Charge oscillations in quantum dots: Renormalization group and Hartree method calculations

We analyze the local level occupation of a spinless, interacting two-level quantum dot coupled to two leads by means of Wilson’s numerical renormalization group method. A gate voltage sweep, causing a rearrangement of the charge such that the system’s energy is minimized, leads to oscillations, and sometimes even inversions, in the level occupations. We find that these oscillations, qualitatively understandable by a simple Hartree analysis, are generic and occur in a wide range of system parameters. By allowing a relative sign in one tunneling matrix element between dot and leads, we extend our findings to more generic models. Experimental applications and the qualitative effect of spin are discussed.

Figure 1

(Color online) Schematic depiction of the model. The sign $s$ between the tunneling amplitudes $V_{\ell R}$ and $V_{\ell L}$ determines whether the two dot levels couple to the same (symmetric) channel ($s=+1$, upper mapping), or to two different channels ($s=-1$, lower mapping), with strength ${\tilde{V}}_i=\sqrt{2}{V}_i$, $i=\ell ,u$.

Figure 2

(Color online) NRG results for the occupation of the lower [$n_{\ell }$, (curves $1^{\prime }–3^{\prime }$)] and the upper [$n_u$, (curves 1–3)] level for fixed $\Delta$ and different values of $\Gamma$, for $s=+1\left(\mathrm{a}\right)$ $+1$ and (b) $s=-1$. The dotted lines indicate level where the lower or upper level crosses the Fermi level, at $ϵ=\Delta ∕2$ and $-U-\Delta ∕2$.

Figure 3

(Color online) Comparison of NRG and SCHA results for $n_u\left(ϵ\right)$, for fixed $\Delta$ and variable $\Gamma$ [(a),(b)] or fixed $\Gamma$ and variable $\Delta$ [(c),(d)]. The naive SCHA used for $s=+1$ works well for $\Gamma ∕\Delta <1$. Insets in (b) and (d): Hartree levels ${ϵ}_{\ell }^H$ (circles) and ${ϵ}_u^H$ (stars), calculated via the SCHA from Eq. (2), for $\Gamma ∕\Delta =0.1$ in (b) and 0.4 in (d). For $\Gamma ∕\Delta =\infty$ the Hartree levels are degenerate (not shown). The arrow in inset (d) indicates the value of ${ϵ}_0$, the local minimum of ${ϵ}_u^H$.

Figure 4

(Color online) Allowing $\gamma \equiv {\Gamma }_u∕{\Gamma }_{\ell }$ to be $\ne 1$ results in an inversion of the occupation for a certain range of $ϵ$. The SCHA results ($n_u$, crosses; $n_{\ell }$, boxes) for $s=-1$ (where $F_{-1}$ is known) agree well with NRG results ($n_u$, solid; $n_{\ell }$, dashed lines). For $\gamma <1$ the Hartree levels cross near the left CB peak, implying an occupation inversion below the corresponding crossings.

Figure 5

(Color online) Same as Fig. 4, but for $\gamma >1$. Now the Hartree levels cross near the right CB peak [see inset in (b) for $\gamma =4$], implying an occupation inversion above the corresponding crossings. Notice the differences in the shape of the occupation curves between the $s=1$ and $-1$ cases.

Figure 6

(Color online) (a) NRG results for the occupations with $s=1$ and $\gamma =30$. (b) A comparison between the SCHA results ($n_u$ crosses, $n_{\ell }$ boxes) and for the NRG results ($n_u$ solid, $n_{\ell }$ dashed lines), for $s=-1$ (where $F_{-1}$ is known) and $\gamma =30$. The agreement between the NRG and the Hartree approximation is remarkably good and does not depend on $\gamma$.

Figure 7

(Color online) NRG results in the presence of spin (with the relative sign $s=-1$). The parameters are identical to those in Fig. 5. The occupation crossing persists in the spinful case. The additional oscillations in the occupations are observed due to the possibility to put two electrons of opposite spin within each level.