A Loop Diagram Approach to the Nonlinear Optical Conductivity for an Electron-Phonon System

ABSTRACT

A loop diagram approach to the nonlinear optical conductivity of an electron-phonon system is introduced. This approach can be categorized as another Feynman-like scheme because all contributions to the self-energy terms can be grouped into topologically-distinct loop diagrams. The results for up to the first order nonlinear conductivity are identical to those derived using the KC reduction identity (KCRI) and the state- dependent projection operator (SDPO) introduced by the present authors. The result satisfies the “population criterion” in that the population of electrons and phonons appear independently or the Fermi distributions are multiplied by the Planck distributions in the formalism. Therefore it is possible, in an organized manner, to present the phonon emissions and absorptions as well as photon absorptions in all electron transition processes. In additions, the calculation needed to obtain the line shape function appearing in the energy denominator of the conductivity can be reduced using this diagram method. This method shall be called the “KC loop diagram method”, since it originates from proper application of KCRI’s and SDPO’s.

A loop diagram approach to the nonlinear optical conductivity of an electron-phonon system is introduced. This approach can be categorized as another Feynman-like scheme because all contributions to the self-energy terms can be grouped into topologically-distinct loop diagrams. The results for up to the first order nonlinear conductivity are identical to those derived using the KC reduction identity (KCRI) and the state- dependent projection operator (SDPO) introduced by the present authors. The result satisfies the “population criterion” in that the population of electrons and phonons appear independently or the Fermi distributions are multiplied by the Planck distributions in the formalism. Therefore it is possible, in an organized manner, to present the phonon emissions and absorptions as well as photon absorptions in all electron transition processes. In additions, the calculation needed to obtain the line shape function appearing in the energy denominator of the conductivity can be reduced using this diagram method. This method shall be called the “KC loop diagram method”, since it originates from proper application of KCRI’s and SDPO’s.

KEYWORDS

Optical Conductivity, Projection Operator, KC Reduction Identity, Electron-Phononinteraction

Optical Conductivity, Projection Operator, KC Reduction Identity, Electron-Phononinteraction

Cite this paper

nullN. Kang and S. Choi, "A Loop Diagram Approach to the Nonlinear Optical Conductivity for an Electron-Phonon System,"*Journal of Modern Physics*, Vol. 2 No. 11, 2011, pp. 1410-1414. doi: 10.4236/jmp.2011.211173.

nullN. Kang and S. Choi, "A Loop Diagram Approach to the Nonlinear Optical Conductivity for an Electron-Phonon System,"

References

[1] L. Hedin and S. Lundqvist, “In Solid State Physics,” Vol. 15, Academic, New York, 1969.

[2] G. Grimvall, “The Electron-Phonon Interaction in Metals,” North-Holland, New York, 1981.

[3] P.-B. Allen and B. Mitrovvich, “In Solid State Physics,” Vol. 37, Academic, New York, 1982.

[4] G. D. Mahan, “Many-Particle Physics,” Plenum, New York, 2000.

[5] B. Hellsing and S. Lundqvist, “Lifetime of Holes and Electrons at Metal Surfaces; Electron-Phonon Coupling,” Journal of Electron Spectroscopy and Related Phenomena, Vol. 129, No. 2-3, 2003, pp. 97-104. doi:10.1016/S0368-2048(03)00056-2

[6] F. Giustino, M. L. Cohen and S. G. Louie. “Small Phonon Contribution to the Photoemission Kink in the Copper Oxide Superconductors,” Nature, Vol. 452, 2008, pp. 975-978. doi:10.1038/nature06874

[7] A. Nechaev, I. Yu. Sklyadneva, V. M. Silkin, P. M. Echenique and E. V. Chulkov, “Theoretical Study of Quasiparticle Inelastic Lifetimes as Applied to Aluminum,” Physical Review B, Vol. 78, No. 8, 2008, p. 085113. doi:10.1103/PhysRevB.78.085113

[8] N. L. Kang, Y. J. Cho and S. D. Choi, “A Many-Body Theory of Quantum-Limit Cyclotron Transition Line- Shapes in Electron-Phonon Systems Based on Projection Technique,” Progress of Theoretical Physics, Vol. 96, No. 2, 1996, pp. 307-316. doi:10.1143/PTP.96.307

[9] N. L. Kang and S. D. Choi, “Scattering Effects of Phonons in Two Polymorphic Structures of Gallium Nitride,” Journal of Applied Physics, Vol. 106, No. 6, 2009, p. 063717. doi:10.1063/1.3226885

[10] H. J. Lee, N. L. Kang, J. Y. Sug and S. D. Choi, “Calculation of the Nonlinear Optical Conductivity by a Quantum-Statistical Method,” Physical Review B, Vol. 65, No. 19, 2002, p. 195113. doi:10.1103/PhysRevB.65.195113

[11] N. L. Kang and S. D. Choi, “Derivation of Nonlinear Optical Conductivity by Using a Reduction Identity and a State-Dependent Projection Method,” Journal of Physics A: Mathematical and Theoretical, Vol. 43, No. 16, 2010, p. 165203. doi:10.1088/1751-8113/43/16/165203

[12] N. L. Kang, YounJu Lee and S. D. Choi, “Generalization of the Nonlinear Optical Conductivity Formalism by Using a Projection-Reduction Method,” Journal of Korean Physical Society, Vol. 58, No. 3, 2011, pp. 538-544. doi:10.3938/jkps.58.538

[13] N. L. Kang and S. D. Choi, “Validity of nth-Order Conductivity by the General Projection-Reduction Method,” Progress of Theoretical Physics, Vol. 125, No. 5, 2011, pp. 1011-1019. doi:10.1143/PTP.125.1011

[14] R. P. Feynman, “Space-Time Approach to Quantum Electrodynamics,” Physical Review, Vol. 76, No. 6, 1949, pp. 769-789. doi:10.1103/PhysRev.76.769

[15] L. Fetter and J. D. L. Walecka, “Quantum Theory of Many-Particle Systems,” McGraw Hill, New York, 1971.

[16] R. D. Mattuch, “A Guide to Feynman Diagrams in the Many-Body Problem,” 2nd Edition, McGraw-Hill, New York, 1976.

[1] L. Hedin and S. Lundqvist, “In Solid State Physics,” Vol. 15, Academic, New York, 1969.

[2] G. Grimvall, “The Electron-Phonon Interaction in Metals,” North-Holland, New York, 1981.

[3] P.-B. Allen and B. Mitrovvich, “In Solid State Physics,” Vol. 37, Academic, New York, 1982.

[4] G. D. Mahan, “Many-Particle Physics,” Plenum, New York, 2000.

[5] B. Hellsing and S. Lundqvist, “Lifetime of Holes and Electrons at Metal Surfaces; Electron-Phonon Coupling,” Journal of Electron Spectroscopy and Related Phenomena, Vol. 129, No. 2-3, 2003, pp. 97-104. doi:10.1016/S0368-2048(03)00056-2

[6] F. Giustino, M. L. Cohen and S. G. Louie. “Small Phonon Contribution to the Photoemission Kink in the Copper Oxide Superconductors,” Nature, Vol. 452, 2008, pp. 975-978. doi:10.1038/nature06874

[7] A. Nechaev, I. Yu. Sklyadneva, V. M. Silkin, P. M. Echenique and E. V. Chulkov, “Theoretical Study of Quasiparticle Inelastic Lifetimes as Applied to Aluminum,” Physical Review B, Vol. 78, No. 8, 2008, p. 085113. doi:10.1103/PhysRevB.78.085113

[8] N. L. Kang, Y. J. Cho and S. D. Choi, “A Many-Body Theory of Quantum-Limit Cyclotron Transition Line- Shapes in Electron-Phonon Systems Based on Projection Technique,” Progress of Theoretical Physics, Vol. 96, No. 2, 1996, pp. 307-316. doi:10.1143/PTP.96.307

[9] N. L. Kang and S. D. Choi, “Scattering Effects of Phonons in Two Polymorphic Structures of Gallium Nitride,” Journal of Applied Physics, Vol. 106, No. 6, 2009, p. 063717. doi:10.1063/1.3226885

[10] H. J. Lee, N. L. Kang, J. Y. Sug and S. D. Choi, “Calculation of the Nonlinear Optical Conductivity by a Quantum-Statistical Method,” Physical Review B, Vol. 65, No. 19, 2002, p. 195113. doi:10.1103/PhysRevB.65.195113

[11] N. L. Kang and S. D. Choi, “Derivation of Nonlinear Optical Conductivity by Using a Reduction Identity and a State-Dependent Projection Method,” Journal of Physics A: Mathematical and Theoretical, Vol. 43, No. 16, 2010, p. 165203. doi:10.1088/1751-8113/43/16/165203

[12] N. L. Kang, YounJu Lee and S. D. Choi, “Generalization of the Nonlinear Optical Conductivity Formalism by Using a Projection-Reduction Method,” Journal of Korean Physical Society, Vol. 58, No. 3, 2011, pp. 538-544. doi:10.3938/jkps.58.538

[13] N. L. Kang and S. D. Choi, “Validity of nth-Order Conductivity by the General Projection-Reduction Method,” Progress of Theoretical Physics, Vol. 125, No. 5, 2011, pp. 1011-1019. doi:10.1143/PTP.125.1011

[14] R. P. Feynman, “Space-Time Approach to Quantum Electrodynamics,” Physical Review, Vol. 76, No. 6, 1949, pp. 769-789. doi:10.1103/PhysRev.76.769

[15] L. Fetter and J. D. L. Walecka, “Quantum Theory of Many-Particle Systems,” McGraw Hill, New York, 1971.

[16] R. D. Mattuch, “A Guide to Feynman Diagrams in the Many-Body Problem,” 2nd Edition, McGraw-Hill, New York, 1976.