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Injectance and a Paradox

This study explores the paradox of injectivity in quantum systems through experimental analysis, shedding light on its implications and insights. The research delves into density of states, phase shifts, and more to uncover the complexity of these phenomena.

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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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