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1 Graphene & Dirac Fermion confinement Zahra Khatibi 1
2 Outline: What is so special about Graphene? applications What is Graphene? Structure Transport properties Dirac fermions confinement Necessity External magnetic field Strain engineering Pseudo magnetic fields Conclusion 2
3 What is so special about Graphene? 3
4 What is so special about Graphene? 4 Andre Geim Konstantin Novoselov K. S. Novoselov, et al., SCIENCE VOL (2004)
5 What is so special about Graphene? Graphene has the strongest material stiffness ever measured (1.0 TPa) has high thermal conductivity is chemically stable can withstand large current densities has very high mobility has ballistic transport over sub-micron scales 5 N.M.R Peres et al., J. Phys.: Condens. Matter 21 (2009)
6 What is so special about Graphene? 6
7 What is so special about Graphene? High performance transistors silicon-based electronics Quantum Von Neumann architecture in single chip graphene-based electronics Two inductors and a transistor Can be used as amplifiers and wireless communication 7
8 What is so special about Graphene? Perfect solar cells Out of 1000 watts of sunlight silicon solar cell: 14 watts graphene-based solar cell: 1.3 watts graphene-based hybrid film on a flexible plastic substrate 8
9 What is so special about Graphene? Better lenses mobile-phone cameras, webcams and auto-focusing eye glasses Spin lens 9 Veselago s lens SCIENCE VOL (2007) Computer simulation of electron charge density
10 What is graphene? 10
11 What is is graphene? (structure) Graphitic materials 11
12 What is graphene? (structure) 12
13 What is graphene? (structure) 13
14 What is graphene? (structure) A honey comb lattice 14
15 What is graphene? (structure) A honey comb lattice 15
16 What is is graphene? (structure) A honey comb lattice 16
17 What is graphene? (structure) A honey comb lattice Sub lattices: A B 17
18 What is graphene? (structure) Band structure t ' : NNN hopping energy 1.42 A r 1 r 2 t 0 t =0 t: NN hopping energy 18 G. Baskaran, S. A. Jafari, Novel Quantum Phenomena in Graphene
19 What is graphene? (structure) Dirac points Six Fermi points at six corners of BZ (only 2 are independent) E/t 19
20 What is graphene? (Transport properties) Klein tunneling 200 ev 285 ev Transparent barriers for some angles No backscattering Lack of localization for smooth potential over atomic scale 20 M.I.Katsnelson et al., nature physics VOL (2006) Transmission probability
21 Dirac fermions confinement 21
22 Dirac fermions confinement (necessity) Not the only but the most important reason Graphene-based field effect transistors (FET) high mobility & ballistic transport at submicron distances structures smaller than 10 nm with higher mobility graphene gapless spectrum? Flow spreading switch off 22
23 External magnetic field 23
24 External magnetic field Inhomogeneous magnetic fields no transmission is possible for 24 A. De Martino et al., PRL 98, (2007) Transmission probability T for a magnetic barrier of width 2d
25 armchair External magnetic field Zigzag states Zigzag 25 K. Wakabayashi et al., PRB (1999)
26 External magnetic field circular dot in magnetic field (10 T) ❶ ❷ ❸ ❶ ❷ ❸ independent particle Hartree bands parabolic potential Energy spectrum of a graphene quantum dot at B = 10 T 26 N.M.R Peres et al., J. Phys.: Condens. Matter (2009)
27 Strain engineering 27
28 Strain engineering (pseudo magnetic field) Disorder changes hopping energy by: distance Pz orbitals angle 28
29 Strain engineering (pseudo magnetic field) Hopping energies modification Change in bond length Change in unit cell area 29
30 Strain engineering (pseudo magnetic field) Inversion symmetry φ r = φ r A = A x (r) + ia y (r) AB BB AA BA 30
31 Strain engineering (pseudo magnetic field) Time reversal is not broken C depends on detailed model of chemical bonding β and g are the coupling to acoustical wave in graphene 31 M.A.H. Vozmedianoa et al., arxiv: v2 (2010)
32 Strain engineering (pseudo magnetic field) strong pseudomagnetic field which guides electrons 32 N. Levy et al., SCIENCE VOL (2010)
33 Strain engineering (pseudo magnetic field) ring-shaped Gaussian deformation B ps sin 3θ Pseudo magnetic field Classical trajectories Current density 33 G. M. M. Wakker et al., PHYSICAL REVIEW B 84, (2011)
34 Strain engineering (pseudo magnetic field) rotationally symmetric strain Made by atomic force microscopy (AFM) tip or by a homogeneous gas pressure vertical displacement 34 K.J Kim et al., PHYSICAL REVIEW B 84, (R) (2011) Pseudo magnetic field
35 Strain engineering (pseudo magnetic field) rotationally symmetric strain Classical trajectories Probability density 35 K.J Kim et al., PHYSICAL REVIEW B 84, (R) (2011)
36 Strain engineering (pseudo magnetic field) Tunable gap in strained graphene 36 Current density T. Low et al., PHYSICAL REVIEW B 83, (2011)
37 Conclusion Graphene is a promising candidate for nano-electronic applications. Confinement is achievable via strain engineering or external fields. 37
38 Thank you 38
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