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Review: Castro-Neto et al, Rev. Mod Phys. Abanin, Lee and Levitov, Solid state communications, 143 77 (2007), part of special volume on graphene. Outline: Review the origin of Dirac spectrum. Special properties of Dirac spectrum: transport, localization and screening. Quantum Hall effect in graphene vs conventional 2DEG. Quantized spin Hall current and spintronics. Nature of the ν=1 Landau level: order by disorder. Review of the topological insulator. 0Å 9Å 13Å Novoselov et al, Science 306, 666 (2004) Key: optical detection. Two Dimensional Crystallites crystal faces zigzag 10 μm armchair not just flakes but graphene crystallites 1 µm Single layer graphene B A t’ Brillouin zone 1.42 A t • 1 electron per π orbital: half-filled • Kinetic energy: n.n. t ~ 3 eV n.n.n. t’ ~ 0.1 eV E=v0| p | massless Dirac spectrum See notes A Momentum Space E=v0| p | massless Dirac spectrum v0= 8x107 cm/sec =c/400 Klein paradox: Transmission thru arbitrarily high barrier. Transmission probability T=1 for forward direction ,(ie reflection R=0 ), due to conservation of pseudo-spin. Katsnelson,Novoselov and Geim,Nature Phys. 2, 620 (2006) Absence of localization for any disorder strength for a single Dirac node! 1. No bound state for arbitrarily deep potential: always scattering state because there is no band bottom. 2. Naively expectation is “symplectic class” which has anti-localization. Recent work by Nomura, Koshino and Ryu (PRL 99, 146806 (2007)) showed that unlike the symplectic case, there is no transition to insulator. 3. If inter-node scattering is small (smooth potential), we may expect positive magneto-resistance. Experimentally, almost no magnetoresistance is observed. Explanation was proposed in terms of buckling of graphene layers. (Morozov et al, PRL 97, 016801 (2006).) More later. Electric Field Effect in Graphene conductivity σ=n(Vg)eμ Hall effect electrons B =2T T =10K 10 2 0 holes 1 T =10K 0 -100 -50 0 50 100 Vg (V) SiO2 Si Au contac ts GRAPHENE 1/ρxy=ne/B -100 1/ρxy (1/kΩ) σ (1/kΩ) 3 -10 -50 0 50 100 Vg (V) simple behaviour; practically constant mobility; no trapped carriers Electric conductivity single layer Universal DC conductivity h Simple argument: σ ~ N(0)D. N(0) ~ 1/τ. D~v^2*τ/2 ρmax (h/4e2) 2 15 devices 1 0 μ (cm2/Vs) 0 4,000 8,000 Mystery of the missing π Puddles of n and p type domains? (Altshuler et al) Interaction effect: Incomplete screening at the Dirac point. Dimensionless coupling constant is (unlike e^2/c for QED) Constant mobility due to partial screening of charged impurity (McDonald) , or buckling of layers. Pertubation theory P h y s . R e v . B . { \ b f 7 2 } , 1 7 4 4 0 6 ( 2 0 0 5 ) Not like a Fermi Liquid! Gonzalez et al. PRL 77, 3589 (96) Strong coupling treatment by 1/N expansion, D. Son PRB75, 235423 (2007). Predicts a power law dispersion E(k) ~ k^(1-α), α~1/N. In the presence of a magnetic field B: Landau levels ωc = eB /(mc) = h /(mlB2 ) l B = hc /(eB) B = 10T hωc ≈ 1K B=10T, E1-E0=1500K Zero mode QHE at room temperature! Existence of a zero eigenstate is a general feature. For example, for arbitrary B(r) , Aharonov and Casher (PRA 19 2461 (1979) showed that E=0 is a normalizable eigenstate. (this is a special case of what is called index theorem. graphene 3 5/ 2 3/ 2 2 1 1/ 2 0 -1/2 -1 -3/2 -5/2 σxy (4e2/h) σxy (4e2/h) 7/ 2 -2 -3 -7/2 -4 ρxx (kΩ) 6 T =4K 4 B =12T 2 0 Y.Zhang et al., Nature 438, 201 (05) -4 -2 0 2 4 n (1012 cm-2) Novoselov et al., Nature 438, 197 (05) 29T 4 2 20 10 0 room temperature -2 σxy (e2/h) ρxx (kΩ) 30 -4 0 -60 -6 -30 0 30 60 Vg (V) Novoselov et al, Science 315, 1379 (2007) IR absorption: Jiang,…P. Kim and H. Stormer, PRL 98,197403 (2007) Edge state picture of QHE. Rxy=1/(N(e2/h)) N=number of edge states that crosses the Fermi level. Edge states propagate in one direction and are not back-scattered. See notes B graphene 3 5/ 2 3/ 2 2 1 1/ 2 0 -1/2 -1 -3/2 -5/2 -2 -3 -7/2 -4 ρxx (kΩ) 6 Spin splitting T =4K 4 B =12T 2 0 -4 -2 0 2 n (1012 cm-2) 4 σxy (4e2/h) σxy (4e2/h) 7/ 2 LLandauer-Buttiker Picture of transport: transmitting and reflecting channels. Application to QHE: Ballistic chiral channels. Contacts: Current leads: Charge reservoir with local chemical potential V. Ιnject current I with energy less than V. I=(e^2/h)V per channel. Absorb all incoming electrons. Voltage leads. Iin=Iout equilibrium voltage V, same for both spins. Spin filtered edge states. Abanin, Lee and Levitov, PRL96,176803(2006) Spin up moves left Spin down moves right Simple example of a topological Hall insulator. (Kane and Mele,PRL2005) where gap is opened by spin-orbit effect. No Hall voltage Rxy=0 , but ideal spin current: Is=2e2V/h. Polarized spin current generator: Simply remove one voltage probe. 4 probe longitudinal resistivity measurement: Backscattering require spin flip. Spin flip scattering will reduce Gxx from 2. Spin orbit coupling is estimated to be very weak. 1.Rashba term 0.5mK 2 Intrinsic term λσz require parallel magnetic field. Mostly likely there are some magnetic impurities. Indeed spin flip rate is so slow that nonlocal spin valve effect has been observed even at room temperature. (Tombros …van Wees, Nature 448, 571 (2007)) Jiang et al PRL99 106802 (2007) Abanin et al , PRL98,196806(2007) We also predict non-local transport. Voltage leads far away can affect current. Generation and detection of spin current: Nonlocal transport. Our entire picture assumes spin splitting is greater than orbital splitting (between K and K’). Otherwise, we have a regular insulator. We estimate the energy difference of K ,K’ for n=0 level to be very small. Nonlocal transport is a key expt test. This raises the question which orbital state is preferred. Tilted field experiment supports our level splitting scheme: Jiang et al PRL99,106802(2007) Distortion of the bonds due to strain or buckling is equivalent to a random gauge field a, which couples to K,K’ nodes with opposite signs. (Iordanskii and Koshelev, 1985) height ≈5Å; size ≈5nm; Consequences of random gauge fields: 1. Equivalent field is quite large (0.1T to 1T). 2. Explains why weak localization is not seen (or suppressed in some samples).(Morozov et al PRL 06) 3. Order out of disorder: select a linear combination of K, K’ to be the occupied ν=-1 state. (Abanin, Lee and Levitov, cond-mat,06) Pseudo-spin picture: K is spin up and K’ spin down. (like the bilayer quantum Hall problem). Prediect XY transition and meeron (1/2 integer excitation) Perhaps not the whole story! Checkelsky, Li and Ong, cond-mat. Topological insulator. Question: can we have quantum Hall like behavior without an external magnetic field? ,ie can we have energy gap in the bulk (insulator) and gapless edge states without breaking T reversal symmetry? Such edge state will lead to quantum spin Hall effect. (which, unlike ordinary QHE, does not break T, as emphasized by SC Zhang.) 1. Haldane (1988) showed that spatially varying B with net flux zero can cause QHE. 2. Kane and Mele (2005) showed that graphene with spin-orbit coupling gives rise to spin-filtered edge states without breaking of time reversal symmetry. Z2 topological classification proposed. 3. SC Zhang and collaborators (2006) proposed experimental realization in Hg-Te quantum well. 4. Expt performed and predictions verified by Konig et al (2007). How to give a energy gap (mass in particle physics language) to graphene? 1. NNN hopping t’ give uniform shift of enegy band. (breaks particle-hole symmetry), but does not open gap. 2. One way is to have different potential energy at A and B sites. Breaks parity but preserves time reversal T. Sign of mass the same for K and K’. 3. Second way is to have complex t’ as suggested by Haldane. Sign of mass is opposite for K and K’ and T is broken. Haldane showed that in case 3, QHE=1 or -1, depending on the sign of m when fermi energy is in the gap. Kane and Mele (PRL 95, 226801(2005)) added spin degrees of freedom to restore time reversal symmetry in the Haldane model. This is produced by the spin-orbit coupling term in graphene. Edge states are time reversed pairs, exactly like the zeroth landau level case we studied, except that there is no magnetic field and the gapless edge modes are protected by time reversal symmetry. Unfortunately spin-orbit gap is very small (10mK) Bernevig, Hughes and Zhang, Science 314,1757(2006) In a quantum well, gap is opened ,4 band model.. Konig et al, Science 318, 766(2007) 3 Dimensional example is also possible: Edge states will be 2Dim Dirac spectrum. These are predicted to show anomalous Hall effect. (Foo and Kane, PRB76,045302 (2007)) Possible examples are grey tin and HgTe under uni-axial stress to open a gap. Also Bi Sb alloys. Other 2D Atomic Crystals 2D boron nitride in AFM 0Å 9Å 16Å 23Å 2D NbSe2 in AFM 0Å 8Å 23Å 1μm 0.5μm 1 μm 2D Bi2Sr2CaCu2Ox in SEM 1μm 1 μm 2D MoS2 in optics Geim et al PNAS 102, 10451 (2005) Special features of graphene: Strong bonding, robust layer structure: Large layer with high mobility. Dirac spectrum: large gating effect and new physics. Weak spin-orbit coupling and spin flip scattering. Here we discussed interesting spin current effects. Other applications: Graphene based electronics and spintronics? superconducting FET. (Delft group, Heersche et al, Nature 446,56(2007)). Resonator, Molecule sensor ……. Bunch et al Science (07) Schedin et al. Nature Mat. (07)