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Mohit Kumar Jat, Shubhankar Mishra, Harsimran Kaur Mann, Robin Bajaj, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), H. R. Krishnamurthy, Manish Jain, Aveek Bid

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This document is the Accepted Manuscript version of a Published Work that appeared in final form in Nano Letters, copyright © 2024 American Chemical Society after peer review and technical editing by the publisher. To access the final edited and published work see https://doi.org/10.1021/acs.nanolett.3c04223.[In Copyright](http://rightsstatements.org/vocab/InC/1.0/)

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[Controlling Umklapp Scattering in a Bilayer Graphene Moiré Superlattice](https://mdr.nims.go.jp/datasets/dad6d437-73bf-4a8d-953a-a33db26eb66f)

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Controlling Umklapp scattering in bilayergraphene moiré superlatticeMohit Kumar Jat1, Shubhankar Mishra1, Harsimran Kaur Mann1, RobinBajaj1, Kenji Watanabe2, Takashi Taniguchi3, H. R. Krishnamurthy1,Manish Jain1 and Aveek Bid1∗1Department of Physics, Indian Institute of Science, Bangalore 560012, India2 Research Center for Electronic and Optical Materials, National Institute for Materials Science,1-1 Namiki, Tsukuba 305-0044, Japan3 Research Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1Namiki, Tsukuba 305-0044, JapanE-mail: aveek@iisc.ac.inAbstractWe present experimental findings on electron-electron scattering in two-dimensional moiréheterostructures with tunable Fermi wave vector, reciprocal lattice vector, and band gap. Weachieve this in high-mobility aligned heterostructures of bilayer graphene (BLG) and hBN.Around half-filling, the primary contribution to the resistance of these devices arises fromelectron-electron Umklapp (Uee) scattering, making the resistance of graphene/hBN moirédevices significantly larger than that of non-aligned devices (where Uee is forbidden). We findthat the strength of Uee scattering follows a universal scaling with Fermi energy and has anon-monotonic dependence on superlattice period. The Uee scattering is electric field tunableand is affected by layer-polarization of BLG. It has a strong particle-hole asymmetry – theresistance when the chemical potential is in the conduction band is significantly lesser than1aveek@iisc.ac.inwhen it is in the valence band, making the electron-doped regime more practical for potentialapplications.KeywordsUmklapp scattering, bilayer graphene, moiré superlattice, layer polarization, Brown-Zak oscillations.In a Galilean-invariant electron liquid, normal electron–electron scattering does not cause a lossof the momentum imparted to the electrons by the driving electric field; consequently, it cannot lead to electrical resistance. A realistic Fermi liquid is, however, not Galilean invariant –a finite coupling to an underlying lattice provides a mechanism for the momentum relaxationof the quasiparticles via the Umklapp process1. Umklapp electron-electron (Uee) scattering isthe fundamental mechanism that allows momentum transfer from electrons to lattice and impartselectrical resistance to the metal2–8. In this process, the crystal lattice gives a momentum kick toa pair of interacting electrons, backscattering them to the other side of the Fermi surface. Theirquasi-momentum is conserved, modulo a reciprocal lattice vector G,k1 + k2 = k3 + k4 +G (1)Here ℏk1,2 and ℏk3,4 are the initial and final quasi-momenta of the two electrons near the Fermilevel, respectively, and G is a non-zero reciprocal lattice vector of the crystal. This stringentconservation constraint, coupled with the lack of tunability of the Fermi wave vector, makesexperimental identification of Uee processes in normal metals challenging7–9. Notable exceptionsare heavy-fermionic systems whose large effective quasiparticle mass leads to an appreciable Uee-mediated resistance at very low temperatures (≈ 100 mK)10.In the limit of nearly free electrons, one can view the Uee scattering as a two-stage process: Inthe first step, an electron-hole pair is excited into a virtual state by an electron, followed by thescattering of one of these particles by the periodic lattice potential. The temperature dependence of2the Uee scattering process at a finite temperature is thus set by the size of the scattering phase space(∝ kBT/EF ) for each electron; only the quasiparticles residing within a width of order kBT aroundthe Fermi energy EF can undergo binary collisions. Consequently, the Uee contribution to the sheetresistance in 2D goes as R□Uee = fnT2 11. fn ∝ E−2F is a material-dependent parameter12–14.Note, however, that Uee need not be the only source of T 2 -resistivity in a material13,15–18. Aclaim that the dominant source of scattering is the Uee process should be backed up by a (1)quantification of the prefactor fn, (2) a demonstration of the scaling of fn ∝ E−2F , and (3) rulingout other competing mechanisms (e.g. electron-phonon scattering18) that can give T -dependentcharge scattering.Graphene-based moiré superlattices 4,6,19–25 provide a system with precise tunability of the reciprocallattice vectors G (via the twist angle between the constituent layers) and the Fermi wave vectorskF (by controlling the carrier density n through electrostatic gating). It thus provides a vast phasespace in which Eqn. 1 may be satisfied, and the scaling of fn versus EF can be verified. Recentcalculations (that treat both the electron–electron Coulomb interaction and the moiré superlatticepotential perturbatively) predict that in aligned heterostructures of Bernal bilayer graphene (BLG)and hBN, Uee scattering processes should be the primary source of resistance5.In this Letter, we experimentally verify that in high-mobility moiré superlattices of BLG and hBN,Uee is the dominant source of resistance near half-filling. Our studies show that the strength ofUee depends non-monotonically on the superlattice period. This is at par with recent theoreticalpredictions5 and in sharp contrast to observations in single-layer graphene-based superlattices4.We illustrate the tunability of strength of the Uee process (quantified by fn) with displacementfield, D and carrier density, n. Additionally, we demonstrate a strong particle-hole asymmetry inthe strength of the Uee process, whose origin can be traced to the moiré potential having a muchstronger effect on the valence band than on the conduction band 4,5. Furthermore, we demonstratethe high tunability of Umklapp resistivity with an external vertical electric field, emphasizingthe potential for precise control over the electronic properties of bilayer graphene superlattices.3Finally, we show that these processes are completely absent in non-aligned devices.High-quality hBN/BLG/hBN heterostructures were fabricated using the dry transfer technique(Supplementary Information, section S1)26–28. The top hBN was aligned at nearly zero degreeswith BLG, and the bottom hBN was intentionally misaligned to a large angle to ensure that amoiré pattern forms only between top hBN and BLG (Fig.1(a)). The device is in hall bar geometry(Fig.1(b)) with dual gates to tune the carrier density n and the vertical displacement field Dindependently via n = [(CtgVtg +CbgVbg)/e+nr] and D = [(CtgVtg −CbgVbg)/2+Dr]. Here Cbg(Ctg) is the back-gate (top-gate) capacitance, and Vbg (Vtg) is the back-gate (top-gate) voltage. nrand Dr are the residual number density and displacement field in the graphene due to impurities.The direction of the negative displacement field (D) is marked schematically in Fig.1(a). In themain text, we provide the data for a device M1 (with twist angle ≈ 0◦ and superlattice wavelength14 nm), unless otherwise mentioned. The data for three more hBN/BLG/hBN superlattice devices,labeled M2, M3 and M4 with twist angle ≈ 0.26◦, 0.47◦ and 1.70◦ respectively, and with superlatticewavelength ≈ 13.64 nm, 12.73 nm and 7.20 nm respectively, are presented in SupplementaryInformation. We also present data for a non-aligned hBN/BLG/hBN device (labeled N1) to comparethe T -dependence of resistance between Uee-allowed (aligned devices) and Uee-forbidden (non-aligned devices) systems.The measured longitudinal resistance Rxx on device M1, at 2 K temperature shows a peak atthe charge neutrality point (CNP), nCNP = 0 and moiré satellite peaks at nM = ±2.30 ×1016 m−2 (Fig. 1(c)). The mobility at CNP is extracted to be 350, 000 m2V−1s−1. Quantum Hallmeasurements at a perpendicular magnetic field of B = 5 T establish that both spin and valleydegeneracies are lifted, indicating the high quality of the device (Supplementary Information,section S5); these measurements are used to calibrate the values of Cbg and Ctg. The anglehomogeneity of the device is ascertained by comparing the Rxx data measured in different configurations(Supplementary Information, section S2).Our results for Rxx as a function of carrier density n and electric field D/ϵ0, shown in Fig.1(d)4ascertain that the values of the moiré gap in carrier density nM are independent of the appliedelectric field. The plot can be divided into four quadrants labeled I-IV. In quadrants I (n > 0,D/ϵ0 > 0) and III (n < 0, D/ϵ0 < 0), at a finite D, the occupied electronic states near the Fermienergy are predominantly localized (marked with a red color oval) in the bottom layer of BLG(away from moiré interface) and are weakly localized (marked with a blue color oval) in the toplayer of BLG (close to moiré interface). This leads to the suppression of moiré effects and lowresistance value of satellite peak in these quadrants. The opposite effect is seen in quadrants II(n > 0, D/ϵ0 < 0) and IV (n < 0, D/ϵ0 > 0), the occupied electronic states are predominantlylocalized in the top layer of BLG (close to moiré interface) leading to the enhancement of moiréeffects and higher resistance of the satellite peaks in these quadrants. Later in this Letter, weexplore the consequences of this displacement-field-induced layer polarization on Uee scattering.The moiré periodicity of the system is estimated from Brown-Zak oscillation measurements at T =100 K (Fig.1(e)). Thermal broadening smears out Landau oscillations at this elevated temperature,and only Bloch oscillations survive 19,29–31. A Fourier spectrum of the oscillations yields the inverseperiodicity or the ‘frequency’ of the oscillations to be Bf = 24.2 T (Fig.1(f)). Observation of onlya single frequency rules out the double alignment of the BLG with hBN 19,32. Using the relationS = h/eBf (S being the real space area of the moiré superlattice cell, h: Planck’s constant, e:electronic charge), the moiré wavelength is calculated to be λ = 14 nm and the carrier densitycorresponding to filling the bands just up to the moiré gaps is 4/S = 2.30 × 1016 m−2; the factorof 4 arises from the two-fold spin-and valley-degeneracy of graphene. This value of carrier densitymatches nM exactly, validating the number density corresponding to the moiré gap obtained fromzero-magnetic field Rxx measurements. The twist angle between BLG and hBN corresponding tothis moiré wavelength is approximately 0◦ indicating near-perfect alignment between the top hBNand the BLG.Fig. 2(a) shows the plots of the zero-magnetic field longitudinal sheet resistance R□ = Rxxw/l (wand l are the width and length of the channel respectively with w/l = 1.5) versus the moiré band5filling fraction n/n0 over a temperature range 5 K < T < 300 K at zero displacement field. Here,n0 = 1/A = nM/4 is the carrier density at one-forth filling of the moiré band. With increasingtemperature, one notices a sharp increase in R□ around n/n0 = −2 (Fig.2(d)); this feature iscompletely absent in non-aligned BLG devices (Supplementary Information, section S3). As weestablish below, this rapid increase in R□ with T arises from the Umklapp scattering in the device.At T = 0, Uee is suppressed, and the resistivity is dominated by disorder scattering12. To mitigatethe effect of static disorder scattering, we henceforth focus on ∆R□(T ) = R□(T )−R□(5 K). Themagnitude of R□(5 K) at n/n0 = −2 is ≈ 14Ω. In Fig. 2(b), we plot ∆R□/T2 versus n/n0 over atemperature range from 30 K to 110 K – the data at all temperatures collapse onto a single curve inthe filling fraction range −2 ≤ n/n0 ≤ −1 (marked by the dotted ellipse) showing that ∆R□ ∝ T 2over this range. This can be better appreciated from the inset, which shows the data over a narrowrange around n/n0 = −2. Fig. 2(c) plots the ∆R□ versus T 2 to better show the electron-holeasymmetry over a range of n/n0. The linearity of the plots of sheet resistance versus T 2 in thiscarrier density regime persists till about T ≃ 110 K, establishing Uee scattering as the sourceof resistance (Fig. 2(c)). This temperature is of the order of the Bloch–Grüneisen temperaturein graphene. Above this T , electron-phonon scattering starts becoming the dominant source ofresistance, and the quadratic relation between ∆R□ and T breaks down33–35. Fig. 2(d) showsa comparison of ∆R□ for the five devices – the strong quadratic T -dependence seen in aligneddevices is completely absent in the non-aligned device where Uee is forbidden from phase-spacearguments. Inset in Fig. 2(d) shows fn versus twist angle for the aligned devices, illustrating thenon-monotonicity of umklapp strength on the twist angle.To understand the number density limits over which Umklapp processes are seen, recall that atvery low n/n0, transport in graphene is dominated by electron-hole puddles33,36,37; this gives apractical lower bound of n/n0 at which e-e scattering is detectable5. A more accurate lower limitis obtained by the constraint that the Uee process imposes on the Fermi wave vector kF (Eqn. 1),this sets a lower bound on |n/n0| equal to π/(2√3) = 0.91. (see Supplementary Information,6section S4). At the other extreme, at high number densities, one begins to encounter electron-hole scattering processes at the principal mini band edges because of the moiré induced van-Hovesingularity (Fig.3(b)), which masks the Umklapp scattering process5.Before proceeding further, we eliminate the other probable causes that are known to lead to a T 2-dependence of the resistance. In a system with different carrier types/masses (as is the case nearthe primary and secondary gaps or van Hove singularities), the transfer of momentum between thetwo carrier reservoirs can lead to a resistivity with T 2 dependence38–40. This consideration guidesus to avoid filling fractions that lead to Fermi levels close to these regions of the moiré bands andconfine our analysis to the filling fraction range −2 ≤ n/n0 ≤ −1, as shown in Fig.3(a). Wenote that, in low-mobility dilute alloys, the thermal motion of impurity ions can also give rise to aT 2-dependent resistance41; this scenario does not apply to our high-mobility heterostructures.A phenomenological treatment, based on the Rice-Kadowaki–Woods scaling analysis13,14 yields:fn ∝ ℏe2(kBEF)2(2)In Fig.3(b), we plot A = fnt (t = 0.8 nm is the thickness of BLG) as a function of the Fermienergy EF along with the compilation of data on several different materials12. A very good matchis obtained, emphasizing the universality of the value of fn.Having established Uee as the source of quasiparticle scattering in bilayer graphene/hBN moirénear half-filling (n/n0 = −2), we now shift our focus on the effect of inter-layer potential asymmetry(tuned using D) on the Umklapp scattering in the quadrant III and IV of Fig. 1(d). Fig. 4(a)plots ∆R□/T2(n/n0 = −2) versus T for several different values of D/ϵ0. We find that thetemperature exponent of the resistance α = dln(∆R□)/dln(T ) ≈ 2 for −0.3 V/nm ≤ D/ϵ0 ≤0.3 V/nm (Fig. 4(b)). In this range of D/ϵ0, we find a substantial increase in the scatteringstrength with increasing D/ϵ0 in conformity with theoretical predictions5 (Fig. 4(c)). Fig. 4(d)plots fn(D,n/n0 = −2) versus D/ϵ0 over the temperature range 60 K– 100 K. These data points7collapse on top of each other with fn growing quadratically with D/ϵ0.Note that fn has a slight asymmetry under sign-reversal of D/ϵ0. To understand this, we recallthat the sign of layer polarization in BLG depends on the direction of D. A positive D–field (asdefined in Fig. 1(a)) increases the potential energy of electronic states in the lower layer of BLGas compared to those in the upper layer of BLG. For negative n, the occupied electronic states aremainly localized in the top layer of the BLG (that forms the moiré with the hBN)5. For the negativeD–field, on the other hand, the occupied electronic states are mainly localized in the bottom layerof the BLG (that does not form the moiré with the hBN). We postulate that the combined effectof this asymmetry of layer polarization on the sign of D and the asymmetry of the moiré potentialinherent in this device architecture ultimately manifests as fn(D) ̸= fn(−D).With further increase in the displacement field, α deviates from two, indicating a suppressionof Umklapp processes for |D/ϵ0| > 0.3 V/nm. We do not have a clear understanding of theorigin of this. One plausible reason can be that at large D, the trigonal warping becomes strong,severely limiting the phase space over which Eqn. 1 may be satisfied42. A related effect of thetrigonal warping is the formation of overlapping electron-hole bands at certain number densities– the scattering between thermally excited electrons and holes then masks Uee processes5,42. Asecond possible cause of the suppression of Uee at high D can be the strong modification of theBLG band by the displacement field (this includes layer-polarization, the opening of a band gap,and enhanced trigonal warping) leading to strong Zitterbewegung, which becomes the relevantscattering mechanism at large |D|43. Further experimental and theoretical studies are required toverify if any of these is indeed the cause for suppression of Umklapp scattering with increasing D.To conclude, our experiments unequivocally establish Umklapp scattering to be the leading sourceof resistance in hBN/BLG superlattices in certain filling fraction ranges. Our findings on hBN/BLGsuperlattice differ from recent studies on hBN/SLG superlattice4 in several significant aspects. InSLG hBN moiré, RUee increases monotonically with increasing superlattice period and chargecarrier density4. In contrast, RUee in BLG moiré superlattice is predicted to have a non-monotonic8dependence on superlattice period5. In this Letter, we have experimentally verified this prediction.Additionally, bilayer-based systems provide strong electric field tunability of the band gap andlayer polarization and thus have an enormous scope for room-temperature applications 44–49. Wehave shown that the strength of Uee increases rapidly with the increasing strength of the displacementfield; this fact must be factored in when designing any D-field controlled superlattice devicearchitectures. Additionally, we find the strength of Uee scattering to be stronger in BLG/hBNsuperlattice than in SLG/hBN superlattice (Supplementary Information, section S6).With the presently available technology, the best quality BLG field effect devices are formed whenencapsulated between a crystalline insulator, like hBN 50–52. As the growth of graphene in hBNleads to aligned layers 53–55, it is imperative to understand the significant sources of Joule heating insuch systems for optimal room-temperature operations. Our present study achieves this and shouldmotivate further studies in related systems like twisted bilayer graphene and twisted bilayers oftransition metal dichalcogenides.While this manuscript was under review, we became aware of a preprint 56 which demonstratesthat at n/n0 = −2, transport in BLG/hBN moiré is dominated by Umklapp scattering.Acknowledgment: A.B. acknowledges funding from U.S. Army DEVCOM Indo-Pacific (Projectnumber: FA5209 22P0166) and Department of Science and Technology, Govt of India (DST/SJF/PSA-01/2016-17). M.J. and H.R.K. acknowledge the National Supercomputing Mission of the Departmentof Science and Technology, India, and the Science and Engineering Research Board of the Departmentof Science and Technology, India, for financial support under Grants No. DST/NSM/R&D_HPCApplications/2021/23 and No. SB/DF/005/2017, respectively. M.K.J. and R.B. acknowledge thefunding from the Prime Minister’s research fellowship (PMRF), MHRD. S.M. acknowledges thefunding from the National post doctoral fellowship (N-PDF), SERB. K.W. and T.T. acknowledgesupport from the JSPS KAKENHI (Grant Numbers 21H05233 and 23H02052) and World PremierInternational Research Center Initiative (WPI), MEXT, Japan.Author contributions: M.K.J., S.M., H.K.M., and A.B. conceived the idea of the study, conducted9the measurements, and analyzed the results. T.T. and K.W. provided the hBN crystals. R.B.,M.J., and H.R.K. developed the theoretical model. All the authors contributed to preparing themanuscript.100 2 401232 3 4 5 6 7 8 9 100204060801000 20 40 6000.40.8Rxx(kΩ)n (1016 m-2)n(1016m-2)Gxx(e2 /h)q/p1.972.663.354.056.858.950.083 0.165 0.248 0.331 0.4141/B (T-1)FFTamplitude(a.u.)Bf (T)24.2 T-nM-nM nM  nM nCNP  nCNPIII VI 5.0 2.5 0.0-2.5-5.0                    -0.8              -0.4                0.0                0.4               0.8(c) (e)(d) (f)10 II I(a)(b)2000Rxx (Ω)D/ε0 ( V/nm     )IRxyRxxVbgVtgmoiréno-moiréBLGD < 0top hBNbottom hBN5 υmFigure 1: Characteristics of the moiré device M1. (a) Schematic of the device layers, indicatingmoiré (no-moiré) superlattice formation between top hBN (bottom hBN) and the BLG. (b) Anoptical image of the device labeled with the measurement configuration (scale bar: 5µm). (c) Plotof the longitudinal resistance Rxx(B = 0) at T = 2 K as a function of n. Dotted gray lines markthe moiré satellite peaks with carrier density nM = ±2.30 × 1016 m−2. (d) 2D map of Rxx as afunction of n and D. Labels I-IV mark the four quadrants in the n−D plane. The four insets showschematically the charge distribution in the two layers of BLG in these four regimes at high D.The red (blue) ovals indicate the layers of BLG with the higher (lower) occupation of the electronicstates. The upper bound on Rxx is set to be 2 kΩ for better visibility of the satellite peak (for thecomplete data, see Supplementary Information, section S2). (e) Plot of Brown-Zak oscillationsGxx versus 1/B for different carrier densities (units of 1016 m−2) measured at T = 100 K. (f) TheFourier spectrum of the Brown-Zak oscillations measured at n = 4.05× 1016 m−2 shows a singleprominent peak at Bf = 24.2 T.11-4 -2 0 2 400.51-4 -3 -2 -1 0-0.2-0.100.10.20 5000 10000 15000 2000001002003004000 50 100 1500150300R(kΩ)n/n05 K 300 K 30 K 110 K(b)(c) (d)∆R/T2(ΩK-2)n/n0n/n0∆R(Ω)T2 (K2)-2.1-1.7-1.41.41.72.1HolesElectrons}}(a)∆R(Ω)T (K)M1M2M3M4N1-2.4 -2 -1.6 -1.200.03∆R/T2(ΩK-2)0 1 201020f n(mΩK-2)θ (degrees)Figure 2: Umklapp scattering at D/ϵ0 = 0 V/nm. (a) Plot of sheet resistance R□ as a functionof filling fraction n/n0 over a range of temperature from 5 K (blue) to 300 K (red). (b) Plot of∆R□/T2 = (R□(T ) − R□(5K))/T 2 versus n/n0 over a range of temperature from 30 K (blue)to 110 K (red). The dotted ellipse marks the region where Umklapp is the dominant scatteringmechanism. The negative value of ∆R□/T2 around n/n0 = −4 is a consequence of the fact thatat these number densities, the value of R□ decreases with increasing T . Inset: Zoomed-in viewof the region around n/n0 = −2. (c) Plot of ∆R□ as a function of T 2 for six different values ofn/n0. (d) Comparison of plots of ∆R□ versus temperature at n/n0 = −2 for four aligned devices(M1, M2, M3 and M4) with twist angle (0◦, 0.26◦, 0.47◦, 1.70◦) and the non-aligned device N1 atn = −1 × 1016 m−2. Inset: Dependence of fn on the moiré twist angle θ (measured in degrees).The dashed line is a guide to the eyes.12� � � � � � � � 000 . 511 0 0 0 1 0 0 0 01 0 � �1 0 � �1 0 � �1 0 01 0 1DOSn / n 0( a ) ( b )A (nΩ cm K-2 )E F  / k B  ( K )B iG r a p h i t eS bP dM o WA lM 2 M 1Figure 3: Universal scaling of Umklapp scattering. (a) Plot of the calculated density of states(DOS) versus n/n0. The shaded area marks the number density range, away from band edges andvan Hove singularities, where Uee processes can be unambiguously detected. (b) Plot of A = fntversus EF/kB. The open circles are the data from the current study on M1 and M2 superlatticedevices. The filled triangles are the data from Ref12.130020040060080040 60 80 1000.050.10.150 0.15 0.30.0150.020.0250.03∆R(Ω)n/n0-0.3 V/nm-0.2 V/nm-0.1 V/nm0.0 V/nm0.1 V/nm0.2 V/nm0.3 V/nm(b)(c)(a)T (K)0.6 V/nm0.5 V/nm0.4 V/nm0.3 V/nm0.2 V/nm0.1 V/nm0.0 V/nm-0.1 V/nm-0.2 V/nm-0.3 V/nm-0.4 V/nm-0.5 V/nm-0.6 V/nmf n(ΩK-2)60 K70 K80 K90 K100 K0 0.3 0.61.41.61.822.2α(d)ΔR / T2(ΩK-2) D (V// nm)ε0 D (V// nm)ε0Figure 4: Electric field dependence of Umklapp scattering. (a) Plots of ∆R□/T2 versus T fordifferent values of D/ϵ0, the data are for n/n0 = −2. The numbers on the plots are values ofthe D/ϵ0. The data have been vertically offset for clarity. (b) Plot of the resistance exponent α(α = dln(∆R□)/dln(T )) versus D/ϵ0 at n/n0 = −2. (c) Plot of ∆R□ versus filling fraction n/n0at temperature T = 80 K for different values of D/ϵ0. (d) Plots of fn = ∆R□/T2 versus D/ϵ0 ata few representative values of T in the Umklapp region at n/n0 = −2.14k1k3k2k4Fermi surfaceGmoiré band filling-2      -1          0        1         2 0.3 0.0-0.3Electric field (V/nm)f n (a.u.)Figure 5: For Table of Contents Only.15References(1) Ashcroft, N. W.; Mermin, N. D. Solid State Physics; Holt-Saunders, 1976.(2) Kaveh, M.; Wiser, N. Electron-electron scattering in conducting materials. Advances inPhysics 1984, 33, 257–372.(3) Hideaki, M.; Hidetoshi, F. Electrical Conductivity of Interacting Fermions. II. Effects ofNormal Scattering Processes in the Presence of Umklapp Scattering Processes. Journal ofthe Physical Society of Japan 1998, 67, 242–251.(4) Wallbank, J. R.; Krishna Kumar, R.; Holwill, M.; Wang, Z.; Auton, G. H.; Birkbeck, J.;Mishchenko, A.; Ponomarenko, L. A.; Watanabe, K.; Taniguchi, T.; Novoselov, K. S.;Aleiner, I. L.; Geim, A. K.; Falko, V. I. 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