Correlations of heat and charge currents in quantum-dot thermoelectric engines

We consider the situation where the dots are weakly coupled to the leads such that transport occurs via sequential tunneling. The electron dynamics is well described by a Markovian master equation for the reduced density matrix [5] with diagonal elements given by the occupation probability of each charge state, ρ = (ρ₀₀,ρ₁₀,ρ₀₁,ρ₁₁)^T.

We are interested in the charge flowing through one of the leads of the conductor (say l = 2) and the heat transferred from the heat source (contact l = 3). The stationary charge and heat currents, $I=\langle \skew3\hat{I}\rangle$ and $J=\langle \skew3\hat{J}\rangle$, zero frequency charge and heat noise, $S_{I\! I}=\langle (I-\skew3\hat{I})^2\rangle$ and $S_{J\! J}=\langle (J-\skew5\hat{J})^2\rangle$, and cross-correlation, $S_{I\! J}=\langle (I-\skew3\hat{I})(J-\skew5\hat{J})\rangle$, as well as higher order cumulants, can be calculated by counting statistics techniques [60, 61]⁶. Here we introduce the charge, χ, and heat, ξ, counting fields⁷ in the master equation, $\dot \rho ={\cal M}(\chi ,\xi )\rho$, with [22]

$$\begin{eqnarray} \fl {\cal M}(\chi,\xi)= \left(\begin{array}{@{}cccc@{}} -\Gamma_{\!s0}^--\Gamma_{\!g0}^- & \Gamma_{\!10}^++{\rm e}^{{\rm i}q\chi}\Gamma_{\!20}^+ & {\rm e}^{{\rm i}\Theta_{30}\xi}\Gamma_{\!g0}^+ & 0\\ \Gamma_{\!10}^-+{\rm e}^{-{\rm i}q\chi}\Gamma_{\!20}^- & \ \ -\Gamma_{\!s0}^+-\Gamma_{\!g1}^- & 0 & {\rm e}^{{\rm i}\Theta_{31}\xi}\Gamma_{\!g1}^+\\ {\rm e}^{-{\rm i}\Theta_{30}\xi}\Gamma_{\!g0}^- & 0 & -\Gamma_{\!s1}^--\Gamma_{\!g0}^+ & \Gamma_{\!11}^++{\rm e}^{{\rm i}q\chi}\Gamma_{\!21}^+\\ 0 & {\rm e}^{-{\rm i}\Theta_{31}\xi}\Gamma_{\!g1}^- & \Gamma_{\!11}^-+{\rm e}^{-{\rm i}q\chi}\Gamma_{\!21}^- & -\Gamma_{\!s1}^+-\Gamma_{\!g1}^+\\ \end{array} \right), \end{eqnarray} \tag{ 3 }$$

where Θ_ln = E_α,n − qV_l is the heat carried by electrons in the tunneling process, and Γ^±_sn = Γ^±_1n + Γ^±_2n. Here, Γ^±_ln are the tunneling rates for electrons tunneling into (−) or out of (+) a quantum dot through barrier l when the other dot contains n electrons. They are given by Fermi's golden rule as Γ⁻_ln = Γ_lnf(Θ_lnβ_l) and Γ⁺_ln = Γ_ln − Γ⁻_ln, with the Fermi function f(x) = (1 + e^x)⁻¹ and β_l = (kT_l)⁻¹. Note that the energy dependence of the rates arises both from the Fermi functions and the tunnel couplings, Γ_ln. We consider the regime kT_l ≫ Γ_ln where broadening of the energy levels can be neglected. Currents are defined as positive when flowing into terminals.

The correlations are given by derivatives of the cumulant generating function, ${\cal F}(\chi ,\xi )$, with respect to the counting fields. ${\cal F}$ is the eigenvalue of ${\cal M}(\chi ,\xi )$ that approaches 0 as χ, ξ → 0. In general, an analytical expression for $\cal F$ cannot be obtained. In this case, the cumulants can still be extracted order by order. Following [54], we make an expansion ${\cal F}(\chi ,\xi )=\sum _{m,p>0}c_{m,p}({\mathrm { e}}^{{\mathrm { i}}\chi }-1)^m({\mathrm { e}}^{{\mathrm { i}}\xi }-1)^p$ such that we get recursively the stationary currents, I = c_1,0, J = c_0,1, zero frequency noises, S_II = I + 2c_2,0 and S_JJ = J + 2c_0,2, and cross-correlations, S_IJ = c_1,1.

It is useful to normalize the auto-correlations such that the charge and heat Fano factors, F_I = S_II/qI and F_J = S_JJ/E_CJ, reflect the sub- or super-Poissonian character of the fluctuations when they are lesser or greater than 1, respectively. We can also define the cross-correlation coefficient, $r=S_{I\! J}/\sqrt {S_{I\! I}S_{J\! J}}$, which is bounded by the Cauchy–Schwartz inequality: −1 ⩽ r ⩽ 1. We have r = ± 1 when the two currents are totally correlated such that their cumulants are proportional to each other.

From the stationary solution of the master equation ${\cal M}(0,0)\bar \rho =0$, we obtain the occupation probabilities of the different charge states. The analytical expressions are given in [21]. Using them, one gets the dot occupations $\langle n_{\mathrm { s}}\rangle =\bar \rho _{10}+\bar \rho _{11}$ and $\langle n_{\mathrm { g}}\rangle =\bar \rho _{01}+\bar \rho _{11}$. Their dependence on the two voltage differences V₁ − V₂ (the applied bias) and V₁ − V₃ (which serves as a gate voltage to dot s) defines the stability diagram of the system. They are also affected by the coupling to the leads, in particular by their asymmetry due to energy-dependent tunneling. We define here three different configurations that will be considered in the following:

• Symmetric configuration: Γ_ln = Γ for all l, n.
• Asymmetric configuration: Γ₁₀Γ₂₁ ≠ Γ₁₁Γ₂₀, where noise-induced transport takes place. For our numerics, we choose it to be as in the symmetric configuration except that Γ₁₁ < Γ.
• Optimal configuration: transitions in the conductor $(0,n_{\mathrm { g}})\leftrightarrow (1,n_{\mathrm { g}})$ with different n_g = 0,1 involve different terminals. Here, we will consider couplings as in the symmetric case except for Γ₁₁ = Γ₂₀ = 0.

In the region delimited by the conditions E_s1 = qV₁ and qV₂, the occupation of dot s is fixed (0 or 1) by Coulomb blockade, see figure 2. For larger bias voltage, the charge fluctuates by tunneling and shot noise is present. Being coupled to only one terminal, the charge of dot g only fluctuates along E_gn = qV₃. We call it the fluctuating gate region whose width is kT₃. In the presence of energy-dependent tunnel couplings, the regions where n_s fluctuates are modified by the presence of the coupled dot, g, see 〈n_s〉^(asym) and 〈n_s〉^(opt) in figure 2, which will affect the non-equilibrium currents.

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For our system to work as an energy harvester, both left–right and particle–hole symmetry have to be broken. Otherwise at zero applied bias, transport is equiprobable in opposite directions and no net current is generated. For the quantum-dot harvester, the symmetry breaking is provided by asymmetric, energy-dependent tunnel couplings Γ_ln [21, 23, 28]. Then, processes carrying an electron from left to right or vice versa after absorbing energy from the hot bath will have different rates (proportional to Γ₁₀Γ₂₁ and Γ₂₀Γ₁₁, respectively), resulting in a finite current generated at zero bias [21]

$$\begin{equation} \fl I=q\frac{(\Gamma_{11}\Gamma_{20}-\Gamma_{10}\Gamma_{21})\Gamma_{g0}\Gamma_{g1}}{8\gamma^3}\sinh\left[\frac{E_{\rm C}}{2}\left(\beta_3-\beta\right)\right]\prod_{\alpha,n}\mathrm{sech}\left(\frac{\Theta_{\alpha,n}\beta_\alpha}{2}\right) \end{equation} \tag{ 4 }$$

if the two conductors are at different temperatures. Here, Θ_α,n = E_α,n − qV_α, with V₁ = V₂ = V_s and V_g = V₃. Through all the paper, we will consider β₃ < β₁ = β₂ = β. The denominator γ³ contains third order products of tunnel couplings. In contrast to the charge current, heat flows into the conductor independently of the asymmetry:

$$\begin{equation} J=E_{\rm C}\frac{\Gamma_{s0}\Gamma_{s1}\Gamma_{g0}\Gamma_{g1}}{8\gamma^3}\sinh\left[\frac{E_{\rm C}}{2}\left(\beta_3-\beta\right)\right]\prod_{\alpha,n}\mathrm{sech}\left(\frac{\Theta_{\alpha,n}\beta_\alpha}{2}\right). \end{equation} \tag{ 5 }$$

The finite charge and heat current cross-correlations are evident already in the proportionality of the two currents (4) and (5). The explicit expression for the equilibrium cross-correlations, as discussed in (1), is

$$\begin{equation} S_{I\! J}^{\rm eq}=2qE_{\rm C}\frac{(\Gamma_{11}\Gamma_{20}-\Gamma_{10}\Gamma_{21})\Gamma_{g0}\Gamma_{g1}}{8\gamma^3}\prod_{\alpha,n}\mathrm{sech}\left(\frac{\Theta_{\alpha,n}\beta_\alpha}{2}\right), \end{equation} \tag{ 6 }$$

which clearly shows that the asymmetry Γ₁₁Γ₂₀ − Γ₁₀Γ₂₁ is necessary in order to correlate the heat and charge dynamics in the linear regime. From (4) and (6) one can easily verify the first equality in the fluctuation–dissipation theorem for crossed charge and heat currents (1), the second one being fulfilled by multiterminal Onsager symmetry [65–68]. For completeness, we have checked that the auto-correlations obey S^eq_II = 2kT∂I/∂ΔV and S^eq_JJ = 2kT∂J/∂(ΔT/T).

As shown in figure 3, the cross-correlations for V₁ = V₂ vanish in the perfectly symmetric case, Γ_ln = Γ and increase with the asymmetry. In the particular case where each charge state of the conductor is coupled to only one lead (e.g. Γ₁₁ = Γ₂₀ = 0), the cross-correlation is maximal, r² = 1, with an electron being transported across s for every energy quantum E_C absorbed from the hot source. We call it the optimal configuration for achieving a tight energy–matter coupling, I^(opt)/q = J^(opt)/E_C, for which the converter reaches the Carnot efficiency [21] and the maximal harvesting thermopower E_C/(qT₃) [22].

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The nonlinear transport is governed by the alignment of the different states with the Fermi levels of the leads. They correspond to the conditions Θ_ln = E_ln − qV_l = 0 (marked by dotted lines in figure 2), where charge fluctuations through the corresponding contact are enhanced. Far from them, the charge of at least one of the dots is well defined and the currents show plateaux as a function of the applied voltages. In our three-terminal system, not only the voltage bias applied to the charge conductor, V₁ − V₂, but also the voltage applied to the heat source, V₁ − V₃, plays a role in the charge current. Its effect is twofold: (i) it serves as a gate voltage to the conductor. This is evident in the charge current for the symmetric configuration, I^(sym), that develops a Coulomb blockade stability diagram; (ii) when |Θ_3n| > kT₃, the charge fluctuations of the gate dot are suppressed (see figure 2). Thus, in asymmetric configurations where, for instance Γ_l1 < Γ_l0, the current through the conductor will be reduced in the region Θ₃₁ < 0 where dot g is almost constantly occupied, see the panel labeled I^(asym) in figure 4. This effect, which involves a competition of fast and slow transport transitions, leading to negative differential conductance, is known as dynamical channel blockade [69].

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The dynamical blocking of the current has visible effects only in configurations with energy-dependent tunneling. In the optimal configuration, where electrons in s need to exchange energy with those in g in order to contribute to the current, its effect is maximal. There, I^(opt) is finite only in the fluctuating gate region (around Θ_3n = 0), where the charge of the hot dot fluctuates and heat flows, as shown in figure 4. Far from this region the heat current vanishes, i.e. the two conductors can be tuned to become thermally isolated by means of the gate voltage V₃.

On the other hand, it is also necessary that the charge of the conductor fluctuates in order for heat to be transferred from the hot source. For the symmetric configuration, the heat flow is therefore suppressed on the Coulomb-blockade plateaux, and shows peaks of J < 0 (out of terminal 3) along the lines Θ_l0 = 0 and Θ_l1 = 0, with l = 1,2, where the Coulomb-blockade borders cross the fluctuating gate region. For larger applied bias, Joule heating in the conductor dominates such that the heat flow is reversed: i.e. transport pumps heat against the temperature gradient. Note that the generation of Joule heat prevents this effect to be used for refrigeration of the charge conductor.

In the optimal configuration, an electron that has tunneled into dot s needs to absorb or relax an energy E_C before it can be transferred to the opposite contact. The sign of the heat flow is hence defined by the sign of the charge current: heat is driven by charge transport. For low voltages, a current flows in the direction defined by the asymmetry, e.g. from left to right if Γ₁₁ = Γ₂₀ = 0. Applying a bias opposite to it, a stall potential q(V₂ − V₁) = E_Cη_c is reached where the charge current generated by heat conversion is compensated by voltage. There, I^(opt) = J^(opt) = 0 and the engine works reversibly [22] at Carnot efficiency, η_c = 1 − T/T₃ [21]. A larger bias reverses the flow of charge and heat, such that heat is transferred to the hot terminal. On the contrary, if the bias is applied in the direction of the generated current, I^(opt) and therefore also J^(opt) will increase in absolute value, as compared with non-optimal configurations. In this way, driving a current through the conductor can serve to increase the heat extraction rate from the hot source. For asymmetric enough configurations, this effect coexists with the heat pumping discussed above for the symmetric configuration, see J^(asym) in figure 4.

Let us now discuss the current–current correlations for the different configurations. In the absence of noise-induced transport (symmetric configuration), the hot gate does not leave any clear feature in the charge noise: S^(sym)_II presents similar characteristics as a (two-terminal) single quantum dot, i.e. a plateau dominated by shot noise (for large bias), and noise suppression in the Coulomb blockade region, cf figure 5. Obviously, the heat noise is also suppressed where J vanishes. More interestingly, the charge–heat cross-correlations, S^(sym)_IJ, are maximal and change sign around the border of the Coulomb blockade region, related to the opening of conduction channels in s and with the change of sign of J^(sym) (see figure 4), respectively.

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The asymmetry in the energy-dependent couplings introduces features in the charge noise that depend on 〈n_g〉. As for the current, and depending on the asymmetry, the charge noise is reduced far from the fluctuating gate region, see figure 5. On the other hand, the noise is maximal along Θ₃₀ = 0 for ΔV >0. It is a consequence of the dynamical channel blockade effect where a competition of slow and fast channels enhances the noise signal. In the optimal configuration, excess noise and cross-correlations only appear in the region where both n_s and n_g fluctuate. Remarkably, in this configuration, I^(opt)/q = J^(opt)/E_C and all the charge and heat correlations are proportional to each other

$$\begin{equation} q^{-2}S_{I\! I}^{(\rm opt)}=E_{\rm C}^{-2}S_{J\! J}^{(\rm opt)}=-(qE_{\rm C})^{-1}S_{I\! J}^{(\rm opt)}, \end{equation} \tag{ 7 }$$

consistently with our previous finding that r² = 1, see section 2.

Additional information is obtained by looking at the noise to signal ratios: the Fano factors F_I and F_J. They are depicted in figure 6 for the different tunneling configurations. Again, the charge statistics for the symmetric configuration is not changed by the presence of the hot terminal. Similar to a single-level two-terminal quantum dot, the Fano factor is close to Poissonian inside the Coulomb blockade region and F^(sym)_I = 1/2 in the presence of transport [70, 71]. We emphasize that higher order (co)tunneling processes which might affect the blocked region and may cause super-Poissonian noise [72, 73] are neglected here. At zero applied voltage, I^(sym) = 0 and hence the Fano factor diverges. Similar divergences are observed in the heat Fano factor along the lines where Joule heating in the conductor compensates the flux coming from the heat source. The heat transfer depends on the characteristics of charge fluctuations in s, which is manifested in the Fano factor. Whereas both the heat current and noise vanish far from the condition Θ_3n = 0, F^(sym)_J distinguishes the Coulomb blockade region and the large bias regime dominated by shot noise. In the large bias regime, q(V₁ − V₂) ≫ kT, it is given by

$$\begin{equation} F_J^{(\rm sym)}=\mathrm{coth}\frac{E_{\rm C}}{2kT_3}-c\mathrm{\ sech}\frac{\Theta_{30}}{2kT_3}\mathrm{\ sech}\frac{\Theta_{31}}{2kT_3}\sinh\frac{E_{\rm C}}{2kT_3}, \end{equation} \tag{ 8 }$$

with c = 1/4 + ΓΓ_g/[2(Γ_g + 2Γ)²] and Γ_g = Γ₃₀ = Γ₃₁. It has a minimum at Θ₃₀ = 0, where it is simply F^(sym)_J = coth(E_Cβ₃/2) − c tanh(E_Cβ₃/2). Note that F^(sym)_J is not necessarily greater than 1. For the regime that we are interested in, with E_C ∼ kT₃, it is, however, super-Poissonian at any voltage.

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Asymmetric configurations, where heat to charge conversion occurs, present peculiar features both in the charge and heat Fano factors. In the previous section, we discussed an increase of charge noise S^(asym)_II in the regions where heat is most effectively transferred (along Θ_3n = 0) in terms of a dynamical blocking of the charge current due to the stabilization of the charge state in the hot dot. Such an effect is known to lead to super-Poissonian noise [69, 74–76] and positive cross-correlations [33] in capacitively coupled conductors. When Γ_l1 ≪ Γ_l0, current only flows when dot g is empty, i.e. n_g acts as a current switch. If the switch rate is slow (for low T₃ and Γ_3n), the charge noise becomes super-Poissonian [77] (not shown). Consistent with telegraph noise, F^(asym)_I diverges as 1/Γ_3n. In the configuration relevant for heat engines, T₃ > T and Γ_3n ∼ Γ, we find F^(asym)_I ≈ 1, cf figure 6. As a counterpart, the heat Fano factor becomes asymmetric and adopts some electron-like structure in the form of plateaux in the Coulomb blockade regions.

An interesting feature is that, due to the noise-induced current, the charge Fano factor is finite at zero applied voltage. The divergence of F_I is shifted to a finite stall voltage applied against the generated current such that I^(asym) = 0. This non-equilibrium state (without charge current) is maintained by a finite heat current. The point at which F_I diverges can thus be considered as a measure of the nonlinear thermovoltage, which depends on the gate voltage, the asymmetry and the temperature gradient, as shown in figure 7.

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The optimal configuration is quite different. As we discussed above, transport is finite only in the region where the charge of the hot dot fluctuates, and charge and heat statistics become identical, with F^(opt)_I = F^(opt)_J. We recall that in this case there is only one tunneling sequence that contributes to charge and heat transport (the one represented in figure 1). As a consequence, rather than a competition of different channels—which led to an increased charge noise in the asymmetric configuration—they cooperate and far from the divergence F_I ⩽ 1. Importantly, in the optimal configuration also the heat noise becomes sub-Poissonian, cf figure 6. In this configuration, the divergence of the Fano factor is shifted to a constant stall voltage, q(V₂ − V₁) = E_Cη_c, where both I^(opt) and J^(opt) vanish, cf figure 7. Note that despite the non-equilibrium situation, no average current flows and entropy production is zero [22]. Therefore, in the optimal case, at the stall voltage, the non-equilibrium state is maintained solely by fluctuations in the charge and heat current.

We emphasize that in a Coulomb-blockade device like the one discussed here, charge fluctuations are antibunched even when noise is super-Poissonian [78]. This applies also to the heat transfer, which requires a sequence of charge tunneling events.