C 60 - Single Molecule Transistor Aniruddha Chakraborty Indian Institute of Technology Mandi, Mandi , Himachal Pradesh, India. 22/09/2010
Plan of the talk 1. What is C 60 - single molecule transistor? 2. Experimental results 3. Our work 4. Conclusions
Park et al. Nature 497, 57 (2000). C 60 - Single Molecule Transistor C 60 molecule Sphere, diameter 0.7 nm. 12 pentagons and 20 hexagons.
Park et al. Nature 497, 57 (2000). Current Vs Voltage Plot at 1.5 K Conductance gap Asymmetric Different step heights 5 meV
‘Two photon’ Process Center of mass motion Energy (0,0) (0,1) (0,2) (0,3) Voltage Current (0,0) (0,1) (0,2) Energy Nuclear Coordinate A B A B t=0
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
The Model Perturbation (electron hopping) ‘Two photon’ Process (Resonance Raman Spectroscopy) Perturbation (Light) Kramers-Heisenberg-Dirac formula Second order Perturbation theory C 60 - Single Molecule Transistor
The Hamiltonian Internal vibrational modes of C 60 are not considered. Position dependence of LUMO energy is neglected.
Perturbation (electron hopping)
Center of mass motion Energy Geometry independent. New Mechanism: distance dependent electron hopping probability
Kramers-Heisenberg-Dirac type formula *Boese et al. Europhys. Lett. 54, 668 (2001). Temperature effect neglected 1.5K =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 electrodes C 60 No experimental information available
Van der Waals interaction between C 60 and Au electrode *Acknowledgement: Hao Tang (CEMES/CNRS, France). Hollow sphere Carbon atoms smeared into a continuum Metal assumed to form a continuum Energy ( eV )
Van der Waals interaction: C 60 trapped between gold electrodes Energy Center of mass motion Approximate Potential Choice of d Best distance – maximum binding energy
Classical Electrodynamics: J. D. Jackson; 3rd ed. (1999). Image interaction Hollow sphere Carbon 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.
Center of mass motion Energy Approximate Potentials
Current Vs Voltage Plot Voltage (meV) Current (arb. units) Qualitative agreement !
Van der Waals interaction between C 60 and Gold electrode Energy ( eV ) Hollow sphere Carbon atoms smeared into a continuum Metal assumed to form a continuum Larger radius – effect of protrusion is less Smaller radius – C 60 won’t stable on top
Van der Waals interaction: C 60 trapped between Gold electrodes Van der Waals interaction: C 60 trapped between Gold electrodes Energy Center of mass motion Choice of d Best distance – maximum binding energy
Image Interaction Classical Electrodynamics: J. D. Jackson; 3rd ed. (1999). Hollow sphere Carbon 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 Center of mass motion Energy
Voltage (meV) Current (arb. units) Current Vs Voltage Plot Qualitative agreement !
Voltage (meV) Current (arb. units) Current Vs Voltage Plot Qualitative agreement !
Contribution from hopping matrix element Voltage Current (0,0) (0,1) (0,2) (0,3)
Electrode geometry & hopping matrix element Voltage Current (0,0) (0,1) A. Chakraborty, Chapter 2, Ph.D thesis, IISC, Bangalore, India, 2005.
Only Qualitative Agreement ! Double well problem! Internal modes! Energy Center of mass motion A. Chakraborty, K. Kumar and K. L. Sebastian, Phys. Rev. B 68, (2003).
Effective Hamiltonian Approach A. Chakraborty (manuscript under preparation) (2010).
Conclusions 1. Two possible mechanisms for vibrational excitations. 2. Our results are in qualitative agreement with experiment. (a) The displacement of equilibrium position. (b) The position dependence of the electron hopping matrix element A. Chakraborty, Nano Devices, 2D electron solvation and curve crossing problems: Theoretical Model Investigations, LAMBERT Academic Publishing, Germany (2010).
Prof. K.L. Sebastian Hao Tang Dr. Keshav Kumar ( University of Pennsylvania, USA ) ( CEMES/CNRS, France ) CSIR ( New Delhi, India ) Acknowledgements ( Indian Institute of Science, India )