Graphene and Quantum Electrodynamics

V.P. Gusynin∗ Bogolyubov Institute for Theoretical Physics NAS of Ukraine, 03680, Kiev, Ukraine (Received January 1, 2013)

A single atomic layer of carbon, graphene, has the low-energy “relativistic- like” gapless quasiparticle excitations which in the continuum approximation are described by quantum electrodynamics in 2+1 dimensions. The Dirac- like character of charge carriers in graphene leads to several unique electronic properties which are important for applications in electronic devices. We study the gap opening in graphene following the ideas put forward by P. I. Fomin for investigation of chiral symmetry breaking and particle mass generation in quantum ﬁeld theory. PACS: 81.05.uW, 03.65.Pm

1. INTRODUCTION general, some residual short-range interactions rep- resented by local four-fermion terms. Graphene, a one-atom-thick sheet of crystalline carbon with atoms packed in the honeycomb lattice, is The vanishing density of states at the Dirac points a remarkable system with many unusual properties ensures that the Coulomb interaction between the that was fabricated for the first time 8 years ago [1]. electrons in graphene retains its long-range character Graphene is the building block for many forms of car- due to vanishing of the static polarization function bon allotropes, for example, graphite is obtained by when the wave vector ~q → 0. The large value of the the stacking of graphene layers that is stabilized by 2 “fine-structure” coupling constant αg = e /~vF ∼ 1 weak van der Waals forces, carbon nanotubes are 6 (vF ≈ 10 m/s is the Fermi velocity) means that a formed by graphene wrapping, and fullerenes can strong attraction takes place between electrons and also be obtained from graphene by replacing several holes in graphene at the Dirac points. For graphene hexagons into pentagonsand heptagons. on a substrate with the effective coupling αg/κ ¿ 1, 8 2 The ability to sustain huge (> 10 A/cm ) elec- κ being a dielectric constant, the system is in a weak- tric currents and a very high electron mobility, µ ∼ coupling regime and exhibits semimetallic properties 5 2 2·10 cm /Vs for suspended graphene, make graphene due to the absence of a gap in the electronic spec- a promising candidate for applications in electronic trum. All the currently proposed applications of devices such as nanoscale field effect transistors. Its graphene are based on that fact. Much less is known 3 thermal conductivity ∼ 5 · 10 W/mK is larger than about suspended graphene where the coupling con- for carbon nanotubes or diamond. Graphene is op- stant is large.In fact, suspended graphene provides a tically transparent: it absorbs πα ≈ 2.3% of white condensed-matter analog of strongly coupled quan- light (α = 1/137 – fine structure constant) that can tum electrodynamics (QED) intensively studied by be important for making liquid crystal displays. It P.I. Fomin and collaborators in the 1970s and 1980s. is the strongest material ever tested with stiffness [8, 9]. The dynamics of the vacuum in QED leads 340 N/m. Because of this graphene remains stable to many peculiar effects not yet observed in nature. and conductive at extremely small scales of order sev- Some QED-like effects such as zitterbewegung (trem- eral nanometers. bling motion), Klein tunneling, Schwinger pair pro- Theoretically, it was shown long time ago [2] that duction, supercritical atomic collapse and symmetry quasiparticle excitations in graphene have a linear broken phase with a gap at strong coupling have a dispersion at low energies and are described by the chance to be tested in graphene. In fact, graphene massless Dirac equation in 2+1 dimensions. The ob- could be used as a bench-top particle-physics labora- servation of anomalous integer quantum Hall effect tory,allowing us to investigate the fundamental inter- in graphene [3, 4] is in perfect agreement with the actions of matter(recently, the Klein tunneling and theoretical predictions [5]–[7] and became a direct supercritical atomic collapse in graphene were ob- experimental proof of the existence of gapless Dirac served experimentally, see papers [10] and [11], re- quasiparticles in graphene. In the continuum limit, spectively). graphene model on a honeycomb lattice, with both on-site and nearest-neighbor repulsions,maps onto a 2+1-dimensional field theory of Dirac fermions in- teracting through the Coulomb potential plus, in ∗Corresponding author E-mail address: [email protected]

ISSN 1562-6016. PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2013, N3(85). 29 Series: Nuclear Physics Investigations (60), p.29-34. 2. DYNAMICAL MASS GENERATION IN where ωD is the Debye frequency, g the electron- QUANTUM FIELD THEORY phonon coupling constant, and νF is the density of states on the Fermi surface. In the BCS theory the Proposals that electron-electron interactions in gap in the quasiparticle spectrum is generated for any graphene could generate an electronic gap were in- value of the coupling constant. The physical reason vestigated and actually preceded the discovery of for zero value of the critical coupling constant in the graphene [12, 13]. The gap opening in graphene is BCS theory is connected with the presence of the an analogue of the phenomenon of dynamical mass Fermi surface. generation which is studied in quantum field theory More refined study of the Schwinger-Dyson equa- since the Nambu-Jona-Lasinio (NJL) paper in 1961 tion for the fermion propagator in QED (Fig. 1), leads [14] The Lagrangian of the NJL model// in the ladder approximation to the solution [17], G £ ¤ µ ¶ L = Ψ¯ iγµ∂ Ψ + (ΨΨ)¯ 2 + (Ψ¯ iγ Ψ)2 (1) const µ 2 5 m ' Λ exp −√ , (8) α − αc is invariant under ordinary phase and chiral transfor- where the critical coupling constant is of order one, mations, αc ∼ 1 Note the essentially nonanalytical dependence Ψ → eiαΨ, Ψ¯ → Ψ¯ e−iα, of the mass on the coupling constant α. Such a behav- (2) Ψ → eiαγ5 Ψ, Ψ¯ → Ψ¯ e−iαγ5 , ior of a mass remains valid in more refined approximation when fermions loops are neglected but all and the chiral invariance forbids the mass generation diagrams with crossed photon lines are taken into ac- in perturbation theory. The self-consistent Hartree- count (the so-called quenched approximation). Tak- Fock equation for a mass (Λ is a cutoff), ing into account the vacuum polarization [18] changes µ ¶ the result to 4π2m m3 m2 = m − ln 1 + (3) 1 GΛ2 Λ Λ2 m ' Λ(α − α )β, α & α , β . . (9) c c 2 has a nontrivial solution m 6= 0 if the coupling con- 4π2 stant exceeds some critical value, G > Gc = Λ2 , leading to a nontrivial chiral condensate < ΨΨ¯ >6= 0. In QED the problem of a mass generation starts since the work by V. Weisskopf [15] who calculated for the first time the one-loop correction to the bare Fig.1. The Schwinger-Dyson equation for the electron mass m0, fermion propagator µ ¶ 3α Λ m = m0 + m0 ln (4) 2π m The solution (8) for the dynamical electron mass combines features of a superconducting gap(non- The total electron mass m vanishes for m0 = 0. On analytical dependence on the coupling constant) and the other hand, solving the self-consistent equation of the NJL gap (the presence of a critical coupling). for the mass, The presence of a magnetic field makes the situa- µ ¶ 3α Λ tion even more interesting. It was shown in Ref. [19] m = m0 + m ln (5) that a magnetic field catalyzes the gap generation 2π m for gapless fermions in relativistic-like systems, and one gets a nontrivial solution, even the weakest attraction leads to the formation µ ¶ of a symmetry-breaking condensate and a gap gen- 3π eration. For example, in the four-dimensional NJL m = Λ exp − , (6) 2α model in an external magnetic field the gap (mass) is generated at any coupling constant [20]: even for zero bare electron mass m = 0. This so- p 1 lution, called the superconducting-type solution be- ∆ ' |eB| exp(− ) , (10) cause of its nonanalytical dependence on the coupling 2ν0G α, was thoroughly studied in the works by P. I. Fomin where B is the strength of a magnetic field and (see review [16] and the references therein) using the ν = |eB|/4π2 is the density of states at the low- Schwinger-Dyson equations for the electron and pho- 0 est Landau level (compare with a superconducting ton propagators and for the vertex function. The gap).This phenomenon is called magnetic catalysis solution (6) should be compared to the solution for and its main features are model independent. The a quasiparticle gap in the Bardeen-Cooper-Schrieffer essence of the magnetic catalysis is that the dynam- (BCS) theory of superconductivity, ics of the electrons in a magnetic field, B,corresponds 2 effectively to a theory with spatial dimension reduced ∆ = ωD exp(− ), (7) by two units (note a close similarity with the role of gνF

30 the Fermi surface in the BCS theory)p if their energy Taking into account the Coulomb interaction be- is much less than the Landau gap |eB|. The zero- tween quasiparticles we come at the effective low- energy Landau level, which plays, in fact, the role energy theory described by the QED-like action (the of the Fermi surface, has a finite density of states spin index is omitted) and this is the key ingredient of magnetic catalysis. Z ½ The magnetic catalysis plays an important role in 1 S = dtd3r (∂ A )2 − A j + quantum Hall effect studies in graphene where it is 2 i 0 0 0 responsible for lifting the degeneracy of the Landau ¾ £ 0 i ¤ levels. Ψ(¯ t, r)(~γ i∂t + ~vF iγ ∂i − ∆0)Ψ(t, r) δ(z) , 3. LOW-ENERGY EFFECTIVE THEORY (13) OF GRAPHENE ¯ 0 The electronic band structure of graphene close to where j0 = eΨ(t, r)γ Ψ(t, r)δ(z) is the charge den- the Fermi level forms the basis of the low-energy ef- sity. Note that the electric field, described by the fective theory of graphene. This band structure is a scalar potential A0 and responsible for interparticle reflection of the hexagonal arrangement of the car- interaction, propagates in a three-dimensional space, bon atoms. As was shown by P. Wallace [2], the while fermion fields, describing electron- and hole- two-dimensional graphene honeycomb lattice has two type quasiparticles, are localized on two-dimensional atoms per unit cell and its tight-binding band struc- planes. This is a typical example of the so-called ture consists of conduction and valence bands. The brane world models studied last years in high en- energy spectrum of electrons as a function of a wave ergy physics.Integration over z-coordinate leads to vector has the form the standard nonlocal Coulomb interaction for quasiparticles in a plane. In Eq. (13) we included also s √ a bare gap ∆0 which can be generated due to the 3kxa kya kya E(~k) = ±t 1 + 4 cos cos + 4 cos2 , interaction with a substrate [21]. In the presence of 2 2 2 (11) this term the dispersionq law for quasiparticles takes ~ ~ 2 2 where t ≈ 2.8 eV is nearest-neighbor hopping energy the form E(k) = ± (~vF k) + ∆0 and two zones are (hopping between different sublattices), in the tight- separated by 2∆0. For ∆0 = 0 the continuum effec- binding lattice Hamiltonian, a = 2.46 A˚ is the lattice tive theory described by the action (13) possesses the constant. The two bands touch each other and cross large “flavor” U(4) symmetry in the spin-valley space. the Fermi level, corresponding to zero chemical po- In the next section we will study the dynamical sym- tential µ = 0, in six K points located at the corners metry breaking of this symmetry and generation of of the hexagonal 2D Brillouin zone, but only two of a quasiparticle gap due to the Coulomb interaction 0 4π them, say K, K = (± 3α , 0), are nonequivalent. Low- when the bare gap is zero. energy excitations near two nonequivalent K-points ~ ~ have a linear dispersion, E(k) = ±~v√F |k| where 4. GAP GENERATION AND ~ = h/2π is the Planck constant, vF = 3ta/2~ ≈ SEMIMETAL-INSULATOR PHASE 106 m/s is the Fermi velocity which is only 300 times TRANSITION IN GRAPHENE less the velocity of light. Electron states at each K- point are described by the spinor, In pristinegraphenean analog of the fine-structure µ ¶ constant αg & 1. Given the strong attraction, ΨKAσ one may expect an instability in the excitonic chan- ΨKσ = , ΨKBσ nel (electron-hole pairing) with subsequent quantum phase transition to a phase with gapped quasipar- which consists of electron wave functions on atoms A, ticles that may turn graphene into an insulator. B of two sublattices, σ = ± describes the real spin, In graphene sheets deposited on a substrate,such a and satisfies the massless Dirac equation, transition is effectively inhibited due to the screen- ¡ 0 1 2 ¢ ~γ i∂t + ~vF (iγ ∂x + iγ ∂y) ΨKσ(x, y) = 0. (12) ing of the Coulomb interaction by the dielectric (αg → αg/κ). This semimetal-insulator transition Two-dimensional Dirac gamma matrices are given in in graphene is widely discussed now in the literature µ terms of the Pauli matrices γ = (τ3, iτ2, −iτ3), µ = since the first study of the problem in Refs. [12, 13]. 0, 1, 2. Spinors at two K-points can be combined into As is well known, in the BCS theory the Bethe- one four-dimensional Dirac spinor Salpeter (BS) equation for an electron-electron bound T state in the normal state of metal has a solution with Ψ = (Ψ , Ψ , Ψ 0 , Ψ 0 ) , σ KAσ KBσ K Aσ K Bσ imaginary energy, i.e., a tachyon. This means that which satisfies the Dirac equation (12) but with 4x4 normal state is unstable and a phase transition to µ gamma matrices γ =τ ˜3 ⊗ (τ3, iτ2, −iτ3) (the Pauli the superconducting state takes place. matrixτ ˜3 acts in the space of two K-points). The In order to analyze excitonic instability in four-component spinor structure accounts for quasi- graphene, we consider the BS equation for an particle excitations of sublattices A and B around the electron-hole bound state. In the random phase ap- two Dirac points (valleys) in the band structure. proximation, the BS equation for an amputed bound-

31 state wave function reads and K(x) is the complete elliptic integral of the first Z kind. The nontrivial solution for the momentum iα χ (q, P ) = g d3kD(k , |~q − ~k|)× dependent gap function ∆(p) exists if the coupling αβ (2π)2 0 · ¸ λ > 1/4 (αgc > 1.62 and its value at p = 0 (gap P P itself) is given by × γ0S(q + χ(k, P )S(k − )γ0 , (14) 2 2 αβ Ã ! π ∆(p = 0) = ΛvF exp −p . (20) where q = (q0, ~q), P = (P0, P~ ) are relative and total λ − 1/4 energies-momenta of bound electron and hole. The Coulomb propagator has the form The essentially nonanalytical dependence of the gap (20) on the coupling constant corresponds to a contin- 1 D(ω, ~q) = , (15) uous phase transition of infinite order. Such a behav- |~q| + Π(ω, ~q) ior is inherent for the Berezinskii-Kosterlitz-Thouless and the one-loop polarization function is phase transition, or the conformal phase transition, and is related to the scale invariance of the problem πe2 ~q2 under consideration. Note, however, that taking into Π(ω, ~q) = p , (16) 2 2 account the finite size of graphene samples should 2κ (~vF ~q) − ω turn this phase transition into a second-order one. which in the instantaneous approximation reduces In the case of frequency dependent polarization to Π(ω, ~q) = παg|~q|/2κ. We consider the following function (dynamical screening) the critical constant is ansatz for the matrix structure of the wave function lower, αgc = 0.92. A dynamical gap is generated only of an excitonic bound state, if αg > αgc. Since for suspended clean graphene the fine-structure constant α = 2.19 is supercritical, the 5 0 5 g χ(q, P ) = χ5(q, P )γ + χ05(q, P )~q~γγ γ . (17) dynamical gap will be generated making graphene an In the random phase approximation with static polar- insulator. Note that for graphene on SiO2 substrate, ization function, i.e., Π(ω = 0, ~q), we find an analyti- the dielectric constant κ=2.8 and αg = 0.78, i.e., the cal solution with pure imaginary energy (tachyon) if system is in the subcritical regime. The value of αgc is rather large that implies that a weak-coupling ap- the coupling constant αg exceeds some critical value α (for details, see Ref. [22]): proach might be quantitatively inadequate for the gc problem of the gap generation in suspended clean µ ¶ 4πn graphene. Therefore, it is instructive to compare our P 2 = −4Λ2 exp −√ + δ , δ ≈ 7.3, (18) 0 4λ − 1 analytical results to lattice MonteCarlo studies, [24] where αgc = 1.08 ± 0.05 that is rather close to our 1 αg λ > λc = , λ = , n = 1, 2, ... analytical findings. 4 2 + πα g Additional short-range four-fermion interactions

In the instantaneous approximation αgc = 2.33, while were also included in the continuum model to account more accurate numerical treatment gives αgc = 1.62. for the lattice simulation results [23]. In spite of being In the supercritical regime the wave function χ5(~q) as small, the induced local interactions can play a signif- a function of a relative momentum behaves asymp- icant role in the critical behavior observed in lattice totically as simulations.Instead of a critical point, we obtained Ãr µ ¶ ! the critical line in the plane of electromagnetic, α, −1/2 1 |~q| and four-fermion,g ˜, coupling constants (Fig. 2), and χ5(|~q|) ∼ |~q| cos λ − ln + const . 4 |P0| found a second-order phase transition separating zero gap and gapped phases with critical exponents close Such oscillatory behavior is typical for the phenom- to those found in lattice calculations. enon known in quantum mechanics as the “fall into We expect that the form of the critical curve in the center” (collapse): in this case the energy of a sys- graphene can be checked in further lattice simula- tem is unbounded from below and there is no ground tions. state. Nodes of the wave function of the bound state signify the existence of the tachyon states with imaginary energy P0. To ﬁnd a stable ground state we solved a gap equation which in the random phase approximation with static polarization has the form [23], Z Λ q∆(q)K(p, q) ∆(P ) = λ p .dq (19) 2 2 0 q + (∆(q)/vF ) Here the symmetric kernel of the equation is µ √ ¶ 2 1 2 pq K = K , Fig.2. Phase diagram π p + q p + q

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ГРАФЕН И КВАНТОВАЯ ЭЛЕКТРОДИНАМИКА В.П. Гусынин

Одноатомный слой углерода, графен, обладает низкоэнергетическими безщелевыми квазичастичными возбуждениями, которые в континуальном приближении описываются квантовой электродинамикой в размерности 2 + 1. Дираковский характер заряженных носителей в графене приводит ко многим уни- кальным электронным свойствам, имеющим важное значение для применений в электронных устрой- ствах. Мы исследуем открытие щели в графене, следуя идеям, предложенным П.И. Фоминым для исследования нарушения киральной симметрии и генерации масс частиц в квантовой теории поля.

ГРАФЕН I КВАНТОВА ЕЛЕКТРОДИНАМIКА В.П. Гусинiн

Одноатомний шар вуглецю, графен, має низькоенергетичнi безщiлиннi квазiчастинковi збудження, якi в континуальному наближеннi описуються квантовою електродинамiкою в розмiрностi 2+1. Дiра- кiвський характер заряджених носiїв у графенi призводить до багатьох унiкальних електронних вла- стивостей, що мають важливе значення для застосувань в електронних пристроях. Ми дослiджуємо вiдкриття щiлини в графенi, слiдуючи iдеям, запропонованим П.I. Фомiним для дослiдження пору- шення киральної симетрiї i генерацiї мас частинок у квантовiй теорiї поля.