Impurity and correlation effects on transport in one-dimensional quantum wires

We study transport through a one-dimensional quantum wire of correlated fermions connected to semi-infinite leads. The wire contains either a single impurity or two barriers, the latter allowing for resonant tunneling. In the leads the fermions are assumed to be noninteracting. The wire is described by a microscopic lattice model. Using the functional renormalization group we calculate the linear conductance for wires of mesoscopic length and for all relevant temperature scales. For a single impurity, either strong or weak, we find power-law behavior as a function of temperature. In addition, we can describe the complete crossover from the weak- to the strong-impurity limit. For two barriers, depending on the parameters of the enclosed quantum dot, we find temperature regimes in which the conductance follows power laws with “universal” exponents as well as nonuniversal behavior. Our approach leads to a comprehensive picture of resonant tunneling. We compare our results with those of alternative approaches.

Figure 1

Schematic plot of the quantum dot situation, where the barriers are modeled by two site impurities.

Figure 2

The cutoff function ${\chi }^{\Lambda }\left(x\right)$ for fixed $\Lambda$ and the negative of its derivative (with respect to $\Lambda$).

Figure 3

Decay of the oscillatory part of the off-diagonal matrix element of the self-energy away from a single hopping impurity at bond $j_0,j_0+1$. Results for $t^{\prime }=0.1$, $j_0=5000$, $N={10}^4$, $U=1$, $n=1∕2$ and different temperatures $T={10}^{-1}$ (solid line), $T={10}^{-2}$ (dotted line), $T={10}^{-3}$ (dashed line), and $T={10}^{-4}$ (dashed-dotted line) are presented. The left panel shows the data on a log-log scale, the right panel on a linear-log scale. For comparison the left panel contains a power law ${(j-j_0)}^{-1}$ (thin solid line).

Figure 4

Temperature dependence of the conductance for a single site impurity with different $V$ as indicated in the legend and $U=0.5$, $n=1∕2$, $N={10}^4$. The dotted line shows a power law with an exponent ${\beta }_s$ obtained from fitting the $V=10$ data.

Figure 5

Dependence of the strong-impurity exponent ${\beta }_s$ on the interaction $U$. For comparison the LSGM prediction $2{\alpha }_B$ (solid line) and the leading order (in $U$) result $2{\alpha }_B^l$ (dashed line) are shown.

Figure 6

Weak-impurity behavior of $1-G\left(T\right)∕(e^2∕h)$ for $V=0.01$, $n=1∕2$, $N={10}^4$ and different $U$ as indicated in the legend.

Figure 7

Dependence of the weak-impurity exponent ${\beta }_w$ on the interaction $U$. For comparison the LSGM prediction $2(K-1)$ (solid line) and the leading order (in $U$) result $-2{\alpha }_B^l$ (dashed line) are shown.

Figure 8

Relative error of the fRG approximations for the TLL parameter from the strong-${\beta }_s$ and weak-${\beta }_w$ impurity exponents at $n=1∕2$ and $n=1∕4$.

Figure 9

One-parameter scaling plot of the conductance. Open symbols represent results obtained for $U=0.5$, $n=1∕2$, and different $T$ and $V$, while filled symbols were calculated for $U=0.851$, $n=1∕4$. Both pairs of $U$ and $n$ lead to the same $K_w=0.85$. The solid lines indicate the asymptotic behavior for small and large $x$.

Figure 10

Flow diagram of the conductance in the plane defined by the real and imaginary part of the auxiliary Green function. The direction of the flow for ${\delta }_N\to 0$ and $U>0$ is indicated by the arrows. For $U<0$ it is reversed. The axis are lines of fixed points. $\Delta$ axis: line of “perfect chain” fixed points with $G=e^2∕h$. $(\tilde{V}+2\Omega )$ axis: line of “open chain” fixed points with $G=0$.

Figure 11

The conductance $G\left(T\right)$ for a site impurity with $V=10$, $U=1$, $n=1∕2$, $N={10}^4+1$, and different impurity positions $j_0$.

Figure 12

Gate-voltage dependence of the conductance for a symmetric double barrier at $T=0$. The parameters are $N_D=6$ and $V_{l∕r}=2$.

Figure 13

Gate voltage dependence of the conductance for an asymmetric double barrier at $T=0$. The parameters are $U=0.5$, $N_D=10$, $t_l=0.1$, and $t_r=0.3$. Only gate voltages close to $V_g^r$ are shown. Note the linear-log scale.

Figure 14

Schematic plot of the different regimes for the scaling of the conductance $G_p$ at resonance for symmetric barriers found using our approximation scheme. The upper left panel is for dots with a large number of lattice sites $N_D$ and high barriers, the upper right for small $N_D$ and high barriers, the lower left for large $N_D$ and low barriers, and the lower right for small $N_D$ and low barriers. The relevant energy scales are the band width $B$, the $U=0$ level spacing of the dot ${\delta }_{N_D}$, the lower bound of the UST regime $T^{*}$ [see Eq. (53)], and the infrared cutoff ${\delta }_N$ associated with the number of lattice sites forming the interacting wire.

Figure 15

Effective exponent (i.e., logarithmic derivative) of $1-G_p\left(T\right)∕(e^2∕h)$ for a small dot with $N_D=1$, $U=1$, $N={10}^4$, and intermediate to weak hopping barriers $t_{l∕r}$. Dashed-dotted line: $2K_w$.

Figure 16

Upper panel: $G\left(T\right)$ for symmetric barriers with $V_{l∕r}=10$ for $U=0.5$, $N={10}^4$, $N_D=100$ at resonance $\mid \Delta V_g\mid =0$ (circle), very close to a resonance with $\mid \Delta V_g\mid =0.001$ (squares), and in a conductance minimum with $\mid \Delta V_g\mid =0.04$ (diamonds). Lower panel: Logarithmic derivative of $G\left(T\right)$. Solid line: ${\beta }_s$; dashed line: ${\beta }_s∕2-1$.

Figure 17

Upper panel: $G\left(T\right)$ for symmetric barriers with $t_{l∕r}=0.2$ for $U=1$, $N={10}^4$, $N_D=4$ at resonance $\mid \Delta V_g\mid =0$ (circle), close to a resonance with $\mid \Delta V_g\mid =0.005$ (squares), and further away from the resonance with $\mid \Delta V_g\mid =0.1$ (diamonds). Lower panel: Logarithmic derivative of $G\left(T\right)$. Solid line: ${\beta }_s$; dashed line: ${\beta }_s∕2-1$.

Figure 18

Upper panel: $G_p\left(T\right)$ for asymmetric barriers with $U=0.5$, $N={10}^4$, $N_D=2$, and $V_l=7$, $V_r=12.3$ (squares) as well as $V_l=4$, $V_r=13.6$ (diamonds). For comparison also the symmetric case with $V_l=V_r=10$ is shown (circles). Lower panel: Logarithmic derivative of $G_p\left(T\right)$. Solid line: ${\beta }_s$; dashed line: ${\beta }_s∕2-1$.

Figure 19

Upper panel: $G_p\left(T\right)$ for asymmetric barriers with $V_l=5$, $V_r=50$, $U=0.5$, $N={10}^4$, $N_D=2$ (circles), and $N_D=20$ (squares). Lower panel: Logarithmic derivative of $G_p\left(T\right)$. Solid line: ${\beta }_s$; dashed line: ${\beta }_s∕2-1$.