Five Lecture Course on Basic Physics of Nanoelectroomechanical Devices

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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

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Lecture 5 Superconducting NEM Devices
Outline

• Superconductivity Basic facts
• Nanomechanically asisted 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 quantum
coherence 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.
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Lecture 5 Superconducting NEM devices
Superconductivity Basic Experimental Facts

1. Zero electrical resistance (1911)
2. 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).
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Lecture 5 Superconducting NEM devices
Part 1Mesoscopic Superconductivity (Basic facts)
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Ground State and Elementary Excitations in
Normal Metals
Lecture 5 Superconducting NEM devices
5/31
Ground state wave function
Energy of an elementary excitation
Enrico Fermi, 1901 - 1954
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Cooper Instability
Lecture 5 Superconducting NEM devices
6/31
The Fermi ground state of free electrons becomes
unstable 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 lattice
ions attracts the ions to the electron. b)
The resulting lattice deformation relaxes slowly
and leaves a cloud of uncompensated
positive charge c) This cloud attracts, in its
turn, another electron leading to an indirect
attraction between electrons. In some metals
this phonon-mediated attrcation can overcome the
repulsive Coulomb interaction between the
electrons
Leon Cooper, b.1930
Herbert Fröhlich, 1905 - 1991
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Superconducting Ground State
Lecture 5 Superconducting NEM devices
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Normal metal

• In the ground state of a normal metal, each
• single-electron state p is occupied if pltpF .
• - 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 (ps,-p-s)
• One says that the BCS state forms a condensate
• of pairs of electrons, so called Cooper pairs.

Superconductor
The complex parameter Deif, which controls the
quantum fluctuations in the occupation of paired
states, determines a new symmetry achieved by
the formation of the super- conducting ground
state. 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).
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Quantum Fluctuations of Cooper-Pair Number
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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
f occur if fluctuations of the pair number is
restricted. This is the case for small samples
where the Coulomb blockade phenomenon occurs.
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Superconducting Current Flow
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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.
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Quasiparticle Excitations in a Superconductor
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• 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 TltltD.
• 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.

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Lecture 5 Superconducting NEM devices
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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.
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Josephson Effect
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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.
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Josephson Coupling
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• 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 their kinetic energy. Similarly,
  letting Cooper pairs be spread over two
  superconductors lower their energy.
• This lowering depends on D and the barrier
  transparency (through the conductance G) and can
  be viewed as a coupling energy caused by Cooper
  pair transfer.

EJ Josephson coupling energy Ic Josephson
critical current
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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
.
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Lecture 5 Superconducting NEM devices
15/31
Lifting of the Coulomb Blockade of Cooper Pair
Tunneling
Single-Cooper-Pair Hybrid
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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(ff1-f2) due to the
flow of Cooper pairs through the Coulomb-blockade
island. The Josephson current is the given as
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Lecture 5 Superconducting NEM devices
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Part 2Nanomechanically assisted superconductivity
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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)
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Lecture 5 Superconducting NEM devices
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How to Avoid Decoherence?
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Lecture 5 Superconducting NEM devices
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Coulomb Blockade of Cooper Pair Tunneling
Single-Cooper-Pair Hybrid
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Lecture 5 Superconducting NEM devices
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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
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Movable Single-Cooper-Pair Box
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Josephson hybridization is produced at the
trajectory turning points since near these points
the CB is lifted by the gates.
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Lecture 5 Superconducting NEM devices
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How Does It Work?
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Lecture 5 Superconducting NEM devices
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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.
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Lecture 5 Superconducting NEM devices
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Shuttling Between Coupled Superconductors
Relaxation suppresses the memory of initial
conditions.
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Lecture 5 Superconducting NEM devices
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Resulting Expression for the Current
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Lecture 5 Superconducting NEM devices
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Average Current in Units I02ef as a Function of
Electrostatic, ?, and Supercondudting, ?, Phases
Black regions no current. The current direction
is indicated by signs.
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Lecture 5 Superconducting NEM devices
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Shuttling of Cooper Pairs
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Mechanically Assisted Superconducting Coupling
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Distribution of Phase Differences as a Function
of Number of Rotations. Suppression of Quantum
Fluctuations of Phase Difference
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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.
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Mesoscopic Nanoelectromechanics
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)
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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
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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
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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
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Lecture 5 Superconducting NEM devices
Quantum limit of detection using an RF-SET
Speaker Professor Robert Shekhter, Gothenburg
University 2009
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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