Transport properties of single-channel quantum wires with an impurity:  Influence of finite length and temperature on average current and noise

The inhomogeneous Tomonaga-Luttinger liquid model describing an interacting quantum wire adiabatically coupled to noninteracting leads is analyzed in the presence of a weak impurity within the wire. Due to strong electronic correlations in the wire, the effects of impurity backscattering, finite bias, finite temperature, and finite length lead to characteristic nonmonotonic parameter dependencies of the average current. We discuss oscillations of the nonlinear current voltage characteristics that arise due to reflections of plasmon modes at the impurity and quasi-Andreev reflections at the contacts, and show how these oscillations are washed out by decoherence at finite temperature. Furthermore, the finite-frequency current noise is investigated in detail. We find that the effective charge extracted in the shot noise regime in the weak backscattering limit decisively depends on the noise frequency $\omega$ relative to $v_F∕gL$, where $v_F$ is the Fermi velocity, $g$ the Tomonaga-Luttinger interaction parameter, and $L$ the length of the wire. The interplay of finite bias, finite temperature, and finite length yields rich structure in the noise spectrum which crucially depends on the electron-electron interaction. In particular, the excess noise, defined as the change of the noise due to the applied voltage, can become negative and is nonvanishing even for noise frequencies larger than the applied voltage, which are signatures of correlation effects.

Figure 1

The upper part of the figure shows a QW with an impurity adiabatically coupled to Fermi liquid leads. In order to allow for a finite bias, the leads are held on different electrochemical potentials ${\mu }_L$ and ${\mu }_R$. The middle part of the figure shows the actual variation of the TLL parameter $g$ along the wire-leads system in the ITLL model, and the lower part of the figure its simplification under the assumption that ${\lambda }_F⪡L_s⪡L$.

Figure 2

Behavior of the backscattering current [in units of $e{(\lambda {\omega }_L^g∕{\omega }_c^g)}^2∕{\hslash }^2{\omega }_L$] as a function of $V$, for two different impurity positions ${\xi }_0=0$ (solid line) and ${\xi }_0=\pm 0.2$ (dashed line).

Figure 3

The backscattering current (same units as in Fig. 2) as a function of voltage and temperature.

Figure 4

The backscattering current (same units as in Fig. 2) as a function of temperature, for a voltage value $\mathit{eV}=5\hslash {\omega }_L$ corresponding to the first current minimum at $T=0$ in Fig. 2.

Figure 5

(Color online) The dimensionless quantity ${\kappa }_{\mathrm{BS}}\left(V\right)$ defined in Eq. (21) as a function of the voltage, for a centered impurity ${\xi }_0=0$. (a) shows the case of strong interaction $g=0.25$, while (b) depicts the case of weak interaction $g=0.75$. The solid lines refer to zero temperature and the dashed lines to the dimensionless temperature $\mathrm{\Theta }=k_{B}T∕\hslash {\omega }_L=2$. The function ${\kappa }_{\mathrm{BS}}\left(V\right)$ oscillates around the value $2g-1$, indicated by the dotted line, which corresponds to the power law exponent of $I_{\mathrm{BS}}$ in the homogeneous TLL model.

Figure 6

(Color online)${\kappa }_{\mathrm{BS}}\left(V\right)$ as a function of the voltage for an off-centered impurity ${\xi }_0=0.25$, and for strong interaction strength $g=0.25$. The three different curves refer to three different values of the dimensionless temperature $\mathrm{\Theta }=k_{B}T∕\hslash {\omega }_L$: $\mathrm{\Theta }=0.5$ (solid), 1 (dashed), and 3 (dotted). The two frequencies mentioned in the text are in this case ${\omega }_{L_1}∕{\omega }_L=2∕3$ and ${\omega }_{L_2}∕{\omega }_L=2$. For low temperatures the voltage dependence is affected by Andreev-type reflections at both contacts. For higher temperatures interference effects survive only for Andreev-type reflections at the closer contact, so that only the frequency ${\omega }_{L_2}$ is present. The average value of the oscillations is $2g-1$ (horizontal solid line), whereas the period of the oscillations $\mathrm{\Delta }V=2\pi \hslash {\omega }_L∕e(1-2\mid {\xi }_0\mid )$ reveals the position of the impurity.

Figure 7

(Color online) The real part of the conductivity (32) of a clean wire (in units of $e^2∕h$), is plotted as a function of $\omega$, for three different positions $x$ of the measurement point, and for interaction strength $g=0.25$; the parameter ${\delta }_x=(\mid x\mid -L∕2)∕L$ defines the distance of $x$ to the closer contact. One can see the oscillatory behavior characterized by the two periods $\mathrm{\Delta }{\omega }_1$ and $\mathrm{\Delta }{\omega }_2$, with $\mathrm{\Delta }{\omega }_1⪡\mathrm{\Delta }{\omega }_2$. As the point $x$ gets closer to the contact the beating period $\mathrm{\Delta }{\omega }_2$ tends to infinity. The dashed curve results from an averaging over the rapidly oscillating component with period $\mathrm{\Delta }{\omega }_1$ according to Eq. (38). The oscillation amplitude of the dashed curve is $\gamma$, i.e., it allows one to determine the interaction strength $g$ from Eq. (18).

Figure 8

(Color online) The noise in the absence of the impurity (in units of $e^2{\omega }_L$), as a function of $\omega ∕{\omega }_L$, for the measurement position ${\delta }_x=0.05$. (a) The whole range of frequencies for three values of the dimensionless temperature $\mathrm{\Theta }=k_{B}T∕\hslash {\omega }_L$ (solid curve $\mathrm{\Theta }=0$, dashed curve $\mathrm{\Theta }=2$, and dotted curve $\mathrm{\Theta }=10$). (b) Enlargement for the range of low frequencies $\omega \in [0,3{\omega }_L]$. (c) The case of low temperatures (solid curve $\mathrm{\Theta }=0$, dashed curve $\mathrm{\Theta }=0.05$, and dotted curve $\mathrm{\Theta }=0.1$).

Figure 9

(Color online) The impurity noise $S_{\mathrm{imp}}$ at equilibrium [in units of $e^2{\omega }_L{({\lambda }^{*}∕\hslash {\omega }_L)}^{2(1-g)}$], as a function of $\omega ∕{\omega }_L$, for the measurement position ${\delta }_x=0.05$, the impurity in the center $({\xi }_0=0)$, and three different values of the dimensionless temperature $\mathrm{\Theta }=k_{B}T∕\hslash {\omega }_L$ (solid curve $\mathrm{\Theta }=0$, dashed curve $\mathrm{\Theta }=2$, and dotted curve $\mathrm{\Theta }=10$). Lower graph: Enlargement for low frequencies. Finite temperature influences significantly the slope of the noise near $\omega =0$.

Figure 10

(Color online) The zero-temperature impurity noise $S_{\mathrm{imp}}$ at equilibrium [in units of $e^2{\omega }_L{({\lambda }^{*}∕\hslash {\omega }_L)}^{2(1-g)}$] as a function of $\omega ∕{\omega }_L$, for the measurement position ${\delta }_x=0.05$, and interaction parameter $g=0.25$. Shown are three cases of the impurity position: ${\xi }_0=0$ (solid curve) a centered impurity, ${\xi }_0=0.25$ (dashed curve) an impurity off-centered by $1∕4$ toward the direction of the measurement point, and ${\xi }_0=-0.25$ (dotted curve) an impurity off-centered by $1∕4$ opposite to the direction of the measurement point.

Figure 11

(Color online) The zero-temperature impurity noise $S_{\mathrm{imp}}$ at equilibrium [in units of $e^2{\omega }_L{({\lambda }^{*}∕\hslash {\omega }_L)}^{2(1-g)}$] as a function of $\omega ∕{\omega }_L$, for the measurement position ${\delta }_x=0.05$, in the case of a centered impurity $({\xi }_0=0)$ and for three different values of the interaction strength: $g=0.25$ (solid curve), $g=0.50$ (dashed curve), and $g=0.75$ (dotted curve).

Figure 12

(Color online) The frequency spectrum of the nonequilibrium impurity noise $S_{\mathrm{imp}}$ [in units of $e^2{\omega }_L{({\lambda }^{*}∕\hslash {\omega }_L)}^{2(1-g)}$], at $T=0$, for three different values of the dimensionless voltage $u=\mathit{eV}∕\hslash {\omega }_L$: $u=2$ (solid curve), 4 (dashed curve), and 8 (dotted curve). The interaction strength is $g=0.25$ and the impurity position is ${\xi }_0=0$. $S_{\mathrm{imp}}$ has a cusp singularity at $\omega =\mathit{eV}∕\hslash$.

Figure 13

(Color online) (a) The frequency spectra of the two contributions $S_A$ (dashed curve) and $S_C$ (dotted curve) to the impurity noise $S_{\mathrm{imp}}=S_A+S_C$ (solid curve) show that the presence of the cusp is due to $S_A$. The value of the dimensionless voltage is $u=2$, and the other parameters are chosen as in Fig. 12. (b) The contribution $S_A$ for four different values of the interaction strength: $g=0.25$ (solid curve), 0.50 (dashed curve), 0.75 (dotted curve), and 0.95 (dash-dotted curve). The interaction modifies the slope to the right and to the left of the singularity at $\omega =\mathit{eV}∕\hslash$.

Figure 14

The periodic function $F\left(\omega \right)$ defined in Eq. (56) is plotted for $g=0.75$ against $\omega ∕{\omega }_L$. While $F\left(0\right)=1$, the average over one period yields the interaction parameter $g$.

Figure 15

(Color online) The dimensionless quantity ${\mathrm{\Delta }}_A$ defined in the text as a function of frequency in the shot noise regime $\mathit{eV}⪢\{k_{B}T,\hslash \omega ,\hslash {\omega }_L\}$. The value of the dimensionless voltage is $u=\mathit{eV}∕\hslash {\omega }_L=100$, and the impurity is located in the middle ${\xi }_0=0$. Differences between the exact value of ${\mathrm{\Delta }}_A$ and its approximation (60) are not visible. (a) refers to the case of weak interaction $g=0.75$ and (b) to the case of strong interaction $g=0.25$. The three different curves in each plot correspond to three different values of the dimensionless temperatures $\mathrm{\Theta }=k_{B}T∕\hslash {\omega }_L$, $\mathrm{\Theta }=0$ (solid), 1 (dashed), and 2 (dotted). Notice the different scales in the two cases: For weak interaction even small temperatures allow one to satisfy the requirement ${\mathrm{\Delta }}_A\simeq 1$, whereas for strong interaction a temperature of the order of $\mathrm{\Theta }\simeq 2\text{–}3$ is required.

Figure 16

(Color online) The frequency spectrum of the full noise $S(x,x,\omega )$ (solid line) in units of $e^2{\omega }_L$ and the approximation (54) (dotted line) are depicted for two systems with the parameters (a) $g=0.75$, $\mathrm{\Theta }=1$, $u=100$, ${\delta }_x=0.05$, ${\xi }_0=0$, $\mathcal{R}=0.2$ and (b) $g=0.25$, $\mathrm{\Theta }=2$, $u=98.83$, ${\delta }_x=0.05$, ${\xi }_0=0$, $\mathcal{R}=0.2$. The dashed curve is obtained after a background subtraction (see text).

Figure 17

(Color) The excess noise at zero temperature and for an impurity in the middle of the wire $({\xi }_0=0)$ is shown [in units of $e^2{\omega }_L{({\lambda }^{*}∕\hslash {\omega }_L)}^{2(1-g)}$] as a function of the dimensionless frequency $\omega ∕{\omega }_L$ and the applied voltage $\mathit{eV}∕\hslash {\omega }_L$ for four values of interaction strength: $g=$ (a) 0.25, (b) 0.5, (c) 0.75, (d) 0.95.

Figure 18

The coefficient $D^{\left(1\right)}(+\mid \xi \mid ,\mathrm{\Theta })$ as a function of the relative impurity position $\xi$ and the dimensionless temperature $\mathrm{\Theta }=k_{B}T∕\hslash {\omega }_L$ for $g=0.25$.

Figure 19

The coefficient $D^{\left(2\right)}\left(\mathrm{\Theta }\right)$ as a function of dimensionless temperature $\mathrm{\Theta }=k_{B}T∕\hslash {\omega }_L$ for $g=0.25$ (solid line), 0.5 (dashed line), and 0.75 (dotted line).