Transient currents and universal time scales for a fully time-dependent quantum dot in the Kondo regime

Using the time-dependent noncrossing approximation, we calculate the transient response of the current through a quantum dot subject to a finite bias when the dot level is moved suddenly into a regime where the Kondo effect is present. After an initial small but rapid response, the time-dependent conductance is a universal function of the temperature, bias, and inverse time, all expressed in units of the Kondo temperature. Two timescales emerge: the first is the time to reach a quasimetastable point where the Kondo resonance is formed as a broad structure of half-width of the order of the bias; the second is the longer time required for the narrower split peak structure to emerge from the previous structure and to become fully formed. The first time can be measured by the gross risetime of the conductance, which does not substantially change later while the split peaks are forming. The second time characterizes the decay rate of the small split Kondo peak (SKP) oscillations in the conductance, which may provide a method of experimental access to it. This latter timescale is accessible via linear response from the steady state and appears to be related to the scale identified in that manner [A. Rosch, J. Kroha, and P. Wölfle, Phys. Rev. Lett. 87, 156802 (2001)].

Figure 1

(Color online) Time dependent conductance vs time for system one. The ordinate is the time dependent conductance $G\left(t\right)\equiv I\left(t\right)∕V$ in units of the open-channel conductance $G_0\equiv 2(e^2∕2\pi \hslash )$. As before, $I\left(t\right)$ is the time dependent current, and $V$ the bias. The abscissa is the time $t$ after the gate switches the dot into the Kondo regime, in units of ${\Gamma }^{-1}$. The numbers in the legend are the temperatures $T$ for the corresponding curves in units of $\Gamma$. All curves are in the linear response region of approaching zero bias $V$.

Figure 2

(Color online) Universality of time-dependent conductance curves. The S1 curves are identical to the corresponding ones in Fig. 7 with the time axis rescaled to inverse $T_K$ units, where here $T_K=0.0022\Gamma$. The S2 are curves for system two (see text), which has a smaller $T_K$.

Figure 3

(Color online) Time dependent conductance vs time. The ordinates are the time dependent conductance $G\left(t\right)\equiv I\left(t\right)∕V$ in units of the open-channel conductance $G_0\equiv 2(e^2∕2\pi \hslash )$. Here $I\left(t\right)$ is the time dependent current, and $V$ the bias. The abscissas are the time $t$ after the gate switches the dot into the Kondo regime, in units of ${\Gamma }^{-1}$. Both panels are for system one (see text), with $T=0.0015\Gamma$ (left panel) and $T=0.005\Gamma$ (right panel). The numbers in the legend are the biases $V$ in units of $\Gamma$ for the corresponding curves in each panel.

Figure 4

(Color online) Initial oscillations at short times. The top (long-dashed) curve in the top panel is the short time version of the system one $(\mid {ϵ}_{\text{dot}}\mid =2\Gamma )$ curve for bias $V=0.01\Gamma$. Thus it represents the very short time behavior of the curves in Fig. 3. The behavior is similar for all biases studied. The other three curves are for other values of $\mid {ϵ}_{\text{dot}}\mid$ whose ratio to $\Gamma$ is indicated in the legend. In each case the time coordinate is rescaled as indicated, to show that the frequency of the oscillation is about equal to the magnitude of virtual resonant level’s energy (measured from the Fermi level). The renormalized value of the latter is a bit larger than the bare value $\mid {ϵ}_{\text{dot}}\mid$. The bottom panel, showing the second derivatives of the curves in the top panel, more clearly identifies the period of oscillation.

Figure 5

(Color online) Comparison of instantaneous conductance $G\left(\tau \right)=I\left(\tau \right)∕V$ as a function of time $\tau$ after the level was moved and $[dQ\left(\tau \right)∕d\tau ]∕V$ as a function of pulse length $\tau$, both expressed in units of $G_0$. The quantum dot is system one. The bias across the dot is $V=0.08\Gamma$ and the temperature is $T=0.0015\Gamma$.

Figure 6

(Color online) The large-time limit of $G\left(t\right)∕G_0$ vs temperature. Since $V\sim 0$, this is the equilibrium value of $G∕G_0$. The points labeled S1 and S2 are from system one and two, respectively (see text). The curve labeled asymptote in the legend represents the large $T$ asymptote within NCA (see text).

Figure 7

Rate $(1∕\tau )$ at which the conductance approaches its final value vs temperature $T$, with zero bias $V=0$. The solid dots are for system one (see text), while the open circles are for system two, which has a different Kondo temperature $T_K$. The solid curve is a straight line of slope $\pi$ through the origin [see Eq. (9)].

Figure 8

Long time or dc conductance vs bias. The ordinate $G∕G_0$ is the dc conductance in units of the open channel conductance $G_0=2e^2∕2\pi \hslash$ for the two spins, while the abscissa is the bias $V$ in units of the Kondo temperature $T_K$. The triangles represent the zero temperature NCA data of Ref. 21 with the abcissa rescaled to correct for the $\sim 30\%$ difference between the two definitions of the Kondo temperature. The open circles, squares, and diamonds represent the long time saturation of our time-dependent conductance curves for system one at the temperatures indicated. The smaller solid symbols represent the values at the same respective temperatures for system two, which is a different Kondo temperature (see text).

Figure 9

Rate (inverse time) at which the conductance approaches its “final value” vs bias $V$, at several temperatures as indicated. The open circles, squares, and diamonds are for system one (see text), while the corresponding solid symbols are for system 2, which has a different Kondo temperature. Our $V=0$ points are shown in the figure at $V∕T_K=0.1$. The solid lines are the predictions of Eq. (10).

Figure 10

(Color online) Time-dependent spectral functions for the time scale relevant for the rise time of the conductance. These curves are for system one with $T=0.0015\Gamma =0.69T_K$, and $V=0.08\Gamma =37T_K$. The curves are at equal time intervals $\Delta t=2.8{\Gamma }^{-1}=0.006T_K^{-1}=0.17(4∕\pi V)$, where the $4∕\pi V$ factor is motivated by the high temperature asymptote of Eq. (10). This factor has the value of unity at $t=6\Delta t$ (thick solid black curve). The lowest curve is for time $(t=2\Delta t)$ after the virtual level parameter was switched to its final value of $-2\Gamma$. By curve $15$ the conductance has for all practical purposes reached its dc value.

Figure 11

The rise in conductance as the Kondo peak forms for bias $V=0.08\Gamma$ at $T=0.0015$. The solid curve is a magnified section of the corresponding curve in Fig. 3 with the horizontal axis shifted as indicated in the legend. The dashed curve is the indicated exponential fit. The rate $R$ obtained in this way characterizes the risetime $(R=1∕\tau )$ of the conductance, and this is the rate used in the preparation of Fig. 9. Of all the curves in Fig. 3 this is the worst case for the exponential fit, because here the SKP oscillations are largest. The first period of these oscillations is partially visible here. Multiple cycles are shown in the larger magnification of Fig. 13. The quantum dot is system one.

Figure 12

(Color online) Time-dependent spectral functions for the time scale relevant for the development of the split Kondo peak and the damping of SKP oscillations. This is a continuation of the snapshots shown in Fig. 10, so all the parameters have the values shown in the caption of that figure. Here, however, the time interval between the successive snapshots is much greater, varying from $5\Delta t$ to $50\Delta t$, as indicated by the labeling in the legend. The quantum dot is system one.

Figure 13

(Color online) SKP oscillations for bias $V=0.08\Gamma$ at $T=0.0015$. The solid curve is a magnified section of the corresponding curve in Fig. 3 with the position of the horizontal axis shifted as indicated in the legend. The dashed curve is a progressively more magnified version of the solid curve, so that more SKP oscillations can be seen. As indicated in Fig. 15, the angular frequency of these oscillations is almost exactly equal to $V$. The quantum dot is system one.

Figure 14

Decay of SKP oscillations vs bias $V$. The empty squares, diamonds, and triangles represent data derived from the damping rate of the SKP oscillations in our calculations of conductance vs time at the temperature indicated, respectively. The quantum dot is system one. The solid circles are the calculations of $2\gamma$ at zero temperature in Ref. 21.

Figure 15

(Color online) Positions of extrema and zeros of SKP oscillations for $V=0.080\Gamma$ as in Fig. 13. The slope of the fitted line gives a frequency of $0.0783\Gamma$. The frequency is always slightly less than $V$, but for our largest $V$’s, we find the difference to be only around 0.1%. The quantum dot is system one.

Figure 16

Determination of the decay rate of the SKP oscillations from the positions of the minima in Fig. 13. The slope of the curve gives a rate of $0.015\Gamma$ for $V=0.08\Gamma$ and $T=0.0015$. These rates are universal functions of $V∕T_K$ and $T∕T_K$, and hence depend on $\Gamma ,E$, and $D$ only through Eq. (2). The quantum dot is system one with a Kondo temperature $T_K=0.0022\Gamma$.