Effect of assisted hopping on thermopower in an 4 1 interacting quantum dot 0 2 b e S. B. Tooski,1,2 A. Ramˇsak2,3, B. R. Bul ka1, and R. Zˇitko2,3 F 1Institute of Molecular Physics, Polish Academy of Sciences, ul. M. Smoluchowskiego 1 17, 60-179Poznan´, Poland 2 2 Joˇzef Stefan Institute, Ljubljana, Slovenia 3 ] Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia l l a E-mail: [email protected] h - s Abstract. Weinvestigatetheelectricalconductanceandthermopowerofaquantum e m dot tunnel coupled to external leads described by an extension of the Anderson impurity model which takes into account the assisted hopping processes, i.e., . t a the occupancy-dependence of the tunneling amplitudes. We provide analytical m understanding based on scaling arguments and the Schrieffer-Wolff transformation, - corroborated by detailed numerical calculations using the numerical renormalization d group(NRG) method. The assisted hopping modifies the coupling to the two-particle n o state, which shifts the Kondo exchange coupling constant and exponentially reduces c or enhances the Kondo temperature, breaks the particle-hole symmetry, and strongly [ affects the thermopower. We discuss the gate-voltage and temperature dependence of 2 the transport properties in various regimes. For a particular value of the assisted v hopping parameter we find peculiar discontinuous behaviour in the mixed-valence 2 3 regime. Near this value, we find very high Seebeck coefficient. We show that, quite 1 generally, the thermopower is a highly sensitive probe of assisted hopping and Kondo 1 correlations. . 1 0 4 1 PACS numbers: 72.15.Jf, 72.15.Qm, 73.63.-b : v i X Submitted to: New J. Phys. r a CONTENTS 2 Contents 1 Introduction 2 2 Model and method 3 2.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Perturbative analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Transport coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4 Different regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Static properties and spectral densities 8 4 Thermopower and conductance 10 4.1 Gate-voltage dependence . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4.2 Temperature dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 5 Conclusions 16 1. Introduction The thermoelectric effect is the conversion of temperature differences to electric voltage and vice-versa. Thermoelectric devices find application in power generation, refrigeration, and temperature measurement [1]. The progress in nanotechnology has led to lower thermal conductivity while retaining the electrical conductivity and Seebeck coefficient [3, 4, 5, 6], which is important for applied use. In basic research, the thermoelectric effect is a tool for revealing the transport mechanisms. For instance, the position of the molecular states relative to the Fermi level can be deduced from the thermoelectric potential of molecular junctions [2]. Thermoelectric properties of nanomaterials are intensively studied [7, 8, 9, 10, 11, 12, 13, 14]. In transport through Coulomb islands some novel effects have been observed: the oscillations of the thermopower [7] and thermal conductance [8, 9] with gate voltage. In the coherent regime, the transport properties of quantum dots (QDs) attached to external leads strongly depend on the correlated many-body Kondo state. Its most notorious signature is the increased conductance at low temperatures. Recently, there has also been growing interest in the thermopower of Kondo correlated quantum dots, which has been measured [15, 16] and theoretically analyzed [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]. Interacting QDs are commonly modelled using the single-impurity Anderson model (SIAM). The most prominent terminthisHamiltonianistheon-siteCoulomb repulsion. The assisted-hopping terms arise as the next-leading effect of the Coulomb interaction after the on-site repulsion [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52]. Such processes are always present in real devices but are commonly neglected in theoretical modelling, despite the fact that they may, in fact, be quite CONTENTS 3 sizeable. A generalized Anderson impurity model with assisted hopping can be formally derived by integrating out high-energy degrees of freedom [36, 37], leading to assised hopping to the retained level in the restricted basis. Such Hamiltonian can also be postulated as a phenomenological model. The assisted-hopping terms first appeared in proposals for describing the properties of mixed-valence bulk systems. The discovery of high-T superconductivity increased the interest in correlated hopping as a possible new c mechanism for superconducting instability and temperature-induced metal-insulator transition [37, 38, 39, 40]. Furthermore, assisted hopping is considered as a significant factor in the stabilization of the ferromagnetism and localization of electrons [52]. In the context of nanodevices, the assisted hopping has been proposed to account for anomalies in the conductance [43] and the unusual gate-voltage dependence of the measured Kondo temperature [44] which does not follow the usual form expected for the standard SIAM. It has also been proposed to explain the conductance increase through local pairing effects [45]. It has been established that for transition-metal complexes the correlated hybridization can be very large, comparable even to the standard single- particle hybridization through the interatomic potential [46]. For bulk systems, the inclusion of an assisted hopping term in the electronic Hamiltonian favors the existence of pairing correlations [47]. In the case of a QD, this tendency towards local pairing quenches thelocal moment, leads to asymmetries inthe conductance ofpeaks associated with the same level [47], and to a change of the thermopower in the sequential regime [13]. In this work, we study the effect of the assisted-hopping term in a generalized SIAM on the thermopower and conductance of the QD by applying the numerical renormalization group (NRG) technique. We will show that the Kondo effect can be either suppressed or enhanced, depending on the gate voltage and the sign of the assisted-hopping term. This results from the renormalization of the level positions and widths, as well as the modification of the effective Kondo exchange coupling, which leads to exponential reduction of the Kondo temperature and consequently to a strong enhancement of the Seebeck coefficient. Another important effect is the particle-hole (p-h) symmetry breaking and the resulting asymmetry in the gate-voltage dependence of system properties. 2. Model and method 2.1. Model We consider a QD described by an extended SIAM H = ǫ n +ǫ n+Un n −V [(1−xn )c† c +H.c.]. (1) k αkσ ↑ ↓ σ¯ αkσ σ αX,k,σ αX,k,σ The first term corresponds to electrons in the leads (left and right, α = L,R), n = c† c is the number operator for electron with wavevector k, spin σ, and αkσ αkσ αkσ energy ǫ = hk|h|ki; h is the one-particle kinetic Hamiltonian. The second term k CONTENTS 4 describes electrons in the QD level with energy ǫ = hc|h|ci. The Coulomb repulsion energy for two electrons with opposite spin in the same level is U = hcc|e2/r|cci. Here n = c†c is the number operator for an electron in QD, and n = n + n . The σ σ σ ↑ ↓ last term describes the coupling between the leads and the QD, V = −hk|h|ci is the single-electron hopping parameter between the impurity and the leads, which is positive for electron hopping. The assisted-hopping parameter X = hkc|e2/r|cci describes the Coulomb-interaction-mediated transfer of an electron from the state |ki in the lead to the QD, when the QD is already occupied by an electron with the opposite spin σ¯. The parameter X calculated for inter-atomic hopping in transition metals [32, 52] and in copper oxides [53, 54] is in the range 0.1−1 eV. In semiconducting quantum dots Meir et al. [43] estimated the ratio X/V = 0.65 and showed that the assisted-hopping can lead to a significant reduction of the tunneling rate through the excited state. In the following, we use the normalized assisted-hopping parameter x = X/V and we assume symmetric coupling to both leads and a flat density of states ρ = 1/(2D), where D = 1 is the half-width of the conduction band. Since the value of x can be of order 1, the assisted hopping can significantly affect the transport properties. An interesting special case occurs for x = 1, when the doubly-occupied state is fully decoupled from the leads and the transport is due solely to the singly-occupied states. The model is non-ergodic at x = 1 and one expects anomalous properties. Some vestigial effects are also expected for values of x near 1. Negative values for x are not excluded, e.g., for the case of transport through molecules with higher angular momentum orbitals which can lead to positive or negative hopping-overlap integrals. 2.2. Perturbative analysis In the roughest mean-field treatment, the effective hybridization in this model is occupancy-dependent, Γ = Γ(1 − xhni)2, where Γ = 2πρV2. This approximation eff is, however, overly simplistic. It is crucially important to account for the different rates of the 0 ↔ 1 and 1 ↔ 2 charge fluctuations. We expect two main effects: (i) different level renormalisations, and (ii) modified Kondo exchange coupling constant. We discuss first the level renormalization in the spirit of the Haldane scaling where high energy charge fluctuations are integrated out [55, 56, 57]. The zero- occupancy level E is renormalized by the processes 0 ↔↑,↓. The singly-occupied 0 level E is renormalized by the processes σ ↔ 0,2. Finally, the doubly-occupied level is 1 renormalized by the processes 2 ↔↑,↓. The renormalization of the lower atomic level ǫ = ǫ can be extracted as the difference E −E , while the renormalization of the upper 1 1 0 atomic level ǫ = ǫ+U is obtained as E −E . The scaling calculations gives for ǫ a 2 2 1 1 shift of 1 Γ[1−f(ω)] Γ(1−x)2f(ω) δǫ = − dω + , (2) 1 π Z (cid:26) ω −ǫ ǫ+U −ω (cid:27) while δǫ = −δǫ . Here f is the Fermi-Dirac distribution. In the limit of standard 2 1 CONTENTS 5 SIAM, i.e., for x = 0, one can perform the integration exactly, the T = 0 result being Γ |ǫ| δǫ = ln . (3) 1 π |ǫ+U| It vanishes for |ǫ| = |ǫ + U|, i.e., at the particle-hole (p-h) symmetric point. In the presence of assisted hopping, the integration becomes more involved and cutoff dependent, thus no simple closed-form expression can be provided. In any case, it is easy to see that the p-h transformation with respect to the point ǫ = −U/2 is no longer a symmetry of the system. To find the Kondo exchange coupling constant J , we performed the Schrieffer- K Wolff transformation, obtaining 2Γ(1−x)2/π 2Γ/π ρJ = − . (4) K ǫ+U ǫ At ǫ = −U/2, this simplifies to ρJ = 4Γ (2−2x+x2), and for x = 0 one recovers K πU the standard result ρJ = 8Γ/πU. Equation (4) succinctly shows the general trend: at K the p-h symmetric point of the standard SIAM, the Kondo coupling is renormalized by the assisted-hopping simply by a multiplicative factor (2 − 2x + x2)/2. This function is a parabola with a minimum at x = 1, where it has value 1/2. Thus at ǫ+U/2 = 0, going from x = 0 to x = 1 the Kondo coupling will be reduced by half, resulting in an exponentially strong suppression of the Kondo temperature [58] 1 T = c(U,Γ) ρJ exp − (5) K K (cid:18) ρJ (cid:19) p K where c(U,Γ) is the U and Γ dependent effective bandwidth. For positive x < 1 we thus expect a rapidly decreasing T for increasing x. For x > 1, J starts to increase K K again, and at x = 2 it has the same value as for x = 0. For x < 0, the Kondo coupling increases with the absolute value of x, thus T is exponentially enhanced. K At other values of the gate voltage, the renormalization effects of the assisted hopping need to be estimated from the more general equation (4). For x = 0, the Kondo coupling attains its minimum value at ǫ = −U/2. Minimizing the expression in equation (4) we find that the minimum is in general shifted to U ǫ = − , (6) 1+|1−x| i.e., to smaller values of the gate voltage for increasing x. A highly curious feature is that at x = 1, we find ǫ = −U, i.e., the gate voltage that corresponds to the edge of the Kondo plateau in the standard x = 0 SIAM. At x = 1, instead, we there find a minimum of J and deep Kondo regime. For x → 1, we will thus expect a spectacular K asymmetry where the Kondo plateau extends right up to the point where the occupancy should suddenly go to 2, because the decoupled upper effective atomic level at ǫ+U falls below the Fermi level. Since the deep Kondo regime is associated with single occupancy, CONTENTS 6 this observation suggests anomalous behaviour for x → 1 for gate voltage near ǫ = −U. This is indeed fully corroborated by the numerical results presented in the following. The system possess an exact symmetry around x = 1, which holds beyond the perturbative derivation of the Kondo coupling J . In particular, all results presented in K this paper exhibit a perfect symmetry with respect to the x = 1 point, i.e., the results calculated at x are identical to those obtained at 2 − x. This can be understood as follows. For n = 0, the hopping matrix element of spin-σ electron is −V, while for σ¯ n = 1 it equals −V(1 − x). Replacing x with 2 − x, for n = 0 the matrix element σ¯ σ¯ remains unchanged, while for n = 1 we find V(1−x), which differs only in sign. The σ¯ sign is, however, immaterial. Each electron hop in one direction must be followed by another in the opposite direction in order to obtain a state which is not orthogonal to the original state. The statistical sum thus only depends on x through the combination V2(1−x)2, hence all system properties are symmetric with respect to x = 1. In numerical calculations presented in this work we set the hybridization strength to Γ/D = 0.02 and we focus on the stronger coupling, U/Γ = 8, where the model enters the Kondo regime near half-filling; for x = 0, T is of order 10−3. For convenience, we K fix the Fermi level at zero, ǫ = 0. F 2.3. Transport coefficients Thermoelectric transport is calculated for a situation in which a small external bias voltage, δV = V −V , and a small temperature gradient δT are applied between left L R and right leads [17]. Left and right leads are then at different chemical potentials µ L and µ and temperatures T and T , with eδV = µ − µ and δT = T − T . To R L R L R L R linear order, the following expressions for the electrical conductance, G(T), and the thermopower (Seebeck coefficient), S(T), are obtained G(T) = e2I (T), (7) 0 |e| I (T) 1 S(T) = − , (8) k T I (T) B 0 where I are the transport integrals n 2 ∂f I (T) = dωωnT (ω) − . (9) n h Z (cid:18) ∂ω(cid:19) Here, edenotestheunitchargeandhthePlanck’sconstant. Thetransmissioncoefficient T (ω) is given by T (ω) = πΓA(ω) with A(ω) = −1Imhha ;a†ii , with the operator π σ σ ω+i0+ a† = (1−xn )c†. This is similar to the standard SIAM, but with a spectral function σ σ¯ σ of the operator a†, rather than c†, as can be seen from the Dyson equation for the σ σ Gk,k′ Green’s function in the leads. It can namely be shown that the T-matrix is given by the correlator of the [H ,c ] objects, where H is the hybridization part of the hyb kσ hyb Hamiltonian (term proportional to V in equation 1). To evaluate the transport integrals we used the numerical renormalization group (NRG) method [59, 60, 61]. This method allows to calculate static, dynamic and CONTENTS 7 2 6 • 1.5 (a) (b) • 1 xor(2−x) • 4 .0 n 1 000-...02461 nc=0.9•63 U=0.16 • •• 2 n)/0c 0.5 0.8 • − 1 Γ=0.02 •• 1 • ( 0 •• 0 -0.2 -0.1 0 0.1 0.2 0 0.03 +U/2 Γ 0.8 x • 0.6 (c) -1 (f) 0 0.2 0.4 2 0.4 0.6 S 0.8 1 (x) 0.2 nc 0 • -0.04 f S · -0.08 S (d) (g) -0.12 0.4 (e) (h) 0.3 2 n 0.2 δ 0.1 0 • -0.2 -0.1 00 0.1 0.2 0 0.5 1 1.5 2 +U/2 n Figure 1. (a) Total impurity charge (occupancy) hni as a function of gate voltage ǫ+U/2forvariousx(and,duetothe symmetry,also2−x). ParametersareΓ=0.02, U = 0.16, and T = 10−5. The arrow indicates critical nc. (b) Critical nc as a function ofthe hybridizationΓ. (c) Localspin squared(static localmoment), (d) spin correlation between the QD level and the leads, and (e) local charge fluctuations, all as a function of the gate voltage(arrowsindicate increasingx). In panels (f), (g), and (h)the samequantities areshownasa functionofthe occupancy hni. Bullets indicate values at hni=nc. transport properties in a reliable and rather accurate way in a wide temperature range. The approach is based on the discretization of the continuum of bath states, transformation to a chain Hamiltonian, and iterative diagonalization. The calculations reportedherehavebeencarriedoutforadiscretization parameterΛ = 2, twist averaging over N = 4 interleaved discretization meshes, and retaining 500 states per NRG step. z CONTENTS 8 2.4. Different regimes Near half-filling, the local-moment (LM) regime corresponds to the formation of a localized spin on the dot at intermediate temperatures (T . T . U). In this regime, K the conductance has logarithmic temperature dependence which is the signature of the emerging Kondo state. At low temperatures, T . T , i.e., in the strong-coupling (SC) K regime, the moment is fully screened and the system is characterized by Fermi-liquid properties. The mixed-valence (MV) regime corresponds to gate voltages where the charge on the dot fluctuates between 0 and 1, or between 1 and 2. The physics is then governed by charge fluctuations and the system acts as a two noninteracting resonant-level model. Finally, in the empty-orbital (EO) regime, n ≈ 0, and the full-orbital (FO) regime, n ≈ 2, there are only thermally activated charge fluctuations and the system behaves as a noninteracting resonant-level model. 3. Static properties and spectral densities Infigure1, thestaticquantitiesatlowtemperatureT = 10−5 areplottedinleftpanelsas a function of ǫ which is shifted by the gate voltage. The results for the standard SIAM, x = 0, are well known: for ǫ+U/2 ≈ 0 the occupancy hni tends to be pinned to 1 near half-filling, see panel (a); an even more pronounced plateau in hni vs. ǫ curves would form in the large U/Γ limit. In this gate-voltage range, the local moment is formed, thus hS2i is large, panel (c), and the antiferromagnetic exchange interaction between the dot level and the leads is signaled by negative values of hS·S i, panel (d). Here S f and S are the spin operators for the QD and the first site in the leads, respectively. f The charge fluctuations hδn2i = h(n−hni)2i = hn2i−hni2 have a (local) minimum near half-filling, but are enhanced in the MV regimes for ǫ ≈ 0 and ǫ+U ≈ 0, panel (e). As expected on the basis of equation (2), the filling of the dot with electrons is significantly affected by the assisted hopping. At fixed gate voltage, the occupancy is increased for ǫ + U . 0 and decreased for ǫ + U & 0 if 0 < x < 2, and vice-versa for x < 0 and x > 2. Another notable effect is the breaking of the p-h symmetry. This is a trivial consequence of different hopping rates for 0 ↔ 1 and 1 ↔ 2 processes in the presence of assisted hopping. The assisted hopping in the range 0 < x < 2 enhances the local moment in the gate voltage range −U/2 . ǫ + U/2 . 0, as visible from the increased values of the hS2i curves. This is a consequence of the reduced Kondo coupling J , see equation (4), K because it makes the local spin more decoupled from the leads. This is also mirrored in the decreasing absolute value of hS·S i. For ǫ+U/2 & 0, the local moment is reduced, f but only very slightly. This small reduction can be thought of as a higher-order effect of the assisted hopping in the regime of small hn i. The correlation hS·S i is also affected σ¯ f more mildly in this gate-voltage range. The charge fluctuations hδn2i are particularly interesting, since they are directly CONTENTS 9 affected by the assisted-hopping term. For 0 < x < 1 we observe that the fluctuation peak at ǫ+U ∼ 0 becomes increasingly narrow with increasing x: the parameter range of the valence-fluctuation region 1 ↔ 2 is shrinking. In fact, it becomes extremely small in the x → 1 limit. The width of the peak at ǫ ∼ 0 is, however, not significantly affected. The particular behaviour in the occupancy can be understood analyzing the single particle Green’s function derived within the Hubbard-I approximation [32], 1−hni/2 hni/2 hhc ;c†ii ≃ + . (10) σ σ ω ω −ǫ−iΓ/2 ω +−ǫ−U −i(1−x)2Γ/2 This result is beyond the mean-field approximation and takes into account the Coulomb blockade, but neglects the spin-flip processes leading to the Kondo resonance (see also Ref. [62]). The spectral function given by (10) has two peaks at ω = ǫ and ω = ǫ+U with different width. The width of the excitation peak at ω = ǫ+U shrinks to zero for x = 1, becoming a delta peak, thus the occupancy jumps when the delta peak crosses ǫ F as the gate voltage is swept, from hni = n ∼ 1 to exactly hni = 2. All other quantities c also exhibit sharp transitions across this occupancy jump. In panel (a) the position of hni = n near the transition is indicated by an arrow. Note that n < 1, which can be c c explained as an effect of the charge fluctuations, since critical 1−n scales as Γ/U, as c is clearly seen from panel (b). In panels (f), (g), and (h) we show static quantities as a function of the occupancy hni. This alternative representation reveals additional effects of the assisted hopping beyond the dominant effect (i.e., the modification of the filling dependence hni vs. ǫ). We note a reduction of the charge fluctuations and an increase of the moment which are rather symmetric with respect to hni = n ∼ 1. This result can be fully accounted c for within the simple Hubbard-I approximation which properly describes the high- energy charge fluctuations. The reduction of the hS·S i correlations is, however, quite f asymmetric. Within a simple approximation, this expectation value is proportional to J hS2i; this explains the overall shape of the curve, and in particular the asymmetry K whpich is due to the asymmetry of J . Note also the x = 1 results: hS2i = 3hni for K 4 hni < n , because hn n i = 0, i.e., because the doubly-occupied state is fully decoupled. c ↑ ↓ Extremely interesting is also the hδn2i vs. hni plot, panel (h). For n < hni < 2, the c gate-voltage dependence is squeezed into a narrow peak [panel (e)] in the x → 1 limit, yet the dependence on hni reveals similarity to results for x < 1. Note that the curve is continuous, because the calculation is performed at finite temperature; strictly at T = 0 there would be a true discontinuity. Finally, we observe that the value at the minimum of the hδn2i curve at n , indicated by the bullet in panel (h), also scales with c Γ/U (scaling not shown here). The generalized local spectral densities A(ω) in the Kondo regime (at ǫ+U/2 = 0) are shown in figure 2 for various x. At low temperatures, T < T , A(ω) is characterized K by three peaks: the Kondo resonance at ǫ , and two atomic resonances: lower peak at F ω = ǫ+δǫ and upper peak ω = ǫ+U +δǫ . 1 2 The narrow Kondo resonance for 0 < x < 2 lies very near ǫ , see panel (b). F Its width is of order T and its reduction for 0 < x < 1 is manifest. For x > 2 (or, K CONTENTS 10 0.6 1 (a) x (b) +U/2 =0 -1 0.8 0 0.4 0.4 0.8 x 1 0.6 ω) -1 0 A( 0.4 0.8 0.4 0.2 1 0.2 0 0 -0.1 -0.05 0 0.05 0.1-3 -2 -1 0 1 2 3 −3 ω ω/10 Figure 2. (a) Spectral function A(ω) for different values of the assisted hopping parameter x at ǫ+U/2 = 0 and T = 10−5. Note the absence of the Kondo peak for x=−1 (or for x=3). (b) Close-up of the Kondo peak region. alternatively, for x < 0), the Kondo resonance decreases rapidly inheight and eventually disappears completely, see the dashed line in panel (b), since the system moves away from the Kondo regime. The atomic resonances contain most of the spectral weight and have widths approximately 2Γ and 2Γ(1−x)2. Accordingly, the lower atomic peak does not depend much on x, while the other becomes narrower with increasing x in the range 0 < x < 1. 4. Thermopower and conductance 4.1. Gate-voltage dependence We now study the influence of the assisted hopping on the transport properties. Thermopower and conductance are plotted in figure 3(a) and (b), respectively, as a function of the gate-voltage for constant high temperature T = 0.01. This temperature isabovetheKondotemperatureforall0 < x < 2. Gischaracterizedbytwoconductance peaks associated with the levels ǫ and ǫ+U, separated by the Coulomb blockade valley. For x = 0, the peaks are symmetric with respect to ǫ + U/2 = 0 due to the p-h symmetry. S is relatively high and its ǫ-dependence is similar to that derived from the master equation for sequential tunneling through a quantum dot [63]. Thermopower is positive for a large negative gate voltage, when transport is through holes. S changes sign at ǫ+U ≈ 0, when G reaches its maximum. Next, S goes through zero at the p-h symmetric point and becomes positive. At ǫ ≈ 0, S changes sign yet again and becomes negative – now the transport is dominated by electrons. The assisted hopping modifies the spectral peaks at ǫ and ǫ + U, their widths being proportional to Γ and Γ(1 − x)2, respectively. One may use the formula S = G S / G for the thermopower in a system with several types of charge n n n n n carriePrs {n}, eacPh of which would separately have a contribution S to thermopower n and G to conductance [64]. In our model, we have two resonant channels with different n