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Collective Modes of the Massless Dirac Plasma

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Collective Modes of the Massless Dirac Plasma Collective modes of the massless Dirac plasma S. Das Sarma and E. H. Hwang Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, Maryland 20742-4111 and Kavli Institute for Theoretical Physics, Santa Barbara, California 93106 (Dated: April 26, 2009) We develop a theory for the long-wavelength plasma oscillation of a collection of charged massless Dirac particles in a solid, as occurring for example in doped graphene layers, interacting via the long-range Coulomb interaction. We find that the long-wavelength plasmon frequency in such a doped massless Dirac plasma is explicitly non-classical in all dimensions with the plasma frequency being proportional to 1/√!. We also show that the long wavelength plasma frequency of the D- dimensional superlattice made from such a plasma does not agree with the corresponding D +1 dimensional bulk plasmon frequency. We compare and contrast such Dirac plasmons with the well- studied regular palsmons in metals and doped semiconductors which manifest the usual classical long wavelength plasma oscillation. PACS numbers: 52.27.Ny; 81.05.Uw; 71.45.Gm A collection of charged particles (i.e. a plasma), elec- ‘!’ appearing manifestly in the long wavelength plasma trons or holes or ions, is characterized by a collective frequency in D = 1, 2, 3 dimension (and in between). mode associated with the self-sustaining in-phase den- By contrast the long wavelength plasma frequency of or- sity oscillations of all the particles due to the restoring dinary electron liquids is classical, and quantum effects force arising from the long-range 1/r Coulomb poten- show up only as nonlocal corrections in higher order wave tial. The classical plasma frequency in three-dimensional vector dispersion of the plasmon mode. This is quite un- 2 1/2 (3D) plasmas [1] is well-known to be ω3 = (4πn3e /m) expected in view of the popular belief that the long wave- where e and m are respectively the charge and the mass length quantum plasmon dispersion is necessarily a clas- of each particle, and n3 is the 3D particle density. (In sical plasma frequency [2–4]. The popular belief seems this paper, we use ωD and nD as the D-dimensional to be true for the usual parabolic energy dispersion, but long-wavelength plasma frequency and particle density not for the linear Dirac spectrum. respectively.) A solid state degenerate plasma [2–4] ex- We start from the fundamental many-body formula ists in metals and doped semiconductors where free car- defining the collective plasmon mode in an electron sys- riers can move around quantum mechanically in the ionic tem: lattice background. Such a degenerate quantum plasma #(q,ω) = 1 v(q)Π(q,ω)=0, (1) has the quantized version of exactly the same collective − mode, the so-called plasmon [2–4], which dominates the where #(q,ω) is the wave vector (q) and frequency (ω) spectral weight of the long-wavelength elementary exci- dependent dynamical dielectric function of the system, tation spectrum of an electron liquid. (We will use the with Π(q,ω) the irreducible polarizability and v(q) the world ‘electron’ generally throughout this paper to in- Coulomb interaction between the electrons in the wave dicate either electron or hole.) The collective plasmon vector space. The zero of the dielectric function in Eq. (1) modes of solid state quantum plasmas have been ex- signifies a self-sustaining collective mode, with the solu- tensively studied experimentally and theoretically over tion of Eq. (1) giving the plasmon frequency as a function the last sixty years in both metals and doped semicon- of wave vector. We first recapitulate the known results ductors. In the present work, we study theoretically for the parabolic dispersion electron system before dis- the collective plasmon mode in a solid state plasma of cussing the novel collective dispersion for massless Dirac massless Dirac fermions, as occurring for example, in 2D plasma. graphene layers. We define the Dirac plasma as a system The Coulomb interaction in the wave vector space is of charged carriers whose energy-momentum dispersion given by the appropriate D-dimensional Fourier trans- is linear, obeying the Dirac equation. form of the Coulomb interaction v(r)=e2/κr Our main qualitative result is that the massless Dirac 4πe2 v(q)= D =3, (2a) plasma is manifestly quantum, and does not have a classi- κq2 cal limit in the form of an !-independent long-wavelength 2πe2 = D =2, (2b) plasma frequency, in a striking contrast to the corre- κq sponding parabolic dispersion electron liquids familiar 2e2 from the extensive study of plasmons in metals and = K0(qa) D =1, (2c) semiconductors [3–5]. The long-wavelength plasmon fre- κ quency of a Dirac plasma is necessarily quantum with where we have introduced a background dielectric con- 2 stant (κ) which, in general, differs from unity in semi- the corresponding massless Dirac plasma, that the long (p) conductor based electron systems, and K0 is the zeroth- wavelength plasmon frequency ω for parabolic systems order modified Bessel function of the second kind. We is completely classical since ‘!’ does not appear in the note that K0(x) ln(x) for x 0, and the length ‘a’ leading term of Eq. (5) in any dimension. (Note that the in the 1D Coulomb∼| interaction| in→ Eq. (2c) characterizes right hand side of Eq. (5) has the explicit dimensionality the typical lateral confinement size of the 1D electron of time inverse, i.e. a frequency, in each dimension since D system (ES) which is obviously necessary in defining a nD has the dimension of (length)− in D-dimension.) 1DES. The second order dispersion correction term in Eq. (5), The irreducible polarizability function Π(q,ω) of an i.e. the O(q2,q3/2,q3) term in D =3, 2, 1 respectively, interacting ES is, in general, unknown since self-energy is fully quantum mechanical (i.e. ‘!’ shows up explicitly and vertex corrections cannot be calculated exactly. A in the non-local wave vector corrections), and is affected great simplification, however, occurs in the long wave- by interaction corrections (both self energy and vertex length limit (q 0) when the dielectric function, and corrections to the irreducible polarizability). consequently, the→ plasmon frequency is determined en- Now we consider plasmons in the D-dimensional mass- tirely by the noninteracting irreducible polarizability, the less Dirac plasma, where the single-particle energy dis- electron-hole ‘bubble’ diagram. The noninteracting irre- persion is linear, i.e. ξk = !vF k vp classically, in ducible polarizability is given by the expression: D =1, 2, 3. The long wavelength| |→ quantum plasmon dispersion is still defined by the set of formula given D d k nF (ξk) nF (ξk+q) by Eqs. (1)–(3) with the explicit form of the nonin- Π(q,ω)=g D − F (k, q), (3) (2π) !ω + ξk ξk+q teracting irreducible polarizability being calculated with ! − ξ = v k in Eq. (3). where ξ is the single particle energy dispersion i.e. k ! F k The long| | wavelength (q 0) form for the noninteract- ξ = 2k2/2m for parabolic systems (and ξ = v k k ! k ! F ing irreducible polarizability→ (Eq. (3)) can be calculated for the massless Dirac plasma), n is the Fermi distri- F for linear energy dispersion relation (i.e. ξ = v k) in bution function, and F (k, q) is the overlap form factor k ! F all dimensions, giving due to chirality. For non-chiral systems F (q, k) = 1. The D 1 D/2 2 factor ‘g’ in Eq. (3) is the degeneracy factor: g = gsgv gv k − 2π q Π(q,ω)= F F + O(q4/ω4), (6) where gs (=2) is the spin degeneracy and gv is the valley D(2π)D Γ( D ) ω2 or pseudospin degeneracy. 2 2 2 Putting ξk = ! k /2m, we can easily calculate Eq. (3) where kF is the Fermi momentum of the system and Γ(x) upto the leading order in wave vector (i.e. the long wave- is the Gamma function. (We note that the chirality factor length limit) to obtain F (k, q) in Eq. (3) does not influence the long wavelength limit.) n q2 Π(q,ω) D + O(q4/ω4). (4) Combining Eqs. (1), (2), and (6), we get the following ≈ m ω2 (l) for the long wavelength plasmon frequency, ωD , in D = Combining Eqs. (1)–(4) we immediately obtain the 1, 2, 3 Dirac plasma: well-known long-wavelength plasma frequency in a D- g dimensional ES: ω(l) = √r v q ln(qa) + O(q3), (7a) 1 s π F | | " 2 2e n1 (l) 1/#4 1/2 3/2 (p) 3 ω = √rs(gπn2) vF q + O(q ), (7b) ω1 = q ln(qa) + O(q ), (5a) 2 κm | | 1 " 6 2# (l) 32πg 1/3 2 (p) 2πn2e 1/2 3/2 ω3 = √rs n3 vF + O(q ), (7c) ω2 = q + O(q ), (5b) 3 " κm $ % 2 where we have introduced the dimensionless fine struc- (p) 4πn3e ω = + O(q2), (5c) 2 3 κm ture constant rs( e /(κ!vF )) for notational simplicity. " Comparing Eqs.≡ (5) and (7) we see that ω(p) and ω(l) (p) have one important similarity and several striking differ- where ωD denotes the long-wavelength (q 0) plas- mon mode in the D-dimensional parabolic dispersion→ ES ences. The similarity is that the plasmon dispersion (i.e. (with the carrier density nD per unit D-dimensional vol- the power law dependence of the plasma frequency on ume) where the one particle energy is given by ξ = wave vector) is the same in the parabolic system and the !2k2/2m p2/2m = mv2/2 classically (where p = !k massless Dirac plasma for all D.
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