All-optical band engineering of gapped Dirac materials
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a r All-optical band engineering of gapped Dirac materials
O. V. Kibis , , , ∗ K. Dini , I. V. Iorsh , , and I. A. Shelykh , Department of Applied and Theoretical Physics, Novosibirsk State Technical University,Karl Marx Avenue 20, Novosibirsk 630073, Russia Science Institute, University of Iceland, Dunhagi 3, IS-107, Reykjavik, Iceland Division of Physics and Applied Physics, Nanyang Technological University 637371, Singapore and ITMO University, Saint Petersburg 197101, Russia
We demonstrate theoretically that the interaction of electrons in gapped Dirac materials (gappedgraphene and transition-metal dichalchogenide monolayers) with a strong off-resonant electromag-netic field (dressing field) substantially renormalizes the band gaps and the spin-orbit splitting.Moreover, the renormalized electronic parameters drastically depend on the field polarization.Namely, a linearly polarized dressing field always decreases the band gap (and, particularly, canturn the gap into zero), whereas a circularly polarized field breaks the equivalence of valleys in dif-ferent points of the Brillouin zone and can both increase and decrease corresponding band gaps. Asa consequence, the dressing field can serve as an effective tool to control spin and valley propertiesof the materials and be potentially exploited in optoelectronic applications.
PACS numbers: 73.22.Pr, 78.67.Wj
I. INTRODUCTION
Advances in laser physics and microwave techniqueachieved in recent decades have made possible the useof high-frequency fields as tools of flexible control of var-ious atomic and condensed-matter structures (so called“Floquet engineering” based on the Floquet theory ofperiodically driven quantum systems ). As a conse-quence, the properties of electronic systems driven byoscillating fields are actively studied to exploit uniquefeatures of composite states of field and matter. Particu-larly, electron strongly coupled to electromagnetic field— also known as “electron dressed by field” (dressedelectron) — has become a commonly used model inmodern physics . Recently, the physical propertiesof dressed electrons were studied in various nanostruc-tures, including quantum wells , quantum rings ,graphene , and topological insulators . Develop-ing this excited scientific trend in the present article,we elaborated the theory of dressed electrons for gappedDirac materials.The discovery of graphene — a monolayer of carbonatoms with linear (Dirac) dispersion of electrons —initiated studies of the new class of artificial nanostruc-tures known as Dirac materials. While graphene by itselfis characterized by the gapless electron energy spectrum,many efforts have been dedicated towards fabricationDirac materials with the band gap between the valenceand conduction bands (gapped Dirac materials). Theelectron energy spectrum of the materials is parabolicnear band edges but turns into the linear Dirac disper-sion if the band gap vanishes. Therefore, electronic prop-erties of gapped Dirac materials substantially depend onthe value of the gap and, consequently, are perspectivefor nanoelectronic applications . Although dressedcondensed-matter structures are in focus of attention fora long time, a consistent quantum theory of the gappedDirac materials strongly coupled to light was not elab- orated before. Since the electronic structure of Diracmaterials differs crucially from conventional condensed-matter structures, the known theory of light-matter cou-pling cannot be directly applied to the gapped Dirac ma-terials. Moreover, it should be noted that gapped Diracmaterials are currently considered as a basis for new gen-eration of optoelectronic devices. Therefore, their opti-cal properties deserve special consideration. This mo-tivated us to fill this gap in the theory. To solve thisproblem in the present study, we will focus on the twogapped Dirac materials pictured schematically in Fig. 1.First of them is the graphene layer grown on a hexago-nal boron nitride substrate , where the band gap canbe tuned in the broad range with an external gate volt-age (see Fig. 1a). The second is a transition metaldichalchogenide (TMDC) which is a monolayer of atom-ically thin semiconductor of the type MX , where M is atransition metal atom (Mo, W, etc.) and X is a chalco-gen atom (S, Se, or Te) (see Fig. 1b). The specificfeature of the TMDC compounds is the giant spin-orbitcoupling which is attractive for using in novel spin-tronic and valleytronic devices . Formally, electronicproperties of these materials near the band edge can bedescribed by the same two-band Hamiltonianˆ H = (cid:18) ε cτs γ ( τ k x − ik y ) γ ( τ k x + ik y ) ε vτs (cid:19) , (1)where k = ( k x , k y ) is the electron wave vector in the layerplane, γ is the parameter describing electron dispersion, ε cτs = ∆ g τ s ∆ cso ε vτs = − ∆ g − τ s ∆ vso g is the bandgap between the conduction band and the valence band, (a) (b) EM wave (dressing field) EM wave (dressing field)
FIG. 1: (Color online) Sketch of the considered gapped Diracmaterials subject to electromagnetic wave (dressing field): (a)Graphene grown on the substrate of hexagonal boron nitride;(b) Transition metal dichalcogenide monolayer MoS . ∆ c,vso is the spin-orbit splitting of the conduction (valence)band, s = ± τ = ± K and K ′ valleys in graphene and the K and − K valleys in TMDC monolayers ). If∆ g = 0 and ∆ c,vso = 0, the Hamiltonian (4) describesTMDC monolayer . In the case of zero spin-orbit split-ting, ∆ c,vso = 0, the Hamiltonian (4) describes gappedgraphene , whereas the case of ∆ g = ∆ c,vso = 0 corre-sponds to usual gapless graphene . It should be notedthat the two-band Hamiltonian (1) describes successfullylow-energy electron states near the band edge. As toomitted terms corresponding to the trigonal-warping de-formation of electron bands in monolayer graphene, theycan be neglected if the Rashba spin-orbit coupling isstronger than the intrinsic spin-orbit coupling . In thepresent paper, we elaborate the theory of electromag-netic dressing for electronic systems described by the low-energy Hamiltonian (1) and demonstrate that both theband gap and the spin splitting can be effectively con-trolled with the dressing field.The paper is organized as follows. In the Section II,we apply the conventional Floquet theory to derive theeffective Hamiltonian describing stationary properties ofdressed electrons. In the Section III, we discuss the de-pendence of renormalized electronic characteristics of thedressed materials on parameters of the dressing field. Thelast two Sections contain conclusion and acknowledge-ments. II. MODEL
Let us consider a gapped Dirac material with theHamiltonian (1), which lies in the plane ( x, y ) at z = 0and is subjected to an electromagnetic wave propagat-ing along the z axis (see Fig. 1). The frequency of thewave, ω , is assumed to be far from all resonant frequen-cies of the electron system. Therefore, the electromag- netic wave cannot be absorbed by electrons near bandedge and should be considered as a dressing field for thestates around k = 0. Considering the electron-field inter-action within the minimal coupling approach, propertiesof dressed electrons can be described by the Hamiltonianˆ H ( k ) = (cid:18) | e | γ ( τ A x − iA y ) / ~ | e | γ ( τ A x + iA y ) / ~ (cid:19) + (cid:18) ε cτs γ ( τ k x − ik y ) γ ( τ k x + ik y ) ε vτs (cid:19) , (4)which can be easily obtained from the Hamiltonian of“bare” electrons (1) with the replacement k → k − ( e/ ~ ) A , where A = ( A x , A y ) is the vector potentialof the dressing field, and e is the electron charge. Itshould be noted that the quantum electrodynamics pre-dicts the quadratic (in the vector potential) additions tothe Hamiltonian (4) . To avoid complication of themodel, we will assume that the considered dressing field isclassically strong and can be described successfully withthe minimal coupling. In what follows, we will show thatthe properties of dressed electrons strongly depend onthe polarization of the dressing field. Therefore, we haveto discuss the solution of the corresponding Schr¨odingerproblem for different polarizations successively. Linearly polarized dressing field.—
Assuming the dress-ing field to be linearly polarized along the x axis, thevector potential can be written as A = (cid:18) E ω cos ωt, (cid:19) , (5)where E is the electric field amplitude, and ω is thewave frequency. Correspondingly, the Hamiltonian of thedressed electron system (4) can be rewritten formally asˆ H ( k ) = ˆ H + ˆ H k , (6)where ˆ H = (cid:18) τ ~ ω/ τ ~ ω/ (cid:19) cos ωt (7)is the Hamiltonian of electron-field interaction,ˆ H k = (cid:18) ∆ g / τ s ∆ cso / γ ( τ k x − ik y ) γ ( τ k x + ik y ) − ∆ g / − τ s ∆ vso / (cid:19) (8)is the Hamiltonian of “bare” electron, andΩ = 2 γ | e | E ( ~ ω ) (9)is the dimensionless parameter describing the strength ofelectron coupling to the dressing field. The nonstationarySchr¨odinger equation with the Hamiltonian (7), i ~ ∂ψ ∂t = ˆ H ψ , (10)describes the time evolution of electron states at the bandedge ( k = 0). The two exact solutions of the Schr¨odingerproblem (10) read as ψ ± = 1 √ (cid:18) ± (cid:19) exp (cid:20) ∓ i Ω τ sin ωt (cid:21) . (11)Since the two wave functions (11) form a complete basisat any fixed time t , we can seek solutions of the nonsta-tionary Schr¨odinger equation with the full Hamiltonian(6) as an expansion ψ k = a ( t ) ψ +0 + a ( t ) ψ − . (12)Substituting the expansion (12) into the Schr¨odingerequation, i ~ ∂ψ k ∂t = ˆ H ( k ) ψ k , (13)we arrive at the expressions i ~ ˙ a ( t ) = (cid:20) ε cτs + ε vτs γτ k x (cid:21) a ( t )+ (cid:20) ε cτs − ε vτs iγk y (cid:21) e i Ω τ sin ωt a ( t ) ,i ~ ˙ a ( t ) = (cid:20) ε cτs + ε vτs − γτ k x (cid:21) a ( t )+ (cid:20) ε cτs − ε vτs − iγk y (cid:21) e − i Ω τ sin ωt a ( t ) . (14)It follows from the conventional Floquet theory of quan-tum systems driven by an oscillating field that thesought wave function (12) must have the form Ψ( r , t ) = e − i ˜ ε ( k ) t/ ~ φ ( r , t ), where the function φ ( r , t ) periodicallydepends on time, φ ( r , t ) = φ ( r , t + 2 π/ω ), and ˜ ε ( k ) is thequasi-energy of an electron. Since the quasi-energy (theenergy of dressed electron) is the physical quantity whichplays the same role in quantum systems driven by an os-cillating field as the usual energy in stationary ones, thepresent analysis of the Schr¨odinger problem (13) shouldbe aimed to find the energy spectrum, ˜ ε ( k ). It followsfrom the periodicity of the function φ ( r , t ) that one canseek the coefficients a , ( t ) in Eq. (12) as a Fourier ex-pansion, a , ( t ) = e − i ˜ ε ( k ) t/ ~ ∞ X n = −∞ a ( n )1 , e inωt . (15)Substituting the expansion Eq. (15) into the Eqs. (14)and applying the Jacobi-Anger expansion, e iz sin θ = ∞ X n = −∞ J n ( z ) e inθ , one can rewrite the equations of quantum dynamics (14)in the time-independent form, ∞ X n ′ = −∞ X j =1 H ( nn ′ ) ij a ( n ′ ) j = ˜ ε ( k ) a ( n ) i , (16) where J n ( z ) is the Bessel function of the first kind, and H ( nn ′ ) ij is the stationary Hamiltonian of dressed electronin the Floquet space with the matrix elements H ( nn ′ )12 = (cid:20) ε cτs − ε vτs iγk y (cid:21) J n ′ − n (Ω τ ) , H ( nn ′ )21 = (cid:20) ε cτs − ε vτs − iγk y (cid:21) J n ′ − n (Ω τ ) , H ( nn ′ )11 = (cid:20) ε cτs + ε vτs γτ k x + n ~ ω (cid:21) δ nn ′ , H ( nn ′ )22 = (cid:20) ε cτs + ε vτs − γτ k x + n ~ ω (cid:21) δ nn ′ , (17)where δ nn ′ is the Kronecker delta. It should be notedthat the Schr¨odinger equation (16) describes still exactlythe initial Schr¨odinger problem (13). Next we will makesome approximations.In what follows, let us assume that the field frequency, ω , is high enough to satisfy the condition (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H (0 n ) ij H (00) ii − H ( nn ) jj (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ n = 0 and i = j . Mathematically, the condition(18) makes it possible to treat nondiagonal matrix ele-ments of the Hamiltonian (17) with n = n ′ as a smallperturbation which can be omitted in the first-order ap-proximation of the conventional perturbation theory formatrix Hamiltonians (see, e.g., Ref. 48). Since the con-dition (18) corresponds to an off-resonant field, the fieldcan be neither absorbed nor emitted by the electrons. Asa consequence, the main contribution to the Schr¨odingerequation (16) under the condition (18) stems from termswith n, n ′ = 0, which describe the elastic interaction be-tween an electron and the field. Neglecting the smallterms with n, n ′ = 0, the Schr¨odinger equation (16) canbe rewritten in the simple form X j =1 H (00) ij a (0) j = ˜ ε ( k ) a (0) i , (19)where H (00) ij is the 2 × U = 1 √ (cid:18) − (cid:19) , we arrive at the effective stationary Hamiltonian ofelectrons dressed by a linearly polarized field, ˆ H eff = U † H (00) U , which is given by the matrixˆ H eff ( k ) = (cid:18) e ∆ g / τ s e ∆ cso / τ ˜ γ x k x − i ˜ γ y k y τ ˜ γ x k x + i ˜ γ y k y − e ∆ g / − τ s e ∆ vso / (cid:19) , (20)where e ∆ g = ∆ g J (Ω) (21)is the effective band gap, e ∆ cso = ∆ cso − ∆ vso cso + ∆ vso J (Ω) (22)is the effective spin splitting of the conduction band, e ∆ vso = ∆ vso − ∆ cso cso + ∆ vso J (Ω) (23)is the effective spin splitting of the valence band, and˜ γ x = γ, ˜ γ y = γJ (Ω) (24)are the effective parameters of electron dispersion alongthe x, y axes. The eigenenergy of the effective Hamilto-nian (20),˜ ε ± τs ( k ) = τ s ( e ∆ cso − e ∆ vso )4 (25) ± vuut" e ∆ g + τ s ( e ∆ cso + e ∆ vso )4 + ˜ γ x k x + ˜ γ y k y , is the sought energy spectrum of electrons dressed by thelinearly polarized field. Mathematically, the unperturbedHamiltonian (1) is equal to the effective Hamiltonian (20)with the formal replacements ∆ g → e ∆ g , ∆ c,vso → e ∆ c,vso , γ x,y → ˜ γ x,y . Therefore, the behavior of a dressed elec-tron is similar to the behavior of a “bare” electron withthe renormalized band parameters (21)–(24). It shouldbe noted that the effective Hamiltonian (20) is derivedunder the condition (18). Taking into account Eqs. (17),the condition (18) can be rewritten as γk, ∆ g ≪ ~ ω .Therefore, the effective Hamiltonian is applicable to de-scribe the dynamics of dressed electron near the bandedge if the photon energy of the dressed field, ~ ω , sub-stantially exceeds the band gap, ∆ g . It should be notedthat such high-energy photons will lead to direct electrontransitions between the valence band and the conductionband. However, the transitions take place very far fromthe band edge and do not effect on the considered renor-malization of low-energy electron states. Circularly polarized dressing field.—
For the case ofcircularly polarized electromagnetic wave, the vector po-tential A = ( A x , A y ) can be written as A = (cid:18) E ω cos ξωt, E ω sin ξωt (cid:19) , (26)where the different chirality indices ξ = ± k = 0, where the Hamiltonian (4) can be writtenin the formˆ H (0) = (cid:18) ε cτs − ( ~ ω Ω τ / e − iτξωt − ( ~ ω Ω τ / e iτξωt ε vτs (cid:19) , (27)which is similar to the well-known Hamiltonian ofmagnetic resonance. The corresponding nonstationary Schr¨odinger equation, i ~ ∂ψ τs (0) ∂t = ˆ H (0) ψ τs (0) , (28)describes the time evolution of electron states at the wavevector k = 0. Solutions of the equation (28) can besought as ψ ± τs (0) = e − i ˜ ε ± τs (0) t/ ~ (cid:18) A ± e − iτξωt/ B ± e iτξωt/ (cid:19) e ± iτξωt/ , (29)where ˜ ε ± τs (0), A ± and B ± are the undefined constants.Substituting the wave function (29) into the Schr¨odingerequation (28) with the Hamiltonian (27), we arrive at thesystem of two algebraic equations, A ± (cid:20) ε cτs − τ ξ ~ ω ∓ − ˜ ε ± τs (0) (cid:21) − B ± ~ ω Ω τ ,A ± ~ ω Ω τ − B ± (cid:20) ε vτs + τ ξ ~ ω ± − ˜ ε ± τs (0) (cid:21) = 0 , (30)which can be easily solved. As a result, the two orthonor-mal exact solutions of the Schr¨odinger problem (28) are ψ ± τs (0) = e − i ˜ ε ± τs (0) t/ ~ e ± iτξωt/ × ∓ h √ Ω + δ ±| δ | √ Ω + δ i / e − iτξωt/ sgn( δ ) h √ Ω + δ ∓| δ | √ Ω + δ i / e iτξωt/ , (31)where˜ ε ± τs (0) = ε cτs + ε vτs ± τ ξ ~ ω ± sgn( δ ) ~ ω p Ω + δ (32)is the quasienergy (energy of dressed electron in the con-duction/valence band) at k = 0, and δ = ε cτs − ε vτs − τ ξ ~ ω ~ ω is the resonance detuning assumed to be nonzero in or-der to avoid the field absorption near the band edge.Correspondingly, the effective stationary Hamiltonian ofdressed electron states at k = 0 can be written in thebasis (31) as ˆ H eff (0) = (cid:18) ˜ ε + τs (0) 00 ˜ ε − τs (0) (cid:19) . (33)In order to find the energy spectrum of dressed electronat the wave vector k = 0, let us restrict the considerationby the case of Ω ≪
1, which corresponds physically tohigh frequencies ω [see Eq. (9)]. Expanding the electronwave function, ψ τs ( k ), on the basis (31), ψ τs ( k ) = a + ( t ) e i ˜ ε + τs (0) t/ ~ ψ + τs (0) + a − ( t ) e i ˜ ε − τs (0) t/ ~ ψ − τs (0) , (34)and substituting the expansion (34) into the Schr¨odingerequation with the total Hamiltonian (4), we arrive at thesystem of equations i ~ ˙ a + ( t ) ≈ ˜ ε + τs (0) a + ( t ) − sgn( δ ) γ ( τ k x − ik y ) a − ( t ) ,i ~ ˙ a − ( t ) ≈ ˜ ε − τs (0) a − ( t ) − sgn( δ ) γ ( τ k x + ik y ) a + ( t ) . (35)The quantum dynamics equations (35) are equal to thestationary Schr¨odinger equation, i ~ ∂∂t (cid:18) a + ( t ) a − ( t ) (cid:19) = ˆ H eff ( k ) (cid:18) a + ( t ) a − ( t ) (cid:19) , whereˆ H eff ( k ) = (cid:18) ˜ ε + τs (0) − sgn( δ ) γ ( τ k x − ik y ) − sgn( δ ) γ ( τ k x + ik y ) ˜ ε − τs (0) (cid:19) . (36)is the effective stationary Hamiltonian of the consideredsystem. The eigenenergy of the Hamiltonian,˜ ε ± τs ( k ) = ˜ ε + τs (0) + ˜ ε − τs (0)2 ± s(cid:20) ˜ ε + τs (0) − ˜ ε − τs (0)2 (cid:21) + ( γk ) , (37)presents the sought energy spectrum of dressed electrons.If ∆ g = ∆ c,vso = 0, Eq. (37) exactly coincides with theknown spectrum of electrons in gapless graphene irra-diated by a circularly polarized light . It follows fromEq. (37) that the renormalized band gap is e ∆ g = τ ξ ~ ω + sgn (∆ g − τ ξ ~ ω ) ~ ω s Ω + (cid:20) ∆ g − τ ξ ~ ω ~ ω (cid:21) ≈ τ ξ ~ ω − p Ω + 1 (cid:18) τ ξ ~ ω − ∆ g Ω + 1 (cid:19) , (38)where the last equality holds under condition ~ ω ≫ ∆ g .The spin splittings in the conduction and valence bandscan be written in simple form for the two limiting cases: e ∆ c,vso = ± ∆ cso − ∆ vso cso + ∆ vso √ , ~ ω ≫ ∆ g , (39)and e ∆ c,vso = ± ∆ cso − ∆ vso cso + ∆ vso (cid:20) − Ω ~ ω ) ∆ g (cid:21) , ~ ω ≪ ∆ g . (40)As expected, the renormalized band gap (38) and spinsplittings (39)–(40) turn into their “bare” values, ∆ g and∆ c,vso , if the dressing field is absent ( E → Elliptically polarized dressing field.—
Assuming thelarge axis of polarization ellipse to be oriented along the x axis, the vector potential of arbitrary polarized electro-magnetic wave, A = ( A x , A y ), can be written as A = E ω (cid:16) cos ωt, sin θ sin ωt (cid:17) , (41) where θ ∈ [ − π/ , π/
2] is the polarization phase: the po-larization is linear for θ = 0, circular for θ = ± π/ θ . Substituting the vec-tor potential (41) into Eq. (4), we can write the totalHamiltonian (4) asˆ H ( k ) = ˆ H k + (cid:16) ˆ V e iωt + ˆ V † e − iωt (cid:17) , (42)where the Hamiltonian ˆ H k is given by Eq. (8) andˆ V = ~ ω Ω4 (cid:18) τ − sin θτ + sin θ (cid:19) . (43)is the operator of electron interaction with the dressingfield (41). Generally, the effective stationary Hamiltonianof an electron driven by an oscillating field can be soughtin the form ˆ H eff ( k ) = e i ˆ F ( t ) ˆ H ( k ) e − i ˆ F ( t ) + i ∂e i ˆ F ( t ) ∂t ! e − i ˆ F ( t ) , (44)where ˆ F ( t ) is the anti-Hermitian operator which is pe-riodical with the period of the oscillating field, ˆ F ( t ) =ˆ F ( t + 2 π/ω ). In the particular case of weak electron-fieldcoupling, Ω ≪
1, this operator and the effective Hamilto-nian (44) can be easily found as power series expansions,ˆ F ( t ) = ∞ X n =1 F ( n ) ( t ) ω n , ˆ H eff ( k ) = ∞ X n =0 ˆ H ( n )eff ( k ) ω n , (45)where F ( n ) ( t ) ∼ Ω n (the Floquet-Magnus expansion ).Substituting the expansions (45) into Eq. (44) and re-stricting the accuracy by terms ∼ Ω , we arrive at theeffective Hamiltonianˆ H eff ( k ) = ˆ H k + h ˆ V , ˆ V † i ~ ω + [[ ˆ V , ˆ H k ] , ˆ V † ] + h.c. ~ ω ) . (46)Taking into account Eqs. (8) and (43), the effective sta-tionary Hamiltonian (46) can be written as a matrix (20),where e ∆ g = ∆ g (cid:20) − Ω θ ) (cid:21) − τ ~ ω Ω θ, (47) e ∆ c,vso = ± ∆ cso − ∆ vso cso + ∆ vso × (cid:20) − Ω (cid:0) θ (cid:1)(cid:21) , (48)˜ γ x = γ (cid:20) − Ω θ (cid:21) , ˜ γ y = γ (cid:20) − Ω (cid:21) (49)are the band parameters renormalized by an ellipticallypolarized dressing field. Correspondingly, the eigenen-ergy of the effective Hamiltonian (46) represents thesought energy spectrum of dressed electrons,˜ ε ± τs ( k ) = τ s ( e ∆ cso − e ∆ vso )4 (50) ± vuut" e ∆ g + τ s ( e ∆ cso + e ∆ vso )4 + ˜ γ x k x + ˜ γ y k y , with the renormalized band parameters (47)–(49). Itshould be stressed that the effective Hamiltonian (20)with the band parameters (47)–(49), which describeselectrons dressed by an arbitrary polarized weak field,is derived under assumption of small coupling constant(9) and high frequency, ω . On the contrary, the effec-tive Hamiltonian (20) with the band parameters (21)–(24) and the effective Hamiltonian (33) are suitable todescribe electrons dressed by linearly and circularly po-larized dressing fields of arbitrary intensity. As a con-sequence, the band parameters (21)–(24) and (38)–(39)turn into the band parameters (47)–(49) for Ω ≪ ~ ω ≫ ∆ g , and θ = 0 , ± π/ III. RESULTS AND DISCUSSION
First of all, let us apply the developed theory to gappedgraphene, assuming e ∆ c,vso = 0 in all derived expressions.The electron dispersion in gapped graphene, ˜ ε ( k ), is plot-ted in Fig. 2 for the particular cases of linearly and cir-cularly polarized dressing field. It the absence of thedressing field, the electron dispersion is isotropic in thegraphene plane (see the solid lines in Figs. 2a and 2b).However, a linearly polarized field breaks the equiva-lence of the x, y axes [see Eq. (25)]. As a consequence,the anisotropy of the electron dispersion along the wavevectors k x and k y appears (see the dashed and dottedlines in Figs. 2a and 2b). In contrast to the linear po-larization, a circularly polarized dressing field does notinduce the in-plane anisotropy [see Eq. (37)]. However,the electron dispersion is substantially different for clock-wise and counterclockwise polarizations (see the dashedand dotted lines in Fig. 2c). Moreover, both linearlyand circularly polarized field renormalizes the band gap(see Fig. 3). Mathematically, the dependence of therenormalized band gap, | e ∆ g | , on the irradiation inten-sity, I ∼ E , is given by Eqs. (21) and (38) [whichare plotted in Fig. 3a] and Eq. (47) [which is plottedin Fig. 3b]. It should be noted that Eq. (38) correctlydescribes the gap for any off-resonant frequencies ω ,whereas Eqs. (21) and (47) are derived under the con-dition ~ ω ≫ ∆ g and, therefore, applicable only to smallgaps. However, the gap can be gate-tunable in the broadrange, ∆ g = 1 −
60 meV . Assuming the gap to be ofmeV scale and the field frequency to be in the terahertzrange, we can easily satisfy this condition. It follows from -1.0 -0.5 0.0 0.5 1.0-1.5-1.0-0.50.00.51.01.5 E n e r gy , ε ( m e V ) Wavevector, k y ( µ m -1 ) (b)(a) ~ -1.0 -0.5 0.0 0.5 1.0-1.5-1.0-0.50.00.51.01.5 E n e r gy , ε ( m e V ) Wavevector, k x ( µ m -1 ) -1.0 -0.5 0.0 0.5 1.0-2.0-1.5-1.0-0.50.00.51.01.52.0 E n e r gy , ε ( m e V ) Wavevector, k ( µ m -1 ) (с) ~~ FIG. 2: (Color online) The energy spectrum of dressed elec-tron, ˜ ε ( k ), near the band edge of gapped graphene (∆ g =2 meV, γ/ ~ = 10 m/s) irradiated by a dressing field withthe photon energy ~ ω = 10 meV and the different intensities, I . In the parts (a) and (b): the dressing field is linearly po-larized along the x axis; the irradiation intensities are I = 0(solid lines), I = 7 . / cm (dashed lines), I = 15 kW / cm (dotted lines). In the part (c): the dressing field is circu-larly polarized; the solid line describes the energy spectrum of“bare” electron ( I = 0), whereas the dotted and dashed linescorrespond to the different circular polarizations ( τ ξ = − τ ξ = 1, respectively) with the same irradiation intensity I = 300 W / cm . Eqs. (21), (32) and (47) that the renormalized gap, e ∆ g ,crucially depends on the field polarization. Particularly,the clockwise/counterclockwise circularly polarized field(polarization indices ξ = ±
1) differently interacts withelectrons from different valleys of the Brillouin zone (val-ley indices τ = ± τ ξ = − τ ξ = 1, the gap first decreases to zero andthen starts to grow (see the solid line in Fig. 3a). Thislight-induced difference in the band gaps for the two dif-ferent valleys is formally equivalent to the appearance ofan effective magnetic field acting on the valley pseudo- P o l a r i za ti on ph a s e , θ Irradiation intensity, I (kW/cm ) ~ | ∆ g / ∆ g | (a) (b) Irradiation intensity, I (W/cm ) B a nd g a p , ~ | ∆ g / ∆ g | Band gap,
FIG. 3: (Color online) Dependence of the band gap in irra-diated gapped graphene (∆ g = 2 meV, γ/ ~ = 10 m/s) onthe irradiation intensity, I , and the polarization, θ , for thephoton energy ~ ω = 10 meV. In the part (a): the dotted linecorresponds to the linearly polarized dressing field, whereasthe dashed and solid lines correspond to the different circu-lar polarizations ( τ ξ = − τ ξ = 1, respectively). In thepart (b): the dashed lines correspond to the polarizations, θ ,which do not change the band gap. spin and, therefore, can be potentially used in valleytron-ics applications. It should be noted that this optically-induced lifting of valley degeneracy has been observed forTMDC in the recent experiments which are in reason-able agreement with the present theory. As to linearlypolarized dressing field, it always quenches the band gapand can even turn it into zero (see the dotted line inFig. 3a). Formally, the collapse of the band gap origi- nates from zeros of the Bessel function in Eq. (21). Sincethe linearly and circularly polarized fields change the gapvalue oppositely, there are field polarizations which donot change the gap. The polarization phases, θ , corre-sponding to such polarizations are marked by the dashedlines in Fig. 3b.Applying the elaborated theory to analyze the renor-malized spin splitting in TMDC monolayers, let us re-strict the consideration by the most examined TMDCmonolayer MoS . The dependence of the spin splittingon the dressing field is described by Eqs. (22)–(23) forthe case of linearly polarized field, Eqs. (39)–(40) for thecase of circularly polarized field, and Eq. (48) for an ar-bitrary polarized field. It follows from analysis of theseexpressions that the most pronounced renormalization ofthe splitting takes place for a circularly polarized field.The dependence of the field-induced renormalization ofthe band gap, e ∆ g , and the spin splitting, e ∆ c,vso , in MoS monolayer on the field intensity, I ∼ E , is plotted inFig. 4 for such a field. It is seen in Fig. 4a that abso-lute values of the field-induced renormalization are of thesame order for both the band gap, e ∆ g − ∆ g , and the spinsplitting, e ∆ c,vso − ∆ c,vso . However, the unperturbed spinsplitting of the conduction band, ∆ cso , is small as com-pared to both the unperturbed band gap, ∆ g , and theunperturbed spin splitting of the valence band, ∆ vso (seeRef. 43). Therefore, the relative field-induced renormal-ization, | e ∆ cso / ∆ cso | , is most pronounced for the spin split-ting of the conduction band (see Fig. 4b). It should bestressed that the renormalized splitting depends on theproduct of the polarization and valley indices, τ ξ = ± g . In contrast to gapped graphene with bandgaps of meV scale, TMDC monolayers have band gaps ofeV scale . As a consequence, the considered terahertzphotons effect on the gaps of TMDC very weakly (in con-trast to the previously considered case of narrow-gappedgraphene). Particularly, it follows from this that the ir-radiation intensity which collapses the spin splitting inTMDCs monolayers (see Fig. 4b) is really large than theintensity collapsing the band gap in gapped graphene (seeFig. 3a).It should be noted that the similar optically-inducedspin splitting was recently observed experimentally inGaAs . However, the one-band energy spectrum of con-duction electrons in GaAs differs crucially from the elec-tron spectrum of gapped Dirac materials describing bythe two-band Hamiltonian (1). Therefore, the knowntheory of optically-induced spin splitting for electronswith simple parabolic dispersion — including both therecent paper and the classical article — cannot by ap-plied directly to the materials under consideration. Onehas to take also into account that optical properties ofTMDC are dominated by excitons . To avoid the in-fluence of excitons on the discussed dressing-field effects,the photon energy, ~ ω , should be less than the bindingexciton energy (which is typically of hundreds of meV inTMDC). ~ ∆ s o / ∆ s o cc S p i n s p litti ng , Irradiation intensity, I (MW/cm ) I (MW/cm ) ~ ∆ s o / ∆ s o cc || || (b) C h a ng i ng g a p ( m e V ) ∆ so - ∆ so ~ c c ∆ so - ∆ so ~ v v ∆ g - ∆ g ~ (a) FIG. 4: (Color online) Dependence of the band gap, e ∆ g , andthe spin splitting of conduction and valence bands, e ∆ c,vso , onthe irradiation intensity for MoS monolayer (∆ g = 1 .
58 eV,∆ cso = 3 meV, ∆ vso = 147 meV, γ/ ~ = 7 . × m/s) irra-diated by a circularly polarized field with the photon energy ~ ω = 10 meV: (a) The field-induced changing of the bandgap and the spin splitting for the circularly polarized fieldwith τ ξ = −
1; (b) The renormalized spin-slitting of conduc-tion band for different field polarizations (the solid and dashedlines correspond to τ ξ = − τ ξ = 1, respectively). From viewpoint of experimental observability of thediscussed phenomena, it should be noted that alldressing-field effects increase with increasing the intensityof the dressing field. However, an intense irradiation canmelt a condensed-matter sample. To avoid the melting,it is reasonable to use narrow pulses of a strong dressingfield. This well-known methodology has been elaboratedlong ago and commonly used to observe various dress-ing effects — particularly, modifications of energy spec-trum of dressed electrons arisen from the optical Starkeffect — in semiconductor structures (see, e.g., Refs. 54– 56). Within this approach, giant dressing fields (up toGW/cm ) can be applied to the structures. It should benoted also that we consider the electromagnetic wave asa purely dressing field which cannot be absorbed by elec-trons. Within the classical Drude theory, the collisionalabsorption of the oscillating field (5) by conduction elec-trons is given by the well-known expression Q = 1 T Z T j ( t ) E ( t )d t = E σ ωτ ) , where T is the period of the field, Q is the period-averaged field energy absorbed by conduction electronsper unit time and per unit volume, j ( t ) is the ohmiccurrent density induced by the oscillating electric field E ( t ) = E sin ωt , σ is the static Drude conductivity,and τ is the electron relaxation time. Evidently, theDrude optical absorption, Q , is negligibly small underthe condition ωτ ≫
1. Thus, an electromagnetic wavecan be considered as a purely dressing field in the high-frequency limit (see, e.g., Ref. 10 for more details). Itshould be stressed that the increasing of temperature de-creases the time τ because of the strengthening of theelectron-phonon scattering. Therefore, the temperatureshould be low enough to meet the aforementioned condi-tion. IV. CONCLUSION
We showed that the electromagnetic dressing can beused as an effective tool to control various electronicproperties of gapped Dirac materials, including the bandgap in gapped graphene and the spin splitting in TMDCmonolayers. Particularly, both the band gap and the spinsplitting can be closed by a dressing field. It is demon-strated that the strong polarization dependence of therenormalized band parameters appears. Namely, a lin-early polarized field decreases the band gap, whereas acircularly polarized field can both decrease and increaseone. It is found also that a circularly polarized fieldbreaks equivalence of valleys in different points of theBrillouin zone, since the renormalized band parametersdepend on the valley index. As a consequence, the elabo-rated theory creates a physical basis for novel electronic,spintronic and valleytronic devices operated by light.
Acknowledgments
The work was partially supported by the RISE projectCoExAN, FP7 ITN project NOTEDEV, RFBR projects16-32-60123 and 17-02-00053, the Rannis projects141241-051 and 163082-051, and the Russian Min-istry of Education and Science (projects 3.1365.2017,3.2614.2017 and 3.4573.2017). O.V.K. and I.V.I. ac-knowledge support from the Singaporean Ministry of Ed-ucation under AcRF Tier 2 grant MOE2015-T2-1-055. ∗ Electronic address: Oleg.Kibis(c)nstu.ru P. H¨anngi, Driven quantum systems, in
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