# Spontaneous voltage peaks in superconducting Nb channels without engineered
asymmetry

Shamashis Sengupta<EMAIL_ADDRESS>Université Paris-
Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France  Miguel Monteverde  Université
Paris-Saclay, CNRS, Laboratoire de Physique des Solides, 91405, Orsay, France
Sara Loucif  Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France
Florian Pallier  Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay,
France  Louis Dumoulin  Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405
Orsay, France  Claire Marrache-Kikuchi  Université Paris-Saclay, CNRS/IN2P3,
IJCLab, 91405 Orsay, France

###### Abstract

Rectification effects in solid-state devices are a consequence of
nonreciprocal transport properties. This phenomenon is usually observed in
systems with broken inversion symmetry. In most instances, nonreciprocal
transport arises in the presence of an applied magnetic field and the
rectified signal has an antisymmetric dependence on the field. We have
observed rectification of environmental electromagnetic fluctuations in plain
Nb channels without any asymmetry in design, leading to spontaneous voltage
peaks at the superconducting transition. The signal is symmetric in the
magnetic field and appears even without an applied field at the critical
temperature. This is indicative of an unconventional mechanism of
nonreciprocal transport resulting from a spontaneous breaking of inversion
symmetry.

Rectification of a.c. signals into a d.c. voltage is known in superconducting
devices incorporating some form of asymmetry in structure, either at atomic or
mesoscopic lengthscales. Rectification effects have their origin in
nonreciprocal transport properties induced by a breaking of inversion
symmetry. In most cases, a breaking of time reversal symmetry by application
of a magnetic field is also required. Nonreciprocal transport in an applied
magnetic field has been observed in non-centrosymmetric superconductors [1,
2], polar superconductors with Rashba-type spin-orbit interaction [3],
topological insulator/superconductor interfaces [4] and amorphous
superconductors attached to a magnetic material [5]. In devices employing the
principle of vortex ratchets, the asymmetry is engineered through artificially
created pinning arrays [6, 7, 8]. The lack of inversion symmetry in certain
types of superconducting devices [9, 10, 11, 12] leads to the observation of
different critical currents for opposite bias polarities. This phenomenon,
called the superconducting diode effect, is presently a very active area of
research. Some systems showing nonreciprocal properties are capable of
rectifying environmental electromagnetic fluctuations into a d.c. voltage.
This has been observed in MoGe (superconductor)/Y3Fe5O12 (insulator magnet)
heterostructures [5] and thin flakes of NbSe2 [2]. A common feature of such
nonreciprocal transport, induced by an applied magnetic field, is that the
observed voltage has an antisymmetric field-dependence [1, 2, 3, 4, 5, 9, 12,
13, 14]. In this work, we report the observation of spontaneous voltage peaks
in plain channels of superconducting Nb without any asymmetry in its design.
The voltage peaks are symmetric with respect to the magnetic field. These
features are observed even without the application of a magnetic field, at the
critical temperature of the transition. This highlights an unconventional
mechanism of rectification in a superconducting device which is different from
the most common examples of this phenomenon.

The experiments were conducted on superconducting channels of Nb in various
geometries, realized on commercially available Si wafers covered with 500 nm
of SiO2. The devices were patterned using electron beam lithography, following
which we evaporated a layer of Nb (thickness between 55 nm to 72 nm). The
superconducting critical temperature ($T_{c}$) of the samples varied from one
batch to another. The $T_{c}$ of niobium depends strongly on the quality of
vacuum in the deposition chamber [15] and average grain size [16]. Further,
contamination from electron beam resist contributed to a significantly lower
$T_{c}$ of narrow channels with widths below a few micrometres. We were thus
able to study superconducting devices with a wide range of critical
temperatures (2.5 K to 7.7 K). Electrical contacts were established with
ultrasonic wire-bonding directly on contact pads of the sample. The electrical
lines inside the cryostat used for transport measurements have a resistance of
about 1 $\Omega$ each, measured from the external connectors at room
temperature down to the sample stage.

Fig. 1a shows the schematic diagram of a sample (named D1) with a simple
geometry. It consists of a central channel of width ($w$) 38 $\mu$m and length
($L$) 245 $\mu$m, connected to two large contact pads. The thickness of the Nb
film is 72 nm. (See Supplemental Material for images.) The variation of
resistance ($R$) measured in two-probe configuration with temperature ($T$) is
shown in Fig. 1b. We determine $T_{c}$ and the upper critical magnetic field
($B_{c2}$) by the midpoint of the resistance drop. The direction of applied
magnetic field was always perpendicular to the plane of the sample.

There are two superconducting transitions in Fig. 1b, corresponding to the
critical temperatures of the large pads (5.1 K) and the central channel (3.9
K). At a temperature of 2.5 K, we observed $B_{c2}$ of 2.8 T for the pads and
1.6 T for the channel (Fig. 1c). We will now focus on the response of the
device in the absence of any applied bias current. For this, we simply
connected a voltmeter across the sample. Fig. 1d shows a schematic diagram of
this measurement. The two contacts pads on the device are labelled T1 and T2.
There are two ways of connecting the voltmeter leads across a two-terminal
device. These are marked as Configurations 1 and 2. We adopt the convention of
denoting the measured voltage as V[T1,T2], where the first (second) contact is
the one connected to the positive (negative) terminal of the voltmeter input.
The voltmeter used for this particular measurement was a Keithley DMM6500
Multimeter (henceforth DMM). Upon sweeping the magnetic field, we observed two
sharp peaks in the measured voltage (Fig. 1e) almost coinciding with $B_{c2}$
of the central channel. These spontaneously arising peaks are not transient
features. The measured voltage is steady in time if the magnetic field is held
constant.

Figure 1: (a) Representation of the geometry of sample D1. (b, c) Observation
of the superconducting transition by measuring the resistance as a function of
temperature (b) and magnetic field (c). The d.c. current applied was 1 $\mu$A.
(d) Schematic diagram of the measurement of voltage across the sample without
any applied bias current using a d.c. voltmeter (VM). (e) Spontaneous voltage
peaks as a function of magnetic field with the circuit of Configuration 1. (f)
Spontaneous voltage peaks as a function of magnetic field with the circuit of
Configuration 2. Figure 2: (a,b) Image of sample D2 in the scanning electron
microscope (SEM). A close-up of the narrow channels of the voltage probes
(dashed rectangle in (a)) is shown in (b). (c) Four-probe measurement of
resistance using a low frequency lock-in technique. (d) Measurement of
V[U1,U4] with a DMM simultaneously with the resistance measurement described
in (c). (e) $R_{2f}$ estimated from the voltage at the second harmonic
frequency measured between U1 and U4. (f,g) Two-probe resistance measurement
as a function of temperature (f) and magnetic field (g) across U7 and U8,
using an applied d.c. current of 78 nA. (h) Measurement of V[U7,U8] without
any applied bias current with a nanovoltmeter as a function of magnetic field.
(i) Measurement of V[U7,U8] without any applied bias current with a
nanovoltmeter as a function of temperature at zero field. (j) The
spontaneously arising voltage V[U7,U8] (open circles) is measured with a
variation of temperature (line). When the temperature reaches 2.4 K, it is
held constant and the voltage is recorded with time. (k) Measurement of
spontaneous voltage across U7 and U8 as a function of magnetic field by
interchanging the polarities of nanovoltmeter leads on the two contacts. (l)
Measurement of V[U7,U8] and V[U8,U7] as a function of the magnetic field.
(Inset) Circuit implemented with a resistance $R_{p}$ in parallel with the
sample.

The measurement of a finite voltage by a voltmeter implies that a current is
flowing through its internal resistance. Therefore, power is continuously
dissipated when the voltage is steady in time. We can obtain insights about
the physical origin of the spontaneous voltage peaks (Fig. 1e) by identifying
the source of energy. Heat flowing to the sample from the thermal bath (sample
stage) cannot be considered as a possibility. This would imply that heat
extracted from a source is converted completely into electrical work, in
violation of the second law of thermodynamics. A more reasonable explanation
is the rectification of environmental electromagnetic fluctuations by the
device at the superconducting transition. This phenomenon, that a
superconducting system acts as a rectifying antenna capable of generating d.c.
electricity from nonequilibrium environmental electromagnetic fluctuations,
has been observed in MoGe/Y3Fe5O12 heterostructures [5] and NbSe2 [2]. The
voltage peaks at the critical field in our devices show clear similarities
with observations reported in these works [17]. The surprising fact, however,
is that we observe a rectification effect at all, since there is no apparent
source of asymmetry in our devices. In Refs. 2 and 5, the voltage peaks are
antisymmetric as a function of the magnetic field. The symmetric-in-field
peaks in our experiments highlight that a different mechanism of rectification
is at work here.

A d.c. voltage requires a preferential direction to determine its polarity.
How this biasing happens is a key question that needs to be understood. If the
polarities develop at random, an even distribution of positive and negative
peaks is expected. However, in most instances, we observed positive peaks in
our measurements (for reasons that will be explained later). Further cross-
checks yielded unexpected results. We interchanged the leads of the voltmeter
on the contact pads to measure V[T2,T1] (Configuration 2 in Fig. 1d). We would
expect V[T2,T1] to have opposite sign of V[T1,T2]. However, upon sweeping the
magnetic field, we still observed (Fig. 1f) positive voltage peaks. This shows
that features of the spontaneous d.c. voltage are not only related to the
superconducting transition in the sample, but also to an influence of the
internal circuitry of the voltmeter. Further discussions on this issue will be
presented later.

We now present results from a second sample D2 (Fig. 2a). This consisted of a
central channel ($w$ = 40 $\mu$m) for applying a current (between contacts C1
and C2) and eight probes for measuring voltage (numbered from U1 to U8). The
thickness of Nb film was 55 nm. The voltage probes consisted of narrow
channels a few micrometres wide (Fig. 2b), with reduced $T_{c}$ ($<$ 3.0 K).
Standard four-probe measurements showed a sharp superconducting transition at
7.7 K in the central channel connecting C1 and C2. (See Supplemental Material
for the characterization of sample D2.) Fig. 2c shows the result of four-probe
resistance measurement using a standard lock-in technique. A current of 1
$\mu$A r.m.s. was applied between C1 and C2. The voltage was measured between
the probes U1 and U4 with a Stanford Research Systems SR830 lock-in amplifier.
Simultaneously, a DMM measured the d.c. voltage between the same contacts U1
and U4 (Fig. 2d). The resistance drop at the superconducting transition at
$B_{c2}$ (Fig. 2c) was accompanied by d.c. voltage peaks (Fig. 2d).

The resistance of a system exhibiting nonreciprocal properties, as a function
of current ($I$), may be written as:

$R=R_{0}+\alpha I$ (1)

where $R_{0}$ is the resistance in the limit of zero current and $\alpha$ is
the parameter describing the nonreciprocity in transport. We neglect higher
order terms in $I$. For such a system, an a.c. current at frequency $f$ with
r.m.s. amplitude $I_{ac}$ results in a r.m.s. voltage
$V_{2f}=\frac{1}{\sqrt{2}}\alpha I_{ac}^{2}$ at the second harmonic frequency
$2f$. (See Supplemental Material for the derivation). An a.c. current
($I_{ac}$ = 5 $\mu$A, $f$ = 13.27 Hz) was applied between contacts C1 and C2
of sample D2. $V_{2f}$ was measured between probes U1 and U4. The
corresponding resistance $R_{2f}=V_{2f}/I_{ac}$ is shown in Fig. 2e. We
observed sharp peaks in $R_{2f}$ at the critical field transition (determined
by resistance measurements in Fig. 2c). This shows that the superconducting
transition is indeed associated with even-in-field nonreciprocal transport
properties and a finite value of the nonreciprocity parameter $\alpha$.

The resistance of the conduction channel between the contacts U7 and U8 was
measured in two-probe configuration as a function of temperature (Fig. 2f).
The narrow segments (marked by a dashed ellipse in Fig. 2b) have a broad
transition with a $T_{c}$ of 2.5 K. The zero resistance state is still not
reached at the lowest accessible temperature of 2.0 K. The critical magnetic
field of these segments at 2.0 K was determined to be 0.6 T from resistance
measurements (Fig. 2g). Fig. 2h shows the measurement of spontaneous voltage
without an applied current, using a Keithley 2182A nanovoltmeter. The
transitions of wide and narrow regions correspond to two voltage peaks around
4.0 T and 0.6 T respectively. The peaks arising at the superconducting
transition of the narrow regions show step-like features. A spontaneous
voltage was also seen to develop at the superconducting transition induced by
varying the temperature at zero magnetic field (Fig. 2i). It is therefore
evident that the phenomenon of rectification is associated with both the
field-induced and temperature-induced transitions. If the temperature is held
constant, the voltage is stable over time (Fig. 2j).

We carried out voltage measurements across the probes U7 and U8 by
interchanging the polarities of the nanovoltmeter leads (Fig. 2k). In both
cases, there were positive peaks. The features of the voltage curves vary in
their details from one set of measurement to another, as can be seen in Fig.
2k, and also in comparison with Fig. 2h. The general feature is that the peaks
always occur across the broad transition seen in resistivity measurements
(Fig. 2g). We now turn to the discussion of the observation that the voltage
peaks do not change sign upon interchanging the voltmeter input connectors.
Voltmeters used in practice are not ideal instruments and may apply small
signals on the sample being measured. One such signal is the input bias
current [18] which flows between its terminals. This effect is negligible
under most practical circumstances. The nanovoltmeter bias current is
typically 60 pA, producing a drop of 60 nV per k$\Omega$ of external
resistance. Other sources of signals from the voltmeter might be stray
voltages due to charges accumulated across the input impedance and
thermoelectric potentials in its internal circuitry. We speculate that it is
precisely such small and undetermined signals which cause the polarity of the
spontaneous d.c. voltage peak across the sample to develop in a specific
orientation, in correlation with the polarity of connectors at the voltmeter
input. To verify this understanding, the circuit outlined in Fig. 2l was
implemented. Here, the device (with resistance $R_{s}$) is connected in
parallel with a resistance $R_{p}$. When $R_{p}\ll R_{s}$, it lessens the
impact of any signal originating within the voltmeter on the sample. The
spontaneous voltage which develops across the sample changes drastically,
because in order to maintain the same voltage drop across a small $R_{p}$, a
far greater amount of power needs to be delivered which places severe
constraints on the physical system. Despite this drawback, this set-up enables
us to test whether lowering the impact of signals from the voltmeter allows
the polarity of spontaneous voltage to develop independently of the
orientation of the voltmeter leads, or not. Upon conducting these measurements
(Fig. 2l) with $R_{p}=$ 67 $\Omega$, we found that the measured voltage
features indeed reversed the sign upon interchanging the voltmeter leads. This
shows that the usual observation of positive voltage peaks (when no $R_{p}$ is
present) is likely to be a consequence of small signals within the voltmeter.
These observations highlight that the polarity of spontaneous voltage at the
superconducting transition is extremely susceptible to small external fields.
It indicates that a mechanism of spontaneous symmetry breaking might be at
work. An analogy may be drawn to the three dimensional Ising model of spins
[19, 20] with attractive interaction, where the spins are allowed to point
either ‘up’ or ‘down’. As is well known, a net magnetization $M$ develops
below a critical temperature due to spontaneous breaking of up-down symmetry.
The preferred direction (‘up’ or ‘down’) may be determined by an
infinitesimally small external magnetic field. In our experiments, the d.c.
voltage seemingly has its origin in rectification effects due to a spontaneous
breaking of inversion symmetry along the Nb channel at the superconducting
transition. A small external electric field is capable of inducing a preferred
direction, which in our measurements is provided by the measuring instrument
(voltmeter).

From our empirical results, we postulate that the rectification effect seen
here is associated with the spontaneous development of an electric
polarization across the length of the sample, resulting in a potential
landscape without inversion symmetry. The origin of such a polarization should
be related to the appearance of dipole moments at the microscopic scale. We
will now discuss a possible mechanism of how such microscopic dipoles can
arise. For this, we will focus on two components of the niobium film - the
system of conducting electrons within which superconductivity develops, and
non-metallic niobium grains which do not contribute to superconductivity.

Moro et al. [21] reported the signature of ferroelectricity in clusters of Nb
atoms. These dipoles are nonclassical in nature [22] and can not be understood
in terms of a polarization built into the structure of the cluster. The dipole
moments in such clusters bear the signature of a spontaneous-symmetry-broken
state. Durkin et al. [23] studied the transport properties of niobium
nanoislands and concluded that superconductivity develops from a mechanism of
rare-region onset with only very large grains contributing to superconducting
order. Bose et al. [16] observed that superconductivity disappeared for
niobium films with an average particle size of 8 nm, consistent with the
Anderson criterion. At low temperatures, insulating behaviour might be
expected in very small grains which are a few nanometres in size [24], as well
as in intergrain boundaries. We hypothesize that the insulating parts of the
niobium films host nonclassical dipole moments similar to those observed in
the clusters in Refs. [21] and [22]. The mutual interaction between these
dipole moments in insulating grains leads to the development of a macroscopic
electric polarization $\mathbf{P_{i}}$. We now turn our attention to the
system of conducting electrons which arises from the larger and more metallic
grains and in which superconductivity develops. Theoretical studies [25, 26]
have shown that the chemical potentials of superconducting and normal states
are different, as a consequence of which there is a transfer of charge from
one electronic subsystem to another if they coexist. Therefore, the presence
of two different phases in the vicinity of the superconducting transition
leads to the development of dipole moments at the interface of superconducting
and normal regions within the niobium film. The total polarization
$\mathbf{P_{el}}$ within the electronic system, on taking the macroscopic
average, should generally be zero. However, when there exists a macroscopic
electric polarization $\mathbf{P_{i}}$ in the background originating from the
insulating grains, $\mathbf{P_{el}}$ can have a finite non-zero value.
Conduction electrons experience the electric field due to the polarization
$\mathbf{P_{el}}$ at the interface of superconducting and normal regions. This
offers a possible mechanism of the asymmetry leading to a rectification
effect.

In summary, we have observed spontaneous voltage peaks at the superconducting
transition of plain Nb devices resulting from a rectification of environmental
fluctuations. The key observations are that the peaks are symmetric in the
magnetic field and are present at the critical temperature without an applied
magnetic field. This signifies an unconventional mechanism of rectification
which is in stark contrast to the commonly observed antisymmetric field-
dependence of the rectified signal. The spontaneously arising asymmetry in our
devices is possibly related to the development of an electric polarization at
the superconducting transition. We can account for this phenomenon based on
the hypothesis that insulating regions of niobium films (nanosized grains and
intergrain boundaries) host non-classical dipole moments. Experimental [27,
28, 29] and theoretical [30, 31] works have previously suggested that there
are many interesting questions regarding the impact of electric fields on
superconductors which are yet to be fully understood. Our experiments show
that rectification effects can reveal the existence of electric polarization
concomitant with superconductivity and provide new directions for research on
these questions.

##  I Acknowledgments

We thank Jana Lustikova, Alejandro Silhanek and Xiaoshan Xu for insightful
discussions. We are grateful to Laurent Bergé and Sophie Guéron for help with
fabrication of devices. This work was supported by public grants from the
French National Research Agency (ANR), project CP-Insulators No.
ANR-2019-CE30-0014-03.

## References

  * [1] R. Wakatsuki, Y. Saito, S. Hoshino, Y. M. Itahashi, T. Ideue, M. Ezawa, Y. Iwasa, and N. Nagaosa, Sci. Adv. 3, e1602390 (2017)
  * [2] E. Zhang, X. Xu, Y.-C. Zou, L. Ai, X. Dong, C. Huang, P. Leng, S. Liu, Y. Zhang, Z. Jia, X. Peng, M. Zhao, Y. Yang, Z. Li, H. Guo, S. J. Haigh, N. Nagaosa, J. Shen, and F. Xiu, Nat. Commun. 11, 5634 (2020)
  * [3] Y. M. Itahashi, T. Ideue, Y. Saito, S. Shimizu, T. Ouchi, T. Nojima, and Y. Iwasa, Sci. Adv. 6, eaay9120 (2020)
  * [4] K. Yasuda, H. Yasuda, T. Liang, R. Yoshimi, A. Tsukazaki, K. S. Takahashi, N. Nagaosa, M. Kawasaki, and Y. Tokura, Nat. Commun. 10, 2734 (2019)
  * [5] J. Lustikova, Y. Shiomi, N. Yokoi, N. Kabeya, N. Kimura, K. Ienaga, S. Kaneko, S. Okuma, S. Takahashi, and E. Saitoh, Nat. Commun. 9, 4922 (2018)
  * [6] J. E. Villegas, S. Savel’ev, F. Nori, E. M. Gonzalez, J. V. Anguita, R. García, and J. L. Vicent, Science 302, 1188 (2003)
  * [7] C. C. de Souza Silva, J. Van de Vondel, M. Morelle, and V. V. Moshchalkov, Nature 440, 651 (2006)
  * [8] J. Van de Vondel, C. C. de Souza Silva, B.Y. Zhu, M. Morelle, and V.V. Moshchalkov, Phys. Rev. Lett. 94, 057003 (2005)
  * [9] F. Ando, Y. Miyasaka, T. Li, J. Ishizuka, T. Arakawa, Y. Shiota, T. Moriyama, Y. Yanase, and T. Ono, Nature 584, 373 (2020)
  * [10] L. Bauriedl, C. Bäuml, L. Fuchs, C. Baumgartner, N. Paulik, J. M. Bauer, K.-Q. Lin, J. M. Lupton, T. Taniguchi, K. Watanabe, C. Strunk, and N. Paradiso, Nat. Commun. 13, 4266 (2022)
  * [11] C. Baumgartner, L. Fuchs, A. Costa, S. Reinhardt, S. Gronin, G. C. Gardner, T. Lindemann, M. J. Manfra, P. E. Faria Junior, D. Kochan, J. Fabian, N. Paradiso, and C. Strunk, Nat. Nanotechnol. 17, 39 (2022)
  * [12] H. Wu, Y. Wang, Y. Xu, P. K. Sivakumar, C. Pasco, U. Filippozzi, S. S. P. Parkin, Y.-J. Zeng, T. McQueen, and M. N. Ali, Nature 604, 653 (2022)
  * [13] D. Cerbu, V. N. Gladilin, J. Cuppens, J. Fritzsche, J. Tempere, J. T. Devreese, V. V. Moshchalkov, A. V. Silhanek, and J. Van de Vondel, New. J. Phys. 15, 063022 (2013)
  * [14] V. V. Pryadun, J. Sierra, F. G. Aliev, D. S. Golubovic, and V. V. Moshchalkov, Appl. Phys. Lett. 88, 062517 (2006)
  * [15] W. De Sorbo, Phys. Rev. 132, 107 (1963)
  * [16] S. Bose, P. Raychaudhuri, R. Banerjee, P. Vasa, and P. Ayyub, Phys. Rev. Lett. 95, 147003 (2005)
  * [17] The relevant figures are Fig. 4f in Ref. 2 and Fig. 3b in Ref. 5.
  * [18] Low Level Measurements Handbook: Precision DC Current, Voltage, and Resistance Measurements, 6th Edition, Keithley Instruments, Cleveland (2004)
  * [19] F. Reif, Fundamentals of Statistical and Thermal Physics, McGraw-Hill, New York (1965)
  * [20] N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group, CRC Press, Boca Raton (2018)
  * [21] R. Moro, X. Xu, S. Yin, and W. A. de Heer, Science 300, 1265 (2003)
  * [22] X. Xu, S. Yin, R. Moro, A. Liang, J. Bowlan, and W. A. de Heer, Phys. Rev. B 75, 085429 (2007)
  * [23] M. Durkin, R. Garrido-Menacho, S. Gopalakrishnan, N. K. Jaggi, J.-H. Kwon, J.-M. Zuo, and N. Mason, Phys. Rev. B 101, 035409 (2020)
  * [24] S. Bose, R. Banerjee, A. Genc, P. Raychaudhuri, H. L. Fraser, and P. Ayyub, J. Phys.: Condens. Matter 18, 4553 (2006)
  * [25] D. I. Khomskii, and F. V. Kusmartsev, Phys. Rev. B 46, 14245 (1992)
  * [26] D. I. Khomskii, and A. Freimuth, Phys. Rev. Lett. 75, 1384 (1995)
  * [27] R. Tao, X. Xu, Y.C. Lan, and Y. Shiroyanagi, Physica C 377, 357 (2002)
  * [28] R. E. Glover III, and M. D. Sherill, Phys. Rev. Lett. 5, 248 (1960)
  * [29] G. De Simoni, F. Paolucci, P. Solinas, E. Strambini, and F. Giazotto, Nature Nanotechnol. 13, 802 (2018)
  * [30] J. E. Hirsch, Phys. Rev. B 68, 184502 (2003)
  * [31] J. E. Hirsch, Phys. Rev. B 69, 214515 (2004)

Supplemental Material

1\. Second harmonic component in voltage with an a.c. bias current

The simplest expression to describe a directional anisotropy in resistance (R)
as a function of the current (I) is

$R=R_{0}+\alpha I,$ (S1)

where $R_{0}$ is the resistance in the limit of zero current and $\alpha$ is
the coefficient describing the anisotropy. Let us consider the case of an a.c.
current of amplitude $\widetilde{I_{ac}}$ at frequency $f$ applied through the
system.

$I=\widetilde{I_{ac}}sin(2\pi ft),$ (S2)

with t denoting time.

The corresponding voltage drop ($V=RI$) is:

$V=\frac{1}{2}\alpha{\widetilde{I_{ac}}}^{2}+R_{0}\widetilde{I_{ac}}sin(2\pi
ft)+\frac{1}{2}\alpha{\widetilde{I_{ac}}}^{2}sin(2\pi.2ft-\pi/2)$ (S3)

Expressed in terms of the r.m.s. current $I_{ac}=\widetilde{I_{ac}}/\sqrt{2}$,
the r.m.s. voltage $V_{2f}$ at the second harmonic frequency ($2f$) is

$V_{2f}=\frac{1}{\sqrt{2}}\alpha{I_{ac}}^{2}$ (S4)

2\. Characterization of sample D1

Fig. S1a shows the image of sample D1 observed with a scanning electron
microscope. The conduction channel between the two contact pads has a length
of 245 $\mu$m and a width of 38 $\mu$m. The topography of the edge of this
channel, measured with an atomic force microscope, is displayed in Fig. S1b.
The thickness of the Nb film is 72 nm.

3\. Transport measurements on sample D2

Sample D2 consists of a long conduction channel of length 1380 $\mu$m
connected to the contacts pads marked as C1 and C2 (Fig. 2a of the main text).
The voltage probes have a width of only a few micrometres near the junction
with the central conduction channel (Fig. 2b of main text). One such junction
was scanned with an atomic force microscope (Fig. S2). The thickness of the
film is 55 nm.

Figure S1: (a) Scanning electron microscope image of sample D1. (b) Image of
one edge of the conduction channel observed with an atomic force microscope.
Figure S2: Image of sample D2 with an atomic force microscope.

Figure S3: (a) Four-probe measurement of the critical superconducting
temperature in sample D2 using a d.c. current bias of 1 $\mu$A. The current
was applied between contact pads C1 and C2. (See Fig. 2a of main text for the
nomenclature of contacts.) The voltage drop was measured between U1 and U4
with a DMM. (b) Measurement of the critical magnetic field at a temperature of
6.6 K in the same setup. The d.c. current applied was 5 $\mu$A. (c,d)
Spontaneous voltage measured on contacts C1 and C2 at 6.6 K using a DMM. The
two plots show the data for two different configurations of the voltmeter
leads, as outlined in Fig. 1d of the main text. Figure S4: Spontaneous voltage
measured between contacts C1 and C2 using a DMM while sweeping the magnetic
field. Figure S5: (a,b) Simultaneous measurement of resistance in four-probe
configuration and spontaneous d.c. voltage. An a.c. current of r.m.s.
amplitude 1 $\mu$A and frequency ($f$) of 13.27 Hz was applied between the
contacts C1 and C2. The voltage drop was measured between U1 and U4 with a
lock-in amplifier. The resistance thus measured is shown in (a). The d.c.
voltage, displayed in (b), was measured between the same contacts (U1 and U4)
with a DMM. (c,d) Simultaneous measurement of resistance and second harmonic
voltage in four-probe configuration. An a.c. current of r.m.s. amplitude 5
$\mu$A ($f$ = 13.27 Hz) was applied between the contacts C1 and C2. One lock-
in amplifier measured the voltage between U1 and U4. The resistance thus
calculated is shown in (c). A second lock-in ampifier measured the second
harmonic voltage $V_{2f}$, shown in (d). Figure S6: Measurement of $V_{2f}$
for different a.c. bias currents. The a.c. current of r.m.s. amplitude
$I_{ac}$ (f=13.27 Hz) was applied between the contacts C1 and C2. The voltage
$V_{2f}$ was measured across U1 and U4. Figure S7: Measurement of spontaneous
d.c. voltage with a 10 k$\Omega$ external resistor in parallel. A resistance
$R_{p}$ of value 10 k$\Omega$ was put across the connectors (at room
temperature outside the cryostat) leading to the contacts U7 and U8 on the
sample. The voltage across these connectors was measured with a nanovoltmeter
while sweeping the magnetic field. The temperature of the Nb device in the
cryostat was 2.0 K. The magnetic field was held constant when it reached 0.600
T and the voltage was recorded with time.

In Fig S3a, we show four probe measurements on the device. The d.c. current (1
$\mu$A) was applied between C1 and C2. The voltage drop was measured between
the probes U1 and U4 using a digital multimeter (DMM). The superconducting
transition occurs at 7.7 K. There is a feature of a small peak (indicated by
an arrow in Fig. S3a) at 7.0 K. A similar feature was also visible (at a
magnetic field of 420 mT) when the superconducting transition was induced by
varying the magnetic field (Fig. S3b) at 6.6 K. The critical field of the
central conduction channel (Fig. S3b) at 6.6 K is 800 mT. The feature at 420
mT (indicated by arrows) possibly arises from rectification of environmental
electromagnetic fluctuations within the voltage probes. The characterization
of voltage probes (using contacts U7 and U8), shown in Fig. 2f of main text,
reveals a transition at 7.0 K. The small peak feature in Fig. S3a occurs at a
similar temperature, suggesting that the cause may be attributed to
rectification along the voltage probes. Ideally, electrical measurements in
the four-probe configuration are not supposed to be affected by the properties
of the voltage probes. It is interesting to note that in the presence of
rectification effects, this ideal scenario is not respected. We therefore need
to be cautious while interpreting the results of transport experiments on
superconducting devices, since such features arising within voltage probes may
add unexpected voltage contributions to the measured signal.

Magnetotransport measurements with a current applied along the central channel
of sample D2 reveal that its mean free path ($l$), Fermi wave vector ($k_{F}$)
and superconducting coherence length ($\xi$) are 3.2 nm, 11.2 nm-1 and 9.1 nm
respectively.

Figs. S3c and S3d show the measurement of spontaneous voltage between contacts
C1 and C2 with a DMM at 6.6 K. The peaks are positive, as we have observed on
most instances. We generally observe that the peak heights show some
differences from one experimental run to another. When measurements are
performed on different days, the magnitude of the peaks can also be quite
different. We attribute these differences to an overall change in the amount
of environmental fluctuations, which is not a controlled parameter. Some
difference can also arise from the behaviour of the sample itself. At a
microscopic level, the spontaneously arising asymmetry in the sample is
possibly related to the redistribution of charges owing to the presence of
superconducting and non-superconducting domains. The configuration of these
domains are likely to be different from one set of measurement to another.

Although in most instances we observe positive peaks, we have observed
negative peaks a few times. One dataset is shown in Fig. S4. The peaks are
always symmetric in the magnetic field.

Figs. 2c and 2d of the main text show the four-probe resistance (measured with
a lock-in technique) and the spontaneous d.c. voltage in sample D2. The
complete dataset is shown in Figs. S5a and S5b. At 2 K (Fig. S5b), there are
features below 1 T (indicated by arrows). These arise from the narrow channels
of the voltage probes (U1 and U4). We do not see these features above 3 K,
since these temperatures are above the critical temperature for these narrow
channels.

In Fig. 2e of the main text, the data for measurement of second harmonic
voltage was displayed. For these measurements, we used two lock-in amplifiers
measuring the voltage on contacts U1 and U4. One lock-in amplifier measured
the voltage at the same frequency while the other measured the voltage
$V_{2f}$ at the second harmonic frequency. The complete dataset is presented
in Figs. S5c and S5d.

It is predicted that the second harmonic voltage $V_{2f}$ will vary as
$I_{ac}^{2}$ (Eq. S4). We performed measurements of $V_{2f}$ for three
different values of $I_{ac}$ at 3 K (Fig. S6). The peak heights are plotted in
the inset of Fig. S6. These indeed follow the $I_{ac}^{2}$ dependence, with
the fit parameter $\alpha$ = 5.2 $\times$ $10^{4}$ $\Omega$/A. An underlying
assumption during this analysis is that the anisotropy parameter $\alpha$ is a
constant. This is found to be generally true when the measurements are done in
quick succession, as in this case. However, when performed in a different
experimental run, the results can be different. This is seen when comparing
the results of Fig. S6 (for $I_{ac}$ = 5 $\mu$A) with the plot in Fig. S5d
(for a temperature of 3 K). The experimental setup was identical in both
cases, but the peak heights are not the same.

It was discussed in the main text that the measurement of a d.c. voltage in a
voltmeter implies that work is being done across its internal impedance. This
consideration led us to conclude that the spontaneous voltage arises from the
rectification of environmental fluctuations. The fact that work is being done
can be demonstrated more clearly by adding a known resistor in parallel with
the sample. We measured the voltage across the probes U7 and U8 with a 10
k$\Omega$ resistor in parallel. The field was ramped till 600 mT (upper
critical field of superconductivity) and then held constant, with the
temperature of the sample at 2.0 K. With the magnetic field set at 600 mT, the
observed voltage was steady over several minutes (Fig. S7) at approximately
3.4 $\mu$V. This voltage corresponds to a steady power dissipation of 1.2 fW
across the 10 k$\Omega$ resistor.

4\. Generation of d.c. voltage due to mixing of d.c. and a.c. signals

The phenomenon of non-reciprocal transport is at the heart of the
rectification effect seen in our experiments. However, the generation of a
d.c. voltage from a.c. fluctuations can occur under a very different scenario
as well, which does not require the presence of any directional asymmetry in
the system. This may happen in isotropic systems with non-linear voltage-
current relation, for which the resistance ($R$) can be expressed as an
expansion in even powers of the current $I$.

$R=R_{0}+\sum_{i=2,4,6...}^{\infty}\eta_{i}I^{i}$ (S5)

In the above expression, $R_{0}$ is the resistance in the limit of zero
current and $\eta_{i}$ denotes the coefficient of the term $I^{i}$ ($i$ being
an even integer). It follows that the voltage $V$=$RI$ is an odd function of
$I$. A simple form of Eq. S5 with the lowest order term in $I$ is:

$R=R_{0}+\eta_{2}I^{2}.$ (S6)

Let us consider a situation where a d.c. current $I_{dc}$ is applied along
with an a.c. current of amplitude $\widetilde{I_{ac}}$ and frequency $f$.

$I=I_{dc}+\widetilde{I_{ac}}sin(2\pi ft)$ (S7)

Since the voltage $V=RI$ involves the term $\eta_{2}I^{3}$, the a.c. current
contributes to the d.c. voltage $V_{dc}$ and generates a voltage
$\widetilde{V_{2f}}$ at frequency $2f$.

$V_{dc}=R_{0}I_{dc}+\eta_{2}I_{dc}^{3}+\frac{3}{2}\eta_{2}I_{dc}\widetilde{I_{ac}}^{2}$
(S8)
$\widetilde{V_{2f}}=\frac{3}{2}\eta_{2}I_{dc}\widetilde{I_{ac}}^{2}sin(2\pi.2ft-\frac{\pi}{2})$
(S9)

The above equations show that, for an a.c. current passing through a
resistance described by Eq. S6, the generation of a d.c. voltage as well as an
a.c. voltage at second harmonic frequency is indeed possible. Similar results
can also be derived starting from the more general expression Eq. S5. It can
be shown that each term involving $I^{i}$ leads to the generation of d.c.
voltage contributions with two terms varying as ${I_{dc}}^{i+1}$ and
${I_{dc}}^{i-1}$ ${\widetilde{I_{ac}}}^{2}$ (assuming
${\widetilde{I_{ac}}}\ll{I_{dc}}$).

Unlike the case of non-reciprocal transport (described by Eq. S3), $V_{dc}$
and $\widetilde{V_{2f}}$ calculated above (Eqs. S8 and S9) have finite values
only when there is a d.c. current $I_{dc}$ in the system. The presence of a
voltmeter connected to the device implies that the input bias current
($I_{B}$) of the instrument is passing through the sample. Further, we can
expect small non-linearities in voltage-current relation (finite $\eta_{i}$)
at the superconducting transition. Therefore, it may be argued that all the
necessary conditions for the generation of a d.c. voltage from electromagnetic
fluctuations (as described in Eq. S8) are satisfied in our experiments. This
implies a trivial explanation of the spontaneous voltage peaks observed in our
devices without any requirement for invoking non-reciprocal transport. We
present below results to show that this is actually not the case.

Fig. S8a shows a diagram of the circuit implemented to measure voltage-current
(V-I) characteristics across the probes U7 and U8 of sample D2. The voltage
across the two contacts was measured with a nanovoltmeter. This instument has
$I_{B}$ $<$ 60 pA. An external current was applied beween the same contacts,
as shown by the current source $I_{S}$ is Fig. S8a. If we assume that the d.c.
voltage peaks at the superconducting transition result from the mixing of d.c.
and a.c. signals (Eq. S8), with $I_{dc}$ being the same as $I_{B}$, we can
expect a reduction of the generated $V_{dc}$ to zero when $I_{B}$ is offset by
an externally applied current in the opposite direction.

Fig. S8b shows the V-I characteristic at a temperature of 2.0 K and applied
magnetic field of 1.8 T. At this value of the magnetic field, the narrow
channels along the path connecting U7 and U8 are close to the onset of the
superconducting transition (see Fig. 2g of the main text). The peak of the
spontaneous voltage is expected to develop at much lower fields (Fig. 2h of
main text). The linear slope of the V-I curve (Fig. S8b) corresponds to a
resistance of 940 $\Omega$.

Following the above measurement, the magnetic field was further reduced
without any externally applied current to induce the superconducting
transition in the device. As expected, a spontaneous voltage was seen to
develop. The magnetic field sweep was stopped at 0.6 T. The spontaneous
voltage ($V_{0}$) observed was 7 $\mu$V. The V-I curve measured at this value
of the magnetic field is displayed in Fig. S8c. The applied current spanned a
wide range from -1 nA to 1 nA. This is much larger than the input bias current
of the nanovoltmeter. The voltage changed very little and was far from
approaching zero. This result demonstrates that the spontaneous voltage
observed in our devices is not arising from the mixing of d.c. and a.c.
signals as described in Eq. S8.

While sweeping the d.c. current, it was found that the voltage measured was
more noisy for faster rates of current sweep. The data of Fig. S8c was
acquired at a sweep rate of 0.8 pA/s. Fig. S8d shows the measurement of V-I
characteristics over a wider current range at a faster sweep rate of 28 pA/s.
A linear fit to the data (Fig. S8d) reveals a resistance of 555 $\Omega$. This
value is consistent with the resistance measurements shown in Fig. 2g of the
main text in the same device, where a d.c. current of 78 nA was used. It
confirms that the evolution of $V_{dc}$ observed in the V-I measurements
occurs entirely due to the resistive voltage drop ($R_{0}I_{dc}$) and does not
involve the mixing of the voltmeter input bias current with environmental
electromagnetic fluctuations.

Figure S8: (a) Circuit implemented to measure voltage-current (V-I)
characteristics across the probes U7 and U8 of sample D2. (b) V-I curve
measured at 2.0 K temperature with an applied magnetic field of 1.8 T. The
linear fit gives a resistance of 940 $\Omega$. (c,d) V-I curves at an applied
magnetic field of 0.6 T. Linear fits to the data correspond to resistance
values of 538 $\Omega$ (c) and 555 $\Omega$ (d). The voltage at zero current
obtained from the fit is 7.1 $\mu$V in (c) and 6.6 $\mu$V in (d).