Five-Lecture Course on the Basic Physics of Nanoelectromechanical 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 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
• interferrence of electrons

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Lecture 4 Quantum Nano-Electromechanics
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Quantum Coherence of Electrons
Classical approach
Heisenberg principle in quantum approach
Formalization of Heisenbergs principle
operators for physical variables
eigenfunctions quantum states
quantum state with definite
momentum In this
state the momentum experiences
quantum fluctuations
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Lecture 4 Quantum Nano-Electromechanics
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Stationary Quantum States
Hamiltonian of a single electron
Stationary quantum states
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Lecture 4 Quantum Nano-Electromechanics
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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
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Lecture 4 Quantum Nano-Electromechanics
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Field Operators
Density Matrix
Louville von Neumann equation
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Zero-Point Oscillations
Lecture 4 Quantum Nano-Electromechanics
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Consider a classical particle which oscillates in
a quadratic potential well. Its equilibrium
position, X0, corresponds to the potential
minimum EminU(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

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Lecture 4 Quantum Nano-Electromechanics
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Quantum Nanoelectromechanics of Shuttle Systems
If then
quantum fluctuations of the grain significantly
affect nanoelectromechanics.
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Lecture 4 Quantum Nano-Electromechanics
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Conditions for Quantum Shuttling

1. Fullerene based SET
2. Suspended CNT

Quasiclassical shuttle vibrations.
STM
L
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Quantum Harmonic Oscillator
Ledder operators
Probability densities ?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.
Eigenstate of oscillator is a ideal gas of
elementary excitations vibrons, which are a
bose particles.
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Lecture 4 Quantum Nano-Electromechanics
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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.
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Hamiltonian
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Theory of Quantum Shuttle
x
The Hamiltonian
Dot
Lead
Lead
Time evolution in Schrödinger picture
Total density operator
Reduced density operator
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Lecture 4 Quantum Nano-Electromechanics
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Generalized Master Equation
density matrix operator of the uncharged shuttle
density matrix operator of the charged shuttle
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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
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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.
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Lecture 4 Quantum Nano-Electromechanics
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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).
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Aharonov-Bohm Effect
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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
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Lecture 4 Quantum Nano-Electromechanics
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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.
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Quantum Nanomechanical Interferometer
Classical interferometer Quantum
nanomechanical interferometer
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Electronic Propagation Through Swinging Polaronic
States
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Model
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Coupling to the Fundamental Bending Mode
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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.
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Tunneling through Virtual Electronic States on
CNT
Lecture 4 Quantum Nano-Electromechanics
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• 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

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Calculation of the Electrical Current
Lecture 4 Quantum Nano-Electromechanics
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Lecture 4 Quantum Nano-Electromechanics
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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.

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Linear Conductance
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Vibrational system is in equilibrium Fo
r 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-current of
0.1-0.3 pA.
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Quantum Nanomechanical Magnetoresistor
Temperature
Magnetic field
R.I. Shekhter et al., PRL 97, 156801 (2006)
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Magnetic Field Dependent Offset Current
Lecture 4 Quantum Nano-Electromechanics
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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
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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
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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
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Lecture 4 Quantum Nano-electromechanics
II. Backflow Current
Backscattering potential
Speaker Professor Robert Shekhter, Gothenburg
University 2009
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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
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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
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Excess Current
Lecture 4 Quantum Nano-electromechanics

• ah?4eFm/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
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Model
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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 .
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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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4-5
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4-6
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Lecture 4 Quantum Nano-electromechanics
Speaker Professor Robert Shekhter, Gothenburg
University 2009
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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
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Magnetic Field Dependent Tunneling
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• In order to proceed it is convenient to make the
  unitary transformation