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Dynamics of Vibrational Excitation Dynamics of Vibrational Excitation in the Cin the C60 60 - Single Molecule - Single Molecule TransistorTransistor
Aniruddha Chakraborty
Department of Inorganic and Physical Chemistry Indian Institute of Science, Bangalore-560012, India. http://www.ipc.iisc.ernet.in/~anirud
Plan of the talk
1. What is C60 - single molecule transistor?
2. Experimental results
3. Our work
4. Conclusions
Park et al. Nature 497, 57 (2000).
C60 - Single Molecule Transistor
C60 molecule
Sphere, diameter 0.7 nm.12 pentagons and 20 hexagons.
Park et al. Nature 497, 57 (2000).
Current Vs Voltage Plot at 1.5K
Conductance gap
Asymmetric
Different step heights
5 meV
‘Two photon’ Process
Center of mass motion
En
ergy
(0,0)(0,1)(0,2)(0,3)
Voltage
Cu
rren
t
(0,0
)
(0,1
)
(0,2
)
En
ergy
Nuclear Coordinate
d
d
Lennard-Jones potential for Au-C interaction:
Theoretical analysis by Park et al.
Lennard-Jones+Coulomb
Park et al. Nature 497, 57 (2000).
Center of mass motion
En
ergy
Chem. Phys. Lett. 214, 569 (1993)
Lennard-Jones
Hollow sphere
Carbon atoms smeared into a continuum
Coulomb interaction
Extra electron is uniformly distributed
Point charge at the center
Why not internal vibrational excitation?Lowest energy mode: 33meV
Why Not?
Why not electronic excitation?Very high energy
Why not rotational excitation?No net dipole moment
Theoretical Analysis by Boese et al.
Boese et al. Europhys. Lett. 54, 668 (2001).
Local system + Bosonic Bath+two electronic reservoirs
Local system= quantum dot+ harmonic oscillator
The Model
Perturbation (electron hopping)
‘Two photon’ Process (Resonance Raman Spectroscopy)
Perturbation (Light)
Kramers-Heisenberg-Dirac formula
Second order Perturbation theory
C60 - Single Molecule Transistor
L
The HamiltonianThe Hamiltonian
Internal vibrational modes of C60 are not considered.Position dependence of LUMO energy is neglected.
PerturbationPerturbation(electron hopping)
Center of mass motion
En
ergyGeometry independent.
Kramers-Heisenberg-Dirac type formula
*Boese et al. Europhys. Lett. 54, 668 (2001).
Temperature effect neglected1.5K =0.13 meV
(a) The displacement of the (a) The displacement of the equilibrium position
Contributing factors to the vibrational excitation
(b) The position dependence of the (b) The position dependence of the electron hopping matrix element
trapped between gold electrodesC60
No experimental information available
Van der Waals interaction between C60 and Au electrode
*Buckingham potential for Au-C interaction
*Acknowledgement: Hao Tang (CEMES/CNRS, France).
6 8 10 12 14
-0.5
0.0
0.5
En
ergy
( e
V )
Chem. Phys. Lett. 214, 569 (1993)
Hollow sphereCarbon atoms smeared into a continuum
Metal assumed to form a continuum
Van der Waals interaction:C60 trapped between gold electrodes
Ene
rgy
Center of mass motion
Approximate Potential
Analysis by Park et al.
Choice of dBest distance – maximum binding energy
Classical Electrodynamics: J. D. Jackson; 3rd ed. (1999).
Image interaction
Hollow sphereCarbon atoms smeared into a continuum
Extra electron is uniformly distributed
Point charge at the center
Force Calculation (convergent Series)
Images placed at larger and larger distances.
d
d
Center of mass motion
En
ergy
Analysis by Park et al.
Approximate Potentials
Current Vs Voltage Plot
0 5 10 15 200
2
4
6
8
Voltage (meV)
Cu
rren
t (a
rb. u
nit
s)
Qualitative agreement !
Van der Waals interaction between C60 and Gold electrode
9 11 13 15 17
-0.25
0.0
0.25
0.5
En
ergy
( e
V )
Hollow sphere
Carbon atoms smeared into a continuum
Metal assumed to form a continuum
Larger radius – effect of protrusion is lessSmaller radius – C60 won’t stable on top
Van der Waals interaction: CVan der Waals interaction: C6060 trapped between Gold electrodestrapped between Gold electrodes
En
ergy
Center of mass motion
Analysis by Park et al.
Choice of dBest distance – maximum binding energy
Image Interaction
Classical Electrodynamics: J. D. Jackson; 3rd ed. (1999).
Hollow sphereCarbon atoms smeared into a continuum
Extra electron is uniformly distributed
Point charge at the center
= +
32760 images
Image Interaction
Force Calculation (convergent Series)
Images from reflection between parallel electrodes : placed at larger and larger distances.
With each reflection the images change sign.
Each reflection on the sphere, reduces the images change.
generated from a set of SIX successive reflections
seven
five
Approximate Potentials
d
d
Center of mass motion
En
ergy
Analysis by Park et al.
0 5 10 15 200
2
4
6
8
Voltage (meV)
Cu
rren
t (a
rb. u
nit
s)
Current Vs Voltage Plot
Qualitative agreement !
0 5 10 15 200
2
4
6
8
Voltage (meV)
Cu
rren
t (a
rb. u
nit
s)
Current Vs Voltage Plot
Qualitative agreement !
Contribution from hopping matrix element
Voltage
Cu
rren
t
(0,0
)
(0,1
)
(0,2
)
(0,3
)
Electrode geometry & hopping matrix element
Voltage
Cu
rren
t
(0,0
)
(0,1
)
*Boese et al. Europhys. Lett. 54, 668 (2001).
Only Qualitative Agreement !
Double well problem!
Internal modes! En
ergy
Center of mass motion
Conclusions
1. Two possible mechanisms for vibrational excitation.
2. Our results are in qualitative agreement with experiment.
A. Chakraborty, K. Kumar and K. L. Sebastian, Phys. Rev. B 68, 085411 (2003).
(a) The displacement of equilibrium position
(b) The position dependence of the electron hopping matrix element
A. Chakraborty, Chapter 2, Ph.D thesis, IISC, Bangalore, India, 2005.
Prof. K.L. Sebastian
Hao Tang
Keshav Kumar ( University of Pennsylvania, USA )
( CEMES/CNRS, France )
CSIR ( New Delhi, India )
AcknowledgemenAcknowledgementsts
( Indian Institute of Science, India )
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