Raman scattering from molecular conduction junctions: Charge transfer mechanism

We present a model for the charge transfer contribution to surface-enhanced Raman spectroscopy (SERS) in a molecular junction. The model is a generalization of the equilibrium scheme for SERS of a molecule adsorbed on a metal surface [B. N. J. Persson. Chem. Phys. Lett. 82, 561 (1981)]. We extend the same physical consideration to a nonequilibrium situation in a biased molecular junction and to nonzero temperatures. Two approaches are considered and compared: a semiclassical approach appropriate for nonresonance Raman scattering, and a quantum approach based on the nonequilibrium Green's function method. Nonequilibrium effects on this contribution to SERS are demonstrated with numerical examples. It is shown that the semiclassical approach provides an excellent approximation to the full quantum calculation as long as the molecular electronic state is outside the Fermi window, that is, as long as the field-induced charge transfer is small.

Figure 1

Diagrams relevant for the charge transfer contribution to SERS. Dashed lines represent Green's functions of the free photon modes (pumping and accepting), wavy lines represent Green's functions of molecular vibration, directed solid lines indicate electronic single-particle Green's functions with arrows indicating electron propagation. Time labels correspond to the integration variables in Eq. (26). The four diagrams represent scattering events of particles and holes, interacting via electromagnetic field and molecular vibration.

Figure 2

Absolute value of the total Stokes scattering amplitude vs gate voltage calculated at $V_{sd}=0$ within (a) the quasiclassical approach [Eqs. (20) and (22)] and (b) the quantum calculation [Eqs. (34)–(38)]. Shown are results for ${\Gamma }_{L,R}=0.25$ (solid line, red), $0.5$ (dashed line, blue), $1.0$ (dashed-dotted line, green), and $2.0$ eV (dotted line, black). See text for other parameters.

Figure 3

Absolute value of the total Stokes scattering amplitude [Eqs. (36, 37, 38)] vs bias calculated at ${\varepsilon }_0-E_F=1.8$ V within (a) the quasiclassical approach [Eqs. (20) and (22)] and (b) the quantum calculation [Eqs. (34)–(38)]. Shown are results for ${\Gamma }_{L,R}=0.25$ (solid line, red), $0.5$ (dashed line, blue), 1 (dashed-dotted line, green), and 2 eV (dotted line, black). Other parameters are as in Fig. 2.

Figure 4

Absolute value of the total Stokes scattering amplitude [Eqs. (34, 35, 36, 37, 38)] vs bias $V_{sd}$ and the incoming frequency ${\nu }_i$ calculated for ${\varepsilon }_0-E_F=1.8$ eV. See text for other parameters.

Figure 5

Absolute value of the total (a) Stokes and (b) anti-Stokes amplitudes, plotted against the bias voltage $V_{sd}$ for a molecular level near ${\varepsilon }_0-E_F=1.8$ V. Shown are noninteracting (dashed line, blue) and SCBA (solid line, red) results. See text for other parameters.

Figure 6

Examples of projections on the Keldysh contour.