We present a theory for Raman scattering by current-carrying molecular junctions. The approach combines a nonequilibrium Green’s function (NEGF) description of the nonequilibrium junction with a generalized scattering theory formulation for evaluating the light scattering signal. This generalizes our previous study [M. Galperin and A. Nitzan, Phys. Rev. Lett. 95, 206802 (2005); J. Chem. Phys. 124, 234709 (2006)] of junction spectroscopy by including molecular vibrations and developing machinery for calculation of state-to-state (Raman scattering) fluxes within the NEGF formalism. For large enough voltage bias, we find that the light scattering signal contains, in addition to the normal signal associated with the molecular ground electronic state, also a contribution from the inverse process originated from the excited molecular state as well as an interference component. The effects of coupling to the electrodes and of the imposed bias on the total Raman scattering as well as its components are discussed. Our result reduces to the standard expression for Raman scattering in the isolated molecule case, i.e., in the absence of coupling to the electrodes. The theory is used to discuss the charge-transfer contribution to surface enhanced Raman scattering for molecules adsorbed on metal surfaces and its manifestation in the biased junction.
REFERENCES
1.
K.
Kneipp
, M.
Moskovits
, and H.
Kneipp
, Topics in Applied Physics
(Springer
, Berlin, Heidelberg
, 2006
), Vol. 103
.2.
3.
J.
Gersten
and A.
Nitzan
, J. Chem. Phys.
73
, 3023
(1980
).4.
J. I.
Gersten
, R. L.
Birke
, and J. R.
Lombardi
, Phys. Rev. Lett.
43
, 147
(1979
).5.
E.
Burstein
, Y. J.
Chen
, C. Y.
Chen
, S.
Lundquist
, and E.
Tosatti
, Solid State Commun.
29
, 567
(1979
).6.
B. N. J.
Persson
, Chem. Phys. Lett.
82
, 561
(1981
).7.
K. A.
Willets
and R. P.
Van Duyne
, Annu. Rev. Phys. Chem.
58
, 267
(2007
).8.
A.
Otto
, I.
Mrozek
, H.
Grabhorn
, and W.
Akemann
, J. Phys.: Condens. Matter
4
, 1143
(1992
).9.
J. R.
Lombardi
, R. L.
Birke
, T.
Lu
, and J.
Xu
, J. Chem. Phys.
84
, 4174
(1986
).10.
J. R.
Lombardi
and R. L.
Birke
, J. Chem. Phys.
126
, 244709
(2007
).11.
J. R.
Lombardi
and R. L.
Birke
, J. Phys. Chem. C
112
, 5605
(2008
).12.
A. C.
Albrecht
, J. Chem. Phys.
34
, 1476
(1961
).13.
S.
Nie
and S. R.
Emory
, Science
275
, 1102
(1997
).14.
K.
Kneipp
, Y.
Wang
, H.
Kneipp
, L. T.
Perelman
, I.
Itzkan
, R. R.
Dasari
, and M. S.
Feld
, Phys. Rev. Lett.
78
, 1667
(1997
).15.
J.
Jiang
, K.
Bosnick
, M.
Maillard
, and L.
Brus
, J. Phys. Chem. B
107
, 9964
(2003
).16.
P. K.
Aravind
, A.
Nitzan
, and H.
Metiu
, Surf. Sci.
110
, 189
(1981
).17.
M.
Inoue
and K. J.
Ohtaka
, J. Phys. Soc. Jpn.
52
, 3853
(1983
).18.
S. R.
Emory
, R. A.
Jensen
, T.
Wenda
, M.
Han
, and S.
Nie
, Faraday Discuss.
132
, 249
(2006
).19.
D. P.
Fromm
, A.
Sundaramurthy
, A.
Kinkhabwala
, P. J.
Schuck
, G. S.
Kino
, and W. E.
Moerner
, J. Chem. Phys.
124
, 061101
(2006
).20.
A.
Nitzan
and M.
Ratner
, Science
300
, 1384
(2003
).21.
M.
Galperin
, M. A.
Ratner
, A.
Nitzan
, and A.
Troisi
, Science
319
, 1056
(2008
).22.
T.
Dadosh
, Y.
Gordin
, R.
Krahne
, I.
Khivrich
, D.
Mahalu
, V.
Frydman
, J.
Sperling
, A.
Yacoby
, and I.
Bar-Joseph
, Nature (London)
436
, 677
(2005
).23.
24.
J.-H.
Tian
, B.
Liu
, X.
Li
, Z.-L.
Yang
, B.
Ren
, S.-T.
Wu
, N.
Tao
, and Z.-Q.
Tian
, J. Am. Chem. Soc.
128
, 14748
(2006
).25.
D. R.
Ward
, N. K.
Grady
, C. S.
Levin
, N. J.
Halas
, Y.
Wu
, P.
Nordlander
, and D.
Natelson
, Nano Lett.
7
, 1396
(2007
).26.
D. R.
Ward
, N. J.
Halas
, J. W.
Ciszek
, J. M.
Tour
, Y.
Wu
, P.
Nordlander
, and D.
Natelson
, Nano Lett.
8
, 919
(2008
).27.
M.
Galperin
and A.
Nitzan
, Phys. Rev. Lett.
95
, 206802
(2005
).28.
M.
Galperin
and A.
Nitzan
, J. Chem. Phys.
124
, 234709
(2006
).29.
Lombardi and co-workers have invoked the Herzberg–Teller intensity borrowing concept as used by Tang and Albrecht in their calculation of the contribution of the charge-transfer mechanism to SERS. In the analysis here, based on Eq. (1), this is unnecessary. This concept was introduced in order to explain the appearance of molecular spectral features that are forbidden by symmetry in the Born–Oppenheimer (BO) approximation. However, the zero-order description of the molecule-metal system is not done in the adiabatic BO, but on the “diabatic” level in which the BO picture is applied to the uncoupled molecule and metal systems, leaving their electronic coupling as a perturbative correction in the Hamiltonian. In this representation there is no need to invoke intensity borrowing in a discussion of photon induced charge-transfer transitions between metal(s) and molecule. The actual coupling needs of course to be estimated, and in the present work, we have based our estimate on a model proposed by Persson (Ref. 6) as discussed at the end of Sec. III.
30.
31.
T.
Holstein
, Ann. Phys. (N.Y.)
8
, 325
(1959
).32.
33.
M.
Galperin
, A.
Nitzan
, and M. A.
Ratner
, Phys. Rev. B
73
, 045314
(2006
).34.
Y.
Meir
and N. S.
Wingreen
, Phys. Rev. Lett.
68
, 2512
(1992
).35.
H.
Haug
and A. -P.
Jauho
, Quantum Kinetics in Transport and Optics of Semiconductors
(Springer
, Berlin
, 1996
).36.
M.
Galperin
, M.
Ratner
, and A.
Nitzan
, Phys. Rev. B
75
, 155312
(2007
).37.
J. -S.
Wang
, J.
Wang
, and N.
Zeng
, Phys. Rev. B
74
, 033408
(2006
).38.
J. -S.
Wang
, N.
Zeng
, J.
Wang
, and C. K.
Gan
, Phys. Rev. E
75
, 061128
(2007
).39.
N.
Mingo
, Phys. Rev. B
74
, 125402
(2006
).40.
T.
Yamamoto
and K.
Watanabe
, Phys. Rev. Lett.
96
, 255503
(2006
).41.
42.
T.
Bornath
, D.
Kremp
, and M.
Schlanges
, Phys. Rev. E
60
, 6382
(1999
).43.
H.
Schoeller
, Lect. Notes Phys.
544
, 137
(2000
).44.
We use the term normal Raman scattering for the process that starts and ends in the molecular ground state, while inverse Raman starts and ends with the molecule in the excited state. The term normal in the present paper is unrelated to the normal used to distinguish from “resonant.”
45.
I. S.
Gradshteyn
and I. M.
Ryzhik
, Table of Integrals, Series, and Products
(Academic
, San Diego
, 2000
).46.
Note that the electron-hole contribution to damping rates in the single electron GFs (43) and (44), as taken into account in Refs. 27 and 28, is dropped here for simplicity. In the present consideration we focus on the effect of damping due to electron-hole excitations on the two-particle Green functions, since it is this process which directly influences the Raman signal. The latter enters via the ansatz, Eq. (50), employed in the calculation of excitation operators correlation function (53)–(55).
47.
Due to the assumed weak coupling between the molecular vibration and the thermal bath, the vibrational state density is almost a delta function. Hence the actual character of the distribution is not important, since the vibration population is defined by one point .
48.
S.
Doniach
, Phys. Rev. B
2
, 3898
(1970
).49.
J.
von Delft
, F.
Marquardt
, R. A.
Smith
, and V.
Ambegaokar
, Phys. Rev. B
76
, 195332
(2007
).50.
In principle, the same ansatz can be used to include damping due to coupling to the laser field; however, we disregard it due to the weak electron-photon coupling assumed.
51.
Equations (53)–(55) are obtained by separating the polarization correlation functions into products of correlation functions involving operators associated with the individual levels (this can be done when these levels do not mix by their interactions with the leads), then applying Wick's theorem to represent these correlation functions as sums of products of single electron GFs for the corresponding levels. The assumption of no level mixing can be relaxed at the price of getting more complex expressions that involve also mixed single electron GFs .
52.
In Eqs. (56)–(58) two energy variables are used for each molecular electronic level. represents the energy variable associated with level .
54.
A. M.
Kelley
, J. Chem. Phys.
128
, 224702
(2008
).55.
Z.
Ioffe
, T.
Shamai
, A.
Ophir
, G.
Noy
, I.
Yutsis
, K.
Kfir
, O.
Cheshnovsky
, and Y.
Selzer
, Nat. Nanotechnol.
3
, 727
(2008
).56.
J. R.
Lombardi
, R. L.
Birke
, L. A.
Sanchez
, I.
Bernard
, and S. C.
Sun
, Chem. Phys. Lett.
104
, 240
(1984
).57.
This result was obtained using the order of magnitude estimate based on Eq. (79), taking the same molecular radiative rate used to estimate the Raman intensity associated with the molecular process. Note, however, that in Eq. (79) is the radiative rate associated with a transition dipole of magnitude ( is the electron charge and molecule-metal distance) that may be considerably larger than a typical molecular radiative rate.
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