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Bridging the gap between nanowires and Josephson junctions: a superconducting device based on controlled fluxon transfer across nanowires

The basis for superconducting electronics can broadly be divided between two technologies: the Josephson junction and the superconducting nanowire. While the Josephson junction (JJ) remains the dominant technology due to its high speed and low power dissipation, recently proposed nanowire devices offer improvements such as gain, high fanout, and compatibility with CMOS circuits. Despite these benefits, nanowire-based electronics have largely been limited to binary operations, with devices switching between the superconducting state and a high-impedance resistive state dominated by uncontrolled hotspot dynamics. Unlike the JJ, they cannot increment an output through successive switching, and their operation speeds are limited by their slow thermal reset times. Thus, there is a need for an intermediate device with the interfacing capabilities of a nanowire but a faster, moderated response allowing for modulation of the output. Here, we present a nanowire device based on controlled fluxon transport. We show that the device is capable of responding proportionally to the strength of its input, unlike other nanowire technologies. The device can be operated to produce a multilevel output with distinguishable states, which can be tuned by circuit parameters. Agreement between experimental results and electrothermal circuit simulations demonstrates that the device is classical and may be readily engineered for applications including use as a multilevel memory.


I. INTRODUCTION
Superconducting devices have played critical roles in technologies such as quantum computing [1], astronomical imaging [2][3][4], magnetometry [5,6], and digital logic [7][8][9][10]. Past superconducting electronics have largely been based on the Josephson junction (JJ) due to its desirable characteristics such as rapid operation speeds >100 GHz and power dissipation on the order of 10 −19 J per switch [11]. Recently, however, superconducting nanowires have emerged as an alternative platform for new electronics. Unlike JJs, superconducting nanowires are dominated by thermal dissipation and a loss of phase coherence, switching from a superconducting state to a high-impedance resistive state when triggered by an external signal, such as a photon or current pulse. This functionality has enabled nanowires to be used in applications where JJs fall short-for instance, the nanocryotron (nTron) is a three-terminal comparatorlike nanowire device that can support high impedances and large fanout [12], whereas JJs lack intrinsic device gain and compatibility with high-impedance environments. The nTron can also be * berggren@mit.edu triggered by single flux quantum (SFQ) pulses to interface between JJs and CMOS circuits [13,14], demonstrating that nanowire electronics have a unique place in computing architectures. Other advantages of nanowires, including their single-layer fabrication, high-output voltages, and scalability, have prompted the development of new devices for use in readout, memory, and sensing.
Despite the growth of nanowire electronics, further advancement into new applications is currently hindered by characteristics that are inherent to nanowires. By switching from the superconducting domain to the resistive domain in response to an external input, nanowires are limited to operations of two states, with the resistive state expelling nearly all current from the high-impedance hotspot and generating a single large output. For the nTron, in which an input gate current triggers the growth of a high-impedance normal domain in the channel, this means that the output voltage is fixed and is only dependent on the gate current exceeding a critical threshold. Similarly, a recently reported superconducting nanowire memory based on persistent current was limited to binary operations of either "0," no current stored in the superconducting loop, or "1," a maximum amount of current stored in the loop [15]. By breaking coherence, hotspot formation also prevents incrementation of the output by successive device switching, as can be done with JJs. Another disadvantage of nanowire devices is that the operation speeds are slow due to the long thermal healing time of the resistive domain, which is often made slower by the electrical time constants of the biasing circuit.
These limitations leave a gap in the family of superconducting electronics for a device that offers the robust interfacing capabilities of a traditional nanowire, but with a more moderated response that allows for modulation of the output, analogous (but not identical) to incrementing in a JJ. Past approaches to creating such a device have focused on Dayem bridge weak links [16,17], where the dimensions of a nanowire in relation to the material's coherence length allow for preservation of phase coherence over a temporary phase-slip center. However, there have been very few experimental demonstrations of these devices in real circuit operations, and the primary goal has largely been to achieve true Josephson behavior in a nanowire rather than to demonstrate a device with the intermediate characteristics of both technologies.
Here, we report on a superconducting-nanowire device based on thermal principles that demonstrates a controlled output. Unlike other nanowire technologies, the device responds proportionally to the magnitude of an input signal and can be operated to achieve multiple discrete states. By using local resistive shunting and a high-inductance superconducting loop, we are able modulate the expulsion of current in an ultranarrow superconducting constriction and to controllably trap flux [18] in quantities of n 0 , where n is an integer less than 10, and 0 is the magnetic flux quantum. We experimentally show that the amount of flux per event n is dictated by circuit parameters and validate these results with electrothermal simulations. The results show that the device may be designed to achieve different n 0 outputs depending on the desired application. We anticipate that this device will serve as a foundation for new nanowire technologies such as a multilevel memory or multilevel-logic-circuit elements.

A. Device fabrication
The devices presented in this work are fabricated using a multistep lithography process. The shunt resistors and alignment marks are first patterned using electron-beam lithography (Elionix F125). A bilayer resist process is employed by first spinning the polymethyl methacrylate (PMMA) copolymer EL6 (6% in ethyl lactate) at 5 krpm for 60 s, then spinning the positive-tone resist gL2000 (Gluon Lab LLC) at 6 krpm for 60 s. The resist is developed in o-xylene and MIBK:IPA in a 1:3 ratio. A 10-nm Ti + 25-nm Au metal bilayer is evaporated, and lift off is achieved in N-Methyl-2-pyrrolidone (NMP) heated to 60°C for 1 h. An approximately 20-nm-thick NbN film is then deposited in an AJA sputtering system following the procedure described in Ref. [19]. The resulting sheet resistance is 150 /sq and the critical temperature is 8.5 K. The nanowire structures are then patterned with electron-beam lithography using gL2000, followed by cold development in o-xylene at 5°C and reactive ion etching in CF 4 (Plasmatherm, rf power of 50 W, chamber pressure of 10 mTorr). The structure is imaged using a scanning electron microcrope (Zeiss) to check for proper alignment. The full process flow may be found in the Supplemental Material [20].

B. Experimental setup
All measurements are performed with the devices submerged in liquid helium at 4.2 K. The devices are adhered to a printed circuit board (PCB), and electrical connections between the devices and gold PCB pads are made using aluminum wire bonds. The PCB ports are connected to room-temperature electronics outside of the liquid helium dewar through CMP cables. Current-voltage characteristics are measured by applying a sinusoidal bias current from an arbitrary waveform generator (Agilent AWG33622A) at a sweep frequency of 10-20 Hz with a 10-k series resistor. The dc output voltage is read by a 2-GHz, real-time oscilloscope (LeCroy 620Zi) after amplification through a low-noise preamplifier (Stanford Research Systems SRS560). The switching current of the readout device (the yTron) is measured by applying a voltage pulse through a 30-dB attenuator to the device and measuring the skew between the oscilloscope trigger rising edge and the time at which a voltage output from the yTron is recorded, signifying a switching event. The bias pulse has a frequency of 500 Hz, width of 650 µs, rising edge of 400 µs, and height of 560 mV. The yTron output is sent through a pulse splitter and a low-noise amplifier (RF Bay LNA-2000, bandwidth: 10 kHz-2000 MHz, gain: 26 dB) before being read by the oscilloscope. The skew is then converted to units of switching current based on the slope of the bias waveform. To apply a dc bias to the constriction, a dc battery source (Stanford Research Systems SIM928) is connected to the constriction through a 100-k series resistor and a dc 1.9-MHz coaxial low-pass filter (MiniCircuits). To apply a pulse to the constriction, a pulsed-voltage waveform of widths ranging from 5 ns to 100 µs and heights ranging from 50 to 550 mV is sent to the constriction input through a 30-dB attenuator. Details of the complete measurement setup may be found in the Supplemental Material [20].

C. Simulation details
The circuit simulator employed in this work uses a superconducting nanowire model implemented in MAT-LAB that includes thermoelectric dynamics of hotspot formation and decay [21]. Physical parameters for the material stack are derived from prior literature [22] and 034006-2 are adjusted to match the experimental results. Flux quantization in the superconducting loop is enforced following each transient, which is found to be sufficient in explaining behavior in the absence of coherent transport events.

III. RESULTS AND DISCUSSION
A. Device characterization Figure 1 summarizes the device architecture and its basic characteristics. The device is comprised of three superconducting nanowire elements: a narrow 60-nm-wide constriction, a storage loop, and a nanoscale readout tool known as the yTron [23]. All three elements are fabricated together on an approximately 20-nm-thick niobium nitride (NbN) film on a silicon oxide substrate using electronbeam lithography. In addition to the nanowire components, a resistive metal shunt is patterned in parallel with the constriction to reduce Joule heating and provide damping, similar to the purpose served in resistively shunted JJs. Initial efforts to fabricate the resistor on top of the NbN film failed due to contact resistance, requiring us to place the resistor layer beneath the NbN film, as was done in similar work on Nb nanowires [24]. Previous attempts to shunt nanowires have found that series inductance between the shunt and the constriction plays a critical role in the effectiveness of the damping; high series inductance produces relaxation oscillations, while increasing the inductance even further makes the resistor completely ineffective [25]. To reduce this effect, the shunt is patterned as close to the constriction as possible and the leads are made wide to reduce the number of squares of material. A simple circuit model for the device is shown in Fig. 1(b). To trap flux in the loop, a bias write current I write is inductively split to the nanowire in the amount of αI write , where α = L loop /(L constriction + L loop ). In this case, L constriction = 284 pH and L loop = 1.87 nH, leading to α = 0.87. Once the sum of αI write and any existing current circulating in the loop surpasses the critical current of the constriction, the nanowire switches and the bias current is diverted away from the constriction to the shunt resistor and the righthand side of the loop. By shunting the majority of the bias current, the resistor allows the nanowire to recover the superconducting state more quickly and limits the amount of current that charges L loop , thus controlling the amount of flux that is trapped once the constriction heals. After the constriction heals, a persistent current circulates in the superconducting loop in quantized units of n 0 /L, where n is an integer, 0 is the magnetic flux quantum, and L is the total loop inductance. Since the geometric inductance of this device is <0.33 f H, the total loop inductance is dominated by the kinetic inductance of the superconducting nanowires. In this device, persistent current is estimated to be quantized as approximately 0.95 µA/fluxon.
The amount of circulating current in the loop can be nondestructively read out using the yTron. As described by McCaughan et al. [23], the yTron is a three-terminal nanoscale device with two adjoining arms whose switching currents depend on one another as a result of current crowding around the intersection point. In our device, the left arm of the yTron forms part of the superconducting loop so that the switching current of the right arm

034006-3
I switch is a function of the amount of circulating current-a higher circulating current in the clockwise direction flowing through the yTron's left arm will result in a higher switching current in the yTron's right arm. Since the two arms of the yTron are electrically disconnected from one another, switching the right arm does not break superconductivity in the left. As a result, the state of the loop is undisturbed by the reading process in which I switch is measured by applying a bias current I read to the yTron's right arm until it switches and generates a voltage, allowing us to nondestructively sense the amount of circulating current in the loop. Figure 1(c) shows the current-voltage characteristics of an isolated shunted nanowire patterned alongside the device with dimensions identical to those of the constriction. The absence of hysteresis, as shown by the lack of separation between the switching and retrapping currents, indicates that the shunt resistor is able to reduce Joule heating through the constriction by effectively diverting the bias current, thereby reducing power dissipation in the nanowire and allowing it to regain the superconducting state more quickly [26]. Observation of the amplified rf output of the device within a bandwidth of 2 GHz does not reveal any relaxation oscillations, suggesting that the shunt inductance is low enough to prevent stable relaxation oscillations at least within the limits of our measurement capabilities [25]. Figure 2 shows the response of the device to an input voltage pulse of varying amplitude and width; the response is compared to that of an otherwise identical device lacking a resistive shunt. For these measurements, an input voltage of widths ranging from 5 ns to 100 µs and heights ranging from 50 to 550 mV is sent to the constriction through a −30 dB attenuator. The change in the amount of stored current in the loop is inferred by measuring the switching current of the yTron readout arm. Before each positive input pulse, a large negative pulse (width = 10 µs, height = −1.3 V) is sent to the constriction to reset the superconducting loop.

B. Demonstration of controlled dynamics
As shown in Fig. 2(a), the amount of stored current in the unshunted device sharply increases with increasing input voltage, but then abruptly drops off, suggesting instability. This response was also observed in the device reported in Ref. [27], and was speculated to be due to overheating of the constriction, causing flux to be lost. In contrast, the response of the shunted device in Fig. 2(b) shows that the amount of flux stored in the loop increases proportionally with input voltage. Unlike the unshunted constriction, in the shunted device there was no sudden loss of stored flux or signatures of unstable oscillations, implying that heating in the constriction is moderated by the presence of the resistive shunt.
To demonstrate the flux-shuttling capabilities of the shunted device, we measure its dependence on the previously written state by ramping a dc bias current on the constriction without resetting the loop and recording the switching current of the yTron at every bias point. As shown in Fig. 3(a), ramping the bias current to the constriction produces either increasing or decreasing steps in the switching current output of the yTron, signifying a sudden addition or subtraction in the amount of trapped flux. The horizontal lines in Fig. 3(a) show that the steps can be categorized into seven distinguishable states, revealing that successive switching of the constriction produces controlled, incremental changes in the amount of circulating current in the loop, rather than storing the maximum amount of current every time. The slight variation in the position of the seven states occurs due to instability in the plateaus, representing when the loop current is nearly maximized (approximately |I c,NW |) and may have lost a small amount of flux (1-2 0 ) to achieve stability. Despite the small shifts at the plateaus, the seven states have wellseparated mean values including consideration of their standard deviations (see Fig. S8 within the Supplemental Material [20]). In contrast, Fig. 3 While the yTron is an effective tool for sensing when there is a change in circulating current, it can be imprecise for extracting the exact amount of circulating current in the loop. Depending on the geometry of the yTron, there may be a nonlinear relationship between the amount of circulating current and the induced change in switching current (see Fig. S6 within the Supplemental Material [20]). Additionally, the sensitivity of the yTron depends on the intersection point between its two arms, which has a radius of curvature <5 nm, leaving room for fabrication variability and thus differences in sensitivity between yTrons of identical design.
To bypass this shortcoming, we use the yTron only to sense when a change in trapped flux occurs and examine the corresponding bias current at each of the points of change in order to infer the magnitude of the loop current. Figure 3(c) shows the bias current at each of the first 14 steps of the plot in Fig. 3(a). The bias current at each step can be used to estimate the amount of circulating current remaining in the loop, given that the transition occurs when the nanowire switches, or when |αI write + I loop | > |I c,NW |. The average zero flux state (I loop ≈ 0) occurs at I c,NW ≈ 20.48 ± 1.45 μA over a set of eight bias ramps. While the seven levels in the yTron switching current of Fig. 3(a) are spaced unevenly, the steps in terms of bias current occur at nearly equal intervals of approximately 5 µA, or roughly 5 0 of circulating current. As a result, it is possible to infer that the loop gains or loses aproximately 5 0 of trapped flux every time the constriction switches. Repeating this measurement over eight ramping cycles with a finer sweep produces an average of 4.77 0 per step in circulating current, with variation from an integer amount (n ≈ 5) expected to be caused by noise in the measurement setup. A discussion of the experimental noise may be found in the Supplemental Material [20].  Thus, despite the nonlinear response of the yTron, movement of a controlled n 0 of flux per step can be validated by examining the bias current at which each of the steps occurs. Table I summarizes the results of repeating this measurement on two other devices with varying circuit parameters. Device 2 has R s = 7.8 and L loop = 1.87 nH, and Device 3 has R s = 7.8 and L loop = 0.66 nH. All other geometries and parameters are kept the same. For each device, the bias write current I write is ramped to ± 40 μA as before in increments of about 10% of the amount of current per fluxoid, or 0.1 0 L, where L is the total inductance of the loop. For L loop = 1.87 nH, I write = 0.1 μA, and for L loop = 0.66 nH, I write = 0.25 μA. At each bias point, the write current is applied for 100 ms and subsequently turned off. The device is allowed to rest for 100 ms before 10 readings of the yTron switching current are measured. Eight complete ramping cycles are recorded for each device. The write current at the first point of each step in switching current is recorded and the difference between sequential steps is calculated. The results from Table I show that increasing R s decreases the number of consistent states and increases the average number of fluxons trapped per switching event, while decreasing the loop inductance may slightly reduce the amount. Plots showing the states of yTron outputs for Devices 2 and 3 may be found in the Supplemental Material [20].

C. Electrothermal simulations
To better understand how the device parameters influence the amount of flux n trapped per switching event, we model the dynamics of the system using circuit simulations that include the electrothermal dynamics of the resistive hotspot [21] and material-specific physical parameters [22]. Figure 4 shows the basic circuit, highlighting the main branches through which current is divided after the constriction switches: the shunt resistor [ Fig. 4(b)], the constriction itself [ Fig. 4(c)], and the loop inductor [ Fig.  4(d)]. In a device with an unshunted constriction (represented here as R s = 1 M for consistency), nearly all of the bias current is diverted to the loop inductor after a switching event, causing the maximum amount of persistent current approximately equal to I c to be stored in the loop once the hotspot collapses and the constriction regains the superconducting state. In contrast, shunting the constriction with R s = 5 allows the majority of the bias current to be diverted instead to the resistor, minimizing the amount of flux trapped through the loop inductor. Figures  4b(i)-4d(i) show the results over a longer timescale, where continuous switching of the shunted constriction brought on by a steadily increasing bias current ramp adds flux to the loop in increments of 5 0 , thus confirming the experimental observations displayed in Fig. 3 for Device 1. Figure 5 displays the amount of flux per switching event resulting from simulating devices of varying circuit parameters. Figure 5(a) shows that the amount of flux increases proportionally with increasing shunt resistance and shunt inductance, which agrees with the experimentally observed shift caused by increasing R s . Figure 5(b) suggests a slightly more complex relationship between R s and L loop , with plateaus occurring due to limitations on the maximum loop current with respect to the critical current of the constriction-for example, if I c = 20 µA, a loop inductance leading to a ratio of 2 µA of circulating current per fluxon cannot have more than 10 fluxons per switching event. While both of these results rely on the electrothermal dynamics included in the simulation, their general shapes stem from current division between the shunt impedance and the loop impedance after the nanowire switches. Details on this relationship may be found in the Supplemental Material [20]. In Fig. 5(c), the simulated trends for varying R s with L loop = 1.85 nH or L loop = 0.65 nH are compared to the experimental results for the three devices listed in Table I. Data points representing the three measured devices show that the electrothermal simulations are in good agreement with the experimental measurements.

D. Discussion on applications
The results of our experiments and correspondence with simulations demonstrate that the device output may be tuned through simple circuit parameters and tailored to meet specific design requirements. As a result, the device is a promising platform for the development of a multilevel memory, with the number of states dictated by the critical current of the constriction and the amount of flux per event. While we demonstrate a maximum of seven states in the case of Fig. 3, the simulations of Fig. 5 show that the number could be increased by changes that reduce the number of fluxons per event, such as further reducing R s . This multilevel operation is significantly different from previously reported superconducting nanowire-based memories, which have thus far been predominantly binary devices [15,27,28]. It has recently been argued that multilevel memory may compensate for the large power consumption of peripheral circuits in superconducting memory arrays by providing a higher information capacity per cell given the same peripheral circuitry; additionally, multilevel memories may allow for increased memory density as the limits of physically shrinking a memory unit are approached [29]. Thus, future investigation of the device presented here as a multilevel memory may advance the scaling of superconducting memory arrays. Other potential applications of the device include use in multilevel logic or integration with other superconducting elements such as photon detectors. While the proof-of-concept device reported here has a rather large size (3 × 25 µm 2 ), the device can be scaled down by introducing a high-kinetic-inductance wiring layer for the loop, given that the geometric inductance is inconsequential. Further scaling improvements could be made by fabricating the loop as a stacked structure, as was suggested with previous nanowire-based memories [15].

IV. CONCLUSIONS
In summary, we develop a superconducting nanowirebased device capable of generating a response that is proportional to the strength of its input. By introducing local resistive shunting through on-chip fabrication, we are able to control the dynamics of a shunted constriction with a high-inductance superconducting loop and display behavior vastly different from its unshunted counterpart. When subjected to a dc bias current ramp, the device produces seven distinguishable states as a result of controlled flux trapping, illustrating that it may be able to be used as a multilevel memory. Through electrothermal circuit simulations and experimental measurements of devices with different circuit parameters, we show how the amount of flux added or subtracted per event-and thus the number of distinguishable states-can be adjusted through device design. We envision that this device can be used as a tunable element for proportional and multilevel operations.

CODE AVAILABILITY
The code that supports the plots within this paper and the electrothermal simulations used to confirm the findings of this study are available from the corresponding author upon reasonable request.