# A superconducting full-wave bridge rectifier

Matteo Castellani  Owen Medeiros  Alessandro Buzzi  Reed A. Foster  Marco
Colangelo  Karl K. Berggren  [<EMAIL_ADDRESS>
###### Abstract

Superconducting thin-film electronics are attractive for their low power
consumption, fast operating speeds, and ease of interface with cryogenic
systems such as single-photon detector arrays, and quantum computing devices.
However, the lack of a reliable superconducting two-terminal asymmetric
device, analogous to a semiconducting diode, limits the development of power-
handling circuits, fundamental for scaling up these technologies. Existing
efforts to date have been limited to single-diode proofs of principle and
lacked integration of multiple controllable and reproducible devices to form
complex circuits. Here, we demonstrate a robust superconducting diode with
tunable polarity using the asymmetric Bean-Livingston surface barrier in
niobium nitride micro-bridges, achieving a 43 % rectification efficiency. We
then realize and integrate several such diodes into a bridge rectifier circuit
on a single microchip that performs continuous full-wave rectification up to 3
MHz and AC-to-DC conversion in burst mode at 50 MHz with an estimated peak
power efficiency of 60 %.

###### keywords:

American Chemical Society, LaTeX

Massachusetts Institute of Technology] Dept. of Electrical Engineering and
Computer Science, Massachusetts Institute of Technology, Cambridge, MA, USA
IR,NMR,UV

Superconducting nanowire single-photon detectors (SNSPD) arrays are expected
to be of key importance to the future development of photonic quantum
computing 1, quantum communication 2, optoelectronic spiking neural networks
3, and high-energy physics 4. Scaling up these systems requires the
integration of cryogenic co-processors for readout and signal processing 5, 6,
7, 8, 9, 10, and interconnection between many modular units as in
superconducting quantum computing platforms 11, 12. Therefore, it is crucial
to build architectures capable of delivering and handling significant power at
cryogenic stages, while dynamically tuning bias levels, minimizing cross-talk
and electrical noise, and maintaining thermal isolation. Any solution to this
problem will have to leverage the intrinsic energy efficiency of thin-film
superconductors, be compatible with standard materials used for
superconducting detectors and electronics, and interface with conventional
circuitry.

In modern integrated circuits, diodes are fundamental components for power
control and distribution. As such, superconducting diodes will be needed for
efficient and compact bias-handling systems needed in cryogenic electronics
and detectors. The main functionality enabled by superconducting diodes would
be AC-to-DC conversion at low temperatures (e.g. using full-wave bridge
rectifiers) to provide stable and tunable DC bias currents through RF signals.
Such an operation might be useful to deliver power from room temperature using
a limited number of RF lines or to interconnect different sub-systems of the
same cryogenic architecture. Other typical functionalities of semiconducting
diodes could be realized with superconductors, such as frequency mixing,
signal modulation, voltage-controlled oscillation, and isolation; benefiting
the development of several cryogenic platforms.

The superconducting diode effect (SDE) manifests in asymmetric critical
currents for opposite current directions in a superconducting device. This
phenomenon has been recently demonstrated using multiple technologies 13 that
primarily rely on the combined effect of a magnetic field and spin-orbit
coupling 14, 15 or the Josephson effect 16, 17, 18. Moreover, field-free
superconducting diodes have also been realized 19, 20.

Despite impressive technical progress in the formation of individual devices,
and the recent demonstration of half-wave rectification at 100 kHz,21 diode
effects have not been explored for near-term use in complex and useful
circuits that require more than a single diode. Past diode demonstrations had
limitations such as: low rectification efficiency ($\eta$), the need for
Tesla-scale applied magnetic field, or the use of complex geometries and
heterostructures requiring materials processing that is incompatible with
conventional superconducting circuits. Thus, the assumption that cryogenic
circuits could one day usefully take advantage of these SDEs remains to be
proven.

Among the variety of proposed SDEs, the vortex diode effect, based on the
asymmetric Bean-Livingston surface barrier 22 in micro-bridges, has a great
potential for realizing diode-based circuits at a large scale because: it
requires the patterning of only a single layer; it operates with milliTesla-
scale fields; it can generate a high-resistance state (useful to interface
with traditional electronics) and; it can be implemented in essentially any
superconductor. In particular, implementing vortex diodes in niobium nitride
thin films opens the door to a wide range of applications considering that
such material has been widely used for high-performance SNSPDs 23,
nanocryotron digital circuits that can be monolithically integrated with
SNSPDs 24, 25, 26, 27, 6, 7, microwave devices 28, 29, spiking neural networks
30, 31, and comparators that interface Josepshon junctions with conventional
electronics 32.

The vortex diode effect was initially proposed by Vodolazov et al.33, and then
observed by Cerbu et al. 34 in aluminum wires with controlled edge defects,
and by Semenov et al. in NbN SNSPDs with $90^{\circ}$ turns 35. However, the
first diode based on this SDE was only recently demonstrated by Hou et al. 36
and used controlled edge defects on straight vanadium micro-bridges under an
applied out-of-plane field of a few milliTesla. In this device, the efficiency
was also boosted from $50\%$ to $65\%$ by introducing an adjacent thin layer
of a ferromagnetic material, which eliminated the need for geometrical defects
and external fields. The vortex diode effect has been recently demonstrated
also using NbTiN nanowires 37 ($\eta=24$ %), and NbN micro-bridges ($\eta=30$
%) 38; however, the NbN-based devices relied on random edge defects introduced
during the fabrication process, thus limiting the device reproducibility.
Overcoming this limitation would help realize a reliable and reproducible NbN
diode useful to design AC-to-DC converters in power distribution
architectures.

In this work, we demonstrate the potential of using vortex diodes for
designing power-handling systems by combining them into a non-trivial circuit:
a full-wave bridge rectifier composed of four integrated operational diodes.
With this circuit, we demonstrated sustained full-wave rectification of
sinusoidal signals at 3 MHz; rectification of a continuous pulsing waveform at
20 MHz; and AC-to-DC conversion at 50 MHz for limited time intervals. We
implemented the vortex diodes using micro-bridges with controlled triangular
edge defects, following Hou et al. 36 but using NbN as the superconducting
material. We characterized both stand-alone diodes and bridge rectifiers by
studying how the rectification efficiency and input current margins change as
a function of the field, frequency, and shape of the signal. We also created
an LTspice model of the diode to estimate the power efficiency of the AC-to-DC
conversion and facilitate the design of larger diode-based circuits. To show
the prospect of using such devices in circuits at a larger scale, we designed
and simulated an architecture, based on the demonstrated AC-to-DC converter,
to bias a network of 10 SNSPDs with dynamically tunable DC currents, by
frequency multiplexing the bias levels on a single RF line as described in the
supplementary material.

##  1 Bridge rectifier with vortex diodes

Figure 1: Superconducting diode and bridge rectifier. (a) Scanning electron
micrograph of a superconducting diode, fabricated on a 14 nm thick film of
niobium nitride. The constriction is 400 nm wide and the notch has an angle of
$45^{\circ}$. The gray regions are niobium nitride, and the dark borders of
the traces show the underlying SiO2 substrate. The ground plane surrounds all
the traces. The right inset depicts the working principle of a vortex diode
when an entering out-of-plane magnetic field is applied, and a current $I^{+}$
passes through the device. Vortex dynamics are governed by three forces: force
due to circulating current formed by the Meissner effect (horizontal blue
arrows); Lorentz force due to $I^{+}$ (horizontal green arrows) and; force due
to the surface barriers (red arrows). The net force pushes vortices from left
to right. (b) Circuit symbols for superconducting diodes with positive or
negative applied field. The field is entering the plane and the notch is on
the left edge of the device. Inverting the field orientation changes the
polarity of the diode. (c) Circuit schematic of the superconducting bridge
rectifier driving a resistive load $R_{\text{L}}$. A smoothing capacitor $C$
(dashed line) can be added in parallel to $R_{\text{L}}$ to obtain a DC output
current for AC-to-DC conversion. (d) Experimental full-wave current
rectification of a 3 MHz sinusoidal signal $i_{\text{IN}}$ (black dashed
traces), with $R_{\text{L}}=50$ $\Omega$ and without smoothing capacitor, and
for both positive (blue trace) and negative (green trace) applied field.
Traces are vertically shifted for clarity. Each trace is the average of 10
different traces acquired in sequence, to increase the signal-to-noise-ratio
(one of the 10 traces is shown in the supplementary material).

Our vortex diodes were 1 µm wide wires patterned on NbN films with three
different thicknesses (9 nm, 13 nm, and 14 nm), deposited on a 300 nm thick
thermal-silicon-oxide (SiO2) on silicon substrate. Figure 1a shows the
scanning electron micrograph of the device and an illustration of the
operating principle. The triangular notch lowers the vortex surface barrier on
the left side due to current crowding 39, while the barrier on the right edge
remains unchanged. A current $I^{+}$ generates a Lorentz force that pushes
vortices towards the center of the wire. When this force is large enough,
vortices cross the wire from the lower barrier on the left, generating a
hotspot (resistive state). When an out-of-plane magnetic field is applied, a
screening current circulates at the edges of the wire (Meissner effect). Since
this Meissner current flows in opposite directions on the two edges, the
surface barrier of one edge is lowered, while the other is increased.
Inverting the field direction inverts the direction of the circulating
current, and thus reverses the surface barrier suppression, inducing the diode
effect with opposite polarity. Figure 1b shows the associated circuit symbols.

We optimized the diode by patterning 18 superconducting devices on a 14 nm
thick film with constant wire width (1 µm) but varying constriction width and
aperture angle of the triangular defect. We observed a SDE in every device at
4.2 K, with consistent diode polarity and a similar rectification efficiency
($\eta$). The maximum value of $\eta$ was obtained using a notch with a
$45^{\circ}$ angle, and a 400 nm wide constriction (figure 1a).

Bridge rectifiers are more power efficient than single-diode rectifiers in AC-
to-DC current conversion because both halves of the input waveform are
rectified (full-wave rectification). Therefore, such circuits are preferable
for power distribution networks. After rectification, the DC current is
usually achieved by using a low-pass filter. Figure 1c shows the circuit
schematic of the superconducting bridge rectifier driven by an ideal current
source and coupled to the low-pass filter. Thanks to the duality between
superconducting diodes and their conventional counterparts, the topology of
the circuit is familiar. The rectifier is composed of four vortex diodes
($D_{1}$, $D_{2}$, $D_{3}$, and $D_{4}$) connected in a superconducting loop.

Firstly, we tested just the bridge circuit with a surface-mount resistor as a
load, obtaining full-wave rectification of various input waveforms while using
differential inputs and outputs to achieve isolation from the ground. We then
added a smoothing capacitor to the output to demonstrate AC-to-DC conversion.
Figure 1d shows full-wave rectification of a 3 MHz sinusoidal signal (without
smoothing capacitor), for two values of the applied magnetic field. The output
current could be tuned by flipping the field, as all the diodes in the circuit
had tunable polarity.

##  2 Rectification efficiency

The rectification efficiency
$\eta=\frac{|I^{+}_{\text{c}}|-|I^{-}_{\text{c}}|}{|I^{+}_{\text{c}}|+|I^{-}_{\text{c}}|}$
is a critical figure of merit to be maximized for optimal rectification. We
measured the I-V characteristic of the diodes to extract the positive and
negative critical currents ($I^{+}_{\text{c}}$ and $I^{-}_{\text{c}}$
respectively). Figure 2a shows the I-V curves of a diode patterned on the 14
nm thick film, for different values of the field. With positive magnetic
fields (entering the plane) the negative critical current $|I^{-}_{\text{c}}|$
increased, and the positive critical current $|I^{+}_{\text{c}}|$ decreased.
Negative magnetic fields caused the opposite effect.

We studied how the critical currents and rectification efficiency vary with
the magnetic field by applying a 1 kHz triangular wave to a device shunted
with 1 M$\Omega$. The upper panel of figure 2b shows that $|I^{+}_{\text{c}}|$
and $|I^{-}_{\text{c}}|$ were equal at zero magnetic field.
$|I^{-}_{\text{c}}|$ linearly increased with an increasing positive field up
to a peak value, while $|I^{+}_{\text{c}}|$ decreased. After the peak,
$|I^{-}_{\text{c}}|$ linearly decreased because the surface barrier on the
right edge became smaller than the barrier on the left. At about $\pm 5.5$ mT,
the relation ceased to be linear and both negative and positive currents
approached similar low values because bulk vortex pinning effects dominated
38. The peak of $|I^{-}_{\text{c}}|$ was higher than the peak of
$|I^{+}_{\text{c}}|$, as observed by Suri et al. 38 The lower panel of figure
2b shows the rectification efficiency $\eta$ as a function of the magnetic
field for two different film thicknesses (14 nm and 9 nm). For the 14 nm thick
device the maximum rectification efficiency of 35 % was reached at 3.3 mT. The
9 nm thick device had a maximum $\eta$ of 36 % at 4 mT.

Figure 2: Characterization of superconducting diodes. (a) I-V curves of a
superconducting diode, fabricated on a 14 nm thick film, for different values
of magnetic field. The field was swept from -2 mT to 2 mT. The curves are
shifted along the x-axis for clarity. (b) Upper panel: positive and negative
critical currents ($|I^{+}_{\text{c}}|$ and $|I^{-}_{\text{c}}|$) as a
function of the magnetic field (field swept from -10.3 mT to 10.3 mT.). Dashed
lines fit the linear regions of the plot. Lower panel: Rectification
efficiency $\eta$ as a function of the magnetic field with a triangular wave
as bias signal, for two different film thicknesses and the same geometry. (c)
Current margins to maintain a superconducting state (upper panel), and
rectification efficiency $\eta$ (lower panel), for a sinusoidal bias signal at
different frequencies. Devices with three different thicknesses $t=9,13,14$ nm
were tested. For each $t$, the applied magnetic field was chosen to maximize
$\eta$ at 10 Hz. In the top panel, the data points above and below zero
correspond to $|I^{-}_{\text{c}}|$ and $|I^{+}_{\text{c}}|$ respectively. The
colors of traces in both panels are associated with the legend.

Demonstrating signal rectification at frequencies in the MHz range would
enable radio-frequency signals to be processed and converted into DC levels,
and would thus be enabling for the future development of this technology.
Therefore, we studied how the rectification efficiency varies as a function of
the frequency of an applied sinusoidal signal. Figure 2c shows the value of
$\eta$ at frequencies between 10 Hz and 5 MHz for three different film
thicknesses (with a 50 $\Omega$ shunt). For all the thicknesses, $\eta$
quickly dropped to zero at frequencies higher than 100 kHz (a similar cutoff
has been observed by Chahid et al. 21). In these measurements, the cutoff
seemed to be independent of the thickness. Nevertheless, $\eta$ increased with
thinner films, as predicted by Vodolazov et al. 33 The highest rectification
efficiency of 43 % was achieved at 10 Hz, for the 9 nm thick film. At the same
frequency (1 kHz) and same thickness (9 nm or 14 nm), we observed different
maxima of $\eta$ for sinusoidal signals with a 50 $\Omega$ shunt (figure 2c)
and triangular waves with a 1 M$\Omega$ shunt (figure 2b). This result shows
that shape of the signal and shunt impedance influence the efficiency.

Using a single vortex diode patterned on the 14 nm thick film, we then
demonstrated half-wave rectification of a triangular waveform at 1 kHz, with a
1 M$\Omega$ load (see supplementary material).

##  3 Margins for full-wave rectification

Figure 3a shows a superconducting bridge rectifier patterned on a 13 nm thick
film. The left branch of the loop is composed of $D_{2}$ and $D_{3}$, the
right branch is composed of $D_{1}$ and $D_{4}$. Both branches had a
characteristic total kinetic inductance ($L_{\text{L}}$ and $L_{\text{R}}$
respectively), and the ratio $L_{\text{L}}/L_{\text{R}}$ influenced the
operations and margins of the device. We fabricated bridge rectifiers with
$L_{\text{L}}/L_{\text{R}}=1$ and $L_{\text{L}}/L_{\text{R}}=10$.

Figure 3: Characterization of superconducting bridge rectifiers. (a) Scanning
electron micrograph of a superconducting bridge rectifier, fabricated on a 13
nm thick film of niobium nitride ($L_{\text{R}}/L_{\text{L}}=1$). The gray
regions are niobium nitride, and the dark borders of the traces show the
underlying SiO2 substrate. The ground plane surrounds all the traces. The
current source connects the IN- port to the IN+ port. The load resistor
$R_{\text{L}}$ is connected between OUT- and OUT+ (see figure 1c). The
geometry of the rectifier with $L_{\text{L}}/L_{\text{R}}=10$ is shown in the
supplementary material. (b) I-V curve of a bridge rectifier for two different
values of $L_{\text{R}}/L_{\text{L}}$, with a 4 mT field ($R_{\text{L}}=50$
$\Omega$). Green regions indicate ideal current margins for the correct
operation of the rectifier with $L_{\text{R}}/L_{\text{L}}=10$. (c) Margins to
obtain correct full-wave rectification, for the input current amplitude of a
periodic sinusoidal signal $i_{\text{IN}}$, as a function of frequency (field:
4 mT). The two dashed lines indicate the ideal minimum and maximum currents to
operate the rectifier according to figure b ($L_{\text{R}}/L_{\text{L}}=10$).
In blue regions, the device correctly rectifies. In red and white regions, the
device does not rectify. (d) Full-wave rectification of periodic current
pulses at 20 MHz, for positive and negative magnetic fields. The measured
input signals $v_{\text{IN}}$ are shown with dashed lines. Traces are
vertically shifted for clarity. Each trace is the average of 100 different
traces acquired in sequence, to increase the signal-to-noise-ratio. An output
pulse was observed for every semi-period.

Figure 3b shows the I-V curves of two devices with different
$L_{\text{R}}/L_{\text{L}}$, while the applied positive field (4 mT) was set
to maximize $\eta$. For both devices, we observed two switching events at
positive and negative currents: $I^{+}_{\text{c1}}$, $I^{+}_{\text{c2}}$,
$I^{-}_{\text{c1}}$, and $I^{-}_{\text{c2}}$. For the positive quadrant of the
plot, when the current through $D_{3}$ and $D_{4}$ was lower than their
$I^{+}_{\text{c}}$ ($I_{\text{IN}}<I^{+}_{\text{c1}}$), the input current
$I_{\text{IN}}$ was split into the two branches. Thus, no hotspots were
generated and no current was provided to the load resistor (zero voltage in
the I-V curve). In this region, for $L_{\text{R}}/L_{\text{L}}=1$, the current
was equally split. For $L_{\text{L}}/L_{\text{R}}=10$, most of the current
flowed into the left branch. When the currents through $D_{3}$ and $D_{4}$
exceeded $I^{+}_{\text{c}}$, $D_{3}$ and $D_{4}$ switched to the resistive
state, and the input current was diverted to $D_{1}$ and $D_{2}$, which
remained in the superconducting state. Therefore, $I_{\text{IN}}$ passed
through the load resistor, and the slope in the I-V curve was $1/R_{\text{L}}$
($I^{+}_{\text{c1}}<I_{\text{IN}}<I^{+}_{\text{c2}}$). When the currents
through $D_{1}$ and $D_{2}$ exceeded $I^{-}_{\text{c}}$
($I_{\text{IN}}>I^{+}_{\text{c2}}$), all the diodes switched, resulting in a
high resistance state in the I-V curve. When the input current was negative,
$D_{3}$ and $D_{4}$ were superconducting and $D_{1}$ and $D_{2}$ were
resistive so that the output current could flow in the same direction as with
positive $I_{\text{IN}}$.

For ideal full-wave signal rectification, the device should operate with a
positive peak input current between $I^{\text{+}}_{\text{c1}}$ and
$I^{\text{+}}_{\text{c2}}$, and a negative peak current between
$I^{\text{-}}_{\text{c2}}$ and $I^{\text{-}}_{\text{c1}}$ in figure 3b. In
theory, the switching currents, and thus the margin of the device, depend on
the rectification efficiency and the geometry of the rectifier with the
following relations:
$|I^{\text{+}}_{\text{c1}}|=|I^{\text{-}}_{\text{c1}}|\approx(1+L_{\text{L}}/L_{\text{R}})|I^{+}_{\text{c}}|$
and
$|I^{\text{+}}_{\text{c2}}|=|I^{\text{-}}_{\text{c2}}|\approx|I^{-}_{\text{c}}|$
(derived by applying a current division between the two branches of the loop).
However, we observed a slight asymmetry in the positive and negative critical
currents and a deviation from the ideal theory. For
$L_{\text{R}}/L_{\text{L}}=10$: $|I^{+}_{\text{c1}}|$ was 161 µA and
$|I^{-}_{\text{c1}}|$ was 155 µA instead of 119 µA; $|I^{+}_{\text{c2}}|$ was
251 µA and $|I^{-}_{\text{c2}}|$ was 245 µA instead of 261 µA. Nevertheless,
the largest margins were obtained when the inductance ratio was highest, as
expected. Therefore, we used the device with $L_{\text{R}}/L_{\text{L}}=10$
for all the following analysis. Given the values of switching currents, the
resulting margins for the current amplitude of an input signal to be rectified
should have been between 161 µA and 245 µA but margins could also depend on
the frequency and shape of the input signal.

We further characterized the rectifier to study how the margins varied as a
function of the frequency of a sinusoidal signal. Figure 3c shows the range of
$i_{\text{IN}}$ amplitudes (the region in blue) that allowed full-wave
rectification, for different input frequencies. Outside of these margins,
either only half-wave rectification was observed, or the output current was
zero. The red region is delimited by the minimum and maximum $i_{\text{IN}}$
obtained where correct operation occurred somewhere in the measured spectrum.
In theory, these values should have been $|I^{+}_{\text{c1}}|$ and
$|I^{-}_{\text{c2}}|$ from figure 3b, but we observed a positive shift of
about 15 µA for both of them, which could be caused by the use different
measurement setups. Moreover, the margins were narrower than predicted from
the I-V curve. In particular, they were discontinuous between 100 Hz and 10
kHz: for example, at 100 Hz, the device worked properly between 175 µA and 176
µA, it did not operate correctly from 176 µA to 218 µA, and it rectified again
between 218 µA and 257 µA. Above 10 kHz the margins were continuous but
gradually reduced from 38 µA at 10 kHz to 3 µA at 3 MHz. The observation of
the 3 MHz frequency limit in both the diode and the bridge rectifier suggests
that the speed constraint was fundamental to the hotspot dynamics of the
diode, and most likely depends on the thermal behavior of the devices during
hotspot growth and relaxation.

The ratio between the peak output current and input current amplitude varied
with frequency. At 100 Hz, with a 240 µA amplitude, the ratio was 0.91, and it
decreased to 0.76 at 3 MHz. From this result, we conclude that a small portion
of input current always flowed through the two diodes in the resistive state
instead of flowing into the load and that this current increased with the
frequency. Most Likely, such current maintained their resistive state, and it
was frequency-dependent because $\eta$ decreased with the frequency (with a
lower $\eta$, more current flowed into the diodes to obtain a switch).

Figure 3d shows that the circuit could also correctly rectify 10 ns wide
pulses at a maximum frequency of 20 MHz, but with a lower efficiency than for
a sinusoidal signal at 3 MHz: the peak output current was about 3 times
smaller than the input peak current. The higher operating speed with pulses is
consistent with the idea that the duty cycle of the waveform, and thus the
heating of the substrate, is critical to correct operation.

We created a simple LTspice model of the vortex diode, based on the
electrothermal model of NbN nanowires 40, to better describe the experimental
behaviors of the bridge rectifier. In simulations, we observed an effect that
might explain the reduction of the margins (see supplementary material).
However, we did not find a clear explanation for the discontinuous operation
between 100 Hz and 10 kHz.

##  4 AC-to-DC conversion

The device could rectify continuous sinusoidal signals up to 3 MHz. However,
we observed rectification at higher frequencies in burst mode. Figure 4a shows
a fully rectified 50 MHz sinusoidal signal applied for 100 ns with a 1 µs
burst period. At this frequency, a longer burst duration caused instability in
the system (only a portion of semi-periods were rectified after 100 µs). We
believe the observed difference in maximum operating frequency between the
continuous and burst modes was due to the system heating up significantly
during the continuous operation, while having time to cool off between bursts
in burst mode.

Figure 4: Operating the bridge rectifier and AC-to-DC converter at 50 MHz. (a)
Full-wave rectification of a sinusoidal signal for 100 ns. The input signal
$i_{\text{IN}}$ is set to burst mode with a 1 µs burst period. The load
$R_{\text{L}}$ is 50 $\Omega$ (100 $\Omega$ surface-mount resistor in parallel
with the 100 $\Omega$ differential mode impedance from the amplifiers). There
is a transient for the lower level of the output signal $i_{\text{OUT}}$
because the input capacitance of the used amplifiers need to be charged. (b)
AC-to-DC conversion with a 500 pF smoothing capacitor (see figure 1c), for 440
ns in burst mode (the burst period is 100 µs). The red dashed line indicates
the value of the output DC current (69 µA) between 340 ns and 520 ns.

Considering that the circuit could operate up to 50 MHz in burst mode, we
demonstrated AC-to-DC conversion by shunting the load $R_{\text{L}}$ with a
500 pF surface-mount smoothing capacitor (dashed line in figure 1c). Figure 4b
shows the output current of the converter through the load resistor. The
device could operate for a 460 ns long burst (burst period: 100 µs), with an
8% ripple whose amplitude was set by the capacitance. The DC value of the
current saturated to 69 µA. The waveform in the figure was averaged over 100
signals to increase the SNR and thus extract the ripple amplitude. A single
trace without averaging is shown in the supplementary material. After a start-
up time of about 300 ns, the capacitor was correctly charged at each semi-
period, for all the acquired signals.

It is worth mentioning that including the smoothing capacitor enabled correct
rectification for a longer burst duration compared to the configuration with a
resistive load. This difference likely comes from the intrinsic dependence on
load impedance for the hotspot growth and, consequently, for thermal
dissipation.

We could not experimentally extract the power efficiency of the circuit
because sensing the input voltage would have altered the operation of the
device. Therefore, we estimated the efficiency by mimicking the AC-to-DC
conversion with our LTspice model of the diode end extracting the value from
the simulation. The estimated peak power efficiency, defined as the ratio
between power dissipated in $R_{\text{L}}$ and total power consumption (410 nW
without considering bias resistors) after the start-up time, was about 60 %
with a 50 $\Omega$ load and a 500 pF smoothing capacitor.

##  5 Conclusions

We have successfully demonstrated a superconducting circuit based on vortex
diodes, showing correct full-wave rectification and characterizing the device
margins. As a proof-of-concept, we achieved efficient AC-to-DC conversion by
shunting the resistive load with a surface-mount smoothing capacitor to obtain
a low-pass filter. However, integrated capacitors would increase the footprint
of the device which is undesirable. Instead, in future work the filter could
be designed using inductors rather than capacitors, exploiting the high
kinetic inductance of superconducting nanowires.

In the supplementary material, we explored using an AC-to-DC converter with an
inductor-based filter to create a bias distribution network for SNSPDs. In our
analysis, we had to assume that the diodes were optimized to operate at 50 MHz
and higher frequencies in continuous mode with ideal margins (and thus this
design is conditional on significant future advancement of the work
demonstrated experimentally so far). Nonetheless, the design demonstrates the
relevance and future potential of superconducting diodes in any cryogenic
system that requires dynamically tunable DC bias currents.

For a NbN micro-bridge with these dimensions, our model suggests that the
maximum frequency should be much higher than 3 MHz or even 20 MHz: relaxation
oscillations in NbN superconducting nanowires have been observed up to 1.25
GHz 41. Thus, it may be possible to optimize geometry, material properties of
the diode, and the load on the device to minimize the hotspot growth and heat
dissipation and reach higher frequencies (e.g. by changing substrate to
improve thermal transfer). Moreover, new circuit topologies might be
introduced to enlarge the operating margins, and thus the tunability range for
the output current.

Improving the rectification efficiency will benefit margins, speed, and device
footprint. In particular, a symmetric rectifier geometry with large margins
would be achievable with $\eta\gg 33$ %. Following our results and the
literature 33, it might be possible to further improve $\eta$ by decreasing
the thickness to 5 nm or less (typical for SNSPDs). We fabricated only wires
with a 1 µm width (smaller than the Pearl length), but using wider wires might
also increase the efficiency. In a wider micro-bridge under the same applied
magnetic field, higher Meissner currents are generated and thus the asymmetric
effect might increase. Moreover, if the width exceeds the Pearl length of the
film, the current flows along the edges, and thus closer to the defect which
might further improve the efficiency of the diode.

For future designs of large-scale diode circuits, the need for an external
magnetic field uniformly applied across the chip will limit the integration
with field-sensitive superconducting devices. Therefore, vortex diodes based
on NbN could be integrated with ferromagnetic thin films to localize the field
and enhance the rectification efficiency, similar to the approach by Hou et
al. 36.

The results of this work can be seen as a fundamental starting point for
future development in the field of integrated power electronics based on
superconducting diodes in thin film platforms.
Note added: During the submission process of this work we became aware of a
related work on superconducting rectifiers 42.

##  6 Methods

###  6.1 Fabrication process

All the tested devices were fabricated with the same process: we deposited the
NbN films on 300 nm thick silicon oxide on silicon substrates in an AJA
sputtering system; then, we patterned the devices with electron-beam
lithography, using a positive-tone resist (ZEP530A) with cold development in
o-Xylene, and CF4 reactive ion etching (RIE) 6.

###  6.2 Measurement setup for I-V curves and half-wave rectification

We performed the measurements in a liquid helium dewar at 4.2 K. The chip was
glued to a PCB and the pads of the devices were connected to the pins through
aluminum wire bonds. The PCB was then placed in a custom cryogenic probe with
28 spring-loaded RF sub-miniature push-on connectors 43. The out-of-plane
magnetic field was applied with a superconducting solenoid mounted on the
cryogenic probe in close proximity to the chip. The solenoid was controlled by
a DC current source.

For each point of the curves in figure 2a and figure 3b, a DC current was
applied with a low-noise battery (SRS SIM928) in series with a 10 k$\Omega$
resistor, and the voltage across the device was measured by a multi-meter
(Keithley 2001) in series with a 1 M$\Omega$ resistor.

In figure 2b, the critical currents for each point were found by averaging
over 500 I-V curves, which were obtained by applying a 1 kHz triangular
waveform with an arbitrary waveform generator (AWG, Agilent AWG33622A). A 10
k$\Omega$ bias resistor was used and the voltage across the diode was measured
with a 2 GHz real-time oscilloscope (LeCroy 620Zi) with 1 M$\Omega$ input
impedance. Half-wave rectification was obtained using the same setup and
averaging over 100 traces (see supplementary material). For figure 2c, we used
the same configuration but the impedance of the oscilloscope was set to 50
$\Omega$ and the biasing signal was a sinusoidal wave. Each point of the curve
is the average of 1000 values.

###  6.3 Measurement setup for full-wave rectification

We performed the measurements in a liquid helium dewar at 4.2 K, as described
in the previous section. For all the full-wave rectification tests, the input
current $i_{\text{IN}}$ was provided with a differential setup: the signal was
sent to the IN+ terminal by the AWG in series with a 5 k$\Omega$ surface-mount
resistor placed on the PCB at 4.2 K; simultaneously, the inverted signal was
sent to IN- using the same configuration. The two input lines were 50 $\Omega$
matched thanks to a surface-mount 50 $\Omega$ resistor to ground in front of
the bias resistors. A 100 $\Omega$ load resistor was placed on the PCB at
$4.2$ K, and wire bonded to the OUT- and OUT+ terminals of the rectifier. The
output current $i_{\text{OUT}}$ was calculated by measuring the voltage across
the load resistor $V($OUT+$)-V($OUT-$)$ with a differential readout: for
figure 1d, 3c, 3d, and 4b, both the nodes were connected to the 50 $\Omega$
input ports of the scope; for figure 4a, both output voltages were filtered by
a bias tee, attenuated by 10 dB, and amplified by two low-noise amplifiers in
series (RF Bay LNA-2500, bandwidth: 10 kHz to 2500 MHz, gain: 25 dB, and RF
Bay LNA-2000, bandwidth: 10 kHz to 2000 MHz, gain: 26 dB). In both
configurations, the total load resistance $R_{\text{L}}$ was calculated
considering the 100 $\Omega$ resistor in parallel with the 100 $\Omega$
differential resistance of the scope (or amplifiers). A more detailed circuit
schematic of the differential measurement setup is shown in the supplementary
material. For all the figures, the shown waveforms were obtained by averaging
multiple traces.

In figure 3c, at each tested frequency, we swept the input current amplitude
from 100 µA to 300 µA with an increment of 0.2 µA and observed if full-wave
rectification was achieved. Data points in the figure are associated with the
current amplitude for which there was a transition between correct and
incorrect operation.

###  6.4 LTspice modeling

We created the LTspice model of the diode by slightly modifying the existing
hotspot-growth model of a superconducting nanowire 40. We kept the same
material parameters, modified the thickness and width, and introduced a
critical current that depends on the current flow direction. We did not model
the hotspot growth in the real geometry of the device. Instead, we considered
the diode as a 7.5-squares long wire with a constant width equal to the
constriction width (400 nm). For simulating the bridge rectifier, we extracted
the number of squares for each branch from the device geometry (15 for the
left branch and 150 for the right branch) to estimate $L_{\text{R}}$ and
$L_{\text{L}}$ (the sheet inductance was set to 30 pH/sq). For the left
branch, $L_{\text{L}}$ was reproduced by the inductances included in the
models of $D_{2}$ and $D_{3}$. For the right branch, two lumped inductors were
added in series with $D_{1}$ and $D_{4}$ to reach the value of $L_{\text{R}}$.

We successfully modeled the time-domain behavior of the bridge rectifier shown
in figure 1d by setting the high critical current and $\eta$ to the maximum
values obtained in figure 3c for the 13 nm thick film (261 µA and 40 %
respectively). The comparison between the model and experimental data is shown
in the supplementary material. The simulated rectifier did not have the same
limitations in frequency and margins as the experiment because the
rectification factor was set to be constant, and substrate temperature changes
caused by self-heating effects, which might alter the hotspot reset time, were
not considered. Nevertheless, the model was accurate enough to estimate the
power efficiency of the bridge rectifier, considering that the power
dissipation mainly depends on the modeled electrothermal behavior of the
wires. We estimated the power efficiency of the AC-to-DC converter by setting
the rectification factor of the model to 18 % to match the value of DC current
obtained in the experiment and the high critical current to 261 µA. In the
supplementary material, we show the comparison between experimental and
simulated output current at 50 MHz in burst mode. The peak power efficiency
was calculated after the start-up time when the output current saturated.

##  7 Acknowledgements

This material is based upon work supported by the U.S. Department of Energy,
Office of Science, Office of Basic Energy Sciences, under Award Number DE-
AC02-07CH11359. O.M. acknowledges support from the NDSEG Fellowship program.
A.B. acknowledges support from Politecnico di Torino. R.A.F. acknowledges
support from the DOE under the National Laboratory LAB 21-2491
Microelectronics grant. M.Colangelo acknowledges support from MIT Claude E.
Shannon award. The data that support the findings of this study are available
from the corresponding author upon reasonable request. The authors have no
conflicts of interest to report. The authors would like to thank Prof.
Christoph Strunk and Prof. Philip Moll for advising in the interpretation of
the results.

##  8 Contributions

M.Castellani conceived the idea, fabricated the devices, supervised the
experiment, analysed the data, and performed the simulations. A.B. and O.M
contributed to the design of the diode geometry. A.B. O.M and R.A.F assisted
in the measurement. All the authors contributed to the discussions and
production of the manuscript.

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