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Masaki Kashiwagi, Toshihiro Taen, Kazuhito Uchida, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Toshihito Osada

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[Weak localization on moiré superlattice in twisted double bilayer graphene](https://mdr.nims.go.jp/datasets/abc2eb6e-9b16-4f96-a873-79796bbcb061)

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Weak localization on moiré superlattice in twisted double bilayer grapheneJapanese Journal of AppliedPhysics     RAPID COMMUNICATION • OPEN ACCESSWeak localization on moiré superlattice in twisteddouble bilayer grapheneTo cite this article: Masaki Kashiwagi et al 2022 Jpn. J. Appl. Phys. 61 100907 View the article online for updates and enhancements.You may also likeGraphene-based heterostructures withmoiré superlattice that preserve the Diraccone: a first-principles studyXiangru Kong, Linyang Li and François MPeeters-Enhanced intervalley scattering inartificially stacked double-layer grapheneM Z Iqbal, Özgür Kelekçi, M W Iqbal et al.-Strain-induced suppression of weaklocalization in CVD-grown grapheneXiaochang Miao, Sefaattin Tongay andArthur F Hebard-This content was downloaded from IP address 144.213.253.16 on 31/10/2022 at 06:54https://doi.org/10.35848/1347-4065/ac934a/article/10.1088/1361-648X/ab132f/article/10.1088/1361-648X/ab132f/article/10.1088/1361-648X/ab132f/article/10.1088/1367-2630/16/8/083020/article/10.1088/1367-2630/16/8/083020/article/10.1088/0953-8984/24/47/475304/article/10.1088/0953-8984/24/47/475304Weak localization on moiré superlattice in twisted double bilayer grapheneMasaki Kashiwagi1* , Toshihiro Taen1, Kazuhito Uchida1, Kenji Watanabe2, Takashi Taniguchi3, and Toshihito Osada1*1Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan2Reserch Center for Functional Materials, National Institute for Materials Science, Tsukuba, Ibaraki 305-0044, Japan3International Center for Materials Nanoarchitectonics, National Institute for Materials Science, Tsukuba, Ibaraki 305-0044, Japan*E-mail: kashiwagi@issp.u-tokyo.ac.jp; osada@issp.u-tokyo.ac.jpReceived August 10, 2022; revised September 15, 2022; accepted September 19, 2022; published online October 10, 2022Negative magnetoresistance owing to weak localization (WL) was investigated in twisted double bilayer graphene (TDBG) as a function of the twistangle. The ratio of the intervalley scattering time to the intravalley scattering time, estimated using the WL formula for bilayer graphene, tended todecrease as the twist angle increased. This feature is qualitatively explained by the enhancement of intervalley scattering due to the reduction ofthe intervalley distance in the moiré Brillouin zone (BZ). This indicates that WL in the TDBG occurs with the reconstructed BZ and originates fromthe scattering process owing to the disorder in moiré superlattice. © 2022 The Author(s). Published on behalf of The Japan Society of AppliedPhysics by IOP Publishing LtdVan der Waals heterostructures1) (VdWHs) of atom-ically thin films with rotational misalignment have amoiré pattern, which induces rich physical proper-ties. Recent progress in the transfer techniques of atomicallythin films enabled the fabrication of arbitrary VdWHs, inother words, various combinations of thin films and rotationalmisalignment.2) Recent studies reported that twisted bilayergraphene (TBG), which consists of two monolayer graphene(MLG) stacked with a twist, shows a correlated insulatingstate,3,4) superconductivity,5,6) and ferromagnetism.7,8)Similar to TBG, twisted double bilayer graphene (TDBG),which is a twisted stack of two bilayer graphene (BLG) films,also shows multiple physical properties.9–11) These physicalproperties depend on the twist angle of TBG or TDBG, andthe twist angle has attracted remarkable attention as a degreeof freedom of the material.Weak localization (WL) and weak antilocalization (WAL)effects are characteristic phenomena that reflect the scatteringprocess in the system. WL is caused by the constructiveinterference between time-reversal scattering processes,forming a standing wave, and exhibits a negative magne-toresistance corresponding to the destruction of interference.In contrast, the WAL corresponds to the destructive inter-ference at a zero magnetic field by the (pseudo-) spinrotation, and exhibits the positive magnetoresistance. InMLG, WAL appears when intravalley scattering is dominantbecause of the pseudo-spin rotation as well as WL whenintervalley scattering is dominant,12,13) whereas only WL isobserved in BLG and multilayer graphene, where bothintravalley and intervalley scattering contribute to WL.14–16)As for TBG systems, the observation of WL suggested thatintervalley scattering is enhanced in TBG compared toMLG.17,18) Scattering by sharp point defects, local deforma-tion, and bending in an artificially stacked graphene systemwere proposed as possible origins of this enhancement withno quantitative analysis, but the related mechanism has notbeen clarified yet. These works considered valleys in theoriginal Brillouin zone (BZ) of each layer.In this study, we investigated the twist angle dependence ofnegative magnetoresistance owing to WL in TDBG, anddiscussed intervalley scattering in the mini BZ of the moirésuperlattice. The electronic structure and properties of twistedstacking systems depend on the twist angle caused by changes inthe moiré pattern. In order to evaluate electron scattering intwisted stacked graphene, we chose TDBG for a more directobservation of WL than in TBG. The BLG has a parabolic bandin the vicinity of the K(K′) valley, and the Berry phasesurrounding the quadratic band contact points at K(K′) is equalto zero (or 2π), leading to WL. However, in MLG with linearDirac cone dispersion, the Berry phase is equal to π, resulting inWAL. In TBG and TDBG, the band contact points at the cornersof the moiré BZ (K̄ and ¯ ¢K points) originate from those at the Kand K′ points in the original BZ of each layer. Therefore, theTDBG is expected to exhibit WL without any mixture of WALif the Berry phase surrounding the K̄ ( ¯ ¢K ) point is zero for arather large twist angle.TDBG samples were fabricated by the “tear and stack”technique.2) Both graphene and hexagonal boron nitride(hBN) were mechanically exfoliated and transferred on aSiO2/doped-Si substrate. The thickness of the flakes wasevaluated by optical contrast under a microscope, which wasconfirmed by Raman spectroscopy in the case of graphene.The schematic structure of our device is shown in Fig. 1(a).The TDBG was fabricated using the following processes:First, the top hBN was picked up using a polydimethylsi-loxane hemisphere coated with polymethylmethacrylate.Subsequently, a part of the BLG was contacted hBN andpicked up with tearing the BLG film. The remainder of theBLG remained on the substrate. In sequence, the substratewas rotated and the hBN/BLG structure was released on theremaining BLG. The TDBG was capped by hBN with athickness of approximately 10–20 nm and after etchingprocess to expose edge of TDBG, Au/Cr electrodes weredeposited by electron beam evaporation.19) An opticalmicroscopy image of the device is shown in Fig. 1(b). Thetwist angles of the devices were θ= 0.93°, 2.93°, 4.84°, and7.15°, respectively. Those twist angles were estimated fromthe microscope images of the mark on substrate taken beforeand after rotation. The accuracy of this method itself isestimated about 0.01̊, although this angle may differ slightlyfrom the actual twist angle. We also prepared a BLG and afour-layer graphene (4LG) sample capped with hBN as aContent from this work may be used under the terms of the Creative Commons Attribution 4.0 license. Any further distribution of thiswork must maintain attribution to the author(s) and the title of the work, journal citation and DOI.100907-1© 2022 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdJapanese Journal of Applied Physics 61, 100907 (2022) RAPID COMMUNICATIONhttps://doi.org/10.35848/1347-4065/ac934ahttps://crossmark.crossref.org/dialog/?doi=10.35848/1347-4065/ac934a&domain=pdf&date_stamp=2022-10-10https://orcid.org/0000-0002-1873-4075https://orcid.org/0000-0002-1873-4075mailto:kashiwagi@issp.u-tokyo.ac.jpmailto:osada@issp.u-tokyo.ac.jphttps://creativecommons.org/licenses/by/4.0/https://doi.org/10.35848/1347-4065/ac934areference. The resistance was measured using the two-terminal method at several temperatures from 1.8 to 30 K,and the carrier density n was tuned to 7.5× 1011 cm−2 for allsamples by adjusting the back-gate voltage. This carrierdensity was determined from the amount of carriers injectedwith the charge neutral point (CNP) as the origin. The CNPwas found at the peak in the back-gate voltage dependence ofresistance [Fig. 1(c)]. From the carrier density and the twistangle, we calculated the filling factor /n = n n4 ,M whichindicates the ratio of carrier occupation in the lowest energyband. Here / /q=n a8 sin 2 3M2 2 is the density of moiréunit cell. As shown in Table I, the filling factor n was lessthan a half in all TDBG samples. Therefore, the carriersoccupied the lowest moiré band around the K̄ and ¯ ¢K points.Figures 2(a)–2(c) show the magnetoconductivity( ) ( ) ( )( ) ( )s s sD = - = -r rB B 0B1 10xx xxof the TDBG withthe twist angles of q = 7.15 , q = 4.84 , and q = 2.93 ,respectively under normal magnetic fields. The magnetocon-ductivity increased as the magnetic field increased for allTDBG samples at low temperatures. This behavior isoriginated from the WL effect. The increase of magnetocon-ductivity became small as the temperature increased andnegligible above T= 30 K.As indicated by the solid curves in Fig. 2, the observedpositive magnetoconductivity was fitted using the followingequation:20–22){ }( ) ( ) ( )( )( )sD= - +p + + +f f fBF F F2 . 1ehBBBB BBB B B2 i i2*Here ( )( ) = + Y +F z zln ,z121 where Y is the digammafunction. =f tfB ,i eD, , 4 i, ,* *where D, tf, t ,i and t* are thediffusion coefficient, dephasing time due to inelastic scat-tering, elastic intervalley scattering time, and elastic intra-valley scattering time, respectively. This equation wasoriginally proposed for the WL in AB-stacking BLGs, whichhas parabolic band dispersions in the vicinity of the K (K′)points with the zero Berry phase surrounding them.According to the continuum model,23,24) the TDBG systemalso has parabolic dispersions around the K̄ and ¯ ¢K points atthe corners of the moiré BZ, which originate from the K (K′)points of each BLG layer. According to recent theories,25–27)the Berry phase surrounding the K̄ and ¯ ¢K points in the AB-AB stacking TDBG is zero, because the Berry curvatureremains zero when applied perpendicular electric field is(a) (b) (c)(d)(e)Fig. 1. (Color online) (a) Schematic of the device structure. (b) Optical microscope image of TDBG device capped by hBN. In the case of this sample, theresistance between terminal 1 and 2 was measured, and back-gate voltage was induced through terminal 4. (c) Back-gate-voltage dependence of longitudinalresistance measured by two-terminal method at 4.4 K in the TDBG sample with θ = 0.93°. (d) Schematics of the valley structure, the intervalley scattering andthe intravalley scattering. Top figure shows the valley structure of MLG, BLG and 4LG. Bottom figure shows the valley structure of TDBG. (e) Schematics ofthe band structure of MLG, BLG, 4LG, and TDBG. Berry phase in the vicinity of K(K′) point is shown respectively.100907-2© 2022 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdJpn. J. Appl. Phys. 61, 100907 (2022) M. Kashiwagi et al.small. Therefore, the WAL contribution should be precluded,and we can safely use the WL formula of Eq. (1) as a fittingfunction.To validate this choice of fitting function, we alsoperformed fitting using the formula for MLG, in which thesign of the third term in Eq. (1) was reversed. This signchange is due to the WAL related to the Berry phase π in theMLG. As shown in Fig. 2(d), Eq. (1) provided a better fitthan the formula for the MLG.The fitting parameters of each sample are listed in Table I.There is a small increasing trend on ti and a decreasing trendon t* as the twist angle of TDBG increased. However, theabsolute values of these parameters depend on the quality ofsamples, such as the amount of scattering impurities, defects,and dislocations present. Therefore, we employed the ratio/t ti* to represent the enhancement of the intervalley scat-tering against intravalley scattering by compensating forsample dependence.The experimental results indicated that /t ti* in TDBG waslarger than that in BLG. This suggests that intervalleyscattering was enhanced in TDBG compared to that inBLG, and this enhancement is similar to that in the TBGreported in previous studies.17,18) On the other hand, it wasalso shown that /t ti* in TDBG decreased as the twist angleof TDBG increased. In previous studies,18) the enhancementof intervalley scattering was ascribed to extrinsic origins, thatis, sharp point defects, local deformation, and bending byartificial stacking of two graphene layers. However, weTable I. Dephasing time tf, intervalley scattering time t ,i and intravalley scattering time t* at T = 4.4 K of four TDBG samples with different twist angles,obtained from the fitting of Eq. (1). In addition, the BLG and 4LG data are presented for reference.TDBG BLG 4LGθ (°) 0.93 2.93 4.84 7.15 — —ΔK 0.016 0.05 0.083 0.12 1 1Mobility (cm2 V−1·s−1) 2.1 × 103 8.3 × 102 6.9 × 103 1.2 × 103 4.6 × 103 1.9 × 103Carrier density (cm−2) 7.5 × 1011 7.5 × 1011 7.5 × 1011 7.5 × 1011 7.5 × 1011 7.5 × 1011Filling factor 0.38 0.038 0.014 0.0063 — —Diffusion coefficient (cm2 s−1) 19 91 7.6 × 102 1.3 × 102 5.0 × 102 2.1 × 102τf (ps) 3.9 3.7 4.2 4.6 4.6 1.7τi (ps) 2 3.2 3.0 4.2 47 22τ* (ps) 0.24 0.2 0.14 0.099 0.4 0.3τ*/τi 0.1 0.06 0.05 0.03 0.009 0.01(a) (b)(c) (d)Fig. 2. (Color online) Magnetic-field dependence of the conductivity of TDBG samples with (a) θ = 7.15°, (b) θ = 4.84°, and (c) θ = 2.93° at severaltemperatures. The solid lines indicate the fitting results of Eq. (1). (d) Comparison between the results using the fitting function for BLG (black solid line) andMLG (gray dashed line). The red dots are the data of TDBG with θ = 0.93° at T = 4.4 K.100907-3© 2022 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdJpn. J. Appl. Phys. 61, 100907 (2022) M. Kashiwagi et al.propose an intrinsic origin based on the formation of a moiréBZ to explain the twist angle dependence of intervalleyscattering in TDBG.In TBG or TDBG with a moiré superlattice, the moiré BZcan be considered in the continuum model, as schematicallyshown in Fig. 3(a). The distance between the K̄ and ¯ ¢K pointsof moiré BZ is given by ( )/ /qD = KK 2 sin 2 .0 Therefore,the size of the moiré BZ depends on the twist angle and issmaller than that of the original BZ. Most electronicstructures and properties can be described by the moiré BZ.The WL on the moiré superlattice can be discussed byconsidering intravalley and intervalley scattering in the moiréBZ with K̄ and ¯ ¢K valleys. Here, intervalley scattering mustbe largely affected by the reduction of the distance betweenthe K̄ and ¯ ¢K valleys in the moiré BZ. When the Fouriercomponent of the scattering potential v(q) is a decreasingfunction of q= ∣q∣, intervalley scattering with∣ ∣= - = Dq K K’ K is considered to be enhanced at smalltwist angles.For a more quantitative evaluation, we considered thelong-range Coulomb-type scattering potential with inverse ofdistance dependence (∼1/r) as an example. When we performtwo-dimensional Fourier transform for Coulomb-type poten-tial as,∬( )∣ ∣{ · }p= -v i dkrk r r121expThe result becomes ( ) /=v k k1 . In this case, the scatteringprobability associated to the WL is proportional to∣ ( )∣ /µv qq 1 ,2 2 and we estimated the twist angle dependencyof the intravalley scattering and intervalley scattering due toCoulomb-type potential. Since we fixed the carrier density,the size of the Fermi pockets at K̄ and ¢K is unchanged withthe twist angle, which means that the values of q related tothe intravalley scattering (time-reversal pair in each intra-valley) are irrespective of the twist angles. The intravalleyscattering probability is therefore not affected by the twistangle. By contrast, the intervalley scattering probability isproportional to the twist angle through the relation∣v ( ¯ ¯-K K’)∣2 µ 1/(ΔK)2. Therefore, we can expect that theratio between intravalley and intervalley scattering times/t ti* is proportional to ( )/ D1 K 2 in the TDBG. Figure 3(b)shows the DK dependence of /t ti* of the TDBG withdifferent twist angles. Here, DK is normalized by∣ ∣D = -K KK ’ ,0 which is the distance between points Kand K′ in the BLG. The solid curve indicates the fitting result,assuming ( )/ /t t µ D1 K .i2* The observed trend of /t ti* withdifferent twist angles is roughly reproduced by this relation.Although requiring a more detailed theoretical explanation,this agreement is consistent, considering the scattering processin moiré BZ. Moreover, this agreement also suggests that theWL in TDBG originates from quantum interference owing tomultiple scattering due to a disorder in moiré superlattice, butnot that in the original BLG lattice scale disorder. For example,the twist angle disorder28) can be considered as the candidateof creating the intervalley scattering.From the above discussion, it can be expected that 4LGshows the highest /t t ,i* because 4LG is considered as TDBGwith θ = 0° in a certain view. However, in our experiment,/t ti* in 4LG was not larger than the other TDBGs. In theresult of fitting to Eq. (1), it is considered that the intervalleyscattering in 4LG is captured in the same valley structure ofMLG or BLG and not of TDBG. Therefore, it does not seemappropriate to consider 4LG as TDBG with θ = 0°, and it issuggested that the difference between the fitting result ofTDBG and 4LG originates from the existence of moirésuperlattice.For the realization of the interference in the moiré BZ, thecharacteristic scattering length must be comparable to orlarger than the scale of the moiré superlattice in a real system.To verify this, we compared the intervalley scattering lengtht=L vi F i with the size of the moiré unit cell( ( ))/ /q=L a 2 sin 2 .0 Here, = pvF mn2*is the Fermi velo-city, m* is the effective mass of the carrier, n is the carrierdensity, and a0 is the the lattice constant of graphene. At atwist angle of q » 1 , the effective mass increased to»m m0.3 ,e* 29) where me is the electron rest mass. In thecase of θ = 0.93°, using this effective mass, the intervalleyscattering length was estimated as Li= 84 nm. Because it islonger than the moiré period, L= 15 nm, electrons can sensethe moiré potential during the intervalley scattering process.For the other twist angle samples with no remarkable massenhancement, we adopted the effective mass of the BLG,=m m0.03 .e* 30) We found that the intervalley scatteringlength was longer than the moiré period in all case. This factjustifies the use of Eq. (1) as the fitting function. In addition,we compared the intravalley scattering length L* and size ofthe moiré unit cell in the same manner. We found that theintravalley scattering length was longer than the moiré periodfor the cases of θ> 1°, except for the case of θ = 0.93°,where the estimated intravalley scattering length L*= 9 nm(a)(b)Fig. 3. (Color online) (a) Schematics of moiré BZ of two TDBG withdifferent twist angles. The edge length of moiré BZ, ΔK, corresponds to thedistance between K (or K′) of two BLG. (b) DK dependence of /t ti* inTDBG samples at =T 4.4 K. DK is normalized by ∣ ∣D = - ¢K KK0 inBLG. The solid line shows the fitting result of ( )/ D1 K .2100907-4© 2022 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdJpn. J. Appl. Phys. 61, 100907 (2022) M. Kashiwagi et al.was less than the moiré period. Although the large error in thefitting parameters may be responsible for this unfavorableresult, most plausible reason is that the fitting function forBLG of Eq. (1) can be hardly applied to a TDBG with a twistangle of less than 1° because the distance between K̄ and ¯ ¢Kpoints becomes very small, and the hybridization between thetwo BLG layers complicates the band structure.26,27)Therefore, for a small twist angle θ< 1°, we need the otherpicture of the WL in a moiré superlattice system with a verylarge period.We note that our result does not exclude the possibility,that the enhancement of the intervalley scattering in twistedstacking graphene is a result of sharp point defects, localdeformation, and bending, as proposed in previous studies,18)in which the valleys in the original BZ of each layer wereconsidered; however, to explain the twist angle dependenceof the enhancement of intervalley scattering, it seemsappropriate to consider that the scattering occurs in themini-moiré BZ rather than the original BZ of each layer.Related to our result, interminivalley scattering was studiedby measurement of high-temperature magnetooscillations inrecent studies,31,32) and strong interminivalley scattering wasobserved.31) According to the result, the intervalley andintravalley scattering times were of similar order in thesmall-angle TBG with θ= 1.65°. This agrees with our resultthat the intervalley and intravalley scattering times becomeclose to comparable values as the twist angle decreases.In summary, we studied the twist angle dependence of theWL in TDBG. Intervalley and intravalley scattering timeswere obtained for several TDBG samples with differenttwist angles using the WL formula for the BLG. The ratio ofthe intervalley scattering time to the intravalley scatteringtime tended to decrease as the twist angle increased, whichcan be explained by the twist angle dependence of thedistance between K̄ and ¯ ¢K points in the moiré BZ. Thisimplies that the WL in the TDBG occurs for the moirésuperlattice with the reconstructed BZ and originates fromthe scattering process owing to the disorder in moirésuperlattice.Acknowledgments This work was partly supported by JSPS KAKENHI,Grant Numbers JP19K14655, JP20H01860, and JP21K18594.ORCID iDs Masaki Kashiwagi https://orcid.org/0000-0002-1873-40751) A. K. Geim and I. V. 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