Inelastic quantum transport in nanostructures: The self-consistent Born approximation and correlated electron-ion dynamics

A dynamical method for inelastic transport simulations in nanostructures is compared to a steady-state method based on nonequilibrium Green’s functions. A simplified form of the dynamical method produces, in the steady state in the weak-coupling limit, effective self-energies analogous to those in the Born approximation due to electron-phonon coupling. The two methods are then compared numerically on a resonant system consisting of a linear trimer weakly embedded between metal electrodes. This system exhibits an enhanced heating at high biases and long phonon equilibration times. Despite the differences in their formulation, the static and dynamical methods capture local current-induced heating and inelastic corrections to the current with good agreement over a wide range of conditions, except in the limit of very high vibrational excitations where differences begin to emerge.

Figure 1

(Color online) Steady-state currents as a function of effective cross section for a single degree of freedom of infinite mass for a variety of biases in the CEID approach. The device length used here was 401 atoms with 100 atoms in each electrode and with the parameters $\Gamma$ and $\Delta$ set to 0.4 and 0.0 eV, respectively. The point at which the current drops to zero corresponds to one of the hopping integrals in the Hamiltonian (24) going to zero.

Figure 2

(Color online) Linear trimer weakly coupled to two one-dimensional electrodes. In the simulations considered here, only the central atom of the trimer is allowed to undergo vibrational motion. $t_1$ is the nearest-neighbor hopping matrix element in the metal electrode, $t_2$ is the electrode-trimer hopping integral, and $t_3$ is the intratrimer hopping integral.

Figure 3

(Color online) Current-voltage spectrum and second derivative of inelastic current for the trimer system in the externally damped limit.

Figure 4

(Color online) Total vibrational energy of a single dynamical ion as a function of time and corresponding electronic current for various electrochemical potential differences in the CEID approach. The current is computed in a bond close to (but outside) the left electrode. Thus, when the propagation under the bias starts, first a train of (electron) current from the left electrode reaches this point; later, the (hole) current from the more distant right electrode arrives together with any reflected parts of the original current from the left—leading to the sharp jumps in local current.

Figure 5

(Color online) Maximal vibrational energy as a function of voltage for the resonant trimer system in the dynamical CEID calculation, the SCBA and lowest-order perturbation theory. Also shown is the vibrational energy obtained within the FGR for a single ion (with the same angular frequency) in a ballistic chain.

Figure 6

(Color online) Asymptotic values of the current as a function of voltage for the resonant trimer system (in the maximally heated limit) for the CEID and SCBA methods. Also illustrated are the steady-state currents in the fully damped limit—illustrating the effect of the heating.

Figure 7

(Color online) Inelastic current as a function of $N_{\text{ph}}/\sqrt{M}$ for a bias of 1 V for a variety of ionic masses.