Phase slips dynamics in gated Ti and V all-metallic supercurrent nano-transistors: a review
PPhase slips dynamics in gated Ti and V all-metallicsupercurrent nano-transistors: a review.
C. Puglia , , ∗ , G. De Simoni , F. Giazotto Dipartimento di Fisica, Universit`a di Pisa, Largo Bruno Pontecorvo 3, I-56127 Pisa,Italy NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore, I-56127 Pisa, ItalyE-mail: ∗ [email protected] February 2021
Abstract.
The effect of electrostatic gating on metallic elemental superconductorswas recently demonstrated in terms of modulation of the switching current and controlof the current phase relation in superconducting quantum interferometers. The lattersuggests the existence of a direct connection between the macroscopic quantum phase( φ ) in a superconductor and the applied gate voltage. The measurement of theswitching current cumulative probability distributions (SCCPD) is a convenient andpowerful tool to analyze such relation. In particular, the comparison between theconventional Kurkij¨arvi–Fulton–Dunkleberger model and the gate-driven distributionsgive useful insights into the microscopic origin of the gating effect. In this review,we summarize the main results obtained in the analysis of the phase slip events inelemental gated superconducting weak-links in a wide range of temperatures between20 mK and 3.5 K. Such a large temperature range demonstrates both that the gatingeffect is robust as the temperature increases, and that fluctuations induced by theelectric field are not negligible in a wide temperature range.
1. Introduction
Although static electric fields are largely ineffective on metals due to their largefree carrier density [1–8], it has been unexpectedly shown that the superconductingproperties of metallic Bardeen-Cooper-Schrieffer (BCS) superconducting wires [9],Dayem bridges [10, 11], and proximity superconductor-normal metal-superconductor(SNS) [12] Josephson junctions (JJ) can be manipulated via the application of a controlgate voltage. The critical current ( I C ) of these systems was observed to decreaseas a function of an increasing control gate voltage, while no clear relation could beestablished with an eventual current flowing between the gate and the superconductor[13]. The gate voltage was found as well to affect the current-phase relation ofsuperconducting quantum interference devices (SQUIDs) [14], on the one hand, throughdirect suppression of the critical current of a gated weak-link. On the other hand, phaseshifts in the SQUID current vs flux relation were observed also for gate voltages so a r X i v : . [ c ond - m a t . s up r- c on ] F e b small that no apparent influence on I C was observed. Such behavior was interpretedas stemming from the voltage-driven occurrence of fluctuations of the superconductingphase ( φ ) [14]. This hypothesis has found qualitative support through the study ofthe evolution of the dynamics of the phase slips, i.e., local random 2 π phase jumps thatresponsible for the superconducting-to-normal state switching [15,16]. The investigationof the switching current ( I S ) probability distribution (SCPD) of a JJ is, indeed, atool to access information on the phase slippage dynamics as a function of externalparameters such as the temperature or externally applied fields [17–20], and allowedus to demonstrate that in metallic superconductors under the action of a gate voltagethe SCPD is remarkably different from that measured as a function of the temperature.This suggests, thereby, that the gate-voltage driven suppression of I C cannot be relatedto a conventional thermal-like overheating of the superconductor, but rather it involvesa different elusive mechanism which is yet to be identified, and fully understood [21–23].In this manuscript, we examine in-depth the modification of the SCPD of gatedall-metallic Dayem nano-bridge devices for temperatures as high as 3.75 K, i.e., in atemperature range much wider than that explored in previous analogous experiments.To this aim, we report on Ti and V gated all-metallic supercurrent nano-transistorsand discuss the evolution of the switching escape rate in the presence of a controlgate voltage. Our data provide another tile in the puzzle of the interaction of electricfields and the BCS condensate, and are particularly relevant, for instance, in view of therealization of electrostatically-driven phase-slip qubit based on gated metallic Josephsonnanojunctions [24].
2. RCSJ model for phase slips in superconducting weak-links
The resistively capacitively shunt junction (RCSJ) model [25] is a convenient tool toanalytically describe phase-slips phenomena in a JJ. It describes the behavior of a JJunder an arbitrary current bias. In this scheme, the Josephson weak-link is replaced bythe parallel of a resistor R , a capacitor C and a non-linear current generator I ( φ ). Forsystems with a sinusoidal current-phase relation (CPR), I ( φ, T ) = I S ( T ) sin( φ ) [26].This phenomenological circuital approach allows to write down the differentialequation I = I S sin ( φ ) + φ πR N ˙ φ + C φ π ¨ φ, where R N is the normal-state resistance of the junction, C its capacitance, φ is themagnetic flux quantum, and I is the circulating current. Such equation is equivalentto the motion of a classical particle of mass m p = C (cid:125) e under the action of a tiltedwashboard potential U W B , as depicted in Fig. 1b [25]. U W B is characterized by a seriesof minima separated by barriers with a height expressed by [27]∆ U ( I, T ) = a E J ( T ) (cid:18) − II S ( T ) (cid:19) b , where a and b take into account the geometry of the systems. I > I QPS U W B φ TAPS I = I/I S SCCPD V ( a . u . ) I (a.u.) I S SCPD
I R C I( φ ) a b c Figure 1. a. RCSJ model in which the junction is represented as the parallel of aresistor, a capacitor and a phase-dependent current generator. b. Particle in a tiltedwashboard potential. The ”jump” between two adjacent minima can occur throughtwo different mechanisms, either thermal activated hopping over the barrier (red) orquantum tunnelling (blue). The ratio II S determines the potential tilting. c. Repeated I vs V measurements. The areas most populated with transitions are the regions whereit is more likely for the bias current to trigger a transition to the normal state. Thered curve represents the probability of transition for bias current I , the SCPD. Theblue curve is obtained via integration of the SCPD and is the cumulative probabilitydistribution, the SCCPD. The motion of a phase particle from one local minimum to the successive one leadsto sharp 2 π variations of φ . Such events are called phase slips (PSs) and are attributedto two distinct mechanisms [18]: thermal hopping [28] and quantum tunnelling. Theformer requires a thermal activation energy, while the latter is only weakly dependent ontemperature. The interplay of the two contributions allows us to define three differentregimes divided by two crossover temperatures T Q < T M : • for T < T Q the quantum tunnelling provides the majority of the phase slip eventsand the system lays in the quantum phase slip (QPS) regime [29, 30]; • for intermediate temperatures, when T Q < T < T M , the thermal energy is largeenough to trigger the hopping of the barrier of the washboard potential, leadingthe system in the thermally-activated phase slip (TAPS) regime; • for higher temperatures, T > T M , the phase slips events occur more than one atonce, and the system shows the multiple phase slip (MPS) regime [31]. In contrastto QPS and TAPS, there are no analytical models for MPS.For a particle in an equilibrium position, we can define an escape rate Γ, whichevaluates the survival time of the particle in the equilibrium point. Kramers’ theory [32]provides a general formula for the escape rate in TAPS [29, 33] and QPS regimes [17, 34]Γ T AP S ( I, T ) = L πξ ( T ) τ GL ( T ) (cid:115) ∆ U ( I, T ) k B T exp (cid:18) − ∆ U ( I, T ) k B T (cid:19) Γ QP S ( I, T, T
QP S ) = L πξ ( T ) τ GL ( T ) (cid:115) ∆ U ( I, T ) k B T QP S exp (cid:18) − ∆ U ( I, T ) k B T QP S (cid:19) , respectively, where L is the length of the junction, ξ ( T ) is the coherence length, τ GL = π (cid:125) k B ( T C − T ) is the Ginzburg-Landau time constant [19] and T QP S is effectivetemperature that allows to compare the height of the barrier ∆ U with the energy term k B T QP S . Since ∆ U is a function of the weak-link bias current, the switching probabilitydue to the occurrence of a phase slip event as a function of I can be reconstructedby acquiring a large number of V vs I characteristics. This procedure shows that thetransition probability is distributed with an asymmetric bell-shaped function around avalue which is conventionally defined the as the critical current I C of the weak-link.The connection between the probability distribution P ( I ) and the phaseslip rate Γ is provided by the Kurkij¨arvi theory (KT) [19, 35–37] via theKurkij¨arvi–Fulton–Dunkleberger (KFD) transformation, which for continuous anddiscrete distributions can be written asΓ( I ) = P ( I ) ν I (cid:20) − (cid:90) I P ( I (cid:48) ) dI (cid:48) (cid:21) − Γ( I N , T ) = P ( I N , T ) ν I − w (cid:80) Nk =0 P ( I k , T ) , where ν I is the slope of the current ramp, w is the bin size of the P ( I, T ) histogram, and P ( I k , T ) is the switching probability in the current interval [ kw, ( k + 1) w ] with k ∈ N .It is worth to notice that the different mechanism for the escaping phenomena leadsto a non-trivial behavior of the distribution width. In particular, below T Q where thetunnelling effects are predominant, the standard deviation σ vs T is expected to betemperature independent because QPSs do not require a thermal activation. On theother hand, when T Q < T < T M , σ is expected to grow as a function of T because ofthe larger available thermal energy that allows hopping events for a wider range of biascurrent values. Finally, for T > T M , the standard deviation is expected to decrease asa function of the temperature, as already observed in similar experiments [18, 28, 30].Such a model, although phenomenological, is very effective and allows to analyzethe effect of the temperature in the dynamics of the phase slip events in mesoscopicJosephson junctions. Moreover, it is suitable to study how such dynamics is modifiedwhen the weak link is subject to the action of electrostatic gating. In the following,we will focus on this specific issue by reporting the results of experiments in whichthe modifications of the phase slips occurrence in nanojunctions was obtained via theapplication of a control gate voltage. Data are then discussed and interpreted in theframework of the conventional theory.
3. Titanium Dayem bridge PS dynamics
Ti wires and Dayem bridges transistors are the systems in which the gating effect onmetallic superconductors was originally demonstrated [9]. In this section, we discuss theeffect of electrostatic gating on these systems in terms of its impact on the switchingcurrent probability distribution. Data shown in the following were acquired on a30-nm-thick ultra high-vacuum e-beam evaporated titanium Dayem bridge (150-nm-long and 120-nm-wide) shaped with a single-step electron beam lithography of a poly-methyl-methacrylate (PMMA) polymeric mask deposited on top of a sapphire (Al O )substrate. In the same evaporation, a 140-nm-wide gate electrode separated by adistance of approximately 80 nm from the constriction was deposited. A pseudo-colorscanning electron micrograph (SEM) of a typical device is shown in Figure 2a. I S ( µ A ) T (mK) Fit T (mK)300 V G (V) I S ( µ A )
400 nm V V G I a b c Figure 2. a. Pseudo-color SEM image of a representative titanium gated (orange)Dayem bridge (purple) with the four-probes bias scheme used for the characterization.b. Critical current I S vs T characteristic fitted with the conventional Bardeen’s formula[38] (black dotted line). The fit parameters are T C (cid:39)
348 mK and I S (cid:39) . µ A. c. I S vs V G characteristics at different bath temperatures between 20 mK and 300 mK.Switching current values were computed averaging over 50 acquisitions. The device shows a normal-state resistance of 550 Ω and a switching current of 6 µ Aat a temperature of 20 mK. The evolution of the switching current as a function of bathtemperature is shown in Figure 2b. I S evolution follows the Bardeen’s law [39] (dottedline in Figure 2b), I S ( T ) = I S (cid:2) − ( T /T C ) (cid:3) / , where I S is the zero-temperatureswitching current, and T C is the Ti film critical temperature. The fit procedure returnsas parameters T C (cid:39)
348 mK and I S (cid:39) . µ A that are in agreement with similar devicesvalues [10]. Similarly, the evolution of the switching current as a function of the gatevoltage applied through the gate electrode is shown in Figure 2c. The curves, acquired atdifferent bath temperatures ranging from 20 mK to 300 mK, show monotonic suppressionof I S , with complete quenching for V CG (cid:39)
34 V. Note the conventional widening of theplateau for which I S is unaffected by the gate voltage as the temperature increases, inagreement with experiments performed on gated elemental superconductors [9–12, 40].The effect of the temperature on the number of phase slip events was assessed byacquiring 10 switching current values biasing the junction with a linear current ramp inthe four-probe scheme shown in Figure 2a at different temperatures ranging from 20 mKto 300 mK. I S values were acquired with a 750 KHz bandwidth input/output analogue-to-digital/digital-to-analogue converter (ADC/DAC) board for the acquisition of thevoltage drop across the junction and the generation of the bias current, respectively. Figure 3. a. SCCPDs computed at several bath temperature ranging from 20 mKto 300 mK. I S was acquired 10 times to reconstruct the SCPD. Dashed grey linesrepresent the fit curves obtained for QPS and TAPS regimes in the framework ofthe KFD theory [41]. b. σ vs T characteristic with the two crossover temperatures T Q (cid:39)
110 mK and T M (cid:39)
150 mK that define the transition between QPS/TAPS andTAPS/MPS regimes, respectively. c. Escape rate computed from the data in panel awith the KFD transform. They show a phase lifetime between 1 µ s and 10 ms. The current input signal consisted of an 8.7 Hz sawtooth wave obtained by applyinga voltage signal generated by the digital board to a 1 MΩ load resistor. The resultingsignal consisted of a positive linear ramp with amplitude 10 µ A, and slope ν I = 133 µ A/sfollowed by a 100 ms zero-current plateau essential for the system to cool down betweentwo consecutive super-to-normal state transitions. I S acquisitions were combinedvia numerical integration to reconstruct the switching current cumulative probabilitydistributions (SCCPDs, the S-curves), as shown in Figure 3a. These curves represent,for a given current I , the probability that the switching current satisfies the condition I S < I .We note that, as conventionally observed in similar systems [18], the S-curves shiftto a lower value of injected current as the temperature T increases. Such observationis equivalent to the decrease of the critical temperature showed in Figure 2b. Thewidth of the SCCPDs ( σ ) can be estimated by computing the standard deviation ofthe switching current probability distribution (SCPD) that is obtained via numericalderivation of the SCCPDs. The conventional behavior of the σ vs T characteristics isassessed in Figure 3b where the three different phase slip regimes can be distinguished,separated by the crossover temperatures T Q (cid:39)
110 mK and T M (cid:39)
150 mK: • QPS regime: for
T < T Q , the standard deviation is constant as a function ofthe temperature since the quantum tunnelling process does not require activationenergy. • TAPS regime: for T Q < T < T M , σ and T are linearly correlated because thetemperature increase provides a growing amount of thermal energy to the systems,facilitating the hopping of the potential barrier. • MPS regime: for
T > T M , the width of the transition decreases, as already observedin previous systems.The escape rate Γ( I, T ) is another relevant quantity to be extracted from SCCPDs.It provides an estimate of the phase lifetime in the Josephson nanojunctions [42, 43],and ranges between 1 µ s (Γ ∼ Hz) and 10 ms (Γ ∼ Hz). The values obtainedfor the escape rate are in agreement with the literature data [18, 19, 44] performed onsuperconducting weak-links.
Figure 4. a. SCCPDs acquired at several gate voltages ranging from 2 V to 30 V.b. σ vs T characteristic with the two crossover voltages V E and V Q that define thetransition between QPS/EAPS and EAPS/MPS regimes, respectively. c. Escape ratescomputed from the data in panel a. They shows a phase lifetime between 10 µ s and100 ms. The effect of electrostatic gating on the shape of SCCPDs is shown in Figure 4a.The curves were acquired at 20 mK for several gate voltage values ranging from 2V to 30 V. The shape of the S-curves is substantially modified by gating. Indeed, for V G < < V G <
14 V. In the range between14 V < V G <
24 V the curves widens significantly due to the increase of the phase slipevents. Finally, at higher gate voltages ( V G >
24 V), the S-curve narrows.The evolution of the shape of SCCPDs is plotted in Figure 4b with the σ vs V G characteristic. Notably, a region of constant standard deviation is observed, statinga minor contribution of gate effect to the phase slips occurrence for low V G values.In analogy with the thermal case, such evolution seems to be equivalent to the QPSregime. By increasing V G the standard deviation grows up to a value of about 200nA. This region is defined as an electrically-activated phase slip (EAPS) regime dueto the increase of gate-induced phase slips. At higher gate voltage, σ decreases andsaturates to approximately 75 nA. Despite resembling the MPS regime, such behaviorcannot be ascribed to conventional heating effects. The evolution of σ vs V G defines twocrossover gate voltages V Q (cid:39) V E (cid:39)
21 V that represent the transition betweenQPS/EAPS and EAPS/MPS regimes, respectively.It is worth to emphasize that the lifetime of the phase particle, represented inFigure 4c as Γ, in the gate-driven regimes ranges between 10 µ s and 100 ms andon average is around one order of magnitude smaller than the thermal rates. Thisindicates a possible action of the electrostatic gating in driving the superconductor farfrom equilibrium. Figure 5. a. σ vs I C curves as a function of temperature at V G = 0 (violet to orange),and vs gate voltage at 20 mK (blue to yellow). b. I C -matched S-curves, brown andorange distributions were acquired for V G = 0 at selected temperatures whereas blueand green characteristics were measured at T = 20 mK for different gate voltage values. The differences between thermal and electric S-curves are highlighted by comparingthe characteristics with the same critical current 2 . , . , . µ A acquired at V G = 0 V(brown to orange) and T = 20 mK (blue to yellow). Figure 5a highlight the differencesin the shape of the distributions and in particular the larger width of the gate-inducedones. Such a behavior is probably due to a gate-driven strong non-equilibrium stateinduced in the superconducting junction. Also, the direct comparison between the σ ( I S ( V G )) and the σ ( I S ( T ))characteristics (reported in Figure 5b) displays a value ofthe standard deviation of the gate-driven curve larger by one order of magnitude in thefirst case. We emphasize that such a strong deviation from the thermal case preventsto exploit the conventional theory [19] to extract an effective electronic temperature inthe weak link, which should be much higher than the Ti critical temperature. A result,therefore, completely devoid of physical meaning. This observation suggests that thegate action affects deeply the system phase dynamics resulting in an increased switchingprobability in a wider bias current range with respect to the corresponding thermal case.This behavior is compatible with the picture of gate-induced phase fluctuations in theDayem bridge [14], and with the formation of a glassy phase configuration along thesupercurrent path [23].
4. Vanadium Dayem bridge PS dynamics
The experiment performed on Ti weak-links allowed us to study the phase slips dynamicsin a temperature range
T <
300 mK where thermal fluctuations and the electron-phononcoupling in the superconductor are strongly suppressed. To assess the influence of thegate-induced phase slip events in a different temperature range the same analysis hasto be performed on devices based on different elemental superconductors with a highercritical temperature. This is the case of gate-controllable vanadium (V) Dayem bridges,in which the higher T C allowed to probe the SCCPD up to bath temperatures above 2K. The vanadium gated device, shown in Figure 6a, consists of a 60-nm-tick, 160-nm-long, 90-nm-wide constriction aligned with a 70-nm-far, and 120-nm-wide side-gate.The metal deposition was performed on a silicon/silicon-dioxide (Si/SiO ) substrate ata rate of 0.36 nm/s in an ultra-high vacuum e-beam evaporator with a base pressureof ∼ − Torr [45–51]. Figure 6a shows the pseudo-color SEM of a representativeV Josephson weak-link along with the four-probes biasing scheme used for the low-temperature characterization [52].The device shows a normal-state resistance of about 110 Ω and switching andretrapping currents of 1.4 mA and 0.35 mA, respectively, at a temperature of 2 K. Theevolution of the switching current as a function of the bath temperature, shown in Figure6b, follows Bardeen’s law (red dotted line in Figure 6b). The fit procedure yielded acritical temperature T C (cid:39) .
62 K and I S (cid:39) . I S , with acomplete quenching of the critical current occurring for V CG (cid:39)
34 V. Also in V devices0
Fit I S ( m A ) T (K) I R T (mK)3.03.32.82.52.3 I S ( m A ) V G (V) IV V G a b c Figure 6. a. Pseudo-color SEM image of a representative vanadium gated (orange)Dayem bridge (purple) with the four-probes bias scheme used for the low temperaturecharacterization. b. I S vs T characteristic fitted with the conventional Bardeen’sformula [38] (red dotted line). c. I S vs V G characteristics measured at differentbath temperatures ranging between 2 K and 3.3 K. The switching current values werecomputed by averaging over 50 acquisitions. the widening of the I S plateau with temperature was observed [9–13, 40]. Figure 7. a. SCCPDs acquired at several bath temperaturse from 3 K to 3.5 K. Thetemperature range is compatible with the MPS regime [19]. b. SCCPDs acquired atseveral gate voltages from 0 to 15 V at 3.0 K. The curves show starkly different shapedue to gate-induced phase slips that widen the zero-to-one probability transition.
We focus now on the SCCPD behavior as a function of bath temperature and the1gate voltage. In particular, we analyze the S-curves with bias current smaller than330 µ A. Notably, the gate-induced characteristics present a somewhat wider transitionthan the thermal ones in the same range of switching current. Such a characteristic,already observed in experiments on Ti weak-links, is present in a temperature intervalas large as
T >
5. Conclusion
In this review, we resumed the results of experiments performed on Ti and V gatedsuperconducting mesoscopic weak links. They shed light on the interplay between theconventional gating and the dynamics of phase slips in elemental superconductors Dayembridges. Firstly, in Ti-based devices, we showed the impact of the temperature on theshape of the S-curves in a regime where both the electron-phonon coupling and thermalfluctuations were strongly suppressed. At the same time, the gate-driven characteristicspresent a wider zero-to-one transition for the same value of the critical current comparedwith the thermal curves. Such SCCPDs comparison is further evidence that the effect ofelectrostatic gating cannot be simply interpreted as a trivial overheating of the junctionvia current injection or Joule heating.Secondly, we performed a similar experiment in a V-based device that presents acritical temperature larger by more than one order of magnitude than the Ti Dayembridges. The higher critical temperature allowed us to explore a temperature regimewhere both thermal fluctuations and the electron-phonon coupling are exponentiallylarger compared to devices operating at sub-kelvin temperatures. Yet, also in thisregime, the comparison between thermal- and gate-driven- cumulative switching currentprobability distributions confirms the results of the previous experiments, highlightingthat electrostatic gating affects the dynamics of the phase slips in a starkly differentway with respect to temperature.
Acknowledgments
We acknowledge the EU’s Horizon 2020 research and innovation program under GrantAgreement No. 800923 (SUPERTED), and the European Research Council under GrantAgreement No. 899315-TERASEC for partial financial support.
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