Interplay of mesoscopic and Kondo effects for transmission amplitude of few-level quantum dots

The magnitude and phase of the transmission amplitude of a multilevel quantum dot is calculated for the mesoscopic regime of level spacing large compared to level width. The interplay between Kondo correlations and the influence by neighboring levels is discussed. As in the single-level case, the Kondo plateaus of magnitude and phase disappear with increasing temperature. At certain gate voltages, “stationary” points are found at which the transmission phase is independent of temperature. Depending on the mesoscopic parameters of the adjacent levels (such as relative sign and magnitude of tunneling matrix elements), the stationary points are shifted to or repelled by the neighboring level.

Figure 1

(Color online) Transmission amplitude per spin, $t_d=\left|t_d\right|e^{i\alpha }$, through a spinful two-level quantum dot for various temperatures and constant couplings. Regimes (i), (ii), indicated in panel (a) only, refer to Kondo valleys or Coulomb blockade valleys, respectively (see text). The levels involved are indicated by their level number 1,2. Level 2 is coupled more strongly to the leads than level 1, resulting in different bare Kondo temperatures, e.g., $T_K^{j=1}<T_K^{j=2}$. We use $\Gamma /U=0.03$ (a,c) and $\Gamma /U=0.08$ (b,d), thus $T_K^{\left(a,c\right)}<T_K^{\left(b,d\right)}$. The minimum value of the $T_K^j$ (in the center of the Kondo valleys) are indicated in the legends. In accordance with Ref. 24, we find shoulders in the phase [see, e.g., the fat arrow and the dashed curve $\left(T/U={10}^{-6}\right)$ in (c) for level 1] and an enhanced sensitivity of the phase to Kondo correlations compared to the magnitude, see e.g., the dashed-dotted $\left(T/U={10}^{-4}\right)$ curves in (d) for level 1 or the dashed curve $\left(T/U={10}^{-6}\right)$ for level 2 in (a). There, the typical $\frac{\pi }{2}$-Kondo plateau in the phase is present, whereas the Kondo plateau in amplitude is not fully developed yet. At certain points in gate voltage, say $V_g^{c_j}$ (as indicated by thin arrows), we find stationary points where the curves for $\alpha$ for all temperatures intersect. The position of $V_g^{c_j}$ is shifted by the presence of a neighboring level, being repelled by or shifted towards the latter for $s=+$ or $-$, compare (a,c) or (b,d), respectively. Depending on the mesoscopic parameter $s=\pm$, the phase either exhibits a sharp drop of $\pi$, accompanied by a zero in the amplitude $\left|t_d\right|$ [$s=+$, see (a,b)], or increases monotonically [$s=-$, see (c,d)] in the Coulomb blockade valleys.

Figure 2

(Color online) Transmission per spin through a spinful two-level quantum dot for both choices of $s=\pm$ and various values of mean couplings $\Gamma$ at fixed temperature $T$, level spacing $\delta$ and relative couplings $\gamma$. Due to the mixing of the levels, no stationary points with respect to $\Gamma$ exist, see text and the arrows in (d). Level numbers and the regime of (i) Kondo and (ii) Coulomb blockade valleys are indicated in panel (a) only.

Figure 3

(Color online) Transmission per spin through a three-level dot for all four possible combinations of $s=\left\{s_1,s_2\right\}$ and various temperatures, for fixed $\Gamma /U=0.03$. Level numbers and the regime of (i) Kondo and (ii) Coulomb blockade valleys are indicated in panel (a) only. The two-channel calculations for (b,c,d) qualitatively capture the physical trends. The asymmetry in the Kondo valleys is determined by both $s$ and $\gamma$. For convenience the figure legends for $\gamma$ display only the total relative coupling of each level. The minimal bare Kondo temperatures are indicated. (a) $s=++$: $\gamma =\left\{0.6,0.6,0.4,0.4,0.5,0.5\right\}$. The case $T/U={10}^{-8}$ is included only for this panel. (b) $s=--$: $\gamma =\left\{0.5,0.3,0.4,0.6,0.5,0.7\right\}$. (c) $s=+-$: $\gamma =\left\{0.4,0.8,0.4,0.4,0.3,0.7\right\}$. (d) $s=-+$: $\gamma =\left\{0.5,0.3,0.3,0.7,0.7,0.5\right\}$.

Figure 4

Schematic depiction of an Aharonov-Bohm interferometer of the type used by the Heiblum group (compare Fig. 1 of Ref. 2): The base region (B), emittor (E), and collector (C) have chemical potentials ${\mu }_0=0$, ${\mu }_1$, and ${\mu }_2$, respectively. In the base region four reflectors (shown in white) and a central barrier (black) define an upper and lower arm forming a “ring” threaded by an applied magnetic flux $\Phi$. The lower arm contains a quantum dot (D), coupled to the base region via tunable left and right tunnel barriers (L,R). The gate voltages $V_g$ or $V_u$ can be used to sweep the dot’s energy levels relative to ${\mu }_0$ or to change the transmission amplitude of the upper arm, respectively. The dashed lines are guide for the eyes, indicating how this devices realizes the geometry depicted in Fig. 5.

Figure 5

(Color online) Abstract depiction of a multiterminal Aharonov-Bohm interferometer, with a multilevel quantum dot embedded in the lower arm, penetrated by a magnetic flux $\Phi$. The different tunnelling amplitudes used in the text are indicated by arrows.