Coulomb screening and collective excitations in biased bilayer graphene arXiv:1001.2762v1 [cond-mat.mes-hall] 15 Jan 2010 Xue-Feng Wang ∗ Department of physics, Soochow University, 1 Shizi Street, Suzhou, China 215006 Tapash Chakraborty Department of Physics and Astronomy, The University of Manitoba, Winnipeg, Canada, R3T 2N2 We have investigated the Coulomb screening properties and plasmon spectrum in a bilayer graphene under a perpendicular electric bias. The bias voltage applied between the two graphene layers opens a gap in the single particle energy spectrum and modifies the many-body correlations and collective excitations. The energy gap can soften the plasmon modes and lead to a crossover of the plasmons from a Landau damped mode to being undamped. Plasmon modes of long lifetime may be observable in experiments and may have potentials for device applications. PACS numbers: 71.10.-w,75.10.Lp,75.70.Ak,71.70.Gm Bilayer graphene (BLG) has attracted much attention due to its unique electronic characteristics, distinct from the Dirac gas in monolayer graphene and the Fermi gas in traditional semiconductor quantum wells [1–3]. In addi- tion, an energy gap between the conduction and valence bands of a BLG can be opened and tuned by introducing an electrostatic potential bias between the two graphene layers [4–11]. This can be easily realized via one or more external gates to perpendicularly bias BLG and make it a potential component for integrated electronics. It is then very intriguing to understand some fundamental properties such as correlation and screening properties of electron gases in a biased BLG. As collective excitations, plasmon modes are a direct result of electronic correlation due to Coulomb interaction between electrons. Experi- mental detection of plasmon modes has recently become feasible and has been used to determine the dynamical behavior of electrons in graphene layers [12–14]. Previously, assuming zero or non-zero spin-orbit inter- action induced energy gap, we have studied the Coulomb screening and collective excitation spectrum of intrin- sic and doped monolayer graphenes at zero and finite temperatures in the random phase approximation (RPA) [15]. Later, Qaiumzadeh and Asgari [16] assumed an un- specified energy gap of arbitrary width for doped mono- layer graphene and studied the corresponding ground- state properties at zero temperature in RPA. They con- cluded that the conductance and charge compressibility decrease with the band gap. Furthermore, a THz source has been proposed based on the stimulated plasmon emis- sion in graphene [18] and the absorption of THz electro- magnetic radiation in gapped graphene has been esti- mated [19]. On the other hand, the Coulomb screening and the collective excitations in zero gap BLG have been studied in our previous work at zero and finite tempera- tures [2] and by Hwang and Das Sarma [3] for the zero temperature case. In this paper we report on our studies The eigenenergy of the above Hamiltonian is E λ k = λU √ 1 + ( ℏ $^{2}$k$^{2}$/m $^{∗}$U ) $^{2}$/ 2 with the eigenfunc- tions Ψ +1 k ( r ) = ( cos( α k / 2) − sin( α k / 2) e i 2 θ$_{k}$ ) e i k $^{·}$r and Ψ − 1 k ( r ) = ( sin( α k / 2) cos( α k / 2) e i 2 θ$_{k}$ ) e i k $^{·}$r for λ = +1 and − 1 respectively. Here θ is the azimuth of the vector k , i.e., tan θ$_{k}$ = k$_{y}$/k$_{x}$ , and α$_{k}$ indicates the ratio of the kinetic energy to the po- tential bias with tan α$_{k}$ = ℏ $^{2}$k$^{2}$/ ( m $^{∗}$U ). The conduction band which touches the valence band at k = 0 in unbi- ased BLG becomes separated from it by an energy gap equal to the potential bias U . This gap converts the BLG from a semimetal into a semiconductor and accordingly modifies the optical and electric properties of the elec- trons inside. For finite U , the density of states of the BLG diverges on the edge of the energy gap | E | = U/ 2. At zero temperature, the carrier density N in a BLG of with k ± = k$_{x}$ ± ik$_{y}$ = − i ∇ x ∓ i ∇ y and k = ( k$_{x}$, k$_{y}$ ) being measured from the K point. The effective mass of the quadratic term is m ∗ = 2 ℏ $^{2}$γ 1 / (3 a 0 γ 0 ) 2 ≈ 0 . 033 m 0 with m 0 the free electron mass, a 0 = 1 . 42 ˚ A the C-C bond length on the graphene layer, γ 0 = 3 . 16 eV the intra-layer coupling, and γ 1 = 0 . 4 eV the direct inter-layer coupling. The second term arises from the electrostatic potential bias U between the two graphene layers separated by a distance d = 3 . 35 ˚ A. The Hamiltonian is obtained by keeping only the linear term of the Tayler expansion on the small energy value in unit of γ 1 , so it is valid for electrons of energy less than 0.4 eV which is adequate in our case. The indirect inter-layer coupling is neglected since it affects only the energy band in the range of less than 2 meV from the middle of the conduction-valence band gap [2]. of the correlations, screening, and the plamson spectrum of electron gases in a biased BLG. In the effective-mass approximation [1], the Hamilto- nian describing electrons of moderate energies in the K valley of a biased BLG reads H$_{K}$ = ℏ 2 2 m ∗ ( 0 k 2 − k 2 + 0 ) + U 2 ( 1 0 0 − 1 ) (1) ∗ Electronic address: xf˙wang1969@yahoo.com