frac{1}{2}\text{ln}\frac{B_{\phi}+\text{2}B_{SO}}{B}+\frac{1}{2}\text{ln}\frac{B_{\phi}}{B}\Bigg{]}$

where $N$ is the number of independent interference channels
Nakamura:2020p1161 , $\Psi$ is the digamma function,
$B_{\phi}$=$\frac{\hbar}{\text{4e}l_{\phi}^{2}}$ ($l_{\phi}$ is the phase
coherence length) and $B_{SO}$=$\frac{\hbar}{\text{4e}l_{SO}^{2}}$ ($l_{SO}$
is the spin-precession length) where $\hbar$ is the reduced Planck’s constant.
Associated characteristics time scales, phase coherence time ($\tau$ϕ) and
spin precession time ($\tau$${}_{\text{SO}}$) are given by
$\tau$ϕ=$\frac{\hbar}{\text{4e}DB_{\phi}}$ and
$\tau$${}_{\text{SO}}$=$\frac{\hbar}{\text{4e}DB_{\text{SO}}}$ where $D$ is
the diffusion coefficient given by $D$=$v_{f}$2$\tau$/2 ($v_{f}$ is the Fermi
velocity and $\tau$ is the elastic scattering time). $v_{f}$ and $\tau$ were
estimated assuming a single band having parabolic dispersion with effective
mass $m$∗= 0.3$m$e Bareille:2014p3586 . Fig. S7a shows the temperature
dependent $\tau$ϕ and $\tau$${}_{\text{SO}}$ obtained from the fitting for
both the Hall bars along [$11\bar{2}$] and [$1\bar{1}0$]. As evident,
$\tau$${}_{\text{SO}}$ is smaller than $\tau$ϕ satisfying the criteria for WAL
BERGMANN:1984p1 ; Caviglia:2010p126803 . Temperature dependence of the number
of independent channels has been plotted in Fig. S7b.

S8. Transverse resistance ($R_{xy}$) as a function of the out-of-plane
magnetic field at different temperatures

Figure S8:  Antisymmetrized transverse resistance as a function of the
magnetic field measured at various temperatures in the metallic phase for 7 nm
AlOx/KTO (111) sample. The direction of $I$ is along the crystallographic axis
[$11\bar{2}$].

Fig. S8 shows $R_{xy}$ vs. $B$ curves recorded at several fixed temperatures.
To eliminate the longitudinal component of resistance ($R_{xx}$), the data has
been antisymmetrized with respect to $B$. As clearly evident, the slope
$R_{xy}/B$ increases with the decrease in temperature. Assuming single band
transport, the Hall coefficient is given by $R_{H}$ = -1/$ne$ (where $n$ is
the carrier density and -$e$ is the electron’s charge). Since, $R_{xy}/B$
=$R_{H}$, an increase in slope $R_{xy}/B$ with lowering of temperature
immediately suggests decreasing $n$ with lowering of temperature. A similar
trend has also been observed for other Hall bar along [$1\bar{1}$0]. In
sections S9 and S10, we discuss the origin of the nonlinear Hall effect in the
present case.

S9. Non-linear Hall effect and evidence for two-band transport

Figure S9: The Hall coefficient $R_{H}$ as a function of out-of-plane magnetic
field at 5 K and 50 K for both the Hall bars along [$11\bar{2}$] and
[$1\bar{1}$0] has been shown in a and b. c. Linear fitting of $R_{xy}$ vs. $B$
data at 5 K for $I$ along [$11\bar{2}$]. To visualize the presence of little
non-linearity in $R_{xy}$, the residual ($R_{xy}$-Fit) is multiplied by 50. d.
$R_{xy}$ vs. $B$ data at 5 K (for $I$ along [$11\bar{2}$]) along with fitting
using two band model.

To extract the value of $n$, we have tried to fit the data with a straight
line assuming one band model. Interestingly, we find that, below 20 K,
$R_{xy}$ vs. $B$ can not be captured using one band approximation due to the
presence of a little non-linearity. This is clearly evident in the $R_{H}$ vs.
$B$ plot shown in Fig. S9a and b for both the Hall bars. Fig. S9c shows the
failure of one band model in describing our Hall data at 5 K. Such nonlinear
effects in $R_{xy}$ could arise from the presence of multi carrier transport
at the interface. In order to verify this, we have considered a minimal two
band model where $R_{xy}$ is given by

$R_{xy}=-\frac{1}{e}\frac{(\frac{n_{1}\mu_{1}^{2}}{1+\mu_{1}^{2}B^{2}}+\frac{n_{2}\mu_{2}^{2}}{1+\mu_{2}^{2}B^{2}})B}{(\frac{n_{1}\mu_{1}}{1+\mu_{1}^{2}B^{2}}+\frac{n_{2}\mu_{2}}{1+\mu_{2}^{2}B^{2}})^{2}+(\frac{n_{1}\mu_{1}^{2}}{1+\mu_{1}^{2}B^{2}}+\frac{n_{2}\mu_{2}^{2}}{1+\mu_{2}^{2}B^{2}})^{2}B^{2}},$

with the constraint $(e.R_{S})^{-1}$ = $n_{1}\mu_{1}+n_{2}\mu_{2}$. Here,
$n_{1}$, $n_{2}$ and $\mu$1, $\mu$2 are the sheet carrier densities and
mobilities of the two types of electrons. As evident from the Fig. S9d, two
band model provides an excellent fit to our Hall data in the whole range of
$B$ strongly indicating the presence of two types of carriers in the system.
Such a two band transport has not been demonstrated so far for KTO (111) based
superconductors.

S10. Temperature dependent sheet carrier density ($n_{S}$), mobility ($\mu$)
and evidence for carrier freezing effect

Figure S10:  a. Temperature dependent $n_{s}$ (for Hall bar along
[11$\bar{2}$] and [1$\bar{1}$0]) obtained from fitting of antisymmetrized
$R$${}_{\text{xy}}$. $n_{1}$ and $n_{2}$ are the density of electrons confined
to the lower and upper band, respectively. b. Temperature dependent mobility
($\mu$) for Hall bar along [11$\bar{2}$] and [1$\bar{1}$0]. $\mu_{1}$ and
$\mu_{2}$ are electron’s mobility confined to the lower and upper band,
respectively. c. The Arrhenius plot of $ln$ ($n_{1}$) for the temperature
range 100 K $-$ 175 K for both the Hall bars.

Fig. S10a shows temperature dependent $n_{1}$ and $n_{2}$ obtained for $I$
along [11$\bar{2}$] and [1$\bar{1}$0]. Surprisingly, $n_{2}$ is found out to
be $\sim$109/cm2 which is 5 orders of magnitude lower than $n_{1}$. Fig. S10b
shows the corresponding variation of mobility. The mobility of low density
carriers ($\mu_{2}$) is found to be higher than that of high density carriers
($\mu_{1}$). Interestingly, a prominent carrier freezing effect
Liu:2011p146802  is observed below 175 K down to 100 K. This is evident from
the Arrhenius plot of $ln$ ($n_{1}$) vs. 1/$T$ shown in the figure Fig. S10c.
A linear fit results in a very shallow defect state, which would be just 3.6
meV below the conduction band. S11. Temperature dependent $I$-$V$
characteristics and critical current ($I_{C}$) for Hall bar along
[11$\bar{\textbf{2}}$]

Figure S11: a. Temperature dependent full cycle $I$-$V$ curves measured in
current bias mode for the Hall bar along [11$\bar{2}$] on 7 nm AlOx/KTO (111)
sample. For full cycle measurement, the current was swept from 0
$\mu$A$\rightarrow$200 $\mu$A$\rightarrow$-200 $\mu$A$\rightarrow$200
$\mu$A$\rightarrow$0 $\mu$A. For the sake of clarity, 0 $\mu$A$\rightarrow$200
$\mu$A and 200 $\mu$A$\rightarrow$0 $\mu$A branches have not been shown in the
plot. 200 $\mu$A$\rightarrow$-200 $\mu$A branch is denoted as backward sweep
and -200 $\mu$A$\rightarrow$200 $\mu$A branch is denoted as forward sweep. b.
$dV$/$dI$ plot at 1.26 K for $I$ along [11$\bar{2}$]. Several spikes in the
derivative above certain current correspond to several discrete jumps in the
voltage drop. d. Temperature dependence of critical current ($I_{C}$).

S12. Temperature dependent $I$-$V$ characteristics and observation of
clockwise hysteresis for Hall bar along [1$\bar{\textbf{1}}$0]

Figure S12: a. Temperature dependent full cycle $I$-$V$ curves measured in
current bias mode for the Hall bar along [1$\bar{1}$0] on 7 nm AlOx/KTO (111)
sample. For the sake of clarity, 0 $\mu$A$\rightarrow$200 $\mu$A and 200
$\mu$A$\rightarrow$0 $\mu$A branches have not been shown in the plot. Fig. b
shows the same plot with all the curves (except at 1.26 K) shifted vertically
for visual clarity. Similar to $I$ along [11$\bar{2}$], clockwise hysteresis
appears above a certain temperature.

S13. $R_{S}$ vs. $T$ data of another sample with 14 nm AlOx thickness.

Figure S13: Temperature dependent $R_{S}$ of another sample with 14 nm AlOx
thickness. Similar to 7 nm AlOx/KTO (111) sample, a little anisotropy is
observed between Hall bars made along [11$\bar{2}$] and [1$\bar{1}$0].
$T$${}_{\textnormal{C}}$ is found to be 1.57 K and 1.51 K for current along
[11$\bar{2}$] and [1$\bar{1}$0], respectively.

S14. Temperature dependent $I$-$V$ characteristics of another sample with 14
nm AlOx thickness for $I$ along [11$\bar{\textbf{2}}$]

Figure S14: Temperature dependent full cycle $I$-$V$ curves measured in
current bias mode ($I$ $\parallel$ [11$\bar{2}$]) on another sample with 14 nm
AlOx thickness. For the sake of clarity, 0 $\mu$A$\rightarrow$200 $\mu$A and
200 $\mu$A$\rightarrow$0 $\mu$A branches has not been shown in the plot.

S15. Temperature dependent $I$-$V$ characteristics and observation of
clockwise hysteresis on another sample with 14 nm AlOx thickness for $I$ along
[1$\bar{\textbf{1}}$0]

Figure S15:  a. Temperature dependent full cycle $I$-$V$ curves measured in
current bias mode for the Hall bar along [1$\bar{1}$0] on another sample with
14 nm AlOx thickness. For the sake of clarity, 0 $\mu$A$\rightarrow$200 $\mu$A
and 200 $\mu$A$\rightarrow$0 $\mu$A branches have not been shown in the plot.
Fig. b shows the same plot with all the curves (except at 1.26 K) shifted
vertically for visual clarity. Similar to $I$ along [11$\bar{2}$], clockwise
hysteresis appears above certain temperature.

## References

  * (1) Ichimiya, A., Cohen, P. I. & Cohen, P. I.  _Reflection high-energy electron diffraction_ (Cambridge University Press, 2004).
  * (2) Middey, S. _et al._ Epitaxial growth of (111)-oriented laalo3/lanio3 ultra-thin superlattices.  _Applied Physics Letters_ 101, 261602 (2012).  https://doi.org/10.1063/1.4773375.  eprint https://doi.org/10.1063/1.4773375.
  * (3) Björck, M. & Andersson, G.  Genx: an extensible x-ray reflectivity refinement program utilizing differential evolution.  _Journal of Applied Crystallography_ 40, 1174–1178 (2007).
  * (4) Zhang, G. _et al._ Spontaneous rotational symmetry breaking in ktao $\\_3$ interface superconductor.  _arXiv preprint arXiv:2111.05650_ (2021).
  * (5) Nakamura, H. _et al._ Robust weak antilocalization due to spin-orbital entanglement in dirac material sr3sno.  _Nature Communications_ 11, 1161 (2020).  https://doi.org/10.1038/s41467-020-14900-1.
  * (6) Bareille, C. _et al._ Two-dimensional electron gas with six-fold symmetry at the (111) surface of ktao3.  _Scientific Reports_ 4, 3586 (2014).  https://doi.org/10.1038/srep03586.
  * (7) Bergmann, G.  Weak localization in thin films: a time-of-flight experiment with conduction electrons.  _Physics Reports_ 107, 1–58 (1984).  https://www.sciencedirect.com/science/article/pii/0370157384901030.
  * (8) Caviglia, A. D. _et al._ Tunable rashba spin-orbit interaction at oxide interfaces.  _Phys. Rev. Lett._ 104, 126803 (2010).  https://link.aps.org/doi/10.1103/PhysRevLett.104.126803.
  * (9) Liu, Z. Q. _et al._ Metal-insulator transition in ${\mathrm{srtio}}_{3-x}$ thin films induced by frozen-out carriers.  _Phys. Rev. Lett._ 107, 146802 (2011).  https://link.aps.org/doi/10.1103/PhysRevLett.107.146802.