Results on graphene in high fields and on Sr 2 RuO 4 in weak fields

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1 Results on graphene in high fields and on Sr 2 RuO 4 in weak fields Joseph G. Checkelsky, Lu Li and N.P.O. Princeton University 1. Introduction 2. Graphene from Kish graphite 3. The zero-energy state 4. Edge modes or QHF Magnetization in p-wave superconductor L. Li, P. Casey, Y. Maeno and NPO 1. M vs H in Sr 2 RuO 4 H c St. Andrews July 2007

2 Quantum Hall Effect in graphene σ xy (4e 2 /h) 7 / 2 5 / 2 3 / 2 1 / / 2-3 / 2-5 / 2-7 / σ xy (4e 2 /h) ρ xx (kω) 6 4 T =4K B =12T n (10 12 cm -2 ) Y.Zhang, Kim et al., Nature 438, 201 (05) Novoselov, Geim et al., Nature 438, 197 (05)

3 Single atomic-layer graphene Graphene sheets peeled off onto Si/SiO 2 wafers Single atomic-layer samples identified Au contacts attached by e-beam lithography Checkelsky, Li, Ong Au contact pads Single atomic-layers

4 Single atomic-layer graphene Etch Contacts 5 μm 20 μm 1 μm

5 Single atomic-layer graphene Gate I+ V- V+ I-

6 Checkelsky, Li, Ong The Quantized Hall Effect in graphene Panel (a) Resistivity R xx of graphene vs gate voltage V g at fields H = 8, 11 and 14 T. R xx peaks at Landau Levels n = 0 and +1 and -1. The peak at n = 0 is singularly large. Temperature fixed at 0.3 K Panel (b) The Hall conductivity σ xy shows step-quantization at universal values 0, 2e 2 /h, 6e 2 /h,

7 Landau Levels in graphene Zero mode B=10T, E 1 -E 0 =1500K QHE at room temperature!

8 Momentum Space E=v 0 p massless Dirac spectrum v 0 = 10 6 m/s =c/300

9 Opening of gap with field

10 Physics at the Dirac Point (n = 0 Landau Level) E(k) k (a) R xx in n = 0 Landau Level increases steeply as T 0. (b) Conductivity shows sublevel split. Hall conductivity displays plateau. (c) Quantum oscillations in conductance at 0.3 K

11 Conductance G 0 at Dirac point μ = 0 Checkelsky, Li, Ong 1. At large H, G 0 falls as T 2 K revealing gap 2. G 0 saturates to G res below 2 K Gapless excitations 3. G res strongly suppressed by H Faster than Gaussian exp(-h 2 ) 4. Phase diagram reveals unusual approach to insulating state a) Fixed H, gapless conductance b) Fixed T, insulating limit at large H

12 Abanin, Lee and Levitov, PRL (2006) Spin-filtered chiral edge states Edge No Hall current R xy =0, but ideal spin current: I s =2e 2 V/h. Also predicts longitudinal charge current, i.e. R xx =h/2e 2. (13 kohms)

13 Incipient ν = 1 Hall steps

14 Contour map of R 0 at Dirac point in H-T plane R 0 is exponentially sensitive to H but T-independent below 2 K

15 Quantum Hall ferromagnet? Layer index Valley index K, K Nomura, MacDonald, PRL06 Goerbig, Moessner, Ducot, PRB06 Alicia, Fisher PRB Coulomb exchange Splits 4-fold degeneracy Of n = 0 Landau Level 2. In high fields (and low disorder), have QHF state.

16 Role of Coulomb Interaction -- quantum Hall ferromagnet Pseudospin in Bilayer QHE systems 1 = Moon, Yang, Girvin, MacDonald 1995 Capacitance U(1) symm. 1 i e φ = iφ ( 1 + e ) 1 2 φ Paramagnet KT trans. 2DXY Ferromagnet Coulomb exchange leads to spontaneous alignment of pseudospins (Hund s rule)

17 Divergent resistance in high field Approaching KT transition? Correlation length ξ = a exp b h 1 Data suggest H c ~ Tesla

18 Magnetization of Sr 2 RuO 4 in field H c Lu Li, P. Casey, Y. Maeno and N. P. Ong

19 Cooper pairs in p-wave superconductor (and 3 He A-phase) d l S Equal spin pairing Dipole-locked (l d) H d l S Magnetic field H drives dipole unlocking transition Half-vortex may be energet. favored

20 A μi = μ Circulation around a half-vortex ( mˆ i nˆ ) i d + l m l m l n l m m Phase rotates by π (non-abelian) d d d d vector rotates by π

21 Torque signal in Sr 2 RuO 4 H c

22 Sharp transition In magnetic fied H ~ 120 Gauss Dipole unlocking Transition?

24 Break in slope at H = 0

25 M H = 1+ h 2 + K ( h = H H J d c 0 )

26 Critical state model Reversible behavior J l = r ( ρ l) s

27 + Ψ = Ψ Ψ ˆ H B M δ δ μ δ = Δ = id d d d id d ˆ ( ) l r l s ρ = J ) ( d J H H h h H M c = + + = K

28 Conclusions 1. The n=0 LL splits into 2 sublevels (σ xy = 0, 2e 2 /h) Incipient step visible at ρ xy ~ h/e 2 2. Observation of gapless excitations inside gap at μ = 0 3. Residual conductance G res decreases with incr. H Faster than Gaussian ~exp(-h 2 ) 4. Suggests important role of Coulomb exchange in strong fields (approach to 2D transition?)

29 End

30 graphene σ xy (4e 2 /h) 7 / 2 5 / 2 3 / 2 1 / / 2-3 / 2-5 / 2-7 / σ xy (4e 2 /h) Spin splitting ρ xx (kω) T =4K B =12T n (10 12 cm -2 )

31 Thickness of graphene layers measured by AFM SEM AFM AFM

32 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 current R xy =0, but ideal spin current: I s =2e 2 V/h. Also predicts longitudinal charge current, ie R xx =h/2e 2. (13 kohms)

33 30 29T 4 ρ xx (kω) room temperature σ xy (e 2 /h) V g (V) -6 Geim et al

Experimental Observation of the Quantum Anomalous Hall Effect in a Magnetic Topological Insulator

Experimental Observation of the Quantum Anomalous Hall Effect in a Magnetic Topological Insulator Chang et al., Science 340, 167 (2013). Joseph Hlevyack, Hu Jin, Mazin Khader, Edward Kim Outline: Introduction: