电声相互作用下AB干涉仪的量子输运特性研究
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摘要
随着纳米技术的迅速发展,生产超小电子器件成为可能,因此量子干涉效应在电子器件中起着非常重要的作用。为了研究电子波的相位相干性,AB干涉仪成功地把AB效应应用到量子器件中,并用之观察到了与电子相位相干性有关Fano效应。电声相互作用对量子相干性有着非常大的影响,很容易使得相干性丢失。
本文运用非平衡格林函数方法研究AB干涉仪的量子输运问题,同时考虑量子点中的电子库仑作用和电子一声子相互作用,研究它们的竞争对量子输运特性及电子相干性的影响。本文首先简单地回顾一些介观物理中重要的概念如量子点、AB效应及应用以及Fano效应。接下来介绍了电声相互作用在理论和实验上相关的研究背景、发展动态和处理方法,也对本文所采用的理论工具非平衡格林函数的概念、常用到的定理、方程、性质进行了介绍。最后,在Aharonov-Bohm环上包含一个量子点的系统中,我们研究电声相互作用和电子间的库仑相互作用对Fano共振、Breit-Wigner共振隧穿输运特性的影响,采用非平衡格林函数的方法获得计算电流的表达式,发现在Fano共振中当电声相互作用大于库仑作用时,由于干涉效应会出现负微分电导特性；而在Breit-Wigner共振中,正的有效库仑作用U使的主峰宽度变窄,负的有效库仑作用U使主峰变宽,当电声相互作用大于库仑作用时,不再出现负的微分电导。
With the rapid development of microfabrication techniques it is feasible to manufacture ultrasmall electronic devices in order to miniaturize integrated circuits. Thus, the quantum conherence become very important in electronic devices.In order to investigate the electron wave phase coherence, AB effect is successfully applied to the quantum device named AB interferometer. AB interferometer can let us observe famous Fano effect induced by electronic phase coherence. Electron-phonon interactions have more effects on the conherece and easily make it disappeared.
In this paper, considering Coulomb interaction and electron-phonon interaction meanwhile, we study the transport properties of the AB interferometer by the nonequilibrium Green's function approach. We first recall some important conceptions in the mesoscopic physics, such as the quantum dot, Fano effect, AB effect and so on. Then, we gives the related research background and development of the electron-phonon effect in theorical and experimental aspects. And The approachs of dealing the electron-phonon interaction are also introduced including the canonical transformation method,the nonequilibrium Green's function etc. Finally, we study the influence that the electron-phonon (e-ph) interaction and the Coulomb interaction have on the Fano resonance and the Breit-Wigner resonance of the transport through the AB interferometer.Using the non-equilibrium Green function,we obtain the expressions of the current and find the negative differential conductance when the e-ph interaction have more effect than the Coulomb interaction in the Fano resonance due to the interference;The positive and effective Coulomb interaction makes the width of main peak narrower, the negative one makes the the width of main peak wider. When the e-ph interaction have more effect than the Coulomb interaction in the Breit-Wigner resonance, the negative differential conductance no longer appear.
本文运用非平衡格林函数方法研究AB干涉仪的量子输运问题,同时考虑量子点中的电子库仑作用和电子一声子相互作用,研究它们的竞争对量子输运特性及电子相干性的影响。本文首先简单地回顾一些介观物理中重要的概念如量子点、AB效应及应用以及Fano效应。接下来介绍了电声相互作用在理论和实验上相关的研究背景、发展动态和处理方法,也对本文所采用的理论工具非平衡格林函数的概念、常用到的定理、方程、性质进行了介绍。最后,在Aharonov-Bohm环上包含一个量子点的系统中,我们研究电声相互作用和电子间的库仑相互作用对Fano共振、Breit-Wigner共振隧穿输运特性的影响,采用非平衡格林函数的方法获得计算电流的表达式,发现在Fano共振中当电声相互作用大于库仑作用时,由于干涉效应会出现负微分电导特性；而在Breit-Wigner共振中,正的有效库仑作用U使的主峰宽度变窄,负的有效库仑作用U使主峰变宽,当电声相互作用大于库仑作用时,不再出现负的微分电导。
With the rapid development of microfabrication techniques it is feasible to manufacture ultrasmall electronic devices in order to miniaturize integrated circuits. Thus, the quantum conherence become very important in electronic devices.In order to investigate the electron wave phase coherence, AB effect is successfully applied to the quantum device named AB interferometer. AB interferometer can let us observe famous Fano effect induced by electronic phase coherence. Electron-phonon interactions have more effects on the conherece and easily make it disappeared.
In this paper, considering Coulomb interaction and electron-phonon interaction meanwhile, we study the transport properties of the AB interferometer by the nonequilibrium Green's function approach. We first recall some important conceptions in the mesoscopic physics, such as the quantum dot, Fano effect, AB effect and so on. Then, we gives the related research background and development of the electron-phonon effect in theorical and experimental aspects. And The approachs of dealing the electron-phonon interaction are also introduced including the canonical transformation method,the nonequilibrium Green's function etc. Finally, we study the influence that the electron-phonon (e-ph) interaction and the Coulomb interaction have on the Fano resonance and the Breit-Wigner resonance of the transport through the AB interferometer.Using the non-equilibrium Green function,we obtain the expressions of the current and find the negative differential conductance when the e-ph interaction have more effect than the Coulomb interaction in the Fano resonance due to the interference;The positive and effective Coulomb interaction makes the width of main peak narrower, the negative one makes the the width of main peak wider. When the e-ph interaction have more effect than the Coulomb interaction in the Breit-Wigner resonance, the negative differential conductance no longer appear.
引文
[1]S.Washburn, etal.Adv. Phys.35,375(1986)
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[3]Yoseph Imry. Introduction to mesoscopic physics [M].Oxford University Press, 2002.
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[5]H. Heinrich, G.Bauer, and F. Kuchar. Physics and Technology of Submicron Structures [M].1988,Springer-Verlag, Berlin.
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[7]R. C.Ashoori.Electrons in Artificial Atoms[J].Nature,1996,379,431.
[8]K. Tsukagoshi.B.W. Alphenaar, and H.Ago. Coherent transport of electron spin in a ferromagnetically contacted carbon nanotube [J].Nature,1999,401:572-574.
[9]J.R. Petta, S.K. Slater, and D. C.Ralph. Spin-Dependent Transport in Molecular Tunnel Junctions [J].Phys. Rev. Lett,2004,93:136601(1)-136601(4).
[10]L. Zhang, C. Wang, Y. Wei, X. Liu, and D. Davidovic. Spin-polarized electron transport through nanometer-scale Al grains [J].Phys. Rev.B,2005,72:155445(1)-(5).
[11]J. Reichert, R. Ochs, D.Beckmann, H. B.Weber, M. Mayor, and H. V. Lohneysen. Driving Current through Single Organic Molecules [J].Phys. Rev. Lett,2002,88: 176804(1)-176804(4).
[12]M. A. Reed, R. T. Bate, K. Bradshaw, W. M. Duncan, W. M. Frensley, J. W. Lee and H.D.Shin. Spatial Quantization in GaAs-AIGaAs Multiple Quantum Dots[M]. J.Vacuum. Sci.Technol. B4,1986,4:358-360.
[13]D. L. Klein, R. Roth, and A. K. L. Lim. A Single-Electron Transistor Made from a Cadmium Selenide Nanocrystal [J].Nature,1997,389,699.
[14]R. G Chambers. Shift of an Electron Interference Pattern by Enclosed Magnetic Flux [J].Phys.Rev. Lett,1960,5:3-5.
[15]A. Yacoby, M. Heiblum, D.Mahalu, and H. Shtrikman. Coherence and Phase Sensitive Measurements in a Quantum Dot [J].Phys. Rev. Lett,1995,74:4047-4050.
[16]V. Moldoveanu and M.Tolea. Controlled dephasing in single-dot Aharonov-Bohm interferometers[J].Phys. Rev. B,2007,75:045309(1):045309(9).
[17]B.Kubala and J. Konig, Aharonov-Bohm interferometry with quantum dots: scattering approach versus tunneling picture [J] Phys.Rev. B,2003,67:205303(1)-205303(8).
[18]J.Gores, D. Goldhaber-Gordon, S.Heemeyer, and M. A. Kastner. Fano resonances in electronic transport through a single-electron transistor[J].Phys. Rev. B,2000, 62:2188-2194.
[19]Kensuke Kobayashi, Hisashi Aikawa, Akira Sano, Shingo Katsumoto, and Yasuhiro Iye.Fano resonance in a quantum wire with a side-coupled quantum dot[J].Phys. Rev. B,2004,70:035319(1)-035319(9).
[20]Jinhee Kim, Jae-Ryoung Kim, Jeong-O Lee, Jong Wan Park, Hye Mi So, Nam Kim, Kicheon Kang, Kyung-Hwa Yoo, and Ju-Jin Kim. Fano Resonance in Crossed Carbon Nanotubes[J].Phys. Rev. Lett,2003,90:166403(1)-166403(4).
[21]A. A. Clerk, X. Waintal, and P. W. Brouwer. Fano Resonances as a Probe of Phase Coherence in Quantum Dots[J].Phys. Rev. Lett,2001,86:4636-4639.
[22]Dipl.-Phys.Jens Koch. Quantum transport through single-molecule devices[M]. 2006, Berlin.
[23]L. H.Yu, Z. K. Keane, J. W. Ciszek, L. Cheng, M. P. Stewart, J.M. Tour, and D. Natelson. Inelastic Electron Tunneling via Molecular Vibrations in Single-Molecule Transistors [J].Phys. Rev. Lett,2004,93:266802(1)-266802(4).
[24]N. B.Zhitenev, H. Meng, and Z. Bao. Conductance of Small Molecular Junctions [J]. Phys.Rev. Lett,2002,88:226801(1)-226801(4).
[25]V.J.Goldman, D. C. Tsui and J. E. Cunningham. Evidence for LO-phonon-emission-assisted tunnelinging in double-barrier heterostrues [J].Phys. Rev.B,1987,36:7635.
[26]M. L. Leadbeater, E. S.Alves, and L. Eave, M. Henini.Magnetic field studies of elastic scattering and optic-phonon emission in resonant-tunneling devices [J].Phys. Rev. B,1989,39:3438-3441.
[27]H. Park, J.Park, A.K.L. Lim, E.H. Anderson, A.P. Alivisatos, and P.L. McEuen. Nanomechanical oscillations in a single-C60 transistor [J].Nature,2000,407:57-60.
[28]Akiko Ueda and Mikio Eto. Resonant tunneling and Fano resonance in quantum dots with electron-phonon interaction [J].Phys Rev B,2006,73:235353(1)-235353(12).
[29]N.B.Zhitenev, H.Meng, and Z. Bao.Conductance of Small Molecular Junctions [J]. Phys. Rev. Lett,2002,88:226801(1)-226801(4).
[30]J. Park, A.N. Pasupathy, J.I. Goldsmith, C. Chang, Y.Yaish, J.R. Petta, M. Rinkoski, J.P. Sethna, H.D.Abruna, P.L. McEuen, and D.C.Ralph. Coulomb blockade and the Kondo effect in single-atom transistors [J].Nature,2002,417:722-725.
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[32]J.Koch and F. V.Oppen. Franck-Condon blockade and giant Fano factors in transport through single molecules [J].cond-mat/0409667v2.
[33]A. Mitra and I. Aleiner, A. J. Millis. Quantal and classical Phonon effects in molecular transistors:treatment[J].Phys.Rev. B,2004,69:245302(1)-245302(21).
[34]Jian-Xin Zhu and A.V.Balatsky. Theory of current and shot-noise spectroscopy in single-molecular quantum dots with a phonon mode[J].Phys Rev B,2003, 67:165326(1)-165326(7).
[35]U. Lundin and R. H. McKenzie. Temperature dependence of polaronic transport through single molecules and quantum dots [J].Phys. Rev. B,2002,66:075303(1)-075303(8).
[36]N. S.Wingreen, K. W. Jacobsen, and J.W. Wilkins. Inelastic scattering in resonant tunneling [J].Phys. Rev. B,1989,40:11834-11850.
[37]David M. T. Kuo and Y. C. Chand. Tunneling current through a quantum dot with strong electron-phonon interaction [J].Phys. Rev. B,2002,66:0853 11(1)-085311(5).
[38]P. S.Cornaglia, D. R. Grempel, and H. Ness. Quantum transport through a deformable molecular transistor[J].Phys. Rev. B,2005,71:075320(1)-075320(10).
[39]P. S.Cornaglia, H. Ness, and D.R. Grempel.Many-Body Effects on the Transport Properties of Single-Molecule Devices [J].Phys. Rev. Lett,2004,93:147201(1)-147201(4).
[40]K. Flensberg.Tunneling broadening of vibrational sidebands in molecular transistors [J].Phys. Rev. B,2003,68:205323(1)-205323(4).
[41]A. S.Alexandrov and A. M. Bratkovsky. Memory effect in a molecular quantum dot with strong electron-vibron interaction [J].Phys.Rev. B,2003,67:235312(1)-235312(8).
[42]Stephan Braig and Karsten Flensberg. Vibrational sidebands and dissipative tunneling in molecular transistors [J].Phys.Rev. B,2003,68:205324(1)-205324(10).
[43]D. C.Langreth, J. T. Devreese and V.E.Van Doren. Linear and Nonlinear Electron Transport in Solids [M].New York, Plenum,1976.
[44]李正中固体物理,第二版,高等教育出版社,2002
[45]Hartmut Haug and Antti-Pekka jauho.Quantum Kinetics in Transport and Optics[M]. Second Edition. Germany, Springer,2007,52-54.
[46]Jurgen Konig and Yuval Gefen. Coherence and Partial Coherence in Interacting Electron Systems[J].Phys Rev Lett,2001,86:3855-3858
[47]Yang Ji and M.Heiblum. Transmission Phase of a Quantum Dot with Kondo Correlation near the Unitary Limit[J].Phys Rev Lett,2002,88:076601(1)-076601(4).
[48]Jurgen Konig and Yuval Gefen. Aharonov-Bohm interferometry with interacting quantum dots:Spin configurations, asymmetric interference patterns, bias-voltage-induced Aharonov-Bohm oscillations, and symmetries of transport coefficients[J].Phys Rev B,2002,65:045316(1)-045316(20).
[49]Bogdan R. Bulka and Piotr Stefanski.Fano and Kondo Resonance in Electronic Current through Nanodevices[J].Phys Rev B,2001,86:5128-515131.
[50]G D. Mahan,Many-Particle Physics[M].3rd ed. NewYork:Plenum Press,2000,33.
[51]Antti-Pekka Jauho, Ned S.Wingreen and Yigal Meir. Time-dependent transport in interacting and noninteracting resonant-tunneling systems[J].Phys Rev B,1994, 50:5528-5544.
[52]Rui-Qiang Wang and Yun-Qing Zhou. Spin-dependent inelastic transport through single molecule junctions with ferromagnetic electrodes[J].Phys Rev B,2007, 75:045318(1)-045318(10).
[53]N.S.Wingreen, K.W.Jacobsen, and J.W. Wilkins.Phys. Inelastic scattering in resonant tunneling [J].Phys. Rev. B,1989,40:11834-11850.
[54]Walter Hofstetter and Jurgen Konig. Kondo Correlations and the Fano Effect in Closed Aharonov-Bohm Interferometers[J].Phys Rev Lett,2001,87:156803(1)-156803(4).
[55]O.Entin-Wohlman,Y. Imry, and A. Aharony.Effects of external radiation on biased Aharonov-Bohm rings[J].Phys. Rev. B,2004,70:075301(1)-075301(15).
[2]阎守胜,甘子钊.介观物理[M].北京大学出版社,1995：20-22.
[3]Yoseph Imry. Introduction to mesoscopic physics [M].Oxford University Press, 2002.
[4]L. P. Levy, G.Dolan, J.Dunsmuir and H.Bouchiatt. Magnetization of mesoscopic copper rings:Evidence for persistent currents [J].Phys. Rev. Lett,1990,64:2074-2077.
[5]H. Heinrich, G.Bauer, and F. Kuchar. Physics and Technology of Submicron Structures [M].1988,Springer-Verlag, Berlin.
[6]Dong Bing, H.L.Cui. Photon-phonon-assisted tunneling through a single-molecule quantum dot[J].Phys. Rev.B,2004,69:205315(1)-205315(9).
[7]R. C.Ashoori.Electrons in Artificial Atoms[J].Nature,1996,379,431.
[8]K. Tsukagoshi.B.W. Alphenaar, and H.Ago. Coherent transport of electron spin in a ferromagnetically contacted carbon nanotube [J].Nature,1999,401:572-574.
[9]J.R. Petta, S.K. Slater, and D. C.Ralph. Spin-Dependent Transport in Molecular Tunnel Junctions [J].Phys. Rev. Lett,2004,93:136601(1)-136601(4).
[10]L. Zhang, C. Wang, Y. Wei, X. Liu, and D. Davidovic. Spin-polarized electron transport through nanometer-scale Al grains [J].Phys. Rev.B,2005,72:155445(1)-(5).
[11]J. Reichert, R. Ochs, D.Beckmann, H. B.Weber, M. Mayor, and H. V. Lohneysen. Driving Current through Single Organic Molecules [J].Phys. Rev. Lett,2002,88: 176804(1)-176804(4).
[12]M. A. Reed, R. T. Bate, K. Bradshaw, W. M. Duncan, W. M. Frensley, J. W. Lee and H.D.Shin. Spatial Quantization in GaAs-AIGaAs Multiple Quantum Dots[M]. J.Vacuum. Sci.Technol. B4,1986,4:358-360.
[13]D. L. Klein, R. Roth, and A. K. L. Lim. A Single-Electron Transistor Made from a Cadmium Selenide Nanocrystal [J].Nature,1997,389,699.
[14]R. G Chambers. Shift of an Electron Interference Pattern by Enclosed Magnetic Flux [J].Phys.Rev. Lett,1960,5:3-5.
[15]A. Yacoby, M. Heiblum, D.Mahalu, and H. Shtrikman. Coherence and Phase Sensitive Measurements in a Quantum Dot [J].Phys. Rev. Lett,1995,74:4047-4050.
[16]V. Moldoveanu and M.Tolea. Controlled dephasing in single-dot Aharonov-Bohm interferometers[J].Phys. Rev. B,2007,75:045309(1):045309(9).
[17]B.Kubala and J. Konig, Aharonov-Bohm interferometry with quantum dots: scattering approach versus tunneling picture [J] Phys.Rev. B,2003,67:205303(1)-205303(8).
[18]J.Gores, D. Goldhaber-Gordon, S.Heemeyer, and M. A. Kastner. Fano resonances in electronic transport through a single-electron transistor[J].Phys. Rev. B,2000, 62:2188-2194.
[19]Kensuke Kobayashi, Hisashi Aikawa, Akira Sano, Shingo Katsumoto, and Yasuhiro Iye.Fano resonance in a quantum wire with a side-coupled quantum dot[J].Phys. Rev. B,2004,70:035319(1)-035319(9).
[20]Jinhee Kim, Jae-Ryoung Kim, Jeong-O Lee, Jong Wan Park, Hye Mi So, Nam Kim, Kicheon Kang, Kyung-Hwa Yoo, and Ju-Jin Kim. Fano Resonance in Crossed Carbon Nanotubes[J].Phys. Rev. Lett,2003,90:166403(1)-166403(4).
[21]A. A. Clerk, X. Waintal, and P. W. Brouwer. Fano Resonances as a Probe of Phase Coherence in Quantum Dots[J].Phys. Rev. Lett,2001,86:4636-4639.
[22]Dipl.-Phys.Jens Koch. Quantum transport through single-molecule devices[M]. 2006, Berlin.
[23]L. H.Yu, Z. K. Keane, J. W. Ciszek, L. Cheng, M. P. Stewart, J.M. Tour, and D. Natelson. Inelastic Electron Tunneling via Molecular Vibrations in Single-Molecule Transistors [J].Phys. Rev. Lett,2004,93:266802(1)-266802(4).
[24]N. B.Zhitenev, H. Meng, and Z. Bao. Conductance of Small Molecular Junctions [J]. Phys.Rev. Lett,2002,88:226801(1)-226801(4).
[25]V.J.Goldman, D. C. Tsui and J. E. Cunningham. Evidence for LO-phonon-emission-assisted tunnelinging in double-barrier heterostrues [J].Phys. Rev.B,1987,36:7635.
[26]M. L. Leadbeater, E. S.Alves, and L. Eave, M. Henini.Magnetic field studies of elastic scattering and optic-phonon emission in resonant-tunneling devices [J].Phys. Rev. B,1989,39:3438-3441.
[27]H. Park, J.Park, A.K.L. Lim, E.H. Anderson, A.P. Alivisatos, and P.L. McEuen. Nanomechanical oscillations in a single-C60 transistor [J].Nature,2000,407:57-60.
[28]Akiko Ueda and Mikio Eto. Resonant tunneling and Fano resonance in quantum dots with electron-phonon interaction [J].Phys Rev B,2006,73:235353(1)-235353(12).
[29]N.B.Zhitenev, H.Meng, and Z. Bao.Conductance of Small Molecular Junctions [J]. Phys. Rev. Lett,2002,88:226801(1)-226801(4).
[30]J. Park, A.N. Pasupathy, J.I. Goldsmith, C. Chang, Y.Yaish, J.R. Petta, M. Rinkoski, J.P. Sethna, H.D.Abruna, P.L. McEuen, and D.C.Ralph. Coulomb blockade and the Kondo effect in single-atom transistors [J].Nature,2002,417:722-725.
[31]Kevin D.McCarthy, Nikolay Prokof ev, and Mark T. Tuominen. Incoherent dynamics of vibrating single-molecule transistors [J].Phys Rev B,2003,67:245415(1)-245415(6).
[32]J.Koch and F. V.Oppen. Franck-Condon blockade and giant Fano factors in transport through single molecules [J].cond-mat/0409667v2.
[33]A. Mitra and I. Aleiner, A. J. Millis. Quantal and classical Phonon effects in molecular transistors:treatment[J].Phys.Rev. B,2004,69:245302(1)-245302(21).
[34]Jian-Xin Zhu and A.V.Balatsky. Theory of current and shot-noise spectroscopy in single-molecular quantum dots with a phonon mode[J].Phys Rev B,2003, 67:165326(1)-165326(7).
[35]U. Lundin and R. H. McKenzie. Temperature dependence of polaronic transport through single molecules and quantum dots [J].Phys. Rev. B,2002,66:075303(1)-075303(8).
[36]N. S.Wingreen, K. W. Jacobsen, and J.W. Wilkins. Inelastic scattering in resonant tunneling [J].Phys. Rev. B,1989,40:11834-11850.
[37]David M. T. Kuo and Y. C. Chand. Tunneling current through a quantum dot with strong electron-phonon interaction [J].Phys. Rev. B,2002,66:0853 11(1)-085311(5).
[38]P. S.Cornaglia, D. R. Grempel, and H. Ness. Quantum transport through a deformable molecular transistor[J].Phys. Rev. B,2005,71:075320(1)-075320(10).
[39]P. S.Cornaglia, H. Ness, and D.R. Grempel.Many-Body Effects on the Transport Properties of Single-Molecule Devices [J].Phys. Rev. Lett,2004,93:147201(1)-147201(4).
[40]K. Flensberg.Tunneling broadening of vibrational sidebands in molecular transistors [J].Phys. Rev. B,2003,68:205323(1)-205323(4).
[41]A. S.Alexandrov and A. M. Bratkovsky. Memory effect in a molecular quantum dot with strong electron-vibron interaction [J].Phys.Rev. B,2003,67:235312(1)-235312(8).
[42]Stephan Braig and Karsten Flensberg. Vibrational sidebands and dissipative tunneling in molecular transistors [J].Phys.Rev. B,2003,68:205324(1)-205324(10).
[43]D. C.Langreth, J. T. Devreese and V.E.Van Doren. Linear and Nonlinear Electron Transport in Solids [M].New York, Plenum,1976.
[44]李正中固体物理,第二版,高等教育出版社,2002
[45]Hartmut Haug and Antti-Pekka jauho.Quantum Kinetics in Transport and Optics[M]. Second Edition. Germany, Springer,2007,52-54.
[46]Jurgen Konig and Yuval Gefen. Coherence and Partial Coherence in Interacting Electron Systems[J].Phys Rev Lett,2001,86:3855-3858
[47]Yang Ji and M.Heiblum. Transmission Phase of a Quantum Dot with Kondo Correlation near the Unitary Limit[J].Phys Rev Lett,2002,88:076601(1)-076601(4).
[48]Jurgen Konig and Yuval Gefen. Aharonov-Bohm interferometry with interacting quantum dots:Spin configurations, asymmetric interference patterns, bias-voltage-induced Aharonov-Bohm oscillations, and symmetries of transport coefficients[J].Phys Rev B,2002,65:045316(1)-045316(20).
[49]Bogdan R. Bulka and Piotr Stefanski.Fano and Kondo Resonance in Electronic Current through Nanodevices[J].Phys Rev B,2001,86:5128-515131.
[50]G D. Mahan,Many-Particle Physics[M].3rd ed. NewYork:Plenum Press,2000,33.
[51]Antti-Pekka Jauho, Ned S.Wingreen and Yigal Meir. Time-dependent transport in interacting and noninteracting resonant-tunneling systems[J].Phys Rev B,1994, 50:5528-5544.
[52]Rui-Qiang Wang and Yun-Qing Zhou. Spin-dependent inelastic transport through single molecule junctions with ferromagnetic electrodes[J].Phys Rev B,2007, 75:045318(1)-045318(10).
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