Five-Lecture Course on the Basic Physics of Nanoelectromechanical Devices

1. 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

2. Lecture 4: Quantum Nano-Electromechanics Outline • Quantum coherence of electrons • Quantum coherence of mechanical displacements • Quantum nanomechanical operations: • a) shuttling of single electrons; • b) mechanically induced quantum • interferrenceof electrons

3. Lecture 4: Quantum Nano-Electromechanics 3/29 Quantum Coherence of Electrons Classical approach Heisenberg principle in quantum approach 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

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

5. Lecture 4: Quantum Nano-Electromechanics 5/29 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

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

7. Zero-Point Oscillations Lecture 4: Quantum Nano-Electromechanics 7/29 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: Amplitude of zero-point oscillations: Classical vs quantum description: the choice is determined by the parameter where d is a typical length scale for the problem. “Quantum” when

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

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

10. Lecture 4: Quantum Nano-Electromechanics 10/29 Quantum Harmonic Oscillator Ledder operators Probability densities |ψn(x)|2for 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. Eigenstate of oscillator is a ideal gas of elementary excitations – vibrons, which are a bose particles.

11. Lecture 4: Quantum Nano-Electromechanics 11/29 Quantum Shuttle Instability Quantum vibrations, generated by tunneling electrons, remain undamped and accumulate in a coherent “condensate” of phonons, which is classical shuttle oscillations. e eV Phase space trajectory of shuttling. From Ref. (3) References: (1) D. Fedorets et al. Phys. Rev. Lett. 92, 166801 (2004) (2) D. Fedorets, Phys. Rev. B 68, 033106 (2003) (3) T. Novotny et al. Phys. Rev. Lett. 90 256801 (2003) Shift in oscillator position caused by charging it by a single electron charge.

12. Hamiltonian Lecture 4: Quantum Nano-Electromechanics 12/29

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

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

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

16. Lecture 4: Quantum Nano-Electromechanics 16/29 Quantum Shuttle Instability Importantconclusion: An electromechanicalinstability is possibleevenif the initial displacement of the shuttle is smallerthanits de Brogliewavelength and quantum fluctuations of the shuttle position can not be neglected. In this situation onespeaks of a quantum shuttleinstability. Nowonce an instabilityoccurs, howcanone distinguish between quantum shuttle vibrations and classical shuttle vibrations? Moregenerally: Howcanonedetect quantum mechanical displacements? This question is important since the detectionsensitivity of modern devices is approaching the quantum limit, where classical nanoelectromechanics is not valid and the principle for detection should be changed.

17. Lecture 4: Quantum Nano-Electromechanics 17/29 How to Detect Nanomechanical Vibrations in the Quantum Limit? Try new principles for sensing the quantum displacements! 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.156801 (2006).

18. Aharonov-Bohm Effect Lecture 4: Quantum Nano-Electromechanics 18/29 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:

19. Lecture 4: Quantum Nano-Electromechanics 19/29 Classical and Quantum Vibrations 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.

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

21. Electronic Propagation Through Swinging Polaronic States

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

23. Coupling to the Fundamental Bending Mode Lecture 4: Quantum Nano-Electromechanics 23/29 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.

24. Tunneling through Virtual Electronic States on CNT Lecture 4: Quantum Nano-Electromechanics 24/29 • 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

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

26. Lecture 4: Quantum Nano-Electromechanics 26/29 Vibron-Assisted Tunneling through Suspended Nanowire • Tunneling through vibrating nanowire is performed in both elasticand 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 independentof both temperature and bias voltage.

27. Linear Conductance Lecture 4: Quantum Nano-Electromechanics 27/29 Vibrational system is in equilibrium For a 1 μm long SWNT at T = 30 mK and H ≈ 20 - 40 T a relative conductance change is of about 1-3%, which corresponds to a magneto-currentof 0.1-0.3 pA.

28. Lecture 4: Quantum Nano-Electromechanics 28/29 Quantum Nanomechanical Magnetoresistor Temperature Magnetic field R.I. Shekhter et al., PRL97, 156801 (2006)

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

31. Quantum Suppression of Electronic Backscattering in a 1-D Channel Lecture 4: Quantum Nano-electromechanics G.Sonne, L.Y.Gorelik, R.I.Shekhter, M.Jonson, Europhys.Lett. (2008), (in press); Speaker: Professor Robert Shekhter, Gothenburg University 2009 31/41

32. CNT with an Enclosed Fullerene Lecture 4: Quantum Nano-electromechanics Shallow harmonic potential for the fullerene position vibrations along CNT Strong quantum fluctuations of the fullerene position, comparable to Fermi wave vector of electrons. Speaker: Professor Robert Shekhter, Gothenburg University 2009 32/41 32

33. py Lecture 4: Quantum Nano-electromechanics Electronic Transport through 1-D Channel • Weak interaction with the impurity • Only a small fraction of electrons scatter back. • Elastic and inelastic backscattering channels y • No backscattering • Left and right side reservoirs inject electrons with energy distribution according to Hibbs with different chemical potentials Speaker: Professor Robert Shekhter, Gothenburg University 2009 33/41 33

34. Lecture 4: Quantum Nano-electromechanics II. Backflow Current Backscattering potential Speaker: Professor Robert Shekhter, Gothenburg University 2009 34/41 34

35. Lecture 4: Quantum Nano-electromechanics Backscattering of Electrons due to the Presence of Fullerene 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 . Speaker: Professor Robert Shekhter, Gothenburg University 2009 35/41

36. Lecture 4: Quantum Nano-electromechanics Pauli Restrictions on Allowed Transitions through Oscillator Bias potential across tube selects allowed transitions through oscillator as fermionic nature of electrons has to be considered. × Speaker: Professor Robert Shekhter, Gothenburg University 2009 36/41

37. Excess Current Lecture 4: Quantum Nano-electromechanics αħω=4εFm/M, excess current independent of confining potential, voltage and temperature. Scales as ratio of masses in the system. Speaker: Professor Robert Shekhter, Gothenburg University 2009 37/41 37

38. Model

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 .

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

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42. 4-6

43. Lecture 4: Quantum Nano-electromechanics Speaker: Professor Robert Shekhter, Gothenburg University 2009 43/41

44. Lecture 4: Quantum Nano-electromechanics 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. Speaker: Professor Robert Shekhter, Gothenburg University 2009 44/41 44

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