General Theoretical Description of Angle-Resolved Photoemission Spectroscopy of Van Der Waals Structures

General theoretical description of angle-resolved photoemission spectroscopy of van der Waals structures

B. Amorim CeFEMA, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal∗

We develop a general theory to model the angle-resolved photoemission spectroscopy (ARPES) of commensurate and incommensurate van der Waals (vdW) structures, formed by lattice mismatched and/or misaligned stacked layers of two-dimensional materials. The present theory is based on a tight-binding description of the structure and the concept of generalized umklapp processes, going beyond previous descriptions of ARPES in incommensurate vdW structures, which are based on continuous, low-energy models, being limited to structures with small lattice mismatch/misalignment. As applications of the general formalism, we study the ARPES bands and constant energy maps for two structures: twisted bilayer graphene and twisted bilayer MoS2. The present theory should be useful in correctly interpreting experimental results of ARPES of vdW structures and other systems displaying competition between diﬀerent periodicities, such as two-dimensional materials weakly coupled to a substrate and materials with density wave phases.

I. INTRODUCTION defined, this picture might breakdown, as the ARPES re- sponse is weighted by matrix elements which describe the The development of fabrication techniques in recent light induced electronic transition from a crystal bound years enabled the creation of structures formed by state to a photoemitted electron state. For bands that stacked layers of different two-dimensional (2D) mate- are well decoupled from the remaining band structure, rials, referred to as van der Waals (vdW) structures [1– the ARPES matrix elements are featureless, and indeed 3]. By combining layers of materials displaying different ARPES can be seen as a direct probe of the band struc- properties, it is possible to engineer devices with new ture. However, exceptions to this can occur and the ma- functionalities, not displayed by the individual layers. trix elements can impose selection rules on the transi- This makes vdW structures very appealing from the ap- tions. Two well known examples where this occurs are plications point of view. As examples, transistors based graphite [15] and graphene [16–22]. In these materials on graphene and hexagonal boron nitride (h-BN) or a part of the Fermi surface is not observed in constant en- semiconducting transition metal dichalcogenide (STMD) ergy ARPES map [23]. This effect is due to the ARPES [4,5] and photodetectors based on graphene and a STMD matrix elements, which suppress the signal from some [6–8] have already been realized. The properties of a parts of the band structure. vdW structure depend not only on the properties of the individual layers, but also on how different layers inter- Another case where the ARPES matrix elements act with each other. Due to the high crystallographic should play an important role is in systems with com- quality of 2D materials, the interlayer interaction de- peting periodicities, such as materials displaying charge pends crucially on the lattice mismatch and misalign- density wave (CDW) phases [24, 25]. In this case it is ment between different layers. This is clearly exempli- easy to understand how the interpretation of ARPES as fied, both experimentally [9] and theoretically [10, 11], a direct probe of the band structure can break down. Let by the observation of negative differential conductance in us suppose that for a given material in the normal, undis- graphene/h-BN/graphene vertical tunneling transistors, torted phase, ARPES accurately maps the electronic where the bias voltage at which peak current occurs is band structure. Suppose that the system undergoes a controlled by the angle between the misaligned graphene transition into a commensurate CDW, with a larger unit electrodes. A necessary step to fully understand and take cell. If the distortion is small, the bands in Brillouin zone advantage of vdW structures is to characterize their elec- will be weakly perturbed apart from back-folding into the tronic properties and how these depend on the lattice new, smaller, Brillouin zone associated with the enlarged mismatch/misalignment. unit cell. If the CDW perturbation is weak, by an adia- Angle-resolved photoemission spectroscopy (ARPES) batic argument, the ARPES mapped bands observed in arXiv:1711.02499v2 [cond-mat.mtrl-sci] 19 Apr 2018 is an extensively used tool to characterize the electronic both phases must be essentially unchanged. This means degrees of freedom of materials [12–14]. In crystals, that the signal of the back-folded bands must be very ARPES is generally understood as a direct probe of the weak, and the observed ARPES bands will mostly fol- electronic band structure over the Brillouin zone of occu- low the bands of the undistorted phase, in an extended pied states. Nevertheless, even in a perfect crystal where zone scheme. The suppression of the back-folded bands the notions of reciprocal space and Brillouin zone are well is encoded by the ARPES matrix elements. This was ex- emplified in Ref. [24] in a simple one-dimensional tight- binding model. These effects may also be relevant when interpreting ARPES experiments in cuprates, for which ∗ [email protected]; [email protected] a hidden density wave state has been proposed [26]. 2

In vdW structures, such competing periodicities natu- with Hbt = Htb† . In the previous equations, the indices rally occur due to the lattice mismatch between different i, j run over lattice sites and the indices α, β run over layers. Therefore, a general theory capable of correctly sublattice, orbital and spin degrees of freedom. The op- taking into account ARPES matrix elements is essen- erator c`,† R ,α creates an electron in state `, R`,i, α , a tial to interpret ARPES data from vdW structures. We `,i | i localized Wannier state in layer `, lattice site R`,i and point out that the modeling of ARPES in twisted bi- sublattice site τ`,α. The Wannier wavefunction in real layer graphene [27] and graphene/ -BN [28] structures h space reads r `, R`,i, α = w`,α (r R`,i τ`,α) where has been considered previously in the literature. How- h | i − − w`,α (r) is the Wannier wavefunction or type α centered ever the models employed relied on effective low energy, t,t b,b on the origin. hαβ (Rt,i, Rt,j) and hαβ (Rb,i, Rb,j) are continuous descriptions of the systems, which are only intralayer hopping terms, which we assume to be invari- valid for small misalignment angles. As the field of vdW ant under translations by lattice vector of the respective structures develops, a more general and flexible approach layer. ht,b (R , R ) and hb,t (R , R ) are interlayer is required. The goal of this work is to develop a general αβ t,i b,j αβ b,i t,j hopping terms, which describe the coupling between the framework to theoretically model ARPES, which is valid two layers. It is convenient to express the electronic op- for both commensurate and incommensurate structures erators in terms of Fourier components formed by arbitrary materials and with arbitrary lattice mismatch/misalignment. 1 X ik` (R`,i+τ`,α) c† = e− · c† , (4) The structure of this paper is as follows. In SectionII, `,R`,i,α `,k`,α √N` we review the description of vdW structures based on k` tight-binding models and generalized umklapp processes developed in Refs. [29, 30]. We use this description of where k` belongs to the Brillouin zone of layer ` and N` vdW structures to compute the ARPES matrix elements is the number of unit cells in layer `. Notice that if G` iG` R`,i for an arbitrary structure in SectionIII. We apply the is a reciprocal lattice vector of layer `, i.e., e · = 1, iG τ then c† = e ` `,α c† . These states bring the general framework to model ARPES in twisted bilayer `,k+G`,α · `,k,α Hamiltonians of the isolated layers to a block diagonal graphene and twisted bilayer MoS2 in SectionIV. Con- clusions are drawn in SectionV. For completeness and form, the convenience of the reader, in AppendixA, we briefly X `,` H = c† h (k ) c , (5) review the derivation of the ARPES intensity within the ` `,k`,α αβ ` `,k`,β non-equilibrium Green’s function approach [31, 32]. In k`,α,β AppendixB, we present some details on the evaluation of `,` P ik` (R`,i+τ`,α τ`,β ) `,` the Fourier transform components of the interlayer hop- where hαβ (k`) = i e− · − hαβ (R`,i, 0). ping in twisted bilayer MoS2. For the interlayer term, we assume a two-center approximation for the hopping elements and write them in terms of their Fourier transform components [30] as (focusing II. TIGHT-BINDING DESCRIPTION OF VDW on the Htb term) STRUCTURES t,b p h (Rt,i, Rb,j) = Ac,tAc,b A first step in modeling ARPES of vdW structures αβ × Z 2q d iq (Rt,i+τt,α Rb,j τb,β ) t,b is to describe the electronic states bound to the struc- e · − − h (q) , (6) ture. Following Refs. [29, 30], we employ a tight-binding × (2π)2 αβ model to describe the bound states. We will focus on bilayer structures, where each layer has a periodic structure where Ac,` is the area of the unit cell of layer `. With this we can write as with Bravais lattice sites given by R`,i , with ` = t, b Htb labeling the top and bottom layers,{ respectively.} The X X iτt,α Gt,i t,b iτb,β Gb,j single-particle Hamiltonian of the structure is written as H = e · h (k + G ) e− · tb αβ t t,i × kt,kb,α,β i,j H = Ht + Hb + Htb + Hbt, (1)

ct,† k +G ,αcb,kb+Gb,j ,βδkt+Gt,i,kb+Gb,j , (7) where Ht and Hb are the tight-binding Hamiltonians of × t t,i the isolated top and bottom layers and H (H ) de- tb bt where G are reciprocal lattice vectors of layer . scribes the hopping of electrons from the bottom (top) `,i ` Hbt is written in a similar way. to the top (bottom) layer. More concretely, the intralayer Equation (7) tells us that states of the two layers with terms are written as crystal momentum kt and kb are only coupled provided X `,` H = h (R , R ) c† c (2) G and G exist, such that the generalized umklapp ` αβ `,i `,j `,R`,i,α t,R`,j ,β t,i b,j i,j,α,β condition kt + Gt,i = kb + Gb,j is satisﬁed [30]. In an extended Brillouin zone scheme, the generalized umklapp for ` = t, b, and the interlayer terms as condition can be satisﬁed if for each Gt,i and Gb,j we X t,b R R † (3) write kt = k + Gb,j and kb = k + Gt,i for any k. This Htb = hαβ ( t,i, b,j) ct,Rt,i,αcb,Rb,j ,β, i,j,α,β fact motivates us to look for eigenstates of the complete 3

Hamiltonian Eq. (1) of the form H ( G , G ) is finite. In this case, the eigenvalues k { b} { t} and eigenstates of Hk ( Gb , Gt ) for k restricted to vdW X n { } { } ψ = φ (Gb,i) c† 0 the Brillouin zone of the commensurate structure pro- k,n t,k,α t,k+Gb,i,α | i i,α vide us the full spectrum and eigenstates of the bilayer X n structure. For an incommensurate structure, the sums + φ (Gt,i) c† 0 , (8) b,k,α b,k+Gt,i,α | i over reciprocal lattice vectors in Eq. (8) involve an infi- i,α nite number of terms and the corresponding Hamiltonian matrix H ( G , G ) is infinite. However, an approx- with φn (G ) and φn (G ) coefficients that are k b t t,k,α b,i b,k,α t,i imation to the{ eigenstates} { } and energies of the incommen- to be determined. We now introduce a convenient n surate structure can still be obtained by suitably trun- compact notation. We define φ`,k (G) as a vector h i cating Hk ( Gb , Gt ), considering a finite number of n n { } { } with entries given by φ`,k (G) = φ`,k,α (G). In reciprocal lattice vectors Gt,i and Gb,i. Even in the case α `,` of a commensurate structure, the supercell might be very the same way we introduce the matrices hk with h i large, leading to a very large matrix Hk ( Gb , Gt ), `,` `,` t,b { } { } entries hk = hαβ (k), hk,G ,G with entries and in that situation it might still be beneficial to com- αβ b t h i pute the eigenstates and energies of the system approx- t,b iGt τt,α t,b iGb τb,β h = e · h (k + Gb + Gt) e− · k,Gb,Gt αβ imately by truncating H ( G , G ). In the follow- αβ k { b} { t} and hb,t defined in a similar way. This allows us ing, we will generally refer to the approximate eigenval- k,Gt,Gb to write the eigenvalue problem which determines the ues Ek,n as band structure, even in the case where we eigenstates and energies of the vdW structure as have an incommensurate structure and the concept of Brillouin zone no longer applies. n n We will now prove a formal relation satisfied by the φt,k ( Gb ) φt,k ( Gb ) Hk ( Gb , Gt ) n { } = Ek,n n { } , solutions of Eq. (9) that will be useful in the next section. { } { } · φ ( Gt ) φ ( Gt ) b,k { } b,k { } (9) Assume that where Ek,n are the energies, φn ( G ) t,k { b} , (15) φn ( G ) n n n t b,k t φ ( G ) = φ (Gb,1) φ (Gb,2) , (10) { } t,k { b} t,k t,k ··· n n n t is an eigenstate of Hk ( Gb , Gt ) with eigenvalue φb,k ( Gt ) = φb,k (Gt,1) φb,k (Gt,2) , (11) { } { } { } ··· Ek,n. Then are vectors formed by the coefficients n G and φt,k,α ( b,i) iGt,j τt n n e− · φt,k ( Gb ) φ (Gt,i) for different Gb,i and Gt,i, and the Hamil- { } , (16) b,k,α φn ( G + G ) tonian matrix is written as b,k { t,j t} " t,t t,b # is an eigenstate of H G G with the same H H k+Gt,j ( b , t ) k+ Gb k, Gb , Gt { } { } H ( G , G ) = { } { } { } , eigenvalue. This allows us to identify k { b} { t} Hb,t Hb,b k, Gt , Gb k+ Gt { } { } { } n G iGt,j τt,α n G (17) (12) φt,k+Gt,j ,α ( b) = e− · φt,k,α ( b) , t,t where H is a block diagonal matrix n n k+ Gb φ (Gt) = φ (Gt,j + Gt) . (18) { } b,k+Gt,j ,α b,k,α   ht,t 0 This statement can be proved by looking at the structure k+Gb,1 t,t ··· of H ( G , G ). First, we notice that t,t  0 h  k+Gt,j b t H = k+Gb,2 , (13) { } { } k+ Gb   { }  . .  h i h i . .. t,t iGt,j τt,α t,t iGt,j τt,β . hk+G +G = e− · hk+G e · , t,j b,i αβ b,i αβ (19) with Hb,b similarly defined, and Ht,b is a k+ Gt k, Gb , Gt h i h i { } { } { } t,b iGt,j τt,α t,b dense matrix hk+G ,G ,G = e− · hk,G ,G +G . t,j b t αβ b t t,j αβ (20)   ht,b ht,b k,Gb,1,Gt,1 k,Gb,1,Gt,2 ··· With these relations, we can write t,b  ht,b ht,b  H = k,G ,Gt,1 k,G ,Gt,2 , (14) k, Gb , Gt  b,2 b,2  { } { }  ···  iGt,j τt . . .. e− · 0 . . . H + ( G , G ) = k Gt,j { b} { t} 0 1

iGt,j τt b,t t,b † e · 0 with H = H . For a commen- Hk ( Gb , Gt + Gt,j ) . (21) k, Gt , Gb k, Gb , Gt 0 1 { } { } { } { } · { } { } · surate structure, there exist Gb,i and Gt,j such that Gb,i = Gt,j and the sums over reciprocal lattice vectors For a commensurate structure, the set of vectors Gt in Eq. (8) become ﬁnite, and consequently the matrix is ﬁnite and periodic modulo reciprocal lattice vectors{ of} 4 the commensurate lattice. Therefore G + G coin- tation matrix P such that P G = G + G . Since { t t,j} { t} { t t,j} cides with Gt apart from a reordering of the vectors. both the Hamiltonian Hk ( Gb , Gt + Gt,j ) and the Assume that{ this} reordering is implemented by a permu- n n { } { t } vector φt,k ( Gb ) φb,k ( Gt,j + Gt ) are reordered in the same way,{ we} can write{ }

iGt,j τt n iGt,j τt n e− · φt,k ( Gb ) e− · 0 φt,k ( Gb ) Hk+Gt,j ( Gb , Gt ) n { } = Hk ( Gb , Gt + Gt,j ) n { } { } { } · φ ( Gt,j + Gt ) 0 1 · { } { } · φ ( Gt,j + Gt ) b,k { } b,k { } iGt,j τt n e− · 0 1 1 φt,k ( Gb ) = P P − Hk ( Gb , Gt + Gt,j ) P P − n { } 0 1 · · · { } { } · · · φ ( Gt,j + Gt ) b,k { } iGt,j τt n e− · 0 φt,k ( Gb ) = P Hk ( Gb , Gt ) n { } 0 1 · · { } { } · φ ( Gt ) b,k { } iGt,j τt n iGt,j τt n e− · 0 φt,k ( Gb ) e− · φt,k ( Gb ) = Ek,n P n { } = Ek,n n { } , (22) 0 1 · · φ ( Gt ) φ ( Gt,j + Gt ) b,k { } b,k { }

proving our statement for the commensurate case. For III. ARPES IN VDW STRUCTURES an incommensurate structure, the set G is inﬁnite { t} and therefore, apart from a reordering, we have that We wish to model an experimental situation where the Gt,j + Gt and Gt coincide[33] and we also obtain incident electromagnetic ﬁeld is monochromatic with fre- Eq.{ (22). This} proves{ } our formal statement for both the quency ω0 > 0, and the electron detector is placed at commensurate and the incommensurate cases. Following position r, far away from the crystal sample, collecting the same argumentation it can also be formally shown electrons emitted with energy E along the direction ˆr. In that this situation, the energy resolved ARPES intensity can be evaluated from [31, 32, 41–45] (see also AppendixA n G n G G (23) φt,k+Gb,j ,α ( b) = φt,k,α ( b + b,j) , for a brief derivation)

n iGb,j τt,α n φ (G ) = e− · φ (G ) . (24) b,k+Gb,j ,α t b,k,α t X IARPES(E, ˆr) f (E ω0 µ) ∝ − − × In this section, the coupling between the two layers a was assumed to only give origin to interlayer hopping 2 1 ME,ˆr;a(ω0) a (E ω0) . (26) terms, not affecting the intralayer Hamiltonians Ht and × | | 2π A − Hb, which were assumed to preserve the translational where µ is the chemical potential, the index a runs over symmetry of the isolated layers. Besides this effect, there crystal bound states, with corresponding wavefunction is also the possibility of one of the layers inducing a po- ψa(r); a (E) is the spectral function, which in the tential to which the electrons in the other layer will be non-interactingA limit reduces to (E) = 2πδ (E ), subjected to [34–36]. The coupling between the layers Aa − α where a is the energy of state a; and ME,ˆr;a(ω0) are the can also lead to structural relaxation, which gives origin ARPES matrix elements. These are given by to a modulation of the intralayer hoppings due to the displacement of the atomic positions [37–40]. Although Z e 3 we will not explore those effects in the present work, we ME,ˆr;a(ω0) = i d r1 ψE,∗ ˆr(r1)( ψa(r1)) − ~ ∇ note that these corrections will have a spatial modulation ψ∗ (r1) ψa(r1) Aω (r1), (27) given by Gt,i Gb,j and can therefore be incorporated − ∇ E,ˆr · 0 in the present− formalism by including off-diagonal blocks where A (r ) is the screened [32] vector potential and in the matrices Ht,t and Hb,b of the form ω0 1 k+ Gb k+ Gb r is the complex conjugate of a photoemitted state { } { } ψE,∗ ˆr( 1) with energy E, which is the solution of the Lippmann- h t,t(b,b) i Schwinger equation Vk+G ,k+G = b(t),i b(t),j αβ Z t(b) ipE r1 3 R = t(b), k + Gb(t),i, α V t(b), k + Gb(t),j, β , (25) ψE,∗ ˆr(r1) = e− · + d r0ψE,∗ ˆr(r0)V (r0)Gfree(E; r0, r1) (28) where t(b) describes the potential or intralayer hopping p + 2 V where pE = pEˆr, with pE = 2m (E + i0 ) /~ , modulation on the top (bottom) layer and `, k, α is the GR (E; r, r ) is the free space electronic Green’s func- | i free 0 state created by the operator c`,† k,α in Eq. (4). tion (which is explicitly given by Eq. (A19)), and V (r) is 5 the crystal potential. Using integration by parts, we can together with the in-plane momentum-conserving Kro- write the ARPES matrix element as necker symbols, we can write 2e Z 3 n G iGt,j τt,α ME,ˆr;a(ω0) = i d r1 ψE,∗ ˆr(r1) ψa(r1) Aω0 (r1), φt,k,α ( b,i) e · = ~ ∇ · n G iGt,j τt,α n 0 (29) = φt,Q⊥+Gt,j Gb,i,α ( b,i) e · = φt,Q⊥,α ( ) , where we have neglected the contribution arising from − (33)

Aω0 (r) = ρ/ (iω0), where ρ is the total charge in the∇ · crystal, which is a common approximation [32, 45, n iGb,j τb,α 46]. In the same spirit, we neglect effects of screening in φb,k,α (Gt,i) e · = A (r ) and assume it to be described by a plane wave n iG τ n ω0 1 G b,j b,α 0 λ λ iq r1 λ = φb,Q⊥+Gb,j Gt,i,α ( t,i) e · = φb,Q⊥,α ( ) , A (r1) = A e e , where A is the amplitude, q is − ω0 ω0 q · ω0 (34) the wavevector, satisfying ω0 = c q with c the speed of λ | | light, and eq is the polarization vector for the λ = s, p and the ARPES matrix elements can be rewritten as polarizations. For simplicity we will also approximate ipE r1 ψ∗ (r1) by a plane wave ψ∗ (r1) e− · [15], which p 2e E,ˆr E,ˆr λ p eλ ' ME,ˆr;k,n(ω0) = Nt Aω0 E q Q,n will greatly simplify the evaluation of ME,ˆr;a(ω0), while ~ · M × providing a non-trivial description of the ARPES matrix X δk Q⊥,Gb,i+Gt,j , (35) × − elements.[47] With these approximations we obtain i,j 2e Z λ p eλ 3r i(pE q) r1 r where we have defined ME,ˆr;a(ω0) Aω0 E q d 1e− − · ψa( 1) ' ~ · z X n iQz τ (30) = φ (0) e− t,α w˜ (Q) MQ,n t,Q⊥,α t,α and the ARPES matrix element becomes proportional to α the Fourier transform of the crystal bound state. s Ac,t X z n 0 iQz τb,α Q (36) In order to obtain the ARPES intensity for a bilayer + φb,Q⊥,α ( ) e− w˜b,α ( ) , Ac vdW structure we need to evaluate Eq. (30) with the ,b α crystal bound state given by Eq. (8). In real space we and assumed that the total area of both layers is the have that same, that is NtAc,t = NbAc,b [30]. It is still necessary to evaluate w˜`,α (Q). Assuming that the Wannier functions are well localized they can be written in a separable form vdW 1 X n i(k+Gb,i) (Rt,j +τt,α) ψ (r) = φ (Gb,i) e · k,n √N t,k,α as [15] t i,j,α mα w (r R τ ) w`,α (r) = R`,α ( r ) l (ˆr) , (37) × t,α − t,j − t,α | | Y α 1 X n i(k+Gt,i) (Rb,j +τb,α) where r is the radial wave function and m r + φb,k,α (Gt,i) e · R`,α ( ) l (ˆ) = √Nb m m | |m Y i,j,α N ( 1)| | P | | (cos θˆ)Φ (φˆ) are real spherical har- l − l r m r w (r R τ ) , (31) monics, where × b,α − b,j − b,α s and the ARPES matrix element Eq. (30) becomes (2l + 1) (l m )! N m = − | | , (38) l 2π (l + m )! 2e λ λ p | | ME,ˆr;k,n(ω0) = Aω pE eq Nt ~ 0 · is a normalization factor, P m(x) is an associated Legen- " l z dre polynomial and Φm (φˆr) is defined as X n iGt,j τt,α iQz τ φ (G ) e · e− t,α × t,k,α b,i  i,j,α cos (mφ ) , m > 0  ˆr Q 1 (39) w˜t,α ( ) δk+Gb,i Q⊥,Gt,j Φm (φˆr) = √2 , m = 0 . r × −  z sin ( m φˆr) , m < 0 Nb X n iGb,j τb,α iQz τ + φ (G ) e · e− b,α | | N b,k,α t,i t i,j,α Using the plane wave expansion [48] # + l ∞ w˜b,α (Q) δk+Gt,i Q⊥,Gb,j , (32) iQ r X X l m m × − e− · = 4π ( i) j (Qr) Qˆ (ˆr) , (40) − l Yl Yl l=0 m= l − where Q = pE q is the transferred momentum, with Qz and Q indicating− the components parallel and perpen- where jl(x) is a spherical Bessel function, we can write ⊥ R 3 iQ r w˜ (Q) as dicular to the z axis, and w˜`,α (Q) = d re− · w`,α(r) `,α is the Fourier transform of the Wannier wave functions. lα mα ˆ ˜ Using the relations given by Eqs. (17), (18), (23) and (24) w˜`,α (Q) = ( i) Q R`,α (Q) , (41) − Ylα 6

+ ˜ R ∞ 2 where R`,α (Q) = 4π 0 drr jlα (Qr) R`,α (r) and we bt,1 have used the orthogonality property of the real spherical bb,2 harmonics. For the case in which R`,α (r) are given by bb,1 hydrogen-like wavefunctions

K 1 bt,2 t 2 u lα lα+1 R`,α (r) = n ,l e− 2 x L (x), (42) 3/2 N α α nα lα 1 Γm nαa − − Γ Mm ∗ m Kb p where n,l = (n l 1)!/ (n + l)!, x = 2r/(nαa ), N − − ∗ a = a0/Z (with a0 the Bohr radius and Z the ef- ∗ ∗ α ∗ fective nuclear charge), and Ln is a generalized Laguerre polynomial; then R˜`,α (Q) can be evaluated analytically and is given by [14, 49]

3 2 ˜ / 2 2lα+2 Figure 1. Brillouin zone and basis vectors of reciprocal lattice R`,α (Q) = 4πa nα,lα nα2 lα! ∗ N × for the top (in red) and bottom (in blue) isolated graphene ylα y2 1 Clα+1 , (43) layers that form a twisted bilayer graphene vdW structure. l +2 nα lα 1 2 − × (y2 + 1) α − − y + 1 The dashed green hexagon shows the Brillouin zone of the Moiré superlattice. The yellow arrows show the path along α which the ARPES mapped band structure of Fig.3 is com- where y = nαQ/a and Cn is a Gegenbauer polyno- ∗ puted. mial. Analytic expressions for R˜`,α (Q) are also avail- able if R`,α (r) are approximated by Slater type [50] or Gaussian type [51] orbitals. operators in Eq. (4). It is also possible to work with an Summarizing the results of this section, we can write alternative convention, where the ARPES intensity corresponding to photoemitted electrons with energy emitted along direction ˆr, E > 0 1 X ik` R`,i c† = e− · c˜† . (47) due to an incident electromagnetic ﬁeld with frequency `,R`,i,α `,k`,α √N` λ k` ω0, wave number q, and polarization vector eq as

The operators c† and c˜† are related via c˜† = 2 `,k`,α `,k`,α `,k`,α 2e λ λ 2 ik` τ`,α r q p e e− · c† . Using the convention of Eq. (47), IARPES (E, ˆ ω0, , λ) Nt Aω0 E q `,k`,α | ∝ ~ · × Eq. (31) would be written as X 2 1 f (E ω0 µ) Q,n Q ,n (E ω0) . × − − |M | 2π A ⊥ − 1 n vdW X n i(k+Gb,i) Rt,j ψ (r) = φ˜ (Gb,i) e · (44) k,n √N t,k,α × t i,j,α

p 2 wt,α (r Rt,j τt,α) + where, pE = pEˆr with pE = 2mE/~ , Q = pE q, × − − − 1 X n i(k+Gt,i) Rb,j + φ˜ (Gt,i) e · √N b,k,α × = b i,j,α MQ,n X z n iQz τt,α lα mα wb,α (r Rb,j τb,α) , (48) = φ (0) e− ( i) Qˆ R˜t,α (Q) + t,Q⊥,α − Ylα × − − α ñ n i(k+Gb,i) τt,α s where φt,k,α (Gb,i) = φt,k,α (Gb,i) e · and Ac,t X z ( + ) τ n 0 iQz τb,α ñ G n G i k Gt,i b,α . With these + φb,Q⊥,α ( ) e− φb,k,α ( t,i) = φb,k,α ( t,i) e · Ac,b × α definitions, Q,n would be obtained from Eq. (45) by re- n M n iQ τ lα mα ˆ ˜ placing φ (0) φ˜ (0) e ⊥ `,α . Second, we ( i) Q Rb,α (Q) , (45) `,Q⊥,α `,Q⊥,α − · × − Ylα → would like to emphasize that the factors of Φmα φQˆ , and in order to include broadening effects we can approx- which are included in Eq. (45) via mα Qˆ , ensure that imate the spectral function by a Lorentzian Ylα the ARPES intensity has the same symmetries under ro- η tations along the z direction as the bilayer vdW structure Q ,n (ω) 2 , (46) ⊥ 2 2 A ' (ω EQ⊥,n) + η . − We would like to point out that the present formalism with η the broadening factor. can also be applied to model ARPES in other system We make two remarks regarding Eq. (45). First, we subject to competing periodicities, such as systems dis- point out that its form depends on the chosen conven- playing CDWs [52] or 2D materials weakly coupled to a tion to define the Fourier components of the electronic substrate. 7

IV. APPLICATIONS 0.8 0.7 We will now apply the general formalism developed 0.6 in the previous section to two system: twisted bilayer

] 0.5 graphene and twisted bilayer MoS2. eV

)[ 0.4 q ( tb

h 0.3 A. ARPES of twisted bilayer graphene 0.2 Graphene has a triangular Bravais lattice, with a unit 0.1 cell containing two carbon atoms, A and B, which form a 0.0 honeycomb structure. We write the basis vectors for the 0 2 4 6 8 10 12 bottom layer as qa ! 1 √3 Figure 2. Plot of the interlayer coupling function htb(q) for a 1 = a , , (49) b, 2 2 bilayer graphene, Eq. (58), as a function of qa, with a the lattice parameter of graphene. The parameters used are given ! 1 √3 in the main text. a 2 = a , , (50) b, −2 2

where t 2.7 eV is the nearest-neighbor hopping and we where a 2.46 Å is the graphene lattice parameter, and ' the positions' of the A and B atoms are given by the have written a`,0 = (0, 0). For the interlayer coupling, t,b tb sublattice vectors we have h (Rt,i, Rb,j) = h (Rt,i + τt,α Rb,j τb,β), αβ − − α, β = A, B, where the function htb(r) can be written in τb,A = (0, 0) , (51) terms of Slater-Koster parameters [56] as a τb,B = (0, 1) . (52) √3 2 2 tb tb r d h (r) = h (r, d) = Vppπ(R) + Vppσ(R) , (57) The top layer is rotated with respect to the bottom one R2 R2 by an angle of θ such that at,i = R(θ) ab,i, for i = 1, 2, · 2 2 and τt,α = R(θ) τb,α + dez, for α = A, B, where R(θ) is where R = √r + d is the distance between the two the rotation matrix· atoms, with r the distance projected in the x y plane. For the dependence of the Slater-Koster param-− cos θ sin θ R(θ) = , (53) eters on R, we use the parametrization of Refs. [30, sin θ −cos θ 0 (R a/√3)/r0 57]: Vppπ(R) = Vppπe− − and Vppσ(R) = 0 (R d)/r0 0 V e− − , where r0 0.184a, V = t and d 3.35 Å is the separation between the two lay- ppσ ' ppπ − ' ers. The' corresponding reciprocal space basis vectors are 2.7 eV and V 0 0.48 eV. From this we can evalu- − ppσ ' given by ate numerically

! Z 4π √3 1 tb tb 1 2 iq r tb bb,1 = , , (54) hαβ(q) = h (q) = d re− · h (r, d) , (58) √3a 2 2 Ac ! √ 4π 3 1 2 tb bb,2 = , , (55) where A = A = √3a /2. We plot the function h (q) √ c c,` 3a − 2 2 in Fig.2. Writing the electronic creation and annihilation oper- for the bottom layer, and by bt,i = R(θ) bb,i for the top layer. In Fig.1, we show the ﬁrst Brillouin· zone of both ators in terms of Fourier components as in Eq. (4), we layers. Also shown is the Brillouin zone of the moiré su- can construct the Hamiltonian for the graphene bilayer perlattice, whose associated reciprocal lattice basis vec- structure in the form of Eqs. (12)-(14), where tors are given by bm,i = bt,i bb,i [53, 54]. − We will describe each individual graphene layer within `,` 0 γ`,k hk = , (59) the nearest-neighbor tight-binding model for pz orbitals γ`,∗ k 0 [55], which reads

2 is written in the A, B sublattice basis, γ`,k = X X P3 ik δ`,i H` = t c† c`,R +a ,B + h.c. , (56) t i=1 e · , with δ`,i = R (2π (i 1) /3) τ`,B the − `,R`,i,A `,i `,j − − · i j=0 nearest-neighbor vectors in each layer, and the interlayer 8 coupling matrices are given by 3

τ eiGt t,A 0 2 ht,b · k,G ,G = iGt τt,B b t 0 e · · tb tb 1 h (k + Gb + Gt) h (k + Gb + Gt) tb tb

k G G k G G [eV] · h ( + b + t) h ( + b + t) · 0 θ = 11.6◦ τ ,n iGb b,A k e− · 0 iGb τb,B , (60) E · 0 e− · 1 − with hb,t = ht,b †. By truncating the 2 k,Gt,Gb k,Gb,Gt − Hamiltonian to the N shortest G and G vectors, G t b 3 Hk ( Gb , Gt ) becomes a 4NG 4NG matrix, from − which{ we} can{ obtain} an approximation× to the energies Kb Mm Kt Γm Mm Γm and eigenstates of the twisted bilayer graphene structure. 4 The obtained model is a tight-binding extension of the continuous models of Refs. [57–60]. From the eigenstates, we can obtain the ARPES intensity using Eqs. (44) and 2 (45). Due to the fact that the function htb(q) decays rapidly for q a 1, the obtained ARPES signal con-

| | [eV] verges rapidly with only a few Gt and Gb vectors. No- 0 θ = 20◦ ,n tice that due to the fact that the model we use only in- k

E 0 √ volves pz 1 orbitals, we have that Φ0 φQˆ = 1/ 2 ≡ Y 2 in Eq. (45) for Q,n. − In Fig.3 we showM the computed ARPES bands along the path indicated in Fig.1, for two different angles: θ = 11.6 , for which ARPES measurements where per- 4 ◦ − formed in Ref. [61], and for θ = 20◦. The thickness of the Kb Mm Kt Γm Mm Γm 2 bands is proportional to the value of Q⊥,0,n , where for simplicity we assumed that there|M is no transferred| Figure 3. Computed ARPES bands for twisted bilayer ◦ momentum along the z direction, Qz = 0, and neglected graphene for two different twist angles: θ = 11.6 (top) and ˜ θ = 20◦ (bottom). The yellow dotted lines show the bands the effect of R`,α (Q) on Q⊥,0,n. The bands where M E , eigenvalues of H ({G } , {G }), for k along the path computed using a truncated Hamiltonian with NG = 7 k,n k b t Kb → Mm → Kt → Γm → Mm → Γm represented by the [including Gb/t = (0, 0)], which was found to be suffi- cient to obtain converged results. Increasing N virtu- yellow arrows in Fig.1. The blue tick bands represent Ek,n G weighted by the ARPES matrix element value, with the thick- ally does not change the thick bands in a visible away, 2 ness corresponding to |Mk,0,n| . The dashed red lines shown although more Ek,n bands do appear, which, neverthe- the dispersion relation for the decoupled graphene layers. It less, have negligible ARPES weight. As anticipated the was assumed Qz = 0. The horizontal dot-dashed lines mark ARPES weighted bands mostly follow the bands of the the energies −1.3, −1.0 and −0.8 eV (for θ = 11.6◦) and −2.3, decoupled system. Constant energy ARPES maps were −1.65 and −0.7 eV (for θ = 20◦) at which the ARPES con- also computed for the energies signaled by the dot-dashed stant energy maps of Fig.4 are computed. In both plots we horizontal lines in Fig.3. The computed constant energy used a number of reciprocal lattice vectors NG = 7. It was maps are shown in Figs.4 (for θ = 11.6◦) and5 (for assumed Qz = 0. θ = 20◦). For comparison, we also show the constant energy maps for the decoupled bilayer structures. The absence of part of the constant energy surface, due to 1eV along the line that bisects the angle between the the ARPES matrix elements, in both the coupled and Dirac' − points of the two layers (Fig. 2(d) of Ref. [61]), decoupled cases is clear. A significant reconstruction of which is shown as the path Γm Mm Γm in Fig.1, the constant energy ARPES maps is observed for ener- while in our model there is only→ one band→ with ARPES gies at which the band structures of the isolated graphene weight around that energy. The absence of one of the layers intersect. ARPES bands in our results is easily understandable: the Good qualitative agreement between our results for the two bands are formed by bands of the top and bottom lay- twist angle of θ = 11.6◦ and the experimental data of ers which are degenerate for the decoupled system along Ref. [61] is observed, especially taking into account the the path Γm Mm Γm (see the merging of two bands simplicity of our model. There is however an observed of the decoupled→ system→ along that line in Fig.3) corre- discrepancy between our results and the experimental sponding, therefore, to bonding and anti-bonding states. data: in Ref. [61] two bands are observed with energy For the anti-bonding state we can check numerically that 9

0.75 (a) = 1.3eV (d) = 1.3eV 2 (a) = 2.3eV (d) = 2.3eV − − − − 0.50 ˚ A] [

˚ A] 0 [ 0.25 y k y k 0.00 2 0.25 − − 2 (b) = 1.65eV (e) = 1.65eV − − 0.75 (b) = 1.0eV (e) = 1.0eV − − ˚ A] 0.50 [ 0 y k ˚ A]

[ 0.25 y

k 2 0.00 − 0.25 2 (c) = 0.7eV (f) = 0.7eV − − − 0.75 ˚ A]

0.50 k ˚ A] [ 0.25 2

y − k 0.00 2 0 2 2 0 2 − − 0.25 kx [A]˚ kx [A]˚ −

1.5 2.0 1.5 2.0 Figure 5. Computed constant energy ARPES map for twisted ◦ kx [A]˚ kx [A]˚ bilayer graphene for a twist angle of θ = 20 for the energies of = −2.3, −1.65 and −0.7 eV measured from the Dirac point. (a)–(c) show the constant energy ARPES maps for two Figure 4. Computed constant energy ARPES map for twisted decoupled graphene layers, while (d)–(f) show the ARPES ◦ bilayer graphene for a twist angle of θ = 11.6 for the energies maps taking into account coupling between the two layers. of = −1.3, −1.0 and −0.8 eV measured from the Dirac The dashed red, blue and green lines represent the same as in point. (a)–(c) show the constant energy ARPES maps for two Fig.4. A broadening of η = 0.05 eV was used. The Hamilto- decoupled graphene layers, while (d)–(f) show the ARPES nian was truncated with NG = 7. It was assumed Qz = 0. maps taking into account coupling between the two layers. The dashed red and blue lines show the limits of the Brillouin zone of the top and bottom layers (respectively), while the layers are aligned) green dashed hexagon represents the moiré Brillouin zone. A broadening of η = 0.05 eV was used. The Hamiltonian was iQz d anti anti truncated with NG = 7. It was assumed Qz = 0. anti e− φ (0) + φ (0) Mk,Qz , ∝ t,k,A t,k,B anti anti + φb,k,A (0) + φb,k,B (0)

anti anti anti anti iQz d/2 φb,k,A (0) = φt,k,B (0) and φb,k,B (0) = φt,k,A (0). = e− 2 sin (Qzd/2) − −anti [62] For Qz = 0, we then obtain k,0,anti φ (0) + anti anti M ∝ t,k,A φb,k,B (0) + φb,k,A (0) , (61) φanti (0) + φanti (0) + φanti (0) = 0, and therefore this × t,k,B b,k,A b,k,B bond band would not be visible in ARPES. There are two pos- while for the bonding state, for which φb,k,A (0) = bond bond bond sible explanations for the fact that the anti-bonding band φt,k,B (0) and φb,k,B (0) = φt,k,A (0), we would obtain is visible experimentally in ARPES: (i) a possible energy iQz d/2 imbalance between the two layers, (ii) effects of finite bond e− 2 cos (Q d/2) Mk,Qz , ∝ z transferred momentum along the out-of-plane direction, anti anti φb,k,A (0) + φb,k,B (0) . (62) Qz. As a matter of fact, a shift in energy between the × Dirac cones of the two layers of ∆E = 0.05 eV is reported Therefore, both bands can have similar visibility in in Ref. [61]. We have checked that this shift in energy in- ARPES if Qzd/2 (1 + 2n) π/4, n N. Besides this ' ∈ deed leads to k,0,anti = 0, but the value is too small to effect, we also expect that a better agreement with the lead to a significantM visibility6 of the anti-bonding band. experimental data would be possible, provided a more ac- The remaining possibility is finite Qz effects. For finite curate modeling of the band structure of the individual Qz, we would obtain (assuming the Dirac points of both layers was employed. 10

B. ARPES of twisted bilayer MoS2 this into account, we can write the Hamiltonian matrix for each layer, in the orbital basis deﬁned with respect to the common reference frame, as Similarly to graphene, monolayer MoS2 has a honeycomb structure, with the A sites occupied by Mo atoms b,b and the B sites occupied by two S atoms, top and bot- hk = h (k) , (67) tom, which lie at planes above and bellow the Mo plane. ht,t = R(θ) h (R( θ) k) ( θ), (68) k · − · ·R − We write the Bravais basis vectors for MoS2 in the same way as for graphene, Eqs. (49) and (50), with a lattice where the matrix R(θ) rotates the orbitals in Eq. (66) constant a 3.16 Å[63]. For an unrotated layer, the Mo along the z axis, and has the block diagonal form and S atoms' occupy the approximate positions inside the   unit cell R(2θ)  R(θ)    τMo = (0, 0) , (63)  11 1  R(θ) =  × R(θ)  , 1 1    1  τStop/bot = a , . (64)  1 1  √3 ±2  × R(θ)  1 For the separation between the two MoS layers (be- 1 1 2 × (69) tween the Mo planes) we use the value for bulk MoS 2 with R(θ) the rotation matrix Eq. (53). For the inter- c 6.14 Å, which corresponds to a separation between 0 layer coupling, we assume that this is dominated by the nearest' S planes of d 2.98 Å. We will describe the elec- hopping between the Stop p orbitals of the bottom MoS tronic properties of the' individual MoS layers, using the 2 2 layer and the Sbot p orbitals of the MoS layer. We write 11 band tight-binding Hamiltonian of Ref. [63], with the 2 the interlayer hoppings in terms of Slater-Koster param- parametrization of Ref. [64], which involves the dx2 y2 , − eters as dxy dxz, dyz, dz2 Mo orbitals (corresponding to the real 2 2 1 1 0 spherical harmonics 2 , 2− 2 , 2− , 2 ) and the px, tb tb top r top r Y Y Y Y Y hpbot,p ( ) = hpbot,p ( , d) py, pz S orbitals (corresponding to the real spherical har- i j i j 1 1 0 i j i j monics , − , ). The Hamiltonian for an unrotated R R R R Y1 Y1 Y1 = V (R) + V (R) δ , layer is given by ppσ R2 ppπ ij − R2

6 i, j = x, y, z (70) X H = c† hn cR, (65) R+an · · where R = (x, y, d) is the separation between the S n=0 atoms. As before, we assume a dependence of the Slater- where Koster parameters on the distance of the form Vppσ (R) = 0 β(R/dSS 1) 0 β(R/dSS 1) Vppσe− − and Vppπ (R) = Vppπe− − . We 0 0 cR† = cR† ,d , cR† ,d , cR† ,d , cR† ,d , cR† ,d , use Vppσ 0.774 eV and Vppπ 0.123 eV, which are the x2−y2 xy xz yz z2 ' − ' interlayer hoppings used in Ref. [63], with dSS 3.49 Å c† top , c† top , c† top , c† bot , c† bot , c† bot , (66) ' R,px R,py R,pz R,px R,py R,pz the interlayer nearest-neighbor separation between S atoms in bulk MoS2. In the absence of ab initio cal- is a vector of creation operators, hn are hoping ma- culations, we assume the value β 3, in accordance ' trices and an are given by an = R (π(n 1)/3) a1 with Harrison’s argument [65] and as previously used − · for n = 1, ..., 6 and a0 = (0, 0). Writing the elec- to model strained MoS2 [66]. The Fourier transform of tb tronic creation and annihilation operators in terms of h top (r, d) can be written as pbot,p Fourier components as in Eq. (4) we obtain the follow- i j ing Hamiltonian matrix in k-space, h k , with entries ( ) tb 6 top q P ik (an+τα τβ ) hpbot,p ( ) = hαβ (k) = n=0 [hn]αβ e− · − , with α, β = i j bot q2 q q dx2 y2 , ..., pz . As in the case of graphene, we will keep  x x y qx  − V1(q) q2 V2(q) q2 V2(q) i q Vz(q) the bottom layer ﬁxed, while rotating the top layer by an − − 2 − =  qxqy qy qy  , angle θ. When describing a MoS2 layer rotated by an an-  q2 V2(q) V1(q) q2 V2(q) i q Vz(q)  − q− − gle θ, it is important to take into account that under the i qx V (q) i y V (q) V (q) − q z − q z zz rotation the orbitals will transform in a non-trivial way (71) (in graphene this does not happen as pz orbitals are invariance under rotations around the z axis). We chose to where the functions Vzz(q), Vz(q), V1(q) and V2(q) are represent the Hamiltonian of both layers in terms of or- shown in Fig.6 and the explicit expressions used to eval- bitals deﬁned with respect to the same common reference uate them are provided in AppendixB. We point out frame, which we choose to be the unrotated frame. It is that the general expression for Vzz(q) is the same as for also with respect to this common reference frame that graphene, Eq. (58). As for graphene, we can see that the the plane wave expansion Eq. (40) is written. Taking Fourier components of the interlayer hoppings decay fast. 11

0.00 0.50 − (a) 0.25 θ = 13.5◦ − 0.50 0.75 − − ] 0.75

eV −

)[ 1.00

q 1.00 (

− [eV] − tb h 1.25 V ,n − zz k

1.50 V E 1.25 − z − 1.75 V1 − V2 1.50 2.00 − − 0 2 4 6 8 10 qa 1.75 − Mb Γ Kb Mb

Figure 6. Plot of the interlayer coupling functions Vzz(q), Vz(q), V1(q) and V2(q) for bilayer MoS2, Eq. (71), as a func- 1.5 (b) (c) tion of qa, with a the lattice parameter of MoS2. The parameters used are given in the main text. ˚ A] [ 0.0 y tb k Having evaluated h bot top (q) we can construct the ma- pi ,pj trices ht,b and hb,t , from which, together with 1.5 k,Gb,Gt k,Gt,Gb − Eqs. (67) and (68), we can build the Hamiltonian matrix 1.5 0.0 1.5 1.5 0.0 1.5 for the twisted bilayer according to Eqs. (12)-(14). Keep- − − kx [A]˚ kx [A]˚ ing the NG shortest Gt and Gb vectors, Hk ( Gb , Gt ) becomes an 11N 11N matrix, from which{ the} { ener-} G × G gies and eigenstates of twisted bilayer MoS2 can be eval- Figure 7. Computed ARPES bands, (a), and constant energy uated and then used to model the ARPES intensity. As map, (b) and (c), for twisted bilayer MoS2 for a twist angle ◦ already pointed out, the tight-binding model for MoS2 of θ = 13.5 . In (a), the bands are computed along the path involves orbitals that do not transform trivially under Mb → Γ → Kb → Mb [shown in (b)]. The yellow dotted rotations around the z axis. Therefore, it is essential to lines show the band structure, Ek,n, while the blue tick bands represent Ek,n weighted by the ARPES matrix element value, keep the factors Φm φ ˆ , Eq. 39, in the ARPES ma- 2 α Q with the thickness corresponding to |Mk,0,n| . The dashed trix element Q,n in order to obtain an ARPES signal red lines show the band structure for the decoupled MoS2 that respectsM the three-fold rotational invariance of the layers. The horizontal dot-dashed green line marks the energy twisted bilayer MoS2 structure. = −1.12 eV, at which the constant energy maps (b) and In Fig.7 we show the computed ARPES bands and (c) are computed. (b) and (c) are, respectively, the constant constant energy maps for a twisted bilayer MoS with a energy maps for decoupled and coupled twisted bilayer MoS2. 2 A broadening of η = 0.02 eV was used. The Hamiltonian for twist angle of θ = 13.5 , for which ARPES measurements ◦ the coupled bilayer structure was truncated with N = 7. In have been performed [67]. The calculations where per- G all plots it was assumed Qz = 0. formed truncating the Hamiltonian matrix with NG = 7. As can be seen in Fig.7(a), by comparing with the bands of the decoupled layers, the interlayer coupling leads to constant energy map for the same θ = 13.5 twist angle. a large splitting of the states at the point, with one ◦ Γ For comparison, the ARPES constant energy map for the of them becoming the valence band maximum. This decoupled layer is shown in Fig.7(b). Once again, it can also occurs for bulk MoS [63] and has been predicted by 2 be seen that the interlayer coupling affects more strongly ab initio calculations for commensurate twisted bilayer the states close to the Γ point having almost no impact structures for several twist angles [67–69]. We checked, on the states close to the K points. It is observed that the that if the interlayer distance is kept fixed, this splitting valence band pocket at the Γ point has very weak visibil- at is virtually independent of the twist angle. From Γ ity in ARPES in agreement with what is experimentally this, it can be inferred that the change in the splitting observed in Ref. [67]. at Γ with angle predicted in [67–69] and observed in the ARPES measurements of [67] is due to the interlayer separation modulation with the twist angle, which we kept fixed. As we can see in Fig.7(a), the effect of the inter- V. CONCLUSIONS layer coupling is negligible at the K point. As in the case of twisted bilayer graphene, the back-folded bands have In this work we developed a general theoretical frame- negligible visibility in ARPES. In Fig.7(c), we show the work to model ARPES of lattice mismatch/misaligned 12 vdW structures. By describing the photoemitted state (mean-field) Hamiltonian as done in this work. In the as a plane wave and the bound electronic states in terms latter case, it is not possible to describe the system at of Bloch waves of the individual layers, while taking into a single-particle level and the electronic spectral func- account generalized umklapp processes, we obtained an tion of crystal bound states, which can be severely re- efficient description of ARPES that can be applied both constructed by the interactions, has to be obtained from to commensurate and incommensurate structures. Being the full Green’s function of the interacting system[52]. based on a tight-binding description of the bound elec- In this case, it is therefore crucial to distinguish between tronic states, the present formalism can deal with arbi- the effects of interactions and the effect of the ARPES trary lattice mismatch/misalignment, going beyond pre- matrix elements (which encode the effect of competition vious low energy, continuum descriptions of both twisted between the periodicities) [13]. The approach presented bilayer graphene and graphene/h-BN structures. We ap- in this work can be used to take into account the latter plied the developed formalism to the cases of twisted bi- effect. layer graphene and twisted bilayer MoS2. The example of graphene showcases the importance of the ARPES matrix elements in two ways: (i) by showing the importance ACKNOWLEDGMENTS of the momentum dependence of the ARPES weight in entangled bands, which is responsible for the absence of The author would like to acknowledge useful discus- part of the constant energy surface map around the K sions with J. L. Lado, J. Fernández-Rossier and Eduardo points, and (ii) by showing that the ARPES weight of V. Castro. The author also acknowledges the hospitality back folded bands is very small, which is a consequence of the International Iberian Nanotechnology Laboratory of the weak coupling between the two layers. As a con- (INL), where early ideas on this problem were developed. sequence, the observed ARPES bands mostly follow the B. A. received funding from the European Union’s Hori- band structure of the decoupled system, except at en- zon 2020 research and innovation programme under grant ergies at which states from both layers are degenerate, agreement No 706538. where a significant reconstruction of the spectrum occurs. By comparing the results of the current model to the experimental data of ARPES of twisted bilayer Appendix A: Non-equilibrium Green’s function graphene at a twist angle θ = 11.6◦[61], we showcased description of ARPES the importance of the transferred momentum between the incoming radiation and the photoemitted state along In this appendix we briefly review the rigorous deriva- the z direction, which significantly affects the visibility of tion of the observed photoemitted current in an ARPES bands in ARPES that correspond to bonding and anti- experiment using the non-equilibrium Green’s function bonding states of the two layers. In the example of MoS2, technique. This approach was first developed in Ref. [31] we once again observed that the ARPES bands mostly and its connection to other formalisms and experimental follow the band structure of the decoupled system. Dif- observations clarified in Ref. [32]. ferently from graphene, we found out that while there is a In order to describe the process of photoemission, we significant shift in energy of the states at the Γ point, the must consider both states that are bound to the crys- reconstruction of the valence band close to the K points tal and unbound, nearly free, states. Therefore, in the due to the interlayer coupling is negligible, which is in model employed, the crystal must not occupy the entire accordance with experimental ARPES data [67]. space. For concreteness we assume that the positions Although this work focused on bilayer vdW structures, of the atoms that form the crystal are restricted to the the formalism can also be applied to structures formed by z 0 region. We assume that the crystal is infinite along the≤ x y plane. The single particle Hamiltonian for the multiple lattice misaligned/mismatch layers. We would − like to point out that the present approach to model electrons reads ARPES can also be extended to other systems where Z 2 3 p H0 = d rΨ†(r) + V (r) Ψ(r), (A1) competition between different periodicities occurs, both 2m commensurate and incommensurate, such as 2D materials placed on top of a weakly coupled substrate or materi- where Ψ†(r) is the electronic creation field operator, V (r) als displaying CDW phases. The coupling to a substrate is the crystal potential, and p = i~ r. We describe the can be described via a substrate surface Green’s function, exciting electromagnetic field in− the∇ Weyl gauge, such which would play a role similar to in the present work. Hb that E(t, r) = ∂tA(t, r) and B(t, r) = A(t, r), The effect of the substrate can also be approximated by where A(t, r) is− the vector potential. The∇ coupling × to a potential which acts on the 2D material. In systems the electromagnetic field is obtained via minimal cou- with CDW phases, two cases must be distinguished: (i) pling p p + eA(t, r). In the presence of the vector a system with weak fluctuations, which can be described potential,→ the Hamiltonian can be written as the sum of at a single-particle level, and (ii) a strongly interacting three terms system, with strong fluctuations. In the former case, it is possible to describe the system with a single-particle H = H0 + H1,A + H2,A, (A2) 13 where H0 is the same as in Eq. (A1) and and the product represents integration over the spa-

Z tial variables and time, and summation over other possi- 3 (1) H1,A = d r Ψ†(r)J (r)Ψ(r) A(r, t), (A3) ble degrees of freedom (sublattice, orbital, spin,...). All − · the Green’s functions in Eq. (A9) are evaluated in the Z 1 3 (2) 2 absence of A(t, r), and are therefore in thermodynamic H2 = d r Ψ†(r)J (r)Ψ(r) A (r, t), (A4) ,A −2 equilibrium. In Eq. (A9) interactions between the emitted state and the remaining hole are neglected, an ap- with proximation that is typically referred to as the sudden ap- (1) e~ proximation. Including these kind of interactions would J (r) = −→ ←− , (A5) −2mi ∇r − ∇r lead to a renormalization of the vertices J(1) and J (2) [31]. e2 For a photodectector placed very far away from the crys- J (2)(r) = , (A6) − m tal sample, only the last term in Eq. (A9) gives a finite contribution. This can be understood if we write G< in the matrix elements in real space of the paramagnetic terms of the eigenstates ψ (r) of Eq. (A1). The creation and diamagnetic currents, respectively. The arrows in a field operator can be written as Ψ (r) = P c ψ (r), the differential operator indicate whether the derivative † a a† a∗ where c creates an electron in state ψ (r) with energy acts on the electronic field operator placed to the right or a† a a, and the lesser Green’s function, for a non-interacting to the left. The expectation value of the current is given system, becomes by 0 < X ia(t t ) 2 G (t, r; t0, r0) = ψ (r)ψ∗(r0)e− − i c† c . ~ e a a a a J(t, r) = lim e ( r r0 ) A(t, r) a h i r0 r − 2mi ∇ − ∇ − m × → (A12) < ( i) GA(t, r; t, r0), (A7) Notice that only crystal bound states are occupied, and × − therefore the sum in previous equations is restricted to with the lesser Green’s function defined as those states due to the occupation function ca† ca . At < the same time, the wavefunction of crystal bound states G (t, r; t0, r0) = i Ψ†(t0, r0)Ψ(t, r) , (A8) A A decays exponentially away from the crystal. Therefore if where the A subscript means the expectation value is one of the arguments of G< is evaluated away from the evaluated taking into account A(t, r). This can be crystal, that term can be safely neglected. This also al- done perturbatively in A(t, r) using the non-equilibrium lows us to discard the diamagnetic term in Eq. (A7), as Green’s function formalism [70, 71]. By expanding it would only contribute to third order in A(t, r). There- the contour-ordered Green’s function in the Schwinger- fore, to second order in the electromagnetic field, the Keldysh contour in powers of A(t, r) and then using Lan- photoemitted current measured away from the crystal is greth’s rules, the lesser Green’s function is given to sec- given by ond order in A(t, r) by

< < < (1) A Z Z GA = G + G J A G e~ 3 3 · J(t, r) = lim ( r r0 ) dt1d r1 dt2d r2 h i r0 r 2m ∇ − ∇ R (1) < → + G J A G R (1) · G (t, r; t1, r1)J (r1) A(t1, r1) · × 1 < (2) 2 A < (1) A + G J A G + G (t1, r1; t2, r2)J (r2) A(t2, r2)G (t2, r2; t, r0). 2 × · (A13) 1 + GR J (2)A2 G< 2 Assuming a monochromatic electromagnetic ﬁeld at fre- + G< J(1) A GA J(1) A GA quency ω0 > 0, · · · · iω t iω t R (1) R (1) < A(t, r) = A (r)e 0 + A (r)e 0 , (A14) + G J A G J A G ω0 − ω0 · · − expressing all quantities in Fourier components in time + GR J(1) A G< J(1) A GA · · and looking at the time-averaged current over one period 3 + A (A9) 2π/ω0, we obtain O where the retarded and advanced Green’s functions are Z Z Z e~ dE 3 3 given by J(r) = lim ( r r0 ) d r1 d r2 h i r0 r 2m ∇ − ∇ 2π → R h G (t, r; t0, r0) = iΘ(t t0) Ψ(t, r), Ψ†(t0, r0) , R (1) < G (E; r, r1)J (r1) Aω (r1)G (E ω0; r1, r2) − − (A10) · 0 − × i A (1) A r r r r J (r2) A ω0 (r2)G (E; r2, r0) + (ω0 ω0) . G (t, ; t0, 0) = iΘ(t0 t) Ψ(t, ), Ψ†(t0, 0) , × · − ↔ − − (A11) (A15) 14

Some further simpliﬁcations can be performed. First, by Equation (A18) for the advanced Green’s function can be using the ﬂuctuation-dissipation theorem, which is valid obtained in a similar way. Inserting Eqs. (A16), (A17) in thermodynamic equilibrium even for an interacting and (A18) in Eq. (A15), the detected photoemitted cur- system, we can write rent becomes

< X Z ∗ G (E; r1, r2) = ψa(r1)ψa∗(r2)if (E µ) a (E) , e dE ~pE i(pE p )r − A J(r) = − ˆr Re e − E a h i (4πr)2 2π m × (A16) R A X h 2 where a (E) = Ga (E) Ga (E) is the electronic spectral f (E ω0 µ) ME,ˆr;a(ω0) a (E ω0) + A − 1 × − − | | A − function in the eigenstate basis, and f(ω) = eβω + 1 − a is the Fermi-Dirac distribution function with µ the chem- + (ω0 ω0)] . (A23) ical potential. Once again, only crystal bound states are ↔ − occupied and therefore, the integration over the spatial where we have defined the ARPES matrix elements as coordinates in Eq. (A15) is mostly confined to the region of the crystal. This allows us to use the asymptotic ex- 2m ME,n;a(ω0) = pression of the retarded and advanced Green’s functions, − ~2 × Z valid for r and r0 far away from the crystal, [32, 41] 3 (1) d r1 ψ∗ (r1)J (r1)ψ (r1) A (r1), (A24) × E,n a · ω0 R 2m 1 ipE r G (E; r, r1) e ψ∗ (r1), (A17) ' − 2 4πr E,ˆr which can also be written as Eq. (27) of the main text. ~ ∗ 2m 1 ∗ 0 i(pE pE )r 2 pE r A ipE r For E < 0, we have that e − = e− | | and there- G (E; r2, r0) 2 ψE,ˆr (r2) e− , (A18) ' − ~ 4πr0 fore this contribution vanishes for a detector far away from the crystal. Therefore, we can restrict the integra- r p 2 2 tion in Eq. (A23) from 0 to + . At the same time, we where pE = 2mE/~ for E > 0 and pE = i 2m E /~ notice that the second term in∞ Eq. (A23) involves states for E < 0 (we choose as zero of energy the threshold to with energy E + ω0 > 0 which are unoccupied, and can have free electron states), ψE,∗ ˆr(r1), given by Eq. (28), is therefore be neglected. This allows us to write the conjugate of an electron diffraction state [32, 41], ˆr R is the unit vector pointing along r, and G (E; r, r0) is Z + free e ∞ dE ~pE the free space retarded Green’s function J(r) = − 2 ˆr f (E ω0 µ) h i (4πr) 0 2π m − − × 0 ipE r r X 2 R 2m 1 e | − | ME,ˆr;a(ω0) a (E ω0) . (A25) Gfree(E; r, r0) = 2 . (A19) × | | A − − ~ 4π r r0 a | − | Equation (A17) can be obtained from the Dyson equation Assuming that the electron detector can resolve the en- for the retarded Green’s function ergy of the photoemitted states and it collects electrons emitted along direction ˆr, the measured ARPES intensity R R is proportional to Eq. (26) of the main text. G (E; r, r0) = Gfree(E; r, r0)+ Z 3 R R + d r1Gfree(E; r, r1)V (r1)G (E; r1, r0). (A20) Appendix B: Fourier transform of interlayer hopping If we are interested in the Green’s function when r is for MoS2 very far away from the crystal, we can use the following R approximation for Gfree(E; r, r0), valid for r r0 , In this appendix, we provide details on how the two- | | | | dimensional Fourier transform of the interlayer coupling ipE r 0 R 2m 1 e ipRˆr r for S p orbitals for twisted bilayer MoS2, Eq. (70), is Gfree(E; r, r0) e− · . (A21) ' − ~2 4π r computed. We wish to evaluate

Inserting this approximation into the Dyson equation Z 2 tb d r iq r (A20), we obtain Eq. (A17) with top q hpbot,p ( ) = e− · i j Ac × 0 Z  2 xy  ipEˆr r 3 ipEˆr r1 R x xd v (r) + v (r) 2 v (r) 2 v (r) 2 ψE,∗ ˆr(r0) = e− · + d r1e− · V (r1)G (E; r1, r0). 1 2 R 2 R 2 R  xy y2 yd  v (r) 2 v (r) + v (r) 2 v (r) 2 . (A22)  2 R 1 2 R 2 R  xd yd d2 Using the Dyson equation for the Green’s function v2(r) R2 v2(r) R2 v1(r) + v2(r) R2 Eq. (A20), we can rewrite the above equation as Eq. (28), (B1) which is nothing more than the Lippmann-Schwinger equation for the scattering of an incoming plane wave where v1(r) = Vppπ (R) and v2(r) = Vppσ (R) Vppπ (R), 0 ipEˆr r 2 2 − state, φ0∗(r0) = e− · , by the crystal potential V (r). with R = √r + d , and Ac is the unit cell area of MoS2. 15

We have three kinds of integrals: Using the properties of Bessel functions, it is possible to write Z 2 iq r 0(q) = d re− · f(r), (B2) I ∂ 1 2 Z J0(qr) = r qi (J0(qr) + J2(qr)) , (B8) 2 iq r ∂qi −2 i(q) = d re− · xif(r), (B3) 2 I ∂ 1 2 Z J0(qr) = δij r (J0(qr) + J2(qr)) 2 iq r ∂qi∂qj − 2 ij(q) = d re− · xixjf(r), (B4) I qiqj 2 + r J 2(qr). (B9) q2 where f(r) is a function only of r = r . Using the fact R 2π ix cos φ | | that 0 dφe− = 2πJ0(x), where J0(x) is the Bessel Using this, we can write Eq. (B1) as Eq. (71), with function of ﬁrst kind and order 0, we can write Z 2π Z V1(q) = drr v1(r)J0(qr), 0(q) = 2π drrf(r)J0(qr). (B5) A I c 1 r2 (B10) The integrals (q) and (q) can be written as + v2(r) 2 (J0(qr) + J2(qr)) Ii Iij 2 R Z 2π Z r2 2 ∂ iq r V (q) = drrv (r) J (qr), (B11) (q) = i d r e− · f(r) 2 2 2 2 i Ac R I ∂qi Z π Z r2 ∂ V (q) = qd drrv (r) [J (qr) + J (qr)] , (B12) = i drrf(r) J0(qr), (B6) z 2 2 0 2 Ac R ∂qi Z 2 2π Z d2 2 ∂ iq r V (q) = drr v (r) + v (r) J (qr). (B13) (q) = d r e− · f(r) zz 1 2 2 0 ij Ac R I − ∂qi∂qj Z ∂2 = drrf(r) J0(qr). (B7) The remaining integration over r has to be performed − ∂qi∂qj numerically.

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