Improving transition voltage spectroscopy of molecular junctions

Transition voltage spectroscopy (TVS) is a promising spectroscopic tool for molecular junctions. The principles in TVS is to find the minimum on a Fowler-Nordheim plot where $\ln (I/V^2)$ is plotted against $1/V$ and relate the voltage at the minimum $V_{\min }$ to the closest molecular level. Importantly, $V_{\min }$ is approximately half the voltage required to see a peak in the $\mathit{dI}/\mathit{dV}$ curve. Information about the molecular level position can thus be obtained at relatively low voltages. In this work we show that the molecular level position can be determined at even lower voltages, $V_{\min }^{(\alpha )}$, by finding the minimum of $\ln (I/V^{\alpha })$ with $\alpha <2$. On the basis of a simple Lorentzian transmission model we analyze theoretical ab initio as well as experimental $I-V$ curves and show that the voltage required to determine the molecular levels can be reduced by $~$$30\%$ as compared to conventional TVS. As for conventional TVS, the symmetry/asymmetry of the molecular junction needs to be taken into account in order to gain quantitative information. We show that the degree of asymmetry may be estimated from a plot of $V_{\min }^{(\alpha )}$ vs $\alpha$.

Figure 1

Top: $I-V$ curve from a symmetric junction and a Lorentzian transmission function with ${\varepsilon }_0-E_F=-1\hspace{0.5em}$eV and $\Gamma =0.2\hspace{0.5em}$eV (shown in the inset). The three lower points on the curve mark the voltages $V_{\min }^{(\alpha )}$ at the minima of Fowler-Nordheim plots $\ln (I/V^{\alpha })$ vs $1/V$. The numbers show the $\alpha$ values. The resonant transmission voltage is also marked. Bottom: Generalized Fowler-Nordheim plots where $\ln (I/V^{\alpha })$ is plotted against $1/V$.

Figure 2

Ratio of the resonant level and minimum voltage, $|{\varepsilon }_0-E_F|/V_{\min }^{(\alpha )}$ vs $\alpha$ obtained for a symmetric ($\eta =0$) and completely asymmetric ($\eta =1/2$) junction.

Figure 3

Atomic structure of a symmetric Au-BDT-Au junction (a) together with an asymmetric Au-Ph-STM junction (c). The corresponding transmission functions are shown in panels (b) and (d), respectively.

Figure 4

Predicted HOMO level of phenyl (Ph) and benzenedithiol (BDT) junctions in Fig. 3 obtained from the TVS spectra using different $\alpha$ values. The horizontal dashed lines mark the HOMO positions as determined from the transmission spectra in Figs. 3b and 3d.

Figure 5

Ratio of HOMO position and minimum voltage, $|{\varepsilon }_H-E_F|/V_{\min }^{(\alpha )}$ vs asymmetry parameter $\eta$. The minimum voltages are obtained with the generalized TVS scheme using $\alpha =2.0$ (a), $\alpha =1.5$ (b), and $\alpha =1.3$ (c). The ab initio calculations follow the Lorentzian model prediction for $\alpha =1.5$ and $\alpha =2.0$, while substantial deviations are seen for $\alpha =1.3$.

Figure 6

(a)$I-V$ curve acquired from Ref. 3 (circles) together with a smooth curve obtained by convolution of the data points with a Gaussian function. (b) Fowler-Nordheim plot (using $\alpha =2$) for the data points and the smooth function. (c) and (d) show similar data obtained from Ref. 8.

Figure 7

Ratio of $V_{\min }^{(\alpha )}/V_{\min }^{(2)}$ obtained from the experimental data (symbols) and from the Lorentzian model (lines). The experimental data follow the model reasonably well for $\eta =0.25$. This asymmetry value is in good agreement with our previous calculations (Ref. 11).

Figure 8

Predicted values of $|{\varepsilon }_H-E_F|$ vs $\alpha$ (a) and vs $V_{\min }$ (b) for anthracene (Beebe data) and terphenyl (Tan data) junctions. The predicted HOMO levels are reasonably constant for $\alpha >1.5$ thus indicating that the Lorentzian model is applicable in analyzing the experimental data. Panel (b) shows that using $\alpha \approx 1.5$, the HOMO level can be determined at $~$$30\%$ lower voltage than required for $\alpha =2$.