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Sun Fu-Li, Wang Yuan-Dong, Wei Jian-Hua, Yan Yi-Jing. Capacitive coupling induced Kondo–Fano interference in side-coupled double quantum dots. Chinese Physics B, 2020, 29(6): 067204  
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Capacitive coupling induced Kondo–Fano interference in side-coupled double quantum dots
Sun Fu-Li1, Wang Yuan-Dong1, Wei Jian-Hua1, †, Yan Yi-Jing2
Department of Physics & Beijing Key Laboratory of Optoelectronic Functional Materials and Micro-nano Devices, Renmin University of China, Beijing 100872, China
Hefei National Laboratory for Physical Sciences at the Microscale & Department of Chemical Physics, University of Science and Technology of China, Hefei 230026, China

 

† Corresponding author. E-mail: wjh@ruc.edu.cn

Project supported by the National Natural Science Foundation of China (Grant Nos. 11774418, 11374363, and 21373191).

Abstract

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.

PACS: ;72.15.Qm;;73.63.Kv;;75.20.Hr;
Keyword:;Kondo effect;;Fano effect;;quantum dot;;capacitive coupling;
1. Introduction

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. 1(a). The Hamiltonian for the isolated DQD is

where represents the electron occupation operator on dot u = 1,2, with spin s = ↑,↓. We set ϵus = ϵd for the on-dot energy. and are the creation and annihilation operators for an electron in the dot-system state. U and U12 denote the intradot and interdot Coulomb interaction parameters, respectively.

Fig. 1. (a) Schematic of the capacitively side-coupled DQD, where QD1 and QD2 couple with each other through the interdot Coulomb interaction U12, but only QD1 couples directly to reservoirs L and R with hybridization ΓL and ΓR, respectively. (b) Schematic view of the removal of degeneracy to the S-O (singly-occupied) and D-O (doubly-occupied) states by U12. At very low temperature, the S-O2 and S-O1 substates go through ϵF as U12 increases, and then interact with the Kondo resonance to induce Kondo–Fano interference.

To focus on the capacitive coupling interaction, we set the interdot tunneling coupling to be zero as shown in Fig. 1. We justify this approximation via a prior report showing that experimentally, the tunneling coupling can be weak enough to be neglected.[25] Further, a similar approximation has been adopted previously by other researchers.[31] In the absence of tunneling coupling, there is only one transport path (L–QD1–R) left for electrons; this path completely avoids the multiple-channel complication. We have also validated our results in the weak tunneling regime.

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 theory, the phase shift can be obtained by different methods and parameter limits.[34] In the local moment regime with half-occupation of each spin, η(ω) = π/2 as a Fermi-liquid feature of Kondo scattering.[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 A, emphasize the calculation of Green’s function in Appendix B, and present an understanding of the ‘background’ in Appendix C.

2. Model and Hamiltonian

The total system-plus-reservoir composite Hamiltonian reads Htotal = Hsys + Hres + HSys-res. For the model DQD of Fig. 1 in this study, the other two parts can be written as

where and are the creation and annihilation operators in the specified reservoir state, with energy ϵαks. The dot system linearly couples with a reservoir by Γα, with transport path of L–QD1–R. In the HEOM method, we calculate Dαuvs(ω) by the definition , where and are the reservoir operators defined by and . In practice, we choose the initial input of the reservoir spectral density function as with effective dot–reservoir coupling of Γα (see Fig. 1), a width of the conduction band of Wα in the α-reservoir, and the chemical potential μα. Of note, Γα is the aforementioned hybridization, , where ρα is the density of states of the reservoir α at its Fermi level (or chemical potential). Here, we set the equilibrium Fermi energy to be . In the presence of bias voltage, μL – μR = V.

According to Dyson’s equation in the textbook scattering theory

where the retarded Green’s function G+(ω), the background part , and the scattering core T(ω) are all complex functions. It is apparent that T(ω) contains all the information of the scattering process from the initial state to the final state G+. T(ω) directly relates to the phase shift through Eq. (2).

By using the HEOM approach, we can obtain any retarded Green’s function of the DQDs numerically (cf. Appendix B). We use G(ω) to represent the retarded Green’s function of QD1, while ignoring the spin index due to the spin-degeneracy. Then, T(ω) can be calculated accurately from Eq. (5)

and accordingly, the phase shift η(ω) can be obtained from Eq (2).

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

where qd(ω) is the asymmetric function of interest. Note that qd(ω) is different from the conventional Fano parameter q. Since q cannot be obtained from our HEOM approach, we choose qd(ω) by following the literature definition. We will discuss this point later.

In this work, we are interested in a DQD [see Fig. 1(a)] with particle–hole symmetry (ϵd = –U/2) and the fixed parameters ΓL = ΓR = 0.5Γ, in units of Γ = 0.4 meV, U = 2.0 meV and WL = WR = 10Γ (W is the bandwidth of the reservoirs). We compare the results at low temperature (T = 0.02 meV ≪ TK) and high temperature (T = 0.3 meV ≫ TK), where the reference TK ≈0.12 meV from U12 = 0 scenario by an analysis expression in Ref. [3].

3. Three regimes
3.1. Overall features

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. 2(a) with yellow lines. We observe that U12 splits the individual S-O and D-O states into three quasi-particles substates {S-O0, S-O1, S-O2} and {D-O0, D-O1, D-O2}, respectively. The resultant Hubbard peaks, S-Om and D-Om, are centered at about ω = ϵd + mU12 and ω = ϵd + U + mU12.

Fig. 2. (a) Color-scale plot of local spectral density of QD1, A(ω), varying with the capacitive interdot coupling U12, at the strong-hybridization (Γ = 0.4 meV) high-temperature (T = 0.3 meV ≫ TK) regime. (b) Color-scale plot of A(ω) varying with U12 at the strong-hybridization, low-temperature (T = 0.02 meV ≪ TK) regime..

Figure 2(a) depicts the variance of the local spectral density of QD1 in color-scale, A(ω), with the capacitive interdot coupling, U12, at the strong-hybridization (Γ = 0.4 meV), high-temperature (T = 0.3 meV ≫ TK) regime. Generally, the numerical result matches well with the analytical one. The peaks in the color-scale plot of Fig. 2(a) correspond to the poles in Eq. (C1).

Figure 2(b) shows the A(ω) at low-temperature, T = 0.02 meV≪ TK, with the same hybridization Γ = 0.4 meV as that shown in Fig. 2(a). In this regime, the strong correlation quantum effect dominates. Moreover, QD1 contains a localized magnetic moment, with strong coupling to the electron reservoirs at T ≪ TK. The Kondo singlet state emerges at the Fermi level ϵF, resulting in a resonant peak at ω = 0. Meanwhile, the capacitive interdot coupling U12 removes the degeneracy in the individual S-O and D-O states of the isolated QD1. The aforementioned elevated S-O2 and S-O1 substates go through the Fermi level successively, as U12 increases. In particular, when the S-O2 or S-O1 substates appear near ϵF, the Kondo resonant peak in A(ω) becomes asymmetric, due to the underlying Fano interference.

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. 2(b): the Kondo-I (KI) regime (U12 < 0.1U), where the interdot Coulomb coupling has little effect, the Kondo–Fano (K–F) regime (0.1U ≤ U12 ≤ 0.6U), where the S-O2 and S-O1 substates significantly interfere with the Kondo state; and the Kondo-II (KII) regime (U12 > 0.6U), where the S-O2 and S-O1 substates are elevated above the Fermi level and the S-O0 substate supports Kondo resonance.

The only difference between Figs. 2(a) and 2(b) occurs when the temperature is far above or below TK, indicating there is a hallmark around the Fermi level without or with the Kondo effect. Since the scattering matrix carries the information about the Kondo effect [[cf. the K–F regime], we are able to regard Fig. 2(a) as the background of Fig. 2(b).

3.2. The Kondo-I regime

In the KI regime, A(ω) has a three-peak structure, as demonstrated in Fig. 3(a). The middle peak is a Kondo resonant peak having a symmetric line shape with width ∼1.75Γ [cf. Fig. 2(b)] that disappears at high-T [cf. Fig. 2(a)]. U12 in the KI regime is so small that it is difficult to resolve the S-O or D-O substates. The three-peak structure contains an S-O state below ϵF, a D-O state located symmetrically above ϵF, and a Kondo resonant state at ϵF. Figure 3(b) emphasizes the Kondo peaks at U12 = 0 and 0.05U. There is little difference between these two peaks in either height or width. We propose that A(ω) keeps its three-peak structure, much like a single QD, when U12 is small.

Fig. 3. (a) The three-peak structure consists of, from left to right, the S-O state, the Kondo resonant state, and the D-O state. Here ϵd = –2.5Γ and ϵd + U = 2.5Γ. (b) Details of the Kondo resonant state around the Fermi level.
3.3. The Kondo–Fano regime

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 4(a)–4(c) plot A(ω) in the vicinity of ϵF in the K–F regime. When U12 grows larger than U12 = 0, the highest point of A(ω) moves away from ω = 0 in the same direction as S-Om, {m = 1, 2}. This phenomenon (the S-O state driving the highest point of A(ω) higher) is clear in a single QD,[2] but has not been investigated in the S-O substates of a DQD.

Fig. 4. The A(ω) in the vicinity of ϵF in the Kondo–Fano regime. The Kondo peak at the Fermi level has (a) constructive interference from the S-O2 substate, (b) competition between constructive and destructive interferences, and (c) destructive interference from the S-O1 substate. (d) A(ω = 0) variation by U12.

In Fig. 4(d), A(ω = 0) increases (constructive interference), decreases (competition), and then falls below its original height (destructive interference). This is due to interference, not simple superposition, as evidenced by distinct asymmetry.

Figure 4(a) shows in detail the constructive interference of the Kondo resonant peak caused by interference from the S-O2 substate. A(ω = 0) reaches its maximum when the S-O2 state arrives at ϵF. It has been demonstrated that this K–F constructive interference is not due to other effects, such as the t-enhanced Kondo effect.[38] Figure 4(b) demonstrates a feature due to competition between constructive and destructive interferences of the Kondo resonant peak. This competition occurs because while the S-O1 state is about appearing, the S-O2 state still has an effect around the Fermi level. At U12 = 0.4, the S-O1 state arrives at ϵF and becomes dominant. Figure 4(c) shows the destructive interference of the Kondo resonant peak caused by interference from the S-O1 state. A kink appears at ω = 0, taking the place of the peak localized near ϵF; we propose that this kink is a signal line shape of the destructive K–F effect. This destructive feature of the line shape at the zero point of the asymmetry factor has previously been reported in different Fano resonance systems in the literature.[39,40] As the S-O1 state moves away from ϵF, the destructive interference becomes weak and the profile around ϵF smoothly changes to a small bump that is symmetric, and the system enters the next regime.

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. 5(a) and 5(b), respectively. As expected, the change in the phase shift π/2 - η(ω = 0) contains the essential physics of the K–F interference over a wide range of U12. Here we see the same trend with U12 as the asymmetry function.

Fig. 5. (a) The phase shift η(ω = 0) and (b) asymmetry function qd(ω = 0) as functions of U12. The changes in η(0) and qd(0) in the K–F regime are shown in solid lines, while those in the KI and KII regimes are shown in dotted lines.

At U12 = 0, the numerical phase shift η(0) is exactly equal to π/2, as shown in Fig. 5(a), This feature is well-known as the phase shift of pure Kondo scattering.[3] When the system enters into the K–F regime, η(0) experiences two maxima corresponding to interference between the Kondo resonance state and each of the S-O2 and S-O1 substates, respectively. The constructive process due to the shift of the S-O2 substate dominates at the first maximum. Destructive interference from S-O1 then takes the dominant position to induce the second maximum before the competition region that results in a small valley of η(0). Unfortunately, the signature of η(0) itself cannot distinguish whether the interference is constructive or destructive, thus spectral analysis (cf. Fig. 4) is always necessary to clarify the microscopic picture.

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. 5(b), which changes nearly synchronously with η(0). At U12 = 0, qd(0) = 0, indicating a symmetry resonance of the Kondo effect without any interference. With the increase of U12, qd(0) decreases after two local maxima in a similar fashion as η(0). At U12 > 0.6U, qd(0) approaches a constant value of –2, indicating the presence of another symmetric Kondo resonance without interference. This Kondo effect is a special characteristic of DQDs, that goes beyond the simple form of Kondo scattering in single QDs, as described in Ref. [35].

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. (7). By fitting our numerical results to A(ω), we obtain q = 263.72→∞ at U12 = 0, consistent with the Fano interference theory, which postulates that large q values lead to peaks in spectral line-shapes.

3.4. The Kondo-II regime

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. 6), we find that the peak height in the KII regime is 1/4 of that in the KI regime. However, these peaks have the same semi-width, meaning they have the same characteristic temperature TK. Our results indicate that there is essentially no difference between the Kondo-I and II regimes other than the height. The Fermi-liquid behavior of the KII regime seems to conflict with our intuition that small local momentum should be over-screened by conduction electrons.

Fig. 6. The A(ω) in the vicinity of ϵF in the KII regime. The insert illustrates A(ω) at U12 = 2U (solid black line) as well as at U12 = 0 (dash gray line) on a detailed scale around the Fermi level. The three-peak structure is partially recovered, and A(ω) at U12 = 2U has a quarter of the height of A(ω) at U12 = 0.

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. 3(a). By considering the triple-splitting of the S-O state (see Figs. 1 and 2), we can understand that the single Kondo peak actually involves the screening of momentum in all three S-O substates. The spin of each is still equal to 1/2, otherwise an over-screening effect would change the essential features of the Kondo peak. It is obvious from Fig. 3 that this does not occur. Therefore, the charge of the electron, not the spin, has been divided by the triple-splitting. We also find that the occupation number in QD1 is going to reduce to 1/4, which proves further that the amount of charge, not the 1/2 spin, is reduced. In summary, in the KII regime, near ϵF, A(ω) partially recovers the three-peak structure, where each peak has the same width and a quarter of the height of the corresponding peak in the KI regime.

4. Differential conductance

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. (A21)]. Through the extended Meier–Tannor parametrization method and multiple-frequency-dispersed hierarchy construction, we can achieve the closed HEOM formalism.[33]

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 7 depicts the numerical dI/dV with Fig. 7(a) in the KI regime, Figs. 7(b) and 7(c) in the K–F regime, and Fig. 7(d) in the KII regime. In Fig. 7(a), the dI/dV curves present a zero-bias peak structure, which is a typical feature of the ordinary Kondo effect.

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. 7(b) and 7(c), one can see two kinds of peak structures in the K–F regime resulting from constructive and destructive Fano interferences. The constructive interference enhances the Kondo effect to a certain degree and consequently reinforces the zero-bias peak of dI/dV. On the other hand, the observed destructive interference suppresses the zero-bias peak gradually. As shown in Fig. 7(c), symmetric peak shoulders appear on each side of the distinctly suppressed main peak at U12 = 0.5U. At this value, the Kondo peak of A(ω) has changed to the kink at ϵF [see Fig. 4(c)]. As U12 increases to a value of U12 = 0.6U, the shoulders grow in intensity and move outwards, while the zero-bias peak becomes clearer and less suppressed.

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. 7(d). The height of the zero-bias peak in the KII regime is about a quarter of that in the KI regime; this is the same ratio as that seen in the spectral function A(ω).

As demonstrated in Fig. 7, the destructive K–F effects can be experimentally investigated by virtue of the peak–shoulder structure of the differential conductance. For constructive interference, we find that the peak becomes slightly broader and higher, but this change is subtle and is difficult to resolve.

5. Summary

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. 2). Generally, U12 splits the singly-occupied (S-O) state and the doubly-occupied (D-O) state into three substates, with the S-Om and D-Om, (m = 1,2) substates elevated linearly and the S-O0 and D-O0 substates unchanged. At temperatures lower than the characteristic Kondo temperature, in the KI regime, the Kondo screening of the 1/2-spin localized in the whole S-O state induces a single Fermi-liquid peak at the Fermi level (ϵF). In the K–F regime, the S-O2 and S-O1 substates shift near ϵF, interfering with the Kondo resonance and appearing as the K–F effect, which first enhances the Kondo peak by constructive interference (from S-O2) and then depresses it by destructive interference (from S-O1). In the KII regime, after U12 elevates the S-Om (m = 1,2) substates away from ϵF, the three-peak structure partially recovers on a small scale and the Kondo screening of the residual spin in the S-O0 restores the Fermi-liquid peak at ϵF, with the same width but 1/4 the height of the peak seen in the KI regime.

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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