Effects of heterostrain and lattice relaxation on optical conductivity of twisted bilayer graphene
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Effects of heterostrain and lattice relaxation on optical conductivity of twisted bilayergraphene
Zhen-Bing Dai,
1, 2
Yan He, and Zhiqiang Li ∗ College of Physics, Sichuan University, Chengdu, Sichuan 610064, China Department of Physics, Sichuan Normal University, Chengdu, Sichuan 610066, China
We present a theoretical study of the effects of heterostrain and lattice relaxation on the opticalconductivity of twisted bilayer graphene near the magic angle, based on the band structures obtainedfrom a continuum model. We find that heterostrain, lattice relaxation and their combination giverise to very distinctive spectroscopic features in the optical conductivity, which can be used to probeand distinguish these effects. From the spectrum at various Fermi energies, important features inthe strain- and relaxation-modified band structure such as the bandgap, bandwidth and van Hovesingularities can be directly measured. The peak associated with the transition between the flatbands in the optical conductivity are highly sensitive to the direction of the strain, which can providedirect information on the strain-modified flat bands.
I. INTRODUCTION
Recent experimental studies have discovered manynew correlated electronic phases [1–9] in twisted bilayergraphene (TBG) associated with the formation of flatelectronic bands near some magic angles [10–12], whichhas generated great interest in this system. The flatbands and intriguing electronic phases created by in-terlayer coupling in the moir´e superlattices exhibit crit-ical dependence on small lattice deformations. Scan-ning tunneling experiments [9] and theoretical calcula-tions [13, 14] have shown that strains with opposite signin the two layers (heterostrain) can significantly modifythe band structure and the observed electronic phases,through strain-induced effective gauge field and changesin stacking and interlayer tunneling. Extrinsic strain isubiquitous in TBG, which is induced during the fabrica-tion process and by the substrate. Moreover, the compe-tition between interlayer van der Waals interaction andintralayer lattice distortion can cause significant atomicscale lattice relaxations in TBG involving local lattice ro-tations with localized strain to favor interlayer commen-surability [15–20] , which induces substantial changes inlattice symmetry and electronic band structure. There-fore, it is of great importance to explore the effects ofstrain and lattice relaxation in TBG.Optical experiment is a powerful technique to in-vestigate the band structures, many body interactionsand various electronic excitations in graphene-based sys-tems [21]. The optical conductivity of TBG has beenstudied theoretically and experimentally in several pre-vious works [22–30]. Earlier theoretical calculations [22–24, 31] have been done for twist angles θ > = 1 . ◦ . Morerecently, the optical conductivity near the magic anglehas been calculated taking into account correlation ef-fects [25] and interlayer tunneling asymmetry [26, 27].So far, the combined effects of heterostrain and lattice ∗ Electronic address: [email protected] relaxation on the optical properties of TBG are yet to beexplored.In this work, we show that the band structure of TBGat the magic angle is strongly affected by strain andlattice relaxation, which leads to very distinctive spec-troscopic features in the optical conductivity, includingthe energy and lineshape of the peaks in the spectrum.Meanwhile, the bandgap, bandwidth and van Hove sin-gularities can be directly measured from the spectrum bychanging the Fermi energy. The peak associated with thetransition between the flat bands are highly sensitive tothe strain, which can provide direct information on thestrain-modified flat bands. Therefore, the spectroscopicfeatures in the optical conductivity can be used to probeand distinguish the effects of strain and lattice relaxation.This paper is organized as follows: Section II summa-rizes the effective continuum model taking into accountthe effects of strain and lattice relaxation, and describesthe procedure used to calculate the optical conductivity.Section III presents our main findings on the band struc-ture, DOS and optical conductivity. A summary andconclusions are reported in Section IV.
II. THEORETICAL BACKGROUND
To investigate the effect of geometric deformation onthe properties of TBG, we at first define the primitive lat-tice vectors of the initial monolayer graphene (MLG) be-fore deformation as A = a (1 ,
0) and A = a (1 / , √ / a ≈ .
246 nm is the lattice constant of MLG.The corresponding reciprocal lattice vectors are G = πa (1 , − / √
3) and G = πa (0 , / √ A , B , A , B ) as H = (cid:18) H U † U H (cid:19) , (1)where A l and B l are the two sublattices of the layer l with l = 1 , H l is the in-tralayer Hamiltonian given by the two-dimensional Weylequation [14, 17] H l = − ~ v F [( I + E Tl )( k − D l ) · ( σ x , σ y ) , (2)where v F is the Fermi velocity of MLG, σ x and σ y arePauli matrices. We take ~ v F /a ≈ . E l includes both strainand rotation in general. D l denotes the positions of thetwo Dirac fermions in momentum space under deforma-tion, which is given by the previous studies [14, 17]. Theeffective interlayer coupling U is given by [34] U = (cid:18) U A A U A B U B A U B B (cid:19) = (cid:18) u u ′ u ′ u (cid:19) + (cid:18) u u ′ ω − ξ u ′ ω ξ u (cid:19) e iξ G M · r + (cid:18) u u ′ ω ξ u ′ ω − ξ u (cid:19) e iξ ( G M + G M ) · r , (3)where G Ml = E T G l ( l = 1 ,
2) represents the reciprocallattice vectors for the moire pattern, ξ = ± u and u ′ describe respectively the amplitudes ofdiagonal and off-diagonal terms in the sublattice space,and ω = e πi/ . For unrelaxed TBG, u = u ′ = 103meV [31]. After atomic scale lattice relaxation, the bandstrcuture and physical properties in TBG were reportedto have been significantly modified [17, 18, 35]. The ef-fect of lattice corrugation (relaxation in the out-of-planedirection) in TBG can be included by choosing u =0.0797eV and u ′ =0.0975 eV in the continuum model [34]. Figure 1. Sketch of TBG with uniform uniaxial strain alongarmchair direction and the resulting moire Brillouin zone. In(a), blue lines and black lines from the top views representthe top layer and bottom layer, respectively. Red arrows inthe side views stand for the axial strain applied to each layer.The solid black lines and dashed green lines in (b) indicatethe moire Brillouin zones of TBG with and without uniformstrain, respectively. The blue arrows denote the reciprocal lat-tice vectors for the moire patterns. The dashed red lines showthe path in the momentum space for the electronic structure.
Here, we focus on the effect of uniaxial heterostrain inTBG observed in scanning tunneling microscope (STM) experiments [7–9]. With uniaxial strain, the deformationmatrix E l can be written as [9, 36] E l = T ( θ ) + S ( ǫ l , φ )= ǫ l (cid:18) − cos φ + ν sin φ (1 + ν ) cos φ sin φ (1 + ν ) cos φ sin φ − sin φ + ν cos φ (cid:19) + (cid:18) cos θ − sin θ sin θ cos θ (cid:19) , (4)where T ( θ ) represents rotation with angle θ , the straintensor S ( ǫ l , φ ) is characterized by the strain magnitude ǫ l and the strain direction φ . In this paper, we assumethat E = −E = E /
2, namely the two layers are strainedand rotated oppositely with the same magnitude. Whena uniaxial heterostrain is applied to TBG, the moire su-perlattice in real space will be distorted from a regulartriangular lattice as schematically shown in Fig. 1(a).Consequently, the resulting Brillouin zone is a distortedhexagon as shown in Fig. 1(b).To study the optical conductivity, we calculate theeigenenergies and eigenstates in k-space by numericallydiagonalizing the Hamiltonian within a limited wavespace [31]. Within the linear response theory, the Kuboformula for optical conductivity is given by [23] σ xx ( w )= e ~ i (2 π ) Z dk x dk y X m,n f ( E m ) − f ( E n ) E m − E n × |h u m | v x | u n i| ~ ω + iη + E m − E n , (5)where f ( E m ) is the Fermi-Dirac distribution function f ( E m ) = 1 / (1 + e ( E m − ε F ) / ( k B T ) ) with E m ( E n ) and | u m i ( | u n i ) representing the eigenenergies and eigenstatesof TBG, respectively. The velocity operator is related tothe Hamiltonian through the relation v x = ~ ∂H∂k x , and η denotes a phenomenological damping parameter. III. RESULTS AND DISCUSSIONSA. Band structure
The effects of heterostrain and lattice relaxation on theband structure of TBG at the magic angle θ = 1 . ◦ areshown in Fig. 2. In the absence of strain, the non-relaxedTBG exhibits flat bands with vanishing band velocitynear the charge neutral point (CNP), leading to a strongpeak in DOS near CNP. However, in the relaxed TBG,a band gap opens up between the flat bands near CNPand the excited bands in both the electron and hole sides.The bandwidth of the flat bands becomes even narrowerand the band velocity is reduced compared to the non-relaxed case. The excited bands and the associated vanHove singularities are moved to higher energies when thelattice relaxation is taken into account. These results aresimilar to previous studies [34]. In the presence of strain,the band structure is strongly affected. Comparing thenon-relaxed cases with and without strain, the energy -0.15-0.10-0.050.050.100.150 E ne r g y ( e V ) K G K’ M G M K(a) No strain e e h h S S S S DOS -0.15-0.10-0.050.050.100.150 E ne r g y ( e V ) (b) 0.6% strainK G K’ M G M K S S S S DOS
Figure 2. Band structure and DOS of TBG without/with the uniform heterostrain and relaxation at twist angle θ = 1 . ◦ .(a) and (b) show the influence of uniform heterostrain on the band structure (left) and DOS (right), in which the red linesand black lines correspond respectively to the cases without and with relaxation. The heterostrain, ǫ = 0 . φ = 30 ◦ . The interband transitions between the saddle point of the lowest band( h and e ) and the band edge of the second band ( h and e ), which are marked by the colored arrows S ( S ) and S . And S represents the transition between two split flat subbands near CNP. separation between the lowest conduction and valencebands is significantly enlarged in the former case [14],which can be seen from the separation of two van Hovesingularities near CNP. While these two bands are quiteflat in most regions of the Brillouin zone, they remainconnected by two Dirac crossings in each valley [14]. Thehigh energy van Hove singularities are generally broad-ened in the strained case compared to the non-strainedcase.The band structure and DOS of strained TBG are in-fluenced by lattice relaxation. As shown in Fig. 2(b), thevan Hove singularities of the lowest conduction and va-lence bands in relaxed TBG are broader compared to thenon-relaxed TBG. Moreover, the excited bands shift tohigher energies under the influence of lattice relaxation.As to the DOS, such a shifting is the most pronouncedfor van Hove singularities associated with the first excitedbands in both electron and hole sides. We note that het-erostrain and lattice relaxation induce strong features ofthe DOS in different energy regions. The most promi-nent feature due to strain is the large splitting of twovan Hove singularities near CNP, whereas the strongestfeature due to lattice relaxation is the blue-shift of vanHove singularities associated with the first excited bands. B. Optical conductivity
In order to reveal the influence of strain and lattice re-laxation, we compare the optical conductivity of TBG fornon-relaxed TBG without strain, non-relaxed TBG withstrain, strained TBG without relaxation, and strainedTBG with relaxation. The optical conductivity are scaled by σ = e / ~ , which is the background conductivity ofmonolayer graphene. Spectra calculated using two dif-ferent damping parameters η are shown in order to illus-trate the effect of electronic mobility of TBG samples,with η = 3 meV ( η = 10 meV [37]) corresponding toa high (low) mobility sample. The optical conductiv-ity spectrum exhibits two groups of peaks in the energyranges of 0 −
100 meV and 150 −
300 meV. The secondgroup of peaks are relatively weak and broad especiallyin low mobility samples ( η = 10 meV), so the changeof these features due to strain and lattice relaxation willbe difficult to detect in realistic samples. On the otherhand, the first group of peaks are strong and show dra-matic changes under the influence of strain and latticerelaxation in all spectra, which are easier to detect inoptical measurements regardless of the mobility of sam-ples. Hence, we will focus on peaks below 100 meV inour discussion.In the non-relaxed TBG without strain [24, 31], theoptical conductivity exhibits two prominent peaks S and S below 100 meV arising from the interband transitionsbetween flat bands and the first excited bands ( h → e , h → e ) as shown in Fig. 2. This assignment isobtained from calculating the contributions to σ xx ( w )from transitions involving different pairs of bands. Thecontribution to Re σ xx ( w ) from h → e transition atfinite energy is a smooth tail without a peak.The optical conductivity spectrum is significantlychanged by lattice relaxation effects (red lines in Fig.3): the S peak becomes much stronger and the S peakshifts to higher energy compared to the nonrelaxed case.The shift of the S peak is due to the blue-shift of vanHove singularities associated with the first excited bands R e s xx / s Transition energy (eV) e=0, non-relaxede=0, relaxede=0.6%, non-relaxede=0.6%, relaxed (a) R e s xx / s Transition energy (eV) e=0, non-relaxede=0, relaxede=0.6%, non-relaxede=0.6%, relaxed (b)
Figure 3. Real part of optical conductivity of TBG for the four typical cases shown in Fig. 2. The phenomenological dampingrate η is chosen to be 3 meV in (a) and 10 meV in (b). In the realistic situation, the value of η depends usually on the electronicmobility of the sample. The arrows indicate the transitions shown in the band structure Fig. 2. The spectra are offset in σ xx by 3 σ apiece for clarity. The temperature is set to be T = 10 K in the numerical calculation. σ = e / ~ . under lattice relaxation. In non-strained low mobilitysamples (Fig. 3(b)), the S and S peaks are broadenedand merge into a single broad peak below 100 meV inthe nonrelaxed case, and lattice relaxation splits thesetwo peaks with a large energy separation. Such a largechange is readily observable in optical measurements ofrealistic samples.The application of strain leads to large modifications inthe optical conductivity. In strained TBG without latticerelaxation (green lines in Fig. 3), the interband transition h → e gives rise to a small S peak because strainsplits these two bands with a finite energy separation inmost regions of the Brillouin zone. Moreover, under theinfluence of strain several interband transitions including h → e , h → e , h → e , h → e , h → e , h → e , h → e produce narrow peaks in overlapping energyranges around 50 meV, giving rise to a very strong peak S m at higher energy compared to the S peak in non-relaxed TBG without strain. Comparing non-relaxed lowmobility samples (Fig. 3(b)), the peak below 50 meV inthe non-strained case shifts to higher energy and gainsa high-energy shoulder with strain. The peaks between150 and 300 meV in non-relaxed cases are also stronglymodified by strain.Further inclusion of lattice relaxation effects instrained TBG makes the S peak substantially stronger(blue lines in Fig. 3), because the separated h and e bands become almost parallel in many regions of the ofthe Brillouin zone under lattice relaxation, leading tovery strong joint DOS at the energy of the S peak.Furthermore, the S m peak in non-relaxed strained TBGshifts to higher energy by lattice relaxation due to theblue-shift of van Hove singularities associated with theexcited bands. In low mobility samples with strain (Fig.3(b)), the non-relaxed case exhibits a single peak with a high-energy shoulder below 100 meV, whereas thereare two well-defined peaks in this energy range in therelaxed case. Such a large contrast will be easy to ob-serve experimentally. The totality of the results in Fig. 3demonstrates that the dramatic changes of peaks in op-tical conductivity can serve as spectroscopic signaturesto probe and distinguish the effects of strain and latticerelaxation and explore the resulting band structure.Important features in the band structure induced bystrain and lattice relaxation can be explored from opti-cal conductivities with various Fermi energies. Fig. 4(a)shows the optical conductivity of relaxed TBG withoutstrain with η =3 meV for different Fermi energies. WhenFermi energy is moved into the bandgap between the flatbands and the neighboring bands, a new peak S ∆ ap-pears below the S peak compared to the spectrum with ε F = 0, which is clearly shown in the inset of Fig. 4(a).The peak S ∆ arises from the e → e transition and be-comes active with ε F in the gap, therefore it providesa direct measure of the bandgap. Moreover, the energyseparation between S ∆ and S is a measure of the totalbandwidth of the two flat bands. As ε F is further in-creased up to the next van Hove singularity around 75meV, the S peak disappears and then the S peak weak-ens significantly due to Pauli blocking. Conversely, low-ering the ε F for a highly doped sample can activate thesetransitions, which is a measure of the related van Hovesingularity. For the strained relaxed TBG (Fig. 4(b)),the lowest energy S peak disappears as the Fermi energyis increased above the e band, due to Pauli blocking. Inthis case, the S peak provides information on the energyseparation of the two flat bands induced by strain. Theresults in Fig. 4 demonstrates that important features inthe band structure such as the bandgap, bandwidth andvan Hove singularities can be directly measured from theoptical conductivity spectrum by changing the Fermi en-ergy. R e s xx / s Transition energy (eV) e F =0 e F =10meV e F =40meV e F =75meV R e s xx / s Energy (meV) -0.10-0.08-0.06-0.04-0.020.020.040.060.080.100 E ne r g y ( e V ) DOS(a) No strain e F =0 e F =15meV R e s xx / s Transition energy (eV)(b) 0.6% strain -0.10-0.08-0.06-0.04-0.020.020.040.060.080.100 E ne r g y ( e V ) DOS
Figure 4. Real part of optical conductivity of relaxed TBGwithout strain (a) and with strain ǫ = 0 .
6% (b), for variousFermi energies ε F . The first significant van Hove singularity isat E ≃
75 meV in the DOS in Fig. 2(a) and at E ≃
15 meV inFig. 2(b), respectively. Various significant optical transitionsare indicated by the colored arrows. The Fermi energies aremarked as the dashed lines in the DOS. The inset in (a) showsthe relative spectrum of Re σ xx at ε F = 10 meV by subtract-ing Re σ xx at ε F = 0 as a function of transition energy. Thephenomenological damping rate and the temperature are setto be η = 3 meV and T = 10 K, respectively. Now we study the dependence of the DOS and opticalconductivity on the direction of the strain φ in relaxedTBG. The DOS with fixed θ = 1 . ◦ , ǫ = 0 .
6% for sev-eral values of φ ∈ [0 , ◦ ) are shown in Fig. 5(b). Thesystem is periodic for φ → φ + 60 ◦ [14]. As φ is var-ied, we observe interesting changes in the structure ofvan Hove singularities within h and e bands (between-20 and 20 meV) evolving from multiple DOS peaks to one prominent peak and then to multiple peaks in bothbands, consistent with previous studies [9, 14, 38]. Thebandwidth of these two bands stays approximately con-stant. As shown in Fig. 5(c), the Re σ xx ( w ) and Re σ yy ( w ) spectra are different for all values of φ . Thisanisotropy is a result of the uniaxial strain consideredin our calculation. Both Re σ xx ( w ) and Re σ yy ( w ) spec-tra are very sensitive to the strain direction. The peak S ( h → e ) around 20 meV in Re σ xx ( w ) and a peakaround 10 meV in σ yy ( w ) become most prominent for φ near 30 ◦ and 40 ◦ and disappear for φ near 0 ◦ and 10 ◦ .We find that the strength of these low energy peaks arerelated to the band structure and DOS. For φ around 30 ◦ and 40 ◦ , the h and e bands are almost parallel in mostregions of the Brillouin zone (Fig. 2(b)), leading to oneprominent DOS peak in their associated van Hove sin-gularities, which gives rise to very pronounced peaks inRe σ xx ( w ) and Re σ yy ( w ) below 20 meV. For φ around0 ◦ − ◦ , the h and e bands are no longer parallel (Fig.5(a)) leading to multiple DOS peaks in the associated vanHove singularities, so the low energy peaks in Re σ xx ( w )and Re σ yy ( w ) disappear due to the lack of well-definedpeaks in the joint DOS. Therefore, the low energy peaksin optical conductivity can provide information on theshape of strain-modified flat bands and the associatedvan Hove singularities.The flat bands h and e are most sensitive to strain asshown in Fig. 5(d), and their energy separation increaseswith strain. The other bands are less affected by strain.Fig. 5(e) displays the evolution of Re σ xx ( w ) with strain.With increasing strain, the S peak ( h → e transition)appears gradually and shifts to higher energy, becauseof the increasing energy separation between the h and e bands. Therefore, this peak can be used to estimatethe magnitude of strain. The S and S peaks in thenon-strained case shift to higher energy with strain andmerge into a peak with shoulders on both sides.In our work, a particular type of lattice relaxation,lattice corrugation in the out-of-plane direction, is in-cluded in our calculations by setting u = 0 . u ′ = 0 . -0.15-0.10-0.050.050.100.150-0.15-0.10-0.050.050.100.150 0.00 0.04 0.08 0.12 0.16 0.200123456789 E ne r g y ( e V ) G K’ M G M K K (a) -0.10 -0.05 0.00 0.05 0.100.00.10.20.30.40.50.6 D O S Energy (eV)f =0(cid:176)f =10(cid:176)f =20(cid:176)f =30(cid:176)f =40(cid:176)f =50(cid:176)(b) R e s aa / s Transition energy(eV)f =0(cid:176)f =10(cid:176)f =20(cid:176)f =30(cid:176)f =40(cid:176)f =50(cid:176)(c) E ne r g y ( e V ) G K’ M G M K K (d) = 0.6% = 0.3%DOS R e s xx / s Transition energy (eV) = 0.6% = 0.3% = 0(e)
Figure 5. Influence of heterostrain on the band structure, DOS and optical conductivity. (a) Band structure of the relaxed TBGwith strain ǫ = 0 . , φ = 0 ◦ at the magic angle. (b) DOS of relaxed TBG at θ = 1 . ◦ with fixed strain ǫ = 0 . φ . (c) Real part of optical conductivity σ αα , where α = x (thick lines) or y (thin lines), of relaxed TBGswith the different directions of strain φ ∈ [0 , ◦ ). The spectra are offset in σ by 3 σ apiece for clarity. (d) Band structure ofrelaxed TBG at θ = 1 . ◦ , φ = 30 ◦ with ǫ = 0 .
3% (red thin lines) and ǫ = 0 .
6% (black thick lines). (e) Real part of longitudinaloptical conductivity of relaxed TBG with φ = 30 ◦ for the various strain magnitude. The first absorption peak associated withthe transition S shows dramatic changes under the influence of strain. The spectra are offset in σ by 2 σ apiece for clarity.The temperature and the phenomenological damping rate are set to be T = 10 K and η = 3 meV, respectively. IV. CONCLUSION
In this paper, we systematically studied the effects ofuniaxial heterostrain and lattice relaxation on the elec-tronic structure and optical conductivity of TBG at themagic angle, which was calculated based on the band structure obtained from a widely used continuum model.We calculated the optical conductivity using differentdamping rates in order to facilitate comparisons with fu-ture optical measurements of TBG samples with differ-ent electronic mobilities. It was found that heterostrain,lattice relaxation and their combination lead to very dis-tinctive spectroscopic features in the optical conductiv-ity, including the energy and lineshape of the peaks in thespectrum, which can be used to probe and distinguish theeffects of strain and lattice relaxation.The heterostrain generically broadens the bandwidthof the nearly flat bands, which had been verified via en-ergy separation of van Hove singularities in STM exper-iments. Moreover, different interband transitions can beactivated or blocked at various Fermi energies, so impor-tant features in the band structure such as the bandgap,bandwidth and van Hove singularities can be directlymeasured from the spectrum by changing the Fermi en-ergy. In the presence of heterostrain, we investigatedthe anisotropy of the conductivity spectra. Remarkably,we found that the absorption peaks associated with thetransition between the flat bands are highly sensitive tothe direction and magnitude of strain, which can pro-vide direct information on the strain-modified flat bands.Therefore, our results can provide some insights into un-derstanding the optical properties of TBG and offer a comparison of the optical spectrum of TBG in future ex-periment for detecting the imperceptible changes of thelattice structure.In realistic TBG samples, the extrinsic strain and thedegree of lattice relaxation generally change spatially.Spatial dependent measurements of the local optical con-ductivity using infrared microscopy can provide impor-tant insights into the local band structure and DOS mod-ified by strain and lattice relaxation.
ACKNOWLEDGMENTS
We are grateful to Guo-Yu Luo, Lu Wen, and Prof.Hao-Ran Chang for valuable discussions. This work wassupported by the National Natural Science Foundationof China under Grant Nos. 11874271 and 11874272.We thank the High Performance Computing Center atSichuan Normal University. [1] Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken,J. Y. Luo, J. D. Sanchez-Yamagishi, K. Watanabe,T. Taniguchi, E. Kaxiras, et al., Nature , 80 (2018).[2] Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi,E. Kaxiras, and P. Jarillo-Herrero, Nature , 43(2018).[3] M. Yankowitz, S. Chen, H. Polshyn, Y. Zhang, K. Watan-abe, T. Taniguchi, D. Graf, A. F. Young, and C. R. Dean,Science , 1059 (2019).[4] A. L. Sharpe, E. J. Fox, A. W. Barnard, J. Finney,K. Watanabe, T. Taniguchi, M. A. Kastner, andD. Goldhaber-Gordon, Science , 605 (2019).[5] X. Lu, P. Stepanov, W. Yang, M. Xie, M. A. Aamir,I. Das, C. Urgell, K. Watanabe, T. Taniguchi, G. Zhang,et al., Nature , 653 (2019).[6] M. Serlin, C. L. Tschirhart, H. Polshyn, Y. Zhang, J. Zhu,K. Watanabe, T. Taniguchi, L. Balents, and A. F. Young,Science , 900 (2020).[7] Y. Xie, B. Lian, B. Jack, X. Liu, C. L. Chiu, K. Watan-abe, T. Taniguchi, B. A. Bernevig, and A. Yazdani, Na-ture , 101 (2019).[8] Y. Jiang, X. Lai, K. Watanabe, T. Taniguchi, K. Haule,J. Mao, and E. Y. Andrei, Nature , 91 (2019).[9] A. Kerelsky, L. J. McGilly, D. M. Kennes, L. Xian,M. Yankowitz, S. Chen, K. Watanabe, T. Taniguchi,J. Hone, C. Dean, et al., Nature , 95 (2019).[10] R. Bistritzer and A. H. MacDonald, Proc. Natl. Acad.Sci. USA , 12233 (2011).[11] E. Suarez Morell, J. D. Correa, P. Vargas, M. Pacheco,and Z. Barticevic, Phys. Rev. B , 121407(R) (2010).[12] J. M. B. Lopes dos Santos, N. M. R. Peres, and A. H.Castro Neto, Phys. Rev. B , 155449 (2012).[13] L. Huder, A. Artaud, T. Le Quang, G. T. de Laissardiere,A. G. M. Jansen, G. Lapertot, C. Chapelier, and V. T.Renard, Phys. Rev. Lett. , 156405 (2018).[14] Z. Bi, N. F. Q. Yuan, and L. Fu, Phys. Rev. B ,035448 (2019).[15] K. Zhang and E. B. Tadmor, J. Mech. Phys. Solids , 225 (2018).[16] S. Carr, D. Massatt, S. B. Torrisi, P. Cazeaux, M. Luskin,and E. Kaxiras, Phys. Rev. B , 224102 (2018).[17] N. N. T. Nam and M. Koshino, Phys. Rev. B , 075311(2017).[18] H. Yoo, R. Engelke, S. Carr, S. Fang, K. Zhang,P. Cazeaux, S. H. Sung, R. Hovden, A. W. Tsen,T. Taniguchi, et al., Nat. Mater. , 448 (2019).[19] J. S. Alden, A. W. Tsen, P. Y. Huang, R. Hovden,L. Brown, J. Park, D. A. Muller, and P. L. McEuen,Proc. Natl. Acad. Sci. USA , 11256 (2013).[20] S. Dai, Y. Xiang, and D. J. Srolovitz, Nano Lett. ,5923 (2016).[21] D. N. Basov, R. D. Averitt, D. van der Marel, M. Dressel,and K. Haule, Rev. Mod. Phys. , 471 (2011).[22] C. J. Tabert and E. J. Nicol, Phys. Rev. B , 121402(R)(2013).[23] T. Stauber, P. San-Jose, and L. Brey, New J. Phys. ,113050 (2013).[24] P. Moon, Y.-W. Son, and M. Koshino, Phys. Rev. B ,155427 (2014).[25] M. J. Calderon and E. Bascones, npj Quantum Mater. , 57 (2020).[26] P. Novelli, I. Torre, F. H. L. Koppens, F. Taddei, andM. Polini, Phys. Rev. B , 125403 (2020).[27] L. Wen, Z. Li, and Y. He, Chin. Phys. B , 017303(2021).[28] V. Hung Nguyen, A. Lherbier, and J. C. Charlier, 2DMater. , 025041 (2017).[29] R. W. Havener, Y. Liang, L. Brown, L. Yang, andJ. Park, Nano. Lett. , 3353 (2014).[30] K. Yu, N. Van Luan, T. Kim, J. Jeon, J. Kim, P. Moon,Y. H. Lee, and E. J. Choi, Phys. Rev. B , 241405(R)(2019).[31] P. Moon and M. Koshino, Phys. Rev. B , 205404(2013).[32] J. M. Lopes Dos Santos, N. M. Peres, and A. H. Cas-tro Neto, Phys. Rev. Lett. , 256802 (2007). [33] M. Kindermann and P. N. First, Phys. Rev. B , 045425(2011).[34] M. Koshino, N. F. Q. Yuan, T. Koretsune, M. Ochi,K. Kuroki, and L. Fu, Phys. Rev. X , 031087 (2018).[35] F. Gargiulo and O. V. Yazyev, 2D Mater. , 015019(2017).[36] F. M. D. Pellegrino, G. G. N. Angilella, and R. Pucci, Phys. Rev. B , 035411 (2010).[37] Y. Luo, R. Engelke, M. Mattheakis, M. Tamagnone,S. Carr, K. Watanabe, T. Taniguchi, E. Kaxiras, P. Kim,and W. L. Wilson, Nat. Commun. , 4209 (2020).[38] Y. Choi, J. Kemmer, Y. Peng, A. Thomson, H. Arora,R. Polski, Y. Zhang, H. Ren, J. Alicea, G. Refael, et al.,Nat. Phys.15