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[2024A00643G_Supplemental Material.docx](https://mdr.nims.go.jp/filesets/fab502f6-e66d-4dde-9177-d92791c2809b/download)

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Geon-Hyoung Park, Wonjun Lee, Sein Park, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Gil Young Cho, Gil-Ho Lee

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© 2024 American Physical Society[In Copyright](http://rightsstatements.org/vocab/InC/1.0/)

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[Controllable Andreev Bound States in Bilayer Graphene Josephson Junctions from Short to Long Junction Limits](https://mdr.nims.go.jp/datasets/4747077a-63d9-4d9a-92c0-fc412baa1ce1)

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Supplemental MaterialControllable Andreev Bound states in Bilayer Graphene Josephson Junctions from Short to Long Junction LimitsGeon-Hyoung Park1, Wonjun Lee1,2, Sein Park1, Kenji Watanabe3, Takashi Taniguchi4, Gil Young Cho1,2,5, and Gil-Ho Lee1,5*1Deparment of Physics, Pohang University of Science and Technology, Pohang, Republic of Korea2Center for Artificial Low Dimensional Electronic Systems, Institute for Basic Science, Pohang, Republic of Korea3Research Center for Electronic and Optical Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan4 Research Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan5Asia Pacific Center for Theoretical Physics, Pohang, Republic of KoreaS1. General information on the bilayer graphene Josephson junction devicesFigure S1. (a) A false-colored scanning electron microscope image of the loop type (LD) and the two-terminal type (TD) Josephson junction devices. The green area represents the bilayer graphene (BLG) layer encapsulated by top and bottom hexagonal boron nitride sheets. The blue electrodes are Ti/Al double layer for inducing Josephson coupling into the BLG and the yellow electrodes are Al tunnel probes. (b)-(c) Temperature dependence of resistance for the Al tunnel probe electrode (b) and the Ti/Al electrode (c), respectively. Each data point excluded offset resistance from the DC measurement lines using two-stage RC filters .S2. Transport measurements of the two-terminal type device (device TD) Figure S2. (a) Two-terminal resistance () of the device TD as a function of backgate voltage Vbg at T = 0.8 K. The inset shows the measurement configuration for (a)-(d). (b) Ballistic charge transport in the BLG. (Upper panel) Fabry-Pérot-like interference occurred at the hole-doped region (blue line). The red curve represents the moving average of the original values. (Lower panel) Background subtracted periodic oscillation from the upper panel. (c) A color map of differential resistance (dV/dI) of the device TD as a function of biasing current I and Vbg. (d) Fraunhofer diffraction pattern of the device TD. The yellow dashed line indicates the offset value of the perpendicular magnetic field, B0 ~ -0.72 G.Since the loop-type device, LD is inappropriate for measuring the voltage drop in the junction, instead, we performed conventional two-terminal measurements on the device TD to confirm the quality of the BLG and JJs. The device TD has an equivalent geometry of conducting channel as the device LD, and both devices share the same BLG sheet. The electrostatic field effect of the BLG of the device TD is shown in Fig. S2(a). We refer to the charge neutral point (CNP) of the BLG where the highest resistance peak exhibits, VCNP ~ -7.2 V. Slightly higher resistance in the p-doped area (Vbg < VCNP) than in the n-doped area (Vbg > VCNP) results from the formation of p-n junctions at the contact interfaces of BLG and Ti/Al [1-4]. Here, we also confirmed the ballistic property of the BLG channel by Fabry-Pérot-like oscillations in the p-doped region [1,2,4]. As shown in Fig. S1(b), the interference occurs when the quasiparticles undergo a series of reflections at the contact interfaces when they satisfy the resonance condition , where  is the effective length of the cavity, m is an integer, and  is the Fermi wavelength. The moving-averaged slowly varying background (red curve in upper panel) is taken for removing the background of the oscillation. For mth  peak, one can use the relation  [5]. The average effective cavity length for selected neighbor peaks in Fig. S2(b) is , which well matches with the designed channel length . For some reasons, the lengths of p-n junctions at the interfaces are estimated shorter (~ 30 nm) than those in other graphene Josephson junctions reported previously [1,2,4]. Figure S2(c) shows the gate dependence of differential resistance (dV/dI) as a function of I and Vbg. The zero resistance observed in the black region indicates the presence of proximitized superconductivity in the normal channel. The  reaches up to ~ 0.21 as the gate voltage increases to 30 V. Here, Ic is the critical current of Josephson junction, RN is the normal resistance of the junction, and  is the proximity superconducting gap. Figure S2(d) shows the Fraunhofer diffraction pattern (FP) of the device TD at Vbg = 30 V. The period of the FP,  is larger than the expected value , which is calculated with the area of the junction  and the magnetic field penetration depth  into the superconductors [6,7]. The origin of the large oscillation period is uncertain; however, it is considered to be related to the increase in the periodicity when the aspect ratio of the junction  is large enough (1.5 for the device TD) [8].S3. Magnetoresistance of the device TDFigure S3. (a) A colormap of Landau fan diagram as a function of backgate voltage Vbg and magnetic field B measured from device TD. (b) A line-cut from (a) at B = 2 T. (c) Backgate dependence of the charge carrier density n extracted from Shubnikov de Haas oscillations (SdHO). All the data were measured at T = 1.6 K.The magnetoresistance of the device TD at a high magnetic field was measured to estimate the charge carrier density of the BLG. All the measurements with a high magnetic field were performed at the Oxford Telsatron fridge with a variable temperature insert (VTI) system after the tunneling spectroscopy measurement of ABS in the main text. The two-terminal conductance in Fig. S3(b) exhibits a pronounced quantized conductance step for BLG,  for , where N is an integer, confirming our graphene sheet is a bilayer. Fig. S3(c) shows the charge carrier density n at different back gate voltages which can be estimated from the relation , where BN is the magnetic field at the amplitude peak of Shubnikov de Haas oscillation (SdHO) and N is an integer. Since , we used the linear regression method to obtain the gate efficiency coefficient  of the device, where  is the permittivity of free space,  is the relative permittivity of the dielectric layer, d is the thickness of the dielectric layer, and e is the elementary charge. Considering each thickness and permittivity of the SiO2 layer (, ) and the bottom hexagonal boron nitride sheet (, ) [9] also expect  .S4. Estimation of the effective mass in bilayer graphene (device TD)Figure S4. (a) Background subtracted SdHO amplitudes at different temperatures with B = 2 T. (b) A temperature dependence of normalized SdHO amplitudes (blue circles) at Vbg = 10.1 V. The red solid curve from the Lifshitz-Kosevich formula well fits the experimental data. (c) Estimated effective masses () normalized by the electron rest mass () at specific backgate voltages (blue squares). The red solid curve is the theoretical fit.Continuing from Fig. S3, one can extract the effective mass of BLG from SdHO at different backgate voltages. We exploited the temperature dependence of SdHO amplitude following Lifshitz-Kosevich (L-K) formula which can be written as the following [10,11]:,where  is the effective mass of BLG,  is the reduced Planck constant, and B is the magnetic field. Figure S4 (b) shows normalized SdHO amplitudes as a function of temperature (blue circles) and the best L-K fit (red solid line) at Vbg = 10.1 V. From the fitting, we estimated effective masses of BLG at different backgate voltages as shown in Fig. S4 (c). The estimated  exhibits 0.03, where  is the electron rest mass. We found that these values are similar to the  from references 10 and 11 using the same method. The red fitting curve is calculated  from the quadratic energy dispersion relation: . The lowest energy band  was calculated using an approximated tight-binding method, neglecting interlayer asymmetry [12]. The following dispersion relation quotes the notations from the Slonczewski-Weiss-McClure model [13-15].,where  is the hopping parameter between interlayer atoms (dimer sites), and  is the band velocity of monolayer graphene, with  representing the hopping parameter between nearest-neighbor intralayer atoms. We used  and  for the best fit to the experimental data in Fig. S4 (c).S5. Additional data for phase dependence of ABS (device LD)  Figure S5. Additional dataset of differential tunnel conductance dI/dV as a function of bias voltage V and magnetic field B. All the data were measured at T = 17 mK.S6. Numerical calculation of ABS in a bilayer graphene Josephson junctionIn this section, we will discuss the details of calculating the spectrum of ABS in a BLG Josephson junction used in the main text. To achieve this, we will first introduce our theoretical model for the system, followed by detailed steps for computing the spectrum specifically using the scattering formalism.We consider a situation in which superconductors are placed on both sides of a BLG strip with length . A tunneling probe is located at one-third along the longitudinal direction of the strip, resembling the configuration of the device LD. The presence of this tunneling probe will be effectively modeled as a scatterer at its position [16]. Next, let us move on to the computation of the spectrum of ABS in BLG JJ. In the ballistic limit, electrons form freely propagating waves within the BLG strip. These waves scatter upon reaching the boundaries of the superconductors [17] or some other impurities and potential walls within the bulk [18-20]. Due to these scatterers, only some resonant waves known as ABS with specific energies are allowed, not all wave numbers [21]. Our goal here is to find such a resonant wave using the scattering formalism [18,22].Formally, the freely propagating waves of BLG JJ can be obtained by solving the Dirac-Bogoliubov-De-Gennes (DBdG) equation [22,23]with the energy  and the chemical potential . Here,  and  are electron and hole waves in the strip, and  is the Hamiltonian of BLG which is given by [12,19,24]with the momentum , the on-site energy , the Fermi velocity , and the interlayer coupling . The y-directional momentum  is a free parameter that can be independently fixed from  via fitting. The Hamiltonian can be simplified by rescaling the coordinates  and . The eigenstates of the Hamiltonian are given as the columns of  up to normalization factorswith the energy . Here,  is given by with  and the wave numbers . The electron and hole waves are coupled by the superconductors at the boundaries of the strip by [22-24]with , , the phase difference  between the superconductors, and the superconducting gap .The spectrum of ABS in the BLG JJ can be obtained by finding the solutions to the resonance condition introduced by the scattering formalism [18,22]:.Here,  is the diagonal matrix whose entries are phases accumulated when the waves propagate to the distance . In addition,  represents the transfer matrix of the effective scatterer, which accounts for the effect of the tunneling probe [16]. We designed this scatterer to mix left and right propagating waves within each layer, considering that the interlayer coupling is relatively small compared to the experimentally relevant range of chemical potential. Formally, the transfer matrix of the scatterer is set by , where and  denote the transmission and reflection components of the scattering matrix of the scatterer. They are related by , and one of them serves as a free parameter determined through fitting. The transmission component  is related to the transparency  of the junction by . Intuitively, the resonance condition finds a bound state that regains the initial state after completing a round trip along the strip with scatterings. Therefore, solutions to the resonance condition can be found by varying the energy  of the bound state and checking if the determinant vanishes.The remainder of this section is dedicated to the discussion of the fitting process. There are two parameters that influence the spectrum of the ABS: the y-directional momentum and the transmission component  of the scattering matrix. We need to identify these parameters in a way that aligns the theoretical expectations with the experimentally measured ones. To this end, we find that low y-directional momentums of  with the width of the junction  adequately explain the experimental data, as shown in Fig. S6(a). The suppression of the large y-directional momentum is consistent with what we intuitively expect: the ABSs with the finite y-directional momentum(scattering with oblique angles at the interface of superconductors and normal metal) require electrons to travel much longer lengths to experience the “constructive interference” (the origin of the ABS). However, the longer the electrons travel, the more decoherence the electrons would feel. This will naturally suppress the formation of such bound states in experiments. Furthermore, we take the ABS with the zero y-directional momentum as the representative of the low y-directional ABSs as their spectral properties are indistinguishable within the experimental resolution of our tunneling probes, as shown in Fig. S6(b). Subsequently, we search the optimal transmission matrix component for each backgate voltage by minimizing the difference between the theoretical and experimentally measured spectra, which nicely reproduces the experimental data.Figure S6. (a) The energy spectrum of Andreev bound states with various y-directional momentum over the experimentally-measured differential conductance at the backgate voltage Vbg = -7.5 V. The transmission matrix component of the effective scatterer is set as t = 0.75. (b) The energy spectrum of Andreev bound states with various y-directional momentums at the given backgate voltages. Calculations are done with the transmission matrix components t = 0.95 and t = 0.75 of the effective scatterers for the backgate voltages Vbg = 30 V and Vbg = -7.5 V, respectively.S7. Temperature dependence of Andreev bound states (device LD) Figure S7. (a) A color map of tunneling differential conductance (dI/dV) as a function of bias voltage and temperature. Red dashed curves represent the theoretical calculation of temperature dependence of ,  and . (b) A schematic of the energy diagram for additional electron tunneling from the tunneling probe to the ABS below the Fermi level () at . (c) A set of voltage bias line cuts at selected temperatures from (a). For visibility, each line cut has y-axis offsets except for the one measured at 20 mK.Figure S7 shows the temperature dependence of ABS at a fixed phase difference  with Vbg = 30 V. In Fig. S7(a), two differential tunnel conductance peaks from  are clearly shown under . We found that the temperature dependence of the superconducting gap from the Bardeen-Cooper-Schrieffer (BCS) theory (red dashed curves) well fits our experimental data [25]. For the convenience in calculation, we used the following interpolating function instead of the original equation [26]:,where  is the Boltzmann constant, and  is the critical temperature of the superconductor. Here, we assume that since the transparency of the channel is 0.95 at Vbg = 30 V (See the discussion about transparencies of the device LD in the main text). At , the dI/dV peaks from ABS disappear, and only the conductance dip structure from  survives.We observed weak dI/dV peaks at  at  which were also found in the previous report with monolayer graphene JJ [27]. As shown in Fig. S7(b), a portion of the electrons in the state can be thermally excited when the temperature is high enough, forming an empty state inside . In that case, tunnel current can additionally occur even below the bias energy . S8. Replicas of Andreev bound statesFigure S8. (a) A set of differential tunnel conductance (dI/dV) color maps as a function of bias voltage V and magnetic field B for selected backgate voltages. (b) (Left panel) Bias dependence of differential tunnel conductance as a function of backgate voltage at a fixed phase difference . (Right panel) A bias voltage line cut from the left panel at Vbg = 26 V. The dashed arrows indicate replicas of the original ABS ().While measuring the tunneling conductance, we observed additional dI/dV peaks even when . In Fig. S8(a), the above gap peaks appear to be replicas of the original peaks including both phase-dependent ABS and the non-phase-dependent proximity gap, . Below Vbg < 15 V, observing these replicas becomes challenging (see Fig. S5). Moreover, extended bias energy measurement in Fig. S8(b) shows that secondary replica peaks (Replica #2) appear around . It seems that the replica peaks have little gate voltage dependence. However, separately from this, we observed gate-dependent dI/dV peaks (slanted dI/dV peaks in Fig. S8(b)) that appear to be the result of unintentional quantum dots inside the tunnel barrier or within the BLG channels [28,29]. To the best of our knowledge, we have not found any reports identical to our observation. 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