Inelastic transport theory from first principles: Methodology and application to nanoscale devices

We describe a first-principles method for calculating electronic structure, vibrational modes and frequencies, electron-phonon couplings, and inelastic electron transport properties of an atomic-scale device bridging two metallic contacts under nonequilibrium conditions. The method extends the density-functional codes SIESTA and TRANSIESTA that use atomic basis sets. The inelastic conductance characteristics are calculated using the nonequilibrium Green’s function formalism, and the electron-phonon interaction is addressed with perturbation theory up to the level of the self-consistent Born approximation. While these calculations often are computationally demanding, we show how they can be approximated by a simple and efficient lowest order expansion. Our method also addresses effects of energy dissipation and local heating of the junction via detailed calculations of the power flow. We demonstrate the developed procedures by considering inelastic transport through atomic gold wires of various lengths, thereby extending the results presented in Frederiksen et al. [Phys. Rev. Lett. 93, 256601 (2004)]. To illustrate that the method applies more generally to molecular devices, we also calculate the inelastic current through different hydrocarbon molecules between gold electrodes. Both for the wires and the molecules our theory is in quantitative agreement with experiments, and characterizes the system-specific mode selectivity and local heating.

Figure 1

Schematic of two generic system setups. (a) To calculate vibrational frequencies and $e\text{−}\mathrm{ph}$ couplings with SIESTA we use a supercell setup with periodic boundary conditions (BCs) in all directions. The cell contains the device region $D$ and possibly some additional atom layers to come closer to a representation of bulk electrodes. The dynamic atoms are a relevant subset of the device atoms for which we determine the vibrations. (b) In the transport setup we apply the TRANSIESTA scheme where the central region $D$ is coupled to fully atomistic semi-infinite electrodes via self-energies, thereby removing periodicity along the transport direction (the periodic BCs are retained in the transverse plane).

Figure 2

Flow diagram for the complete analysis of the inelastic transport properties of an atomic structure.

Figure 3

(Color online) Vibrational frequencies calculated for some simple molecules (${\mathrm{Au}}_2$ and ${\mathrm{Pt}}_2$, acetylene ${\mathrm{C}}_2{\mathrm{H}}_2$, ethylene ${\mathrm{C}}_2{\mathrm{H}}_4$, and ethane ${\mathrm{C}}_2{\mathrm{H}}_6$). The results obtained directly from SIESTA are shown together with those of our scheme (typical/accurate) based on the correction (13). The different calculational settings are described in the text. For comparison the experimentally measured values of the frequencies are also given (Refs. 81, 82, 83) To indicate the accuracy of the calculations the numerical values for the zero-frequency modes (translation/rotation) are included, where negative values correspond to imaginary frequencies.

Figure 4

(Color online) Convergence of calculated vibrational frequencies for a four-atom Au wire with the most important DFT settings. For each of the two choices for the vibrational region (as indicated with boxes) the reference calculation—carried out with SZP, a $200\mathrm{Ry}$ real space grid energy cutoff, and $0.02\mathrm{\AA }$ finite displacements—and other three separate calculations (with one of the settings improved at a time) yield essentially the same results for the phonon energy $\hslash {\omega }_{\lambda }$ versus mode index $\lambda$.

Figure 5

The lowest order diagrams for the phonon self-energies to the electronic description. The “Hartree” (a) and “Fock” (b) diagrams dress the electron Green’s functions (double plain lines). The phonon Green’s functions (single wiggly lines) are assumed to be described by the unperturbed ones, i.e., we ignore the $e\text{−}\mathrm{ph}$ renormalization of the phonon system.

Figure 6

(Color online) Schematic representation of the energy phase space available for scattering processes due to the Pauli principle. Phonon emission (a) and absorption (b) between scattering states originating from the left and right contacts. (c) and (d) correspond to phonon absorption between scattering states in the same contact.

Figure 7

(Color online) Universal functions (55, 56) giving symmetric and asymmetric phonon contributions to the conductance in the LOE, respectively. The differential conductance $dI∕dV$ and the second derivative $d^{2}I∕d{V}^2$ are shown (in arbitrary units) for one phonon mode for three different temperatures (a) $k_{B}T∕\hslash {\omega }_{\lambda }=0.02$, (b) $k_{B}T∕\hslash {\omega }_{\lambda }=0.06$, and (c) $k_{B}T∕\hslash {\omega }_{\lambda }=0.10$.

Figure 8

(Color online) Generic gold wire supercells containing 3 to 7 atoms bridging pyramidal bases connected to stacked Au(100) layers. As indicated on the figure, the electrode separation $L$ is defined as the distance between the plane in each electrode containing the second-outermost Au(100) layer.

Figure 9

(Color online) Energetic, geometric, and conductive properties of atomic gold wires: (a) Kohn-Sham total energy (cohesive energy) vs electrode separation, (b) bond angles and bond lengths, (c) phonon energies, and (d) elastic transmission at the Fermi energy calculated both for the $\Gamma$ point (colored open symbols) as well as with a $5\times 5$ $\mathbf{k}$-point sampling of the two-dimensional Brillouin zone perpendicular to the transport direction (black stars).

Figure 10

(Color online) Generic transport setup in which a relaxed wire geometry—here a seven-atom wire with $L=29.20\mathrm{\AA }$—is coupled to semi-infinite electrodes. As indicated on the figure the vibrational region is taken to include the atoms in the pyramidal bases and the wire itself, whereas the device region (describing the $e\text{−}\mathrm{ph}$ couplings) includes also the outermost surface layers.

Figure 11

(Color online) Elastic and inelastic differential conductance calculated at $T=10.0\mathrm{K}$ in a reduced device region for the seven-atom wire shown in Fig. 10. The small variation in elastic conductance with bias (dotted curve) relates to a weak energy dependence of the elastic transmission function at the $\Gamma$ point around ${\epsilon }_F$. The full SCBA calculation (circles) follows this trend and shows on top of it symmetric drops characteristic for phonon scattering. The LOE calculation (line) does not include the elastic variation but gives basically the same predictions for the inelastic signals as the SCBA with the elastic background signal subtracted (dashed curve). This illustrates the agreement between the LOE and SCBA approaches for the inelastic contribution.

Figure 12

(Color online) The differential conductance $G$ and its derivative $dG∕dV$ calculated with the LOE approach for the three- to seven-atom gold wires in the externally damped limit. The electrode separation $L$ is indicated next to the conductance curves. As shown in Fig. 10 the device region includes the outermost electrode layer whereas the dynamic atoms are pyramidal bases plus wire. The temperature of the leads is $T=4.2\mathrm{K}$.

Figure 13

(Color online) Inelastic signals plotted as a function of the electrode separation $L$. Each mode is represented by a dot with an area proportional to the corresponding conductance drop. On the $y$ axis we show (a) the phonon mode energy, (b) a measure of the longitudinal component of the mode, (c) a measure of the ABL character, and (d) a measure of the localization to the wire atoms only. The straight lines in plot (a) are linear interpolations to the most significant signals (the slopes are given too).

Figure 14

(Color online) ABL-mode broadening due to coupling to bulk phonons. The spectrum $B_{\lambda }\left(\epsilon \right)$ corresponds to the important ABL-mode for a seven-atom wire $(L=29.20\mathrm{\AA })$. By fitting the calculated points with a Lorentzian we extract a full-width half maximum (FWHM) broadening of $2{\gamma }_{\text{damp}}^{\lambda }=8\mu \mathrm{eV}$ and a frequency shift of $\delta {\omega }_{\lambda }=-6\mu \mathrm{eV}$. The inset shows the calculated total density of states for bulk Au (full line), as well as a decomposition in the direction of the transport (dashed red curve), and in the transverse direction (dotted blue curve).

Figure 15

(Color online) Comparison between theory and experiment (Ref. 15) for the inelastic conductance of an atomic gold wire. The measured characteristics (noisy black curves) correspond to different states of strain of wire (around 7 atoms long). The calculated results (smooth colored lines) are for the seven-atom wire at $L=29.20\mathrm{\AA }$ using different values for the external damping as indicated in the right side of the plot (in units of $\mu \mathrm{eV}∕\hslash$). The dashed curve is the calculated result in the externally undamped limit $({\gamma }_{\text{damp}}^{\lambda }=0)$. The lower plot shows the numerical derivative of the conductance. Note the indication of a secondary phonon feature below $5\mathrm{meV}$ in all curves. The temperature is $T=4.2\mathrm{K}$ and the lock-in modulation voltage $V_{\mathrm{rms}}=1\mathrm{mV}$ (in both theory and experiment).

Figure 16

(Color online) Calculated IETS spectrum for an OPE molecule compared to the experimental data from Ref. 21 (scaled by a factor of 2). Each of the three inelastic scattering peaks arise from different kinds of vibrations localized on the molecule.

Figure 17

(Color online) Calculated IETS spectra for (a) an OPV molecule and (b) an OPE molecule. The chemical structure of these hydrocarbon molecules are shown in the insets. The two plots show that the simple LOE scheme predicts the same IETS spectrum as the full SCBA (if one neglects the elastic variation).