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) plasmon 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. This is indeed required is momentum).→ The long wavelength plasmon frequen- under very general principles since for any Coulomb sys- cies for parabolic dispersion systems given in Eq. (5) are, tem, the long wavelength plasmon dispersion is set by of course, well-known and have been verified experimen- the continuity equation (or equivalently, by particle con- (3 D)/2 tally extensively [4–6]. Our purpose of deriving Eq. (5) is servation) to be ωD(q 0) q − as is obeyed by (p) (l) → ∼ the explicit demonstration, to be contrasted below with both ωD and ωD . 3

(l) The most striking qualitative feature of ωD in Eq. (7), clusion of the inter-layer or inter-ribbon Coulomb inter- (p) action, which will necessarily couple all the layers (or the in sharp contrast with the usual ωD in Eq. (5), is that ‘!’ appears explicitly in the leading term, not just the sub- ribbons) due to the long range nature of the Coulomb leading nonlocal corrections. A simple dimensional anal- potential. This changes the fundamental collective mode (l) 1/2 equation (Eq (1)) to an infinite matrix equation: ysis of Eq. (7) shows that ωD O(!− ) in all dimen- 0 ∼ sions in contrast to the O(! ) purely classical behavior δ v (q,ω)Π (q,ω) =0, (8) (p) | ll! − ll! l! | of ωD in Eq. (5). There is no classical plasma frequency in the massless Dirac plasma, i.e. the long-wavelength where Πl =Πis the irreducible polarizability of each 2D plasma frequency for ES with linear dispersion explicitly layer (or 1D ribbon), which is exactly the same polarizability considered in Eqs. (3) and (4). In Eq. (8), v is depends on !, and is therefore, by definition, nonclassi- ll! cal. This absence of a classical long wavelength plasma the Coulomb interaction between the l and the l" layer frequency in the Dirac plasma is a direct manifestation or ribbon in the periodic array, which is given by of the relativistic Dirac nature of the underlying quan- 2 2πe qd l l! tum description, and such a Dirac plasma does not have v = e− | − | D =2, (9a) ll! κq a classical plasma frequency. Note that the nonclassical 2 nature of the long wavelength plasma oscillation of the 2e vll = [K0(qa)+K0(qd l l" )] D =1, (9b) Dirac plasma is independent of the chirality or gapless- ! κ | − | ness of graphene, and arises primarily from the linear where ‘d’ is the superlattice period (to be distinguished Dirac spectrum. from the length ‘a’ in D = 1 which defines the lateral Associated with the appearance of 1/√! in the long- width of each ribbon). wavelength plasma frequency of the Dirac plasma are sev- The periodic invariance of the superlattice and the as- eral other interesting properties distinguishing it from sociated Bloch’s theorem allow an immediate solution of the standard parabolic dispersion Schrödinger plasma: the infinite-dimensional determinantal equation defined (1) The density dependence of the Dirac plasmon is dif- by Eq. (8), leading to the following collective plasmon ferent from the regular plasmon [7, 8] — in particu- bands for the superlattice structure: lar, the density dependence is weaker in the sense that (l) 1/3 1/4 0 ω˜2s(q; k)=ω2(q)S2(q, k) D =2, (10a) ωD n ,n ,n in D = 3, 2, 1 respectively in ∝ (p) ω˜1s(q; k)=ω1(q)S1(q, k) D =1, (10b) contrast with ωD √n in all dimensions. In general, the plasmon frequency∝ in the Dirac plasma is given by whereω ˜Ds is the plasmon band frequency for the su- (l) (D 1)/2D ω n − . (2) The 1D Dirac plasmon frequency perlattice (D = 2 for the multilayer and D = 1 for the D ∝ is curiously density independent. (3) The quantum cou- multiribbon periodic arrays) and ωD is the corresponding pling parameter (i.e. the effective fine structure constant) 2D (D = 2) and 1D (D = 1) plasmon modes discussed shows up explicitly in the long-wavelength Dirac plasmon in Eqs. (5) and (7). The wave vector q in Eq. (10) is the (l) frequency, ω √rs. (4) The long wavelength Dirac same conserved 2D or 1D wave vector in each individual D ∝ (l) 1/2 2D layer or 1D nanoribbon defining the plasmon disper- plasmon ωD goes as !− for all dimensions whereas (p) 1/2 sion relation ωD(q) whereas the additional wave vector the long wavelength regular plasmon ωD goes as nD in all dimensions – we are unsure whether this curious k is a new continuous parameter defining the superlat- dichotomy has any deep significance or not. tice plasmon band (arising from the periodicity in the array structure). The band wave vector k is restricted Before concluding, we consider another interesting and to the first superlattice Brillouin zone, k π/d, in the peculiar feature of the Dirac plasmon distinguishing it reduced zone scheme. For the 2D layer superlattice,≤ if from the regular plasmon. We consider collective modes each layer is assumed to lie in the x-y plane, then k = qz of periodic arrays of 2D Dirac plasma layers (for exam- is along the superlattice direction of the z-axis. For the ple, a graphene superlattice made of parallel 2D graphene 1D ribbon superlattice, if each ribbon is assumed to be sheets in the direction transverse to the 2D graphene along the x-axis (i.e. q = qx) with a width of ‘a’ defining plane) and of 1D Dirac plasma nanoribbons (i.e. a the ribbon in the y-direction, then k = qy is along the graphene superlattice made of identical 1D graphene superlattice direction of the y-axis. nanoribbons placed parallel to each other in the 2D The function SD in Eq. (10) is a form factor arising plane). Collective plasmon modes of such 2D [9] and from the Coulomb coupling between all the layers and 1D [10] superlattices have been theoretically studied in the ribbons forming the periodic array, and is given by the context of regular parabolic systems, and have been q l l! d iqz l l! d experimentally observed in doped GaAs multi-quantum S2 = e− | − | − | − | D =2, (11a) well and multi-quantum wire structures. &l! The main physics to be considered in describing the S1 = [K0(q l l" d) cos(lqyd)+K0(qa)] D =1. (11b) collective plasmon modes of such superlattices is the in- | − | &l! 4

Combining the above equations for superlattice plas- all on the carrier density, and only the following substi- mons, we get the following long wavelength (q 0) plas- tution provides a correspondence betweenω ˜(l)(q, k = 0) → 1s mon bands for 2D and 1D arrays in parabolic and Dirac (l) and ω2 (q 0): plasma systems, respectively: → 2 1/2 n˜2 = g/(πd ), (16) (p,l) (p,l) sinh(qd) ω˜2s (q)=ω2 (q) , (12a) cosh(qd) cos(qzd) which is a constant for all carrier density. This absence ' − ( of correspondence betweenω ˜(l) (k = 0) and ω(l) is the ω˜(p,l)(q)=ω(p,l)(q) Ds D+1 1s 1 direct consequence of the density dependence of the irre- 1/2 ∞ ducible polarizability, Eq. (6), (i.e., the density response K (qa)+2 K (nqd) cos(q nd) . (12b) × 0 0 y function). In linear response the density response func- ) n=1 * & tion should depend linearly on the total density of the Eq. (12) above defines plasmon bands for superlattice ES as it does for the ordinary parabolic ES. However, for arrays made out of periodic 2D layers and 1D ribbons in the Dirac plasma the density response function is given (D 1)/D parabolic and linear plasma systems. by Π n − . This peculiar density dependence of (l) the polarizability∝ is a manifestation of the quantum na- An interesting quantum feature ofω ˜Ds(q, k) is appar- ent when one looks at the long-wavelength plasmon (q ture of the Dirac plasma and gives rise to the lack of → 0) at the band edge k = 0, and comparesω ˜(l) (q, k = 0) correspondence between the band-edge plasmon at k =0 Ds in the D-dimensional superlattice and the corresponding withω ˜(p)(q, k = 0). We get Ds bulk plasmon in (D + 1)-dimension. 4πn˜ e2 1/2 n In summary, we have found that the long-wavelength ω˜(p)(q; q = 0) = 3 withn ˜ = 2 , (13a) 2s z κm 3 d plasma frequency of a massless Dirac plasma with lin- $ % ear carrier energy dispersion is non-classical with an ex- 2 1/2 (p) 2πn˜2e q n1 plicit 1/√! appearing in the plasma frequency. This is in ω˜ (q; qy = 0) = withn ˜2 = , (13b) 1s κm d sharp contrast with the widespread expectation that the $ % and long-wavelength plasmon is a classical plasma oscillation – in fact, a massless Dirac plasma has no classical anal- 1/4 (l) 1/4 n˜3 ogy. In addition, the long-wavelength Dirac plasma fre- ω˜ (q; qz = 0) = √rs(4πg) vF , (14a) 2s d quency depends explicitly on the coupling constant (“the $ % g fine structure constant”). We have also shown that the ω˜(l)(q; q = 0) = √r v √q. (14b) 1s y s d F long wavelength plasma mode of a D-dimensional super- " lattice of massless Dirac plasma does not reduce to the We note that Eq. (13) for the usual parabolic electron corresponding (D + 1)-dimensional bulk plasmon, as one plasma has the appropriate physical limit at the band- would have expected intuitively. All of these peculiar re- edge k = 0, where the D-dimensional superlattice plas- sults follow from the fact that a massless Dirac plasma is mon should have the precise character of the correspond- fundamentally non-classical since the energy dispersion ing (D + 1)-dimensional bulk plasmon in the long wave- E = vp characterizing a system with constant velocity (p) (p) length limit, and indeedω ˜2s (q, k = 0) andω ˜1s (q, k = 0) (but variable momentum) simply cannot happen in clas- are identical to the corresponding 3D and 2D plasmons sical physics. We believe that our predictions can be (in Eq. (5)) respectively withn ˜3 = n2/a andn ˜2 = n1/a. tested in doped graphene layers and multilayers, and in This is exactly what one expects since the D-dimensional doped graphene ribbons and multiribbons arrays using superlattice “loses” its discrete periodic structure for electron scattering [11], light scattering [12], or infrared k = 0 and simply becomes the (D + 1)-dimensional reg- [13] spectroscopies. But the real importance of our re- ular plasmon at long wavelength. sults is conceptual as we establish a strange quantum However, this correspondence does not happen for the behavior in the graphene world of a Dirac plasma where Dirac plasma, i.e., the D-dimensional superlattice plas- the long wavelength plasmon is explicitly non-classical mon for k = 0 does not become the corresponding (D+1)- in contrast to plasmons in ordinary semiconductors and dimensional bulk plasma frequency as one expects intu- metals whose long wavelength limit is necessarily a clas- (l) itively. In particular,ω ˜2s (q, k = 0) would agree with the sical plasma frequency. (l) This work is supported by US-ONR, NSF-NRI, SWAN, corresponding 3D Dirac plasmonω ˜3 (q) (in Eq. (7)) only if we define the corresponding effective 3D density to be and DOE-Sandia.

1/4 2 3/4 n˜3 = (9πg/16) n2/d , (15) rather than the intuitive deﬁnition+ n ˜3, = n2/d. For the 1D superlattice Dirac plasmon, the situation is qualita- [1] J. D. Jackson, Classical Electrodynamics, (John Wiley & (l) tively diﬀerent sinceω ˜1s (q, k = 0) does not depend at Sons, New York, 1999). 5

[2] P. M. Platzmann and P. A. Wolﬀ, Waves and Interactions [9] S. Das Sarma and J. J. Quinn, Phys. Rev. B 25, 7603 in Solid State Plasmas, (Solid State Physics, Academic (1982). Press, 1973). [10] S. Das Sarma and Wu-yan Lai, Phys. Rev. B 32, 1401 [3] D. Pines and P. Nozieres, The Theory of Quantum Liq- (1985). uids, (W. A. Benjamin, New York, 1966). [11] T. Eberlein et al., Phys. Rev. B 77, 233406 (2006); A [4] G. D. Mahan, Many Particle Physics, (Plenum Publisher, Nagashim et al., Solid State Commun. 83, 581 (1992). New York, 2000). [12] A. Pinczuk et al., Phys. Rev. Lett. 56, 2092 (1986); D. [5] T. Ando, A. B. Fowler, and F. Stern, Rev. Mod. Phys. Olego et al., Phys. Rev. B 25, 7867 (1982); A. Pinczuk 54, 437 (1982). et al., Phys. Rev. Lett. 47, 1487 (1981); M. A. Eriksson [6] Q. P. Li and S. Das Sarma, Phys. Rev. B 43, 11768 et al., Phys. Rev. Lett. 82, 2163 (1999); A. R. Goni et (1991). al., Phys. Rev. Lett. 67, 3298 (1991). [7] E. H. Hwang and S. Das Sarma, Phys. Rev. B 75, 205418 [13] S. J. Allen et al., Phys. Rev. Lett. 38, 980 (1977). (2007). [8] L. Brey and H. Fertig, Phys. Rev. B 75, 125434 (2007).