Dynamic response of a spin-$\frac{1}{2}$ Kondo singlet

We present a study of spin-1/2 Kondo singlets in single electron transistors under a microwave frequency bias excitation. We compare time-averaged conductance $G$ to predicted universal response with respect to microwave frequency, oscillation amplitude, and the Kondo temperature and find a nonadiabatic response when the microwave photon energy $hf$ is comparable to the Kondo temperature $k_B{T}_K$. We show that our measurements are qualitatively consistent with the predictions for the radiation-induced decoherence rate of the Kondo spin.

Figure 1

Electron micrograph of the SET showing the labeling of dc voltages on the “side,” ‘top,” “bottom,” and ‘gate” gates that define the SET. The oscillating voltage with amplitude $V_{\text{ac}}$ and microwave frequency $f$ is supplied by a coaxial line coupled capacitively to the source-drain path. The dc (static) portion of the SET drain-source bias, $V_{\text{ds}}$, is supplied by a separate filtered computer-controlled dc circuit (not shown).

Figure 2

(a), (b) Differential conductance map in SET1 (a) and SET2 (b). Dashed lines indicate the gate voltages $V_G$, $-673$ mV for SET1 and $-558$ mV for SET2, used in the experiment. (c), (d) Cor-responding static conductance $G_{\text{dc}}$ traces. (e), (f) A schematic illustrating three different regimes depending on the excitation voltage $V_{\text{ac}}$ in the adiabatic limit, and the corresponding changes in the zero-bias time-averaged conductance $G$: (1) low $V_{\text{ac}}$: $G$ drops with increasing $V_{\text{ac}}$, reaches a minimum (2), and begins to increase again (3). For explanations, see text.

Figure 3

Predicted boundaries between the adiabatic and nonadiabatic regimes in a Kondo-correlated SET at zero static bias and different ac bias amplitudes $V_{\text{ac}}$ and bias oscillation frequencies ${\omega }^{\prime }$. Reproduced with permission from Ref. [20], Fig.3. Equations (76) and (77) in [20] describe the adiabatic regime and are derived from a form mathematically identical to Eq. (1) in this paper. Equation (74) [Eq. (2) in this paper] and Eq. (73) in [20] describe the high-frequency nonadiabatic regime.

Figure 4

Zero-bias conductance $G$ as a function of power at the end of the microwave line. (a) SET 1. (b) SET2.

Figure 5

(a) Conductance as a function of $V_{\text{ac}}$. Conversion from microwave power to $V_{\text{ac}}$ for each frequency is done by matching the high-power rise of $G$, regimes (2) and (3) shown in Figs. 2 and 2, to the adiabatic calculation (solid black line). (b) Conductance as a function of frequency at a fixed $V_{\text{ac}}=20\mu \text{V}$. The $V_{\text{ac}}$ value [(a), dashed line] is chosen in the range where the curves show the strongest sensitivity to microwave frequency. Vertical arrow marks frequency $f_K$ at which the photon energy matches the Kondo temperature, $f_K=k_B{T}_K/h$. The conductance suppression is maximized at frequencies of order $f_K$. For the analysis of Kondo temperatures, see Supplemental Material [23].

Figure 6

(a) Conductance as a function of $V_{\text{ac}}$ for SET2. Figure notations are the same as for SET1 (see caption for Fig. 5).

Figure 7

Universality check: peak conductance $G\left(V_{\text{ac}}\right)$ scaled by $G_0$ as a function of the predicted spin decoherence rate, scaled by the Kondo temperature. For SET1 (a) and SET2 (b). The dotted lines show the predicted universal dependence of $G$ on the spin decoherence rate $\tau$.