Electronic Properties of Single-layered Transition Metal Dichalcogenides

Joao Eduardo BrazCeFEMA, Department of Physics, Instituto Superior Tecnico,

Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal(Dated: November 2015)

Transition metal dichalcogenides (TMDCs) have stood alongside graphene since the discovery of2D materials, but stayed outside the limelight until recently. Though similar to graphene, thesematerials possess a band gap and, due to the high atomic number of transition metals, displaysizable effects of spin-orbit coupling. Starting from an effective tight-binding Hamiltonian used toderive a low-energy description of the band structure, we attempt to gain insight into the effectsof electronic interactions to the single-electron band structure and into the existence of a possiblebroken symmetry phase. Regarding the former, computing the self-energy correction to the barepropagator in the Hartree-Fock approximation yielded estimates for the renormalization of the gapin both the undoped and hole doped system. As for the latter, a Hubbard model was used todetermine the possible existence of a spin and valley polarized phase, which is predicted to displayan anomalous Hall response.

I. INTRODUCTION

Two dimensional (2D) materials have faced a chal-lenging existence, in theory as well as in laboratories. Ithas been almost 80 years since both Landau and Peierlsseparately argued that 2D crystalline systems are ther-modynamically unstable and could not exist [1, 2]. Assuch, 2D materials were presumed not to exist with-out a 3D base, until the 2004 experimental discovery ofgraphene along with other free-standing 2D materials[3, 4].

Although graphene attracted the spotlight amongthe brand new class of 2D materials, certainly owingboth to its vast new landscape of physics as well asto its potential for device applications [5], it has beenin exceedingly good company since as early as 2005[4, 6], namely alongside BN, Bi2Sr2CaCu2Ox, NbSe2

and MoS2. The last two belong in the family of tran-sition metal dichalcogenides (TMDCs), which have in-creasingly attracted interest in the past few years .

Due to its truly outstanding properties, graphene hasattracted widespread interest from physicists, chemistsand engineers alike since its very discovery. However,it shows limitations concerning some electronic applica-tions, namely due to the absence of a band gap, andthe existence and robustness of different phases such asintrinsic magnetism, superconductivity and topologicalphases.

Faced with the shortcomings of single-layer grapheneand motivated by the prospect of intriguing new physics,interest has begun to shift to its 2D companions,namely to the family of transition-metal dichalcogenides(TMDCs) [6–14]. This work shall focus primarily ongroup-VI transition metals , namely molybdenum (Mo)and tungsten (W), compounded with two chalcogenssulphur (S), selenium (Se). From here, TMDCs shall

Figure 1: (a) Unit cell of bulk 2H-MoS2. (b) Top viewof the MoS2. Ri are the vectors connecting the Mo

atoms. (c) Schematic drawing of the band structure atthe band edges located at the K points. [11]

be referred to generically as MX2, where M representsthe transition metal and X the chalcogen.

A monolayer can be seen as a triangular lattice ofM atoms. Each M atom becomes coordinated with sixX atoms. Note that this structure is not as strictlytwo-dimensional as, say, graphene: the X atoms areat an angle with the M -atom plane [7, 11, 14]. Thelattice is hexagonal with sites M and X. Figure 1.bshows a top-down view of the monolayer crystal struc-ture. Single-layered TMDCs have a vast landscape ofphysics and potential applications to explore. In par-ticular, the many-body physics of these systems, andits implications to their electronic structure and non-interactive ground state, are yet to be known and may

render desirable properties to these materials, from apoint of view of both fundamental physics and appli-cations. As such, the basic consequences of electron-electron interactions within a low-energy theory are theprimary focus of this work.

II. BAND STRUCTURE

A. An effective three-band tight-bindingHamiltonian

We follow the method employed by Liu et al. [15] forobtaining a minimal symmetry-based three-band tight-binding (TB) model for monolayers of group-VI TMDCs(MX2) with trigonal prismatic coordination. The basisbehind this approach is twofold:

On one hand, previous theoretical studies and calcu-lations revealed that orbital contributions to the bandedges stem mostly from M atoms’ d-orbitals, namely,d3z2−r2 ≡ dz2 , dxy and dx2−y2 , with negligible contribu-tions from X atoms’ p-orbitals.

On the other hand, monolayers of MX2 with trigo-nal prismatic coordination have D3h point-group sym-metry, and the three sets of M atom d-orbitals dz2,dxy, dx2−y2 and dxz, dyz constitute basis of irre-ducible representations of this symmetry group. Thismeans that an orbital will only mix with orbitals withinits set under any symmetry operation of the group. Animportant symmetry of this group is the reflection bythe x − y plane (denoted σh), which implies that hy-bridization can only take place between orbitals of thefirst two sets.

The computation of the TB Hamiltonian matrix willbe based on the basic relation for the single-electronHamiltonian,

g H g† = H , (1)

i.e. that the Hamiltonian, essentially the free-particle operator plus the potential from the lat-tice, is invariant under any symmetry operation g ∈E, C3, C

23 , σv, σ

′

v, σ′′

v , where E is the identity, C3 isrotation by 2π/3 around the z axis and σv is the reflec-tion through the plane defined by the z axis and theangular bisector of R1 and R6. The matrix elements ofH yield the hopping integrals between sites, which read

Eijµν(Rl) =⟨φiµ(r)

∣∣ H ∣∣φjν(r−Rl)⟩, (2)

with the basis ⟨x∣∣φ1

1(r)⟩

= dz2(x− r) ,⟨x∣∣φ2

1(r)⟩

= dxy(x− r) ,⟨x∣∣φ2

2(r)⟩

= dx2−y2(x− r) .

The symmetry operations, when acted upon an orbital,produce a linear combination of the elements of the ba-sis set to which the operated orbital belongs. This resultallows us to derive a generating equation which will al-low us to arrive at the hopping matrix for any latticevector. Consider a lattice vector gRl related to Rl bya symmetry operation of the subset of the lattice. Thematrix elements of its hopping matrix are given by

Eij(gRl) = Di(g)Eij(Rl)Dj(g)†. (3)

Taking the lattice vector R1 as a reference and relatingwith R4, we find it to be

E(R1) =

t0 t1 t2−t1 t11 t12

t2 −t12 t22

,

and then, employing Eq. (3) to generate the hoppingmatrices of the six nearest neighbour lattice vectors, wearrive at the tight-binding Hamiltonian, which, in recip-rocal space, reads

Htb(k) =

h0 h1 h2

h∗1 h11 h12

h∗2 h∗12 h22

, (α, β) =

(1

2kxa,

√3

2kya

),

h0 = ε1

+ 2t0(cos 2α+ 2 cosα cosβ) ,

h11 = ε2 + 2t11 cos 2α ,

+ (t11 + 3t22) cosα cosβ

h22 = ε2 + 2t22 cos 2α

+ (3t11 + t22) cosα cosβ ,

h1 = −2√

3t2 sinα sinβ

+ 2it1(sin 2α+ sinα cosβ) ,

h2 = 2t2(cos 2α− cosα cosβ)

+ 2√

3it1 cosα sinβ ,

h12 =√

3(t22 − t11) sinα sinβ

+ 4it12 sinα(cosα− cosβ) .

Numerical values for orbital energies and hopping inte-grals were taken from Ref. [15].

Additionally, intrinsic spin-orbit coupling induces asizable spin-split at the valence band edge. The spin-orbit coupling is included in the tight-binding Hamilto-nian as

Hsoc(k) = 1⊗Htb(k) + ∆E

=

(Htb(k) + λ

2L0 00 Htb(k)− λ

2L0

)

L0 =

0 0 00 0 2i0 −2i 0

.

,

2

B. Continuum limit and Berry curvature

The continuum limit can be obtained simply by firstperforming transforming coordinates as k = τK+q andexpanding all trigonometric functions up to 2nd orderin q1 and q2, making use of the trigonometric identitiesfor the sum of two variables in the argument. The τKvector, with τ = ±, corresponds to the K and K ′ pointsof the Brillouin zone where band-gap is located, as de-picted in Fig. 1, with K = 4π/(3a). The two inequiv-alent valleys, connected by time-reversal symmetry, arelocated in the vicinity of these points and are the stageof the low-energy physics in this system.

We are left with the problem of diagonalizing the ex-panded TB Hamiltonian followed by the projection ontothe appropriate subspace of the eigenbasis, since our in-terest is to arrive at a spinor description (a Dirac Hamil-tonian) of charge carriers on the valence and conductionbands. We can employ a systematic method which, es-sentially, consists of treating the higher order terms inthe expanded TB Hamiltonian as a perturbation.

To first order, we arrive at the Dirac Hamiltonian forlow-energy electrons

HD(k) = at(τ3 σ1 k1 + σ2 k2) + ∆σ3 , (4)

with a ∼ 3.2 ÷ 3.3 A, the lattice constant, t ∼ 1.2 ÷1.6 eV the valence-conduction band effective hopping,∆ ∼ 0.7 ÷ 0.9 eV the half-gap, where σi, i = 1, 2, 3, arePauli matrices acting on the space of band indices, andτ3 is a Pauli matrix acting on the space of valley indices.This Hamiltonian yields a spectrum

E(1)λ = λEk = λ

√∆2 + a2t2k2, λ =

+ , if c

− , if v.

(5)In turn, the 2nd order matrix reads

H(2)eff (q) =

(εc + κca

2q2 t aqτ+ + u a2q2τ−

t aqτ− + u a2q2τ+ εv + κva

2q2

),

(6)with κc ∼ −(0.22÷0.30) eV, κv ∼ 0.25÷0.30 eV and u ∼−0.037÷ 0.009 eV. Furthermore, the 1st order effectivespinful Hamiltonian reads

HD(k) = at(τ3 σ1 k1 + σ2 k2) + ∆σ3 + λ τ3 (1− σ3)s3 ,

with λ ∼ 0.07 ÷ 0.23 eV, where s3 is the adimensionalsz-spin-component operator. From the spectrum of thisHamiltonian we can infer that the inclusion of the spinindex amounts to affine transformations of the half-gapand of the Fermi energy,

∆ 7−→ ∆ατ = ∆− ατλ/2 ,εF 7−→ εF + ατλ/2 .

(7)

Table I: Numerical values of the effective masses toboth 1st and 2nd order for the conduction band,

degenerate valence band and spin-split valence band.(Units: me)

M X2 MoS2 WS2 MoSe2 WSe2order 1st 2nd 1st 2nd 1st 2nd 1st 2nd

mc 0.3380.401

0.2690.291

0.3590.450

0.2800.312

mv 0.428 0.316 0.477 0.343m>

v 0.323 0.405 0.237 0.273 0.336 0.438 0.238 0.283m<

v 0.353 0.453 0.300 0.360 0.382 0.518 0.321 0.407

Clearly, the valence band becomes spin-split at its edge,and the sign ατ implies a spin-valley coupling. In thelow-energy limit, this lifting of spin degeneracy trans-lates into a pair of high-lying and low-lying cones ateach valley, a distinction that will be employed furtheron. From the spectra of the 1st and 2nd order spinfulHamiltonians, we can obtain estimates for the effectivemasses of charge carriers:

E(n)σ,ν = εσ,ν + σ

~2q2

2m(n)σ,ν

+O(aq)3

⇒

mνσme

=(

~2

2me a2

)2∆ν

t2 , n = 1

mνσme

=(

~2

2me a2

)2∆ν

t2+σκσ 2∆ν , n = 2,

where ν = ατ . Numerical values are shown in Ta-ble I. To both 1st and 2nd orders, the eigenstates canbe generically written as

|k, σ, τ, α〉 = e−iσϕτk

√1

2

(1 +

∆ατk

Eα,τk

)|ψσ(τK), α〉

+ σ

√1

2

(1−

∆ατk

Eα,τk

)|ψσ(τK), α〉 ,

(8)

where, in 2nd order, we have

∆ατk = ∆ατ − κ−a2k2 ,

Eα,τk =√

(∆ατk )2 + (t2 + u2a2k2)a2k2 + 2τ tua3k3 cos 3θ ,

with κ− = (κv − κc)/2 and with σ band, τ valley and αspin indices, respectively.

The concept of Berry phase has found a unifyingand fundamental role in the understanding of condensedmatter systems, besides all other branches of quantumphysics, having proven that phenomena pertaining tothe response of electrons to many qualitatively diverseperturbations is, in fact, a consequence of the geometryof Bloch bands [16]. The Berry curvature, a more fun-damental quantity than the Berry phase, can be seen asa curvature in reciprocal space.

3

We can rewrite the eigenvectors of Eq. (8) as

|k, σ, τ〉 = cosθk2e−iσϕ

τk |ψσ(τK)〉+ σ sin

θk2

∣∣ψσ(τK)⟩.

Considering only the valence band, the Berry curvaturereads

Ωvµν(k) = i 〈∂νv(k) | ∂µv(k)〉 − i 〈∂µv(k) | ∂νv(k)〉

=1

2εµν sin θk ∂αθk ∂βϕk ε

αβ , (9)

which, for the valence band of spinful TMDCs, yields

Ωστµν(k) = εµντ

2

a2t2

(∆στ )2

[1 +

a2t2k2

(∆στ )2

]−3/2

. (10)

The general formula for the transverse conduction

σvµν =e2

~

∫BZ

d2k

(2π)2nF (εnk − µ) Ωvµν , (11)

reduces to an integral of the curvature in the undopedsystem. Plugging in the general form of the curvature,Eq. (9),and integrating over the entire Brillouin zoneyields

σvµν = εµνστH = εµντ3

e2

2h.

So, although single Dirac cones have a topological in-variant, the material is topologically trivial, since bothvalleys will contribute the same amount with oppositesign. However, it is still possible to selectively excitevalley-polarized charge carriers [15, 17].

III. PERTURBATION THEORY ANDSELF-ENERGY CORRECTION TO THE

ELECTRONIC STRUCTURE

A. 1st order self-energy in the continuum limit

Considering a long-ranged Coulomb interaction, dueto the expected feebleness of screening in a gapped 2Dmaterial, the action of the imaginary time path integralis written as

S[ψ, ψ] =∑k

ψk,α (−i~ωn + ξk,α)ψk,α

+1

2β~∑k,k′,q

ψk−q,µ ψk′+q,λ Vµµ′νν′

k,k′;q ψk′,λ′ ψk,µ′ ,

(12)

where k ≡ (iωn,k) and q ≡ (iνn,q) stand for fermionicand bosonic imaginary-time 4-momenta, respectively.

kFff

q

ak+qff

kF =

[G0 Σ(1) G0

](k)

Figure 2: Feynman diagram contributing the 1st orderexchange self-energy operator.

The self-energy in the Hartree-Fock approximation iseasy to obtain from a diagrammatic interpretation ofthe action. Since the Hartree contribution cancels outwith contributions from the background of positive ions,we are left with the exchange self-energy, reading

Σαα′(k) = −1

~∑q,λ

Vαλλα′

k+q,k;q nF (ελk+q − µ) ,

which can be obtained from the diagram of Figure 2,followed by a Matsubara frequency summation over theinternal propagator. The matrix elements of the inter-action can be rewritten as

Vαλλα′

k+q,k;q = v(q)∑β,β′

gαβ (k)gλβ(k + q)gλβ′(k + q)gα′

β′ (k) ,

where the g functions are obtained from diagonalizationof the Hamiltonian and v(q) is the 2D Fourier transformof the Coulomb potential. In the low energy limit, theg functions are, according to Eq. (8),

gα,τ,σβ (k) = e−iαϕτk

√1

2

(1 +

∆στk

Eσ,τk

)δαβ

+ α

√1

2

(1−

∆στk

Eσ,τk

)δαβ .

Since the Coulomb interaction conserves spin and isexpected to not be energetic enough to induce valleytransitions, we need only consider band transitions andthe self-energy operator turns out to be a 2 × 2 ma-trix. The matrix elements involve k-dependent integralswhich have no closed form and, due to the divergenceassociated with the unscreened Coulomb potential, aredifficult to converge numerically in general. For k = 0,however, they simplify greatly and it is possible to esti-mate the renormalization of the gap.

B. Gap renormalization

Undoped system

In this case, the band-gap becomes renormalized as

∆στ∗ = ∆στ + δ∆στ ,

4

where the renormalization of the half-gaps δ∆στ aregiven by

δ∆στ =α

2

~cεra

∫ Λ

0

adq

∫ 2π

0

dθ

2π

∆στq

Eσ,τq,

where ∆στq and Eσ,τq are the q-dependent coefficients of

Eq. (8). Setting the cutoff at Λ = 1/a and the dielectricconstant at εr = 1, we find δ∆> ∼ 1.6 eV and δ∆< ∼1.7 eV, where > and < label the high- and the low-lyingcones of the valence band, respectively. These valuesare clearly an overestimation: the renormalizations areabout twice the value found for the bare gap 2∆, whichare of order ∼ 1.5 eV.

Hole-doped system

The self-energy has now an additional subtractivecontribution, which translates into an additional renor-malization,

δ∆στh (kF ) =

α

2

~cεra

∫ kF

0

adq

∫ 2π

0

dθ

2π

q

q + qTF

∆στq

Eσ,τq.

(13)The additional factor in the integral is, essentially, thescreened Coulomb potential, since the doped system al-lows for a response in charge fluctuation. The parame-ter qTF is the Thomas-Fermi vector, obtained from theThomas-Fermi approximation as the static low-energylimit of RPA [18], reading

qTF =2e2

εrε0

D>

Auc=

e2

ε0~2

m>v

πεr,

wher Auc = a2√

3/2 is the area of the unit cell, D>

and m> are the DOS in the parabolic approximationand the effective mass of charge carriers, respectively, ofthe high-lying cones of the valence band, as implied byEq. (7). In the next section, the DOS will be shown tobe

D> =2π

ABZm>

~2=

√3

4π

(a~

)2

m>v . (14)

Notice that qTF defines a scale for the Fermi momen-tum and, thus, for the density, to which we can assigna low-density regime for kF qTF and a high-densityregime for kF qTF . We then have a threshold be-tween these regimes at kF = qTF which, in turn, setsthe intermediate density ρTF , given by

ρTF =q2TF

2π∼ (0.7÷ 1.7)× 1016 cm−2 ,

which can be obtained using the spectrum of theparabolic approximation and the expression for the DOS

above. The range of values is beyond the experimentallyrealizable doping density, as we will show in the next sec-tion. Thus, we can safely consider kF qTF so that wecan expand Eq. (13) to 2nd order, yielding

δ∆στh =

α

4

~cεr

k2F

qTF=

α

4εr

hc

qTFρ ,

which scales linearly with the hole surface density. Notethat the result does not depend on στ and thus isthe same for all cones. The coefficient is evaluated toδ∆h/ρ ∼ (0.07÷ 0.11)× 10−14 eVcm2, at εr = 1, whichfor a density ∼ 1014 cm−2 is comparable to the correc-tions obtained to second order of the Hamiltonian and,thus, experimentally accessible with techniques such asARPES.

IV. BROKEN SYMMETRY PHASES -HUBBARD MODEL

A. Intra- and inter-orbital couplings, validity ofthe Hubbard model

We introduce the the Hubbard model for these sys-tems starting with the spinful lattice Hamiltonian fromwhich the action of Eq. (12) is derived. The Hubbardmodel is obtained by considering only terms which ac-count for on-site, density-density interactions, such thatthe many-body term reads

V =1

2

∑i

∑α,β

∑σ,σ′

Uσσ′

αβ ni,ασni,βσ′ ,

where the matrix elements of U are U , for intra-orbitalCoulomb interaction, and U ′ = U − 2J for the inter-orbital Coulomb interaction, where J is the inter-orbitalexchange interaction and J ′ = J the inter-orbital pair-hopping amplitude. The relations between these pa-rameters hold for any combination of 3d orbitals [19].Consequently, there are two Hubbard terms (one forintra-orbital and another for inter-orbital interactions)as well as a anti-Hubbard term (due to inter-orbital in-teractions).

To greatly simplify the analysis, we note that, al-though J is generally much smaller than U [19], it is notentirely negligible, in the sense that U ′ < U holds but,also, U ′ ' U to a good approximation, so that, besideshaving a contribution ratio of two terms to one, the Hub-bard interaction is expected to be favoured. This allowus to set an effective interaction by dropping the anti-Hubbard and, then, approximate it by setting U = U ′,thus reading

Veff ' U∑i

ni↑ni↓ ,

5

where niσ =∑α ni,ασ and α stands for a orbital in-

dex. Although this approximation will almost certainlyyield inaccurate quantitative predictions, we believe theessential physics is accounted for.

B. Low-energy theory

Within the low-energy limit we can transform theHamiltonian from orbital to band space without intro-ducing momentum-dependence in the many-body term,an approximation which holds especially well in thehomogeneous mean-field treatment taken further on.Moreover, we consider the hole-doped low-energy sys-tem, so that we need only regard the contributions fromthe valence band cones. The action describing this the-ory thus reads

S[ψτσ, ψτσ] =∑σ,τ

∑k

ψτk,σ(−i~ωn + ξτk,σ

)ψτk,σ

+U

Nβ~∑τ,τ ′

∑q

ρτq,↑ ρτ ′

−q,↓ ,

where ξτk,σ = ετk,σ − µ, ετk,σ is the low-energy spectrumof the spin-split valence band, τ and τ ′ are valley in-dices, and k and q are imaginary-time 4-momenta andthe fields ρτq,↑ are given by

ρτq,σ =∑k

ψτk+q,σ ψτk,σ , ∀σ,q .

The decoupling of the quartic term is taken on the elec-tron densities by introducing an Hubbard-Stratonovichtransformation,

∏q

exp

− U

Nβ~2

∑τ,τ ′

ρτq,↑ρτ ′

−q,↓

=

∫Dφτσ

∏q

exp

−NβU∑τ,τ ′

φτq,↑φτ ′

−q,↓

+iU

~∑τ,τ ′

(ρτq,↑φ

τ ′

−q,↓ + φτq,↑ρτ ′

−q,↓

) ,thus introducing the real bosonic decoupling fields φτσ.After integrating out the fermionic fields and reexponen-tiating, the φ-field action reads

S[φτσ]/~ = NβU∑q

∑τ,τ ′

φτq,↑φτ ′

−q,↓

− Trlnβ~

[G−1∗ +

iU

~∑τ ′

φτ′

],

(15)

with ~(G−1∗

)τk,σ

= i~ωn − ετk,σ + µ .

Saddle-point solution

Since the configurations that minimize the action areexpected to be yield the dominant contributions to thegrand-partition function, we look for the saddle-pointsolution, which reads

φτq,σ =i

Nβ~∑k

[(G−1∗

)τk,σ

δq,0 +iU

~∑τ ′

φτ′

q,σ

]−1

.

Furthermore, introducing a space- and time-homogeneous solution φτq,σ = inτσ δq,0 , ∀τ, σ , andperforming the Matsubara frequency summations yields

nτσ =1

N

∑k

nF

(ετk,σ − µ+ U

∑τ ′

nτ′

σ

). (16)

The saddle-point solutions are the cone-resolved elec-tron densities. For the remainder of this work we willconsider only a low doping regime, in which only thehigh-lying cones ↑-A and ↓-B are affected by charge im-balance. In this case, the spin density nA↑ − nB↓ reads

nA↑ − nB↓ =1

N

∑k

[nF

(εAk,↑ − µ+ U

∑τ ′

nτ′

↓

)

−nF

(εBk,↓ − µ+ U

∑τ ′

nτ′

↑

)]

= D>

∫ −∆>

ε>Λ

dε

[nF

(εAk,↑ − µ+ U

∑τ ′

nτ′

↓

)

−nF

(εBk,↓ − µ+ U

∑τ ′

nτ′

↑

)]T→0= D>U

(∑τ ′

nτ′

↑ −∑τ ′

nτ′

↓

)= D>U

(nA↑ − nB↓

),

where D> is the DOS of the high-lying cones in thevalence band, within the parabolic approximation, asgiven by Eq. (14), ∆> = ∆ − λ is the maximum ofthe high-lying cones, working with the zero of energy atthe mid-point of the gap, and ενΛ is the cone-dependentenergy cutoff.

The saddle-point solution fails to account for a time-reversal symmetry-broken system. We must then take acloser look at the action in the vicinity of the transition.But, before moving on, note that there is a relation be-tween the Fermi level µ for U 6= 0 and the Fermi levelµ0 for U = 0, in virtue of charge conservation, which

6

reads

n = nA↑ + nB↑ + nA↓ + nB↓

= 2D>(µ0 − ε>Λ ) + 2D<(−∆< − ε<Λ )

(16)= 2D>

(µ− U

2

∑τ,σ

nτσ − ε>Λ

)+ 2D<(−∆< − ε<Λ ) .

(17)Noting that

∑σ n

Aσ =

∑σ n

Bσ =

∑τ n

τ↑ =

∑τ n

τ↓ = n/2

holds in the normal phase, the equation above yields

µ = µ0 +Un

2. (18)

This relation holds for the normal phase as well as forthe broken symmetry phase in the low doping regime,as we will show. Since the zero of energy is at the mid-point of the gap, we haveµ, µ0 < 0.

Action expansion

Consider the Hubbard-Stratonovich action that pre-cedes the effective action of Eq. (15),

S[φτσ, ψτσ, ψτσ]/~ =

= NβU∑q

∑τ,τ ′

φτq,↑φτ ′

−q,↓

−∑k,k′

∑τ,σ

ψτk′,σ

[(G−1∗

)τk,σ

δk′,k +iU

~∑τ ′

φτ′

k−k′,σ

]ψτk,σ ,

(19)and introduce the ansatz φτq,σ = inτσ δq,0 + δφτq,σ , ∀τ, σ ,essentially, nτσ the saddle-point solutions for the nor-mal phase - the cone-wise electron densities in the nor-mal phase - perturbed by a contribution δφτq,σ for which,

by hypothesis,∣∣δφτq,σ∣∣ nτσ holds. Absorbing the

saddle-point contribution into the inverse of the propa-gator G∗, we arrive at the bare propagator in the non-interactive system G0,

~(G−1

0

)τk,σ

= i~ωn − ετk,σ + µ− U∑τ ′

nτ′

σ

= i~ωn − ετk,σ + µ0 .

Integrating out the fermionic fields, reexponentiatingand factoring out the bare propagator yields a new φ-field action S′,

S′[δφτσ]/~ =

= −NβU(n

2

)2

−NβU n2

∑q

∑τ

(δφτq,↑ + δφτ−q,↓

)+NβU

∑q

∑τ,τ ′

δφτq,↑δφτ ′

−q,↓ − Trln

[1 +

iU

~G0

∑τ

δφτ

].

(20)

Given the aforementioned considerations regarding themagnitude of the δφτq,σ, we Taylor expand thefermionic contribution, yielding

F [δφτσ] =

=U

~∑σ,τ

∑k1,k2

i(G0

)τ,σk1,k2

(∑τ ′

δφτ′)σk2,k1

+1

2

U

~∑

k1,k2,k3,k4

3∏m=1

(G0

)τ,σkm,km+1

(∑τ ′

δφτ′)σkm+1,km+2

,where

(G0

)τ,σk′,k

=(G0

)τk,σ

δk′,k and(δφ)τ ′,σ

k′,k= δφτ

′

k−k′,σ

and with k5 = k1, in the 2nd order term. Note that,within the validity of the parabolic approximation, thisexpansion is exact.

We now make a homogeneous ansatz δφτq,σ ≡iητσ δq,0 , ∀τ, σ , where the ητσ are variational param-eters to be determined by minimizing the action. Theexpanded fermionic contribution now reads

F (ητσ) =

= −U~∑σ,τ

∑k

∑τ ′ ητ

′

σ

iωn − ετk,σ + µ0

− 1

2

(U

~

)2∑σ,τ

∑k

( ∑τ ′ ητ

′

σ

iωn − ετk,σ + µ0

)2

.

Eq. (19) along with the variational ansatz gives impliesa fermionic propagator in the broken symmetry phasegiven by

~(G−1

)τk,σ

= i~ωn − ετk,σ + µ0 − U∑τ ′

ητ′

σ ,

which, in turn, implies that the spectrum in the brokensymmetry phase is

Eτk,σ = ετk,σ + U∑τ ′

ητ′

σ

= ετk,σ −σU

2

∑τ ′,σ′

σ′ητ′

σ′ +U

2

∑τ ′,σ′

ητ′

σ′ ,(21)

which we shall to refer to as magnetic bands (or cones).Knowing this structure allows us to impose a constrainton the variational parameters ητσ in virtue of chargeconservation. Proceeding as in Eq. (17) we find∑

τ,σ

ητσ = 0 , (22)

which is the central conservation law for the system inthe broken symmetry phase. In light of our ansatz, this

7

constraint means that charge variations among the fourcones must total to zero. As a result, the fermionicaction simplifies to

F (ητσ) =

= −1

2

(U

~

)2∑σ,τ

∑k

∂nF (ετk,σ − µ0)

∂ετk,σ

(∑σ,τ

σητσ2

)2

T→0= N

∑σ,τ

Dστ

∫ −∆στ

−εστΛ

dε δ (µ0 − ε)

(∑σ,τ

σητσ2

)2

= ND>

(∑σ,τ

σητσ2

)2

,

where the first equality is a result of Cauchy’s integralformula upon taking the sum on Matsubara frequencies.The effective action of Eq. (20) thus becomes

S′(ξ)

Nβ~U=S′(ητσ)Nβ~U

+(n

2

)2

=(1−D>U

)(ξ2

)2

.

where ξ =∑σ,τ ση

τσ = ηA↑ − ηB↓ is the order parame-

ter. This result implies the action inverts its concavity,according to a Stoner-like condition, at Uc = 1/D> ∼11÷ 20 eV, becoming negative for U > Uc. Beyond thatpoint, the system will maximize the order parameter byfilling the ↑-A cone to its capacity. This condition is setby matching the maximum of the cone with the Fermilevel, reading

−∆> = µ0 +Uξ

2⇒ ξU = 2

|µ0| −∆>

U.

This is the mean-field solution for ξU , which can be in-cluded as a constraint in the action above by a Lagrangemultiplier, thus yielding the physical action

S(ξ)

Nβ~U= −

(U

Uc− 1

)(ξU −

ξ

2

)ξ

2, (23)

valid for U > Uc and ξ > 0. Considering the conserva-tion law Eq. (22) and that ηA↓ = ηB↑ = 0 holds, in this

doping regime, we can determine the solutions ηA↑,U and

ηB↓,U in the broken symmetry phase, yielding

ηA↑,U = −ηB↓,U =|µ0| −∆>

U. (24)

These solutions confirm our ansatz: they are propor-tional (by a factor of D>U) to the variation of chargedensity in each cone and, indeed, such a variation issmaller in magnitude that the total charge density of acone. Furthermore, it as well confirms the validity ofEq. (18) for the broken symmetry phase in this regime,

Figure 3: Magnetic bands, according to Eq. (21), for(a) the normal phase, D>U < 1, and (b) the magnetic

phase, D>U > 1. In (b) we have assumed ξ > 0.

since the charge transferred from the upwards shiftedcone is identically the same as the charge transferredto the downwards shifted one, so that any shift of theFermi level identically cancels, remaining the same asin the normal phase. The physical picture is shown inFigure 3.

Anomalous Hall effect in the spin-valley-polarized phase

This phase has physical implications beyond mag-netism, in virtue of being not only spin polarized butalso valley polarized, most notably, the existence of ananomalous Hall conductivity, rooted in the topologicalfeatures of the Dirac cones. Retrieving the Berry curva-ture of Eq. (10) and computing the transverse conduc-tivity with Eq. (11), using the spectrum provided by themagnetic cones, Eq (21), yields the conductivity

2hσvAHe2

= 1−

[1 +

8πD>

√3

(t

∆>

)2

(|µ0| −∆>)

]− 12

.

It is easy to see that, using the magnetic bands, themagnetization density (i.e. per unit cell) is given bym = nA↑ − nB↓ = 2D>(|µ0| −∆>), the same as the holedensity. Thus, for doping only slightly below the conemaximum, we can Taylor expand the expression above,yielding

2hσvAHe2

=2π√

3

(t

∆>

)2

m =

(at√π

∆>

)2

ρ .

Thus, for low doping, the transverse conductivity scaleslinearly with ρ, the hole surface density.

8

Experimental realization

The existence of an anomalous Hall conductivity inthe broken symmetry phase provides an experimentallyaccessible test to its realization in these materials. Butthere is a limitation regarding the doping capacity, inthe sense that not all regimes may be experimentallyaccessible due to fundamentally physical limitations.

Due to the geometry of 2D materials, such as single-layered TMDCs, it is possible to induce charge carriersby electric field effect. The relation between the appliedelectric field and the induced hole surface density is,thus,

E =eρ

ε,

with ε the permittivity of the medium. This implies agate voltage such that Vg = E l, where l is the inter-plane distance. The set-up consists, essentially, of thecapacitor composed of the single-layered TMDC and of awafer of Si+. The inter-planar medium is a SiO2 wafer ofdielectric constant εr = 3.9 and with a typical thicknessl ∼ 300 nm [20].

This technique allows for inducing charge carrierswithout introducing disorder but it is limited regard-ing the intensity of the applied electric and, thus, of theinduced charge: at a certain magnitude of the electricfield, the samples will be massively ionized and, ulti-mately, annihilated. An upper bound for this limit canbe determined in terms of the electric field necessary toionize hydrogen, which can be estimated as

EH =e

4πε0a20

,

where a0 is the Bohr radius. In turn, this implies a holesurface density and a gate voltage respectively given by

ρH =εr

4πa20

= 1.11× 1016 cm−2 ,

Vg,H =el

4πε0a20

= 1.54× 105 V ,

where εr is the relative permittivity of the medium.Note that, from µ0 = µA = −∆ down to µ0 =µB = −∆<, the ↓-A valley will contribute as well, sincethat valley rises in energy enough to be energeticallyfavourable to lose some of its electrons to ↑-A, whereasfrom µB down, all cones become affected by charge im-balance. µA and µB are the thresholds to what we callthe intermediate and the high doping regimes, respec-tively. Assuming that, at that energy level we are withinthe validity of the parabolic approximation, we can es-timate the threshold µA in terms of hole surface densityby

ρA =2D>λ

Auc∼ (1.2÷ 2.7)× 1015 cm−2 .

Clearly, these values are only one order of magnitudebelow the order of ρH , so that we can expect the inter-mediate and high doping regimes to be experimentallyuntenable.

However, it is interesting, if only academically, to geta sense of the physics in these regimes.

Dependence on the Fermi level

Before proceeding, some remarks are in order.Regarding the parabolic approximation, we can ex-

pect it to break down as we go further into the high-lying cones, thus requiring a higher-order correction tothe constant DOS. Moreover, as we go into high-dopingregimes, charge imbalances may become of the order ofthe cone-wise electron densities in the normal phase, sothat the Taylor expansion requires higher-order terms.This reinforces the need for a corrected DOS, since termshigher than 2nd order vanish for a constant DOS.

The central conservation law, Eq. (22), must hold forall regimes, so that Eq. (18) must become corrected byµ − Un/2 = µ0 + δµ, where the additional term is avariational parameter. We note, also, that Eq. (23) isformally the same for all regimes, depending only on Ucand ξU . For the intermediate doping regime, we haveUc = 1/D>, same as for low doping, and a mean-fieldsolution

ξU = ξU =2

U

D>(|µ0| −∆>) +D<λ

D> +D<.

As for the high doping regime, we find Uc = 1/(D> +D<) ∼ 5÷ 9 eV, and the solution

ξU =2

U

D>(|µ0| −∆>) +D<(|µ0| −∆<) +D<λ

D> +D<.

We note, also, that the yielded physical observables,namely, magnetization density and transverse conduc-tivity, are continuous across all regimes.

V. CONCLUSIONS AND FUTURE WORK

A group theoretical framework for tight-binding Ob-taining a three-band effective tight-binding Hamiltonianusing a group theoretical framework as proved to be avaluable strategy, as it has managed to provide a low-energy theory very accurately while avoiding complica-tions due to an unnecessarily increased number of or-bitals.

Self-energy correction The values found for the gaprenormalization in the undoped system are not consis-tent with previous results [7] but, still, are within a rea-sonable order of magnitude. This is an indicator that

9

there may exist screening effects due to high-energy elec-trons unaccounted for in this low-energy model. Al-though these electrons are unable to effectively screenthe potential and cut its long range short, since theyare constrained within the Fermi sea, they may providea substantial increase of the dielectric constant in thesematerials.

Such is the situation in graphene: for the Fermi levelat the Dirac point, the DOS vanishes and the Coulombinteraction is not screened. It turns out that he σ-orbitalelectrons are responsible for an increased dielectric con-stant [21]. We can expect this to happen, even more dra-matically, in TMDCs: not only do these materials have11 valence electrons per unit cell, the large atomic num-ber of transition metals means there is a large amountof core electrons in the system, thus possibly yielding asubstantial dielectric constant.

Furthermore, this result indicates that electron-electron interactions are an important contribution to-wards the gap, likely more preponderant than the lat-tice potential. As such, since electron-electron interac-tions are susceptible to the dielectric constant of thesubstrate, we expect a tunable gap. Although the sub-strate affects also the lattice potential, it is expected toaffect the electronic long-range interactions to a largerdegree.

As for the the hole doped system, the results are morereliable, since the additional contribution is not cutoff-dependent, but are still expected to be an overestimate,for the reasons argued above.

Despite the simplicity of the Hartree-Fock approxima-tion, the divergent nature of the unscreened Coulombpotential has rendered the problem numerically in-tractable, so far. Furthermore, we have found the com-putable results to be highly-sensitive to the cutoff. At-tenuating the dependence on the cutoff is a priority, andwe expect to be able to solve this problem by computingthe results self-consistently. The divergence, however, isa problem requiring a solution of its own.

Note that obtaining accurate estimates for the self-energy correction is not only and end in itself, but is alsoof great importance for improving the accuracy of thequantitative results of the Hubbard model, especiallythe corrections in the hole doped case.

Hubbard model The values found for the criticalHubbard coupling appear to be higher than currentestimates for the interaction strength, ranging in 2 ÷10 eV [22], especially for W compounds. There are rea-sons to believe, however, that the strength of the on-site Coulomb interaction (or, put simply, of the Hub-bard coupling) might cover these critical values, basedon studies carried out for graphene [21], where the en-ergy of the partially screened on-site Coulomb interac-

tion is found to be 9.3 eV.

TMDCs, in turn, due to their richer electronic struc-ture, are expected to have significantly larger values forthe Coulomb interaction, so that these values do notstand as a compromise the realization of this phase.Note, also, that the values used for the effective massare an estimate, since they are obtained from the non-interacting, undoped electronic structure and, as such,are not accurate for the system studied here. Further-more, the same results obtained with the formalism usedcan be obtained using a variational mean-field approachbased on the Gibbs-Bogoliubov-Feynman inequality.

Although the physics is expected to be correct, theapproximations taken to arrive at the effective Hub-bard model used may have rendered much of the quan-titative predictions inaccurate, particularly, underesti-mated, since we have discarded the anti-Hubbard term.Besides, mean-field is known to systematically underes-timate critical parameters.

We believe it is possible to carry out a more accuratemean-field treatment of the full Hubbard model (includ-ing the anti-Hubbard term) in these systems withoutgoing beyond an analytical approach. Although we arefaced with the problem of having 3 bands coupled to2 spin degrees of freedom, a dimensionality which mostlikely renders the subsequent matricial manipulationsintractable, the fact is that the valence band, at its edge,is composed solely of orbitals dxy and dx2−y2 , thus effec-tively reducing the dimensionality of the problem. This,along with other reasonable approximations, is expectedto make the problem analytically tractable while yield-ing more accurate results and physics.

It might as well be worthwhile to go beyond theparabolic approximation, even within the low dopingregime. Although the quantitative corrections are ex-pected to not be significative, it might be interesting tolook at the physics yielded by a 4th order term, namely,regarding the existence of meta-stable solutions. Ofcourse, all analytical results are to be corroborated withresults from numerical simulations of the system.

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