Electronic Raman Scattering in Twistronic Few-Layer Graphene

We study electronic contribution to the Raman scattering of two- to four-layer graphene with the top layer twisted by a small angle, $\theta<2^{\circ}$, relatively to the 2D crystal underneath. We find that the Raman spectrum features two peaks produced by van Hove singularities in moir\'{e} minibands characteristic for twistronic graphene, one related to direct hybridization of Dirac states, and the other resulting from band folding caused by moir\'{e} superlattice. The positions of both peaks strongly depend on the twist angle, so that their detection can be used for non-invasive measurements of twist angle, even in the hBN-encapsulated structures.

Here, we show that this can be achieved using Raman spectroscopy of electronic excitations. Raman scattering with phonons has stood out as a powerful, non-invasive method to inspect carbon materials [22], providing information about defects, doping, strain and the number of layers in the film. Twist of graphene layers was shown to lead to resonant enhancement of the G peak [23,24], the width and position of the 2D peak [25], as well as of processes involving phonons folded onto the mBZ [26][27][28] and layer breathing and shear modes [29,30], however, with a limited accuracy in determining the twist angle. Below, we demonstrate that Raman spectroscopy of the interband electronic excitations (ERS) [31][32][33][34][35][36][37][38][39][40][41][42] can be used to detect twist-angle-dependent features in the electronic spectrum of material. Below, we study the electronic minibands and electronic contributions to the Raman spectra for (1+N )-layer graphene stacks in which the top layer is twisted by a small angle 0.8 • < θ < 2 • with respect to N = 1, 2, 3 layers underneath, as shown in Fig. 1(a). We find that ERS is formed by transitions from the n-th valence to the n-th conduction moiré superlattice (mSL) miniband (selection rules complementary to those for optical absorption [43,44]) and features two spectral peaks. One, at a lower Raman shift, is caused by the resonant hybridization of electronic states of the twisted monolayer and the underlying N -layer stack [45,46]. Another, higher-energy peak is due to the anti-crossing of bands, backfolded by mSL. Both peaks are related to van Hove singularities in the mSL minibands. We trace the peaks positions as a function of the twist angle and estimate their quantum efficiency, I ∼ 10 −11 , to be in the measurable range [31][32][33][34][35][36][37].
To model twistronic graphene, we use a hybrid k · p theory-tight-binding model, where we describe electrons' states in each flake using the k · p expansion around ±K and ±K Brillouin zone corners [see Fig. 1(b)] and the interlayer hybridization using tunneling Hamiltonian [46,47]. The HamiltonianĤ 1+N X for an electron in a (1 + N X) structure (X = B for Bernal or X = R for rhombohedral stacking of the N ≡ 3 crystal) and with momentum p = (p x , p y ) measured from the centre of the valley ±K in the N -layer bottom flake iŝ Here, K = 4π 3a (1, 0), a is the graphene lattice constant, K =R θ K [R θ clockwise rotates by angle θ, see Fig. 1(b)],0 n×m is the n × m matrix of zeros, andĤ 1,θ andĤ N X account for rotated monolayer graphene and a perfectly stacked N -layer crystal, respectively: with σ x , σ y , σ z the Pauli matrices; ∆K = K −K is a mismatch between Brillouin zone corners (valleys) in the top and bottom crystals and v is the monolayer Dirac velocity. The interlayer coupling block inĤ 2 in Eq. (2) iŝ whereπ = p x + ip y , γ 1 = 0.39 eV is the 'direct' interlayer coupling between the dimer sites, see Fig. 1(a), and v 3 and v 4 are velocities related to interlayer skew couplings γ 3 ≈ 0.3 eV and γ 4 ≈ 0.04 eV [48]. In trilayer (and N > 3) crystals,T B =T † butT R =T . Finally, the block T (θ) in Eq. (1) captures the interlayer coupling between the N -layer crystal and the twisted top monolayer and we write it following previous works [46,47], T n+1 =Î 2 + cos 2πn 3 σ x + ξ sin 2πn 3 σ y .
In the electronic Raman scattering (ERS), photon of energy Ω arrives at the sample (here, we assume normal incidence of light) and scatters to a photon with energy Ω = Ω − ω, leaving behind an electron-hole pair with energy ω. In contrast to classical plasmas, where the amplitude of such process is controlled by contact interaction (AA ∂ 2Ĥ ∂pi∂pj , A and A are the vector potentials of the incoming and outgoing light, respectively), in graphene, the dominating contribution comes from a twostep process, described by the Feynman diagrams shown in Fig. 1(c). It corresponds to absorption (emission) of a photon with energy Ω (Ω ) transferring an electron with momentum p from an occupied state in the valence band into a virtual intermediate state (energy is not conserved at this stage), followed by emission (absorption) of a second photon with energy Ω (Ω) [31,32], with an amplitude [37] Here, the main contribution to the Raman signal comes from n − → n + minisubband transitions, where n s denotes the n-th miniband on the valence (s = −1) or conduction (s = 1) side. This ERS signal can be filtered out from the phonon G-line by selecting the crosspolarized component of the Raman signal. The overall lineshape g(ω) of inelastic photon scattering with Raman shift, ω = Ω − Ω , is described using Fermi's golden rule, where f p,n s is the occupation factor of a state |p, n s with momentum p and energy p,n s in the band n s . In Fig. 2 (r.h.s.), we present the ERS intensity map for a (1 + 1) twisted bilayer graphene in a twist angle range 0.8 • < θ < 2 • , neglecting its atomic reconstruction [49]. Two bright features stand out in this plot, corresponding to transitions between the minibands 1 − → 1 + and 2 − → 2 + . For two selected cuts at θ = 1.8 • and θ = 1.1 • , the peaks come from transitions indicated by arrows in the corresponding minibands plots. The first peak, 1 − → 1 + , is due to transitions from or to flat regions of electronic dispersion resulting from hybridization of the Dirac cones of the two layers (the Dirac points can still be identified in the dispersion for θ = 1.8 • as touching points of the minibands shown in red), which are responsible for a van Hove singularity (vHs) in the density of states [45,46]. This vHs can be regarded as a direct evidence of interlayer hybridization of the Dirac band in graphene monolayers. As the twist angle is decreased, this ERS peak moves to lower energies. The second peak, 2 − → 2 + , is due to the transitions between the flat regions of the second valence and conduction minibands, indicated by orange arrows in the miniband structure for θ = 1.1 • , with the overall intensity prodived by the mBZ section painted in yellow in the left-most inset below the ERS map. The other two panels show the real-space distribution of the saddle point states across the moiré supercell in the top and bottom monolayers. For the 2 − → 2 + peak, both the initial and final states are close to vHs, while for the 1 − → 1 + peak this is the case only for either the initial or final state. Hence, for the same twist angle, the 2 − → 2 + feature is larger than 1 − → 1 + .
In Fig. 3, we show the ERS intensity map for (1 + 2) twistronic graphene and an exemplary miniband structure for θ = 1.1 • . Similarly to (1+1) graphene, the dominant contributions come from the 1 − → 1 + and 2 − → 2 + electronic transitions. The two peaks also have the same origins: the first one is due to direct hybridization of the monolayer and bilayer states between the valleys K and K while the second is formed by states backfolded by moiré superlattice. However, for a given twist angle, the peaks appear at lower Raman shifts than in the (1 + 1) structure. This is because the unperturbed dispersion in bilayer is flatter than in a monolayer -as a consequence, the anti-crossings and hence the vHs form at lower energies as compared to the (1 + 1) stack. To the right of the miniband spectrum, we show (in yellow) the regions of the mBZ which contribute to the 2 − → 2 + ERS peak.
In Fig. 4, we plot ERS maps of two monolayer-ontrilayer structures, (1 + 3B) and (1 + 3R), together with exemplary miniband dispersions for θ = 1.1 • . Bernalstacked trilayer graphene, Fig. 4(a), hosts both a bilayerand monolayer-like low-energy bands [50] which hybridize with the top monolayer states and produce multiple overlapping contributions with a structrue less pronounced than in both the (1 + 1) and (1 + 2) systems. In contrast, rhombohedral trilayers only host one low-energy band, with a cubic dispersion in the vicinity of the valley centre [51]. This low-energy band is localised on the top and bottom surfaces of the crystal and is significantly affected by mSL produced by the top layer twist, leading to a pair of clear spectral features, Fig. 4(b), as in (1 + 1) and (1 + 2) twistronic graphene.
Overall, our studies of electronic Raman scattering in (1 + N ) twistronic graphene (for N = 1, 2, 3) identify two clear features in the Raman spectrum, corresponding to electron-hole excitations involving van Hove singularities of moiré superlattice minibands, with the Raman shift dependent on the twist angle between the layers. Currently, accurate determination of the twist angle, especially in the small-angle regime, requires timeconsuming microscopic investigations of the moiré pattern or magneto-transport measurements to determine the size of the mBZ. We suggest that, based on our results, calibration of the positions of the Raman features and the twist angle in the structures with known θ would allow to identify the orientations of the component crystals in other samples, enabling a non-invasive method to determine the twist angle, even in structures encapsulated with other materials, where the graphene/graphene moiré pattern is not directly accessible for tunnelling spectroscopy studies [1,2,5,6,[9][10][11][12][13][14].