1. Dynamics of Vibrational Excitation in the C 60 - Single Molecule Transistor Aniruddha Chakraborty Department of Inorganic and Physical Chemistry Indian Institute of Science, Bangalore-560012, India. http://www.ipc.iisc.ernet.in/~anirud

2. Plan of the talk 1. What is C 60 - single molecule transistor? 2. Experimental results 3. Our work 4. Conclusions

3. 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.

4. Park et al. Nature 497, 57 (2000). Current Vs Voltage Plot at 1.5K Conductance gap Asymmetric Different step heights 5 meV

5. ‘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

6. 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 Energy Chem. Phys. Lett. 214, 569 (1993) Lennard-Jones

7. Hollow sphere Carbon atoms smeared into a continuum Coulomb interaction Extra electron is uniformly distributed Point charge at the center

8. 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

9. 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

10. 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

11. The Hamiltonian Internal vibrational modes of C 60 are not considered. Position dependence of LUMO energy is neglected.

12. Perturbation (electron hopping)

13. Center of mass motion Energy Geometry independent.

14. 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

15. trapped between gold electrodes C 60 No experimental information available

16. Van der Waals interaction between C 60 and Au electrode *Buckingham potential for Au-C interaction *Acknowledgement: Hao Tang (CEMES/CNRS, France). 681012 14 -0.5 0.0 0.5 Energy ( eV ) Chem. Phys. Lett. 214, 569 (1993) Hollow sphere Carbon atoms smeared into a continuum Metal assumed to form a continuum

17. Van der Waals interaction: C 60 trapped between gold electrodes Energy Center of mass motion Approximate Potential Analysis by Park et al. Choice of d Best distance – maximum binding energy

18. 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.

19. Center of mass motion Energy Analysis by Park et al. Approximate Potentials

20. Current Vs Voltage Plot 05101520 0 2 4 6 8 Voltage (meV) Current (arb. units) Qualitative agreement !

21. Van der Waals interaction between C 60 and Gold electrode 9111315 17 -0.25 0.0 0.25 0.5 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

22. 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 Analysis by Park et al. Choice of d Best distance – maximum binding energy

23. 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 = +

24. 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

25. Approximate Potentials Center of mass motion Energy Analysis by Park et al.

26. 05101520 0 2 4 6 8 Voltage (meV) Current (arb. units) Current Vs Voltage Plot Qualitative agreement !

27. 05101520 0 2 4 6 8 Voltage (meV) Current (arb. units) Current Vs Voltage Plot Qualitative agreement !

28. Contribution from hopping matrix element Voltage Current (0,0) (0,1) (0,2) (0,3)

29. Electrode geometry & hopping matrix element Voltage Current (0,0) (0,1) *Boese et al. Europhys. Lett. 54, 668 (2001).

30. Only Qualitative Agreement ! Double well problem! Internal modes! Energy Center of mass motion

31. 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.

32. Prof. K.L. Sebastian Hao Tang Keshav Kumar ( University of Pennsylvania, USA ) ( CEMES/CNRS, France ) CSIR ( New Delhi, India ) Acknowledgements ( Indian Institute of Science, India )