Abstract
The performance of thermoelectric energy harvesters can be improved by nanostructures that exploit inelastic transport processes. One prototype is the threeterminal hopping thermoelectric device where electron hopping between quantumdots are driven by hot phonons. Such threeterminal hopping thermoelectric devices have potential in achieving high efficiency or power via inelastic transport and without relying on heavyelements or toxic compounds. We show in this work how output power of the device can be optimized via tuning the number and energy configuration of the quantumdots embedded in parallel nanowires. We find that the staircase energy configuration with constant energystep can improve the power factor over a serial connection of a single pair of quantumdots. Moreover, for a fixed energystep, there is an optimal length for the nanowire. Similarly for a fixed number of quantumdots there is an optimal energystep for the output power. Our results are important for future developments of highperformance nanostructured thermoelectric devices.
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Introduction
Thermoelectric energy harvesting has the reverse effect as opposed to the thermoelectric refrigerator^{1,2}, and has been studied extensively in recent decades^{3,4,5,6}. Right from the invention of the Seebeck and Peltier effects up to now, people have been using doped semiconductor materials with the aim of increasing the electrical conductivity and reducing thermal conductivity for a higher figure of merit. Figure 1a,b show conventional twoterminal Seebeck thermoelectric energy harvester in its normal and unfolded geometries, respectively. The configuration in Fig. 1b is very similar to pn junction for solar cells. However, the metallic contacts in the middle remove the junction barrier and enables elastic thermoelectric transport. Although Fig. 1b reveals the similarity and difference between a thermoelectric engine and a solar cell, the mechanism that accounts for the significant difference of the two devices in efficiency (i.e., solar cells have much higher efficiency compared with thermoelectric engines) has not been uncovered. It was found only recently that a pn junction thermoelectric engine based on hotphononassisted interband transition can have considerably augmented thermoelectric efficiency and output power compared to conventional twoterminal thermoelectric devices with the same material. The intrinsic mechanism that distinguishes thermoelectric engines and solar cells in their efficiency and output power is then revealed as akin to inelastic transport processes in a threeterminal geometry^{7,8} (Fig. 1c).
Other prototypes of threeterminal inelastic thermoelectric devices include Coulomb coupled quantumdots (QDs)^{9,10}, phononassisted hopping in QD chains (or localized states in 1D or 2D systems)^{11,12,13,14,15}, and inelastic thermoelectric transport across an electronic cavity promoted by mismatched resonant tunneling at the twosides of the cavity^{16,17,18,19}. In those devices thermal energy from the third, insulating terminal of phonon or electronic bath is converted to electrical energy between the source and the drain (and vice versa). It was found that to optimize the performance (efficiency and output power) the energy of the two QDs has to be above and below the chemical potential around 3k_{B}T, respectively. Experimental developments on threeterminal inelastic thermoelectric devices in mesoscopic systems at lowtemperature were established recently^{20,21,22}, attracting more and more researches in the field^{8}.
In this work we focus on phononassisted hopping thermoelectric transport in a threeterminal thermoelectric energy harvester. Our main concern is to optimize the output power of such a device by tuning the energy configuration of a chain of QDs embedded in a nanowire. Such a scheme can be used to form a macroscopic thermoelectric device, since many parallel nanowires can be assembled together and the nanoscale thermoelectric engines can be connected in series. We compare the power factor (density) P = σS^{2} for these configurations consisting of many serially connected nano thermoelectric engines along the x direction (while in yz directions there are many parallel nanowires, see Fig. 2). Specifically, we focus on two configurations: (1) in each nanoengine there is only a single pair of QDs; (2) in each nanoengine there are a chain of N QDs with staircase configuration of energy (see Fig. 1d). We emphasize that the density of QDs along x direction is the same for all these situations and the only difference here is the energy configuration. We show that the power factor is largest for staircase energy configuration (the main focus of this paper). Particularly we study the dependence of power factor on the number N and the energystep dE for each hopping. We find that for a given dE there is an optimal number N that maximizes the power factor. Our findings reveal important information for future design of inelastic thermoelectric devices.
ThreeTerminal Thermoelectric Transport for Nanowire QuantumDots
The threeterminal thermoelectric energy harvesting device reported here is composed of two electrodes on the left and right sides under roomtemperature environment, and a central region comprising QDs heated by the external phonon bath (Fig. 1d). By absorbing the phonon energy electrons hop from one QD to another which leads to electrical current against the voltage gradient. These are the processes that convert thermal energy from phonon bath to electrical energy^{7}. For realistic devices operating at roomtemperature and above, electron hopping is efficiently assisted by scattering with optical phonons. Optical phonon scattering transfer a considerable amount of energy, ranging from 10 meV to 120 meV for various materials^{23}. In addition, electronopticalphonon scattering time can be as short as 0.1 ps, leading to collision broadening as large as 10 meV^{24}. These features make opticalphononassisted hopping thermoelectric transport as promising mechanism for powerful and efficient thermoelectric energy conversion.
We shall consider many serially connected nano thermoelectric engines along the x direction, while in yz directions there are many parallel nanowires, as shown in Fig. 2. Specifically, each QD is of length l_{qd} = 6 nm along x direction. The distance between adjacent QDs is d = 6 nm. The probability of finding an electron outside the QD decays exponentially with the distance away from it with a characteristic length ξ = 2 nm. A simplified treatment based on Fermi golden rule yields the following hopping transition rate from QD i to QD j,
where the factor of two comes from spindegeneracy, α_{ep} = 10 meV characterizes the strength of electronphonon scattering, f_{i} and f_{j} are the probability of finding electron on QDs i and j, respectively. The x coordinates of the two QDs are x_{i} and x_{j}, respectively, while their energies are E_{i} and E_{j}, respectively. Here the phonon distribution function is given by
In our thermoelectric energy harvester, the phonon bath has temperature higher than the electrodes, i.e., T_{p} > T. The heat from the phonon bath is then converted into electricity. The electron distribution in each QD can be described by a Fermi distribution
From the above, the electric current flowing from QD i to QD j is given by
with e being the charge of a single electron. The linear conductance of electric conduction between QDs i and j is given by
where is the transition rate at equilibrium and the superscripts 0 denote equilibrium distributions. Hence each pair of QDs form a resistor with conductance G_{ij}. Hopping conduction is mapped to conduction in network of resistance (i.e., the MillerAbrahams network^{25}). Such method is generalized to threeterminal hopping conduction in refs 11,12. Thermoelectric transport through the system is calculated by solving the Kirchhoff current equation, i.e., the total current flowing into QD i is equal to the total current flowing out of QD j for the MillerAbrahams network^{12}. In this fashion the electrochemical potentials at each QD. i.e., μ_{I}’s, are determined numerically via the method presented in ref. 12.
We shall consider a chain of N QDs with staircase energy configuration. Each energy step is dE = E_{i+1} − E_{i} (we focus on the situation with dE > 0). The total energy difference is ΔE = (N − 1)dE. The first QD has energy E_{1} = −ΔE/2, while the last QD has energy E_{N} = ΔE/2. Here the energy we referred to is the energy of the lowest two degenerate electronic levels (i.e., spinup and spindown) of the QD. Higher levels in the QDs are ignored due to their much higher energies as we consider small QDs here. N = 2 is the case with a single pair of QDs in a nanothermoelectric engine. The energy configuration is chosen to have particlehole symmetry, which has been proven to be best for thermoelectric performance as shown in refs 16,18.
It is noted that differing from variable range hopping between randomly localized states in nanowires or higher dimensional systems, the staircase energy configuration always favours the nearest neighbour hopping. This is because hopping to farther neighbour QDs costs larger energy gap and longer distance simultaneously. In contrast, in variable range hopping, the nearby neighbours may have larger energy differences compared to farther localized states. Optimization of the hopping distance in 1D localized system leads to the Mott’s law^{26} in noninteracting electron systems (with slight modifications^{27}).
It is necessary to mention that the Fermi golden rule requests that the energy difference dE between the two electronic states must be the same as the optical phonon energy (i.e., microscopic energy conservation). We are interested only in the range with dE ∈ (10, 120) meV, which can be realized in IIIV, IIVI, VI semiconductors. Considering the electronopticalphonon scattering rate for most of those semiconductors are around 0.1 ps. Our model calculation hence captures the main physics of the system. We mention that acousticphonon scattering near the Debye frequency is also very efficient. In our calculation, modifying dE may need to be fulfilled by changing the materials for nanowireQDs. Nevertheless, our study reveals for a given dE (i.e., a given material) the number of QDs in a single nanowire that optimizes the power factor, as well as how such an optimal number varies with dE. These information are useful for future material design of nanowireQD thermoelectric devices.
Beside the inelastic hopping conduction, there is also elastic transport through the system. The elastic transport defines pure electron quantum tunneling mechanism between QDs and electrodes. A resonant tunneling mechanism is exploited to describe such conduction process. Note that since we consider QDs with considerably large energy differences (much larger than coupling between quantum dots) sequential tunneling between QDs is suppressed. The dominant contribution comes from single QD resonant tunneling^{12}, where each QD forms one of such resonant tunneling conduction channel independently. Hence the elastic conduction contributes to the electric current via
Here V is the voltage across the source and drain electrodes, h is the Planck constant, G_{el} denotes the elastic conductance, f^{0}(ε) = 1/[exp(ε/(k_{B}T)) + 1] (we set the electrochemical potential at equilibrium as energy zero). The tunnel coupling between the QD i and the left (right) electrode is γ_{Li} (γ_{Ri}). We shall set the coordinate of the left electrode as x = 0, while the right electrode has x = L_{tot} with L_{tot} = Nl_{qd} + (N − 1)d + 2l_{b} where l_{b} is the distance between the first (last) QD and the left (right) electrode. The tunnel coupling is hence γ_{Li} = t_{0} exp(−x_{i}/ξ) and γ_{Ri} = t_{0} exp(−(L_{tot} − x_{i} − l_{qd})/ξ) with t_{0} = 100 meV that characterizes hybridization energy of closely coupled QDs. We emphasize that the elastic current I_{elas} does not vary with the temperature of the phonon bath T_{p} since it originates purely from the quantum tunneling instead of coupling with phonons. In fact, in our thermoelectric engine, elastic conduction dissipates the electric energy generated by inelastic hopping into Joule heating.
Thermoelectric transport in our system the linearresponse regime can be described by the coupled electric and heat conduction equation
where G = G_{in} + G_{el} with G_{in} being the inelastic conductance. It was shown in ref. 12 that where is the average energy of electrons entering into the right (left) electrode. For instance, hopping thermoelectric transport in a single pair of QDs gives L = G_{in} (E_{2} − E_{1})/e. Hopping for a chain of N QDs with staircase energy configuration yields L = G_{in} (E_{N} − E_{1})/e. We emphasize that elastic tunneling does not contribute to the Seebeck effect here, which is the essential difference between threeterminal and conventional thermoelectric effects. The Seebeck coefficient for the phonondriven threeterminal thermoelectric effect is then
For our system to work as an thermoelectric energy harvester, the inelastic conduction should dominate over the elastic conduction. The inelastic, elastic and total conductivity are plotted in Fig. 3(a) for a nano thermoelectric harvester with a single pair of QDs as functions of energy step dE for dE ∈ (10, 120) meV. Both the inelastic and elastic conductivity decreases with increasing dE. The elastic conductivity is reduced as the first QD has lower energy below the electrochemical potential while the second QD is higher above the electrochemical potential, leading to less effective conduction. The inelastic conductivity is also reduced due to the larger thermal activation energy dE and exponentially decreased phonon number . Therefore at very large energy difference dE the elastic conductivity may be more important. In reality, the elastic conduction also dominates in the small dE regime, which we ignored in this study. For a chain of QDs with N = 10, the results in Fig. 3(b) shows that the elastic conduction is much reduced, since tunneling over a longer distance is exponentially suppressed. The conductivity is then dominated by inelastic hopping in long chains of QDs.
Next we examine the conductivity as a function of the length of the chain. We show the results in Fig. 3(c,d) for dE = 10 meV and 30 meV, respectively. As the number of QDs increases both the inelastic hopping conductivity and elastic tunneling conductivity decreases. However, the elastic conductivity decreases much rapidly. The initial decay of hopping conductivity is subexponential, since increase the number of hopping is similar to increase the number of resistors. However, for large N, as the energy of the first (last) few QDs is much lower (higher) than the electrochemical potential, the hopping rates are suppressed by the exponentially small availability (occupation) of the final (initial) state. The decrease of conductivity at large N is hence exponential. Such exponential decrease become stronger for larger dE = 30 meV as shown in Fig. 3(d).
Power Factor for Different Energy Configurations
We then study the power factor P = σS^{2} for various energy step dE and length of the QDs chain N. We remark again that for all situations the density of QDs is the same, according to our geometry of the nanowire QDs. The conductivity is calculated via σ = Gl/A where l and A are the length and area of a single nano thermoelectric engine. Here the area is determined by the density of nanowires as A^{−1} = 10^{15} m^{−2} ^{28,29}. By focusing on the scale independent conductivity σ and power factor σS^{2} we are able to discuss ways of optimizing the power factor by engineering each nano thermoelectric element. In this way, the variation of the power factor σS^{2} is a sole consequence of the energy configuration (rather than geometry) in each nano thermoelectric engine.
In Fig. 4(a) we show the dependences of power factor σS^{2} and the Seebeck coefficient S on the energy step dE for a nano device with a single pair of QDs. It is found that the Seebeck coefficient S increases monotonically with the energy step dE, which is consistent with Eq. (10). As a consequence of competition between the conductivity and the Seebeck coefficient, the power factor is optimized around dE = 3k_{B}T. For a chain of N = 10 QDs, the power factor is maximized at a much lower dE, as shown in Fig. 4(b). This is due to the more rapid decay of the conductivity as shown in Fig. 3(b).
Similarly, the dependence of the number of QDs N for a given energy difference dE also has a peak, as shown in Fig. 4(c,d). In Fig. 4(c) we plot the power factor σS^{2} and the conductivity σ as functions of the number of QDs N in a single nano device for dE = 10 meV. The power factor is maximized at N = 21. This maximum also appears as a consequence of the competition of the conductivity and the Seebeck coefficient when the number N is increased. The Seebeck coefficient increases as the total energy difference ΔE = (N − 1)dE increases, while the conductivity decays exponentially with the number of QDs for large N. Since such exponential decay of conductivity is more severe for larger energy step dE, the maximum appears at a smaller number of QDs N for dE = 30 meV, as shown in Fig. 4(d).
To have a global view of the dependence of the power factor on the energy configuration of QDs, we plot the σS^{2} for various N and dE in Fig. 5. It is seen that for each dE there is an optimized N at which the power factor is maximized. For smaller dE the optimal N is larger. More importantly, the maximal power factor is greater. Our study thus reveal the optimal energy configurations for powerful threeterminal thermoelectric energy harvester. In reality, it is important to find the energy dE that optimize the phononassisted hopping rate and the conductivity. Fixing such a dE one can find an optimal number of QDs N that form the maximal output power for a given material.
Conclusion
We study the optimization of energy configurations of thermoelectric energy harvester assembled by many nano thermoelectric elements. Each nano device contains N QDs of staircase energy configuration with energy step dE. It is found that such energy configuration is better than the situation studied before: each nano thermoelectric element contains only N = 2 QDs. More importantly, we find that for each given energy step dE there is an optimal number of QDs N that maximizes the power factor. Such optimization yields higher output power when dE is smaller. Finally, we argue that our design is also better in thermoelectric power factor than hopping in a chain of QDs with random energy configuration. This is because the conductivity of such a random energy QDs chain is lower than that of the nano device with a single pair of QDs (when its parameters are optimized). On the other hand, the Seebeck coefficient is fluctuating around zero^{12}, yielding relatively low Seebeck coefficient. Therefore, the power factor can be relatively lower as compared to the situation of assembled thermoelectric energy harvester with each nanoscale element contains a single pair of QDs. Our design of staircase energy configuration can have much better output power than the siglepair of QDs nano device. Therefore, our study is valuable for future design of nanostructured thermoelectric devices. Future study should also include the effect of parasitic heat conduction due to, e.g., phonon thermal conductivity, which may reduce the figure of merit, although it is usually much smaller in nanowires than in bulk materials.
Methods
The power factor is calculated by computing the conductivity σ and Seebeck coefficient S. From Eq. (10), the essential quantities of interest are the conductivity for both inelastic and elastic transport processes. The conductivity for elastic processes are calculated via Eqs (7) and (8). The hopping conductivity is calculated via solving the Kirchhoff current equation for MillerAbrahams resistor network numerically. The key quantities to be calculated are the “local voltage” V_{i} = μ_{i}/e for all i labeling the QDs. According to ref. 12 the equations to be solved are
where
The left electrode has voltage V_{L} = V/2, while the right electrode has voltage V_{R} = −V/2. The conductance G_{iL} (G_{iR}) is finite only when i labels the first (last) QD, and . The hopping conductance between QDs are given by Eqs (5) and (6). By setting V = 0.01 and solving the above equation, we obtain V_{i} for all i. The electrical current flowing through the system due to hopping is then calculated via using the numerically obtained V_{1}. The inelastic conductance of the nanowireQDs system is then .
Additional Information
How to cite this article: Li, L. and Jiang, J.H. Staircase Quantum Dots Configuration in Nanowires for Optimized Thermoelectric Power. Sci. Rep. 6, 31974; doi: 10.1038/srep31974 (2016).
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Acknowledgements
LL appreciates the support of college of engineering, Swansea University. This work was supported by the UK EPSRC (EP/H004742/1). JJ would like to thank the faculty startup funding of Soochow University for support. All authors are grateful for the encouragement from Prof. Yoseph Imry at Weizmann Institute of Science, Israel.
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Li, L., Jiang, JH. Staircase Quantum Dots Configuration in Nanowires for Optimized Thermoelectric Power. Sci Rep 6, 31974 (2016). https://doi.org/10.1038/srep31974
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DOI: https://doi.org/10.1038/srep31974
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