Transcript Relaxation and Nonequilibrium Dynamics in Single

Relaxation and Nonequilibrium
Dynamics in Single-Molecule
Devices
Yuval Vinkler
Racah Institute of Physics, The Hebrew University
In Collaboration with:
Avi Schiller, The Hebrew University
Natan Andrei, Rutgers
Experimental Motivation:
Vibrational Modes in Nano Devices
Suspended nano-tubes
Nano-tube
electrode
Single-molecule Transistors
electrode
Molecule
electrode
electrode
Displacement-Coupling Hamiltonian
V
electrode
Molecule
q
Configurational coordinate


H   k ck†ck  Vk d †ck  ck†d 
 0b†b  g b†  bd †d
Interaction via displacement
Hopping term
Energy scales and limit of interest
D
Conduction-electron Bandwidth
tunnel  0 Vk2
Half the tunneling rate
g 2 / 0
Polaronic shift
We obtained an asymptotically exact solution,
in the regime where
tunnel  g , g 2 / 0
Solution of the Model
H   k ck†ck  Vk d †ck  ck†d  
 0b†b  g b†  bd †d
Continuum Limit
Diagonalization
allows
exact calculation
Localized
level is absorbed
in the continuum
of dynamical response
to abrupt change
Abelian
Bosonization
in the
system parameters
Hamiltonian is quadratic
in bosons
Exactly Solvable!
Solving the LippmannSchwinger eq.
H   kk†k
1st Quenching Scenario: Turning on the interaction
Interaction is turned
on at time t=0
Characteristic decay
time depends both
on coupling and
vibrational energy
g2

0
2
4tunnel
Vibrational mode
softens with
increasing coupling
1st Quenching Scenario: Turning on the interaction
Thermalization to
equilibrium
Equilibrium expectation
value independent of
the initial state of the
vibrational mode
2nd Quenching Scenario: Abrupt Change of Frequency
At t=0 the phonon
frequency is abruptly
changed by 0
Thermalization to a
new equilibrium
Relaxation time and
amplitude of
oscillations depend
on 0
Equilibrium
expectation value
independent of
quench scenario
Response to AC Drive
For t > 0, phonon
position is forced by
an AC drive
H AC   b †  b  sin t 
Transient oscillations
are governed by 0
Long-time oscillations
at frequency 2 . No
linear harmonic with
frequency  !
Conclusions
Analytical, non-perturbative, and asymptotically exact solution
of quench and ac dynamics of a single-molecule device.
Complete real-time dynamics of the localized phonon was
calculated under different quenching scenarios and ac drive.
Relaxation to new thermal equilibrium independent of the
initial phonon state and details of the quench scenario.
Relaxation time is linear in the frequency, quadratic in the
coupling, and inversely proportional to the tunneling rate
squared.
Outlook
Nonequilibrium steady state and I-V characteristics