Incoherent scattering can favorably influence energy filtering in nanostructured thermoelectrics

www.nature.com/scientificreports OPEN Received: 27 April 2017 Accepted: 5 July 2017 Published: xx xx xxxx Incoherent scattering can favorably influence energy filtering in nanostructured thermoelectrics Aniket Singha & Bhaskaran Muralidharan Investigating in detail the physics of energy filtering through a single planar energy barrier in nanostructured thermoelectric generators, we reinforce the non-trivial result that the anticipated enhancement in generated power at a given efficiency via energy filtering is a characteristic of systems dominated by incoherent scattering and is absent in ballistic devices. In such cases, assuming an energy dependent relaxation time τ(E) = kEr, we show that there exists a minimum value rmin beyond which generation can be enhanced by embedding nanobarriers. For bulk generators with embedded nanobarriers, we delve into the details of inter sub-band scattering and show that it has finite contribution to the enhancement in generation. We subsequently discuss the realistic aspects, such as the effect of smooth transmission cut-off and show that for r > rmin, the optimized energy barrier is just sufficiently wide enough to scatter off low energy electrons, a very wide barrier being detrimental to the performance. Analysis of the obtained results should provide general design guidelines for enhancement in thermoelectric generation via energy filtering. Our non-equilibrium approach is typically valid in the absence of local quasi-equilibrium and hence sets the stage for future advancements in thermoelectric device analysis, for example, Peltier cooling near a barrier interface. An important direction in the context of electronic engineering to enhance the performance of nanostructured thermoelectric generators1–10, is to utilize the physics of electronic energy filtering through nanoscale barriers and nanoinclusions11–24. To put it simply, energy filtering aims to provide a unidirectional flow of electrons from the hot contact to the cold contact while prohibiting the reverse flow of electrons, which occurs typically when lower energy electrons are scattered off due to the interface potentials9, 20–22, 25. In the case of semiconductors, there is the flexibility of varying the equilibrium Fermi energy via appropriate doping. In such a case, a ‘good thermoelectric’ as schematized in Fig. 1(a) is ideally achieved by tuning the Fermi energy near the conduction band edge so that the resulting transport is devoid of electrons below the Fermi energy. This off-resonant conduction typifies good thermoelectric behavior and has been the object of several initial proposals1–3, 5, 8. Such an approach however leads to a drastic reduction in the conductivity while enhancing the Seebeck coefficient. We will refer this as Approach A. Energy filtering, as schematized in Fig. 1(b), on the other hand, strives to achieve a desirable performance via engineering nano-barriers12, 15, 16, 18–23, 26–29. In this case, the Fermi level resides inside the conduction band and the principal aim is to enhance the Seebeck coefficient by reflecting low energy electrons from the energy barrier. This approach also promotes a unidirectional flow of electrons from the hot contact to the cold contact. We will refer to this as Approach B. Ideally, we can say that energy filtering is successful if the latter yields a better performance than the former, especially with a thinner barrier. While recent works throw some light on this topic18–24, 26–29, we believe that a few aspects about energy filtering require attention: (1) Most of the current work is based on a linear response analysis of the Seebeck coefficient despite the fact that the regions in the vicinity of the barrier are strongly out of equilibrium. Linear response analysis typically masks the crucial transport physics that determines the delivered terminal power output and efficiency of the generator30–36. (2) The role of various scattering mechanisms contributing to the physics of energy filtering is still unclear. (3) A generalized picture of the physics of energy filtering independent of various material parameters is also unclear. In this paper, our focus is hence to develop a general and intuitive understanding of energy filtering and to systematically point out the role played by various scattering mechanisms. Besides that our work uses the non-equilibrium Green’s function method which accounts for the non-equilibrium nature of transport and directly evaluates the power and efficiency to provide an overall picture of the device operation. Department of Electrical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai, 400076, India. Correspondence and requests for materials should be addressed to A.S. (email: ) Scientific ReportS | 7: 7879 | DOI:10.1038/s41598-017-07935-w 1 www.nature.com/scientificreports/ Figure 1. (a,b) Schematics depicting the two common approaches of improving the thermoelectric performance. (a) Approach A: Enhancing the thermoelectric generation by tuning the position of Fermi energy near the conduction band edge and (b) Approach B: enhancing the thermoelectric performance by energy filtering via embedding a nano barrier within the thermoelectric generator. The hot and cold contacts are assumed to be macroscopic bodies in equilibrium with the quasi Fermi energy μ at temperature TH and TC respectively. (c,d) The device used for simulation, with a device region of length 20 nm comprising (c) embedded nanowire thermoelectric generator (d) bulk thermoelectric generator. The shaded region in both the cases represents the embedded barrier (in case of Approach B). (e) The band profile of the device region of length 20 nm, embedded with a Gaussian energy barrier of height Eb = 150 meV and width σw = 2.7 nm. The brown dotted line shows the Fermi energy of the device for the case Eb − μ0 = 2kBT. (f) Schematic of the voltage controlled model used to simulate the power-efficiency trade-off points. Our goal here is to present important clarifications on the aforesaid aspects. As a principal contribution, we clarify that the power generation enhancement via energy filtering in both nanowire and bulk devices is dependent on a specific property of the scattering mechanism, which we call the order of scattering. It is first shown that for the ballistic case, both Approach A and Approach B lead to an identical performance and hence energy filtering is of limited use. On the other hand, in the diffusive limit for both nanowire and bulk thermoelectric generators, it is shown that the type of scattering mechanism is the principal deciding factor in order to gauge the advantage gained via Approach B. Assuming an energy dependent relaxation time τ(E) = kEr, we then show that there is a minimum value of the exponent r (termed rmin), beyond which energy filtering via Approach B leads to a better enhancement in the generated power compared to Approach A. For such cases, the generated power at a given (...truncated)