Single-electron transistors studied by microwave and far-infrared absorption: Theoretical results and experimental proposal

We present theoretical results on microwave and far-infrared (FIR) absorption of single-electron transistors obtained within exact numerical diagonalization for finite clusters. They show that both the microwave and the FIR spectra consist of two maxima, whose origin can be understood physically. Our results on microwave absorption provide a physically intuitive qualitative interpretation of the Kondo splitting observed by Kogan et al. [Science 304, 1293 (2004)]. The present results on the FIR absorption supplement and provide a physical insight into previous results obtained by means of the numerical renormalization group. Based on our theoretical results, we propose to conduct FIR experiments to determine the charging energy and other relevant parameters.

Figure 1

(Color online) Dependence on the dot energy ${\epsilon }_d$ of the four relevant (a) absorption frequencies $\omega$ (in meV) and (b) absorption intensities $\mu$ (in arbitrary units) of the optical transitions with significant spectral intensities. In (a), deeper within the Kondo plateau, the exact frequencies ${\omega }_{3,4}$ are well approximated by ${\omega }_{3,4}^0$. The dashed lines are the analytical continuations of the full lines. The parameter values are: $t=0.5\text{eV}$, $t_d=0.2\text{meV}$, and $U=8\text{meV}$.

Figure 2

(Color online) Multielectronic configurations with significant contributions to the ground state ${\Psi }_0$ and the excited states ${\Psi }_{1,2,3,4}$ important for ac absorption. For each configuration, we show the electrons at the Fermi levels of the left and right electrodes and on the dot (red, blue, and green, respectively). In either electrode, the single-electron states below the Fermi level are occupied.

Figure 3

(Color online) ${\epsilon }_d$ dependence of the weights $p_{1,2}^0\equiv {\left|⟨u_{1,2}|{\Psi }_0⟩\right|}^2$ of the states $u_{1,2}$ entering the linear combination of Eq. (6) for clusters with $N=3,7,11$ sites. Parameter values as in Fig. 1. For all $N$’s, the deviation from unity of the sum $p_1^0+p_2^0$ (at most $\sim {10}^{-3}$) is invisible within the drawing accuracy.

Figure 4

(Color online) Curves for $p_1^1\equiv {\left|⟨g_1|{\Psi }_1⟩\right|}^2$ and $p_3^1\equiv {\left|⟨g_3|{\Psi }_1⟩\right|}^2$ similar to Fig. 3. For all $N$’s, the $p_1^1$ curve cannot be distinguished within the drawing accuracy from that for $p_3^3\equiv {\left|⟨g_3|{\Psi }_3⟩\right|}^2$ and the $p_3^1$ curve from that for $p_1^3\equiv {\left|⟨g_1|{\Psi }_3⟩\right|}^2$. For all $N$’s, the deviation from unity of the sum $p_1^1+p_3^1$ (at most $\sim {10}^{-3}$) is invisible within the drawing accuracy.

Figure 5

(Color online) Results on the (a) higher (FIR) frequency ${\omega }_{3,4}$ and (b) lower (rf/microwave) frequency ${\omega }_{1,2}$ ac absorption for several cluster sizes $N$ and same parameter values as in Fig. 1. In panel (a), the triangles and circles are for clusters with $N=3$ and 7, respectively, and solid lines are for clusters with $N=11$. The latter cannot be distinguished within the drawing accuracy from those of asymmetric clusters, wherein the dot is attached to a single electrode with 7, 9, 11, and 13 sites. In the Kondo regime, ${\omega }_{3,4}$ are only slightly size dependent while ${\omega }_{1,2}$ are strongly size dependent. Notice the logarithmic scale on the ordinate in panel (b).