Precise electronic and valleytronic nanodevices based on strain engineering in graphene and carbon nanotubes

Transcription

1 Precise electronic and valleytronic nanodevices based on strain engineering in graphene and carbon nanotubes European Graphene Forum 2017, Paris Nikodem Szpak 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 electronic transport in deformed graphene Why? in 1μm 2 : over 10 7 atoms! ( ab initio) regular graphene: extrapolate microscopic calculations; irregular??? Idea: Quantum currents in elastically deformed graphene Waves propagating in curved space continuous limit field equations Source: NASA Motivation: tailor-made mesoscopic structures special properties nano-devices... 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: Effective metric: g ij ij ij ( x) ( x) Effective magnetic potential: s s s K ( x) K ( 1) xx ( x) yy ( x), 2 xy ( x) 2 Effective magnetic field: B( x) 2 ( x) ( x) ( x) x xy y xx y yy

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 E = 0.2 t 0, r 0 = 200 d 0, h 0 = 1.00 r 0 E = 0.3 t 0, r 0 = 200 d 0, h 0 = 1.25 r 0 N. Szpak, Univ. Duisburg-Essen Current flow paths in deformed graphene... Current flow vs geodesics Waves propagate along classical trajectories for the curved space! Varying bump height: E = 0.2 t 0 r 0 = 150 d 0, h 0 = 0.50 r 0 r 0 = 150 d 0, h 0 = 0.75 r 0 r 0 = 150 d 0, h 0 = 1.00 r 0 Crossing of trajectories focusing of waves:

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 lensing of the current flow Maxwell lense: position dependent refraction index n(x,y) Now: both charge signs both valleys Continuous model prediction

14 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)

15 Geometrical lensing pressure nanosensor Bump height

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

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

18 Conclusions Current flow in deformed graphene Hˆ = <n, m> T + n m aˆ n aˆ, m + n V n aˆ + n aˆ n Dirac eq. + pseudomagnetic field in continuous curved space 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 Conclusions: deformation curvature, pseudo-b Perturbed lattice QFT in curved space Hˆ = <n, m> T + n m aˆ n aˆ, m + n V n aˆ + n aˆ 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

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

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

22 Current flow vs geodesics Waves propagate along classical trajectories for the curved space! Varying bump height: E = 0.2 t 0 r 0 = 150 d 0, h 0 = 0.50 r 0 r 0 = 150 d 0, h 0 = 0.75 r 0 r 0 = 150 d 0, h 0 = 1.00 r 0

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