Staircase Quantum Dots Configuration in Nanowires for Optimized Thermoelectric Power

The performance of thermoelectric energy harvesters can be improved by nanostructures that exploit inelastic transport processes. One prototype is the three-terminal hopping thermoelectric device where electron hopping between quantum-dots are driven by hot phonons. Such three-terminal hopping thermoelectric devices have potential in achieving high efficiency or power via inelastic transport and without relying on heavy-elements 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 quantum-dots embedded in parallel nanowires. We find that the staircase energy configuration with constant energy-step can improve the power factor over a serial connection of a single pair of quantum-dots. Moreover, for a fixed energy-step, there is an optimal length for the nanowire. Similarly for a fixed number of quantum-dots there is an optimal energy-step for the output power. Our results are important for future developments of high-performance nanostructured thermoelectric devices.

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 y-z directions there are many parallel nanowires, see Fig. 2). Specifically, we focus on two configurations: (1) in each nano-engine there is only a single pair of QDs; (2) in each nano-engine 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 energy-step 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.  Three-Terminal Thermoelectric Transport for Nanowire Quantum-Dots The three-terminal thermoelectric energy harvesting device reported here is composed of two electrodes on the left and right sides under room-temperature 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 room-temperature 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, electron-optical-phonon scattering time can be as short as 0.1 ps, leading to collision broadening as large as 10 meV 24 . These features make optical-phonon-assisted 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 y-z 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 spin-degeneracy, α ep = 10 meV characterizes the strength of electron-phonon 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 i j i j j i with e being the charge of a single electron. The linear conductance of electric conduction between QDs i and j is given by where Γ ij 0 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 Miller-Abrahams network 25 ). Such method is generalized to three-terminal 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 Miller-Abrahams 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 Δ E = (N − 1)dE. The first QD has energy E 1 = − Δ E/2, while the last QD has energy E N = Δ E/2. Here the energy we referred to is the energy of the lowest two degenerate electronic levels (i.e., spin-up and spin-down) 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 nano-thermoelectric engine. The energy configuration is chosen to have particle-hole 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 non-interacting 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 III-V, II-VI, VI semiconductors. Considering the electron-optical-phonon 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 acoustic-phonon scattering near the Debye frequency is also very efficient. In our calculation, modifying dE may need to be fulfilled by changing the materials for nanowire-QDs. 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 nanowire-QD 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 linear-response regime can be described by the coupled electric and heat conduction equation where E R E ( ) L 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 three-terminal and conventional thermoelectric effects. The Seebeck coefficient for the phonon-driven three-terminal thermoelectric effect is then B in in el B 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 N p 0 . 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 sub-exponential, 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 Δ E = (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 three-terminal thermoelectric energy harvester. In reality, it is important to find the energy dE that optimize the phonon-assisted 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 nano-scale element contains a single pair of QDs. Our design of staircase energy configuration can have much better output power than the sigle-pair 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