Current flow paths in deformed graphene and carbon nanotubes

Transcription

1 Current flow paths in deformed graphene and carbon nanotubes DPG Tagung 2017, Dresden Nikodem Szpak Erik Kleinherbers Ralf Schützhold Fakultät für Physik Universität Duisburg-Essen Thomas Stegmann Instituto de Ciencias Físicas Universidad Nacional Autónoma de México PROJECT (DFG): Curvature, Defects, Geometry in Graphene and Optical Lattices

2 Efficient model of transport in deformed graphen in 1μm2: over 107 atoms! regular graphene OK, but irregular? Quantum currents in elastically deformed graphene Waves in curved continuous space continuous limit Source: NASA Applications: tailor-made mesoscopic structures special properties better understanding of structural perturbations

3 Graphene Tight-binding Hamiltonian: Linear dispersion arround K points:

4 Graphene at low energies Tight-binding Hamiltonian: Linear dispersion arround K points: long wave regime (low energy excitations) Dirac Hamiltonian:

5 Graphene deformation Tight-binding Hamiltonian: Surface deformation h(x,y) e.g. height function: Position-dependent hopping:

6 Graphene deformation Tight-binding Hamiltonian: Surface deformation h(x,y) e.g. height function: Position-dependent hopping: Cones (locally) shifted and deformed! shift pseudo-magnetic potential deformation metric and spin connection

7 Dirac equation in curved space Dirac Hamiltonian: Local frame vectors: ij ij ij g ( x ) 2 (x) Effective metric: Effective magnetic potential: s K ( x ) ( 1) s xx ( x ) yy ( x ), 2 xy ( x ) Effective magnetic field: B( x ) 2 x xy ( x ) y xx ( x ) y yy ( x )

8 Geometrical optics: waves trajectories Dirac Hamiltonian: Eikonal approximation geodesics:

9 Current flow vs geodesics Effect of both, curvature and pseudo-magnetic field: Continuous space approximation Lattice simulation (NEGF)

10 Current flow vs geodesics Waves propagate along classical trajectories for the curved space! Varying bump height: r0 = 150 d0, h0 = 0.50 r0 E = 0.2 t0 r0 = 150 d0, h0 = 0.75 r0 r0 = 150 d0, h0 = 1.00 r0 Crossing of trajectories focusing of waves: E = 0.2 t0, r0 = 200 d0, h0 = 1.00 r0 E = 0.3 t0, r0 = 200 d0, h0 = 1.25 r0

11 Geometrical lensing of the current flow Maxwell lense: effectively position dependent refraction index n(x,y) Continuous model prediction

12 Geometrical lensing of the current flow Lattice simulations (NEGF)

13 Geometrical valley separation K left & right Valley separation: K left & right, K center K center Injection at K Injection at K+K Injection at K Lattice simulations (NEGF)

14 Geometrical lensing pressure nanosensor Bump height

15 Bent nanotubes Strain induced metric and pseudo-magnetic field: Surface parameterization by pair of angles (θ,φ) 2 Effective metric: g g ij 0 0 ( R cos ) g Pseudo-magnetic vector potential: 2 0 cos Ai R 0 Pseudo-magnetic field: B B0 sin

16 Bent nanotubes: Dirac equation on torus Dirac Mathieu equation Mathieu f s: Analytical current: (m=0 mode) Numerical current (NEGF):

17 Conclusions Hopping in perturbed lattice Dirac + pseudomagnetic in continuous curved space Hˆ = <n, m> Tn,m aˆ n+ aˆ m + Vn aˆ n+ aˆ n n T. Stegmann and N.S., Current flow paths in deformed graphene, New J. Phys. 18 (2016) PROJECT: Curvature, Defects, Geometry in Graphene and Optical Lattices

18 Conclusions: deformation curvature, pseudo-b Perturbed lattice QFT in curved space Hˆ = <n, m> Tn,m aˆ n+ aˆ m + Vn aˆ n+ aˆ n n T. Stegmann and N.S., Current flow paths in deformed graphene, New J. Phys. 18 (2016) PROJECT: Curvature, Defects, Geometry in Graphene and Optical Lattices

19 Particles (E>0) and antiparticles (E<0) K vs K valleys Two types of valleys: K K K K Magnetic K field B at different valleys! K

20 Stationary current NEGF method Tight-binding Hamiltonian: Green's function: Self energy: Local current: Correlation function: Inscattering function: at boundary to emulate infinite surface discretization of Dirac current and Green s funct. representati (x ) of solutions with source

21 Current flow vs geodesics Waves propagate along classical trajectories for the curved space! Varying bump height: r0 = 150 d0, h0 = 0.50 r0 E = 0.2 t0 r0 = 150 d0, h0 = 0.75 r0 r0 = 150 d0, h0 = 1.00 r0

22 Geometrical valley separation Two (or more) bumps even stronger K / K separation...