Nanoscience, MCC026 2nd quarter, fall Quantum Transport, Lecture 1/2. Tomas Löfwander Applied Quantum Physics Lab

Transcription

1 Nanoscience, MCC026 2nd quarter, fall 2012 Quantum Transport, Lecture 1/2 Tomas Löfwander Applied Quantum Physics Lab

2 Quantum Transport Nanoscience: Quantum transport: control and making 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 (the nano 10-9 m) properties and behavior of nanostructures that 1) cannot be understood starting from classical (non-quantum) physics 2) does not depend on materials or atomic structure per se, rather, on a set of principles that holds in certain regimes Example: contuctance G of the order GQ=2e2/h the quantum of conductance

3 Energy scales Temperature k B T, note k B ~0.087 mev/k Voltage applied to the device Level spacing δ, ~0.038 ev nm 2 Charging energy E ch =e 2 /C, ~0.1 mev (for C=10-15 F) Fermi energy E F, ~ several ev for 3D metals ~14 mev (GaAs 2DEG), ~ mev (Si 2DEG)

4 System size L Length scales Fermi wave length λ F, ~20 nm (GaAs), ~100 nm (Si) L < λ F, wave character pronounced L > λ F, electron billiards + interference Elastic scattering length l imp, ~100 nm - 10 µm (GaAs), ~100 nm (Si) L < l imp, ballistic motion L > l imp, diffusive motion Phase coherence length l φ, ~100 nm (GaAs, Si) at 1K (scales as 1/ T) L < l φ, mesoscopic regime L > l φ, macroscopic regime

5 Outline of Quantum Transport Lectures Examples of nanostructures: 2DEG, graphene, nanotubes, molecules Transport of electrons: key concepts Size quantization, energy levels Density of states Semiclassical transport: Boltzmann approach Scattering approach to electron transport in nanostructures Sharvin resistance Conductance quantization and the Landauer formula Tunneling, resonant tunneling Master equation approach to electron transport in nanostructures single-level quantum dot without Coulomb interactions Coulomb interaction and the multi-electron picture single-level quantum dot with Coulomb interactions Coulomb blockade

6 Basic concepts in these lectures size quantization matters in nanoscale devices Schrödinger equation the concept of the scattering (Landauer) approach to coherent quantum transport in nanoscale devices reservoir (= lead, contact) scattering region how to sum over scattering states (quantum numbers) the role of Coulomb interaction in ultra small devices (simplest case: incoherent transport regime) Coulomb blockade

7 Literature Electron Transport in Mesoscopic Systems S. Datta Cambridge University Press 1995 Quantum Transport: Atom to Transistor S. Datta Cambridge University Press 2005 Quantum Transport: Introduction to Nanoscience Y.V. Nazarov and Y.M. Blanter Cambridge University Press 2009

8 Low-dimensional structures - examples 2D quantum contacts (B. van Wees et al PRL 60, 848, 1988) 1D quantum contacts 0D quantum contacts (C. Dekker et al, Nature 386, 474, 2003) (S. Kubatkin et al, Nature 425, 698, 2003) 2DEG= 2 Dimensional Electron Gas single-walled carbon nanotubes extended molecular electron orbits conductance quantization quantum wires quantum dots

9 The 2DEG Quantum point contact Quantum dot

10 Low-dimensional materials 2D graphene K.S. Novoselov, A.K. Geim et al, Science 306, 666 (2004) 1D Carbon Nanotubes S. Iijima, Nature 354, 56 (1991)

11 Carbon LEGO The Rise of Graphene, A.K. Geim and K.S. Novoselov, Nature Materials 6, 183 (2007)

12 Transport: key concepts electronics cooling, heating, thermoelectricity Measurements of charge, heat, and spin currents How do we compute charge currents? spintronics

13 Transport: key concepts electronics cooling, heating, thermoelectricity Measurements of charge, heat, and spin currents How do we compute charge currents? spintronics distribution function = probability of state p being occupied sum over states (quantum numbers) current carried by state p

14 Transport: key concepts How do we compute charge currents? When we know the energy dispersion, e.g. integral over energies density of states spectral current distribution function density of states = # of states per energy interval & unit volume

15 Size quantization Textbook examples of size quantization: (a) particle in a box in one dimension energy levels: follows from Schrödinger s eq. with hard wall boundary conditions discrete wave numbers: (a) particle in a harmonic potential energy levels:

16 Density of states The previous considerations of energy levels can be extended to more complicated structures and to 2D and 3D. But the energy levels become more dense and degenerate. More useful concept: local density of states Examples: (a) Free electrons (3D) with dispersion (b) compare with 2D (quantization in one of three dimensions) (c) compare with 1D (quantization in two of three dimensions)

17 Density of states The previous considerations of energy levels can be extended to more complicated structures and to 2D and 3D. But the energy levels become more dense and degenerate. More useful concept: local density of states Examples: (a) Free electrons (3D) with dispersion (b) compare with 2D (quantization in one of three dimensions) (c) compare with 1D (quantization in two of three dimensions) blackboard

18 Density of states Collection of results: 0D: discrete spectrum

19 Density of states examples Graphene Linear DOS near the Fermi level Carbon nanotubes Semiconducting or metallic (zero or finite DOS near the Fermi level)

20 Semiclassical transport: Boltzmann approach Macroscopic regime: L Conductance σ = conductivity (material specific) recall Ohm s law j = σe What determines the current? V j =? group velocity current density: Boltzmann s eq:

21 Semiclassical transport: Boltzmann approach Linear response: local equilibrium (symmetric) small correction (antisymmetric) => => =>

22 Semiclassical transport: Boltzmann approach Ohm s law: with diffusion constant (material parameter): Einstein relation

23 The Fermi-Dirac distribution function determines the occupation of the energy levels: The Fermi energy of metals are typically ~5 ev to 10 ev. Room temperature (300 K) corresponds to ~0.026 ev.

24 The Fermi-Dirac distribution function determines the occupation of the energy levels: The Fermi energy of metals are typically ~5 ev to 10 ev. Room temperature (300 K) corresponds to ~0.026 ev.

25 Size quantization: the 2DEG z-direction confinement Method of separation of variables: 3 f 1 2 2DEG

26 Sharvin resistance Consider a pinhole - e.g. a 2DEG with a constriction (here viewed from above): (ballistic regime) Under applied voltage: local equilibrium deep in the reservoar, d w

27 Sharvin resistance The current can be computed within frame work of the scattering approach: net current = difference between right and left going currents

28 Sharvin resistance quantum of conductance # of conducting channels

29 Sharvin resistance quantum of conductance # of conducting channels Sharvin

30 Conductance of one channel y x Separation of variables for one transverse mode in the channel: assume reflectionless contacts wavefunction: E k -E 0 µ L energy: µ R net current = difference between right and left going currents k x

31 Conductance of one channel y E k -E 0 x µ R µ L k x remember: peculiar cancellation in 1D

32 Conductance of one channel y E k -E 0 x µ R µ L k x Thus: = the quantum of conductance note:

33 1D conductance: many channels y (wider contact) x Separation of variables for many transverse modes in the channel: wavefunction: E k energy: µ L consider M open channels: µ R conductance: k x

34 1D conductance: many channels y (wider contact) x M open channels conductance: E k Note: finite resistance in a perfect sample. Why? µ L Note: where is the energy dissipated? µ R k x

35 1D conductance: many channels y (wider contact) x conductance: Sharvin s resistance! =contact resistance E k Note: finite resistance in a perfect sample. Why? µ L Note: where is the energy dissipated? µ R k x

36 1D conductance: many channels y (wider contact) x conductance: Sharvin s resistance! =contact resistance E k Note: finite resistance in a perfect sample. Why? µ L Note: where is the energy dissipated? µ R in the reservoirs! k x

37 Observation of opening of modes