Out-of-Equilibrium Kondo Effect in a Mesoscopic Device

We study the nonequilibrium regime of the Kondo effect in a quantum dot laterally coupled to a narrow wire. We observe a split Kondo resonance when a finite bias voltage is imposed across the wire. The splitting is attributed to the creation of a double-step Fermi distribution function in the wire. Kondo correlations are strongly suppressed when the voltage across the wire exceeds the Kondo temperature. A perpendicular magnetic field enables us to selectively control the coupling between the dot and the two Fermi seas in the wire. Already at fields of order 0.1 T only the Kondo resonance associated with the strongly coupled reservoir survives.

Figure 1

(a) Splitting of the Kondo resonance due to finite bias voltage across the quantum dot. The resulting split peaks in the density of states are aligned with the Fermi energies of the leads. (b) Additional splitting due to a double-step distribution function in the source reservoir. (c) Scanning electron micrograph of the device and measurement scheme. Light grey corresponds to metallic gates, dark grey to etched regions.

Figure 2

(a),(c),(e) Differential conductance, $dI/dV$, vs bias voltage, $V$, across the dot. The plunger gate is at $V_g=-40\mathrm{m}\mathrm{V}$. The voltage $\Delta V$ between source 1 and source 2 is set to 0 (a), $-150$ (c), and $150\mu \mathrm{V}$ (e). Inset to (a): linear conductance, $G$, vs $V_g$ around the Kondo valley (see arrow). (b),(d),(f) $dI/dV$ vs ($V,V_g$) on gray scale. Dark gray corresponds to large $dI/dV$. $\Delta V=0$ (b), $-150\mu \mathrm{V}$ (d), and $150\mu \mathrm{V}$ (f). Dotted lines identify the edges of the Coulomb diamond. The Kondo anomaly shows up as a single vertical line at zero bias in (a) and as a vertical split line (see arrows) in (d) and (f).

Figure 3

(a) $dI/dV$ vs $V$, for $\Delta V$ between $-300\mu \mathrm{V}$ (top trace) and $300\mu \mathrm{V}$ (bottom trace) in steps of $50\mu \mathrm{V}$. Curves are vertically offset by $0.2e^2/h$. The voltage scale is shifted by $0.20\Delta V$ to compensate for the voltage drop on $R_1$. The zero-bias Kondo peak at $\Delta V=0$ (thick trace) splits and weakens at finite $\Delta V$. Dotted lines are drawn to emphasize the splitting. (b) $V$ position of the Kondo peaks at $B=0$ (solid circles), $B=-0.2\mathrm{T}$ (open squares), and $B=0.2\mathrm{T}$ (open diamonds). (c) Height of the Kondo peaks, normalized to the $\Delta V=0$ value, vs $\Delta \tilde{\mu }\equiv ({\mu }_{S2}-{\mu }_{S1})/e{w}_0$. $w_0$ is the full width at half maximum of the Kondo peak at $\Delta V=0$. For $B=0$ (solid circles), peak heights are obtained from fitting to a double Lorenztian and a linear background. For each $\Delta V$ we plot only the height of the largest peak. For $B=-0.2\mathrm{T}$ (open squares) and 0.2 T (open triangles), peak heights are obtained from fitting to a Lorentzian and a 5th order polynomial background.

Figure 4

(a) White dotted lines depict electron motion through the wire in the presence of a perpendicular magnetic field $B=-0.2\mathrm{T}$. The Lorentz force pushes right movers down, closer to the dot, and left movers up, away from it. An opposite scenario occurs for $B=0.2\mathrm{T}$ as shown in (b). (c),(d) $dI/dV$ vs $V$, for $\Delta V$ between $-300\mu \mathrm{V}$ (top trace) and $300\mu \mathrm{V}$ (bottom trace) in steps of $50\mu \mathrm{V}$. Curves are offset vertically by $0.2e^2/h$ and horizontally by $0.20\Delta V$. The inset to (c) shows that most of the current through the dot comes from the strongly coupled lead (i.e., source 1 for $B=-0.2\mathrm{T}$). The dot-dashed (dotted) trace is $dI/dV$ vs $V$, measured by the lock-in technique with the ac voltage excitation, $\tilde{v}$, applied only to the left (right) reservoir. This allows us to directly measure the contribution from the left (right) reservoir. The horizontal axis runs from $-100$ to $100\mu \mathrm{V}$, the vertical axis from 0 to $e^2/h$. The inset to (d) shows $dI/dV$ vs $V$ for $B=0.2\mathrm{T}$, $\Delta V$ from $-150$ (top) to $150\mu \mathrm{V}$ (bottom) in steps of $50\mu \mathrm{V}$, and a more negative voltage $V_{ch}=-1.5\mathrm{V}$ on the top gate [see Fig. 1c]. Horizontal scale: from $-150$ to $150\mu \mathrm{V}$. Vertical scale: from 0 to $2e^2/h$. Traces are vertically offset by $0.3e^2/h$.