† Corresponding author. E-mail:
Project supported by the National Natural Science Foundation of China (Grant Nos. 11774418, 11374363, and 21373191).
We report capacitive coupling induced Kondo–Fano (K–F) interference in a double quantum dot (DQD) by systematically investigating its low-temperature properties on the basis of hierarchical equations of motion evaluations. We show that the interdot capacitive coupling U12 splits the singly-occupied (S-O) state in quantum dot 1 (QD1) into three quasi-particle substates: the unshifted S-O0 substate, and elevated S-O1 and S-O2. As U12 increases, S-O2 and S-O1 successively cross through the Kondo resonance state at the Fermi level (ω = 0), resulting in the so-called Kondo-I (KI), K–F, and Kondo-II (KII) regimes. While both the KI and KII regimes have the conventional Kondo resonance properties, remarkable Kondo–Fano interference features are shown in the K–F regime. In the view of scattering, we propose that the phase shift η(ω) is suitable for analysis of the Kondo–Fano interference. We present a general approach for calculating η(ω) and applying it to the DQD in the K–F regime where the two maxima of η(ω = 0) characterize the interferences between the Kondo resonance state and S-O2 and S-O1 substates, respectively.
Kondo resonance is a many-body process corresponding to a localized magnetic moment screened by itinerant electrons. It was originally proposed to explain the abnormal increase of resistance at low temperature in metals with dilute magnetic impurities.[1] Approximately 20 years ago, the Kondo effect was unveiled in a quantum dot (QD) system; this system was shown to provide a flexible and controllable physical implementation of the Kondo effect.[2] For a QD system in the Coulomb blockade regime, when the temperature T is below the characteristic temperature TK, a sharp peak appears at the Fermi level ϵF in the spectral function of the system with an odd-number of localized electrons.[3]
Theoretically, a Kondo peak with particle–hole symmetry in the spectral function has the following three features: the height is 1/πΓ (Γ is the hybrid function) at zero temperature and decreases as the temperature increases, the width is proportional to TK, and the lineshape can be fitted by a Lorentz profile.[3,4] However, in experimental conditions, the Kondo resonant peak may be affected by other channels and taken on an asymmetric lineshape. Li et al. reported an asymmetric Kondo resonant peak in a magnetic impurity embedded within the first atomic surface layer and attributed it to destructive Kondo–Fano (K–F) interference therein.[5] Subsequently, others have reported experimental Kondo resonance line-shapes that are asymmetric due to K–F interference.[6–8]
The K–F interference plays an important role in side-coupled double quantum dots (DQDs). For example, Sasaaki et al. observed a Fano resonance that arose from the interference between discrete levels in one dot and the Kondo effect in the other dot as a continuum.[8] They proved that the Kondo resonance is partially suppressed by destructive Fano interference, indicating the presence of a novel Fano–Kondo competition. Later theory work by Žitko et al. suggested that this phenomenon could be interpreted as a two-stage Kondo effect.[9]
Both the aforementioned experiment and theory on side-coupled DQDs have focused on the interdot tunneling coupling that directly mixes discrete and continuous levels. There could be other interactions that induce K–F interference in side-coupled DQDs. Such interactions would result in some interesting new spectral features. This issue is the main motivation of the present work.
In general DQDs, there are two basic kinds of interdot interactions. One is the tunneling type that can be described with the transfer coupling parameter, This type is related to the two-channel Kondo effects,[10,11] two-stage Kondo effects,[9,12–18] and others. The other interaction type is the capacitive type, and can be described by interdot Coulomb repulsion. This interaction is responsible for the single-electron switch and other transport properties.[19–28] Hatano et al. recognized that capacitive Coulomb repulsion strongly affects charging diagrams[29] and Ruiz–Tijerina et al. treated it as an effective means of gating.[30] However, the possible K–F interference due to capacitive interactions has not been discussed previously.
In this work, the capacitive coupling induced K–F interference will be elucidated with a representative side-coupled DQD under quantum transport, as depicted in Fig.
![](cpb_29_6_067204/cpb_29_6_067204_ieqn1.gif)
![](cpb_29_6_067204/cpb_29_6_067204_ieqn2.gif)
![](cpb_29_6_067204/cpb_29_6_067204_ieqn3.gif)
To focus on the capacitive coupling interaction, we set the interdot tunneling coupling to be zero as shown in Fig.
Krychowski et al. studied the capacitive coupling effect without tunneling coupling using finite-U slave-boson mean-field theory (SBMFT).[32] SBMFT can qualitatively describe features of the Fermi-liquid ground state, but it does not simulate Hubbard peaks in spectral functions; thus, cannot yet describe K–F interference. As will be demonstrated, the shift of Hubbard peaks and their interplay with the Kondo peak play essential roles in K–F interference which is the main concern of the present work. Therefore, rather than SBMFT, we have adopted the hierarchical equations of motion (HEOM) approach, which is a powerful method for universal characterization of strongly correlated impurity systems. HEOM is especially useful for describing both Kondo and Hubbard peaks.[4,33]
The phase shift η(ω) is essential to describing the K–F interference. According to standard scattering theory, η(ω) directly relates to the scattering core (T matrix) T(ω) by[3]
In a single QD system, the change in the phase shift due to the K–F interference was first noticed by Luo et al. when they investigated the interference between Kondo resonance and the broadened QD level in the mixed valence regime.[35] To avoid the difficult calculation, Luo et al. suggested a simple form of T(ω) from Kondo scattering; they found that this phase shift is necessary to accurately reproduce the results of scanning tunneling microscopic experiments.[36] Since then, the phase shift has been widely used in literature, but only as a fitting parameter. In the present paper, we present a general approach to calculate η(ω) accurately, and then apply it to thoroughly investigate K–F interference in a DQD, where charge and spin degrees of freedom strongly interact with each other.
In the next section, we introduce the model Hamiltonian and the relevant parameters. Following, we demonstrate the three regimes (Kondo-I, Kondo–Fano, and Kondo-II regimes), characterized by different features of spectral A(ω), phase shift of K–F interference η(ω), and asymmetry function qd(ω). We discuss these regimes separately in three subsections. Moreover, we compare the results obtained from differential conductance measurements. Finally, we conclude this work. We briefly describe our HEOM approach in Appendix
The total system-plus-reservoir composite Hamiltonian reads Htotal = Hsys + Hres + HSys-res. For the model DQD of Fig.
![](cpb_29_6_067204/cpb_29_6_067204_ieqn4.gif)
![](cpb_29_6_067204/cpb_29_6_067204_ieqn5.gif)
![](cpb_29_6_067204/cpb_29_6_067204_ieqn6.gif)
![](cpb_29_6_067204/cpb_29_6_067204_ieqn7.gif)
![](cpb_29_6_067204/cpb_29_6_067204_ieqn8.gif)
![](cpb_29_6_067204/cpb_29_6_067204_ieqn9.gif)
![](cpb_29_6_067204/cpb_29_6_067204_ieqn10.gif)
![](cpb_29_6_067204/cpb_29_6_067204_ieqn11.gif)
![](cpb_29_6_067204/cpb_29_6_067204_ieqn12.gif)
![](cpb_29_6_067204/cpb_29_6_067204_ieqn13.gif)
According to Dyson’s equation in the textbook scattering theory
![](cpb_29_6_067204/cpb_29_6_067204_ieqn14.gif)
![](cpb_29_6_067204/cpb_29_6_067204_ieqn15.gif)
By using the HEOM approach, we can obtain any retarded Green’s function of the DQDs numerically (cf. Appendix
Another important physics quantity that we describe in this report is Fano’s asymmetric function. The existence of interdot Coulomb interaction U12 will affect the single QD Kondo scattering and lead to the K–F effect. As proved in Ref. [35], there is a convenient way to illustrate the resultant asymmetric line shape only using the scattering background
In this work, we are interested in a DQD [see Fig.
Now we examine the effect of capacitive interdot coupling, U12, on the spectral function of QD1, A(ω), evaluated via the HEOM implementation of linear response theory.[4,37]
As a reference, we have derived the Green’s function of the isolated DQD [cf. Eq. (C1)] and plotted the positions of its poles in Fig.
Figure
Figure
Therefore, the capacitively side-coupled DQD at T ≪ TK experiences three distinct Kondo regimes, corresponding to the relative value of U12 with respect to the intradot Coulomb coupling strength U. For the particle–hole symmetry (ϵd = –U/2) in the DQD examined here, these three regimes are divided by the vertical-dashed lines in Fig.
The only difference between Figs.
In the KI regime, A(ω) has a three-peak structure, as demonstrated in Fig.
Further increasing U12 drives the symmetric Kondo peak in A(ω) to gradually change into an asymmetric Kondo–Fano peak, indicating that the DQD has entered the K–F regime. Figures
In Fig.
Figure
We are now in the position to demonstrate our numerically phase shift η(ω) in the K–F regime, where charge and spin degrees of freedom strongly interact with each other. The U12-dependent η(ω = 0) and the asymmetry function qd(ω = 0) are shown in Figs.
At U12 = 0, the numerical phase shift η(0) is exactly equal to π/2, as shown in Fig.
In our DQD, the Kondo resonance should be a discrete channel with a narrow peak profile in A(ω) at ϵF; and the background provides the open channel. Since the former is a robust many-body state localized around ω = 0, the change in interference is determined mainly by the latter. This has been verified by the asymmetric function qd(0) shown in Fig.
In our calculations, qd(0) = 0 corresponds to the symmetric line shape around the Fermi level, and shows a different feature from that of the Fano asymmetric factor q. We note that the difference results from the definition of qd in Eq. (
As the S-Om {m = 1,2} states move further away from ϵF at U12 > 0.6U, the zero-point peak of A(ω) tends to not change in intensity. On a small scale around the Fermi level, the three-peak structure partially recovers.
In this side-coupled DQD, we assume that no electron could transport to or from QD2, and we set ϵd = –1/2U, where the correlation effect is the most significant.[41] Consequentially, for the quasi-particles, the weight of the S-O0 (D-O0) substate is a quarter of that of the total S-O (D-O) state. Similarly to the S-O (D-O) state, the S-O0 (D-O0) substate broadens from the environment, it is obvious that they have the same width. At large U12, without the S-O1 and S-O2 states, the position of the S-O0 substate is fixed, and an effective localized magnetic moment is maintained in QD1.
In the KII regime, at low temperature, and with invariant QD1 coupling to the reservoirs, the necessary conditions exist for Kondo resonance.A quarter-localized 1/2 spin in QD1 leads to a Kondo peak at ϵF in A(ω). By carefully comparing the Kondo peak at U12 = 0 and 2U (cf. insert of Fig.
We explain this as follows. Let us first consider the KI regime, where Fermi-liquid behavior is characterized by the symmetric Kondo resonance peak localized at ϵF, as shown in Fig.
The spectral function A(ω) is capable of revealing detailed dynamical properties over a wide range of parameters. However, in QD experiments, A(ω) cannot be observed conveniently. Conductance is more favourable for experimental measurement; therefore, we calculate the differential conductance dI/dV under nonequilibrium conditions taking advantage of the HEOM evaluation.
We set the initial total system at equilibrium at μα = μeq = 0 (α = L and R). Application of a voltage to the left and right reservoirs, resulting in current flow into the α-reservoir Iα(t), causes the system to leave equilibrium [details in Eq. (
As a result, the current from L to R can be denoted as I(t) = IL(t) = –IR(t). The differential conductance dI/dV can then be obtained from the steady current I = I(t→∞) as a function of bias V.
Figure
![]() | Fig. 7. (a) The Kondo-I regime, (b) the Kondo–Fano regime with constructive resonance, (c) the Kondo–Fano regime with destructive resonance, and (d) the Kondo-II regime. |
In Figs.
The Fano interference disappears gradually, until finally the shoulders cannot be observed at –0.8 mV < V < 0.8 mV; the single peak structure is restored at V = 0. At this point, the system comes into the KII regime, as shown in Fig.
As demonstrated in Fig.
We have reported a capacitive coupling induced Kondo–Fano interference in a side-coupled DQD by systematically investigating its low-temperature properties on the basis of HEOM evaluations. We found that as the interdot Coulomb coupling U12 increases, the system excitations evolve from the Kondo-I regime to the K–F regime, and from the K–F regime to the Kondo-II regime (cf.\,Fig.
We have also presented a general approach to calculate the phase shift η(ω) of the K–F interference numerically. We then applied this approach to our DQD in the K–F regime, where charge and spin degrees of freedom strongly interact with each other. We have shown that the change of the phase shift contains the essential physics of the K–F interference over a wide range of U12. When the system enters into the K–F regime, η(ω = 0) experiences two maxima corresponding to the interference between the Kondo resonance state and the S-O2 and S-O1 substates, respectively.
In order to facilitate experimental observation, we have also calculated the differential conductance dI/dV under nonequilibrium conditions. We found that both the KI and KII regimes are characterized by the single zero-bias peak of dI/dV, analogously to the ordinary Kondo systems. We have shown that the peak–shoulder structure of dI/dV at small bias can serve as an observable feature of the Kondo–Fano effect with destructive interference in the K–F regime.
In reality, the capacitive interdot coupling may be accompanied by interdot charge transfer. It is expected that the additional transport path of L–QD2–R will inevitably affect the Kondo–Fano features at low temperature This will be addressed in our future work.
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