Abstract

We propose an optical weighing technique with a sensitivity down to a single atom through the coupling between a surface plasmon and a suspended graphene nanoribbon resonator. The mass is determined via the vibrational frequency shift on the probe absorption spectrum while the atom attaches to the nanoribbon surface. We provide methods to separate out the signals of the ultralow frequency vibrational modes from the strong Rayleigh background, first based on the quantum coupling with a pump-probe scheme. Owing to the spectral enhancement in the surface plasmon and the ultralight mass of the nanoribbon, this scheme results in a narrow linewidth (GHz) and ultrahigh mass sensitivity (30  yg). Benefitting from the low noises, our optical mass sensor can be achieved at room temperature and reach ultrahigh time resolution.

© 2018 Chinese Laser Press

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References

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    [Crossref]
  22. R. Gillen, M. Mohr, and J. Maultzsch, “Symmetry properties of vibrational modes in graphene nanoribbons,” Phys. Rev. B 81, 205426 (2010).
    [Crossref]
  23. M. K. Schmidt, R. Esteban, A. Gonzalez-Tudela, G. Giedke, and J. Aizpurua, “Quantum mechanical description of Raman scattering from molecules in plasmonic cavities,” ACS Nano 10, 6291–6298 (2016).
    [Crossref]
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    [Crossref]
  29. C. Chen, S. Rosenblatt, K. I. Bolotin, W. Kalb, P. Kim, I. Kymissis, H. L. Stormer, T. F. Heinz, and J. Hone, “Performance of monolayer graphene nanomechanical resonators with electrical readout,” Nat. Nanotechnol. 4, 861–867 (2009).
    [Crossref]
  30. A. M. van der Zande, R. A. Barton, J. S. Alden, C. S. Ruiz-Vargas, W. S. Whitney, P. H. Q. Pham, J. Park, J. M. Parpia, H. G. Craighead, and P. L. McEuen, “Large-scale arrays of single-layer graphene resonators,” Nano Lett. 10, 4869–4873 (2010).
    [Crossref]
  31. V. Singh, S. Sengupta, H. S. Solanki, R. Dhall, A. Allain, S. Dhara, P. Pant, and M. M. Deshmukh, “Probing thermal expansion of graphene and modal dispersion at low-temperature using graphene nanoelectromechanical systems resonators,” Nanotechnology 21, 165204 (2010).
    [Crossref]
  32. B. R. Goldsmith, J. G. Coroneus, V. R. Khalap, A. A. Kane, G. A. Weiss, and P. G. Collins, “Conductance-controlled point functionalization of single-walled carbon nanotubes,” Science 315, 77–81 (2007).
    [Crossref]
  33. K. L. Ekinci, Y. T. Yang, and M. L. Roukes, “Ultimate limits to inertial mass sensing based upon nanoelectromechanical systems,” J. Appl. Phys. 95, 2682–2689 (2004).
    [Crossref]
  34. C. Jiang, B. Chen, J. J. Li, and K. D. Zhu, “Mass sensing based on a circuit cavity electromechanical system,” J. Appl. Phys. 110, 083107 (2011).
    [Crossref]

2016 (3)

R. Chikkaraddy, B. Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535, 127–130 (2016).
[Crossref]

P. Roelli, C. Galland, N. Piro, and T. J. Kippenberg, “Molecular cavity optomechanics as a theory of plasmon-enhanced Raman scattering,” Nat. Nanotechnol. 11, 164–169 (2016).
[Crossref]

M. K. Schmidt, R. Esteban, A. Gonzalez-Tudela, G. Giedke, and J. Aizpurua, “Quantum mechanical description of Raman scattering from molecules in plasmonic cavities,” ACS Nano 10, 6291–6298 (2016).
[Crossref]

2014 (2)

S. Chun, Y. Kim, H. Jin, E. Choi, S. B. Lee, and W. Park, “A graphene force sensor with pressure-amplifying structure,” Carbon 78, 601–608 (2014).
[Crossref]

W. Zhu and K. B. Crozier, “Quantum mechanical limit to plasmonic enhancement as observed by surface-enhanced Raman scattering,” Nat. Commun. 5, 5228 (2014).
[Crossref]

2013 (3)

D. Wang, W. Zhu, M. D. Best, J. P. Camden, and K. B. Crozier, “Directional Raman scattering from single molecules in the feed gaps of optical antennas,” Nano Lett. 13, 2194–2198 (2013).
[Crossref]

R. Zhang, Y. Zhang, Z. C. Dong, S. Jiang, C. Zhang, L. G. Chen, L. Zhang, Y. Liao, J. Aizpurua, Y. Lou, J. L. Yang, and J. G. Hou, “Chemical mapping of a single molecule by plasmon-enhanced Raman scattering,” Nature 498, 82–86 (2013).
[Crossref]

J. J. Li and K. D. Zhu, “All-optical mass sensing with coupled mechanical resonator systems,” Phys. Rep. 525, 223–254 (2013).
[Crossref]

2012 (3)

J. J. Li and K. D. Zhu, “Weighing a single atom using a coupled plasmon-carbon nanotube system,” Sci. Tech. Adv. Mater. 13, 025006 (2012).
[Crossref]

M. D. Sonntag, J. M. Klingsporn, L. K. Garibay, J. M. Roberts, J. A. Dieringer, T. Seideman, K. A. Scheidt, L. Jensen, G. C. Schatz, and R. P. Van Duyne, “Single-molecule tip-enhanced Raman spectroscopy,” J. Phys. Chem. C 116, 478–483 (2012).
[Crossref]

J. Chaste, A. Eichler, J. Moser, G. Ceballos, R. Rurali, and A. Bachtold, “A nanomechanical mass sensor with yoctogram resolution,” Nat. Nanotechnol. 7, 301–304 (2012).
[Crossref]

2011 (1)

C. Jiang, B. Chen, J. J. Li, and K. D. Zhu, “Mass sensing based on a circuit cavity electromechanical system,” J. Appl. Phys. 110, 083107 (2011).
[Crossref]

2010 (5)

A. M. van der Zande, R. A. Barton, J. S. Alden, C. S. Ruiz-Vargas, W. S. Whitney, P. H. Q. Pham, J. Park, J. M. Parpia, H. G. Craighead, and P. L. McEuen, “Large-scale arrays of single-layer graphene resonators,” Nano Lett. 10, 4869–4873 (2010).
[Crossref]

V. Singh, S. Sengupta, H. S. Solanki, R. Dhall, A. Allain, S. Dhara, P. Pant, and M. M. Deshmukh, “Probing thermal expansion of graphene and modal dispersion at low-temperature using graphene nanoelectromechanical systems resonators,” Nanotechnology 21, 165204 (2010).
[Crossref]

M. Sadeghi and R. Naghdabadi, “Nonlinear vibrational analysis of single-layer graphene sheets,” Nanotechnology 21, 105705 (2010).
[Crossref]

R. Gillen, M. Mohr, and J. Maultzsch, “Symmetry properties of vibrational modes in graphene nanoribbons,” Phys. Rev. B 81, 205426 (2010).
[Crossref]

R. Narula, R. Panknin, and S. Reich, “Absolute Raman matrix elements of graphene and graphite,” Phys. Rev. B 82, 045418 (2010).
[Crossref]

2009 (3)

C. Chen, S. Rosenblatt, K. I. Bolotin, W. Kalb, P. Kim, I. Kymissis, H. L. Stormer, T. F. Heinz, and J. Hone, “Performance of monolayer graphene nanomechanical resonators with electrical readout,” Nat. Nanotechnol. 4, 861–867 (2009).
[Crossref]

A. K. Naik, M. S. Hanay, W. K. Hiebert, X. L. Feng, and M. L. Roukes, “Towards single-molecule nanomechanical mass spectrometry,” Nat. Nanotechnol. 4, 445–450 (2009).
[Crossref]

E. Gil-Santos, D. Ramos, A. Jana, M. Calleja, A. Raman, and J. Tamayo, “Mass sensing based on deterministic and stochastic responses of elastically coupled nanocantilevers,” Nano Lett. 9, 4122–4127 (2009).
[Crossref]

2008 (4)

H. Y. Chiu, P. Hung, H. W. C. Postma, and M. Bockrath, “Atomic-scale mass sensing using carbon nanotube resonators,” Nano Lett. 8, 4342–4346 (2008).
[Crossref]

K. Jensen, K. Kim, and A. Zettl, “An atomic-resolution nanomechanical mass sensor,” Nat. Nanotechnol. 3, 533–537 (2008).
[Crossref]

A. Sakhaee-Pour, M. T. Ahmadian, and R. Naghdabadi, “Vibrational analysis of single-layered graphene sheets,” Nanotechnology 19, 085702 (2008).
[Crossref]

J. P. Camden, J. A. Dieringer, Y. Wang, D. J. Masiello, L. D. Marks, G. C. Schatz, and R. P. Van Duyne, “Probing the structure of single-molecule surface-enhanced Raman scattering hot spots,” J. Am. Chem. Soc. 130, 12616–12617 (2008).
[Crossref]

2007 (2)

B. R. Goldsmith, J. G. Coroneus, V. R. Khalap, A. A. Kane, G. A. Weiss, and P. G. Collins, “Conductance-controlled point functionalization of single-walled carbon nanotubes,” Science 315, 77–81 (2007).
[Crossref]

J. S. Bunch, A. M. van der Zande, S. S. Verbridge, I. W. Frank, D. M. Tanenbaum, J. M. Parpia, H. G. Craighead, and P. L. McEuen, “Electromechanical resonators from graphene sheets,” Science 315, 490–493 (2007).
[Crossref]

2004 (2)

M. LaHaye, O. Buu, B. Camarota, and K. Schwab, “Approaching the quantum limit of a nanomechanical resonator,” Science 304, 74–77 (2004).
[Crossref]

K. L. Ekinci, Y. T. Yang, and M. L. Roukes, “Ultimate limits to inertial mass sensing based upon nanoelectromechanical systems,” J. Appl. Phys. 95, 2682–2689 (2004).
[Crossref]

2001 (1)

V. Giovannetti and D. Vitali, “Phase-noise measurement in a cavity with a movable mirror undergoing quantum Brownian motion,” Phys. Rev. A 63, 023812 (2001).
[Crossref]

1997 (1)

S. M. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275, 1102–1106 (1997).
[Crossref]

1992 (1)

A. Otto, I. Mrozek, H. Grabhorn, and W. Akemann, “Surface-enhanced Raman scattering,” J. Phys. Condens. Matter 4, 1143–1212 (1992).
[Crossref]

1985 (1)

M. Moskovits, “Surface-enhanced spectroscopy,” Rev. Mod. Phys. 57, 783–826 (1985).
[Crossref]

Ahmadian, M. T.

A. Sakhaee-Pour, M. T. Ahmadian, and R. Naghdabadi, “Vibrational analysis of single-layered graphene sheets,” Nanotechnology 19, 085702 (2008).
[Crossref]

Aizpurua, J.

M. K. Schmidt, R. Esteban, A. Gonzalez-Tudela, G. Giedke, and J. Aizpurua, “Quantum mechanical description of Raman scattering from molecules in plasmonic cavities,” ACS Nano 10, 6291–6298 (2016).
[Crossref]

R. Zhang, Y. Zhang, Z. C. Dong, S. Jiang, C. Zhang, L. G. Chen, L. Zhang, Y. Liao, J. Aizpurua, Y. Lou, J. L. Yang, and J. G. Hou, “Chemical mapping of a single molecule by plasmon-enhanced Raman scattering,” Nature 498, 82–86 (2013).
[Crossref]

Akemann, W.

A. Otto, I. Mrozek, H. Grabhorn, and W. Akemann, “Surface-enhanced Raman scattering,” J. Phys. Condens. Matter 4, 1143–1212 (1992).
[Crossref]

Alden, J. S.

A. M. van der Zande, R. A. Barton, J. S. Alden, C. S. Ruiz-Vargas, W. S. Whitney, P. H. Q. Pham, J. Park, J. M. Parpia, H. G. Craighead, and P. L. McEuen, “Large-scale arrays of single-layer graphene resonators,” Nano Lett. 10, 4869–4873 (2010).
[Crossref]

Allain, A.

V. Singh, S. Sengupta, H. S. Solanki, R. Dhall, A. Allain, S. Dhara, P. Pant, and M. M. Deshmukh, “Probing thermal expansion of graphene and modal dispersion at low-temperature using graphene nanoelectromechanical systems resonators,” Nanotechnology 21, 165204 (2010).
[Crossref]

Bachtold, A.

J. Chaste, A. Eichler, J. Moser, G. Ceballos, R. Rurali, and A. Bachtold, “A nanomechanical mass sensor with yoctogram resolution,” Nat. Nanotechnol. 7, 301–304 (2012).
[Crossref]

Barrow, S. J.

R. Chikkaraddy, B. Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535, 127–130 (2016).
[Crossref]

Barton, R. A.

A. M. van der Zande, R. A. Barton, J. S. Alden, C. S. Ruiz-Vargas, W. S. Whitney, P. H. Q. Pham, J. Park, J. M. Parpia, H. G. Craighead, and P. L. McEuen, “Large-scale arrays of single-layer graphene resonators,” Nano Lett. 10, 4869–4873 (2010).
[Crossref]

Baumberg, J. J.

R. Chikkaraddy, B. Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535, 127–130 (2016).
[Crossref]

Benz, F.

R. Chikkaraddy, B. Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535, 127–130 (2016).
[Crossref]

Best, M. D.

D. Wang, W. Zhu, M. D. Best, J. P. Camden, and K. B. Crozier, “Directional Raman scattering from single molecules in the feed gaps of optical antennas,” Nano Lett. 13, 2194–2198 (2013).
[Crossref]

Bockrath, M.

H. Y. Chiu, P. Hung, H. W. C. Postma, and M. Bockrath, “Atomic-scale mass sensing using carbon nanotube resonators,” Nano Lett. 8, 4342–4346 (2008).
[Crossref]

Bolotin, K. I.

C. Chen, S. Rosenblatt, K. I. Bolotin, W. Kalb, P. Kim, I. Kymissis, H. L. Stormer, T. F. Heinz, and J. Hone, “Performance of monolayer graphene nanomechanical resonators with electrical readout,” Nat. Nanotechnol. 4, 861–867 (2009).
[Crossref]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, 2008).

Bunch, J. S.

J. S. Bunch, A. M. van der Zande, S. S. Verbridge, I. W. Frank, D. M. Tanenbaum, J. M. Parpia, H. G. Craighead, and P. L. McEuen, “Electromechanical resonators from graphene sheets,” Science 315, 490–493 (2007).
[Crossref]

Buu, O.

M. LaHaye, O. Buu, B. Camarota, and K. Schwab, “Approaching the quantum limit of a nanomechanical resonator,” Science 304, 74–77 (2004).
[Crossref]

Calleja, M.

E. Gil-Santos, D. Ramos, A. Jana, M. Calleja, A. Raman, and J. Tamayo, “Mass sensing based on deterministic and stochastic responses of elastically coupled nanocantilevers,” Nano Lett. 9, 4122–4127 (2009).
[Crossref]

Camarota, B.

M. LaHaye, O. Buu, B. Camarota, and K. Schwab, “Approaching the quantum limit of a nanomechanical resonator,” Science 304, 74–77 (2004).
[Crossref]

Camden, J. P.

D. Wang, W. Zhu, M. D. Best, J. P. Camden, and K. B. Crozier, “Directional Raman scattering from single molecules in the feed gaps of optical antennas,” Nano Lett. 13, 2194–2198 (2013).
[Crossref]

J. P. Camden, J. A. Dieringer, Y. Wang, D. J. Masiello, L. D. Marks, G. C. Schatz, and R. P. Van Duyne, “Probing the structure of single-molecule surface-enhanced Raman scattering hot spots,” J. Am. Chem. Soc. 130, 12616–12617 (2008).
[Crossref]

Ceballos, G.

J. Chaste, A. Eichler, J. Moser, G. Ceballos, R. Rurali, and A. Bachtold, “A nanomechanical mass sensor with yoctogram resolution,” Nat. Nanotechnol. 7, 301–304 (2012).
[Crossref]

Chaste, J.

J. Chaste, A. Eichler, J. Moser, G. Ceballos, R. Rurali, and A. Bachtold, “A nanomechanical mass sensor with yoctogram resolution,” Nat. Nanotechnol. 7, 301–304 (2012).
[Crossref]

Chen, B.

C. Jiang, B. Chen, J. J. Li, and K. D. Zhu, “Mass sensing based on a circuit cavity electromechanical system,” J. Appl. Phys. 110, 083107 (2011).
[Crossref]

Chen, C.

C. Chen, S. Rosenblatt, K. I. Bolotin, W. Kalb, P. Kim, I. Kymissis, H. L. Stormer, T. F. Heinz, and J. Hone, “Performance of monolayer graphene nanomechanical resonators with electrical readout,” Nat. Nanotechnol. 4, 861–867 (2009).
[Crossref]

Chen, L. G.

R. Zhang, Y. Zhang, Z. C. Dong, S. Jiang, C. Zhang, L. G. Chen, L. Zhang, Y. Liao, J. Aizpurua, Y. Lou, J. L. Yang, and J. G. Hou, “Chemical mapping of a single molecule by plasmon-enhanced Raman scattering,” Nature 498, 82–86 (2013).
[Crossref]

Chikkaraddy, R.

R. Chikkaraddy, B. Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535, 127–130 (2016).
[Crossref]

Chiu, H. Y.

H. Y. Chiu, P. Hung, H. W. C. Postma, and M. Bockrath, “Atomic-scale mass sensing using carbon nanotube resonators,” Nano Lett. 8, 4342–4346 (2008).
[Crossref]

Choi, E.

S. Chun, Y. Kim, H. Jin, E. Choi, S. B. Lee, and W. Park, “A graphene force sensor with pressure-amplifying structure,” Carbon 78, 601–608 (2014).
[Crossref]

Chun, S.

S. Chun, Y. Kim, H. Jin, E. Choi, S. B. Lee, and W. Park, “A graphene force sensor with pressure-amplifying structure,” Carbon 78, 601–608 (2014).
[Crossref]

Collins, P. G.

B. R. Goldsmith, J. G. Coroneus, V. R. Khalap, A. A. Kane, G. A. Weiss, and P. G. Collins, “Conductance-controlled point functionalization of single-walled carbon nanotubes,” Science 315, 77–81 (2007).
[Crossref]

Coroneus, J. G.

B. R. Goldsmith, J. G. Coroneus, V. R. Khalap, A. A. Kane, G. A. Weiss, and P. G. Collins, “Conductance-controlled point functionalization of single-walled carbon nanotubes,” Science 315, 77–81 (2007).
[Crossref]

Craighead, H. G.

A. M. van der Zande, R. A. Barton, J. S. Alden, C. S. Ruiz-Vargas, W. S. Whitney, P. H. Q. Pham, J. Park, J. M. Parpia, H. G. Craighead, and P. L. McEuen, “Large-scale arrays of single-layer graphene resonators,” Nano Lett. 10, 4869–4873 (2010).
[Crossref]

J. S. Bunch, A. M. van der Zande, S. S. Verbridge, I. W. Frank, D. M. Tanenbaum, J. M. Parpia, H. G. Craighead, and P. L. McEuen, “Electromechanical resonators from graphene sheets,” Science 315, 490–493 (2007).
[Crossref]

Crozier, K. B.

W. Zhu and K. B. Crozier, “Quantum mechanical limit to plasmonic enhancement as observed by surface-enhanced Raman scattering,” Nat. Commun. 5, 5228 (2014).
[Crossref]

D. Wang, W. Zhu, M. D. Best, J. P. Camden, and K. B. Crozier, “Directional Raman scattering from single molecules in the feed gaps of optical antennas,” Nano Lett. 13, 2194–2198 (2013).
[Crossref]

Demetriadou, A.

R. Chikkaraddy, B. Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535, 127–130 (2016).
[Crossref]

Deshmukh, M. M.

V. Singh, S. Sengupta, H. S. Solanki, R. Dhall, A. Allain, S. Dhara, P. Pant, and M. M. Deshmukh, “Probing thermal expansion of graphene and modal dispersion at low-temperature using graphene nanoelectromechanical systems resonators,” Nanotechnology 21, 165204 (2010).
[Crossref]

Dhall, R.

V. Singh, S. Sengupta, H. S. Solanki, R. Dhall, A. Allain, S. Dhara, P. Pant, and M. M. Deshmukh, “Probing thermal expansion of graphene and modal dispersion at low-temperature using graphene nanoelectromechanical systems resonators,” Nanotechnology 21, 165204 (2010).
[Crossref]

Dhara, S.

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M. D. Sonntag, J. M. Klingsporn, L. K. Garibay, J. M. Roberts, J. A. Dieringer, T. Seideman, K. A. Scheidt, L. Jensen, G. C. Schatz, and R. P. Van Duyne, “Single-molecule tip-enhanced Raman spectroscopy,” J. Phys. Chem. C 116, 478–483 (2012).
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J. Chaste, A. Eichler, J. Moser, G. Ceballos, R. Rurali, and A. Bachtold, “A nanomechanical mass sensor with yoctogram resolution,” Nat. Nanotechnol. 7, 301–304 (2012).
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A. K. Naik, M. S. Hanay, W. K. Hiebert, X. L. Feng, and M. L. Roukes, “Towards single-molecule nanomechanical mass spectrometry,” Nat. Nanotechnol. 4, 445–450 (2009).
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R. Chikkaraddy, B. Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535, 127–130 (2016).
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J. S. Bunch, A. M. van der Zande, S. S. Verbridge, I. W. Frank, D. M. Tanenbaum, J. M. Parpia, H. G. Craighead, and P. L. McEuen, “Electromechanical resonators from graphene sheets,” Science 315, 490–493 (2007).
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M. D. Sonntag, J. M. Klingsporn, L. K. Garibay, J. M. Roberts, J. A. Dieringer, T. Seideman, K. A. Scheidt, L. Jensen, G. C. Schatz, and R. P. Van Duyne, “Single-molecule tip-enhanced Raman spectroscopy,” J. Phys. Chem. C 116, 478–483 (2012).
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M. K. Schmidt, R. Esteban, A. Gonzalez-Tudela, G. Giedke, and J. Aizpurua, “Quantum mechanical description of Raman scattering from molecules in plasmonic cavities,” ACS Nano 10, 6291–6298 (2016).
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A. Otto, I. Mrozek, H. Grabhorn, and W. Akemann, “Surface-enhanced Raman scattering,” J. Phys. Condens. Matter 4, 1143–1212 (1992).
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A. K. Naik, M. S. Hanay, W. K. Hiebert, X. L. Feng, and M. L. Roukes, “Towards single-molecule nanomechanical mass spectrometry,” Nat. Nanotechnol. 4, 445–450 (2009).
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R. Chikkaraddy, B. Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535, 127–130 (2016).
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A. K. Naik, M. S. Hanay, W. K. Hiebert, X. L. Feng, and M. L. Roukes, “Towards single-molecule nanomechanical mass spectrometry,” Nat. Nanotechnol. 4, 445–450 (2009).
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C. Chen, S. Rosenblatt, K. I. Bolotin, W. Kalb, P. Kim, I. Kymissis, H. L. Stormer, T. F. Heinz, and J. Hone, “Performance of monolayer graphene nanomechanical resonators with electrical readout,” Nat. Nanotechnol. 4, 861–867 (2009).
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R. Zhang, Y. Zhang, Z. C. Dong, S. Jiang, C. Zhang, L. G. Chen, L. Zhang, Y. Liao, J. Aizpurua, Y. Lou, J. L. Yang, and J. G. Hou, “Chemical mapping of a single molecule by plasmon-enhanced Raman scattering,” Nature 498, 82–86 (2013).
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H. Y. Chiu, P. Hung, H. W. C. Postma, and M. Bockrath, “Atomic-scale mass sensing using carbon nanotube resonators,” Nano Lett. 8, 4342–4346 (2008).
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E. Gil-Santos, D. Ramos, A. Jana, M. Calleja, A. Raman, and J. Tamayo, “Mass sensing based on deterministic and stochastic responses of elastically coupled nanocantilevers,” Nano Lett. 9, 4122–4127 (2009).
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M. D. Sonntag, J. M. Klingsporn, L. K. Garibay, J. M. Roberts, J. A. Dieringer, T. Seideman, K. A. Scheidt, L. Jensen, G. C. Schatz, and R. P. Van Duyne, “Single-molecule tip-enhanced Raman spectroscopy,” J. Phys. Chem. C 116, 478–483 (2012).
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R. Zhang, Y. Zhang, Z. C. Dong, S. Jiang, C. Zhang, L. G. Chen, L. Zhang, Y. Liao, J. Aizpurua, Y. Lou, J. L. Yang, and J. G. Hou, “Chemical mapping of a single molecule by plasmon-enhanced Raman scattering,” Nature 498, 82–86 (2013).
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C. Chen, S. Rosenblatt, K. I. Bolotin, W. Kalb, P. Kim, I. Kymissis, H. L. Stormer, T. F. Heinz, and J. Hone, “Performance of monolayer graphene nanomechanical resonators with electrical readout,” Nat. Nanotechnol. 4, 861–867 (2009).
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B. R. Goldsmith, J. G. Coroneus, V. R. Khalap, A. A. Kane, G. A. Weiss, and P. G. Collins, “Conductance-controlled point functionalization of single-walled carbon nanotubes,” Science 315, 77–81 (2007).
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B. R. Goldsmith, J. G. Coroneus, V. R. Khalap, A. A. Kane, G. A. Weiss, and P. G. Collins, “Conductance-controlled point functionalization of single-walled carbon nanotubes,” Science 315, 77–81 (2007).
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K. Jensen, K. Kim, and A. Zettl, “An atomic-resolution nanomechanical mass sensor,” Nat. Nanotechnol. 3, 533–537 (2008).
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C. Chen, S. Rosenblatt, K. I. Bolotin, W. Kalb, P. Kim, I. Kymissis, H. L. Stormer, T. F. Heinz, and J. Hone, “Performance of monolayer graphene nanomechanical resonators with electrical readout,” Nat. Nanotechnol. 4, 861–867 (2009).
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S. Chun, Y. Kim, H. Jin, E. Choi, S. B. Lee, and W. Park, “A graphene force sensor with pressure-amplifying structure,” Carbon 78, 601–608 (2014).
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P. Roelli, C. Galland, N. Piro, and T. J. Kippenberg, “Molecular cavity optomechanics as a theory of plasmon-enhanced Raman scattering,” Nat. Nanotechnol. 11, 164–169 (2016).
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M. D. Sonntag, J. M. Klingsporn, L. K. Garibay, J. M. Roberts, J. A. Dieringer, T. Seideman, K. A. Scheidt, L. Jensen, G. C. Schatz, and R. P. Van Duyne, “Single-molecule tip-enhanced Raman spectroscopy,” J. Phys. Chem. C 116, 478–483 (2012).
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C. Chen, S. Rosenblatt, K. I. Bolotin, W. Kalb, P. Kim, I. Kymissis, H. L. Stormer, T. F. Heinz, and J. Hone, “Performance of monolayer graphene nanomechanical resonators with electrical readout,” Nat. Nanotechnol. 4, 861–867 (2009).
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M. LaHaye, O. Buu, B. Camarota, and K. Schwab, “Approaching the quantum limit of a nanomechanical resonator,” Science 304, 74–77 (2004).
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S. Chun, Y. Kim, H. Jin, E. Choi, S. B. Lee, and W. Park, “A graphene force sensor with pressure-amplifying structure,” Carbon 78, 601–608 (2014).
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R. Zhang, Y. Zhang, Z. C. Dong, S. Jiang, C. Zhang, L. G. Chen, L. Zhang, Y. Liao, J. Aizpurua, Y. Lou, J. L. Yang, and J. G. Hou, “Chemical mapping of a single molecule by plasmon-enhanced Raman scattering,” Nature 498, 82–86 (2013).
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R. Zhang, Y. Zhang, Z. C. Dong, S. Jiang, C. Zhang, L. G. Chen, L. Zhang, Y. Liao, J. Aizpurua, Y. Lou, J. L. Yang, and J. G. Hou, “Chemical mapping of a single molecule by plasmon-enhanced Raman scattering,” Nature 498, 82–86 (2013).
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J. P. Camden, J. A. Dieringer, Y. Wang, D. J. Masiello, L. D. Marks, G. C. Schatz, and R. P. Van Duyne, “Probing the structure of single-molecule surface-enhanced Raman scattering hot spots,” J. Am. Chem. Soc. 130, 12616–12617 (2008).
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J. P. Camden, J. A. Dieringer, Y. Wang, D. J. Masiello, L. D. Marks, G. C. Schatz, and R. P. Van Duyne, “Probing the structure of single-molecule surface-enhanced Raman scattering hot spots,” J. Am. Chem. Soc. 130, 12616–12617 (2008).
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R. Gillen, M. Mohr, and J. Maultzsch, “Symmetry properties of vibrational modes in graphene nanoribbons,” Phys. Rev. B 81, 205426 (2010).
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A. M. van der Zande, R. A. Barton, J. S. Alden, C. S. Ruiz-Vargas, W. S. Whitney, P. H. Q. Pham, J. Park, J. M. Parpia, H. G. Craighead, and P. L. McEuen, “Large-scale arrays of single-layer graphene resonators,” Nano Lett. 10, 4869–4873 (2010).
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R. Gillen, M. Mohr, and J. Maultzsch, “Symmetry properties of vibrational modes in graphene nanoribbons,” Phys. Rev. B 81, 205426 (2010).
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J. Chaste, A. Eichler, J. Moser, G. Ceballos, R. Rurali, and A. Bachtold, “A nanomechanical mass sensor with yoctogram resolution,” Nat. Nanotechnol. 7, 301–304 (2012).
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A. Otto, I. Mrozek, H. Grabhorn, and W. Akemann, “Surface-enhanced Raman scattering,” J. Phys. Condens. Matter 4, 1143–1212 (1992).
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M. Sadeghi and R. Naghdabadi, “Nonlinear vibrational analysis of single-layer graphene sheets,” Nanotechnology 21, 105705 (2010).
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A. Sakhaee-Pour, M. T. Ahmadian, and R. Naghdabadi, “Vibrational analysis of single-layered graphene sheets,” Nanotechnology 19, 085702 (2008).
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A. K. Naik, M. S. Hanay, W. K. Hiebert, X. L. Feng, and M. L. Roukes, “Towards single-molecule nanomechanical mass spectrometry,” Nat. Nanotechnol. 4, 445–450 (2009).
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R. Narula, R. Panknin, and S. Reich, “Absolute Raman matrix elements of graphene and graphite,” Phys. Rev. B 82, 045418 (2010).
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S. M. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275, 1102–1106 (1997).
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R. Chikkaraddy, B. Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535, 127–130 (2016).
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A. Otto, I. Mrozek, H. Grabhorn, and W. Akemann, “Surface-enhanced Raman scattering,” J. Phys. Condens. Matter 4, 1143–1212 (1992).
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R. Narula, R. Panknin, and S. Reich, “Absolute Raman matrix elements of graphene and graphite,” Phys. Rev. B 82, 045418 (2010).
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V. Singh, S. Sengupta, H. S. Solanki, R. Dhall, A. Allain, S. Dhara, P. Pant, and M. M. Deshmukh, “Probing thermal expansion of graphene and modal dispersion at low-temperature using graphene nanoelectromechanical systems resonators,” Nanotechnology 21, 165204 (2010).
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A. M. van der Zande, R. A. Barton, J. S. Alden, C. S. Ruiz-Vargas, W. S. Whitney, P. H. Q. Pham, J. Park, J. M. Parpia, H. G. Craighead, and P. L. McEuen, “Large-scale arrays of single-layer graphene resonators,” Nano Lett. 10, 4869–4873 (2010).
[Crossref]

Park, W.

S. Chun, Y. Kim, H. Jin, E. Choi, S. B. Lee, and W. Park, “A graphene force sensor with pressure-amplifying structure,” Carbon 78, 601–608 (2014).
[Crossref]

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A. M. van der Zande, R. A. Barton, J. S. Alden, C. S. Ruiz-Vargas, W. S. Whitney, P. H. Q. Pham, J. Park, J. M. Parpia, H. G. Craighead, and P. L. McEuen, “Large-scale arrays of single-layer graphene resonators,” Nano Lett. 10, 4869–4873 (2010).
[Crossref]

J. S. Bunch, A. M. van der Zande, S. S. Verbridge, I. W. Frank, D. M. Tanenbaum, J. M. Parpia, H. G. Craighead, and P. L. McEuen, “Electromechanical resonators from graphene sheets,” Science 315, 490–493 (2007).
[Crossref]

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A. M. van der Zande, R. A. Barton, J. S. Alden, C. S. Ruiz-Vargas, W. S. Whitney, P. H. Q. Pham, J. Park, J. M. Parpia, H. G. Craighead, and P. L. McEuen, “Large-scale arrays of single-layer graphene resonators,” Nano Lett. 10, 4869–4873 (2010).
[Crossref]

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P. Roelli, C. Galland, N. Piro, and T. J. Kippenberg, “Molecular cavity optomechanics as a theory of plasmon-enhanced Raman scattering,” Nat. Nanotechnol. 11, 164–169 (2016).
[Crossref]

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H. Y. Chiu, P. Hung, H. W. C. Postma, and M. Bockrath, “Atomic-scale mass sensing using carbon nanotube resonators,” Nano Lett. 8, 4342–4346 (2008).
[Crossref]

Raman, A.

E. Gil-Santos, D. Ramos, A. Jana, M. Calleja, A. Raman, and J. Tamayo, “Mass sensing based on deterministic and stochastic responses of elastically coupled nanocantilevers,” Nano Lett. 9, 4122–4127 (2009).
[Crossref]

Ramos, D.

E. Gil-Santos, D. Ramos, A. Jana, M. Calleja, A. Raman, and J. Tamayo, “Mass sensing based on deterministic and stochastic responses of elastically coupled nanocantilevers,” Nano Lett. 9, 4122–4127 (2009).
[Crossref]

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R. Narula, R. Panknin, and S. Reich, “Absolute Raman matrix elements of graphene and graphite,” Phys. Rev. B 82, 045418 (2010).
[Crossref]

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M. D. Sonntag, J. M. Klingsporn, L. K. Garibay, J. M. Roberts, J. A. Dieringer, T. Seideman, K. A. Scheidt, L. Jensen, G. C. Schatz, and R. P. Van Duyne, “Single-molecule tip-enhanced Raman spectroscopy,” J. Phys. Chem. C 116, 478–483 (2012).
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P. Roelli, C. Galland, N. Piro, and T. J. Kippenberg, “Molecular cavity optomechanics as a theory of plasmon-enhanced Raman scattering,” Nat. Nanotechnol. 11, 164–169 (2016).
[Crossref]

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C. Chen, S. Rosenblatt, K. I. Bolotin, W. Kalb, P. Kim, I. Kymissis, H. L. Stormer, T. F. Heinz, and J. Hone, “Performance of monolayer graphene nanomechanical resonators with electrical readout,” Nat. Nanotechnol. 4, 861–867 (2009).
[Crossref]

Rosta, E.

R. Chikkaraddy, B. Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535, 127–130 (2016).
[Crossref]

Roukes, M. L.

A. K. Naik, M. S. Hanay, W. K. Hiebert, X. L. Feng, and M. L. Roukes, “Towards single-molecule nanomechanical mass spectrometry,” Nat. Nanotechnol. 4, 445–450 (2009).
[Crossref]

K. L. Ekinci, Y. T. Yang, and M. L. Roukes, “Ultimate limits to inertial mass sensing based upon nanoelectromechanical systems,” J. Appl. Phys. 95, 2682–2689 (2004).
[Crossref]

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A. M. van der Zande, R. A. Barton, J. S. Alden, C. S. Ruiz-Vargas, W. S. Whitney, P. H. Q. Pham, J. Park, J. M. Parpia, H. G. Craighead, and P. L. McEuen, “Large-scale arrays of single-layer graphene resonators,” Nano Lett. 10, 4869–4873 (2010).
[Crossref]

Rurali, R.

J. Chaste, A. Eichler, J. Moser, G. Ceballos, R. Rurali, and A. Bachtold, “A nanomechanical mass sensor with yoctogram resolution,” Nat. Nanotechnol. 7, 301–304 (2012).
[Crossref]

Sadeghi, M.

M. Sadeghi and R. Naghdabadi, “Nonlinear vibrational analysis of single-layer graphene sheets,” Nanotechnology 21, 105705 (2010).
[Crossref]

Sakhaee-Pour, A.

A. Sakhaee-Pour, M. T. Ahmadian, and R. Naghdabadi, “Vibrational analysis of single-layered graphene sheets,” Nanotechnology 19, 085702 (2008).
[Crossref]

Schatz, G. C.

M. D. Sonntag, J. M. Klingsporn, L. K. Garibay, J. M. Roberts, J. A. Dieringer, T. Seideman, K. A. Scheidt, L. Jensen, G. C. Schatz, and R. P. Van Duyne, “Single-molecule tip-enhanced Raman spectroscopy,” J. Phys. Chem. C 116, 478–483 (2012).
[Crossref]

J. P. Camden, J. A. Dieringer, Y. Wang, D. J. Masiello, L. D. Marks, G. C. Schatz, and R. P. Van Duyne, “Probing the structure of single-molecule surface-enhanced Raman scattering hot spots,” J. Am. Chem. Soc. 130, 12616–12617 (2008).
[Crossref]

Scheidt, K. A.

M. D. Sonntag, J. M. Klingsporn, L. K. Garibay, J. M. Roberts, J. A. Dieringer, T. Seideman, K. A. Scheidt, L. Jensen, G. C. Schatz, and R. P. Van Duyne, “Single-molecule tip-enhanced Raman spectroscopy,” J. Phys. Chem. C 116, 478–483 (2012).
[Crossref]

Scherman, O. A.

R. Chikkaraddy, B. Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535, 127–130 (2016).
[Crossref]

Schmidt, M. K.

M. K. Schmidt, R. Esteban, A. Gonzalez-Tudela, G. Giedke, and J. Aizpurua, “Quantum mechanical description of Raman scattering from molecules in plasmonic cavities,” ACS Nano 10, 6291–6298 (2016).
[Crossref]

Schwab, K.

M. LaHaye, O. Buu, B. Camarota, and K. Schwab, “Approaching the quantum limit of a nanomechanical resonator,” Science 304, 74–77 (2004).
[Crossref]

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M. D. Sonntag, J. M. Klingsporn, L. K. Garibay, J. M. Roberts, J. A. Dieringer, T. Seideman, K. A. Scheidt, L. Jensen, G. C. Schatz, and R. P. Van Duyne, “Single-molecule tip-enhanced Raman spectroscopy,” J. Phys. Chem. C 116, 478–483 (2012).
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V. Singh, S. Sengupta, H. S. Solanki, R. Dhall, A. Allain, S. Dhara, P. Pant, and M. M. Deshmukh, “Probing thermal expansion of graphene and modal dispersion at low-temperature using graphene nanoelectromechanical systems resonators,” Nanotechnology 21, 165204 (2010).
[Crossref]

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V. Singh, S. Sengupta, H. S. Solanki, R. Dhall, A. Allain, S. Dhara, P. Pant, and M. M. Deshmukh, “Probing thermal expansion of graphene and modal dispersion at low-temperature using graphene nanoelectromechanical systems resonators,” Nanotechnology 21, 165204 (2010).
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Solanki, H. S.

V. Singh, S. Sengupta, H. S. Solanki, R. Dhall, A. Allain, S. Dhara, P. Pant, and M. M. Deshmukh, “Probing thermal expansion of graphene and modal dispersion at low-temperature using graphene nanoelectromechanical systems resonators,” Nanotechnology 21, 165204 (2010).
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M. D. Sonntag, J. M. Klingsporn, L. K. Garibay, J. M. Roberts, J. A. Dieringer, T. Seideman, K. A. Scheidt, L. Jensen, G. C. Schatz, and R. P. Van Duyne, “Single-molecule tip-enhanced Raman spectroscopy,” J. Phys. Chem. C 116, 478–483 (2012).
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A. M. van der Zande, R. A. Barton, J. S. Alden, C. S. Ruiz-Vargas, W. S. Whitney, P. H. Q. Pham, J. Park, J. M. Parpia, H. G. Craighead, and P. L. McEuen, “Large-scale arrays of single-layer graphene resonators,” Nano Lett. 10, 4869–4873 (2010).
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R. Zhang, Y. Zhang, Z. C. Dong, S. Jiang, C. Zhang, L. G. Chen, L. Zhang, Y. Liao, J. Aizpurua, Y. Lou, J. L. Yang, and J. G. Hou, “Chemical mapping of a single molecule by plasmon-enhanced Raman scattering,” Nature 498, 82–86 (2013).
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R. Zhang, Y. Zhang, Z. C. Dong, S. Jiang, C. Zhang, L. G. Chen, L. Zhang, Y. Liao, J. Aizpurua, Y. Lou, J. L. Yang, and J. G. Hou, “Chemical mapping of a single molecule by plasmon-enhanced Raman scattering,” Nature 498, 82–86 (2013).
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J. J. Li and K. D. Zhu, “All-optical mass sensing with coupled mechanical resonator systems,” Phys. Rep. 525, 223–254 (2013).
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J. J. Li and K. D. Zhu, “Weighing a single atom using a coupled plasmon-carbon nanotube system,” Sci. Tech. Adv. Mater. 13, 025006 (2012).
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C. Jiang, B. Chen, J. J. Li, and K. D. Zhu, “Mass sensing based on a circuit cavity electromechanical system,” J. Appl. Phys. 110, 083107 (2011).
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W. Zhu and K. B. Crozier, “Quantum mechanical limit to plasmonic enhancement as observed by surface-enhanced Raman scattering,” Nat. Commun. 5, 5228 (2014).
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D. Wang, W. Zhu, M. D. Best, J. P. Camden, and K. B. Crozier, “Directional Raman scattering from single molecules in the feed gaps of optical antennas,” Nano Lett. 13, 2194–2198 (2013).
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K. L. Ekinci, Y. T. Yang, and M. L. Roukes, “Ultimate limits to inertial mass sensing based upon nanoelectromechanical systems,” J. Appl. Phys. 95, 2682–2689 (2004).
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C. Jiang, B. Chen, J. J. Li, and K. D. Zhu, “Mass sensing based on a circuit cavity electromechanical system,” J. Appl. Phys. 110, 083107 (2011).
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Nanotechnology (3)

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Nat. Commun. (1)

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C. Chen, S. Rosenblatt, K. I. Bolotin, W. Kalb, P. Kim, I. Kymissis, H. L. Stormer, T. F. Heinz, and J. Hone, “Performance of monolayer graphene nanomechanical resonators with electrical readout,” Nat. Nanotechnol. 4, 861–867 (2009).
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Phys. Rep. (1)

J. J. Li and K. D. Zhu, “All-optical mass sensing with coupled mechanical resonator systems,” Phys. Rep. 525, 223–254 (2013).
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V. Giovannetti and D. Vitali, “Phase-noise measurement in a cavity with a movable mirror undergoing quantum Brownian motion,” Phys. Rev. A 63, 023812 (2001).
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J. J. Li and K. D. Zhu, “Weighing a single atom using a coupled plasmon-carbon nanotube system,” Sci. Tech. Adv. Mater. 13, 025006 (2012).
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[Crossref]

B. R. Goldsmith, J. G. Coroneus, V. R. Khalap, A. A. Kane, G. A. Weiss, and P. G. Collins, “Conductance-controlled point functionalization of single-walled carbon nanotubes,” Science 315, 77–81 (2007).
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Figures (6)

Fig. 1.
Fig. 1. (a) Schematic diagram of the suspended graphene nanoribbon placed in the surface plasmonic cavity with presence of a strong pump beam and a weak probe beam; G points the direction of the gravity. The Ne atoms are deposited onto the surface of the graphene sheet in a special evaporator. (b) Displacement pattern of the atoms in the graphene nanoribbon due to the fundamental in-plane flexural resonance mode.
Fig. 2.
Fig. 2. Energy level diagram of the SGR-plasmon optomechanical system, where M and c denote the number states of mechanical mode and plasmon cavity photon, respectively. The three pictures correspond to the physical processes of (a) Stokes scattering, (b) Rayleigh scattering, and (c) anti-Stokes scattering.
Fig. 3.
Fig. 3. Strength of Rayleigh scattering on the probing absorption spectrum as a function of the probe-pump detuning δc for different quality factors of the plasmon. We set Ep=0; other parameter values are Ωs=0.1  THz, γ=0.5  GHz.
Fig. 4.
Fig. 4. Plot of absorption spectrum as a function of probe-pump detuning with R=103  Å4·amu1, g=200  GHz, Qc=10, and Δp=0 for I=1,2,and3kW/cm2, respectively. Other parameter values are the same as in Fig. 3.
Fig. 5.
Fig. 5. Pump intensity dependence of the ratio between Raman and Rayleigh scattering strength with different optomechanical coupling rate g.
Fig. 6.
Fig. 6. Absorption spectra of the probe pulse as a function of δ before (black line) and after the binding events of one Ne atom (blue line) and 10 atoms (red line). The frequency shifts induced by additional masses can be well distinguished in the spectra. Here we choose R=103  Å4·amu1, I=1  kW/cm2. Other parameters used are the same as in Fig. 4.

Tables (1)

Tables Icon

Table 1. Parameters of the Plasmon Optomechanical System Used in the Mass Measurement

Equations (23)

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g=Ψ·xZPF=αqωcε0Vc2ωm.
Ωj=14π3ϵ0λ32ηκ|Ej|,j=p,s.
H=Hc+Hm+Hint+Hopt.
H=Δpc+c+ωma+agc+c(a++a)iΩp(cc+)iΩs(ceiδtc+eiδt),
dcdt=(iΔp+κ)c+ignc+Ωp+Ωseiδt+κa^in,
d2ndt2+γdndt+ωm2n=2ωmgc+c+ξ^(t),
a^in(t)a^in+(t)=δ(tt),
a^in+(t)a^in(t)=0.
ξ^+(t)ξ^(t)=κωdωωm2πeiω(tt)[1+coth(ω2kBT)].
c0=Ωp(iΔp+κ)ign,n0=2g|c|2ωm.
Ωp2=ω0[κ2+(Δpg2ω0ωm)2].
c=c0+δc,n=n0+δn.
δc˙=κδc+ig(n0δc+c0δn)+Ωp+Ωseiδt,
δn¨+γδn˙+ωm2δn=2ωmgc02.
δc=c+eiδt+ceiδt,
δn=n+eiδt+neiδt.
c+=Ωs[B(OL)+2ig2ωmω0]B(O2L2)+4iLg2ωmω0,
cout(t)=cout0+cout+eiδt+couteiδt=2κ(c0+c+eiδt+ceiδt).
I=12ϵ0C|Ep|2.
Δm=2mωmΔω.
δω=γkBTΔfEc.
δm2mωmδω=2mγωm(kBTEc)1/2.
Qmax=mωmvPA,

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