Kondo-peak splitting and resonance enhancement caused by interdot tunneling in coupled double quantum dots

In this paper, we report our recent work on the interdot tunneling-dependent Kondo effect for two serially coupled quantum dots (QDs) by adopting the hierarchical equations of motion approach. As is known, for such a system, when interdot tunneling $t$ increases, a continuous crossover from a Kondo singlet to a spin singlet is observed rather than a quantum phase transition, even though an induced antiferromagnetic interaction exists due to the interdot tunneling $t$. Our study focuses on the influence of the tunneling $t$ between two QDs at a finite temperature and especially on the smearing of the quantum phase transition. In this study, we find that, at small $t$, there is an enhanced Kondo effect, which results in the most noticeable Kondo effect appearing at finite $t$ rather than at $t=0$. When $t$ is larger, the Kondo peak splits into two subpeaks. Because of the weak coupling to the leads, the splitting space exactly corresponds to the energy difference between the original degenerated local spin singlet and the local spin triplet of the two isolated QDs. The Kondo peak can be preserved when the energy difference between the local spin singlet and the local spin triplet is smaller than the enhanced Kondo temperature. Finally, interdot tunneling also introduces a doubly occupied charge singlet, which causes charge fluctuations in the ground state at finite temperature and thus, therefore, smears the phase transition. This can also explain why no phase transition is observed in experiments in other QDs systems, even though the phase transition is always predicted in theory.

Figure 1

(a) Schematic diagram of the change in the ground state when $t$ increases from 0. The blue patterns show the ground states, while the red patterns show the first excited state. Because of the tunneling $t$, an effective antiferromagnetic exchange interaction $J$ appears, which is the value indicated by the black formula. At finite $t$, the ground state is an orbital singlet, which is a combination of a local spin singlet and a charge singlet, with probabilities of $\alpha$ and $\beta$, respectively, whose value changes are also given as $t$ varies from 0 to $\infty$. (b) The splitting split space of the Kondo peaks in the main panel and the spectral function $A\left(\omega \right)$ of the DQDs in the inset, with $A\left(\omega \right)$ labeled by their values of interdot tunneling $t$ in the middle. The scatter combined with the line show that the space of the two split Kondo peaks is exactly linear with respect to $J$. The temperature $T=0.1$, the lead bandwidth $W=6.67$, and $U_1=U_2=-2{\epsilon }_1=-2{\epsilon }_2=4$. (In units of $\mathrm{\Delta }$ with $\mathrm{\Delta }=0.3\mathrm{meV}$.)

Figure 2

The reduced density matrix elements for LSS, LST, and CS and the tunneling term between LSS and CS. The increase or decrease in the reduced density matrix element is linear with respect to the induced antiferromagnetic interaction $J$, which is a good corroboration of the linear splitting in the Kondo peak at finite $t$ shown in Fig. 1. The parameters are the same as in Fig. 1. In finite $J$ (i.e., finite $t$), the CS state enters the new ground state and becomes increasingly important when $J$ increases. In this figure, the coordinate scale is also given by $J$, which is quadratic to $t$ as in the formula shown in Fig. 1. (In units of $\mathrm{\Delta }$.)

Figure 3

The width variation in the Kondo peak before it splits as $t$ increases. The parameters are the same as those in Fig. 1. The black squares are the half width at $\frac{1}{2}A(0,t)$, while the red circles are the half width at $\frac{1}{2}A(0,0)$. The value of $t$ at which the single peak starts to split is shown by the dashed border. Because of the EKE, the Kondo peak splitting occurs at larger $t$, while the original $t$ decided by $T_K/2$ is shown by the black arrow. The difference between these HWs at the two maxima is shown by the blue triangles. The unfilled region shows the rapid heightening of the Kondo peak. (In units of $\mathrm{\Delta }$.)

Figure 4

The height $A(0,t)$ of the Kondo peak scaled by $A(0,0)$ as a function of $t$. The curves are $T=0.06$, 0.10, and 0.20 with $U=4$ from left to right, labeled by circles of different colors. The curves are $U=3$, 4, and 5 with $T=0.10$, labeled by squares, circles, and diamonds, respectively. We can see that $T=0.06$ has the most dramatic increase and decrease and that the smallest $t$ is at the maximum height $A(0,t)$ of the Kondo peak. The Coulomb interaction $U$ has almost no influence on the height and therefore on $t$ at the maximum $A(0,t)$ of the Kondo peak. In the calculations, we kept the parameters ${\epsilon }_d=-U/2$ and set the bandwidth to $W=7.0$ for both leads. (In units of $\mathrm{\Delta }$.)

Figure 5

$t^{*}$ of the maximized $A(0,t^{*})$, shown by black squares, and $t^{**}$ at which $A(0,t^{**})=A(0,0)$, shown by red diamonds, versus temperature $T$. In the inset, the choice of $t^{*}$ and $t^{**}$, i.e., the cross points of $A(0,t)$ and horizontal dotted lines with corresponding colors, are indicated schematically by the black and red arrows. The corresponding $t^{*}$ and $t^{**}$ at $T=0$ are extrapolated, and join together at $t=0.18$. The cubic B-spline interpolation/extrapolation curves are shown by the black and red dashed lines. The same result is obtained from the linear and cubic B-spline extrapolations. This may be a potential phase transition from the Kondo singlet to the spin singlet (OSS) at zero temperature. However, at finite temperature, there is a region of coexistence for the Kondo singlet and OSS and the phase transition is replaced by a continuous crossover. The parameters are the same as those in Fig. 4, except $U=4$. (In units of $\mathrm{\Delta }$.)

Figure 6

The zero bias differential conductivity (ZBDC) at different $t$. A red vertical dashed line is used to mark the position of the maximum ZBDC. There is a different value of $t$ in ZBDC comparing to the $t^{*}$ value of the maximum $A(0,t)$ in the spectral function, which are at $t=0.47$ and $t=0.23$, respectively. This is because the ZBDC not only depends on the quantum states of the DQDs but also on the interdot tunneling and other factors. The parameters are the same as those in Fig. 5, except $T=0.1$. (In units of $\mathrm{\Delta }$.)