Atomistic theory for the damping of vibrational modes in monoatomic gold chains

We develop a computational method for evaluating the damping of vibrational modes in monatomic metallic chains suspended between bulk crystals under external strain. The damping is due to the coupling between the chain and contact modes and the phonons in the bulk substrates. The geometry of the atoms forming the contact is taken into account. The dynamical matrix is computed with density-functional theory in the atomic chain and the contacts using finite atomic displacements while an empirical method is employed for the bulk substrate. As a specific example, we present results for the experimentally realized case of gold chains in two different crystallographic directions. The range of the computed damping rates confirms the estimates obtained by fits to experimental data [T. Frederiksen et al., Phys. Rev. B 75, 205413 (2007)]. Our method indicates that an order-of-magnitude variation in the harmonic damping is possible even for relatively small changes in the strain. Such detailed insight is necessary for a quantitative analysis of damping in metallic atomic chains and in explaining the rich phenomenology seen in the experiments.

Figure 1

Two ways of partitioning the central part of the chain-substrate system. (Top) The chain is the part of the system that only contains one atom in a plane parallel to the surface and the base is what connects the two-dimensional surface to the chain. (Bottom) The pyramid is the base plus the chain atom closest to the base. The central chain is the remaining part of the chain after removing one atom at each end.

Figure 2

Parameters used for calculating the dynamical matrix. Only nearest-neighbor coupling is shown. The coupling elements labeled “EM” are found by the empirical model and the coupling elements labeled “DFT” by DFT. Onsite elements are determined from the coupling elements (see Appendix ) and are not shown in the figure.

Figure 3

Adding atoms to two surfaces. (Top) The forces between surface atoms within next-nearest-neighbor distance $4.08\text{Å}$ of the added atoms are perturbed by the presence of the added atoms. (Bottom) The device region is where the coupling between the atoms is different from the values for the two unperturbed surfaces. The coupling between the device region and the leads is considered to be unperturbed.

Figure 4

Distances and angles used to define the average bond length, $B=⟨b_j⟩$, and the average bond angle, $T=⟨{\theta }_j⟩$, respectively.

Figure 5

(Color online) Force as a function of average bond length, $T=⟨{\theta }_j⟩$.

Figure 6

(Color online) Average bond angle as a function of the average bond length. The long chains adopt a zigzag structure at low $B$ while the short chains remain linear.

Figure 7

(Color online) Local energies of a four-atom chain between two (100) surfaces at different strains. The largest eigenvalues are connected by a line to guide the eye.

Figure 8

Local energies for selected blocks of $\mathbf{K}$ (pyramid/central chain) plotted vs the average bond length in the four-atom chain. Since the central chain in this case consists of two atoms we can classify the eigenvectors as LO: longitudinal optical, LA: longitudinal acoustic, TO: transverse optical (doubly degenerate), or TA: transverse acoustic (doubly degenerate).

Figure 9

Projected DOS onto a representative selection of the vibrational modes of the device region that has a large overlap with the added $\left(\text{chain}+\text{base}\right)$ region.

Figure 10

The vibrational modes for chains with three to seven atoms between two 100 surfaces. The center of the disks is positioned at the peak of the projection of vibrational DOS on the mode in question. The area of a disk is proportional to the $Q_{\lambda }$ but is limited to what corresponds to a $Q$ factor of 250. The gray level that ranges from light gray to black in four steps signifies that $s_{\lambda }∊\left[0,2\right[$ (light gray), $\left[2,4\right[,\left[4,6\right[,\left[6,8\right[$, or $\left[8,\infty \right[$ (black).

Figure 11

The vibrational modes for five-atom chains between two (111) surfaces. (Top) Symmetric pyramids. (Bottom) Asymmetric pyramids (one atom added to one of the pyramids). The area of a disk is proportional to the $Q_{\lambda }$ but is limited to what corresponds to a $Q$ factor of 250. The gray level that ranges from light gray to black in four steps signifies that $s_{\lambda }∊\left[0,2\right[$ (light gray), $\left[2,4\right[,\left[4,6\right[,\left[6,8\right[$, or $\left[8,\infty \right[$ (black).

Figure 12

Modes of the device region with a different DFT/EM interface. The label designates the region treated with DFT, where the “added” region is the one used in the main part of the calculations and “all” is fully ab initio. See Fig. 11 to see what the color and size signify. In this plot modes with $s_{\lambda }∊\left[0,2\right[$ are suppressed.

Figure 13

Example $\text{Re}\mathbf{D}$ and $n\left(ϵ\right)$, dashed and solid line, respectively, for a Green’s function with two poles at 1 and 2 with a 0.1 broadening. We see that the values where the real part is zero only correspond to peaks in the density if the slope is positive.