Five-Lecture Course on the Basic Physics of Nanoelectromechanical Devices Lecture 1: Introduction to nanoelectromechanical systems (NEMS) Lecture 2: Electronics and mechanics on the nanometer scale Lecture 3: Mechanically assisted single electronics Lecture 4: Quantum nano-electro-mechanics Lecture 5: Superconducting NEM devices

Lecture 3: Mechanically Assisted Single Electronics Electric charge quantization in solids (important historical events) Self-assembled nanocomposites – new materials with special electronic and mechanical properties Nanoelectromechanical coupling due to tunneling of single electrons Shuttling of single electrons in NEM-SET devices Electromechanics of suspended carbon nanotubers (CNT) Outline

Millikan’s Oil-Drop Experiment (Nobel Prize in 1923) The electronic charge as a discrete quantity: In 1911, Robert Millikan of the University of Chicago published* the details of an experiment that proved beyond doubt that charge was carried by discrete positive and negative entities of equal magnitude, which he called electrons. The charge on the trapped droplet could be altered by briefly turning on the X-ray tube. When the charge changed, the forces on the droplet were no longer balanced and the droplet started to move. *Phys. Rev. 32, (1911) See also Lecture 3: Electronics and Mechanics on the Nanometer Scale 3/35

Electrically Controlled Single-Electron Charging Experiment: L.S.Kuzmin, K.K.Likharev, JETP Lett. 45, 495(1987): T.A.Fulton, C.J.Dolan, PRL, 59,109(1987); L.S.Kuzmin, P.Delsing, T.Claeson, K.K.Likharev, PRL,62,2539(1989 ); P.Delsing, K.K.Likharev, L.S.Kuzmin & T.Claeson, PRL, 63, 1861, (1989) Theory: R.Shekhter., Soviet Physics JETP 36, 747(1973); I.O.Kulik, R.Shekhter, Soviet Physics JETP 41, 308(1975); D.V.Averin, K.K.Likharev, J.Low Temp.Phys. 62, 345 (1986) Dot e 2 /C Lecture 3: Electronics and Mechanics on the Nanometer Scale 4/35

Gate Dot Electron Attraction to the gate Repulsion at the dot Cost At the energy cost vanishes ! Coulomb Blockade Single-electron transistor (SET) Lecture 3: Electronics and Mechanics on the Nanometer Scale 5/35

Single Molecular Transistors with OPV5 and Fullerenes Nature 425, 628 (2003 ) Lecture 3: Electronics and Mechanics on the Nanometer Scale 6/35

Self-Assembled Metal-Organic Composites Molecular manufacturing – a way to design materials on the nanometer scale Encapsulated 4nm Au particles self-assembled into a 2D array supported by a thin film, Anders et al., 1995 Scheme for molecular manufacturing Lecture 3: Electronics and Mechanics on the Nanometer Scale 7/35

Materials properties: Electrical – heteroconducting Mechanical – heteroelastic Electronic features: Quantum coherence Coulomb correlations Electromechanical coupling Basic Characteristics Lecture 3: Electronics and Mechanics on the Nanometer Scale 8/35

H. Park et al., Nature 407, 57 (2000) Quantum ”bell”Single C 60 Transistor A. Erbe et al., PRL 87, (2001); D. Scheible et al. NJP 4, 86.1 (2002) Here: Nanoelectromechanics caused by or associated with single-charge tunneling effects Nanoelectromechanical Devices Lecture 3: Electronics and Mechanics on the Nanometer Scale 9/35

CNT-Based Nanoelectromechanics A suspended CNT has mechanical degrees of freedom => study electromechanical effects on the nanoscale. V. Sazonova et al., Nature 431, 284 (2004) B. J. LeRoy et al., Nature 432, 371 (2004) Lecture 3: Electronics and Mechanics on the Nanometer Scale 10/35

Fullerene Empty CNT Lecture 3: Electronics and Mechanics on the Nanometer Scale 11/35

Shuttling of Single Electrons in NEM-SET Devices 12/35 Lecture 3: Electronics and Mechanics on the Nanometer Scale 12/35

Millikan’s Set-up on a Nanometer Length Scale If W exceeds the dissipated power an instability occurs Gorelik et al., PRL, 80, 4256(1998) Lecture 3: Electronics and Mechanics on the Nanometer Scale 13/35

The Electronic Shuttle Lecture 3: Electronics and Mechanics on the Nanometer Scale 14/35

Circuit Model for the Shuttle Electrostatic energy of the charged grain Electronic tunneling rate: Lecture 3: Electronics and Mechanics on the Nanometer Scale 15/35

Formulation of the Problem Lecture 3: Electronics and Mechanics on the Nanometer Scale 16/35

Nanoelectromechanical Instability Weak electromechanical coupling: Lecture 3: Electronics and Mechanics on the Nanometer Scale 17/35

Stable Shuttle Vibrations pumping dissipation W Lecture 3: Electronics and Mechanics on the Nanometer Scale 18/35

Different Scenarios for a Shuttle Instability Only one stable mechanical regime is possible Two regimes of locally stable mechanical operations ”SOFT” instability ”HARD” instability Lecture 3: Electronics and Mechanics on the Nanometer Scale 19/35

”Soft” Onset of Shuttle Vibrations Lecture 3: Electronics and Mechanics on the Nanometer Scale 20/35

”Hard” Onset of Shuttle Vibrations Lecture 3: Electronics and Mechanics on the Nanometer Scale 21/35

Shuttling of Electronic Charge Lecture 3: Electronics and Mechanics on the Nanometer Scale 22/35

Electromechanical Instabilities of Suspended CNTs L. M. Jonsson et al. Nano Lett. 5, 1165 (2005) Lecture 3: Electronics and Mechanics on the Nanometer Scale 23/35

Model The model system. An STM-tip, biased at –V/2 is placed above a suspended CNT with diameter D and length L. One end of the nanotube is connected to an electrode biased at V/2. A gate electrode with potential Vg is used to control the electronic levels on the nanotube. Only deformation of the tube in the plane is considered and the shape of the nanotube is given by u(z; t). The nanotube is considered to be a metallic island between the STM and the electrode. Tunnelling from the STM to the nanotube depends on tube deformation whereas the tunneling matrix elements connecting the nanotube and the electrode are constant. Lecture 3: Electronics and Mechanics on the Nanometer Scale 24/35

E-the Youngs modulus; I-moment of inertia of the cross section Boundary conditions: Formulation of the Problem: Mechanics of Suspended CNT Vibrational modes: Lecture 3: Electronics and Mechanics on the Nanometer Scale 25/35 Ansatz:

Normal Vibrations We include a finite number of eigenmodes (flexural vibrations) The frequencies of the modes are determined by The fundamental mode Mechanics is described by a collection of harmonic oscillators Frequencies of different modes are incommensurable! Lecture 3: Electronics and Mechanics on the Nanometer Scale 26/35

The electrostatic energy E c should be included in the Hamiltonian of the vibrating CNT. A deformation u(z,t) causes an extra charge Q on the tube. To linear order in Q the electrostatic energy Ec({u(z,t)},Q) is: In the limit of an STM size d much smaller than the length L of the CNT one can model K(z) as: (E - electric field in the vicinity of the STM tip) Formulation of the Problem: Electrostatics of Suspended CNT Lecture 3: Electronics and Mechanics on the Nanometer Scale 27/35

Equation for the mechanical vibrations: Coupling of electrons to a set of vibrators Vibrations in different modes are coupled γ- damping of the mechanical vibrations Nanoelectromechanics of Suspended CNT Modified circuit model for the charge dynamics Lecture 3: Electronics and Mechanics on the Nanometer Scale 28/35

Linearization with respect to X n (t) Weak electromechanical coupling: The dynamics of different vibration modes decouple Nanoelectromechanical Instability Lecture 3: Electronics and Mechanics on the Nanometer Scale 29/35

Optimal Conditions for NEM Instability Optimal conditions: Γ = ω 1 Symmetric tunneling rates Lecture 3: Electronics and Mechanics on the Nanometer Scale 30/35

Instabilities of Different Modes In the classical limit we find solutions on the form δ n > 0 implies an instability. Threshold dissipation for mode n: Different modes can be treated independently in the weak electromechanical limit! L. M.Jonsson et al., New J.Phys. 9, 90 (2007) Lecture 3: Electronics and Mechanics on the Nanometer Scale 31/35

Development of a Multimode Instability Classical approach Weak NEM coupling: Low frequency limit: - Each mode is described by a driven and damped h.o. - Kinetic equation for charge Lecture 3: Electronics and Mechanics on the Nanometer Scale 32/35

Nonlinear Multimode Dynamics Development of two unstable modes Ansatz+ Averaging Lecture 3: Electronics and Mechanics on the Nanometer Scale 33/35

If at t=0 the condition holds: then for t>0 monotonically increases approaching value: and other amplitudes decay exponentially when t →∞: Selective Pumping of N Vibrational Modes Evolution of the vibrations is described by vector: Only N points in the vector space are attracting focuses: THEOREM Lecture 3: Electronics and Mechanics on the Nanometer Scale 34/35

Self-organization of Shuttle Vibrations with Three Unstable Modes Lecture 3: Electronics and Mechanics on the Nanometer Scale 35/35