Quantum Confinement in Graphene

Transcription

1 Quantum Confinement in Graphene from quasi-localization to chaotic billards MMM dominikus kölbl / 27

2 Outline some facts about graphene quasibound states in graphene numerical calculation of edge effects size effects in graphene quantum dots charge sensing with graphene point contacts MMM dominikus kölbl / 27

3 Conventional Quantum Dots QD in 2DEGs, CNT, nanowires Coulomb blockade single electron regime excited states MMM dominikus kölbl / 27

4 Conventional Quantum Dots long term goal: spin Qbits / quantum computing spin coherence large T 1, T 2 (relaxation, dephasing) limiting effects: spin-orbit coupling hyperfine interaction SO in CNTs: MMM dominikus kölbl / 27

5 Graphene one-atomic layer of graphite hexagonal carbon lattice simple exfoliation technique Novoselov et al., Science 306(2004) MMM dominikus kölbl / 27

6 Graphene sublattices A,B tight binding: nearest + next-nearest E linear at dirac-point (K,K ) low E: linear dispersion E > 1eV: trigonal warping MMM dominikus kölbl / 27

7 Graphene density of states: linear in E and K,K semi-metalic MMM dominikus kölbl / 27

8 Graphene Hamiltonian around Dirac point: with Dirac equation: two component wavefunction: rotation θ +2π : ψ ψ Berry s phase π : spinor with respect to pseudo-spin MMM dominikus kölbl / 27

9 Graphene projection of momentum along pseudo-spin: chirality / helicity = good quantum number (low E) electrons : positive helicity holes: negative helicity MMM dominikus kölbl / 27

10 Graphene Klein paradox no scattering at potential barriers T=1 (i.e. θ =0) angle dependence how to confine in graphene? MMM dominikus kölbl / 27

11 Confinement in Graphene confinement with sharp potential: Hamiltonian: wavefunction components for E < V A. Matulis, F.M. Peeters; PRB 77(2008) MMM dominikus kölbl / 27

12 local density of states: Confinement in Graphene DOS in barrier confinement modulation V=12 - quasi-localized due to interference (and not due to tunneling barriers) - orbital momentum dependence - stronger localization for narrow peaks: trapping time τ ~ (FWHM) -1 MMM dominikus kölbl / 27

13 Confinement in Graphene quasi-bound states zero orbital momentum: total internal reflection for E = V orbital momentum m = 2: interference below and above V MMM dominikus kölbl / 27

14 QD cut out of a Graphene sheet: Numerics tight binding simulation edge roughness / impurities d = 20nm averaged DOS: - localization peaks - disappearing for rough edges Libisch et al.,arxiv: v2(2008) MMM dominikus kölbl / 27

15 Numerics nearest neighbour level spacing: - phase transition from regular to chaotic billards - Dirac billards more stable due to time-reversal symmetry MMM dominikus kölbl / 25

16 Graphene QDs `all graphene` device: side gates source contact drain contact side gate Ponomarenko et al., Science 18(2008) MMM dominikus kölbl / 27

17 Graphene QDs 250nm central island: T= 0.3K periodic Coulomb resonances ΔV g = 16 mev C g = e/δv g 10 af plate capacitance estimate: C g 2 ε 0 (ε+1) D 20 af screening of the contact regions MMM dominikus kölbl / 27

18 Graphene QDs elevated temperature: T = 4K - typical V-shape with fluctuations (quantum interference effects) - more than 1000 (!) CB-oscillations inset: -Coulomb diamonds: charging energy E c = 3 mev (d = 250 nm) - smeared barrier transparency (with T) MMM dominikus kölbl / 27

19 Graphene QDs smaller QD: d 15nm - non-periodic CB-peaks - domination of confinement energy: ΔE = E c + δe 40meV unique to massless fermions: δe ν F h/2d ( m 0: δe h 2 /8mD 2 ) MMM dominikus kölbl / 27

20 chaotic Dirac billard peak spacing histograms for different D: - random position for D < 100 nm - strong level repulsion - Gaussian unitary ensemble quantum chaos in Dirac billiards - at relatively large diameters - manely due to edge roughness MMM dominikus kölbl / 27

21 Quantum Point Contacts - large gaps for narrow constrictions - mesoscopic fluctuations in conductance - QPC transparencies limit the CB region of the quantum dots MMM dominikus kölbl / 27

22 QD + QPC Quantum dot: d = 200nm Point Contact: d = 45nm CB in `transport gap Dot QPC Stampfer et al., Appl.Phys. Lett.(2008) MMM dominikus kölbl / 27

23 QD + QPC charge sensing detection of single charging events T=1.7K charging energy: E c = 4.3 mev lever arms α = 0.06 / 0.34 MMM dominikus kölbl / 27

24 Quantum Confinement smaller dot size: d = 140 nm E c 10 mev sharp Coulomb resonances at 200 mk (electron temp.) strong variations indicating confinement effect MMM dominikus kölbl / 27

25 Quantum Confinement excited states: Δ = mev peak broadening due to coupling to the leads inelastic co-tunneling E c varies with charge numbers Δ(N) = ħν F /(d N) N ~ O (10) MMM dominikus kölbl / 27

26 Quantum Confinement energy shift in perpendicular magnetic field flux quantum per dot area B c = 270 mt it s not a Zeeman effect g 50 (g 2) B = 4 T: bulk LL formation positive slopes: electron regime (?) MMM dominikus kölbl / 27

27 Conclusion quantum confinement of massless charges possible charge sensing of orbital states with QPC excited state measurements next steps: single electron regime magnetic field dependence electron-hole symmetry MMM dominikus kölbl / 27