We predict the existence of transverse electric (TE) plasmons in bilayer
graphene. We find that their plasmonic properties are much more pronounced in
bilayer than in monolayer graphene, in a sense that they can get more localized
at frequencies just below $\hbar\omega=0.4$~eV for adequate doping values. This
is a consequence of the perfectly nested bands in bilayer graphene which are
separated by $\sim 0.4$~eV

Jablan, Marinko

Buljan, Hrvoje

Soljačić, Marin

Optics Express

English

arXiv

summary:Properties of Lipschitz and d.c. surfaces of finite codimension in a Banach space and properties of generated $\sigma $-ideals are studied. These $\sigma $-ideals naturally appear in the differentiation theory and in the abstract approximation theory. Using these properties, we improve an unpublished result of M. Heisler which gives an alternative proof of a result of D. Preiss on singular points of convex functions

Zajíček, Luděk

Czech digital mathematics library

On Lipschitz and d.c. surfaces of finite codimension in a Banach space

Institute of Mathematics AS CR, v. v. i.

We predict the existence of transverse electric (TE) plasmons in bilayer graphene. We find that their plasmonic properties are much more pronounced in bilayer than in monolayer graphene, in a sense that they can get more localized at frequencies just below h̄ω = 0.4 eV for adequate doping values. This is a consequence of the perfectly nested bands in bilayer graphene which are separated by ∼ 0.4 eV.United States. Dept. of Energy (S3TEC Award No. DE-SC0001299

Soljacic, Marin

DSpace@MIT

arXiv:1102.5690v1  [physics.optics]  28 Feb 2011Transverse electric plasmons in bilayer grapheneMarinko Jablan and Hrvoje BuljanDepartment of Physics, University of Zagreb, Bijenicˇka c. 32, 10000 Zagreb, CroatiaMarin Soljacˇic´Department of Physics, Massachusetts Institute of Technology,77 Massachusetts Avenue, Cambridge MA 02139, USAWe predict the existence of transverse electric (TE) plasmons in bilayer graphene. We find thattheir plasmonic properties are much more pronounced in bilayer than in monolayer graphene, in asense that they can get more localized at frequencies just below ~ω = 0.4 eV for adequate dopingvalues. This is a consequence of the perfectly nested bands in bilayer graphene which are separatedby ∼ 0.4 eV.Plasmons are self-sustained collective electron excitations which are of great interest both for fundamental physicsand potential technological applications. Plasmon is a paradigmatic quantum many-body phenomenon studied incondensed matter physics [1]. Closely related excitations are surface plasmons which hold promise as a possible toolfor controlling light at subwavelength scales [2–5], giving rise to the field of nanophotonics, and they also play animportant role in metamaterials [6–9]; plasmons in essentially two-dimensional (2D) structures are similar in thisrespect to surface plasmons. These reasons are a great motivation for studying plasmonic excitations and theirproperties in novel materials. Two such materials are monolayer [10, 11] and bilayer (see e.g., [11, 12]) graphene.Graphene is a 2D sheet made of carbon atoms organized in a honeycomb lattice [10, 11], whereas bilayer grapheneconsists of two such layers stacked on top of each other in a certain way [11, 12]. While there are only a few studiesof plasmons in bilayer graphene [13–16], these collective excitations have attracted substantially more attentionin monolayer graphene [17–23]. Several years ago it was predicted that graphene, besides the ordinary longitudinalplasmons [transverse magnetic (TM) modes] [17–20, 23], also supports unusual transverse plasmons [transverse electric(TE) modes] [19]. These excitation are possible only if the imaginary part of the conductivity of a thin sheet of materialis negative [19]. On the other hand, such a conductivity requires some complexity of the band structure of the materialinvolved. For example, TE plasmons cannot occur if the 2D material possesses a single parabolic electron band. Fromthis perspective, bilayer graphene, with its rich band structure and optical conductivity (e.g., see [24] and referencestherein), seems as a promising material for exploring the possibility of existence of TE plasmons. Here we predict theexistence of TE plasmons in bilayer graphene. We find that their plasmonic properties are much more pronounced inbilayer than in monolayer graphene, in a sense that the wavelength of TE plasmons in bilayer can be smaller than inmonolayer graphene at the same frequency.Throughout this work we consider bilayer graphene as an infinitely thin sheet of material with conductivity σ(q, ω).We assume that air with ǫr = 1 is above and below bilayer graphene. Given the conductivity, by employing classicalelectrodynamics, one finds that self-sustained oscillations of the charge occur when (see [19] and references therein)1 +iσ(q, ω)√q2 − ω2/c22ǫ0ω= 0 (1)for TM modes, and1−µ0ωiσ(q, ω)2√q2 − ω2/c2= 0 (2)for TE modes. The TM plasmons can considerably depart from the light line, that is, their wavelength can beconsiderably smaller than that of light at the same frequency. For this reason, when calculating TM plasmons it isdesirable to know the conductivity as a function of both frequency ω and wavevector q. However, it turns out thatthe TE plasmons (both in monolayer [19] and bilayer graphene, as will be shown below) are quite close to the lightline q = ω/c, and therefore it is a good approximation to use σ(ω) = σ(q = 0, ω). Moreover, these plasmons areexpected to show strong polariton character, i.e., creation of hybrid plasmon-photon excitations. At this point it isworthy to note that if the relative permittivity of dielectrics above and below graphene are sufficiently different, sothat light lines differ substantially, then TE plasmon will not exist (perhaps they could exist as leaky modes).The conductivity σ(ω) = ℜσ(ω) + iℑσ(ω) is complex, and plasmon dispersion is characterized by the imaginarypart ℑσ(ω), whereas ℜσ(ω) determines plasmon losses, or more generally absorption of the sheet. From Eq. (2) itfollows that the TE plasmons exist only if ℑσ(ω) < 0 [19].In order to calculate the imaginary part of the conductivity, we employ Kramers-Kronig relations and the calculationof absorption by Nicol and Carbotte [24], where ℜσ(ω) [see Eqs. (19)-(21) in Ref. [24]] was calculated by using the2−1 0 1−101k/q0ε(k)/γFIG. 1: The band-structure of bilayer graphene. The two upper bands (as well as the two lower bands) are perfectly nestedand separated by γ ∼ 0.4 eV; q0 = γ/~vF . Horizontal line depicts one possible value of the Fermi level, and arrows denotesome of the possible interband electronic transitions. See text for details.Kubo formula. The optical conductivity has rich structure due to the fact that the single-particle spectrum of grapheneis organized in four bands given by [24],ǫ(k)γ= ±√14+(~vF kγ)2±12, (3)where vF = 106 m/s, the parameter γ ≈ 0.4 eV is equal to the separation between the two conduction bands (whichis equal to the separation between the valence bands). The band structure (3) is calculated from the tight bindingapproach, where vF is connected to the nearest-neighbour hopping terms for electrons to move in each of the twographene planes, and the distance between Carbon atoms in one monolayer (see Ref. [24]), whereas γ is the hoppingparameter corresponding to electrons hoping from one layer to the other and vice versa [24]. The two graphene layersare stacked one above the other according to the so-called Bernal-type stacking (e.g., see Ref. [11]). We emphasizethat the perfect nesting of bands gives rise to the stronger plasmon like features of TE plasmons in bilayer than inmonolayer graphene. The four bands are illustrated in Fig. 1 along with some of the electronic transitions whichresult in absorption. Absorption depends on γ and the Fermi level µ; the latter can be changed by applying externalbias voltage.The imaginary part of the conductivity can be calculated from ℜσ(ω) by using the Kramers-Kronig relationsℑσ(ω) = −2ωπP∫∞0ℜσ(ω′)ω′2 − ω2dω′, (4)which yieldsℑσ(ω)σ0= f(Ω, 2µ) + g(Ω, µ, γ)+ [f(Ω, 2γ) + g(Ω, γ,−γ)]Θ(γ − µ)+ [f(Ω, 2µ) + g(Ω, µ,−γ)]Θ(µ− γ)+γ2Ω2[Ωπ(2µ+ γ)+ f(Ω, 2µ+ γ)]+γ2Ω2[Ωπγ+ f(Ω, γ)]Θ(γ − µ)+γ2Ω2[Ωπ(2µ− γ)+ f(Ω, 2µ− γ)]Θ(µ− γ)+a(µ)πΩ+2Ωb(µ)π(Ω2 − γ2), (5)3wheref(x, y) =12πlog∣∣∣∣x− yx+ y∣∣∣∣ ,g(x, y, z) =z2π(x− z) log |x− 2y|+ (x+ z) log |x+ 2y| − 2x log |2y + z|x2 − z2,a(µ) =4µ(µ+ γ)2µ+ γ+4µ(µ− γ)2µ− γΘ(µ− γ),b(µ) =γ2[log2µ+ γγ− log2µ− γγΘ(µ− γ)], (6)σ0 = e2/2~, Θ(x) = 1 if x ≥ 0 and zero otherwise, and Ω = ~ω. Here we assume zero temperature T ≈ 0, whichis a good approximation for sufficiently doped bilayer graphene where µ ≫ kBT . Formulae (5) and (6) are used todescribe the properties of TE plasmons.In Figure 2 we show the real and imaginary part of the conductivity for two different values of the Fermi level:µ = 0.4γ and µ = 0.9γ (we focus on the electron doped system µ > 0). Because plasmons are strongly dampedby interband transitions, it is instructive at this point to discuss the kinematical requirements for the excitation ofelectron-hole pairs. If the doping is such that µ < γ/2, a quantum of energy ~ω (plasmon or photon) with in-planemomentum q = 0 can excite an electron-hole pair only if ~ω > 2µ (excitations from the upper valence to the lowerconduction band shown as red dot-dashed line in Fig. 1). If µ > γ/2, the (q = 0, ω)-quantum can excite an electron-hole pair only for ~ω ≥ γ (excitations from the lower to the upper conduction band shown as green solid lines in Fig.1 occur at ~ω = γ). If the plasmon/photon has in-plane momentum q larger than zero, then interband transitionsare possible for smaller frequencies (see blue dashed lines in Fig. 1). There is a region in the (q, ω)-plane whereelectron-hole excitations are forbidden due to the Pauli principle (e.g., see figures in Refs. [14–16]). Because plasmonsare strongly damped by these interband transitions (this is Landau damping), in our search for the TE plasmons,we focus on their dispersion curve in the regime where electron-hole pair formation is inadmissible (via first-ordertransition).0 1 2−202ω/ω0σ/σ0(a)0 1 2−202ω/ω0σ/σ0(b)FIG. 2: The real (red dotted lines) and imaginary (blue solid lines) part of the conductivity of bilayer graphene for two valuesof doping: µ = 0.4γ (a), and µ = 0.9γ (b). The conductivity is in units of σ0 = e2/2~, and the frequency is in units of ω0 = γ/~.The δ-functions in ℜσ(ω) at ω = 0 (intraband transitions) and ω = γ/~ (transitions from the lower to the upper conductionband depicted as green solid arrows in Fig. 1) are not shown (see [24]).In Figure 3 we show the plasmon dispersion curves for µ = 0.4γ and µ = 0.9γ; in the spirit of Ref. [19], we show∆q = q− ω/c as a function of frequency ω. Plasmons are very close to the light line and thus one can to a very goodapproximation write the dispersion curve as∆q ≈ω8ǫ20c3ℑσ(ω)2. (7)To the left (right) of the vertical red dotted line in Fig. 3, plasmon damping via excitation of electron-hole pairs is(is not) forbidden.40.6 0.7 0.80123x 10−6ω/ω0∆ q/q 0(a)0.9 0.95 1024x 10−4ω/ω0∆ q/q 0(b)FIG. 3: The plasmon dispersion curve ∆q = q − ω/c vs. ω for µ = 0.4γ (a), and µ = 0.9γ (b) is shown as blue solid line.To the right of the vertical red dotted lines plasmons can be damped via excitation of electron-hole pairs, whereas to the leftof this line these excitations are forbidden due to the Pauli principle. Black dashed line in (b) (which closely follows the blueline) corresponds to Eq. (8). The wave vector is in units of q0 = γ/~vF , and the frequency is in units of ω0 = γ/~.For µ = 0.4γ, ℑσ(ω) is smaller than zero for ω in an interval of frequencies just below 2µ. From the leading termin ℑσ(ω) we find that departure of the dispersion curve from the light line is logarithmically slow: ∆q0<µ<γ/2 ∝[log |~ω − 2µ|]2. The same type of behavior occurs in monolayer graphene [19].However, for µ = 0.9γ, one can see the advantage of bilayer over monolayer graphene in the context of TE plasmons.The conductivity ℑσ(ω) is smaller than zero in an interval of frequencies below γ. In this interval, the most dominantterm to the conductivity is the last one from Eq. (5), that is,∆qγ/2<µ<γ ≈ωσ202π2ǫ20c3[~ωb(µ)γ2 − (~ω)2]2. (8)This approximation is illustrated with black dashed line in Fig. 3, and it almost perfectly matches the dispersioncurve. Note that the singularity in ℑσ(ω) at ~ω = γ is of the form 1/(γ − ~ω), whereas the singularity at ~ω = 2µ islogarithmic (as in monolayer graphene [19]). As a consequence, the departure of the dispersion curve from the lightline in bilayer graphene is much faster for µ > γ/2 than for µ < γ/2, and it is faster than in monolayer grapheneas well [note the two orders of magnitude difference between the abscissa scales in Figs. 3(a) and (b)]. Thus, weconclude that more pronounced plasmonic features of TE plasmons (shrinking of wave length which is measured asdeparture of q from the light line) can be obtained in bilayer graphene. The term in ℑσ(ω) which is responsible for TEplasmons for µ > γ/2 corresponds (via Kramers-Kronig relations) to the absorption term b(µ)δ(~ω − γ) [24], whicharises from the transitions from the first to the second valence band (shown as green solid arrows in Fig. 1), whichare perfectly nested and separated by γ. Thus, this unique feature of bilayer graphene gives rise to TE plasmons withmore pronounced plasmon like features than in monolayer graphene.Before closing, let us discuss some properties and possible observation of TE plasmons. First, note that since theelectric field oscillations are both perpendicular to the propagation vector q, and lie in the bilayer graphene plane, theelectric current j = σ(ω)E is also perpendicular to q. Thus, j · q = 0, and the equation of continuity yields that thecharge density is zero (i.e., one has self-sustained oscillations of the current). In order to excite plasmons of frequencyω with light of the same frequency, one has to somehow account for the conservation of the momentum which is largerfor plasmons. Since the momentum mismatch is relatively small, the standard plasmon excitation schemes such asthe prism or grating coupling methods (e.g., see [2] and references therein) could be used for the excitation of theseplasmons.In conclusion, we have predicted the existence of transverse electric (TE) plasmons in bilayer graphene. Since theyexist very close to the light line, these plasmons are expected to show strong polariton character, i.e., mixing withphoton modes. However, due to the perfectly nested valence bands of bilayer graphene, their dispersion departs muchmore from the light line than in monolayer graphene.We acknowledge most useful comments from Guy Bartal and Mordechai Segev from the Technion, Israel. Thiswork was supported in part by the Croatian Ministry of Science (Grant No. 119-0000000-1015), the Croatian-Israeliscientific cooperation program funded by the Ministries of Science of the State of Israel and the Republic of Croatia.This work is also supported in part by the MRSEC program of National Science Foundation of the USA under Award5No. DMR-0819762. M.S. was also supported in part by the S3TEC, an Energy Frontier Research Center funded bythe U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Award No. de-sc0001299.[1] D. Pines, P. Nozieres, The Theory of Quantum Liquids (Benjamin, New York, 1966).[2] W.L. Barnes, A. Dereux, T.W. 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Transverse electric plasmons in bilayer graphene

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