A Static structure factors

Pseudo-zero-mode Landau levels and collective excitations in bilayer graphene

Abstract

Bilayer graphene in a magnetic field supports eight zero-energy Landau levels, which, as a tunable band gap develops, split into two nearly-degenerate quartets separated by the band gap. A close look is made into the properties of such an isolated quartet of pseudo-zero-mode levels at half filling in the presence of an in-plane electric field and the Coulomb interaction, with focus on revealing further controllable features in bilayer graphene. The half-filled pseudo-zero-mode levels support, via orbital level mixing, charge carriers with nonzero electric moment, which would lead to field-induced level splitting and the current-induced quantum Hall effect. It is shown that the Coulomb interaction enhances the effect of the in-plane field and their interplay leads to rich spectra of collective excitations, pseudospin waves, accessible by microwave experiments; also a duality in the excitation spectra is revealed.

pacs:
73.43.-f,71.10.Pm,77.22.Ch

I Introduction

Graphene, a monolayer of graphite, attracts a great deal of attention, both experimentally (1); (2); (3) and theoretically, (4); (5); (6); (7); (8) for its unusual electronic transport, characteristic of relativistic” charge carriers, massless Dirac fermions. Dirac fermions give rise to quantum phenomena reflecting the particle-hole picture of the vacuum state, such as Klein tunneling, (9) and, especially in a magnetic field, such peculiar phenomena (10); (11); (12); (13); (14) as spectral asymmetry and induced charges, which are rooted in the chiral anomaly (i.e., a quantum conflict between charge and chirality conservations). Graphene provides a special laboratory to test such consequences of quantum electrodynamics. Actually, the half-integer quantum Hall (QH) effect and the presence of the zero-energy Landau level observed (1); (2) in graphene are a manifestation of spectral asymmetry.

Bilayer graphene is equally exotic (15); (16); (17) as monolayer graphene. It has a unique property that the band gap is controllable (18); (19); (20); (21); (22) by the use of external gates or chemical doping; this makes bilayers richer in electronic properties. In bilayer graphene interlayer coupling modifies the intralayer relativistic spectra to yield quasiparticles with a parabolic energy dispersion. (16) The particle-hole structure still remains in the ”chiral” form of a Schroedinger Hamiltonian, and there arise eight zero(-energy)-mode Landau levels (two per valley and spin) in a magnetic field.

Zero-mode Landau levels, specific to graphene, deserve attention in their own right. They would show quite unusual dielectric response (23); (24) while they carry normal Hall conductance per level; for bilayer graphene direct calculations (25) indicate that the zero-modes show no dielectric response for a zero band gap but the response grows linearly with the band gap. In bilayer graphene, with a tunable band gap, the zero-mode levels split into two quartets separated by the band gap at different valleys. Such an isolated quartet of ”pseudo”-zero-mode levels remains nearly degenerate and, as noted earlier, (25) the level splitting is enhanced or controlled by an in-plane electric field or by an injected current; this opens up the possibility of the current-induced QH effect for the pseudo-zero-mode sector around half filling, i.e., at filling factor .

The purpose of this paper is to further examine the properties of the pseudo-zero-mode levels, especially, coherence and collective excitations in the presence of an external field and the Coulomb interaction. The pseudo-zero-mode levels at half filling support, via orbital level mixing, charge carriers with nonzero electric dipole moment, which is responsible for field-induced level splitting and the current-induced QH effect. Along this line our discussion comes in contact with the works of Barlas et al. (26) and Abergel et al. (27) who, from the viewpoint of QH ferromagnets, studied the interaction-driven QH effect in the nearly-degenerate octet of zero-mode levels in bilayer graphene. Our paper partly extends their analysis by revealing an interesting interplay between the in-plane field and Coulomb exchange interaction, which leads to rich spectra of collective excitations, accessible by microwave experiments. We shall find a duality in the excitation spectra and, under certain circumstances, an instability in pseudospin textures.

In Sec. II we briefly review some basic features of the pseudo-zero-mode levels in bilayer graphene. In Sec. III we construct a low-energy effective theory for the half-filled pseudo-zero-mode sector. In Sec. IV we examine the spectrum and collective excitations in it. In Sec. V we study the microwave response of collective excitations. Section VI is devoted to a summary and discussion.

Ii Bilayer graphene

Bilayer graphene consists of two coupled hexagonal lattices of carbon atoms, arranged in Bernal stacking. The electron fields in it are described by four-component spinors on the four inequivalent sites and in the bottom and top layers, and their low-energy features are governed by the two inequivalent Fermi points and in the Brillouin zone. The intralayer coupling is related to the Fermi velocity  m/s (with nm) in monolayer graphene. The interlayer couplings and are one-order of magnitude weaker than ; numerically, (28)  eV, eV and  eV.

Actually, interlayer hopping via the dimer sites modifies the intralayer relativistic spectra to yield spectra with a quadratic dispersion and the characteristic cyclotron energy  meV, with a magnetic field in Tesla. The low-energy branches, thereby, are essentially described by two-component spinors on the sites (with the high energy branches being separated by a large gap ). The effective Hamiltonian is written as (16)

 H = ∫d2x[ψ†(H+−eA0)ψ+χ†(H−−eA0)χ], Hξ = H0+Has, H0 = ωc(−(a†)2+λa−a2+λa†), Has = ξU2(1−za†a−(1−zaa†)), (1)

together with coupling to electromagnetic potentials . Here, assuming placing graphene in a uniform magnetic field , we have rescaled the kinetic momenta with the magnetic length and defined and so that ; we set to supply a strong magnetic field normal to the sample plane. The field refers to the valley with while refers to the valley with . For simplicity we ignore weak Zeeman coupling and suppress the electron spin indices.

In the linear kinetic term, leading to ”trigonal warping”, represents direct interlayer hopping via , with .

The takes into account a possible layer asymmetry, caused by an interlayer voltage , which leads to a tunable (18); (19); (20) gap between the conduction and valence bands. The terms in represent a kinetic asymmetry related to the charge depleted from the dimer sites; note that it is very weak, with .

While the linear spectra are lost, the bilayer Hamiltonian still possesses a key feature of relativistic field theory, the particle-hole (or chiral) structure of the quantum vacuum: When the tiny asymmetry is ignored, the spectrum of , in general, is symmetric about zero energy , apart from possible spectra that evolve from the zero-energy modes of . Indeed, for the spectrum of consists of an infinite tower of Landau levels of paired positive and negative energies,

 ϵn=snωc√|n|(|n|−1)+(U/2ωc)2−14ξzU, (2)

labeled by integers , and (or ); specifies the sign of .

In addition, there arise nearly-degenerate Landau levels carrying the orbital index and , with spectrum

 ϵ|n|=0 = ξU/2, ϵ|n|=1=ξ(U/2)(1−z). (3)

From now on we take, without loss of generality, and denote and to specify these pseudo-zero-mode levels. There are four such pseudo-zero-mode levels (or two levels per spin) at each valley and they reside on different layers; the quartet on sites at valley is separated from the the quartet on sites by a band gap . Each quartet remains degenerate, apart from fine splitting.

The presence of the pseudo-zero-mode levels and their double-fold degeneracy (per spin and valley) both have a topological origin, and are consequences of spectral asymmetry, or the nonzero index of the Hamiltonian ,

 Index[Hξ|U→0]=∫d2x2×(eB/2π), (4)

which is tied to the chiral anomaly in 1+1 dimensions. This degeneracy is unaffected by the presence of trigonal warping alone, (25) but is affected by nontrivial diagonal components in . Indeed, the kinetic asymmetry leads to tiny level splitting and an electric field can also enhance the splitting. In other words, the pseudo-zero-mode levels have an intrinsic tendency to be degenerate, but this at the same time implies that their fine structure or the way they get mixed depends sensitively on the environment.

The main purpose of the present paper is to study such controllable features of the isolated pseudo-zero-mode quartet in the presence of external fields and Coulomb interactions. We are particularly interested in the properties of such a quartet at half filling, where mixing of the zero-modes leads to nontrivial coherence effects and collective excitations. For definiteness we focus on the sector at valley, i.e., around filling factor (or when the spin is resolved), and ignore the presence of other levels which are separated by relatively large gaps; the case is treated likewise. We ignore the effect of trigonal warping, which causes only a negligibly small level splitting (25) of , apart from a common level shift of .

To project out the sector let us make the Landau level structure explicit by the expansion , with the field operators obeying . The charge density is thereby written as

 ρ−p=γp∞∑k,n=−∞gkn(p)∫dy0ψ†keip⋅rψn, (5)

where ; stands for the center coordinate with uncertainty . For the sector the relevant coefficients are given by (25) , , , and , with .

Let us put the and modes into a two-component spinor , and define the pseudospin operators in the orbital space,

 Sμ−p = γp∫dy0Ψ†12σμeip⋅rΨ, (6)

where () with Pauli matrices and . The charge density projected to the sector is thereby written as

 ¯ρ−p=2(w0pS0−p+w3pS3−p+w1pS1−p+w2pS2−p), (7)

where , , , and . Note that acts as a vector under rotations about the axis .

The Hamiltonian of Eq. (1) is projected into the sector to yield

 △¯H=−∫d2xeA0¯ρ+∫dy0mΨ†σ3Ψ, (8)

where ; the common energy shift of the sector has been isolated from . One can equally write as

 △¯H=2∑pPμpSμ−p, (9)

with being the Fourier transform of ,

 P0 = −m−eA0−14eℓ2∇⋅E, P3 = m−14eℓ2∇⋅E, (P2,P1) = eℓ√2(Ex,Ey), (10)

where denotes the in-plane electric field.

The Coulomb interaction also simplifies via projection. The pseudo-zero-modes at each valley essentially lie on the same layer, apart from a negligibly small admixture of . One may thus retain only the intralayer interaction for ,

 ¯HC=12∑pvp:¯ρ−p¯ρp:, (11)

where is the Coulomb potential with the fine-structure constant and the substrate dielectric constant ; . The normal-ordered charges in are rewritten as

 :¯ρ−p¯ρp: = ¯ρ−p¯ρp−△, △ = 2γ2p{|wμp|2S00+2w0pw3pS30} (12)

under a symmetric integration over .

The pseudospin operators obey the SU(2) algebra, (29); (30)

 [Sap,Sbk]=c(p,k)iϵabcScp+k−is(p,k)δabS0p+k, [S0p,S0k]=−is(p,k)S0p+k, [S0p,Sak]=[Sap,S0k]=−is(p,k)Sap+k, (13)

where runs over , and

 Missing or unrecognized delimiter for \Big (14)

; for set in .

Iii Pseudospin textures

In this section we study the properties of the pseudo-zero-mode levels at half filling, using the projected Hamiltonian and the charge algebra (13), with focus on orbital mixing of the zero-modes. Let us suppose that such a half-filled state is given by a classical configuration where the pseudospin points in a fixed direction in pseudospin space, i.e., and with the total number of electrons .

Note that corresponds to the filled level with the vacant level while represents the filled level. The direction would, in general, vary in response to the external field , and, as tilts from , the and levels start to mix. For selfconsistency we assume that represents a uniform in-plane electric field and that it leads to a homogeneous state of uniform density (with filling factor for the spin-degenerate state or for the spin-resolved state). We thus consider all classical configurations with and single out the ground state or the associated by minimizing the energy . For we use the parametrization , , , with and . We denote expectation values for short.

Let us first substitute the charge and pseudospin, and , into . (Here , and equals the total area so that .) This yields the response of the classical state to an external probe, with

 nμPμ = −eA0+m(cosθ−1)−E∥⋅de, de = −eℓ√2(n2,n1)=−eℓ√2sinθ^n∥, (15)

where is an in-plane unit vector.

Note that the half-filled pseudo-zero-mode state has an in-plane electric dipole moment of strength per electron, proportional to the in-plane component of the pseudospin. Mixing of the and modes gives rise to this dipole and its in-plane direction is related to the relative phase between them; i.e., under U(1) transformations , rotates,

 (S1+iS2)→eiϕrel(S1+iS2) or ϕ→ϕ+ϕrel. (16)

This tells us that a change in relative phase caused by a change in direction of the dipole is physically observable.

In the absence of the Coulomb interaction, naturally points in the direction and, as a result, an in-plane field, coupled to the the electric dipole, works to enhance the pseudo-zero-mode level splitting, (25)

 △ϵ=2√m2+e2ℓ2E2∥/2. (17)

We shall discuss below how the Coulomb interaction modifies this.

The calculation of the Coulomb energy requires the knowledge of pseudospin structure factors , which, for the present half-filled state with pseudospin , are given by

 ⟨SμpSνq⟩=14ρ0[sμνγ2pδp+q,0+nμnνρ0δp,0δq,0], (18)

where and run from 0 to 3, with and ; with and

 s33 = sin2θ, s11 = cos2θcos2ϕ +sin2ϕ, s22 = cos2θsin2ϕ +cos2ϕ, s31 = −sinθcosθcosϕ+isinθsinϕ, s32 = −sinθcosθsinϕ−isinθcosϕ, s12 = −sin2θsinϕcosϕ+icosθcos2ϕ. (19)

See Appendix A for details. Actually, the normal-ordered correlation functions take simpler form

 ⟨ :SμpSνq:⟩=−14ρ0nμnν(γ2pδp+q,0−ρ0δp,0δq,0), (20)

with which one can cast the Coulomb energy in the form

 ⟨¯HC⟩=−12Ne∑pvpe−q2/2Cθ(q2/4), (21)

where and . From we have omitted a constant which is removed when the neutralizing positive background is taken into account. Integrating over yields a typical scale of the Coulomb exchange energy,

 V1=∑pvpe−ℓ2p2/2=αϵbℓ√π2, (22)

The effective Hamiltonian is conveniently written in space as with

 Heff = −eA0+E(θ), E(θ) = −12V1+m(cosθ−1)+Esinθ (23) +132V1(1−cosθ)2,

where . In the present notation (with ) stands for energy per electron in state with pseudospin . The Coulomb correlation energy consists of a negative uniform component [relative to the zero of energy ] and a polarization dependent component which alone favors , i.e., the filled level, and which varies continuously by an amount as sweeps in pseudospin space. The Coulomb interaction thus significantly enhances the pseudo-zero-mode level splitting.

The stable configuration of the half-filled zero-mode state is determined by minimizing with respect to , or . Obviously depends on through , which attains a maximum when , or . Accordingly it is convenient, without loss of generality, to suppose that the in-plane field and lie along the axis, and . With this choice the ”1”, ”2” and ”3” axes in pseudospin space coincide with the , and axes in real space, respectively. We adopt this choice and set in what follows.

One can now look for possible ground-state configurations by writing down a phase diagram as a function of and . For clarity of exposition we leave it for a later stage and here study collective excitations over a given ground state . We focus on a special class of low-energy collective excitations, pseudospin waves, that are rotations about the energy minimum in pseudospin space.

As is familiar from the case of quantum Hall ferromagnets, (30) such a collective state is represented as a texture state

 |~G⟩=e−iO|G⟩|n, (24)

where the operator with

 O=∑pγ−1pΩapSa−p (25)

locally tilts the pseudospin from to by small angle . Repeated use of the charge algebra (13) then allows one to express the texture-state energy in terms of the structure factors in Eq. (18), and this yields an effective Hamiltonian as a functional of or its -space representative .

The angle variables may be normalized so that classically. As we shall see below, the effective theory is expressed in terms of the following components of ,

 η ≡ Ω1cosθ−Ω3sinθ, ζ ≡ Ω2, (26)

along the two orthogonal axes (i.e., the tilted ”1” axis and the ”2” axis) perpendicular to the pseudospin , with normalization . (Here we have chosen and set , as remarked above; the case, if needed, is readily recovered. (31)) Actually they refer to the following induced pseudospin components in the excited state ,

 η ∼ −γ−1p⟨~G|S2|~G⟩, ζ ∼ γ−1p⟨~G|S1cosθ−S3sinθ|~G⟩, (27)

as seen from the induced pseudospin .

We expand to second order in and retain all powers of derivatives acting on to study the spectrum over a wide range of wavelengths. The calculation is outlined in Appendix B. The result is

 ⟨~G|¯H|~G⟩ = ρ0∫d2x(Heff+Hcoll), Hcoll = 12ηΓηη+12ζΓζζ+ζWpη+Hch+δH, Γη = −E/sinθ+Fp, Γζ = E′′(θ)+Gp, δH = E′(θ){ζ+Ω1η/(2sinθ)}; (28)

, etc. Here

 Fp = Vc2[12√π2+ξq+P−λq], Gp = Vc2[12√π2+ξq−bqsin2θ−P−λqcos2θ], Wp = Missing or unrecognized delimiter for \Big (29)

with , and

 ξq = νe−q2/2q2−∫∞0dze−z2/2(1−12z2)J0(zq), λq = νe−q2/2q2−12∫∞0dze−z2/2z2J2(zq), bq = νe−q2/2(q/2−q3/8) +18∫∞0dze−z2/2(1−4z2+z4)J0(zq), τq = νe−q2/2q/2 (30) +2q∫∞0dze−z2/2z(1−14z2)J′0(zq),

where ; substitution is understood in the representation. For , , and while they all tend to zero for . See Appendix C for more explicit forms of the integrals involved in these functions.

In , refers to a topological charge (to be detected with a constant potential )

 ρ0∫d2xHch=−e∫d2xA0ν8πϵabcϵij(∂iΩa)(∂jΩb)nc, (31)

where and . With promoted to , involves the winding number, which implies (30) that possible topologically-nontrivial semiclassical excitations (Skyrmions) associated with , in general, carry electric charge of integral multiples of . Note also that , involving a term linear in , disappears for a stable-state configuration for which .

The the kinetic term for is supplied from the electron kinetic term as Berry’s phase,

 ⟨~G|i∂t|~G⟩ = −14ρ0∑kncϵabcΩa−k∂tΩbk (32) = ρ02∫d2xζ∂tη,

to , apart from surface terms. This shows that is canonically conjugate to . One can now write the effective Lagrangian for the collective excitations as

 L = 12ζ∂tη−Hcoll[η,ζ]. (33)

Note here that this Lagrangian is written as

 ρ0∫d2xL=⟨G|eiO(i∂t−¯H)e−iO|G⟩. (34)

This representation realizes and systematizes the single-mode approximation (29) (SMA) within a variational framework. (32) The present theory thus embodies nonperturbative aspects of the SMA.

Upon elimination of , Eq. (33) leads to an alternative form of the effective Lagrangian for as follows:

 LΦ = 12(∂tΦ)2−12Φ(Mp)2Φ, Mp = 2√ΓηΓζ−|Wp|2, (35)

where we have set .

The spectrum of collective excitations is in general anisotropic in at low energies and depends critically on the stable-state configuration . In particular, the leading long-wavelength correction in starts with the direct Coulomb interaction of , which leads to the spectrum,

 Mp ≈ √(2κηκζ)2+Vcνℓ|p|{κ2ηcos2θp2y+κ2ζp2x}, κ2η ≡ Γη|p=0=−E/sinθ, κ2ζ ≡ Γζ|p=0=E′′(θ). (36)

The excitation gap at zero wave vector is thus given by . In contrast, recovers isotropy and the standard excitonic behavior (33) at short wavelengths,

 Mp→∞≈V1/2+κ2η+κ2ζ, (37)

with , and for ; .

It is worth noting here that the Coulomb interaction alone yields , i.e., a flat direction in space. This implies that, unlike in ordinary bilayer QH systems, there is no cost of interlayer capacitance energy for the pseudo-zero-mode sector which essentially resides in the same layer. Coherence is thus easier to form in this sector of bilayer graphene.

If we set , our precisely reproduces the excitation spectrum derived in Ref. [(26)] by assuming spatial isotropy; actually, the effective Hamiltonian of Ref. [(26)] is apparently different from our but the spectrum turns out to be the same. It is our use of general textures that allows to handle spatially anisotropic situations as well.

Iv possible ground states and collective excitations over them

In this section we study the spectrum of the half-filled pseudo-zero-mode state and the associated collective excitations. Let us first gain a rough idea of the strengths of and relative to the Coulomb correlation energy . A naive estimate

 △Ec=18√π2α