Phase-dependent microwave response of a graphene Josephson junction

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Phase-dependent microwave response of a graphene Josephson junction © Authors, 2022 Published version Haller, R.; Fülöp, G.; Indolese, D.; Ridderbos, J.; Kraft, R.; Cheung, L. Y.; Ungerer, J. H.; Watanabe, K.; Taniguchi, T.; Beckmann, D.; Danneau, R.; Virtanen, P.; Schönenberger, C. Haller, R., Fülöp, G., Indolese, D., Ridderbos, J., Kraft, R., Cheung, L. Y., Ungerer, J. H., Watanabe, K., Taniguchi, T., Beckmann, D., Danneau, R., Virtanen, P., & Schönenberger, C. (2022). Phasedependent microwave response of a graphene Josephson junction. Physical Review Research, 4(1), Article 013198. https://doi.org/10.1103/PhysRevResearch.4.013198 2022 PHYSICAL REVIEW RESEARCH 4, 013198 (2022) Phase-dependent microwave response of a graphene Josephson junction R. Haller ,1,*G. Fülöp ,1,2D. Indolese,1J. Ridderbos ,1R. Kraft,3,4L. Y. Cheung,1J. H. Ungerer,1,5K. Watanabe ,6 T. Taniguchi ,7D. Beckmann ,8R. Danneau ,8P. Virtanen ,9and C. Schönenberger 1,5,† 1Department of Physics, University of Basel, Klingelbergstrasse 82 CH-4056, Switzerland 2Department of Physics, Budapest University of Technology and Economics and MTA-BME Momentum Nanoelectronics Research Group, H-1111 Budapest, Budafoki út 8, Hungary 3Institute of Nanotechnology, Karlsruhe Institute of Technology, D-76021 Karlsruhe, Germany 4Institute of Physics, Karlsruhe Institute of Technology, D-76131 Karlsruhe, Germany 5Swiss Nanoscience Institute, University of Basel, Klingelbergstrasse 82 CH-4056, Switzerland 6Research Center for Functional Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan 7International Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan 8Institute of Quantum Materials and Technologies, Karlsruhe Institute of Technology, D-76021 Karlsruhe, Germany 9Department of Physics and Nanoscience Center, University of Jyväskylä, P.O. Box 35 (YFL), University of Jyväskylä FI-40014, Finland (Received 30 July 2021; revised 18 August 2021; accepted 6 February 2022; published 14 March 2022) Gate-tunable Josephson junctions embedded in a microwave environment provide a promising platform to in situ engineer and optimize novel superconducting quantum circuits. The key quantity for the circuit design is the phase-dependent complex admittance of the junction, which can be probed by sensing a radio frequency SQUID with a tank circuit. Here, we investigate a graphene-based Josephson junction as a prototype gate-tunable element enclosed in a SQUID loop that is inductively coupled to a superconducting resonator operating at 3 GHz. With a concise circuit model that describes the dispersive and dissipative response of the coupled system, we extract the phase-dependent junction admittance corrected for self-screening of the SQUID loop. We decompose the admittance into the current-phase relation and the phase-dependent loss, and as these quantities are dictated by the spectrum and population dynamics of the supercurrent-carrying Andreev bound states, we gain insight to the underlying microscopic transport mechanisms in the junction. We theoretically reproduce the experimental results by considering a short, diffusive junction model that takes into account the interaction between the Andreev spectrum and the electromagnetic environment, from which we estimate lifetimes on the order of ∼10 ps for nonequilibrium populations. DOI: 10.1103/PhysRevResearch.4.013198 I. INTRODUCTION For Josephson junctions (JJs), in which the superconducting electrodes are linked with a short normal-conducting region, the coherent superconducting interaction is promoted by so-called Andreev bound states (ABSs) [1]. The material and geometrical properties of the weak link together with the superconducting phase difference ϕacross the JJ define the energy of the ABSs [2]. Their structure and occupation dynamics determine the inductive and dissipative microwave response, i.e., the admittance of the JJ [3,4]. In particular, the inductive response relates to the time-averaged dispersion of the populated ABSs and reflects the phase dependence of the supercurrent Is(ϕ) across the junction [5,6], which is known as the current-phase relation (CPR). However, the dissipative response relates to the fluctuations in the ABS population *roy[email protected] †http://www.nanoelectronics.unibas.ch/ Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. resulting in temporal changes of the supercurrent [7,8]. The microscopic source for those dynamics are thermally activated or microwave induced short-lived ABS excitations [9]. Conclusively, the junction admittance, which is the key quantity to engineer high-frequency Josephson circuits, is highly dependent on the underlying microscopic processes. The junction admittance can be probed as a function of phase by embedding a JJ in an radio frequency (rf) SQUID that couples to a resonator [9–13]. The rf SQUID acts as a magnetic flux-tunable complex impedance in the circuit that shifts and broadens the resonate behavior, from which one can infer the phase-dependent inductive and dissipative response of the junction [14]. The strong demand for in situ controllable junctions in microwave applications has raised the attention to JJs consisting of gate-tunable weak links [15]. Here, we determine the full complex admittance of a Josephson weak link made of graphene, which is a twodimensional (2D) material with a linear band structure and excellent gating properties. Although graphene JJs have already demonstrated their compatibility in different superconducting high-frequency circuits, such as bolometers [16,17], transmon qubits [18,19], and tunable microwave cavities [20], only few experiments have addressed the determination of their phase-dependent 2643-1564/2022/4(1)/013198(11) 013198-1 Published by the American Physical Society R. HALLER et al. PHYSICAL REVIEW RESEARCH 4, 013198 (2022) (a) 50 m NbTiN SiOx 1 mm (b) Iflux Vbg Al Al (c) 50 m 1 m G Graphite hBN SiOx Al intrinsic Si FIG. 1. Graphene rf SQUID inductively coupled to a superconducting transmission line resonator. (a) Optical image of the NbTiN λ/4-resonator consisting of a meandered coplanar transmission line with the shorted end (current anti-node) on top, seen also at the bottom of image (b), and the open end (current node) at the bottom, shown in the zoom-in. (b) Optical image of the monolayer graphene (G) Josephson junction (JJ) embedded in an Al loop forming the rf SQUID. The DC current Iflux creates a flux inside the loop (blue line), which allows to phase bias the junction. The inductive coupling to the resonator induces a small oscillating probe flux δ (red lines). The gate voltage Vbg applied on the bottom graphite sheet tunes the charge carrier density in G. (c) Scanning electron micrograph and cross-sectional schematics of the hBN-encapsulated G-JJ with Al side-contacts of width W=1μm and length L=400 nm. admittance [13,21]. While Ref. [13] has been focusing on the phase-dependent dissipation of the junction under the influence of external irradiation and Ref. [21]ontheinductive behavior, we here investigate both the inductive and dissipative response simultaneously by studying the inherent photonic phase-dependent interplay between the sensing resonator and the graphene JJ. We present a classical, comprehensive circuit model to infer the full complex junction admittance from the reflective response of a graphene rf SQUID coupled to a superconducting microwave resonator operating at ∼3 GHz. We further translate this to the CPR and the phase-dependent dissipation as a function of gate voltage, under consideration of the self-screening effect that arises due to the finite inductance of the SQUID loop. We describe our observations within the framework of ABSs and find remarkable agreement between the experimental results and the theoretically predicted microwave response of a short, diffusive junction. II. DEVICE The device is presented in Fig. 1and consists of a graphene JJ embedded in a superconducting loop, which inductively couples to a coplanar transmission line (CTL) resonator. The resonant structure and supply lines are etched into NbTiN (80nm)sputteredonanintrinsicSi/SiOx(500 μm/170 nm) substrate. The meandered CTL shown in Fig. 1(a) is shorted to ground on one side, and interrupted by a coupling capacitor on the other. Both of these terminations act as microwave mirrors of the opposite type, and thereby form a superconducting λ/4-resonator with a fundamental bare resonance frequency fbare =3.098 GHz. The graphene JJ, shown in Fig. 1(c), is made of a van der Waals heterostructure consisting of a monolayer graphene encapsulated in hexagonal boron nitride (hBN). The lower hBN layer (47.5 nm) separates the graphene flake from the bottom graphite gate. A thermally evaporated Ti/Al (5 nm/90 nm) lead contacts the graphene from both sides [22] and encloses the junction in a loop, thus forming a graphene rf SQUID, which is inductively coupled to the current anti-node of the resonator as illustrated in Fig. 1(b). The galvanic grounding of the loop defines the reference potential for the gate voltage Vbg applied on the bottom graphite structure. The DC current Iflux controls the magnetic flux inside the loop and therefore tunes the external phase difference ϕext =2π/0across the rf SQUID, where 0=h/2eis the superconducting flux quantum with hbeing the Planck constant and ethe elementary charge. Consider the Supplemental Material (SM) for details about the device fabrication [23]. In the subsequent experiment we perform reflectance measurements on the port denoted by in Fig. 1(a) and investigate the resonant circuit as a function of Vbg and Iflux, from which we later infer the CPR and the phase-dependent loss of the graphene JJ. III. REFLECTOMERTY The coupled microwave circuit is probed by reflectometry in a dry dilution refrigerator, in which the device is surrounded by a permalloy shield. With a vector network analyzer we measure the complex reflection coefficient as a function of probe frequency fand Iflux. We ensure a quasiequilibrium sensing by setting the probe power to an averaged intracavity occupation of ∼100 photons, which corresponds to an oscillating probe flux δ ≈0/100 inside the SQUID loop. Additionally, we tune the charge carrier density in the graphene layer by applying a gate voltage in the range Vbg =[−9,9] V. The conversion from Vbg to charge carrier density as well as the measurement scheme and the calibration of the probe power can be found in the SM [23]. The reflective response at Vbg =6 V presented in Fig. 2 is exemplary for the whole measurement set. Clear periodic shifts of the resonance frequency f0as a function of Iflux can be observed in Figs. 2(a) and 2(b). We encounter no phase jumps and relate the external phase ϕext =noddπ(=nevenπ)to points of minimal (maximal) resonance frequencies [10,14]. Besides f0, the resonance lineshape also changes as seen in Figs. 2(c) and 2(d) when comparing line cuts at ϕext =−π and ϕext =0. As we will show, both the modulation in f0 and the altered lineshape are the consequence of the phasedependent complex admittance of the graphene JJ. To characterize the JJ from the reflective response, we fit ||and arg() simultaneously for each combination of Vbg and Iflux with the complex resonance curve of a loaded λ/4resonator expressed according to Ref. [24]as =min +2jQ f−f0 f0 1+2jQ f−f0 f0−1ejφ+1.(1) Thus, we can deduce f0and assess the broadening of the resonance curve. The latter is determined by the total quality factor Q=1/(Q−1 load +Q−1 i+Q−1 c), which in turn, consists 013198-2 PHASE-DEPENDENT MICROWAVE RESPONSE OF A … PHYSICAL REVIEW RESEARCH 4, 013198 (2022) FIG. 2. Flux dependence of the reflection coefficient at Vbg =6 V. (a, b) Colormaps of ||and arg() as a function of probe frequency fand DC flux current Iflux. The horizontal top axis represents the conversion to the external phase ϕext across the rf SQUID. (c, d) ||and arg()atϕext =[−π,0] overlaid with fits to Eq. (1) (solid lines), from which we obtain the resonance frequency f0, asymmetry angle φ, coupling quality factor Qcand effective quality factor Qeas listed below: ϕext f0φQcQe −π3.09755 GHz 0.224 23 400 19 400 0 3.09821 GHz 0.235 23 700 >200 000 of three different dissipation sources: (i) The inverse load quality factor Q−1 load describes loss generated by the rf SQUID, (ii) the inverse internal quality factor Q−1 idescribes loss inherent to the properties of the CTL, and (iii) the inverse coupling quality factor Q−1 cdescribes loss to the measurement environment. Here, Q−1 load and Q−1 iare merged to an effective quality factor Qe=1/(Q−1 load +Q−1 i). Furthermore, we define min =(Qc−Qe)/(Qc+Qe) and introduce the angle φ, which accounts for an asymmetric line shape. The fits to Eq. (1)atϕext =−πand ϕext =0, shown in Figs. 2(c) and 2(d) as solid lines, reveal an overall shift of 660 kHz in f0and a drastic change in Qe, while Qc and φremain similar. At ϕext =−π, we obtain Qe=19 400 and Qc=23400, whereas at ϕext =0, we find Qe>200 000 and Qc=23700. Consequently, the resonator is undercoupled (Qe<Qc)atϕext =−π, but overcoupled (Qe>Qc)at ϕext =0, which explains the distinct resonance lineshapes [25]. Since Qican be treated as a constant with Qebeing a lower bound, we conclude that Qi>200 000. This large value allows us to treat the CTL as lossless (Q−1 i=0) such that Qe≈Qload. The SM provides further insights to the resonance curve fitting [23]. The observed flux tunable microwave response in terms of f0and Qload is the direct manifestation of phase-dependent microscopic processes in the graphene JJ [10], which will be discussed in detail in Secs. VII and VIII within the framework of ABSs. In the following section we model the electrical properties of the graphene JJ with lumped elements and explain their effect on the resonant behavior with the circuit of a loaded λ/4-resonator. FIG. 3. Circuit schematic of a rf SQUID coupled to a λ/4resonator. The resonator couples inductively to the rf SQUID with strength Mand connects to the reflectometry setup via capacitance Cc. The rf SQUID is modeled as a loop with self-inductance Lloop in series with the JJ, which in turn, is modeled as a variable Josephson inductance LJin parallel with a variable shunt resistance Rs.This forms a variable load impedance Zload, which tunes the reflective response . IV. CIRCUIT MODEL The inductively coupled rf SQUID acts as a variable load impedance Zload attached to the resonator, which tunes the reflective response. We express Zload according to the circuit schematic depicted in Fig. 3. The rf SQUID is modeled as a loop with self-inductance Lloop in series with the JJ. The mutual inductance Mquantifies the coupling strength to the resonator, which is built from a CTL with characteristic impedance Zr. The JJ itself is represented by a variable Josephson inductance LJin parallel with a variable shunt resistance Rs. For this arrangement the load impedance terminating the resonator is detailed in the SM and reads [23] Zload =ω2M2 jωLloop +(Gs+jBJ)−1,(2) where ω=2πfis the angular frequency, Gs=1/Rsis the shunt conductance and BJ=−1/(ωLJ) is the susceptance. Note that Y=Gs+jBJis the complex admittance of the JJ. The influence of Zload on the λ/4-resonator is twofold: First, the imaginary part of Zload causes a shift of the resonance frequency as derived in the SM [23], δf0=f0−fbare =− 2 πZr Im(Zload)fbare,(3) with respect to the unloaded resonance frequency fbare. Second, the real part of Zload gives rise to dissipation in the resonant circuit, which can be expressed according to the derivations presented in the SM as [23] Qload =πZr 4Re(Zload).(4) From Eq. (2) one recognizes that the junction variables, Gsand BJaffect both Re(Zload ) and Im(Zload). Consequently, δf0and Qload would need to be considered simultaneously to evaluate them. However, it turns out that, due to the obtained relatively large Qload values, one is allowed to set Gs→0 in Eq. (3), which simplifies the relation as shown in the SM to [23] δf0≈8 π2 M2 Lp(LJ+Lloop)fbare,(5) where Lpis the parallel LC-equivalent inductance of the λ/4-resonator. This means that the shift of the resonance 013198-3 R. HALLER et al. PHYSICAL REVIEW RESEARCH 4, 013198 (2022) FIG. 4. Evaluation of the CPR. (a) Colormap of the resonance frequency shift δf0=f0−fbare with fbare =3.098 GHz as a function of gate voltage Vbg and external phase ϕext.(b)δf0at Vbg =6 V as a function of ϕand ϕext, respectively overlaid with the fits to Eq. (5) (solid lines), from which the CPR is deduced. (c) Presents the CPR atVbg =6 V, corrected for the self-screening of the SQUID (blue) and uncorrected (dashed), in comparison with the sine function (dotted). In panels (b, c) arrows illustrate the correction introduced by the nonlinear mapping from ϕext to ϕ. (d) Corrected CPR inferred from panel (a) as a function of Vbg.(e)δf0at the charge neutrality point (Vbg =−0.44 V) as a function of ϕoverlaid with the fit and in panel (f) the corresponding CPR. frequency mainly originates from the Josephson inductance LJ, whereas the broadening of the resonance originates from the dissipation in the JJ specified by the shunt conductance Gs. Since the inverse Josephson inductance is a measure of the change in the supercurrent Is(ϕ) with respect to the phase ϕ across the junction [6], LJ(ϕ)−1=2π 0 ∂Is(ϕ) ∂ϕ ,(6) we can express the resonance frequency shift and the behavior of LJ(ϕ) with the current-phase relation (CPR). To quantify the CPR and Gsfrom the resonator response, we perform finite-element simulations [26] based on the device geometry, to acquire Lloop =211 pH and M=30.83 pH. Moreover, we find Zr=69.5from the aspect ratios of the CTL [27] in combination with the resonant behavior of the circuit and deduce Lp=4.55 nH. The evaluation of Zrand Lp can be found in the SM [23]. V. CURRENT-PHASE RELATION In this section we extract the CPR by fitting the periodic shift of the resonance frequency under consideration of self-screening effects. The coupling strength between the superconducting leads is determined by the Cooper pair transmission probability and defines the shape of the CPR. For small coupling or low transmission probability the CPR is sinusoidal, whereas the CPR becomes forward-skewed for increased coupling. Due to the semiconducting properties in graphene JJs, the coupling strength and therefore the CPR skewness can be tuned with the gate voltage [20,28–31]. To capture the nonsinusoidal behavior, we express the CPR as Fourier series [32] Is(ϕ)= k (−1)k−1Aksin(kϕ),(7) with kbeing the harmonic order and Akthe corresponding amplitude. To extract the CPR from the measured resonance frequency modulations, we need to relate the external phase ϕext to the phase difference ϕacross the JJ. This is not straightforward, since if a supercurrent flows within the rf SQUID, there is a phase drop over the loop inductance Lloop in addition to the phase drop over the JJ, which leads to a nonlinear relation between the internal phase ϕand the external phase ϕext— known as the screening effect [33]: ϕ=ϕext −2π 0 LloopIs(ϕ).(8) Here, we obtain the CPR for each gate voltage by solving the set of equations Eqs. (5)–(8) in a self-consistent way by using an iterative fitting method. The basis for this method is the resonance frequency shift as a function of ϕext, which is presented for the entire gate range in Fig. 4(a). At each fitting iteration we include Fourier amplitudes Akup to the 10th-harmonic and allow for small changes in fbare. Details about the method can be found in the SM [23]. In Fig. 4(b) we illustrate the effect of screening by comparing δf0as a function of ϕand ϕext, respectively—for the example at Vbg =6 V. The corresponding CPRs, deduced from fitting the modulations in δf0with respect to phase, shown as solid lines in Fig. 4(b), are presented in Fig. 4(c). The screening consideration causes a distortion of the phase around πas indicated by arrows. Omitting this effect results in an apparent enhancement of the skewness [34]. Even after correcting for screening, we find a substantially forward-skewed CPR, visualized by the comparison with a sinusoidal behavior. Although screening effects are small in this case, we emphasize that they can have a significant impact on the evaluated skewness, especially for large Isand Lloop. In Fig. 4(d) we map the extracted CPR as a function of Vbg. The smallest CPR amplitude is found at Vbg =−0.44 V, 013198-4 PHASE-DEPENDENT MICROWAVE RESPONSE OF A … PHYSICAL REVIEW RESEARCH 4, 013198 (2022) FIG. 5. Characteristics of the CPR as function of gate voltage Vbg. The step size inVbg is reduced close to the CNP (Vbg =−0.44 V). (a) Critical current Icand Fourier amplitudes Ak. (b) Skewness parameter Sand ratios Ak/A1. The theoretical skewness value for a short, diffusive system under ideal conditions S=0.255 is illustrated with the pink mark. (a, b) Systematic error bars in Icand Sare generated by modifying Mby ±3% and Lloop by ±5% in the CPR evaluation. The amplitudes Akfor k⩾5 are negligibly small and omitted in the figures. which we attribute to the charge neutrality point (CNP) of graphene. Here, resonance frequency modulations of only ±10 kHz can still be clearly resolved as seen in Fig. 4(e), which demonstrates the sensitivity of the microwave circuit. The CPR at the CNP, shown in Fig. 4(f), is slightly skewed and has a maximal supercurrent of Ic=6.3nA. In the following, we quantify the CPR and its skewness by two commonly used ways: (i) by the skewness parameter S=(2ϕmax/π )−1, where ϕmax is the phase maximizing the CPR to the critical current Ic[29], and (ii) by directly providing the set of Fourier amplitudes Ak[32]. The latter description is more precise, since it captures the entire CPR lineshape, whereas the S-parameter together with Icdo not uniquely characterize the CPR, but are more intuitive. In Fig. 5we employ both of these characterizations to illustrate the gate dependence of the CPR. We observe a rapid enhancement of Icup to ∼200 nA for gating toward positive voltages (n-doped), whereas toward negative voltages (p-doped) the increase is weaker and reaches only ∼50 nA as seen in Fig. 5(a). Because A1closely follows Ic,theCPR is mainly determined by the 2π-periodic sinusoidal contribution for all Vbg. However, the small additions from higher harmonics lead to a forward-skewed CPR. From Fig. 5(b) it FIG. 6. Evaluation of the shunt conductance Gs.(a)Theload quality factor Qload in logarithmic scale as a function of Vbg and ϕ, deduced from resonance curve fittings. (b) Gsin logarithmic scale obtained by using Eq. (4) with Qload and the CPR results. (c) Phase dependence of Gsfor different gate voltages. (d) Gate dependence of Gsfor phase biasing conditions ϕ=noddπ. These correspond to the local dissipation maxima, indicated by the horizontal arrows in panel (b). appears that the skewness saturates in both doping regimes with a slight reduction around the CNP. For the n-doped side, the skewness saturates around S≈0.22, whereas on the p-doped side the skewness is less pronounced, saturating around S≈0.12. The ratios Ak/A1follow the same trend. The asymmetric behavior in Icand Swith respect to Vbg are attributed to the presence of n-doped contact regions inducing additional scattering potentials. The JJ is therefore more transparent in the nnn situation compared to the npn case [29,35]. We speculate that the minimal skewness of S≈0.05 close to the CNP originates from the formation of electronhole puddles [36] in the graphene flake, which further enhance the scattering probability. VI. PHASE-DEPENDENT LOSS Having extracted the CPR from the resonance frequency shift, we now deduce the phase-dependent dissipative part of the graphene JJ; namely, the shunt conductance Gs.We can infer Gsfrom Eq. (4), in which we express the susceptance BJwith the CPR according to Eq. (6) and make use of Qload obtained from the reflectance curve analysis presented in Sec. III. 013198-5 R. HALLER et al. PHYSICAL REVIEW RESEARCH 4, 013198 (2022) From Fig. 6(a), we observe that around the 0-points (ϕ=nevenπ) the dissipation in the microwave circuit stemming from the rf SQUID is minor (Qload >200 000) for all Vbg. However, at the π-points (ϕ=noddπ), the dissipation becomes significantly larger and gate dependent with a minimal quality factor of Qload ≈9800. This behavior is reflected in Gs, which is mapped in Fig. 6(b) as a function of Vbg and ϕ. Around the 0-points, we deduce low conductance values Gs⩽0.1m−1, which refers to weak dissipation according to the parallel junction circuit model used here. In contrast, at the π-points, a pronounced Lorentzian-shaped dissipation peak develops, as seen in Fig. 6(c). The dissipation onsets are located symmetrically around the π-points and are weakly gate dependent. However, the peak heights are strongly influenced by Vbg and reach a maximal value of Gs≈10 m−1at large n-doping. Although the amplitude of the peak appears to fluctuate as a function of Vbg, the height replicates for the three different π-points measured here, as illustrated in Fig. 6(d). This demonstrates the stability of the gate-tunable potential landscape in graphene. To explain the dissipative response of the JJ, the underlying phase-dependent transport processes need to be consider, which are discussed in the next section. VII. THEORY OF ANDREEV BOUND STATES In the following we relate the CPR and the phasedependent dissipation to the microscopic concept of Andreev bound states (ABSs) formed within the JJ. Coherent Andreev reflections of quasiparticles at the graphene-superconductor interfaces lead to the formation of ABSs [37]. These quasiparticle states transfer Cooper pairs across the junction in form of counter propagating electronhole pairs [38]. Due to the electron-hole symmetry, the ABSs come in pairs; one state has negative energy E− n⩽0 and the other has positive energy E+ n=−E− n, where ndenotes a specific transport channel. The spectral gap δEquantifies the minimal transition energy between states with negative and states with positive energies. Each occupied state carries current proportional to the derivative of its energy with respect to phase. The sum over the set of all channels defines the total supercurrent [39], which can be expressed as Is(ϕ)=2π 0 n f(E± n)∂E± n ∂ϕ ,(9) where f(E± n) is a functional describing the occupation probability of the nth ABS. In equilibrium the functional is given by the Fermi-Dirac distribution. At zero temperature and in the absence of photons, all ABSs with negative energies are occupied [f(E− n)=1], whereas all ABSs with positive energies are empty [f(E+ n)=0]. In this situation the system is in the ground state and the occupation of the ABS spectrum is constant. Therefore, the supercurrent Isis free of any fluctuations. By virtue of the fluctuation-dissipation theorem [40], there is no dissipation and the effective junction shunt conductance assumes Gs→0. When finite electronic temperatures Tand/or the absorption of photons from the electromagnetic environment are considered, the situation becomes different; thermal activation and/or microwave-induced transitions will drive the system out of the ground state. The excitation-relaxation dynamics give rise to fluctuations in the ABS population, and correspondingly, in the supercurrent as well. Consequently, there is dissipation and a finite shunt conductance Gsappears [8]. When the spectral gap closes (δE→0) already small temperatures Tand small photon energies hf will trigger fluctuations. We note that the fluctuations are determined by the temperature, the photon absorption and emission rates and as well by the relaxation time τrel of a nonthermal distribution toward a thermal one, which we express in the following as the energy γ=¯h/(2τrel). In conclusion, this means that in general, both the inductive and dissipative part of a JJ depend on the ABS spectrum and the population dynamics within this spectrum. Inherent to wide junctions—like the graphene JJ investigated here—is that there are various possible transport channels leading to many ABSs and hence to a dense ABS spectrum [41]. The phase dependence of the ABS spectrum is determined by the geometry of the JJ and its material properties, i.e., the superconducting gap in the leads and the inverse transport time in the normal region that relates to the Thouless energy ET. In the ballistic transport limit ET=¯hvF/L, where vFis the Fermi velocity in the normal region and Lis the junction length. In the diffusive limit ET=¯hD/L2, where D=vFlmfp/2 is the diffusion coefficient determined by the elastic scattering mean-free path lmfp.An important characteristics of JJs is whether they are in the “short” or “long” junction limit. The former case is realized when ET, while the latter holds in the opposite limit. The condition for the short junction limit can also be expressed with the coherence length ξ, which needs to be longer than L. In the ballistic case the coherence length reads ξ=¯hvF/ and in the diffusive case ξ=√¯hD/. For JJs in the short junction limit the ABS energies are given by E± n(ϕ)=±1−τnsin2(ϕ/2), where τnis the transmission probability of the nth channel. Thus, the ABS spectrum strongly depends on the transparency distribution, which further defines the transport regime. For diffusive transport the transmission coefficients are continuously distributed following Dorokhov’s bimodal distribution [43], which describes that there are many channels with low transmission (τn→0), but also many with high transmission probabilities (τn→1). Consequently, a dense ABS spectrum emerges as illustrated in Fig. 7(a) with a spectral gap δE=2|cos(ϕ/2)|that closes (δE→0) toward the π-points and maximally opens (δE=2) toward the 0points. In long, diffusive junctions the spectral gap evolves as δE≈2×3ET|cos(ϕ/2)|[13]. To evaluate the dynamics of an ABS spectrum and translate it to lumped element quantities, we make use of theoretical works that predict the phase-dependent linear microwave response in terms of the susceptance BJand the shunt conductance Gs[3,4]. For the following theoretical analysis we consider a diffusive multichannel JJ in the short junction limit at finite temperature coupled to a photonic environment of energy hf. Note that in the experiment the photonic environment is provided by the driven microwave resonator. The justification of the above mentioned junction classification for 013198-6 PHASE-DEPENDENT MICROWAVE RESPONSE OF A … PHYSICAL REVIEW RESEARCH 4, 013198 (2022) FIG. 7. ABS spectrum and theoretical microwave response for a short, diffusive JJ. (a) Spectrum of a short JJ with multiple channels of different transparencies. (b) Microwave-induced transitions between states triggered by the absorption of a photon with energy hf. (c) The finite lifetime of states described by the relaxation rate γcauses a spectral broadening of the ABS energies and hence blurs the transition condition. (d) Theoretically predicted dissipative and inductive response: Gs(blue, left axis) and BJ(red, right axis) normalized by the conductance value at ϕ=πas a function of ϕ for different γ/ETratios. The normalization values for increasing γ/ETread: Gs(π)/GN=45,11,8.6,7,5.9,5,4.4,where GNis the normal state conductance. Here: /ET=0.1, hf/ET=0.01, and kT/ET=0.008. the graphene JJ investigated here is discussed in the beginning of Sec. VIII. We first focus on the dissipative response when long-lived excitations (γ→0) are considered. In this situation a sharp onset in Gs(ϕ) emerges as seen by the solid gray line in Fig. 7(d). The dissipation occurs in the phase range, where the spectral gap becomes smaller than the excitation energy δE⩽hf, thus allowing microwave-induced cross-gap transitions. The width and the height of the dissipation peak depends on characteristic energy scales, which are denoted in the figure caption. Not only transitions across the gap lead to dissipation; all possible absorption processes, including intraband excitations E+ n→E+ m, contribute to it, whereas the transition probability scales according to Fermis Golden rule with the available density of states [13]. Figure 7(b) depicts a microwave-induced transition of a quasiparticle from an arbitrary initial state to an available final state. The fact that the ABSs have a finite lifetime causes a spectral broadening of the energies. This results in a blurring of the transition condition (δE⩽hf) as sketched in Fig. 7(c). Therefore, increasing γ, i.e., shortening the lifetime, broadens the dissipation peak as seen by the blue lines in Fig. 7(d). Importantly, the lifetime broadening also affects the susceptance, in particular the phase conditions for BJ=0 shift away from the π-point, which is equivalent to a reduction of the CPR skewness. Note that BJfor γ→0 shown in dashed gray appears different, because it is rescaled with a large conductance value Gs(π). A representation of Fig. 7(d) without normalization is shown in the SM [23]. The influence of temperature on the microwave response is theoretically discussed, and together with experimental results, presented in the SM [23]. In short, environmental perturbations, namely, temperature and electromagnetic irradiation, cause dynamical variations in the population of ABS spectra on the timescale of the nonequilibrium occupation lifetime, which influence the susceptance BJlikewise the CPR and give rise to dissipation captured by the shunt conductance Gs. VIII. COMPARISON WITH THEORY Finally, we compare the experimental results of the graphene junction investigated here with theoretical predictions based on the assumption of a short, diffusive multichannel JJ. The condition for the short junction limit (ξ>L) seems reasonably valid, since the superconducting coherence length of similar devices is reported to be ξ≈500 nm [41,42] and the junction under investigation has a length L=400 nm. The assumed predominant diffusive transport is supported by multiple observations: (i) the small discrepancy between the experimentally determined skewness at large n-doping (S≈0.22) and the one predicted theoretically (S=0.255) [44], (ii) the lack of Fabry-Pérot oscillations in the gate dependence of the CPR presented in Fig. 5indicates suppressed ballistic transport [29], and (iii) the randomly evolving shunt conductance Gsseen in Fig. 6hints at universal conductance fluctuations, which are expected for diffusive systems. We believe that here the diffusive character of the device is stemming from scattering processes at the graphene edges, which are significant due to a small width to length ratio (W/L≈2), and hence reduce the amount of ballistic channels. One theoretical prediction, which was not explicitly pointed out above, is that the inductive and dissipative response (BJ,Gs) scale linearly with the normal state conductance GN[3,4], which is tunable with the gate voltage in our experiment. From Fig. 8(a) one can verify this relation, since the relation between the experimentally deduced values of the susceptance BJand conductance Gsobtained at ϕ=πfor all different Vbg—clearly follows a linear trend. Furthermore, the ratio BJ/Gsis the inverse loss tangent describing the quality of the Josephson inductance [14], where a larger ratio implies a more ideal behavior of the inductance. We attribute the cone-shaped spread in Fig. 8(a) around the mean ratio (BJ(π)/Gs(π)≈7) to altered ABS spectra and modified relaxation dynamics at different gate voltages. In the next step, we search for the best match between the theoretically predicted and the experimentally deduced phase-dependent microwave response by considering both the inductive and the dissipative properties of the JJ. To this end, we numerically generate sets of BJand Gswith different characteristic parameters. In particular, we vary the ratios kT/ET and γ/ETto account for a finite electronic temperature and to capture the effect of lifetime broadening. We have fixed the Thouless energy to ET=10and the photon energy to hf =/10: the first condition ensures the short junction 013198-7 R. HALLER et al. PHYSICAL REVIEW RESEARCH 4, 013198 (2022) FIG. 8. Experimental observations in comparison with theoretical predictions for a short, diffusive JJ. (a) Experimentally obtained susceptance BJversus shunt conductance Gsat ϕ=πfollows a mean ratio of ∼7 indicated with the dashed line. (b, c) Normalized measured Gs(dotted blue, left axis) and BJ(dotted red, right axis) overlaid with the normalized theoretical predictions for Gs(solid) and BJ(dashed), for which /ET=0.1andhf/ET=0.01 are fixed, but kT/ETand γ/ETare variable. The best fitting parameter ratios are indicated. For Vbg =−6 V (6 V) the normalizations read Gs(π)=0.98 m−1(5.19 m−1) for the experimental traces and for the theoretical traces Gs(π)/GN=4.4(7). limit, whereas the second one compares favourably well to the expected experimental relation between the photon energy of the resonator and the superconducting gap of the contact material. Because in our experimental setting we are only sensitive to the low-energy ABS spectrum, the precise choice of ETis irrelevant as long as the short junction limit can be assumed. In Figs. 8(b) and 8(c) we compare the normalized theoretical and experimental values for Vbg =−6 V and Vbg =6V. The experimental values Gs(blue dots) and BJ(red dots) are normalized with the shunt conductance at ϕ=π, denoted by Gs(π)[45]. Close overlap between theory and experiment can be found for both gate voltages with the same temperature (kT/ET=0.008), but distinct relaxation rates γ. At Vbg =−6 V we observe differences between the model and the experimental data even with the best match (γ/ET=0.03). This is especially evident at the flanks of the dissipation peak and the susceptance at the π-point. We attribute this mismatch to an inappropriate choice of transport regime for this gate voltage, because here the additional pn-junctions at the interfaces effectively elongate the quasiparticle trajectories. Consequently, the JJ tends to be in the long-junction limit causing a compression of the ABS spectrum. However, we stress that we observe striking agreements between the theoretical predictions with γ/ET=0.015 and the experimental data at Vbg =6 V. Apparently, the model of a short, diffusive junction reproduces simultaneously the inductive and dissipative response of the graphene JJ for this doping configuration. By evaluating the best fitting ratios kT/hf =0.8 and γ/hf =1.5 with the resonance frequency f=3.098 GHz, we deduce an electronic temperature T=120 mK and obtain a relaxation time τrel =17 ps. A similar equilibration time (τrel =7 ps) is reported for an equivalent short, diffusive Al-graphene JJ probed at mK temperatures and large n-dopings [46]. We stress that the ABS spectrum of a short, diffusive junction might not be the only spectrum, which in a similar theoretical model could reproduce the experimentally observed response. In particular, in a wide JJ the ABS spectrum can be built from quasiparticles with long and short trajectories leading to more complex ABS structures than discussed here [41]. Therefore, the relaxation time given above should be interpreted as an order of magnitude estimation. IX. CONCLUSION We have measured the reflective response of a microwave resonator inductively coupled to a graphene-based rf SQUID as a function of flux-bias and charge carrier density. We developed a concise circuit model to infer the CPR and the phase-dependent dissipation of the graphene JJ from the changes in the resonance frequency and broadening. We hereby obtain the full complex admittance of the junction, which is the key parameter to design Josephson microwave circuits. Our comprehensive investigation demonstrates the impact of the environment on the performance of JJs in terms of finite temperature and microwave photons. If the environment provides energies larger than the spectral gap, short-lived excitations appear in the ABS spectrum, which induce fluctuations in the supercurrent leading to dissipation. The comparison between the experimentally deduced microwave response at high electron density and the one predicted by theory for a short, diffusive junction model, yields striking agreement, from which we estimate a relaxation time on the order of ∼10 ps. This fast thermal relaxation makes graphene-based JJs unique candidates for highly sensitive and fast boloand calorimeters [16,17,47]. Furthermore, the device architecture and measurement protocols presented in this work are well-suited to explore the fundamental properties of other JJs, such as junctions made of 2D/3D topological insulators or Dirac and Weyl semimetals [48]. Particularly, the topological nature of these JJs can be 013198-8