Faraday rotations, ellipticity and circular dichroism in the magneto-optical spectrum of moiré superlattices
FFaraday rotations, ellipticity and circular dichroism in themagneto-optical spectrum of moir´e superlattices ∗ J. A. Crosse , and Pilkyung Moon , , † Arts and Sciences, New York University Shanghai, Shanghai, 200122, China. NYU-ECNU Institute of Physics at NYU Shanghai, Shanghai, 200062, China. Department of Physics, New York University, New York, 10003, USA.February 22, 2021
Abstract
We study the magneto-optical conductivity of a number of Van der Waals heterostructures, namely,twisted bilayer graphene, AB-AB and AB-BA stacked twisted double bilayer graphene and monolayergraphene and AB-stacked bilayer graphene on hexagonal boron nitride. As magnetic field increases, theabsorption spectrum exhibits a self-similar recursive pattern reflecting the fractal nature of the energy spec-trum. Whilst twisted bilayer graphene displays only weak circular dichroism, monolayer graphene andAB-stacked bilayer graphene on hexagonal boron nitride show specifically strong circular dichroism, owingto strong inversion symmetry breaking properties of the hexagonal boron nitride layer. As, the left and rightcircularly polarized light interact with these structures differently, plane polarized incident light undergoesa Faraday rotation and gains an ellipticity when transmitted. The size of the respective angles is on theorder of a degree.
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1. Introduction
Van der Waals heterostructures [1–5], stacks of 2 𝐷 monolayers with relative twist angles between them,offer a novel way to engineer materials with specific electronic properties and, hence, are a promising platformfor high-performance electronics. As the optical properties of a material are directly related to its electronicproperties, these types of heterostructure potentially offer a method for optical control as well. One immediateconcern would be optical depth. However, although only a few atoms in thickness, the quantum efficiencyof these structures can be remarkably high - monolayer graphene displays a universal absorption of 2 .
3% [6]and recent experiments in hexagonal Boron Nitride encapsulated graphene have shown 36% absorption for ∗ Project supported by the National Natural Science Foundation of China (Grant No. xxxxxxxx) and Science and TechnologyCommission of Shanghai Municipality (Grant No. 19ZR1436400) † Corresponding author. E-mail: [email protected] a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b y L M1 L M2 L M1 L M2 (a) (b) K (+1,1) K (+1,2) K (+1,1) K (-1,1) K (-1,2) G M2 G M1 G M1 G M2 -- KK' Γ M (c) (d) K (-1,2) K (+1,2) K (-1,1) Figure 1: The moir´e superlattice generated by (a) two layers with different orientations and (b) two layers withdifferent lattice constants. Brillouin zone folding in moir´e superlattice generated by (c) two layers with differentorientations and (d) two layers with different lattice constants. The large red and blue hexagons represent theBrillouin zone of the two layers. The small hexagons represent the moir´e Brillouin zone. The inset shows thehigh-symmetry points of the moir´e Brillouin zone.mid-infrared frequencies when a perpendicular magnetic field is applied [7]. Furthermore, single and multilayergraphene has also been shown to display giant Faraday rotations of up to 6 deg . when subjected to a magneticfield [8], another feature that could potentially be exploited for optical control. These results lead one toconjecture as to whether van der Waals heterostructures subject to an applied magnetic field might be suitablematerial for optical device design.In addition to potential technological application, van der Waals heterostructures also provide a direct probeof fundamental physics. When a material is subjected to an applied magnetic field the electron band structuresplits up into a series of discrete Landau levels. At higher magnetic fields, specifically when the magnetic lengthbecomes smaller than the lattice constant, the individual Landau levels split into a series of sublevels resultingin the fractal-like Hofstadter spectrum [9]. In traditional materials, the magnetic field required to realize theHofstadter spectrum is unfeasibly large - on the order of 10 T. However, in van der Waals heterostructuresthe lattice constant of the moir´e interference pattern, which is generated when adjacent layers have differentorientations and/or different lattice constant (see Fig. 1), can be very large and hence the Hofstadter regimecan be realized at reasonable magnetic fields [10,11]. This led to it’s indirect observation via measurementsof the Hall conductivity [12–15]. The optical conductivity offers a way to directly observe the energy spacingbetween bands and, hence, could be used to observe the fractal nature of the Hofstadter spectrum directly [16].2 th Layer3rd Layer2nd Layer1st Layer Upper BilayerLower Bilayer
AB-BA Stacked Bi-Bilayer Graphene B B B B A A A A AB-AB Stacked Bi-Bilayer Graphene A A A A B B B B A A B B Twisted Bilayer Graphene A V N B V B hBN-Monolayer Graphene A A V N V B hBN-Bilayer Graphene Figure 2: Cross-section through the various van der Waals heterostructure considered here.Here, we present the magneto-optical spectrum, computed via the optical conductivity, of a variety of vander Waals Heterostructures. The specific structures we consider are: twisted bilayer graphene (TBG), AB-ABand AB-BA stacked twisted double bilayer graphene (TDBG) and monolayer graphene and AB-stacked bilayergraphene on hexagonal boron nitride (MLG-hBN and BLG-hBN respectively). In all but the first structurethe materials display strong circular dichroism, where the optical conductivity is significantly different for thetwo different polarizations of circularly polarized light. Some materials also display valley dichroism, wherethe contribution to the optical conductivity from each valley is significantly different. This would allow opticalexcitation of a single valley and, hence, may provide a suitable platform for ‘valleytronic’ devices [17]. We alsocompute the circular dichroism induced Faraday rotations and ellipticities experienced by transmitted plane-polarized light. At high magnetic fields the Hofstadter spectrum is clearly visible in the optical conductivityand hence would correspond to a direct probe of the fractal nature of bands.
2. Theoretical Methods
In order to study the optical conductivity of van der Waals heterostructures we will use an effective con-tinuum model. This model is valid when the moir´e lattice constant is much larger than the graphene latticeconstant [18–25]. In such systems, the separation between the Dirac points is sufficiently large that inter-valleymixing can be safely neglected and the full Hamiltonian separates into two independent Hamiltonians, eachdescribing the electronic properties of a single valley. 3 .1. Atomic Structure of Monolayer Graphene
Monolayer graphene (MLG) consists of a hexagonal lattice of carbon atoms, each contributing a singlep-orbital electron to the electronic structure. In the following, we take the lattice vectors of the unrotatedgraphene layers to be a = 𝑎 (1 ,
0) and a = 𝑎 (1 / , √ /
2) with the graphene lattice constant 𝑎 = 0 .
246 nm. Intwisted structures the two layers are rotated by an angle 𝜃 with respect to each other. After rotation, the latticevectors in each layer read a ( 𝑙 ) 𝑖 = 𝑅 ( ∓ 𝜃/ a 𝑖 where 𝑅 ( 𝜃 ) is the rotation matrix and 𝑙 ∈ , ∓ 𝜃 , respectively. Accordingly, the unrotated reciprocal lattice vectors are given by a * = (2 𝜋/𝑎 )(1 , − / √
3) and a * = (2 𝜋/𝑎 )(0 , / √
3) and the rotated reciprocal lattice vectors in each layer by a * ( 𝑙 ) 𝑖 = 𝑅 ( ∓ 𝜃/ a * 𝑖 . The Diracpoints of the two graphene layers are located at K ( 𝜉,𝑙 ) = − 𝜉 [2 a * ( 𝑙 )1 + a * ( 𝑙 )2 ] /
3, with 𝜉 = ± 𝐾 and 𝐾 ′ valleys. The Hamiltonian for each valley reads 𝐻 ( 𝜉,𝑙 ) 𝑀𝐿𝐺 ( k ( 𝜉,𝑙 ) ) = ⎛⎝ − (cid:126) 𝑣 𝐹 𝑘 ( 𝜉,𝑙 ) − − (cid:126) 𝑣 𝐹 𝑘 ( 𝜉,𝑙 )+ ⎞⎠ . (1)Here, 𝑣 𝐹 is a Fermi velocity, which we take to be ≈ × m / s in MLG, 𝑘 ( 𝜉,𝑙 ) ± = 𝑒 ± 𝑖𝜉𝜂 ( 𝑙 ) ( 𝜉𝑘 ( 𝜉,𝑙 ) 𝑥 ± 𝑖𝑘 ( 𝜉,𝑙 ) 𝑦 ), where 𝜂 ( 𝑙 ) is the angle between a ( 𝑙 )1 and the 𝑥 -axis (which in this case is 𝜂 (1) / (2) = ± 𝜃/ k ( 𝜉,𝑙 ) = k − K ( 𝜉,𝑙 ) ]. In the following we will consider two possible stackings of bilayer graphene (BLG), AB- and BA- stacking.In isolation, these two stackings are identical - the two forms being related by a 180 degree in-plane rotation.However, when part of a van der Waals heterostructure these two stackings are distinguishable because theatoms in the BLG layer are in different locations with respect to the atoms of the adjoining layers. The BLGHamiltonian for each valley reads 𝐻 ( 𝜉,𝑙 ) 𝐴𝐵 ( k ( 𝜉,𝑙 ) ) = ⎛⎝ 𝐻 ( 𝜉,𝑙 ) 𝑀𝐿𝐺 ( k ( 𝜉,𝑙 ) ) + ∆ 𝐵 𝑔 † ( k ( 𝜉,𝑙 ) ) 𝑔 ( k ( 𝜉,𝑙 ) ) 𝐻 ( 𝜉,𝑙 ) 𝑀𝐿𝐺 ( k ( 𝜉,𝑙 ) ) + ∆ 𝐴 ⎞⎠ , (2)for AB-stacking and 𝐻 ( 𝜉,𝑙 ) 𝐵𝐴 ( k ( 𝜉,𝑙 ) ) = ⎛⎝ 𝐻 ( 𝜉,𝑙 ) 𝑀𝐿𝐺 ( k ( 𝜉,𝑙 ) ) + ∆ 𝐴 𝑔 ( k ( 𝜉,𝑙 ) ) 𝑔 † ( k ( 𝜉,𝑙 ) ) 𝐻 ( 𝜉,𝑙 ) 𝑀𝐿𝐺 ( k ( 𝜉,𝑙 ) ) + ∆ 𝐵 ⎞⎠ , (3)for BA-stacking. Here, 𝐻 ( 𝜉,𝑙 ) 𝑀𝐿𝐺 ( k ( 𝜉,𝑙 ) ) is the MLG Hamiltonian given in Eq. (1), 𝑔 ( k 𝑙 ) = ⎛⎝ (cid:126) 𝑣 𝑘 ( 𝜉,𝑙 )+ 𝛾 (cid:126) 𝑣 𝑘 ( 𝜉,𝑙 ) − (cid:126) 𝑣 𝑘 ( 𝜉,𝑙 )+ ⎞⎠ , (4)is the intra-bilayer coupling and ∆ 𝐴 = ⎛⎝ ∆ 00 0 ⎞⎠ , ∆ 𝐵 = ⎛⎝ ⎞⎠ , (5)are the on-site potential of the lattice sites that are vertically aligned with the lattice sites of the adjacentlayer of the bilayer. Numerically, we take this to be ∆ = 0 .
05 eV. 𝑣 𝐹 is, again, the Fermi velocity which, we4ake to be ≈ . × m / s in BLG to make the results consistent with previous work [26,27]. The remainingparameters are 𝛾 = 0 . 𝐴 and 𝐵 sites in the upper and lower layers ofthe graphene bilayer and 𝑣 = 1 . × m / s and 𝑣 = 0 . × m / s, which are responsible for the trigonalwarping and electron-hole asymmetry, respectively [26,27,29]. When two 2 𝐷 materials with different lattice constants and/or different orientations are stacked on top ofeach other, the mismatch between the lattices of the two layers leads to a moir´e interference pattern. In thelow energy regime, the long wavelength of the electrons are more strongly affected by the long period moir´esuperlattice rather than the short period of the individual lattices of each layer. The dominance of the moir´esuperlattice leads to a dramatic increase in the size of the material’s unit cell and a correspondingly largereduction in the size of the Brillouin zone. The reciprocal lattice vectors for the moir´e Brillouin zone are givenby G M 𝑖 = a * (1) 𝑖 − a * (2) 𝑖 , from which the real space moir´e lattice vectors can be found via G M 𝑖 · L M 𝑗 = 2 𝜋 . Themoir´e lattice constant is given by 𝐿 M = 𝑎/ [2 sin( 𝜃/ 𝐴 = | L M1 × L M2 | =( √ / 𝐿 . The interlayer interaction can be found by considering the points in the Brillouin zone coupled bythe generalized Umklapp process [23]. By considering only the three strongest processes (the next strongestprocesses have coupling strengths two orders of magnitude smaller), the form of the coupling between adjacentgraphene layers can be found to be 𝑈 = ⎛⎝ 𝑢 𝑢 ′ 𝑢 ′ 𝑢 ⎞⎠ + ⎛⎝ 𝑢 𝑢 ′ 𝜔 − 𝜉 𝑢 ′ 𝜔 𝜉 𝑢 ⎞⎠ 𝑒 𝑖𝜉 G M1 · r + ⎛⎝ 𝑢 𝑢 ′ 𝜔 𝜉 𝑢 ′ 𝜔 − 𝜉 𝑢 ⎞⎠ 𝑒 𝑖𝜉 ( G M1 + G M2 ) · r , (6)where 𝜔 = 𝑒 𝜋𝑖/ and 𝑢 = 0 . 𝑢 ′ = 0 . 𝐴 ↔ 𝐴 , 𝐵 ↔ 𝐵 ) and opposing ( 𝐴 ↔ 𝐵 ) sublattices, respectively [25]. Thus, one canwrite down the general form of the Hamiltonian for TBG 𝐻 ( 𝜉 ) 𝑇 𝐵𝐺 = ⎛⎝ 𝐻 ( 𝜉, 𝑀𝐿𝐺 ( k ( 𝜉, ) 𝑈 † 𝑈 𝐻 ( 𝜉, 𝑀𝐿𝐺 ( k ( 𝜉, ) ⎞⎠ , (7)and, similarly, for TDBG 𝐻 ( 𝜉 ) 𝑖𝑗 = ⎛⎝ 𝐻 ( 𝜉, 𝑖 ( k ( 𝜉, ) 𝑈 † × 𝑈 × 𝐻 ( 𝜉, 𝑗 ( k ( 𝜉, ) ⎞⎠ , (8)where 𝑖, 𝑗 ∈ 𝐴𝐵, 𝐵𝐴 and 𝑈 × = ⎛⎝ 𝑈 ⎞⎠ . (9) Hexagonal Boron Nitride (hBN) is another 2 𝐷 material that has a hexagonal lattice. However, in thismaterial the two triangular sublattices consist of different atomic species. As a result, inversion symmetry isbroken and hBN exhibits a 4 . 𝜃 , when the K-points in the adjacent layers are close to each other, one finds that 𝐻 ℎ𝐵𝑁 = ⎛⎝ 𝑉 𝑁 𝑉 𝐵 ⎞⎠ , (10)with 𝑉 𝑁 = 3 .
34 eV and 𝑉 𝐵 = − .
40 eV. Using this simplified Hamiltonian in Eq. (10) and the parameters 𝑢 = 𝑢 = 𝑢 ′ = 0 .
152 eV in Eq. (6), one can identify the potential at 𝐸 ≈ 𝑉 ℎ𝐵𝑁 = 𝑈 † ( − 𝐻 ℎ𝐵𝑁 ) 𝑈, = 𝑉 ⎛⎝ ⎞⎠ + ⎧⎨⎩ 𝑉 𝑒 𝑖𝜉𝜓 ⎡⎣⎛⎝ 𝜔 − 𝜉 𝜔 − 𝜉 ⎞⎠ 𝑒 𝑖𝜉 G M1 · r + ⎛⎝ 𝜔 𝜉 𝜔 𝜉 𝜔 − 𝜉 ⎞⎠ 𝑒 𝑖𝜉 G M2 · r + ⎛⎝ 𝜔 − 𝜉 𝜔 − 𝜉 ⎞⎠ 𝑒 − 𝑖𝜉 ( G M1 + G M2 ) · r ⎤⎦ + h . c . ⎫⎬⎭ , (11)with 𝑉 = − 𝑢 (︂ 𝑉 𝑁 + 1 𝑉 𝐵 )︂ , (12) 𝑉 𝑒 𝑖𝜓 = − 𝑢 (︂ 𝑉 𝑁 + 𝜔 𝑉 𝐵 )︂ . (13)The numerical values of the parameters are given by 𝑉 ≈ . 𝑉 ≈ . 𝜓 ≈ − .
29 (rad) [11].Thus, the Hamiltonians for MLG-hBN and BLG-hBN heterostructures are 𝐻 ( 𝜉 ) 𝑀𝐿𝐺 − ℎ𝐵𝑁 = 𝐻 ( 𝜉 ) 𝑀𝐿𝐺 ( k ) + 𝑉 ℎ𝐵𝑁 , (14)and 𝐻 ( 𝜉 ) 𝐵𝐿𝐺 − ℎ𝐵𝑁 = 𝐻 ( 𝜉 ) 𝐴𝐵 ( k ) + 𝑉 ℎ𝐵𝑁, × (15)respectively, with 𝑉 ℎ𝐵𝑁, × = ⎛⎝ 𝑉 ℎ𝐵𝑁
00 0 ⎞⎠ . (16) In the absence of a magnetic field, the band structure of the material can be found by diagonalizingthe Hamiltonain in the basis of the Bloch wavefunctions for each layer. To find the band structure when aperpendicular magnetic field is applied, the Hamiltonain can be diagonalized in the basis of single particlemonolayer graphene Landau levels (in the following the Zeeman effect is neglected) [10,11,27]. In general, in thepresence of a magnetic field, the periodicity of the lattice is lost owing to the spatial dependence of the vectorpotential, which, in the Landau gauge, reads A = (0 , 𝐵𝑥, / Φ = 𝑝/𝑞 where 𝑝 and 𝑞 are co-prime integers, Φ = 𝐵𝐴 is the flux through the unit cell and Φ = ℎ/𝑒 is the quantum ofmagnetic flux) - in the Landau gauge with the y-axis of the coordinate system parallel to L , one can introducea periodic magnetic unit cell with lattice vectors ˜ L = 𝑞 L and ˜ L = L [30,31]. Hence, one can find ‘magnetic’6loch conditions for this enlarged unit cellΨ k ( r + ˜ L ) = 𝑒 𝑖 k · ˜ L 𝑒 − 𝑖 ( 𝑒/ (cid:126) )( A − B × r ) · ˜ L Ψ k ( r ) , (17)Ψ k ( r + ˜ L ) = 𝑒 𝑖 k · ˜ L Ψ k ( r ) . (18)Therefore, it is only at specific values of the magnetic field that the band structure can be found.Within each layer, one can construct a wavefunction that obeys the magnetic Bloch conditions in Eqs. (17)and (18) from the Landau levels of monolayer graphene. The effective continuum Landau levels for the +ve( 𝜉 = +1) and -ve ( 𝜉 = −
1) in monolayer graphene read [32,33]Ψ (+ ,𝑙 ) 𝑛,𝑘 𝑦 ( r ) = 𝐶 𝑛 𝑒 𝑖𝑘 𝑦 𝑦 ⎛⎝ − 𝑖 sgn( 𝑛 ) 𝜑 | 𝑛 |− ,𝑘 𝑦 ( 𝑥 ) − 𝑒 𝑖𝜂 ( 𝑙 ) 𝜑 | 𝑛 | ,𝑘 𝑦 ( 𝑥 ) ⎞⎠ 𝑒 𝑖 K (+ ,𝑙 ) · r , (19)Ψ ( − ,𝑙 ) 𝑛,𝑘 𝑦 ( r ) = 𝐶 𝑛 𝑒 𝑖𝑘 𝑦 𝑦 ⎛⎝ 𝑒 𝑖𝜂 ( 𝑙 ) 𝜑 | 𝑛 | ,𝑘 𝑦 ( 𝑥 ) − 𝑖 sgn( 𝑛 ) 𝜑 | 𝑛 |− ,𝑘 𝑦 ( 𝑥 ) ⎞⎠ 𝑒 𝑖 K ( − ,𝑙 ) · r , (20)respectively, with the upper and lower components of the vector referring to the 𝐴 and 𝐵 sublattices respectively.Here, 𝑛 is the Landau level index, 𝑘 𝑦 is the wave vector in the 𝑦 -direction and 𝜂 ( 𝑙 ) is, again, the angle between a ( 𝑙 )1 and the 𝑥 -axis. The single particle Landau level is defined in terms of the Hermite polynomial, 𝐻 𝑛 ( 𝑧 ), as 𝜑 | 𝑛 | ,𝑘 𝑦 ( 𝑥 ) = (2 𝑛 𝑛 ! √ 𝜋𝑙 𝐵 ) − / 𝑒 − 𝑧 / 𝐻 𝑛 ( 𝑧 ) with 𝑧 = ( 𝑥 + 𝑘 𝑦 𝑙 𝐵 ) /𝑙 𝐵 and 𝑙 𝐵 = √︀ (cid:126) / ( 𝑒𝐵 ) [33,34]. The normalizationcoefficient reads 𝐶 𝑛 = 1 for 𝑛 = 0 and 𝐶 𝑛 = 1 / √ 𝑛 ̸ = 0. To find a wavefunction that satisfies Eqs. (17)and (18) one needs to combine Landau levels at different 𝑘 𝑦 viaΨ ( 𝜉,𝑙 ) 𝑛,𝑚 = ∞ ∑︁ 𝑗 = −∞ 𝛼 𝑗 Ψ ( 𝜉,𝑙 ) 𝑛,𝑘 𝑚𝑦 ( r ) , (21)with 𝛼 = 𝑒 𝑖 ( k − K ( 𝜉,𝑙 ) ) · (˜ L − 𝑞 ˜ L / 𝑒 𝑖𝜋𝑝𝑞 ( 𝑗 +1) / 𝑒 𝑖𝜋𝑞𝑚 , (22) 𝑘 𝑚𝑦 = 𝑘 𝑦 − 𝐾 ( 𝜉,𝑙 ) 𝑦 − 𝜋𝐿 𝑦 ( 𝑝𝑗 + 𝑚 ) , (23)and the 𝑚 index running from 0 to 𝑝 − 𝐻 ( 𝜉 ) 𝑛,𝑛 ′ ,𝑚,𝑚 ′ ,𝑙,𝑙 ′ = ⟨ Ψ ( 𝜉,𝑙 ′ ) 𝑛 ′ ,𝑚 ′ | 𝐻 ( 𝜉 ) 𝜇 | Ψ ( 𝜉,𝑙 ) 𝑛,𝑚 ⟩ , (24)with 𝜇 indicating the specific heterostructure. The diagonal elements of the matrix Hamiltonian reduces to 𝐻 ( 𝜉 ) 𝑛,𝑛 ′ ,𝑚,𝑚 ′ ,𝑙,𝑙 = 𝜀 𝑛 𝛿 𝑛,𝑛 ′ 𝛿 𝑚,𝑚 ′ , (25)where 𝜀 𝑛 = (cid:126) 𝜔 𝐵 sgn( 𝑛 ) √︀ | 𝑛 | is the single particle Landau level energy with 𝜔 𝐵 = √︀ 𝑣 𝐹 𝑒𝐵/ (cid:126) [33,34]. Theinter-layer matrix elements of 𝑈 can be evaluated using the identity [35] ⟨ 𝜑 | 𝑛 ′ | ,𝑘 ′ 𝑦 ( 𝑥 ) 𝑒 𝑖𝑘 ′ 𝑦 𝑦 | 𝑒 𝑖 G · r | 𝑒 𝑖𝑘 𝑦 𝑦 𝜑 | 𝑛 | ,𝑘 𝑦 ( 𝑥 ) ⟩ = 𝛿 𝑘 ′ 𝑦 ,𝑘 𝑦 + 𝐺 𝑦 √︂ 𝜆 !Λ! (︂ 𝐺 𝑥 + 𝑖𝐺 𝑦 | G | )︂ 𝑛 − 𝑛 ′ × (︂ 𝑖 | G | 𝑙 𝐵 √ )︂ | 𝑛 − 𝑛 ′ | 𝑒 −| G | 𝑙 𝑏 / 𝑒 − 𝑖𝑙 𝐵 𝐺 𝑥 ( 𝑘 ′ 𝑦 + 𝑘 𝑦 ) / 𝐿 | 𝑛 − 𝑛 ′ | 𝜆 (︂ | G | 𝑙 𝐵 )︂ , (26)7here 𝜆 = min( 𝑛, 𝑛 ′ ), Λ = max( 𝑛, 𝑛 ′ ) and 𝐿 𝛼𝜆 ( 𝑥 ) is an associated Laguerre polynomial. This matrix Hamiltonianis unbounded in both 𝑛 and 𝑗 . However, by applying magnetic Bloch conditions one can see that the state with 𝑗 = 1 and 𝑚 = 0 is equivalent to the state with 𝑗 = 0 and 𝑚 = 𝑝 which is just the state 𝑗 = 0 and 𝑚 = 0 withthe addition of a phase. Thus, one only needs to consider a single cycle of 𝑚 ∈ [0 , 𝑝 −
1] with the appropriateperiodic boundary conditions. The 𝑛 index relates to the energy of the Landau level basis but one can truncatethe Hamiltonian at an energy at which the Landau level only weakly affect the low energy spectrum. Thiscutoff energy must be significantly larger that the interlayer coupling characterized by the coupling constants 𝑢 and 𝑢 ′ . This bounded matrix can then be diagonalized to find the electronic structure of the heterostructurein question. Once the band structure of the material has been obtained, the optical conductivities for left ( 𝜎 + ) and right( 𝜎 − ) circularly polarized light can be found from [16,36,37] 𝜎 ± = 𝑒 (cid:126) 𝑖𝑆 ∑︁ 𝛼,𝛽 𝑓 ( 𝜀 𝛼 ) − 𝑓 ( 𝜀 𝛽 ) 𝜀 𝛼 − 𝜀 𝛽 |⟨ 𝛼 | ˆ 𝑣 ± | 𝛽 ⟩| 𝜀 𝛼 − 𝜀 𝛽 + (cid:126) 𝜔 + 𝑖𝜂 , (27)where 𝑆 is the area of the system. Here, ˆ 𝑣 ± = ˆ 𝑣 𝑥 ± ˆ 𝑣 𝑦 / √ 𝑣 𝑗 = − ( 𝑖/ (cid:126) )[ˆ 𝑟 𝑗 , ˆ 𝐻 ] ( 𝑗 ∈ 𝑥, 𝑦 ) is the velocityoperator and 𝜂 = 0 .
01 meV is a phenomenological broadening. 𝜀 𝑖 and | 𝑖 ⟩ ( 𝑖 ∈ 𝛼, 𝛽 ) are the eigenenergies andeigenstates which are found from the diagonalization of the Hamiltonian in Eq. (24). The function 𝑓 ( 𝜀 ) is theFermi distribution. In this study we take the temperature to be absolute zero, hence 𝑓 ( 𝜀 ) = 1 for all statesbelow the charge neutral point and 𝑓 ( 𝜀 ) = 0 for all states above the charge neutral point. Note that the Peierlssubstitution, which is used to include the the effects of the applied magnetic field, replaces the momentum withthe gauge invariant canonical momentum (cid:126) k → (cid:126) k + 𝑒 A , where A is the vector potential. However, if we thenchoose the Landau gauge A = (0 , 𝐵𝑥, 𝑇 then the velocity operators remain unchanged in the presence of amagnetic field. Furthermore, the interlayer coupling, as it is not a function of the momentum, also does notaffect the velocity operators. Thus, the velocity operators are only dependent on the form of the Hamiltonianin each individual layer. Circular dichroism is where the optical response of a material to the two polarization of circularly polarizedlight is different and can be quantified by the parameter 𝐷 𝑐 = Re [ 𝜎 + ] − Re [ 𝜎 − ]Re [ 𝜎 + ] + Re [ 𝜎 − ] . (28)This feature can have a significant effect on incident electromagnetic radiation. Furthermore, as the magneto-optical conductivity for circularly polarized light is a complex variable, it has both real and imaginary partsand these parts also affect the propagation of light in different ways. Specifically, for incident plane polarizedlight, which consists of an equal mixture of the two circularly polarized components, a difference in the realpart of the magneto-optical conductivity leads to a change in the relative magnitude of the two components,generating an ellipticity, and a difference in the imaginary part leads to a change in the relative phase of each8omponent, leading to a Faraday rotation of the polarization direction. These effect can be used to probe thecircular dichromatic properties of a material.For an electromagnetic plane wave incident on linearly responding 2 𝐷 electron gas (i.e. the electromagneticresponse is of the form 𝑗 = 𝜎 · E ), Maxwell’s equations give the transmission of circularly polarized light as [38] 𝑡 ± = 2 𝑛 𝑢 𝑛 𝑢 + 𝑛 𝑠 + ( 𝜎 ± /𝑐𝜀 ) , (29)where 𝑛 𝑠 and 𝑛 𝑢 are the refractive indices of the substrate and encapsulating layer respectively (in the followingwe will consider free standing structures so 𝑛 𝑠 = 𝑛 𝑢 = 1). If circular dichroism is present in the material, thenthe magneto-optical conductivity for the two circular polarizations will be different and, hence, the transmissionproperties of these two polarizations will also be different. This difference can be probed by using incident planepolarized light, which will undergo a Faraday rotation, 𝜃 , and obtain an ellipticity, 𝛿 , the magnitudes of whichare given by 𝜃 = 12 (Arg [ 𝑡 + ] − Arg [ 𝑡 − ]) ≈ 𝑛 𝑢 + 𝑛 𝑠 ) 𝑐𝜀 (Im [ 𝜎 + ] − Im [ 𝜎 − ]) , (30) 𝛿 = | 𝑡 + | − | 𝑡 − || 𝑡 + | + | 𝑡 − | ≈ 𝑛 𝑢 + 𝑛 𝑠 ) 𝑐𝜀 (Re [ 𝜎 + ] − Re [ 𝜎 − ]) , (31)respectively. The approximation on the right hand side valid for the case of 𝑛 𝑠 + 𝑛 𝑢 > 𝜎 𝑥𝑦 /𝑐𝜀 and where onehas taken terms to first order in the conductivity only [39,40]. Thus, the ellipticity is a probe of the real partand the Faraday rotation probes the imaginary part of the circular dichroism.Finally, some materials presented here display Valley dichroism, in that incident light interacts more stronglywith one valley than the other. Here, and in a similar manner to the circular dichroism parameter, the Valleydichroism paramter is defined to be 𝐷 𝑣 = Re [ 𝜎 ¯ 𝐾 ] − Re [ 𝜎 ¯ 𝐾 ′ ]Re [ 𝜎 ¯ 𝐾 ] + Re [ 𝜎 ¯ 𝐾 ′ ] , (32)where 𝜎 𝐾 ( ′ ) is the magneto-optical conductivity for the 𝐾 ( ′ ) valley only.
3. Results and Discussion
Under the influence of a perpendicular magnetic field the continuous band structure of MLG splits upinto a series of discrete Landau levels with the energy and angular momentum of each level given by 𝜀 𝑛 = (cid:126) 𝜔 𝐵 sgn ( 𝑛 ) √︀ | 𝑛 | and 𝐿 𝑛 = (cid:126) | 𝑛 | , where 𝜔 𝐵 = √︁ 𝑣 𝑓 𝑒𝐵/ (cid:126) is the cyclotron frequency and 𝑛 is the Landau levelindex. Left and right circularly polarized light carries angular momentum of ± (cid:126) , respectively, and hence candrive transitions between certain Landau levels, specifically those related by | 𝑛 | → | 𝑛 | + 1 for left circularlypolarized light and | 𝑛 | → | 𝑛 | − .00.20.4-0.4-0.2 E n e r g y ( e V ) K K'M Γ θ = 2.65 deg. (a) E n e r g y ( e V ) Magnetic Field (B/B ) B = 169 T T r a n s i t i o n E n e r g y ( e V ) (a) (b) (c) K,K' Γ +0 +1 +2 Γ - K:0 K:1K:-2 K:3K:-1 K:2 K:0 0 Γ :0 Γ :1 + - Γ :1 Γ :2 + - Γ :2 Γ :3 + - Magnetic Field (B/B )
Figure 3: (Color online) The electronic and magneto-optical properties of TBG with a twist angle of 2 .
65 deg . (a) The band structure at 𝐵 = 0 T. The black and red lines correspond to the bands originating from the 𝐾 and 𝐾 ′ valleys, respectively. (b) Energy spectrum as a function of magnetic field, 𝐵 . The black and redlines correspond to the bands originating from the 𝐾 and 𝐾 ′ valleys, respectively. The blue line indicates thecharge neutral point. The labels mark various Landau levels with their index, 𝑛 and the charge pocket fromwhich they originate indicated. (c) The magneto-optical spectrum for left circularly polarized light ( 𝜎 + ). Thelabels mark various transitions with their initial and final index, 𝑛 , and the charge pocket from which theyoriginate indicated. The collection of low energy transitions highlighted by the white dashed box originate fromtransitions between the sub-bands of the 𝑛 = 0 Landau level, which is marked in the energy spectrum in (b) bythe black dashed box.displays linear bands emanating from Dirac cones located at the corners of the moir´e Brillouin zone. The bandstructure in the presence of a magnetic field is shown in Fig. 3(b). At weak magnetic fields the continuousband structure splits up into the same series of discrete Landau levels. However, owing to the reduction inBrillouin zone size one can observe both the electron-like Landau levels (or hole-like Landau levels at -veenergies) associated with the 0 eV charge pockets at ¯ 𝐾 and ¯ 𝐾 ′ and the hole-like Landau levels (or electron-likeLandau Levels at -ve energies) associated with the 0 .
25 eV charge pocket at ¯Γ. At higher magnetic fields, thebands evolve to display a more complicated fractal structure known as the Hofstadter spectrum. The transitionbetween the semi-classical Landau level structure and the Hofstadter spectrum occurs when the magnetic length, 𝑙 𝐵 , becomes smaller than the moir´e lattice constant, 𝐿 𝑀 , which occurs at about 𝐵/𝐵 = 𝑝/𝑞 = √ / 𝜋 ≈ . 𝐾 and ¯ 𝐾 ′ charge pockets, which increase in energy as the magnetic field is increased. At higher energies one sees transi-tion peaks between Landau levels of the ¯Γ charge pocket. These decrease in energy as magnetic field increases.Transitions between the ¯ 𝐾 , ¯ 𝐾 ′ and ¯Γ charge pockets are negligible because the momentum of the photon is10uch smaller than the momentum of the electrons and hence the 𝑘 of the electron is only negligibly changed inthe excitation process. At higher energies the magneto-optical conductivity becomes more complicated as moreenergy transitions become possible.At higher magnetic fields the individual Landau levels split into 2 𝑝 subbands, where 𝑝 is given by 𝐵/𝐵 = 𝑝/𝑞 and the 2 refers to the number of layers. This fractal band structure leads to the appearance of a fractalstructure for the magneto-optical conductivity and is a directly observable consequence of the appearance ofthe Hofstadter spectrum. Each subband is identified with a different, additional angular momentum componentand hence the angular momentum of the subbands are now given by 𝐿 𝑛 = (cid:126) ( | 𝑛 | + | 𝑚 | ), where 𝑚 is the indexfor the second-generation Landau levels. As a result, transitions within Landau level 𝑛 are now allowed becausethe difference in angular momentum of the subbands allows the material to absorb the angular momentumof the photon. This leads to the very low energy transitions observable in the bottom left of Fig. 3(c) [16].Reference [16] also highlighted the appearance of transitions between | 𝑛 | → | 𝑛 | + 3 𝑚 ∓ 𝐶 𝑥 and 𝐶 𝑦 symmetry. The former ensures that the Landau level energiesfrom a magnetic field 𝐵 are the same as those from a magnetic field − 𝐵 . The latter ensures that the Landaulevel energies in one valley under a magnetic field 𝐵 are the same as those from a magnetic field − 𝐵 in theother valley. Together these symmetries ensure that one has valley degenerate Landau levels. In addition, theTBG Hamiltonian obeys the symmetry Σ − 𝐻 ( 𝜉 ) 𝑇 𝐵𝐺
Σ = − 𝐻 ( − 𝜉 ) 𝑇 𝐵𝐺 , (33)where Σ = ⎛⎝ 𝜎 𝑥 − 𝜎 𝑥 ⎞⎠ , (34)and, hence, the band structure displays particle-hole symmetry (although in real materials this symmetry isbroken by higher order effects and, hence, may not be precisely satisfied). Particle-hole symmetry mean thatcircular dichroism weak in TBG because the left circularly polarized light transition between − 𝑛 → 𝑛 + 1 hasa similar transition energy to the right circularly polarized light transition between − ( 𝑛 + 1) → 𝑛 . Breaking ofexact particle-hole symmetry will lead to larger circular dichroism. TDBG has two different stackings, AB-AB and AB-BA, the latter of which is related to the former by a 180degree in-plane rotation of one of the upper layer (see Fig. 2). Despite having a similar atomic structure andsimilar band structure at 𝐵 = 0 T their energy spectra under the influence of a magentic field (and, hence, theirmagneto-optical conductivites) differ dramatically. The reason for the difference is related to the symmetries ofthe underlying lattice. AB-AB stacked TDBG obeys 𝐶 𝑥 symmetry, which ensures that the energy spectrumunder positive and negative B fields are identical. Time reversal symmetry ensures that the energy spectrumof one valley under a positive B field is the same as that of the opposing valley under a negative B field.The combination of these two symmetries means that the energy spectrum of AB-AB stacked TDBG is valley11 .30.20.10.000.020.04-0.04-0.02 E n e r g y ( e V ) K K'M Γ θ = 1.33 deg. B = 42.5 T Magnetic Field (B/B ) (a) (b) E n e r g y ( e V ) B = 42.5 T
Magnetic Field (B/B ) (c) T r a n s i t i o n E n e r g y ( e V ) Magnetic Field (B/B ) (d) T r a n s i t i o n E n e r g y ( e V ) Figure 4: (Color online) The electronic and magneto-optical properties of AB-AB TDBG (a) The band structureat 𝐵 = 0 T. The black and red lines correspond to the bands originating from the 𝐾 and 𝐾 ′ valleys, respectively.(b) Energy spectrum as a function of magnetic field, 𝐵 . The black and red lines correspond to the bandsoriginating from the 𝐾 and 𝐾 ′ valleys, respectively. The blue line indicates the charge neutral point. (c) and(d) The magneto-optical spectrum for left ( 𝜎 + ) and right ( 𝜎 − ) circularly polarized light, respectively.degenerate. In contrast, AB-BA stacked TDBG obeys 𝐶 𝑦 symmetry, which, like time reversal symmetry,ensures that the energy spectrum of one valley under a positive B field is the same as that of the opposingvalley under a negative B field. Thus, valley degeneracy is not symmetry protected and one sees a dramaticdifference between the energy spectrum of the two valleys [27]. As a result the magneto-optical spectrum ofAB-AB and AB-BA TDBG are also different.Bilayer graphene also introduces a number of new effects that are not present in monolayer graphene. Theparameter, 𝑣 , in the Hamiltonian of both forms of TDBG leads to electron-hole asymmetry. The appearance ofthis term in the Hamiltonian induces a self coupling in each Landau level resulting in a level dependent energyshift. Thus, the energy of the transitions between − 𝑛 → 𝑛 + 1 driven by left circularly polarized light as andthe transitions between − ( 𝑛 + 1) → 𝑛 driven by right circularly polarized light transition have different energiesensuring that circular dichroism is present. The parameter also allows for a new series of transitions within12 single Landau level and, hence, transitions with the selection rule | 𝑛 | → | 𝑛 | are now allowed, even outsidethe Hofstadter spectrum regime. The parameter, 𝑣 , leads to trigonal warping. This also directly affects theselection rules as this term in the Hamiltonain hybridizes the | 𝑛 | and | 𝑛 | + 3 Landau levels [28]. As a result, anextra set of transitions between | 𝑛 | → | 𝑛 | + 3 𝑚 ∓
1, are now allowed.First consider AB-AB stacked TDBG. Unlike TBG, at low magnetic fields TDBG does not display a clearset of regularly spaced Landau levels. This is because the 𝐵 = 0 T band structure does not display the simplelinear or parabolic structure one finds in other materials. This can be seen in Fig. 4(a). Figure 4(b) showsthe band structure for AB-AB TDBG in the presence of a magnetic field. The central region of Landau levelsbetween − . < 𝐸 < . 𝐾 , ¯ 𝐾 ′ and ¯Γ whilstthe Landau levels at 𝐸 < − . 𝐸 > . 𝑘 , thehigh energy transitions are exclusively caused by transitions near ¯Γ whereas the lower energy transitions canoccur at any of the high symmetry points ¯ 𝐾 , ¯ 𝐾 ′ and ¯Γ. The main feature of the low magnetic field regime isa comparative absence of optical transitions in the region between 20 meV and 40 meV. The large collectiontransitions at lower energies stem from transitions within the central region of densely packed Landau level.The higher energy transitions come exclusively from transitions between the Landau levels at 𝐸 < − . 𝐸 > . | 𝑛 | → | 𝑛 | + 1 transitions and right circularly driven | 𝑛 | → | 𝑛 | − 𝐾 and ¯ 𝐾 ′ valleys. As before, the central region of Landau levels between − . < 𝐸 < . 𝐾 , ¯ 𝐾 ′ and ¯Γ whilst the Landau levelsat 𝐸 < − . 𝐸 > . 𝐸 < − . 𝐸 > . .000.020.04-0.04-0.02 E n e r g y ( e V ) K K'M Γ θ = 1.33 deg. B = 42.5 T Magnetic Field (B/B ) (a) (b) E n e r g y ( e V ) B = 42.5 T
Magnetic Field (B/B ) (c) T r a n s i t i o n E n e r g y ( e V ) Magnetic Field (B/B ) (d) T r a n s i t i o n E n e r g y ( e V ) Figure 5: (Color online) The electronic and magneto-optical properties of AB-BA TDBG (a) The band structureat 𝐵 = 0 T. The black and red lines correspond to the bands originating from the 𝐾 and 𝐾 ′ valleys, respectively.(b) Energy spectrum as a function of magnetic field, 𝐵 . The black and red lines correspond to the bandsoriginating from the 𝐾 and 𝐾 ′ valleys, respectively. The blue line indicates the charge neutral point. (c) and(d) The magneto-optical spectrum for left ( 𝜎 + ) and right ( 𝜎 − ) circularly polarized light, respectively. MLG-hBN is a heterogeneous material, which means that, unlike TBG and TDBG, the lattice obeysneither 𝐶 𝑥 nor 𝐶 𝑦 symmetry. Furthermore, the potential from the hBN layer does not obey the symmetry inEq. (33). Hence, neither valley degeneracy nor particle-hole symmetry are enforced. The lack of both valleydegeneracy and particle-hole symmetry can be clearly seen in Figs. 6(a) and (b), which show, respectively, thezero-field band structure and the band structure under the influence of a magnetic field. In this case one wouldexpect to observe both significant circular and valley dichroism. Figures 6(c) and (d) show the magneto-opticalconductivity for left and right circularly polarized light and, as expected, they differ significantly.The magento-optical spectrum, again, displays strong peaks at energies which are resonant with transitionsbetween Landau levels. The effective theory models the interaction with the hBN layer using the potential in Eq.(11) which is derived from the interlayer interaction, Eq. (6), and the dispersionless low energy Hamiltonian for14he hBN layer, Eq. (10). Thus, the potential in Eq. (11) is not a function of the electron momentum and, hence,does not contribute to the magneto-optical conductivity directly - the potential only acts to change energiesof the Landau levels themselves. Thus, the selection rules for this material are the same as for monolayergraphene - | 𝑛 | → | 𝑛 | + 1 for left circularly polarized light and | 𝑛 | → | 𝑛 | − | 𝑛 | → | 𝑛 | + 3 𝑚 ∓ 𝐸 ≈ .
17 eV observedin Ref. [7] is not included in the calculation and in reality one would expect to observe a similar discontinuityin both here and in BLG-hBN, which is discussed below. As the energy increases the difference between theparticle and hole bands becomes more dramatic as the influence of the hBN layer increases. However, as theenergy increases the number of bands becomes large and hence the difference is obscured by the shear numberof possible transitions.At higher magnetic fields the circular dichroism becomes more discernable as the gaps between the Landaulevels broaden in energy. The distinction becomes pronounced at
𝐵/𝐵 = 𝑝/𝑞 ≈ . → − → * in Fig. 6(b) and (d)]. In the former one sees a narrower collection of transitions compared tothe broader collection for the latter, the difference stemming from the particle-hole asymmetry of the bandstructure. BLG-hBN, like MLG-hBN is a heterogeneous material which is subject to the potential from the hBN layer.This means that, again, the lattice obeys neither 𝐶 𝑥 nor 𝐶 𝑦 symmetry nor the symmetry in Eq. (33) and,hence, neither valley degeneracy nor particle-hole symmetry are enforced, as can be seen from Fig. 7(a) and (b).However, the presence of the extra graphene layer means that both the particle-hole and valley asymmetriesare far more pronounced. This is because the hBN potential only affects one of the two layers in the bilayerand hence inversion symmetry of the bilayer is more strongly broken [11]. The result is a complicated energyspectrum when a magnetic field is applied with a striking differences between the contribution from the twovalleys. At low magnetic fields the ¯ 𝐾 valley displays Landau level like band structure whereas the ¯ 𝐾 ′ valleyalready displays a fractal-like bands. The reason for this is that the wavefunction of the zero energy Landaulevel is localized on upper layer of the graphene bilayer in the ¯ 𝐾 valley and on the lower layer of the graphenebilayer in the ¯ 𝐾 ′ valley. Hence, the bands in the ¯ 𝐾 ′ valley are much more strongly affected by the presenceof the hBN layer adjacent to the lower layer and, thus, the hofstadter spectrum develops more rapidly in this15alley.The presence of the bilayer also introduces many features that were previously found in both AB-AB andAB-BA TDBG. The interlayer interaction within the bilayer introduces both trigonal warping and particle-holesymmetry breaking via the 𝑣 and 𝑣 terms of Eq. (4), respectively, the former of which leads to the | 𝑛 | → | 𝑛 | and | 𝑛 | → | 𝑛 | + 3 𝑚 ∓ 𝐵 = 0 T, BLG-hBN exhibits a large gap of around ≈
40 meV. This is causedby the proximity of the hBN layer, which induces a potential in the adjoining layer resulting in a potentialasymmetry across the two BLG layers. This, clearly, results in an absence of low energy transitions across mostof the magneto-optical spectrum until the increasing magnetic field raises the valence band energy in the positivevalley (and lowers the conduction band energy in the negative valley) such that the bands rise above (dropsbelow) the charge neutral point, at which point low energy transitions between the subbands of the Hofstadterspectrum become feasible. This occurs at 𝑝/𝑞 ≈ .
9. As the energy increases, one observes the appearanceof transitions peaks once the energy is enough to drive transitions across the gap. At low magnetic fields onesees the distinct peaks associated with transitions between individual landau levels. However, in this case thesepeaks do not display the usual ∝ √ 𝐵 dependence as the bands around the ¯ 𝐾 and ¯ 𝐾 ′ are almost flat as opposedto linear as in MLG (or MLG-hBN). At higher magnetic fields the Landau levels obtain a linear characteristicreminiscent of non-relativistic Landau levels from parabolic bands, before developing into a the more complexfractal structure of the Hofstadter spectrum.Figures 7(c) and (d) show the magneto-optical spectrum for left and right circularly polarized light, re-spectively. Both show a complicated series of transitions that are a combination of the Landau level like bandsoriginating from the ¯ 𝐾 valley and the fractal like bands originating from the ¯ 𝐾 ′ valley. Thus, unlike the caseof MLG-hBN, where the valley dichroism was small owing to the small separation of the transition peaks,BLG-hBN offers the opportunity for optically induced valley polarization since there is a significant differencebetween the spectra of the two valleys and there are large regions in both spectra where the dominant transitionsoriginate from a single valley. Note, again, that the discontinuity in the magneto-optical spectrum caused bythe strong optical phonon mode of hBN at 𝐸 ≈ .
17 eV, which was observed in Ref. [7], is not included in thecalculation but one would expect to observe it here as well as in MLG-hBN, which was discussed above.Figures 7(c) and (d) also highlight a number of the strong transitions. At low magnetic fields, left circularlypolarized light displays a number of strong, narrow transitions that stem from ¯ 𝐾 valley with the the 0 → − → 𝑣 trigonal warping term in the Hamiltonian. The transition obeys the | 𝑛 | → | 𝑛 | +3 𝑚 − 𝑚 = 1. At higher magnetic fields left circularly polarized light still displaysprominent strong transitions, though broadened by the development of the fractal hofstadter spectrum. Onealso sees the appearance of low energy 0 → 𝐾 ′ valley and a few weak transitions thatstem from the ¯ 𝐾 valley. A notable transition here is the − → * in Fig. 7(c)] thatis allowed because of the appearance of the Hofstadter subbands.16 .5. Circular Dichroism, Faraday Rotations and Ellipticity Owing to the strong breaking of inversion symmetry by the hBN layer both MLG-hBN and BLG-hBNdisplay distinct spectra when illuminated by left or right circularly polarized light. Figure 8(a) shows thecircular dichroism ( 𝐷 𝑐 ) for MLG-hBN as defined by Eq. (28). At low energies and low magnetic fields onesees a slight circular dichroism owing to the slight difference in energy of the particle-like and hole-like bands.Significant circular dichroism over a large energy region/magnetic field range is only present at higher energiesand higher magnetic fields when the differences between the particle and hole bands becomes pronounced.The dichroism characteristics of the material can be probed by measuring the Faraday rotation and elliptic-ity of transmitted plane polarized light, whose magnitude is related to the difference in the real and imaginaryparts of the right and left circularly polarized magneto-optical conductivity via Eqs. (30) and (31), respectively.Figure 8(c) show the Faraday rotation angle and Fig. 8(d) show the ellipticity as a function of magnetic field andtransition energy. Here we have used a phenomenological broadening of 𝜂 = 10 meV (rather than the value of 𝜂 = 0 .
01 meV used in to compute the magneto-optical conductivity) to match the observed disorder broadeningin experiments on similar systems [8]. This broadening washes out the details of the Hofstadter spectrum butleaves the overall shape intact. Figure 8(e) and (f) show the Faraday rotation angle ellipticity for low energiesand a range of magnetic fields. Near the resonant transitions one observes large Faraday rotations and ellipticityon the order of a degree.Finally it is worth noting that, owing to the difference in the band structure for the 𝐾 and 𝐾 ′ valleys,valley dichroism will be present in this material. Figure 8(b) shows the valley dichroism ( 𝐷 𝑣 ) for MLG-hBN asdefined in Eq. (32). Owing to minor differences in the band energies at the 𝐾 and 𝐾 ′ valleys one sees slightdichroism effects near the energies of the major transitions. However, the significant differences in the bandsfrom the two valleys occurs at energies deep in the valence band side of the spectrum and mainly originatesfrom the charge pockets at the ¯Γ-point. Thus, the transitions energies needed to observe the difference arelarge and, hence, the effect is somewhat obscured by the numerous high energy transitions between Landaulevels originating from the ¯ 𝐾 -points. At higher magnetic fields one sees larger regions where a valley polarizedresponse is possible although a large energy is still required to access these regions.Like its monolayer counterpart, BLG-hBN also displays inversion symmetry breaking and, hence, strongcircular dichroism and likewise, displays a similar Faraday rotation and ellipticity when illuminated by planepolarized light. Again, Fig. 9(a) shows the circular dichroism ( 𝐷 𝑐 ) for MLG-hBN as defined by Eq. (28). Onecan see for a large region where the interaction with right left circularly polarized light is strongly favoured.Figure 9(c) show the Faraday rotation angle and Fig. 9(d) show the ellipticity as a function of magnetic fieldand transition energy with a phenomenological broadening of 𝜂 = 10 meV [8], which, again, obscures the detailsof the Hofstadter spectrum. Figure 9(e) and (f) show the Faraday rotation angle ellipticity for low energiesand a range of magnetic fields. The magnitudes of these effects is about the same order of magnitude as inMLG-hBN - about a degree.Finally, Fig. 9(b) shows the valley dichroism as defined by Eq. (32). The valley dichroism is significantlygreater than that of MLG-hBN and is addressable over a wider range of energies and magnetic fields. Thus,BLG-hBN may offer a better platform for valleytronic applications compared to it’s monolayer counterpart.17 . Summary Here we have studied the magneto-optical properties of five van der Waals heterostructures. In the lowmagnetic field regime all structures show transitions with selection rules | 𝑛 | → | 𝑛 |± | 𝑛 | → | 𝑛 |± 𝑚 ∓ | 𝑛 | → | 𝑛 | which is allowedvia trigonal warping and electron-hole asymmetry effects, respectively. As TBG does not break inversionsymmetry, it only displays weak circular or valley dichroism. In contrast ABAB-TDBG, ABBA-TDBG, MLG-hBN and BLG-hBN all break inversion symmetry and hence display significant circular dichroism and all butABAB-TDBG also showing valley dichroism. Differences in the real and imaginary parts of the of the magneto-optical conductivity for left and right circularly polarized light lead to a Faraday rotation and ellipticity inincident plane polarized light. MLG-hBN and BLG-hBN both exhibit Faraday rotations and ellipticities ofabout a degree.
5. Acknowledgements
J.A.C was supported by the National Science Foundation of China Research Grant No. 12050410228.P.M. acknowledges the support by National Science Foundation of China (Grant No. XXX) and Science andTechnology Commission of Shanghai Municipality (Shanghai Natural Science Grants, Grant No. 19ZR1436400).J.A.C. and P.M. were supported by the NYU-ECNU Institute of Physics at NYU Shanghai. This research wascarried out on the High Performance Computing resources at NYU Shanghai.
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K, (K)(K')
K:0 K:1K:-2 K:3K:-1 K:2 K:0 0K:0 0 K' K:-1 K:0K:-3 K:2K:-2 K:1 * + * + B = 24.4 T T r a n s i t i o n E n e r g y ( e V ) (c) Magnetic Field (B/B )0.10.4 Γ - Γ - Figure 6: (Color online) The electronic and magneto-optical properties of MLG-hBN (a) The band structure at 𝐵 = 0 T. The black and red lines correspond to the bands originating from the 𝐾 and 𝐾 ′ valleys, respectively.(b) Energy spectrum as a function of magnetic field, 𝐵 . The black and red lines correspond to the bandsoriginating from the 𝐾 and 𝐾 ′ valleys, respectively. The blue line indicates the charge neutral point. The labelsmark various Landau levels with their index, 𝑛 , and the charge pocket from which they originate indicated.The transitions marked + and * are the transitions that most significantly contribute to optical dichroism. (c)and (d) The magneto-optical spectrum for left ( 𝜎 + ) and right ( 𝜎 − ) circularly polarized light, respectively. Thelabels mark various transitions with their initial and final index, 𝑛 and the charge pocket from which theyoriginate indicated. The collection of low energy transitions highlighted by the white dashed box originate fromtransitions between the sub-bands of the 𝑛 = 0 Landau level, which is marked in the energy spectrum in by theblack dashed box. The energy regions marked * and + are the transitions that most significantly contribute tooptical dichroism and correspond to the transitions marked in (b).21 Γ K'K θ = 0 deg.0.000.10-0.10 E n e r g y ( e V ) -0.050.05 B = 24.4 T Magnetic Field (B/B ) (a) (b)
B = 24.4 T (c) Magnetic Field (B/B ) B = 24.4 T (d) Magnetic Field (B/B )0.100.150.200.000.05 T r a n s i t i o n E n e r g y ( e V ) E n e r g y ( e V ) -0.050.05 T r a n s i t i o n E n e r g y ( e V ) K K' K
K:-2 K:1K:-1 K:1 K':0 K':0K:-1 K:2K':0 K':0K:0 K:1 +2 K:-2 K:3K:-1 K:3 * + Figure 7: (Color online) The electronic and magneto-optical properties of BLG-hBN (a) The band structure at 𝐵 = 0 T. The black and red lines correspond to the bands originating from the 𝐾 and 𝐾 ′ valleys, respectively.(b) Energy spectrum as a function of magnetic field, 𝐵 . The black and red lines correspond to the bandsoriginating from the 𝐾 and 𝐾 ′ valleys, respectively. The blue line indicates the charge neutral point. The labelsmark various Landau levels with their index, 𝑛 , and the charge pocket from which they originate indicated.(c) and (d) The magneto-optical spectrum for left ( 𝜎 + ) and right ( 𝜎 − ) circularly polarized light, respectively.The labels mark various transitions with their initial and final index, 𝑛 , and the charge pocket from which theyoriginate indicated. The transitions marked + and * in (c) and (d) show a transition owing to trigonal warpingeffects and the appearance of subbands respectively. 22 = 24.4 T T r a n s i t i o n E n e r g y ( e V ) (d) Magnetic Field (B/B )0.10.4B = 24.4 T T r a n s i t i o n E n e r g y ( e V ) (c) Magnetic Field (B/B )0.10.4 Transition Energy (eV) E ll i p t i c i t y ( d e g . ) F a r a d a y A n g l e ( d e g . ) (e) (f) -2.0 0.0 2.0 Angle (deg.) -2.0 0.0 2.0
Angle (deg.)
B = 24.4 T T r a n s i t i o n E n e r g y ( e V ) (b) Magnetic Field (B/B )0.10.4B = 24.4 T T r a n s i t i o n E n e r g y ( e V ) (a) Magnetic Field (B/B )0.10.4 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 -0.5
Figure 8: (Color online) (a) Circular and (b) valley dichroism for MLG-hBN. (c) Faraday rotation angle and(d) Ellipticity. Low energy (e) Faraday rotation and (f) ellipticity for a variety of magnetic field values. Notethat for (c)-(f) the phenomenological broadening parameter is taken to be 10 meV.23 = 24.4 T (d) Magnetic Field (B/B )B = 24.4 T (c) Magnetic Field (B/B ) Transition Energy (eV) E ll i p t i c i t y ( d e g . ) F a r a d a y A n g l e ( d e g . ) (e) (f) -2.0 0.0 2.0 Angle (deg.) -2.0 0.0 2.0
Angle (deg.)B = 24.4 T (b) Magnetic Field (B/B )B = 24.4 T (a) Magnetic Field (B/B )0.100.150.200.000.05 T r a n s i t i o n E n e r g y ( e V ) T r a n s i t i o n E n e r g y ( e V ) T r a n s i t i o n E n e r g y ( e V ) T r a n s i t i o n E n e r g y ( e V ) -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 -0.51.0 0.02.0 1.00.5