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Jesse Balgley, Jinho Park, Xuanjing Chu, Ethan G. Arnault, Martin V. Gustafsson, Leonardo Ranzani, Madisen Holbrook, Yangchen He, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Daniel Rhodes, Vasili Perebeinos, James Hone, Kin Chung Fong

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[Crystalline superconductor-semiconductor Josephson junctions for compact superconducting qubits](https://mdr.nims.go.jp/datasets/52804611-dabb-4dca-abf9-6690a0231203)

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Crystalline superconductor-semiconductor Josephson junctions for compact superconducting qubitsJesse Balgley,1, ∗ Jinho Park,1, ∗ Xuanjing Chu,2, ∗ Ethan G. Arnault,3 Martin V. Gustafsson,4Leonardo Ranzani,4 Madisen Holbrook,5 Yangchen He,6 Kenji Watanabe,7 TakashiTaniguchi,7 Daniel Rhodes,6, 8 Vasili Perebeinos,9 James Hone,1 and Kin Chung Fong4, †1Department of Mechanical Engineering, Columbia University, New York, NY 10027, USA2Department of Applied Physics and Mathematics, Columbia University, New York, NY 10027, USA3Department of Electrical Engineering and Computer Science,Massachusetts Institute of Technology, Cambridge, MA 021394RTX BBN Technologies, Quantum Engineering and Computing Group, Cambridge, MA 02138, USA5Department of Physics, Columbia University, New York, NY 10027, USA6Department of Materials Science and Engineering, University of Wisconsin–Madison, Madison, WI 53706, USA7National Institute for Materials Science, 1-2-1 Sengen, Tsukuba, Ibaraki 305-0044, Japan8Department of Physics, University of Wisconsin–Madison, Madison, WI 53706, USA9Department of Electrical Engineering, University at Buffalo,The State University of New York, Buffalo, New York 14260, USA(Dated: August 11, 2025)The narrow bandgap of semiconductors allows for thick, uniform Josephson junction barriers, potentiallyenabling reproducible, stable, and compact superconducting qubits. We study vertically stacked van der WaalsJosephson junctions with semiconducting weak links, whose crystalline structures and clean interfaces offera promising platform for quantum devices. We observe robust Josephson coupling across 2–12 nm (3–18atomic layers) of semiconducting WSe2 and, notably, a crossover from proximity- to tunneling-type behaviorwith increasing weak link thickness. Building on these results, we fabricate a prototype all-crystalline merged-element transmon qubit with transmon frequency and anharmonicity closely matching design parameters. Wedemonstrate dispersive coupling between this transmon and a microwave resonator, highlighting the potential ofcrystalline superconductor-semiconductor structures for compact, tailored superconducting quantum devices.I. INTRODUCTIONState-of-the-art transmon qubits rely on Josephson junc-tions (JJs) with amorphous insulating weak links. However,their large bandgaps necessitate extremely thin barriers, com-promising uniformity, reproducibility, and precise control ofjunction critical current which controls qubit frequency andcharge sensitivity. Pinholes and grain boundaries in these ma-terials further constrain JJ sizes, and transmons based on suchJJs typically require large footprint shunt capacitors for idealtransmon capacitances [1].In contrast, semiconducting weak links offer precise controlof junction properties over a wide range [2], and their reducedbandgaps allow for thicker barriers, which are less prone topinhole formation compared to typical insulating barriers [3].This enables larger-area JJs with built-in capacitance, forminga smaller-footprint “merged-element” transmon (MET) [4].Recent advances in vertical JJs with semiconducting weaklinks have achieved fine control of junction critical currentover a wide range [5] but relied on amorphous barriers thathost high densities of microwave-active two-level systems, aprimary factor limiting superconducting qubits lifetimes [6].Crystalline materials, characterized by their lack of grainboundaries, atomically pristine interfaces, and reduced two-level fluctuator densities [7], offer a promising alternativeplatform for high-quality quantum devices. Beyond epitaxial∗ These authors contributed equally to this work.† Present address: k.fong@northeastern.edu, Northeastern Universitycrystalline materials [8–10], layered van der Waals (vdW) ma-terials, which can be peeled apart into atomically thin layers,present distinctive features to create low-loss and novel devices[11, 12]. For example, they may be readily encapsulated asprotection against oxidation, and can be “stacked” together incomplex heterostructures of different vdW materials [13] withhighly ordered internal interfaces [14]. Their layered struc-ture facilitates unprecedented uniformity and reproducibilitycompared to deposited thin films. To date, however, studiesof vdW JJs have primarily focused either on vertical junctionsincorporating weak links ≤ 6 vdW layers thick [15–20], or onlateral junctions with monolayer vdW weak links [21–25].In this work, we systematically investigate DC electronictransport in vertical vdW JJs with semiconducting weak linksin the 2–12 nm thickness range, aimed at advancing a materialsplatform for transmon qubits. As we vary the thickness of thesemiconductor, we observe a crossover from proximity-type totunneling-type behavior around 7 atomic layers. These find-ings inform the optimal semiconductor weak link thicknessesfor compact METs. As a proof of concept, we fabricate aprototype all-vdW MET and demonstrate its operation as atwo-level system via dispersive readout.II. EXPERIMENTAL RESULTSWe study vertical crystalline JJs consisting of NbSe2, ananisotropic 𝑠-wave vdW superconductor (critical tempera-ture 7.2 K, superconducting gap 1.3 meV [26]), sandwich-ing WSe2, a vdW semiconductor with a 1.2 eV indirect bulkbandgap [27]. This is ∼ 6 times smaller than the bandgap of2(d)(c)0 5 10 15 20WSe2 layers (#)100103106RnA (Ω·µm2 )10-3100103j s0 (µA/µm2 )20 μm(a)0 100 2000123V (mV)I (µA)RnIs0Ir0hBN NbSe2 WSe2(b)VFIG. 1. Josephson junction characterization. (a) Optical image of anNbSe2/WSe2/NbSe2 JJ encapsulated in hBN flakes. (b) 𝐼-𝑉 curve ofa JJ with a 9-layer-thick WSe2 weak link, with dashed lines markinghow 𝐼𝑠 , 𝐼𝑟 , and 𝑅𝑛 are obtained. Blue (red) trace indicates increasing(decreasing) bias current. (Note, the discrepancy between the retrap-ping current seen here and in Fig. 2(c) is due to varying sweep rates[29].) Inset, schematic of a vdW JJ. (c) Switching current density at𝑇 = 20 mK, 𝑗𝑠0, and (d) normal-state-resistance–area product, 𝑅𝑛𝐴,as functions of WSe2 weak link thickness. The black dashed linein (c) is an exponential fit to the data to guide the eye. The solidblack line in (d) results from the resistivity simulations within theThomas-Fermi model.aluminum oxide and ∼ 5 times smaller than that of the vdWinsulator hexagonal boron nitride (hBN), which acts as a tun-nel barrier even in the single-layer limit [28]. The lower tunnelbarrier height in WSe2 allows for a thicker, more uniform weaklink. Consequently, the junction critical current can be con-trolled in small increments by changing the number of vdWlayers in the WSe2 weak link. We exfoliate flakes of WSe2and NbSe2 from bulk crystals and build stacks with them afteridentifying flakes with suitable geometries (see Appendix A).We encapsulate the JJs in flakes of hBN to protect them fromsurface contamination and oxidation (Fig. 1(a)).In total, we study DC electronic transport in twenty JJs withWSe2 thicknesses ranging from 3 to 18 layers. An example𝐼-𝑉 curve of a JJ with a 9-layer-thick WSe2 weak link takenat temperature 𝑇 = 20 mK is shown in Fig. 1(b). Sweeps ofincreasing (blue trace) and decreasing (red trace) bias currentallow us to extract key JJ parameters: the switching current𝐼𝑠0, the retrapping current 𝐼𝑟0, and the normal state resistance𝑅𝑛. The subscript “0” indicates values at 𝑇 = 20 mK. InFig. 1(c) & (d) we plot the switching current density 𝑗𝑠0 =𝐼𝑠0/𝐴 and the product of the normal state resistance and JJarea, 𝑅𝑛𝐴, respectively. 𝐴 is defined by the overlapping areaof the two NbSe2 flakes sandwiching the WSe2 weak link. Both𝑗𝑠0 and 𝑅𝑛𝐴 obey exponential dependencies over roughly sixorders of magnitude as a function of the WSe2 thickness. Thetrends in these quantities are robust despite some scatter in thedata, which we attribute to uncertainty in the JJ geometry (seeAppendix A). As we will show, the trend of 𝑗𝑠0 versus WSe2thickness enables accurate prediction of the 0 → 1 transitionenergy of a transmon qubit.To illustrate the impact of the semiconductor weak linkthickness on the electronic properties of vertical JJs, inFig. 2(a) & (b) we plot example DC 𝐼-𝑉 transport charac-teristics of current-biased junctions with 7-layer-thick (7L)and 13-layer-thick (13L) WSe2 weak links, respectively. Blue(red) traces indicate increasing (decreasing) bias current, alltaken at 𝑇 = 20 mK. In the 13L device, the retrapping cur-rent (𝐼𝑟0 = 0.2 µA) is two orders of magnitude smaller thanthe switching current (𝐼𝑠0 = 24 µA) red (see Appendix A),whereas those of 7L (574 µA and 830 µA, respectively) arewithin a factor of two of each other [30]. The ratio 𝐼𝑠0/𝐼𝑟0provides a measure of the JJ 𝐼-𝑉 hysteresis, which we plotfor all devices in Fig. 2(c). In the range of 3–7 layers,𝐼𝑠0/𝐼𝑟0 is saturated around 1. Above 7 layers, the hystere-sis increases exponentially with WSe2 thickness. 𝐼𝑠0/𝐼𝑟0 ≈ 1corresponds to minimal hysteresis typical of proximity-typeJJs [31]. Though proximity JJs possesses a conductive weaklink between the superconducting electrodes, which in princi-ple should yield a nonhysteretic 𝐼-𝑉 , finite circuit capacitancesand Joule heating in the normal state can contribute to finitehysteresis [31]. On the other hand, pronounced hysteresis ischaracteristic of tunneling-type JJs with insulating weak links[32, 33]. These two different hysteretic behaviors suggest thatthere is a crossover from proximity- to tunneling-type JJ as theWSe2 thickness increases.Further evidence of such a crossover is provided by the tem-perature dependence of the switching and retrapping currents,measured in fourteen devices with WSe2 thicknesses rangingfrom 3 to 13 layers. In Fig. 3(a), we plot the temperature-dependent switching current 𝐼𝑠 (which we use as a proxyfor the critical current 𝐼𝑐), multiplied by 𝑅𝑛. The solidblack line represents the Ambegaokar-Baratoff (A-B) limit ofan ideal tunnel junction: 𝐼𝑐𝑅𝑛 = (𝜋/2)Δ tanh [Δ/2𝑘𝐵𝑇]/𝑒[34]. Here, Δ is the superconducting gap at temperature𝑇 , for which we use the BCS interpolation formula Δ =Δ0 tanh [(𝜋𝑘𝐵𝑇𝑐/Δ0)√︁𝑇𝑐/𝑇 − 1] [35]. Δ0 = 1.764𝑘𝐵𝑇𝑐 ≈1 meV is the zero-temperature gap value proportional to thejunction critical temperature 𝑇𝑐 = 6.8 K. Systematically, wefind that the 𝐼𝑠𝑅𝑛 products of all junctions with weak linkthicknesses ≤ 7 layers (light to dark blue circles) exceed theA-B limit, whereas those with weak links > 7-layers thick (yel-low to red circles) undershoot the predicted value. The 𝐼𝑐𝑅𝑛product of a JJ exceeds the A-B limit when the transparencyof the weak link is finite, i.e., in a proximity-type junction[36–38]. On the other hand, 𝐼𝑐𝑅𝑛 in a JJ with an ideal tunnelbarrier should correspond precisely to the A-B limit. However,superconducting proximity effects or a finite boundary resis-tance at the superconductor-weak link interface can lead to areduction in 𝐼𝑐𝑅𝑛 to a value below the A-B limit [39–41]. Thegrouping of 𝐼𝑠𝑅𝑛 above and below the A-B limit around 7 lay-ers of WSe2 again suggests either proximity- or tunneling-typebehavior occurs, depending on the WSe2 thickness.3(a) (b) (c)0 5 10 15 20100102104I s0/I r0WSe2 layers (#)-200 0 200I (μA)-202V (mV)13L WSe2tunneling-type-1000 0 1000-6-3036V (mV)7L WSe2proximity-typeI (μA)FIG. 2. Hysteresis in a vertical superconductor-semiconductor Josephson junction. (a), (b) DC 𝐼-𝑉 curves for current-biased JJs with 7-layer-thick and 13-layer-thick WSe2 weak links, respectively. Blue (red) traces indicate sweeps from negative to positive (positive to negative) currentbias. (c) 𝐼𝑠0/𝐼𝑟0 as a function of WSe2 weak link thickness.We corroborate the crossover between two Josephson junc-tion regimes by investigating the temperature-dependent re-trapping current, plotted in Fig. 3(b) normalized by the base-temperature switching current as 𝐼𝑟/𝐼𝑠0. In JJs with WSe2weak links thinner than 8 layers (light to dark blue circles)𝐼𝑟/𝐼𝑠0 monotonically decreases with increasing temperature.For ≥ 8-layer-thick WSe2 (yellow to red circles) 𝐼𝑟/𝐼𝑠0 is non-monotonic, wherein the retrapping current, constant at lowtemperatures, first rises as the temperature increases beforeeventually decreasing for temperatures above 5 K.We may understand the nonmonotonic temperature depen-dence of the retrapping current in tunneling-type JJs by re-calling that the current-voltage relation of a JJ is related tothe evolution of the difference in the phases of the order pa-0 2 4 6T (K)I r/I s00101234I sRn (mV)A-B limit3L5L5L5L7L7L8L9L10L11L12L12L12L13L(b)(a)FIG. 3. Temperature dependence of Josephson junction properties.(a), Temperature dependence of 𝐼𝑠𝑅𝑛. (b), Temperature dependenceof retrapping current 𝐼𝑟 normalized by 𝐼𝑠0.rameter between the two superconductors. In the resistivelyand capacitively shunted junction (RCSJ) model [32, 33], thecurrent-biased junction is modeled as a parallel combination ofan ideal Josephson element, a capacitor, and a resistor, and thephase difference across the junction behaves as a particle sub-ject to a tilted washboard potential. The damping of the motionof this “phase particle” is inversely proportional to the junctionresistance [32, 33]. In a tunneling-type JJ with low damping(high junction resistance), the particle can escape from thepotential well when the current bias is increased, leading toa transition from the zero-voltage state to the resistive state.However, due to its inertia, the particle may not retrap backinto the zero-voltage state unless the current is reduced signif-icantly, creating hysteresis. At low temperatures, the damp-ing is constant and the tunnel junction exhibits pronouncedhysteresis. According to the quasiparticle tunneling modelof Chen, Fisher, & Leggett [42], as the temperature increases,thermal excitation facilitates quasiparticle tunneling across thejunction, increasing the effective damping. This increases theretrapping current and reduces the hysteresis. Close to thecritical temperature, the superconducting gap diminishes sub-stantially, reducing both the critical and retrapping currents.This process explains the qualitative behavior of 𝐼𝑟/𝐼𝑠0 in JJswith WSe2 thickness ≥ 8 layers. In contrast, in proximity-typeJJs where the weak link can readily conduct quasiparticles be-tween the superconducting electrodes, the change in dampingdue to thermal activation of quasiparticles is negligible. Thus,the retrapping current in proximity-type JJs has a monotonictrend as a function of temperature, as seen for WSe2 weak linksthinner than 8 layers.In NbSe2/WSe2/NbSe2 JJs, a crossover from proximity-typeto tunneling-type behavior with weak link thickness is plausi-ble if we consider the relative band alignment between the twomaterials. Ab initio calculations estimate a difference betweenthe NbSe2 work function and the ionization potential of WSe2of 1.07 eV, with the Fermi energy of NbSe2 intersecting the va-lence band of WSe2 [43, 44]. Such a band alignment leads to aninterfacial charge transfer between the two materials [45, 46],which can form an accumulation region of hole-doped WSe2.When the WSe2 weak link is thinner than the accumulation4METΦIfluxrf gate(c)(b)(a) (e) fdrive (GHz)|S21| (a.u.)f01f02/25.1 5.2 5.3 5.4051015drivepower(dB)(f)AlNb20 μmNbSi(d)ΦJJ1JJ210 μm0 5 10 15 20weak link layers (#)100102104f 01 (GHz)WSe2(calculated)hBN(calculated)WSe2(measured)-60 -30 0power (dB)0123f -  6.48 GHz (MHz)-0.5 0 0.5flux bias (mA)-35-30-25-20|S21 | (dB)FIG. 4. van der Waals merged-element transmon. (a) Calculated MET 0 → 1 transition frequency 𝑓01 for varying numbers of WSe2 (greencircles) and hBN (pink crosses) weak links, and the measured 𝑓01 (blue star) for the MET shown in (c). (b) MET circuit schematic. (c) Opticalimage of a vdW MET. Capacitive coupling wires, flux bias line, and rf drive line are false-colored orange, green, & red, respectively. (d)Zoom-in of the vdW stack in (c) showing the SQUID flux loop shaded blue with two JJ areas outlined in white. (e) Single-tone spectroscopyas a function of readout power (left) and flux bias current at low microwave power (right). The vertical white dashed line in the left panelindicates the power at which readout was performed in the right panel. (f) Two-tone spectroscopy. Drive pulse power increases from purple toviolet to pink to gold traces, which are vertically offset for clarity.regions formed by each NbSe2 electrode, the entire weak linkwill be doped and conductive, forming a proximity-type JJ.When the WSe2 is thicker than the accumulation regions, thecentral layers will remain undoped, providing a tunnel barrierfor the JJ. In the superconducting state, we infer from the datathat ≈ 3 layers (2 nm) of WSe2 are doped by each NbSe2 elec-trode, leading to proximity-type behavior in JJs with WSe2weak links thinner than 7 layers, and tunneling-type behav-ior when the WSe2 is thicker than 7 layers. While this chargetransfer can give way to a superconducting proximity effect be-tween the NbSe2 and the doped WSe2 layers, and subsequentlythe reduction in 𝐼𝑐𝑅𝑛 demonstrated in Fig. 2(c), the tunnelinggap alone does not dictate the amount of band bending andproximitization. Instead, the band alignment between the twomaterials determines whether the Fermi energy of the super-conductor will intersect the semiconductor bands or create apotential which induces band bending, as seems to be the casebetween NbSe2 and WSe2.The relative band alignment between the two materials alsodictates the strength of the exponential dependence of 𝑗𝑠0 and𝑅𝑛𝐴 on weak link thickness. We employ the Thomas-Fermimodel to calculate the normal state resistance for this materialsystem, the result of which is shown in Fig. 1(d) (see AppendixB). The slope of the resistance in the semi-logarithmic plot ofFig. 1(d) is determined by the Fermi level position inside thetunnel junction, which we determine to be 28.4 meV above thevalence band of WSe2 based on the fit to the data. Consider-ing the individual material properties and relative band align-ments, we find quantitative agreement between the measuredand calculated tunneling resistance in NbSe2/WSe2/NbSe2 JJs.Having characterized the effects of semiconducting weaklink thickness on the properties of a vdW JJ, we predict theproperties of an NbSe2/WSe2/NbSe2-based MET to determinewhat weak link thicknesses will result in a qubit 0 → 1 tran-sition frequency 𝑓01 in the range of 1–15 GHz with sufficientanharmonicity between subsequent qubit energy levels. Us-ing the exponential fit of 𝑗𝑠0 from Fig. 1(c) and the expectedgeometric junction parallel-plate capacitance, we calculate theexpected MET frequencies of JJs with 1–20 layers of WSe2weak link (Fig. 4(a)). Here, we use 𝑓01 =(√8𝐸𝐽𝐸𝐶 − 𝐸𝐶)/ℎ,where ℎ is Planck’s constant, 𝐸𝐽 = Φ0𝐼𝑐/2𝜋 is the Josephsonenergy with Φ0 = ℎ/2𝑒 the superconducting flux quantum,𝑒 the elementary charge, and 𝐸𝐶 = 𝑒2/2𝐶 the charging en-ergy [1]. For comparison, we plot the calculated 𝑓01 for JJswith hBN weak links using previously measured tunneling re-sistance values [28] and the Ambegaokar-Baratoff relation toestimate the critical current. Compared to WSe2, hBN showsmuch coarser variation in 𝑓01 with layer number due to itslarge bandgap, increasing by roughly an order of magnitudewith each added vdW layer and requiring extremely thin weaklinks which are difficult to isolate and construct JJs with. Incontrast, we find that JJs with 15–20-layer-thick WSe2 weak5links can produce METs with frequencies between 1–15 GHz.To test these predictions, we fabricate a prototype all-vdW-material MET, comprising an NbSe2/17-layer-thickWSe2/NbSe2 JJ. The MET readout circuit, depicted schemati-cally in Fig. 4(b), consists of a hanger-type microwave readoutresonator capacitively coupled on one end to a microwavefeedline with characteristic impedance 𝑍0 = 50 Ω. The JJis fashioned into a SQUID loop using reactive ion etchingin a lithographically defined window and is capacitively cou-pled to the readout resonator at one end, and to ground atthe other, by lithographically defined leads. We include ad-ditional flux bias and microwave drive lines for tuning andcontrol. Fig. 4(c) shows an optical image of the vdW MET,magnified in Fig. 4(d).In this device, we confirm dispersive interaction between thereadout resonator and the MET by measuring the transmissioncoefficient 𝑆21 of the feedline near the readout resonator fre-quency as a function of power and flux bias current (Fig. 4(e)).We observe a distinct shift in resonator frequency below a criti-cal power level, indicating the onset of dispersive coupling to atwo-level system. At a readout power of −27 dB — below thiscritical threshold — the resonator frequency shows a periodicresponse to magnetic flux threaded through the SQUID loop,affirming that it is interacting with the frequency-tunable vdWSQUID. In Fig. 4(f), we plot two-tone spectroscopy of the de-vice taken at the flux-bias “sweet spot” where the change in thereadout resonator frequency versus flux is minimized. Here,we input to the feedline a continuous weak probe tone nearthe readout resonator center frequency, as well as a microwavepulse at frequency 𝑓drive, intended to create excitations in thetwo-level system. We record the change in |𝑆21 | at the probefrequency that results as we vary the 𝑓drive and the drive am-plitude. At low amplitudes (purple data), we observe a sharppeak in the response at 𝑓drive ≈ 5.30 GHz, which we identifyas 𝑓01. As we increase the drive amplitude (purple to violetto pink to gold) the peak at 𝑓01 broadens while a second peakat ≈ 5.18 GHz emerges. We designate this to be 𝑓02/2, halfthe 0 → 2 transition frequency, emerging as the result of atwo-photon excitation process. From this, we obtain the trans-mon anharmonicity 𝛼 = 𝑓21 − 𝑓01 = −242 MHz. We calculate𝐸𝐽/𝐸𝐶 = 66 for this qubit, placing it well within the transmonregime [1]. The measured 𝑓01 is within 10% of the expectedvalue, validating that the DC transport characterization of vdWJJs with semiconducting weak links can be used to accuratelydesign transmon qubits.III. CONCLUSIONIn summary, we have studied DC electronic transport incrystalline vertical Josephson junctions made of supercon-ducting NbSe2 and semiconducting WSe2 weak links withthicknesses varying from 3–18 vdW layers. Trends in thejunction switching current, retrapping current, and hystere-sis as functions of WSe2 thickness and temperature reveal acrossover from proximity-type to tunneling-type junction be-havior around 7 layers of WSe2. We attribute this crossover tothe relative band alignment between the superconductor andsemiconductor. However, the Thomas-Fermi model does notpredict such a crossover, raising fundamental questions aboutthe nature of the proximitization between van der Waals su-perconductors and semiconductors. The observation of thiscrossover underscores the importance of the choice of materi-als used in superconductor-semiconductor quantum devices.Following from this, we comment on how the switching andretrapping current can provide information about the internalloss of the JJ. In tunnel JJs, the magnitude of hysteresis isunderstood to be related to the junction quality factor, 𝑄, as𝑄∗ = (4/𝜋)𝐼𝑐/𝐼𝑟 ≈ 𝑄 ≡ 𝜔𝑝𝑅𝐶, a measure of the loss of theJJ at the junction plasma frequency𝜔𝑝 (equivalent to the METtransition frequency here) [32]. For NbSe2/WSe2/NbSe2 JJs,we find 𝑄∗ exponentially increases with WSe2 thickness upto ≈ 104 for an 18-layer-thick weak link. While this quantityshould only be taken as a lower bound for the junction qualityfactor, as Joule heating during the DC measurement may leadto a reduced hysteresis in tunnel JJs, it can serve as a proxyfor comparing the quality of different materials. In AppendixC, we show that by substituting WSe2 for MoS2, a vdW semi-conducting weak link with a similar tunnel barrier height butdifferent band alignment, we observe less damping and higher𝑄∗ for the same thicknesses of weak link.Using the exponential dependence of the JJ critical currentdensity on weak link thickness, we calculated the frequenciesof merged-element transmon qubits based on these crystallineJJs and validated the accuracy of these predictions by fab-ricating a prototype all-crystalline-material-based MET. Im-portantly, these qubits offer a remarkably small footprint withqubit transition frequencies that can range from a few GHzto millimeter-wave frequencies [? ]. Our results support thatcrystalline superconductor-semiconductor JJs are a promisingplatform for compact superconducting qubits with distinctivedesign flexibility and tunability.ACKNOWLEDGMENTSThis work was primarily supported by the Army ResearchOffice under Contract W911NF-22-C-0021 (NextNEQSTSuperVan-2). Synthesis and characterization of WSe2 crys-tals (M.H.) and the use of facilities and instrumentation forsample assembly (J.B.) were supported by National ScienceFoundation through the Columbia University, Columbia NanoInitiative, and the Materials Research Science and Engineer-ing Center (DMR-2011738). Theoretical modeling (V.P.)was supported by the National Science Foundation (GrantNo. 2235276). Synthesis of boron nitride (K.W. and T.T.)was supported by the Elemental Strategy Initiative conductedby the MEXT, Japan (Grant No. JPMXP0112101001) andJSPS KAKENHI (Grant Nos. JP19H05790 and JP20H00354).J.H. acknowledges support from the Gordon and Betty MooreFoundation’s EPiQS Initiative, Grant GBMF10277. J.P. ac-knowledges support from the education and training programof the Quantum Information Research Support Center, fundedthrough the National Research Foundation of Korea (NRF)by the Ministry of Science and ICT (MSIT) of the Koreangovernment (No. 2021M3H3A1036573).6APPENDIX A: METHODSWSe2 crystals are synthesized using the two-step flux syn-thesis method [48]. Synthesis of MoS2 was performed byreacting Mo (99.997%) and S (99.999%) (1:2+150 mg excessS) in a eutectic flux of CsCl and NaCl, as described in Ref. [51]under vacuum (∼10−5 Torr) in a quartz ampoule. The ampoulewas heated to 1000 ◦C over 24 h. At peak temperature, a tem-perature gradient was applied with the hot end of the ampoule(1000 ◦C) at the Mo and S precursor. This resulted in the crys-tals primarily being deposited at the cold end (900 ◦C). Theampoules were then naturally cooled to room temperature byshutting the furnace off. Crystals were extracted by washingin DI water to remove the salt flux. The as-extracted crystalswere then reloaded into a new quartz ampoule with excess Sin a Mo:S ratio of 1:100 and heated to 900 ◦C for two daysand then naturally cooled. Crystals were once more extractedand excess S was removed by following the chalcogen filteringmethod as described in Ref. [48].We mechanically exfoliate hBN, WSe2, and MoS2 flakesfrom bulk crystals in air and use atomic force microscopy(AFM) to scan flakes for cleanliness and determine their thick-ness. While AFM is a sensitive and commonly used probe ofvdW materials, the presence of physisorbed organic moleculesor water on or under exfoliated flakes, air gaps, or instrumen-tal offset can impede the accurate determination of vdW flakethickness using atomic force microscopy. This leads to a typi-cal uncertainty of around ± 1 atomic layer [49] for our WSe2and MoS2 flakes. Other techniques to determine thickness likesecond harmonic generation may be used in future works tomore accurately determine the layer number [50].NbSe2 flake exfoliation and vdW device stacking are per-formed in a glovebox under a N2-rich atmosphere (< 0.5 ppmO2 & H2O) to minimize oxidation during fabrication. Duringstacking, bubbles which commonly form in vdW heterostruc-tures can lead to further uncertainty in the Josephson junctiongeometry, as the effective junction area defined by the overlapof the two NbSe2 electrodes may be reduced by the presenceof bubbles. We use an MMA/PMMA bilayer resist for e-beamlithography. CF4 reactive ion etching is performed to modifythe device area, and CHF3 etching is used to etch regions ofthe encapsulating hBN and expose NbSe2 for electrical con-tact. Before metal deposition, oxidized surface NbSe2 layersare removed in situ using argon ion-milling. Then, we deposit3-nm-thick titanium sticking layer and 40–60 nm aluminumleads by e-beam evaporation in the same chamber [11].We can control the device area within the precision of elec-tron beam lithography in order to achieve desired critical cur-rents or fabricate SQUID loops from a single junction. Forexample, for the MET device shown in Fig. 4, we stacked a 23µm2 JJ. We patterned a 0.5 µm slit in the middle of the junc-tion to split it into two equal-area junctions. A reactive ionetch comprising a combination of CF4 and Ar gases was usedto etch through the entire stack. Since the etch is anisotropic(predominantly vertical), the timing of the etch is not very im-portant as long as it etches all the way through at least the topNbSe2 flake. This process is reproducible and was also usedto reduce the junction area of thinner WSe2 weak link devices(3–7 layers), whose enormous critical current densities requirevery small junction areas (≲ 1 µm2) ensure the critical currentof the junction is smaller than that of the NbSe2 flakes them-selves. This is possible thanks to the sub-100-nm resolutionof the electron beam lithography system.We characterize our Josephson junctions (JJs) using 4-terminal DC transport measurements in a dilution refrigera-tor (Bluefors BF-LD400) with a base temperature below 20mK. The bias current is swept through the junctions while thevoltage difference across them is measured. This bias currentis applied through a load resistor, whose resistance is muchgreater than the junction resistance, in series with a voltagesource. The voltage across the junction is amplified and thenread by a digital multimeter. To prevent high-frequency noisefrom exciting the junctions, a two-stage low-pass filter with acutoff frequency of approximately 20 kHz is installed on allDC lines at the 4 K stage.Temperature control of the junctions mounted on the coldfinger is achieved using a PID feedback loop implementedthrough an AC resistance bridge (Lakeshore Model 372) anda 50 Ω heater mounted on the mixing chamber plate. To raisethe temperature above 1.2 K, the cooling power is intentionallyreduced by collecting most of 3He-4He mixture and loweringpumping speed by turning off the turbo pump. All measure-ments are performed after the temperature has been stabilizedfor at least 20 minutes.MET devices are mounted on the cold finger of the samedilution refrigerator with a base temperature below 20 mK.To reduce decoherence from external magnetic fields, the coldfinger is enclosed in Cryoperm magnetic shielding. On theinput side, the total line attenuation ranges from 70 dB to84 dB, depending on the resonance frequency, with a 40 dBattenuator placed on the mixing chamber to further protect thequbit from thermal radiation. On the output side, a circulatoris installed to block noise from external sources and preventthe input signal from reflecting back into the qubit. The outputsignal is amplified by low-temperature and room-temperatureamplifiers. The qubit transition of the MET device, as shown inFig. 4(f), is characterized using a two-tone pulse measurement.A fixed DC current of 55 µA is applied to generate magneticflux, positioning the transmon at its least susceptible point toflux noise. Both the cavity readout and qubit control pulsesare fed through the input port of the transmission line.APPENDIX B: ADDITIONAL 𝐼-𝑉 CURVES, DESCRIPTIONOF CRITICAL AND RETRAPPING CURRENTEXTRACTION, AND DISCUSSION OF 𝐼-𝑉CHARACTERISTICSWe provide additional JJ 𝐼-𝑉 curves to illustrate how thecritical and retrapping currents are extracted in junctions withlow damping and large hysteresis. In Fig. 5(a) we plot the same𝐼-𝑉 curve for a Josephson junction (JJ) with a 13-layer-thickWSe2 weak link shown in Fig. 2. In Fig. 5(b) we zoom inon the low-current-bias regime to show the small measuredretrapping current.In Fig. 5(c) we show an 𝐼-𝑉 curve for a JJ with 17-layer-thick7(a)-200 0 200I (μA)-202V (mV)I (μA)V (mV)-1 0 1-0.500.5(b)13L WSe2 13L WSe20 100 200I (nA)01234V (mV)-2 0 2I (nA)-1-0.500.51V (mV)17L WSe2 17L WSe2(c) (d)FIG. 5. Additional 𝐼-𝑉 curves. (a) 𝐼-𝑉 curve of a JJ with a 13-layer-thick WSe2 weak link (same as shown in Fig. 2. Blue (red) traceindicates increasing (decreasing) bias current sweep. (b) Zoom-in of(a) showing small retrapping current. (c) 𝐼-𝑉 curve of a JJ with a 17-layer-thick WSe2 weak link. The vertical black dashed line indicatesthe value of 𝐼 at which d𝐼/d𝑉 is maximized (i.e. the superconductinggap edge), which we use as a proxy for the critical current. (d) Zoom-in of (c).WSe2. In this particular device, both the switching current andretrapping current are very small, and the former fluctuatesgreatly between successive measurements, making estimationof the critical current using the switching current difficult. Weattribute these fluctuations to noise in our measurement systemwhich excites the junction to the normal state at current biaseswell below the critical current, which junctions with smallercritical currents are naturally more susceptible to. Specifically,the noise in the system would need to rise above the poten-tial barrier of the washboard potential, Δ𝑈 = 2𝐸𝐽 , where𝐸𝐽 = Φ0𝐼𝑐/2𝜋 is the Josephson energy. Using the measured𝐼𝑠0 of ∼ 2 nA in place of 𝐼𝑐, this barrier would correspond to avalue of ∼ 0.1 K at zero bias. However, by taking the point ofthe 𝐼-𝑉 curve with minimal slope (i.e., the d𝐼/d𝑉 maximum),we can estimate the superconducting gap edge, as in a super-conducting tunnel junction. Since junctions that do not sufferfrom premature switching due to fluctuations typically switchat the gap edge, we use the current bias at which d𝐼/d𝑉 is max-imized as a proxy for the expected switching current 𝐼𝑠0 and,ultimately, the critical current in junctions that exhibit pre-mature switching. Explicitly, these are our 14–18-layer-thickWSe2 JJs, whose low critical current densities and relativelysmall junction areas yield low critical currents. In the case ofthe 17L junction, the estimated 𝐼𝑠0 would then become 36 nA,corresponding to a zero-bias Δ𝑈 of 1.7 K. The switching rate,Γ, is given by Γ ∝ exp(Δ𝑈/𝑘𝐵𝑇esc), where 𝑇esc is the noisetemperature of the system. We can model our device as beingin the thermally activated regime with an extracted𝐶𝐽 = 53 fFand measured 𝑅𝑛 = 2700 Ω. We find that, at a bias of 2 nA(corresponding to the measured 𝐼𝑠0), a 𝑇esc ≈ 64 mK is suffi-cient to switch the device at a Γ ≈ 0.1 Hz, which is comparableto the integration time of our DC measurement. These detailshighlight the difficulty of measuring DC transport in JJs withsmall critical currents.To measure the wide-range 𝐼-𝑉 shown in Fig. 5(c) we apply avoltage across a 10 MΩ load resistor to supply the bias current.However, for the narrower range in Fig. 5(d) to observe the sub-nA retrapping current, we use a 1:1000 voltage divider to biasa 1 MΩ load resistor, giving us finer resolution in bias current.The use of a battery-powered instrumentation amplifier in thebiasing circuit prevents ground loops among the instrumentsand reduces noise to help supply such small biases to ourdevices.We also comment on the curvature observed in the subgapregion of these 𝐼-𝑉 curves, distinct from the commonly ob-served sharp transitions of Al/AlOx/Al JJs. While a gradualretrapping such as those shown in Fig. 5 can suggest finite sub-gap conduction, we point out that there are other factors whichcan influence the curvature of the subgap regime. For one,measurement of the subgap resistance of a Josephson junctioncan elucidate the density of states (DOS) of the superconduc-tor near the gap edge and even in-gap states. However, wedo not observe signatures of Andreev reflection or Andreevbound states, even in JJs with the thinnest WSe2 weak links,implying there are minimal to no in-gap states. Additionally,proximitization at the superconductor-weak link interface canalso influence the DOS near the gap edge and lead to somecurvature of the 𝐼-𝑉 in the subgap region. However, it must beremembered that Joule heating is induced in the normal state,which persists into the subgap region, not only influencing theretrapping behavior but also increasing the effective junctiontemperature and broadening the DOS near the gap edge. Sincethe superconducting gap of NbSe2 (∼ 1 meV) is approximatelyfive times greater than that of Al (∼ 200 µeV), the Joule heat-ing near the gap edge can be significantly greater in JJs withNbSe2 electrodes, leading to curvature of the 𝐼-𝑉 in this regiondue to thermal broadening of the DOS. For this reason, it maybe unreliable to extract subgap resistances for these junctionssince they can be reduced due to Joule heating.APPENDIX C: COMPARISON OF DIFFERENTCRYSTALLINE SEMICONDUCTOR WEAK LINKSTo emphasize the impact of the band alignment of the con-stituent materials on the tunneling properties and dampingin a crystalline Josephson junction, we measure DC trans-port in vdW JJs with NbSe2 electrodes and semiconductingMoS2 weak links. As depicted schematically in Fig. 6(d), thevalence band maximum of MoS2 is lower than that of WSe2[44], and so the band bending effect at the interface with NbSe2should be weaker when used as a JJ weak link than when us-ing WSe2. Meanwhile, the bandgaps and effective masses ofthe two materials are similar and so, according to the Thomas-80 5 10 15 20weak link layers (#)I s0/I r0100102104 WSe2MoS2(b)(a)0 5 10 15 20RnA (Ω·µm2 )100103106WSe2MoS20 5 10 15 20j s0 (µA/µm2 )10-3100103WSe2MoS2-3.0-4.5-6.0WSe2 MoS2NbSe2Energy (eV)(d)(e)0 2 4 601234I sRn (mV)3L5L9LMoS20 2 4 601I r/I s03L5L9LT (K)MoS2(c) (f)FIG. 6. Comparison of different semiconducting crystalline weaklinks. (a), (b), (c) 𝐼𝑠0/𝐼𝑟0, 𝑗𝑠0, and 𝑅𝑛𝐴, respectively, versus weaklink thickness for Josephson junctions with NbSe2 electrodes andWSe2 (green symbols) or MoS2 (orange symbols) weak links. Dashedlines in (a) are fits to the data to guide the eye. Solid lines in (b) arethe result of resistivity simulations within the Thomas-Fermi model.(d) Schematic band alignment of bulk WSe2 (green-shaded bars) andMoS2 (orange-shaded bars) relative to the NbSe2 work function (bluedashed line), along with their monolayer values (red outlines) [43, 44].(e), (f) Temperature dependencies of 𝐼𝑠𝑅𝑛 and 𝐼𝑟 /𝐼𝑠0, respectively,for JJs with MoS2 weak links. Solid black line in (e) is the A-B limit.Fermi model, their tunneling barrier heights should be roughlythe same. Our DC transport results affirm this prediction, with𝑗𝑠0 and 𝑅𝑛𝐴 having nearly the same exponential dependenceon weak link thickness for both WSe2 and MoS2 (Fig. 6(a) &(b)). However, fewer layers of MoS2 are required to achievethe same tunneling properties than with WSe2, suggesting thattunneling-type behavior is onset at a smaller thickness. Wecompare 𝐼𝑠0/𝐼𝑟0 for WSe2 and MoS2 weak links in Fig. 6(c)and find that the junctions with MoS2 have lower loss for thin-ner weak links as indicated by a higher𝑄∗. In Fig. 6(e) we plot𝐼𝑠𝑅𝑛 for JJs with 3-, 5-, and 9-layer-thick MoS2 weak links, andwe find that the crossover from above to below the A-B limit(black line) occurs between 3 and 5 layers, once again suggest-ing a crossover from proximity- to tunneling-type behavior.Meanwhile, the temperature dependence of 𝐼𝑟/𝐼𝑠0, plotted inFig. 6(f) is less indicative of a crossover but nonmonotonicityappears to enhance with increasing layer number. Regardless,the data suggest that by changing the band alignments of theconstituent JJ materials, we may change the tunneling strengthsufficiently to yield different amounts of Josephson couplingand damping for the same weak link thicknesses and tunnelbarrier heights.APPENDIX D: THOMAS-FERMI MODEL FORSELF-CONSISTENT POTENTIAL IN THE TUNNELINGREGIONWe employ the Thomas-Fermi model to simulate charge car-rier density in layers of WSe2 or MoS2 between NbSe2 elec-trodes by solving the Poisson equation for the self-consistentpotential 𝑉 (𝑥),𝜖𝑑2𝑉 (𝑥)𝑑𝑥2 = −𝜌(𝑥), (1)where 𝜖 is the dielectric constant in the perpendicular to thesemiconductor plane direction, one of two fitting parameters.The best fit of 𝜖 = 5.5 for both WSe2 and MoS2 is consistentwith values reported in literature [52–54]. The charge carrierdensity 𝜌(𝑥) is found from the 2D carrier density on eachWSe2 layer 𝑛𝑖 separated by distance 𝑑 = 6.5 Å,𝜌(𝑥) =𝑁∑︁𝑖=1𝑛𝑖 exp(−(𝑥 − 𝑑 · 𝑖)2/𝜎2)/(√𝜋𝜎), (2)where we employ Gaussian broadening with𝜎 = 3 Å to mimicthe finite width of the electron cloud between the crystallinesemiconductor layers. The 2D charge carrier density 𝑛𝑖 de-pends on the electrostatic potential 𝑉𝑖 = 𝑉 (𝑑 · 𝑖) in the middleof the monolayers, which is found self-consistently via𝑛𝑖 (𝑉𝑖) = 𝑘𝐵𝑇 · 2𝑚ℎ𝜋ℏ2 · ln(1 + 𝑒 (−𝑒𝑉𝑖+Δ𝑊 )/𝑘𝐵𝑇1 + 𝑒 (𝑒𝑉𝑖−𝐸𝑔−Δ𝑊 )/𝑘𝐵𝑇), (3)where 𝑚ℎ = 0.53 𝑚𝑒 [55] (0.64 𝑚𝑒 [56]) is the hole ef-fective mass in WSe2 (MoS2) in units of electron mass 𝑚𝑒,Δ𝑊 = 1.07 eV (0.34 eV) [43, 44] is the workfunction mis-match between NbSe2 and WSe2 (MoS2), and 𝐸𝑔 is thebandgap of the semiconductor. We choose the Fermi energyin the metal as zero of energy and the boundary condition of𝑉 (𝑥 = 0) = 𝑉 (𝑥 = 𝐿) = 0 V, where 𝐿 = 𝑑 · (𝑁 + 1), implyhole doping of the semiconductor near the contacts.Once the self-consistent potential𝑉 (𝑥) is obtained for junc-tions with different numbers of layer 𝑁 , we use WKB approx-imation to calculate tunneling conductance according to:𝐺 = 𝐺𝑀∫ +∞−∞𝑇 (𝐸)(−𝜕 𝑓 (𝐸, 𝐸𝐹 , 𝑇)𝜕𝐸)𝑑𝐸,𝑇 (𝐸) = exp(−2∫ 𝐿0√︂2𝑚ℎ (𝐸 − 𝑒𝑉 (𝑥) − Δ𝑊)ℏ2 𝑑𝑥)(4)where 𝐺𝑀 = 1.6 S·µm−2 (0.026 S·µm−2) is a fit parameterwhich has the meaning of a product of ballistic conductance ofNbSe2 and an additional scattering barrier at the NbSe2/WSe2(NbSe2/MoS2) interface due to the wavefunction mismatchin two materials. The resulting resistance 1/𝐺 simulated at𝑇 → 0 K limit is plotted in Fig. 1(d) and Fig. 6(b).9[1] J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster,J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, and R. 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