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Thermal hopping and retrapping of a Brownian particle in the tiltedperiodic potential of a NbN/MgO/NbN Josephson junction

Luigi Longobardi,1,2, ∗ Davide Massarotti,3, 2 Giacomo Rotoli,1 Daniela Stornaiuolo,2 GianpaoloPapari,4 Akira Kawakami,5 Giovanni Piero Pepe,3, 2 Antonio Barone,3, 2 and Francesco Tafuri1, 2

1Seconda Universita degli Studi di Napoli,Dipartimento di Ingegneria dell’Informazione, via Roma 29, 81031 Aversa (Ce) Italy

2CNR-SPIN UOS Napoli, Complesso Universitario diMonte Sant’Angelo via Cinthia, 80126 Napoli (Na) Italy

3Universita degli Studi di Napoli ”Federico II”,Dipartimento di Scienze Fisiche, via Cinthia, 80126 Napoli (Na) Italy

4NEST, CNR-NANO, and Scuola Normale Superiore di Pisa ,Piazza dei Cavalieri 7, 56126 Pisa, Italy

5Advanced ICT Research Institute, National Institute of Information and Communications Technology,588-2 Iwaoka-cho, Nishi-ku, Kobe 651-2492, Japan

(Dated: November 23, 2011)

We report on the occurrence of multiple hopping and retrapping of a Brownian particle in a tiltedwashboard potential. The escape dynamic has been studied experimentally by measuring the switch-ing current distributions as a function of temperature in a moderately damped NbN/MgO/NbNJosephson junction. At low temperatures the second moment of the distribution increases in agree-ment with calculations based on Kramers thermal activation regime. After a turn-over temperatureT ∗, the shape of the distributions starts changing and width decreases with temperature. We an-alyze the data through fit of the switching probability and Monte Carlo simulations and we find agood agreement with a model based on a multiple retrapping process.

PACS numbers: 05.40.Jc, 74.50.+r, 85.25.Cp

I. INTRODUCTION

Research on superconducting quantum sys-tems with potentials for qubits applications hasboosted interest on several complementary as-pects of coherence and dissipation. There is agrowing evidence of the occurrence of a mod-erately damped regime (MDR) in supercon-ducting Josephson junctions (JJs) of variousmaterials1–6. If we use the Q = ωpRC parame-ter as a measure of dissipation in a junction7,8,where ωp = (2eIc/~C)1/2 represents the plasmafrequency at zero bias current, Ic is the junctioncritical current and C and R are the junctioncapacitance and resistance, a MDR is presentfor 1 < Q < 5. This regime is quite distinctfrom the well-known case of underdamped sys-tems (Q > 10)9,10, and apparently quite com-mon in junctions characterized by low Ic. Inview of a more and more relevant use of nan-otechnologies in quantum superconducting elec-

tronics and therefore of low values of Ic, studieson MDR can offer novel insights on dissipativeeffects on Josephson junctions, and inspire ap-propriate designs to respond to specific circuitrequirements.

Our analysis is based on measurements on lowcritical current density (Jc) NbN/MgO/NbNjunctions. NbN is a material of great interest forsensor applications, as documented by severalworks both on junctions and nanowires11–14,and it guarantees both fast non equilibriumelectron-phonon relaxation times τ < 10ps andhigher gap values, when compared with tradi-tional junction technologies based on Nb, Aland Pb15. Low-Jc NbN/MgO/NbN devicesmay contribute to set a more comprehensiveNbN platform, and constitute a non-trivial ex-tension to thicker barriers of the more estab-lished high-Jc NbN junctions, usually designedfor superconducting digital circuits16,17.

In this work we demonstrate the occurrence

2

of a phase diffusion regime (PDR) induced bylow-Jc in large area junctions. Areas are oneor two order of magnitudes larger than thoseof junctions where PDR has been previouslyobserved2,3, and Josephson (EJ ) and Coulomb(EC) energies entering in the tunneling processwill scale with size in a different manner becauseof the diverse JC and specific capacitance (Cs)values. PDR prevails over thermal activationat temperatures above a threshold T ∗ of about1.5K . Experimental results are very close totheoretical predictions1,5 such to provide a re-liable estimation of Q = 2.7. We compare ourresults with the model of Fenton andWarburton(FW)5 which condenses ideas on phase diffusionof the last 20 years, offering a reliable method-ology to evaluate levels of dissipation in MDR,in analogy with what well established for under-damped junctions9,10. In particular FW modelprovides additional criteria to study low Q junc-tions to prove their MDR nature based on theasymmetry of the switching current histogramsor its skewness γ, i.e. the ratio m3/σ

3 wherem3 is the third central moment of the distribu-tion, which are closely followed by experimen-tal data. A physical picture emerges of mod-erately damped junctions, with a damping sub-stantially independent of the frequency and ableto sustain macroscopic quantum tunneling18 atlower temperatures.

II. THEORETICAL BACKGROUND

The significance of the analysis of phase dif-fusion phenomena extends to the more generalproblem of the motion of a Brownian parti-cle in a periodic potential, which is behind theResistively Shunted Junction (RSJ) model forthe dynamics of the phase difference across aJosephson junction7,8. It can describe manydifferent physical phenomena, i.e. transport oncrystalline surfaces19, rotating dipoles in exter-nal fields20, charge density waves21, and particleseparation by electrophoresis22. Kramers23 firststudied the one-dimensional escape dynamic ofa quantum Brownian particle in presence ofdamping, in the low and high damping limits,

RIC

C IB

U

ϕ

ωp

Runnin

g S

tate

Thermal

Activation

Retrapping

FIG. 1: Tilted periodic potential and the equivalentcircuit of the RSJ model.

and subjected to a tilted potential U(z)

U(z) = V (z)− Fz (1)

where V (z) = V (z + P ) denotes a periodic po-tential of period P and F is an external staticforce. The dynamics of the phase difference ϕof the superconducting order parameter acrossthe junction of a current-biased JJ is equivalentto the Brownian motion of a particle of ”mass”C (the junction capacitance) in a tilted periodicpotential7,8

U(ϕ) = −EJ

(

cosϕ+I

Icϕ

)

(2)

In this case the external static force is con-trolled by the bias current I and is given byEJ(I/Ic) where EJ = ~Ic/2e is the Josephsonenergy. The motion of the particle along the po-tential is also subject to friction, whose strengthcan be characterized by the above mentionedjunction quality factor Q. In this model the su-perconducting branch of the junction current-voltage (I-V) characteristic corresponds to theconfinement of the particle in one well of the po-tential. The escape from this metastable statecorresponds to the appearance of a finite volt-age across the junction. As it is shown in fig.1, in case of low damping the escaped particlegains sufficient energy to roll down the potentialin the so-called running state, meanwhile if the

3

damping is sufficiently high, escape due to ther-mal hopping does not necessarily lead to runwaydown the tilted potential2,3. Following an eventof escape the particle may travel down the po-tential for a few wells and then be retrapped inone of the following minima of the potential24.At low bias the process of escape and retrappingmay occur multiple times generating extensivediffusion of the phase until an increase of thetilt of the potential, due to a change in the biascurrent, raises the velocity of the particle andthe junction can switch to the running state5.This phenomenon of phase diffusion is more

evident when studying the temperature depen-dance of the switching probability25:

P (I) =Γ(I)

dI/dtexp

[

−

∫ I

0

Γ(I ′)

dI ′/dtdI ′

]

(3)

where the rate of escape due to thermal activa-tion is given by Kramers formula23

ΓTA(I) = atωp(I)

2πexp

(

−∆U(I)

kBT

)

(4)

being ∆U the height of the energy barrier be-tween consecutive potential wells and the pref-actor is at = 4/[(1 + QkbT/1.8∆U)1/2 + 1]2.For underdamped junctions the width σ of theswitching distributions has a monotone depen-dence on temperature as σ ∝ T 2/3. In thecase of moderately damped junctions, due tothe retrapping process the switching dynamicis modified; below a turn-over temperature T ∗

the width of the switching distributions followsthe usual σ ∝ T 2/3 , while for T > T ∗, σ isreduced with increasing temperature. In thispaper we also report on the observation of thisbehavior which we fit using the FW model.

III. SAMPLES FABRICATION

PROCESS AND EXPERIMENTAL

SETUP

In NbN/MgO/NbN JJs a 1 nm thick barrierprovides Jc of about 3A/cm2, which is the low-est value ever reported for NbN based junctions.For circular junctions with a diameter of 10µm,

Ic is about 1-2 µA and falls under the crite-ria of the moderately damped regime, as exten-sively discussed below. In the trilayer, epitaxi-ally grown at ambient temperature on a single-crystal MgO substrate26, the NbN base (BE)and counter electrode (CE) are both 200nmthick and were deposited using DC magnetronsputtering with a Nb target in a mixture of 5parts argon and 1 part nitrogen gas. The MgObarrier is about 1.0nm thick and was depositedby rf sputtering. This step was followed by areactive ion etch (RIE) for junction definitionand by the deposition of a MgO insulating layerpatterned by a lift-off process. The process isconcluded with the deposition of a 350nm NbNwiring layer which was patterned and definedby RIE. The realized junctions have a supercon-ducting transition temperature of about 16.6 Kfor both electrodes. A more detailed descrip-tion of the fabrication process can be found ina paper from Kawakami et al.26.

The junctions have been tested through mea-surements of the I-V characteristics (see fig.2)and of switching current distribution (see nextsection). From the magnetic field dependenceof the critical current (shown in the inset offig. 2) we estimated the London penetrationdepth at 300mK to be about λL = 190nm whichis in good agreement with previously measuredvalues for epitaxially grown NbN27,28, thereforefor a barrier of thickness t=1 nm the Joseph-son penetration depth turns out to be7 λJ =(~c2/8πeJcd)

1/2 = 150µm, where d = 2λL + t.

To study the escape rates of aNbN/MgO/NbN Josephson junction wehave thermally anchored the sample to themixing chamber of a He3/He4 Oxford dilutionrefrigerator and performed measurements of thejunction switching current probability. Figure3 shows a block diagram of the experimentalsetup including room temperature electronicand filtering. The room temperature circuitshave been optimized to minimize the effectof unwanted noise. In order to avoid groundloops and noise pick up, the whole experimentis designed to have a single ground29,30. Allconnections to the chip are floating and areonly capacitively coupled to ground through

4

Data

Theory

FIG. 2: Current-Voltage characteristic of a 10µmdiameter JJ measured at 290mK. The inset showsthe magnetic field dependence of the Josephson tun-nel current, with magnetic field values normalizedto the first minimum.

the filters. All grounded signal sources passthrough battery powered unity gain isolationamplifiers that effectively disconnects thissignal from the earth ground. The currentpaths are all designed to be symmetric withrespect to the chip to reduce the effect of com-mon mode noise31. This allows for sufficientdecoupling from the ground while keeping theamplifier from saturating due to charging byproviding a return current path. The amplifiersare designed to have 100 MΩ of resistancebetween their inputs and the common of thebattery circuit. All signals entering the fridgeare isolated, shielded and filtered allowing thedilution refrigerator itself to act as an rf-shieldfor the cold portion of the experiment. In oursystem we use a room temperature electro-magnetic interference filter stage followed bylow pass RC filters with a cut-off frequency of1.6MHz anchored at 1.5 K. Further filtering isprovided by a combination of copper powder32

and twisted pair filters33 thermally anchored atthe mixing chamber of the dilution refrigerator.

IV. DATA AND ANALYSIS

The signal sequence used to measure theswitching current distribution (SCD) is shownin the inset of fig.4. The bias current of thejunction is ramped at a constant sweep rate

ISO

AMP

ISO

AMP

AWG

Threshold

DetectorFast Meter

RC Filters

300 K

1.5 K

B field

IB

Josephson Junction Chip

EMI

Filters

Cu Powder

Filters50 mK

20 mKTwisted

Pair Filters

FIG. 3: (color online) Schematic of measurementelectronics including thermal anchoring and the var-ious stages of filtering.

dI/dt = 122µA/s, the voltage is measured usinga low noise differential amplifier and is fed intoa threshold detector which is set to generate apulse signal when the junction switches fromthe superconducting state to the finite volt-age state. This signal is used to trigger a fastvolt meter to record the value of the switchingcurrent34. This procedure is repeated at least104 times at each temperature, which allows tocompile a histogram of the switching currents.In fig.4, we report the SCD curves collectedover a wide range of temperatures in absenceof an externally applied magnetic field. Dis-

5

t

Vthreshold

ISWI

B

V

FIG. 4: Switching current probability distributionat B = 0G for different bath temperatures. Theinset shows the signal sequence used to acquire thedata.

tinctive fingerprints of phase diffusion, due tomultiple hopping and retrapping, can be foundin the temperature dependence of the width σof the SCD curves, which is shown in Fig. 5.The most striking effect observable in Fig.5 isthe appearance of an anticorrelation betweenthe temperature and the width of the switch-ing distributions2–4. At low temperatures theσ follow the expected T 2/3 dependence, devi-ations are evident in proximity and above a”critical temperature” T ∗ where the tempera-ture derivative of σ(T ) becomes negative. Ex-perimental data (upper frame in Fig.5) are wellreproduced by the expected values (lower framein Fig.5), calculated on the basis of the physicalarguments of phase diffusion.The simulations shown in Fig.5 in particular,

are based on the recent work on phase diffusionby Fenton and Warburton5. The phase differ-ence ϕ(t) is a solution of the following Langevindifferential equation:

ϕtt + ϕt/Q+ ι+ ιN = 0 (5)

Times t are normalized to 1/ωp; ι is the biascurrent normalized to critical current Ico and ιNis a Gaussian correlated thermal noise current,i.e.:

〈ιN (t), ιN (t′)〉 =√

2πkBT/QIcoΦoδ(t− t′).

Stochastic dynamics is simulated by integrat-ing the above Langevin equation by a Bulirsh-Stoer integrator using as noise generator thecernlib routine RANLUX35. Simulations havebeen carried out for different temperatures anddissipation values.

The multiplicity of switching modes betweenthe running and the trapped states raises aproblem of how to define an escape event. In oursimulations the condition to define the switch isV (ι, T ) ≥ V (ι, 0)/2, where V represents the av-erage velocity of the phase-particle in the wash-board potential. In other words the particlespends in the running state more than 50% ofobservation time. Typical runs for simulationsof Eq.(5) will last from 4 · 106 to 6 · 106 nor-malized time units, i.e., 6 · 105 to 9 · 105 plasmaperiods. Observation time for each point gen-erated in the I-V characteristics is 2 · 104 timeunits, which is a long enough time to ensure thatthe average time spent in running/zero voltagestate does not vary as a function of the ob-servation time1. To obtain the SCD we havesimulated a number of escape events between3000 and 5000, which is similar to the numberof counts experimentally measured. Simulatedcurves of σ vs T for different values of the mag-netic field are plotted in Fig. 5. The magneticfield works as a knob to tune T ∗ and providesan additional validity test for the estimate ofQ = 2.7 ± 0.1. From fitting of the SCD andof the moments of the distributions at temper-atures below T ∗ we have estimated the valueof the zero temperature critical current to beIco = 1.91 ± 0.03µA. We have also estimatedvalues for the junction capacitance and plasmafrequency and obtained C = 0.3pF and ωp ≃22 GHz18.

The quality of the fitting procedure is evenmore significant if we consider that we do nothave any degree of freedom associated to a pos-sible frequency dependence of Q as occurring inother experiments1,3,6. For instance the proce-dure used for our analysis is different from thatused by Mannik et al.3, where retrapping prob-ability is calculated independently for differentdissipations. An analysis of full escape rate ismade by combining the 0 → V escape probabil-

6

0.0 0.5 1.0 1.5 2.0 2.510

20

30

40

10

20

30

40

(nA

)

Temperature (K)

FIG. 5: Top frame: temperature dependence of thestandard deviation, σ, of the switching distributionsfor B = 0G (squares), B = 3.05G (circles), andB = 6.09G (triangles). Bottom frame: a numericalsimulation of the data. Vertical dotted lines havebeen inserted in correspondence of the values of T ∗

including error bars. Data and numerical simula-tions are in good agreement in the whole tempera-ture range and for all magnetic fields within errorbars..

FIG. 6: Temperature dependence of mean cur-rent, I, of the switching distributions for B = 0G(squares), B = 3.05G (circles), and B = 6.09G (tri-angles). The solid lines are the calculated valuesof I considering thermal activation and no retrap-ping. The inset shows the value of T ∗ vs the appliedmagnetic field.

ity with retrapping probability which allows toextract the resistance by fitting the escape ratecurves.

In Fig. 6 the mean values I of the SCD (datapoints) are plotted along with the expected val-ues (solid lines) without taking into account re-trapping effects. Due to the onset of retrap-ping events it is necessary to provide a largertilt to the energy potential to allow the systemto switch to the running state. Discrepanciesat higher T demonstrate that the experimen-tal values of I at higher temperatures (aboveT ∗) are greater than the predicted values, whichonly consider the effects of thermal activation.Due to the dependence of T ∗ with the externalmagnetic field, as shown in the inset of Fig. 6,this latter effect is particularly evident for thedata taken in presence of a 6.09G static mag-netic field, where the onset of retrapping occursat a lower temperature.

Phase diffusion also appears in the escaperates Γ, shown in figure 7 as a function of the ra-tio between the barrier height and the thermalenergy. The escape rates are calculated fromthe switching distributions using eq. 3. In thethermal activation regime the distributions areasymmetric and skewed to the left, and Γ val-ues all fall onto the same line, as it is the casefor the reported data from T=0.3K to 1.56K.Retrapping processes cause a progressive sym-metrization of the switching distribution, as itcan be seen from the inset in the bottom leftcorner of Fig. 7, and a bending in the Γ vsu = ∆U/kBT . We use the same procedure pre-viously described to evaluate the numericallysimulated escape rates Γ as function of reducedbarrier height u. Numerical data have been ob-tained by a polynomial fit of numerical escaperates in order to compare it with experiments.The same value of the Q factor is obtained byfitting the Γ(u) curves, shown in dashed lines inFig. 7.

The symmetrization of the switching distri-bution due to the interplay between escape andretrapping events can be clearly observed byplotting, as a function of temperature, the skew-ness of the distributions γ36. For the data withno external magnetic field, we report such plot

7

0.3K

Psw

Psw

I

I

FIG. 7: Escape rates (symbols) as a function of thebarrier height at zero magnetic field, for tempera-tures near T ∗, and numerical simulations (dashedlines). Below T ∗ the fit has been calculated usingKramers formula for thermal activation (solid line).The inset shows the experimental value of the skew-ness of the switching distribution.

in the inset of fig. 7. For the lowest tem-peratures we obtain γ = −1, which is consis-tent with the case of switching current distri-butions in the quantum or thermal regime. Asthe temperature increases the distributions be-come more and more symmetric as γ tends tozero. It should be noted that for these data thetemperature T ∗ at which the derivative of σ(T )changes sign is equal to 1.62± 0.03 K and thatthe skewness starts increasing already at about1.2K, which is a clear indication that the on-set of retrapping phenomena occurs well belowT ∗.5

V. DISCUSSION AND CONCLUDING

REMARKS

A change in the sign of the derivative ofthe second moment of the distribution at aturn-over temperature T ∗ and a modificationof the shape of the distributions at tempera-ture around T ∗ are robust signatures of thephase diffusion regime, and also occur in ourNbN/MgO/NbN junctions, as discussed in theprevious section. A non exhaustive list ofJosephson devices (including ours) that have

displayed a similar PDR behavior is reportedin table I along with the most relevant deviceparameters.

The papers on which table I is based report onsimilar experimental results but their interpre-tation differs in few assumptions, as properlypointed by Fenton and Warburton5 . For in-stance Kivioja et al.2 interpreted their resultswithin the semiclassical model of Larkin andOvchinnikov37,38Since in dc-squids there are fewenergy levels and the hypothesis of continu-ous energy spectrum is not valid, they used amodel which takes account both phase diffu-sion and level quantization. On the other handthis model, which assumes separated levels inthe metastable well, is not properly valid fora single Josephson junction since the numberof energy levels is large and the separation issmaller than their width. Mannik et al.3 andBae et al.6 calculated the retrapping probabil-ity through Monte Carlo simulations and in-cluded frequency dependent damping. The au-thors expressed the net escape rate as a sumof probabilities of multiple escape-retrappingevents based on thermal escape rate and re-trapping probability. The probability of re-trapping is considered as a time-independentquantity which is in contrast with the work ofBen-Jacob et al.39 in which retrapping is mod-eled by a rate and therefore by a probabilityincreasing proportionally to the time spent inthe running state. The very good fitting of ex-perimental curves obtained in this work usingthe FW approach5 confirms the occurrence of amultiple-retrapping regime with a large numberof escapes of duration of Γ−1

R , and in particularit confirms that the time dependence of the re-trapping probability cannot be ignored. If thefast scattering time τ plays a role in diffusiveprocess is a topic of further investigations.

As it can be seen from table I, our experimentconfirms that, independently of the physical sizeof the device, all the junctions exhibiting phasediffusion over a large range of materials and ge-ometry have a low critical current, 2 < Q < 5and 12 < ∆U(I)/kBT

∗ < 18, which are there-fore the relevant parameters signaling the in-surgence of multiple escape and retrapping in a

8

TABLE I: Comparison of device parameters.

Author Device Structure R (Ω) Ico (µA) Area (µm2) Q(I = 0) ∆U(I)/kBT ∗ EJ/EC

This work NbN/MgO/NbN JJ 65 1.91 78.5 2.7 17 14760Kivioja et al.2 Al/AlOx/Al dc SQUID 500 0.2 1.0 3.9 14 515Kivioja et al.2 Al/AlOx/Al JJ 230 0.63 2.6 3.6 18 2110Mannik et al.3 Nb/AlOx/Nb dc SQUID 70 4.25 1.0 2.4 15 9850Mannik et al.3 Nb/AlOx/Nb dc SQUID 70 2.9 2.9 3.3 17 19420Bae et al.6 a Bi-2212 Intrinsic JJ 62 1.26 7.3 2.2 14 10710Yu et al.40 Nb/AlOx/Nb JJ 1800 0.122 0.4 4.8 NA 62Yu et al.40 Nb/AlOx/Nb JJ 315 0.48 1.5 3.3 12 950

aIn this paper the authors estimated the fit parameters

to be temperature dependent. Here we report the values

at the lowest experimental temperature T=1.5K.

washboard potential.The possibility to have extremely low critical

current density can be functional to investigatephase dynamics at extreme conditions. An ex-ample is given by a recent experiment on sub-micron Nb/AlOx/Nb junctions40. Data showan anomalous σ(T ) dependence with a negativedσ/dT over the entire temperature range anda saturation at low temperatures. This regimecan be achieved by engineering junctions withlow critical current, such that the turn over tem-perature T ∗ is lower or comparable to the quan-tum crossover temperature Tcross

9. In this casethe enhancement of σ when increasing the tem-perature, characteristic of the thermal regime,is not observed. Junctions with intrinsically lowcritical current density, such as the one reportedin the present work, could represent an interest-ing term of comparison to study these kind ofunconventional regimes using standard microm-eter junctions.In conclusions we have proved that low Jc

NbN/MgO/NbN JJs are characterized by atransition from thermal activation regime tophase diffusion. This is consistent with what

has been observed in other types of junctionswith similar values of Q and ∆U(I)/kBT

∗. Theexperimental results are well described by a nu-merical model involving a frequency indepen-dent damping which demonstrates an efficientway to estimate the dissipation in moderatelydamped JJs.

VI. ACKNOWLEDGMENTS

This work was supported in part by STREP”MIDAS-Macroscopic Interference Devices foratomic and Solid State Physics: Quantum Con-trol of Supercurrents” and by a Marie Curie In-ternational Reintegration Grant number 248933”hybMQC” within the 7th European Commu-nity Framework Programme. We also acknowl-edge partial support by MIUR PRIN 2009 un-der the project ”SuFET based on nanowiresand HTS” and by ISCRA the Italian Super-Computing Resource Allocation through grantIscrB NDJJBS 2011.

∗ Electronic address:llongobardi@ms.cc.sunysb.edu

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