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J. Tang, M. T. Wei, A. Sharma, E. G. Arnault, A. Seredinski, Y. Mehta, [K. Watanabe](https://orcid.org/0000-0003-3701-8119), [T. Taniguchi](https://orcid.org/0000-0002-1467-3105), F. Amet, I. Borzenets

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[Overdamped phase diffusion in hBN encapsulated graphene Josephson junctions](https://mdr.nims.go.jp/datasets/645e9706-b4ac-4b75-8ce3-a0d90d1fa973)

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Overdamped phase diffusion in hBN encapsulated graphene Josephson junctionsPHYSICAL REVIEW RESEARCH 4, 023203 (2022)Overdamped phase diffusion in hBN encapsulated graphene Josephson junctionsJ. Tang ,1 M. T. Wei ,2 A. Sharma,1,3 E. G. Arnault,4 A. Seredinski ,5 Y. Mehta,6 K. Watanabe ,7T. Taniguchi,7 F. Amet,6 and I. Borzenets 1,3,*1Department of Physics, City University of Hong Kong, Kowloon, Hong Kong SAR2Joint Quantum Institute, University of Maryland, Maryland 20742, USA3Department of Physics and Astronomy, Texas A&M University, Texas 77843, USA4Department of Physics, Duke University, Durham, North Carolina 27708, USA5Department of Sciences, Wentworth Institute of Technology, Boston, Massachusetts 02115, USA6Department of Physics and Astronomy, Appalachian State University, Boone, North Carolina 28607, USA7Advanced Materials Laboratory, National Institute for Materials Science, Tsukuba 305-0044, Japan(Received 29 December 2020; revised 3 May 2022; accepted 13 May 2022; published 10 June 2022)We investigate the zero-bias behavior of Josephson junctions made of encapsulated graphene boron nitrideheterostructures in the long ballistic junction regime. For temperatures down to 2.7 K, the junctions appear non-hysteretic with respect to the switching and retrapping currents IC and IR. A small nonzero resistance is observedeven around zero-bias current and scales with temperature as dictated by the phase diffusion mechanism. Byvarying the graphene carrier concentration we are able to confirm that the observed phase diffusion mechanismfollows the trend for an overdamped Josephson junction. This is in contrast with the majority of graphene-basedjunctions which are underdamped and shorted by the environment at high frequencies.DOI: 10.1103/PhysRevResearch.4.023203Graphene-based superconductor-normal metal-supercond-uctor Josephson junctions (JJs) have been a popular mediumof choice for studying the fundamentals [1–11] as well asapplications [12–17] of superconducting devices for morethan a decade. However, the full spectrum and consequencesof the interactions between the graphene Josephson junctionand the environment have not been fully mapped. For exam-ple, the observed critical current IC of graphene Josephsonjunctions is consistently suppressed compared to theoreti-cal predictions; leading to postulations that the junctions areseverely underdamped [5,6,9,10,18], despite the relatively lowhysteresis between the switching IS and the retrapping IRcurrents. The effect of a junction’s environment on its dynam-ics can be directly investigated by looking at the statisticaldistribution of the switching current IS [7–9,19,20], however,such measurements add significant complexity and are nottypically practical for device characterization. Alternatively,the damping regime of the junction can be assessed via themeasurement of zero-bias resistance arising from the phasediffusion mechanism [5,18,21–25]. Analysis of the phase dif-fusion mechanism can be performed via the common I–V biasmeasurements, and among other things, allows one to extractthe Josephson energy without measuring the critical current.*Corresponding author: borzenets@tamu.eduPublished by the American Physical Society under the terms of theCreative Commons Attribution 4.0 International license. Furtherdistribution of this work must maintain attribution to the author(s)and the published article’s title, journal citation, and DOI.Previous works have shown that the vast majority of graphene-based JJs are underdamped [5,6,9,10]; this is often attributedto the large capacitance generated by the bonding pads andleads. Therefore, a scheme for designing and characterizingoverdamped graphene Josephson junctions is worthwhile.In this paper we report on Josephson junctions madefrom hexagonal boron nitride (hBN) encapsulated graphenewith molybdenum rhenium (MoRe) alloy superconductingcontacts [14,15,26]. These devices are governed by ballisticelectron transport and have been found to be in the interme-diate to long-junction regime [9] (see Appendices A and C).Here, the MoRe contacts terminate shortly after the activeregion and are connected to the bonding pads via thin goldleads. Moreover, the MoRe contacts exhibit a high level ofoxidation, introducing disorder, decreasing the Cooper pairdensity and, in turn, increasing the kinetic inductance. Thiskinetic inductance acts to insulate the Josephson junctionfrom the effect of the large bonding pad capacitance (seeAppendix B). The junctions were measured at temperaturesbetween 2.7 and 7 K where a clear phase diffusion governedzero-bias resistance can be observed [5,18,21–25]. However,for these devices, when changing the carrier concentrationvia the backgate, the zero-bias resistance follows the trendexpected for overdamped junctions [5,25,27]. Thus, we con-clude that we have demonstrated ballistic graphene Josephsonjunction in the overdamped regime.Graphene is made using the exfoliation method [28] andis encapsulated in hBN using the pickup method [26]. UsingCHF3/O2 plasma, the hBN-graphene-hBN stack is etchedthorough in order to make quasi-one-dimensional electricalcontacts with superconducting electrodes [14]. MoRe alloyelectrodes are deposited onto the device using DC sputtering2643-1564/2022/4(2)/023203(7) 023203-1 Published by the American Physical Societyhttps://orcid.org/0000-0002-4275-2855https://orcid.org/0000-0002-2279-3721https://orcid.org/0000-0002-5463-6392https://orcid.org/0000-0003-3701-8119https://orcid.org/0000-0002-1577-8312http://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevResearch.4.023203&domain=pdf&date_stamp=2022-06-10https://doi.org/10.1103/PhysRevResearch.4.023203https://creativecommons.org/licenses/by/4.0/J. TANG et al. PHYSICAL REVIEW RESEARCH 4, 023203 (2022)FIG. 1. The inset: The measurement schematic of the device.Differential resistance dV/dI as a function of DC bias current IBias.The current is swept from negative to positive. Panel (a) showsdV/dI curves for different temperatures. The gate voltage here is setto 35 V. Panel (b) shows dV/dI curves for different applied gate volt-ages. The temperature here is 3 K. Note that there is no observablehysteresis between the switching and the retrapping current. (Thecurves appear symmetrical about zero bias). Moreover, all curvesfeature a measurable zero-bias resistance R0 arising from the phasediffusion mechanism.with the approximate thickness of 100–120 nm. (Previouswork has demonstrated that MoRe alloy electrodes in aquasi-one-dimensional contact configuration result in electri-cal contact transparencies of up to 90% [9,14].) The bondingpads and thin metal leads making contact to MoRe are madeof Cr/Au (5/110 nm). Here, we present data on the deviceof length L = 500 nm (the distance between MoRe contacts)and the width W = 3 μm (see Appendix A).The device is measured in a home-made cryocooler witha base temperature of 2.5 K, which is isolated via both aheat shield and RC filters placed at the low-temperature stage.Josephson junction resistance is measured using the lock-inmethod with a four-probe geometry [Fig. 1(a) inset]. Thejunction is biased by a variable DC current with a small ACexcitation of 5 nA. The voltage across the junction is amplifiedusing a custom differential preamp prior to being fed into thelock in. The gate voltage applied to the back of the 300-nmSiO2 oxide layer is used to control the carrier density ofgraphene. Figure 1 presents the differential resistance dV/dIas a function of applied DC bias current IBias. All the curvesshow two transition points as the bias current is swept from alarge negative value to a large positive value. The absolutevalue of the current on the negative side below which thejunction becomes superconducting is the retrapping current IR,i.e., |IBias| = IR. On the positive side, the junction transitionsfrom the superconducting to the normal state at the switchingcurrent IS [18]. Figure 1(a) shows resistance versus IBias fordifferent temperatures with the backgate voltage set to 35 V.Figure 1(b) shows the resistance versus bias current for dif-ferent gate voltages taken at 3 K. In both cases, the switchingand retrapping currents IS and IR follow the expected trends:falling exponentially with increasing temperature and increas-ing with gate voltage away from the Dirac point with the holeconduction regime exhibiting a suppressed critical current dueto the effects of contact doping [5,6,9]. Moreover, the trendof IS with respect to temperature T follows that expected forballistic Josephson junctions in the long junction regime (seeAppendix C) [9].For all tested gates and temperatures we do not observea difference between IS and IR, i.e., there is no observablehysteresis. The vast majority of previously reported grapheneJosephson junctions exhibit hysteresis between the switchingand retrapping currents (with IS > IR) even for temperaturesabove 3 K. Whereas certain works attribute the existence ofhysteresis to self-heating of the junction [13,29], it has alsobeen shown that most of the graphene Josephson junctionsexhibit underdamped behavior [5,6,9,10]. However, it is stillpossible that hysteresis has been smeared out due to tempera-ture [18] as opposed to overdamped junction behavior.Therefore, we now further investigate the junction dynam-ics directly via the characterization of device behavior in thephase diffusion regime. Phase diffusion manifests itself asan observable nonzero resistance below the critical current(even at IBias = 0), arising from phase slips that are caused bythermal noise. The rate of these phase slips down the prototyp-ical titled washboard potential, and, therefore, the measuredzero-bias resistance is governed by the junction dynamicswhich dictate the energy dissipation rate [18]. Indeed, we areable to observe a resistance in our devices, at zero bias, anddown to 3 K in temperature. We define the measured zero-biasdifferential resistance as R0. In order to confirm that R0 arisesfrom the phase diffusion mechanism, we study the evolutionof this resistance with respect to temperature. Regardless ofthe junction damping dynamics the trend behavior of R0 withrespect to temperature should have the following dependence[5,18,21–25,27]:R0(T ) ∝ 2EJkBTe−2EJ /kBT . (1)Here, EJ = h̄IC/2e is the Josephson energy. (The aboveexponential dependence holds when 2EJ/kBT > 1 [18].)Reworking Eq. (1), we can arrive at the proportionality re-lationship: ln (R0T ) ∝ −2EJ/kBT . In Fig. 2(a) we plot thevalue R0T versus inverse temperature T on a semilogarithmicscale. Indeed, we find that the relationship is nearly linear,consistent with theory. From here, knowing the temperature,we can extract EJ from the slope of the curves. The fittedJosephson energy versus gate voltage is plotted in Fig. 2(b).For single layer graphene Josephson junctions governed byballistic transport it is expected that the value of EJRN isconstant with respect to VG [9]; and we find not only thatindeed this is the case (see Fig. 2 inset), but also that thisvalue matched well with the expected energy scale extracted023203-2OVERDAMPED PHASE DIFFUSION IN hBN … PHYSICAL REVIEW RESEARCH 4, 023203 (2022)FIG. 2. (a) The product of temperature and the zero-bias re-sistance R0T as a function of inverse temperature 1/T shown forseveral different gate voltages plotted on a semilogarithmic scale.The linear dependence of the curves shown here confirms that theresistance R0 arises from the phase diffusion mechanism. (b) TheJosephson energy EJ versus gate voltage VG. Here EJ is calculatedfrom the slope of the curves shown in panel (a). The dashed linesrepresent a square-root relationship between EJ and VG (with anoffset), the expected relationship for single layer graphene Josephsonjunctions in the long ballistic regime. (The inset) The product of theJosephson energy and normal resistance EJ RN . For ballistic singlelayer graphene Josephson junctions, this product is expected to beconstant with respect to gate voltage VG.from the trend of IS versus T (see Appendix C). Here RNis the normal resistance, i.e., the resistance of the junctionwhen it is in normal state. (The measured RN can be found inFig. 3.)Having found the Josephson energy, we can determine thelast parameter governing R0. This final parameter is differentdepending on the damping dynamics of the junction. Previoustheoretical works have defined three different regimes: Foroverdamped junctions, the governing parameter is RN , thenormal resistance [25,27]. For underdamped junctions, R0depends on the plasma frequency ωp ∝ √EJ/C [23]. (HereC is the capacitance of the junction.) Finally, if the junctionis underdamped at low frequencies, but becomes overdampedat the plasma frequency (due to being shorted by the envi-ronment), we have R0 ∝ Z0 [21]. Here Z0 is the real partof the high-frequency impedance caused by the junction’senvironment [21]. We find that analyzing our devices as if theywere underdamped or damped by the environment does notFIG. 3. Gate voltage dependence of measured normal resis-tance RN (black line) compared to the resistance calculated fromthe theoretically predicted relationship between RN and R0: RN =R0/[I0(EJ/kBT )]−2. (I0 is the modified Bessel function.) RN is mea-sured at IBias � IS and is averaged for several data points for negativeand positive biases. Here we present RN measured at 3.5 K, however,at these temperatures the resistance shows negligible temperaturedependence. The data and calculated result match well, supportingthe claim that the measured device is an overdamped junction.produce a good match between measured data and theoreticalexpectation. (See Appendix D).Now, we confirm that our devices are indeed in the over-damped regime by comparing the measured normal resistanceRN with that backcalculated from the the zero-bias resistanceR0. Following the full expression in Ref. [27] we havelimIBias→0V/IBiasRN=[I0(12γ)]−2. (2)Here, I0 is the modified Bessel function, and γ = Ich̄/ekBT .(V is the voltage measured across the junction.) ForIBias approaching zero, the equation simplifies to RN =R0/[I0(EJ/kBT )]−2. Figure 3 shows the measured normal re-sistance RN versus gate voltage VG plotted together with theresistance calculated from Eq. (2) for different temperatures. Itcan be seen that we have a good match between the measuredand the theoretical results, in particular, for high values of RN .The damping of the junction is typically determined bythe quality factor Q with Q = RN (2eICC/h̄)1/2 = 2eh̄ RN√EJC.(C being the capacitance of the junction.) A Q < 0.85 re-sults in an overdamped junction [30]. If we assume thatthe net junction capacitance includes the contribution of the100 × 100-μm bonding pads that couple to each other viathe backgate below the 300-nm thick SiO2 layer, we arriveat C ≈ 600 fF. Taking the above capacitance, and the lowestmeasured values of EJ = 0.3 meV and RN = 100 �, we ar-rive at a minimum quality factor of Q = 1.6: the underdampedregime. This estimate is contradictory to the observationsabove. In order for the junction to be overdamped, it needsto be adequately isolated from the bonding pads. At firstglance, one may suspect that the resistance from the goldleads would be capable of sufficiently isolating the junction[13,31]. However, measurements of our leads show only 10 sof Ohms of resistance from the gold. Numerically solving theresistively capasitively shunted junction equations, whereas023203-3J. TANG et al. PHYSICAL REVIEW RESEARCH 4, 023203 (2022)including the effects of the bonding pads and leads showthis is not enough to isolate the junction. In fact, the leadresistance would need to be near 500 � to sufficiently sup-press hysteretic effects in the junction [11,32]. Instead, webelieve that the junction is isolated through a series in-ductance, which, at the plasma frequency of our device(∼35 GHz) would need to be ∼2.2 nH. This inductance canbe explained by the kinetic inductance of the disordered MoReleads. Indeed, the room-temperature resistance of the MoRe isin excess of 10 k� indicating a high level of oxidation. Theoxidation would introduce disorder, decreasing the Cooperpair density and, in turn, increasing the kinetic inductance.In fact, we estimate that given the lead room-temperatureresistance, and the elevated temperature as compared to thesuperconducting gap, our leads have an inductance in excessof 10 nH (see Appendix B).In conclusion, we have investigated the phase diffusionregime in hBN encapsulated graphene Josephson junctiongoverned by ballistic electron transport. The observed trendof the measured zero-bias resistance R0 with respect to car-rier concentration conforms well to theory describing phasediffusion in an overdamped junction regime. This is a conclu-sive confirmation of overdamped behavior in graphene-basedJosephson junction. We attribute this behavior to effectiveisolation of the Josephson junction from the capacitive con-tribution of the bonding pads. The isolation arises from aninductive connection within the device layout. In workingtowards applications of graphene-based superconducting de-vices in the field of quantum information [12–17] (e.g.,quantum entanglers or parafermion-based qubits), a full de-scription of the device behavior is required. This includes thedesign of the ture Josephson energy as well as controllinginteraction with the environment. Our paper is an importantstep towards such environmental control.ACKNOWLEDGMENTSI.V.B. acknowledges funding from the Texas A&M Uni-versity. J.T., A.Sh., and I.V.B. acknowledge CityU NewResearch Initiatives/Infrastructure Support from Central(APRC): 9610395, and the Hong Kong Research GrantsCouncil Projects: (ECS) Projects No. 2301818 and (GRF)No. 11303619. Lithographic fabrication and characterizationof the samples performed by E.G.A. and A.S. and were sup-ported by the Division of Materials Sciences and Engineering,Office of Basic Energy Sciences, U.S. Department of Energy,under Award No. DE-SC0002765.APPENDIX A: DEVICE DESIGNAND CHARACTERIZATIONThe optical image of the device presented in the main textis shown in Fig. 4(a). Graphene is made using the exfoliationmethod [28], prior to encapsulation in (hBN the graphene isverified to be single layer using Raman spectroscopy [33].The measured Raman spectrum is shown in Fig. 4(b). AnAFM image is taken of the completed hBN-graphene-hBNstack (not shown), and an area free of bubbles and defectsis chosen for further processing. Using CHF3/O2 plasma, thehBN-graphene-hBN stack is etched thorough in order to makeFIG. 4. (a) Optical image of the device presented in the main text.(Scale bar presented for reference). Graphene encapsulated in hBNacts as the normal metal portion of the device. The superconductoris made from MoRe alloy (yellow arrow) with the superconduct-ing leads defining a junction of 500-nm length. The MoRe leadsterminate ∼50 μm past the active area of the device. The bondingpads and leads connecting to the MoRe region are made of Cr/Au,thickness 5nm/110 nm). An example of the transition between MoReand Cr/Au is highlighted by the green arrow. The red arrow indicatedthe junction presented in the main text. (b) The Raman spectrum ofthe graphene region used in the device.quasi-one-dimensional electrical contacts with superconduct-ing electrodes [14]. MoRe alloy electrodes are deposited ontothe device using DC sputtering with the approximate thick-ness of 100–120 nm. [Yellow arrow in Fig. 4(a)]. The MoRecontacts define the Josephson junction dimensions. However,the MoRe leads are terminated ∼50 μm past the active area ofthe device. [Green arrow in Fig. 4(a)]. The bonding pads andthin metal leads making contact to MoRe are made of Cr/Au(5/110 nm).APPENDIX B: ESTIMATION OF THE ISOLATINGINDUCTANCEAs described in the main text in order for the junction to bein the overdamped regime, the junction must be isolated fromthe capacitance generated by the bonding pads and leads. Inour device, we attribute the isolation to a large series kineticinductance of the disordered MoRe film. Previous work [34],has shown that alloyed thin films can have substantial kinetic023203-4OVERDAMPED PHASE DIFFUSION IN hBN … PHYSICAL REVIEW RESEARCH 4, 023203 (2022)inductance Lk . Lk can be estimated using Ref. [34],LK = lwRsqh2π2�1tanh(�2kBT) , (B1)where l and w are the length and width of the film, Rsqis the normal-state resistance, � is the gap, and T is thetemperature. We measure the normal-state resistance of thefilm at room temperature to be Rsq � 10 k�, an indicationthe film is highly disordered. Based on the optical imaging inFig. 4, the most likely region to host the disordered film isthe dark gray region denoted by the yellow arrow. Therefore,the length and width are roughly 10 and 1 μm, respectively.The gap of MoRe is ∼1.3 meV. At 3 K, we estimate that thisinductance is on the order of 10 nH; which at the plasmafrequency of 35 GHz would result in an impedance of over2 k�: substantially more than required to successfully isolatethe junction from the bonding pads and leads.APPENDIX C: SWITCHING CURRENT IS DEPENDENCEON TEMPERATURE TIn Josephson junctions, the measured switching current ISis suppressed from the expected maximum critical current ICwith IS found to be decreasing with increasing temperature T[18]. Measuring the trend of IS with respect to temperatureT , one can determine whether the junction is in the diffusive[6,35,36] or ballistic transport regime [9,37–43]. Addition-ally, by sweeping gate voltage VG, one can also determinewhether the junction is made of single layer or multilayergraphene. For the case of a Josephson junction governed byballistic electron transport and with junction length L � ξ (thesuperconducting coherence length) we expect that IS (T ) ∝exp(−kBT/δE ) [9,37–40]. The value of δE ≈ h̄vF /2πL is re-lated to the Andreev bound states energy-level spacing (E0 =π h̄vF /L) [18,37,38,44,45]. Here vF is the the Fermi velocity;and in single layer graphene vF is a constant with respect tocarrier density. Therefore, δE is expected to be independentof the applied gate voltage VG (as long as the junction re-mains ballistic). The trend of the switching current IS versustemperature T can be seen in Fig. 5(a), plotted on a semilog-arithmic scale for several values of VG. A nearly linear decaycan be observed. Fitting the data to the above exponentialdependence, we can extract the value of δE . The guidelinesof the fit result are shown as dashed lines in Fig. 5(a), whereasthe fitted value of δE is presented in Fig. 5(b). One cansee that δE is independent with respect to gate voltage VG(aside from the expected deviations close to the Dirac point).Moreover, the extracted value is consistent with the designedlength L of the junction [9]. As the temperature approacheszero, the switching current approaches the critical currentwith ICRN = C δE . (Here RN is the normal resistance.) It hasbeen found empirically, that the dimensionless proportionalityconstant C is ≈ 1 to 2, and is suppressed by the coefficient oftransmission τ across the junction [9]. In Fig. 5(b), alongsideδE , we plot the value of ICRN . The critical current IC isbackcalculated from the fitted Josephson energy EJ in themain text. Indeed, we find a good match between the twovalues. This suggests that our device is single layer grapheneand is governed by ballistic electron transport. Moreover, thatFIG. 5. (a) Switching current IS versus temperature T plotted ona semilogarithmic scale. Plots for several values of gate voltage areshown. The plot shows a nearly linear decay with respect to T , fromthe slope of the decay one can fit the relevant energy scale δE . Thefit results are plotted in panel (b), and as dashed guide lines in panel(a). (b)The fitted energy scale δE with respect to gate voltage VG. AδE that is independent of gate is expected for single layer graphene.Alongside, is plotted the value ICRN (critical current times normalresistance). The two sets of data are very similar as is expected.our fitted Josephson energy is close to the actual EJ of thedevice.Had the device been in the diffusive regime, IS would begoverned by the Thouless energy, which has a more compli-cated relationship with respect to gate voltage. (A detailedinvestigation into the switching current and the governingparameters in garphene Josephson junctions can be found inRef. [9] for ballistic devices and Ref. [6] for diffusive devices).The device shown in the main text largely follows the designparameters of those presented in Ref. [9].APPENDIX D: COMPARISON TO UNDERDAMPEDAND DAMPED BY ENVIRONMENT REGIMESThe main text presents the analysis of phase diffusion inthe overdamped regime. Here we follow the analyses as ifthe Josephson junction was underdamped or damped by theenvironment at the plasma frequency. In following previousworks, define the prefactor to the exponential in the main textEq. (1) as a variable R′0 with R′0 ≡ R0e(2EJ /kBT ) [5]. For un-derdamped junctions, R′0 ∼ (h/e2)h̄ωp/kBT [23], where ωp ∝√EJ/C is the plasma frequency. Note that C is the capacitanceof the junction and is a constant.) However, if the junction is023203-5J. TANG et al. PHYSICAL REVIEW RESEARCH 4, 023203 (2022)FIG. 6. The variable R′0 versus the Josephson energy EJ . Theblue scatter data represent results obtained from the measured data.The black and green lines represent best-fit lines for the theoreticalexpectation in the case the Josephson junction is underdamped ordamped by the environment (respectively). The mismatch betweenthe measured data and theoretical predictions suggest than neither ofthese cases apply.underdamped at low frequencies but becomes overdamped atthe plasma frequency (due to being shorted by the environ-ment), we have R′0 = 2πZ0EJ/kBT [21]. Here Z0 is the realpart of the high-frequency impedance caused by the junction’senvironment [21]. Typically Z0 is found to be ∼200–250 �and can also be treated as a constant. (Note, that the aboverelationship holds only for EJ > kbT [21]). The measuredR′0 versus the Josephson energy EJ is shown in Fig. 6 (datataken at 4.0 K). Alongside, we plot the best-fitting resultsobtained for each of the two cases mentioned above. 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