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Berry curvature dipole with different strain and displacement fields a The Vz(out of plane displacement field) and nh(number of holes per unit cell) dependence of Dx(x component Berry dipole) with strain ϵ = 0.5% along the zigzag edge direction. The color bar shows the magnitude of Dx. The critical Vz at which the phase transition occurs is denoted by the horizontal dashed line. Dx is strongly enhanced near the phase transition point and nh ≈ 2 when the Fermi energy is near the bottom of the top valence band. bDx, Dy as a function of Vz with fixed nh. The values of nh ( = 1.95) are denoted by the vertical line 1 in a. For b, an uniaxial strain is applied along the zigzag edge direction. c Same parameters as b but the uniaxial strain is applied along the armchair edge direction. d, e The Berry curvatures Ω of top valence band of strained tWSe2 in the deformed Brillouin zone before and after the topological phase transition respectively. In d, Vz = 3 meV, in e, Vz = 5 meV. The strain is applied along the zigzag edge direction. The temperature is set to 1.5K.
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Recently, it has been pointed out that the twisting of bilayer WSe2 would generate topologically non-trivial flat bands near the Fermi energy. In this work, we show that twisted bilayer WSe2 (tWSe2) with uniaxial strain exhibits a large nonlinear Hall (NLH) response due to the non-trivial Berry curvatures of the flat bands. Moreover, the NLH effect...
Citations
... Theoretically, it is well understood that the linear AHE response can originate from the electronic Berry curvature monopole [5][6][7]. More recently, it is pointed out that the higher-order multipoles of Berry curvature can give rise to even richer nonlinear AHE responses, both in theoretical [18,20,[26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42] and experimental [1, 12, 15-17, 19, 21-24, 43-45] activities of topological quamtum materials. The nonlinear AHE describes the phenomenon when an AC current with frequency ω is injected, a voltage at integer multiples of the input frequency emerges along the orthogonal direction, and this signal is nonlinear with respect to the current and/or external fields. ...
The anomalous Hall effect (AHE), which provides a bridgeway between the geometry of quantum wavefunctions and transport measurements, has been a key focus of intensive studies. In addition to the well-studied linear AHE, governed by the electronic Berry curvature, nonlinear AHE originated from higher-order Berry-curvature multipoles has also been observed in recent studies. Inspired by the 3rd order AHE and its room temperature sign switching in kagome antiferromagnet FeSn [S. Sankar, R. Liu, X. Gao, et al, to be published (2023)], we investigate the generic sign structure of Berry-curvature induced 3rd order AHE in topological magnetic material. We find that in contrast to the linear Hall coefficient, whose sign is determined by the broken time-reversal symmetry, the sign of the 3rd order Hall coefficient is dictated by the interplay between time-reversal symmetry breaking, magnetic order, and spin-orbit couplings. Our calculations give a possible solution for the ``sign problem" of the 3rd order AHE response in the phase space spanned by the in- and out-of-plane magnetization, the spin-orbital coupling strength and chemical potential. We further propose realistic experiment setups to systematically reveal the sign structure in the 3rd order AHE response via continuously rotating the magnetic field directions.
... Moiré systems also offer a promising platform for exploring the valley and topological physics. In addition to large Berry curvature hotspots, valley-Chern number based topological phase transitions 23 have also been demonstrated in moiré systems such as TBG 24 , TDBG 10,15,[25][26][27][28][29] , multi-layered twisted systems 30,31 , and twisted transition metal dichalcogenides 32,33 . ...
... However, the nonlinear anomalous (NLA) Hall effect induced by the Berry curvature dipole (BCD) [36][37][38][39][40][41] can probe the Berry curvature in systems with TRS [ Fig. 1(a)]. The NLA Hall effect requires the breaking of space inversion symmetry (SIS), and it appears in crystals with reduced symmetry 32,33,[42][43][44][45][46][47][48][49][50][51] . In addition to the NLA Hall charge response to the electric field, there are also thermoelectric and thermal NLA responses that have a similar origin. ...
... In this paper, we demonstrate that the electrical, ther- moelectric and thermal NLA Hall responses in strained TDBG can probe the topological phase transition, as they undergo a sign reversal across the transition. We present a systematic study of the family of NLA Hall effects: the electrical NLA Hall, the thermoelectric NLA Nernst [ Fig. 1 32,46,55 in TDBG. In addition to tuning the electronic structure, the perpendicular electric field in TDBG also induces topological phase-transition of the valley-Chern type. ...
Nonlinear anomalous Hall effect is the Berry curvature dipole induced second-order Hall voltage or temperature difference induced by a longitudinal electric field or temperature gradient. These are the prominent Hall responses in time-reversal symmetric systems. These band-geometry induced responses in recently realized twistronic platforms can probe their novel electronic band structure and topology. Here, we investigate the family (electrical, thermoelectric, and thermal) of second-order nonlinear anomalous Hall effects in the moiré system of twisted double bilayer graphene. We combine the semiclassical transport framework with the continuum model of twisted double bilayer graphene to demonstrate that the nonlinear anomalous Hall signals can probe topological phase transitions in moiré systems. We show that the whole family of nonlinear anomalous Hall responses undergo a sign reversal across a topological phase transition. Our study establishes a deeper connection between valley topology and nonlinear Hall effects in time-reversal symmetric systems.
... systems undergo a transition between topological phases and these transitions are hard to detect, unlike phase transitions of the order parameter. Recent proposals suggest that topological transitions are accompanied by simultaneous changes in the BCD 14,15 . Moiré systems are known to be natural candidates to host topological bands 16 . ...
... As one moves from regime I to regime II, the touching of the flat bands with the higher moiré dispersing bands leads to a change in the valley Chern numbers of the flat bands 9,10 , which in turn should be reflected in the change in sign of the BCD 15 . ...
... The scaling analysis using temperature as a parameter also gives a similar magnitude of BCD (Extended Data Fig. 3). A high BCD magnitude has been theoretically predicted in strained twisted systems 15,24,[30][31][32] . The origin of high BCD in graphene-based moiré systems, in the vicinity of the CNP, can be understood using the model of strained twisted bilayer graphene as a tilted Dirac system 40 . ...
Topological aspects of the electron wave function—including the Berry curvature and Chern number—play a crucial role in determining the physical properties of materials. Although the Berry curvature and its effects in materials have been studied1,2, detecting changes in the Chern number can be challenging, particularly changes in the valley Chern type. In this regard, twisted double bilayer graphene3–7 has emerged as a promising platform to gain electrical control over the Berry curvature hotspots⁸ and the valley Chern numbers of topological flat bands9,10. In addition, strain-induced breaking of the threefold rotation symmetry leads to a non-zero first moment of Berry curvature (called the Berry curvature dipole)¹¹. Here we show that a sign change in the Berry curvature dipole detects topological transitions in the bands. In twisted double bilayer graphene, the perpendicular electric field simultaneously tunes the valley Chern number and Berry curvature dipole, providing a tunable system to probe the topological transitions. Furthermore, we find hysteresis in the transport response that is caused by switching of the electric polarization. This holds promise for next-generation Berry-curvature-based memory devices. Our technique can be emulated in three-dimensional topological systems to probe topological transitions governed by parameters such as pressure or anisotropic strain.
... Moiré systems also offer a promising platform for exploring the valley and topological physics. In addition to large Berry curvature hotspots, valley-Chern number based topological phase transitions 23 have also been demonstrated in moiré systems such as TBG 24 , TDBG 10,15,[25][26][27][28][29] , multi-layered twisted systems 30,31 , and twisted transition metal dichalcogenides 32,33 . ...
... However, the nonlinear anomalous (NLA) Hall effect induced by the Berry curvature dipole (BCD) [36][37][38][39][40][41] can probe the Berry curvature in systems with TRS [ Fig. 1(a)]. The NLA Hall effect requires the breaking of space inversion symmetry (SIS), and it appears in crystals with reduced symmetry 32,33,[42][43][44][45][46][47][48][49][50][51] . In addition to the NLA Hall charge response to the electric field, there are also thermoelectric and thermal NLA responses that have a similar origin. ...
... In this paper, we demonstrate that the electrical, ther- moelectric and thermal NLA Hall responses in strained TDBG can probe the topological phase transition, as they undergo a sign reversal across the transition. We present a systematic study of the family of NLA Hall effects: the electrical NLA Hall, the thermoelectric NLA Nernst [ Fig. 1 32,46,55 in TDBG. In addition to tuning the electronic structure, the perpendicular electric field in TDBG also induces topological phase-transition of the valley-Chern type. ...
Nonlinear anomalous Hall effect is the Berry curvature dipole induced second-order Hall voltage or temperature difference in response to a longitudinal electric field or temperature gradient. These are the prominent Hall responses in time reversal symmetric systems. Here, we investigate the family of second-order nonlinear anomalous Hall effects, the electrical, thermoelectric and thermal nonlinear Hall effects in the moir\'e system of twisted double bilayer graphene. We demonstrate that the nonlinear anomalous Hall signals can be used to probe the topological phase-transitions in moir\'e systems, induced by a perpendicular electric field. Specifically, we show that the whole family of nonlinear anomalous Hall responses undergo a sign reversal across a topological phase-transition.
... The theoretical predictions of the nonlinear Hall effect in a material can be achieved by calculating the Berry curvature dipole from band structure. Multiple materials [5][6][7][8][9][10][11][12][13][14][15][16][17] have been predicted or validated to possess strong nonlinear Hall effect, such as the Weyl semimetals [5][6][7][8] , the giant Rashba material bismuth tellurium iodine (BiTeI) under pressure 10 , the monolayer WTe 2 and MoTe 2 with an external electric field 11 , and the strained twisted bilayer graphene 12 or WSe 2 13,14 . The experimental observations of the nonlinear Hall effect are mainly limited to the two-dimensional systems 3,4,[14][15][16] . ...
... It is clear that the Berry curvature dipole D xz and D yz exhibit drastic oscillating behavior near the Fermi level, and D xz and D yz can switch their signs dramatically within a very narrow energy region. On the other hand, we find that the magnitude of Berry curvature dipole in our considered twisted bilayer WTe 2 is much larger than that in previous reports 3,4,[11][12][13][14][15][16] . For example, the peak of D yz locates near the Fermi level, which is calculated to be~1400 Å. ...
In a system with broken inversion symmetry, a second-order nonlinear Hall effect can survive even in the presence of time-reversal symmetry. In this work, we show that a giant nonlinear Hall effect can exist in twisted bilayer WTe 2 system. The Berry curvature dipole of twisted bilayer WTe 2 ( θ = 29.4°) can reach up to ~1400 Å, which is much larger than that in previously reported nonlinear Hall systems. In twisted bilayer WTe 2 system, there exist abundant band anticrossings and band inversions around the Fermi level, which brings a complicated distribution of Berry curvature, and leads to the nonlinear Hall signals that exhibit dramatically oscillating behavior in this system. Its large amplitude and high tunability indicate that the twisted bilayer WTe 2 can be an excellent platform for studying the nonlinear Hall effect.
... Mathematically, the BCD is an integral of the product of local Berry curvature and velocity over the Fermi surface, so metals with prominent Berry curvature near the Fermi surface are ideal platforms to observe this effect. Under this guiding principle, as band degeneracies are natural sources of divergent Berry curvature, three-dimensional Weyl semimetals [21][22][23][24][25][26][27][28][29][30], two-dimensional transition-metal dichalcogenides [31][32][33][34][35][36][37][38][39][40][41], strained graphene [42][43][44] and topological insulators close to the phase boundary [45], which have either tilted gapless Weyl cones or tilted gapped Dirac cones, have been actively studied both theoretically and experimentally [46,47]. As the nonlinear Hall effect is an effect related to Fermi surface, it is noteworthy that doping is necessary for its observation in pristine gapped systems, such as topological insulators. ...
In a time-reversal invariant system, while the anomalous Hall effect identically vanishes in the linear response regime due to the constraint of time-reversal symmetry on the distribution of Berry curvature, a nonlinear Hall effect can emerge in the second-order response regime if the inversion symmetry is broken to allow a nonzero Berry curvature dipole (BCD) on the Fermi surface. In this work, we study the nonlinear Hall effect of the BCD origin in two-dimensional doped insulators and semimetals belonging to the symmetry class AI which has spinless time-reversal symmetry. Despite that the class AI does not host any strong topological insulator phase in two dimensions, we find that they can still be classified as topologically obstructed insulators and trivial insulators if putting certain constraint on the Hamiltonians. When the insulator gets closer to the phase boundary of the two distinct phases, we find that the BCDs will become more prominent if the doping level is located near the band edge. Moreover, when the insulator undergoes a phase transition between the two distinct phases, we find that the BCDs will dramatically change their signs. For the semimetals without inversion symmetry, we find that the BCDs will sharply reverse their signs when the doping level crosses the Dirac points. With the shift of the locations of Dirac points in energy, the critical doping level at which the BCDs sharply reverse their signs will accordingly change. Our study reveals that class AI materials can also have interesting geometrical and topological properties, and remarkable nonlinear Hall effect can also appear in this class of materials even though the spin-orbit coupling is negligible. Our findings broaden the scope of materials to study the nonlinear Hall effect and provide new perspectives for the application of this effect.
... An important stepping stone towards the understanding of these phases is the characterization of the quantum geometric and dispersive features of the flat bands. Because of inversion symmetry breaking, TBG and similar twisted bilayers exhibit intriguing nonlinear phenomena [47][48][49][50][51][52][53] such as the bulk photovaltaic effect (BPVE) [1,54] and nonlinear anomalous Hall effect [55,56]. These nonlinear probes are ideally suited for the investigation of quasiparticle properties in flat bands as they are not suppressed by a vanishing Fermi velocity [18]. ...
The detection of terahertz (THz) radiation promises intriguing applications in biology, telecommunication, 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.
In this work, we establish a theoretical analysis of the emergence of layer-contrasted Nernst response perpendicular to the direction of the temperature gradient in twisted moiré layers, called layer Nernst effect (LNE). This phenomenon arises from the trigonal warping of the Fermi surface along with a layer-contrasted pseudomagnetic field. Interestingly, the Fermi surface's warping explicitly breaks intravalley inversion symmetry, which leads to an imbalance between left- and right-moving carriers, thus resulting in a nonvanishing LNE. We then validate our theoretical scheme by applying it to twisted bilayer graphene (TBG). Importantly, we find that the LNE coefficient in TBG can reach values as high as 103 A/(m·K), surpassing those of previously known materials by at least one order of magnitude. These results provide a theoretical foundation for utilizing TBG and other twisted moiré layers as promising platforms to explore layer caloritronics and develop thermoelectric devices.
