Energy-Momentum Dispersion Relation of Plasmarons in Bilayer Graphene

PHYSICAL REVIEW B 88, 165420 (2013)

Energy-momentum dispersion relation of plasmarons in bilayer graphene

P. M. Krstajic´1 and F. M. Peeters2 1Institute of Microelectronic Technologies and Single Crystals (IHTM), University of Belgrade, Njegosevaˇ 12, 11000 Belgrade, Serbia 2Departement Fysica, Universiteit Antwerpen, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium (Received 20 August 2013; published 22 October 2013) The relation between the energy and momentum of plasmarons in bilayer graphene is investigated within the Overhauser approach, where the electron-plasmon interaction is described as a ﬁeld theoretical problem. We ﬁnd

that the Dirac-like spectrum is shifted by E(k) ∼ 100 ÷ 150 meV depending on the electron concentration ne and electron momentum. The shift increases with electron concentration as the energy of plasmons becomes larger. The dispersion of plasmarons is more pronounced than in the case of single layer graphene, which is explained by the fact that the energy dispersion of electrons is quadratic and not linear. We expect that these predictions can be veriﬁed using angle-resolved photoemission spectroscopy (ARPES).

DOI: 10.1103/PhysRevB.88.165420 PACS number(s): 73.22.Pr, 73.20.Mf, 71.10.−w

I. INTRODUCTION into a field theoretical problem. By doing so, one is able to evaluate the correction to the band structure which comes Coulomb interaction and plasmarons in both single layer as a consequence of the interaction of charge carriers with graphene1–4 and bilayer graphene5,6 have recently attracted plasmons. As far as the interaction between plasmons and a lot of interest. One of the reasons is that it was found charge carriers is concerned, we generalize the Overhauser experimentally that, for instance in monolayer graphene,4 approach8,9 to the two-dimensional electron gas in bilayer the accepted view of linear (Dirac-like) spectrum does not graphene. provide a sufficiently accurate picture of the charge carrying We organize the paper as follows. In Sec. II we present excitations in this material. The concept of a quasiparticle the theoretical model and give pertinent expressions for the named “plasmaron,” was introduced which is in fact a bound interaction and the coupling between electrons and plasmons state of charge carriers with plasmons. The motivation behind in bilayer graphene. In the subsequent section, Sec. III,the the interest in this kind of studies is that exploring the numerical calculations of the energy correction due to the physics of interaction between electrons and plasmons may interaction with plasmons are presented for various doping lead to realizations of “plasmonic” devices which merge levels, i.e., charge carrier density. The influence of the doping photonics and electronics. In earlier experiments, this more level is analyzed and discussed. Finally, we summarize our complicated picture of the band structure was not observed results and present the conclusions in Sec. IV. because of the low quality and low mobility of old samples. The interest in similar phenomena in bilayer graphene is equally high. II. THEORETICAL MODEL Coulomb interaction and electronic screening was probed If the relevant energy scale in bilayer graphene is smaller in bilayer and multilayer graphene using angle-resolved than the interlayer hopping parameter t⊥,onemayusethelow photoemission spectroscopy (ARPES) in Ref. 7. Recently, 6 energy limit. In this limit, the problem can be reduced to the Sensarma et al. investigated plasmarons and the quantum effective two-band model and the corresponding Hamiltonian spectral function in bilayer graphene theoretically. The authors reads10 of that reference predicted a broad plasmaron peak away 2 †2 from the Fermi surface. Similar findings were reported in vF 0 π H0 =− , (1) Ref. 5 where thermal Green’s functions in both random phase t⊥ π 2 0 approximation and self-consistent GW approximation were used to determine the spectral function. Bilayer graphene where vF is the Fermi velocity and π = px + ipy . The eigen- (0) = 2 2 shares some properties with both graphene and the two- values of Eq. (1) are well known and read Es sh¯ k /2me dimensional electron gas found in common semiconduc- where s =±1. Here we introduced the effective mass me = 2 ≈ tors. While the energy dispersion in graphene is linear in t⊥/(2vF ) in the low energy limit, and me 0.034m0. Please momentum, in bilayer graphene it is nearly quadratic. The note that we are interested in energies larger than 1–5 meV advantage of bilayer graphene over usual semiconductors such that the usual warping is not important. It is well known is that its charge carrier density can be controlled by the that graphene structures may sustain quanta of collective application of a gate voltage over orders of magnitude charge excitations of the electron gas, i.e., plasmons, due to and this from electrons to holes. Furthermore, the band the restoring force of the long-range 1/r Coulomb interaction. gap can be tuned to meet requirements for several device However in contrast to the usual two-dimensional electron applications. gas, the “Dirac plasma” is manifestly of quantum nature.11 In this paper, we employ an alternative approach in order For instance, in single√ layer graphene the plasma frequency to determine the energy spectrum of bilayer graphene. The is proportional to 1/ h¯ , and does not have a classical limit approach is based on second order perturbation theory of independent of the Planck constant. As far as bilayer graphene the electron-plasmon interaction, and the problem is cast is concerned the plasma frequency (in the long wavelength

1098-0121/2013/88(16)/165420(4)165420-1 ©2013 American Physical Society P. M. KRSTAJIC´ AND F. M. PEETERS PHYSICAL REVIEW B 88, 165420 (2013) limit) is given by12 degeneracy factors for spin and valley degrees of freedom. The

1 previous relation, Eq. (9), is obtained from the general relation 2 2 √ 2 2πnee (q,ω) = 1 + vc(q) (q,ω), where vc(q) = 2πe /(κq)isthe ωq = q, (2) κme Fourier transform of the two-dimensional Coulomb interaction, and (q,ω) is the 2D polarizability. Finally, the actual where κ is the dielectric constant of the material and is related plasmon frequency is given by9 to the one of the substrate, κ = (1 + κs )/2 ≈ 2.5 for SiO2 substrate. The excitations of the electron gas are represented (q) ω2 = ω2 . (10) by a scalar field previously described by Overhauser8 for the q q (q) − 1 3D electron gas. The corrections in the electron spectrum are calculated analogously as for the polaron problem, where now Now, we are ready to evaluate the correction in the a test charge interacts with the plasmon field. The interaction of energy spectrum due to the interaction between electrons and an electron displaced from the graphene layer with plasmons plasmons. This will be carried out by employing second order was treated in our earlier work,9 and the interaction term of perturbation theory, and for the case of bilayer graphene it the Hamiltonian is given by reads 1 |V |2 V † =− q = √q · + E0(k) P , (11) Hint exp(iq r)(aq a−q), (3) hω¯ + E (k − q) − E (k) q q 0 0 q · where the electron-plasmon interaction matrix element is13 where P ( ) stands for the principal value. The cutoff value for the momentum q was taken to be qc = 1/a0, where a0 is the 2πe2 lattice constant. Note that this correction is given within non- Vq = √ λq. (4) κq degenerate Rayleigh-Schrodinger¨ perturbation theory (RSPT). However, for certain values of the plasmon wave vector q a Its value can be determined using the f -sum rule applied to the degeneracy occurs when E0(k) = hω¯ q + E0(k − q). Because case of interest. The starting point is the fact that the expecta- of this degeneracy, improved Wigner Brillouin perturbation tion value of the double commutator n|[n−q,[nq,H ]]|0 can 13 14 theory (IWBPT) can be employed to tackle this problem. be evaluated in two different ways. First, it is known that the The main idea behind this method is to ensure enhanced relation n|C|m=(En − Em)n|A|m holds for an arbitrary = convergence when the denominator in Eq. (11) approaches commutator with the Hamiltonian, C [H,A]. Second, it can zero, which is realized by adding the term (k) = E(k) − easily be proven that E0(k)[E0(k) is evaluated within RSPT], | | = | | | |2 n [n−q,[nq,H ]] 0 2 hω¯ n0 n nq 0 , (5) |V |2 E(k) =−P q . (12) n hω¯ + E (k − q) − E (k) − (k) q q 0 0 wherehω ¯ n0 = En − E0. Further, the explicit evaluation of the double commutator yields Equation (12) should be solved self-consistently since E appears on both sides of the equation. Note that E(k) = E(k) h¯ 2q2 hω¯ |n|n |0|2 = N . (6) due to the isotropic nature of the spectrum. In the following n0 q 2m n e section the value of E(k) will be calculated numerically The f -sum rule then reduces to for concrete values of the doping level, permittivity, and other parameters of the material. As pointed out in Ref. 3, h¯ 2q2 2 = the plasmon excitation in graphene of the Dirac sea remains hω¯ qλq N . (7) 2me pretty much well defined even when it penetrates the interband Upon combining relations in Eqs. (4)–(7) we arrive at the particle-hole continuum. This is because the transitions near expression for the interaction matrix element the bottom of the interband particle-hole continuum have almost parallel wave vectors k and k + q and therefore carry 2 negligible charge-fluctuation weight. A similar conclusion = 2πe hn¯ e Vq , (8) holds for bilayer graphene. In practice, the damping can κ 2meω q be important for very large momentum q, but then the where ne is the electron concentration, ne = N/. Note that contribution to the energy shift, i.e., to the integral in Eq. (12) is ωq is not the bare plasmon frequency but is altered by the small. polarization of the electron gas. We need the dielectric function of the electron in order to determine ωq. It can be shown that III. NUMERICAL RESULTS in the long wavelength limit (q → 0) and within the random phase approximation (RPA) the dielectric function can be The numerical calculations are carried out for doped bilayer approximated by the following relation15 graphene, with varying electron concentration. First we give q in Fig. 1 the results for the energy correction E(k)for = + s 17 12 −2 (q) 1 , (9) three levels of doping: ne = 3 × 10 cm (solid curve), q − − 5 × 1012 cm 2 (dashed curve), and 1013 cm 2 (dotted curve). 12 where qs is the screening wave vector and given by qs = As can be seen from the figure, the shift increases with the 2 2πe /κD0, while D0 is the density of states of bilayer electron momentum. The increase with k is more rapid than 16 2 18 graphene, D0 = gs gvme/(2πh¯ ). Here gs and gv are the in the case of single layer graphene. This is qualitatively

165420-2 ENERGY-MOMENTUM DISPERSION RELATION OF ... PHYSICAL REVIEW B 88, 165420 (2013)

-0.10 0.00

-0.12 -0.05

-0.14 -0.10 E(k) [eV] Δ E(k=0) [eV]

-0.16 Δ

-0.15 -0.18

0246810 0.0 0.1 0.2 0.3 0.4 Electron concentration n [x1012 cm-2] Electron wave vector k [nm-1] e = FIG. 1. The correction to the energy E(k) vs electron momen- FIG. 2. The correction to the energy E(0) for k 0vsthe 12 −2 electron concentration ne. tum k for three values of the doping level ne = 3 × 10 cm (solid curve), 5 × 1012 cm−2 (dashed curve), and 1013 cm−2 (dotted curve). −8 −5 −2 a =−4.7 × 10 , b = 2.53 × 10 [ne is given cm and similar to what was found in Refs. 5 and 6 where a broad peak E(0) in eV]. It would be instructive to determine the was attributed to plasmarons. While the√ explicit dependence effective mass of the plasmaron band defined by E(k) = ∝ 2 2 on the concentration is the same Vq ne, the interaction E0(k) + E(k) = h¯ k /2m . Figure 3 shows the dependence matrix element is related to the doping level also through of the effective mass ratio m /me on the electron concentration. the plasmon frequency. The latter in single layer graphene The ratio starts from a value around 2 and drops rapidly to 1/4 × 12 −2 √is mainly proportional to ne while in bilayer graphene is 1.2, while for ne > 2 10 cm it converges slowly to 1. ne. Further, the effective plasmon frequency is modulated The low and high density behavior fits are shown by dashed through the polarization of the surrounding electron gas. lines in Fig. 3 and were fitted respectively to the following −12 2 −26 4 One should not forget that a property of bilayer graphene, expressions: 2.1 − 1.2 × 10 cm · ne + 5.1 × 10 cm · 2 × 11 −2 − × −14 2 · + important for the present analysis, is the fact that the coupling ne for ne 8 10 cm and 1.2 3.9 10 cm ne −1/2 × −27 4 · 2 × 11 −2 parameter is a function of the carrier density rs ∝ ne 2.4 10 cm ne,forne > 8 10 cm . Note that (while in single layer graphene it is independent of ne). More the large dispersion of the plasmaron band was also re- precisely, the strength of the Coulomb interaction is tunable vealed in recent theoretical investigation of the spectral 6 and depends on the level of doping. As for the comparison with function, where it was determined that plasmarons do have a earlier theoretical findings, we got E(k = kF ) = 0.12 eV for broad peak. 12 −2 ne = 3 × 10 cm while the authors of Ref. 6 obtained the value E(k = kF ) = 0.18 eV. IV. CONCLUSION Unlike the case of polarons in conventional semiconductors, here it is not straightforward to derive any approximate In this paper we investigated the interaction between an analytical relation for E(k)atsmallk. This is due to the electron and the collective excitation of the electron gas, i.e., fact that plasmons here have a more complicated dispersion plasmons, in bilayer graphene by using a field-theoretical approach. This interaction is modeled by generalizing the relation, and the fact that the interaction strength Vq depends on q in a nontrivial manner. Thus we will treat Eq. (11) numerically and one may write for small k 2.2 2 4 E(k) = E(0) + αk + βk . (13) 2.0 e We fitted Eq. (13) to our numerical results within the range 1.8 −1 12 −2 0

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Overhauser approach8 to the 2DEG in this material. We Further, we investigated the inﬂuence of the doping level evaluated the energy correction, that is the shift in the energy on the shift E(0), and it is shown that it increases with ne spectrum as a result of this interaction. We employed second which is more pronounced than in the case of single layer order perturbation theory in order to determine the energy of graphene.18 The difference with single layer graphene lies the plasmaron, which is a composite quasiparticle, i.e., a bound in the actual dependence of the interaction strength Vq on state of an electron with plasmons. the electron concentration. This should be revealed in future The motivations behind the present study were the increased angle-resolved photoemission spectroscopy. interest in the spectral function of bilayer graphene,5,6 and the prediction of the existence of a broad plasmaron peak away,6 ACKNOWLEDGMENTS but near the Fermi surface. First we evaluated the correction to the energy as a result of the interaction between electron This work was supported by the Flemish Science Foun- and plasmons. The shift is appreciable and lies in the range dation (FWO-Vl), the ESF-EuroGRAPHENE project CON- of 100–150 meV depending on the electron concentration and GRAN, and by the Serbian Ministry of Education and Science, electron wave vector. within the Project No. TR 32008.

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