Graphene and Quantum Electrodynamics

Transcription

1 Graphene and Quantum Electrodynamics V.P. Gusynin Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine International Jubilee Seminar "Current problems in Solid State Physics" dedicated to the memory of Associate member of National Academy of Sciences of Ukraine E. A. Kaner and 55th anniversary of discovery of Azbel-Kaner cyclotron resonance, November 16 19, 2011, Kharkov, Ukraine November 16, 2011 V.P. Gusynin (BITP, Kiev) Graphene and Quantum Electrodynamics Kharkov 1 / 36

2 Outline Introduction: Why graphene is so interesting? Dirac theory of graphene QED form of graphene Lagrangian Landau levels and Quantum Hall effect Supercritical Coulomb center: resonant states Excitonic instability and gap generation Conclusion: Possible future applications V.P. Gusynin (BITP, Kiev) Graphene and Quantum Electrodynamics Kharkov 2 / 36

3 Why graphene is so interesting? Atomic structure. Graphene is an one-atom-thick layer of graphite packed in the honeycomb lattice a two-dimensional crystal. Free suspended graphene shows rippling of the flat sheet with amplitude of about one nanometer. The ability to sustain huge (> 10 8 A/cm 2 ) electric currents make it a promising candidate for applications in devices such as nanoscale field effect transistors. Graphene exhibits the highest electronic quality among all known materials. It has a high electron mobility at room temperature, µ 15000cm 2 /V s. In free suspended graphene µ cm 2 /V s: the quasiparticles in graphene take less than 0.1 ps to cover the typical distance between source and drain electrods in a transistor. V.P. Gusynin (BITP, Kiev) Graphene and Quantum Electrodynamics Kharkov 3 / 36

4 Why graphene is so interesting? Easy control of charged carriers. Anomalous quantum Hall effect: σ xy = ±4(n + 1/2) e2 h. Thermal properties. The thermal conductivity κ W/m K is larger that for carbon nanotubes or diamond. Optical properties. Graphene has high optical transparency: it absorbs only πα 2.3% of white light (α = 1/137 - fine structure constant). This can be important for making liquid crystal displays. Mechanical properties. Graphene is the strongest material ever tested, stiffness - 340N/m. Because of this graphene remains stable and conductive at extremely small scales of order several nanometers. V.P. Gusynin (BITP, Kiev) Graphene and Quantum Electrodynamics Kharkov 4 / 36

5 Graphene as a bench top QED The charge carriers in graphene are described by the massless Dirac equation what leads to several unusual properties: the Klein tunneling zitterbewegung (trembling motion) the Schwinger pair production Casimir-like interactions between adsorbates on graphene supercritical atomic collapse symmetry broken phase with a gap at strong coupling Only the Klein tunneling has been verified experimentally: A. Young, P. Kim, Nature Physics, 5, 222 (2009). V.P. Gusynin (BITP, Kiev) Graphene and Quantum Electrodynamics Kharkov 5 / 36

6 Qusiparticle zoo V.P. Gusynin (BITP, Kiev) Graphene and Quantum Electrodynamics Kharkov 6 / 36

7 Reviews A.C. Neto, F. Guinea and N.M. Peres. Drawing conclusions from graphene. Physics World, 19, 33 (2006). M. Katsnelson, Graphene: carbon in two dimensions. Materials today, 10, 20 (2007). A.Geim and K. Novoselov. The rise of graphene. Nature Materials 6, 183 (2007). V.P. Gusynin, S.G. Sharapov, J.P. Carbotte. AC conductivity of graphene: from tight-binding model to 2+1-dimensional quantum electrodynamics. Int.J.Mod.Phys.B21, (2007). A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, A. K. Geim. The electronic properties of graphene. Rev. Mod. Phys. 81, 109 (2009). V.P. Gusynin (BITP, Kiev) Graphene and Quantum Electrodynamics Kharkov 7 / 36

8 One atomic layer of graphene In 2004, the Manchester group (A. Geim, K. Novoselov et al.) obtained graphene by mechanical exfoliation of graphite. They used cohesive tape to repeatedly split graphite crystals into increasingly thinner pieces. This micromechanical "peeling" created flakes, some of which were unexpectedly just one layer thick. The key for the success was the use of high-throughput visual recognition of graphene on a properly chosen substrate, which provides a small but noticeable optical contrast. Photograph in normal white light of multilayer graphene flake K. Novoselov et al.,science 306, 666 (2004) V.P. Gusynin (BITP, Kiev) Graphene and Quantum Electrodynamics Kharkov 8 / 36

9 Allotropes of carbon Well-known forms of carbon all are derived from graphene. V.P. Gusynin (BITP, Kiev) Graphene and Quantum Electrodynamics Kharkov 9 / 36

10 Field Effect Experiment Field effect transistor Longitudinal conductivity as a function of gate voltage. V g > 0 means that we increase the number of electrons with increasing V g, while we control holes for V g < 0. Close to zero V g, this film is a compensated semimetal. Novoselov et al., Nature 308, 197 (2005). σ min = 4e2 πh minimal conductivity E.V. Gorbar, V.P.G., V.A. Miransky, I.A. Shovkovy, Phys. Rev. B66, (2002). V.P. Gusynin (BITP, Kiev) Graphene and Quantum Electrodynamics Kharkov 10 / 36

11 Orbitals of graphene Carbon has 6 electrons, 2 are core electrons 4 are valence electrons: one 2s and three 2p orbitals single 2s and two 2p orbitals hybridize forming three σ bonds in the x-y plane remaining 2p z orbital ( π orbital) is perpendicular to the x y plane V.P. Gusynin (BITP, Kiev) Graphene and Quantum Electrodynamics Kharkov 11 / 36

12 Lattice of graphene H = t n,δ,σ a n,σ b n+δ,σ + h.c., t 3eV, a = 2.46Å is the lattice constant, σ = ±1 is the spin. H = t k,σ ( a k,σ b k,σ ) ( 0 δ e ikδ 0 δ eikδ ) ( ak,σ b k,σ ) The π bands structure of a single graphene sheet is 3kx a E(k x, k y ) = ±t cos cos k ya k ya cos2 2. V.P. Gusynin (BITP, Kiev) Graphene and Quantum Electrodynamics Kharkov 12 / 36

13 Band structure of graphene b 2 a b 1 K K K a 2 3 A b 2 Ε t 0 B 2 ak y Two bands touch each other and cross the Fermi level in six K points located at the corners of the hexagonal 2D Brillouin zone. ak x (a) Graphene hexagonal lattice can be described in terms of two triangular sublattices, A and B. (b) Hexagonal and rhombic extended Brillouin zone (BZ). Two non-equivalent K points in the extended BZ, K = K +. P.R. Wallace, PR 71, 622 (1947). V.P. Gusynin (BITP, Kiev) Graphene and Quantum Electrodynamics Kharkov 13 / 36

14 Low-energy excitations in graphene The low-energy excitations at two inequivalent K, K = (0, ±4π/3a) points have a linear dispersion E k = ± v F k with v F = ( 3/2 )ta 10 6 m/s. v F plays role of the velocity of light c v F = c/300. Each K ( point is described ) by its own spinor: ψkaσ ψ K,σ = ψ KBσ K, K points are also called Dirac points. Pseudospin direction is linked to an axis determined by electronic momentum: for conduction and valence band electrons τ p p = ±1. V.P. Gusynin (BITP, Kiev) Graphene and Quantum Electrodynamics Kharkov 14 / 36

15 Hamiltonian for K point The Hamiltonian for K point ( d 2 k H K = v F (2π) 2ψ Kσ σ=±1 0 k x ik y k x + ik y 0 ) ψ Kσ, where the momentum k = (k x, k y ) is already given in a local coordinate system associated with a chosen K point. Introduce ψ Kσ = ψ Kσ γ0 and 2 2 γ matrices γ 0,1,2 = (τ 3, iτ 2, iτ 1 ): H K = v F σ=±1 d 2 k (2π) 2 ψ Kσ (t,k)( γ 1 k x + γ 2 k y )ψ Kσ (t,k). Spinors can be also combined in one four-component Dirac spinor Ψ T σ = (ψ KAσ, ψ KBσ, ψ K Bσ, ψ K Aσ). Ψ σ = Ψ σ γ0 and 4 4 γ-matrices γ ν = τ 3 (τ 3, iτ 2, iτ 1 ) belonging to a reducible representation of Dirac algebra. G. Semenoff, PRL 53, 2449 (1984); D. DiVincenzo and E. Mele, PRB 29, 1685 (1984). V.P. Gusynin (BITP, Kiev) Graphene and Quantum Electrodynamics Kharkov 15 / 36

16 Introducing a gap H K = σ=±1 ( d 2 k (2π) 2ψ Kσ v F (k x ik y ) v F (k x + ik y ) ) ψ Kσ, where the momentum k = (k x, k y ) is already given in a local coordinate system associated with a chosen K point. The presence of a gap 0 changes the spectrum: E(k) = ± 2 v 2 F k For B = 0 a gap can be induced by interaction with substrate. S.Y. Zhou et al., Nature Mat. 6, 770 (07). Observation of the gap opening in single-layer epitaxial graphene on a SiC substrate at the K point. (a) Structure of graphene in the real and momentum space. (b) ARPES intensity map taken along the black line in the inset of (a). V.P. Gusynin (BITP, Kiev) Graphene and Quantum Electrodynamics Kharkov 16 / 36

17 QED form of Lagrangian L = d 2 r Ψ σ (t,r) [γ 0 (i t + µ) + v F ( ) ] γ 1 i x + γ 2 i y Ψ σ (t,r) e2 d 2 r d 2 r Ψ σ (t,r)γ 0 1 Ψ σ (t,r) 2κ r r Ψ σ (t,r )γ 0 Ψ σ (t,r ). µ sgn(v g ) V g [ 3600K, 3600K] when V g [ 100V, 100V]; µ > 0 electrons. V.P. Gusynin (BITP, Kiev) Graphene and Quantum Electrodynamics Kharkov 17 / 36

18 QED form of Lagrangian L = d 2 r Ψ σ (t,r) [γ 0 (i t + µ) + v ( ) ] F γ 1 i x + γ 2 i y Ψ σ (t,r) e2 d 2 r d 2 r Ψσ (t,r)γ 0 1 Ψ σ (t,r) 2κ r r Ψ σ (t,r )γ 0 Ψ σ (t,r ). µ sgn(v g ) V g [ 3600K, 3600K] when V g [ 100V, 100V]; µ > 0 electrons. is a possible excitonic gap (Dirac mass) generated due to Coulomb interaction. Magnetic field favors gap opening: Magnetic catalysis phenomenon: I. Krive, S, Naftulin, PRD 46, 2737 (1992); V.P. G., V.A. Miransky, I.A. Shovkovy PRL 73, 3499 (1994). V.P. Gusynin (BITP, Kiev) Graphene and Quantum Electrodynamics Kharkov 17 / 36

19 Symmetry The Lagrangian L possesses flavor U(4) symmetry 16 generators read (spin valley) σ α 2 I 4, σα 2i γ 3, σα 2 γ5, and σα 2 γ3 γ 5 The Zeeman term breaks U(4) down to U(2) + U(2) Dirac mass breaks U(2) s down to U(1) s V.P. Gusynin (BITP, Kiev) Graphene and Quantum Electrodynamics Kharkov 18 / 36

20 QED form of Lagrangian In graphene the quasiparticles are localized on two-dimensional plane while a gauge field responsible for interaction propagates in a three-dimensional bulk (braneworld model). [ S = d 3 rdt 1 4 F µν 2 A 0j c A j j 0 j i + Ψ σ (r, t) ( γ 0 (i t + µ) + i v F γ i i ) Ψ σ (r, t)δ(z) ], = e Ψ σ (r, t)γ 0 Ψ σ (r, t)δ(z), = ev F Ψσ (r, t)γ i Ψ σ (r, t)δ(z). Integration over z-coordinate gives a nonlocal interaction for quasiparticles in a plane: [ S = d 2 rdt 1 4 F 1 ab F ab + Ψ σ (r, t) ( γ 0 (i t ea 0 + µ) 2 v F γ(i ea/c) ) ] Ψ σ (r, t). V.P. Gusynin (BITP, Kiev) Graphene and Quantum Electrodynamics Kharkov 19 / 36

21 Landau levels Free electrons in a magnetic field: Landau levels (LL). L.D. Landau, Z.fur Physik, 64, 629 (1930); M.P. Bronstein, Ya.I. Frenkel, ЖРФХО, 62, 485 (1930); I. Rabi, Z. fur Physik, 49, 507 (1928). In nonrelativistic case the distance between LL coincides with the cyclotron energy, ω c = e B m ec 1.35K B[Tesla]. For the relativistic-like system (graphene), E n = ± 2n vf 2 eb /c, n = 0, 1,.... the energy scale, characterizing the distance between Landau levels, 2eB is ω L = v F 424K B[Tesla] for v c F 10 6 m/s! This is why QHE effect is observed at room temperature! V.P. Gusynin (BITP, Kiev) Graphene and Quantum Electrodynamics Kharkov 20 / 36

22 Dirac Landau levels B = 0 and zero gap, = 0 a B = 0 and finite gap, 0 b B 0 and finite gap, 0 c B 0 B 0 B 0 E k Μ Μ K E k ±ħv Fk K ' Μ K E k ± 2 ħ 2 v 2 F k 2 K ' K E n ± 2 2 nħeb v 2 F c E 0 K ' E 0 E k E n (a) The low-energy linear-dispersion. µ = 0 is a neutral point. k (b) A possible modification of the spectrum by the finite gap. µ is shifted from zero by the gate voltage. k (c) Landau levels E n. Notice that for K point E 0 = and for K point E 0 =. Then the degeneracy of n = 0 level is half of the degeneracy of LL with n 0. k V.P. Gusynin (BITP, Kiev) Graphene and Quantum Electrodynamics Kharkov 21 / 36

23 Quantum Hall effect in graphene - theory Σ h e Σ xx Σ xy Σ xy classic Ν sgn Μ The Hall conductivity σ xy and the diagonal conductivity σ xx measured in e 2 /h units as a function of the filling ν. The straight line classical dependence σ xy = ec ρ /B. σ xy = e2 ν, ν = 4(n + 1 ) = ±2, ±6,..., n = 0, ±1,... h 2 The n = 0 Landau level is special: E n = ± 2vF 2 neb /c E 0 = 0 and its degeneracy is half of the degeneracy of LL with n 0. V.P. Gusynin, S.G. Sharapov, PRL 95, (2005). V.P. Gusynin (BITP, Kiev) Graphene and Quantum Electrodynamics Kharkov 22 / 36

24 Hall effect in graphene - experiment a 2 a 1 s2 s3 s1 K.S. Novoselov, A. Geim et al., Nature 438, 197 (2005) Hall conductivity σ xy = (4e 2 /h)ν and longitudinal resistivity ρ xx of graphene as a function of their Observation of IQHE is the ultimate proof of the existence of Dirac quasiparticles in graphene. It is a consequence of the honeycomb lattice concentration at B =14T. Inset: σ xy in two-layer graphene where the quantization sequence occurs at integer ν. The latter shows that the half-integer QHE is exclusive to ideal graphene. V.P. Gusynin (BITP, Kiev) Graphene and Quantum Electrodynamics Kharkov 23 / 36

25 Supercritical Coulomb center: resonant states [ v F (iσ 1 x + iσ 2 y ) + σ 3 + V(r)] Ψ(r) = EΨ(r), V(r) = Ze2 Ze2 e2 θ(r R) θ(r r), α =. κr κr κ v F The Dirac equation is solved in terms of the Whittaker functions W µ,ν (ρ). The spectrum (Balmer-like formula): [ ] 1/2 { Z 2 α 2 n = 0, 1, 2, 3,..., j > 0, E nj = 1 + (, j 2 (Zα) 2 + n) 2 n = 1, 2, 3,..., j < 0. The lowest energy level E 0,j=1/2 = 1 (2Zα) 2. For Zα > 1/2 the energy becomes imaginary, i.e., the fall into the center phenomenon occurs (similar to the Dirac equation in dimensions). V.P. Gusynin (BITP, Kiev) Graphene and Quantum Electrodynamics Kharkov 24 / 36

26 Supercritical Coulomb center: resonant states For finite size R of the Coulomb center discrete levels exist for Zα > 1/2: The energy levels in the regularized Coulomb potential. The lowest energy level E 0,1/2 dives into continuum for Zα > Z c α: Z c α π 2 log 2, R 1. (cr ) V.P. Gusynin (BITP, Kiev) Graphene and Quantum Electrodynamics Kharkov 25 / 36

27 Gapless graphene Zα < 1/2 no discrete or resonant states, E (0) n Zα > 1/2 we find quasidiscrete levels with imaginary energy: [ ] = ( i)R 1 πn exp, n = 1, 2,.... Z 2 α 2 1/4 The resonant states with E n (0) describe the spontaneous emission of positively charged holes when electron bound states dive into the lower continuum. The finite gap (mass) results in decreasing the width of resonance, E n = E (0) n + 2 ( i), E n (0) and, therefore, increases stability of the system. V.P. Gusynin (BITP, Kiev) Graphene and Quantum Electrodynamics Kharkov 26 / 36

28 Excitonic instability The Bethe-Salpeter equation for electron-hole bound state χ: χ(q, p) = iλ 2π 2 S(p) = d 3 k q k γ0 S(k + p)χ(k, p)s(k p)γ 0, γ0 p 0 γp + p0 2 p2 2 + i0, λ = α 2(1 + πα/2). V.P. Gusynin (BITP, Kiev) Graphene and Quantum Electrodynamics Kharkov 27 / 36

29 Excitonic instability Matrix structure of the excitonic wave function in case of gapless quasiparticles ( = 0): χ(q) = χ 5 (q)γ 5 + χ 05 (q)q i γ i γ 0 γ 5, γ 5 = iγ 0 γ 1 γ 2 γ 3. For scalar functions we get the system of differential equations χ q χ 5 +λχ 5 + p 0 χ 05 q 2 p 2 0 with the boundary conditions = 0, χ q χ λ 2 q 2 χ 05 + p 0 χ 5 q 2 (q 2 p 2 0 ) = 0, q 2 χ 5 (q) q=0 = 0, (qχ 5 (q)) q=λ = 0, q 4 χ 05 (q) q=0 = 0, (q 3 χ 05 (q)) q=λ = 0, Λ = t/v F, t 3eV. V.P. Gusynin (BITP, Kiev) Graphene and Quantum Electrodynamics Kharkov 28 / 36

30 Excitonic instability For the coupling constant larger than some critical one we find the solution with pure imaginary energy (tachyon), ( ) p0 2 = Λ2 exp 4πn δ, δ 7.3, λ > 1/4 (α > α c ). λ 1/4 In the supercritical regime the function the wave function χ 5 (q) behaves asymptotically as χ 5 (q) q 1/2 cos( λ 1/4ln q + const). The tachyon states play here the role of the quasistationary states in the problem of supercritical Coulomb center resulting in the vacuum instability. In fact, the tachyon instability can be viewed as the field theory analogue of the fall into the center phenomenon and the critical coupling α c is an analogue of the critical coupling Z c α in the problem of the Coulomb center. O.V. Gamayun, E.V. Gorbar,V.P.G., Phys. Rev. B80, (2009). V.P. Gusynin (BITP, Kiev) Graphene and Quantum Electrodynamics Kharkov 29 / 36

31 Excitonic instability and gap generation The gap equation in the random phase approximation has the form: (p) = λ Λ 0 dqq (q)k(p, q) q2 + ( (q)/v F ) 2, λ = α g 2(1 + πα/2), where α = e 2 /( v F ) 2.19 is the fine structure constant for graphene, v F 10 6 m/s. The kernel K(p, q) = 2 ( ) 1 2 pq π p + q K, p + q and K(x) is full elliptic integral of first kind. The nontrivial solution exists if the coupling λ > 1/4 (α > 1.62): ( ) π (p = 0) Λv F exp. λ 1/4 The strong-coupling limit of graphene is an insulator. V.P. Gusynin (BITP, Kiev) Graphene and Quantum Electrodynamics Kharkov 30 / 36

32 Excitonic instability and gap generation In the random phase approximation with static vacuum polarization we find α crit = 1.62 is the quantum critical point of metal-insulator phase transition: O. Gamayun, E. Gorbar, V. G., Phys. Rev. B81, (2010). When the dependence on the frequency is included D(ω,q) = 1 πe2 q 2, Π(ω,q) = q + Π(ω,q) 2κ 2 vf 2q2 ω 2, the quantum critical point is α crit = Monte-Carlo simulations give: α crit = J. Drut and D. Son, Phys. Rev. B77, (2008); J. Drut, T. Lahde, Phys. Rev. B79, (2009). See, also S. Hands, C. Strouthos, Phys. Rev. B78, (2008). V.P. Gusynin (BITP, Kiev) Graphene and Quantum Electrodynamics Kharkov 31 / 36

33 Including residual lattice interactions When comparing the results of lattice simulations with analytical calculations one should have in mind that a theory in a continuum described by the QED 2+1 Lagrangian considered on a lattice contains in fact more interactions terms. L = L QED3 + G 2 [ (ΨΨ) 2 + (Ψiγ 5 Ψ) 2], G a, a is the lattice constant. Gap equation is modified Λ (p) = 0 G ΨΨ + λ 0 dk k (k) K(p, k). k2 + 2 (k) V.P. Gusynin (BITP, Kiev) Graphene and Quantum Electrodynamics Kharkov 32 / 36

34 Phase diagram The critical line separating the spontaneously broken and unbroken phases (g = G Λ/π): { g = 1 8 ( λ/λ c ) 2, for λ λc = 1 4, g < 1 8, λ = λ c = 1 4. V.P. Gusynin (BITP, Kiev) Graphene and Quantum Electrodynamics Kharkov 33 / 36

35 Critical indices Along the part of the critical line, λ < λ c = 1/4, we have second order phase transition (0) τ ν, ν = 1/2ω, ω = λ c λ, where τ (τ 0) is the deviation from the critical line. Other critical indices can be determined: ΨΨ τ β, β = 1 ( ) 3 2ω 2 ω, ΨΨ 1/δ 0, δ = 3/2 + ω τ=0 3/2 ω, 0 ΨΨ τ γ, γ = 1, η = 2 2ω. 0 The found critical exponents satisfy the hyperscaling relations 2β + γ = Dν, 2βδ γ = Dν, δ 1 δ + 1 = 2 η D, at space-time dimension D = 3 and are close to those found in Monte Carlo simulations by J.Drut and T.Lähde. V.P. Gusynin (BITP, Kiev) Graphene and Quantum Electrodynamics Kharkov 34 / 36

36 Conclusion: Possible future applications. Graphene is already positioned as the next big thing for many technologies, such as computers, displays, biosensors, and flexible electronics, e.t.c. Graphene becomes a real alternative for building atomic-scale, all-carbon based electronics. New composite materials, graphene paper ( ). On April 7, 2011 IBM company announced the creation of the world s fastest graphene transistor with the frequency 155 GHz. The graphene transistor exhibited excellent temperature stability from room temperature down to minus 268 degrees Celsius. This accomplishment is a key milestone in an effort to develop next-generation communication devices. Bilayer graphene promises even more perspectives for applications in electronic devices due to the possibility of inducing and controlling the energy gap by gates voltage. Graphene supercapacitors are already fabricated. V.P. Gusynin (BITP, Kiev) Graphene and Quantum Electrodynamics Kharkov 35 / 36

37 Conclusion: Possible future applications Graphene wires have higher electron mobility, better thermal conductivity, higher mechanical strength and reduced capacitance. Researchers at Samsung company, in Korea, have created a flexible graphene sheet of the size 70 cm that can be used as a touch screen. Optical modulators, photodetectors, graphene batteries for the cell phone ( fully recharged during 10 min.). Gas sensors capable of detecting a single molecule. THANK YOU FOR ATTENTION! V.P. Gusynin (BITP, Kiev) Graphene and Quantum Electrodynamics Kharkov 36 / 36