Super-micron-scale atomistic simulation for electronic transport with atomic vibration: Unified approach from quantum to classical transport

We develop a powerful simulation method that can treat electronic transport in a super-micron-scale open system with atomic vibration at finite temperature. As an application of the developed method to realistic materials, we simulate electronic transport in metallic single-walled carbon nanotubes from nanometer scale to micrometer scale at room temperature. Based on the simulation results, we successfully identify two different crossovers, namely, ballistic to diffusive crossover and coherent to incoherent crossover, simultaneously and with equal footing, from which the mean free path and the phase coherence length can be extracted clearly. Moreover, we clarify the scaling behavior of the electrical resistance and the electronic current in the crossover regime.

Figure 1

Simulation model consists of three regions: the central region, the left lead, and the right lead. The central region is composed of a scattering region with atomic vibration and two buffer layers without vibration, each of which is composed of two unit cells. The central region is sandwiched between the two semi-infinite leads without atomic vibration.

Figure 2

Time $t$ dependence of the cumulative average ${\overline{j}}_{{\epsilon }_{\mathrm{F}}}\left(t\right)$ at 300 K for a (5,5)-SWNT from 0 to 200 ps. ${\overline{j}}_{{\epsilon }_{\mathrm{F}}}\left(t\right)$ arrives at a steady current as $t\to \infty$ for every $L$.

Figure 3

System length $L$ dependence of the electrical resistance $R$ of an armchair SWNT at 300 K. The error bars originate from the standard error of $j_{{\epsilon }_{\mathrm{F}},n}\left(t\right)$ from 160 to 200 ps per 1 ps. The dotted line represents $R=(R_0/2)L/L_{\mathrm{m}}$ in the diffusive limit at $L/L_{\mathrm{m}}\to \infty$, where $L_{\mathrm{m}}$ is the mean free path, and the dash-dotted line represents $R=R_0/2$ in the ballistic limit at $L/L_{\mathrm{m}}\to 0$. The inset shows that the diameter $d_t$ dependence of $L_{\mathrm{m}}$ at 300 K as $L_{\mathrm{m}}=366.6d_t$. The error bars originate from those of $R$.

Figure 4

System length $L$ dependence of the coherent component $J_{{\epsilon }_{\mathrm{F}}}\left({\omega }_{\mathrm{F}}\right)$ for an armchair SWNT at 300 K. $J_{{\epsilon }_{\mathrm{F}}}\left({\omega }_{\mathrm{F}}\right)$ behaves as $J_{{\epsilon }_{\mathrm{F}}}\left({\omega }_{\mathrm{F}}\right)=2exp(-L/L_{\varphi })$ (solid line), where $L_{\varphi }$ is the phase coherence length. The dotted line, $1-L/L_{\varphi }$, shows the asymptotic behavior of $J_{{\epsilon }_{\mathrm{F}}}\left({\omega }_{\mathrm{F}}\right)$ for $L/L_{\varphi }\to 0$. The inset shows the diameter $d_t$ dependence of $L_{\varphi }$ at 300 K as $L_{\varphi }=352.1d_t$.

Figure 5

Relationship between the mean free path $L_{\mathrm{m}}$ and the phase coherence length $L_{\varphi }$ for an armchair SWNT at 300 K. The solid line, shown as $L_{\varphi }=0.958L_{\mathrm{m}}$, is always under the dotted line shown as $L_{\varphi }=L_{\mathrm{m}}$. This means that $L_{\mathrm{m}}$ is always longer than $L_{\varphi }$.

Figure 6

System length $L$ dependence of the electrical resistance $R$ for an SWNT with various diameters at 300 K. The error bars originate from the standard error of $j_{{\epsilon }_{\mathrm{F}},n}\left(t\right)$ from 160 to 200 ps in the steady state per 1 ps. The dotted lines represent the resistance $R=(R_0/2)L/L_{\mathrm{m}}$ in the diffusive limit $L\gg L_{\mathrm{m}}$, and the dash-dotted line represents the resistance $R=R_0/2$ in the ballistic limit $L_{\mathrm{m}}\gg L$.

Figure 7

System length $L$ dependence of the coherent component $J_{{\epsilon }_{\mathrm{F}}}\left({\omega }_{\mathrm{F}}\right)$ for an armchair SWNT with various diameters $d_t$ at 300 K. The dotted lines, $1-L/L_{\varphi }$, show the asymptotic behavior of $J_{{\epsilon }_{\mathrm{F}}}\left({\omega }_{\mathrm{F}}\right)$ for $L\to 0$.