Current flow paths in deformed graphene and carbon nanotubes

Transcription

1 Current flow paths in deformed graphene and carbon nanotubes Cuernavaca, September 2017 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 graphene in 1μm 2 : over 10 7 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 at low energies Tight-binding Hamiltonian: Linear dispersion arround K points: long wave regime (low energy excitations) Dirac Hamiltonian: Deformation of graphene: coupling of the Dirac field to curvature?

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

7 Graphene deformation Tight-binding Hamiltonian: Surface deformation h(x,y) e.g. height function: Position-dependent hopping: Difference between strain and bond bending!

8 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

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

10 Dirac equation in curved space N. Szpak, Univ. Duisburg-Essen Current flow paths in deformed graphene t t t t t A , , 2 1 t t t t t t t t g ij Effective metric from frame: Effective magnetic field from potential: ab i a i a ij x e x e x g ) ( ) ( ) ( ) ( rot ) ( x A x B

11 Geometrical optics: waves trajectories Dirac Hamiltonian: Square Klein-Gordon eq. for scalar Eikonal approximation: ~ exp( i) Introduce velocity field: v Integrate trajectories = geodesics + EM field

12 Geometrical optics: waves trajectories Eikonal approximation geodesic trajectories:

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

14 Current flow vs geodesics Effect of both, curvature and pseudo-magnetic field: Numerical boundary (NEGF) Lattice simulation (NEGF)

15 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!

16 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

17 Current flow vs geodesics Crossing of trajectories focusing of waves: 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

18 Current flow vs geodesics Finite width contact ~ single slit diffraction

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

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

21 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)

22 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 Place contacts: source + several drains

23 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 Measure valley polarization locally

24 Measurement of valley polarization K1 AC injection K2

25 Measurement of valley polarization K1 ZZ injection K2

26 Measurement of valley polarization K1 K1 K1 K2 AC injection K2

27 Measurement of valley polarization K1 K1 K1 K2 ZZ injection K2

28 Measurement of valley polarization K1 K2 K1 ZZ injection at K1-K2 K2 Fourier transform (k space)

29 Deformation / pressure nanosensor K K-K K

30 Geometrical lensing pressure nanosensor Bump height

31 Geometrical valley separation Two (or more) bumps

32 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)

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

34 Conclusions Hopping in perturbed lattice Dirac + pseudomagnetic in continuous 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

35 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:

36 Density of states