Computational methods for 2D materials modelling
A. Carvalho, P.E. Trevisanutto, S. Taioli, A. H. Castro Neto
CComputational methods for 2D materials modelling
A. Carvalho † Centre for Advanced 2D Materials, National University of Singapore, 6 Science Drive 2, 117546 Singapore ∗ andFondazione Bruno Kessler, Via Sommarive, 18, 38123 Povo TN, Trento, Italy P. E. Trevisanutto † European Centre for Theoretical Studies in Nuclear Physics and Related Areas (ECT*-FBK)and Trento Institute for Fundamental Physics and Applications (TIFPA-INFN),Via Sommarive, 14, 38123 Povo TN, Trento, Italy
S. Taioli † European Centre for Theoretical Studies in Nuclear Physics and Related Areas (ECT*-FBK)and Trento Institute for Fundamental Physics and Applications (TIFPA-INFN),Via Sommarive, 14, 38123 Povo TN, Trento, Italy andPeter the Great St. Petersburg Polytechnic University,Polytechnicheskaya 29, St. Petersburg 195251, Russia
A. H. Castro Neto
Centre for Advanced 2D Materials, National University of Singapore, 6 Science Drive 2, 117546 Singapore ∗ andMaterials Science and Engineering, National University of Singapore, 9 Engineering Drive 1, Singapore 117575 Materials with thickness ranging from a few nanometers to a single atomic layer present unprece-dented opportunities to investigate new phases of matter constrained to the two-dimensional plane.Particle-particle Coulomb interaction is dramatically affected and shaped by the dimensionality re-duction, driving well-established solid state theoretical approaches to their limit of applicability.Methodological developments in theoretical modelling and computational algorithms, in close in-teraction with experiments, led to the discovery of the extraordinary properties of two-dimensionalmaterials, such as high carrier mobility, Dirac cone dispersion and bright exciton luminescence, andinspired new device design paradigms. This review aims to describe the computational techniquesused to simulate and predict the optical, electronic and mechanical properties of two-dimensionalmaterials, and to interpret experimental observations. In particular, we discuss in detail the par-ticular challenges arising in the simulation of two-dimensional constrained fermions, and we offerour perspective on the future directions in this field. [Note: for a version with third-partyfigures, please contact the authors]I. INTRODUCTION
The behaviour of quantum particles, either bosons orfermions, is affected by the dimensionality of the spaceaccessible to them. Developments in nanotechnology andnanofabrication over the last few years made it possi-ble to use dimensionality as a parameter, which can beharnessed by precise control of the atomic structure. Inparticular, the fabrication of two-dimensional (2D) ma-terials – materials where electrons, phonons or other par-ticles are constrained to a 2D manifold – have revealed aplethora of new phenomena that significantly enrichedour knowledge in condensed matter physics. Startingfrom the synthesis of graphene, the epitome of 2D materi-als, several novel layered crystal families, such as metallicand semiconducting dichalcogenides, silicene, germanene,and phosphorene, have been discovered. These findingsare a potentially disruptive innovation in optoelectronics, ∗ Electronic address: [email protected][ † ] These authors contributed equally to this work spintronics, electromechanics, energy storage, and ther-moelectrics.To investigate the properties of these novel 2D ma-terials, which are dramatically modified by the reducedscreening and quantum confinement, numerical simula-tions based on first principles, multiscale and more re-cently machine learning [1, 2] approaches represent anessential tool to interpret and guide experiments. 2D ma-terials can be sub-nanometer thick and may have onlya few atoms per unit cell, making it possible to modelthem with great precision. In fact, surfaces and inter-faces are atomically sharp and can be represented accu-rately by atomistic models. Nevertheless, the coexistenceof atomic and solid state features of these 2D architec-tures have pushed the existing theories to their limits andcreated a quest for new computational modelling strate-gies to assess accurately and efficiently both ground andexcited state properties.This manuscript thus reviews (addressing, in partic-ular, the experimental reader) state-of-the-art computa-tional methods used to model 2D materials at differentlevels of theory. We discuss the theoretical foundations ofthe numerical methods used to characterise ground and a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n excited-state properties of 2D materials, their respectiverange of applicability, and we show a few illustrative ex-amples of the computation of electronic, optical, trans-port and mechanical properties. Finally, we reflect uponfuture challenges and directions of this rapidly evolvingfield.The article is structured as follows: in section II, weshortly review density functional theory (DFT) for theelectronic ground state; in section III, we outline compu-tational approaches based on many-body perturbationtheory (MBPT) for electronic excited states; in sectionIV, we discuss several methods to carry out charge trans-port simulations; and finally section V is devoted to com-putational approaches for the simulation of mechanicalproperties of 2D materials. II. DENSITY FUNCTIONAL THEORY
DFT is one of the most used computational methodsin materials research and 2D materials in particular. It isimplemented in numerous codes and has achieved a highlevel of reproducibility.[3]. We will briefly introduce thefundamentals of DFT and discuss some of the achieve-ments and challenges in 2D and layered materials.
A. Fundamentals
DFT is rooted on the two Hohenberg–Kohn theorems[4]. It is an exact formalism, based on a variational prin-ciple, that allows one to find the energy and electron den-sity n ( r ) of the ground state | Ψ (cid:105) of a quantum system.This is conceptually attractive because, unlike the wave-function, the charge density is an observable. However,in practice, the determination of the electron density ofreal systems relies on approximations to describe the ex-change and correlation interactions between electrons[5].In particular, the exchange interaction, whereby many-body systems must be antisymmetric under exchange,stems from Pauli’s exclusion principle and acts upon elec-trons with the same spins. The latter force leads e.g. tospin alignment in ferromagnets or to the Hund’s rule inatoms.The remainder of the Coulomb interaction after in-cluding static exchange, is the electron-electron correla-tion [5, 6]. The effect of this potential becomes visiblee.g. in the dissociation of the H molecule, where it isresponsible for the electrons ending up on different atomsupon adiabatic bond stretching [7]. We notice that in thecontext of DFT, the correlation energy is defined differ-ently from other fields of computational chemistry andcondensed matter physics [8, 9]. B. Correlation in 2D materials
In 2D materials, the charge density can be easilychanged by back-gating, enabling correlations to betuned experimentally in an unprecedented way. Ingraphene, for example, correlation becomes importantwhen the Fermi energy is close to the Dirac point, and thedensity of free charges is low [10]. For strongly correlatedelectronic phases, where electrons are no longer weaklyinteracting, the DFT treatment no longer provides a validphysical picture. An example of this DFT failure is thedescription of the electronic states in bilayer graphenetwisted at a small angle, forming Moir´e superlattices.At specific twist angles, the electronic band dispersionclose to the Fermi level is nearly flat on the reciprocalspace. When the flat bands are partially filled, the re-sulting electronic states are highly correlated and can dis-play orbital magnetism, superconductivity, or quantisedanomalous Hall effect.[11–15]. Other correlated systems,such as low-density electron gas phases in graphene, arestill object of research. Accurate numerical treatmentof correlation, such as the random phase approximation(RPA), will be examined in sec. III.Alternatively, the Reduced Density Matrix FunctionalTheory (RDMFT), which upgrades DFT by using the re-duced density matrix rather than just the spatial density,has proven the ability to deal with strongly correlatedsystems dominated by static correlations, and in par-ticular to calculate fundamental band gaps of semicon-ductors/insulators and in transition metal oxides [16, 17]where DFT fails.
C. Correlation and van der Waals forces
Correlation is also in part responsible for the van derWaals (vdW) bonding [18]. When two-dimensional layerspair up, a charge density redistribution, arising from in-duced transient dipoles associated with electronic chargefluctuations, generates an attractive force between layers.It is known that local or semi-local exchange-correlationfunctionals developed for 3D solids yield incorrect vdWbinding energies and geometries. However, a correctdescription of vdW systems can be achieved with lowcomputational overhead by adding a semi-empirical dis-persion potential, such a pair-wise force field optimizedfor several popular DFT functionals, to the conventionalKohn-Sham DFT energy (see Fig. 1) [19–24]. Correlationenergy expression to obtain dispersion-corrected func-tionals to deal with weak interactions can be derived alsousing the Adiabatic-Connection Fluctuation–Dissipationtheorem approach (see for example ref. [25]) within theRPA approximation. These functionals are able to de-termine the relative thermodynamic stability of differentvdW layered superlattices. Conversely, one can arguethat the discovery of 2D materials has contributed tothis important methodological development. vdw.png
FIG. 1:
Van der Waals functionals.
Comparison of the interlayer distance yield by different levels of treatment of the vander Waals interaction for bulk graphite and boron nitride. Experimental reference values at low temperature are representedby the horizontal lines. Values are from REF.[24]
D. Choice of basis set for 2D materials
The majority of DFT codes expand the KS eigenfunc-tions into a plane wave (PW) basis set [26, 27], owingto the fact that in principle the accuracy can be indef-initely increased by simply increasing the energy cutoff( E cutoff ). However, the presence of vacuum between 2Dlayer replicas and inter-layer spacing raise the computa-tional cost, as the number of PWs increases rapidly at agiven E cutoff . We will come back to this replica problemin sec. III.Owing to the hybrid character of 2D materials, theideal basis set to deal with layered systems, however,is one in which its functions are periodic in-plane whileatomic-like along the perpendicular direction [28, 29]. E. Electronic structure
The electronic band structure can be calculated(with little formal justification) from the Kohn-Sham(KS) orbital energies (cid:15) l ( k ) as a function of the crystalmomentum k , as well as the Fermi energy from thenormalisation constraint. However, we notice that thesingle electron energies (cid:15) l ( k ) describe the dispersion forthe auxiliary system obtained by creating an effectivepotential that is felt by non-interacting electrons. Thus,in metallic systems where e–e interactions dominate, farfrom the Landau Fermi liquid picture, the one-electron energies are not a good representation of the many-bodyelectron system. Still, in some instances, they may beused due to their conceptual simplicity.An immediate application of DFT one-electron ener-gies is the calculation of the bandgap, which can bemeaningfully determined from the difference in energiesbetween the lowest unoccupied and the highest occupiedKS states [30, 31]. Also this approach is prone to very-well known failures, such as the bandgap underestima-tion when a local density approximation is used.[4] Moreaccurate methods for reckoning the electronic fundamen-tal band gap and optical spectra of 2D materials will bediscussed in Section III. III. AB-INITIO MANY-BODY PERTURBATIONTHEORY IN 2D MATERIALSA. The GW approximation
Excited-state properties and the interpretation of elec-tron spectroscopy experimental measurements in con-densed matter physics rely on the concept of quasiparticle(QP), which is represented as a fundamental excitationof bare particle plus its polarization cloud. QPs are noteigenstates of the Hamiltonian and thus acquire finitelife-times. In this subsection, we will limit our discussionto particular QPs, the electronic polarons (as originally excitons.png
FIG. 2:
Origin of the excitons in the optical absorp-tion spectra of monolayer MoS . a | LDA (dashed bluecurve) and GW (solid red curve) band structure. Arrows in-dicate the optical transitions. b | Absorption spectra without(dashed red curve) and with (solid green curve) electron-holeinteractions, calculated by solving the BSE. c | Same as b butusing an ab-initio broadening based on the electron-phononinteractions. Panels a-c are adapted from REF. [32]. Panel d is from REF. [33]. defined by L. Hedin [34]), which can be imagined as elec-trons (or holes) dressed by their electronic clouds.Within the ab-initio many-body perturbation theory(Ai-MBPT) framework for electronic structure calcula-tions, the central quantity is the one-particle Green’sfunction G (at zero temperature), which is defined asthe inverse of the ground state expectation value of theinteracting Hamiltonian of the N electron system ( | N (cid:105) ).Using this definition, one can write the Dyson equationas follows: G − ( r , r , ω ) ≡ G − ( r , r , ω ) − Σ( r , r , ω ) (1)where G − ( r , r , ω ) is the inverse of the non-interactingGreen’s function and Σ( r , r , ω ) is the self-energy.Σ( r , r , ω ) is a non–Hermitian, energy–dependent, andnon-local quantity, which “dresses” the bare particle withthe whole many-body electron interaction cloud. To de-scribe some important features of Ai-MBPT, it is usefulto write G , through the Lehmann spectral representation[34–36], in termsof the f s , the transition amplitudes from the N -electron system state | N (cid:105) to the excited state s of the N ± | N ± s (cid:105) In Ai-MBPT, the quantum states | N (cid:105) to | N ± s (cid:105) are determined from either the non-interacting DFT or a tight-binding (TB) approximation(see sec. IV A). The QP energies E QPs and the amplitudes f s are obtained from the homogeneous equation (1): (cid:0) E QPs − h − Σ( ω ) (cid:1) f s ( r ) = 0 (2)where the real part of the eigen-energies E QPs delivers theQP band structures, while the imaginary part the QP damping (by means of the imaginary part of Σ). Both f s and E QPs are directly related to the the spectral functionof particles and, therefore to the intrinsic losses in photo-emission spectroscopy (PES) and inverse PES.GW is the most successful and popular approximationto the self–energy in Ai-MBPT. With respect to DFT cal-culations, GW improves the agreement with the exper-imental data of the simulated electronic band structure[35], and reads (the symbol “*” stands for the convolu-tion product):Σ( r , r , ω ) = i π G ( r , r , ω ) ∗ W ( r , r , ω ) e − iηω (3)where W is the screened Coulomb interaction that is re-lated to the bare Coulomb potential w via the dielectricfunction (cid:15) : W ( r , r , ω ) = (cid:90) d r (cid:15) − ( r , r , ω ) w ( r , r , ω ) (4) (cid:15) ( r , r , ω ) measures the tendency of a medium to bepolarized under the action of an external electromagneticfield, it is related to the energy loss spectrum [37] andenergy loss in charge transport Monte Carlo (see sec. IV).We notice that the GW self-energy is the HF exchangepotential in Eq. (3) if the screened Coulomb potential W is replaced by the bare Coulomb potential w . B. The Bethe–Salpeter equation
Besides one-particle properties, two-particle propertiesalso play a prominent role in the study of excited-statefeatures of 2D materials, namely excitonic effects in ab-sorption spectra. A typical example is the electron-hole(e-h) QP: the exciton. Similarly to QPs theory, the two-body Green’s function L is determined by solving theBethe–Salpeter equation (BSE): L − vc,v (cid:48) c (cid:48) ( ω ) ≡ H BSE vc,v (cid:48) c (cid:48) ( ω )= ( E QP c − E QP v ) δ v,v (cid:48) δ c,c (cid:48) + I ehcv,c (cid:48) v (cid:48) ( ω ) (5)where v and c are valence and conduction one-particlestates, H BSE vc,v (cid:48) c (cid:48) ( ω ) ≡ (cid:104) vc | ˆ H | v (cid:48) c (cid:48) (cid:105) is the effective two-body Hamiltonian, and I eh cv,c (cid:48) v (cid:48) is the kernel of this Dyson-type equation, which includes the e-h interactions. InAi-MBPT, an important assumption for semiconductorsand insulators is to approximate the kernel I with thestatic screened potential I ≈ W .In bulk materials, these properties are calculated fromthe macroscopic dielectric function (cid:15) M ( ω ) which is re-lated to the imaginary part of L ( q is a reciprocal latticevector) through: (cid:15) M ( ω ) = lim q → (1 − v ( q ) Im L ( q, ω )) (6)where the long wavelength limit of the interactionCoulomb potential v must be considered. In 2D materi-als the macroscopic dielectric function reduces to 1 and,to determine the optical properties, the longitudinal in-plane polarisability has to be considered [28]: χ P ( ω ) ≈ lim q → q L ( q, ω ) (7)Alternatively, one can obtain Eq. (6) by averaging W over the material rather than the simulation supercell[38].Figure 2 shows how GW+BSE can be used to inter-pret the exciton features in MoS , including the opticalabsorption threshold, exciton binding energy, and spinand momentum-related selection rules. C. The screened potential W in 2D materials The screened potential W enters in both these ap-proaches and is by far the most difficult to estimate ,amongst the many-body terms. It is calculated startingfrom the non interacting polarization function P , whichis determined from the ground (DFT or TB) state energyand wave function. The minimal basis set necessary toobtain accurate results comparable with experiments isusually very large. This issue is increased in quasi-2D ma-terials, where the crystalline periodicity is present onlyin the planar directions whereas in the z direction (nor-mal to the plane) the wave functions have a polyatomicmolecular character. This hybrid (both crystalline andmolecular) behaviour induces, on the plane of the layers,electric field fluctuations that are not screened at largedistances: the e-e and e-h interactions are thus muchstronger than in 3D materials, producing excitons withnotably high binding energy [39–49] and, in presence ofdefects, extremely stable [48].By using periodic Ai-MBPT plane-wave codes, quasi-2D materials calculations present the replica issue previ-ously mentioned (see sec. II D). Owing to the ineffectiveCoulomb interaction screening of the bare w , the requireddistance between the replicas can be very large, makingcalculations unfeasible, as the mesh grid for reciprocalspace integration has to be increased. To overcome thisproblem, the Coulomb interaction is truncated beyond acertain cutoff distance in the z -direction [38, 50–52]. Aslightly different approach [53, 54] consists in replacing w by a Keldysh-like potential to simulate the in-planescreening of a 2D layer in vacuum, W D ( q ) = 2 πe | q | (1 + α D q ) (8)where e is the charge and q is a reciprocal lattice vectorin the plane. This potential is often used in connectionwith TB calculations [55, 56].Due to their high binding energy, excitons in quasi-2Dmaterials are stable and localized. Qubits [57], single-photon light emitter devices using the self-trapping ef-fects [58], excitonic devices or cavity polaritons [59],and valleytronics [60] are some of the potential applica-tions exploiting these peculiar many-body effects in 2D materials[39–49].Finally, 2D materials are often encapsulated or lyingover substrates. Several works were devoted to deal withsubstrate effects on W [52, 55, 61–63]. D. Second, third, and higher harmonic generations
The strong optical response of 2D materials to exter-nal electromagnetic fields generates peculiar effects alsoin non-linear regime, giving rise to an optical responseat multiples of the exciting frequency ω . Graphene andTMD semiconductors show non-trivial topological char-acteristics [60, 64, 65], which have been investigated withhigher harmonic generation calculations [66]. To dealwith these physical mechanisms, two different main ap-proaches are developed in the context of Ai-MBPT: thedirect integration of the time-dependent macroscopic po-larization and the perturbative approach for the non-linear susceptibilities.The macroscopic polarization P ( t ) is the key observ-able of interest for optical properties. This can be ex-pressed in terms of one-particle reduced density matrixas: P ( t ) = e (cid:90) r ˆ ρ ( r , t ) d r = eA (cid:88) cv k r cv k ρ cv k ( t ) , (9)where e is the charge of the electron, and A is the 2Dcrystal area. The time-dependent density matrix is de-fined as ρ mnk ( t ) ≡ (cid:104) ˆ a † mk ( t )ˆ a nk ( t ) (cid:105) by the creation (anddestruction) operators ˆ a † (ˆ a ). In the length gauge [67],the dipole matrix elements r mnk ≡ (cid:104) mk | ˆ r | nk (cid:105) are sepa-rated in two components,the intraband (or Berry Connection [68]) r i and theinterband dipole operators r e where the indices m and n are the bands.In the high harmonic generation optical responses, thee-h interactions are dominant. An ai-MBPT descriptionis provided by the time-dependent (TD) BSE [69]. TheTD-BSE is originally derived from the Kadanoff-Baymequation and it can be related to the Linblad masterequation for open quantum systems [70, 71]. If the inter-action with the external electric field F is written in thelength gauge ( H int = e F · r ) TD-BSE reads i (cid:126) ∂∂t G k ( t ) = (cid:104) h k + e F ( t ) · r + Σ k [ G k ( t )] , G k ( t ) (cid:105) , (10)where the self-energy Σ is introduced either in the HFapproximation or in the static GW self-energy (COH-SEX), and [ ., . ] is the commutator. In absence of theself-energy (and then, the e-h interactions), the “semi-conductor Bloch equations” (SBE) is retrieved from theTD-BSE when considering the connection between theGreen’s function and the density matrix: G nm k ( t ) ≡ G nm k ( t, t ) = iρ nm k ( t ). In the strong-field resonant dy-namics and attosecond optoelectronics, the SBE is cur-rently implemented to calculate the band population (di-agonal terms) and coherences (the off-diagonal terms,see subsection III E) [72–74]. Analogous formulas forthe density evolution can be retrieved within the time-dependent DFT (TD-DFT) framework, leading to equa-tions analogous to the SBE and TD-BSE (see [66] for anexhaustive review on the topic).From Eq. (10) the non linear responses such as thesecond, third, and higher harmonic generations can beestimated with the perturbative derivation of the har-monic generation response functions P = χ (1) F + χ (2) F F + χ (3) F F F + ... (11)where χ (1) , χ (2) , χ (3) are, respectively the macroscopicfirst, second and, third, order susceptibilities. These highorder responses can be analytically derived by expandingthe Eq. (10) in terms of one-particle Green’s function G nm k ( ω ), G nm k (2 ω ), G nm k (3 ω ) [55].Alternatively, macroscopic polarization P can be ob-tained through Eq. (9) and with the Green’s functiontime-integration scheme of the eq.10. The linear response(i.e. the BSE (5)) and the higher order generation re-sponses χ ( i ) are retrieved by numerical Fourier transformof P ( ω ) [69, 75].Ai-MBPT calculations of the optical response of boronnitride show that excitons also dominate the nonlinearoptical properties.[55]. Moreover, crystal local field ef-fects, whose importance was already displayed for bulkand surfaces [66], dramatically shape the SHG and THGspectra in MoS and h-BN monolayers [76]. Finally, dif-ferently from the linear response regime, the interbandand intraband transitions cannot be separated as shownin the SHG and THG calculations in the spectrum ofbilayer graphene [77], and in the THG calculations ofsemiconductor BP [78], MoS and h-BN [55] SLs. Real-time simulations have recently been used to better de-termine the model used to extract the SHG coefficientfrom the experimental data, such as in the case of the2D monochalcogenides GaSe and InSe [79]. E. Trions, Exciton–polarons, and Biexcitonformations
Controlling the optical response of the material by dop-ing the substrate or by gating is a possibility in 2D ma- terials that has had no parallel in traditional optical ma-terials. Recently, 2D TMDs optical absorption experi-ments [80, 81] have shown that, in the presence of freecarriers, the most prominent excitonic features split intotwo distinctive peaks. The second peak was initially at-tributed to trions, i.e. the three–body QP characterisedby a bound state of an exciton plus an hole or electron,and later to the dressed excitons, the exciton-polarons ,that is the many–body generalization of the trion boundand unbound states. It has been theoretically shown thatat low doping, the ground state corresponds to the trionand becomes an exciton-polaron at higher doping. [82–84]. From an ab-initio point of view, this problem wastackled by extending the Ai-MBPT to include this three-body effective Hamiltonian (e.g. in carbon nanotubes[85, 86]), or to describe the optical spectrum lineshape(e.g. of MoS [87]). In a (two-electrons) plus hole sys-tem, eeh (the electron and two holes, hhe , is analogous)the effective Hamiltonian is approximated as an exten-sion of the BSE (5): the particle-particle interactions aredescribed by the single BSE (an then mediated by theScreened many–body interaction W ) correlations withthe third particle non interacting. The three-body cor-relation kernel term is disregarded. This Hamiltonian,with a combination of GWA and Configuration Interac-tion (CI) approaches, has been applied to the case of WS and MoS monolayers [88].These calculations provide quantitative binding ener-gies of the trion resonances both in a free-standing layerand in a more realistic case with substrate (or encapsu-lation) that enhances the environmental screening. Moreimportantly, the analysis of transitions has shown thenature of trions [87] (Fig. 5).Recently in ref.[89], ultrafast pump-probe experimentson monolayer WSe showed the biexcitons signatures anda fine structure in excellent agreement with the theory.The dynamics-controlled truncation theory for biexcitonsappear in the THG susceptibility. The biexciton spec-trum was modeled with SBE–like equation of motion forthe four–particle correlation function: B c (cid:48) v (cid:48) cvk,k (cid:48) ( q, t ) ≡ (cid:104) ˆ a † c (cid:48) ( − k (cid:48) − q ) ˆ a v (cid:48) k (cid:48) ˆ a † c ( − k ) ˆ a v (cid:48) ( k + q ) (cid:105) ( t ) − (cid:104) ˆ a † c (cid:48) ( − k (cid:48) − q ) ˆ a v (cid:48) k (cid:48) (cid:105) ( t ) (cid:104) ˆ a † c ( − k ) ˆ a nv ( k + q ) (cid:105) ( t ) (12)where the interaction Hamiltonian is described as theextension of trion and two-body BSE Hamiltonian forthe 4-particle B c (cid:48) v (cid:48) cvk,k (cid:48) but in the HF approximation. F. Dark and bright exciton formation,photolomuniscence and biexciton fine structures inmonolayer TMDs.
Besides bright excitons, TMDs exhibit dark excitonsthat modify the dynamics, the coherence lifetime and,at last, the photo–luminescence. In a series of articles[90–93], the SBE-like equations for the excitonic densitymatrices has been extended to include phonon and pho-ton scattering events and describe the exciton dynam-ics. To summarize, the exciton band structure is cal-culated from DFT and GW calculations. The Heisen-berg equation of motion for the coherent exciton po-larization P q ( t ) ≡ (cid:80) k ψ ∗ k )ˆ a † c ( k + q ) ˆ a v ( k − q ) ( t ) ( ψ k the ex-citon wave-function) is derived. The total Hamilto-nian in the SBEs includes the ”free” exciton Hamilto-nian, incoherent (4-particle) exciton density formation N Q ( t ) ≡ (cid:80) k,k (cid:48) (cid:104) ˆ a † c ( k − αq ) ˆ a v ( k + βq ) ˆ a † v ( k (cid:48) + βq ) ˆ a c ( k (cid:48) − αq ) (cid:105) ( t )through the phonon-exciton coupling, and photon-exciton interactions in the low excitation regime. Onthe same reference, the equation of motion for the inco-herent exciton occupation density N Q is written includ-ing in the Hamiltonian the non-radiative decay of P Q ,phonon scattering processes (in and out), and the spon-taneous photon emission. Between the main results, thetheory predicted that a carrier relaxation in TMDs on50 fs time scale, an order of magnitude faster than inquantum wells whereas the incoherent part of the pho-toluminescence was estimated to decay on a timescale offew tens of nanoseconds. IV. QUANTUM TRANSPORT
In layered structures, band dispersion and electrontransport are both strongly affected by reduced screeningand quantum confinement. This results in a wide rangeof band gaps and charge carrier mobilities (see Fig. 3(a)).On the other hand, highly anisotropic in-plane and out-of-plane electrical properties [102] emerge. For example,in bulk MoS , the in-plane mobility ( µ x,y ) exceeds theout-of-plane mobility ( µ z ) by a factor of 10 [94], essen-tially due to the vdW interlayer spacing acting as a tun-neling barrier (see Fig. 3(a)). In this section, we considertransport modelling approaches and their applicability indifferent regimes. A. The tight-binding approach
Tight-Binding (TB) [103] is an approximate quantum-mechanical method to calculate the electronic structureof solids, using a basis set of atomic orbitals φ i ( r ), andit is widely used for transport modelling in 2D materi-als. TB assumes that inside a solid, electrons are tightlybound to the atoms, and that their single particle wavefunctions | ψ kn (cid:105) in the crystal are quasi-atomic.The TB scheme has been very successful in modellinggraphene, and is easily extendable to other hexagonal 2Dmaterials [95, 104]. In the case of graphene, the DiracHamiltonian of chiral massless particles of quantum elec-trodynamics (QED) in two dimensions can be obtainedby considering only first NN interactions between p z or- bitals and by expanding linearly in the momentum thematrix elements of the TB Hamiltonian in the proximityof the K (or K (cid:48) ) point. This results in a 2 × H| K = v F ˆ σ · p , (13)where ˆ σ are the Pauli matrices, and v F = √ γ a/ (2 (cid:126) ) isthe Fermi velocity. The latter resembles the QED spinHamiltonian where the Fermi velocity plays the role ofthe light velocity of massless particles. By including in-teractions up to third NN π − π ∗ orbitals, it is possibleto recover the band asymmetric behaviour far from theDirac cones.The TB model is computationally very efficient as itcan easily scale up to several million atoms [103]. Forexample, the TB approach was used to calculate theLDOS in 2D graphene pseudospheres (see Fig. 3(b))[95, 104] having a few millions carbon atoms with anumber of Stone-Wales defects (reproduced in the in-set of Fig. 3(c)). The presence of bumps and asym-metries of the bands around the Fermi level, differentfrom graphene’s linear dispersion denotes the presence ofpenta-heptagonal defects related to the negative curva-ture [95]. TB can also be used to determine the band-structure of twisted bilayer graphene, that shows inter-esting flat-band effects when the twist angle is small andthe respective unit cell very large – too large to be treat-able by conventional methods such as DFT (Fig. 4).However, TB has limited use in systems where d elec-trons contribute substantially to the bands involved intransport, or in general when there is substantial orbitalhybridisation. B. Quantum transport: Landauer-B¨uttiker
The key quantity in the study of charge transport in2D materials is the coherence length l φ , which measuresthe length scale of the single-electron wavefunction phasechange owing to phase-relaxing processes. l φ determinesthe crossover between ballistic and diffusive transportregimes. In graphene, l φ ≈ µ m.In the coherent regime, the standard modelling setupto study mesoscopic transport consists of a system con-nected to a thermal reservoir by two reflectionless macro-scopic contacts that define the temperature and thechemical potential of the incoming electrons (see Fig.3(d)). Using this layout, one derives the so-calledtwo-probe Landauer-B¨uttiker (LB) conductivity formula.This formalism can be used both within the quasi-particleFermi liquid framework and within correlated electronsschemes, when the Coulomb interaction between theflowing electrons breaks down the Landau picture.In the LB theory, the conduction through the device isrepresented in terms of scattering processes that the elec-trons undergo after being injected from the left lead intothe device, and before entering the reservoir by crossing figrev.png FIG. 3:
Transport and mechanical properties of 2D materials . a | Mobility vs. bandgap of selected 2D materialscompared, with those of 3D semiconductors. ML, SL, 3L stand for few-layer, single-layer and tri-layer, respectively, and BPstands for black phosphorus [94]. b | Up panel: local density of states (LDOS) symmetry breaking due to curvature effectsin a pseudosphere graphene membrane with N = 2 , ,
976 carbon atoms, evaluated using a TB approach. Bottom: LDOSprojected over the two nonequivalent graphene sublattices A and B near the Fermi energy. c | Graphene membrane shaped as aBeltrami’s pseudosphere. d | Set-up of a typical Landauer-B¨uttiker (LB) transport simulation. e | Logic flow of the LB ab-initio approach. f | Reflection electron energy loss spectra (REELS) of a highly oriented pyrolytic graphite (HOPG) sample for severalprimary beam kinetic energies simulated by MC. g | Stress–strain curves of liskene, tilene, and flakene along the different straindirections from first-principles simulations. Panels b-c are from REF.[95]. Panel f is from REF.[96]. Panel g is from REF.[97]. the right lead. Within the LB scheme the conductancereads: I = 2 eh (cid:90) Tr [Γ L G r † D Γ R G rD ][ f ( E − µ L ) − f ( E − µ R )] dE (14)where T ( E ) = 4 Tr [Γ L G r † D Γ R G rD ] is the lead-to-leadtransmission probability of an electron of energy E ,the integral is over all available energies, and the fac-tor 2 counts the spin multiplicity. The intrinsic energylinewidths Γ L/R = i [Σ rL/R − Σ r † L/R ] of the left and rightleads account for the finite lifetime of the electrons mov-ing from the central region, where the conductance is de-scribed in terms of the retarded Green’s function (rGF)of the device ( G rD ), to the leads.The main tasks of the LB approach consist thusin the accurate evaluation of the self-energies associ-ated with the left and right electrodes, and of the Green’s function (GF) of the central region (the de-vice). These tasks can be accomplished via two dif-ferent computational schemes: (i) GF evaluation recur-sive methods, similar to the techniques used in con-nection to TB Hamiltonians;[105–107]. (i) ab-initio approaches[108, 109]. A schematic logic flow of the stepsinvolved in the ab initio simulation of the LB approachto transport is reported in Fig. 3(e). Figure 6 illustratesan ab-initio calculation of device voltage-current charac-teristics based on the LB approach. C. Quantum transport: Kubo approach
A second, widely used quantum mechanical approachfor coherent, noninteracting electrons is based on theKubo formalism [111]. The Kubo approach is a broadly twisted-bands.png
FIG. 4:
Tight-binding bandstructure of twisted bilayer graphene. a | Moir´e lattice in real space and b | its reciprocalspace. Primitive cells for the angle ( θ ) of interest have thousands of atoms and are often too large to be modelled by DFT. c-e | Scanning tunneling microscopy images at different angles ( θ g − g ) and related periodicity λ g − g . The images are for twistedbilayer graphene over BN, but the BN is not visible. Band structures for fully relaxed twisted bilayer graphene obtained with atight-binding model, for angles slightly above (1.08 ◦ ), very close (1.02 ◦ ), and slightly below (0.93 ◦ ) the ‘magic’ angle at whichthe bands close to the Fermi level become flat. Panels a-b are from REF. [98]. Panels c-e are from REF. [99]. Panels f-h arefrom Ref. [100]. applicable technique, based on the linear response of amaterial to an externally applied electric field and on thefluctuation–dissipation theorem. In quantum transport[112–114], this theorem relates the conductivity σ ( ω ),ie. the dissipative out-of-equilibrium response at fre-quency ω (which can be derived from the current density J ( ω ) = σ ( ω ) E ( ω ), where E ( ω ) is the applied electricfield), with the correlation function of the charge car-rier velocities, which measures the fluctuations that thesystem undergoes by applying e.g. an external (weak)electric field. In 2D materials, these fluctuations cor-respond to electronic transitions between states of thesystem at equilibrium induced by an oscillating field E ( ω ) = E cos( ωt ), which are connected to the total power absorbed per unit time and volume P = J ˙ E = σ (cid:104) E ˙ E (cid:105) = σE / . (15)Using this theorem, at first-order perturbation theory inthe electric field one obtains the Kubo conductivity [111]: σ ( ω ) = π (cid:126) e Ω (cid:88) m,n | < m | v pl | n > | δ ( E m − E n − (cid:126) ω )[ f ( E n ) − f ( E m )](16) where Ω is the sample volume, E m , E n are the energies ofthe levels m, n , f ( E n,m ) is the Fermi–Dirac distribution, e is the electric charge, and v pl is the projection of thevelocity operator v on the axial direction within the 2Dplane.0 Trions.pdf
FIG. 5:
MoS monolayer spectrum from Ai-MBP show-ing excitonic spectrum (red) and trion contribution (blue). a | Negative trions and b | positive trions exhibit resonant states(close to the A and B exciton). The A − trion is split intothree separate peaks, labeled A ( − )(1 , , . From REF. [101]. device.png FIG. 6:
Multi-scale simulation of a MoS transistordevice . The simulation of a ‘planar barristor’ is based onDFT, Green’s function formalism and Landauer-B¨uttiker for-mula. a | Transistor scheme in single gate (SG) and doublegate (DG) configuration. b | Transfer characteristics in lin-ear/semilogarithmic scale of the DG planar barristor, and ofthe SG planar barristors with different oxide thicknesses. I DS , V G and V DS stand for the drain-source current, gate voltageand drain-source voltage, respectively. From REF. [110] The calculation of the Kubo conductivity is compu-tationally very expensive owing to the numbers of or-bitals that must be included in the simulation. Efficientreal-space implementations have been proposed in Refs.[95, 113, 115]. The Kubo approach has been applied forexample to simulate transport characteristics of severalcarbon-based materials [112, 116–121] (Fig. 3d,e).Finally, the Kubo and the LB approaches can be rec-onciled and are actually equivalent in some given con-ditions, most notably if the quantum transmission atthe system/electrode interface is perfect. The Kubo ap-proach is better suited to investigate the transport prop-erties of disordered materials, characterized by localiza-tion phenomena in the low-temperature limit. For ex-ample, it has been used to compute the low-energy de-pendence of the electronic conductivity of pristine anddisordered graphene [122]. It can be directly comparedto experimental four-points transport measurements. Atvariance, the LB transport formalism is better suited tostudy charge transport through a system connected toexternal electrodes. It has a direct connection with two-point transport measurements.
D. Semiclassical approaches: Boltzmann TransportEquation
In diffusive regime, where electrons are characterisedby lifetimes τ D longer than the decoherence time l φ , par-ticles undergo a sequence of phase-breaking scatteringevents, which can be dealt with at semiclassical level.A typical example of such sequential process is theCoulomb blockade [123], where electrons are resonantlytunnelled one by one at only some values of the gatevoltage and blocked otherwise. Conductance shows a setof sharp peaks corresponding to the resonant energies,which can be tuned by changing the gate voltages. Thisprinciple can be used in single-electron transistors [124].Typically, the sequential regime is modelled by rateequations, such as the following Boltzmann transportequation (BTE): (cid:20) ∂∂t + v k · ∇ r + F · ∇ k (cid:21) f k ( r , t ) = ∂f k ( r , t ) ∂t | coll (17)in which the quantum particle dynamics, under the exter-nal force-field F , is described via a distribution functionor particle density f k ( r , t ). The term in the right handside of Eq. 17 accounts for particle-particle collisions thatdrive the system toward equilibrium.Typically the solution of the BTE [115] is found withinthe Relaxation Time Approximation (RTA), which as-sumes that ∂f k ( r , t ) ∂t | coll = − g k τ k , (18)where g k = f k − f eqk represents the variation of the distri-bution function f k ( r , t ) from the equilibrium Fermi-Dirac1distribution f eqk ( r , t ) at temperature T . The relaxationtime τ k is the time that the system takes to relax to f eqk ( r , t ) after the external force-field is switched off.The time-scale τ k is typically assumed to be inverselyproportional to the probability of scattering from the mo-mentum state | k (cid:48) (cid:105) to | k (cid:105) , as given by Fermi’s golden rule.Finally, the calculation of the Boltzmann conductivity σ within RTA is derived from the current density as: σ = − e π (cid:90) kdk (cid:18) ∂f eqk ( r , t ) ∂(cid:15) k (cid:19) τ k v k (19)where (cid:15) k can be obtained via TB model simulations.Note that the BTE is a semiclassical equation, neglectingquantum interference between particles. E. Semiclassical approaches: Monte-Carlo
A semiclassical Monte Carlo (MC) statistical methodcan be used to evaluate the carrier transport when theDe Broglie wavelength of the electrons or holes is signifi-cantly smaller than their mean separation. This methodconsiders electrons or holes as point-like particles, whosemotion within the sample is defined by classical trajec-tories [125, 126]. However, the quantum nature of thecharged particles is accounted for in the calculation ofthe elastic and inelastic scattering cross sections. TheMC approach deals with several different possible scat-tering mechanisms: (i) elastic scattering with atomic nu-clei, producing angular deviation of the electron/hole tra-jectories; (ii) inelastic e-e interaction leading to electronenergy loss, secondary electron generation and angulardeviation; (iii) electron-phonon interaction, typically in-troduced in a semi-empirical fashion at fixed energy loss;and (iv) trapping phenomena that end the trajectory(e.g. polaron quasi-particle excitations).The electron path is generally assumed to be describedby a Poisson-like law, so the step length (∆ s ) betweentwo subsequent collisions is given by:∆ s = − l tot ln( r ) , (20)where l tot = N σ is the total mean free path and r is auniformly distributed random number in the range [0 : 1].The electron transport in the sample is simulated by gen-erating a statistically significant number of trajectories,usually in excess of 10 , to achieve a low noise-to-signalratio.Using this MC approach it is possible to simulatethe anisotropic features of electron transport in layeredmaterials. An example is the calculation of the plas-monic spectrum in highly oriented pyrolytic graphite(HOPG)[96] (see Fig. 3(f)).The e-ph interaction matrix elements, whose squarevalues are directly related to the carrier-phonon scat-tering rate via Fermi’s golden rule, can be evaluatedusing a DFT/DFPT framework. Here density func-tional perturbation theory [127] (DFPT) is used to ob-tain the phononic dispersion relations, based on the DFT electronic structure. This approach has been usede.g. to estimate the temperature-dependent intrinsicscattering rates as a function of electron/hole energyin layered semiconductors. For example, such calcu-lations have been used to clarify the reason for thelower carrier mobilities measured in transition metaldichalcogenides (TMDCs), compared to graphene, andthe regimes where the mobility is defect-limited [128]or phonon-limited[129]. However, the multivalley na-ture of the band structure requires the inclusion of spin-dependent functionals to account for the spin-orbit bandsplitting, which is strong in TMDs and can affect chargetransport, particularly in the valence band. Additionally,many-body approaches beyond DFT should be used foran accurate assessment of the band energies, which canhave a severe bandgap underestimation in DFT. V. MECHANICAL PROPERTIES
While the optical, electrical, and transport propertiesof 2D materials are extensively investigated for their po-tential application in microelectronics, many of the cur-rent commercial applications of 2D materials rely on theirmechanical strength. In addition, they are also expectedto play a pivotal role in flexible electronics [130], andtherefore their mechanical properties are of major in-terest. Theoretical approaches for studying mechanicalproperties can be categorised into three classes: (1) con-tinuum mechanics (CM), e.g. based on finite elements(FEM) methods, and peridynamics [131]; (2) atomisticsimulation based on classical molecular dynamics (MD);(3) ab-initio simulations based on DFT or TB.DFT yields accurate elastic constants, especially thosecorresponding to in-plane deformations, as covalent bondenergies are quantitatively more accurate than vdW en-ergies. DFT is typically applied to defect-free materialswith a limited number of atoms in the unit cell, lowerthan 100. The elastic constants can be calculated fromthe total energy at 0 K, for selected deformations. For ex-ample, Fig. 3(g) illustrates the simulation, using DFT,of the stress-strain characteristics of several 2D carbonallotropes with density lower than graphene[97].Nevertheless, the mechanical properties of real materi-als are often dominated by defects, such as dislocations,and vary considerably from those of the perfect crystals.Thus, models at larger scale are needed. One alterna-tive are semi-empirical TB calculations,[132] which werefor example used to simulate armors based on carbon orhybrid layers [131]. In that study, a size-scale transitionfrom the nano- to the microscale in the impact behaviourwas found by adopting a multiscale approach based onFEM and CM models.For larger computational systems, in a space-time scaleof ≈ µ m and ≈ VI. OUTLOOK
The revolutionary advances that 2D materials haveknown over the last ten years have been greatly sup-ported by computational modelling. Conversely, the de-velopment of computational methods is often driven bythe need to explain new experimental results. A majormotivation that drives experimental and theoretical ef-forts is to find new materials and devices able to achievesuperior information density and higher energy efficiencywith respect to conventional electronics, or to supportflexible electronics. In parallel however, many groups in-vestigating 2D materials have been driven by the possi-bility of studying new physical phenomena. This in turnbrings new questions to be answered by theoretical andcomputational means.For example, the description of effects induced by cor-relation in confined systems remains a challenge, owingto the difficulty in describing both the stronger electron-electron interaction and the more ineffective screeningof external electromagnetic fields with respect to the 3Dcounterpart. Computational modelling is still hamperedby the widespread use of basis sets adopting 3D periodicboundary conditions, which are not naturally adapted to2D systems. This approach leads to an unnecessary com-putational expenditure and, specially in GW, to reducedaccuracy. Furthermore, pristine 2D materials are unsta-ble against the creation of flexural phonons and magnons(for ferromagnetic systems), as a result of the Mermin-Wagner-Hohenberg theorem. It is not surprising thatthese collective excitations are still challenging to modelin 2D, owing to the difficulty in accounting for substratedampening effects, or the numerical instability resultingfrom the lack of dampening in free layers.Further methodological advances are also desirable inthe computational treatment of the excitonic phenom-ena that have no counterpart in 3D. This is mainlydue to the lack of dielectric screening outside the plane,which stabilises excitons, trions, biexctions, etc. Addi-tionally, in vdW structures, the possibility of formingquasi-particles localised in different layers, such as inter-layer excitons, opens up intriguing possibilities. For ex- ample, in heterostructures with layers stacked at a fi-nite twist-angle, leading to Moir´e superlattices, it is pos-sible to choose both the momentum displacement be-tween the electron and hole bands and the confining po-tential. The computational description of the resultingMoir´e excitons, including new selection rules [146, 147]and trapped/delocalised phases [148], is an extremely ac-tive area of research.First-principles approaches able to treat electronic ex-citations, based on TD-BSE and TD-DFT, demonstratedtheir potential to interpret experimental outcomes. Nev-ertheless, ab-initio
TD-BSE is computationally expen-sive and TD-DFT suffers from a kernel that takes intoaccount accurately the excitonic effects. In this regard,the computational development is focusing on numericalapproaches that are able to accelerate the calculations.The ab-initio description of exciton-phonon polaronQPs, characterised by Coulomb correlations renormal-ized by lattice dynamics via polaronic effects, is still inits infancy. In particular, to achieve a complete theoret-ical treatment to predict the full excitonic dispersion in2D materials is a complex challenge, as one would needto include in one unified framework several microscopicingredients, such as spin–orbit coupling, exchange inter-actions, many-body correlations, and polaronic effects.Non-equilibrium exciton dynamics and its effects in theoptical and electronic properties of 2D materials, suchas the carrier-density dependent optical spectra, are verychallenging for first principles calculations but can betackled using density matrix theory.[149–151]Twisted Moir´e multilayers, as well as other 2D sys-tems, such as FeSe monolayers, have recently revealed theemergence of many interesting electronic phases, includ-ing orbital magnetism, superconductivity, or quantisedanomalous Hall effect [11–15, 152]. However, an adequatecomputational modelling of highly correlated electronicphases in 2D materials is still lacking [153, 154], particu-larly in presence of external magnetic fields, such as thoseroutinely applied to probe these phases in experiments.Ferromagnetic systems are also an emerging areawithin 2D materials research. Although there is atpresent no doubt that 2D ferromagnetic materials are sta-ble, there is still lack of information on the role of bound-ary conditions, substrate, etc. in stabilising magnetism.Similar remarks apply to 2D ferroelectrics: for example,it is not completely understood the critical thickness de-pendence of ferroelectric polarization, as well as the rele-vant experimental conditions. Since ferromagnetism andferroelectricity are very dependent on defects and bound-ary conditions, we believe that computational studies willbe paramount to unravel these questions. Other impor-tant emerging areas that we must mention are related tothe modelling of magnetic excitations, such as skyrmions.We point out also the need for developing computa-tional methods to calculate macroscopic observables thatare scalable and computationally effective in terms ofmemory requirements and CPU usage, notably for mod-elling realistic systems. There is still ample room for de-3velopment of first-principles based multiscale approachesto calculate macroscopic observables and to link the prop-erties of 2D materials to the measurable electrical re-sponse of real devices, such as carrier mobilities, ther-mal conductivity, ionic conductivity, spin lifetime, etc.While computer simulations and experimental measure-ments have quantified the intrinsic carrier mobility ofquite a few 2D materials, its assessment in presence ofsubstrates and external fields, at real-scale geometries, isstill challenging for computer simulations. Further, manysystems of technological interest are not crystalline butamorphous (such as graphene oxide), or are molecular 2Dlayers (such as 2D water).Finally, emerging areas of computational research in2D materials modelling aim to machine learning modelsand inverse design[1, 2, 155–157]. Machine learning is being used e.g. to revolutionise the fitting of classicalinter-atomic potentials or other approximate models byreplacing the considerable human effort with an artifi-cial neural network [158, 159]. This may change classicalmodelling approaches and make them promptly availablein the area of application.
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