Abstract
The influence of static disorder on a quantum phase transition (QPT) is a fundamental issue in condensed matter physics. As a prototypical example of a disordertuned QPT, the superconductor–insulator transition (SIT) has been investigated intensively over the past three decades, but as yet without a general consensus on its nature. A key element is good control of disorder. Here, we present an experimental study of the SIT based on precise insitu tuning of disorder in dualgated bilayer graphene proximitycoupled to two superconducting electrodes through electrical and reversible control of the band gap and the charge carrier density. In the presence of a static disorder potential, Andreevpaired carriers formed close to the Fermi level in bilayer graphene constitute a randomly distributed network of proximityinduced superconducting puddles. The landscape of the network was easily tuned by electrical gating to induce percolative clusters at the onset of superconductivity. This is evidenced by scaling behavior consistent with the classical percolation in transport measurements. At lower temperatures, the solely electrical tuning of the disorderinduced landscape enables us to observe, for the first time, a crossover from classical to quantum percolation in a single device, which elucidates how thermal dephasing engages in separating the two regimes.
Introduction
The superconductor–insulator transition^{1,2} (SIT) in disordered twodimensional (2D) superconductors exhibits a zerotemperature separatrix between the superconducting and insulating phases^{3,4}. The associated scaling behavior^{5,6} reveals the intrinsic nature of the quantum criticality. Cooper pairs exist even in the insulating phase, which is evidenced by direct observations of the superconducting gap^{7,8}, as well as the earlier observation of a giant magnetoresistance^{4,9}, magnetooscillations^{10}, and superfluid correlations^{11}. These observations strongly suggest that the loss of pair coherence due to disorder drives the SIT. Several underlying mechanisms for the SIT have been suggested^{1}, including the dirty boson picture based on Anderson localization of Cooper pairs^{12}, classical^{13} and quantum percolation^{14} of superconducting clusters.
In the dirty boson model^{5,12}, which assumes that fluctuations of the pair amplitude are negligible, phase fluctuation of the superconducting order parameter destroys global superconductivity. However, some recent reports have pointed out that strong disorder induces amplitude fluctuations to form superconducting islands in the insulating phases, even in homogenously disordered thin films^{7,15,16,17,18} (which is also relevant to the Higgs amplitude mode^{19,20}). These amplitude fluctuations may drive the universality class of the SIT from the disordered boson class to a percolation universality class, governed by the loss of global connection of disordered superconducting islands.
This issue, concerning the interplay between disorder and superconductivity, is underscored by recent experiments, which have reported classical or quantum percolation critical behavior at the SIT in systems with varying degrees of disorder^{21,22}. The relevance of these two different transitions, classical and quantum percolation, which are governed by different critical exponents, is determined by the question^{14} whether percolating clusters are formed between superconducting islands via either superconducting paths carrying phasedisrupting currents (classical percolation) or via coherent quantumtunneling links (quantum percolation). Thus, precise control of disorder is crucial to elucidate the interplay between disorder and thermal dephasing, which is responsible for the classicaltoquantum crossover behaviors, and in particular, to differentiate the disorderinduced geometrical effects on the SIT from generic density modulations.
Varying the thickness^{1,3} of or annealing^{23} superconducting thin films has been adopted in previous experiments to change the level of disorder. However, this can result in variations in the carrier density as well as the disorder landscape in a noncontrollable way. Electrostatic gating has also been employed for 2D superconducting systems as a means of controlling the carrier density while preserving the spatial disorder on an atomic scale in superconducting films^{24}, heterostructures of complex oxides^{25}, highT_{c} superconductor^{26}, and graphene^{27}. Here, we utilize electrostatic gating for accurate and reversible tuning of the disorderinduced landscape at energies close to the Fermi level by modulating both the carrier density and the band gap independently, rather than simple carrier density modulation with an uncontrolled fixed disorder.
Compared with deeply buried 2D electronic systems of semiconducting heterostructures or oxide interfaces, graphene is more chemically inert and easily accessible using contact probes. However, the carriers are not strongly localized in monolayer graphene (MLG), even at the chargeneutral point (CNP), where the nominal carrier density vanishes. This is accounted for by the presence of electron–hole puddles^{28,29} produced by the disorder potential arising from charged defects on the substrate and/or chemical residues introduced during device fabrication. Since MLG has zero band gap, sufficiently doped bipolar conducting puddles may touch each other [Fig. 1(a)], making the boundaries transmissible by carriers via Klein tunneling. In contrast, in bilayer graphene (BLG), a band gap E_{g} opens when an electric field is applied perpendicular to the graphene, separating the charged puddles [Fig. 1(b)], and the transport behavior becomes percolative. This feature of BLG allows a high degree of independent control of both the band gap and the carrier density in a wide range^{30}, as shown in Fig. 1(c), and provides flexibility in designing novel devices with controlled conductive behavior by finetuning the distance between puddles. Normal percolative transport has been reported in 2D electron gas systems in the lowcarrierdensity regime^{31}, and in MLG nanoribbons with a finite band gap close to the CNP^{32}. Similar behavior was observed in this study for the gapped BLG with normal electrodes (see Materials and Methods). As the Andreevpaired carriers were induced by the proximity effect in our dualgated BLG device, the system precisely simulates a percolative SIT via the puddles of the pairs, the geometry of which is determined by disorder tuning at the Fermi level.
Results
Gatecontrol of superconducting and insulating states
Figure 1(d) shows a schematic diagram of the configuration of the dualgated BLG device. A pair of Pb superconducting electrodes was closely attached to a mechanically exfoliated BLG layer, which was sandwiched between the top and bottom gates (see Methods). A scanning electron microscopy image of the device is shown in Fig. 1(e), together with the measurement configuration. The distance (L) between electrodes is 0.46 μm and the width (W) of the BLG is 7.0 μm. The contact resistances between BLG and Pb electrodes were negligibly small compared to the zerobias junction resistance, R (Supplemental Information, section 1 and Fig. S1). The BLG became superconducting as Andreevpaired carriers formed due to the proximity effect of the superconducting electrodes, along with the consequent Josephson coupling between them^{33,34,35}. The voltages of the bottom gate, V_{b}, and the top gate, V_{t}, induced displacement fields D_{b} = ε_{b}(V_{b} − V_{b,0})/d_{b} and D_{t} = −ε_{t} (V_{t} − V_{t,0})/d_{t}, along the direction, where ε’s are the dielectric constants, d’s are the thicknesses of the dielectric layers, and V_{b,0} (V_{t,0}) is the chargeneutral voltage offset of the bottom (top) gate due to the initial doping. The difference D_{density} = D_{b} − D_{t} controls the carrier density (or the chemical potential), while the average, D_{gap} = (D_{b} + D_{t})/2, breaks the inversion symmetry of the BLG, opening up a band gap^{36} (Supplemental Information, section 2 and Fig. S3).
Figure 2(a) shows the square resistance of the junction, R_{sq} = R × (W/L), as a function of D_{density} and D_{gap} measured at a base temperature of T = 50 mK. The superconducting and insulating states, marked by black and red symbols, respectively, were determined from the current–voltage (I–V) characteristics at each set of D_{density} and D_{gap}. The two phases are separated coincidently by the quantum resistance of Cooper pairs, R_{Q} = h/4e^{2} (green contour line) as observed in other systems. On the weakly insulating side, the system exhibited nonlinear insulating I–V characteristics, as shown in Fig. 2(b), the zerobias conductance of which is consistent with 2D Mott variable range hopping conduction, G(T) ~ exp[−(T^{*}/T)^{1/3}], where T^{*} is a characteristic temperature (see Methods). On the superconducting side, R_{sq} eventually vanished, and a dissipationless supercurrent branch emerged, as shown in Fig. 2(c), which resulted from the proximity Josephson coupling (Supplemental Information, section 3 and Fig. S4).
Finitesize scaling analysis on the temperaturedependent behavior
The temperature dependence of R_{sq} at different D_{density} ranging from insulating to superconducting phases is shown in Fig. 3(a). It shows no signs of the reentrance or kink of the resistance at temperatures down to 50 mK, which was commonly observed in granular films. Below the crossover temperature T_{0} denoted by the broken line, R_{sq} saturated, presumably due to Joule heating of charge carriers. The shift of T_{0} to lower temperatures when the heating was reduced (i.e., when R_{sq} was smaller) is consistent with the Jouleheating interpretation. In Fig. 3(b), the curves of R_{sq} vs D_{density} at different temperatures converge on a single point (i.e., D_{density,c} ~ −0.3 Vnm^{−1}) with a corresponding critical square resistance of R_{sq,c} ~ 1.1R_{Q}, which is close to the universal value predicted by the dirty boson model for a low dissipative system.
The SIT behavior is interpreted as a quantum phase transition (QPT), as confirmed by R_{sq} vs D_{density} data converging to a single finitesize scaling curve^{5,6} of the form R_{sq} = R_{sq,c} f(xT^{−1/vz}) close to the critical point [Fig. 3(c,d)]. Here, f is a scaling function and x D_{density} − D_{density,c} or x D_{gap} − D_{gap,c} is a tuning parameter. The correlation length exponent ν and the dynamical critical exponent z characterize the universality class of the QPT. The data for 400 < T < 600 mK exhibit the best collapse, with a criticalexponent product of vz = 1.44, which is close to the value v_{cl} = 4/3 for classical percolation in 2D^{37} [an exponent of z = 1 has been assumed for a system with charged particles^{1}, which also appears to be valid in our study, as discussed separately in the biasfieldtuned critical point]. However, at lower temperatures (i.e., 200 < T < 375 mK), the best collapse was found with vz = 2.59, which is consistent with a quantum percolation transition in 2D with the value v_{q} = 7/3 (semiclassically one expects^{38} v_{q} = v_{cl} + 1). The best estimates of v_{q} in the literature^{39} lie in the range 2.3–2.5. We will see below that these values were consistently found in several sweeps with different carrier densities and band gaps. Interestingly, there was a classicaltoquantum crossover at T_{1} ~ 400 mK, which will be discussed later in a more quantitative manner. Theoretical studies have predicted^{14,22} such a crossover from quantum to classical percolation due to decoherence at a finite temperature. Observations of similar crossover behavior have been reported^{22} for quantum Hall insulator transitions. However, no estimation was provided for the associated change in the electron temperature, T_{el}, introduced by the biasinduced Joule heating.
Estimation of T_{el} and the classicaltoquantum crossover
Since Joule heating may seriously affect the behavior of the QPT, in particular, close to the lowest measurement temperature, we carried out an indepth quantitative analysis of T_{el}. T_{el} saturated to a temperature T_{0} as the phonon temperature T_{ph} (i.e., the measurement temperature) approached the base temperature, i.e., T_{el} = T_{0} when T_{ph} = 50 mK. The dissipative power P = I^{2}R at T_{ph} = 50 mK was estimated from the saturated resistance R and the rootmeansquare (r.m.s.) bias current of I = 1 nA, which exhibited a powerlaw dependence on T_{0}, as shown in Fig. 4(a), along with the fit to with T_{ph} = 50 mK. The fitting parameters were the electron–phonon coupling exponent θ = 2.8 ± 0.1 and the coefficient A = 77 ± 14 fW∙K^{−2.8}, where T_{0} was estimated to be T_{0} = 160 mK at the SIT point of R_{sq} ~ 1.1R_{Q} (Supplemental Information, section 4 and Fig. S5). The value of θ was consistent with the recently observed value in MLG^{40} in millikelvin regime.
With the electron temperature described by , we now discuss the temperature dependence of the critical exponents in detail. The exponent product vz can be evaluated from the slope of a double logarithmic plot of vs T, as shown in Fig. 4(b), for each gate sweep of the D_{density}tuned (sweep 1, 2, S1, and S2) and D_{gap}tuned (sweep 3 and S3) SIT. Note that, in this plot, the heating effect is excluded by replacing the measurement temperature by the electron temperature with T_{0} = 160 mK. For all gate sweeps, for T > 400 mK, the slope is described well by classical percolation (red line), whereas for T < 400 mK, the slope is consistent with the quantum percolation model (blue line). Successful elimination of the Joule heating effect in this study made it possible to identify a crossover between classical and quantum percolation, with the temperature as a tuning parameter for the decoherence.
Finitesize scaling analysis for bias electric field
Similar to the temperature dependence, the bias current (I) dependence of R_{sq} is also differentiated into two phases as shown in the inset of Fig. 5(a), such that R_{sq} decreases (increases) with lowering I in the superconducting (insulating) phase. Here, we emphasize that the value of θ satisfies the ‘safety’ criterion^{6} 2/θ > z/(z + 1), where z = 1, for the intrinsic fluctuations being dominant in the Jouleheating effect. This allowed fitting of the criticalexponent product v(z + 1) for both the classical and quantum percolation regions from the scaling behavior as a function of the bias electric field. The finitesize scaling analysis with the electric field (E) in Fig. 5(a,b) provides additional information of v(z + 1), because the E dependence of R_{sq} has the form R_{sq} = R_{sq,c} g[xE^{−1/v(z+1)}] near the critical point^{6,41}. Here, g is another scaling function. Similar to the Tvarying scaling in Fig. 3(c,d), Evarying scaling also gives two different values of v(z + 1) depending on the bias current range. For I = 9–15 nA, the best scaling was obtained with v(z + 1) = 2.66, which is close to the value of classical percolation [v(z + 1) = 8/3]. But, for the lower bias current range of I = 3–9 nA the best fit was obtained with v(z + 1) = 4.56, which is close to the value of quantum percolation [v(z + 1) = 14/3].
Products of critical exponents, vz and v(z + 1)
v(z + 1), together with vz from the Tvarying scaling, allows an independent determination of the critical exponents^{41} of v and z. We investigated several critical points for both of D_{density}–driven and D_{gap}–driven SIT as indicated in Fig. 2(a). For each gate sweep, we performed scaling analysis for both the temperature and electrical field dependences to evaluate the criticalexponent products of vz and v(z + 1), respectively. We summarized all the criticalexponent products in Fig. 5(c,d) for both of the classical and quantum percolation regimes. The corresponding scaling results for the classical percolation regime are shown in supporting information (Supplemental Information, section 5 and Figs S7 and S8). At higher temperatures (T > 400 mK) or for higher electric fields (I > 9 nA), with the averaged values of vz = 1.44 ± 0.13 and v(z + 1) = 2.81 ± 0.31 for all different gate sweeps (sweeps 1–3 and S1–S3), we get v = 1.37 ± 0.34 and z = 1.05 ± 0.27. This result supports the SIT of charged bosons (Cooper pairs) in the classical percolation universality class, which is consistent with the percolative transport nature of carriers through charged puddles in BLG at T = 4.2 K. At lower temperatures (T < 400 mK) or lower electric fields (I < 9 nA), the averaged values of vz = 2.83 ± 0.33 and v(z + 1) = 5.25 ± 0.63 give v = 2.42 ± 0.71 and z = 1.17 ± 0.37, which support the quantum percolation universality class for SIT.
Discussion
It is rather surprising that the BLG layer in the narrow region between the superconducting electrodes show the finitesize scaling behaviour of a 2D SIT, which is usually observed in homogeneous 2D systems. We believe that the temperature range of our transport measurements was sufficiently low as to allow the critical behaviour of the correlation length as a function of temperature. The observed temperaturedependent finitesize scaling was then governed by the temporal scale associated with the system temperature without apparent influence of the spatial scale of our device on the transition characteristics. Apparently, the spatial correlation length remained limited at finite temperatures (i.e., shorter than the spacing between the superconducting electrodes) as to neglect the effects arising from possible inhomogeneity of carrier transport or finite size of our system.
Our BLG devices provide a unique method to investigate the underlying mechanisms of SITs via accurate and reversible control of disorder. Electrical gating changed the average spacing between proximityinduced superconducting puddles to drive the QPT as Andreevreflected bound pairs at the Fermi level establish longrange coherence via percolative paths to yield the critical powerlaw behavior of percolation with negligible thermal intervention. At lower temperatures than the range of classical percolation behavior, direct control of the disorder enabled us to estimate the effective electron temperature and consequently to identify the crossover between classical and quantum percolation in a single device. Previously, these two regimes have only been obtained in separate systems belonging to weak and strong disorder regimes^{21}. Our proximitycoupled BLG system demonstrates that it is an exceptionally useful platform to study disorderinduced QPTs.
Methods
Device fabrication
Fabrication of the bilayergraphene Josephsonjunction devices relied on mechanical exfoliation of graphene^{42} on a highly doped silicon substrate, which was capped with a 300nmthick silicon oxide layer to form a bottom gate dielectric (d_{b} = 300 nm, ε_{b} = 3.9). Bilayer graphene was identified via optical contrast (Supplemental Information, section 6 and Fig. S9). Superconducting electrodes were defined using electron beam lithography and thermal evaporation of Pb_{0.9}In_{0.1} onto the bilayer graphene. Indium was added to minimize the grain size and the surface roughness^{35}. The junction area was covered with crosslinked poly(methyl methacrylate) (PMMA)^{43,44}, which formed a dielectric layer for the top gate (d_{t} ≈ 43 nm, ε_{t} = 4.5). A Ti/Au topgate electrode stack (where the layers were 5 and 145nmthick, respectively) was deposited and accurately aligned to cover most (~90%) of the junction area (Supplemental Information, Fig. S2). This allowed uniform gate control over the entire junction area. The thickness of the top gate dielectric, d_{t}, was determined from the shift of the resistance maximum of V_{t} by the modulation of V_{b}. V_{b,0} and V_{t,0} were determined by comparing the band gap, which was estimated from the temperature dependence in Fig. 6.
Lownoise measurements
The sample was maintained in thermal contact with the mixing chamber of a dilution fridge (Oxford Kelvinox AST) and cooled to a base temperature of 50 mK. Electrical measurement lines were filtered by a combination of twostage lowpass RC filters (with a cutoff frequency of ~30 kHz) mounted at the mixing chamber and pifilters (with a cutoff frequency of ~10 MHz), which were at room temperature. We used a conventional lockin technique with a bias current amplitude 1 nA r.m.s. at a frequency of 13.33 Hz for the temperaturedependent measurements, and a directcurrent bias for the biasfielddependent measurements.
Temperature dependence of conductance at CNP
At the charge neutrality point (D_{density} = 0), the Fermi level is placed in the middle of the bandgap E_{g}. Then, the conduction occurs with thermally activated carriers, providing the temperature (T) dependence of conductance, G_{TA}(T) = G_{TA,0} exp(−E_{g}/2k_{B}T), with Boltzmann constant k_{B}. However, in disordered bilayer graphene, bandgap is filled with the localized states such as conducting electron and hole puddles so that the carriers can hop across these states. Hopping transport is more pronounced at lower temperatures where the thermal activation (TA) is exponentially suppressed. As shown in Fig. 6(a), lowtemperature conductance agrees with variable range hopping (VRH) model in two dimensions, G_{VRH}(T) = G_{VRH,0} exp[−(T_{h}/T)^{1/3}], whereas hightemperature data agree with the TA conduction. The measurement was done at temperatures above ~7 K, with the Pb electrodes in the normal state. The charge neutrality point for the top gate was estimated to be V_{t,0} = −6.0 V, where the D_{gap} dependence of resultant fitting parameter E_{g} agrees with the theoretical prediction of selfconsistent tightbinding calculation as shown in Fig. 6(b). Similar TA+VRH transport properties were experimentally investigated in dualgated bilayer graphene^{45}. We could not directly determine V_{t,0} as it was beyond the chargeleakage voltage of the top gate. However, the uncertainty in the determination of V_{t,0} gives additional offsets to D_{gap} only but does not affect the scaling analysis discussed in the text.
Percolation transport in gapped bilayer grapheme
Carrier density inhomogeneity in twodimensional (2D) GaAs semiconducting systems induces the percolative metal–insulator transition (MIT) in the low carrier density regime^{31,46}. Similarly, graphene which has inhomogeneous charge puddles is also expected to exhibit the percolative MIT if a bandgap is introduced to separate the electron band from the hole band. For example, S. Adam et al.^{32} fabricated graphene into a nanoribbon structure to open a bandgap in graphene and demonstrated a 2D MIT of the classical percolation universality class. There is also theoretical prediction of percolation behavior for bilayer graphene with a finite bandgap^{47}. In our case, a vertical electric field opened a bandgap in bilayer graphene. We investigated transport properties of bilayer graphene in the presence of the superconducting proximity effect and analyzed them in the frame of percolative superconductor–insulator transition. To support the percolative transport characteristics in gapped bilayer graphene in the absence of superconductivity, we fabricated and performed control experiments with a device consisting of dualgated bilayer graphene in contact with nonsuperconducting Ti/Au electrodes. Optical image of the device and the measurement configuration are shown in Fig. 7(a). While injecting current (I = 1 nA r.m.s.) from I^{+} to I^{−}, voltage drop between V^{+} and V^{−} was measured as a function of bottom (V_{b}) and top (V_{t}) gate voltages at the base temperature of T = 4.2 K. According to the definition of D_{density} and D_{gap}, a resistance map is plotted as a function of D_{density} and D_{gap} in Fig. 7(b). D_{density} represents the carrier density (n = 5.52 × 10^{12} cm^{−2} × D_{density}∙V^{−1}nm) accumulated by the electrical gates, while D_{gap} determines opening of bandgap (E_{g}) in the bilayer graphene. D_{density} dependence of conductance (G) at a fixed D_{gap} = −0.8 V/nm [along the red line in Fig. 7(b)] is plotted in Fig. 7(c) on loglog scale. The bandgap is estimated to be E_{g} ~ 90 meV according to the selfconsistent tightbinding model^{30,48}. There appears three transport regimes depending on the D_{density} in both electron and hole sides. In a highly doped state (D_{density} > 0.5 V/nm), Fermi level exceeds the bandgap (E_{F} > 100 meV) so that the system is expected to be in the Boltzmann transport regime^{49} where . In the range of 0.1 V/nm < D_{density} < 0.5 V/nm, best fits to the critical behavior give exponents δ^{h} = 1.25 ± 0.02 in the hole side and δ^{e} = 1.25 ± 0.05 in the electron side, where n_{c} is the critical carrier density. They are close to the theoretical prediction δ = 4/3 for 2D classical percolation universality class. Near the charge neutrality point, D_{density} < 0.1 V/nm, G deviates from the percolation behavior and does not converge to zero but becomes saturated. This is because electron and hole puddles remain conducting even though the average carrier density vanishes at D_{density} = 0. Figure 7(d) shows the same data and corresponding fitting lines of Fig. 7(c) on linear scale. The linear relation between G and n in the Boltzmann transport regime (blue lines) and the crossover between percolation and Boltzmann transport regimes (arrows) are more pronounced.
Additional Information
How to cite this article: Lee, G.H. et al. Continuous and reversible tuning of the disorderdriven superconductorinsulator transition in bilayer graphene. Sci. Rep. 5, 13466; doi: 10.1038/srep13466 (2015).
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Acknowledgements
This work was supported by the National Research Foundation (NRF) through the SRC Center for Topological Matter (Grant No. 20110030046 for HJL), the GFR Center for Advanced Soft Electronics (Grant No. 2012M3A6A5055728 for HJL), the Basic Science Researcher Program (Grant No. 20100012134 for MCC), the Max Planck POSTECH/KOREA Research Initiative Program (Grant No. 20110031558 for KSP), and the CRI Program at SKKU (Grant No. 2012R1A3A2048816 for KSP), funded by the Ministry of Science, ICT and Future Planning. Work at BGU was supported by a grant from the Israel Science Foundation.
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Author notes
 GilHo Lee
Current address: Department of Physics, Harvard University, Cambridge, MA 02138, USA.
 Dongchan Jeong
Current address: Semiconductor R&D Center, Samsung Electronics Co. LTD., Hwasung 445701, Republic of Korea.
Affiliations
Department of Physics, Pohang University of Science and Technology, Pohang 790784, Republic of Korea
 GilHo Lee
 , Dongchan Jeong
 & HuJong Lee
Department of Physics, Sungkyunkwan University, Suwon 440746, Republic of Korea
 KeeSu Park
Department of Physics, BenGurion University of the Negev, Beer Sheva 84105, Israel
 Yigal Meir
Department of Applied Physics, Hanyang University, Ansan 426791, Republic of Korea
 MinChul Cha
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Contributions
D.J., G.H.L., and H.J.L. conceived the idea for the project. G.H.L. and D.J. fabricated the devices and carried out the experiments. All authors analyzed the data. M.C.C., Y.M., and K.S.P. provided theoretical consultation on the scaling analysis. M.C.C. and H.J.L. supervised the project. All authors contributed to the discussion and writing the manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding authors
Correspondence to MinChul Cha or HuJong Lee.
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