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Five-Lecture Course on the Basic Physics of Nanoelectromechanical Devices

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

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Five-Lecture Course on the Basic Physics of Nanoelectromechanical Devices

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  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 5: Superconducting NEM Devices Outline • Superconductivity – Basic facts • Nanomechanicallyasisted superconductivity Lecture 4 was about quantum nano-electromechanics, where the quantum effects appeared because of the small size of the movable part. In this lecture we explore how macroscopic quantumcoherence in the source and drain leads (being superconducting), would affect the nano-electromechanics. In the first half of this lecture we will review some basic facts of superconductivity, while in the second half we will consider the nano-electromechanics of superconducting devices.

  3. Lecture 5: Superconducting NEM devices Superconductivity: Basic Experimental Facts • Zero electrical resistance (1911) • Ideal diamagnetism (Meissner effect, 1933) Heike Kammerlingh-Onnes (1853 - 1926) Walther Meissner (1882-1974) A magnet levitating above a high-temperature superconductor, cooled with liquid nitrogen. A persistent electric current flows on the surface of the superconductor, acting to exclude the magnetic field of the magnet (the Meissner effect).

  4. Lecture 5: Superconducting NEM devices Part 1Mesoscopic Superconductivity (Basic facts)

  5. Ground State and Elementary Excitations in Normal Metals Lecture 5: Superconducting NEM devices 5/31 Elementary excitation pF Ground state wave function Energy of an elementary excitation Fermi sphere in momentumspace Enrico Fermi, 1901 - 1954

  6. Cooper Instability Lecture 5: Superconducting NEM devices 6/31 The Fermi ground state of free electrons becomesunstable if an – even infinitesimally weak – attractive interaction between the electrons is switched on. This is the Cooper Instability (1955). The result is a radical rearrangment of the ground state. a) The interaction between an electron and the latticeions attracts the ions to the electron. b) The resultinglattice deformation relaxesslowly and leaves a cloud of uncompensated positive charge c) This cloudattracts, in itsturn, anotherelectronleading to an indirectattractionbetweenelectrons. In somemetals this phonon-mediatedattrcationcanovercome the repulsive Coulomb interaction between the electrons Phonon mediated electron-electron attraction a) b) c) Leon Cooper, b.1930 Herbert Fröhlich, 1905 - 1991

  7. Superconducting Ground State Lecture 5: Superconducting NEM devices 7/31 Normal metal • In the ground state of a normal metal, each • single-electron state p is occupied if p<pF . • - In the BCS state, on the other hand, the occupation • of all states p have quantum fluctuations and vp is • the probability amplitude for state p to be occupied. • In addition the occupation of state p is coupled to • the occupation of state –p, so that the single- • electron states fluctuate in pairs (p;s,-p;-s) • One says that the BCS state forms a condensate • of pairs of electrons, so called Cooper pairs. Superconductor The complexparameterDeif, whichcontrols the quantum fluctuationsin the occupation of pairedstates, determines a new symmetryachieved by the formation of the super- conductinggroundstate. It is called the superconducting order parameter and has to be calculated self consistently using the condition that the ground state energy is minimized. This leads to the self consistency equation above (last equation on this slide).

  8. Quantum Fluctuations of Cooper-Pair Number Lecture 5: Superconducting NEM devices 8/31 Ground state with a given number (n) of Cooper pairs Cooper-pair number operator ( ) It can be proven that: It follows that and consequently: from which the uncertainty relation follows. Note the analogy with the momentum p and coordinate x of a quantum particle. Quantum fluctuations of the superconducting phase foccur if fluctuations of the pair number is restricted. This is the case for small samples where the Coulomb blockade phenomenon occurs.

  9. Superconducting Current Flow Lecture 5: Superconducting NEM devices 9/31 In contrast to nonsuperconducting materials where the flow of an electrical current is a nonequilibrium phenomenon, in superconductors an electrical current is a ground state property. A supercurrent flows if the superconducting phase is spatially inhomogeneous. Its density is defined as: How to arrange for a spatially nonhomogeneous supeconducting phase? One way is just to inject current into a homogeneous sample. Another way is to switch on an external magnetic field. H S f1 f2

  10. Quasiparticle Excitations in a Superconductor Lecture 5: Superconducting NEM devices 10/31 • We have discussed ground state properties of a superconductor. What about its excited states? These may contribute at finite temperatures or when the superconductor is exposed to external time dependent fields. Similary to a normal metal, low energy excited states of a superconductor can be represented as a gas of non-interacting quasiparticles. The energy spectrum for a homogeneous superconductor takes the form • Important features: • The spectrum of the elementary excitations has a gap which is given by the superconducting order parameter D. This is why the number of quasiparticles nF(ep) is exponentially small at low temperatures T<<D. • It is important that at such low temperatures a superconductor can be considered to be a single large quantum particle or molecule which is characterized by a single (BCS) wave function. • A huge amount of electrons is incorporated into a single quantum state. This is not possible for normal electrons due to the Pauli principle. It is the formation of Cooper pairs by the electrons that makes it possible.

  11. Lecture 5: Superconducting NEM devices 11/31 Parity Effect The BCS ground state is a superposition of states with different integer numbers of Cooper pairs. It does not contain contributions from states with an odd number of electrons. What happens if we force one more electron into a superconductor?The BCS state would not be the ground state of such a system. What will it be? The only option is to put the extra electron into a quasiparticle state. Then the ground state would correspond to the lowest-energy quasiparticle state being occupied (see figure). Now the ground state energy depends on the parity of the electron number N (parity effect). Note that the BCS ground state energy does not depend on the superconducting phase f. Next we will see that quantum tunneling of Cooper pairs will remove this degeneracy.

  12. Josephson Effect Lecture 5: Superconducting NEM devices 12/31 Mesoscopic effects in normal metals are due to phase coherent electron transport, i.e. the phase coherence of electrons is preserved during their propagation through the sample. Is it possible to have similar mesoscopic effects for the propagation of Cooper pairs? To be more precise: What would be the effect if Cooper pairs are injected into a normal metal and are able to preserve their phase coherence? One possibility is to let Cooper pairs travel from one superconductor to another through a non-superconducting region. This situation was first considered by Brian Josephson, who in 1961 showed that it would lead to a supercurrent flowing through the non-superconducting region (Josephson effect, Nobel Prize in 1973). This was the beginning of the era of macroscopic quantum coherent phenomena in solid state physics.

  13. Josephson Coupling Lecture 5: Superconducting NEM devices 13/31 ”Black box” • There is a small but finite probability for a phase coherent transfer of electrons between the two superconductors. • Temperature is much smaller than D so quasiparticles can be neglected. Therefore only Cooper pairs can transfer charge. • Due to the Heisenberg uncertainty principle spatial delocalization of quantum particles reduces quantum fluctuations of their momentum and hence lower theirkineticenergy. Similarly, letting Cooper pairs be spread over two superconductors lower their energy. • This lowering depends on D and the barriertransparency (through the conductanceG) and can be viewed as a coupling energy caused by Cooper pair transfer. BCS; f1 BCS; f2 EJ: Josephson coupling energy Ic: Josephson critical current

  14. Charging Effects in Small Superconductors Lecture 5: Superconducting NEM devices 14/31 Another situation where the degeneracy of the ground state energy of a superconductor with respect to the superconducting phase f occurs in small superconductors, where charging effects (Coulomb blockade) are important. Still ignoring the elementary excitations in the superconductor we express the charging energy operator as This operator is nondiagonal in the space of BCS wave functions with different phases. This leads to quantum fluctuations of the phase f whose dynamics is governed by the Hamiltonian .

  15. Lecture 5: Superconducting NEM devices 15/31 Lifting of the Coulomb Blockade of Cooper Pair Tunneling Single-Cooper-Pair Hybrid

  16. Single-Cooper-Pair Transistor Lecture 5: Superconducting NEM devices 16/31 The device in the picture incorporates all the elements we have considered: tunnel barriers between the central island and the leads form two Josephson junctions, while the small dot is affected by Coulomb-blockade dynamics. The Hamiltonian which includes all these elements is expressed in terms of the given superconducting phases in the leads, f1, f2, and the island-phase operator f: The lowest-energy eigenvalue of this Hamiltonian gives the coupling energy E(f=f1-f2) due to the flow of Cooper pairs through the Coulomb-blockade island. The Josephson current is the given as

  17. Lecture 5: Superconducting NEM devices 17/31 Part 2Nanomechanically assisted superconductivity

  18. How Does Mechanics Contribute to Tunneling of Cooper Pairs? Lecture 5: Superconducting NEM devices 18/31 Is it possible to maintain a mechanically-assisted supercurrent? L.Gorelik et al. Nature 2001; A. Isacsson et al. PRL 89, 277002 (2002)

  19. Lecture 5: Superconducting NEM devices 19/31 How to Avoid Decoherence? • To preserve phase coherence only few degrees of freedom must be involved. • This can be achieved or provided: • No quasiparticles are produced • Large fluctuations of the charge are suppressed by the Coulomb blockade:

  20. Lecture 5: Superconducting NEM devices 20/31 Coulomb Blockade of Cooper Pair Tunneling Single-Cooper-Pair Hybrid

  21. Lecture 5: Superconducting NEM devices 21/31 Single Cooper Pair Box Coherent superposition of two succeeding charge states can be created by choosing a proper gate voltage which lifts the Coulomb Blockade. Nakamura et al., Nature 1999

  22. Movable Single-Cooper-Pair Box Lecture 5: Superconducting NEM devices 22/31 Josephson hybridization is produced at the trajectory turning points since near these points the CB is lifted by the gates.

  23. Lecture 5: Superconducting NEM devices 23/31 How Does It Work?

  24. Lecture 5: Superconducting NEM devices 24/31 Possible Setup Configurations Supercurrent between the leads kept at a fixed phase difference. Coherence between isolated remote leads created by a single Cooper pair shuttling.

  25. Lecture 5: Superconducting NEM devices 25/31 Shuttling Between Coupled Superconductors Relaxation suppresses the memory of initial conditions.

  26. Lecture 5: Superconducting NEM devices 26/31 Resulting Expression for the Current

  27. Lecture 5: Superconducting NEM devices 27/31 Average Current in Units I0=2ef as a Function of Electrostatic, , and Supercondudting, , Phases Black regions – no current. The current direction is indicated by signs.

  28. Lecture 5: Superconducting NEM devices 28/31 Shuttling of Cooper Pairs

  29. Lecture 5: Superconducting NEM devices 29/31 Mechanically Assisted Superconducting Coupling

  30. Lecture 5: Superconducting NEM devices 30/31 Distribution of Phase Differences as a Function of Number of Rotations. Suppression of Quantum Fluctuations of Phase Difference

  31. Lecture 5: Superconducting NEM devices 31/31 General conclusion from the course: Mesoscopic effects in the electronic subsystem and quantum coherent dynamics of the mechanical displacements qualitatively modify the NEM operating principles, bringing new functionality determined by quantum mechanical phases and the discrete charge of the electron.

  32. MesoscopicNanoelectromechanics Single electrons Gorelik et al., PRL, 80, 4256 (1998) Single spins Fedorets et al., PRL, 95, 057203 (2005) Nanopolarons Shekhter et al.,PRL, 97, 156801 (2006) Cooper pairs Gorelik et al., Nature, 411, 454 (2001)

  33. Sensing of a Quantum Displacements Lecture 5: Superconducting NEM devices Resolution of the mechanical displacements is limited by a quantum uncertainty principle causing a quantum zero point fluctuations of the nanomechanical subsystem. The amplitude of such vibrations is For a micron size mechanical beam this amplitude is estimated as 0.001 Å. This quantity sets a certain scale for a modern NEM devices. Possibility to achieve such a limit in sensitivity would offer a new nanomechanical operations where quantum nanomechanics and quantum coherence would dominate device performance. How far are we from achieving a quantum limit of NEM operations? I am going to illustrate how the mesoscopic NEM offer the way to approach this limit. Speaker: Professor Robert Shekhter, Gothenburg University 2009

  34. Coulomb Blockade Electrometry Lecture 5: Superconducting NEM devices Charge Quantization (experiment) :two cases :superconducting and normal garain NEM-SET Device Picture This is about a quantum limit Displacement detection Displacement sensitivity M.Devoret et al, Nature 406, 1039 (2000) Speaker: Professor Robert Shekhter, Gothenburg University 2009

  35. Rf-SET-Based Displacement Detection Lecture 5: Superconducting NEM devices Picture : Fig.14 and describtion of the idea of experiment on page 185 From M.Blencowe, Phys.Rept., 395, 159 (2004) Speaker: Professor Robert Shekhter, Gothenburg University 2009

  36. Lecture 5: Superconducting NEM devices Quantum limit of detection using an RF-SET Speaker: Professor Robert Shekhter, Gothenburg University 2009

  37. Sensitivity of the Displacement Measurement (experiment) Lecture 5: Superconducting NEM devices Fif.19 on page 195 of Blencowe Phys.Repts. Speaker: Professor Robert Shekhter, Gothenburg University 2009

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