Graphene-Based Josephson-Junction Single-Photon Detector

Evan D. Walsh,1,2 Dmitri K. Efetov,1,3 Gil-Ho Lee,4,5 Mikkel Heuck,1 Jesse Crossno,2

Thomas A. Ohki,6 Philip Kim,4 Dirk Englund,1,* and Kin Chung Fong6,†1Department of Electrical Engineering and Computer Science,

Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA2School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA

3ICFO-Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology,08860 Castelldefels, Barcelona, Spain

4Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA5Department of Physics, Pohang University of Science and Technology,

Pohang, 790-784, Republic of Korea6Raytheon BBN Technologies, Quantum Information Processing Group, Cambridge,

Massachusetts 02138, USA(Received 3 March 2017; revised manuscript received 1 June 2017; published 24 August 2017)

We propose to use graphene-based Josephson junctions (GJJs) to detect single photons in awide electromagnetic spectrum from visible to radio frequencies. Our approach takes advantageof the exceptionally low electronic heat capacity of monolayer graphene and its constrictedthermal conductance to its phonon degrees of freedom. Such a system could provide high-sensitivityphoton detection required for research areas including quantum information processing and radioastronomy. As an example, we present our device concepts for GJJ single-photon detectors in boththe microwave and infrared regimes. The dark count rate and intrinsic quantum efficiency arecomputed based on parameters from a measured GJJ, demonstrating feasibility within existingtechnologies.

DOI: 10.1103/PhysRevApplied.8.024022

I. INTRODUCTION

Detecting single light quanta enables technologiesacross a wide electromagnetic (EM) spectrum. In theinfrared regime, single-photon detectors (SPDs) areessential components for deep-space optical communi-cation [1] and quantum key distribution via fiber net-works [2]. At frequencies on the order of terahertz,single-photon detectors will allow the study of galaxyformation through the cosmic infrared backgroundwith an estimated photon flux

superconducting-normal-superconducting (SNS) Josephsonjunction (JJ) as a threshold sensor to detect singlephotons across an extremely wide spectrum. Since thefirst observation of the superconducting proximity effectin graphene [15], many advances have been made in thefabrication and performance of graphene-based JJs(GJJs) [16–21]. To emphasize feasibility with existingmaterials and fabrication technologies, we use measuredparameters from a GJJ to calculate the performance ofour proposed SPD. Our modeling suggests that a lowdark count probability with an intrinsic quantum effi-ciency approaching unity is achievable.

II. DEVICE CONCEPT AND INPUT COUPLING

Our proposed device is a hybrid of the calorimeterand the SNS JJ. SNS JJs have been recognized fortheir use as superconducting transistors [22] and assensitive bolometers [23–25]. The concept is to achievecontrol of the supercurrent by perturbing the Fermidistribution of the normal constituent in the junctionthrough Joule heating [26]. Compared with metalsand semiconductors, graphene is a more favorableweak-link material with its high electronic mobility,sensitive thermal response, and field-tunable chemicalpotential. When an absorbed photon raises the electrontemperature in the graphene sheet, the calorimetriceffect can trigger the JJ to switch from the zero-voltageto resistive state (see Fig. 1). We can describe thisheating with a quasiequilibrium temperature Te of thegraphene electrons as they thermalize quickly throughelectron-electron interactions [27,28]. A GJJ SPD canbe achieved by efficient photon absorption (discussed inthe present section), appreciable temperature elevation(Sec. III), and sensitive transition of the GJJ (Sec. IV).A practical challenge concerns the efficient absorption

of EM radiation by the graphene sheet. UsingBoltzmann transport, the high-frequency conductivityof monolayer graphene can be described in a Drudeform [29]. At low frequencies, graphene is essentially alump resistor with impedance depending on its dimen-sions and gate-tunable charge-carrier density. Quarter orhalf-wave resonators, stub or inductor-capacitor (LC)impedance matching networks [9], taper transformers,and log periodic antennas [30] can be employed to

achieve an input coupling approaching unity. Broadband50-Ω matching is achievable with highly doped mono-layer graphene. At optical frequencies, normal-incidencelight absorption is given by πα≃ 2%, where α is thefine structure constant, due to the universal ac conduc-tivity [31]. However, EM waves can be absorbedefficiently by evanescent wave coupling, with lightgrazing across the graphene sheet, using waveguidesand photonic crystal (PC) structures [32]. Figures 2(a)and 2(b) illustrate the proposed GJJ SPDs using imped-ance-matched resonators at microwave and infraredfrequencies, respectively.To couple microwave photons and apply a dc current

bias simultaneously, the GJJ is embedded in a four-terminal geometry as shown in Fig. 2(a). Supercurrentflows from the narrowly gapped (vertical direction inthe figure) superconducting contacts through the mono-layer graphene (orange) by the proximity effect. Theinductive chokes following these JJ contacts isolatethe microwave coupling and permit fast JJ switching.For microwave operation, one quarter-wave microwaveresonator is in contact on each side of the graphenesheet along the direction of wider separation betweenthe superconducting terminals (horizontal direction inthe figure). Together, they form a half-wave resonatorwith the dissipative graphene sheet at the microwavecurrent antinode. High microwave absorption can beachieved by impedance matching the half-wave reso-nator while the temporal mode of the single photondetermines the optimal quality factor for single-photondetection [33].For infrared photodetection, a dielectric photonic crystal

cavity can provide the impedance-matching element toreach near-unity light absorption by the graphene sheet, asillustrated in Fig. 2(b). Infrared radiation passes from aridge waveguide into a PC nanocavity, via a short sectionof PC waveguide. The evanescent cavity field couples tothe graphene monolayer, positioned over the cavity, whilethe JJ is located at the other end. Graphene can becritically coupled to the cavity [32] so that all incidentlight is absorbed. Our PC cavity design uses a thin air slotto concentrate the EM field into the graphene sheet; thisair slot (and the graphene absorber) can be deeplysubwavelength. Figure 2(c) shows the EM field concen-tration into the PC cavity air slot. Using a finite-differencetime-domain simulation tool (Lumerical), we calculate thereflected, absorbed, and scattered power for a broadbandoptical input pulse from the ridge waveguide [plotted inFig. 2(d)]. For a graphene-cavity quality factor of 800, thecalculated power spectrum indicates a peak grapheneabsorption of 93% for 1550-nm wavelength photons.Since the optical losses in silicon are comparativelynegligible, the remaining losses are due to optical scatter-ing, which can likely be eliminated by further numericaloptimization.

Superconductor Superconductor

Graphene

Te

Photon

FIG. 1. Device concept to detect single photon using agraphene-based Josephson junction.

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III. GRAPHENE THERMAL RESPONSE

Upon absorbing a single photon, the thermal response ofthe graphene electrons can be characterized by the thermaltime constant τth, heat capacity Ce, and thermal conduct-ance Gth to the reservoir. Because of the fast electron-electron interaction time, the photon energy can quicklythermalize among the graphene electrons and establish aquasiequilibrium in typically tens of femtoseconds [27,28].Therefore, both the heat injection from the photon and theinitial temperature rise can be considered instantaneouswhen compared with the thermal time constant of thegraphene electrons.This initial temperature rise is determined using the

electronic heat capacity of the monolayer graphene. In thedegeneracy regime,

Ce ¼ AγT ð1Þ[34], where A is the area of the graphene sheet and γ ¼ð4π5=2k2Bn1=2Þ=ð3hvFÞ is the Sommerfeld coefficient withkB and h being Boltzmann’s and Planck’s constants,respectively. This is in contrast to nondegenerate Diracfermions where Ce ∝ T2 [35]. In graphene, the electronFermi energy EF has a Dirac-like dispersion relation, i.e.,EF ¼ℏvFkF, where ℏ¼h=2π, vF¼106ms−1 is the grapheneFermivelocity, and kF ¼

ffiffiffiffiffiffiπn

pis the Fermimomentum,with

n being the charge-carrier density. Thus, for the typical

charge-carrier density ranging from 1010 to 1012 cm−2,EF ishigher than 10 meV so that the Fermi temperature is about135 K, justifying the use of Eq. (1). Ce can be tuned using agate voltage reaching a minimum at the charge neutralitypoint, where it is limited by the residual puddle density [36].We plot Ce in Fig. 3(a) at a carrier density n0 ¼ 1.7 ×1012 cm−2 that we will use in the modeling. Compared withthat of a metallic nanowire used in photon detection at thesame temperature [3], the electronic heat capacity of agraphene sheet of 1-μm2 areawould bemore than 3 orders ofmagnitude smaller. This dramatic improvement is due to theshrinking density of states DðEÞ ¼ 2jEjA=πðℏvFÞ2, with Ebeing the energy measured from the CNP, in monolayergraphene.We can estimate the initial temperature Tpeak of the

hot electrons by equating the integrated internal energyR TpeakT0

CeðTÞdT to the photon energy such that

Tpeak ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2hfpÞ=ðγAÞ þ T20

qð2Þ

[10], where T0 is the base temperature and fp is the photonfrequency. Figure 3(b) plots the temperature rise for variousphoton frequencies and charge-carrier densities at 0.025and 3 K. The temperature rise is higher for a lower charge-carrier density or with a higher-energy photon. Here, weassume a full conversion of the photon energy to internalenergy in the graphene electrons. This assumption

(f)(e)

−100 −80 −60 −45 −30 −15 −5 0

1530 1532 1534 1536 1538 15400

0.2

0.4

0.6

0.8

1

0.0 0.2 0.4 0.6 0.8 1.01552 1554 1556 1558 1560

0

20

40

60

80

100

(d)

(c)

quarter-waveresonator

resonator

Josephsonjunction quarter-wave

resonator

dc

dc

input

(a)

(b) Josephsonjunction

superconductingelectrodes

photoniccavity

waveguide

graphenephotoniccrystal

FIG. 2. Device schematic for (a) microwave and (b) infrared single-photon detection. (a) The graphene flake is located at thecurrent antinode of a half-wave microwave resonator for maximizing input efficiency. Two stages of inductors and capacitors form ahigh-impedance network at microwave frequency for the dc measurement of the GJJ. (b) A graphene sheet lies on top of a PC cavity toincrease its absorption through critical coupling. Light is coupled into the cavity through an in-chip waveguide. (c) Simulation results forcritical coupling. Top view of the PC structure overlaid with the mode profile jEj2 using a decibel scale normalized to a maximum of 1.(d) Spectra showing the wavelength dependence of the reflection Rr, absorption Ta, and scattered power Ts. (e) Cross-sectionaland (f) planar view of close-ups of the cavity mode jEj2 using a linear scale. The parameters of the structure are as follows: membranethickness, h ¼ 250 nm; lattice period, a ¼ 0.27λ; hole radius, r ¼ 0.31a; cavity slot width, ws ¼ 0.032λ; cavity hole shifts, d1 ¼0.365a and d2 ¼ 0.153a; waveguide width, ww ¼ 2ðW

ffiffiffi3

pa=2 − rÞ, whereW ¼ 1.04. The holes at the termination of the PC waveguide

are shifted along the x axis by s1 ¼ 0.44a, s2 ¼ 0.27a, and s3 ¼ 0.1a.

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is justified by pump-probe experiments from which it isinferred that up to 80% of absorbed photon energy iscascaded down to heat electrons [27]. This efficient energyconversion is due to the domination of the electron-electronscattering process over the coupling to the optical phonons.For lower-energy photons at microwave frequencies, theheat leakage to optical phonons is negligible as the energyscale is further below the optical phonon energy.The heat capacity also determines the root-mean-square

fluctuations in energy of the graphene sheet ΔE, shown inFig. 3(c) for n ¼ n0. This intrinsic noise of the calorimeteris given by [37]

ΔE ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiCekBT2

qð3Þ

and describes the thermodynamic fluctuations of the electronsin graphene as a canonical ensemble in thermal equilibriumwith a reservoir. ΔE sets the SPD energy resolution for ameasurement time much longer than τth. The fluctuationpower spectral density at spectral frequency f rolls off asffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4τth=ð1þ 4π2τ2thf2Þ

psince τth determines the time scale of

the energy exchange between the ensemble and reservoir. Inprinciple, widening the measurement bandwidthB can allowdetection of the sharp temperature increase due to a singlephoton, thus circumventing the limitations of calorimetryimposed by these intrinsic fluctuations [38]. However, it canalso expose the JJ to high-frequency noise and increase thethermal conductance of the radiation channel. In this report,we shall focus on the small bandwidth regime in which theSPD requires a photon energy larger than ΔE. The colorgradient orange region in Fig. 3(c) highlights the requirementof operating temperature for a given photon frequency toavoid both the energy fluctuation and thermal noise. Relatedto the energy fluctuations are temperature fluctuations withroot mean square ΔT given by

ΔT ¼ ΔE=Ce ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikBT2=Ce

q: ð4Þ

Curiously, whenCe decreases to kB,ΔT=T approaches unity[Fig. 3(c)]. The possible modification of Boltzmann-Gibbstatistics to describe the fluctuations when the degrees of

freedom in the system are close to one is beyond the scope ofthis report [39]. However, temperature fluctuations can affectJJ transitions and will be included in the performancecalculation in Sec. V.Figure 4(a) depicts the thermal pathways of a graphene

sheet [9,40]. At low temperatures, the absorbed photonenergy in the graphene electrons can dissipate through threemajor channels: electronic heat diffusion, photon emission,and electron-phonon coupling. The electron heat diffusion isthe heat-transfer channel out of the graphene sheet to theelectrodes. However, at the superconductor-graphene con-tact, Andreev reflection can suppress the thermal diffusionand quench this thermal conductance channel [40,41].Photon emission from the graphene sheet to its EM

environment can also be an effective cooling channel at lowtemperatures [42,43]. For a small measurement bandwidthB, such that B < kBT=h this radiation thermal conductanceGrad is given by

Grad ≃ r0kBB; ð5Þwhere r0 is the impedance matching factor. We can reducethis heat-transfer channel by narrowing down the meas-urement bandwidth or deliberately mismatching the normalJJ resistance away from the amplifier input impedance.However, this may trade off the photon-counting speed andJJ voltage measurement signal-to-noise ratio, respectively.For 1-MHz measurement bandwidth and r0 ¼ 1, Grad ≃1.4 × 10−17 W=K [Fig. 4(b)]. This cooling channel is onlysignificant at about 0.01Kwhen it is numerically comparableto the thermal conductance due to the EP coupling GEP.Internal energy can be transferred from electrons

to phonons by scattering [34,44]. Similar to Stefan-Boltzmann blackbody radiation, this heat transfer is a highpower law in temperature, originating from the integral ofbosonic and fermionic occupancies and density of states inthe Fermi golden-rule calculation. However, we can lin-earize this function to extract a thermal conductance GEP.The Bloch-Grüneisen temperature TBG ¼ 2ℏskF=kB,

where s ¼ 26 km s−1 is the speed of sound in graphene,

100

0 (K)

Ce

(kB/

m2 )

101

102

103

10−1

100

101

10−2

E/h

(GH

z)

T/T

T0 (K)10

−110

010

110

−210

−2

10−1

100

E/h

kBT/h

10−1

100

101

102

103

104

100

101

102

104

T0 = 25 mK

100

10−1

10−2

T0 = 3 K

1010

cm–2

1011

cm–2

cm–2

peak

T

T-

T0

(K)

(GHz)10

110

310

5

1.7 x 1012

fp

operatingregimecm

-21.7 x 10

12=

T/T

(a) (c)(b)

FIG. 3. (a) The specific heat of a 1-μm2 graphene sheet as a function of base temperature. (b) The initial temperature rise vs frequencywith electron density n0 ¼ 1.7 × 1012 cm−2 (solid, used for modeling), 1011 cm−2 (dashed), or 1010 cm−2 (dotted) at a base temperatureof 25 mK (blue) or 3 K (purple). For an infrared detector T0 ¼ 3 K will suffice but for microwave detection a lower T0 is required toacquire a noticeable temperature change. (c) Energy resolution and temperature fluctuation of a 1-μm2 graphene sheet at n0, representingthe intrinsic noise of the calorimeter.

EVAN D. WALSH et al. PHYS. REV. APPLIED 8, 024022 (2017)

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marks the temperature when the Fermi momentum of theelectrons is comparable to that of the graphene acousticphonons. For the carrier density we consider here, T < TBGand the heat transfer from electrons to acoustic phonons ingraphene is given by [9,34,40,44–46]

PEP ¼ ΣAðTδe − Tδ0Þ; ð6Þwhere Σ is the electron-phonon coupling parameter. Thepower δ is determined by the disorder in graphene. Disordereffects dominate the electron-phonon coupling when thetypical phonon momentum is smaller than 1=lMFP, theinverse of the electron mean free path (MFP). Thus for Thigher (lower) than Tdis ¼ hs=kBlMFP, the electron-phononcoupling is in the clean (disordered) limit. For cleangraphene, δ ¼ 4 and Σ ¼ π5=2k4BD2n1=2=ð15ρmℏ4v2Fs3Þ,whereas for disordered graphene, δ ¼ 3 and Σ ¼2ζð3Þk3BD2n1=2=ðπ3=2ρmℏ3v2Fs2lMFPÞ, where D≃ 18 eVis the deformation potential, ρm¼7.4×10−19kgμm−2 is themass density of graphene, and ζ is the Riemann zetafunction. In the linear response regime, when ðTe − T0Þ ≪T0, the Fourier law is recovered: PEP ≃GEPðTe − T0Þ with

GEP ¼ δΣATδ−10 as the electron-phonon thermal conduct-ance [Fig. 4(b)]. The total cooling power, including radiationand electron-phonon coupling, is PEP þ GradðTe − T0Þ. Wecan equate this rate of heat transfer to CeðdT=dtÞ such that

ΣAðTδe − Tδ0Þ þ r0kBBðTe − T0Þ≃ −Ce dTedt : ð7ÞIn the linear response regime, Eq. (7) reduces to a simpleresistor-capacitor (RC) circuit with a thermal time constantτth ¼ Ce=ðGEP þGradÞ [Fig. 4(c)]. Effectively, the weakelectron-phonon coupling in graphene helps to maintain theheat in the electrons for a longer period of time. τthdetermines the intrinsic dead time of the graphene SPDso that an infrared detector operating at a few kelvin couldcount photons at a rate up to GHz, while a microwavedetector operating at tens of millikelvin could have a countrate in the MHz range. When GEP ≫ Grad, thermal timeconstants increase rapidly as the operating temperaturedecreases because Ce ∝ T, while GEP has a higher temper-ature power law, i.e., Tδ−1 in monolayer graphene. In thisregime τth is independent of carrier density because bothCeand GEP are proportional to n1=2. In contrast, whenGEP ≪ Grad, τth decreases with decreasing T [Fig. 4(c),green and orange curves] becauseCe decreases whileGrad isconstant in T for a given bandwidth. Grad also has no ndependence so that τth ∝

ffiffiffin

p, allowing for the possibility to

quickly reset the graphene SPD through its gate voltage.We solve Eq. (7) numerically to find TðtÞ for the abso-

rption of a single 26-GHz photon by a 1-μm2 device oper-ating at 25mK and plot the result in Fig. 4(d). Because of thehigh-temperature power law in the electron-phonon heattransfer for ðTe − T0Þ ≫ T0, Te drops faster than exponen-tial, followed by a slow decay at the time constant τth.For this modeling, we employ lMFP ¼ 120 nm at n0,

deduced from the electrical transport measurements on thetypical GJJ devices that we fabricate (see Sec. IV). Thiselectrical transport mean free path would put the disordertemperature higher than the operating temperatures weconsider here. Hereafter, we will use the graphene thermalresponse in the disorder limit, i.e., δ ¼ 3.

IV. GRAPHENE-BASED JOSEPHSON JUNCTION

The detection of single photons relies on the GJJtransition from the zero-voltage to resistive state. Thisswitching is a probabilistic process described by the escaperate Γ of the JJ phase particle from the tilted washboardpotential in the resistively and capacitively shunted junction(RCSJ) model [47]. The rate of switching depends inti-mately on the JJ critical current Ic, which takes on differentforms as a function of temperature depending on whetherthe JJ is short or long and whether it is diffusive or ballistic.Superconductor-graphene-superconductor JJs have beenstudied in these different regimes [16–21]. However,experimental values, such as Ic and the parameters inthe long diffusive junction, have fallen short of theoretical

(b)

(c) (d)

graphene

e

ph

elecamp

base

(a)

EP

diff

rad

250

200

150

100

0

T (

mK

)

420t ( s)

50

T 3 EP + radT 4 EP + rad

T 3 EP + rad

T 4 EP + rad

T

τ

0 (K)10

−110

010

110

−2

10−4

10−6

10−8

10−10

T 3 EP

T 4 EP

10−3

10−1

101

103

105

Gth

[fW/(

Km

2 )]

T0 (K)10

−110

010

110

−2

T3 EP

T4 E

P

rad

1 3 5

FIG. 4. (a) Thermal diagram depicting three heat-transfer path-ways from graphene electrons (e) to the thermal reservoir (base) viagraphene EP coupling (ph), to the electrical contacts (elec) viadiffusion, or to the electrical environment, such as an amplifier(amp), via photon emission (rad). The thermal conductances in thelinear response regime are denoted as GEP, Gdiff , and Grad,correspondingly. (b) The thermal conductance vs base temperaturefor clean graphenewith a T4 EP coupling law (blue), for disorderedgraphene with a T3 EP coupling law (red), and for the radiationchannel (purple). (c) The thermal time constant τth ¼ Ce=Gth in thelinear response regime for clean [green (blue) including (excluding)radiation] and disordered [orange (red) including (excluding)radiation] graphene. (d) The transient thermal response of thegraphene sheet upon absorbing a 26-GHz microwave photon forclean (green) and disordered (orange) graphene including radiation.

GRAPHENE-BASED JOSEPHSON-JUNCTION SINGLE- … PHYS. REV. APPLIED 8, 024022 (2017)

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expectations due to impurity doping. To emphasize thefeasibility of GJJ SPDs under currently realizable param-eters, we model the device performance based on exper-imentally determined values (summarized in Table I) of theGJJ shown in the inset of Fig. 5(b) instead of analyzingthe GJJ from a purely theoretical standpoint.

The measured graphene monolayer is encapsulatedbetween atomically flat and insulating boron nitride(∼30-nm thick) using a dry-transfer technique [36]. Thedoped silicon substrate serves as a back-gate electrode. Thesuperconducting terminals consist of 5-nm-thick titaniumand 60-nm-thick niobium nitride (NbN) deposited afterreactive ion etch and electron beam deposition of 5-nmtitanium to form the etched one-dimensional contact [48].The distance between the superconducting electrodes L isabout 200 nm, forming a proximitized JJ with monolayergraphene as the weak link.The GJJ is mounted at the mixing chamber of a dilution

refrigerator with a base temperature of 25 mK. Theelectrical transport measurement is performed through astandard four-terminal configuration with the silicon sub-strate as the back gate to control the carrier density ingraphene. All dc measurement wires are filtered by a two-stage low-pass RC filter mounted at the mixing chamberwith an 8-kHz cutoff frequency. Ib is set by a dc voltageoutput through a 1-MΩ resistor while the voltage mea-surements are taken by a data acquisition board after a low-noise preamplifier with a 10-kHz low-pass filter.Figure 5(a) shows the resistance as a function of gate

voltage. CNP is observed to occur at VG ¼ −5 V. Thechosen operating carrier density n0 corresponds to VG ¼20 V for a 300-nm-thick dielectric material composed ofsilicon dioxide and hexagonal boron nitride. At this gatevoltage, Rn is measured to be 63 Ω. The mobility iscalculated as μ ¼ L=ðneRnWÞ, the mean free path aslMFP ¼ ℏμðπnÞ1=2=e, and the diffusion coefficient asDe ¼ vFlMFP=2. At n0 this yields μ ¼ 8000 cm2V−1 s−1,lMFP ¼ 120 nm, and De ¼ 0.06 m2 s−1. The slope of con-ductance σ ¼ L=ðRnWÞ versus n gives similar results formobility using μ ¼ ð1=eÞðdσ=dnÞ. lMFP ≃ 120 nm is com-parable to the channel length L≃ 200 nm so this device isin neither a purely ballistic nor purely diffusive regime.When the bias current through the JJ increases, it switchesfrom a supercurrent to a resistive state at a switching currentIs depending on the gate voltage, as shown in Fig. 5(a).Although Is and Rn vary for different gate voltages, theirproduct approaches a constant in both the highly electron-and hole-doped regimes [Fig. 5(b)]. Consistent with theexperimental results in Refs. [17,18,20], the IsRn productfor the niobium-based GJJ has a smaller value than thatcalculated from the superconductor critical temperature,probably due to impurity doping.Figure 6(a) shows typical I-V characteristics for VG ¼

20 V at 25 and 200mK, which correspond to T0 and Tpeak at26 GHz, respectively [Fig. 4(d)], measured by ramping upthe bias current at a rate of 0.1 μA=s. In order to measure theescape rate, we repeat the I-V measurement 100 times ateach base temperature and find the average switchingcurrent, hIsðTÞi [Fig. 6(a), inset], which at this sweep rateis typically about 90%of Ic. From the histogramof these 100GJJ switching events, we obtain the probability density of

TABLE I. List of device parameters to model the graphene-based Josephson-junction single-photon detector in this report.

Modeling parameters

Graphene area 5 μm × 200 nmJJ channel length L 200 nmJJ channel width W 1.5 μmElectron density n0 1.67 × 1012 cm−2

Electronic mobility μ 8000 cm2 V−1 s−1JJ normal resistance Rn 63 ΩMean free path lMFP 120 nmElectronic heat capacity CeðT0Þ 6.3 kBDisorder temperature Tdis 10.4 KBloch-Grüneisen temperature TBG 90.5 KIcðT0ÞRn product IcðT0ÞRn 223 μeVThouless energy ETh 990 μeVJJ coupling energy EJ0ðT0Þ 7.25 meVPlasma frequency ωp0ðT0Þ 156 GHzMcCumber parameter βSM 0.2NbN superconducting gap Δ0 1.52 meV

600

400

200

0

Rn

()

20151050−5−10VG (V)

3

2

1

0

IS(

A)

250

200

150

100

50

0

I SR

n(

V)

20151050−5−10VG (V)

(a)

(b)

FIG. 5. (a) Measured Rn (blue) and Is (red) as a function of VG;(b) IsRn versus VG. Inset: Optical micrograph of the measuredGJJ whose device performance parameters are used in this report.The blue colored region is the graphene channel encapsulated byh-BN and the region emphasized by orange colored lines are theNbN electrodes. The scale bar (white) is 1.5 μm.

EVAN D. WALSH et al. PHYS. REV. APPLIED 8, 024022 (2017)

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the switching current PðIsÞ for each measured temperature.Figure 6(c) showsPðIsÞ at 25 and 200mK.We can derive thephase particle escape rate Γ from the PðIsÞ data using [49]:

PðIsÞ ¼�ΓðIsÞ=

�dIdt

���1 −

ZIs

0

PðI0ÞdI0�; ð8Þ

where dI=dt is the bias current ramping speed.Figure 5(c), right panel, shows the extracted Γ data

which can be described by [50]

Γ ¼ A exp�−

ΔUkBTesc

�; ð9Þ

where ΔU ¼ 2EJ0ðffiffiffiffiffiffiffiffiffiffiffiffiffi1 − γ2JJ

p− γJJcos−1γJJÞ is the energy

barrier of the washboard potential and Tesc is the “escapetemperature” which sets the energy scale competing withΔU for the phase particle to escape from the washboardpotential. Here, EJ0 ¼ ℏIc=ð2eÞ is the Josephson couplingenergy and γJJ ¼ Ib=Ic is the normalized bias current.In the thermal activation (TA) regime, Tesc is simply thetemperature of the device while in the macroscopic quan-tum tunneling (MQT) regime

Tesc ¼ ℏωp=�7.2kB

�1þ 0.87

Q

��; ð10Þ

where Q ¼ ωpRnCJJ is the JJ quality factor with ωp ¼ωp0ð1 − γ2JJÞ1=4 being the JJ plasma frequency, ωp0 ¼½2eIc=ðℏCJJÞ�1=2 being the zero-bias JJ plasma frequency,and CJJ being the effective junction capacitance. While thegeometrical capacitance between the superconducting ter-minals is estimated to be subfemtofarad lower bounded bythe parasitic capacitance to the substrate, there is aneffective capacitance due to electronic diffusion given byCJJ ¼ ℏ=RnETh [16]. ETh ¼ ℏDe=L2 is the Thoulessenergy [51], where De ¼ vFlMFP=2 is the diffusion con-stant. ETh is the characteristic energy scale of a diffusiveGJJ and is estimated to be ∼11 K for the device charac-terized for our modeling, which gives CJJ ∼ 11 fF. Usingthe effective capacitance, we estimate ωp0 of this GJJ to be2π × 156 GHz. The Stewart-McCumber parameter βSM ¼Q2 is about 0.2. This implies that the GJJ should be anoverdamped junction with its phase particle retrappedquickly after switching to the resistive state. However,the measured junction is hysteretic probably because theresistive state bias current self-heats the junction at lowtemperatures [41].Using ωp and Q above, we can estimate this GJJ to be in

the MQT regime for temperatures below 470 mK [16].With the prefactor A of Eq. (9) given by

A ¼

8>><>>:

ωp2π

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 1

4Q2

q− 1

2Q

�ðTAÞ

12ωpffiffiffiffiffiffiffiffiffiffi3ΔU2πℏωp

qðMQTÞ;

ð11Þ

we fit the extracted Γ values with Ic as the only fittingparameter (all other parameters depending on Rn, CJJ, andIb) and find the fitting consistent with the MQT process asexpected. The solid lines in Fig. 6(c) show the MQT ΓwithIc equal to 3.57 and 3.43 μA at 25 and 200 mK, respec-tively. The experimentally determined Γ as a function of Iband IcðTÞ will help us to calculate the SPD performancein Sec. V.

V. PHOTON-DETECTION PERFORMANCE

The GJJ can be set to detect single photons by setting abias current Ib [as an example, Ib ≃ 3.03 μA in Fig. 6(c)]below the switching current, so that when photons raise thegraphene electron temperature, the GJJ may switch tothe resistive state as its critical current quenches. As theelectrons cool, the GJJ will switch back to the super-conducting state for a junction with no hysteresis. For ahysteretic junction, the detector can be reinitialized usingthe bias current or gate voltage, as both the switching andretrapping current depend on the gate voltage. The GJJ canfunction as an SPD when either the MQTor the TA process

10.50T (K)

–4

–2

0

2

4C

urre

nt (

A)

P

–200 0 200Voltage ( V)

0 1

Is (T )

Vamp

3.10

3.05

3.00

2.95

2.90

Ib(

A)

3020100

3.15

Ib

25 mK

200 mK

〈Is〉

〈Is〉

10–2

–1

–1

(Hz)10

010

2

0.4

0.2

0.0

(a) (c)

(b)

FIG. 6. (a) Measured GJJ I-V characteristics for electrontemperature at T0 ¼ 25 mK (green) and Tpeak ¼ 200 mK(orange) with a carrier density of 1.7 × 1012 cm−2. Inset: Themeasured average switching current vs temperature. (b) The JJvoltage and the expected voltage noise from an amplifier at1-MHz bandwidth. The blue dotted line depicts, for a given biascurrent in a photon event, the order of magnitude of the JJ voltageV ¼ IbRn. (c) The switching probability (left) and escape rate(right) of the phase particle as a function of the JJ current bias.The solid line in the left panel is the best-fit probabilitydistribution to the data assuming the escape mechanism isMQT. Solid lines in the right panel are given by Eq. (9) withIc ¼ 3.565 μA (25 mK) and Ic ¼ 3.425 μA (200 mK).

GRAPHENE-BASED JOSEPHSON-JUNCTION SINGLE- … PHYS. REV. APPLIED 8, 024022 (2017)

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dominate the JJ transition because its operation depends onthe change of Γ as the electron temperature increases.However, the phase diffusion process [16] is not desirablebecause the finite subgap resistance can diminish thesignal-to-noise ratio of the GJJ voltage signal readoutsignificantly.The voltage drop across the GJJ upon detection is

given by IbRn, which is about 190 μV for the measureddevice [marked by the vertical dashed line in Fig. 6(a)].The inaccuracy in measuring this voltage with a shortaveraging time will be dominated by the amplifier noisebecause the Johnson noise spectral density

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4RnkBT

pis

only about 10 pV=Hz1=2, while the shot noise RnffiffiffiffiffiffiffiffiffiffiffiffiffiF2eIb

pis about 33 pV=Hz1=2 for Ib ¼ 3.03 μA and the Fano factorF ≃ 0.29 [52,53]. Using a typical low-noise voltageamplifier with a power spectral density of 1 nV=Hz1=2,the amplifier voltage noise at a 1-MHz measurementbandwidth is 1 μV. The signal-to-noise ratio of the GJJreadout is large as depicted in Fig. 6(c), where two well-separated Gaussian peaks centered at Vb ¼ 0 (“0” nophoton state) and Vb ¼ IbRn ¼ 190 μV (“1” photon state)with the FWHM corresponding to the amplifier voltagenoise. A Josephson coupling energy that gives an IcRnproduct of about 100 μV will be sufficient for making aGJJ SPD.The performance of a GJJ SPD can be calculated by the

probabilistic JJ transition at an elevated temperature. Wechoose to benchmark the GJJ SPD by its intrinsic quantumefficiency η and dark count probability Pdark for single-photon detection at 26 GHz with an improved performanceexpected for higher-energy photons. We note that forinfrared photons, the peak temperature will be higher thanthe superconducting gap energy of niobium nitride, result-ing in heat leakage that can reduce the peak temperatureand shorten the duration of the heat pulse. A calculation ofthe spectral current [22] will probably be required tounderstand the GJJ under high-energy photon excitationand is beyond the scope of this work. For microwave

photons, the temperature rise is much smaller than the gapenergy and the detector remains in the quasiequilibriumregime.We can calculate Pdark using Γ. The total escape prob-

ability in a measurement integration time tmeas is given by1 − exp½− R tmeas0 ΓðτÞdτ�. In the absence of incident photons,Γ is equal to the dark count rateΓdark andwould be a constantΓ0ðT ¼ T0Þ if the temperature were fluctuation free. Toinclude the effect of temperature fluctuation, we use theaveraged Γ, i.e., hΓi ¼ R∞0 dTpðTÞΓ(IcðTÞ), where pðTÞ isa Gaussian distribution centered at T0 with a standarddeviation of ΔT. Since Γ increases quickly as a functionof T, hΓi ≥ Γ0. Pdark equals 1 − expð−hΓitmeasÞ and can besignificantly higher than 1 − expð−Γ0tmeasÞ depending onthe size of dIc=dT and ΔT=T0. At T0 ¼ 25 mK and Ib ¼3.28 μA (γJJ ≃ 0.91), Pdark for tmeas ¼ 1 μs is about 0.07using an estimatedΔT=T0 ¼ 0.8 [Fig. 3(c)] and the fitted Γfrom Sec. IV.Upon photon absorption, the electron gas heating and the

subsequent cooling from Tpeak result in a time-dependent Γthat can be used to calculate η. Using the TeðtÞ in Fig. 4(d),IsðTÞ in Fig. 6(a), and fitted ΓðIcÞ, we calculate both thecritical current and Γ as they recover to their nominal valuesafter the photon incidences at t ¼ 0 [see Fig. 7(a)]. Wecalculate the intrinsic quantum efficiency η of the GJJ SPDfrom

η ¼ 1 − exp�−Z

tmeas

0

Γ(IcðτÞ)dτ�: ð12Þ

The detection efficiency increases with the measurementtime, but so doesPdark. To balance between these competingeffects, we benchmark the GJJ SPD performance by takingthe measurement integration time to be 1 μs such thatΓ(IcðtmeasÞ)≃ 2hΓiT¼T0 . Figure 7(b) plots η and Pdark asa function of γJJ for three different dIc=dT values. With thedIc=dT of themeasuredGJJ at−1.1 μA=K [measured in thelinear region of hIsðTÞi above about 100 mK in Fig. 6(a)],

(c)(b)(a)

0.001

0.01

0.1

1

Pro

babi

lity

1.000.950.900.850.80JJ

0.001

0.01

0.1

1

10−12

10−9

10−6

10−3

100

Dark Count Probability

10−3

103

Dark Count Rate (Hz)

dIc /dT

−1−2−3

3.6

3.4

3.2

3.02.01.00.0

10-3

101

105

109

MQ

T(H

z) (µA/K)

dIc /dT

−1−2−3

(µA/K)

JJ = 0.85

Red:Blue:

I c(

A)

1.50.5t ( s)

10−5

10−1

101

105

dIc /dT

−1−2−3

(µA/K)

FIG. 7. (a) Change of critical current and phase slipping probability due to a single microwave photon, impinging at t ¼ 0 s andT0 ¼ 0.025 K, deduced from the graphene thermal properties and measured GJJ parameters. (b) Intrinsic efficiency η and dark countprobability at a 1-MHz count rate versus bias current. The vertical gray dashed line indicates the bias current for which η > 0.99 for themeasured dIc=dT. (c) Trade-off between the intrinsic quantum efficiency and the dark count rate at various bias currents from (b). Graycircle corresponds to gray line from (b).

EVAN D. WALSH et al. PHYS. REV. APPLIED 8, 024022 (2017)

024022-8

we can set γJJ ¼ 0.91 (vertical gray dashed line) to reach anintrinsic quantum efficiency > 0.99 while maintainingPdark ≃ 0.07.We plot the trade-off between η and Pdark in Fig. 7(c) by

eliminating the common parameter γJJ of Fig. 7(b).Favorable SPD performance occurs in the plateau regionwhere η approaches unity while Pdark remains small. ThisSPD regime is feasible within the parameters of existingGJJs that we can fabricate. The operating point circled inFig. 7(c) corresponds to the same bias current and inte-gration time setting as the vertical dashed line in Fig. 7(b).To optimize the SPD performance or to operate at a

different frequency, we argue that the design should focuson the dependence of the GJJ critical current on temper-ature. Although a lower operating temperature and asmaller area of monolayer graphene can enhance thetemperature rise due to a smaller electronic heat capacity,the rate of improvement can quickly diminish because theheat loss from the electrons is dominated by the coupling tophonons and is faster than exponential when Te − T0 > T0.The time integral of Γ in Eq. (11) can only increasemarginally to enhance the intrinsic quantum efficiencyinsignificantly. However, the phase particle escape rate ofthe JJ can increase by orders of magnitude with increaseddIc=dT because of the exponential dependence of ΔU inboth MQT and TA processes, as suggested by the initialnumerical values of Γ in Fig. 7(a). This behavior contrib-utes drastically to the SPD intrinsic efficiency, allowing alower JJ current bias to further reduce the dark countprobability as shown in Figs. 6(b) and 6(c) by the dashedand dotted lines for double and triple the measured dIc=dT,respectively. Merely doubling dIc=dT lowers Pdark corre-sponding to η ¼ 0.99 by 5 orders of magnitude while an8-orders-of-magnitude improvement is possible by triplingdIc=dT. The operation of a GJJ SPD at a higher temper-ature up to 4 K is possible using a ballisticlike GJJ such asthat demonstrated in Ref. [21] with dIc=dT on the orderof −1 μA=K.

VI. CONCLUSION

In conclusion, we have introduced a device schemefor ultrabroadband single-photon detection based on theextremely small electronic specific heat in graphene, whichresults from the vanishingly small electronic density ofstates near the Dirac point and its linear band structure. Ourmodel analysis shows that single-photon detection,together with very small dark count rates, is possibleacross a wide spectral region, from the microwave tonear-infrared light. Efficient light absorption into thegraphene absorber can be achieved by impedance-matchingstructures, such as metallic or dielectric resonators consid-ered here for microwave and optical frequencies. Usingexperimental parameters from a fabricated device, wemodel that a GJJ SPD operating at 25 mK could reach asystem detection efficiency higher than 99%, together with

one-tenth dark count probability for a 26-GHz photon. Thisdevice performance should improve for higher photonenergies. Inductive readout [8,23–25] can be used in thefuture to increase the SPD operation bandwidth by avoid-ing Joule heating when the GJJ switches to a resistive state.We have only explored a small set of possible parameters inthis report. Further optimization of the GJJ SPD willdepend on application-specific performance trade-offs.Heat leakage to the superconducting electrodes as wellas position and time dependence of the heat propagationwill need to be included in the future for a more realisticprediction and a better understanding of the fundamentallimits of the GJJ SPD. The rapid progress in integratinggraphene and other van der Waals materials with estab-lished electronics platforms, such as complementary metal-oxide-semiconductor (CMOS) chips, provides a promisingpath towards single-photon-resolving imaging arrays,quantum information processing applications of opticaland microwave photons, and other applications benefitingfrom quantum-limited detection of low-energy photons.

ACKNOWLEDGMENTS

We thank L. Levitov, S. Guha, B.-I. Wu, J. Habif, andM. Soltani for valuable discussions. E. D.W. was supportedin part by the Office of Naval Research (No. N00014-14-1-0349). The work at Harvard was supported by the Scienceand Technology Center for Integrated Quantum Materials,NSF Grant No. DMR-1231319. Numerical simulationswere supported in part by the Center for Excitonics, anEnergy Frontier Research Center funded by the U.S.Department of Energy, Office of Science, Office ofBasic Energy Sciences under Award No. DE-SC0001088. The works of T. A. O. and K. C. F. werefunded by the Internal Research and Development inRaytheon BBN Technologies.

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