Twisted bilayer graphene — electronic and optical properties

Gonçalo Filipe Santos Catarina1, ∗

1CeFEMA, Department of Physics, Instituto Superior Técnico,Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal

(Dated: February 2017)

Van der Waals heterostructures are promising new materials which have been drawing increas-ingly attention [1–4]. In this work, we focus on the theoretical description of one of the simpleststackings, the twisted bilayer graphene, which can be seen as one of the fundamental pieces formore complex assemblies. Starting from a single-particle theory, based on a low-energy continuummodel [5], we address the optical properties, namely the optical conductivity (within the linearresponse theory) and the spectrum of graphene surface plasmon-polaritons (using a semi-classicaltreatment). As our own contribution, we highlight the introduction of a new and improved methodfor computing the Drude conductivity and the obtention of original results for the plasmonicresponse (transverse magnetic modes) of this system.Keywords: twisted bilayer graphene, low-energy continuum model, optical conductivity, surfaceplasmon-polaritons.

I. INTRODUCTION

Two-dimensional (2D) crystals have attractedwidespread attention of the scientific community,especially since their first experimental isolation in2004 [6]. Amongst them, graphene —a 2D honeycomblattice of carbon atoms— has stood out, leading tothe development of a research area of its own [7–9].In parallel with the efforts being made on grapheneand other 2D materials, a new research field dealingwith van der Waals (vdW) heterostructures —artificialthree-dimensional (3D) structures made by stacking 2Dcrystals together (fig. 1)— has recently emerged [1–3].Within this approach, the aim is to design heterostruc-tures that exhibit tailored properties for technologicalapplications. In this work, the focus is on one of thesimplest stackings, the twisted bilayer graphene (tBLG)—a graphene sheet on top of other, with a twist angleθ—, which can be seen as one of the fundamental piecesfor more complex assemblies.

The complex geometry of the tBLG affects significantlyits electronic properties, making even the single-particlemodels quite involved [10]. This geometry is character-ized by the twist angle θ, which manifests itself in theappearance of a moiré pattern revealing the periodicity(or quasiperiodicity) of the crystal structure [11]. Whilethe moiré exists for any θ, the superstructure, that is,strickly periodic repetition of some large multiatomic su-percell, occurs only for the so-called commensurate an-gles [12]. For commensurate structures, numerical stud-ies based on density functional theory (DFT) have beenperformed [13–15]. However, since the unit cell of thetBLG superlattice contains a large number of sites, es-pecially at small θ, these ab initio calculations incur asignificant computational cost and are therefore rather

∗Electronic address: goncalo.catarina@tecnico.ulisboa.pt

FIG. 1: Illustration of a vdW heterostructure. Source: ref. [2].

unpractical. To avoid this difficulty, semi-analytical the-ories have been developed in order to describe the low-energy electronic properties of the tBLG [5, 12, 16–18].

In general, the study of light-matter interactions is atopic of interest in science, with a wide variety of ap-plications, for example in the field of photonics. Forgraphene, and in particular for the tBLG, the responseto an applied electromagnetic field is characterized bythe optical conductivity, which has been computed us-ing analytical [19, 20] and numerical [21] models. Overthe last few years, graphene plasmonics has also emergedas a new topic [22], especially after the experimental re-alization achieved by the end of 2011, when Ju et al.[23] showed the possibility of exciting graphene surfaceplasmon-polaritons (GSPPs) in the THz spectral rangeby shinning electromagnetic radiation onto a periodicgrid of graphene micro-ribbons. The major advantageof using graphene as the surface for the propagationof these plasmon-polaritons relates essentially with thestrong confinement [24, 25] and the easy tunability [26–28] of the collective excitations. Within a semi-classicaltreatment [29], the dispersion relation of GSPPs depends

mailto:goncalo.catarina@tecnico.ulisboa.pt

2

explicitly on the optical conductivity, wherefore the studyof their spectrum —which, for the tBLG system, is an ab-sent topic in literature— follows as a direct application.

In this work, we start by reproducing, in section II, thecontinuum model developed by Bistritzer and MacDon-ald [5], which is valid for θ . 10◦ and independent of thestructure being commensurate or incommensurate. Insection III, we address the optical conductivity (withinthe linear response theory), comparing our results andmethods with the current literature. Section IV containsoriginal results for the transverse magnetic (TM) modesof GSPPs, which were obtained making use of the pre-vious calculations. Finally, in section V, we present ourmain conclusions and proposals for future work.

II. LOW-ENERGY CONTINUUM MODEL

A. Folded description for single layer graphene

Before moving to the derivation of the continuummodel, we first look into the folded description of thesingle layer graphene (SLG), since it will provide us abetter understanding of that approach.

We start with SLG basics. The geometry (fig. 2) isdefined by the following hexagonal lattice which describesthe positions for the unit cells of this system:

RRRn1,n2 = n1aaa1 + n2aaa2, n1, n2 ∈ Z,

in which the primitive vectors aaa1 and aaa2 are given by

aaa1 =(

1/2,√

3 /2)√

3 d, aaa2 =(−1/2,

√3 /2

)√3 d.

Given this lattice, we define the corresponding kkk-space(fig. 3) by writing the reciprocal lattice as

GGGm1,m2 = m1bbb1 +m2bbb2, m1,m2 ∈ Z,

where the reciprocal primitive vectors bbb1 and bbb2 are givenby

bbb1 =(√

3 /2, 1/2) 4π

3d, bbb2 =

(−√

3 /2, 1/2) 4π

3d.

Within the tight-binding model, the SLG Hamiltonianfor hopping −t (t = 2.97eV [30]) between first nearestneighbors (NN) is given by [31]

H(kkk) =

[0 −tf(kkk)

−tf∗(kkk) 0

], (1)

where

f(kkk) = eikkk.δδδ(1 + e−ikkk.aaa1 + e−ikkk.aaa2),

with δδδ = (0, d) being the vector that links the 2 atoms inthe same unit cell.

a1a2

d =1.4

2 Å

x

yA

B

FIG. 2: SLG geometry. The honeycomb structure can beseen as two interpenetrating hexagonal lattices, A (blue) andB (red). Its experimental structural parameter, the carbon-carbon distance, is d = 1.42Å [8]. The dashed green linemarks a unit cell of this system, which contains 2 atoms. Thecoordinate system is centered at a carbon of sublattice A.

The folded description lays in the observation that wecan opt for larger direct lattices, as long as they stillcapture the system’s periodicity. In the simpler case inwhich we choose unit cells 3p (p ∈ N) times larger thanthe original one, which corresponds to folding the BZ to1/3p of its previous size, we obtain

aaa(p)1 =

{(1/2,√

3 /2)√

3 p+1d if p is even(√3 /2, 1/2

)√3 p+1d if p is odd

,

aaa(p)2 =

{(−1/2,

√3 /2

)√3 p+1d if p is even(

−√

3 /2, 1/2)√

3 p+1d if p is odd,

bbb(p)1 =

{(√3 /2, 1/2

)4π√3 p3d

if p is even(1/2,√

3 /2)

4π√3 p3d

if p is odd,

bbb(p)2 =

{(−√

3 /2, 1/2)

4π√3 p3d

if p is even(−1/2,

√3 /2

)4π√3 p3d

if p is odd.

With this established, we can write the general Hamilto-nian for the folded case as

H(p)kkk =

H

(p−1)kkk 0 0

0 H(p−1)kkk+bbb

(p)1

0

0 0 H(p−1)kkk+bbb

(p)2

,where we note that H(p) has informations of all Hamil-tonians back to the original/unfolded one, H(0). Thepicture of this construction is shown in fig. 4. In short,the interpretation is that we can represent the exact sameband spectrum in a folded BZ description by increasingthe number of bands.

3

b1b2

Γ

M

KK'

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

kx (Å-1)

ky(Å

-1)

FIG. 3: SLG reciprocal space. The blue circles representpoints in the reciprocal lattice; just like the direct lattice,the reciprocal one is also hexagonal, though rotated and witha different lattice parameter. The green primitive unit cellmarks the first Brillouin zone (BZ); some relevant points areplotted in it: Γ = (0, 0), M =

(1, 1/√

3)

π√3 d

, K = ( 4π3√3 d, 0),

K′ = −K.

B. Geometry and moiré pattern

A completely arbitrary arrangement for the tBLG canbe achieved in the following manner: we start with astacking AB, then rotate the second layer by θ (anti-clockwise and about the origin) and finally translate itby τττ . This way, each layer is described by the followinglattice:

RRR(1)n1,n2 = n1aaa1 + n2aaa2,

RRR(2)n1,n2 = Rθ (n1aaa1 + n2aaa2 − δδδ) + τττ

= Rθ

(RRR(1)n1,n2 − δδδ

)+ τττ ,

where Rθ =

[cos(θ) − sin(θ)sin(θ) cos(θ)

]is the rotation matrix. In

fig. 5, we present a particular case of a tBLG geometry,highlighting the moiré pattern arising from its crystallo-graphic alignment. This moiré pattern can be predictedby taking the moiré reciprocal vectors as

bbbm1 = bbb1 − bbbθ1, bbbm2 = bbb2 − bbbθ2,

and determining the corresponding ones in direct spaceby the definition,

aaami .bbbmj = 2πδi,j .

Here, we have introduced the notation vvvθ ≡ Rθvvv, wherevvv is an arbitrary 2D vector.

b1(0)

b2(0)

b1(1)

b2(1)

1

2

22

2

2 2

3

3

3

3

3

3

FIG. 4: Reciprocal space folding picture for SLG. The greendashed line marks the original BZ, while the purple line marksthe BZ for a p = 1 folding. Regions labeled by 1, 2 and 3correspond to the first, second and third BZs for the foldedcase. When folding, the information from regions 2 and 3 iscomprised to the new BZ (region 1).

C. Rotated Dirac Hamiltonian

Expanding the Hamiltonian from eq. (1) around thetwo nonequivalent Dirac points K and K ′ = −K, weobtain, to the first nonvanishing order,

H±K(qqq) = ~vF[

0 ±qx − iqy±qx + iqy 0

]= ±~vF |qqq|

[0 e∓iθqqq

e±iθqqq 0

],

where vF =3dt2~ is the Fermi velocity (~ is the reduced

Planck constant), kkk = ±K + qqq for small qqq in this expan-sion and θqqq = arg(qx + iqy) is the momentum orientationrelative to the x axis.

For a rotated SLG, we get

H±K(qqq, θ) = ±~vF |qqq|[

0 e∓i(θqqq+θ)

e±i(θqqq+θ) 0

], (2)

where qqq is now measured from the rotated Dirac points,±Kθ.

D. Interlayer hopping term

We start by defining the matrix element,

Tα,βkkk,k′k′k′

= 〈ψ(1)kkk,α|H⊥ |ψ(2)

k′k′k′,β〉 ,

which describes a process where an electron with mo-mentum k′k′k′ in layer 2, sublattice β, hops to a momentumstate kkk in layer 1, sublattice α. In the tight-binding ap-

4

layer 1 layer 2

a1

m

a2

m

FIG. 5: tBLG geometry for θ = 5◦ and τττ = 0 (top view).Vectors aaam1 and aaa

m2 mark the basis for the (readily visible)

large-scale hexagonal moiré pattern. The typical interlayerdistance is d⊥ = 3.35Å [10].

proximation, we have

|ψ(1)kkk,α〉 =1√N

∑n1,n2

eikkk.(RRR(1)n1,n2

+δδδ(1)α ) |RRR(1)n1,n2 + δδδ(1)α , α〉 ,

|ψ(2)k′k′k′,β〉 = 1√

N

∑n1,n2

eik′k′k′.

(RRR(2)n1,n2

+δδδ(2)β

)|RRR(2)n1,n2 + δδδ

(2)β , β〉 ,

where N stands for the total number of unit cells and

δδδ(1)α = δδδα, δδδ(2)β = δδδ

θβ ,

with δδδA = 0 and δδδB = δδδ. Invoking the two-center ap-proximation,

〈RRR(1)n1,n2 + δδδ(1)α , α|H⊥ |RRR

(2)n′1,n

′2

+ δδδ(2)β , β〉

= t⊥

(RRR(1)n1,n2 + δδδ

(1)α −RRR

(2)n′1,n

′2− δδδ(2)β

),

we can calculate the following matrix element (which is,for now, completely general):

Tα,βK+qqq1,Kθ+qqqθ2

=∑

k,l,m,n

t⊥(K + qqq1 +GGGk,l)

Ag.u.c.×

× ei[GGGk,l.δδδα−GGGm,n.(δδδβ−δδδ)] δK+qqq1+GGGk,l,Kθ+qqqθ2+GGGθm,n ,

where Ag.u.c. =3√

3 d2

2 is the area of a SLG unit celland t⊥(kkk) is the Fourier transform (FT) of the inter-layer tunneling amplitude, t⊥(rrr). We stress that we havealready set τττ = 000 since the spectrum is known to be τττ -independent for θ 6= 0 [5]. Taking a look at this expres-sion, we notice that the moiré pattern reveals itself in the

Re{ t⟂(kx,ky) } (eV Å2)

0 0.2 0.4 0.6

-4 -2 0 2 4

-4

-2

0

2

4

kx (Å-1)

ky(Å

-1)

(a)

5 10 15 20kx (Å

-1)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Re{ t⟂(kx,0) }

(eV Å2)

(b)

FIG. 6: FT for the interlayer hopping in tBLG (real part).Although not plotted, the imaginary part was shown to benegligible, as expected. In (a), the full 2D FT is shown andan apparently circular symmetry, t⊥(kkk) ≈ t⊥(|kkk|), is observed.In (b), t⊥(kkk) is plotted along the kx axis, with the blackdashed line crossing it at the Dirac point K.

Kronecker delta, δK+qqq1+GGGk,l,Kθ+qqqθ2+GGGθm,n , which dictates

that the momentum difference in the interlayer hoppingmust be a vector resulting from the union of all vectorsGGG and GGGθ.

The continuum low-energy model is obtained by con-sidering wave vectors in both layers relative to their re-spective Dirac points with small deviations compared tothe BZ dimensions: |qqq1|, |qqq2| � |K|. Notice that, bydoing so, a K expansion is implicit at this point. Themodel’s usefulness rests on the numerical results obtainedfor t⊥(kkk) (fig. 6), which indicate that it should fall tozero very rapidly with |kkk| on the reciprocal lattice scale.Keeping the dominant terms from the FT, we get

TK+qqq1,Kθ+qqqθ2= Tqqqb δqqqθ2−qqq1,qqqb

+Tqqqtr δqqqθ2−qqq1,qqqtr+Tqqqtl δqqqθ2−qqq1,qqqtl

,

(3)

with

Tqqqb =t⊥(K)

Ag.u.c.T1, Tqqqtr =

t⊥(K)

Ag.u.c.T2, Tqqqtl =

t⊥(K)

Ag.u.c.T3,

T1 =

[1 11 1

], T2 =

[eiφ 1e−iφ eiφ

], T3 =

[e−iφ 1eiφ e−iφ

],

φ = 2π3 , and

qqqb = K −Kθ, qqqtr = qqqb + bbbm2 , qqqtl = qqqb − bbbm1 .

The parameter t⊥(K) is fixed by taking the limit θ → 0and comparing the resulting interlayer hopping term,3 t⊥(K)Ag.u.c.

δα,Aδβ,Bδqqq1,qqq2 , with the known results for the AB

stacked bilayer graphene [10]; this yields the relation3 t⊥(K)Ag.u.c.

= tAB⊥ , where tAB⊥ = 330meV is the hopping term

between an A site from layer 1 and the NN B site fromlayer 2 in the AB stacked configuration.

The interpretation of this model is more elegant whenwe move to the reference frame where layer 1 is rotated by−θ/2 and layer 2 by θ/2 (it suffices to rotate our previouscoordinate system by θ/2). The geometrical picture forthis interlayer hopping is explained in fig. 7.

5

qb

qtr

qtl

θ/2

q1

q2θ

(a)

qb

qtrqtl

b1m

b2m

(b)

FIG. 7: Momentum-space geometrical picture for the inter-layer hopping on a tBLG. (a) The green dashed line marks thefirst BZ for an unrotated SLG; the red (blue) circles mark thethree equivalent Dirac points K for layer 1 (2). Crystal mo-mentum conservation is attained when qqqθ2−qqq1 = qqqb, qqqtr, qqqtl; inthis reference frame, the three momentum transfers have mod-ulus |qqqj | = 2|KKK| sin(θ/2) and directions (0,−1) for j = b (bot-tom), (

√3 /2, 1/2) for j = tr (top right), and (−

√3 /2, 1/2)

for j = tl (top left). (b) The three equivalent Dirac points inthe first BZ result in three distinct hopping processes (matrixelements) in reciprocal space; when we capture all “orders”of hopping (possible hopping processes after previous ones),we obtain this kkk-space honeycomb structure, which capturesthe periodicity of the moiré pattern. The purple dashed linemarks a moiré unit cell in reciprocal space.

E. Hamiltonian matrix construction

In order to determine the electronic spectrum, we makeuse of the Schrödinger equation, which for our system

reads

(H1 +H2 +H⊥) |ψkkk〉 = E |ψkkk〉 ,

where Hi is the Hamiltonian for layer i (i = 1, 2), E isthe energy and

|ψkkk〉 =∑α,i

c(i)α (kkk) |ψ(i)kkk,α〉

is the total wave function, with c(i)α (kkk) standing for a

complex constant of unit modulus. The interlayer hop-ping matrix elements are modelled by eq. (3) whereas,for the intralayer ones, we use each layer’s (rotated)Dirac Hamiltonian, eq. (2). We thus realize that wecan never get a closed system of equations by apply-

ing bras, 〈ψ(i)kkk,α| ≡ 〈kkk, i, α|: if we start, for example,with the bra 〈K + qqq, 1| (dropping the index of sublat-tice), we will have, aside from the diagonal term, recip-rocal space hopping terms (matrix elements) with states|Kθ + qqq + qqqb, 2〉, |Kθ + qqq + qqqtr, 2〉 and |Kθ + qqq + qqqtl, 2〉(we will call them the first NN in reciprocal space, inlack of better terminology); in turn, each one of thesewill have hopping terms with the former one and withtwo new states (second NN), and so on (recall the pictureshown in fig. 7b). To clarify this matrix construction, wepresent the simpler example in table I, in which we startwith the bra 〈K + qqq, 1| and consider just first NN.

|K + qqq, 1〉 |Kθ + qqq + qqqb, 2〉 |Kθ + qqq + qqqtr, 2〉 |Kθ + qqq + qqqtl, 2〉HK(qqq,−θ/2) Tqqqb Tqqqtr Tqqqtl 〈K + qqq, 1|

T †qqqb HK(qqq + qqqb, θ/2) 0 0 〈Kθ + qqq + qqqb, 2|

T †qqqtr 0 HK(qqq + qqqtr, θ/2) 0 〈Kθ + qqq + qqqtr, 2|

T †qqqtl 0 0 HK(qqq + qqqtl, θ/2) 〈Kθ + qqq + qqqtl, 2|

TABLE I: tBLG K-Hamiltonian matrix elements for NN = 1.

In this matrix construction, we point out the simi-larities with what was shown for the folded SLG (sec-tion II A). In fact, if we eliminate the interlayer hoppingterms (i.e., the non-diagonal matrix elements), we arebasically using a folded description that explicitly cap-tures the moiré periodicity to some extent (dependingon the truncation). In real space, the interpretation isthat we are using an enlarged unit cell with the moiréperiodicity. Obviously, this is not as elegant as the caseof the 1/3p SLG folding which could always be writtenas a finite dimension matrix that captured the period-

icity for all momenta. The major difference is that weare now using a Dirac approximation. For this reason,we should also be aware that we will obtain high-energybands lacking physical meaning; however, this should notconstitute a major problem since we are not interestedin them anyway. When we add the hopping terms, wesee that, the more we add, the less important we expectthem to be. Nevertheless, we cannot always stick to firstNN hopping terms only, because, depending on the angle,the low-energy bands may still depend on high-order NNhopping processes. This is the balance we have to test

6

θ

Γ(1)= Γ(2)

K(1)

K(2)K'

(1)

K'(2)

(a)

Km = ,K(1)

K'(2)

K'm = ,K'(1)

K(2)

Γm

Mm

(b)

FIG. 8: K and K′ bands on a tBLG. (a) Blue/red hexagonsdescribe the BZs for layers 1/2. Dashed purple/greenhexagons represent moiré unit cells in reciprocal space forK/K′ expansions. (b) Moiré BZ with relevant points plottedin it.

numerically in order to truncate our (in principle infinite)matrix. With all the approximations, this model is ex-pected to be very accurate up to energies of ∼ 1eV, whichcan still capture the first low-energy bands for θ . 10◦.

F. Electronic spectrum

Before moving to the band spectrum determination,it is crucial to recall that our Hamiltonian was obtained

within a low-energy expansion around K. Yet, noth-ing prevents us to choose the other nonequivalent Diracpoint, K ′, for which the deductions are completely anal-ogous. Therefore, in order to describe the complete elec-tronic properties of this material, we should always con-sider both contributions, usually called K and K ′ bands.The way to represent both K and K ′ bands in the sameBZ is sketched in fig. 8. We notice that, in a K expansion,the wave vector qqq is measured from K(1) (kkk = K(1) + qqq)while, in a K ′ expansion, we measure it from K ′(1).Therefore, in order to match both moiré unit cells in re-ciprocal space (purple and green), we identify the pointsK(1) and K ′(2) as the same point in the moiré BZ, suchthat the path Km → K ′m → Mm → Km becomes equiv-alent. By doing so, we are making a correspondenceHKtBLG(qqq) ↔ HK

′tBLG(qqq + qqqb) in the Hamiltonians obtained

within K and K ′ expansions.

Results for electronic spectrum, density of states(DOS) and carrier density are plotted in figs. 9, 10. Look-ing at the spectrum, we observe an apparent symmetryfor positive and negative bands. We also see a renormal-ization of the Fermi velocity, which is explored in moredetail in refs. [5, 16]. Addressing fig. 10, we concludethat, by varying the twist angle, van Hove singularitiescan be brought to accessible energies, which is one of themain features of the tBLG [32]. For the carrier density,we highlight that we start to lose the “signature behav-ior” of the decoupled bilayer graphene (BLG) when wereach small angles.

FIG. 9: Electronic spectrum and DOS for tBLG with θ = 5◦. Solid and dashed lines in the spectrum are for K and K′

expansions, respectively; the color code clarifies the situation in which both bands are superimposed.

7

-1000 -500 0 500 10000.00

0.05

0.10

0.15

0.20

0.25

E (meV)

DO

Sp

erg

.u.c

.(e

V-

1)

decoupled BLG

θ = 9°

θ = 5°

θ = 1.8°

(a) DOS.

-400 -200 200 400μ - μ0 (meV)

-2

-1

1

2

n (1013 cm-2)θ = 9°

θ = 5°

θ = 1.8°

decoupled BLG

(b) Carrier density.

FIG. 10: Profiles of DOS and carrier density for different angles of a tBLG. In (a), the DOS is normalized to the SLG unit cell,since the size of the moiré unit cells varies with the angle. In (b), the carrier density, n, is shown as a function of the Fermilevel, µ, with µ0 defining the reference Fermi level corresponding to the half-filling situation.

III. OPTICAL CONDUCTIVITY

A. Linear response theory

Within the linear response theory, the (dynamical) to-tal conductivity tensor, σa1a2(ω) (ai = x, y), can be split-ted into two terms,

σa1a2(ω) = σDa1a2(ω) + σ

rega1a2(ω),

where σD(ω) is the Drude conductivity —an intrabandcontribution (in which momentum is not conserved) thatreflects the response of the electrons to a static appliedelectric field— and σreg(ω) is the regular conductivity—an interband contribution which corresponds to elec-tronic band transitions (within the same kkk) with en-ergy ~ω, induced by an applied harmonic electric field,EEE ∼ e−iωt (t is the time, ω is the angular frequency).

For the Drude term, our derivation yielded

σDa1a2(ω) =i

π

Da1a2~ω + iΓ

,

where Γ is an empirical broadening parameter (usuallyinterpreted as a scatering rate) that accounts for dis-order effects (impurities, electron-electron interactions,substrate, etc...) and Da1a2 is the Drude weight tensor,given by

Da1a2 =8πσ0

NAu.c.

∑kkk∈BZ,λ1

[〈λ1, kkk| j2a1a2 |λ1, kkk〉nF

(�λ1 (kkk)

)+

+∑

λ2 6=λ1

〈λ1, kkk| ja1 |λ2, kkk〉 〈λ2, kkk| ja2 |λ1, kkk〉×

×nF(�λ1 (kkk)

)− nF

(�λ2 (kkk)

)�λ1 (kkk)− �λ2 (kkk)

]. (4)

In the last expression, σ0 = e2/(4~) is the graphene uni-

versal conductivity (e being the elementary charge), Au.c.is the area of a unit cell and nF is the Fermi-Dirac func-tion for some Fermi level µ and temperature T . More-

over, we have jai ≡∂H0(kkk)∂kai

and j2a1a2 ≡∂2H0(kkk)∂ka1∂ka2

, where

H0 is a (general) spin-independent tight-binding Hamil-tonian matrix for a periodic system, with eigenvalues�λ(kkk) and eigenvectors |λ,kkk〉 for momentum kkk and bandλ. We also stress that eq. (4) already takes into accountthe spin degeneracy. As for the valley degeneracy (forthe tBLG case), we clarify that it is implicit in the sumover kkk ∈ BZ, which should be done into two sums overqqq in moiré BZs centered around K and K ′.

For the regular contribution, we obtained

σrega1a2 (ω) =−8σ0iNAu.c.

∑kkk∈BZ,

λ1,λ2 6=λ1

〈λ1, kkk| ja1 |λ2, kkk〉 〈λ2, kkk| ja2 |λ1, kkk〉×

×nF(�λ1 (kkk)

)− nF

(�λ2 (kkk)

)[�λ1 (kkk)− �λ2 (kkk)

] [�λ1 (kkk)− �λ2 (kkk) + ~ω + iΓ

] . (5)

B. Alternative methods

Eqs. (4) and (5) must work when we have the completeHamiltonian defined in the full BZ (this was actually ver-ified by computing the results for the SLG). However, aswe shall discuss, for effective Hamiltonians (which is thecase we are interested in), they might not be the mostappropriate. In this section, we thus provide alternativemethods to compute these quantities.

When determining the Drude weight, we expect thatall the dependency comes from the electrons near theFermi level, which are the ones that can flow in responseto the static applied electric field. Yet, this is not ex-plicit in eq. (4), which indicates that there should be anunderlying annulment of the other terms. In fact, withsome manipulation of this equation, we can arrive to

Da1a2 =−8πσ0NAu.c.

∑kkk∈BZ,λ

∂�λ(kkk)

∂ka1

∂�λ(kkk)

∂ka2

∂nF (�)

∂�(�λ(kkk)) , (6)

which is clearly a better method, since the derivative ofthe Fermi-Dirac is very sharp around � ∼ µ.

Regarding the regular conductivity (eq. (5)), we ob-serve that the real part is strongly constrained to eigen-

8

T = 300K

0 200 400 600 800 1000μ - μ0 (meV)0

2

4

6

8

Dxx

σ0(eV)

decoupled BLG

θ = 9°

θ = 5°

θ = 1.8°

(a)

θ = 1.8°

-2 -1 0 1 2n (1013 cm-2)

0.5

1.0

1.5

Dxx

σ0(eV)

T = 50K

T = 100K

T = 150K

T = 200K

T = 250K

T = 300K

(b)

FIG. 11: Drude weight results for tBLG (2nd method): (a) as a function of the Fermi level, for different angles; (b) as a functionof the carrier density, for different temperatures. The outcomes were isotropic, i.e., Dxx = Dyy, Dxy = Dyx = 0. In (b), theblack dashed line is for decoupled BLG at T = 300K. The results for decoupled BLG (tBLG continuum model with t⊥ = 0)were verified to match the results for the SLG, multiplied by 2 (2×SLG).

states within ~ω of the Fermi level. Hence, this compu-tation should not be problematic and we will keep thismethod. For the imaginary part, we see that, even forsmall ω, we do not have an argument to avoid a sum-mation over all the bands. Therefore, following the workdone in ref. [20], we may think of using the Kramers-Kronig (KK) relations [33] to compute the imaginary partusing the results for the real part,

Im {σreg(ω)} = −2ωπP

∫ +∞0

dsRe {σreg(s)}s2 − ω2 . (7)

Looking at this expression, there is yet no clear advan-tage in using this strategy, since the integral extends toinfinity. Moreover, this integral is ill defined, since athigh frequencies our continuum model for the tBLG isexpected to yield a constant Re {σreg(ω)} = 2σ0. Wecan thus perform a regularization of eq. (7) by invokingthe following property (which we verified numerically):

P

∫ +∞0

ds1

s2 − ω2= 0.

The final regularized definition then reads

Im {σreg(ω)} = −2ωπP

∫ +∞0

dsRe {σreg(s)} − 2σ0

s2 − ω2 , (8)

which we can now evaluate by introducing a finite cutoffΛ for which Re {σreg(Λ)} ' 2σ0. We also remark that,although our model yields a constant for high frequencies,it is not problematic in the range of frequencies in whichwe are interested in.

C. Results

Some of the results obtained for the Drude weight intBLG systems are summarized in fig. 11. As antecipated,only the 2nd method (eq. (6)) worked well. We observesymmetric outcomes for electron or hole dopping; thisreflects the apparent symmetry in positive and negative

bands discussed in section II F. By looking at fig. 11a,along with fig. 10a, we conclude that the Drude weightcurve changes drastically (compared with SLG or decou-pled BLG) when we cross the van Hove singularities; thistendency coincides with what was found in ref. [20]. Theeffect of increasing the temperature is the smootheningof this behavior (fig. 11b).

In figs. 12, 13, we show representative results that al-low us to analyze the regular conductivity in tBLG sys-tems. All conductivity results obtained were isotropic.Looking at fig. 12a, we first notice the expected depen-dency on both the Fermi level and temperature: transi-tions with ~ω . 2µ are Pauli blocked and the decreas-ing of T accentuates this behavior. In addition, we ob-serve a low-energy peak (marked with a green arrow),which we interpret as the dominant transitions shown infig. 12b. Notice that there are other transitions (red andorange arrows) which we would expect to be dominant,since they connect different van Hove singularies; how-ever, these ones are optically inactive, in agreement withwhat was found in refs. [19, 21]. This optical selectionrule occurs due to a symmetry in the effective Hamilto-nian which makes the matrix elements from eq. (5) nullfor bands with symmetric energies and in the Mm pointsonly [21]. From fig. 13a, we highlight the fact that theresults obtained for the decoupled tBLG —tBLG witht⊥ = 0— match perfectly the results for 2×SLG (resultsfor SLG, multiplied by 2). Although this was triviallyexpected, it was only achieved when we used the 2nd

method for computing the imaginary part of the regularconductivity (eq. (8)); therefore, this served as a bench-mark test for the validity of this method. Moreover, weremark that we now have a region with a big deep onIm {σreg(ω)} occuring at lower frequencies, which will bean important feature in section IV. Regarding fig. 13b,we emphasize that, for small angles, we start to lose the“signature” behavior of the curves because of the pres-ence of multiple low-energy van Hove singularities.

9

0 500 1000 1500 2000ℏω (meV)

1

2

3

4

5

Re {σxxreg}

σ0

μ = 0meV

μ = 100meV

μ = 200meV

(a)

Km K'm Γm Mm Km-2000

-1000

0

1000

2000

E(m

eV)

(b)

FIG. 12: tBLG with θ = 9◦: (a) real part of the regular conductivity; (b) spectrum. In (a), the dotted blue line corresponds to T = 100K, thedashed blue line to T = 200K and the solid lines to T = 300K; the black dashed line is for decoupled tBLG (or 2×SLG) at T = 300K and µ = 0.The broadening parameter Γ was set as Γ = 16meV in agreement with ref. [23].

θ = 5°, T = 300K, μ = 0meV

200 400 600 800 1000 1200 1400ℏω (meV)

-4

-2

0

2

4

σxxreg/σ0

Real part Imaginary part

(a)

T = 300K, μ = 0meV

0 500 1000 1500 2000ℏω (meV)0

1

2

3

4

5

Re {σxxreg}

σ0 decoupled BLG

θ = 9°

θ = 5°

θ = 1.8°

(b)

FIG. 13: Regular conductivity results for tBLG. In (a) and (b), the dashed lines are for decoupled BLG or 2×SLG.

IV. SPECTRUM OF GRAPHENE SURFACEPLASMON-POLARITONS — TRANSVERSE

MAGNETIC MODES

We consider a system consisting of a single graphenesheet cladded between two semi-infinite dielectric me-dia, characterized by the real dielectric constants (rel-ative permittivities) εr1 and ε

r2, as depicted in fig. 14. We

stress that, although the tBLG is not truly a 2D surface,its thickness is still negligible and we can view it as amonolayer for these purposes [7].

Assuming, for each medium j = 1, 2, a solution ofMaxwell’s equations in the form of a TM wave, confinedto the neighborhood of the graphene sheet (with dampingparameter κj), propagating along the x̂̂x̂x-direction, and ofthe typical harmonic form, we obtain the following dis-persion relation [29]:

εr1κ1(q, ω)

+εr2

κ2(q, ω)+ i

σ(ω)

ωε0= 0, (9)

which results from imposing the continuity of the tan-gential component of the electric field and the disconti-nuity of the tangential component of the magnetic fieldacross the interface. In this equation, ε0 is the vacuumpermittivity, σ(ω) ≡ σxx = σyy (valid for unstrained

FIG. 14: Illustration of a single graphene sheet sandwiched be-tween two semi-infinite insulators with relative permittivities �j ≡εrj (in our notation). Medium 1 occupies the z < 0 half-space andmedium 2 the z > 0; the graphene sheet is located at the z = 0plane. Source: ref. [29].

graphene) is the total conductivity, q ≡ q1 = q2 is themomentum of the electromagnetic wave propagating ineach medium (which must be conserved due to transla-tional invariance) and

κj(q, ω) =

√q2 −

ω2εrjc2

,

where c is the speed of light. Eq. 9 describes the dis-persion relation, ω(q), of graphene TM surface plasmon-polaritons. Notice that this is an implicit equation, soit needs to be solved numerically. Nonetheless, we easilysee that it is only solvable when Im {σ(ω)} > 0.

10

ε1r = 1, ε2

r = 1

0.5 1.0 1.5 2.0Re{q} (μm-1)

2

4

6

8

10

12f (THz) μ = 0meV

μ = 100meV

μ = 200meV

(a)

ε1r = 1, ε2

r = 1, Re{q} = 0.5μm-1

0.2 0.4 0.6 0.8 1.0n (1013 cm-2)

1

2

3

4

5

6

7f (THz) Im{q} (μm

-1)

0.3

0.4

0.5

0.6

0.7

0.8

(b)

FIG. 15: Spectrum of TM GSPPs in tBLG with θ = 9◦: dependency on the Fermi level/carrier density. The black dots in (a) mark thefixed parameter in (b). The frequency, f , is given by f = ω/(2π).

ε1r = 1, ε2

r = 1

0.5 1.0 1.5 2.0Re{q} (μm-1)

2

4

6

8

f (THz) μ = 0meV

μ = 100meV

μ = 200meV

(a)

ε1r = 1, ε2

r = 1, Re{q} = 0.5μm-1

0.5 1.0 1.5n (1013 cm-2)

1

2

3

4

5f (THz) Im{q} (μm

-1)

0.5

0.6

0.7

0.8

0.9

1.0

(b)

FIG. 16: Spectrum of TM GSPPs in tBLG with θ = 1.8◦: dependency on the Fermi level/carrier density. The black dots in (a) mark thefixed parameter in (b). The frequency, f , is given by f = ω/(2π).

In figs. 15, 16, we present an analysis for two differ-ent twist angles. For θ = 9◦, we see that the “signa-tures” of the curves do not differ a lot from those ofthe SLG [29] (which we also have computed as bench-mark). This happens due to two main reasons: 1) whitinthis low-frequency range —from the THz up to the mid-infrared—, we do not capture any hybridization effect inthe conductivity (see figs. 11a, 12a); 2) within these typ-ical doping levels [34], the curves n(µ) are also very close(see fig. 10b). For θ = 1.8◦, the exact opposite occurs andit leads to the plot of fig. 16b, which we highlight sinceit is totally distinct from all the results obtained before.As an immediate application, we can think of using theseresults as an alternative method for determining θ.

V. CONCLUSIONS AND FUTURE WORK

In this work, besides the understanding of the most re-cent models for the tBLG, we introduced a new methodand improved to compute the Drude weight —eq. (6).In addition, original results for the plasmonic responseof this system were achieved (see figs. 15, 16, with em-phasis for fig. 16b). Nevertheless, a more extensive study

on the behavior of these curves with the variation of θremains to be done, in order to investigate more promis-ing applications. We leave for future work the study ofthe transverse electric (TE) modes, which are the onesthat, in contrast to the TM modes, exist only whenIm {σ(ω)} < 0 [29].

Finally, since this is an extended abstract of a master’sdissertation, it is worth mentioning that, here, due to thelimitations in the number of pages, we did not tackleone of the chapters of the thesis, which concerns theeffects of electron-electron interactions (an absent topicin literature for the tBLG), in particular the self-energycorrection for the band renormalization due to the longrange (screened) Coulomb repulsion. Although we werenot able to compute the self-energy corrections yet, weprovided a discussion on the difficulties encountered andstrategies to solve them in future works.

Acknowledgements

The author thanks João Viana for the helpful discus-sions and acknowlegdes the remarkable supervision pro-vided by profs. Eduardo Castro and Nuno Peres.

11

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IntroductionLow-energy continuum modelFolded description for single layer grapheneGeometry and moiré patternRotated Dirac HamiltonianInterlayer hopping termHamiltonian matrix constructionElectronic spectrum

Optical conductivityLinear response theoryAlternative methodsResults

Spectrum of graphene surface plasmon-polaritons — transverse magnetic modesConclusions and future workAcknowledgementsReferences