Equations of macroscopic electrodynamics for two-dimensional crystals
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a r Equations of macroscopic electrodynamics for two-dimensional crystals
S. A. Mikhailov ∗ Institute of Physics, University of Augsburg, D-86135 Augsburg, Germany (Dated: March 5, 2019)The electrodynamics of two-dimensional (2D) dielectric and conducting layers cannot be describedby such three-dimensional macroscopic quantities as the dielectric constant ǫ or the refractive index n . By means of the proper averaging of the microscopic Maxwell equations we derive generalmacroscopic electrodynamic equations for 2D crystals and discuss some of their consequences. Equations of the macroscopic electrodynamics of bulk(three-dimensional, 3D) materials are derived from the microscopic
Maxwell equationsdiv e = 4 πρ, div h = 0 , rot e = − c ∂ h ∂t , rot h = c ∂ e ∂t + πc j (1)by their averaging over “physically infinitesimal” volumeelements , i.e. over dimensions small as compared tothe macroscopic scales, such as the wavelength of radia-tion and the sample dimensions, but large as comparedto the inter-atomic distances. Here e , h , ρ and j arethe microscopic electric and magnetic fields, charge andcurrent densities. As a result of the averaging one de-fines the macroscopic electric field E = ¯ e , the polar-ization vector P = χ E , which has the meaning of thedipole moment of a volume element, the electric induc-tion D = E + 4 π P , the dielectric susceptibility χ , thedielectric constant ǫ = 1 + 4 πχ and other macroscopicquantities. The dimensionless quantities χ and ǫ (weuse the more physical Gaussian system of units), whichcan in general be functions of the frequency ω and thewave-vector q of the electromagnetic field, fully deter-mine the linear response of the medium to the electro-magnetic field. In the nonlinear optics one defines thehigher order susceptibilities, e.g., the third order tensor χ (3) ijkl which determines the cubic term in the Taylor ex-pansion of the function P ( E ).The discovery of graphene – a one-atom-thick layerof carbon atoms – triggered great interest to this, as wellas other two-dimensional (2D) materials , both metal-lic (graphene), dielectric (e.g. BN) and semiconducting(e.g. MoS ). The notions of the dielectric susceptibil-ity χ , the dielectric constant ǫ and the refractive index n = √ ǫ are no longer applicable to these materials sincethe averaging of microscopic fields over the physicallyinfinitesimal volume elements is impossible in the direc-tion perpendicular to the 2D layer. However, by analogywith 3D materials, many authors continue to character-ize the electrodynamic response of graphene and other2D crystals, especially their nonlinear properties, by 3Dquantities such as the third susceptibility χ (3) or the non-linear refractive index n . A basic inadequacy of such anapproach is evident: the refractive index n characterizesthe change of the phase velocity of the wave propagatinginside the material , but it makes no sense to talk aboutthe propagation of waves inside a one-atom-thick layer.Thus a fundamental question arises, how to write down the macroscopic electrodynamic equations foratomically-thin (in fact, two-dimensional) materials andwhich physical quantity should be used to characterizetheir linear and nonlinear electrodynamic and opticalproperties.In a 3D dielectric the averaging of the microscopiccharge density over the physically infinitesimal volume elements leads to the definition of the polarization vector P , ρ → ¯ ρ = − div P , which has the meaning of the dipolemoment per unit volume and has the dimension [ e /cm ].In two dimensions such an averaging can be performedonly over a physically infinitesimal surface element, ρ → δ ( z )¯ ρ ( r k ) = − δ ( z )div P k ( r k ) , (2)where the subscripts k and 2 indicate 2D vectors or op-erators, the vector P k = ( P x , P y ) is the dipole momentof a surface element (the dimension [ e /cm]), and we con-sider the polarization in the direction parallel to the layer.The corresponding contribution to the polarization cur-rent (the dimension statampere/cm) is j k = ∂ P k /∂t. (3)In the presence of external charges ρ ex ( r , t ) and currents j ex ( r , t ) (which can be three-dimensional) the macro-scopic Maxwell equations for a 2D nonmagnetic mediumassume the formdiv E = − π div P k δ ( z ) + 4 πρ ex , (4)div H = 0 , (5)rot E = − c ∂ H ∂t , (6)rot H = 1 c ∂ E ∂t + 4 πc ∂ P k ∂t δ ( z ) + 4 πc j ex (7)Equations (4) – (7) should be completed by a relation be-tween the 2D polarization vector P k and the electric field E ( z = 0). If we ignore a possible spontaneous ferroelec-tric polarization of a 2D crystal, predicted in Ref. , sucha relation for centrosymmetric dielectric crystals has theform P k ; α = χ (1) , αβ E β + χ (3) , αβγδ E β E γ E δ , (8)where the Greek indexes take only the values { x, y } andthe fields are taken at z = 0. For conducting crystals itis more convenient to use the current-field relation j k ; α = σ (1) , αβ E β + σ (3) , αβγδ E β E γ E δ . (9)The 2D quantities χ and σ are similar to thecorresponding 3D ones but are measured in differentunits: for example, the first- and third-order 2D sus-ceptibilities χ (1) , and χ (3) , are measured in cm andcm /statvolt , respectively, in contrast to the corre-sponding 3D quantities which are dimensionless ( χ (1) )and measured in cm /statvolt ( χ (3) ). In general, the(linear and nonlinear) physical quantities χ and σ are complex functions of the frequency ω and the wave-vector q k of the electromagnetic field. They are relatedto each other by formulas which can be obtained from(3).The electrodynamics of 2D materials should thus bestudied using the system of equations (4) – (7), and theirelectrodynamic properties should be described by the 2Dquantities χ or σ . Describing, for example, Kerreffect in graphene one should directly relate the experi-mentally measured quantities to the linear and nonlinearcomponents of χ or σ , see, e.g., Refs. . The useof unphysical (for 2D crystals) quantities ǫ , n and n is inappropriate since they cannot be rigorously definedfor 2D materials. All said above refers to any materialconsisting of a single or a few atomic layers, including,e.g., tilted Dirac cone 2D systems and thin films oftopological insulators .Let us consider some consequences of the above equa-tions. The linear and nonlinear dynamic conductivitiesof conducting graphene and carbon nanotubes have beentheoretically studied in many papers, . Electrody-namic properties of dielectric 2D materials have been dis-cussed to a lesser extent. Below we discuss some prop-erties of a model dielectric 2D crystal characterized bythe linear susceptibility χ (1) , . If a 2D crystal has ahexagonal lattice like graphene, the susceptibility χ (1) , can be calculated within the tight-binding approxima-tion assuming different on-site energies of electrons sit-ting on atoms of different sublattices. This model welldescribes the hexagonal boron nitride (BN) and was the-oretically studied, for example, in Refs. under thename of “gapped graphene”. The spectrum of electronsand holes near the Dirac points in such a model reads E l k = ( − l p ∆ + (¯ hv F k ) , l = 1 , , (10)where 2∆ = E gap is the energy gap and v F is the effectiveFermi velocity of electrons. Assuming that the Fermienergy lies in the gap and that the temperature is small, T ≪ ∆, one can get the following analytical expressionfor the function χ (1) , ( ω ) χ (1) , ( ω ) = e g s g v π ∆ F (cid:18) ¯ h | ω | (cid:19) , (11)where g s and g v are the spin and valley degeneracies( g s g v = 4) and F (Ω) = 38 (cid:18) | Ω | ln (cid:12)(cid:12)(cid:12)(cid:12) | Ω | − | Ω | (cid:12)(cid:12)(cid:12)(cid:12) − (cid:19) + i π | Ω | −
1) 1 + Ω Ω . (12) Ω F ( Ω ) reim FIG. 1. The frequency dependence of the real and imaginaryparts of the function F (Ω), defined by Eq. (12). The frequency dependence of real and imaginary partsof the function F (Ω), Eq. (12), is shown in Figure1. The susceptibility has a logarithmic divergence atthe frequency ¯ h | ω | = 2∆ = E gap corresponding to theinter-band transition between the valence and conduc-tion bands. At low frequencies Ω ≪ F (Ω)tends to unity, F (Ω →
0) = 1, so that the 2D staticsusceptibility is χ (1) , ω → = e π ∆ . (13)It dramatically grows when the band gap decreases. If∆ = 1 eV ( E gap = 2∆ = 2 eV), the static susceptibility(13) equals χ (1) , ω → = 0 .
153 nm; if the gap lies in theterahertz range it is several orders of magnitude larger.The third-order nonlinear susceptibility χ (3) , can becalculated within the tight binding approximation, in asimilar way as for graphene. Within the Dirac Hamil-tonian approach it was done in Ref. .If an electromagnetic wave is normally incident on a 2Ddielectric layer characterized by the susceptibility (11)the transmission coefficient is determined by the for-mula T ( ω ) = | − πiωχ (1) , ( ω ) /c | − . Its frequencydependence is shown in Figure 2. At small frequencies, ω ≪ c ∆ /e , the function T ( ω ) decreases quadraticallywith the growing frequency, T ( ω ) = " (cid:18) e ω c ∆ (cid:19) − ≃ − Dω , (14)with the coefficient D ∝ / ∆ determined by the bandgap. At frequencies higher that 2∆ the inter-band ab-sorption is switched on and the transmission coefficientfalls down. Notice that at the absorption edge ¯ hω ≃ ≃ . Ω T r an s m i ss i on FIG. 2. The transmission coefficient of a wave passingthrough a 2D dielectric layer with the susceptibility χ (1) , ,Eq. (11), as a function of frequency Ω = ¯ hω/ R φ (r) , a r b . un i t s θ =0 θ=π /2unscreened θ -0.4-0.200.2 C ( θ ) FIG. 3. The potential (16) as a function of R = r/ πχ (1) , for θ = 0 (the direction normal to the 2D plane) and θ = π/ /r .Inset shows the function C ( θ ), Eq. (18), in the interval from θ = 0 to θ = π/ Now consider the static screening of a point charge Qδ ( r ) = Qδ ( r k ) δ ( z ) placed in the plane of a 2D dielectric.The Poisson equation for the electric potential has theform∆ φ + 4 πχ (1) , ∆ φδ ( z ) = − πQδ ( r k ) δ ( z ) , (15) Its solution, φ ( r ) = Q πχ (1) , Z ∞ f ( ξR, θ )1 + ξ dξ, (16)is shown in Figure 3; here f ( t, θ ) = e − t cos θ J ( t sin θ ), J is the Bessel function, R = r/ πχ (1) , and θ is the anglebetween the vector r and the z -axis. At a large distancefrom the charge, R ≫
1, Eq. (16) gives the unscreenedCoulomb potential φ ( r ) ≈ Q/r . At smaller distances
R < ∼ r -dependence of the screened Coulomb potential islogarithmic, φ ( r ) ≈ Q πχ (1) , (cid:18) ln 2 πχ (1) , r + C ( θ ) (cid:19) , (17)and depends on the angle θ , C ( θ ) = Z ∞ f ( t, θ ) t dt − Z − f ( t, θ ) t dt. (18)In the directions perpendicular ( θ = 0) and parallel ( θ = π/
2) to the 2D plane the integrals in (18) are calculatedanalytically, C (0) = − γ and C ( π/
2) = ln 2 − γ ( γ =0 . . . . is the Euler constant). At arbitrary angles θ the function C ( θ ) is shown in the Inset to Figure 3. The2D dielectric substantially screens the field of the pointcharge at the distance smaller than or of order of thesusceptibility length χ (1) , which lies between ∼ . µ mand ∼ . two-dimensional quan-tities σ or χ . The three-dimensional quantities suchas χ (3);3D , the dielectric function ǫ , the refractive index n = √ ǫ (linear and nonlinear) cannot be properly de-fined and should not be used in the electrodynamics of2D crystals.The work has received funding from the EuropeanUnions Horizon 2020 research and innovation programmeGraphene Core 2 under Grant Agreement No. 785219. ∗ [email protected] L. D. Landau and E. M. Lifshitz,
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