No Access Submitted: 23 July 2018 Accepted: 27 September 2018 Published Online: 19 October 2018
J. Chem. Phys. 149, 154112 (2018);
Charge transport through molecular junctions is often described either as a purely coherent or a purely classical phenomenon, and described using the Landauer–Büttiker formalism or Marcus theory (MT), respectively. Using a generalised quantum master equation, we here derive an expression for current through a molecular junction modelled as a single electronic level coupled with a collection of thermalised vibrational modes. We demonstrate that the aforementioned theoretical approaches can be viewed as two limiting cases of this more general expression and present a series of approximations of this result valid at higher temperatures. We find that MT is often insufficient in describing the molecular charge transport characteristics and gives rise to a number of artefacts, especially at lower temperatures. Alternative expressions, retaining its mathematical simplicity, but rectifying those shortcomings, are suggested. In particular, we show how lifetime broadening can be consistently incorporated into MT, and we derive a low-temperature correction to the semi-classical Marcus hopping rates. Our results are applied to examples building on phenomenological as well as microscopically motivated electron-vibrational coupling. We expect them to be particularly useful in experimental studies of charge transport through single-molecule junctions as well as self-assembled monolayers.
The authors thank James Thomas and Bart Limburg for useful discussions and Núria Aliaga-Alcalde and co-workers for providing us with the results of the DFT calculations from Ref. 1515. E. Burzurí, J. O. Island, R. Díaz-Torres, A. Fursina, A. González-Campo, O. Roubeau, S. J. Teat, N. Aliaga-Alcalde, E. Ruiz, and H. S. van der Zant, ACS Nano 10, 2521 (2016). J.K.S. thanks the Clarendon Fund, Hertford College, and EPSRC for financial support. E.M.G. acknowledges funding from the Royal Society of Edinburgh and the Scottish Government, and J.A.M. acknowledges funding from the Royal Academy of Engineering. This project was supported by a grant from the John Templeton Foundation. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation.
  1. 1. A. Nitzan and M. A. Ratner, Science 300, 1384 (2003)., Google ScholarCrossref
  2. 2. S. V. Aradhya and L. Venkataraman, Nat. Nanotechnol. 8, 399 (2013)., Google ScholarCrossref
  3. 3. X. Cui, A. Primak, X. Zarate, J. Tomfohr, O. Sankey, A. Moore, T. Moore, D. Gust, G. Harris, and S. Lindsay, Science 294, 571 (2001)., Google ScholarCrossref
  4. 4. L. Venkataraman, J. E. Klare, C. Nuckolls, M. S. Hybertsen, and M. L. Steigerwald, Nature 442, 904 (2006)., Google ScholarCrossref
  5. 5. B. Stipe, M. Rezaei, and W. Ho, Science 280, 1732 (1998)., Google ScholarCrossref
  6. 6. G. Sedghi, V. M. García-Suárez, L. J. Esdaile, H. L. Anderson, C. J. Lambert, S. Martín, D. Bethell, S. J. Higgins, M. Elliott, N. Bennett et al., Nat. Nanotechnol. 6, 517 (2011)., Google ScholarCrossref
  7. 7. M. A. Reed, C. Zhou, C. Muller, T. Burgin, and J. Tour, Science 278, 252 (1997)., Google ScholarCrossref
  8. 8. T. Böhler, J. Grebing, A. Mayer-Gindner, H. v. Löhneysen, and E. Scheer, Nanotechnology 15, S465 (2004)., Google ScholarCrossref
  9. 9. M. L. Perrin, R. Frisenda, M. Koole, J. S. Seldenthuis, J. A. C. Gil, H. Valkenier, J. C. Hummelen, N. Renaud, F. C. Grozema, J. M. Thijssen et al., Nat. Nanotechnol. 9, 830 (2014)., Google ScholarCrossref
  10. 10. H. Park, J. Park, A. K. Lim, E. H. Anderson, A. P. Alivisatos, and P. L. McEuen, Nature 407, 57 (2000)., Google ScholarCrossref
  11. 11. H. S. van der Zant, Y.-V. Kervennic, M. Poot, K. O’Neill, Z. de Groot, J. M. Thijssen, H. B. Heersche, N. Stuhr-Hansen, T. Bjørnholm, D. Vanmaekelbergh et al., Faraday Discuss. 131, 347 (2006)., Google ScholarCrossref
  12. 12. F. Prins, A. Barreiro, J. W. Ruitenberg, J. S. Seldenthuis, N. Aliaga-Alcalde, L. M. Vandersypen, and H. S. van der Zant, Nano Lett. 11, 4607 (2011)., Google ScholarCrossref
  13. 13. J. A. Mol, C. S. Lau, W. J. Lewis, H. Sadeghi, C. Roche, A. Cnossen, J. H. Warner, C. J. Lambert, H. L. Anderson, and G. A. D. Briggs, Nanoscale 7, 13181 (2015)., Google ScholarCrossref
  14. 14. P. Gehring, J. K. Sowa, J. Cremers, Q. Wu, H. Sadeghi, Y. Sheng, J. H. Warner, C. J. Lambert, G. A. D. Briggs, and J. A. Mol, ACS Nano 11, 5325 (2017)., Google ScholarCrossref
  15. 15. E. Burzurí, J. O. Island, R. Díaz-Torres, A. Fursina, A. González-Campo, O. Roubeau, S. J. Teat, N. Aliaga-Alcalde, E. Ruiz, and H. S. van der Zant, ACS Nano 10, 2521 (2016)., Google ScholarCrossref
  16. 16. C. Jia, A. Migliore, N. Xin, S. Huang, J. Wang, Q. Yang, S. Wang, H. Chen, D. Wang, B. Feng et al., Science 352, 1443 (2016)., Google ScholarCrossref
  17. 17. C. Jia, J. Wang, C. Yao, Y. Cao, Y. Zhong, Z. Liu, Z. Liu, and X. Guo, Angew. Chem. 52, 8666 (2013)., Google ScholarCrossref
  18. 18. Q. Xu, G. Scuri, C. Mathewson, P. Kim, C. Nuckolls, and D. Bouilly, Nano Lett. 17, 5335 (2017)., Google ScholarCrossref
  19. 19. M. Galperin, M. A. Ratner, and A. Nitzan, J. Phys.: Condens. Matter 19, 103201 (2007)., Google ScholarCrossref
  20. 20. A. Riss, S. Wickenburg, L. Z. Tan, H.-Z. Tsai, Y. Kim, J. Lu, A. J. Bradley, M. M. Ugeda, K. L. Meaker, K. Watanabe et al., ACS Nano 8, 5395 (2014)., Google ScholarCrossref
  21. 21. S. Braig and K. Flensberg, Phys. Rev. B 68, 205324 (2003)., Google ScholarCrossref
  22. 22. K. Flensberg, Phys. Rev. B 68, 205323 (2003)., Google ScholarCrossref
  23. 23. J. Koch and F. von Oppen, Phys. Rev. Lett. 94, 206804 (2005)., Google ScholarCrossref
  24. 24. J. Koch, F. von Oppen, and A. Andreev, Phys. Rev. B 74, 205438 (2006)., Google ScholarCrossref
  25. 25. R. Härtle and M. Thoss, Phys. Rev. B 83, 125419 (2011)., Google ScholarCrossref
  26. 26. R. Härtle and M. Thoss, Phys. Rev. B 83, 115414 (2011)., Google ScholarCrossref
  27. 27. M. Galperin, M. A. Ratner, and A. Nitzan, Nano Lett. 5, 125 (2005)., Google ScholarCrossref
  28. 28. A. Zazunov, D. Feinberg, and T. Martin, Phys. Rev. B 73, 115405 (2006)., Google ScholarCrossref
  29. 29. M. Kilgour and D. Segal, J. Chem. Phys. 143, 024111 (2015)., Google ScholarScitation, ISI
  30. 30. S. H. Choi, C. Risko, M. C. R. Delgado, B. Kim, J.-L. Brédas, and C. D. Frisbie, J. Am. Chem. Soc. 132, 4358 (2010)., Google ScholarCrossref
  31. 31. D. Taherinia, C. E. Smith, S. Ghosh, S. O. Odoh, L. Balhorn, L. Gagliardi, C. J. Cramer, and C. D. Frisbie, ACS Nano 10, 4372 (2016)., Google ScholarCrossref
  32. 32. D. Segal and A. Nitzan, Chem. Phys. 281, 235 (2002)., Google ScholarCrossref
  33. 33. A. Nitzan, Chemical Dynamics in Condensed Phases: Relaxation, Transfer and Reactions in Condensed Molecular Systems (Oxford University Press, 2006). Google ScholarCrossref
  34. 34. J. Repp, P. Liljeroth, and G. Meyer, Nat. Phys. 6, 975 (2010)., Google ScholarCrossref
  35. 35. E. A. Osorio, M. Ruben, J. S. Seldenthuis, J. M. Lehn, and H. S. van der Zant, Small 6, 174 (2010)., Google ScholarCrossref
  36. 36. A. Pasupathy, J. Park, C. Chang, A. Soldatov, S. Lebedkin, R. Bialczak, J. Grose, L. Donev, J. Sethna, D. Ralph et al., Nano Lett. 5, 203 (2005)., Google ScholarCrossref
  37. 37. D. Secker, S. Wagner, S. Ballmann, R. Härtle, M. Thoss, and H. B. Weber, Phys. Rev. Lett. 106, 136807 (2011)., Google ScholarCrossref
  38. 38. M. Esposito and M. Galperin, Phys. Rev. B 79, 205303 (2009)., Google ScholarCrossref
  39. 39. H. Wang, I. Pshenichnyuk, R. Härtle, and M. Thoss, J. Chem. Phys. 135, 244506 (2011)., Google ScholarScitation, ISI
  40. 40. A. J. White and M. Galperin, Phys. Chem. Chem. Phys. 14, 13809 (2012)., Google ScholarCrossref
  41. 41. C. Schinabeck, A. Erpenbeck, R. Härtle, and M. Thoss, Phys. Rev. B 94, 201407 (2016)., Google ScholarCrossref
  42. 42. W. Dou, C. Schinabeck, M. Thoss, and J. E. Subotnik, J. Chem. Phys. 148, 102317 (2018)., Google ScholarScitation, ISI
  43. 43. N. A. Zimbovskaya, J. Chem. Phys. 126, 184901 (2007)., Google ScholarScitation, ISI
  44. 44. R. Gutiérrez, S. Mandal, and G. Cuniberti, Phys. Rev. B 71, 235116 (2005)., Google ScholarCrossref
  45. 45. H. Kim and D. Segal, J. Chem. Phys. 146, 164702 (2017)., Google ScholarScitation, ISI
  46. 46. J. K. Sowa, J. A. Mol, G. A. D. Briggs, and E. M. Gauger, Phys. Rev. B 95, 085423 (2017)., Google ScholarCrossref
  47. 47. N. A. Zimbovskaya and A. Nitzan, J. Chem. Phys. 148, 024303 (2018)., Google ScholarScitation, ISI
  48. 48. R. Härtle, C. Benesch, and M. Thoss, Phys. Rev. Lett. 102, 146801 (2009)., Google ScholarCrossref
  49. 49. F.-R. F. Fan, J. Yang, L. Cai, D. W. Price, S. M. Dirk, D. V. Kosynkin, Y. Yao, A. M. Rawlett, J. M. Tour, and A. J. Bard, J. Am. Chem. Soc. 124, 5550 (2002)., Google ScholarCrossref
  50. 50. J. C. Love, L. A. Estroff, J. K. Kriebel, R. G. Nuzzo, and G. M. Whitesides, Chem. Rev. 105, 1103 (2005)., Google ScholarCrossref
  51. 51. Y. Dubi, J. Phys. Chem. C 118, 21119 (2014)., Google ScholarCrossref
  52. 52. I. Lang and Y. A. Firsov, Sov. Phys. JETP 16, 1301 (1963). Google Scholar
  53. 53. G. D. Mahan, Many-Particle Physics (Springer Science & Business Media, 2013). Google Scholar
  54. 54. S. M. Barnett and P. M. Radmore, Methods in Theoretical Quantum Optics (Oxford University Press, 2002), Vol. 15. Google ScholarCrossref
  55. 55. Y. Yan, Phys. Rev. A 58, 2721 (1998)., Google ScholarCrossref
  56. 56. M. Esposito and M. Galperin, J. Phys. Chem. C 114, 20362 (2010)., Google ScholarCrossref
  57. 57. J. Jin, J. Li, Y. Liu, X.-Q. Li, and Y. Yan, J. Chem. Phys. 140, 244111 (2014)., Google ScholarScitation, ISI
  58. 58. D. Malz and A. Nunnenkamp, Phys. Rev. B 97, 165308 (2018)., Google ScholarCrossref
  59. 59. M. Galperin, A. Nitzan, and M. A. Ratner, Phys. Rev. B 73, 045314 (2006)., Google ScholarCrossref
  60. 60. M. Leijnse and M. Wegewijs, Phys. Rev. B 78, 235424 (2008)., Google ScholarCrossref
  61. 61. J. K. Sowa, J. A. Mol, G. A. D. Briggs, and E. M. Gauger, Phys. Chem. Chem. Phys. 19, 29534 (2017)., Google ScholarCrossref
  62. 62. C. Flindt, T. Novotnỳ, and A.-P. Jauho, Europhys. Lett. 69, 475 (2004)., Google ScholarCrossref
  63. 63. N. S. Wingreen, K. W. Jacobsen, and J. W. Wilkins, Phys. Rev. B 40, 11834 (1989)., Google ScholarCrossref
  64. 64. A.-P. Jauho, N. S. Wingreen, and Y. Meir, Phys. Rev. B 50, 5528 (1994)., Google ScholarCrossref
  65. 65. L. Glazman and R. Shekhter, Sov. Phys. JETP 67, 163 (1988). Google Scholar
  66. 66. C. S. Lau, H. Sadeghi, G. Rogers, S. Sangtarash, P. Dallas, K. Porfyrakis, J. Warner, C. J. Lambert, G. A. D. Briggs, and J. A. Mol, Nano Lett. 16, 170 (2015)., Google ScholarCrossref
  67. 67. S. Wu, G. Nazin, X. Chen, X. Qiu, and W. Ho, Phys. Rev. Lett. 93, 236802 (2004)., Google ScholarCrossref
  68. 68. R. Volkovich, R. Härtle, M. Thoss, and U. Peskin, Phys. Chem. Chem. Phys. 13, 14333 (2011)., Google ScholarCrossref
  69. 69. J. S. Seldenthuis, H. S. Van Der Zant, M. A. Ratner, and J. M. Thijssen, ACS Nano 2, 1445 (2008)., Google ScholarCrossref
  70. 70. V. Barone, J. Bloino, M. Biczysko, and F. Santoro, J. Chem. Theory Comput. 5, 540 (2009)., Google ScholarCrossref
  71. 71. F. Duschinsky, Acta Physicochim. URSS 7, 551 (1937). Google Scholar
  72. 72. J. Roden, W. T. Strunz, K. B. Whaley, and A. Eisfeld, J. Chem. Phys. 137, 204110 (2012)., Google ScholarScitation, ISI
  73. 73.Throughout this section, we have used a significantly smaller Γ (1 meV) than extracted in Ref. 1515. E. Burzurí, J. O. Island, R. Díaz-Torres, A. Fursina, A. González-Campo, O. Roubeau, S. J. Teat, N. Aliaga-Alcalde, E. Ruiz, and H. S. van der Zant, ACS Nano 10, 2521 (2016). Using the latter value (10 meV) washes away the vibrational features of the kind reported in Ref. 1515. E. Burzurí, J. O. Island, R. Díaz-Torres, A. Fursina, A. González-Campo, O. Roubeau, S. J. Teat, N. Aliaga-Alcalde, E. Ruiz, and H. S. van der Zant, ACS Nano 10, 2521 (2016).
  74. 74. D. Gelbwaser-Klimovsky, A. Aspuru-Guzik, M. Thoss, and U. Peskin, Nano Lett. 18, 4727 (2018)., Google ScholarCrossref
  75. 75. J. K. Sowa, J. A. Mol, G. A. D. Briggs, and E. M. Gauger, J. Phys. Chem. Lett. 9, 1859 (2018)., Google ScholarCrossref
  76. 76. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, 2002). Google Scholar
  77. 77. V. May and O. Kühn, Charge and Energy Transfer Dynamics in Molecular Systems (John Wiley & Sons, 2008). Google Scholar
  78. 78. R. A. Marcus and N. Sutin, Biochim. Biophys. Acta 811, 265 (1985)., Google ScholarCrossref
  79. 79. R. A. Marcus, J. Chem. Phys. 24, 966 (1956)., Google ScholarScitation, ISI
  80. 80. C. E. Chidsey, Science 251, 919 (1991)., Google ScholarCrossref
  81. 81. Y. Zeng, R. B. Smith, P. Bai, and M. Z. Bazant, J. Electroanal. Chem. 735, 77 (2014)., Google ScholarCrossref
  82. 82. H. Finklea, K. Yoon, E. Chamberlain, J. Allen, and R. Haddox, J. Phys. Chem. B 105, 3088 (2001)., Google ScholarCrossref
  83. 83. M. C. Henstridge, E. Laborda, N. V. Rees, and R. G. Compton, Electrochim. Acta 84, 12 (2012)., Google ScholarCrossref
  84. 84. A. Migliore and A. Nitzan, J. Am. Chem. Soc. 135, 9420 (2013)., Google ScholarCrossref
  85. 85. A. Migliore, P. Schiff, and A. Nitzan, Phys. Chem. Chem. Phys. 14, 13746 (2012)., Google ScholarCrossref
  86. 86. A. M. Kuznetsov and I. G. Medvedev, Phys. Rev. B 78, 153403 (2008)., Google ScholarCrossref
  87. 87. A. M. Kuznetsov, I. G. Medvedev, and J. Ulstrup, J. Chem. Phys. 131, 164703 (2009)., Google ScholarScitation, ISI
  88. 88. L. Yuan, L. Wang, A. R. Garrigues, L. Jiang, H. V. Annadata, M. A. Antonana, E. Barco, and C. A. Nijhuis, Nat. Nanotechnol. 13, 322 (2018)., Google ScholarCrossref
  89. 89. P. R. Bueno, T. A. Benites, and J. J. Davis, Sci. Rep. 6, 18400 (2016)., Google ScholarCrossref
  90. 90. A. Migliore and A. Nitzan, ACS Nano 5, 6669 (2011)., Google ScholarCrossref
  91. 91. J. O. Thomas, B. Limburg, J. K. Sowa, K. Willick, J. Baugh, G. A. D. Briggs, E. M. Gauger, H. L. Anderson, and J. A. Mol, “Environment-Dependent Electron Transfer in Single-Porphyrin Transistors” (unpublished). Google Scholar
  92. 92. N. J. Kay, S. J. Higgins, J. O. Jeppesen, E. Leary, J. Lycoops, J. Ulstrup, and R. J. Nichols, J. Am. Chem. Soc. 134, 16817 (2012)., Google ScholarCrossref
  93. 93. J. Zhang, A. M. Kuznetsov, I. G. Medvedev, Q. Chi, T. Albrecht, P. S. Jensen, and J. Ulstrup, Chem. Rev. 108, 2737 (2008)., Google ScholarCrossref
  94. 94. B. Capozzi, Q. Chen, P. Darancet, M. Kotiuga, M. Buzzeo, J. B. Neaton, C. Nuckolls, and L. Venkataraman, Nano Lett. 14, 1400 (2014)., Google ScholarCrossref
  95. 95. B. Choi, B. Capozzi, S. Ahn, A. Turkiewicz, G. Lovat, C. Nuckolls, M. L. Steigerwald, L. Venkataraman, and X. Roy, Chem. Sci. 7, 2701 (2016)., Google ScholarCrossref
  96. 96. V. Fatemi, M. Kamenetska, J. Neaton, and L. Venkataraman, Nano Lett. 11, 1988 (2011)., Google ScholarCrossref
  97. 97. D. C. Milan, O. A. Al-Owaedi, M.-C. Oerthel, S. Marqués-González, R. J. Brooke, M. R. Bryce, P. Cea, J. Ferrer, S. J. Higgins, C. J. Lambert et al., J. Phys. Chem. C 120, 15666 (2015)., Google ScholarCrossref
  98. 98. S. Fatayer, B. Schuler, W. Steurer, I. Scivetti, J. Repp, L. Gross, M. Persson, and G. Meyer, Nat. Nanotechnol. 13, 376 (2018)., Google ScholarCrossref
  99. 99. N. A. Zimbovskaya, Transport Properties of Molecular Junctions (Springer, 2013), Vol. 254. Google ScholarCrossref
  100. 100. M. Galperin, A. Nitzan, and M. A. Ratner, Mol. Phys. 106, 397 (2008)., Google ScholarCrossref
  101. 101. A. Mitra, I. Aleiner, and A. Millis, Phys. Rev. B 69, 245302 (2004)., Google ScholarCrossref
  1. © 2018 Author(s). Published by AIP Publishing.