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Injectance and a Paradox

Injectance and a Paradox. By Supervisor Urbashi Satpathi Dr. Prosenjit Singha De0. OUTLINE. Experimental background Motivation PLDOS, Injectivity, Emissivity, Injectance, Emitance, LDOS, DOS Paradox and its practical implication. Phase shift.

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Injectance and a Paradox

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  1. Injectance and a Paradox By Supervisor Urbashi SatpathiDr. Prosenjit Singha De0

  2. OUTLINE • Experimental background • Motivation • PLDOS, Injectivity, Emissivity, Injectance, Emitance, LDOS, DOS • Paradox and its practical implication

  3. Phase shift Schematic description of experimental set up [ R. Schuster, E. Buks, M. Heiblum, D. Mahalu, V. Umansky and Hadas Shtrikman , Nature385, 417 (1997) ]

  4. Collector voltage, VCB

  5. Analyticity • Hilbert Transform relates the amplitude and argument

  6. What information we can get from phase shift ?

  7. Phase drop • Apparently does not follow Friedel sum rule (FSR) • However if carefully seen w.r.t Fano resonance (FR) can be understood from FSR • Besides there is a paradox at FR that can have tremendous practical implication.

  8. Larmor precision time (LPT) • Injectivity

  9. Why injectivity is physical?

  10. Local density of states • Density of states This is an exact expression.

  11. In semi-classical limit • Hence , , is FSR (semi classical). [ M. Büttiker, Pramana Journal of Physics 58, 241 (2002) ]

  12. and, is semi classical injectance. [ C. R. Leavens and G. C. Aers, Phys. Rev. B 39, 1202 (1989), E. H. Hauge, J. P. Falck, and T. A. Fjeldly, Phys. Rev. B 36, 4203 (1987) ]

  13. Why semi classical? Incident wave packet Scattered wave packet

  14. Considering no reflected part (E>>V), and no dispersion of wave packet, is stationary phase approximation. i.e. in semi classical case, density of states is related to energy derivative of scattering phase shift.

  15. The paradox : General case The confinement potential, The scattering potential, is symmetric in x-direction

  16. The Schrödinger equation of motion in the defect region is, • In the no defect region, , where, and, , is the energy of incidence.

  17. w For symmetric potentials, For, where, where, and, At resonance,

  18. The system The potential at X, Injectance from wave function is, Internal wave function Modes of the quantum wire

  19. , where, , and

  20. Injectance from wave function is, • Semi classical injectance is, , ,

  21. and

  22. and

  23. and

  24. and

  25. and

  26. and

  27. Conclusion • There is a paradox at Fano resonance • The semi classical injectivity gets exact at FR • Useful for experimentalists

  28. Leggett's conjecture for a mesoscopic ring • P. Singha Deo Phys. Rev. B {\bf 53}, 15447 (1996). • 2. Nature of eigenstates in a mesoscopic ring coupled to a side branch. • P. A. Sreeram and P. Singha Deo Physica B {\bf 228}, 345(1996. • Phase of Aharonov-Bohm oscillation in conductance of mesoscopic systems. • P. Singha Deo and A. M. Jayannavar. Mod. Phys. Lett. B {\bf 10}, 787 (1996). • Phase of Aharonov-Bohm oscillations: effect of channel mixing and Fano resonances. • P. Singha Deo Solid St. Communication {\bf 107}, 69 (1998). • Phase slips in Aharonov-Bohm oscillations • P. Singha Deo Proceedings of International Workshop on $``$Novelphysics in low dimensional electron systems", organized byMax-Planck-Institut Fur Physik Komplexer Systeme, Germanyin August, 1997.\\Physica E {\bf 1}, 301 (1997). • Novel interference effects and a new Quantum phase in mesoscopicsystems • P. Singha Deo and A. M. Jayannavar, Pramana Journal of Physics, {\bf 56}, 439 (2001). Proceedings of the Winter Institute on Foundations of Quantum Theoryand Quantum Optics, at S.N. Bose Centre,Calcutta, in January 2000. • Electron correlation effects in the presence of non-symmetry dictated nodes • P. Singha Deo Pramana Journal of Physics, {\bf 58}, 195 (2002) • Scattering phase shifts in quasi-one-dimension • P. Singha Deo, Swarnali Bandopadhyay and Sourin Das International Journ. of Mod. Phys. B, {\bf 16}, 2247 (2002) • Friedel sum rule for a single-channel quantum wire • Swarnali Bandopadhyay and P. Singha Deo Phys. Rev. B {\bf 68} 113301 (2003) • 10. Larmor precession time, Wigner delay time and the local density of states in a quantum wire. • P. Singha Deo International Journal of Modern Physics B, {\bf 19}, 899 (2005) • 11. Charge fluctuations in coupled systems: ring coupled to a wire or ring • P. Singha Deo, P. Koskinen, M. Manninen Phys. Rev. B {\bf 72}, 155332 (2005). • 12. Importance of individual scattering matrix elements at Fano resonances. • P. Singha Deo} and M. Manninen Journal of physics: condensed matter {\bf 18}, 5313 (2006). • 13. Nondispersive backscattering in quantum wires • P. Singha Deo Phys. Rev. B {\bf 75}, 235330 (2007) • 14. Friedel sum rule at Fano resonances • P Singha Deo J. Phys.: Condens. Matter {\bf 21} (2009) 285303. • 15. Quantum capacitance: a microscopic derivation • S. Mukherjee, M. Manninen and P. Singha Deo Physica E (in press). • 16. Injectivity and a paradox • U. Satpathy and P. Singha Deo International journal of modern physics (in press).

  29. Thank you

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