Momentum Shift Current at Terahertz Frequencies in Twisted Bilayer Graphene

Momentum shift current at terahertz frequencies in twisted bilayer graphene

Daniel Kaplan, Tobias Holder, and Binghai Yan∗ Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 7610001, Israel

The detection of terahertz (THz) radiation promises intriguing applications in biology, telecommu- nication, and astronomy but remains a challenging task so far. For example, semiconductor infrared detectors (e.g., HgCdTe) utilize photo-excited electrons across the bandgap and hardly reach the far-infrared terahertz regime because they are vulnerable to thermally excited carriers. In this work, we propose the THz sensing by the bulk photovoltaic effect (BPVE) in the twisted bilayer graphene (TBG). The BPVE converts light into a coherent DC at zero bias or an open-circuit voltage. As a quantum response from the wave function’s geometry (different from the p-n junction), BPVE is more robust against temperature excitation and disorders. We predict that the TBG (bandgap of several meV) exhibits a sizeable BPVE response in a range of 0.2 – 1 THz. Beyond the ordinary shift current scenario, BPVE in TBG comes from a momentum-space shift of flat bands. Our work provides a pathway to design twisted photonics for resonant terahertz detection.

INTRODUCTION voltaic efficiency for energy harvesting, and have further applications in medical imaging, single-photon circuits and novel electronic devices [27, 28]. The bulk photovaltaic effect (BPVE) refers to the generation of a dc current from a homogeneous solid upon Twisted bilayer graphene (TBG) exhibits flat bands irradiation with light. The so-called shift current [1–4] and narrow band gaps of several meV, and thus pro- is one of the most important mechanisms for the BPVE vides an ideal platform to investigate the THz BPVE. which has been appreciated for a long time for its po- It has attracted an immense amount of attention fol- tential in photovoltaic and photonic applications. So lowing the recent discovery of correlated insulating and far, the BVPE has been studied predominantly for near- superconducting phases at small twist angle (magic angle) infrared and optical frequencies, but it may prove useful θ ∼ 1◦ [29, 30]. While the root cause for this exciting be- to overcome the so-called ’terahertz gap’ [5] in the field havior is widely accepted to be tied to a highly quenched of terahertz photonics [6]. This gap is the perceived band structure [31–33], opinions differ about the mecha- lack of robust, tunable and broad-frequency materials for nisms associated with the various correlated phases which terahertz detection. For instance, usual semiconductor have since been documented in the system [34–46]. An infrared detectors [7] harvest the photon-excited conduc- important stepping stone towards the understanding of tion electrons, which are competing with the thermal these phases is the characterization of the quantum geo- excitation across the energy gap, and thus cannot work in metric and dispersive features of the flat bands. Because the THz range. Recent developments attempt to detect of inversion symmetry breaking, TBG and similar twisted plasmons [8–10] stimulated by the THz radiation. bilayers exhibit intriguing nonlinear phenomena [47–53] such as the bulk photovaltaic effect (BPVE) [1, 54] and We point out that the shift current is quite robust nonlinear anomalous Hall effect [55, 56]. These nonlinear against thermally excited carriers, because it is a coherent probes are ideally suited for the investigation of quasipar- bulk effect driven by the quantum geometry [11–18] and ticle properties in flat bands as they are not suppressed works even in metallic systems such as Weyl semimet- by a vanishing Fermi velocity [18]. als [11, 19–26]. Most importantly, the created current does not require any DC bias or an interface to become With regards to semiconductors, the shift current had accessible to measurement. Therefore, the thermal occu- originally been interpreted as a real-space shift of wave pation of the conduction band at finite temperatures does function centers upon excitation. It has gained renewed not present a source of background current, meaning that interest in recent studies of Weyl semimetals [57, 58], be- the shift current does not compete with thermal currents, cause the shift is believed to get enhanced by the Berry save for thermal noise. While the shift current is indeed curvature near Weyl points [11, 19–21, 23–26]. Very re-

arXiv:2101.07539v1 [cond-mat.mes-hall] 19 Jan 2021 diminished by occupying the conduction band, this effect cently, we proposed to instead view the shift current as a is only moderate in the terahertz regime. For example, result of the anomalous quasiparticle acceleration, which even at room temperature the thermal occupation across is determined by the quantum geometric properties [18]. a 4meV gap will lead to a shift current signal with a mag- In the latter interpretation, the small effective mass in a nitude of still 5% of its zero temperature value. Thus, Weyl semimetal leads to strong acceleration in the field of the shift current can be useful for higher temperature light and hence generates large photocurrent, consistent terahertz sensing based on noncentrosymmetric materials with the shift current interpretation [11]. Flat bands, as that exhibit resonant BPVE response in the meV window. for example in TBG, represent a kind of opposite limit The shift current may substantially improve detection (large quasiparticle mass) compared to Weyl bands (nearly capabilities for terahertz radiation, increase the photo- massless quasiparticles). Using the language of a quasipar- 2

Distinct from the nonlinear anomalous Hall eﬀect, the shift current of TBG originates from the properties of the quantum geometry of the band structure that is un- related to the Berry curvature dipole. The magnitude of the obtained current, tunability of the Hall response, the broadness of the resonances, and the terahertz regime in which they are all observed hold promise of the uti- lization of TBG devices in terahertz technologies, such photodetection and photovoltaic DC current generation.

RESULTS

A. Shift current theory

In the clean limit, the Bloch states produce an intrinsic Figure 1. A schematic comparison of the wavefunction in dc-current response to linearly polarized light [2]. This real-space and momentum-space of electronic states at the nonlinear conductivity σaa;c is commonly formulated [2, 3] Fermi surface for three types of gapless materials. For a (s) as Weyl semimetal, states are well localized at few points in the momentum space but quite extended in the real space. In aa;c contrast, for a flat band all momenta are equally occupied, σ(s) (0; ω, −ω) = leading to sharply localized peaks of the wavefunction in real 3 Z πe X c a 2 space. The ordinary metal with a large Fermi surface repre- 2 fmnSmn|rmn| δ(ω ± εmn), (1) sents the intermediate region between the Weyl semimetal and ~ k mn the flat-band system. If the shift current is viewed as a result of the anomalous acceleration, the Weyl semimetal exhibits where m, n are the band indices, εmn = εm − εn the a large real-space shift while the flat band displays a major band energy difference, fmn = f(εm) − f(εn) the Fermi a a momentum-space shift. distribution function difference, and rmn ≡ hm|r |ni the dipole transition matrix element. Here, a, c represents the light field and current directions, respectively. Smn ticle shift, it is therefore less obvious to deduce the shift is the so-called shift vector, current for a flat-band system. In contrast, in the semi- c c c c classical acceleration picture, a state can be displaced (i.e. Smn = (rmm − rnn) + ∂kc arg rmn, (2) accelerated) in either real-space and momentum-space, c c as illustrated in Fig.1. Although the real-space shift is where (rmm −rnn) is the shift of the wave function centers, indeed small in real-space due to the large quasiparticle which is gauge dependent. mass, for flat bands the acceleration (i.e. the displacement The second term in Eq.2 is the phase derivative of c in momentum space) can still be very large. Therefore, rmn, which ensures gauge-invariance for the shift vector. we are motivated to investigate the shift current of TBG Smn has then been interpreted as the real-space shift of from the acceleration point of view. the mass center of a quasiparticle upon excitation from A previous study has reported a non-linear conductiv- band m to n [2]. As we will show now, the phase term in ity above 30meV due to the orbital magnetization at 3/4 (2) gives rise to a shift in momentum space. Therefore, filling [47]. In this work, we study shift current generation the shift current is actually the result of a shift of the below 10meV in magic angle TBG across the single parti- quasiparticle excitation in both real space and momentum cle gaps. In a numerical analysis, find a large photocurrent space, which is naturally captured using the language of response for light in the low terahertz regime. This is the anomalous quasiparticle acceleration [18]. consistent with the acceleration picture outlined above. To this end, we express the nonlinear conductivity in To further support such a connection, we formally express the form of geometric properties of the band structure as the response in terms of the real-space shift current and follows, momentum-space shift current. Due to the flat-band dis- 3 Z persion, the dominant contribution is constituted by the aa;c πe aac aac σ(s) (0; ω, −ω) = − 2 Rshift + Kshift (3) momentum-space shift, which we explicitly show to be ~ k dissimilar to an ordinary dispersive material (e.g., MoS2). 3

(b) 15 (a) + 10

5 ]

0 u. [ a. (c) - 5

- 10 0 - 15

Figure 2. Band structure, Berry curvature and nonlinear conductivity in the mini-Brillouin zone of TBG. (a) Band structure of TBG with inversion-breaking at twist angle θ = 1.05◦. Solid lines show the unstrained case, the dashed lines are at strain = 0.001. (b) The Berry curvature distribution of all filled bands at half filling for valley K. The colorbar is in logarithmic scale. The Berry curvature in each valley is C3z symmetric, which leads to a vanishing Berry curvature dipole in each valley yy;x separately. (c) Modulus of the nonlinear conductivity as given by σ(s) [Eq. (8)], in logarithmic scale. At frequency ω = 1.4 meV, just above the gap, the resonant features are fairly broad in momentum space, a result of the flatness of the dispersion. .

aac X c c a 2 Rshift = fmn(rmm − rnn)|rmn| δ(ω ± εmn) (4) mn aac X h a ac c aa i ac Kshift = fmn 2irmnλnm − irmnλnm δ(ω ± εmn) − ∂ka Ω |εmn=±ω (5) mn ac X h i a ac c aa a ac i ∂ka Ω |εmn=±ω = fmn 2 rmnΩnm − irmnλnm + irmnλnm δ(ω ± εmn) (6) mn

ab where we use the symmetrized derivative λnm = persion, it can be shown that the general expressions for 1 b a aac aac 2 ∂ka rnm + ∂kb rnm . The two pieces Rshift and Kshift the shift current present here can be connected by a pull- correspond to the contributions from the first term back mapping to the Christoffel symbols of the geometric and the second term, respectively, of the shift vector connection on the generalized Bloch sphere [17]. While aac in Eq. (2). The second term in Kshift is the Berry this reinforces the association of the shift current with a curvature dipole, and the first term is related to the semiclassical acceleration, for the general multi-band case other geometric quantity, the quantum metric, which discussed here, the more convenient expression is the one characterizes the quantum distance between two states. presented in Eq. (3) in terms of λab. aac Thus, Kshift originates directly from the quantum geometry of the wave function in k-space, and it represents the anomalous acceleration in momentum space [18]. ab B. Shift current of TBG The quantity λnm encodes the skewness of the acceleration: The Berry curvature dipole involves deriva- c a b b a tives of the type ∂ (rmnrnm − rmnrnm), which does We model TBG using a modified form of the Bistrizer- not cover the entire motion in momentum space. The Macdonald continuum model [33, 48]. We attach two remaining terms involving λab in Eq. (5) can be con- monolayer graphene sheets, with first rotating them with nected to skew-symmetric derivatives of the structure respect to one another with an angle θ = 1.08o, and then c a b a c b c b a b c a (∂ rmn)rnm − rmn(∂ rnm) + (∂ rmn)rnm − rmn(∂ rnm). introducing inter-layer coupling. By construction, this Note that in the special case of a two-band Dirac dis- model is endowed with C3z symmetry, and since both monolayers are inversion symmetric, the model as a whole 4

200 (a) (b)

100

- 100

- 200 0 2 4 6 8 10 0.8 1.0 1.2 1.4 1.6 1.8 2.0

yy;x Figure 3. (a) σ of TBG with ε = 0%, with C3 symmetry intact. The conductivity is plotted for 3 values of µ = −5, 0, 6 yy;x yy;x meV, respectively. (b) Decomposition of Eq. (3) into the Rshift and Kshift pieces, respectively. Solid lines depict TBG, while the dashed lines show MoS2, for comparison. The shift current is essentially zero for x < 1, has a large resonance around x ≈ 1 and then decays quickly for values above that. The normalized fraction does not show qualitative changes at x ≈ 1 but consistently conforms with the semiclassical intuition that the shift receives contributions from both the dispersive acceleration and the anomalous acceleration, where the latter is dominant for a flat-band dispersion. The maximal conductivity for µ = −5.0meV is attained at ω = 1.4meV, with value σyy;x = +198µAV−2nm, and for µ = 6.0meV it is −168µAV−2nm. has inversion symmetry. Inversion symmetry breaking within the three gaps that are opened by the staggered is introduced by a staggered potential ∆, as it typically potential. Within our numerical accuracy, the current arises from encapsulation of the bilayer in hBN. On is indeed found to have no contributions from the Berry yy;x yy;x the other hand, C3z is not necessarily broken because it curvature dipole, i.e. σ(s2) = σ(s) . Irrespective of this, requires some uniaxial strain of magnitude ε in the bottom for frequencies which cross the single-particle gap the layer. A detailed description of the model is given in the shift current reaches giant values nearly 200 µAV−2nm, supplementary material. The resulting band structure in far exceeding predicted values for other materials [13, 59]. the mini-Brillouin zone of valley K is shown in Fig.2a, both for the unstrained and strained TBG. The flat bands near half filling (µ = 0 meV) are gapped at Γ and K due As we pointed out, from the quasiparticle shift it is to inversion symmetry breaking. Fig.2b shows the Berry not immediately obvious where such a giant non-linear curvature in the mini-Brillouin zone in unstrained TBG, response could originate from. For this reason, we exam- which is C3-symmetric. We note that the Berry charges ine the different contributions in Eq. (8). Fig.2c shows 0 near the K and K -points in the original dispersion have the momentum-space structure of the integrand in Eq. opposite signs and cancel. (8) at µ = 0, ω = 1.4meV, i.e. just above the band gap While the shift vector S in Eq. (2) itself is not expected of the flat bands. The largest contributions are from the to be further decomposable into gauge invariant pieces, residual dispersion around Γ and from the flat parts of the expression that enters the shift current can indeed be the band structure around Ks, both of which are fairly decomposed. Writing σ(s) = σ(s1) + σ(s2), with broad in momentum space. Fig.3b illustrates the relative contribution of the pieces Kaa;c and Raa;c in Eq. (8), as 3 Z shift shift aa;c πe ac a function of the distance from the gap at 0. Over a σ(s1) = 2 ∂ka Ω |εmn=ω (7) aa;c aa;c ~ k large range of frequencies, it holds that Kshift Rshift, πe3 Z meaning that the shift current is produced mostly by the σaa;c = − (Raac + Kaac ) − σaa;c. (8) (s2) 2 shift shift (s1) phase of the Berry connection and not the shift of wave- ~ k function centers. For comparison, we performed the same Here, Eq. (7) contains the contributions from the Berry breakdown for the much more dispersive, gapped material a ac curvature dipole ∂ Ω , which vanishes in the presence MoS2, which has very similar symmetry properties [Fig. aa;c aa;c of C3z symmetry. This symmetry renders only two com- 3b]. In this latter case, Kshift Rshift near the band ponents of σab;c independent. For simplicity we focus edge, which supports the notion that the shift current in the main text on σyy;x, which encodes the transverse is resulting from a displacement in both real space and nonlinear conductivity response. The other independent momentum space, with the latter being greatly enhanced component is σxx;y (Supplementary Figs. 2 and 3); the for a flatband dispersion. As we mentioned in the be- symmetry analysis can be found in the SI. In Fig.3a ginning, these statements might seem questionable upon we present the total shift current in the unstrained case, regauging, a possible shortcoming on which we comment for three values of the chemical potential which reside in the discussion. 5

(a) (b) (c) 300

200

100

1.0 1.2 1.4 1.6 1.8 - 100 0.75 - 200 0.5 - 300 0.25

0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 10

Figure 4. Shift current response of strained TBG, σyy;x, for 3 chemical potentials within the gaps created by the staggered potential, and uniaxial strain ε = 0.1%. (a) Berry curvature dipole contribution, Eq. (7). The largest values are clipped to preserve readability for the total conductivity. (b) Momentum and position shifts contribution, Eq. (8) (c) Total conductivity σyy;x at T = 0K (solid lines) and for T = 300K (dashed lines, shown x10 for clarity). For chemical potential µ = 0 the shift yy;x −2 −2 yy;x yy;x peaks at σ = 315µAnmV (T = 0K) and 15µAnmV (T = 300K), respectively. Inset: relative size of Rshift and Kshift contributing to the total conductivity as a function of the distance from the ﬂat-band band gap, xEg (here µ = 0), cf. Fig.3b.

C. Effect of strain the gaps of the dispersive bands are strongly suppressed, the large density of states in the flat bands supports a strong signal for ω = 4meV, with an amplitude of still In the presence of uniaxial strain, the symmetry group ~ 5% of its value at T = 0. of TBG is reduced to C1. All components of the conductivity are now independent, but for clarity we continue to yy;x examine only the transverse contribution σ . Since the DISCUSSION Berry curvature dipole contribution as given by Eq (7) is no longer zero, we present both its contribution (Fig.4a) and the remaining terms in Eq.8 for σaa;c (Fig.4b). The The shift vector S has previously been connected to the (s2) real-space shift of the center-of-mass coordinate between total shift current is depicted in Fig.4c, with the relative two eigenstates upon excitation from the conduction band sizes of Raa;c and Kaa;c shown in the inset. shift shift m into the valence band n. To give some intuition, we At moderate uniaxial strain of size = 0.1%, the con- expand the real-space representation of the periodic eigen- ductivity σ(s1) due to the Berry curvature dipole becomes functions |unki in terms of local Wannier orbitals |wnRi comparable in size to σ(s2), but it has consistently the with center coordinate R [60] opposite sign. Thus, while the total conductivity unsupris- ingly increases due to the reduced symmetry in the system, X −ik(R−r) hr|unki = e hr|wnRi. (9) it has less accentuated resonances for the transitions at the R band edges, with values up to 300 µAV−1nm across the flatband gap. More unexpectedly, the shift current does Then, in momentum space the Berry connection is given not seem to profit from imbalanced Berry charges, as their by contribution either subtracts from the remaining current, Z or is almost negligible for transitions across the flatbands. a X ik(R−R0) a rmn = e dV hwnR0 |r |wnRi. (10) Our results establish that a large anomalous acceleration RR0 cell ac in the quasiparticle motion can arise even if ∂ka Ω = 0, c c c c i.e. its existence does not rely on the presence of Berry Evaluating the shift vector Smn = rmm −rnn +∂kc arg rmn a a a charges in the system. This is an important distinction based on this representation yields rmm − rnn = Rmm − a between the bulk photovoltaic effect and the anomalous Rnn for the direct difference. This is supplemented by

Hall effect in TBG. We further observe that in all cases the phase derivative ∂ka arg rmn, whose integral over the the shift current at frequencies ω 2meV, corresponding Brillouin zone is a multiple of 2π. If only two Wannier to transitions between dispersive bands, is negligible small orbitals have a significant overlap, the modulus |S| is a a compared to the resonances around the band edges. In clearly bounded by |Rmm−Rnn| < a, with lattice constant other words, the giant shift current shown in Fig.3 is a. If several orbitals overlap, the phase factors in the not tied to the topological properties of the flat bands sum Eq. (10) become important, with slope of growth but rather to their non-dispersive nature. Figure3c also in momentum space being at most a. Then, the phase shows the shift current evaluated at room temperature, derivative is expected to contribute similarly at O(a) to T = 300K (dashed lines). While the transitions across the shift vector. This already indicates that interpretation 6 of the shift current as a result of the wavefunction shift is is a two-band semimetal with one band crossing. There, narrow to some extent. For ease of illustration, imagine a mostly smooth gauge is at the same time periodic (i.e. a set of Landau levels in symmetric gauge. Their center- without phase jump at the Brillouin zone boundaries), of-mass coordinate R can be moved around freely in thus completely eliminating the phase contribution from exchange for acquiring an additional phase factor. Indeed, the integrated shift vector. This is the expected result for generic flat bands it is to be expected that there is a for a quasiparticle with vanishing effective mass which is gauge choice which makes the shift S only depend on the changes position in an applied electric field. phase, because the center of the Wannier functions can be Before closing, we remark that the difficulties in separat- repositioned with an appropriate gauge transformation. ing real-space and momentum-space effects of the acceler- This is inconsistent with the interpretation of the shift ation into gauge invariant pieces are intrinsic to the more current as the real space shift of the wavefunction center complicated semiclassical motion arising at second order upon absorption of a photon, as the wavefunction only in the applied field. In particular, the analogous splitting suffers a phase shift. of the quasiparticle velocity into the regular (dispersive) If the shift current is instead viewed as the anoma- and anomalous velocity has the important distinction lous acceleration that a quasiparticle undergoes due to that these two components of the velocity are orthogonal the interaction with the electric field at second order, to each other, making them linearly independent. Such the photogalvanic response follows as a straightforward a decomposition is not straightforward for the acceler- generalization of the linear response formalism involving ation, because it describes the changes to both regular the anomalous velocity [18, 61], thus removing the direct and anomalous velocity components in both normal and inference of a current from a real-space displacement. In- perpendicular direction, thus mixing them. However, we stead, both Rshift and Kshift appear as the result of the believe that the conclusions outlined above can be made same acceleration that changes both the position and the more rigorous by deriving a lower bound for Rshift and wavevector of the quasiparticle. As shown in the last Kshift, which will be the subject of a future work. section, this is consistent with our numerical findings for In summary, we report a giant photogalvanic current Rshift and Kshift using Bloch wavefunctions for a mostly for TBG which is irradiated by linear polarized light in smooth gauge choice within each band [cf. Fig.3b]. the terahertz range. The resonance profile we observe While one might object against inspecting gauge- in both the strained and unstrained cases suggest that dependent quantities, we emphasize that Rshift and Kshift TBG is a promising candidate for THz detection and can still contain valuable information about the quasipar- circuits, even at room temperature. As the root cause ticle dynamics in the sense that while they are not unique, behind the large response we identified the anomalous this does not at all imply that they are arbitrary. For this, acceleration due to the skew symmetric properties of the recall that all types of topological bands (including flat quantum geometry of the band structure as encoded in ab bands) have no uniquely defined center-of-mass coordinate λmn, which always appear in the shift current, but are within the unit cell, because the topological nature of the greatly amplified in TBG due to the flat-band dispersion. bands prevents such an assignment. However, from this The latter also turned out to be particularly important for it does not follow that the momentum space integral of retaining a large shift conductivity at higher temperatures. Rshift can take arbitrary values, because it contains much We expect the transverse dc-current reported here to more specific information about the relative positional be accessible using current samples and measurement difference between two bands, summed for all momenta. techniques [63]. The line of reasoning developed in this Indeed, it was already pointed out a long time ago [3, 62] work can potentially shed light on the quantum geometry that for an arbitrary band structure one cannot generally of the band structure in similarly twisted van-der-Waals a a expect to find a gauge such that rmm − rnn consistently materials with nearly flat bands, for example MoTe2 or vanishes for all momenta and all bands. WSe2 [64]. Summarizing these observations, we therefore pose that Acknowledgements We thank J. S. Hofmann, a useful indicator for band flatness is that the integrated R. M. Ribeiro, and R. Queiroz for useful discussions. B.Y. a a positional difference rmm − rnn between Bloch wavefunc- acknowledges the financial support by the Willner Family tions can be made substantially smaller than the inte- Leadership Institute for the Weizmann Institute of Sci- grated phase contribution ∂ka arg rmn. We further conjec- ence, the Benoziyo Endowment Fund for the Advancement ture that for highly dispersive bands a similar statement of Science, Ruth and Herman Albert Scholars Program should hold about the smallness of the integrated phase for New Scientists, the European Research Council (ERC contribution. A paradigmatic example in the latter case Consolidator Grant No. 815869, “NonlinearTopo”).

∗ [email protected] [2] R. von Baltz and W. Kraut, “Theory of the bulk photo- [1] V. I. Belinicher and B. I. Sturman, “The photogalvanic voltaic eﬀect in pure crystals,” Phys. Rev. B 23, 5590– eﬀect in media lacking a center of symmetry,” Sov. Phys. 5596 (1981). Usp. 23, 199–223 (1980). [3] J. E. Sipe and A. I. Shkrebtii, “Second-order optical 7

response in semiconductors,” Phys. Rev. B 61, 5337– mat.mes-hall]. 5352 (2000). [18] Tobias Holder, Daniel Kaplan, and Binghai Yan, “Conse- [4] Steve M. Young and Andrew M. Rappe, “First Principles quences of time-reversal-symmetry breaking in the light- Calculation of the Shift Current Photovoltaic Effect in matter interaction: Berry curvature, quantum metric, and Ferroelectrics,” Phys. Rev. Lett. 109, 116601 (2012), diabatic motion,” Phys. Rev. Research 2, 033100 (2020), arXiv:1202.3168 [cond-mat.mtrl-sci]. arXiv:1911.05667 [cond-mat.mes-hall]. [5] Wojciech Knap, Mikhail Dyakonov, Dominique Coquil- [19] Liang Wu, S. Patankar, T. Morimoto, N. L. Nair, E. The- lat, Frederic Teppe, Nina Dyakonova, Jerzy L usakowski, walt, A. Little, J. G. Analytis, J. E. Moore, and Krzysztof Karpierz, MacIej Sakowicz, Gintaras Valusis, J. Orenstein, “Giant anisotropic nonlinear optical re- Dalius Seliuta, Irmantas Kasalynas, Abdelouahad El Fa- sponse in transition metal monopnictide Weyl semimetals,” timy, Y. M. Meziani, and Taiichi Otsuji, “Field effect Nat. Phys. 13, 350–355 (2017), arXiv:1609.04894 [cond- transistors for terahertz detection: Physics and first imag- mat.mtrl-sci]. ing applications,” in Journal of Infrared, Millimeter, and [20] Qiong Ma, Su-Yang Xu, Ching-Kit Chan, Cheng-Long Terahertz Waves, Vol. 30 (Springer, 2009) pp. 1319–1337. Zhang, Guoqing Chang, Yuxuan Lin, Weiwei Xie, Tomás [6] Masayoshi Tonouchi, “Cutting-edge terahertz technology,” Palacios, Hsin Lin, Shuang Jia, Patrick A. Lee, Pablo Nat. Photonics 1, 97–105 (2007). Jarillo-Herrero, and Nuh Gedik, “Direct optical detec- [7] Michael A Kinch, “Fundamental physics of infrared detection of Weyl fermion chirality in a topological semimetal,” tor materials,” Journal of Electronic Materials 29, 809–817 Nat. Phys. 13, 842–847 (2017), arXiv:1705.00590 [cond- (2000). mat.mtrl-sci]. [8] Kamalodin Arik, Sajjad AbdollahRamezani, and Amin [21] Gavin B. Osterhoudt, Laura K. Diebel, Mason J. Gray, Khavasi, “Polarization Insensitive and Broadband Tera- Xu Yang, John Stanco, Xiangwei Huang, Bing Shen, Ni Ni, hertz Absorber Using Graphene Disks,” Plasmonics 12, Philip J. W. Moll, Ying Ran, and Kenneth S. Burch, 393–398 (2017). “Colossal mid-infrared bulk photovoltaic effect in a type- [9] V. M. Muravev and I. V. Kukushkin, “Plasmonic de- I Weyl semimetal,” Nat. Mater. 18, 471–475 (2019), tector/spectrometer of subterahertz radiation based on arXiv:1712.04951. two-dimensional electron system with embedded defect,” [22] P. Hosur and X. Qi, “Recent developments in transport Appl. Phys. Lett. 100, 082102 (2012). phenomena in Weyl semimetals,” C. R. Phys. 14, 857– [10] Denis A. Bandurin, Dmitry Svintsov, Igor Gayduchenko, 870 (2013), arXiv:1309.4464 [cond-mat.str-el]. Shuigang G. Xu, Alessandro Principi, Maxim Moskotin, [23] Katsuhisa Taguchi, Tatsushi Imaeda, Masatoshi Sato, Ivan Tretyakov, Denis Yagodkin, Sergey Zhukov, Takashi and Yukio Tanaka, “Photovoltaic chiral magnetic effect Taniguchi, Kenji Watanabe, Irina V. Grigorieva, Marco in Weyl semimetals,” Phys. Rev. B 93, 201202 (2016). Polini, Gregory N. Goltsman, Andre K. Geim, and [24] Ching-Kit Chan, Netanel H. Lindner, Gil Refael, and Georgy Fedorov, “Resonant terahertz detection using Patrick A. Lee, “Photocurrents in Weyl semimetals,” graphene plasmons,” Nat. Commun. 9, 5392 (2018), Phys. Rev. B 95, 041104 (2017), arXiv:1607.07839 [cond- arXiv:1807.04703 [cond-mat.mes-hall]. mat.mes-hall]. [11] T. Morimoto and N. Nagaosa, “Topological nature of [25] Fernando de Juan, Adolfo G. Grushin, Takahiro Mori- nonlinear optical effects in solids,” Sci. Adv. 2, e1501524– moto, and Joel E. Moore, “Quantized circular photo- e1501524 (2016), arXiv:1510.08112 [cond-mat.mes-hall]. galvanic effect in Weyl semimetals,” Nat. Commun. 8, [12] Liang Z. Tan, Fan Zheng, Steve M. Young, Fenggong 15995 (2017), arXiv:1611.05887 [cond-mat.str-el]. Wang, Shi Liu, and Andrew M. Rappe, “Shift current [26] Yang Zhang, Hiroaki Ishizuka, Jeroen van den Brink, bulk photovoltaic effect in polar materials—hybrid and Claudia Felser, Binghai Yan, and Naoto Nagaosa, “Pho- oxide perovskites and beyond,” npj Comput. Mater. 2, togalvanic effect in Weyl semimetals from first principles,” 16026 (2016). Phys. Rev. B 97, 241118 (2018), arXiv:1803.00562 [cond- [13] A. M. Cook, B. M. Fregoso, F. de Juan, S. Coh, and J. E. mat.str-el]. Moore, “Design principles for shift current photovoltaics,” [27] R. A. Lewis, “A review of terahertz detectors,” J. Phys. Nat. Commun. 8, 14176 (2017). D 52, 433001 (2019). [14] G. B. Ventura, D. J. Passos, J. M. B. Lopes dos Santos, [28] J. P. Guillet, B. Recur, L. Frederique, B. Bousquet, J. M. Viana Parente Lopes, and N. M. R. Peres, “Gauge L. Canioni, I. Manek-Hönninger, P. Desbarats, and covariances and nonlinear optical responses,” Phys. Rev. P. Mounaix, “Review of terahertz tomography techniques,” B 96, 035431 (2017), arXiv:1703.07796 [cond-mat.other]. Journal of Infrared, Millimeter, and Terahertz Waves 35, [15] D. J. Passos, G. B. Ventura, J. M. Viana Parente 382–411 (2014). Lopes, J. M. B. Lopes dos Santos, and N. M. R. Peres, [29] Yuan Cao, Valla Fatemi, Shiang Fang, Kenji Watanabe, “Nonlinear optical responses of crystalline systems: Results Takashi Taniguchi, Efthimios Kaxiras, and Pablo Jarillo- from a velocity gauge analysis,” Phys. Rev. B 97, 235446 Herrero, “Unconventional superconductivity in magic- (2018), arXiv:1712.04924 [cond-mat.other]. angle graphene superlattices,” Nature 556, 43–50 (2018), [16] Daniel E. Parker, Takahiro Morimoto, Joseph Orenstein, arXiv:1803.02342 [cond-mat.mes-hall]. and Joel E. Moore, “Diagrammatic approach to nonlinear [30] Yuan Cao, Valla Fatemi, Ahmet Demir, Shiang Fang, optical response with application to Weyl semimetals,” Spencer L. Tomarken, Jason Y. Luo, Javier D. Phys. Rev. B 99, 045121 (2019), arXiv:1807.09285 [cond- Sanchez-Yamagishi, Kenji Watanabe, Takashi Taniguchi, mat.str-el]. Efthimios Kaxiras, Ray C. Ashoori, and Pablo Jarillo- [17] Junyeong Ahn, Guang-Yu Guo, and Naoto Nagaosa, Herrero, “Correlated insulator behaviour at half-filling in “Low-frequency divergence and quantum geometry of magic-angle graphene superlattices,” Nature 556, 80–84 the bulk photovoltaic effect in topological semimetals,” (2018), arXiv:1802.00553 [cond-mat.mes-hall]. arXiv , arXiv:2006.06709 (2020), arXiv:2006.06709 [cond- [31] J. M. B. L. Santos, N. M. R. Peres, and A. H. 8

Castro Neto, “Graphene Bilayer with a Twist: Elec- magnetism in a moirésuperlattice,” Nature 579, 56–61 tronic Structure,” Phys. Rev. Lett. 99, 256802 (2007), (2020), arXiv:1905.06535 [cond-mat.mes-hall]. arXiv:0704.2128 [cond-mat.mtrl-sci]. [43] U. Zondiner, A. Rozen, D. Rodan-Legrain, Y. Cao, [32] J. M. B. L. Santos, N. M. R. Peres, and A. H. Castro R. Queiroz, T. Taniguchi, K. Watanabe, Y. Oreg, F. von Neto, “Continuum model of the twisted graphene bilayer,” Oppen, Ady Stern, E. Berg, P. Jarillo-Herrero, and Phys. Rev. B 86, 155449 (2012), arXiv:1202.1088 [cond- S. Ilani, “Cascade of phase transitions and Dirac revivals mat.mtrl-sci]. in magic-angle graphene,” Nature 582, 203–208 (2020), [33] Rafi Bistritzer and Allan H. MacDonald, “Moirébands in arXiv:1912.06150 [cond-mat.mes-hall]. twisted double-layer graphene,” PNAS 108, 12233–12237 [44] Dillon Wong, Kevin P. Nuckolls, Myungchul Oh, Biao (2011), arXiv:1009.4203 [cond-mat.mes-hall]. Lian, Yonglong Xie, Sangjun Jeon, Kenji Watanabe, [34] Aaron L. Sharpe, Eli J. Fox, Arthur W. Barnard, Takashi Taniguchi, B. Andrei Bernevig, and Ali Yaz- Joe Finney, Kenji Watanabe, Takashi Taniguchi, dani, “Cascade of electronic transitions in magic-angle M. A. Kastner, and David Goldhaber-Gordon, twisted bilayer graphene,” Nature 582, 198–202 (2020), “Emergent ferromagnetism near three-quarters filling in arXiv:1912.06145 [cond-mat.mes-hall]. twisted bilayer graphene,” Science 365, 605–608 (2019), [45] Matthew Yankowitz, Shaowen Chen, Hryhoriy Polshyn, arXiv:1901.03520 [cond-mat.mes-hall]. Yuxuan Zhang, K. Watanabe, T. Taniguchi, David Graf, [35] M. Serlin, C. L. Tschirhart, H. Polshyn, Y. Zhang, Andrea F. Young, and Cory R. Dean, “Tuning super- J. Zhu, K. Watanabe, T. Taniguchi, L. Balents, and conductivity in twisted bilayer graphene,” Science 363, A. F. Young, “Intrinsic quantized anomalous Hall effect 1059–1064 (2019), arXiv:1808.07865 [cond-mat.mes-hall]. in a moiréheterostructure,” Science 367, 900–903 (2020), [46] S. L. Tomarken, Y. Cao, A. Demir, K. Watanabe, arXiv:1907.00261 [cond-mat.str-el]. T. Taniguchi, P. Jarillo-Herrero, and R. C. Ashoori, [36] Xiaobo Lu, Petr Stepanov, Wei Yang, Ming Xie, Mo- “Electronic Compressibility of Magic-Angle Graphene Su- hammed Ali Aamir, Ipsita Das, Carles Urgell, Kenji perlattices,” Phys. Rev. Lett. 123, 046601 (2019), Watanabe, Takashi Taniguchi, Guangyu Zhang, Adrian arXiv:1903.10492 [cond-mat.mes-hall]. Bachtold, Allan H. MacDonald, and Dmitri K. Efetov, [47] Jianpeng Liu and Xi Dai, “Anomalous Hall effect, “Superconductors, orbital magnets and correlated states magneto-optical properties, and nonlinear optical proper- in magic-angle bilayer graphene,” Nature 574, 653–657 ties of twisted graphene systems,” npj Comput. Mater. (2019), arXiv:1903.06513 [cond-mat.str-el]. 6, 57 (2020), arXiv:1907.08932 [cond-mat.mes-hall]. [37] Guorui Chen, Lili Jiang, Shuang Wu, Bosai Lyu, [48] Wen-Yu He, David Goldhaber-Gordon, and K. T. Law, Hongyuan Li, Bheema Lingam Chittari, Kenji Watan- “Giant orbital magnetoelectric effect and current-induced abe, Takashi Taniguchi, Zhiwen Shi, Jeil Jung, Yuanbo magnetization switching in twisted bilayer graphene,” Nat. Zhang, and Feng Wang, “Evidence of a gate-tunable Commun. 11, 1650 (2020). Mott insulator in a trilayer graphene moirésuperlattice,” [49] Jin-Xin Hu, Cheng-Ping Zhang, Ying-Ming Xie, and Nat. Phys. 15, 237–241 (2019), arXiv:1803.01985 [cond- K. T. Law, “Nonlinear Hall Effects in Strained mat.mes-hall]. Twisted Bilayer WSe2,” arXiv , arXiv:2004.14140 (2020), [38] Yuhang Jiang, Xinyuan Lai, Kenji Watanabe, Takashi arXiv:2004.14140 [cond-mat.mes-hall]. Taniguchi, Kristjan Haule, Jinhai Mao, and Eva Y. An- [50] Meizhen Huang, Zefei Wu, Jinxin Hu, Xiangbin Cai, drei, “Charge order and broken rotational symmetry in En Li, Liheng An, Xuemeng Feng, Ziqing Ye, Nian Lin, magic-angle twisted bilayer graphene,” Nature 573, 91–95 Kam Tuen Law, and Ning Wang, “Giant nonlinear Hall (2019), arXiv:1904.10153 [cond-mat.mes-hall]. effect in twisted WSe2,” arXiv , arXiv:2006.05615 (2020), [39] Youngjoon Choi, Jeannette Kemmer, Yang Peng, Alex arXiv:2006.05615 [cond-mat.mes-hall]. Thomson, Harpreet Arora, Robert Polski, Yiran Zhang, [51] Maximilian Otteneder, Stefan Hubmann, Xiaobo Lu, Hechen Ren, Jason Alicea, Gil Refael, Felix von Oppen, Dmitry A Kozlov, Leonid E Golub, Kenji Watanabe, Kenji Watanabe, Takashi Taniguchi, and Stevan Nadj- Takashi Taniguchi, Dmitri K Efetov, and Sergey D Perge, “Electronic correlations in twisted bilayer graphene Ganichev, “Terahertz photogalvanics in twisted bilayer near the magic angle,” Nat. Phys. 15, 1174–1180 (2019), graphene close to the second magic angle,” Nano Lett. arXiv:1901.02997 [cond-mat.mes-hall]. 20, 7152–7158 (2020). [40] Alexander Kerelsky, Leo J. McGilly, Dante M. [52] Yizhou Liu, Tobias Holder, and Binghai Yan, “Chiral- Kennes, Lede Xian, Matthew Yankowitz, Shaowen Chen, ity induced Giant Unidirectional Magnetoresistance in K. Watanabe, T. Taniguchi, James Hone, Cory Dean, the Twisted Bilayers,” arXiv , arXiv:2010.08385 (2020), Angel Rubio, and Abhay N. Pasupathy, “Maximized arXiv:2010.08385 [cond-mat.mes-hall]. electron interactions at the magic angle in twisted bilayer [53] Cheng-Ping Zhang, Jiewen Xiao, Benjamin T. Zhou, graphene,” Nature 572, 95–100 (2019). Jin-Xin Hu, Ying-Ming Xie, Binghai Yan, and K. T. [41] Yonglong Xie, Biao Lian, Berthold Jäck, Xiaomeng Liu, Law, “Giant nonlinear Hall effect in strained twisted bi- Cheng-Li Chiu, Kenji Watanabe, Takashi Taniguchi, layer graphene,” arXiv e-prints , arXiv:2010.08333 (2020), B. Andrei Bernevig, and Ali Yazdani, “Spectroscopic arXiv:2010.08333 [cond-mat.mes-hall]. signatures of many-body correlations in magic-angle [54] Robert W Boyd, Nonlinear optics (Elsevier, San Diego, twisted bilayer graphene,” Nature 572, 101–105 (2019), United States, 2003). arXiv:1906.09274 [cond-mat.mes-hall]. [55] J. E. Moore and J. Orenstein, “Confinement-Induced [42] Guorui Chen, Aaron L. Sharpe, Eli J. Fox, Ya-Hui Zhang, Berry Phase and Helicity-Dependent Photocurrents,” Shaoxin Wang, Lili Jiang, Bosai Lyu, Hongyuan Li, Kenji Phys. Rev. Lett. 105, 026805 (2010), arXiv:0911.3630 Watanabe, Takashi Taniguchi, Zhiwen Shi, T. Senthil, [cond-mat.mes-hall]. David Goldhaber-Gordon, Yuanbo Zhang, and Feng [56] Inti Sodemann and Liang Fu, “Quantum Nonlinear Hall Wang, “Tunable correlated Chern insulator and ferro- Effect Induced by Berry Curvature Dipole in Time- 9

Reversal Invariant Materials,” Phys. Rev. Lett. 115, [64] Fengcheng Wu, Timothy Lovorn, Emanuel Tutuc, and 216806 (2015), arXiv:1508.00571 [cond-mat.mes-hall]. A. H. MacDonald, “Hubbard model physics in transition [57] B. Yan and C. Felser, “Topological Materials: Weyl metal dichalcogenide moirébands,” Phys. Rev. Lett. Semimetals,” Annu. Rev. Condens. Matter Phys. 8, 121, 026402 (2018). 337–354 (2017), arXiv:1611.04182 [cond-mat.mtrl-sci]. [65] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. [58] N. P. Armitage, E. J. Mele, and A. Vishwanath, “Weyl Novoselov, and A. K. Geim, “The electronic properties and Dirac semimetals in three-dimensional solids,” Rev. of graphene,” Rev. Mod. Phys. 81, 109–162 (2009), Mod. Phys. 90, 015001 (2018), arXiv:1705.01111 [cond- arXiv:0709.1163. mat.str-el]. [66] Vitor M. Pereira, A. H. Castro Neto, and N. M. R. Peres, [59] Tonatiuh Rangel, Benjamin M. Fregoso, Bernardo S. “Tight-binding approach to uniaxial strain in graphene,” Mendoza, Takahiro Morimoto, Joel E. Moore, and Jef- Phys. Rev. B 80, 045401 (2009). frey B. Neaton, “Large bulk photovoltaic effect and spon- [67] F. Guinea, M. I. Katsnelson, and A. K. Geim, “Energy taneous polarization of single-layer monochalcogenides,” gaps and a zero-field quantum Hall effect in graphene Phys. Rev. Lett. 119, 067402 (2017). by strain engineering,” Nat. Phys. 6, 30–33 (2010), [60] David Vanderbilt, Berry Phases in Electronic Structure arXiv:0909.1787 [cond-mat.mes-hall]. Theory: Electric Polarization, Orbital Magnetization and [68] Nguyen N. T. Nam and Mikito Koshino, “Lattice re- Topological Insulators (Cambridge University Press, 2018). laxation and energy band modulation in twisted bilayer [61] D. Xiao, M.-C. Chang, and Q. Niu, “Berry phase effects graphene,” Phys. Rev. B 96, 075311 (2017), 1706.03908. on electronic properties,” Rev. Mod. Phys. 82, 1959– [69] M. Monteverde, C. Ojeda-Aristizabal, R. Weil, K. Ben- 2007 (2010), arXiv:0907.2021 [cond-mat.mes-hall]. naceur, M. Ferrier, S. Guéron,C. Glattli, H. Bouchiat, [62] J. E. Sipe and J. Zak, “Geometric phase for electric J. N. Fuchs, and D. L. Maslov, “Transport and Elastic polarization along ’rational’ directions in crystals,” Phys. Scattering Times as Probes of the Nature of Impurity Lett. A 258, 406–412 (1999). Scattering in Single-Layer and Bilayer Graphene,” Phys. [63] Aaron M. Burger, Radhe Agarwal, Alexey Aprelev, Ed- Rev. Lett. 104, 126801 (2010). ward Schruba, Alejandro Gutierrez-Perez, Vladimir M. [70] M. Tinkham, Group Theory and Quantum Mechanics Fridkin, and Jonathan E. Spanier, “Direct observation (Dover Publications, 2003). of shift and ballistic photovoltaic currents,” Sci. Adv. 5, eaau5588 (2019).

Supplemental Material

I. CONTINUUM MODEL CONSTRUCTION

The continuum model for TBG [33, 48] is constructed by joining together two monolayer graphene layers at zero eﬀective separation between them. We choose the following real space unit vectors, for each graphene layer, √ ! √ ! √ 1 3 √ 1 3 a = 3d , , a = 3d , − (S1) 1 2 2 2 2 2

In this description, the A and B sublattices are located, respectively, at vA = (0, 0), vB = d(0, 1). The reciprocal lattice vectors are, √ ! √ ! 4π 3 1 4π 3 1 b = , , b = − , , (S2) 1 3d 2 2 2 3d 2 2

√ 4π 3 1 and the Brillouin zone corners hosting the low energy states are at Ku = ± 3d 2 , 2 . Within the BM model, the bilayer system is symmetric under C3 by construction, and inversion and time-reversal (when both Ku valleys of the original graphene monolayers are included). In order to break inversion symmetry, we introduce a coupling to a substrate (for example, hBN), which lifts inversion symmetry but leaves C3 symmetry intact. This allows for a ﬁnite shift current which is entirely independent of the Berry curvature dipole, because the latter is set to zero by C3 symmetry. The Hamiltonian of the bilayer system is therefore given by,

H = Ht + Hb + Htb, (S3) 10 where t,b,int denote the top, bottom layer, and interlayer hopping respectively. The top layer has the following continuum Hamiltonian, for a given momentum q,

X † Ht,u(q) = ~vf uat,s,u(q)R+q · σat,s,u(q), (S4) s where s, u designate the spin and valley degrees of freedom; i.e., s =↑, ↓, u = ±1, for the K,K0 valleys of the original graphene monolayers. at/b,s,u is the annihilation operator for an electron with spin s, and valley u, on the A/B sub-lattices of the top/bottom layers. R± is the rotation matrix for the top/bottom layers, given by: θ θ θ ◦ R± = R ± 2 = cos( 2 ) ∓ iσy sin( 2 ), acting on the sub-lattice space, with θ ≈ 1.05 denoting the twist angle. The Fermi velocity is ~vf = 0.596eV nm [65], and σ = (σx, σy, σz) are Pauli matrices. The momentum q is measured relative to the valley cenetered at Ku. Inversion symmetry breaking and uniaxial strain are introduced in the bottom layer. Throughout this work, the strain is applied along the zigzag direction of the bottom graphene sheet. The Hamiltonian of the bottom layer then takes the form, X † Hb,u(q) = ~vf at,s,u(q) R−(1 + )(q + uA) · σ s + ∆σz at,s,u(q), (S5)

−1 0 Here, is the uniaxial strain matrix, which has the form = 0 ν . A is the pseudo-gauge ﬁeld resultant from the application of strain [66, 67]. ∆ = 17 meV, is the staggered potential generated by alignment with an hBN layer. = 0, 0.1% representing the strain-less and strained cases, respectively. With strain Htb is changed accordingly, as discussed in Methods.

II. METHODS

The Bistritzer-MacDonald continuum model contains an inter-layer coupling term, which couples two momenta q, q0, √ √ 0 3 1 3 1 if q − q = {q1,u, q2,u, q3,u}, where q1,u = |q0| (0, −1), q2,u = |q0| 2 , 2 , q3,u = |q0| − 2 , 2 , are the Moir´elattice 8π sin( θ ) vectors with q = √ 2 , and d = 1.42A˚ is the carbon-carbon bond length in graphene [33]. In this framework, the 0 3 3d interlayer coupling is included via the Htb term in Eq. S3 as,

X † 0 Htb = at,s,u(q)(T1,u(q, q )+ (S6) q,q0,s,u 0 0 0 T2,u(q, q ) + T3,u(q, q ))ab,s,u(q ). In the presence of the strain deﬁned in the main text, the coupling matrices (acting on the valley index) become, t 1 1 T = δ 0 (S7) 1,u 3 1 1 q−q ,q1,u −iu 2π 1−2ν2 ! t 1 e 3 ( ) T = δ 0 (S8) 2,u iu 2π 1−2ν2 q−q ,q2,u 3 e 3 ( ) 1

iu 2π 1−2ν2 ! t 1 e 3 ( ) T = δ 0 . (S9) 3,u −iu 2π 1−2ν2 q−q ,q3,u 3 e 3 ( ) 1 Throughout, we take t = 0.33 eV. Accordingly, the lattice vectors of the Moir´esuperlattice are deformed in the presence of strain. These have the form,

4π θ θ q1,u = u √ cos , (2 + ) sin (S10) 3 3d 2 2 2π √ q = u 3 cos θ − 3(2 − ν) sin θ , (S11) 2,u 9d 2 2 √ θ θ 3ν cos 2 − 3(2 + ) sin 2 2π √ q = −u 3 sin θ (2 − ν) − 3 cos θ , (S12) 3,u 9d 2 2 √ θ θ 3ν cos 2 + 3(2 + ) sin 2 11

Figure S1. The Brillouin zone of the top (red) and bottom (blue) graphene sheets, with and without strain. Left: Two undistorted (ε = 0) grahpene layers are rotated one with respect to the other, forming a folded mini Brillouin zone (mBZ), when rotated by an angle ±θ/2. The K point of each layer shifts to Kt and Kb for the top and bottom layers respectively. The separation between them is denoted by q0 = R+Ku − R−Ku. When the mBZ is refolded onto the center of the original √ 3 1 Brillouin zone, Kt/b = |q0| 2 , ± 2 . Middle: Upon introduction of uniaxial strain on the bottom layer, the Brillouin zone deformes by expanding one, and contracting in the other direction. The K points transform according to Eq. S13. Right: when the strained Brillouin zone is rotated with respect to an unstrained one, a deformed mBZ is formed, as shown here. Consequently, tunneling vectors which depend on the positions of Kt/b in the mBZ are modiﬁed, as shown in Methods. The angle formed 0 −1 |q|0 o between the unstrained q0 and the strained vector q is given by θ = cos 0 ≈ 3.16 . 0 |q |0

The Dirac points transform under strain as,

T K¯ u = (1 − ε )Ku − uA (S13)

We introduce a pseudo-gauge ﬁeld which stems from the underlying two-center approximation for the tunneling matrix [68]. It is given by,

β A = − (1 + ν, 0) (S14) d with ν = 0.165, β = 1.57 obtained from monolayer graphene. Finally, the diagonalization of the Hamiltonian H in Eq. S3 is accomplished by recasting it in the form,

X † H = As,u(q)hu(q)As,u, (S15) q,s,u

T where now As,u(q) = [ab,s,u(q), at,s,u(q + q1,u), at,s,u(q + q2,u), at,s,u(q + q3,u)] is the inﬁnite-component operator vector satisfying the constraints on momentum transfer. hu has the following truncated structure, after applying the momentum transfer relations ensuring non-zero T tunnelling,

  Hb,u(q) T1,u T2,u T3,u  T † H(1) 0 0   1,u t,u  hu(q) =  † (2)  , (S16)  T2,u 0 Ht,u 0  † (3) T3,u 0 0 Ht,u

(i) with Ht,u = Ht,u(q + qi,u). In this work, we used 81 sites in reciprocal space for the construction of the hamiltonian, which results in a Hamiltonian which is 324 × 324. The integrals appearing in Eqs. (7),(8) are computed using a 3 discretized grid of 600 × 600 in the (kx, ky) plane of the mini Brillouin zone. 10 frequency point samplings in ω are carried out uniformly. Convergence is checked against the case with C3 symmetry, where veriﬁcation is done by comparing σxx;x with −σyy;x; and σyy;y with −σxx;y. All equalities were veriﬁed to within 5%. The delta functions of Eqs. (7),(8) are broadened with a width Γ = 0.02meV for T = 0K and Γ = 0.1meV at T = 300K. This corresponds to transport lifetimes observed in bilayer graphene in the clean limit [69]. 12

200

100

- 100

- 200

0 2 4 6 8 10

Figure S2. Conductivity σxx;y as a function of frequency, with ε = 0 for 3 values of the chemical potential. Compare with σyy;x in Fig.3. Here, σxx;y is negative for both µ = 6.0, −5.0meV, and has the peak values σxx;y = −108, −121µAnmV−2, respectively. For µ = 0, σxx;y = 28µAnmV−2. A sizeable conductivity, |σxx;y| > 50µAnmV−2 is obtained for a wide range of frequencies between ω = 1.5 − 4.5meV.

III. SYMMETRY PROPERTIES OF THE RESPONSE

In the main text, we observe that the response tensor σab;c has only two independent components, and that the a bc Berry curvature dipole, ∂ Ω vanishes. We proceed to prove this. The generator of C3z is given by [70], √ − 1 − 3 0 √2 2 M = 3 1 . (S17)  2 − 2 0 0 0 1

The Berry curvature dipole (BCD), Dabc is a gauge invariant material property of the system, and has the form Dabc = bc ∂ka Ω , making it a rank-3 pseudotensor. Firstly, we observe that this quantity is anti-symmetric in (b, c). Applying abc P αβγ abc acb Neumann’s principle, we enforce D = αβγ MaαMbβMcγ D . Using the anti-symmetry of D = −D , we focus axy P αβγ P 1 αxy 3 αyx P αxy only on non-trivial components, D = MaαMxβMyγ D = Maα D − D = MaαD . αβγ aα √ 4 4 √ aα xxy 1 xxy 3 yxy yxy 3 xxy 1 yxy Since a = x, y, we obtain the following set of equations, D = − 2 D − 2 D ,D = 2 D − 2 D . This xxy yxy set admits only the solution D = D = 0, as required, demonstrating that the BCD is zero, under C3z. For the ab;c general rank-3 symmetric conductivity tensor σ , we derive analogous symmetry constraints under C3z. This results in two independent components overall,

σxx;x = −σyy;x = −σxy;y = −σyx;y (S18) σyy;y = −σxx;y = −σyx;x = −σxy;x. (S19)

We further note that under linear-polarized light, the conductivity tensor exhibits a special permutation symmetry, σab;c = σba;c.

IV. ADDITIONAL DATA FOR THE TRANSVERSE COMPONENTS

For completeness, we provide the remaining transverse component σxx;y of the shift current, which agrees qualitatively yy;x and in part quantitatively with the component σ discussed in the main text. We recall that in general, with C3z symmetry the conductivity tensor has only 2 independent components, therefore the longitudinal components can be deduced straightforwardly from the data presented here. With ε = 0, the remaining independent component is σxx;y, also a transverse component. This is presented in Fig. S2, for 3 chemical potential values. Note that the contribution of Eq. (7) is zero, due to C3z symmetry. With ﬁnite strain, all terms in Eq. (3) of the main text contribute to the current. For consistency, we again present the transverse component, σxx;y, for 3 chemical potential values, in the same way as in Fig.3 of the main text. With the introduction of strain, the Berry curvature dipole contribution is no longer zero (Fig. S3a). Although the net conductivity is not substantially enhanced by the introduction of strain (cf. Fig. S3c), a broad resonance in σxx;y 13

(a) (b) (c) 300

200

100

- 100

- 200

- 300

0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 10

Figure S3. Contributions to the conductivity σxx;y, with strain ε = 0.001, the same value used in Fig.4 of the main text. (a) Berry curvature dipole, Eq. (7). (b) Rshift + Kshift, as in Eq. (8). (c) Total conductivity. While the introduction of strain produces a giant Berry curvature dipole, the magnitude of Eq. (8) also increases albeit with opposite sign. Consequently, the total conductivity remains comparable to the unstrained case. For µ = −5.3, 0.0, 6.7 meV, the maximal values obtained for the conductivity are σxx;y = −96, 100, −102 µAnmV−2, respectively. We note that for µ = 6.7meV, the response proﬁle is exceptionally broad, and is the conductivity is almost constant for the range ω = 2 − 6meV.

1500 (a) (b) (c) 1000

500

- 500

- 1000

0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 10

Figure S4. Contributions to the conductivity σxx;y with ε = −0.001, for 3 chemical potential values. (a) BCD term. (b) Kshift + Rshift term. (c) Total conductivity. . appears at chemical potential µ = 6.7meV. Note that while the introduction of strain induces a large Berry curvature dipole, it is still smaller than the remaining contributions according to Eq. (8), meaning that the sign of the total conductivity is determined by the latter part of the response. To examine whether our results depend of the sign of the applied strain (i.e., whether the strain is compressive or tensile), we show in Fig. S4 the conductivity σyy;x for strain with negative (i.e., compressive) magnitude, = −0.001. While the detailed frequency dependence is indeed sensitively dependent on the strain, both the magnitude of the response and ts resonance structure are very comparable.