Nanoscience quantum transport

Transcription

1 Nanoscience quantum transport Janine Splettstößer Applied Quantum Physics, MC2, Chalmers University of Technology Chalmers, November 2 10

2 Plan/Outline 4 Lectures (1) Introduction to quantum transport (2) Quantum transport: Graphene; effects of a magnetic field (3) ff. magnetic field effects; 1- and 0-dimensional devices (example carbon nanotubes) (4) Strongly confined systems Literature: "Electronic transport in mesoscopic systems", S. Datta, Cambridge Studies (2009). see also lectures and lecture material by Datta: "Many-Body Quantum Theory in Condensed Matter Physics", H. Bruus and K. Flensberg, Oxford Graduate Texts (2004). "Quantum Transport", Yu. V. Nazarov and Ya. M. Blanter, Cambridge University Press (2009). "Scattering Matrix Approach to Non-Stationary Quantum Transport", M. V. Moskalets, Imperial College Presss (2012). Contact: janines@chalmers.se, MC2 building, C520/521.

3 AN ELECTRON IS A

4 Quantum transport Nanoscience: Control and design of useful things at the atomic scale Goal/dream: build devices atom by atom Interdisciplinary: Physics, Chemistry, Biology, Materials Science, Engineering technology working on small spatial scales ( nano 10 9 m) Quantum transport: Properties and behavior of nanostructures that 1) cannot be understood starting from classical (non-quantum) physics 2) do not depend on materials or atomic structure per se, rather, on a set of principles that holds in certain regimes

5 Mesoscopic physics Example: wire with impurities (diffusive) Length of the wire: Geometric length Elastic scattering length: typical length between two scattering events (without energy exchange). In GaAs: 100nm 10µm. Coherence length: length scale on which phase coherence is lost phase has changed by about 2π. In GaAs: 100nm. Inelastic scattering length: length scale on which energy of the order k B T has been exchanged. Mesoscopic device:

6 Energy scales Temperature k B T, Boltzmann constant k B mev/k Bias voltage applied across the device ev, with electron charge e = C Level spacing δ in strongly confined structures, µev mev Charging energy E ch = e 2 /C, 0.1 mev (for C = F) Fermi energy E F or µ, several ev for 3D metals, 14 mev (GaAs 2DEG).

7 How to make a mesoscopic device? Typically a "small" device... nanometer to micrometer range Low-dimensional (two- or even one- or zero-dimensional devices) Low temperature (Kelvin to milli Kelvin regime) Clean-room conditions

8 Examples semiconductor heterostructures homepage, Katja Nowack.

9 Examples Hall samples edge states can take the role of "wave guides" Y. Ji, Y. Chung, D. Sprinzak, M. Heiblum, D. Mahalu, and Hadas Shtrikman, Nature 422, 415 (2003).

10 Examples Carbon-based devices

11 Examples Carbon-based devices Clean, two-dimensional graphene devices B. Terrés, L. A. Chizhova, F. Libisch, J. Peiro, D. Jörger, S. Engels, A. Girschik, K. Watanabe, T. Taniguchi, S. V. Rotkin, J. Burgdörfer, and C. Stampfer, Nature Communications 7, (2016).

12 Scattering theory for electronic transport Common features of mesoscopic devices: Possibly complex, phase-coherent device Coupled to electronic reservoirs Transport between reservoirs, detection of currents Each reservoir stays in equilibrium during device operation

13 Scattering theory for electronic transport Equilibrium reservoirs Perfect leads Scatterer Perfect leads Equilibrium reservoirs Scatterer Incoming scattering states Outgoing scattering states Characterized by a scattering matrix with transmission and reflection amplitudes. Occupied following the equilibrium occupation of the reservoir with µ α, k B T α. Nonequilibrium states

14 Example for a different 2d-system: Graphene Graphene: monolayer of carbon (ideal, stable 2d-material) Hybridization of carbon atoms in a graphene sheet: Graphene made of mainly 12 C atoms (with nuclear spin 0) Carbon has 6 electrons occupying the orbitals 1s 2, 2s 2, and 2p 2 sp 2 hybridization (see figure) hexagonal graphene structure angle 2π/3 between sp 2 orbitals; form covalent bonds with neighbouring orbitals Electrons in p z shell occupy a π or π band...

15 Lattice in real space Triangular Bravais-lattice with 2-atom basis: B A Wigner-Seitz cell of the honeycomb lattice is rhombic containing two atoms (A and B). Lattice vectors: a 1 = a ( ) 3, a 2 = a ( 3 3 ) δ 1 δ 2 δ 3 with atomic spacing a = 1.42Å. a 2 a a 1 Vectors connecting next neighbours: ( ) 0 δ 1 = a, δ 1 2 = a ( ) 3, 2 1 δ 3 = a ( ) 3 2 1

16 Lattice in momentum space (reciprocal lattice) Reciprocal hexagonal lattice with two inequivalent "Dirac-points" K and K k y Obtain lattice vectors of the reciprocal lattice from a i b j = 2πδ ij : b 1 = 2π ( ) 3, b 2 = 2π ( 3 3a 1 3a 1 ) b 2 b 1 M Γ-point: center of Brillouin zone. K Γ K kx Coordinates with respect to it: K = 4π ( ) 1 3 3a 0 K = 4π ( ) 1 3 3a 0 M = π ( ) 3 = 1 3a 1 2 b 2

17 Energy spectrum Bandstructure calculation within a tight-binding approximation yields: E ± = t g(k)± t f(k) with transfer integrals t and t between nearest and next nearest neighbours and g(k) and f(k) given by [ 3a g(k) = 4 cos 2 ky ] [ 3a cos 2 kx [ ] 3a ky a f(k) = 2 cos 2 kx i e 2 + e iky a ] [ ] + 2 cos 3akx A. H. Castro Neto et al.: Rev. Mod. Phys. 81, 109 (2009) Linearized spectrum in the vicinity of the K-points (valleys): E ± = ±v F k with Fermi ( light!) velocity v F = 3at/2

18 Carbon nanotubes 3 different classes of nanotubes: zigzag armchair chiral

19 Contacted carbon nanotubes Contacted nanotube on a substrate: Contacted, suspended nanotube: P. Jarillo-Herrero et al.: Nature 429, 389 (2004). R. Leturcq et al.: Nat. Phys. 5, 327 (2009).

20 Construct your own carbon nanotubes C h a2 a1 a2 a1

21 Coulomb diamonds of Carbon nanotubes Coulomb diamonds displaying four-fold level degeneracy Excited (vibrational) states and Franck-Condon physics R. Leturcq et al.: Nat. Phys. 5, 327 (2009).