A gate- and flux-controlled supercurrent diode effect

Superconducting electronics (SCE) aspires to substitute semiconductor technology thanks to its superior operation speed and improved energy efficiency.^1,2 To this end, several superconducting counterparts of the widespread semiconductor electronic devices have been developed. Indeed, supercurrent transistors have been realized in the form of cryotrons,³ nanocryotrons,⁴ rapid single flux quanta (RSFQs),⁵ superconducting field-effect transistors (SuFETs),⁶ Josephson field-effect transistors (JoFETs),⁷ and fully metallic gated devices.^8–10 Furthermore, information can be stored in superconducting memories by exploiting kinetic inductance,¹¹ phase-slips in Josephson junctions,¹² superconducting quantum interference devices (SQUIDs),¹³ and superconductor/ferromagnet hybrid structures.¹⁴ In contrast, devices implementing non-reciprocal Cooper pairs transport are still in their infancy. Indeed, the first supercurrent diode (SD), i.e., a device showing different positive (⁠ $I S +$⁠) and negative (⁠ $I S −$⁠) values of the switching current [see Fig. 1(a)], has been only recently demonstrated in artificial superconducting superlattices.^15,16 Other realizations rely on proximitized two-dimensional electron systems,^17–21 van der Waals heterostructures,^22–24 superconducting transition metal dichalcogenides,²⁵ and magic angle twisted bilayer graphene.²⁶ All these systems showed limited supercurrent rectification efficiencies [ $η = ( | I S + | − | I S − | ) / ( | I S + | + | I S − | )$] reaching values $η ∼ 1 %$⁠. Despite the great interest in the basic science, the exploitation of these exotic systems in SCE practical architectures seems, however, hardly plausible. Differently, the SD effect shown in conventional superconductor thin films immersed in moderate magnetic fields^27,28 suffers from flux trapping and slow tunability of the rectification absolute value and polarity.

At a given value of $Φ e x , I S +$ (⁠ $I S −$⁠) can be calculated by maximizing (minimizing) I_pass with respect to $φ 1$⁠. The combination of an asymmetry of the junctions critical current (⁠ $δ I C = I C 1 − I C 2 ≠ 0$⁠) and a finite screening parameter β causes a shift $δ Φ = β Φ 0 π δ I C I C ¯$ of the $| I S + |$ and $| I S − |$ maxima in the opposite direction on the $Φ e x$ axis. As a consequence, the positive and negative branches of the switching current are different [ $δ I S ( Φ e x ) = | I S + ( Φ e x ) | − | I S − ( Φ e x ) | ≠ 0$], and the device operates as a SD, as shown in Fig. 1(c) for $I C 1 = 0.5 I C 2$ and β = 6. We stress that $δ I S$ is positive or negative depending on the considered half-period of the $I S ( Φ e x )$ interference pattern.

The SQUID-based geometry allows us to design SDs able to tune the rectification through an external magnetic flux and the asymmetry in the critical current of the Josephson junctions embedded in the ring. The latter could be controlled by employing gated superconductor/normal conductor/superconductor (SNS) Josephson junctions with a low charge carrier material as a normal conductor, such as graphene³³ or a semiconductor nanowire.³⁴ This solution would need hetero-junctions between different materials with non-scalable fabrication protocols. This limitation can be easily bypassed by exploiting the gate-induced suppression of the supercurrent in conventional metallic superconductors.^8–10 This effect has also been demonstrated in nano-constriction Josephson junctions (Dayem bridges, DBs)^35–37 and monolithic interferometers.³⁸ Despite the microscopic mechanism at the origin of the above gating effect is still under debate,^{8–10,39–42} it can be exploited to tune the unbalance between $I C 1$ and $I C 2$⁠. In particular, we will focus on SDs constituted by gated DB-based SQUIDs.

Figure 2(a) shows a false color scanning electron micrograph of a typical flux- and gate-tunable SD. The devices were fabricated by single step electron beam lithography (EBL) followed by the evaporation of a titanium thin film (thickness $∼ 30$ nm and critical temperature $T C , T i ≃ 420$ mK) in the ultra-high vacuum chamber (base pressure $∼ 10 − 11$ Torr) of an electron beam evaporator. The superconducting ring (red) is interrupted by two DB Josephson junctions (length and width $∼ 150$ nm). The critical current $I C 1$ of DB₁ can be controlled by a voltage (V_g) applied to a local gate electrode (green) placed at a distance of about 30 nm. The current-vs-voltage (IV) characteristics of the SD were recorded in a 4-wire configuration in a filtered He³-He⁴ dilution refrigerator. Figure 2(b) shows the modulation of the device IV characteristics with $Φ e x$ recorded at temperature T = 50 mK. The triangular magnetic flux pattern of $I S +$ (⁠ $I S −$⁠) is shifted on the magnetic flux axis of a quantity $δ Φ e x + ≃ 5 × 10 − 2 Φ 0$ (⁠ $δ Φ e x − ≃ − 5 × 10 − 2 Φ 0$⁠), thus indicating a large L_R value and a sizeable asymmetry between the critical currents of the two junctions. Indeed, we extracted $β ∼ 30$ and $I C 1 ≃ 1.03 I C 2$ from a fit with the RSJ model of Eq. (1), where the DB Josephson junctions are described by the zero-temperature Kulik-Omel'Yanchuck CPR for a diffusive short junction (KO-1)^31,32 $I i , K O = I C i cos ( φ i 2 ) ar tanh [ sin ( φ i 2 ) ]$(with i = 1, 2).

In order to quantify the rectification performance of our SD, we analyze the switching current interference patterns of our device by varying the voltage applied to the gate electrode. Figure 3(a) shows the absolute value of $| I S + |$ and $| I S − |$ of the SD recorded at T = 50 mK for selected values of V_g. The pristine device (V_g = 0) exhibits $δ I S > 0$ (⁠ $δ I S < 0$⁠) in the flux range $0 < Φ e x < 0.5 Φ 0$ (⁠ $− 0.5 Φ 0 < Φ e x < 0$⁠). The application of a gate voltage at DB₁ results in the decrease in $I C 1$⁠; thus, the absolute value of both polarities of the switching current decreases. This implies that the power consumption of the device lowers with increasing V_g, since the SD is able to rectify smaller signals. Indeed, the minimum SD power dissipation $P ( V g ) = R N × | I S + , − ( V g ) | 2$ (with $R N = 5 Ω$ the normal-state resistance) takes the values $P ( 0 ) ≃ 2.6$ nW and $P ( 11 V ) ≃ 850$ pW. Furthermore, the maximum of $| I S + |$ (⁠ $| I S − |$⁠) moves toward negative (positive) values of $Φ e x$⁠. As a consequence, the positive (negative) branch of the switching current dominates in the negative (positive) $Φ e x$ half-period. This behavior is highlighted by the most relevant figure of merit of a SD: the supercurrent rectification efficiency [ $η = δ I S / ( | I S + | + | I S − | )$]. Indeed, η changes its polarity by increasing V_g, as shown in Fig. 3(b). Despite the increase in the noise due to V_g,^{8,9,35–38,43} the operation of the SD improves with increasing gate voltage. Furthermore, the $η ( Φ e x )$ characteristics transform from a square wave at low values of the gate voltage to a triangular wave at $V g ≥ 10$ V due to the change of the $I S ( Φ e x )$ interference patterns.

Figure 3(c) shows the dependence of the rectification efficiency on the gate voltage for selected values of $Φ e x$⁠. For all values of the external magnetic flux, the maximum value of both polarities of the rectification efficiency increases with increasing gate voltage until reaching its maximum value $η ≃ ± 6 %$ for V_g = 11 V at $Φ e x = ∓ Φ 0 / 4$⁠. The rectification efficiency is almost symmetric in the polarity of V_g, since the critical current suppression in a metallic superconductor does not depend on the sign of the gate voltage, i.e., it is bipolar in the gate voltage.^8–10 These rectification values are several times larger than those obtained in other exotic superconducting systems^15–26 and smaller than the values reported for metallic superconductors immersed in moderate magnetic fields.²⁷ The main advantage of our SD is the possibility to both tune the value of η and invert its polarity through $Φ e x$ and V_g. This opportunity enables the possibility to exploit our device in environments where appreciable magnetic fields are detrimental. Our SD guarantees, in principle, high speed for the change in the rectification magnitude thanks to the fast gate control (as compared to the slower magnetic field control). Furthermore, the maximum frequency of the rectified signal is limited by the superconducting energy gap of the material forming the SD. At the moment, the maximum operation frequency experimentally demonstrated in a Nb₃Sn SD is $∼ 100$ KHz (theoretical estimation of 100 GHz).²⁸ In our case, the highest operation frequency is $f = 2 Δ 0 , T i / h ≃ 31$ GHz, where $Δ 0 , T i = 64 μ$eV is the zero-temperature energy gap of our titanium thin film and h is the Planck constant.

The temperature strongly affects the SD effect shown by our device, since the critical currents of the two junctions (⁠ $I C 1$ and $I C 2$⁠) are suppressed by the temperature. On the one hand, this implies that the value of β lowers with increasing temperature;³⁰ thus, the shift of the I_S interference along the $Φ e x$ axis decreases.³⁸ As a consequence, the rectification efficiency of the SD is expected to reduce (⁠ $η → 0$ for $δ Φ e x → 0$⁠). On the other hand, the critical currents (⁠ $I C 1$ and $I C 2$⁠) of two junctions can show a slightly different temperature dependence. In particular, $δ I C$ can increase with temperature, resulting in the increase in the shift of the interference pattern along the $Φ e x$ axis (⁠ $δ Φ ∝ δ I C$⁠). As a consequence, both $δ I S$ and η could increase with T in a specific temperature range. Therefore, the rectification efficiency could have a non-trivial dependence on the temperature and gate voltage.

In order to study the full temperature behavior of our SD, we measured its $I S + , − ( Φ e x )$ characteristics as a function of V_g at values of the base temperature ranging from 50 to 350 mK. Figure 4(a) shows the dependence of η on $Φ e x$ recorded at V_g = 10 V for T = 100 (blue), T = 200 (orange), and T = 300 mK (red). On the one hand, the polarity of the rectification of the SD does not change with temperature. On the other hand, the $η ( Φ e x )$ characteristics change from a triangular wave to a square wave with increasing temperature, since the decrease in β entails a $I S ( Φ e x )$ behavior similar to that measured for low values of V_g at the base temperature. As a consequence, the maximum value of the rectification of our SD at V_g = 10 V decreases with increasing temperature. This behavior is highlighted in Fig. 4(b), where η is plotted vs T for selected values of V_g at $Φ e x = − Φ 0 / 4$ (top) and $Φ e x = Φ 0 / 4$ (bottom). Indeed, $η ( V g = 10 V )$ monotonically decreases with temperature, until its full suppression at T = 350 mK. Interestingly, the behavior of the SD for V_g = 6 V is qualitatively different. Specifically, the SD rectification changes its polarity with increasing temperature, since the shift along the flux axis due to the application of V_g becomes negligible. Instead, η does not change sign for V_g = 0, but it shows a non-monotonic temperature dependence suggesting that $δ I C$ increases with increasing T. Finally, we stress that the SD efficiently works up to $T ≃ 300$ mK, i.e., about $∼ 70 %$ of the critical temperature of the titanium thin film.

In summary, we have designed and realized a device demonstrating an original supercurrent diode effect controlled via an external magnetic flux and a voltage applied to a gate electrode capacitively coupled to one of the junctions of the interferometer. To realize such a SD, we exploit two main ingredients: (i) a SQUID with a large screening parameter β^29,30 and asymmetric Dayem bridge Josephson junctions³¹ and (ii) gate control of the supercurrent in metallic superconductors.^8–10 Our SD shows rectification efficiencies reaching about 6% for both supercurrent directions with the possibility to change the polarity of η by both $Φ e x$ and V_g. Moreover, we have demonstrated supercurrent rectification up to T = 300 mK, that is around 70% of the critical temperature of the superconductor composing the device (Ti, $T C ≃ 420$ mK). Despite the well-known $Φ e x$ shift of $I S +$ and $I S −$ in these structures,^1,29,30 our SD is expected to be more attractive for applications than hybrid superconductor/semiconductor structures,^15–26 thanks to the higher rectification efficiency, the absence of interfaces between different materials, and the ease of the fabrication protocol. Yet, although the only other fully metallic supercurrent diode developed so far shows larger values of η,²⁷ our SD has the additional advantage to be controlled both with $Φ e x$ and V_g. In particular, a gate-controlled device guarantees, in principle, a higher switching speed than a magnetic field-tunable system. The operation speed and temperature of our structure can easily be increased by realizing monolithic interferometers made of vanadium, niobium, or niobium nitride that show large energy gap,¹ big kinetic inductance,²⁹ and possibility to tune their critical current through gating.^9,37,43

Finally, our SD shares materials composition and structure with widespread SCE architectures¹ such as cryotrons,³ nanocryotrons,⁴ RSFQs,⁵ all-metallic gated devices,^8–10 and memories.^11–14 Therefore, our flux- and gate-controlled SD geometry could find a number of different applications in superconducting electronics for the realization of high-speed and low-dissipation supercomputers such as rectifiers, logic gates, waveform clippers, and clamping circuits.

The authors wish to acknowledge the EU's Horizon 2020 Research and Innovation Program under Grant Agreement Nos. 800923 (SUPERTED) and 964398 (SUPERGATE) for partial financial support.