Charge-Carrier Transport in Graphene

Transcription

1 Charge-Carrier Transport in Graphene P.V. Buividovich, O.V. Pavlovsky, M.V. Ulybyshev, E.V. Luschevskaya, M.A. Zubkov, V.V. Braguta, M.I. Polikarpov ArXiv: ; ArXiv: Introduction: QCD and graphene Charge carriers in graphene and effective field theory Calculations on the hypercubic lattice Calculations on hexagonal lattice Workshop on QCD in strong magnetic fields November 2012, Trento, Italy

2 Charge-Carrier Transport in Graphene P.V. Buividovich, O.V. Pavlovsky, M.V. Ulybyshev, E.V. Luschevskaya, M.A. Zubkov, V.V. Braguta, M.I. Polikarpov ArXiv: ; ArXiv: Introduction: QCD and graphene Charge carriers in graphene and effective field theory Calculations on the hypercubic lattice Calculations on hexagonal lattice Workshop on QCD in strong magnetic fields November 2012, Trento, Italy

3 Charge-Carrier Transport in Graphene P.V. Buividovich, O.V. Pavlovsky, M.V. Ulybyshev, E.V. Luschevskaya, M.A. Zubkov, V.V. Braguta, M.I. Polikarpov ArXiv: ; ArXiv: Introduction: QCD and graphene Charge carriers in graphene and effective field theory Calculations on the hypercubic lattice Calculations on hexagonal lattice Workshop on QCD in strong magnetic fields November 2012, Trento, Italy

4 Charge-Carrier Transport in Graphene P.V. Buividovich, O.V. Pavlovsky, M.V. Ulybyshev, E.V. Luschevskaya, M.A. Zubkov, V.V. Braguta, M.I. Polikarpov ArXiv: ; ArXiv: Introduction: QCD and graphene Charge carriers in graphene and effective field theory Calculations on the hypercubic lattice Calculations on hexagonal lattice Workshop on QCD in strong magnetic fields November 2012, Trento, Italy

5 Charge-Carrier Transport in Graphene intight binding model P.V. Buividovich, O.V. Pavlovsky, M.V. Ulybyshev, E.V. Luschevskaya, M.A. Zubkov, V.V. Braguta, M.I. Polikarpov ArXiv: ; ArXiv: Introduction: QCD and graphene Charge carriers in graphene and effective field theory Calculations on the hypercubic lattice Calculations on hexagonal lattice Workshop on QCD in strong magnetic fields November 2012, Trento, Italy

6 QCD and Graphene

7 Carbon atom

8 Elementary structure

9 Some allotropes of carbon: a) diamond; b) graphite; c)lonsdaleite; d f) fullerenes (C 60, C 540, C 70 ); g) amorphous carbon; h) carbon nanotube.

10 Fullerene (Buckminsterfullerene) C 60 Richard Buckminster Fuller The Montreal Biosphère by Buckminster Fuller, 1967

11 Fullerene C 540 Richard Buckminster Fuller The Montreal Biosphère by Buckminster Fuller, 1967

12 Nanotube

13 Graphene

14 The Nobel Prize in Physics for 2010 was awarded to Andre Geim and Konstantin Novoselov "for groundbreaking experiments regarding the two-dimensional material graphene

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18 Nonrelativistic particle E mv 2 2

19 Relativistic particle E m c p c

20 Relativistic particle E m c p c Massless particle E cp

21 Relativistic particle Massless particle E E m c p c cp c Graphene E vfp; vf ; 300

22 Relativistic particle Massless particle E E m c p c cp c Graphene E vfp; vf ; 300 g g crit Pure graphene is the insulator! g

23 Hexagonal lattice = Triangle lattice + Triangle lattice On such lattice nonrelativistic electrons are equivalent to massless four component Dirac fermions moving with v F c 300 the effective charge is: ; g

24 Graphene lattice and Brillouin zone

25 Wallace, P. R. (1947). "The Band Theory of Graphite". Physical Review 71 (9): 622. Semenoff, G. W. (1984). "Condensed- Matter Simulation of a Three-Dimensional Anomaly". Physical Review Letters 53 (26): 2449.

26 We can vary the effective coupling in graphene! Graphene in the dielectric media g g Graphene on substrate g 2 g 1 graphene substrate 2 crit if g g ( 1.11) graphene is the conductor 1

27 Effective theory of charge carriers in 1. Massless four component Dirac fermions 2. Fermi velocity is graphene vf c / The effective charge is g We can vary the effective charge if we vary the dielectric permittivity of the substrate g 2 g 1

28 Effective field theory for graphene After transformation we can neglect A i ; 300 g v F

29 On substrate g 2 g 1 (2+1)D fermions (3+1)D Coulomb

30 Simulation of the effective graphene theory Approach 1, hypercubic lattice J. E. Drut, T. A. Lahde, and E. Tolo ( ) W. Armour, S. Hands, and C. Strouthos ( ) P.V. Buividovich, O.V. Pavlovsky, M.V. Ulybyshev, E.V. Luschevskaya, M.A. Zubkov, V.V. Braguta, M.I. Polikarpov (2012) (2+1)D fermions (3+1)D Coulomb

31 Simulation of the effective graphene theory Approach 2, 2D hexagonal lattice and rectangular lattice in z and time dimensions R. Brower, C. Rebbi, and D. Schaich ( ) P.V. Buividovich, M.I.P. (2012)

32 Approach 2, 2D hexagonal lattice, Hamiltonian ^ ^ ^ H H H tb I

33 Approach 2, 2D hexagonal lattice, Hamiltonian Lattice geometry ^ ^ ^ H H H tb I Coulomb interaction

34 Approach 2, 2D hexagonal lattice, Hamiltonian Lattice geometry ^ ^ ^ H H H tb I Coulomb interaction

35 Approach 2, 2D hexagonal lattice, Hamiltonian ^ ^ ^ H H H tb I v F c 300 { ˆ ˆ a, X, a ', Y} ' X, Y

36 Approach 2, 2D hexagonal lattice, Hamiltonian ^ ^ ^ H H H tb I Coulomb interaction

37 Approach 2, 2D hexagonal lattice, Hamiltonian ^ ^ ^ H H H tb I

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39 Fermion condensate as a function of substrate dielectric permittivity Approach 1 Approach 2 Hypercubic lattice Hexagonal lattice

40 Conductivity as a function of substrate dielectric permittivity Approach 1 Approach 2 Hypercubic lattice Hexagonal lattice

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42 Perpendicular magnetic field H graphene H substrate Graphene changes its properties when an external magnetic field is applied, we can numerically simulate all that

43 Fermion condensate as the function of substrate dielectric permittivity at finite magnetic field

44 Substrate dielectric permittivity - Magnetic field phase diagram Approach 1 (preliminary)??????

45 What can be done in the field theory approach Magnetic field Finite temperature Impurities layers Conductivity n E v F Viscosity Entropy Optical properties Critical indices Conductivity of nanotube

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47 Parallel magnetic field (Aleiner, Kharzeev, Tsvelik 2007) graphene H H substrate Graphene changes its properties when an external magnetic field is applied, we can numerically simulate all that

48 Trajectory of the magnetic head graphene Ferromagnetic substrate magnetic head Along the trajectory of the magnetic head graphene becomes the conductor! We can draw (construct) chips! All that we can simulate on computers

49 Mobius carbon is a topological insulator? ArXiv:

50 Dependence of conductivity on the radius of nanotube and on magnetic field 2R B

51 Graphyne Competition for Graphene: Graphynes with Direction-Dependent Dirac Cones Daniel Malko, Christian Neiss, FrancescVines, and Andreas Gorling PRL 108, (2012) PHYSICAL REVIEW LETTERS 24 FEBRUARY 2012

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53 Discussion Sessions Tuesday 13 November 16:30 - Gerald Dunne and Yoshimasa Hidaka Landau-level structure in QCD. Questions: To what extent is the Landau-level picture applicable in QCD as an interacting theory? Is the LLL approximation valid for strong magnetic fields, and does physics reduce to a 1+1 dimensional theory here? If so, does the Mermin-Wagner theorem become effective? Is this visible in the Dirac eigenmodes? Wednesday 14 November 16:30 - Andreas Schmitt and Ingo Kirsch Introduction to the ads/qft approach and comparison to lattice. Questions: Is there any input from the lattice side, that could be used to fix certain free parameters of the holographic approach, and what are the observables that we can compare?

54 Discussion Sessions Thursday 15 November 16:30 - Dmitri Kharzeev and Vladimir Skokov Chiral magnetic effect. Questions: Is there consensus about CME signatures in experimental results from heavy ion colliders? What are the possible interpretations of these data? What are the motivated theoretical suggestions for ALICE to measure from the CME-interested community (becoming especially actual after ALICE upgrade)? Is it resonable to look at higher order sin-harmonics and their correlators? Is it reasonable to study their averages not over set of all event but over some subsets?

55 09:00-09:40 Edward Shuryak QCD topology near and above T_c Friday 16 November Talk of Eduardo Fraga => Wednesday 09:40-10:20 Discussion session Maxim Chernodub, Yoshimasa Hidaka, Arata Yamamoto Superconducting vacuum in strong magnetic field, does it exist or not? 10:20-10:40 Coffee break 10:40-11:20 Final discussion All participants of the workshop

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