Boosting thermoelectric efficiency using time-dependent control

Thermoelectric efficiency is defined as the ratio of power delivered to the load of a device to the rate of heat flow from the source. Till date, it has been studied in presence of thermodynamic constraints set by the Onsager reciprocal relation and the second law of thermodynamics that severely bottleneck the thermoelectric efficiency. In this study, we propose a pathway to bypass these constraints using a time-dependent control and present a theoretical framework to study dynamic thermoelectric transport in the far from equilibrium regime. The presence of a control yields the sought after substantial efficiency enhancement and importantly a significant amount of power supplied by the control is utilised to convert the wasted-heat energy into useful-electric energy. Our findings are robust against nonlinear interactions and suggest that external time-dependent forcing, which can be incorporated with existing devices, provides a beneficial scheme to boost thermoelectric efficiency.

The on-going advances of nano-structure engineering have re-energized the search for high-efficiency thermoelectric devices [1][2][3] . Till date, almost all studies on thermoelectricity are focused on finding high efficiency materials guided by the near equilibrium thermodynamic quantities like the Seebeck coefficient and the thermoelectric figure of merit ZT. These passive searches have reached saturation and the thermoelectric efficiency achieved thus far is still insufficient from a practical standpoint 4 . This is primarily because, in the near equilibrium regime, the thermoelectric efficiency is limited by various thermodynamic constraints, namely, the second law of thermodynamics which imposes an unavoidable entropy flow and the Onsager reciprocal relation that connects the Seebeck and Peltier effects.
In order to achieve high thermoelectric efficiency an active approach to overcome these thermodynamic obstacles is the need of the hour. A possible mechanism overcoming these thermodynamic constraints is to apply a time-dependent forcing to drive the system far from equilibrium. Unlike bulk materials, many nano-systems, such as quantum dots 5,6 , single-electron-transistors 7 , and molecular junctions [8][9][10][11][12] , can strongly interact with an externally applied control force. These systems have been the subject of intense theoretical investigations to better understand the mechanisms underlying electron 13 and heat 14,15 transport in presence of a time-dependent control. Several applications such as overall device efficiency 16 , thermopower 11 , thermal refrigeration 14 , electron pumping 17 , and heat pumping 18,19 have also been studied in these systems to figure out the role of an external control. Despite these advances the study of dynamic thermoelectric efficiency has been highly non-trivial due to the breaking of thermodynamic constraints and the non-unique definitions of the thermodynamic quantities such as the Seebeck coefficient and the figure of merit ZT.
In this study we propose a dynamic theory of thermoelectric efficiency to overcome the present thermodynamic limitations such as the Onsager reciprocal relation. The main idea being that a strong external time-dependent control breaks the time-translational invariance of the system and pushes it in the far from equilibrium regime. This in turn causes a breakdown of the celebrated Onsager symmetry relation that allows for the possibility to boost the resulting efficiency. The boost for the thermoelectric efficiency (This efficiency should not be confused with the overall efficiency; the latter also contains the time-dependent, external driving input power in the denominator; then yielding an overall efficiency that does not exceed unity.) can be as large as four times the near equilibrium value and vitally a large fraction of the supplied input energy is constructively utilized to enhance the thermoelectric performance. Thus, our novel approach makes available an extra knob to engineer high thermoelectric efficiency in nano-devices.

Results
Dynamic theory of thermoelectricity. Since in the far from equilibrium regime the Seebeck coefficient and the figure of merit ZT are ill-defined we establish a thermoelectric formalism based on the underlying time-dependent currents. This objective can be achieved through the evaluation of the Onsager transport matrix which relates the electron or heat current with the temperature or chemical potential bias. In the conventional, near equilibrium, formalism the transport-matrix coefficients are autonomous and constrained by various thermodynamic laws. However, a time-dependent control pushes the system far from equilibrium and results in the transport coefficients depending on the entire history of the applied protocol. Moreover the presence of a time-dependent control force causes the charging and discharging of the nano-system resulting in a time-varying current (known as the displacement current) 11,20 , which will vanish exactly in the near-equilibrium scenario. In order to construe the above mechanism we consider a two-probe transport set-up consisting of a system connected to a left and right lead as depicted in the inset of Fig. 1(a). When the system is subjected to an external driving force F(t) the left lead electron and heat current will be functions of the thermodynamic forces and the entire history of the applied protocol, F(t′ ), t 0 ≤ t′ ≤ t, with the starting time t 0 of the force protocol of otherwise arbitrary strength. In the linear response regime for the thermodynamic forces, namely, temperature difference Δ T/T and chemical-potential difference Δ μ/e small, we Taylor-expand the currents at each instance of time as, Above, the coefficients = 0 arises only in the presence of an external control and is called the displacement current. Since the displacement current doesn't change sign under reversal of thermodynamic forces we can write the above Taylor expansion as a transport-matrix equation that reads, The super-script α takes values L and R corresponding to the leads. The elementary charge e > 0 and represents displacement current kernel acting on the history of the applied protocol such that the displacement current . For an undriven system, i.e. F(t) = 0 ∀t, the displacement currents vanish, leaving only the biased currents, yielding a nonequilibrium steady state. Specifically, the transport matrix reduces to a commonly known, time-independent 2 × 2 Onsager matrix =  11 12 21 22 and the transport coefficients are obeying the constraints of near equilibrium thermodynamic steady state transport; namely the Onsager reciprocal relation are valid, imposing that L 21 /L 12 = 1. Likewise, the second law of thermodynamics ensures a positive thermal conductance, or det(L) > 0 3 . The primary result in this work is to obtain the transport coefficients under a time-dependent control. This can be achieved in the following manner: (i) We assume small thermodynamic forces for the temperature bias and the potential difference so that the relationship w.r.t to these forces stays linear. (ii) The currents are evaluated (see below) at any time instant t as a function of the two small thermodynamic forces. (iii) Then, setting Δ α μ/e = 0 the slope of the electron-(heat-) current w.r.t Δ α T/T yields L 12  In order to investigate the consequences of the time-dependent control on the thermoelectric efficiency we bias the system with a temperature difference Δ T, connect a load of resistance  R to the system and calculate the amount of power consumed by the load. We assume that the load is a pure resistor that cannot lead to charging effects due to the passage of electron current. Therefore, the amount of current passing through the load is related to the bias and the transport matrix L. After accounting for the back-action from the load, the biased electron current reads I e (t) 1 being the resistance of the system. Hence the thermoelectric efficiency ratio of the heat-work conversion, reads 21,22 , with the factor − 1/2 indicating that half of the heat flows back to the hotter lead. In the nonequilibrium steady state this efficiency ratio will reduce to the standard formalism 2 where, det(L)R M /T represents the thermal conductance, L 21 R M is the Peltier coefficient, and the efficiency is directly related to the figure of merit ZT provided that the Onsager reciprocal relation L 21 /L 12 = 1 is satisfied. Since all the transport matrix coefficients [forming the numerator and denominator of Eq. (4)] are affected by the time-dependent control, a priori it is not clear if the control will have an overall enhancing or diminishing effect on the dynamic thermoelectric efficiency. Equation (4) represents one of our main results, generalizing the conventional thermoelectric theory to a dynamic one which can be applied to far-equilibrium non-steady-state regime. Note that the thermoelectric efficiency is defined in the absence of the time-dependent, external driving input power in the denominator. In distinct contrast to an overall efficiency which would contain this additional driving input power, the thermolelectric efficiency in Eq. (4) can formally exceed unity, cf. in Fig. 1.
Non-interacting electrons. As a proof of concept, we first consider a single electron quantum dot in the regime of strong Coulomb blockade. The time-dependent external force F(t) causes charging and discharging on the system and the total Hamiltonian reads, is the tunnelling Hamiltonian between the quantum dot and the leads, and the Hamiltonian of the quantum-dot system is S 0 This quantum resonant model has been extensively studied in the context of single-electron-transistors 6,7,23-25 , molecular junctions 26 , and nano-wires 27,28 . The energy level of the dot can be controlled either via a time-dependent gate voltage 11 , or via long-wavelength electromagnetic fields such as microwaves 12,29 or lasers 28,30 .
The result of heat-work conversion efficiency ratio of this model under step-like control is shown in Fig. 1. From Fig. 1(a) we detect large enhancements in the efficiency, upto a factor of 4 compared to the steady-state efficiency, as soon as the step-pulse is applied. After some relaxation time the values eventually saturate to the new steady state. Interestingly, the magnitude of L 21 /L 12 [ Fig. 1(b)] shows a profile similar to the efficiency ratio indicating that the breakdown of the Onsager reciprocal relations L 21 /L 12 ≠ 1 and the efficiency enhancement are closely intertwined. Physically, when L 21 is not bounded by L 12 , the contribution of the particle flow to the heat current can increase under the influence of external driving. As a result the efficiency is boosted via increasing the useful heat (due to particle flow) while limiting the waste heat (due to entropy flow). To substantiate this claim we plot det(L) in Fig. 1(c). Because the det(L) is proportional to the entropy flow we see that it decreases in the regime of efficiency enhancements. Importantly, for sufficiently strong driving (s = 0.5Γ ) we detect a regime with negative values for det(L), indicating a reversal of the entropy flow, even though the overall heat current still flows from the hot lead to the cold one.
Harvested power. Supplementary to the colossal boosts in the heat-work conversion efficiency it is also crucial that most of the input power due to the control force is properly utilized to enhance the heat-work conversion efficiency. We analyse this using the harvested and input power where  w harvest is the power harvested from the enhancement of efficiency that is defined as the difference of the useful power consumed on the load under time-dependent control (first term) and the power from the steady-state contribution (second term).
 w input is the input power via the external control force, where the factor, − F(t)/e, represents the input voltage and ( ) I t 2 e D is the total displacement current. Figure 2(a) depicts that the harvested power can be much larger than the input power due to the control forcing. This feature occurs, even in the linear response regime, in the system-parameter regime when the steady-state efficiency is low, due to the low electron conductance, but the Seebeck coefficient itself remains large. Thus, the presence of time-dependent control constructively facilitates the movement of electrons and boosts the thermoelectric efficiency.
Resistor-capacitor model. In order to better understand the displacement current in the high temperature and weak control limit, we propose an elementary resistor-capacitor model as illustrated in the inset of Fig. 2(b). The time-dependent control is acting on the gate with capacitance C g which can induce charging or discharging of the capacitor. This variation leads to a current generation which flows from the capacitance towards the leads which are represented as the two sink sources (the ground connection) in the circuit. The current generated solely depends on the time-dependent control and does not require The intuitive picture for the displacement current above is based solely on circuit law considerations. Hence a priori it is not clear if such a model is able to describe correctly a fully quantum mechanical system. In order to justify that this indeed is the case we use the parameters R and τ from Eq. (8) as variables and fit the equation to the fully quantum mechanical displacement current obtain via nonequilibrium Green's function. Figure 2(b) shows the NEGF calculation as a solid line and the fit via the red dots. The perfect fit gives us the parameters τ ≈ ħ/Γ which further strengthens our resistance-capacitance circuit model. This is because in an open dissipative quantum system one expects the relaxation time of the system to be inversely proportional to the sum of the coupling strengths of each lead Γ −1 see Ref. 31. Thus, the verification of our resistance-capacitance model gives an intuitive picture of the displacement current.
Electron-phonon interaction. One of the main challenges for experimental devices to obtain enhanced efficiencies is the unavoidable presence of nonlinear interactions mainly arising due to the involvement of stray phonon modes 32 , which can drastically alter the thermoelectric efficiency. Hence, we consider a quantum dot interacting with a single phonon mode giving rise to the following electron-phonon interaction Hamiltonian, Here, a † and a are creation and annihilation operators of the phonon, ω 0 is the phonon angular frequency, λ is the electron-phonon interaction strength and F(t) represents the time-dependent control of the coherently driven quantum dot. The model manifests itself under various physical scenarios like in a nano-mechanical resonator 6,7,33 , molecular junction 26,34 , and standard lattice vibration model 35 . Recently it was shown that a small amount of nonlinearity in this model can greatly suppress the steady-state efficiency 33 . Thus, the model serves as a perfect test bed to establish the robustness of our approach. In Fig. 3 we depict the results for the interacting electron model. In case of the delta shape and square wave driving we modulate the system for sometime and then let it relax to reach its nonequilibrium steady state. Clearly the enhancement in the efficiency (as seen from the bottom row of Fig. 3) is observed even for a relative strong nonlinear interaction λ as long as the system dynamics is time-dependent. The long-time limit, when the transient effects are wiped out, is easily recovered for the delta shaped and square wave driving giving η(∞)/η ss = 1 [not depicted in Fig. 3]. Analogous to the non-interacting electron model, the enhancements are closely related to the breakdown of the Onsager reciprocal relation L 21 /L 12 and the second law of thermodynamics det(L). Interestingly, external forcing alone is not sufficient to enhance the systems efficiency as seen from the case with a periodic sinusoidal driving where the efficiency even decreases when L 21 /L 12 < 1. Thus, we speculate that although driving is a necessary condition to allow the breakdown of stringent constraints it does not suffice to enhance the efficiency of the device. One possible sufficient condition for an enhancement is the abrupt variation in the driving field which causes a sudden change of the charge occupation in the system. As a result the displacement current will be large (fourth row of Fig. 3).

Discussion
The performance of modern thermoelectric devices is measured in terms of its efficiency to convert waste heat energy into useful electric energy. Till date, most efforts towards efficiency enhancement are based on the search for suitable materials guided by the near-equilibrium thermodynamic constraints. These efforts have led to considerable improvements in the heat-work conversion efficiency but the quest for commercially feasible efficiency has been futile so far.
In the present study, we provide an active and complimentary approach to the material search avenue. This is achieved by introducing a time-dependent control which pushes the system far from equilibrium and provides a rich playground without thermodynamic limitations. The control force can be fused with existing high efficiency devices and would allow us to further boost their efficiency by a factor of 4. The enhancements are robust and persist even in presence of nonlinear interactions indicating its usefulness in existing experimental set-ups.
Thus, our work opens up a whole new arena where we shift attention away from a material design perspective and focus on the non-trivial far from equilibrium regime which leads to smart device design. Overall the method presented herein provides a rigorous stepping stone which can have wide ranging impact for fields such as thermoelectric cooling [36][37][38][39] , solar thermoelectric generation [40][41][42] , and can be extended to the field of spin caloritronics to efficiently pump spin-currents using thermal gradients 43 .

Methods
Nonequilibrium Green's function. For non-interacting electron in a quantum dot under step-like , an exact solution of the electron and heat currents can be obtained using the Landauer formalism via the NEGF approach 11,44 , reading Given the biased and displacement currents the NEGF formalism allows us to obtain a closed form expression of the efficiency η(t) in the weak system-bath coupling and weak control limit. In order to achieve this goal we first simplify the efficiency given by Eq. (4)  . Their ratio, f 2 (t − t 0 )/f 1 (t − t 0 ), monotonically increases from 0 to 1 as (t − t 0 ) → ∞, thus providing no temporal boost in the efficiency arising from the second term in Eq. (17). The third contribution, i.e., η ∝ ( − )  t t 0 , arises due to the contribution that breaks the Onsager symmetry in the biased heat current [second term of Eq. (15)] and is responsible for the temporal boost in the thermoelectric efficiency (in the leading order of s). This further strengthens our claim that it is indeed the breaking of Onsager symmetry that leads to a boost in thermoelectric efficiency.
Equations of motion for the resistor-capacitance model. We further elucidate on the resistor-capacitor model as shown in the inset of Fig. 2(b). Consider that the capacitor has a charge Q then the voltage on its upper plate will be the sum of the voltage across the resistances and the voltage across the capacitor 46  The protocol F(t) begins at t 0 (0 < t 0 < t) and ends at time t and the displacement current depends on the complete history of the protocol. , where Q = eTr[ρ(t)d † d]. The above quantum master equation formalism is valid in the weak system-bath coupling limit and holds true for arbitrary control strength s and electron-phonon interaction strength λ. Moreover, we do not resort to the secular (or rotating wave) approximation and the master equation is kept semi-non-Markovian since time t is explicitly present in the integral-limits of the transition matrix. The approach is robust to deal with nonlinear interactions exactly and thus allows to obtain the transport-matrix coefficients for strongly nonlinear systems.