Prediction of Ferromagnetic Correlations in Coupled Double-Level Quantum Dots

Numerical results for transport properties of two coupled double-level quantum dots (QDs) strongly suggest that under appropriate conditions the dots develop a novel ferromagnetic (FM) correlation at quarter filling (one electron per dot). In the strong coupling regime (Coulomb repulsion larger than electron hopping) and with interdot tunneling larger than tunneling to the leads, an $S=1$ Kondo resonance develops in the density of states, leading to a peak in the conductance. A qualitative “phase diagram,” incorporating the new FM phase, is presented. In addition, the necessary conditions for the FM regime are less restrictive than naively believed, leading to its possible experimental observation in real QDs.

Figure 1

Schematic representation of the two double-level QDs system and the hopping amplitudes of Eqs. (1, 3).

Figure 2

Results for the strong interdot tunneling regime ($t^{\prime \prime }/t^{\prime }\gg 1$) showing the presence of a ferromagnetic state at $1/4$ filling and concomitant Kondo effect: (a) Conductance (solid line), charge occupancy per level and per spin (dashed line), and total spin $S_{\mathrm{T}}$ in the four levels of the dots (dotted line) vs gate potential $V_g$ for two coupled double-level QDs. The values for the parameters are indicated. Note that the maximum in the conductance corresponds to a spin 1 Kondo peak that occurs when there is exactly one electron per dot and the total spin $S_{\mathrm{T}}\approx 1.0$. The other two peaks occur when there are one and three electrons in the two-dots system, corresponding to $S_{\mathrm{T}}=1/2$ Kondo resonances. (b) DOS indicating the presence of a Kondo resonance associated with the $S_{\mathrm{T}}=1$ Kondo peak in (a). The numbers on the right side indicate the total number of electrons inside the two quantum dots. $V_g$ takes the values 0.0, $-0.5$, $-0.83$, $-1.12$, and $-1.5$ from the top to the bottom curve. The Fermi level is located at $\omega /t=0.0$.

Figure 3

(a) Conductance $G$ vs gate potential $V_g$ for different values of interdot tunneling [$t^{\prime \prime }=1.0$, 0.6, 0.4, 0.2, and 0.08, see convention in (b)]. $V_g$ varies across the region where the fourth electron is charged into the double-dot system. Note that the conductance at $V_g=-5.0$ gradually increases from zero (at strong tunneling, $t^{\prime \prime }=1.0$) to the maximum value (at weak tunneling, $t^{\prime \prime }=0.08$). This variation indicates a transition from two $S=1$ spins (in each dot) forming a global singlet to two uncorrelated $S=1$ spins, each forming a Kondo resonance with the lead conduction electrons. The inset shows the variation with $t^{\prime \prime }$ of the total spin inside the dots. (b) Variation with $t^{\prime \prime }$ of the spin-spin correlation between the total spin in the double-dot and a conduction electron located in the first site of the leads.

Figure 4

Variation of the conductance with level separation $\Delta V$. The level separation is indicated for each curve. (a) $t^{\prime \prime }=1.0$: The gate potential decreases down to the particle-hole symmetric value, to highlight how the $S_{\mathrm{T}}=1$ Kondo peak varies with $\Delta V$. Note that all graphs have the same horizontal axis scale, except for the upper one (scale indicated on top). (b) $t^{\prime \prime }=0.08$: The gate potential varies down to the lowest value (total charging of the dots), to highlight the variation of the central peak (at the particle-hole symmetric point). Here also the upper graph has a different horizontal scale. A discussion of how the data interpolate between double- and single-level dots is given in the text.

Figure 5

Qualitative phase diagram for the strong interdot tunneling regime ($t^{\prime \prime }/t^{\prime }\gg 1$). Horizontal axis indicates the level occupancy (controlled by $V_g$) and vertical axis indicates the level separation (controlled by $\Delta V$). In the $1/2$-filling region, displayed in the left side, one goes from an AF coupling between two $S=1$ spins (for $\Delta V<U$) to a situation where the two lower levels are completely occupied (for $\Delta V>U$). In both cases there is no Kondo effect. On the other hand, in the $1/4$-filling region (right side), for $\Delta V<t^{\prime \prime }$ one has the novel $S_{\mathrm{T}}=1$ FM Kondo region, which gives way to an AF region (with no Kondo effect) once $\Delta V>t^{\prime \prime }$. The regime with three electrons in the two dots is very narrow as a function of $V_g$ and it is not shown.

Figure 6

Schematic representation of the main result in this Letter.