Powerful energy harvester based on resonant-tunneling quantum wells

Energy harvesters collect energy from the environment and use it to power small electronic devices or sensors [1]. A wide variety of energy harvesters have now been proposed that convert ambient energy to electrical or mechanical power, e.g. from vibrations, electromagnetic radiation or by relying on thermoelectric effects. The latter systems turn out to be particularly useful for converting heat on a computer chip back into electrical power, thereby reducing both the power consumption of the chip as well as the need to actively cool it.

The main challenge of current research on thermoelectric energy harvesters is to find setups that are both powerful and efficient at the same time. Artificially fabricated nanoscale structures are promising candidates for highly efficient thermoelectrics. Twenty years ago, Hicks and Dresselhaus [2, 3] demonstrated that mesoscopic one-dimensional wires as well as quantum wells have thermoelectric figures of merit that are greatly enhanced compared to the bulk values. Mahan and Sofo [4] later showed that the best thermoelectric properties occur in materials that are good energy filters, i.e. have sharp spectral features.

A paradigmatic realization of such spectral features is given by quantum dots with sharp, discrete energy levels. The thermopower of quantum dots in the Coulomb-blockade regime coupled to two electronic reservoirs at different temperatures has been studied both theoretically [5] and experimentally [6–11]. Later, the thermopower of open quantum dots [12] and carbon nanotube quantum dots [13, 14] was investigated. Thermoelectric effects have also been studied for resonant tunneling through a single quantum dot [15]. Compared to a weakly coupled quantum dot in the Coulomb-blockade regime [16], the power is enhanced while at the same time the efficiency is reduced due to the finite level width of the resonant state. More complicated resonant tunneling configurations have been proposed to optimize the efficiency at finite power output [17, 18].

Recently, there has been a growing interest in thermoelectrics with three-terminal structures [19–29]. Four-terminal configurations (two Coulomb-coupled conductors subject to currents) have been of interest in the discussion of nonequilibrium fluctuations [30–34]. In addition, reciprocity relations for multi-terminal thermoelectric transport have been analyzed [35–39]. Compared to conventional two-terminal setups, they offer the advantage of separating the heat and charge current flow. Furthermore, they naturally operate in a conventional thermocouple-like fashion. While a system of two Coulomb-coupled quantum dots in the Coulomb-blockade regime was shown to work as an optimal heat-to-current converter that can reach Carnot efficiency, the resulting currents and output powers are limited by the fact that transport only proceeds via tunneling of single electrons [19]. A related setup where the Coulomb-blockade dots are replaced by chaotic cavities connected via quantum point contacts with a large number of open transport channels to the electronic reservoirs turns out to deliver much larger currents. Nevertheless, the resulting output power is similar to the Coulomb-blockade setup since the thermoelectric performance is determined by the weak energy dependence of a single partially open transport channel [20]. This problem can be overcome by a heat engine based on resonant-tunneling quantum dots. Such a system yields a large output power of 0.1 pW for a temperature difference of 1 K between the hot and the cold reservoir while at the same time it reaches an efficiency at maximum power of about 20% of the Carnot efficiency at maximum power. In addition, the device can be scaled to macroscopic dimensions by parallelization based on the use of self-assembled quantum dots [21]. Similar setups have also been investigated both theoretically [40, 41] and experimentally [42] in their dual role as refrigerators.

Here, we analyze the performance of a three-terminal energy harvester based on resonant quantum wells. Our work is motivated by a number of advantages that we expect a quantum-well structure to have over a quantum-dot setup. Firstly, quantum wells should be able to deliver larger currents and, therefore, larger output powers because of the transverse degrees of freedom. The available phase space for electrons that can traverse the well is large. Secondly, a quantum-well structure might be easier to fabricate than a system of self-assembled quantum dots that should all have similar properties in order to yield a decent device performance, although there is good tolerance to fluctuations in dot properties [21]. Finally, due to the large level spacing of narrow quantum wells, they are ideally suited for room-temperature applications. Apart from these advantages, we also aim to investigate how the less optimal energy-filtering properties of quantum wells compared to quantum dots deteriorate the efficiency of heat-to-current conversion (quantum wells transmit any electron with an energy larger than the level position whereas quantum dots transmit only electrons with an energy exactly equal to the level energy).

Our paper is organized as follows. In section 2, we introduce the model of the quantum-well harvester. We then present our results for the power and efficiency in section 3 analyzing both the linear and the nonlinear transport regime. Finally, we give some conclusions in section 4.

The system we consider is schematically shown in figure 1. It consists of a central cavity connected via quantum wells to two electronic reservoirs. In the following, we assume the quantum wells to be noninteracting such that charging effects can be neglected in a simplified model. We will revisit the effects of interactions relevant in the nonlinear regime with a more realistic treatment in the future.

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The electronic reservoirs, $r=\mathsf {L,R}$, are characterized by a Fermi function $f_r(E)=\{\exp [(E-\mu _r)/(k_{\mathsf {B}} T_{\mathsf {c}})]+1\}^{-1}$ with temperature $T_{\mathsf {c}}$ and chemical potentials μ_r. The cavity is assumed to be in thermal equilibrium with a heat bath of temperature $T_{\mathsf {h}}$. The nature of this heat bath is not relevant for our discussion and depends on the source from which we want to harvest energy. Strong electron–phonon and electron–electron interactions within the cavity relax the energy of the electrons entering and leaving the cavity toward a Fermi distribution $f_{\mathsf {C}}(E)=\{\exp [(E-\mu _{\mathsf {C}})/(k_{\mathsf {B}} T_{\mathsf {h}})]+1\}^{-1}$ characterized by the cavity temperature T_h and the cavity's chemical potential μ_C.

The cavity potential μ_C, as well as its temperature T_h (or, equivalently, the heat current J injected from the heat bath into the cavity to keep it at a given temperature T_h), have to be determined from the conservation of charge and energy, I_L + I_R = 0 and J^E_L + J^E_R + J = 0. Here, I_r denotes the current flowing from reservoir r into the cavity. Similarly, J^E_r denotes the energy current flowing from reservoir r into the cavity.

The charge and energy currents can be evaluated within a scattering matrix approach as [43]

$$\begin{equation} I_r=\frac{e\nu_2\mathcal A}{2\pi\hbar}\int {\rm d}E_\perp {\rm d}E_z T_r(E_z)\left[f_r(E_z+E_\perp)-f_{\mathsf{C}}(E_z+E_\perp)\right] \end{equation} \tag{ 1 }$$

and

$$\begin{equation} J^{\mathsf{E}}_{r}=\frac{\nu_2\mathcal A}{2\pi\hbar}\int {\rm d}E_\perp {\rm d}E_z (E_z+E_\perp)T_r(E_z)\left[f_r(E_z+E_\perp)-f_{\mathsf{C}}(E_z+E_\perp)\right]. \end{equation} \tag{ 2 }$$

Here, ν₂ = m_*/(πℏ²) is the density of states of the two-dimensional electron gas inside the quantum well with the effective electron mass m_*. $\mathcal A$ denotes the surface area of the well. E_z and E_⊥ are the energy associated with motion in the well's plane and perpendicular to it, respectively. The transmission of quantum well r is given by [44]

$$\begin{equation} T_r(E)=\frac{\Gamma_{r1}(E)\Gamma_{r2}(E)}{(E-E_{nr})^2+[\Gamma_{r1}(E)+\Gamma_{r2}(E)]^2/4}. \end{equation} \tag{ 3 }$$

Here, Γ_r1(E) and Γ_r2(E) denote the (energy-dependent [43]) coupling strength of the quantum well to the electronic reservoir r and the cavity, respectively. The energies of the resonant levels (more precisely the subband thresholds) within the quantum well are given by E_nr. For a parallel geometry with well width L, the resonant levels are simply given by the discrete eigenenergies of a particle in a box, E_nr = (πℏn)²/(2m*L²). In the following, we always restrict ourselves to the situation of weak couplings, Γ_r1,Γ_r2 ≪ k_BT_c,k_BT_h, whose energy dependence can be neglected. Furthermore, we assume that the level spacing inside the quantum wells is large such that only the lowest energy state is relevant for transport. In this case, the transmission function reduces to a single delta peak, $T_r(E)=2\pi {\Gamma _{1r}\Gamma _{2r}}/({\Gamma _{1r}+\Gamma _{2r})}\delta (E_z-E_{1r})$. This allows us to analytically solve the integrals in the expressions (1) and (2) for the currents and yields

$$\begin{equation} I_r=\frac{e\nu_2\mathcal A}{\hbar}\frac{\Gamma_{r1}\Gamma_{r2}}{\Gamma_{r1}+\Gamma_{r2}}\left[k_{\mathsf{B}} T_{\mathsf{c}}K_1\left(\frac{\mu_r-E_r}{k_{\mathsf{B}} T_{\mathsf{c}}}\right)-k_{\mathsf{B}} T_{\mathsf{h}} K_1\left(\frac{\mu_{\mathsf{C}}-E_r}{k_{\mathsf{B}} T_{\mathsf{h}}}\right)\right], \end{equation} \tag{ 4 }$$

as well as

$$\begin{equation} \fl J^{\mathsf{E}}_r=\frac{E_{r}}{e}I_r +\frac{\nu_2\mathcal A}{\hbar}\frac{\Gamma_{r1}\Gamma_{r2}}{\Gamma_{r1}+\Gamma_{r2}}\left[(k_{\mathsf{B}} T_{\mathsf{c}})^2K_2\left(\frac{\mu_r-E_r}{k_{\mathsf{B}} T_{\mathsf{c}}}\right)-(k_{\mathsf{B}} T_{\mathsf{h}})^2K_2\left(\frac{\mu_{\mathsf{C}}-E_r}{k_{\mathsf{B}} T_{\mathsf{h}}}\right)\right], \end{equation} \tag{ 5 }$$

where for simplicity we denote the energy of the single resonant level in the quantum wells as E_r. We furthermore introduced the integrals $K_1(x)=\int _0^\infty {\rm d} t(1+{\mathrm { e}}^{t-x})^{-1}=\log (1+{\mathrm { e}}^x)$ and $K_2(x)=\int _0^\infty {\rm d} t\; t(1+{\mathrm { e}}^{t-x})^{-1}=-{\mathrm { Li}}_2(-{\mathrm { e}}^x)$ with the dilogarithm ${\mathrm { Li}}_2(z)=\sum _{k=1}^\infty \frac {z^k}{k^2}$. The heat current is made up from two different contributions. While the first one is simply proportional to the charge current, the second term breaks this proportionality. We remark that in the case of quantum dots with sharp levels, the latter term is absent [21].

In the following, we first analyze the system in the linear-response regime and then turn to the nonlinear situation. We assume that both quantum wells are intrinsically symmetric, i.e. Γ_L1 = Γ_L2 ≡ (1 + a)Γ, Γ_R1 = Γ_R2 ≡ (1 − a)Γ. Here, Γ denotes the total coupling strength whereas −1 ⩽ a ⩽ 1 characterizes the asymmetry between the coupling of the left and the right well.

3.1. Linear response

We start our analysis by a discussion of the linear-response regime. To simplify notation, we introduce the average temperature T = (T_h + T_c)/2 and the temperature difference ΔT = T_h − T_c. To linear order in the temperature difference ΔT and the bias voltage eV =μ_R − μ_L applied between the two electronic reservoirs, the charge current through the system is given by

$$\begin{equation} I_{\mathsf{L}}=-I_{\mathsf{R}}=\frac{e\nu_2\mathcal A\Gamma}{2\hbar}g_1\left(\frac{E_{\mathsf{L}}}{k_{\mathsf{B}} T},\frac{E_{\mathsf{R}}}{k_{\mathsf{B}} T}\right)\left[-eV-k_{\mathsf{B}}\Delta T g_2\left(\frac{E_{\mathsf{L}}}{k_{\mathsf{B}} T},\frac{E_{\mathsf{R}}}{k_{\mathsf{B}} T}\right)\right], \end{equation} \tag{ 6 }$$

with the auxiliary functions

$$\begin{equation} g_1(x,y)=\frac{1-a^2}{2+(1-a){\rm e}^x+(1+a){\rm e}^y} \end{equation} \tag{ 7 }$$

and

$$\begin{equation} g_2(x,y)=x-y+(1+{\rm e}^x)\log(1+{\rm e}^{-x})-(1+{\rm e}^y)\log(1+{\rm e}^{-y}). \end{equation} \tag{ 8 }$$

At V =0, a finite current driven by ΔT ≠ 0 flows in a direction that depends on the position of the resonant levels. If, e.g., E_R > E_L, electrons will be transferred from the left to the right lead.

The power delivered by the heat-driven current against the externally applied bias voltage eV is simply given by P = I_LV . It vanishes at zero applied voltage. Furthermore, it also vanishes at the so-called stopping voltage V_stop where the heat-driven current is exactly compensated by the bias-driven current flowing in the opposite direction. In between these two extreme cases, the output power depends quadratically on the bias voltage and takes its maximal value at half the stopping voltage. Here, the maximal output power is given by

$$\begin{equation} P_{\mathsf{max}}=\frac{\nu_2\mathcal A\Gamma}{2\hbar}\left(\frac{k_{\mathsf{B}}\Delta T}{2}\right)^2 g_1\left(\frac{E_{\mathsf{L}}}{k_{\mathsf{B}} T},\frac{E_{\mathsf{R}}}{k_{\mathsf{B}} T}\right)g_2^2\left(\frac{E_{\mathsf{L}}}{k_{\mathsf{B}} T},\frac{E_{\mathsf{R}}}{k_{\mathsf{B}} T}\right). \end{equation} \tag{ 9 }$$

The effiency η of the quantum-well heat engine is defined as the ratio between the output power and the input heat. The latter is given by the heat current J injected from the heat bath, i.e. we have η = P/J. For a bias voltage V =V_stop/2 that delivers the maximal output power, the heat current is given by

$$\begin{equation} J=\frac{\nu_2\mathcal A\Gamma}{2\hbar}(k_{\mathsf{B}} T)^2 \frac{\Delta T}{T} g_3\left(\frac{E_{\mathsf{L}}}{k_{\mathsf{B}} T},\frac{E_{\mathsf{R}}}{k_{\mathsf{B}} T}\right), \end{equation} \tag{ 10 }$$

where the function g₃(x,y) that satisfies 0 < g₃(x,y) < 2π²/3 is given in the appendix for completeness. Hence, the efficiency at maximum power is simply given by

$$\begin{equation} \eta_{\mathsf{maxP}}=\frac{\eta_{\mathsf{C}}}{4}\frac{g_1\left(\frac{E_{\mathsf{L}}}{k_{\mathsf{B}} T},\frac{E_{\mathsf{R}}}{k_{\mathsf{B}} T}\right)g_2^2\left(\frac{E_{\mathsf{L}}}{k_{\mathsf{B}} T},\frac{E_{\mathsf{R}}}{k_{\mathsf{B}} T}\right)}{g_3\left(\frac{E_{\mathsf{L}}}{k_{\mathsf{B}} T},\frac{E_{\mathsf{R}}}{k_{\mathsf{B}} T}\right)} \end{equation} \tag{ 11 }$$

with the Carnot efficiency $\eta _{\mathsf {C}}=1-\frac {T_{\mathsf {c}}}{T_{\mathsf {h}}}\approx \frac {\Delta T}{T}$.

We now discuss the output power and the efficiency in more detail, first focusing on a symmetric system, a = 0. In figure 2, we show the power as a function of the level positions E_L and E_R. It is symmetric with respect to an exchange of E_L and E_R. The maximal output power of approximately $P_{\mathsf {max}}\approx \frac {\nu _2\mathcal A\Gamma }{2\hbar }\left (\frac {k_{\mathsf {B}}\Delta T}{2}\right )^2$ arises when one of the two levels is deep below the equilibrium chemical potential, −E_L/R ≫ k_BT while the other level is located at about E_R/L ≈ 1.5k_BT. An explanation for this will be given below.

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Similarly to the power, the efficiency is also symmetric under an exchange of the level positions. It takes its maximal value of η ≈ 0.1η_C in the region E_L,E_R > 0 where the output power is strongly suppressed. For these parameters, energy filtering is efficient but the number of electrons that can pass through the filter is exponentially suppressed. For level positions that maximize the output power, the efficiency is slighty reduced to η_{max P} ≈ 0.07η_C. This efficiency is much smaller than the efficiency at maximum power of a quantum-dot heat engine with couplings small compared to temperature. The latter lets only electrons of a specific energy pass through the quantum dot. Hence, charge and heat currents are proportional to each other. In this tight-coupling limit, the efficiency at maximum power in the linear-response regime is given by η_C/2 [45]. In contrast, the quantum wells transmit electrons of any energy larger than the level position, because any energy larger than the ground state energy can be expressed as E_⊥ + E_z, where E_z is the z-component and E_⊥ the perpendicular component of the electron's kinetic energy. Consequently, even high-energy electrons can traverse the barrier, provided most of the energy is in the perpendicular degrees of freedom, and E_z matches the resonant energy. Therefore, they are much less efficient energy filters.

We now aim to understand why the efficiency at maximum power of the quantum-well heat engine is still only about a factor of three less than the efficiency at maximum power of a quantum-dot heat engine with level width of the order of k_BT [15, 21]. The latter configuration has been shown to yield the maximal output power [21]. To this end, we analyze the situation depicted in figure 1. The right quantum well acts as an efficient energy filter because the number of electrons larger than E_R is exponentially small. The energy filtering at the left quantum well relies on a different mechanism. In order for an electron of energy E to enter the cavity, we need to have f_L(E) > 0 such that the reservoir state is occupied. At the same time, we also require f_C(E) < 1 such that a free state is available in the cavity. These conditions define an energy window of the order k_BT which explains why the quantum-well heat engine has an efficiency comparable to that of a quantum-dot heat engine with level width k_BT.

We now turn to the discussion of an asymmetric system, a ≠ 0. In this case, both the output power and the efficiency are no longer invariant under an exchange of the two level positions. Instead, we now find that power and efficiency are strongly reduced for E_L < 0 and E_R > 0 if a > 0 (for a < 0, the roles of E_L and E_R are interchanged). In contrast, for E_L > 0 and E_R < 0, power and efficiency are even slightly enhanced compared to the symmetric system. This naturally leads to the question of which combination of level positions and coupling asymmetry yields the largest output power. To this end, in figure 3 we plot the power as a function of the asymmetry a and the level position E_L. We find that the maximal power occurs for a ≈ 0.46 and E_L ≈ 2k_BT while −E_R ≫ k_BT. The resulting power is about 20% larger than for the symmetric setup. At the same time, the efficiency at maximum power is also increased compared to the symmetric system to η ≈ 0.12η_C, i.e. it is nearly doubled. We remark that the maximal efficiency that can be obtained for the asymmetric system is given by η ≈ 0.3η_C. However, similar to the symmetric setup, this occurs in a regime where the output power is highly suppressed.

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We now estimate the output power for realistic device parameters. Using m_eff = 0.067m_e, T = 300 K, Γ = k_BT and a = 0.5, we obtain P_max = 0.18 W cm⁻² for a temperature difference ΔT = 1 K. Hence, the quantum-well heat engine is nearly twice as powerful as a heat engine based on resonant-tunneling quantum dots [21]. We remark that materials with higher effective mass yield correspondingly larger output powers. In addition, the quantum-well heat engine offers the advantages of being potentially easier to fabricate. As typical level splittings in quantum wells are in the range of 200–500 meV [46, 47], narrow quantum wells might also be promising candidates for room-temperature applications though leakage phonon heat currents become of relevance then. Finally, we remark on the robustness with respect to fluctuations in the device properties. For the optimal configuration discussed above, fluctuations of E_R do not have any effect as long as −E_R ≫ k_BT. Fluctuations of E_L by as much as k_BT reduce the output power by about 20% as can be seen in figure 3. Hence, our device turns out to be rather robust with respect to fluctuations, similarly to the quantum-dot based setup in [21].

3.2. Nonlinear regime

We now turn to the performance of the heat engine in the nonlinear regime. Nonlinear thermoelectrics has recently received increasing interest [48–50]. We numerically optimized the bias voltage V , the asymmetry of couplings a as well as the level positions E_L,R in order to maximize the output power. The resulting optimized parameters are shown in figure 4 as a function of the temperature difference ΔT. While the optimal asymmetry a ≈ −0.46 is independent of ΔT, the optimal bias voltage grows linearly in ΔT. The right level position E_R decreases only slightly upon increasing ΔT. The left level position should be chosen as −E_L ≫ k_BT, independent of ΔT. It is not shown in figure 4 as our numerical optimization procedure results in large negative values for E_L that vary randomly from data point to data point because the dependence of the power on E_L is only very weak in this parameter regime.

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The resulting maximal power grows quadratically in the temperature difference, cf figure 4. It is approximately given by $P_{\mathsf {max}}=0.3 \frac {\nu _2\mathcal A\Gamma }{2\hbar }(k_{\mathsf {B}}\Delta T)^2$, independent of T. Interestingly, for a given value of ΔT, we obtain the same output power both in the linear and in the nonlinear regime. However, as the efficiency at maximum power grows linearly with the temperature difference, it is preferrable to operate the device as much in the nonlinear regime as possible. In the extreme limit ΔT/T = 2, the quantum-well heat engine reaches η_maxP = 0.22η_C, i.e. it is as efficient as a heat engine based on resonant-tunneling quantum dots while delivering more power [21]. We remark that the efficiency at maximum power is below the upper bound η_C/(2 − η_C) given in [51].

We investigated a heat engine based on two resonant quantum wells coupled to a hot cavity. In the linear-response regime we found that our device can yield a power that is nearly twice as large as that of a similar heat engine based on resonant tunneling through quantum dots. At the same time, the efficiency of the quantum-well heat engine is only slightly lower than that of the quantum-dot heat engine. In addition, a device based on quantum wells offers the advantage of being easy to fabricate and the perspective of room-temperature operation. Finally, we also analyzed the performance in the nonlinear regime. There, we found that for a given temperature difference the system yields the same output power as in the linear regime but with an increased efficiency.

In this paper, we focused on the discussion of a setup with noninteracting quantum wells. An interesting question for future research is: how does the inclusion of charging effects in the wells relevant in particular in the nonlinear regime affect the performance of quantum-well heat engines?

We acknowledge financial support from the European STREP project Nanopower, the Spanish MICINN Juan de la Cierva program and MAT2011-24331 and ITN grant no. 234970 (EU). This work was supported by the US NSF grant no. DMR-0844899.

In this appendix, we give the explicit expression for the function g₃(x,y) that enters the expression (10) for the heat current. It is given by

$$\begin{eqnarray} &&\fl g_3(x,y)=\frac{2\pi^2}{3}-\frac{1}{2}(x-y)g_1(x,y)\left[x-y-2g_2(x,y)\right] -2(1+a){\rm Li}_2\left(\frac{1}{1+{\rm e}^{-x}}\right)\nonumber\\ &&-2(1-a){\rm Li}_2\left(\frac{1}{1+{\rm e}^{-y}}\right) -2(1+a)\log(1+{\rm e}^{x})\log(1+{\rm e}^{-x})\nonumber\\ &&-2(1-a)\log(1+{\rm e}^{y})\log(1+{\rm e}^{-y}) -g_1(x,y)(1+{\rm e}^x)(1+{\rm e}^y)\log(1+{\rm e}^{-x})\nonumber\\ &&\times \log(1+{\rm e}^{-y}) -g_1(x,y)\log^2(1+{\rm e}^{-x})\left[{\rm e}^x\sinh x+\frac{1+a}{1-a}{\rm e}^x(1+{\rm e}^y)\right]\nonumber\\ &&-g_1(x,y)\log^2(1+{\rm e}^{-y})\left[{\rm e}^y\sinh y+\frac{1-a}{1+a}{\rm e}^y(1+{\rm e}^x)\right], \end{eqnarray} \tag{ A.1 }$$

and satisfies the bounds 0 < g₃(x,y) < 2π²/3.