Kondo temperature dependence of the Kondo splitting in a single-electron transistor

A Kondo peak in the differential conductance of a single-electron transistor is measured as a function of both the magnetic field and the Kondo temperature. We observe that the Kondo splitting decreases logarithmically with the Kondo temperature and that there exists a critical magnetic field $B_{\mathrm{C}}$ below which the Kondo peak does not split, in qualitative agreement with theory. However, we find that the magnitude of the prefactor of the logarithm is larger than predicted and is independent of $B$, in contradiction with theory. Our measurements also suggest that the value of $B_{\mathrm{C}}$ is smaller than predicted.

Figure 1

(a) Examples of Coulomb blockade diamonds when the artificial atom is strongly coupled to the leads. The Kondo effect shows up as sharp resonances near $V_{\mathrm{ds}}=0$ in the odd valleys. The inelastic cotunneling thresholds in the even valleys indicate the energies of the excited states. These data are at different electrode voltages and during a different cool down of the device from those in the rest of the paper. (b) The evolution of a Kondo peak as a function of magnetic field. Note that the splitting does not begin until $B=2.4\mathrm{T}$. (c) $\mathit{dI}∕{\mathit{dV}}_{\mathrm{ds}}$ vs $V_{\mathrm{ds}}$ for $B=0\mathrm{T}$ and for other values of $B$ marked by the dash-dot lines in (b). For $B<2.4\mathrm{T}$ there is only one Kondo peak; above the threshold it splits. The traces have been offset by $0.08e^2∕h$ for clarity.

Figure 2

(a) $\mathit{dI}∕{\mathit{dV}}_{\mathrm{ds}}$ vs $V_{\mathrm{g}}$ at $V_{\mathrm{ds}}=0$ (denoted $G$) for a Coulomb charging peak and half a Coulomb valley at two different temperatures. The solid line is a fit to Eqs. (2, 3) and is discussed in the text. (b) $G$ as a function of temperature for the value of $V_{\mathrm{g}}$ marked by the dashed vertical line in (a). Fitting these data to Eq. (3) gives $T_{\mathrm{K}}=1.57\pm 0.05\mathrm{K}$. The inset shows an electron micrograph of a SET similar to the one we have studied. (c) The dots show $T_{\mathrm{K}}$ determined as in (b) as a function of $V_{\mathrm{g}}$. The dotted line is the result of fitting these data to Eq. (2). The dashed line shows $T_{\mathrm{K}}$ as a function of $V_{\mathrm{g}}$ using the results of the fit shown in (a). The solid black line shows $T_{\mathrm{K}}$ for the parameter values $\chi =0.020{\left(\mathrm{mV}\right)}^{-2}$ and $T_{\mathrm{K},0}=250\mathrm{mK}$.

Figure 3

(a) A Kondo feature in a 7.5-T magnetic field. The Kondo peak at zero bias has clearly split into two peaks. One Coulomb charging peak is visible at $V_{\mathrm{g}}=-52\mathrm{mV}$; the other peak is not visible. (b) ${\Delta }_{\mathrm{K}}$ as a function of $V_{\mathrm{g}}$. The splitting is a maximum in the center of the valley, where the Kondo temperature is lowest. The gray line shows the result of fitting the data to a parabola. The dashed horizontal line indicates the Zeeman energy $\mid g\mid {\mu }_{B}B$. Note that the splitting in the middle of the valley exceeds $\mid g\mid {\mu }_{B}B$. (c) $\mathit{dI}∕{\mathit{dV}}_{\mathrm{ds}}$ vs $V_{\mathrm{ds}}$ for the values of $V_{\mathrm{g}}$ marked by the lines in (a). The traces have been shifted horizontally so that the peaks at negative $V_{\mathrm{ds}}$ are aligned. The circles mark the positions of the peaks at positive $V_{\mathrm{ds}}$, showing that the splitting between the peaks changes with gate voltage.

Figure 4

(a) Splitting of the Kondo peak in the center of the Coulomb valley as a function of the magnetic field. The solid line shows $\Delta =\mid g\mid {\mu }_{B}B$ for $\mid g\mid =0.16$ determined by inelastic cotunneling. Above $\sim 4\mathrm{T}$ we find that ${\Delta }_{\mathrm{K}}>\mid g\mid {\mu }_{B}B$. Below $B=2.4\mathrm{T}$, no splitting is observed. (b) The curvature of ${\Delta }_{\mathrm{K}}$ vs $V_{\mathrm{g}}$ parabolas like that in Fig. 3b. The prediction of Moore and Wen (Ref. 8) (dashed line) is that the curvature should decrease with field.

Figure 5

(a) Drain-source sweeps in the middle of a Kondo valley at different magnetic fields. The onset of splitting is clear at $B=2.4\mathrm{T}$ but not visible at 2.2 T. The curves have been offset by $0.11e^2∕h$ for clarity. (b) A Kondo valley with a 5-T field applied. The lines indicate the positions of the $\mathit{dI}∕{\mathit{dV}}_{\mathrm{ds}}$ vs $V_{\mathrm{ds}}$ traces shown in (c). (c) $\mathit{dI}∕{\mathit{dV}}_{\mathrm{ds}}$ vs $V_{\mathrm{ds}}$ at various gate voltages. In the center of the valley where $T_{\mathrm{K}}$ is a minimum, the splitting is clear and symmetric. Closer to the charging peak the Kondo temperature is higher and the splitting is less pronounced but still visible. (d) The dots mark the maximum $T_{\mathrm{K}}$ at which splitting is observed as a function of field. These data are extracted from data like those in (b) and (c) by measuring the gate voltage at which the splitting is barely visible. The conversion of this gate voltage to a Kondo temperature is explained in the text. The square data point at $B=2.4\mathrm{T}$ marks the measurement of $B_{\mathrm{C}}$ from the data in (a). Predictions for $B_{\mathrm{C}}$ of Costi (Ref. 7) (solid line) and Moore and Wen (Ref. 8) (dashed line) are shown.