Resistance saturation in semi-conducting polyacetylene molecular wires

Abstract

Realizing the promises of molecular electronic devices requires an understanding of transport on the nanoscale. Here, we consider a Su-Schrieffer-Heeger model for semi-conducting trans-polyacetylene molecular wires in which we endow charge carriers with a finite lifetime. The aim of this exercise is two-fold: (i) the simplicity of the model allows an insightful numerical and analytical comparison of the Landauer and Kubo linear-response formalism; (ii) we distill the prototypical characteristics of charge transport through gapped mesoscopic systems and compare these to bulk semiconductors. We find that both techniques yield a residual differential conductance at low temperatures for contacted polyacetylene chains of arbitrary length-in line with the resistivity saturation in some correlated narrow-gap semiconductors. Quantitative agreement, however, is limited to not too long molecules. Indeed, while the Landauer transmission is suppressed exponentially with the system size, the Kubo response only decays hyperbolically. Our findings inform the choice of transport methodologies for the ab initio modelling of molecular devices.

Keywords: Electronic correlations; Landauer and Kubo approach; Mesoscopic systems; Transport properties.

Fig. 1

a Representations of a two-terminal junction, with a PA chain bridging Au electrodes through thiol (SH) anchoring groups. b Mapping to an SSH model, single (C−C) and double (C$=$C) bonds correspond to lattice spacing $a_1$ and $a_2$, respectively (with $d=a_1+a_2$ the length of the unit cell) while ${\mathrm{\Gamma }}_{\mathcal{L}}$ and ${\mathrm{\Gamma }}_{\mathcal{R}}$ describe the left and right molecule-lead coupling, respectively. c Evolution of the HOMO-LUMO gap as a function of the PA length N (unit cells) converging to the bulk Peierls gap ${\mathrm{\Delta }}_{\infty }$ (dashed line). d Distribution of the MO eigenenergies ${ϵ}_i$ for a $N=32$ PA chain. e Spectral function of bulk PA and a $N=32$ PA chain. f Site-resolved spectral function for selected C atoms i across the PA chain, at the edges and in the middle, as shown in (g). The grey area in (d, e, f) highlights the spectral gap $\mathrm{\Delta }$. For clarity, the spectra in (e, f) are obtained with a broadening $\eta =0.1$ eV

Fig. 2

Schematic representation of the electron transport across the PA wire within the (a) Landauer and (b) Kubo formalism. The transmission amplitude, $\text{Tr}(M{M}^{†})$, consists of processes M and $M^{†}$ that can be associated with paths in real-space (vertical axis). The Landauer transmission is given by a single path (i.e., from $\ell$ to r). The Kubo transmission is made up of many individual connections, that are built from nearest-neighbor hopping $t_{i,i\pm 1}$ and spectral functions $\Im G_{\mathit{ij}}$. The trace constrains each path to start and end at a given site i. In both panels, colorful circles highlight the sites involved at each step, while colorful horizontal lines highlight local processes

Fig. 3

Representative Green’s function connecting (a, b) the same sublattice, ${\mathbf{G}}_{\ell \ell }$, and (c, d) the opposite sublattice, ${\mathbf{G}}_{\ell r}$, of C atoms in the PA chain. Sublattice one (two) consists of the C atoms with a double bond to their right (left). Clearly visible is that, at the Fermi level ($E_F=0$), the dominant spectral ingredient to the Kubo and the Landauer transmission are the even components $\Im {\mathbf{G}}_{\ell \ell }$ and $\Re {\mathbf{G}}_{\ell r}$, respectively. Noteworthy, only the former is strongly dependent on the scattering rate $\mathrm{\Gamma }$, heralding differences between the Kubo and the Landauer transmission

Fig. 4

a Transmission at the Fermi energy $T(E_F)$ as a function of PA length (unit cells N) and electron scattering rate $\mathrm{\Gamma }$, within the Landauer and the Kubo formalisms. The exponential scaling $\propto exp(-\beta N)$ and the $\mathrm{\Gamma }$-driven hyperbolic scaling $\propto \mathrm{\Gamma }/N^2$ are highlighted. Within Kubo, the exponential scaling disappears for $\mathrm{\Gamma }\gtrsim {\mathrm{\Gamma }}_{\mathcal{L}}=2.5\times {10}^{-3}$ eV as, then, transport is dominated by electron–electron scattering processes. Cuts of $T(E_F)$ at fixed $\mathrm{\Gamma }$ for the (b) Landauer, (c) Landauer with vertex corrections, and (d) Kubo response. Note that in (b) all curves overlap, as Landauer is independent of $\mathrm{\Gamma }$. In the inset of (c) we show the extra contribution from the vertex corrections, i.e., the difference between the main panels (c) and (b), which instead depends on $\mathrm{\Gamma }$. The filled circles in (d) correspond to the crossover length $N_c$ between the exponential and hyperbolic regimes in the Kubo response

Fig. 5

Resistance R(T) within the (a) Landauer without (solid line) and with (dashed line) vertex corrections, and (b, c) Kubo formalism. a, b The saturation value below the length-dependent crossover temperature $R(T<T^{\ast })$ is dominated by $T(E_F)$. The filled circles in (a, b) correspond to the crossover temperature $T^{\ast }$, estimated from the resistance inflection point $d^{2}R(T)/d{T}^2=0$. c For $\mathrm{\Gamma }\to 0$, the Kubo resistance converges towards the Landauer result