Transverse resistance due to electronic inhomogeneities in superconductors

Shamashis Sengupta,1, ∗ Alireza Farhadizadeh,2, † Joe Youssef,1 Sara Loucif,1 Florian Pallier,1 Louis Dumoulin,1
Kasturi Saha,3 Sumiran Pujari,4 Magnus Óden,2 Claire Marrache-Kikuchi,1 and Miguel Monteverde5, ‡
1
Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France
2
Department of Physics, Chemistry and Biology (IFM),
Linköping University, Linköping SE-581 83, Sweden
3
Department of Electrical Engineering, Indian Institute of Technology Bombay, Mumbai 400076, India
4
Department of Physics, Indian Institute of Technology Bombay, Mumbai 400076, India
5
Université Paris-Saclay, CNRS, Laboratoire de Physique des Solides, 91405, Orsay, France
arXiv:2407.16662v1 [cond-mat.supr-con] 23 Jul 2024

Phase transitions in many-body systems are often associated with the emergence of spatial inho-
mogeneities. Such features may develop at microscopic lengthscales and are not necessarily evident
in measurements of macroscopic quantities. In this work, we address the topic of distribution of
current paths in superconducting films. Typical lengthscales associated with superconductivity are
in the range of nanometres. Accordingly, measurements of electrical resistance over much larger dis-
tances are supposed to be insensitive to details of spatial inhomogeneities of electronic properties.
We observe that, contrary to expectations, current paths adopt a highly non-uniform distribution
at the onset of the superconducting transition which is manifested in the development of a finite
transverse resistance. The anisotropic distribution of current density is unrelated to the structural
properties of the superconducting films, and indicates the emergence of electronic inhomogeneities
perceivable over macroscopic distances. Our experiments reveal the ubiquitous nature of this phe-
nomenon in conventional superconductors.

The typical lengthscales which govern spatial varia- its path. The longitudinal resistance is Rxx = Vxx /I. In
tions of superconducting order are the coherence length a homogeneous planar conductor in the shape of a rect-
and the magnetic penetration depth. These parameters angle (Fig. 1a) with a current applied along its length,
are of central importance for understanding the penetra- symmetry requirements determine that the lines of cur-
tion of magnetic flux and the structure of the mixed state rent density are parallel to each other. On the other
in type-II superconductors [1, 2]. Electronic properties hand, if we consider an inhomogeneous conductor (Fig.
of mesoscopic samples are profoundly affected by the re- 1b) consisting of two different regions with resistivities
duction in sample size when it reaches nanometric dimen- ρ1 and ρ2 (ρ2 ≪ ρ1 ), the current lines are attracted to-
sions, comparable to the lengthscales characterizing fluc- wards the region of lower resistivity leading to deviations
tuations of the order parameter [3–8]. In contrast, we do from a symmetric distribution. The bending of current
not expect the presence of microscopic inhomogeneities paths results in a transverse voltage drop VT between the
to be visible in measurements of macroscopic quantities, points A and Z on opposite boundaries, in the direction
for example in the resistance of superconducting films perpendicular to the macroscopic flow of current I. The
when the distance between electrical probes is a few or- observation of a finite transverse resistance, RT = VT /I,
ders of magnitude larger than the coherence length. It is implies the existence of macroscopic inhomogeneities.
natural to treat the system as an isotropic conductor in
this regime. In this work, we describe transport experi- We patterned devices on commercially available
ments in superconducting films of Nb and NbN which re- Si/SiO2 substrates in the shape of Hall bars (Fig. 2a)
veal that the impact of inhomogeneities in the electronic using electron beam lithography, followed by the evap-
system is actually discernible in the form of a transverse oration of Nb. The morphology of evaporated Nb films
resistance in the direction perpendicular to the macro- consist of amorphous and nanocrystalline regions[9]. The
scopic flow of current. Furthermore, the electronic in- electrical properties were measured using a standard low
homogeneities appear to be an emergent phenomenon, frequency lock-in technique. Fig. 2b shows the charac-
without any clear dependence on the morphology of the terization of Rxx as a function of temperature. In this
films. Our observations imply that the current density device (named D1Nb), the voltage probes were a distance
adopts non-uniform patterns, revealing unique aspects of (L) 814 µm apart. The width (w) of the channel was 40
emergent electronic properties which are not expected in µm and its thickness (t) was 55 nm. The critical tem-
such standard superconductors. perature (Tc ) of the superconducting transition at zero
magnetic field is 7.6 K. Figs. 2c and 2d show the result of
We will begin the discussion of our experiments by simultaneous measurement of Rxx and RT as a function
outlining the relation between anisotropy in a conductor of the magnetic field (H), at different values of tempera-
and transverse resistance (Fig. 1). The resistance of a ture (T ). The magnetic field was applied perpendicular
conducting film is characterized by applying an electrical to the plane of the sample in all our experiments. The
current (I) and measuring the voltage drop (Vxx ) along upper critical magnetic field (Hc2 ) is determined from the
 2

a I b I The transverse resistance RT was measured between the

contacts A and Z, 3.5 mm apart. The Tc of this sample,
�1 named S1NbN, was 11.2 K (Fig. 3b). Figs. 3c and 3d
Z A show the results of Rxx and RT measurement as a func-
A Z
�2 �2 tion of the magnetic field. ξGL is estimated to be 2.7 nm
from the Rxx (H) data in Fig. 3c. In this system too, we
observed (Fig. 3d) finite peaks in RT around the critical
magnetic field, indicating that the distribution of cur-
rent density is asymmetric with respect to the direction
I I of macroscopic current flow.
Average values of the transverse (⟨ET ⟩) and longitudi-
FIG. 1. Current paths in inhomogeneous conductors. nal (⟨Exx ⟩) electric fields generated by an applied current
(a) The lines of current density are uniformly distributed in
are estimated from the measured voltages (VT and Vxx
an isotropic planar conductor. (b) In an inhomogeneous con-
ductor with regions of two different resistivities ρ1 and ρ2 respectively) divided by the distances between contacts.
(ρ2 ≪ ρ1 ), the current lines bend towards the region of lower For sample D1Nb, ⟨ET ⟩/⟨Exx ⟩ reaches a maximum of
resistivity. This leads to a voltage drop between the points 2.9% near the upper critical field at T = 2.0 K (Figs. 2c
A and Z, in the direction perpendicular to the net flow of and 2d). In the case of sample S1NbN, the ratio reaches
current. 5.9% at T = 11.0 K (Figs. 3c and 3d).
An observation of great significance for the following
discussion is that the features of the peaks in RT (H)
plots are very different at different temperatures, as can
midpoint of the resistance (Rxx ) drop. It reaches upto be seen in Figs. 2d and 3d. The magnitude changes
3.9 T at T = 2.0 K. The Ginzburg-Landau coherence a lot, and often the sign as well. These features vary
1
length is estimated[10, 11] as ξGL = [ 2πHΦc20 (0) ] 2 , where distinctly from one sample to another[12], even for the
Hc2 (0) = 0.69Tc dH dT |T =Tc and Φ0 is the flux quantum.
c2
same material (either Nb or NbN). We will now turn to
For this sample, ξGL = 9.4 nm. At magnetic fields much the discussion of possible mechanisms of the described
higher than Hc2 , the usual Hall effect is visible in the nor- phenomenon.
mal state leading to the linear variation of RT with the Theoretical studies [13–15] had anticipated the exis-
magnetic field. The estimated carrier density is 4.6×1022 tence of novel effects in transport properties of supercon-
cm−3 . The most interesting feature in the plot RT (H) is ductors due to the motion of vortices. The Hall effect
the occurrence of prominent even-in-field peaks (Figs. 2d in many superconductors[16–19] show distinctly differ-
and 2e) when the superconducting transition takes place. ent features at the onset of the transition compared to
Fig. 2d summarizes the most important point of this its normal state. However, these effects give rise to volt-
work, that there is a finite transverse resistance at the ages with an antisymmetric dependence on the magnetic
superconducting transition implying the current density field and are thus unrelated to the even-in-field transverse
is not uniform along the width of the channel between resistance in our experiments.
the transverse voltage probes. It can be inferred that Finite transverse resistances have been seen in cer-
there are inhomogeneous regions with a distribution of tain superconducting films when a current was applied
resistivity values inside the conductor, their impact be- at zero magnetic field [20–22]. In some instances, an
ing palpable even for large distances (w = 40 µm) be- explanation based on vortex-antivortex interactions [21]
tween the voltage probes. This distance is three orders had been suggested. This effect disappears when a small
of magnitude larger than ξGL . In order to check the magnetic field [22] is applied. Since the peaks in RT in
robustness of this phenomenon, we did similar exper- our samples occur at large fields of few Teslas, such a
iments on another conventional superconductor, NbN. mechanism based on vortex-antivortex interaction does
This time, we studied an unpatterned plain NbN film not offer a suitable explanation.
with t = 205 nm, deposited on MgO(011) substrate using There are reports of superconductors showing trans-
DC magnetron sputtering. The film structure consists of verse resistance peaks at the critical magnetic field[23, 24]
nanosized cuboid domains (see Supplementary Informa- which are symmetric with respect to the field polarity.
tion [12] for details on characterization of the sample). Segal et al. [24] provided a simple explanation for such
Electrical contacts were made by ultrasonic wire-bonding even-in-field peaks based on the consideration that the
following the schematic in Fig. 3a. Resistance measure- superconducting film has non-uniform morphology with
ments were done by applying d.c. currents using a Quan- a spatial distribution of superconducting parameters (Tc
tum Design Model 6000 Physical Property Measurement and Hc2 ). The analysis using a resistor network model
System Controller. Current was injected between the shows that RT has a peak proportional in magnitude to
contacts I+ and I−, 2.1 mm apart. The voltage Vxx was sgn(H) ∂R∂Hxx (H)
when the transition is induced by vary-
measured between contacts V + and V −, 0.7 mm apart. ing the magnetic field. If it is induced by varying the
 3

a c e 120
60
VT 120
2K 40

RT (m�)
Rxx (�)
100
20
I 2K
60
80
A

Rxx (�)
Z 3K 0
4K
60 5K
6K 0 -20
7K -5.0 -4.0 -3.0
40 8K �0H (T)
L Vxx
20
120
10
0 3K

RT (m�)
C -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
0

Rxx (�)
I Magnetic field (T)
60
-10
w d 60
2K -20
3K
4K 0
b 120
40 5K
6K
-4.5 -3.5
�0H (T)
-2.5

7K
RT (m�)

100 8K
20
120 0
Rxx (�)

80 4K

RT (m�)
Rxx (�)
60 0
0T -15
40 60
1T
20 2T -20
3T
0 -30
2.5 5.0 7.5 10.0 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 0
-3.5 -2.5 -1.5
Temperature (K) Magnetic field (T) �0H (T)

FIG. 2. Transverse resistance in Hall bar devices of superconducting Nb. (a) Schematic diagram of a Hall bar device
for measuring Vxx and VT . (b) Measurement of Rxx as a function of temperature at different values of the magnetic field
to characterize the superconducting transition. An a.c. current of 25 µA r.m.s. amplitude was applied for the measurement.
(c,d) Rxx and RT were measured simultaneously as a function of magnetic field by applying an a.c. current of 50 µA r.m.s.
amplitude. Peaks appeared in RT coinciding with the superconducting transition seen in Rxx at the upper critical field. (e)
Plots of Rxx and RT from (c) and (d) are shown together for comparison, highlighting the variation in magnitude and sign of
the peaks in RT at different temperatures. These variations signify the differences in flow patterns of electrical current near
the critical field transition at different temperatures.

temperature, RT shows a peak proportional to ∂R∂T xx (T )

. since the local superconducting properties are rooted in
It provides a trivial explanation of even-in-field trans- the structure of the film itself. They are expected to have
verse resistance in terms of current paths guided by in- the same sign, and also the lineshapes should not vary
homogeneities ingrained in the sample structure. Exper- greatly due to its dependence on sgn(H) ∂R∂H xx (H)
. This
iments with magnetic imaging techniques have revealed is not what we observe in our experiments. The shapes
the non-uniform distribution of electrical current in Nb of the RT (H) plots change drastically for different tem-
networks[25], which was attributed to disorder-driven peratures. More importantly, there is a change in sign
variations of local Tc . Complex and non-uniform cur- of the peaks (Figs. 2d and 3d), despite the fact that
rent flow patterns induced by defects were seen in poly- sgn(H) ∂R∂Hxx (H)
is positive at all temperatures (Figs. 2c
crystalline TlBa2 Ca2 Cu3 Ox superconducting films using and 3c). Therefore, we find that the hypothesis of inho-
magneto-optical imaging[26]. We will now see whether mogeneities in the material structure is unable to provide
the aforementioned model based on structural inhomo- an explanation of the results, even qualitatively.
geneities in the morphology of the film can explain our
Xu et al. [27] recently reported the observation of
results. Let us consider the case of a non-uniform distri-
anomalous even-in-field transverse resistance peaks in β-
bution of superconducting parameters across the width of
Bi2 Pd superconducting films. The transverse resistance
a sample, for example, along the line joining probes A and
was interpreted as a manifestation of chirality inherent
Z in Fig. 2a. We assume that the right hand side (closer
in the system, and postulated to arise from topological
to Z) has on an average a slightly larger critical tempera-
surface states of β-Bi2 Pd. Such features have also been
ture and critical field. This should cause current lines to
seen in the iron-based superconductor Ba0.5 K0.5 Fe2 As2
bend towards the right, leading to a positive voltage peak
[28], possibly arising from an electronic nematic state.
around Hc2 for a field-induced transition. This argument
None of these explanations can be applied in our case
holds for all temperatures. Therefore, peaks in RT (H)
since Nb and NbN films are not supposed to have either
should not differ much from one temperature to another,
topological states or nematic ordering. It is the simplic-
 4

a ingrained in the film morphology. Second, it cannot be

b understood simply by taking into account order param-
6 eter variations over nanometric dimensions comparable
I+
to the coherence length, as these effects are expected to

(�)
V+ 4
Z
disappear upon macroscopic averaging and thus remain

xx
A
V- unnoticeable in resistance measurements. All these con-

R
2
I- siderations prompt us to invoke the concept of emergence
0
to describe the electronic inhomogeneities at the super-
7 9 11 13 15
conducting transition. Patterns of emergent behaviour
Temperature (K)
in complex systems constitute fascinating topics of re-
c search in many branches of science [29]. In solid-state
6 physics, emergent properties are known in strongly cor-
(�)

related materials in which various types of physical inter-

4
actions lead to the spontaneous development of electronic
xx

11.0 K
R

2 10.5 K inhomogeneities at the nanometre scale[30] and novel

0
10.0 K electronic states[31, 32]. Concerning the specific topic
-8 -6 -4 -2 0 2 4 6 8 of superconductors, emergent behaviour has been seen
Magnetic field (T)
[33, 34] in disordered films close to the superconductor-
d insulator transition (SIT), characterized by nanosized do-
0.4 mains much larger than typical lengthscales of structural
0.2 inhomogeneities. There are very few cases where emer-
R (�)

0
-0.2
gent behaviour leads to structures at macroscopic length-
T

-0.4 11.0 K scales. One example is the percolation network emerging

-0.6 10.5 K in superconductor-graphene hybrid devices [35, 36] close
-0.8 10.0 K
to a SIT. In contrast, the devices studied by us are far
-8 -6 -4 -2 0 2 4 6 8
Magnetic field (T) from the limit of a SIT. The Ioffe-Regel parameter, de-
fined as the product of Fermi wavevector (kF ) and mean
FIG. 3. Transverse resistance in plain films of super- free path (l), provides an estimate of disorder in a conduc-
conducting NbN. (a) Electrical contacts are established by tor and is expected to be close to 1 for samples near the
wire-bonding on the plain film. The current is injected be- SIT. The value of kF l is 2.3 for the NbN sample S1NbN,
tween I+ and I−. The current density distributes along the and 30 for the Nb sample D1Nb. These samples cover a
plane (dashed lines). Vxx is measured between V + and V −. broad range of kF l values extending well into the clean
VT is measured between A and Z. A d.c. current of 10 µA metallic regime for Nb, very far from a SIT. Therefore,
was applied. (b) Variation of Rxx with temperature showing
the superconducting transition. (c,d) Rxx and RT were mea-
the phenomenon of emergent electronic inhomogeneities
sured simultaneously as a function of magnetic field. A finite spanning macroscopic distances, which we infer to be the
transverse resistance appears at the critical field transition. reason behind the development of the transverse resis-
The peaks in RT do not have the same sign at all tempera- tance, is unrelated to the physics of SIT. In this respect,
tures. the results reported here represent quite unique observa-
tions regarding the properties of superconductors.
The critical field transition in superconducting films is
ity of these two systems which makes the observation of associated with the melting of the vortex lattice. The
transverse resistance quite intriguing, suggesting that it melting transition has been visualized at microscopic
reflects a phenomenon deeply rooted in the physics of su- scales by scanning tunnelling microscopy in conventional
perconducting criticality itself, irrespective of the struc- superconducting films[37, 38]. The transverse resistance
tural and material properties. observed in our samples is related to the flow pattern
Since the observation of transverse resistance in Nb of electrical current in this regime. In order to obtain a
and NbN films cannot be related to particular features deeper understanding of the phenomenon, it is necessary
of structure and morphology, it seems reasonable to con- to develop a model concerning the statistical mechan-
clude that the underlying phenomenon involves the de- ics of large-scale pattern formation within the electronic
velopment of electronic inhomogeneities of an emergent system and the consequences on charge transport. The
nature. The current density distribution, which deter- features of transverse resistance (seen in Figs. 2d and 3d)
mines the measured RT , follows from the complex spa- are reproducible even when the samples are heated and
tial variation of superconducting order in the vicinity cooled across the critical temperature several times. This
of the phase transition. The essential points inferred indicates that the patterns of macroscopic inhomogene-
from our experiments are the following. First, this elec- ity are reproducible as a function of magnetic field and
tronic texture does not correspond to inhomogeneities temperature, at least in a statistical sense. Two impor-
 5

tant factors influencing the melting transition are ther- Y. Ootuka, Experimental Evidence for Giant Vortex States
mal fluctuations and the distribution of static quenched in a Mesoscopic Superconducting Disk, Phys. Rev. Lett.
disorder in the superconducting films [39]. It seems un- 93, 257002 (2004)
likely that these alone can explain the peaks in RT (H), [4] A. K. Geim, I. V. Grigorieva, S. V. Dubonos, J. G. S.
Lok, J. C. Maan, A. E. Filippov, and F. M. Peeters, Phase
which remain finite even upon macroscopic averaging and transitions in individual sub-micrometre superconductors,
change signs at different temperatures. The non-trivial Nature 390, 259 (1997)
problem here is to understand what other factors might [5] A. K. Geim, S. V. Dubonos, J. G. S. Lok, M. Henini,
be relevant. This would require further research in the and J. C. Maan, Paramagnetic Meissner effect in small
future. superconductors, Nature 396, 144 (1998)
[6] A. S. Mel’nikov, and V. M. Vinokur, Mesoscopic super-
We summarize the implications of this work for our conductor as a ballistic quantum switch, Nature 415, 60
conceptual understanding of the superconducting transi- (2002)
tion. The current density distribution in a homogeneous [7] V. A. Schweigert, F. M. Peeters, and P. Singha Deo, Vor-
planar conductor is expected to be symmetric with re- tex Phase Diagram for Mesoscopic Superconducting Disks,
spect to the direction of macroscopic current flow. We Phys. Rev. Lett. 81, 2783 (1998)
[8] J. J. Palacios, Metastability and Paramagnetism in Su-
observe a finite transverse resistance at the onset of the
perconducting Mesoscopic Disks, Phys. Rev. Lett. 84,
superconducting transition, revealing that the condition 1796 (2000)
of homogeneity is not respected. This indicates the pres- [9] M. Durkin, R. Garrido-Menacho, S. Gopalakrishnan, N.
ence of macroscopic inhomogeneities, palpable even when K. Jaggi, J.-H. Kwon, J.-M. Zuo, and N. Mason, Rare-
the voltage probes are separated by a distance at least region onset of superconductivity in niobium nanoislands,
few thousand times the characteristic Ginzburg-Landau Phys. Rev. B 101, 035409 (2020)
coherence length. Measurements with a variation of both [10] E. Helfand, and N. R. Werthamer, Temperature and
Purity Dependence of the Superconducting Critical Field,
the temperature and magnetic field reveal that the inho-
Hc2 . II, Phys. Rev. 147, 288 (1966)
mogeneities are not related to the material structure and [11] N. R. Werthamer, E. Helfand, and P. C. Hohenberg,
morphology of the film, representing instead an emergent Temperature and Purity Dependence of the Superconduct-
electronic phenomenon. We have demonstrated its occur- ing Critical Field, Hc2 . III. Electron Spin and Spin-Orbit
rence in two different superconducting systems using two Effects, Phys. Rev. 147, 295 (1966)
different contact geometries. Our work provides experi- [12] See Supplementary Information for characterization of
mental evidence for a phenomenon that is unexpected in device properties and data on more samples.
[13] P. G. de Gennes and J. Matricon, Collective Modes of
macroscopic samples. The central problem here concerns
Vortex Lines in Superconductors of the Second Kind, Rev.
the nature of current flow patterns in superconductors Mod. Phys. 36, 45 (1964)
and the lengthscales that are involved. An important [14] J. Bardeen, and M. J. Stephen, Theory of the motion
goal for future research would be to arrive at an explana- of vortices in superconductors, Phys. Rev. 140, A1197
tion for the occurrence of the transverse resistance at the (1965)
critical field transition, as well as its strong dependence [15] P. Nozières and W. F. Vinen, The motion of flux lines in
on temperature, from the perspective of microscopic the- type II superconductors, Philos. Mag.-J. Theor. Exp.
Appl. Phys. 14, 667 (1966)
ory. In recent times, imaging techniques have been used
[16] S. J. Hagen, C. J. Lobb, R. L. Greene, M. G. Forrester,
to map current flow patterns in a variety of systems, for and J. H. Kang, Anomalous Hall effect in superconductors
probing the hydrodynamic transport of electrons in ul- near their critical temperatures, Phys. Rev. B 41, 11630
trapure conductors [40], preferential paths created by do- (1990)
main structure at oxide interfaces[41], and fast vortices in [17] S. J. Hagen, A. W. Smith, M. Rajeswari, J. L. Peng,
superconductors[42] to give a few examples. Application Z. Y. Li, R. L. Greene, S. N. Mao, X. X. Xi, S. Bhat-
of such techniques may provide complementary informa- tacharya, Qi Li, and C. J. Lobb, Anomalous flux-flow Hall
effect: Nd1.85 Ce0.15 CuO4−y and evidence for vortex dy-
tion regarding the transverse resistance observed in our
namics, Phys. Rev. B 47, 1064 (1993)
transport experiments, and yield new insights about mi- [18] A. W. Smith, T. W. Clinton, C. C. Tsuei, and C. J. Lobb,
croscopic aspects of the phenomenon. Sign reversal of the Hall resistivity in amorphous Mo3 Si,
Phys. Rev. B 49, 12927 (1994)
[19] D. I. Khomskii and A. Freimuth, Charged Vortices in
High Temperature Superconductors, Phys. Rev. Lett.
75, 1384 (1995)
∗
shamashis.sengupta@ijclab.in2p3.fr [20] M.S. da Luz, C.A.M. dos Santos, C.Y. Shigue, F.J.H. de
†
alireza.farhadizadeh@liu.se Carvalho Jr., and A.J.S. Machado, Transverse voltage in
‡
miguel.monteverde@universite-paris-saclay.fr superconductors at zero applied magnetic field, Physica
[1] A. A. Abrikosov, Nobel Lecture: Type-II superconductors C 469, 60 (2009)
and the vortex lattice, Rev. Mod. Phys. 76, 975 (2004) [21] T. L. Francavilla, and R. A. Hein, The Observation of
[2] M. Tinkham, Introduction to Superconductivity, McGraw a Transverse Voltage at the Superconducting Transition of
Hill Book and Co. (New York), Second Edition 1996 Thin Films, IEEE Trans. Magn. 27, 1039 (1991)
[3] A. Kanda, B. J. Baelus, F. M. Peeters, K. Kadowaki, and [22] I. Yu. Antonova, V. M. Zakosarenko, E. V. Il’ichev, V. I.
 6

Kuznetsov, and V. A. Tulin, Observation of a nonmono- Larkin, and V. M. Vinokur, Vortices in high-temperature
tonic transverse voltage induced by vortex motion in a su- superconductors, Rev. Mod. Phys. 66, 1125 (1994)
perconducting thin film, JETP Lett. 54, 505 (1991) [40] A. Aharon-Steinberg, T. Völkl, A. Kaplan, A. K. Pariari,
[23] V. V. Guryev, S. V. Shavkin, V. S. Kruglov, and P. V. I. Roy, T. Holder, Y. Wolf, A. Y. Meltzer, Y. Myasoedov,
Volkov, Superconducting transition of Nb–Ti tape studied M. E. Huber, B. Yan, G. Falkovich, L. S. Levitov, M.
by transverse voltage method, Physica C 567, 1353546 Hücker, and E. Zeldov, Direct observation of vortices in
(2019) an electron fluid, Nature 607, 74 (2022)
[24] A. Segal, M. Karpovski, and A. Gerber, Inhomogeneity [41] B. Kalisky, E. M. Spanton, H. Noad, J. R. Kirtley, K.
and transverse voltage in superconductors, Phys. Rev. B C. Nowack, C. Bell, H. K. Sato, M. Hosoda, Y. Xie, Y.
83, 094531 (2011) Hikita, C. Woltmann, G. Pfanzelt, R. Jany, C. Richter,
[25] X. Wang, M. Laav, I. Volotsenko, A. Frydman, and H. Y. Hwang, J. Mannhart, and K. A. Moler, Locally en-
B. Kalisky, Visualizing Current in Superconducting Net- hanced conductivity due to the tetragonal domain struc-
works, Phys. Rev. Appl. 17, 024073 (2022) ture in LaAlO3 /SrTiO3 heterointerfaces, Nat. Mater.
[26] A. E. Pashitski, A. Gurevich, A. A. Polyanskii, D. C. 12, 1091 (2013)
Larbalestier, A. Goyal, E. D. Specht, D. M. Kroeger, J. [42] L. Embon, Y. Anahory, Ž.L. Jelić, E.O. Lachman, Y.
A. DeLuca, and J. E. Tkaczyk, Reconstruction of Cur- Myasoedov, M.E. Huber, G.P. Mikitik, A.V. Silhanek,
rent Flow and Imaging of Current-Limiting Defects in M.V. Milošević, A. Gurevich, and E. Zeldov, Imaging of
Polycrystalline Superconducting Films, Science 275, 367 super-fast dynamics and flow instabilities of superconduct-
(1997) ing vortices, Nat. Commun. 8, 85 (2017)
[27] X. Xu, Y. Li, and C. L. Chien, Anomalous transverse
resistance in the topological superconductor β-Bi2 Pd, Nat.
Commun. 13, 5321 (2022)
[28] Y. Lv, Y. Dong, D. Lu, W. Tian, Z. Xu, W. Chen, X.
Zhou, J. Yuan, K. Jin, S. Bao, S. Li, J. Wen, L. F. Chib-
otaru, T. Schwarz, R. Kleiner, D. Koelle, J. Li, H. Wang,
and P. Wu, Anomalous transverse resistance in 122-type
iron-based superconductors, Sci. Rep. 9, 664 (2019)
[29] E. Bonabeau, J.-L. Dessalles, and A. Grumbach, Char-
acterizing Emergent Phenomena (1): A Critical Review,
Rev. Int. Syst. 9, 327 (1995)
[30] E. Dagotto, Complexity in Strongly Correlated Electronic
Systems, Science 309, 257 (2005)
[31] Y. Yang, and D. Pines, Emergent states in heavy-electron
materials, Proc. Natl. Acad. Sci. 109, E3060 (2012)
[32] H. Y. Hwang, Y. Iwasa, M. Kawasaki, B. Keimer, N.
Nagaosa, and Y. Tokura, Emergent phenomena at oxide
interfaces, Nat. Mater. 11, 103 (2012)
[33] A. Kamlapure, T. Das, S. C. Ganguli, J. B. Parmar,
S. Bhattacharyya, and P. Raychaudhuri, Emergence of
nanoscale inhomogeneity in the superconducting state of
a homogeneously disordered conventional superconductor,
Sci. Rep. 3, 2979 (2013)
[34] C. Carbillet, S. Caprara, M. Grilli, C. Brun, T. Cren,
F. Debontridder, B. Vignolle, W. Tabis, D. Demaille, L.
Largeau, K. Ilin, M. Siegel, D. Roditchev, and B. Leri-
don, Confinement of superconducting fluctuations due to
emergent electronic inhomogeneities, Phys. Rev. B 93,
144509 (2016)
[35] L. B. Ioffe, and M. E. Gershenson, Emergent inhomo-
geneity, Nat. Mater. 11, 567 (2012)
[36] A. Allain, Z. Han, and V. Bouchiat, Electrical con-
trol of the superconducting-to-insulating transition in
graphene–metal hybrids, Nat. Mater. 11, 590 (2012)
[37] I. Guillamón, H. Suderow, A. Fernández-Pacheco, J.
Sesé, R. Córdoba, J. M. De Teresa, M. R. Ibarra and S.
Vieira, Direct observation of melting in a two-dimensional
superconducting vortex lattice, Nat. Phys. 5, 651 (2009)
[38] I. Roy, S. Dutta, A. N. Roy Choudhury, S. Basistha, I.
Maccari, S. Mandal, J. Jesudasan, V. Bagwe, C. Castel-
lani, L. Benfatto, and P. Raychaudhuri, Melting of the
Vortex Lattice through Intermediate Hexatic Fluid in an
a-MoGe Thin Film, Phys. Rev. Lett. 122, 047001
(2019)
[39] G. Blatter, M. V. Feigel’man, V. B. Geshkenbein, A. I.
 1

Supplementary Information

1. Sample preparation and measurement setup

For the fabrication of Nb devices, we used commercially available Si wafers with a 500 nm thick layer of insulating
SiO2 as the substrate. The devices were patterned using electron-beam lithography. We used resist bilayers of
MMA/MAA (methyl methacrylate/methacrylic acid) and PMMA (polymethyl methacrylate). After the patterning
of Hall bar devices with lithography, electron-beam-induced evaporation was performed at a pressure of typically
9×10−8 Torr with the sample holder at room temperature. The sample holder was rotating at 15 revolutions per
minute. The lift-off process was completed by removing the resist layers with acetone.

NbN thin films were prepared using an ultrahigh vacuum DC magnetron sputtering system with a base pressure of
less than 4×10−10 Torr. Single crystals of MgO (011) with dimensions of 10 mm × 10 mm × 0.5 mm were used as
substrates. The offcut of MgO (011) substrates was less than 0.5◦ . The MgO substrates were cleaned following the
method described in Ref. R1. They were then annealed in the deposition chamber for 30 minutes at 800◦ C under
ultrahigh vacuum before beginning the film growth. The NbN films were reactively sputter-deposited from a
single-element, 3-inch Nb target (99.7% purity) at 800◦ C with N2 partial pressure of 0.6 mTorr. The working
pressure was maintained at 4.5 mTorr. The substrate holder was rotating at 10 revolutions per minute. After
deposition, the samples were cooled down to room temperature at a rate of 5◦ C per minute.

Electrical contacts were established on the Nb and NbN samples using a wedge bonder. The samples were inserted
in the cryostat of a Quantum Design Physical Property Measurement System (PPMS) for low temperature
measurements. The electrical transport properties were measured using two different methods. The first method is a
standard lock-in technique, carried out with Stanford Research Systems SR830 lock-in amplifiers. The second
method involves the use of the Quantum Design Model 6000 PPMS Controller, which applies a d.c. current for
determining the resistance. We used the first method for sample D1Nb, while the second one was used for sample
S1NbN. Results from experiments on other samples of Nb and NbN using the lock-in technique are presented later.

2. Analysis of experimental data on transverse resistance

We have measured transverse resistance both in samples patterned in the shape of Hall bars and in plain films with
electrical contacts established by wedge bonding. The underlying principle of all these measurements is that any
deviation of current paths from a symmetric distribution (with the direction of macroscopic current flow being the
axis of symmetry) may show up as a resistance RT in the transverse direction. Ideally, the line joining transverse
voltage probes should be exactly perpendicular to the direction of applied current. However, this situation is most
often not realized in practice since it is hard to avoid small misalignments of the voltage probes. Such misalignments
cause a component of longitudinal resistance (Rxx ) to appear in the measured signal. This has to be subtracted
from the measured signal (RT,m ) of the transverse resistance. The estimate of RT is given by

RT = RT,m − (g ∗ Rxx ) (S1)

In the above equation, g is a numerical factor characterizing the misalignment of transverse voltage probes. It is
determined only by the position of the contacts, and has no dependence on temperature, applied magnetic field and
current.

As we have discussed in the main article, if we assume that the deviation of current paths result mainly due to
structural non-uniformities of the sample, the lineshape of transverse resistance as a function of the magnetic field is
predicted[R2] to follow the function

∂Rxx (H)
Λ(H) ∝ sgn(H) (S2)
∂H
The function Λ(H) predicts the lineshape of transverse resistance, but not the sign of its peak. The sign may be
either positive or negative depending upon the non-uniform distribution of superconducting parameters in the
 2

sample relative to the position of voltage probes. A consequence of this model is that RT (H) is predicted to have
the same sign (either positive or negative) at all temperatures, since Rxx (H) is a monotonic function around the
superconducting critical field at any given temperature. We can compute Λ(H) from the plots of Rxx (H) measured
at different values of temperature. For the samples discussed in the main article (D1Nb and S1NbN), we find that
the observed nature of RT (H) is very different from Λ(H). This discrepancy is the primary reason for interpreting
the occurrence of the transverse resistance as a manifestation of emergent electronic inhomogeneities.

In a film of NbN with a thickness of 9 nm only, we have found that RT (H) indeed shows some similarities to Λ(H).
This data is presented in Section 4.3. Therefore, we note that the phenomenon of transverse resistance due to
structural inhomogeneities reported by Segal et al.[R2] need not be entirely absent in our samples. However, any
such effects, even if they are present, are overshadowed by a much larger contribution from electronic
inhomogeneities as the film thickness becomes larger.

a I+ b
X1

0.5

0.4

� (��-m)
0.3

A Z 0.2

0.1
200 �m

X2 0.0
I- 50 100 150 200 250
T (K)

FIG. S1. Resistivity measurement on sample D1Nb. (a) Image of the device captured with a scanning electron micro-
scope. (b) Resistivity (ρ) estimated from four-probe resistance measurements as a function of temperature (T ). The current
applied was 5 µA.

3. Experiments on Nb samples

We present below the results of experiments on different samples of superconducting Nb films. We will begin with
the sample D1Nb, which has aready been discussed in the main article. The procedure of analyzing the raw data
will be described in detail. Results from two other samples (D2Nb and D3Nb) will also be presented. All the
resistance measurements were conducted by a low-frequency lock-in technique.

3.1 Sample D1Nb

An image of the device is shown in Fig. S1a. The variation of resistivity (ρ) of this sample as a function of
temperature (T ) is shown in Fig. S1b.

Fig. S2 displays the same data presented in Figs. 2c and 2d of the main article. We will now describe how RT is
estimated following Eq. S1. For these experiments, current was applied between the contacts I+ and I− (Fig. S1a).
Longitudinal resistance (Rxx ) was measured between X1 and X2, while the transverse resistance (RT,m ) was
measured between A and Z. The variation of Rxx as a function of magnetic field (H) is shown in Fig. S2a. These
plots are used to estimate the critical field Hc2 at different temperatures and the Ginzburg-Landau coherence length
ξGL . We obtained ξGL = 9.4 nm. As we have discussed earlier, if we take into account the deviation of current paths
owing to structural inhomogeneties of the sample only, the lineshape of transverse resistance RT is predicted to
follow the function Λ(H), which is shown in Fig. S2b.
 3

a b 12
120
10
100

� (arb. unit)
2K 8 2K
80
Rxx (�) 3K
4K
3K
6 4K
60 5K 5K
6K 6K
7K 4 7K
40 8K 8K

20 2

0 0
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
�0H (T) �0H (T)

c 60 d 60
2K g = 0.00005 2K
3K 3K
40 4K 40 4K
5K 5K

RT (m�)
RT,m (m�)

6K 6K
7K 7K
20 8K 8K
20

0
0

-20
-20

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
�0H (T) �0H (T)

FIG. S2. Analysis of transverse resistance measurement in sample D1Nb. (a) The longitudinal resistance Rxx was
measured as a function of magnetic field (H) at different temperatures, using a low-frequency a.c. current of 50 µA. (b) The
function Λ(H) is plotted, following Eq. S2. (c) The raw data of transverse resistance measured between the probes A and
Z (marked in Fig. S1a). (d) Plots of transverse resistance RT as a function of magnetic field, after the background signal
proportional to Rxx (H) is subtracted from the RT,m (H) plots following Eq. S1, with g=0.00005. This figure is the same as
Fig. 2d in the main article.

a X2 A X1 b
100

2.0 K
(�)

2.5 K
3.0 K
I- 50
xx

4.0 K

I+ 5.0 K
R

6.0 K
7.0 K
7.8 K
200 �m
0
-5 -4 -3 -2 -1 0 1 2 3 4 5
�0H (T)
Z

c d
1.5 10
2.0 K
� (arb. unit)

2.5 K 0
3.0 K
R (m�)

4.0 K
1.0 5.0 K -10 2.0 K
6.0 K 2.5 K
7.0 K 3.0 K
T

7.8 K -20 4.0 K

5.0 K
0.5 6.0 K
-30 7.0 K
7.8 K
-40
0
-5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5
�0H (T) �0H (T)

FIG. S3. Measurement of transverse resistance in sample D2Nb. (a) Image of the device captured with a scanning
electron microscope. Its thickness is 55 nm. (b) The longitudinal resistance (Rxx ), measured between the probes X1 and
X2, as a function of magnetic field (H) at different temperatures. A low-frequency a.c. current of 50 µA was applied for
the measurement. (c) The function Λ(H) is estimated from the Rxx (H) data. (d) Plots of transverse resistance (measured
between contacts A and Z) as a function of magnetic field.
 4

50 �m A

I- I+

X Z

b
20

15
(�)

10
xx
R

7.8 K
7.2 K
5 6. 2K
5.0 K
4.0 K
3.0 K
2.0 K
0
-5 -4 -3 -2 -1 0 1 2 3 4 5
�0H (T)

c
0.04

0.02

0.00

-0.02
R (�)

-0.04
T

7.8 K
-0.06
7.2 K
-0.08 6. 2K
5.0 K
-0.10 4.0 K
3.0 K
-0.12 2.0 K
-0.14
-5 -4 -3 -2 -1 0 1 2 3 4 5
�0H (T)

FIG. S4. Measurement of transverse resistance in sample D3Nb. (a) An SEM image of the device. (b) The longitudinal
resistance (Rxx ), measured between the probes Z and X, as a function of magnetic field (H) at different temperatures. A
low-frequency a.c. current of 5 µA was applied for the measurement. (c) The transverse resistance (measured between A and
Z) is shown as a function of magnetic field at different temperatures.

The raw data of measured transverse resistance RT,m (H) is presented in Fig. S2c. Since the probes of the Hall bar
device are not perfectly aligned, there is a small contribution of the longitudinal voltage drop (proportional to Rxx )
which appears in RT,m (H). This becomes evident if we look at the high-field behaviour when the Hall effect in
normal state is visible (Fig. S2c). The Hall voltage is an odd function of H. But we see that the magnitudes of the
transverse resistance are not exactly equal at positive and negative values of the magnetic field. The background
signal proportional to Rxx (H) is subtracted from the RT,m (H) plots following Eq. S1, with g=0.00005. The
resulting plots of RT (H) are shown in Fig. S2d (which is identical to Fig. 2d of the main article). It is evident that
the peaks in RT (H) do not follow the trend of Λ(H).

From the Hall resistance data at high magnetic fields, the mean free path (l) is estimated to be 2.8 nm using the free
electron model. This is an approximate estimate, given the fact that niobium has a complicated Fermi surface [R3]
with positive Hall coefficient.
 5

3.2 Sample D2Nb

Sample D2Nb was prepared with a similar design and under identical conditions as sample D1Nb. Fig. S3a shows
an image of the device. Current was applied between I+ and I−. The variation of four-probe resistance (Rxx ) as a
function of magnetic field (H) is shown in Fig. S3b.

Fig. S3c plots Λ(H) at different values of temperature using the data of Rxx (H). The transverse resistance observed
at the critical field transition is shown in Fig. S3d. The peaks in RT (H) in this case are found to have generally the
same sign at all temperatures, except for T = 6.0 K. The peaks become more prominent at low temperatures, which
contrasts with the predicted trend in Fig. S3c.

3.3 Sample D3Nb

Sample D3Nb is a Hall bar device of Nb with a narrower channel than the samples previously discussed. It has a
width on 20 µm. The thickness is 60 nm. An SEM image is shown in Fig. S4a. The critical temperature (Tc ) of this
device is 7.9 K.

The variation of Rxx as a function of magnetic field is shown in Fig. S4b. Fig. S4c shows the results of RT (H)
measurements. This sample exhibits multiple reversals of the sign of RT peaks. Starting from low temperature (T =
2.0 K) with a negative peak, one reversal to positive sign occurs at T = 4.0 K. It becomes negative again at T = 5.0
K. A further reversal to positive is seen at T = 7.2 K. The current density distribution across the width of the
channel (between the probes A and Z) seems to vary in an erratic manner as the temperature is changed.

4. Experiments on NbN samples

The NbN samples were deposited as plain films on sapphire and MgO substrates. The resistance measurements were
conducted following the schematic outlined in Fig. 3a of the main article. Experiments on the sample S1NbN have
been presented in the main article. We discuss below certain aspects of this sample, as well as results from two other
samples.

FIG. S5. Morphology of NbN film deposited on MgO(011). (a) Result of X-ray diffraction (XRD) experiment. (b) A
top-view scanning electron micrograph.
 6

4.1 Sample S1NbN

The NbN film S1NbN was prepared on a MgO(011) surface. The deposition procedure was described in Section 1.
Fig. S5 shows the morphology of a NbN film on MgO(011) substrate. θ-2θ X-ray diffraction (XRD) patterns were
obtained (Fig. S5a) using a PANalytical X’Pert PRO diffractometer with Cu-Kα radiation (λ = 1.54060 Å) at 45
kV and 40 mA. The scans were carried out in the Bragg-Brentano configuration. Cu-Kβ was removed using a Ni
filter. Fixed slits of 1/2◦ were used as anti-scatter and divergence slits. The step size of θ for this measurement was
0.03◦ . Notably, a minor NbN 111 peak can only be seen on a logarithmic scale. The dominance of the NbN 011 peak
over NbN 111 by over 100-fold suggests a limited percentage of randomly oriented NbN crystals. Fig. S5b shows an
image of a film captured with an SEM. The microsctructure consists of nanoscale cuboid domains. The arrangement
of NbN cuboids is in accordance with the in-plane orientation of MgO (011) as depicted in the figure.

a
6
(�)

4
xx
R

0
20 40 60 80 100 120 140 160 180 200 220 240 260
T (K)

b c 8
6 � (arb. unit) 11.0 K
6 10.5 K
(�)

4 10.0 K
4
xx

11.0 K
R

2 10.5 K 2
10.0 K
0 0
-8 -6 -4 -2 0 2 4 6 8 -8 -6 -4 -2 0 2 4 6 8
�0H (T) �0H (T)
d 0.2
e
0.0 0.4
RT,m (�)

-0.2 0.2
RT (�)

-0.4 11.0 K 0
-0.6 10.5 K -0.2
11.0 K
-0.8 10.0 K -0.4
10.5 K
-0.6
-1.0 10.0 K
-0.8
-1.2
-8 -6 -4 -2 0 2 4 6 8
-8 -6 -4 -2 0 2 4 6 8

�0H (T) �0H (T)

FIG. S6. Resistance measurements on sample S1NbN. (a) The resistance Rxx measured as a function of temperature.
The superconducting transition occurs at 11.2 K. These measurements were done using the bridge of a Quantum Design Model
6000 Physical Property Measurement System. A d.c. current of 40 µA was applied. (b) The resistance Rxx measured as a
function of magnetic field using a d.c. current of 10 µA. (c) The function Λ(H) is plotted. (d) The raw data of transverse
resistance measured between the transverse voltage probes. (e) Plots of transverse resistance as a function of magnetic field
after subtracting a background proportional to Rxx . The parameter g, used for the analysis in Eq. 1, is 0.1235. This figure is
the same as Fig. 3d in the main article.

Resistance measurements on the sample S1NbN over a large range of temperatures are shown in Fig. S6a. The
measurements were conducted using the Quantum Design Model 6000 PPMS Controller, which applies a d.c. current
for measuring the resistances. The resistivity of the film at 15 K is 7.0 µΩ-m. We will now describe the procedure of
analyzing the data concerning the transverse resistance on this sample. The plots of Rxx as a function of magnetic
field are shown in Fig. S6b (identical to Fig. 3c of the main article). These plots are used to estimate Λ(H) in Fig.
S6c. The measured data for RT,m (H) is shown in Fig. S6d. There is a contribution of longitudinal voltage drop,
 7

a b

FIG. S7. Structure of NbN films on Al2 O3 (0006) substrate. (a) Cross-sectional TEM image reveals the detailed
interface structure between NbN and sapphire. The inset presents the corresponding selected area diffraction pattern. (b)
Scanning electron micrograph shows triangular domains.

because of the misalignment of the transverse resistance probes. RT is estimated by correcting for this contribution
using Eq. S1, with g=0.1235. These plots are shown in Fig. S6e, which is the same as Fig. 3d of the main article.

Resistance
h measurements
i reveal very broad transitions at the critical magnetic field (Fig. S6b) with extremely large
value of d(µdT
0 Hc2 )
= 6.0 T/K. We obtain µ0 Hc2 (0) = 46 T and ξGL = 2.7 nm. Such large values of Hc2 (0) in
T =Tc
NbN films have been reported earlier and attributed to the presence of small column void microstructures[R4]. In
our sample, the high critical field is probably related to the occurrence of nanoscale cuboidal domains (Fig. S5b).

4.2 Sample S2NbN

Sample S2NbN is a NbN film with a thickness of 60 nm. It was prepared on single crystal c-plane sapphire (0006)
under similar conditions as those used for growth on MgO with a slight difference in the cleaning procedure. In this
case, the substrates underwent ultrasonic cleaning in acetone and ethanol for 10 minutes, prior to loading into the
chamber.

The morphology of the films on sapphire was examined using transmission electron microscopy (TEM) utilizing an
FEI Tecnai G2 TF 20 UT microscope operating at 200 kV. Cross-sectional samples for TEM were first manually
polished to approximately 60 µm thickness and then subjected to Ar+ ion milling at 5 keV with a 5◦ tilt, while
being rotated in a Gatan precision ion polishing system. Fig. S7a represents the high-resolution TEM image of NbN
grown on c-plane sapphire. The interface between NbN and sapphire appears to be sharp and well-defined,
indicating the absence of an interdiffusion layer. The NbN layer is relatively uniform and crystalline, as shown by
the regular pattern of lattice fringes visible throughout the layer. However, the fringes are less distinct compared to
the high-quality sapphire, indicating the possible presence of some domains.

Fig. S7b shows the top-view SEM image of a typical epitaxial NbN film grown on c-plane sapphire. The film
consists of triangular domains caused by different symmetries of sapphire and NbN[R5]. Based on this figure, the
sizes of the domains are mostly in the range of 5 to 10 nm which can be tuned by varying the deposition conditions.
However, the overall morphology of triangular domains remains consistent.

The sample S2NbN has a Tc of 7.5 K. The results of resistance measurement as a function of magnetic field are
presented in Fig. S8a. The normal state resistance is 21 Ω, corresponding to a resistivity of 3.1 µΩ-m.

The peaks observed in RT (H) (Fig. S8b) at the critical field transition reduce considerably in magnitude as the
temperature is lowered from 7.25 K to 6.00 K. There is an anomalous behaviour at T = 5.00 K. Here the RT (H)
curve becomes very flat with a low value, indicating that the current density is more evenly distributed across the
 8

a 20

15

(�)
xx
10

R
4.00 K
5.00 K
5 6.00 K
7.00 K
7.25 K
8.00 K
0
-8 -6 -4 -2 0 2 4 6 8
�0H (T)

b 0

4.00 K
-1 5.00 K
6.00 K
R (�)

7.00 K
7.25 K
T

8.00 K

-2

-3

-8 -6 -4 -2 0 2 4 6 8
�0H (T)

FIG. S8. Experiments with Sample S2NbN. (a) The resistance Rxx measured as a function of magnetic field using a
lock-in technique. An a.c. current of 10 µA r.m.s. was applied. (b) Plots of transverse resistance as a function of magnetic
field.

film at T = 5.00 K, compared to other temperatures. Upon lowering the temperature further to T = 4.00 K, the
peak becomes larger in magnitude again.

4.3 Sample S3NbN

Sample S3NbN is a film of NbN deposited on MgO(011) substrate. It has a thickness of 9 nm and is the thinnest of
all films measured by us. The critical temperature was found to be 8.5 K. The critical field measurements are shown
in Fig. S9a. The normal state resistance measured in four-probe configuration is 309 Ω. This corresponds to a
resistivity of 9.9 µΩ-m.

The transverse resistance measured in this film offers important insights about the phenomenon. The expected
variation of RT (H), following the lineshape given by Λ(H), is depicted in Fig. S9b. Unlike the previously reported
samples, we see here that the measured RT (shown in Fig. S9c) actually follows the predicted trend of Λ(H) to a
fair degree. The lineshapes of experimentally observed RT (H) (in Fig. S9c) are quite similar at all temperatures.
Although the magnitude varies a bit at different temperatures, the difference is not very large. Thus, we see here a
behaviour which conforms well to the prediction of the model of Segal et al.[R2]. In other words, the impact of
inhomogeneous distribution of superconducting parameters intrinsic in the structure of the film is the dominant
factor for the transverse resistance in this sample. This provides clearly a different case than all the other samples
measured by us. Since this is also the thinnest sample with t = 9 nm, we conclude that the impact of emergent
electronic inhomogeneities on the development of the transverse resistance disappears in the limit of small film
thickness. In this regime the structural non-uniformities play the dominant role in guiding the paths of electric
current.
 9

a 300

250

200

(�)
150

xx
R
100 5.00 K
6.00 K
7.00 K
50 7.50 K
7.75 K
8.00 K
0
-8 -6 -4 -2 0 2 4 6 8
�0H (T)
b 200
� (arb. unit)

5.00 K 7.50 K
150 6.00 K 7.75 K
7.00 K 8.00 K
100

50

0
-8 -6 -4 -2 0 2 4 6 8
�0H (T)

c 5.00 K
6.00 K
7.00 K
7.50 K
7.75 K
8.00 K
25

20
R (�)

15
T

10

0
-8 -6 -4 -2 0 2 4 6 8
�0H (T)

FIG. S9. Experiments with Sample S3NbN. (a) The resistance Rxx measured as a function of magnetic field using a
lock-in technique. A current of 1 µA r.m.s. amplitude was applied. (b) Λ(H) is plotted . (c) Plots of transverse resistance as
a function of magnetic field.

∗
shamashis.sengupta@ijclab.in2p3.fr
†
alireza.farhadizadeh@liu.se
‡
miguel.monteverde@universite-paris-saclay.fr
[R1] A. Le Febvrier, J. Jensen, and P. Eklund, Wet-cleaning of MgO (001): Modification of surface chemistry and effects on
thin film growth investigated by x-ray photoelectron spectroscopy and time-of-flight secondary ion mass spectroscopy, J.
Vac. Sci. Technol. A 35, 021407 (2017)
[R2] A. Segal, M. Karpovski, and A. Gerber, Inhomogeneity and transverse voltage in superconductors, Phys. Rev. B 83,
094531 (2011)
[R3] T. P. Beaulac, F. J. Pinski, and. P. B.Allen, Hall coefficient in pure metals: Lowest-order calculation for Nb and Cu,
Phys. Rev. B 23, 3617 (1981)
[R4] M. Ashkin, J. R. Gavaler, J. Greggi, and M. Decroux, The upper critical field of NbN films. II, J. Appl. Phys. 55, 1044
(1984)
[R5] A. Iovan, A. Pedeches, T. Descamps, H. Rotella, I. Florea, F. Semond, and V. Zwiller, NbN thin films grown on silicon by
molecular beam epitaxy for superconducting detectors, Appl. Phys. Lett. 123, 252602 (2023)