From tunneling to contact: Inelastic signals in an atomic gold junction from first principles

The evolution of electron conductance in the presence of inelastic effects is studied as an atomic gold contact is formed evolving from a low-conductance regime (tunneling) to a high-conductance regime (contact). In order to characterize each regime, we perform density-functional theory (DFT) calculations to study the geometric and electronic structures, together with the strength of the atomic bonds and the associated vibrational frequencies. The conductance is calculated by, first, evaluating the transmission of electrons through the system and, second, by calculating the conductance change due to the excitation of vibrations. As found in previous studies [Paulsson et al., Phys. Rev. B 72, 201101(R) (2005)], the change in conductance due to inelastic effects permits us to characterize the crossover from tunneling to contact. The most notorious effect is the crossover from an increase in conductance in the tunneling regime to a decrease in conductance in the contact regime when the bias voltage matches a vibrational threshold. Our DFT-based calculations actually show that the effect of vibrational modes in electron conductance is rather complex, in particular, when modes localized in the contact region are permitted to extend into the electrodes. As an example, we find that certain modes can give rise to decreases in conductance when in the tunneling regime, opposite to the above-mentioned result. Whereas details in the inelastic spectrum depend on the size of the vibrational region, we show that the overall change in conductance is quantitatively well approximated by the simplest calculation where only the apex atoms are allowed to vibrate. Our study is completed by the application of a simplified model where the relevant parameters are obtained from the above DFT-based calculations.

Figure 1

(Color online) Generic setup for the calculation of structural properties of the atomic gold junction. The periodic supercell consists of a $4\times 4$ representation of two Au(100) surfaces sandwiching two pyramids pointing toward each other. The characteristic electrode separation $L$ is measured between the second-topmost surface layers, since the surface layer itself is relaxed and hence deviates on the decimals from the bulk values. The interatomic distance between the apex atoms is denoted as $d$.

Figure 2

(Color online) Total-energy differences and the numerical derivatives as a function of the electrode separation. The lower part of the figure describes the strain on the unit cell along the transport direction. The onset of chemical interactions is clearly seen around $L=16.0\mathrm{\AA }$ where the force experience a significant increase. (a), (b), and (c) are three representative electrode separations of the three regimes considered in this paper.

Figure 3

(Color online) Vibrational frequencies versus electrode displacement. The connected data series refers to the situation where only the two apex atoms are vibrating (resulting in the six vibrational modes indicated in the plot); circles symbolize the two longitudinal modes (CM and ABL) and diamonds the four (pairwise degenerate) transversal modes. The asterisks are the corresponding vibrational frequencies when also the pyramid bases are considered active. The three regimes are clearly identifiable: (a) concerted apex vibrations, (b) crossover where the stretch modes become degenerate, and (c) independent apex vibrations.

Figure 4

(Color online) Transmission $\tau$ (filled circles) and apex-apex distance $d$ (asterisks) versus electrode separation $L$. In the tunneling regime, the transmission decays exponentially with separation as indicated with the dashed line (corresponding to a tunneling parameter $\Lambda =1.54{\mathrm{\AA }}^{-1}$). The point at (a) corresponds well to the contact region of transmission 1 and closest apex separation, (b) is near half transmission and the instability in apex separation, and (c) is finally the tunneling regime, where the apex atoms are independent.

Figure 5

(Color online) Second derivative of the current versus bias voltage for three characteristic situations: (a) contact, (b) crossover, and (c) tunneling. In each situation, we consider different active vibrational regions: the two apex atoms only (thick solid line), the ten pyramid atoms (thick dashed curve), and both pyramids and first-layer atoms (thin dotted curve). The signal broadening is due to temperature $(T=4.2\mathrm{K})$.