Electron tunneling into a quantum wire in the Fabry-Pérot regime

We study a gated quantum wire contacted to source and drain electrodes in the Fabry-Pérot regime. The wire is also coupled to a third terminal (tip), and we allow for an asymmetry of the tip tunneling amplitudes of right-moving and left-moving electrons. We analyze configurations where the tip acts as an electron injector or as a voltage probe and show that the transport properties of this three-terminal setup exhibit very rich physical behavior. For a noninteracting wire we find that a tip in the voltage-probe configuration affects the source-drain transport in different ways, namely, by suppressing the conductance, by modulating the Fabry-Pérot oscillations, and by reducing their visibility. The combined effect of electron-electron interaction and finite length of the wire, accounted for by the inhomogeneous Luttinger liquid model, leads to significantly modified predictions as compared to models based on infinite wires. We show that when the tip injects electrons asymmetrically the charge fractionalization induced by interaction cannot be inferred from the asymmetry of the currents flowing in source and drain. Nevertheless interaction effects are visible as oscillations in the nonlinear tip-source and tip-drain conductances. Important differences with respect to a two-terminal setup emerge, suggesting new strategies for the experimental investigation of Luttinger liquid behavior.

Figure 1

(Color online) Sketch of the setup. A quantum wire is connected to two metallic electrodes, denoted as source (S) and drain (D) at voltages $V_{\text{S}}$ and $V_{\text{D}}$, respectively. A third sharp electrode, denoted as tip (T), at voltage $V_{\text{T}}$, injects electrons into the wire at position $x_0$. A gate (G) is also present and held at a gate bias voltage $V_{\text{G}}$. The contact resistances are accounted for by two deltalike scatterers with strengths ${\lambda }_1$ and ${\lambda }_2$, and the electron tunneling amplitudes between the tip and the wire are denoted by ${\gamma }_{\pm }$.

Figure 2

(Color online) Zero-temperature differential conductance as a function of the source-drain bias for a noninteracting wire characterized by contact impurity strengths ${\lambda }_1={\lambda }_2=0.1$ and a Fermi wave vector ${\kappa }_W=0.3$. The tip is located in the middle of the wire and the tip voltage $V_{\text{T}}$ is adjusted to fulfill the condition $I_{\text{T}}=0$. Tunneling is symmetric $\left(\chi =0\right)$, and the tunneling strength has the values: $\gamma =0$ (solid line), $\gamma =0.1$ (dashed-line), $\gamma =0.5$ (dashed-dotted line), and $\gamma =1$ (dotted line). The gate is grounded $\left(V_{\text{G}}=0\right)$, and the bias is applied symmetrically $\left(V_{\text{S}/\text{D}}=\pm V/2\right)$.

Figure 3

(Color online) Zero-temperature differential conductance as a function of the source-drain bias for a noninteracting wire with contact impurity strengths ${\lambda }_1={\lambda }_2=0.1$ in presence of a tip with an applied voltage $V_{\text{T}}$ adjusted to fulfill the condition $I_{\text{T}}=0$. Tunneling is totally asymmetric $\left(\chi =1\right)$, and the tunneling strength has the values $\gamma =0.3$ (dotted line) and $\gamma =0.7$ (dashed line). The solid line represents the case with $\gamma =0$. The result is independent of the tip position $x_0$. The gate is grounded $\left(V_{\text{G}}=0\right)$, and the bias is applied symmetrically $\left(V_{\text{S}/\text{D}}=\pm V/2\right)$.

Figure 4

(Color online) Zero-temperature differential conductance as a function of the source-drain voltage for a noninteracting wire for several values of the tip position $x_0=0$ (solid line), $x_0=0.17$ (dotted line), and $x_0=0.41$ (dashed line). Tunneling has amplitude $\gamma =1$ and is symmetric $\left(\chi =0\right)$. The contact impurities have equal strengths ${\lambda }_1={\lambda }_2=0.1$. The gate is grounded $\left(V_{\text{G}}=0\right)$, and the bias is applied symmetrically $\left(V_{\text{S}/\text{D}}=\pm V/2\right)$.

Figure 5

(Color online) Differential conductance of the two-terminal setup (in the absence of a tip) as a function of the source-drain voltage for several values of the interaction parameter $g=1$ (solid line), $g=0.75$ (dashed line), and $g=0.25$ (dotted line). The contact impurities have equal strengths ${\lambda }_{B,1}^{\ast }={\lambda }_{B,2}^{\ast }=0.1$. The gate voltage is $V_{\text{G}}=0$, and the bias is applied symmetrically $\left(V_{\text{S}/\text{D}}=\pm V/2\right)$.

Figure 6

(Color online) The differential conductance $d{I}_{\text{M}}/dV$ [in units of $\left(e{\omega }_L^{\ast }/2\pi \right)$] of the two-terminal setup (in the absence of a tip) is shown as a function of the dimensionless source-drain bias $u$ and the dimensionless gate voltage $u_{\text{G}}$. The strengths of the contact impurities are equal and are characterized by ${\lambda }_{F,i}=0.1$ and ${\lambda }_{B,i}^{\ast }=0.25$. In panels (a), (b), and (c) the source-drain voltage is applied symmetrically, $u_{\text{S}/\text{D}}=\pm u/2$, and the interaction parameter is $g=0.25$, 0.34, and 0.46, respectively. The dashed-dotted lines in panels (b) and (d) are a guide for the eyes to identify the periodic pattern of the Fabry-Pérot oscillations determined by the periods $\Delta u$ and $\Delta u_{\text{G}}$. Their ratio yields the value of $g$, as shown in the table for the three cases. In panel (d) the source-drain voltage is applied asymmetrically ($u_{\text{S}}=u$ and $u_{\text{D}}=0$) to a wire with interaction strength $g=0.34$. When compared with panel (b) for the same interaction strength, the asymmetric bias twists the pattern.

Figure 7

(Color online) Electron injection into an interacting wire. Panel (a) [(b)] shows the tunneling differential conductance $G_{\text{TS}}$ $\left(G_{\text{TD}}\right)$ between the tip and source (drain) electrode as a function of the dimensionless tip-source (tip-drain) bias for a wire with interaction strength $g=0.25$. The various curves refer to different values of the tunneling asymmetry ($\chi =0$ solid line, $\chi =0.6$ dotted line, and $\chi =1$ dashed line). The tunneling strength is ${\gamma }^{\ast }=0.01$, and the tip is located in the middle of the wire. Panels (c) and (d) are the same as panels (a) and (b) but the tip is located near a contact at $x_0=0.45L$.

Figure 8

(Color online) The effect of the tip in the voltage-probe configuration on the zero-temperature conductance is shown as a function of the source-drain bias. Panel (a) the case of symmetric tunneling $\left(\chi =0\right)$. Panel (b) the case of an asymmetry in tunneling $\left(\chi =1/2\right)$. Different curves refer to different values of the interacting strength: noninteracting ($g=1$, solid curve), weakly interacting ($g=0.7$, dashed curve), moderately interacting ($g=0.4$, dotted curve), and strongly interacting ($g=0.25$, dashed-dotted curve). The bare tunneling strength is $\gamma =0.5$ and the dimensionless cut-off parameter is ${\alpha }_W={10}^{-3}$.

Figure 9

(Color online) Zero-temperature nonlinear tip-drain conductance $G_{\text{DT}}$ as a function of the tip bias for tunneling amplitude ${\gamma }^{\ast }={10}^{-2}$ and symmetric tunneling $\chi =0$. The upper panels (a), (b), and (c) are related to a tip in the middle of the wire $\left(x_0=0\right)$, whereas the lower ones (d), (e), and (f) to a tip located at $x_0=0.45L$. Panels pairs (a) and (d), (b) and (e), and (c) and (f) describe the case of a wire with weak $\left(g=0.75\right)$, moderate $\left(g=0.5\right)$, and strong $\left(g=0.35\right)$ interaction strength, respectively. In each panel the different curves refer to different contact impurity strengths. The solid curves describe the case of ideal contacts where the oscillations are purely Andreev type. The dashed (dotted) lines refer to finite contact impurity strength ${\lambda }_{B,1}^{\ast }={\lambda }_{B,2}^{\ast }=0.1$ $\left(=0.2\right)$. For a weakly interacting wire the conductance oscillations are mostly due to the conventional interference between backscattering at the contacts and tip tunneling and Andreev-type oscillations become visible only for extremely low contact resistance. In contrast, for stronger interaction strength a finite contact resistance is sufficient for the oscillations to be attributed to Andreev-type processes. The inset of panel (e) shows the definition of the average amplitude referred to in the text.

Figure 10

Keldysh contour along the time axis.