Raman scattering from biased molecular conduction junctions: The electronic background and its temperature

The existence of background in the surface-enhanced Raman scattering from molecules adsorbed on metal surfaces has been known since the early studies about this phenomenon and is usually attributed to transitions between electronic states of the metal substrate. This paper reformulates the theory of this phenomenon in the framework of the nonequilibrium Green function formalism, which makes it possible to extend it to the case of Raman scattering from nonequilibrium (biased and current-carrying) molecular junctions. Following recent experiments, we address, in particular, the Raman-scattering measurement of current-induced electronic heating. The Raman temperature, defined by fitting the ratio between the Stokes and the anti-Stokes Raman signals to a Boltzmann factor is compared to another measure of electronic heating obtained by assuming that, close to the molecule-metal contact, the electronic distribution is dominated by the transmission process. We find that the Raman temperatures considerably exceed this upper bound to the metal-electron heating. In agreement with this observation, we show that the Raman temperature reflects the electronic nonequilibrium in the molecular bridge itself. We also show that the Raman-temperature concept breaks down at large biases.

Figure 1

A two-level model for a molecular conduction junction.

Figure 2

Sketch of electronic-scattering events corresponding to expressions in Eq. (9). Shown are (a) Rayleigh and (b)–(d) three types of electronic Raman-scattering events. In (a), the light-scattering process may be accompanied by an electron that goes from a state of energy $E^{\left(1\right)}$ to a state of energy $E^{\left(2\right)}$ then returns to the same state $E^{\left(1\right)}$, which involves twice the Fermi factors $f\left(E^{\left(1\right)}\right)[1-f\left(E^{\left(2\right)}\right)]$ as seen in Eq. (9a). In (b), the scattering process is accompanied by an electron starting from a state of energy $E_i^{\left(1\right)}$ and ending in state $E_f^{\left(1\right)}$ [hence, the appearance of the Fermi product $f\left(E_i^{\left(1\right)}\right)[1-f\left(E_f^{\left(1\right)}\right)]$ in Eq. (9b)], having gone through an intermediate state of energy $E^{\left(2\right)}$, which can contribute if empty (implying the square of the corresponding Fermi factor ${[1-f\left(E^{\left(2\right)}\right)]}^2$). In (c), the inelastic contribution results from an electron going from a state of energy $E^{\left(1\right)}$ to energy $E_i^{\left(2\right)}$ while another electron goes from energy $E_f^{\left(2\right)}$ to fill back the hole formed in $E^{\left(1\right)}$. This involves the occupation probability product $f{\left(E_i^{\left(1\right)}\right)}^2[1-f\left(E_i^{\left(2\right)}\right)]f\left(E_f^{\left(2\right)}\right)$. Finally, process (d) results from two independent electronic transitions: The destruction of the incoming photon is accompanied by an electron moving from $E_i^{\left(1\right)}$ to $E_f^{\left(1\right)}$ while the creation of the outgoing photon is accompanied by an electron going from state $E_i^{\left(2\right)}$ to state $E_f^{\left(1\right)}$.

Figure 3

(a) The effective temperatures, Eq. (15), calculated for symmetric junctions with ambient temperatures $T=0$ and 300 K. The blue full and dashed lines, respectively, show the left and right electronic temperatures for the case $T=0\mathrm{K}$. The red dotted and dashed-dotted lines are similar results for $T=300\mathrm{K}$. The inset shows the corresponding $I/V$ curves, for which the $T=0$ and $T=300\mathrm{K}$ results overlap. (b) Same results for the asymmetric junction case. (c) Similar results for a tunneling junction characterized by a square barrier of height 5 eV (above the Fermi energy of the unbiased junction) and width $3\AA$.

Figure 4

(a) The electronic Raman intensity, $J_{{\nu }_i\to {\nu }_f}\sim J_{i\to f}{\nu }_f^3$, calculated for the symmetric junction, ${\Gamma }_m^L={\Gamma }_m^R=0.25,m=1,2$, displayed as a function of the Raman shift ${\nu }_i-{\nu }_f$ for $V_{sd}=0$ (left panels) and $V_{sd}=2\mathrm{V}$ (right panels) and for ambient temperatures $T=0\mathrm{K}$ (upper panels) and $T=300\mathrm{K}$ (lower panels). The solid (red) line shows the overall light-scattering intensity, whereas, the other lines correspond to the different contributions: Eqs. (9b) (dashed, blue), (9c) (dotted, black), and (9d) (dashed-dotted, magenta). (b) Same as (a), for the asymmetric junction, ${\Gamma }_m^L=0.25\mathrm{eV},{\Gamma }_m^R=0.0025\mathrm{eV}$, and $m=1,2$.

Figure 5

The electronic Raman intensity for $|{\nu }_i-{\nu }_f|=0.125\mathrm{eV}$ shown as a function of the bias potential for the symmetric junction ${\Gamma }_m^L={\Gamma }_m^R=0.25\mathrm{eV},m=1,2$. Line notations are as in Fig. 4.

Figure 6

The effective Raman temperature estimated from the S/AS ratio of the electronic Raman spectrum, plotted as a function of the inelastic shift for different voltage biases. Left and right panels show results for the symmetric and asymmetric junctions as defined by the choice of the $\Gamma$ parameters. Upper and lower panels correspond to ambient temperatures $T=0$ and $T=300\mathrm{K}$, respectively.

Figure 7

Nonequilibrium effective temperatures at low bias $\left|e\right|V_{sd}={\mu }_L-{\mu }_R\ll {\varepsilon }_2-{\varepsilon }_1$. The bias is applied symmetrically, ${\mu }_L=\left|e\right|V_{sd}/2$ and ${\mu }_R=-\left|e\right|V_{sd}/2$. Both panels compare the Raman temperature (circles, red), Eq. (16), and the effective electronic temperatures of contacts $L$ (triangles, blue) and $R$ (squares, blue) obtained from Eq. (15) displayed against the voltage bias. Two effective molecular electronic temperatures: one defined in Eq. (17) (diamonds, magenta) and another defined in Eq. (18) (asterisks, magenta) are also shown in the plots. Note that these estimates overlap in panel (a). In panel (a) (reproduced from Ref. 117), we consider the symmetric junction ${\Gamma }_m^L={\Gamma }_m^R=0.25\mathrm{eV}$ for both molecular states. Panel (b) shows results for the asymmetric junctions ${\Gamma }_m^L=0.25$ and ${\Gamma }_m^R=0.0025\mathrm{eV}$. The electronic Raman temperature was calculated both for leads at thermal equilibrium ($T=300\mathrm{K}$) and for leads characterized by the distributions (13). The results are identical in the symmetrical case and are almost indistinguishable in the asymmetric system [full and dashed red lines in panel (b)].