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Elías Portolés, Marta Perego, Pavel A. Volkov, Mathilde Toschini, Yana Kemna, Alexandra Mestre-Torà, Giulia Zheng, Artem O. Denisov, Folkert K. de Vries, Peter Rickhaus, [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), J. H. Pixley, Thomas Ihn, Klaus Ensslin

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[Quasiparticle and superfluid dynamics in Magic-Angle Graphene](https://mdr.nims.go.jp/datasets/c1566cb3-e6b6-4837-8375-82fd4f0d656f)

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Quasiparticle and superfluid dynamics in Magic-Angle GrapheneArticle https://doi.org/10.1038/s41467-025-58325-0Quasiparticle and superfluid dynamics inMagic-Angle GrapheneElías Portolés 1 , Marta Perego1, Pavel A. Volkov 2,3 , Mathilde Toschini1,Yana Kemna1, Alexandra Mestre-Torà 1, Giulia Zheng 1, Artem O. Denisov 1,Folkert K. de Vries1, Peter Rickhaus1, Takashi Taniguchi 4, Kenji Watanabe 5,J. H. Pixley 6,7, Thomas Ihn 1,8 & Klaus Ensslin 1,8Magic-Angle Twisted Bilayer Graphene (MATBG) shows a wide range of cor-related phases which are electrostatically tunable. Despite a growing knowl-edge of the material, there is yet no consensus on the microscopicmechanisms driving its superconducting phase. A major obstacle to progressin this direction is that key thermodynamic properties, such as specific heat,electron-phonon coupling and superfluid stiffness, are challenging tomeasuredue to the 2D nature of the material and its relatively low energy scales. Here,we use a gate-defined, radio frequency-biased, Josephson junction to probethe electronic dynamics of MATBG. We demonstrate evidence for two pro-cesses determining the low-frequency dynamics across the phase diagram:thermalization of electronic quasiparticles through phonon scattering andinductive response of the superconducting condensate. A phenomenologicalapproach allows us to relate the experimentally observed dynamics to severalthermodynamic properties of MATBG, including electron-phonon couplingand superfluid stiffness. Our findings support anisotropic or nodal super-conductivity in MATBG and demonstrate a broadly applicable method forstudying properties of 2D materials with out-of-equilibrium nanodevicedynamics.Magic-angle twisted bilayer graphene (MATBG) is one of the mostprominent examples of a two-dimensional strongly correlated mate-rial. It consists of two layers of graphene twisted with respect to eachother by an angle of 1.1 degrees. In reciprocal space, this results in thehybridization of the Dirac cones of the different layers, leading to theso-called flat bands. When the Fermi energy of the material is tuned,through field-effect, to such bands, electron-electron correlationsdetermine the electronic state of the material. This results in a phasediagram1,2 that has drawn considerable attention due to the presenceof insulating, topological and superconducting phases3. Thecombination of superconductivity and field-effect tunability make it apromising platform for versatile superconducting electronics.Despite these remarkable discoveries, several questions remainopen about the nature of the superconducting state and even less isunderstood about what is the driving mechanism behind it. The mostcentral and pressing issues include whether the superconductingmechanism of MATBG is electronic or phonon-driven4–7 and whetherits superconducting gap is nodal or not8,9. Furthermore, the electron-phonon coupling has been suggested as the origin of the observed(putatively) universal linear-in-T resistance4,10. Characterizing theReceived: 15 October 2024Accepted: 19 March 2025Check for updates1Laboratory for Solid State Physics, ETH Zurich, Zurich, Switzerland. 2Department of Physics, University of Connecticut, Storrs, CT, USA. 3Department ofPhysics, Harvard University, Cambridge, MA, USA. 4Research Center for Materials Nanoarchitectonics, National Institute for Materials Science,Tsukuba, Japan. 5Research Center for Electronic and Optical Materials, National Institute for Materials Science, Tsukuba, Japan. 6Department of Physics andAstronomy, Center for Materials Theory, Rutgers University, Piscataway, NJ, USA. 7Center for Computational Quantum Physics, Flatiron Institute, New York,NY, USA. 8Quantum Center, ETH Zurich, Zurich, Switzerland. e-mail: eliaspo@phys.ethz.ch; pavel.volkov@uconn.eduNature Communications |         (2025) 16:4273 11234567890():,;1234567890():,;http://orcid.org/0000-0001-7202-777Xhttp://orcid.org/0000-0001-7202-777Xhttp://orcid.org/0000-0001-7202-777Xhttp://orcid.org/0000-0001-7202-777Xhttp://orcid.org/0000-0001-7202-777Xhttp://orcid.org/0000-0001-7586-9788http://orcid.org/0000-0001-7586-9788http://orcid.org/0000-0001-7586-9788http://orcid.org/0000-0001-7586-9788http://orcid.org/0000-0001-7586-9788http://orcid.org/0009-0000-1010-2922http://orcid.org/0009-0000-1010-2922http://orcid.org/0009-0000-1010-2922http://orcid.org/0009-0000-1010-2922http://orcid.org/0009-0000-1010-2922http://orcid.org/0000-0003-3503-4940http://orcid.org/0000-0003-3503-4940http://orcid.org/0000-0003-3503-4940http://orcid.org/0000-0003-3503-4940http://orcid.org/0000-0003-3503-4940http://orcid.org/0000-0001-9095-1579http://orcid.org/0000-0001-9095-1579http://orcid.org/0000-0001-9095-1579http://orcid.org/0000-0001-9095-1579http://orcid.org/0000-0001-9095-1579http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0002-3109-640Xhttp://orcid.org/0000-0002-3109-640Xhttp://orcid.org/0000-0002-3109-640Xhttp://orcid.org/0000-0002-3109-640Xhttp://orcid.org/0000-0002-3109-640Xhttp://orcid.org/0000-0002-5587-6953http://orcid.org/0000-0002-5587-6953http://orcid.org/0000-0002-5587-6953http://orcid.org/0000-0002-5587-6953http://orcid.org/0000-0002-5587-6953http://orcid.org/0000-0001-7007-6949http://orcid.org/0000-0001-7007-6949http://orcid.org/0000-0001-7007-6949http://orcid.org/0000-0001-7007-6949http://orcid.org/0000-0001-7007-6949http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-025-58325-0&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-025-58325-0&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-025-58325-0&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-025-58325-0&domain=pdfmailto:eliaspo@phys.ethz.chmailto:pavel.volkov@uconn.eduwww.nature.com/naturecommunicationselectron-phonon coupling and the anisotropy of the superconductinggap is the first step towards answering these questions. However, the2D nature of MATBG, the small moiré Brillouin zone, and the relativelylow energy scales make the use of many standard techniques forinvestigating bulk materials, such as calorimetry, angle-resolved pho-toemission spectroscopy (ARPES) or neutron scattering, challengingor impossible. Similar concerns are relevant for all 2D super-conductors, where in many cases the mechanism and character ofsuperconductivity also remain undetermined11–14.Superconducting mesoscopic devices have proven to be a usefulcharacterization tool of the material they are built of15–17. In particular,Josephson junctions (JJs) have been used as a probe of electronicthermalization rates17, and the superfluid density, through character-izing the kinetic inductance in thin-film devices16. In the case ofMATBG, superconducting devices have already proven instrumentalfor probing the charge of the Cooper pairs18,19, the long-range coher-ence of the superconducting condensate19 and its orbital magneticproperties20.Here, we use a gate-defined Josephson junction (JJ) inMATBG18,20,21to extract electron-phonon coupling, thermodynamic, and superfluidproperties of MATBG across its phase diagram. Biasing the junctionwith a combination of DC and AC currents we probe the dynamics ofboth the electronic quasiparticles and the superfluid of MATBG. Weshow that themeasured timescales governing the junction’s transitionbetween resistive and superconducting states are directly related tothe microscopic properties of the material, such as electronic coolingpower due to phonons, specific heat, and superfluid density. The gate-tunability of the device allows us to probe these quantities across thedensity-tuned phase diagram of MATBG, for chemical potential bothwithin and outside the flat bands. These measurements allow to con-strain the electron-phonon coupling of MATBG, and the current biasdependence of the superfluid density is incompatible with isotropicpairing.Our experimental setup and theoretical models can be applied toother two-dimensional materials, establishing a tool for the study oftwo-dimensional superconductors at cryogenic temperatures. Theyadvance the state of the art forMATBG, directly probing the superfluiddensity, rather than spectroscopic8 properties of the superconductingstate. They also allow for the characterization of the thermal relaxationacross the whole phase diagram of MATBG and at cryogenic tem-peratures. Previous studies either accessed only parts of the phasediagram 22 or were limited to higher temperatures23, where correlationeffects are suppressed1,2.ResultsOverviewOur device is a JJ electrostatically defined in MATBG, with a twist angleof 1.06∘ ±0.04∘, also studied in reference18 (Fig. 1a). The global carrierdensity n, tuned by the back gate, is set to n = −1.73 × 10−12 cm−2, atwhich the bulk has its highest critical current, 250nA (See SI). Twolayers of top gates, separated by a layer of Al2O3, tune the local densityin the central region, allowing us to fine-tune the details of the junction.For each value of electron density in the central region (Fig. 1b) weanalyze the current-voltage (I/V) characteristic. For densities in thecentral region close to nj = −2 we observe a gradual onset of resistanceabove a critical current value, consistent with bulk superconductivity(see also discussion of Fig. 1f below). For all other densities, we uni-versally observe a hysteretic I/V trace with two characteristic voltagejumps ΔV, as shown in Fig. 1c. The two jumps correspond to switchingfrom the superconducting to the resistive state (increasing currentbias, blue line) and retrapping back (decreasing current bias, bluedashed). Together with Shapiro stepmeasurements18 this indicates theformation of a weak superconducting link between the left and rightparts of the device, where the weak link region can switch betweenresistive and superconducting states. From the band structure ofMATBG2 (see inset of Fig. 1(d), for a schematic), the weak link region isexpected to be metallic except for a narrow range of voltages placingthe chemical potential into the gap between the flat and dispersivebands. Such assessment is consistentwith the observation of a positiveexcess current, Iex,24,25 in the resistive state of a large portion of thephase diagram (green curve in Fig. 1d). In analogy to conventionalsuperconductors17, the dynamic response of such metallic weak linksshould give access to the dynamics of the electronic quasiparticles andthe superconducting condensate in MATBG. We probe the dynamicsof our weak links by adding a small AC current component to the DCcurrent flowing through the junction. Sweeping the frequency acrossthree orders of magnitude (0.1–100MHz), we focus on the changes inthe I/V characteristics, as shown in Fig. 1e. At low frequencies, the ACdrive brings the two hysteresis branches closer together, which can beunderstood as follows. The abrupt character of switching and retrap-ping with DC bias suggests that the junction will undergo a changewhenever the total current IDC + IRF(t) reaches the critical value forswitching (Isw) or retrapping (Ire). Consequently, one expects theswitching to occur prematurely at Isw − IRF, and the retrapping to occurat a higher DC bias, Ire + IRF , reducing the size of the hysteresis loop.For increasing frequency, the effect of AC bias gradually dis-appears (Fig. 1e), with a different rate for switching and retrapping.This indicates that both processes, in fact, do not occur instanta-neously and are characterized each by a certain rate, which we denoteas Γre and Γsw. We note that under switching (retrapping) rate wemeanthe characteristic scale for the switching (retrapping) current depen-dence on AC bias frequency, with a precise definition of that scalegiven below. At highest frequencies, the AC drive effect is absent,indicating that neither switching nor retrapping processes are fastenough to occur over one AC drive period. The switching andretrapping rates that can be extracted from Fig. 1e reflect the prop-erties of superconducting MATBG. We now turn to their physicalinterpretation.Modeling the weak linkWe can first rule out switching and retrapping driven only by thedynamics of the superconducting phase difference across the junc-tion, exemplified by, e.g., the Resistively and Capacitively ShuntedJunction (RCSJ) model24. In that case, the characteristic frequency isfixed by the Josephson relation to 2eΔV/ℏ. For our weak links it is of theorder of 10GHz, several orders of magnitude larger than the fre-quencies used in our experiments. The RCSJ model also predicts theswitching rate to be smaller than the retrapping one, inconsistent withexperimental observations (see additional discussion in Supplemen-tary Information). We therefore conclude that our experimentalobservations require a mechanism beyond the RCSJ model to explainthe switching and retrapping charateristics.Such an alternative mechanism, for both the retrapping and thehysteresis in metallic weak links is the heating of the electrons in thejunction, followed by their thermalization17,26. In this case the retrap-ping branch at I < Isw is characterized by a higher temperature than theswitching one due to the Joule heating in the resistive state (Fig. 2a, b).This overheating reduces the weak link critical current for the retrap-pingbranch, leading to a hysteresis.Most importantly, retrappingbackinto the superconducting state requires the electronic temperature toequilibrate to base temperature, a process, depicted in Fig. 2c, that hasbeen directly demonstrated in superconductor-normal metal-super-conductor junctions17.While there are several mechanisms for energy dissipation ingraphene, at low temperatures the dominant one is the couplingbetween electrons and acoustic phonons. In particular, thermalizationcan occur via diffusion of hot electrons into the leads, emission ofblackbody photons or interaction of electrons with acoustic phonons(as the optical ones are frozen out)27. Thefirstmechanism is suppressedby the presence of a superconducting gap28 in the leads in our case,Article https://doi.org/10.1038/s41467-025-58325-0Nature Communications |         (2025) 16:4273 2www.nature.com/naturecommunicationswhile the secondone has been estimated to be negligible inMATBG29,30.This suggests that the dominant heat loss mechanism is via coupling tophonons, in agreement with conventional SNS junctions17,26.The above mechanism on its own, however, still implies thatswitching occurswith the Josephson rate 2eΔV/ℏ, which is inconsistentwith our observations, as detailed above. To understand the switchingdynamics in our devices we now turn to the casewithout a central gatevoltage, i.e., where the sample is homogeneously superconducting atthe optimal density. We observe a frequency-dependent IV char-acteristic (Fig. 1f), despite the absence of a weak link. Note that there isno hysteresis, ruling out overheating as its origin.In addition to these observations, it has been shown that asupercurrent can flow in MATBG in narrow superconducting pathsseparated by normal regions31. The normal region thus forms a resis-tive shunt Rbulk coupled in parallel to the superconducting regions(purple shaded path in Fig. 2a). At a non-zero frequency ω, thesuperfluid impedance is purely inductive due to the inertia of theCooper pairs (blue shaded mechanism in Fig. 2d) and given byZsc = jωLkin, with the kinetic inductance Lkin / m*nse2, where m* is theeffective mass, e the electron charge, and ns is the superfluid density.At frequencies larger than RbulkLkin, the impedance of the superconductingbranch becomes higher than the resistance of the normal bulk and theAC current flows through the non-superconducting regions (purpleshaded mechanism in Fig. 2d). Intriguingly, Lkin in MATBG is expectedto be large19 due to two unique properties: extremely low electrondensities, and high effective mass1. This explains our observation of arather low characteristic switching rate in Fig. 1f. The samemechanismapplies for MATBG weak links - the kinetic inductance of bulk MATBGis then coupled in series to the junction (Fig. 2a).Using the ideas outlined above, we construct a model to describethe non-equilibrium dynamics of the Josephson junction. Importantly,this model allows us to relate the observed switching and retrappingrates,Γsw and Γre, to themicroscopic and thermodynamic properties ofMATBG.The dynamicsof the current-biased junction aredescribed by:IscðtÞ � Iex = I JðTÞ sinðφÞ+_ _φ2eRJð1ÞFig. 1 | Device response to AC and DC biases. a Schematics of the device. Thedevice is depicted by a cross section schematics. The vertical red dashed linesrepresent the central (C) region highlighted in (b). b Top-view simplified sche-matics with the gold contacts on the side, connected by a stripe of magic-angletwisted bilayer graphene (MATBG). The central region, of length 100nm, is high-lighted. The leads are superconducting (SC). c I/V characteristic of the junction at adensity of −2.8 × 1012 cm−2. In blue, solid line, a trace for increasingDCbias is shown.The dashed line represents the trace for decreasing bias. The green line is anextrapolation of the resistive part of the characteristic at0 voltage. The green arrowhighlights what is defined as excess current. d Switching (blue), retrapping (blue,dashed) and excess (green) currents as a function of density in the central region.The colors on the x-axis correspond to the filling of the band structure schematicsshown in the inset. The upper part indicates whether the I/V characteristic shows ajunction-like or bulk superconductor-like behavior. The black star, circle andsquare indicate, respectively, the densities at which is taken the data shown in(e, c, f). e I/V traces of the junction for AC bias of increasing frequency and fixedamplitude (red arrow) as a function of DC bias (horizontal axis). Solid lines showpositive bias directions while dashed ones show negative directions. Curves areoffset vertically for readability. f I/V traces at bias AC amplitude 1.4 nA and fre-quencies of 0.1MHz and 20MHzwhen the sample is tuned to all-bulk configurationsee panel (d).Article https://doi.org/10.1038/s41467-025-58325-0Nature Communications |         (2025) 16:4273 3www.nature.com/naturecommunicationsCel_T =1RJ_ _φ2e� �2� GthT ð2ÞIðtÞ � Iex = Isc � Iex +Lkin _Isc +_ _φ2eRbulkð3ÞWe highlight that this description has not been previously used toanalyze either the MATBG18,19,21 or conventional17 Josephson junctions;in what follows belowwe show that it allows to describe the Josephsonjunction data in a self-contained way without invoking results of othermeasurements29–31.Equation (1) describes a Josephson junction with a phase differ-ence φ, a temperature-dependent critical current IJ(T), and a fixedexcess current value Iex shuntedby resistanceRJ (Fig. 2(a), dashedbox).For results in the main text we assume RJ ≪ Rbulk, the general case isdiscussed in Supplementary Information.Wenote that the formof IJ(T)has not been determined experimentally; we assume that it is adecreasing function of temperature with a single characteristic scaleTJthat can be estimated to be of the order 0.1 K based on the dis-appearance of interference in superconducting quantum interferencedevices (SQUIDs)19. In the main text, we focus on an empirical modelI J = IJð0Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� T=T Jq� θð1� T=T JÞ that correctly captures the high-frequency asymptotic behavior of the retrapping current; we providea discussion of different models and their general properties inthe Supplementary Information.Equation (2) describes the evolution of the electronic tempera-ture T with respect to the base temperature. The left-hand siderepresents the total power dissipated in the link, Cel being the elec-tronic heat capacity. On the right-hand side, the first term correspondsto Joule heating, while the second one is the electronic heat loss (Gth)attributed, as discussed above, to electron-phonon interactions. Theprocesses relevant for the description of the Josephson effect occur atT ≈ TJ (see Supplementary Information), such that the value of thethermal conductivityGth canbe approximatedby its value atT =TJ. Thefinal equation describes the shunting of the junction by the resistivequasiparticles of bulk MATBG (Fig. 2 (a,d)). The current Isc(t) is the fullexternal current driven through the weak link.Remarkably, wefind that themodel defined by Eq. (1)-(3) capturesall of the behaviors observed in the experiment. As an example, weconsider a highly nonlinear regime where the RF amplitude is largerthan the hysteresis Isw � Ire. For a range of DC bias values the junctionspends part of the ACperiod in the resistive regime andpart of it beingsuperconducting, resulting in a double step in voltage, as shown inFig. 3a. Such voltage values are the average between the resistive andsuperconducting voltages weighted by the percentage of the timespent by the junction in each regime. Fig. 3d shows a simulated trace inthe same regime, demonstrating remarkable agreement between themodel and the experiment. Aswe increase the frequencyof the currentbias across the junction we recover the regular hysteresis (Fig. 3a, b,black line). The model captures the evolution of the I/V traces as thebias frequency increases, as is shown in Fig. 3e. Even finer details of theexperimental data, discussed in the Supplementary Information arecaptured by the model. These comparisons confirm that our modelaccurately describes the dynamics of our junction.To extract the retrapping and switching rates, Γre and Γsw, for agiven density from the experimental data, we fit the evolution of theretrapping and switching currents as a function of bias frequency. Ananalysis of the data, discussed in the Supplementary Information,demonstrates that both currents asymptotically approach a constanthigh-frequency value as 1/ω. To fit the results at all frequencies, we usethe following functions: IswðωÞ= Isw,1 � IRFΓsw=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΓ2sw +ω2qandIreðωÞ= Ire,1 + IRFΓre=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΓ2re +ω2q. That allows to characterize the corre-sponding rates (see Fig. 3(c), gray lines). The model described in Eqs.(1),(2),(3), reproduces correctly the asymptotic behavior of theswitching current, while for the retrapping current the result dependson the particular form of IJ(T) (see Supplementary Information). For afixed density in the junction, we extract the switching and retrappingcurrents for all frequencies and fit the results. In the example shown inFig. 3(c), for a density of − 4.5 × 1012cm−2, we extract Γre = 0.52MHz andΓsw = 2.75MHz. Therefore, the weak-link dynamics of our junctionRJRBulkLLead LLead(a)(c) (d)ΓreZ = jLleadsωZ = RbulkRB lkRBulk(b)VdcIdcIr Iswweak linkT0 Tr+DCACACDC+DCACZZZ == Rbulk+DCAACDC++DCCACCFig. 2 | Model for the switching and retrapping mechanisms. a Equivalentscheme of the MATBG junction for densities inside the lower flat band: the weak-link region modelled as a resistively shunted junction is coupled in series with thekinetic inductance of the leads. The superconducting regions (blue and red) arefurther shunted by normal regions (purple). b Illustration of switching andretrapping mechanism and hysteresis origin in MATBG junctions. Ire and Iswrepresent, respectively, the retrapping and switching currents. The retrappingbranch of IV characteristic (red) is characterized by an increased electronic tem-perature Tr due to Joule heating, suppressing the critical current. Retrapping intothe superconducting state requires cooling the electrons (c) to base temperaturecharacterized by a rate dependent on electronic cooling power Gth. Switching rate(blue), on the other hand, is only limited by the shunting kinetic inductance of thebulkMATBG (a,d). c Electronic thermal relaxation inMATBGoccurs via coupling toacoustic phonons. Two electronic quasiparticles in the junction release theirthermal energy to the phononbath and become cold enough tomediate Josephsoncoupling. This coupling is represented by an orange dashed line. Γre represents thethermalization rate. d Due to their inertia, Cooper pairs in a thin superconductor(blue region) prevent the transmission of RF signals at high frequencies. Instead,the AC current at frequencies above ωL =RbulkLleadsis rerouted through non-superconducting regions of the sample (purple region) and does not affect thejunction. Z represents the impedance seen by the signal, Rbulk the resistance of thebulk and Lleads the inductance of the leads.Article https://doi.org/10.1038/s41467-025-58325-0Nature Communications |         (2025) 16:4273 4www.nature.com/naturecommunicationsgives us access to the quasiparticle thermalization rate and kineticinductance of MATBG (Fig. 2c, d).Physical interpretation of the frequency dependenceWe now provide a physical interpretation of the rates, Γsw and Γre thatallows us to connect them to the properties of MATBG. We begin withthe switching rate Γsw. From Eq. (3) we identify the switching rate asΓsw = Rbulk/Lkin ∝ ns (see additional discussion in SupplementaryInformation). Assuming that the resistance of normal regions Rbulkdoes not strongly depend on T or bias strength, Γsw−1 ∝ Lkin, whichallows to probe the superfluid stiffness of MATBG.Before discussing the thermalization rate of theweak link,wenotethat for ω≫ Γsw the AC part of the current does not reach the junctionat all: Isc ≈ IDC. Thus, for Γre > Γsw, the kinetic inductance would set therate for both switching and retrapping. However, as shown in Fig. 4, wehave Γre strictly smaller than Γsw for all densities (note the differenty-axis in Fig. 4a, c), confirming that we can interpret the former as athermalization rate.Retrapping rate and thermalizationThe equation governing thermalization in the device in Eq. (2) containstwo implicit frequency scales: γ � GthCeland k � I2J RJCelT J. Importantly, thehysteresis size for DC driving depends on their ratio γ/k (which isproportional to Gth, but independent of Cel), while the retrapping rateΓre depends on both, allowing in principle, to determine both scales.Theseobservations allow for a qualitative discussion of the results(Fig. 4(a,b)) across theMATBG phase diagram. The noticeable peaks inΔI/Is, Fig. 4(b), occur near the band insulator (BI) and charge neutralitypoints (CNP) and indicate a suppressed thermal conductanceGth. Thiscan be attributed to a lower density of states near these points com-pared to other concentrations, expected from previousexperiments32,33 and theoretical analysis34. In contrast, Γre (Fig. 4(a)),which also depends on Cel (Fig. 4(a)) shows weaker features at theseconcentrations, indicating a simultaneous reduction of Gth and Cel,again consistent with a suppressed density of states. Remarkably, theminimum of Γre occurs for densities within the dispersive band. Apotential explanation for this behavior is an increased resistanceRJ dueto the mismatch between flat-band electrons outside the junction anddispersive ones within it. Deeper into the dispersive band, this effectcan be offset by an increased Gth.The quantitative nature of Γre depends on the particular form ofIJ(T); for the square-root model introduced above and γ/k < 1/2 weobtain an analytical result for the retrapping rate: Γre =k sin2πγk2π .Furthermore, the ratiobetweenDC retrapping current and switching isIre=Isw =ffiffiγkq(see Supplementary Information). We stress that theobserved Ir and Is are rather close to one another,which results in γ andk being effectively of the same order ofmagnitude. For larger values ofγ the model predicts an 1/ω2 dependence of the retrapping currentunder AC bias; therefore the model should not be applicable forIre=Isw > 1=ffiffiffi2p. However, in the absence of directmeasurements of IJ(T),we will use this model to estimate Cel and Gth.Weobserve, across thewholedensity range, three sets of values ofΓre: 0.5MHz, 1MHz and 1.5MHz, corresponding to the chemicalpotential of the link tuned to the dispersive band, lower flat band andupper flat band, respectively. The change in the hysteresis width,Fig. 4(b) is relatively smaller. Using the analytical formula given above,we can estimate for ΔI/Isw ≈ 0.5 that γ ≈ 0.8−2.3MHz. Reference22,where laser-mediated heating of a MATBG sample allows for theextraction of the same quantity, reports a value of γ ≈ 2MHz. The factthat such different methods for extracting the thermalization rateagree on the obtained value strengthens both of them as reliablecharacterization tools.This result already provides an important insight into the low-temperature behavior of electron-phonon coupling in MATBG whencontrasted with those at higher temperatures. In particular, the cool-ing rate has been found to be of the order of hundreds of GHz above5 Kwith a very weak temperature dependence23, attributed to effectivemoiré umklapp scattering35 (that is related to folding of the acousticFig. 3 | Extraction of the switching and rertapping rates. a I/V traces of thejunction at bias frequencies of 0.1MHz (red) and 100MHz (black) in the regimewhere the effective AC amplitude is higher than the hysteresis. ΔIre and ΔIswhighlight the change in retrapping and switching currents, respectively, betweenthe two bias frequencies. b I/V traces of the junction at bias frequencies of 8MHz(red) and 100MHz (black) in the regimewhere the effective AC amplitude is higherthan the hysteresis. Themismatch in retrapping current between the red and blackcurves is probably due to a charge jump (note it is of the order of a few pA).c Switching and retrapping currents as a function of AC bias frequency.d–f Numerical simulations of our device in the same regime as the data shown in(a–c). The grey dashed line in (c) is a fit to the functional forms provided inequations (1)–(3).Article https://doi.org/10.1038/s41467-025-58325-0Nature Communications |         (2025) 16:4273 5www.nature.com/naturecommunicationsphonons by the moiré lattice) explaining the linear-in-temperatureresistivity4,10,35. The strong difference with our result at T ~ TJ ≈ 100mKsuggests a suppression of the cooling rate much stronger than linear-in-temperature. This result is consistent with electron-phonon scat-tering at 100mK being in the Bloch-Gruneisen regime where umklappscattering is suppressed35 and resistivity from electron-phonon scat-tering should follow a stronger power-law dependence ontemperature36. This excludes electron-phonon scattering as the originof linear-in-temperature resistance at low temperatures37.In the case of superconductivity, themost relevant quantity whendiscussing electron-phonon coupling is the dimensionless couplingconstant, whichwe note here λ. The temperature relaxation rate at lowtemperatures is related to the strength of the coupling to acousticphonons38–40. While this coupling does not take the contribution ofoptical phonons into account, it is expected to be of the same order ofmagnitude as the full coupling constant38. To obtain an estimate weuse a Dirac electron model4,39,40, motivated by the theoretical34 andexperimental evidence32,33 for their presence in MATBG bands, inparticular in the vicinity of CNP (consistent with the peak in ΔI/Is dis-cussed above). One finds that γ = GthCel= λ 16π25ðkBTel Þ2_2skF, where s is theacoustic phonon velocity. Using Tel ~ TJ ~ 0.1 K from the extinctiontemperature of SQUIDoscillations19, s≈ 20km/sec (the value for single-layer graphene41 is expected to be close to that in MATBG42), kF =ffiffiffiffiffiffiffiπnpfor n ~ 1 × 10−12 cm−2 and γ ~ 1MHz we obtain λ ~ 10−3. Several commentsare in order regarding this estimate of λ. We begin by stressing thisestimate may not be directly comparable to other transport mea-surements (e.g., resistivity) as our estimates of λ stem from theelectron-phonon cooling rate. While the relation between the wayelectron-phonon coupling enters the cooling rate, the resistivity, andthe superconducting pairing is established in theDiracmodel4,39,40, it isyet to be determined (and may be different) in the MATBG bands.Moreover, the potential inhomogeneity of the twist angle across thesample may reduce the average thermal relaxation rate, since regionsthat are away from themagic angle are expected to have lower densityof states and thus slower thermal relaxation. Additionally, the aboveestimate is assuming the system is in the Bloch-Gruneisen regimeT ≪ cskF; it has been however demonstrated35,43 that the crossover tothis regime may be quite different in MATBG than in single-layer gra-phene and, in particular, it occurs atmuch lower temperatures. In fact,the strong reduction of the thermal relaxation rate in our experimentswith respect to the values at 5 K23 may present a first demonstration ofthis crossover occurring in MATBG. We note that this estimateassumes coupling to lowest-energy acoustic phonons only, and doesnot address optical or any other phonons that are frozen out at theexperimental temperatures. Finally, the Dirac model should beapplicable only around certain fillings; in our case, a signature of Diracphysics is observed near the CNP in enhanced ΔI/Is indicating lessefficient heat relaxation due to lower density of states near aDirac point.We can further estimateGth andCel taking IJ − Iexc ~ 5 nA,ΔV ~ 20 μVfrom Fig. 3. The result is Gth ~ 250 fW/K and Cel ~ 5 × 1019 J/K. From thejunction area and n ~ 1 × 1012 cm−2 one expects above 103 electrons, withthe usual Sommerfeld expression Cel =π22 kN kBTEF� 10�19 kBTEFJ/K. Inusual metals, kBTEF≪1, while in our case this implies kBTEF� 1, that may berelated to large residual entropy of interacting states of MATBG44,45.Both Gth and Cel are much higher than those expected in monolayergraphene40, consistent with strongly suppressed bandwidth and elec-tron velocities of MATBG G / v�2F ,C / vF . However, the γ = GthCelwe find0.00.51.01.5Γ re(MHz)−4 −3 −2 −1 0 1n (cm−2) 1e1205101520Γ sw(MHz)0.000.250.500.751.00ΔI/I slower dispersive band lower flat band upper flat band(a)(b)(c)CNPBI bulk SCJFig. 4 | Switching and retrapping rates and relative hysteresis across the phasediagramofMATBG. aRetrapping rate as a functionof junctiondensity. The regionbetween vertical dashed lines represents the range of densities in which we have abulk superconductor (SC). b Relative hysteresis as a function of junction density.We observe a peak at the charge neutrality point and another one at the bandinsulator between lower dispersive and flat bands. c Switching rate as a function ofjunction density. The red shaded areas highlight the regions in density where atransition between bands takes place. They correspond to the Charge NeutralityPoint (CNP) and Band Insulator (BI).Article https://doi.org/10.1038/s41467-025-58325-0Nature Communications |         (2025) 16:4273 6www.nature.com/naturecommunicationsin MATBG at T ≈ TJ are of the same order as those predicted formonolayer graphene40.Switching rate and superfluid stiffnessWe now discuss the switching rate Γsw (Fig. 4c), related to the super-fluid stiffness in the bulk of our MATBG device. Importantly, the ACmeasurements are still performed at a finite DC bias, thus, our mea-surements reveal the superfluid density at a finite current bias,ns(IDC) ≈ ns(Isw). Since ns(Isw) is a decreasing function of current, thesteep increase in Γsw at the edges of the lower flat band is explained bythe decreasing critical current of the junction (Fig. 1 (d)). On the con-trary, the decrease of Γsw for densities in the top flat and dispersivebands, is unexpected - at such low critical currents ns(Isw) ≈ ns(0)should be density-independent and large. We suggest that thisobservation can be explained by the kinetic inductance of proximity-induced superconductivity in the junction region. Being veryweak, theproximity-induced superfluid has an extremely large kinetic induc-tance that is in parallel to the smaller one from the bulk TBG, effec-tively shunting it. Furthermore, since Γsw ∝ Rbulk, changes in theresistance with concentration can also affect its magnitude. In parti-cular, this provides a plausible explanation of the peak of Γsw nearCNP,where the normal-state resistance peaks.Let us now return to the densities within the lower flat band,where Γsw is related to the superfluid density of bulk MATBG. Thedependence of Γsw ∝ ns as a function of IDC, shown in Fig. 5, givesimportant information about the nature of the superconducting gap inMATBG. Current biasing a superconductor produces a Dopplershift46,47 of the quasiparticle bands in a superconductor, see inset ofFig. 5 (ΔE = vs × ℏk, where vs is the superfluid velocity and k thequasiparticle momentum). For an isotropic superconductor, depictedin the inset of Fig. 5, this does not affect the quasiparticle occupationsuntil a critical value of bias current is reached. As a result, ns(I)dependence is highly nonlinear with an abrupt drop close to the cri-tical current48. For a highly anisotropic or nodal superconductor,across its nodal axis in real space, the quasiparticle band structurepresents cones instead of a gap in density (Fig. 5, inset). A small shiftoriginating from a finite bias current, leads to a finite generation ofquasiparticle pairs, thus reducing the superfluid density beforebreaking down the superconducting condensate. As Fig. 5 shows, therelation between superfluid density and bias current is linear in thecase of MATBG in the range Idc ∈ [0.6Ic, 0.95Ic]. This result is incon-sistent with the behavior expected of an isotropic superconductinggap, ruling in favor of a highly anisotropic or nodal pairing statein MATBG.DiscussionIn conclusion,wehave developed amethod for characterizing electrondynamics in twisted bilayer graphene by combining electrostatic andradiofrequency current bias of an electrically-defined Josephsonjunction. This has resulted in a study of out-of-equilibrium dynamicalproperties of electrons at cryogenic temperatures in MATBG. Ourresults demonstrate the presence of two distinct characteristic time-scales guiding the dynamics of electrons, attributed to the thermali-zation of electrons and the kinetic inductance of the superconductingcondensate, respectively. A phenomenological model, capturing theseprocesses, is found to describe the data well and allows to relate themeasurement outcomes to the physical properties of the electrons inthe material for a wide range of densities. In particular, we discuss theestimates for electron-phonon coupling (that may have bearing ontheories of superconductivity4–7 or strangemetal36,37), specific heat andsuperfluid density. The current bias dependence of the deducedsuperfluid stiffness points towards the superconducting gap of thematerial being anisotropic. The technique we developed in this workcan be applied to a wide range of gate-tunable superconducting 2Dmaterials, introducing a general way to access importantthermodynamic quantities, such as specific heat and superfluid stiff-ness. In addition to being a valuable addition to experimental probesof 2Dmaterials,wedemonstrated a controllabledriving of a correlatedelectronic system, opening the path to the realization of out of equi-librium states of electrons.While writing this manuscript the authors became aware of twoworks where, by different experimental means, some of the quantitiesstudied in this work are also probed49,50.MethodsFabrication details and measurement setupTo fabricate the device, we begin by assembling a so called ‘stack’ oftwo dimensional materials. Such flakes are exfoliated mechanicallyusing a micro manipulator, polydimethylsiloxane/polycarbonatestamps, a moving stage and an optical microscope. The stack consistsof a bottom layer of few-layer graphite, which is used as a global back-gate, 24 nm-thick hexagonal boron nitride (hBN), the magic anglegraphene, and hBN again, this time with a thickness of 27 nm. Thetwisting and stacking are performed following the standard proce-dures of the field, with a cutting step of the graphene flake51. Thegraphene-cutting step is performed with a tungsten needle with a tipdiameter of 2.5 μm. Once the stack is in place we move on to thelithographyphaseof the fabricationprocess.Gold contacts are definedby a combination of electron-beam lithography, reactive ion etchingand electron-beam evaporation. We use chromium as adhesion layerfor the gold (10/70 nm). Top gates are defined by electron-beamlithography and evaporation. The graphene is finally etched to definethemesa. A layer of aluminiumoxide isdeposited through atomic layerdeposition. Finally, we define another layer of local gold gates with achromium adhesion layer (10/110 nm). As a last remark, the deviceused for this work is the same one as the one presented in reference18.In this work, there are however twomodifications with respect tothe setup of the aforementioned reference. The first one is that the ACbias is not sent to the central gate but to one of the leads, using a bias Tto be able to send both AC and DC signals to the same contact. Theother difference is that, because for the ACmeasurements we presentin this study we need a higher degree of precision than for the onespresented in reference18, we must ensure that the AC amplitudereaching the device is neither frequency-dependent nor sample-Fig. 5 | Superconducting stiffness in arbitrary units as a functionof bias currentto critical current ratio. Inset, bottom left: Schematics of Bogoliubov-de Gennesquasiparticle band structure of a superconductor with isotropic gap. The solidstraight line represents the Fermi energy at zero current bias. The dashed linerepresents the Fermi energy at a non-zero current bias. Inset, top right: Schematicsof quasiparticle band structure of a superconductor with anisotropic gap. The redandblue areas represent respectively thehole andelectronpockets that format theband edges under a finite current bias.Article https://doi.org/10.1038/s41467-025-58325-0Nature Communications |         (2025) 16:4273 7www.nature.com/naturecommunicationsresistance-dependent. The details of such procedure are given in thefollowing section.Radiofrequency biasingWhen applying a radiofrequency bias to our junction two differentaspectsmustbe taken into account. Thefirstone is the evolution of theamplitude as a function of its frequency. Indeed, having a frequency-dependent amplitude reaching our junction would make it impossibleto disentangle such effects from the physical mechanisms taking placeat the junction level.We performed simulations of our circuit using thesoftware LTspice and obtain an evolution in amplitude of our radio-frequency bias reaching the junction of less than 5%, for frequenciesranging from 100 kHz to 100MHz. We thus conclude that, in com-parison to the experimental results, these effects can be neglected.Because the junction changes its state during the acquisition of anI/V trace, we must also ensure that the change in resistance triggeredby the RF biasing does not significantly affect the AC current flowingthrough the device. In order to achieve that, we place a 100 kΩ resistorin series between the RF feed line and the device. Like for the previouseffect, this leads to a variation in AC amplitude reaching the device ofthe order of a few percents across the whole resistivity range of thedevice. We thus can also neglect this effect taking into account theprecision of the claims made in our analysis. Supplementary Fig. 1shows a schematics of the electronics set-up. We neglect in this sche-matics the capacitances from the MATBG to the gold or graphite gateelectrodes because they are smaller than 30 fF18. They thus present, atthe frequencies at which we bias our sample (up to 100MHz), animpedance orders of magnitude higher than any other impedance inthe set-up.Data availabilityThedata that support the findings of this study, togetherwith the codefor plotting the figures, is available online through the ETH ResearchCollection at https://doi.org/10.3929/ethz-b-000722186.References1. Cao, Y. et al. Unconventional superconductivity in magic-anglegraphene superlattices. Nature 556, 43–50 (2018).2. Cao, Y. et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. Nature 556, 80–84 (2018).3. Lu, X. et al. 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Kim, K. et al. van der waals heterostructures with high accuracyrotational alignment. Nano Lett. 16, 1989–1995 (2016).AcknowledgementsWe thank Peter Märki, Wister Huang and the staff of the ETH cleanroomfacility FIRST for technical support.We thank LandryBretheau for helpfuland detailed discussions on radiofrequency biasing of superconductingdevices, Sankar das Sarma for discussions on electron-phonon couplingestimates and members of the quantum e-leaps consortium for com-ments on our data. We acknowledge financial support by the EuropeanGraphene Flagship Core3 Project, H2020 European Research Council(ERC) Synergy Grant under Grant Agreement 951541, the EuropeanUnion’s Horizon 2020 research and innovation program under grantagreement number 862660/QUANTUM E LEAPS, the European Innova-tion Council under grant agreement number 101046231/FantastiCOF,NCCR QSIT (Swiss National Science Foundation, grant number 51NF40-185902). K.W. and T.T. acknowledge support from the JSPS KAKENHI(Grant Numbers 21H05233 and 23H02052), the CREST (JPMJCR24A5),JST and World Premier International Research Center Initiative (WPI),MEXT, Japan. E.P. acknowledges support of a fellowship from “la Caixa”Foundation (ID 100010434) under fellowship code LCF/BQ/EU19/11710062. J.H.P. acknowledges NSF Career Grant No. DMR-1941569.P.A.V. acknowledges support by a Quantum-CT Quantum RegionalPartnership Investments (QRPI) Award. This workwas initiated and partlyperformed at the Aspen Center for Physics, which is supported byNational Science Foundation grant PHY-2210452.Author contributionsP.R. and E.P. fabricated the device. F.K.d.V., E.P., and P.R. characterizedthe device. T.T. and K.W. supplied the hBN crystals. E.P. and M.P. per-formed the measurements for this experiment. E.P., P.A.V., M.P., M.T.,and Y.K. analyzed the datawith input fromA.M-T, A.O.D., andG.Z.; P.A.V.developed the theoretical model and performed the numerical andanalytical calculations. K.E., T.I., and E.P. conceived and designed theexperiment. J.H.P., T.I., and K.E. supervised the project. E.P. and P.A.V.wrote the manuscript with comments from all authors.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-025-58325-0.Correspondence and requests for materials should be addressed toElías Portolés or Pavel A. Volkov.Peer review information Nature Communications thanks Jianpeng Liu,and the other, anonymous, reviewer for their contribution to the peerreview of this work. 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To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.© The Author(s) 2025Article https://doi.org/10.1038/s41467-025-58325-0Nature Communications |         (2025) 16:4273 9https://doi.org/10.1038/s41467-025-58325-0http://www.nature.com/reprintshttp://creativecommons.org/licenses/by-nc-nd/4.0/http://creativecommons.org/licenses/by-nc-nd/4.0/www.nature.com/naturecommunications Quasiparticle and superfluid dynamics in Magic-Angle Graphene Results Overview Modeling the weak link Physical interpretation of the frequency dependence Retrapping rate and thermalization Switching rate and superfluid stiffness Discussion Methods Fabrication details and measurement setup Radiofrequency biasing Data availability References Acknowledgements Author contributions Competing interests Additional information