A'M'Dykhne, RSC TRINITI, MPTI

1
Broken Symmetry and Coherence of Molecular
Vibrations in tunnel transitions

• A.M.Dykhne, RSC TRINITI, MPTI
• A.G.Rudavets, MPTI
•  

2
Plan

• Tunneling Optical Trap
• TOT schematics
• Events History
• TOT Potentials
• Flying Up and Down on substrate
• TOT - element of atomic Microchips
• Bouncing ball C60
• Current of Detailed balance
• The Breit Wigner Scattering Approximation
• Dissipative tunneling
• Field splitting and broadening of resonance level

3

• Current Voltage Curves
• Shuttling Instability
• Reduced Shot noise of shuttling
• Self-consistent charge
• Charging regimes
• Hamiltonian in adiabatic approximation
• Broken symmetry and instability Tunneling
  Electron Terms
• Density Of States
• Current simulation
• Coherence of Electron transport via Double Wells
• Conclusions

4
Tunneling Optical Trap

• Electro optical trapping U-aE2 , Eev- evanescent
  wave, Eel - electrostatic source-drain field
• The gap 0.1-1 m between SD electrodes allows the
  electron transport through the resonance states
  of TOT atoms only
• The atoms in TOT are quite isolated, ultra-cold
  and manageable to BEC transition
• Surface acts as evaporative knife to assist
  cooling by selectively absorbing higher energy
  atoms. The surface elastically reflects cold
  remnant. BEC Franklin Bell !
• TOT current is controlled by evanescent light
  that plays role of gate electrode.

BEC
current
S
D
5
TOT schematics

• Evanescent wave attracts atoms detuned from the
  resonance to red wing. Induced dipole potential
  creates gradient force of light pressure.
• Blue detuned light forms repulsive force. The
  forces combination cage atoms near surface
• (Grimm 2002).
• In TOT, the charged terminals play the role of
  red detuned light attracting particles.
• The volume confinement is obtained by combining
  the optical (repulsive) and electrostatic
  (attractive) potentials between source-drain
  leads. Only sum of the fields is capable of
  caging transversal and lateral atomic motions.
• TOT is designed to circumvent Earnshow theorem
  that forbids trapping in free space with pure
  electrostatic fields.

6
Events History

• The physics of resonant light pressure has been
  recognized long ago by A. Kazantsev, 1972. He has
  Unfortunately passed in very productive age. In
  this April he would be of seventy.
• Double Evanescent Wave trap for atoms on glass
  surface has been proposed by Ovchinnikov,
  Shulga, and Balykin, J.Phys.BAt.Mol. Opt. Phys.
  24, 3173,(1991)
• Bezryadin A, Dekker C, Schmid G. Electrostatic
  trapping of single conducting nanoparticles
  between nanoelectrodes, Appl. Phys. Lett. 71,
  12731275 (1997).
• Scanning Electron Microscope View
• The picture shows 14 nm gap between
• Pt leads bridged by a single Pd
  nanoparticle.
• Polarized particles are attracted to maximum
• field and self-assembled to wire the tips.

7
TOT Potentials

• Typical potential experienced by atoms in TOT is
  the sum of electrostatic, optical and van der
  Waals or Casimir-Polder (CP) potentials.
• UCPc4/R4 c4 1nK/mm4 for Rb on isolator
• At 0.5mm from surface, CP compares with
  polarization potential U-aE2 produced by 30 mV
  biased 1mm separated SD terminals

8
Flying Up and Down on substrate

• Rb ground state DC polarizability
  a80mHz/(V/cm)2
• 1 mk separated and 10mV biased S-D leads form 800
  Hz deep trap. It is capable of isolation up to
  103 -104 atoms in quantum degenerate regime (10
  nK cold).
• Repulsive gradient (Kazantsev) potential balances
  attraction of ultracold atoms to leads. For this
  goal it is enough to focus 1mW laser beam on the
  substrate.
• Typical potential between alkaly atom and surface
  (from Grimms data) shown in the picture.

9
TOT -as future element for atomic Microchips

• ? Smallest density of states
• ? Negate leakage currents, noises.
• ? High degree isolation
• ?Coherence
• ? Quantum Information Processes Double Wells ?
  Josephson tunneling
• ? Matter wave Interferometry
• ? Addressing quantum states both
• optically and electrically
• ? Key principles are already (ok!) tested

Nowadays atomic Microchips are reminiscent first
generation electronic tubes, with their
significant heat scattering, energy consumption
and volumes occupation.
10
Bouncing ball C60

• In 1999 Prof. McEuens group reported
  Nanomechanical oscillations in a single-C 60
  transistor, Nature, 407, 57, (2000).
• ? Current-voltage curves at T1.5 K.
• ? 5 meV plateau-type gap correlates with slosh
  mode that accompanies resonance tunneling via
  LUMO state. Why 2 steps present and Nor more ?
• ?Competing theories quenching Franck-Condon
  transitions versus shuttling instability (GF
  ME)
• Our goals are (a) to develop scattering approach
  to SET, explain twin features and (b) extend it
  to TOT

11
Current of Detailed balance

• Landauers mantra - transport is transmission
  TS2.
• Multi channels n summarizing at zero temperature
  present Poison shot noise
• At weak transparency Tltlt1 it is to current
  itself
• Fano factor F1 is known as Schottky noise

12
The Breit Wigner Scattering Approximation

• The energies E nearest to level Ed (LUMO or HOMO
  ) contribute mainly to conductance. The
  Breit-Wigner approximation is to be viable to
  electron S-matrix
• The level Ed includes confinement energy e0 ,
  chemical potential m of electrons in mean field
  (U -Coulomb charging), and mechanical corrections
  due to translations, vibrations and rotations.
• Let ?0 wFermi 8eV, work function Ai 5ev. At
  r1 nm from lead tunnel broadening ? 1 mev
• Vibronic frequency (5meV) and temperature
  (1K0.1meV) far exceed broadening ?
• n gtgtTempgtgtG
• The Hamiltonian of mechanical motion fixes p, x
  adiabatic variables, slow compared to electron
  dynamics

13
Dissipative tunneling

• Irreversibility of the open system arises from
  non correlated electrons leaving or entering
  leads to broad resonance states
• We distinguish 2 regimes
• Virtual transition via resonance center in Raman
  like scattering (with no charging states and no
  molecular conformations).
• In sequential tunneling, charge accumulates on
  resonance center. It can be accompanied by
  Coulomb blockade, broken symmetry of molecular
  vibrations and Landau bifurcation of confining
  potential. Damping and dissipation prevent
  shuttle mechanism to take effect.

14
Field splitting and broadening of resonance level

• The vibrator subjects to alternative potential
  while it travels from one lead to another to be
  excited
• Tunneling spectrum inherits effective
• temperature Teff eV/2, for eVgtw, as
• follows from FluctuationDissipation
• Theorem and Schottky formula for shot
• noise.
• Alternative Stark shift produces level
• splitting and broadening as it does
• Autler-Townes effect in Doppler-
• broadened system.

15
Current Voltage Curves

• Breit-Wigner approximation to electron
  scattering with adiabatic variables provides
  versatile mechanisms contrasting theory and
  experiment. Using resonance scattering
  description we modeled vibrational gap and
  plateau-type behavior of IV-curve.
• Negative
  differential resistance is
  explained by field splitting
  effect.
• Presented curve neglects the charging,
  Ohmic dissipation and relaxation, which
  must spread serrated IV steps.

16
Shuttling Instability

• Prof Robert Shekhter with colleagues Phys.Rev.
  Lett., (1998) puts forward idea of shuttling
  (vibrational) instability of electron tunneling.
• It is due to electron affinity to atoms with
  which the charge may travel in classically
  forbidden region.
• Current across the double tunnel barrier is
  assisted by mechanical oscillator which builts
  in soft host and subjects to vibrations excited
  by the current itself.
• For tunnel transport the shuttle instability
  plays a role of Cooper instability.
• Soon after it recognizing the idea of shuttling
  was flying on its owing wings

Franklin Bell
17
Reduced Shot noise of shuttling
FDouble Barrier1/2
Fano factor FS/2eI

• Better transparency - quietly noise for quiet
  electronic sea

18
Self-consistent charge

• The integral over spectrum is transformed into
  nonlinear equation for the distribution function
  f depending on adiabatic variables x and p.
  Connection between charge transport in space x,p
  (current) and mean charge itself is central goal
  of tunneling spectroscopy
• As a matter of fact, the tunnel broadening
  is negligible in comparison with thermal
  spreading kTgtgtG. Function f can be found by a
  library routine (Newton-Raphson or globally
  convergent one) as shown below
• The equation has nice properties important for
  adiabatic motion. First f is guaranteed to lie
  in interval 0, 1. Second, it is switched fast
  from state 0 to 1.

19
Charging regimes

• At small biases the mean charge state is almost
  of Gaussian shape.
• At threshold bias ( Vt vibr. frequency) the
  charge approaches to maximum allowed by the Pauli
  principle and then saturates.
• The charging energy creates barrier for
  adiabatic motion of nuclei. The barrier width
  compares with the value of zero oscillations
  amplitude for bias less than the threshold Vt.

20
Hamiltonian in adiabatic approximation
Electromechanics is mainly controlled by string
constants - rigidity, elasticity and Coulombic
energy of charging. The former responsible for
small amplitude oscillations around equilibrium
position. For C60 it is 0.04A. Coulomb
Blockade is accompanied by barrier formation that
increases potential and makes small oscillations
unfavorable at old equilibrium. (Orthodox
Coulomb blockade requires the dissipation
mechanisms to be present )
21
Broken symmetry and instability

• Expected potential for TOT system as function of
  applied bias.
• The bias makes equilibrium motion unstable.
  Initial trajectories will change their own fixed
  points. The external field makes positive work.
  First to double well.
• This potential symmetry will be broken again with
  increasing voltage. It is due to persistent
  Coulomb blockade that produces triple well
  system.

22
Tunneling Electron Terms

• To verify the picture we have performed a
  numerical experiment based on solution of
  Schrodinger equation with the goal to find total
  electronic energy (kineticpotential parts )
  depending on bias and fixed position of charges.
• Potential has been taken to include the charge
  screening in metal electrodes from solution of
  Poisson equation.
• Quantum simulations of electron tunneling in
  double barrier system composed from biased leads
  and field induced dipole. Tunneling terms present
  a total electronic energy as function of fixed
  atomic positions. This extends BO strategy into
  classically forbidden region of electron motion.
  Potential energy of slow particle dynamics is
  crucial to TOT design. The well begins to appear
  in the center. Then it grows in width and
  simultaneously in depth. The well results in
  shuttling regime of conductivity.

23
Density Of States

• We keep track of electronic density of states.
• Typically in our simulation we took 500 energy
  channels from Fermi sea in both leads.
• Friedel oscillations in the leads and resonance
  tunneling peak in center of DB junction has been
  observed to correlate with tunnel terms at
  symmetry breaking point.

24
Current simulation

• The tunneling current across dipole molecule as
  function of it center position and bias between
  leads. The maximum of conductivity is at DB
  center.
• Averaging the partial currents with Gibbs
  distribution in given tunneling terms produce IV
  curve of step like shape typical for shuttling
  instability.

25
Coherence of Electron transport via Double Wells

• The particle transport in Double well potential
  can be described by linearized Schrodinger eq.
  (because exact treatment or even mean field
  Hartree approximation is often impractical). We
  introduce the overlap integral D (from the
  otential) to describe tunneling in classically
  forbidden region. The same type eq. holds true
  for the left and right well (I.e. if L?R and R?
  L).
• Charge transported i.e. current is presented by
  the L/R overlapping wave functions y which
  coherence length must be of the barrier width
• This primary quantization demonstrates that the
  current grows as ground state splitting. It
  oscillates with the same frequency due to
  tunneling. This provides a dissipation mechanism.
  Frequency is controlled by the external
  potential bias that creates additional asymmetry
  of left/right wells. The transport across them is
  easy formalized in second quantization approach.
  

26
Conclusions

• TOT permits to circumvent dissipative tunneling
  roadblocks
• Broken symmetry of molecular vibrations is
  responsible for cure the junctions of shuttling
  instability
• Instability induces Coherence
• We explained plateau-type and Negative
  Differential Resistance features of
  nanomechanical oscillator due to field splitting
  and broadening of vibronic resonances
• We point out to experimentally accessible
  Franklin Bell made of BEC.