Theory of length-dependent conductance in one-dimensional chains

The length-dependent conductance of molecular wires and quantum dot array has been theoretically investigated based on finite size one-dimensional model. It has been clarified that the dependences of the conductance on the length of the chain have qualitatively different characteristics even for noninteracting electrons as the Fermi energy of the lead is varied relative to the energy spectra of atoms. It has also been found that the relative magnitude of the transfer integrals in the chain and the electrode, and those between the electrode and the chain plays a crucial role on the conductance. We have found length dependence in these cases have more physics besides the even-odd oscillatory behavior, which has been studied in a special case of the atomic wire system. The dephasing effects and the effects of strong correlation in the dot have been briefly investigated based on the Hubbard chain. A simple formula for the conductance given here may be useful to analyze band calculation results and experimental results.

Figure 1

(Color online) A schematic top view of the nanocontact system composed of a conductor and electrodes.

Figure 2

The local density of state per spin of the extended chain at the terminal atom; i.e., $-(1∕\pi )\mathrm{Im}{g}_{11}\left(E\right)$. The contact effect is included. (a), (b), (c), and (d) are the results of the one, two, three, and four atomic chains. Arrows denote the positions of the delta function peaks of the corresponding local density of state of the isolated chains.

Figure 3

(Color online) The length dependence of the conductance $G_F$ in unit of $2e^2∕h$ for various values of the energy difference between the chemical potential of the electrode and the atomic energy expressed in terms of a parameter $\chi ={\Delta }_F∕2t$, where ${\Delta }_F=E_F-ϵ$ for the choice of parameters $t=-1.0$, $t_L=-1.0$, $t_M=-10.0$, then ${\tilde{t}}_L∕t=0.1$ and in the case where atoms in the electrodes and those of the chain are the same and the atomic site energy is taken as origin of energy, i.e., $ϵ=\xi =0$. The results from $N=1$ to $N=12$ are shown together with those up to $N=100$ in the inset.

Figure 4

(Color online) The length dependence of the conductance $G_F$ in unit of $2e^2∕h$ for various values of the energy difference between the chemical potential of the electrode and the atomic energy expressed in terms of a parameter $\chi ={\Delta }_F∕2t$, where ${\Delta }_F=E_F-ϵ$ for the choice of $t=-1.0$, $t_L=-\sqrt{10}$, $t_M=-10.0$, then ${\tilde{t}}_L=t$ and in the case where atoms in the electrode and those of chain are the same and the atomic site energy is taken as origin of energy, i.e., $ϵ=\xi =0$. The results from $N=1$ to $N=12$ are shown together with those up to $N=100$ in the inset.

Figure 5

(Color online) The length dependence of $G_F$ in unit of $2e^2∕h$ for $t=-1.0$, $t_M=-10.0$, ${\Delta }_F∕2t=0.0$, and $\xi =0$ with various values of $t_L$. ${\tilde{t}}_L=-1.0$ being realized when $t_L=-\sqrt{10}\simeq -3.16$.

Figure 6

(Color online) The local electron density per spin at the $i\text{−}\text{th}$ site in the chain $-(1∕\pi ){\int }^{E_F}\mathrm{Im}{g}_{ii}^R\left(E\right)dE$ as a function of $\chi =\mid {\Delta }_F∕2t\mid$. $E_F$ is the chemical potential of the electrode. (a) The plots for $N=50$. (b) The plots for $N=51$, $N$ is the total number of atoms in the chain. $i$ is the atomic site index in the chain. The parameter set taken here is same as that in Fig. 4.

Figure 7

(Color online) The length dependence of $G_F$ in unit of $2e^2∕h$ for $t=-1.0$, $t_M=-10.0$, ${\Delta }_F∕2t=0.0$, and $\mid \xi \mid =9.0$ with various values of $t_L$. In the cubic lattice model for the electrode, $t_L=-3.44$ satisfies the condition $\mid {\tilde{t}}_L∕t\mid =4{\pi }^2∕\int d{\vec{k}}_{\perp }\sin k_{\Lambda }^{\prime }a\simeq 1.18$.

Figure 8

(Color online) The length dependence of $G_F$ in unit of $2e^2∕h$ for $t=-1.0$, $t_L=-3.44$, $t_M=-10.0$, and ${\Delta }_F∕2t=0.0$ with various values of $\xi$. In the cubic lattice model for the electrode, the value of $t_L$ satisfies the condition $\mid {\tilde{t}}_L∕t\mid =4{\pi }^2∕\int d{\vec{k}}_{\perp }\sin k_{\Lambda }^{\prime }a\simeq 1.18$.

Figure 9

(Color online) Dephasing effect on the conductance $G_F$ (in unit of $2e^2∕h$) in terms of a parameter $\chi ={\Delta }_F∕2t$, where ${\Delta }_F=E_F-ϵ$ for the choice of parameters $t=-1.0$, $t_L=-1.0$, $t_M=-10.0$, $ϵ=0$, $\kappa =0.01$, and $\xi =0$.

Figure 10

(Color online) The conductance $G\left({\Delta }_F\right)$ (in unit of $2e^2∕h$) versus the energy difference between the chemical potential of the electrode and the atomic energy ${\Delta }_F=E_F-ϵ$ for various $U$ in the case of $N=8$. (a) The solid line denotes the result of $U=0.2$, while the dotted line with open circles denotes the result of $U=0$. (b) The result for $U=4$. Here $t=-1.0$, $t_L=-\sqrt{10}$, $t_M=-10.0$, $ϵ=0$, and $\xi =0$ for the both cases.

Figure 11

(Color online) The conductance $G\left({\Delta }_F\right)$ (in unit of $2e^2∕h$) versus the energy difference between the chemical potential of the electrode and the atomic energy ${\Delta }_F=E_F-ϵ$ for various $U$ in the case of $N=8$. (a) The solid line denotes the result of $U=0.2$, while the dotted line with open circles denotes the result of $U=0$. (b) The result for $U=4$. Here $t=-1.0$, $t_L=-1.0$, $t_M=-10.0$, $ϵ=0$, and $\xi =0$ for the both cases.

Figure 12

(Color online) $G\left(E_F\right)$ (in unit of $2e^2∕h$) versus the chemical potential of the electrode $E_F$ for various chain length $N$ for $U=4$, $t=-1.0$, $t_L=-\sqrt{10}$, $t_M=-10.0$, $ϵ=-U∕2$, and $\xi =0$. The center of the Coulomb gap is located at ${\Delta }_F=E_F-ϵ=U∕2$, i.e., $E_F=0$. (a) The linear plot for $N=1$, 2 and 3. (b) The semilog plot for $N=1$, 5, and 10.

Figure 13

(Color online) $I\text{−}V$ curve for $N=1$ (the curve with open stars), $N=2$ (the curve with open circles), $N=3$ (the curve with closed triangles), and $N=8$ (the curve with reversed open triangles) quantum dot array. For $U=4$, $t=-1.0$, $t_L=-\sqrt{10}$, $t_M=-10.0$, $ϵ=-U∕2$, and $\xi =0$. In the inset, the same plot is given in the semilog scale.