Thermoelectric properties of a double-dot system in serial configuration within the Coulomb blockade regime

Studies of thermoelectric transport in tailored nanoscale systems attract significant attention due to its fundamental and applied perspectives.^1–6 In general, thermoelectric charge and heat transport is governed by simultaneous action of electric and thermal driving forces. The combined effect of these forces depends on several factors, including coupling of the “bridge” part of the system (most commonly, a molecule, single or multiple quantum dots, or carbon-based nanostructure) to electrodes^7–15 and Coulomb interactions between electrons. Besides other things, the latter lead to oscillations of electron conductance and thermal conductance,^16–19 violations of the Wiedemann–Franz law,^20,21 and quantum interference effects.^22–29 Also, thermoelectric transport may be strongly affected by electron–phonon interactions^30–40 and by the interplay between charge/energy transfer and spin transport occurring when “the bridge” is sandwiched between ferromagnetic electrodes.^22,41 Advances in control and the measurements of heat transferred through mesoscopic and nanoscale systems^42–44 open up possibilities for manufacturing novel nanoelectronic devices such as heat-to-electric energy converters,^45–48 cooling systems,^2,3,49,50 and heat rectifiers.^51–56 

In the present work, we concentrate on thermoelectric properties of a double quantum dot placed in between nonferromagnetic electrodes. More specifically, we theoretically analyze the effect of interdot Coulomb interactions on thermoelectric transport in such systems. Methods of quantum dots fabrication are sufficiently well advanced.^57,58 Usually, sets including several dots are manufactured by applying a suitable electrostatic potential to GaAs/AlGaAs superstructures or heavy molecules.^59–66 In such sets, the dots may be arranged in a serial configuration^59–63 with hard^59–61 or soft^62,63 separating barriers or in a parallel geometry.^64–66 

Transport properties of double quantum dots were (still are) rather intensely studied. Specifically, electron transport through a double quantum dot in a serial configuration was discussed in several works. Various manifestations of Kondo effect in such systems were analyzed^67–70 along with the effects of ferromagnetic^69,71 and superconducting⁷² electrodes, the effects of spin–orbit interactions and those of the charge and spin pumping,⁷³ and the influence of Coulomb interactions between electrons on the efficiency of heat-to-electric energy conversion.⁷⁴ Also, it was shown that double dots in a serial configuration might undergo quantum dynamical phase transitions.⁷⁵ 

Nevertheless, in studies of transport properties of double and/or triple quantum dots, more attention so far was given to dots in parallel geometry. Here, we study thermoelectric transport through a double dot in a serial configuration schematically shown in Fig. 1. In the considered system, single-level dot 1 is coupled to the left electrode and another single-level dot (dot 2) to the right one. Electron tunneling between the dots is characterized by a coupling parameter β. Also, we take into account Coulomb repulsion between electrons on the same dot, as well as interdot Coulomb interactions.

This paper is organized as follows: In Sec. II, we describe the employed model and the formalism. We apply obtained results to briefly analyze the effect of interdot Coulomb interactions on the Seebeck effect and on thermally induced charge current in the considered system in Secs. III and IV. In Sec. V, we concentrate on the heat transport. It is shown that interdot Coulomb interactions may result in a significant heat current rectification. Conclusions are given in Sec. VI. In  Appendixes A and  B, we describe derivation of the relevant Green’s functions.

Studies of electron transport through quantum dots are simplified by the fact that phonon contributions to their electrical and thermal conductances are often small due to mismatch between frequencies of phonon modes associated with the dots embedded in their matrices and modes associated with the environment, preventing an overlap between the two.^76,77 On these grounds, we omit electron–phonon interactions and assume that the coherent electron transmission is the predominant mechanism governing electron transport. Within Anderson model, the pair of quantum dots coupled in series and put between electrodes is described by the Hamiltonian

Here, the contribution from the electrodes has the form

where α = {L, R} labels the left and right electrodes, ϵ_α,k,σ are single electron energies in the electrodes, and $Cα,k,σ†,Cα,k,σ$ are creation and annihilation operators for electron states k, σ (the index σ labeling spin-up and spin-down electrons). The term H_d is associated with the dots,

In this expression, i = 1, 2, E_0i are electron energies on the dots active in the transport process, U_i are intradot charging energies, U describes the interdot Coulomb interactions, $ni,σ=di,σ†di,σ$⁠, and $di,σ†,di,σ$ are creation and annihilation operators for electrons on the dots.

The last term

describes the couplings of the dots to the electrodes where factors t_α,k,σ represent coupling strengths.

Basing on the Hamiltonian (1), we derive expressions for the retarded [G^r(E)] and advanced [G^a(E)] electron Green’s functions for the double-dot within the Coulomb blockade regime where charging energies significantly exceed thermal energies kT_α (T_α being the electrodes temperatures and k being the Boltzmann constant) as well as energies characterizing couplings of the dots to electrodes. We use the equations of motion (EOM) method, as described in  Appendix A. The resulting expression for G^r agrees with the corresponding result of Ref. 78. Note that in a nonmagnetic system, G^r,a(E) do not depend on the electron spin orientation provided that electron transport is spin conserving.

In further analysis, we restrict ourselves to the limit of strong intradot Coulomb interactions, assuming that U_i ≫ U, E_i, β and to moderately biased systems (|eV| < U_i). Under such conditions, each dot in a double-dot system may contain only a single electron that can participate in transport. Energy levels that could receive a second electron become shifted beyond the conduction window and remain empty for electrons tunnel from these levels to the electrodes. Therefore, strong intradot Coulomb interactions do not affect electron transport through the double dot, and only much weaker interactions between electrons on different dots remain relevant. Thus, the double dot may be treated as a system where moderate/weak Coulomb interactions occur, and the expression for G^r(E) for a double-dot symmetrically coupled to electrodes may be reduced to the form

Here, i ≠ j,

and the parameter Γ describes the coupling of the double-dot to the electrodes in a symmetrically coupled system (⁠$ΓL=ΓR=12Γ$⁠). Equations (5) and (6) were derived applying the wide band approximation for electrodes and treating Γ as a constant independent of the tunnel energy. Note that dependencies of coupling parameters Γ^α on the tunnel energy E may bring changes into the Green’s functions energy dependencies and the characteristics of electron transport, but analysis of these changes is beyond the scope of the present work.

Electron occupation numbers on the dots $ni$ are determined by the following equation:

where the lesser Green’s function G^<(E). The approximation for G^<(E) is derived in  Appendix B. Under the considered conditions, it may be approximated as G^<(E) = i∑_αf_αG^r(E)Γ^αG^a(E) and f_α (α = {L, R}) are Fermi distribution functions for the electrodes with chemical potentials μ_α and temperatures T_α, and 2 × 2 matrices Γ^α for a symmetrically coupled double-dot have the form

We compute $ni$ by self-consistently solving Eq. (7). The occupation numbers are controlled by several factors including the electrodes temperatures and chemical potentials, electron energies on the dots, the interdot charging energy U, coupling strengths Γ and β, and magnitudes and polarities of the gate and bias voltages [the latter originates from the difference in the electrodes chemical potentials in the Fermi functions included in the expression for G^<(E)]. Electron occupation on the dots strongly depends on the interplay between interdot electron tunnelings and Coulomb interactions. When electron tunneling between the dots is weak, the dots occupations are mostly determined by positions of the on-dot electron energies on the energy scale and are considerably affected by Coulomb interactions between electrons on different dots. For strong interdot tunneling, the effect of Coulomb interactions is much less pronounced.

In further analysis, we focus on the case of weakly coupled double-dots to better understand effects originating from Coulomb interactions.

For the past three decades, a significant effort was applied to study thermoelectric transport in nanoscale systems.^{19,45–47,49,51,78,79} To study these properties, we assume that the system is biased by both the voltage ΔV = μ_L − μ_R and the temperature gradient ΔT = T_L − T_R. Within the linear response regime, the charge current I and the energy current J may be presented as⁸⁰ 

and

Here, coefficients L_n have the form

where μ is the chemical potential of electrodes in the absence of bias, f is the Fermi function with the chemical potential μ and temperature T, and τ(E) = Tr[G^a(E)Γ^RG^r(E)Γ^L] is the electron transmission function. When a double-dot in a serial configuration is symmetrically coupled to electrodes, the transmission accepts the form

Equating the charge current to zero, one finds the expression for thermopower (Seebeck coefficient),

The efficiency of the energy conversion is characterized by a dimensionless thermoelectric figure of merit ZT, which is commonly defined as

Thus, ZT is expressed in terms of thermopower, electronic electric conductance G, and thermal conductance κ, which includes both electronic and phonon contributions. In the considered case of a double quantum dot, we can omit the contribution from phonons to the thermal conductance, assuming that electron–phonon interactions are weak because of the above-mentioned mismatch between phonon modes frequencies. Then, the figure of merit may be presented in the form

Both thermopower and ZT depend on the positions of the energy levels on the dots, on the coupling of the dots to the electrodes, and on electron–electron interactions. In general, these transport characteristics may be strongly affected by quantum interference phenomena, but we do not expect these effects to be manifested in the presently analyzed double-dot in a serial configuration.

Now, we consider electron transport solely driven by a difference between T_L and T_R assuming a symmetrical distribution of the temperature difference between and electrodes: $TL,R=T±12ΔT$ [$T=12(TL+TR)$]. The transport occurs when an energy level of a considered system linking the electrodes is situated near their Fermi level μ. If this level is below μ, it works as a transport channel for holes, and if it is situated above μ, charge carriers mainly participating in transport are electrons. The Seebeck coefficient becomes zero when a certain level crosses E = μ, and electron and hole fluxes counterbalance each other. It changes sign when the type of charge carriers (electrons/holes) changes.

In the absence of electron–electron interactions, a double-dot including two single-level dots possesses two energy levels E₁ and E₂, which differ from original electron energy levels on the decoupled dots E_0i due to the interdot coupling. In a weakly coupled double-dot, E_i are close to E_0i. Intradot electron–electron interactions split each level in two. In the considered limit of very strong intradot electron–electron interactions (U_1,2 ≫ U), the levels at E = E_i + U_i are shifted beyond the conduction window. Thus, within this limit, actual transport channels may be associated with levels whose energies equal E_i and/or E_i + U. As follows from Eq. (6), two energy levels of a double-dot system affected by interdot Coulomb interactions may be approximated as $Ẽi≈Ei$ when both dots are nearly empty and as $Ẽi≈Ei+U$ when they are nearly full. When one dot (dot i) is nearly full and another one (dot j) is nearly empty, we obtain $Ẽi≈Ei$ and $Ẽj≈Ej+U$⁠.

In Fig. 2, we show dependencies of S and ZT on the gate voltage V_g, which shifts energy levels on the dots and drags them through the vicinity of the electrodes’ Fermi energy where (due to the presence of a thermal gradient) they become transport channels. The figure is plotted assuming that E₀₁ is located on the energy scale above the chemical potential of electrodes μ and E₀₂ is placed below μ. Therefore, we may expect dot 1 to be nearly empty (⁠$Ẽ1≈E1+U$⁠) and dot 2 to be nearly full (⁠$Ẽ2≈E2$⁠). Accordingly, the thermopower sign changes three times. This happens at V_g ≈ −E₂, at V_g ≈ −(E₁ + U), and at the point $Vg≈−12(E1+E2)$⁠. At this value of the gate potential, the energy levels $Ẽ1$ and $Ẽ2$ are symmetrically arranged with respect to the Fermi level of electrodes and simultaneously serve as transport channels for electrons (⁠$E=Ẽ1$⁠) and for holes (⁠$E=Ẽ2$⁠). Both channels equally contribute to the transport, and therefore, electron and hole currents are counterbalanced and S becomes zero.

Two doubled peaks appear at each ZT vs V_g curve (see inset) at $Vg≈−Ẽi$ in agreement with the thermopower behavior. Peak values of ZT are considerably greater than 1, indicating a potential efficiency of the considered system as a heat-to-electric energy convertor. Note that the peaks heights do not depend on the value of charging constant U. The strength of interdot Coulomb interactions determines a position of the peak emerging at V_g ≈ −(E₁ + U) but not its magnitude.

When the temperature difference across the system increases, it may switch to a nonlinear regime of operation. Then, a linear relationship between ΔT and thermally induced voltage is violated and thermopower becomes a function of ΔT.^{9,22,31,34,36,72} Here, we focus on the thermocurrent I_th, which also is an important quantity characterizing thermoelectric charge transport through nanoscale systems. It could be defined as a difference between the charge current driven by both thermal and electric biases and the current solely driven by the bias voltage at ΔT = 0,⁸¹ 

where the charge current is given by the following standard Landauer expression:^5,73

According to this definition, the thermocurrent I_th represents a contribution to the tunnel current originating from the thermally excited flow of charge carriers. We first consider I_th in an unbiased double-dot, which is solely driven by a thermal gradient ΔT. The effect of interdot Coulomb interactions is better pronounced when both E₁ and E₂ are situated below the electrodes Fermi energy μ = 0 and both dots are nearly full. Then, the thermally induced current reaches maxima at $U≈−Ẽi≈−(Ei+U)$ when corresponding energy levels appear in a close vicinity of E = μ and may serve as transport channels for charge carriers, as shown in Fig. 3. Note that in the considered case, the charge carriers involved in transport are holes, so the current takes on positive values at positive ΔT (that is, when T_L > T_R). If E_i < μ (dot i is nearly full) and E_j > μ (dot j is nearly empty; i ≠ j), a single peak located at U ≈ −E_i should appear at each I_th vs U curve. Finally, if both dots are nearly empty (E_i > μ; i = 1, 2), I_th vs U curves become featureless, as demonstrated in the inset. The thermocurrent magnitude slowly decreases as interdot Coulomb interactions strengthen.

When a double-dot is biased by an applied voltage V, the effect of thermal gradient is important for moderately biased systems where the thermal and electric driving forced are comparable. The interplay between these forces may result in peaks and dips at the current–voltage curves, indicating that a certain energy level $E=Ẽi$ crosses the boundaries of the conduction window. The specifics of these features are mostly determined by the levels’ positions, as illustrated in Fig. 4. One may come to this conclusion by comparing the curves displayed in the main body of this figure with those shown in the inset. Note that I_th is only a small part of a total charge current flowing through a system biased by both applied voltage and thermal gradient even for a moderate electric bias. At sufficiently strong bias, the effect of thermal driving forces is negligible and I_th becomes zero.

In this section, we study heat currents flowing through the system when it is biased by the applied temperature gradient ΔT. Again, we assume a symmetric distribution of the temperature between the electrodes, so T remains constant. We compute the heat current using the general expression⁸² 

Here, the difference in the chemical potentials of electrodes Δμ = μ_L − μ_R equals eΔV_th where the potential difference ΔV_th originates from the Seebeck effect and could be found by assuming that I = 0 (the open circuit condition) and then solving Eq. (9) for a given temperature difference ΔT. As before, we consider a double-dot in a serial geometry, and the electron transport occurs within the Coulomb blockade regime. Using the expression Eq. (18), we omit from consideration phonon mechanisms of energy transfer and concentrate on the electron contribution to the heat current.

In the same way as the thermally induced charge current I_th, the heat current J at a fixed ΔT is mostly controlled by positions of the double-dot energy levels $Ẽi$ on the energy scale. These, in turn, are controlled by the strength of interdot Coulomb interactions. This is illustrated in Fig. 5. As shown in this figure, interdot Coulomb interactions noticeably suppress the heat transfer through the considered system and may bring asymmetry into I_th vs ΔT curves, which indicates heat current rectification. The asymmetry becomes stronger as U increases.

To better understand a mechanism controlling the heat current rectification in the considered system, we need to more thoroughly analyze the specifics of electron transport. At T_L = T_R and μ = 0 and for chosen values of E_0i (E₀₁ = −6 meV, E₀₂ = 4 meV), dot 1 is probably nearly full (⁠$Ẽ1≈E1<0$⁠) and dot 2 is nearly empty [$Ẽ2≈(E2+U)>0$]. Suppose that the thermal gradient drives charge carriers to the right (T_L > T_R, that is, ΔT > 0). Then, the energy level $E=Ẽ2$ does not participate in transport, whereas the level $E=Ẽ1$ being close enough to the electrodes Fermi level serves as a transport channel for holes. When the thermal gradient is reversed (ΔT < 0), dot 2 may become occupied. In this case, both energy levels $E=Ẽi≈(Ei+U)$ are situated beyond the conduction window created by the thermal gradient, and electron transport between the electrodes ceases to exist.

It was suggested in some earlier works^83–86 that nanoscale rectifiers may operate as such due to broken mirror symmetry in a nanoscale junction whose terminals are linked by a chain of elements with different spectral densities. In the considered system, we have a chain-like linker including two elements, and a difference between the spectral densities originates from interdot Coulomb interactions and from difference in electron energies on the decoupled elements (dots), which give rise to different spectral densities associated with the dots in a weakly coupled double-dot.

The rectification ratio for heat currents R_J may be defined as

where J_± are the heat currents corresponding to the forward (ΔT > 0) and reversed (ΔT < 0) bias created by the thermal gradient. The rectification becomes more pronounced at strong interactions between electrons on the dots. Also, R_J and that depends on the on-dot electron energies, as follows from the above discussion. Assuming that the dot attached to the hot electrode is occupied, R_J is determined by the occupation on the dot attached to the cool electrode. When this dot is also occupied, interdot Coulomb interactions affect the electron transport, and their effect vanishes when this dot remains empty. The dependence of the rectification ratio on the electron energies on the dots is displayed in Fig. 6. As shown in this figure, at ΔT > 0, a pronounced rectification appears at certain values of the on-dot energies and sufficiently strong Coulomb repulsion between electrons on different dots. The rectification ratio may reach values of the order of 10, thus satisfying the least rigorous condition for diod operations. Therefore, the considered system may operate as a heat diod.

In the present work, we theoretically analyzed thermoelectric transport through a double-dot in a serial configuration placed between electrodes. Such systems are being manufactured and studied both theoretically and experimentally. We analyzed electron transport through such a junction within a Coulomb blockade regime when the coupling of the dots to the electrodes was weaker than the characteristic thermal energy kT associated with the system. Effects of electron–phonon interactions on the electron transport were omitted based on the assumption that phonon contributions to both electrical and thermal conductance of the system were small and coherent electron transport predominated. Transport characteristics were computed using the Green’s functions formalism within the limit of strong intradot Coulomb interactions where intradot charging energies U_i greatly exceed interdot charging energy U.

It was shown that interdot Coulomb interactions may significantly affect the thermopower of the considered system as well as thermally induced charge current flowing through the latter. The effect of these interactions strongly depends on electron energies on the dots and interdot coupling strength, which affect electron occupation numbers on the dots. Note that the thermoelectric figure of merit in the considered system may reach values noticeably greater than 1, thus indicating that it may serve as a heat-to-electric energy converter.

Also, we showed that electron–electron interactions between the dots may result in rectification of heat currents flowing through the considered system. By varying occupation numbers on the dots by means of the applied bias and gate voltages and/or thermal gradient, one may control the rectification ratio. A model of a heat diod based on a double-dot in a parallel geometry and operating due to interdot Coulomb interactions was suggested in some earlier works.^51,52,87 Here, we demonstrate that a satisfactory operating heat rectifier may be manufactured basing on a double-dot in a serial configuration. For an appropriate choice of electron energies on the dots and sufficiently strong interdot Coulomb interactions, the rectification ratio may reach values of the order of 10.

We believe that the presented results provide a step toward further understanding and modeling of thermoelectric and heat transport through quantum dots and molecules and could be important for manufacturing nanoscale heat rectifiers.

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

This work was supported by NSF-DMR-PREM 1523463. The author thanks G. M. Zimbovsky for help with the manuscript preparation

Within the Coulomb blockade regime, the couplings between the dots and the electrodes are supposed to be weak, and the EOMs for $Gji,σr(E)≡≪dj,σ;di,σ†≫$ have the form

and

where {i, j} = {1, 2}, (i ≠ j). Assuming that dot 1 is coupled to the left electrode and dot 2 to the right one, the parameters Γ_i,σ are further labeled $ΓσL,R$ and are given by

Within a wide band approximation, parameters $ΓσL,R$ may be treated as constants independent of energy E. Assuming that the double dot is symmetrically coupled to the electrodes and coupling strengths do not depend on the electron spin orientation, then $ΓσL=ΓσR=12Γ$⁠. Also, we assume that intradot charging energies are equal: $U1=U2=Ũ$⁠.

Two-particle Green’s functions obey EOMs,

A simplified form for the term $β≪dj,σ;di,σ†≫$ is obtained using the following approximation to decouple relevant four-operator averages,⁸² 

Three-particle Green’s functions that appear in Eqs. (A4) and (A5) are determined by EOMs,

Here, $Ni=di,σ†di,−σdi,σ†di,σ$ are two-particle occupation numbers and $G(4)r(E)$ is a four-particle Green’s function approximated as

In a nonmagnetic system, $ni,−σ=ni,σ=12ni$⁠, and Green’s functions are independent of electron spin orientations provided that spin-flip processes are neglected. Using Eqs. (A4)–(A9), Eq. (A1) may be transformed to the form

where

In these expressions, $P1=1−nj+Nj$⁠, $P2=nj−2Nj$⁠, $P3=Nj$⁠, Π₁ = 0, Π₂ = U, Π₃ = 2U, and the index j satisfies the conditions j = {1, 2} and j ≠ i.

We consider the limit of strong intradot Coulomb interactions when $Ũ≫U,E0i$ and a moderately biased system $(|eV|<Ũ)$⁠, and therefore, we may expect that the double dot energy levels shifted in the energy scale due to intradot interactions remain beyond the conduction window. Electrons from these elevated levels tunnel to the electrodes leaving them empty, so it is highly likely that each single-level dot contains no more than a single electron and two-particles occupation numbers may be set to zero. Also, all terms in Eq. (A11) including $Ũ$ in their denominators remain small inside the conduction window and do not contribute to the electron transmission. Therefore, one can simplify the expressions for υ_i by omitting the terms proportional to $Ni$ as well as those including the intradot charging energies. Then, one arrives at the following approximation for υ_i:

We may transform Eq. (A2) in a similar way and get

Solving Eqs. (A10) and (A13), we obtain

where again i, j = 1, 2 and i ≠ j.

Expressions for G^<(E) and G^>(E) may be derived using the EOM technique as well as the expression for G^r(E).^78,88,89 The results are

Here, I is the identity matrix and

where f_α(E) and α = {L, R} are Fermi distribution functions for the electrodes. These results satisfy the identity G^<(E) − G^>(E) = G^r(E) − G^a(E).

As mentioned in the text, at U_i ≫ U, |e|V, E_i, β, the electron transport is not affected by strong intradot Coulomb interactions because each dot may contain a single electron. Therefore, within the Coulomb blockade regime, we may treat the considered double dot as a system where only moderate/weak interactions occur between the dots and the electrodes and between the dots themselves. Thus, we may use the relationship^82,89

For a serial arrangement of the dots and within a wide band approximation, the matrices Γ^α have the form

Note that beyond the Coulomb blockade regime and at sufficiently strong interdot Coulomb interactions, matrix elements Γ_α should not be approximated by constants. They become dependent on both tunnel energy E and charging energy U.⁸⁹ 

Substituting Eqs. (B4) and (B5) into Eq. (B1), we arrive at the Keldysh equation,

where

This result shows that in the case considered in the present work, the EOM technique leads to the same approximation for the lesser Green’s function as the diagrammatic method.

Charge current flowing through the double dot is given by a general expression,

Using Eqs. (B4), (B6), and (B7), one may reduce this expression to the simple formula used in the text [see Eq. (17)].