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Exciton Formation in Graphene Bilayer PHYSICAL REVIEW B 78, 045401 ͑2008͒ Exciton formation in graphene bilayer Raoul Dillenschneider1,* and Jung Hoon Han2,3,† 1Department of Physics, University of Augsburg, D-86135 Augsburg, Germany 2Department of Physics, BK21 Physics Research Division, Sungkyunkwan University, Suwon 440-746, Republic of Korea 3CSCMR, Seoul National University, Seoul 151-747, Republic of Korea ͑Received 19 September 2007; revised manuscript received 4 March 2008; published 1 July 2008͒ Exciton instability in graphene bilayer systems is studied in the case of a short-ranged Coulomb interaction and a finite voltage difference between the layers. Self-consistent exciton gap equations are derived and solved numerically and analytically under controlled approximation. We obtain that a critical strength of the Coulomb interaction exists for the formation of excitons. The critical strength depends on the amount of voltage differ- ence between the layers and on the interlayer hopping parameter. DOI: 10.1103/PhysRevB.78.045401 PACS number͑s͒: 71.35.Ϫy, 71.20.Mq, 78.67.Pt I. INTRODUCTION quantum electrodynamics ͑QED͒ deduced from the linear en- ergy spectrum of the graphene monolayer. Graphene, layers of two-dimensional honeycomb array of In the case of a bilayer, additional excitonic channels be- carbon atoms, has attracted much interest these last few years come possible as the excess electrons and holes from the two due to its recent experimental accessibility1–3 and a wide layers can form a “real-space” exciton. In this paper, we variety of interesting properties.4–6 Both the single-layer and consider the possibility of an excitonic instability in the bi- the multilayer graphenes are studied intensely. Much of the ased graphene bilayer in the framework of Hartree-Fock peculiar properties of the graphene layers arise from the en- theory. A conventional Hartree-Fock treatment had been used ergy spectrum near the so-called Dirac nodal points and the in the past to understand the exciton formation in semicon- nontrivial topological structure of the wave functions around ductors with success.14 It is shown that the exciton can be them.1,7 formed if the strength of the Coulomb interaction U is larger As the engineering application of the graphene layers at- than the threshold value Uc, which, for realistic graphene tracts increasing significance, we need to explore, experi- parameters, is comparable to the intralayer hopping energy. mentally and theoretically, ways to enrich graphene’s electri- The threshold Uc is, in turn, bias dependent and can be tuned cal properties and to control them. One way to achieve some to a minimum value for an optimal bias Vo. Moreover, a control over the electrical properties is to change the number reduction of the interlayer hopping, perhaps through interca- of layers and/or the bias applied across the layers. A recent lation, is shown to greatly reduce the threshold value Uc. experimental realization of the biased graphene bilayer is In identifying excitonic channels, we consider two pos- such an example.8–10 By applying a gate bias across the two- sible scenarios. One is the pairing through the shortest- layered graphene, the authors of Refs. 8 and 10 have ob- distance neighbors between the layers ͑a-d dimer in Fig. 2͒, served a tunable energy gap varying with the bias ͑see Fig. 1 and the other, through the second shortest-distance neighbors for the bilayer graphene energy bands in the presence of between the layers ͑a-c and b-d dimers in Fig. 2͒. For each bias͒. The bias can also potentially control the formation of scenario we identify the threshold interaction strength Uc, excitons. Since the applied bias leads to the charge imbal- 0.3 ance in the two layers, it is natural to suspect that the Cou- ++ E k/t lomb attraction of the excess electrons and holes on opposite +- layers would lead to an exciton instability similar to the situ- 0.2 E k/t ations considered in an earlier literature.11–13 If so, it will provide an additional control over the graphene as the for- 0.1 mation of excitons is known to affect the electrical properties 12,14 /t significantly. k 0 Recent works on the exciton instability in a single-layer E graphene are based on the Dirac Hamiltonian -0.1 description.15,16 The exciton gap is derived and solved through a self-consistent equation similar to the one appear- -- ing in the chiral symmetry-breaking phenomenon.17 It was -0.2 E k/t -+ shown that an exciton can be formed under a strong long- E k/t 18 ranged particle-hole interaction. Exciton can also be -0.3 Γ <-- K --> M formed in a single-layer graphene through the mechanism of magnetic catalysis of dynamical mass generation, as pointed FIG. 1. Energy spectrum for the graphene bilayer with / / Ϯ− / Ϯ+ / out in Ref. 19. This work showed that the magnetic catalysis t=2.9 eV, tЌ t=0.052, and V t=0.05. Ek t in full line and Ek t can induce exciton condensation even for weak particle-hole in dashed line. See text for definition of the energy branches labeled 20,21 ϮϮ coupling. These results are obtained in the framework of by Ek . 1098-0121/2008/78͑4͒/045401͑9͒ 045401-1 ©2008 The American Physical Society RAOUL DILLENSCHNEIDER AND JUNG HOON HAN PHYSICAL REVIEW B 78, 045401 ͑2008͒ ϮϮ Ϯ ͱ␧2 2 2 Ϯ ͱ 4 ␧2͑ 2 2͒ ͑ ͒ Ek = k +2tЌ + V 2 tЌ + k tЌ + V . 3 ␧ The bare kinetic energy k within the monolayer reads ␧ ͉͚3 ik.e␣͉ k =t ␣=1e , where e␣ are the nearest-neighbor vectors of ͑ ͒ ͑ / ͱ / ͒ the graphene monolayer: e1 = 1,0 , e2 = −1 2, 3 2 , and ͑ / ͱ / ͒ e3 = −1 2,− 3 2 . The two independent nodal points K1͑2͒, where the bare ␧ ͑ 4␲ ͒ ͑ ͒ electron spectrum k vanishes, are chosen as K1 = 0,3ͱ3 FIG. 2. Graphene bilayer Bernal stacking . The a-d dimer is ͑ ͒ ͑ ͒ depicted as dashed lines. K2 =−K1 in the basis ex ,ey in the Brillouin zone. The sum ͚ ik.e␣ ͑ / ͒͑ ͒ ␣e is approximately given by − 3 2 ky −ikx near K1 ͑ / ͒͑ ͒ and its dependence on the bias and the interlayer hopping and by 3 2 ky +ikx near K2. +− parameter. The bottom of the lower conduction band, Ek , occurs at k ␧2 ͑␧2͒ 2͑ 2 2 ͒/͑ 2 2 ͒ This work is divided into the following sequence. Section points where k = k m =V V +2tЌ V +tЌ , with the II describes the graphene bilayer and its model Hamiltonian, /ͱ 2 2 energy Em =tЌV tЌ +V . The energy gap separating the va- including the short-range Coulomb interaction across the lay- lence and conduction bands is twice this value. The energy ers. Two excitonic channels, which we will consider in this difference between the two conduction bands or the two paper, are introduced. In the following two sections, each of valence bands is ͱV2 +2t2 −V at ␧ =0 and these possibilities is examined in detail using the appropriate Ќ k ͱ͓ ͑ 4 4 ͒ 2 2 ͔/͑ 2 2 ͒ /ͱ 2 2 ␧ ͑␧ ͒ 4 V +tЌ +9V tЌ V +tЌ −VtЌ V +tЌ at k = k m. gap equations and their solutions. The work is summarized in 2 These two quantities approach tЌ /V and 2V, respectively, as Sec. V. Some of the technical aspects are summarized in the / →ϱ Appendix. V tЌ . Generally, the presence of both interlayer hopping and the bias is essential in producing the gaps separating the II. FORMULATION OF THE EXCITON PROBLEM various bands, as depicted in Fig. 1. In describing the exciton formation, we propose to use the Graphene bilayer is a two honeycomb array stacked in a interlayer interaction truncated to the second-nearest neigh- Bernal arrangement, as depicted in Fig. 2. In each layer the bors as electrons can hop between nearest-neighbor carbon atoms through ␲ orbitals with energy t, which is typically assumed ͑ ͒ ͑ ͒ VC = U1͚ na,ind,i + U2͚ na,inc,i−e␣ + nb,ind,i−e␣ . 4 to be at 2.9 eV. In a Bernal stacking, electrons are allowed to i i␣ do interlayer hopping through the a-d dimers with the hop- † ping energy given as 2tЌ and with the tЌ /t given as approxi- The local electronic densities are given by na,i =ai ai, etc. The mately 0.052. Here a dimer is defined as the pair of carbon total Hamiltonian then reads H=H0 +VC. The U1 and U2 atoms from the adjacent layers stacked along the c axis. terms are responsible for the exciton formation across the In writing down the Hamiltonian appropriate for the a-d dimer ͑nearest neighbor͒, and the a-c and b-d dimers graphene bilayer, we denote the electron operators for the ͑second-nearest neighbor͒, respectively. At this point, several mean-field decoupling strategies two sublattices in the lower layer by ai and bi, and those in present themselves. The average ͗a†d ͘ might be a candidate the upper layer by ci and di. We assume a symmetric doping i i due to the external bias ϮV with excess electrons and holes order parameter for the exciton pairing but this quantity is on the lower and upper graphene layers, respectively. We are nonzero even in the absence of any interlayer interaction, interested in the formation of the same-spin electron-hole provided the interlayer tunneling tЌ remains nonzero. Only exciton here; hence the spin degree of freedom ␴ will be when tЌ =0 does this average become the exact order param- dropped.
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