- Email: [email protected]
Role of metallic substrate on the plasmon modes in double-layer graphene structures
Solid State Communications ∎ (∎∎∎∎) ∎∎∎–∎∎∎
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Q1 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Role of metallic substrate on the plasmon modes in double-layer graphene structures G. Gonzalez de la Cruz n Departamento de Física CINVESTAV-IPN, Apartado Postal 14-740, 07000 México D.F, México
art ic l e i nf o
a b s t r a c t
Article history: Received 23 December 2014 Received in revised form 25 February 2015 Accepted 30 March 2015 Communicated by E.Y. Andrei
Novel heterostructures combining different layered materials offer new opportunities for applications and fundamental studies of collective excitations driven by interlayer Coulomb interactions. In this work, we have investigated the influence of the metallic-like substrate on the plasmon spectrum of a double layer graphene system and a structure consisting of conventional two-dimensional electron gas (2DEG) immersed in a semiconductor quantum well and a graphene sheet with an interlayer separation of d. Long-range Coulomb interactions between substrate and graphene layered systems lead a new set of spectrum plasmons. At long wavelengths (q-0) the acoustic modes (ω q) depend, besides on the carrier density in each layer, on the distance between the first carrier layer and the substrate in both structures. Furthermore, in the relativistic/nonrelativistic layered structure an undamped acoustic mode emerges for a certain interlayer critical distance dc. On the other hand, the optical plasmon modes emerging from the coupling of the double-layer systems and the substrate, both start at finite frequency at q ¼0 in contrast to the collective excitation spectrum ω q1/2 reported in the literature for doublelayer graphene structures. & 2015 Published by Elsevier Ltd.
Keywords: A. Graphene B. Two-dimensional electron gas C. Double-layers graphene structures D. Collective modes
1. Introduction A revolution of material science is coming since graphene was exfoliated successfully from graphite by Novoselov and Geim [1]. The electrons in graphene behave like massless Dirac-Fermions, which results in a extraordinary properties of carriers (electrons and holes) with ultra-high-mobility and long mean free path, gatetunable carrier densities, anomalous quantum Hall effects, fine structure constant defined optical transmission, and so on [2]. Owing to the two-dimensional nature of the collective excitations, plasmons excited in graphene are confined much more strongly than those in conventional noble metals. One of the most important advantages of graphene would be the tunability plasmons, as the carrier densities can be easily controlled by electrical gating and doping. Consequently, graphene can be applied as terahertz metamaterial and it can be tuned conveniently even for an encapsulated device. On the other hand, graphene can help to tune the surface plasmons in conventional metals, such as Au, which makes it promising plasmonic materials. Refs. [3–7] focus on the recent progress of graphene plasmonics and its technological applications. A plasmon is a collective mode of a charge–
n
Corresponding author. E-mail address: bato@fis.cinvestav.mx
density oscillation in the fee-carrier system, which is present both in classical and quantum plasmas. Studying the collective plasmon excitation in the electron gas has been among the very first theoretical quantum mechanical many-body problems studied in solid-state physics. The collective plasmon modes of monolayer graphene have been extensively studied theoretically [8–10] and experimentally [11–13] and are obtained by the zeros of the corresponding frequency and wave vector dependent dynamical dielectric function. The long-wavelength plasma oscillations are essentially fixed by the particle number conservation, and can be calculated using the random-phase approximation. The plasma dispersion frequency shows a q1/2 behavior and it is nonclassical (it depends explicitly on ℏ), this explicit quantum nature of longwavelength graphene plasmon is a direct manifestation of its linear Dirac-like energy-momentum dependence, which has no classical analogy. Based on monolayer graphene, novel double-layer structures have been recently realized experimentally [14,15] where massless fermions of two separate layers are coupled only through manybody Coulomb interactions. It is known [16,17], that when two graphene layers are put in close proximity with an oxide or semiconductor between them to prevent interlayer tunneling, the two-dimensional plasmon are coupled by the interlayer Coulomb interaction leading to the formation of two branches of longitudinal collective excitation spectra called the optical plasmon ω(q)q1/2
http://dx.doi.org/10.1016/j.ssc.2015.03.021 0038-1098/& 2015 Published by Elsevier Ltd.
Please cite this article as: G.G.d.l. Cruz, Solid State Commun (2015), http://dx.doi.org/10.1016/j.ssc.2015.03.021i
67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97
G.G.d.l. Cruz / Solid State Communications ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
and the acoustic plasmon with ω(q)q where the density fluctuation in each component oscillates in-phase (optical plasmon) and out-of-phase (acoustic plasmon), respectively, relative to each other. These collective modes of the double-layer structures, which have been directly compared with the conventional double-layer twodimensional electron gas (2DEG) [18], play important roles in the many body properties such as screening and drag [18,21]. In addition to the collective modes of two parallel twodimensional graphene layers, special attention has been addressed on the double-layer system considering their plasmon modes for doped (extrinsic) [22,23] or undoped (intrinsic) double layers [24] at finite temperature. Recently, particular interest has been paid to investigated plasmons and Coulomb drag in hybrid double-layer systems composed by a doped graphene sheet deposited on the surface of the semiconductor in close proximity to the 2DEG, long range Coulomb interactions between massive electrons and massless Dirac fermions lead to a new set of optical and acoustic intrasubband plasmons [25]. In particular, plasmons excitations of a system of coupled relativistic and nonrelativistic two-dimensional electron gas was considered by Balram et al. [26]. They found that the strength of the interaction between different charge carriers (that is, tunneling between graphene and 2DEG), play a significant role in determining the number of plasmon modes as well as their dispersions under certain parameters regimes. Such coexistence Dirac/Schrodinger hybrid electron systems have been directly experimentally observed [27,28] and they open a new research opportunities for fundamental studies of electron–electron interactions effects in two spatial dimensions. It has been shown that the surface plasmons in graphene can be significant influenced by many-particle effects involving interactions between electrons and plasmons (plasmarons). A deep understanding of the coupling between charge carriers and plasmons in graphene is highly desirable because of large potential of this material in the context of plasmonics [3]. Graphene on metals is of interest as a route to synthesizing high quality graphene and for electrical contacts to devices. Graphene–metal systems can be roughly classified into two different of film and substrate binding i.e., weak or strong hybridization between graphene π and metal d bands [29, 30]. The coupling between plasmons in graphene and electron gas in metal-like substrate were investigated both, numerically [31] and theoretically [32,33]. In this article, we investigate the role of metallic-like substrates on the collective excitations of double layer two-dimensional electron systems. We concentrate on coupled system of twodimensional Dirac fermions and 2DEG confined in a quantumwell at T¼ 0 K, tunneling between carrier in different layers has been neglected. Plasmon modes in the hybrid double-layer system are calculated within the self-consistent-field linear approximation taking into account the interaction with the substrate. The interaction between the double-layer two-dimensional electron system and the substrate is discussed by replacing the dielectric constant of the substrate by a frequency dependent dielectric function.
2. Theoretical model Firstly, we study the coupling between the two-dimensional double layer graphene system and the substrate. The model system under consideration corresponding to two graphene electron layers separated a distance d is shown in Fig. 1. The graphene electron layers system occupy a half-space z 4 Δ, of background dielectric constant εs. The substrate with dielectric constant ε0 occupies the space z o Δ.
Fig. 1. A double-layer electronic system separated by a distance d, embedded in a dielectric environment with dielectric constant εs. A substrate with dielectric constant ε0 occupies the space to the left from the first electron layer at z ¼ Δ.
Within the self-consistent-field linear approximation theory (SCF) and assuming the relaxation time be infinite, each electron is assumed to move in the self-consistent field arising from the external field plus the induced field of all the electrons, then the electron density in the nth (n ¼0,1) graphene electron layer, ρn(q,ω) is given by X ρn ðq; ωÞ ¼ V n;m ðqÞΠ m ðq; ωÞρm ðq; ωÞ þ Π n ðq; ωÞϕext ð1Þ n ðq; ωÞ m
where
h i V n;m ðqÞ ¼ vq e qjn mjd þ αe ðn þ mÞd
ð2Þ
is the interlayer Coulomb interaction and the second term proportional to εs ε0 2Δq α¼ e ð3Þ εs þ ε0 gives the modified Coulomb interaction between the two Dirac electron layers due to the image charge and Πn(q,ω) is the twodimensional polarizability for layer n calculated in Refs. [8,9], 2 ϕext n ðq; ωÞ is the external potential, and vq ¼2πe /εsq represents the two-dimensional Coulomb electron interaction. In order to study the collective excitation spectrum, it is convenient to write Eq. (1) for zero external potential and considering that charge fluctuation ρn ðq; ωÞ a 0 in each layer, the plasmon dispersion of the coupled, spatially-separated twodimensional graphene layers are obtained by the zeros of the dielectric constant Dðq; ωÞ ¼ ½1 vq ð1 þαÞΠ 1 ðq; ωÞ½1 vq ð1 þ αe 2qd ÞΠ 2 ðq; ωÞ v2q ð1 þ αÞ2 e 2qd Π 1 ðq; ωÞΠ 2 ðq; ωÞ
ð4Þ
The collective modes occur between the intraband and interband single particle excitations where Π n ðq; ωÞ is real and positive and a decreasing function of frequency [8,9]. It has been demonstrated that the two-component plasma has two branches of longitudinal oscillation spectrum which are solutions of Eq. (4). In the higher frequency branch, the two carriers oscillate in phase with the long-wavelength limit at the square root, while in the lower branch the carriers oscillate out of phase, exhibiting at longwavelength a linear dispersion. In order to obtain the expression for the acoustic plasmon oscillations, we proceed as Santoro and Giuliani [34]. We first introduce for q 0 the power expansion ωðqÞ ¼ c1 q þ c2 q2 þ c3 q3 þ ⋯
ð5Þ
for the plasmon dispersion relation and define a function FðqÞ ¼ Dðq; c1 q þ c2 q þc3 q þ ⋯Þ
ð6Þ
where D is defined in Eq. (4). For q 0, F(q) can in turn be written
Please cite this article as: G.G.d.l. Cruz, Solid State Commun (2015), http://dx.doi.org/10.1016/j.ssc.2015.03.021i
67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
G.G.d.l. Cruz / Solid State Communications ∎ (∎∎∎∎) ∎∎∎–∎∎∎
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
in terms of the Laurent–Taylor expansion FðqÞ ¼ f 1 q
1
2
þ f 0 þ f 1q þ f 2q þ ⋯
ð7Þ
where the fi are suitable coefficients which are derived from the expression Π n ðq; ωÞ. The plasmon condition Eq. (4) is then satisfied by requiring that all the coefficients fi vanish independently. The coefficient f 1 depends only on c1 and by equaling its expression to zero we arrive at the following equation for the phase velocity: pffiffiffi c1 1 8de2 π ¼ pffiffiffiffiffiffiffiffiffiffiffiffi; ξ ¼ ð8Þ p ffiffiffi 1=2 1=2 γ þn2 Þγεs 8de2 π þ ðn1 1 ξ2 here ni, i¼ 1,2 is the carrier density in each two-dimensional graphene layer. It follows from Eq. (8) that the ratio c1/γ is greater than unity for any value of the parameters n1, n2, and d and only depends on the dielectric constant of the material where the layers are immersed, it is independent of the substrate dielectric constant. Analogous results were found for inhomogeneous dielectric background graphene double-layer structures in Ref. [19]. Similar analysis can be carried out also for the optical plasmon. In such case, the energy is determined by substituting the long wavelength limit for the two-dimensional polarizability Π n ðq; ωÞ, it leads pffiffiffi pffiffiffiffiffi pffiffiffiffiffi 4e2 π ð n1 þ n2 Þ γq ð9Þ ω2 ðqÞ ¼ ð1 þ εÞεs As can be seen the plasmon energy is independent of the spacing between the two layers and it depends on the ratio ε ¼ ε0 =εs of the background dielectric constants between the substrate and the semiconductor dielectric constant. Double-layer graphene collective excitations can be influenced by the dielectric properties of the substrate, and here we study the effects of the Coulomb interaction of the double-layer graphene plasma with the semi-infinite plasma of the substrate whose local dielectric constant is given by ε0 ðωÞ ¼ εn ð1 ω2p =ω2 Þ where ωp is the plasma frequency of the carriers in the substrate and ε* is the static dielectric background of the substrate. The coupling between plasmons in single-layer graphene and substrates were also considered in Ref. [32,33]. If εs is constant, the plasmon dispersion can be calculated from the zeros of Eq. (4) using the same power expansion as in Eqs. (5) and (7), the hybridization of the plasmon frequencies in the long wavelength limit lead an acoustic plasmon mode with energy given as ωac ðqÞ ¼ cs q
ð10Þ
and the group velocity satisfies the following linear equation: " sffiffiffiffiffiffiffiffiffiffiffiffi #" sffiffiffiffiffiffiffiffiffiffiffiffi # c2s c2s 2 pffiffiffiffiffiffiffiffi 2 pffiffiffiffiffiffiffiffi 1 8e π S Δ γε 1 8e π S ðd þ ΔÞ n n γεs a s a 1 2 γ2 γ2 p ffiffiffiffiffiffiffiffiffiffi 2 ð11Þ 64e2 n1 n2 πSa Δ2 ¼ 0 with Sa ¼
cs γ
sffiffiffiffiffiffiffiffiffiffiffiffi! c2s 1 γ2
3
graphene layer at z¼ 0 and it only depends on the Fermi momenpffiffiffiffiffiffiffiffi tum of the second layer defined as k2 ¼ πn2 2 cs ð8de2 k2 þ γεs Þ2 8de2 k2 þ γεs ¼ ð13Þ γ γεs γεs ð16de2 k2 þγεs Þ showing that the acoustic plasma modes are affected by the fully coupled plasma modes present in the system of Fig. 1. On the other hand, the optical plasma modes emerging from the coupling of the double-layer graphene system and the substrate are obtained as ω2op ðqÞ ¼
2ε 2 ω þ bq; 1 þε s
ω2s ¼ ω2p =2
where 4e2 1 1 pffiffiffiffiffiffiffiffiffiffi ðpffiffiffiffiffi þ pffiffiffiffiffiÞ n1 n2 ð1 þ εÞ b ¼ pffiffiffi n2 π γð1 þ εÞ2 εs n1
ð13Þ
the point to be made here is that the optical plasmon dispersion start at finite frequency for q¼0 and changes linearly with q. Note that ω2op ðqÞ depends on both, the dielectric constant of the substrate and the dielectric constant εs but is independent of Δ and the distance d between the graphene layers. Fig. 2 shows an schematic representation of both the acoustic ωac and optic dispersion relation in the region where the imaginary part of the two-dimensional graphene polarizability vanish. The collective excitations in this region are not damped for q⪡kf. Moreover, as q increase, it is expected that the dispersion relation of the double layer graphene decay into the continuum of single particle excitation region. Finally we compute the plasmon dispersion of a conventionally two-dimensional electron gas (2DEG) and a sheet graphene layer spatially separated a distance d from the 2DEG and coupled dynamically with a substrate described with a frequency dependent dielectric constant, see Fig. 1. The double-layer massless Dirac/Scrodinger hybrid electron system was studied in Refs. [26,27] for static dielectric constant. According with the geometry shown in Fig. 1 where the 2DEG layer is located at z ¼0, the acoustic group velocity can be calculated analytically following the procedure explained in the text. After solve a similar equation as Eq. (4) where nowΠ 1 ðq; ωÞ represents the two-dimensional polarizabilty of the 2DEG [35], and Π 2 ðq; ωÞ the two-dimensional polarizability of the Dirac electron layer, we find the following equation for the group velocity cs: " sffiffiffiffiffiffiffiffiffiffiffiffiffi #" sffiffiffiffiffiffiffiffiffiffiffiffi # c2s c2s 2 2 pffiffiffiffiffiffiffiffi 1 2e m S Δ γε 1 8e π S ðd þ ΔÞ εs n s a 1 b 2 γ2 v21 p ffiffiffiffiffiffiffiffi 16e4 m1 n2 π Sa Sb Δ2 ð14Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 where Sb ¼ cs =v1 cs =v1 1 and v1, m1 the Fermi velocity and the electron effective mass in 2DEG, respectively. Sa has been
ð12Þ
Note that acoustic energy dispersion is still linear on the wavevector, however, when the interaction of the double-layer graphene with the substrate is taken into account, the phase velocity depends on the distance from the first Dirac electron layer to the substrate surface, and in the uniform limit Δ⪢d we recover the expression given by Eq. (8). We also see from this analytic expression that cs is independent of the dielectric constant of the substrate ε0 and depends only on the environment dielectric constant where the two-spatially graphene layers are immersed. On the other hand in the limit when Δ 0, the acoustic plasmon group velocity is independent of the electron density of the
Fig. 2. Schematic representation of the dispersion relation of the acoustic (dotted line) and the optical (started line) in the region where they are free Landau damping.
Please cite this article as: G.G.d.l. Cruz, Solid State Commun (2015), http://dx.doi.org/10.1016/j.ssc.2015.03.021i
67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
G.G.d.l. Cruz / Solid State Communications ∎ (∎∎∎∎) ∎∎∎–∎∎∎
4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61
defined in Eq. (12). Similar results are obtained for the group velocity following the analysis of Eq. (11). An undamped acoustic plasmon emerges for d 4dc where dc is the value of the layers separation for which cs ¼γ if the Dirac velocity is larger than the Fermi velocity and cs ¼ v1 if the Fermi velocity is larger than the Dirac velocity. By using these values of cs in Eq. (14) we obtain 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v1 Δεs > > γ 2 =v21 1 γ 4 v1 < ð2e2 m1 Δ þ εs Þv1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi γ 2 =v21 1 2e2 m1 γΔ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dc ¼ ð15Þ γ 2 εs pffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > v21 =γ 2 1 γ o v1 : 2 pffiffiffiffiffiffi 2 2 8e
n2 π γðv1 =γ
v1 =γ 1Þ
Note that the critical value of d does not depend on n2 when γ4 v1 but depends on the dielectric constant εs and it is proportional to the distance Δ between the 2DEG and the substrate, on the other hand dc depends on the carrier densities of both layers n1 and n2 for γov1 but it is independent of Δ. The analytical analysis of the long-wavelength optical plasmon mode for the hybridized double-layer electron densities is simpler since this mode is finite for tending q to zero. We obtain an analytic result using the well known mathematical expression for 2DEG and graphene polarizabily. To leading order in q in the longwavelength q-0 limit, the frequency of the spatially separated graphene and 2DEG layers optical plasmon can be found analytically. The result is pffiffiffiffiffiffiffiffi 2ε 2 4e2 n2 π γ 4e2 n1 π ω2op ðqÞ ¼ ωs þ qþ q ð16Þ 1þε εs ð1 þ εÞ m1 εs ð1 þ εÞ this plasmon dispersion has the same form as the optical plasmon frequency obtained in the presence of Coulomb coupling between the double-layer graphene systems and the substrate. It is seen however, that in contrast with the previous treatment, the second term in Eq. (16) proportional to wavevector q can be recognized as the contribution of the plasmon frequency of the electron gas in an isolated graphene sheet coupling with the surface plasmons of the substrate as calculated in Refs. [32,33]. The third term represent the contribution to the optical mode due to the Coulomb coupling between the 2DEG with the substrate. Note that the optical frequency is independent of Δ and the interlayer electronic distance. 3. Conclusions We have presented an analysis of the electronic collective modes of systems composed of two Coulomb-coupled graphene layers and relativistic and nonrelativistic electron layers separated spatially by a distance d. In addition, we formulated the theory of the coupling between the plasmon dispersion of both layered systems and substrate surface plasmon excitation. Our study focuses on the longwavelength limit in which qd is small. We have derived analytic expressions for both frequencies, the low-energy linearly dispersion acoustic plasmon mode and for the high-energy optical plasmon pffiffiffi mode that has finite frequency at q¼0 and has q dispersion at long wavelengths. In this limit we find that the acoustic mode is proportional of γq and only depends on the dielectric constant εs and the electron density of the layer at position z¼d, whereas the optical pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mode is proportional to ωs 2ε=1 þ ε at q¼0; i.e., the optical mode pffiffiffi differs drastically of the q dispersion at q-0 for both layered system of two dimensional electron gas. This behavior is a consequence of the
Coulomb coupling with charge density fluctuation (plasmon) of the substrate. On the other hand, we found that the critical distance at which the acoustic mode emerges above the electron–hole pair continuum for the electronic double layer 2DEG/graphene system, is proportional to Δ when γ4v1 and clearly, it predicts a critical distance equal to zero in the particular case when the distance from the 2DEG layer to the substrate is zero, whereas dc only depends on the graphene Fermi velocity and carrier density for γov1.
62 63 64 65 66 67 68 69 70 71 72 Uncited reference Q2 73 74 [20]. 75 76 77 Acknowledgments 78 79 This work was partially supported by the Instituto de Ciencia y Q3 Q4 80 Tecnologia del Distrito Federal (ICYTDF) Mexico. 81 82 83 References 84 85 [1] K.S. Novoselov, A.K. geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubanos, 86 I.V. Grigorieva, A.A. Firsov, Science 306 (2004) 666. [2] A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, A.K. Geim, Rev. Mod. 87 Phys. 81 (2009) 101. 88 [3] X. Luo, T. Qui, W. Lu, Z. Ni, Mater. Sci. Engeen: Rep. 74 (2013) 351. 89 [4] A.N. Grigorenko, M. Polini, K.S. Novoselov, Nat. Photon. 6 (2012) 749. [5] A. Politano, G. Chiarello, Front. Mater. 1 (2014) 1. 90 [6] F. Javier Garcia de Abajo, ACS Photon. 1 (2014) 135. 91 [7] T. Stauber, J. Phys. Condens. Matter 26 (2014) 123201. 92 [8] E.H. Wang, S. DasSarma, Phys. Rev. B 75 (2007) 205418. [9] B. Bunch, T. Stauber, F. Sols, F. Guinea, New. J. Phys. 8 (2006) 318. 93 [10] P.K. Pyotovkiy, 〈arXiv:0808.0931v2〉. 94 [11] J.H. Strait, P. Nene, W.M. Chan, C. manolatau, S. Tiwari, F. Rana, J.W. Kevek, 95 P. MacEven, Phys. Rev. B 87 (2013) 241410. [12] Long Ju, et al., Nat. Nanotechnol. 6 (2011) 630. 96 [13] X. Zhu, W. Yan, P.U. Jepsen, O. Hansen, N.A. Mortensen, S. Xiao, Appl. Phys. Lett. 97 102 (2013) 131101. 98 [14] M. Liu, X. Yin, X. Zhang, Nano Lett. 12 (2012) 1482. 99 [15] L. Britnell, R.V. Gorbacev, R. Jalil, B.D. Belle, F. Schedin, A. Michenko, T. Georgiou, M.I. Katsnelson, L. Eaves, S.V. Morsov, N.M.R. Peres, J. Leist, 100 A.K. geim, K.S. Novoselov, L.A. Ponomarenko, Science 335 (2012) 947. 101 [16] E.H. Wang, S. DasSarma, Phy. Rev. B 80 (2009) 205405. 102 [17] T. Stauber, G. Gomez-Santos, Phys. Rev. B 85 (2012) 075410. [18] Wang Tong Tse, Ben Yu Kuang Hu, S. DasSarma, Phys. Rev. B 76 (2007) 081401. 103 [19] B. Baharami, T. Vazifehshenas, Phys. Lett. A 376 (2012) 3518. 104 [20] R.E. Profumo, R. Asgari, M. Polini, A.H. MacDonald, Phys. Rev. B 85 (2012) 105 085443. [21] S.M. Badalyan, F.M. Peeters, Phys. Rev. B 86 (2012) 121405. 106 [22] T. Vazifehsenos, T. Amalaki, M. Farmanabar, F. Parhizgar, Phys. Lett. A 374 107 (2010) 4899. 108 [23] S.M. Badalyan, F.M. Peeters, 〈arXiv:1112.5886〉. [24] S. DasSarma, Q. Li, Phys. Rev. B 87 (2013) 235418. 109 [25] A. Principi, M. Carrega, R. Asgari, V. Pellegrini, M. Polini, 〈arXiv:1206.3896〉, 2013. 110 [26] A.C. Balram, J.A. Hutasoit, J.K. Jain, 〈arXiv:1405.4014〉, 2014. 111 [27] A. Gamucci, D..Spirito, M. Carrega, B. Karmakar, A. Lombardo, M. Bruna, A.C. 112 Ferrari, L.N. Pfeiffer, K.W. West, M. Polini, V. Pellegrini, 〈arXiv:1401:0902〉, 2014. [28] A. Benz, S. Campaine, M.W. Moseley, J.J. Wierer, A.A. Allerman, J.R. Wendt, 113 I. Brener, ACS Photon. 1 (2014) 906. 114 [29] T. Langer, D.F. Forrster, C. Busse, T. Michely, H. Pfnur, C. Tegenkamp, New J. 115 Phys. 13 (2011) 053006. [30] E. Starodub, A. Bostwick, L. Moreschini, S. Nie, F. El Gabaly, K.F. McCarty, 116 E. Rotenberg, Phys. Rev. B 83 (2011) 125428. 117 [31] J. Yan, K.S. Thygesen, K.W. Jacobsen, Phys. Rev. Lett. 106 (2011) 146803. 118 [32] N.J.M. Horing, Phys. Rev. B 80 (2009) 193401. [33] A. Principi, R. Asgari, M. Polini Solid, State Commun. 151 (2011) 1627. 119 [34] G.E. Santoro, G. Giuliani, Phys. Rev. B 37 (1988) 937. 120 [35] F. Stern, Phys. Rev. Lett. 18 (1967) 546. 121
Please cite this article as: G.G.d.l. Cruz, Solid State Commun (2015), http://dx.doi.org/10.1016/j.ssc.2015.03.021i
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Q1 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Role of metallic substrate on the plasmon modes in double-layer graphene structures G. Gonzalez de la Cruz n Departamento de Física CINVESTAV-IPN, Apartado Postal 14-740, 07000 México D.F, México
art ic l e i nf o
a b s t r a c t
Article history: Received 23 December 2014 Received in revised form 25 February 2015 Accepted 30 March 2015 Communicated by E.Y. Andrei
Novel heterostructures combining different layered materials offer new opportunities for applications and fundamental studies of collective excitations driven by interlayer Coulomb interactions. In this work, we have investigated the influence of the metallic-like substrate on the plasmon spectrum of a double layer graphene system and a structure consisting of conventional two-dimensional electron gas (2DEG) immersed in a semiconductor quantum well and a graphene sheet with an interlayer separation of d. Long-range Coulomb interactions between substrate and graphene layered systems lead a new set of spectrum plasmons. At long wavelengths (q-0) the acoustic modes (ω q) depend, besides on the carrier density in each layer, on the distance between the first carrier layer and the substrate in both structures. Furthermore, in the relativistic/nonrelativistic layered structure an undamped acoustic mode emerges for a certain interlayer critical distance dc. On the other hand, the optical plasmon modes emerging from the coupling of the double-layer systems and the substrate, both start at finite frequency at q ¼0 in contrast to the collective excitation spectrum ω q1/2 reported in the literature for doublelayer graphene structures. & 2015 Published by Elsevier Ltd.
Keywords: A. Graphene B. Two-dimensional electron gas C. Double-layers graphene structures D. Collective modes
1. Introduction A revolution of material science is coming since graphene was exfoliated successfully from graphite by Novoselov and Geim [1]. The electrons in graphene behave like massless Dirac-Fermions, which results in a extraordinary properties of carriers (electrons and holes) with ultra-high-mobility and long mean free path, gatetunable carrier densities, anomalous quantum Hall effects, fine structure constant defined optical transmission, and so on [2]. Owing to the two-dimensional nature of the collective excitations, plasmons excited in graphene are confined much more strongly than those in conventional noble metals. One of the most important advantages of graphene would be the tunability plasmons, as the carrier densities can be easily controlled by electrical gating and doping. Consequently, graphene can be applied as terahertz metamaterial and it can be tuned conveniently even for an encapsulated device. On the other hand, graphene can help to tune the surface plasmons in conventional metals, such as Au, which makes it promising plasmonic materials. Refs. [3–7] focus on the recent progress of graphene plasmonics and its technological applications. A plasmon is a collective mode of a charge–
n
Corresponding author. E-mail address: bato@fis.cinvestav.mx
density oscillation in the fee-carrier system, which is present both in classical and quantum plasmas. Studying the collective plasmon excitation in the electron gas has been among the very first theoretical quantum mechanical many-body problems studied in solid-state physics. The collective plasmon modes of monolayer graphene have been extensively studied theoretically [8–10] and experimentally [11–13] and are obtained by the zeros of the corresponding frequency and wave vector dependent dynamical dielectric function. The long-wavelength plasma oscillations are essentially fixed by the particle number conservation, and can be calculated using the random-phase approximation. The plasma dispersion frequency shows a q1/2 behavior and it is nonclassical (it depends explicitly on ℏ), this explicit quantum nature of longwavelength graphene plasmon is a direct manifestation of its linear Dirac-like energy-momentum dependence, which has no classical analogy. Based on monolayer graphene, novel double-layer structures have been recently realized experimentally [14,15] where massless fermions of two separate layers are coupled only through manybody Coulomb interactions. It is known [16,17], that when two graphene layers are put in close proximity with an oxide or semiconductor between them to prevent interlayer tunneling, the two-dimensional plasmon are coupled by the interlayer Coulomb interaction leading to the formation of two branches of longitudinal collective excitation spectra called the optical plasmon ω(q)q1/2
http://dx.doi.org/10.1016/j.ssc.2015.03.021 0038-1098/& 2015 Published by Elsevier Ltd.
Please cite this article as: G.G.d.l. Cruz, Solid State Commun (2015), http://dx.doi.org/10.1016/j.ssc.2015.03.021i
67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97
G.G.d.l. Cruz / Solid State Communications ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
and the acoustic plasmon with ω(q)q where the density fluctuation in each component oscillates in-phase (optical plasmon) and out-of-phase (acoustic plasmon), respectively, relative to each other. These collective modes of the double-layer structures, which have been directly compared with the conventional double-layer twodimensional electron gas (2DEG) [18], play important roles in the many body properties such as screening and drag [18,21]. In addition to the collective modes of two parallel twodimensional graphene layers, special attention has been addressed on the double-layer system considering their plasmon modes for doped (extrinsic) [22,23] or undoped (intrinsic) double layers [24] at finite temperature. Recently, particular interest has been paid to investigated plasmons and Coulomb drag in hybrid double-layer systems composed by a doped graphene sheet deposited on the surface of the semiconductor in close proximity to the 2DEG, long range Coulomb interactions between massive electrons and massless Dirac fermions lead to a new set of optical and acoustic intrasubband plasmons [25]. In particular, plasmons excitations of a system of coupled relativistic and nonrelativistic two-dimensional electron gas was considered by Balram et al. [26]. They found that the strength of the interaction between different charge carriers (that is, tunneling between graphene and 2DEG), play a significant role in determining the number of plasmon modes as well as their dispersions under certain parameters regimes. Such coexistence Dirac/Schrodinger hybrid electron systems have been directly experimentally observed [27,28] and they open a new research opportunities for fundamental studies of electron–electron interactions effects in two spatial dimensions. It has been shown that the surface plasmons in graphene can be significant influenced by many-particle effects involving interactions between electrons and plasmons (plasmarons). A deep understanding of the coupling between charge carriers and plasmons in graphene is highly desirable because of large potential of this material in the context of plasmonics [3]. Graphene on metals is of interest as a route to synthesizing high quality graphene and for electrical contacts to devices. Graphene–metal systems can be roughly classified into two different of film and substrate binding i.e., weak or strong hybridization between graphene π and metal d bands [29, 30]. The coupling between plasmons in graphene and electron gas in metal-like substrate were investigated both, numerically [31] and theoretically [32,33]. In this article, we investigate the role of metallic-like substrates on the collective excitations of double layer two-dimensional electron systems. We concentrate on coupled system of twodimensional Dirac fermions and 2DEG confined in a quantumwell at T¼ 0 K, tunneling between carrier in different layers has been neglected. Plasmon modes in the hybrid double-layer system are calculated within the self-consistent-field linear approximation taking into account the interaction with the substrate. The interaction between the double-layer two-dimensional electron system and the substrate is discussed by replacing the dielectric constant of the substrate by a frequency dependent dielectric function.
2. Theoretical model Firstly, we study the coupling between the two-dimensional double layer graphene system and the substrate. The model system under consideration corresponding to two graphene electron layers separated a distance d is shown in Fig. 1. The graphene electron layers system occupy a half-space z 4 Δ, of background dielectric constant εs. The substrate with dielectric constant ε0 occupies the space z o Δ.
Fig. 1. A double-layer electronic system separated by a distance d, embedded in a dielectric environment with dielectric constant εs. A substrate with dielectric constant ε0 occupies the space to the left from the first electron layer at z ¼ Δ.
Within the self-consistent-field linear approximation theory (SCF) and assuming the relaxation time be infinite, each electron is assumed to move in the self-consistent field arising from the external field plus the induced field of all the electrons, then the electron density in the nth (n ¼0,1) graphene electron layer, ρn(q,ω) is given by X ρn ðq; ωÞ ¼ V n;m ðqÞΠ m ðq; ωÞρm ðq; ωÞ þ Π n ðq; ωÞϕext ð1Þ n ðq; ωÞ m
where
h i V n;m ðqÞ ¼ vq e qjn mjd þ αe ðn þ mÞd
ð2Þ
is the interlayer Coulomb interaction and the second term proportional to εs ε0 2Δq α¼ e ð3Þ εs þ ε0 gives the modified Coulomb interaction between the two Dirac electron layers due to the image charge and Πn(q,ω) is the twodimensional polarizability for layer n calculated in Refs. [8,9], 2 ϕext n ðq; ωÞ is the external potential, and vq ¼2πe /εsq represents the two-dimensional Coulomb electron interaction. In order to study the collective excitation spectrum, it is convenient to write Eq. (1) for zero external potential and considering that charge fluctuation ρn ðq; ωÞ a 0 in each layer, the plasmon dispersion of the coupled, spatially-separated twodimensional graphene layers are obtained by the zeros of the dielectric constant Dðq; ωÞ ¼ ½1 vq ð1 þαÞΠ 1 ðq; ωÞ½1 vq ð1 þ αe 2qd ÞΠ 2 ðq; ωÞ v2q ð1 þ αÞ2 e 2qd Π 1 ðq; ωÞΠ 2 ðq; ωÞ
ð4Þ
The collective modes occur between the intraband and interband single particle excitations where Π n ðq; ωÞ is real and positive and a decreasing function of frequency [8,9]. It has been demonstrated that the two-component plasma has two branches of longitudinal oscillation spectrum which are solutions of Eq. (4). In the higher frequency branch, the two carriers oscillate in phase with the long-wavelength limit at the square root, while in the lower branch the carriers oscillate out of phase, exhibiting at longwavelength a linear dispersion. In order to obtain the expression for the acoustic plasmon oscillations, we proceed as Santoro and Giuliani [34]. We first introduce for q 0 the power expansion ωðqÞ ¼ c1 q þ c2 q2 þ c3 q3 þ ⋯
ð5Þ
for the plasmon dispersion relation and define a function FðqÞ ¼ Dðq; c1 q þ c2 q þc3 q þ ⋯Þ
ð6Þ
where D is defined in Eq. (4). For q 0, F(q) can in turn be written
Please cite this article as: G.G.d.l. Cruz, Solid State Commun (2015), http://dx.doi.org/10.1016/j.ssc.2015.03.021i
67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
G.G.d.l. Cruz / Solid State Communications ∎ (∎∎∎∎) ∎∎∎–∎∎∎
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
in terms of the Laurent–Taylor expansion FðqÞ ¼ f 1 q
1
2
þ f 0 þ f 1q þ f 2q þ ⋯
ð7Þ
where the fi are suitable coefficients which are derived from the expression Π n ðq; ωÞ. The plasmon condition Eq. (4) is then satisfied by requiring that all the coefficients fi vanish independently. The coefficient f 1 depends only on c1 and by equaling its expression to zero we arrive at the following equation for the phase velocity: pffiffiffi c1 1 8de2 π ¼ pffiffiffiffiffiffiffiffiffiffiffiffi; ξ ¼ ð8Þ p ffiffiffi 1=2 1=2 γ þn2 Þγεs 8de2 π þ ðn1 1 ξ2 here ni, i¼ 1,2 is the carrier density in each two-dimensional graphene layer. It follows from Eq. (8) that the ratio c1/γ is greater than unity for any value of the parameters n1, n2, and d and only depends on the dielectric constant of the material where the layers are immersed, it is independent of the substrate dielectric constant. Analogous results were found for inhomogeneous dielectric background graphene double-layer structures in Ref. [19]. Similar analysis can be carried out also for the optical plasmon. In such case, the energy is determined by substituting the long wavelength limit for the two-dimensional polarizability Π n ðq; ωÞ, it leads pffiffiffi pffiffiffiffiffi pffiffiffiffiffi 4e2 π ð n1 þ n2 Þ γq ð9Þ ω2 ðqÞ ¼ ð1 þ εÞεs As can be seen the plasmon energy is independent of the spacing between the two layers and it depends on the ratio ε ¼ ε0 =εs of the background dielectric constants between the substrate and the semiconductor dielectric constant. Double-layer graphene collective excitations can be influenced by the dielectric properties of the substrate, and here we study the effects of the Coulomb interaction of the double-layer graphene plasma with the semi-infinite plasma of the substrate whose local dielectric constant is given by ε0 ðωÞ ¼ εn ð1 ω2p =ω2 Þ where ωp is the plasma frequency of the carriers in the substrate and ε* is the static dielectric background of the substrate. The coupling between plasmons in single-layer graphene and substrates were also considered in Ref. [32,33]. If εs is constant, the plasmon dispersion can be calculated from the zeros of Eq. (4) using the same power expansion as in Eqs. (5) and (7), the hybridization of the plasmon frequencies in the long wavelength limit lead an acoustic plasmon mode with energy given as ωac ðqÞ ¼ cs q
ð10Þ
and the group velocity satisfies the following linear equation: " sffiffiffiffiffiffiffiffiffiffiffiffi #" sffiffiffiffiffiffiffiffiffiffiffiffi # c2s c2s 2 pffiffiffiffiffiffiffiffi 2 pffiffiffiffiffiffiffiffi 1 8e π S Δ γε 1 8e π S ðd þ ΔÞ n n γεs a s a 1 2 γ2 γ2 p ffiffiffiffiffiffiffiffiffiffi 2 ð11Þ 64e2 n1 n2 πSa Δ2 ¼ 0 with Sa ¼
cs γ
sffiffiffiffiffiffiffiffiffiffiffiffi! c2s 1 γ2
3
graphene layer at z¼ 0 and it only depends on the Fermi momenpffiffiffiffiffiffiffiffi tum of the second layer defined as k2 ¼ πn2 2 cs ð8de2 k2 þ γεs Þ2 8de2 k2 þ γεs ¼ ð13Þ γ γεs γεs ð16de2 k2 þγεs Þ showing that the acoustic plasma modes are affected by the fully coupled plasma modes present in the system of Fig. 1. On the other hand, the optical plasma modes emerging from the coupling of the double-layer graphene system and the substrate are obtained as ω2op ðqÞ ¼
2ε 2 ω þ bq; 1 þε s
ω2s ¼ ω2p =2
where 4e2 1 1 pffiffiffiffiffiffiffiffiffiffi ðpffiffiffiffiffi þ pffiffiffiffiffiÞ n1 n2 ð1 þ εÞ b ¼ pffiffiffi n2 π γð1 þ εÞ2 εs n1
ð13Þ
the point to be made here is that the optical plasmon dispersion start at finite frequency for q¼0 and changes linearly with q. Note that ω2op ðqÞ depends on both, the dielectric constant of the substrate and the dielectric constant εs but is independent of Δ and the distance d between the graphene layers. Fig. 2 shows an schematic representation of both the acoustic ωac and optic dispersion relation in the region where the imaginary part of the two-dimensional graphene polarizability vanish. The collective excitations in this region are not damped for q⪡kf. Moreover, as q increase, it is expected that the dispersion relation of the double layer graphene decay into the continuum of single particle excitation region. Finally we compute the plasmon dispersion of a conventionally two-dimensional electron gas (2DEG) and a sheet graphene layer spatially separated a distance d from the 2DEG and coupled dynamically with a substrate described with a frequency dependent dielectric constant, see Fig. 1. The double-layer massless Dirac/Scrodinger hybrid electron system was studied in Refs. [26,27] for static dielectric constant. According with the geometry shown in Fig. 1 where the 2DEG layer is located at z ¼0, the acoustic group velocity can be calculated analytically following the procedure explained in the text. After solve a similar equation as Eq. (4) where nowΠ 1 ðq; ωÞ represents the two-dimensional polarizabilty of the 2DEG [35], and Π 2 ðq; ωÞ the two-dimensional polarizability of the Dirac electron layer, we find the following equation for the group velocity cs: " sffiffiffiffiffiffiffiffiffiffiffiffiffi #" sffiffiffiffiffiffiffiffiffiffiffiffi # c2s c2s 2 2 pffiffiffiffiffiffiffiffi 1 2e m S Δ γε 1 8e π S ðd þ ΔÞ εs n s a 1 b 2 γ2 v21 p ffiffiffiffiffiffiffiffi 16e4 m1 n2 π Sa Sb Δ2 ð14Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 where Sb ¼ cs =v1 cs =v1 1 and v1, m1 the Fermi velocity and the electron effective mass in 2DEG, respectively. Sa has been
ð12Þ
Note that acoustic energy dispersion is still linear on the wavevector, however, when the interaction of the double-layer graphene with the substrate is taken into account, the phase velocity depends on the distance from the first Dirac electron layer to the substrate surface, and in the uniform limit Δ⪢d we recover the expression given by Eq. (8). We also see from this analytic expression that cs is independent of the dielectric constant of the substrate ε0 and depends only on the environment dielectric constant where the two-spatially graphene layers are immersed. On the other hand in the limit when Δ 0, the acoustic plasmon group velocity is independent of the electron density of the
Fig. 2. Schematic representation of the dispersion relation of the acoustic (dotted line) and the optical (started line) in the region where they are free Landau damping.
Please cite this article as: G.G.d.l. Cruz, Solid State Commun (2015), http://dx.doi.org/10.1016/j.ssc.2015.03.021i
67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
G.G.d.l. Cruz / Solid State Communications ∎ (∎∎∎∎) ∎∎∎–∎∎∎
4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61
defined in Eq. (12). Similar results are obtained for the group velocity following the analysis of Eq. (11). An undamped acoustic plasmon emerges for d 4dc where dc is the value of the layers separation for which cs ¼γ if the Dirac velocity is larger than the Fermi velocity and cs ¼ v1 if the Fermi velocity is larger than the Dirac velocity. By using these values of cs in Eq. (14) we obtain 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v1 Δεs > > γ 2 =v21 1 γ 4 v1 < ð2e2 m1 Δ þ εs Þv1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi γ 2 =v21 1 2e2 m1 γΔ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dc ¼ ð15Þ γ 2 εs pffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > v21 =γ 2 1 γ o v1 : 2 pffiffiffiffiffiffi 2 2 8e
n2 π γðv1 =γ
v1 =γ 1Þ
Note that the critical value of d does not depend on n2 when γ4 v1 but depends on the dielectric constant εs and it is proportional to the distance Δ between the 2DEG and the substrate, on the other hand dc depends on the carrier densities of both layers n1 and n2 for γov1 but it is independent of Δ. The analytical analysis of the long-wavelength optical plasmon mode for the hybridized double-layer electron densities is simpler since this mode is finite for tending q to zero. We obtain an analytic result using the well known mathematical expression for 2DEG and graphene polarizabily. To leading order in q in the longwavelength q-0 limit, the frequency of the spatially separated graphene and 2DEG layers optical plasmon can be found analytically. The result is pffiffiffiffiffiffiffiffi 2ε 2 4e2 n2 π γ 4e2 n1 π ω2op ðqÞ ¼ ωs þ qþ q ð16Þ 1þε εs ð1 þ εÞ m1 εs ð1 þ εÞ this plasmon dispersion has the same form as the optical plasmon frequency obtained in the presence of Coulomb coupling between the double-layer graphene systems and the substrate. It is seen however, that in contrast with the previous treatment, the second term in Eq. (16) proportional to wavevector q can be recognized as the contribution of the plasmon frequency of the electron gas in an isolated graphene sheet coupling with the surface plasmons of the substrate as calculated in Refs. [32,33]. The third term represent the contribution to the optical mode due to the Coulomb coupling between the 2DEG with the substrate. Note that the optical frequency is independent of Δ and the interlayer electronic distance. 3. Conclusions We have presented an analysis of the electronic collective modes of systems composed of two Coulomb-coupled graphene layers and relativistic and nonrelativistic electron layers separated spatially by a distance d. In addition, we formulated the theory of the coupling between the plasmon dispersion of both layered systems and substrate surface plasmon excitation. Our study focuses on the longwavelength limit in which qd is small. We have derived analytic expressions for both frequencies, the low-energy linearly dispersion acoustic plasmon mode and for the high-energy optical plasmon pffiffiffi mode that has finite frequency at q¼0 and has q dispersion at long wavelengths. In this limit we find that the acoustic mode is proportional of γq and only depends on the dielectric constant εs and the electron density of the layer at position z¼d, whereas the optical pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mode is proportional to ωs 2ε=1 þ ε at q¼0; i.e., the optical mode pffiffiffi differs drastically of the q dispersion at q-0 for both layered system of two dimensional electron gas. This behavior is a consequence of the
Coulomb coupling with charge density fluctuation (plasmon) of the substrate. On the other hand, we found that the critical distance at which the acoustic mode emerges above the electron–hole pair continuum for the electronic double layer 2DEG/graphene system, is proportional to Δ when γ4v1 and clearly, it predicts a critical distance equal to zero in the particular case when the distance from the 2DEG layer to the substrate is zero, whereas dc only depends on the graphene Fermi velocity and carrier density for γov1.
62 63 64 65 66 67 68 69 70 71 72 Uncited reference Q2 73 74 [20]. 75 76 77 Acknowledgments 78 79 This work was partially supported by the Instituto de Ciencia y Q3 Q4 80 Tecnologia del Distrito Federal (ICYTDF) Mexico. 81 82 83 References 84 85 [1] K.S. Novoselov, A.K. geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubanos, 86 I.V. Grigorieva, A.A. Firsov, Science 306 (2004) 666. [2] A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, A.K. Geim, Rev. Mod. 87 Phys. 81 (2009) 101. 88 [3] X. Luo, T. Qui, W. Lu, Z. Ni, Mater. Sci. Engeen: Rep. 74 (2013) 351. 89 [4] A.N. Grigorenko, M. Polini, K.S. Novoselov, Nat. Photon. 6 (2012) 749. [5] A. Politano, G. Chiarello, Front. Mater. 1 (2014) 1. 90 [6] F. Javier Garcia de Abajo, ACS Photon. 1 (2014) 135. 91 [7] T. Stauber, J. Phys. Condens. Matter 26 (2014) 123201. 92 [8] E.H. Wang, S. DasSarma, Phys. Rev. B 75 (2007) 205418. [9] B. Bunch, T. Stauber, F. Sols, F. Guinea, New. J. Phys. 8 (2006) 318. 93 [10] P.K. Pyotovkiy, 〈arXiv:0808.0931v2〉. 94 [11] J.H. Strait, P. Nene, W.M. Chan, C. manolatau, S. Tiwari, F. Rana, J.W. Kevek, 95 P. MacEven, Phys. Rev. B 87 (2013) 241410. [12] Long Ju, et al., Nat. Nanotechnol. 6 (2011) 630. 96 [13] X. Zhu, W. Yan, P.U. Jepsen, O. Hansen, N.A. Mortensen, S. Xiao, Appl. Phys. Lett. 97 102 (2013) 131101. 98 [14] M. Liu, X. Yin, X. Zhang, Nano Lett. 12 (2012) 1482. 99 [15] L. Britnell, R.V. Gorbacev, R. Jalil, B.D. Belle, F. Schedin, A. Michenko, T. Georgiou, M.I. Katsnelson, L. Eaves, S.V. Morsov, N.M.R. Peres, J. Leist, 100 A.K. geim, K.S. Novoselov, L.A. Ponomarenko, Science 335 (2012) 947. 101 [16] E.H. Wang, S. DasSarma, Phy. Rev. B 80 (2009) 205405. 102 [17] T. Stauber, G. Gomez-Santos, Phys. Rev. B 85 (2012) 075410. [18] Wang Tong Tse, Ben Yu Kuang Hu, S. DasSarma, Phys. Rev. B 76 (2007) 081401. 103 [19] B. Baharami, T. Vazifehshenas, Phys. Lett. A 376 (2012) 3518. 104 [20] R.E. Profumo, R. Asgari, M. Polini, A.H. MacDonald, Phys. Rev. B 85 (2012) 105 085443. [21] S.M. Badalyan, F.M. Peeters, Phys. Rev. B 86 (2012) 121405. 106 [22] T. Vazifehsenos, T. Amalaki, M. Farmanabar, F. Parhizgar, Phys. Lett. A 374 107 (2010) 4899. 108 [23] S.M. Badalyan, F.M. Peeters, 〈arXiv:1112.5886〉. [24] S. DasSarma, Q. Li, Phys. Rev. B 87 (2013) 235418. 109 [25] A. Principi, M. Carrega, R. Asgari, V. Pellegrini, M. Polini, 〈arXiv:1206.3896〉, 2013. 110 [26] A.C. Balram, J.A. Hutasoit, J.K. Jain, 〈arXiv:1405.4014〉, 2014. 111 [27] A. Gamucci, D..Spirito, M. Carrega, B. Karmakar, A. Lombardo, M. Bruna, A.C. 112 Ferrari, L.N. Pfeiffer, K.W. West, M. Polini, V. Pellegrini, 〈arXiv:1401:0902〉, 2014. [28] A. Benz, S. Campaine, M.W. Moseley, J.J. Wierer, A.A. Allerman, J.R. Wendt, 113 I. Brener, ACS Photon. 1 (2014) 906. 114 [29] T. Langer, D.F. Forrster, C. Busse, T. Michely, H. Pfnur, C. Tegenkamp, New J. 115 Phys. 13 (2011) 053006. [30] E. Starodub, A. Bostwick, L. Moreschini, S. Nie, F. El Gabaly, K.F. McCarty, 116 E. Rotenberg, Phys. Rev. B 83 (2011) 125428. 117 [31] J. Yan, K.S. Thygesen, K.W. Jacobsen, Phys. Rev. Lett. 106 (2011) 146803. 118 [32] N.J.M. Horing, Phys. Rev. B 80 (2009) 193401. [33] A. Principi, R. Asgari, M. Polini Solid, State Commun. 151 (2011) 1627. 119 [34] G.E. Santoro, G. Giuliani, Phys. Rev. B 37 (1988) 937. 120 [35] F. Stern, Phys. Rev. Lett. 18 (1967) 546. 121
Please cite this article as: G.G.d.l. Cruz, Solid State Commun (2015), http://dx.doi.org/10.1016/j.ssc.2015.03.021i