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 4: Quantum Nano- Electromechanics Quantum coherence of electrons Quantum coherence of mechanical displacements Quantum nanomechanical operations: a) shuttling of single electrons; b) mechanically induced quantum interferrence of electrons Outline

Formalization of Heisenberg’s principle: operators for physical variables eigenfunctions – quantum states quantum state with definite momentum In this state the momentum experiences quantum fluctuations Quantum Coherence of Electrons Classical approachHeisenberg principle in quantum approach Lecture 4: Quantum Nano-Electromechanics 3/29

Stationary Quantum States Hamiltonian of a single electron: Stationary quantum states: Lecture 4: Quantum Nano-Electromechanics 4/29

Second Quantization Spatial quantization  discrete quantum numbers Due to quantum tunneling the number of electrons in the body experiences quantum fluctuations and is not an integer One therefore needs a description that treats the particle number N as a quantum variable Wave function for system of N electrons: Creation and annihilation operators fermions bosons Lecture 4: Quantum Nano-Electromechanics 5/29

Field Operators Lecture 4: Quantum Nano-Electromechanics 6/29 Density Matrix Louville – von Neumann equation

Zero-Point Oscillations Consider a classical particle which oscillates in a quadratic potential well. Its equilibrium position, X=0, corresponds to the potential minimum E=min{U(x)}. A quantum particle can not be localized in space. Some “residual oscillations" are left even in the ground states. Such oscillations are called zero point oscillations. Classical motion: Quantum motion: Classical vs quantum description: the choice is determined by the parameter where d is a typical length scale for the problem. “Quantum” when Lecture 4: Quantum Nano-Electromechanics 7/29 Amplitude of zero-point oscillations:

Quantum Nanoelectromechanics of Shuttle Systems If then quantum fluctuations of the grain significantly affect nanoelectromechanics. Lecture 4: Quantum Nano-Electromechanics 8/29

Conditions for Quantum Shuttling 1. Fullerene based SET 2. Suspended CNT Quasiclassical shuttle vibrations. STM L 9 9 Lecture 4: Quantum Nano-Electromechanics 9/29

Quantum Harmonic Oscillator Probability densities |ψ n (x)| 2 for the boundeigenstates, beginning with the ground state (n = 0) at the bottom and increasing in energy toward the top. The horizontal axis shows the position x,and brighter colors represent higher probability densities. Ledder operators Eigenstate of oscillator is a ideal gas of elementary excitations – vibrons, which are a bose particles. Lecture 4: Quantum Nano-Electromechanics 10/29

Quantum Shuttle Instability Quantum vibrations, generated by tunneling electrons, remain undamped and accumulate in a coherent “condensate” of phonons, which is classical shuttle oscillations. Shift in oscillator position caused by charging it by a single electron charge. References: (1) D. Fedorets et al. Phys. Rev. Lett. 92, (2004) (2) D. Fedorets, Phys. Rev. B 68, (2003) (3) T. Novotny et al. Phys. Rev. Lett (2003) Phase space trajectory of shuttling. From Ref. (3) eV e Lecture 4: Quantum Nano-Electromechanics 11/29

Hamiltonian Lecture 4: Quantum Nano-Electromechanics 12/29

The Hamiltonian: Time evolution in Schrödinger picture : Theory of Quantum Shuttle Reduced density operator Total density operator Dot x Lead Lecture 4: Quantum Nano-Electromechanics 13/29

Generalized Master Equation density matrix operator of the uncharged shuttle density matrix operator of the charged shuttle Free oscillator dynamicsDissipationElectron tunnelling Approximation : At large voltages the equations for are local in time: Lecture 4: Quantum Nano-Electromechanics 14/29

After linearisation in x (using the small parameter x o /l) one finds: Result: an initial deviation from the equilibrium position grows exponentially if the dissipation is small enough: Shuttle Instability Lecture 4: Quantum Nano-Electromechanics 15/29

Quantum Shuttle Instability Important conclusion: An electromechanical instability is possible even if the initial displacement of the shuttle is smaller than its de Broglie wavelength and quantum fluctuations of the shuttle position can not be neglected. In this situation one speaks of a quantum shuttle instability. Now once an instability occurs, how can one distinguish between quantum shuttle vibrations and classical shuttle vibrations? More generally: How can one detect quantum mechanical displacements? This question is important since the detection sensitivity of modern devices is approaching the quantum limit, where classical nanoelectromechanics is not valid and the principle for detection should be changed. Lecture 4: Quantum Nano-Electromechanics 16/29

Try new principles for sensing the quantum displacements! How to Detect Nanomechanical Vibrations in the Quantum Limit? Consider the transport of electrons through a suspended, vibrating carbon nanotube beam in a transverse magnetic field H. What will the effect of H be on the conductance? Shekhter R.I. et al. PRL 97(15): Art.No (2006). Lecture 4: Quantum Nano-Electromechanics 17/29

Aharonov-Bohm Effect The particle wave, incidenting the device from the left splits at the left end of the device. In accordance with the superposition principle the wave function at the right end will be given by: The probability for the particle transition through the device is given by: Lecture 4: Quantum Nano-Electromechanics 18/29

In the classical regime the SWNT fluctuations u(x,t) follow well defined trajectories. In the quantum regime the SWNT zero-point fluctuations (not drawn to scale) smear out the position of the tube. Classical and Quantum Vibrations Lecture 4: Quantum Nano-Electromechanics 19/29

Quantum Nanomechanical Interferometer Classical interferometer Quantum nanomechanical interferometer Lecture 4: Quantum Nano-Electromechanics 20/29

Electronic Propagation Through Swinging Polaronic States 21

Model 22 Lecture 4: Quantum Nano-Electromechanics 22/29

Coupling to the Fundamental Bending Mode Only one vibration mode is taken into account CNT is considered as a complex scatterer for electrons tunneling from one metallic lead to the other. Lecture 4: Quantum Nano-Electromechanics 23/29

Tunneling through Virtual Electronic States on CNT Strong longitudinal quantization of electrons on CNT No resonance tunneling though the quantized levels (only virtual localization of electrons on CNT is possible) Effective Hamiltonian Lecture 4: Quantum Nano-Electromechanics 24/29

Calculation of the Electrical Current Lecture 4: Quantum Nano-Electromechanics 25/29

Vibron-Assisted Tunneling through Suspended Nanowire Tunneling through vibrating nanowire is performed in both elastic and inelastic channels. Due to Pauli-principle, some of the inelastic channels are excluded. Resonant tunneling at small energies of electrons is reduced. Current reduction becomes independent of both temperature and bias voltage. Lecture 4: Quantum Nano-Electromechanics 26/29

Vibrational system is in equilibrium For a 1 μm long SWNT at T = 30 mK and H ≈ T a relative conductance change is of about 1-3%, which corresponds to a magneto- current of pA. Linear Conductance Lecture 4: Quantum Nano-Electromechanics 27/29

Quantum Nanomechanical Magnetoresistor Lecture 4: Quantum Nano-Electromechanics 28/29 Temperature Magnetic field R.I. Shekhter et al., PRL 97, (2006)

Magnetic Field Dependent Offset Current Lecture 4: Quantum Nano-Electromechanics 29/29

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Quantum Suppression of Electronic Backscattering in a 1-D Channel G.Sonne, L.Y.Gorelik, R.I.Shekhter, M.Jonson, Europhys.Lett. (2008), (in press); 31 Lecture 4: Quantum Nano-electromechanics Speaker: Professor Robert Shekhter, Gothenburg University /41

CNT with an Enclosed Fullerene Shallow harmonic potential for the fullerene position vibrations along CNT Strong quantum fluctuations of the fullerene position, comparable to Fermi wave vector of electrons. 32 Lecture 4: Quantum Nano-electromechanics Speaker: Professor Robert Shekhter, Gothenburg University /41

Electronic Transport through 1-D Channel y 33 Weak interaction with the impurity Only a small fraction of electrons scatter back. Elastic and inelastic backscattering channels I.No backscattering Left and right side reservoirs inject electrons with energy distribution according to Hibbs with different chemical potentials pypy 33 Lecture 4: Quantum Nano-electromechanics Speaker: Professor Robert Shekhter, Gothenburg University /41

II. Backflow Current 34 Backscattering potential 34 Lecture 4: Quantum Nano-electromechanics Speaker: Professor Robert Shekhter, Gothenburg University /41

The probability of backscattering sums up all backscattering channels. The result yields classical formula for non-movable target. However the sum rule does not apply as Pauli principle puts restrictions on allowed transitions. Backscattering of Electrons due to the Presence of Fullerene 35 Lecture 4: Quantum Nano-electromechanics Speaker: Professor Robert Shekhter, Gothenburg University /41

× Bias potential across tube selects allowed transitions through oscillator as fermionic nature of electrons has to be considered. Pauli Restrictions on Allowed Transitions through Oscillator 36 Lecture 4: Quantum Nano-electromechanics Speaker: Professor Robert Shekhter, Gothenburg University /41

Excess Current αħω=4ε F m/M, excess current independent of confining potential, voltage and temperature. Scales as ratio of masses in the system. 37 Lecture 4: Quantum Nano-electromechanics Speaker: Professor Robert Shekhter, Gothenburg University /41

Model 38

39 Backscattering of Electrons on Vibrating Nanowire. The probability of backscattering sums up all backscattering channels. The result yields classical formula for non-movable target. However the sum rule does not apply as Pauli principle puts restrictions on allowed transitions.

× The applied bias voltage selects the allowed inelastic transitions through vibrating nanowire as fermionic nature of electrons has to be considered. Pauli Restrictions on Allowed Transitions Through Vibrating Nanowire 40

4-5 41

4-6 42

43 Lecture 4: Quantum Nano-electromechanics Speaker: Professor Robert Shekhter, Gothenburg University /41

44 General conclusion from the course: Mesoscopic effects in electronic system and quantum dynamics of mechanical displacements qualitatively modify principles of NEM operations bringing new functionality which is now determined by quantum mechnaical phase and discrete charges of electrons. 44 Lecture 4: Quantum Nano-electromechanics Speaker: Professor Robert Shekhter, Gothenburg University /41

Magnetic Field Dependent Tunneling In order to proceed it is convenient to make the unitary transformation Lecture 4: Quantum Nano-Electromechanics 45/29