Nanostructured potential well/barrier engineering for realizing unprecedentedly large thermoelectric power factors

This work describes through semiclassical Boltzmann transport theory and simulation a novel nanostructured material design that can lead to unprecedentedly high thermoelectric power factors, with improvements of more than an order of magnitude compared to optimal bulk material power factors. The design is based on a specific grain/grain-boundary (potential well/barrier) engineering such that: i) carrier energy filtering is achieved using potential barriers, combined with ii) higher than usual doping operating conditions such that high carrier velocities and mean-free-paths are utilized, iii) minimal carrier energy relaxation after passing over the barriers to propagate the high Seebeck coefficient of the barriers into the potential wells, and importantly, iv) the formation of an intermediate dopant-free (depleted) region. The design consists thus of a three-region geometry, in which the high doping resides in the center/core of the potential well, with a dopant-depleted region separating the doped region from the potential barriers. It is shown that the filtering barriers are optimal when they mitigate the reduction in conductivity they introduce, and this can be done primarily when they are clean from dopants during the process of filtering. The potential wells, on the other hand, are optimal when they mitigate the reduced Seebeck they introduce by: i) not allowing carrier energy relaxation, and importantly ii) by mitigating the reduction in mobility that the high concentration of dopant impurities cause. Dopant segregation, with clean dopant-depletion regions around the potential barriers, serves this key purpose of improved mobility towards the phonon-limited mobility levels in the wells. Using quantum transport simulations as well as semi-classical Monte Carlo simulations we also verify the important ingredients and validate this clean-filtering design.


I. Introduction
Thermoelectric (TE) materials have made dramatic progress over the last several years. The thermoelectric figure of merit ZT, which quantifies the ability of a material to convert heat into electricity, has more than doubled compared to traditional values of ZT~1, reaching values above ZT~2 in several instances across materials and temperature ranges [1,2,3,4,5,6,7,8,9,10,11,12]. The figure of merit is determined by ZT=σS 2 T/(κe+κl), where σ is the electrical conductivity, S is the Seebeck coefficient, T is the absolute temperature, and κe and κl are the electronic and lattice parts of the thermal conductivity, respectively. The recent improvements in ZT are mostly attributed to drastic reductions of the lattice thermal conductivity in nanostructured materials and nanocomposites which has reached amorphous limit values at κl = 1-2 W/mK and below [1,3,13,14], as well as complex phonon dynamics materials [4]. With such low thermal conductivities, however, any further benefits to ZT must be achieved through the improvement of the thermoelectric power factor (PF) σS 2 , for which no noticeable progress has so far been achieved.
The lack of progress in the power factor improvement is attributed to the adverse interdependence of the electrical conductivity and Seebeck coefficient via the carrier density, which proves very difficult to overcome. The most commonly explored direction in improving the power factor is the 'conventional' energy filtering approach in nanocomposites and superlattices, in which built-in potential barriers block the cold low energy carriers, while allow the hot high energy carriers to flow [15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]. This energetic preference increases the Seebeck coefficient. On the other hand, this approach has still not been widely applied because the conductivity is also reduced in the presence of potential barriers. Current research efforts in improving the power factor have thus diverted into many other directions, including: i) taking advantage of the density of states in low-dimensional materials through quantum confinement [31], or in bulk materials that include low-dimensional 'like' features [32,33], ii) bandstructure engineering and band-convergence strategies [32,33,34,35,36], iii) modulation doping and gating [37,38,39,40,41,42,43,44,45], iv) introducing resonances in the density of states [46], and even more recently v) concepts that take advantage of the Soret effect in hybrid porous/electrolyte materials [47]. These approaches target improvements either in the Seebeck coefficient or the electrical conductivity, with the hope that the other quantity will not be degraded significantly, and sometimes they report moderate power factor improvements. However, no significant developments that lead to meaningful improvements, wider applicability, or generalization to many materials have been achieved by these methods either.
Recent efforts by us and others, however, both theoretical and experimental, have revisited the energy filtering concept, and targeted designs that provide simultaneous improvements in both the electrical conductivity and the Seebeck coefficient in order to largely improve the power factor σS 2 . Experimental works have indeed verified that it is possible to achieve very high power factors in nanostructured Si-based materials after undergoing specific treatment [48,49,50,51,52]. Measured data for PF improvements of over 5× compared to bulk values, were adequately explained using Boltzmann transport theory [48,49]. In those cases the grain boundaries of the nanocrystalline material serve as potential barriers resulting in energy filtering and high Seebeck coefficients. On the other hand, the carrier mean-free-path for scattering (and mobility) in the grains was improved due to the fact that dopants were primarily placed in the middle regions of the grains, rather than uniformly spread in the materialsee later on Fig. 3a.
The regions around the grain boundary potential barriers were depleted of dopants and allowed higher local carrier mobility and higher mean-free-paths, which resulted in higher grain conductivities and overall material conductivities [48]. Thus, compared to other approaches, recent evidence suggests that energy filtering could be engineered in such a way as to lead to large PF improvements.
In light of the strong evidence of such exceptionally high power factors demonstrated, as well as the recent surge in efforts to use energy filtering and design the grain/grain-boundary system efficiently in a variety of materials [28,29,30,53], in this work, we re-examine theoretically energy filtering under degenerate conditions and dopant segregation. Using Boltzmann transport theory, we provide an in-depth investigation of a generalized grain/grain-boundary design concept, and explore the ultimate upper limit that can potentially be achieved under realistic conditions. The design is based on a number of concepts/'ingredients', whose contributions are examined individually, and then all combined until the final upper limit is reached. To further examine the validity of those 'ingredients' we also employ more sophisticated quantum transport simulations and Monte Carlo simulations. We show that power factors even up to the exceptionally high value of PF ~ 50 mW/mK 2 can be achieved once a nanostructure is properly designed.
The paper is organized as follows: Section II describes the main features of the proposed design, based on what we refer to as '3-region, clean-filtering'. Section III describes and validates the semiclassical transport model against p-type Si (the semiconductor material for which the largest improvements were experimentally achieved, but without loss in generality for the material choice). We then explore the power factor improvements that the individual design 'ingredients' lead to as they are added to the design one by one. Section IV increases the values of the design ingredients to very high, but still realistic levels, to provide an upper limit estimate for the power factor. Section V then discusses the validity of the assumptions/'ingredients' using the Non-Equilibrium Green's Function (NEGF) quantum transport method and Monte Carlo simulations. Finally, Section VI concludes the work.

II. Approach
To model TE transport in the structure we examine, we start with the Boltzmann transport equation (BTE) formalism, and calibrate our model to match the mobility of ptype bulk Si so that we remain within realistic exploration boundaries. Within the BTE, the conductivity and Seebeck coefficient can be extracted as [54,55]: case (blue line). By using 0 7.4 nm   for phonons, the desired bulk low-density mobility for p-type Si at 300 K is achieved (μ = 450 cm 2 /V-s). Our results agree particularly well at the carrier concentrations of interest, around p = 5×10 19 cm -3 . The difference between the phonon-limited mobility and the phonon plus ionized impurity scattering mobility, which can be up to an order of magnitude, is a central aspect of our design as will be explained below.
With regards to thermoelectric performance, Fig. 1b, 1c, and 1d show comparison of the phonon-limited to the phonon plus IIS conductivity, Seebeck coefficient, and thermoelectric PF, respectively for the p-type bulk Si where our model was calibrated on.
The power factor σS 2 peaks at high carrier concentrations around 10 19 -10 20 /cm 3 , and in most cases this is achieved by doping, which introduces strong IIS and severely limits the carrier mobility and electronic conductivity. The power factor would be at least a factor of >2× higher in the absence of IIS, which is the motivation behind modulation doping and gating methods [37,38,39,40,41,42,43,44,45]. To date, however, in such studies, the improvement over doped materials was only modest, in the best case the power factor values were similar to those of the doped materials. We show below, however, how this can be utilized in a very efficient way.
Energy filtering through the introduction of potential barriers, is the most commonly used approach to impede the flow of low energy carriers, and increase the Seebeck coefficient. As indicated in the schematic of Fig. 2a, electrons transport in the material through alternating potential wells and potential barriers. High energy electrons gain energy to jump over the barriers and make it to the next well region. Once the electrons pass over the barrier and enter the potential well, there they tend to lose energy, usually through the emission of optical phonons (Fig. 2a), and relax to the Fermi level of the well within a few optical phonon emission mean-free-paths, λE. Low energy electrons are blocked. In the sections below, we explain the model for electronic transport in the nanostructures we consider, under different geometrical conditions, which turn out to influence the power factor. We describe the equations that describe electronic transport in the potential wells and afterwards in the potential barriers, determining energy filtering.
Well region -Filtering and electron flow in the potential well: When λE is small (λE << LW, where LW is the length of the well), transport in the barrier/well regions is essentially independent of each other (Fig. 2b). In this case the total Seebeck coefficient and conductivity can be thought of as a simple combination of the individual quantities in each region, weighted by the size (volume fraction) of each region ('W'-well, 'B'-barrier) as [37]: where in 3D bulk materials the weighting factor νi is the volume of each region. The derivation of a generalized composite Seebeck coefficient equation as above can be found in the Appendix. The 'local' TE coefficients of the wells in this case are computed by simply considering the bulk value of these regions as: and those of the barriers by: In this case, PF improvements cannot be easily achieved, as the improvement in the Seebeck coefficient is limited to the volume that the barriers occupy, and in those regions the resistance is significantly increased (exponential drop in density).
Well region -Energy non-relaxing case: In the opposite scenario with respect to carrier energy relaxation, i.e. λE > LW, electrons can flow over the barriers without (fully) relaxing their energy when they enter the wells (Fig. 2c). In this scenario, only carriers with energies above the barrier height VB are contributing to transport. The Seebeck coefficient is benefitted in this case as it is by definition the average energy of the current flow <E> as [26]: (5a) The fact that the flow is high in energy throughout the channel (barriers and wells) essentially results in propagating the high Seebeck of the barrier into the well. In this energy non-relaxing case the TE coefficients in the well region can then be computed by only considering transport above VB as: Here, the only thing that changes from the treatment of each region independently, is that the integral for the conductivity of the well begins from VB, rather than from the band edge of the well EC = 0 eV.
In typical semiconductors, λE is in the order of several nanometers to a few tenths of nanometers. Therefore, nanostructures in which the wells are a few tenths of nanometers, operate between the two cases of Fig . We note that in reality for narrow barriers the 'local' bandstructure could be different compared to the bandstructures of the constituent materials, and quantum mechanical reflections will appear because of a degree of well/barrier state mismatch, which would add to interface resistance. These can be mitigated somewhat by employing smoothened barrier edges and oblique potentials (as in Fig. 3) rather than sharp ones [23,24], and we address this issue later on. The TE coefficients on the barrier region under thermionic emission conditions can then be computed at first order by allowing only the carriers from the wells with energies higher than the barrier heights VB over the barrier (ignoring contact resistance at this point) as:

III. Nanostructured grain/grain-boundary design for very high PFs
Designing optimal attributes for the filtering barrier: After describing the basic transport features and equations in SL structures, we begin our investigation by simulations of the optimized attributes for the potential barrier, and then for the potential welltargeting PF improvements. Figure 4, shows row-wise the TE coefficients electrical conductivity σ, Seebeck coefficient S, and power factor, respectively. Columnwise, it shows the change in these coefficients after each step of the design process we consider. We begin with the pristine channel, whose TE coefficients are shown by the black-dashed lines in all sub-figures for comparison, in which case the doping is uniformly distributed in the entire channel. The basis structure we consider has a well size LW = 30 nm, a barrier length LB = 2 nm, and barrier height VB = 0.15 eV (geometrical features which experimentally showed large power factors [48,49,51]).
As a first step, in the data of the first column of Fig. 4 (Fig. 4a-4c), we introduce a potential barrier, and consider independent transport in the well and barrier regions (i.e. full energy relaxation in the wells), as in the schematic of Fig. 2b. The TE coefficients for this system are given by Eq. 4. For the case where the entire structure is considered doped, the results shown by the green lines, indicate strong reduction in σ, a slight increase in S as expected, but finally a strong reduction in the power factor, at least for carrier concentrations up to ~10 21 .
In a second step, we consider the possibility that the potential barrier is free of dopants. Note that although the potential wells need to be doped to achieve the required carrier density, the barriers do not. In fact, other than them being formed due to bandedge discontinuities between the two materials (well and barriers), they can also be created by junctions of highly-doped/intrinsic regions, as shown in Fig. 3c-3d. The rationale behind the undoped barrier regions originates from the much higher phononlimited mobility and mean-free-paths for scattering compared to those of the ionized impurity-limited transport (as shown in Fig. 1a). Here, the TE coefficients for the barrier can be computed as in Eq. 4c-4d, but the scattering time is only determined by electronphonon scattering. In the case of an undoped barrier ( Fig. 4a-4c red lines), σ is degraded much less, S increases slightly, and the power factor experiences a slight increase. Thus, as the barrier formation increases the material's resistance, by keeping it 'clean' from dopants, at least the mean-free-paths are longer, and a portion of the conductivity is restored, mitigating the reduction in conductivity. This allows higher mobility carriers, limited locally by phonon scattering alone, rather than by the much stronger ionized impurity scattering (see Fig. 1a).
The third barrier attribute we investigate is the case where the carriers undergo thermionic emission above the barrier (blue lines), i.e. the case where the barrier size LB is much narrower compared to the momentum relaxation mean-free-path of the carriers.
Under the assumption of thermionic emission, the carriers from the wells that are not filtered and flow over the barriers, as indicated in the left of Fig. 2e, although subject to the doping in the wells, they occupy high velocity states. In this case, similarly to the undoped barrier case, slight power factor improvements are observed (blue line in Fig.   4c). The transport details over the barrier, however, are different in the two cases. In the undoped barrier case the resistance is mitigated by the use of the higher mobility 'dopantfree' barrier region. In the thermionic case, the resistance is mitigated by the use of higher velocity carriers coming from the well, rather that the lower velocity carriers from the top of the barrier. The key outcome here, is that 'clean' barrier regions or thermionic transport over them, can restore the conductivity reduction that the potential barrier causes (for reasonable VB values). We now proceed with examining the optimal attributes of transport in the potential well.
Designing optimal attributes for the well -(a) avoid energy relaxation: We next investigate electronic transport in the well region. At this stage we consider that the well region is a uniformly, highly doped region. From here on we assume the undoped barrier model, where charge carriers relax on the barriers. This is the conservative worst case scenario compared to the thermionic emission assumption, but we also examine thermionic emission as an upper limit scenario later on as well. As illustrated earlier in Fig. 2d, the current in a large portion of the well adjacent to the barriers propagates at larger energies (before it relaxes at lower energies), indicating both larger Seebeck coefficient and larger carrier velocities (and conductivity). In fact, it is these regions that provide power factor improvements in superlattices and nanocomposites, because the high Seebeck coefficient of the barriers propagates into the wells, and high energies keep the conductivity still high. The optimal design is achieved when the carriers do not relax their energy into the well region after they overpass the barrier as in Fig. 2c. In practice, the barriers are placed at distances short enough to enforce less relaxation, but long enough to reduce the resistance introduced. Thus, an optimal compromise is achieved under semi-relaxation conditions. The advantage of energy semi-relaxation to the Seebeck coefficient is described by us and others in several works [27,26,62,63,64]. 4d shows the mobility of p-type Si (dashed line), and the mobility of the carriers that travel above VB alone, which in this case is more than double (solid line). The conductivity is significantly lower compared to the pristine channel for lower densities ( Fig. 4d), but as the Fermi level is raised to the VB level, the conductivity increases substantially. At those carrier densities the Seebeck coefficient ( Fig. 4e) is still high, and therefore the overall power factor is improved compared to bulk by ~2× (Fig. 4f). This again illustrates the benefits of filtering at degenerate conditions [23,48].
Thus, to summarize the design of the filtering well/barrier system up to this point, one seeks for material designs which: i) Allow for momentum non-relaxation on the top of the barrier, or 'clean' preferentially undoped barriers that restore/mitigate the conductivity reduction introduced by the potential barriers. ii) Allow for energy nonrelaxation in the well, which raises the low Seebeck of the wells to the high values of the barriers. Those criteria impose restrictions in the design, shape, and importantly the 'cleanliness' of the barrier (from dopants for example), to achieve large momentum relaxation lengths, and potential well sizes comparable to λE (or somewhat larger).
Filtering well/barrier optimizationa novel concept consisting of three regions: Moving forwards, a novel filtering design geometry is introduced, and in the following sections its performance is investigated. Although for the analysis we still employ the semi-classical Boltzmann transport formalism, later on quantum mechanical simulators and Monte Carlo simulators are also utilized to further validate some of the design 'ingredients'. A 2D top-down view of the proposed geometry is illustrated in Fig. 3a, where we now have rectangular domains of wells depicted by the blue colored regions and barriers depicted by the grey colored regions. The wells can represent highly doped regions, or regions where the band energy is in general lower (such that wells are formed).
The red colored regions in between the heavily doped regions and the barrier regions are part of the wells (as in a nanocomposite material, for example), although for those regions we consider that the doping is different; for the purposes of this analysis, these regions are undoped. A simplified schematic of the potential profile in a 1D cross section of the material is shown in Fig. 3b, with the middle, doped regions being lowered in energy, the barriers regions' energy residing higher, and the potential of the middle region connecting the two extremes. Here we do not explore the details of the formation of the band profile in this region, but our goal is to illustrate the design principle.
In practice, the oblique potential profile in the middle region can be, for example,  Figure 3d shows the same profiles, but in this case the doping in the left/right regions is raised to 5 × 10 20 /cm 3 (the corresponding elevated Fermi level is depicted in Fig. 3d). Clearly, appropriate barriers for energy filtering are formed, with their shape and height controlled by the dopant values and intrinsic region length. In the rest of the paper, for simplicity, we assume that the potential profile in the undoped regions begins from the edge of the doped region, and ends to the edge of the barrier region. (In principle, however, the potential barrier can be formed in a n++/i/n++ structure without the need of different barrier material itself). Due to the large differences in the doping, the depletion region indeed is shifted mostly in the undoped region, so it does effectively begin from the edge of the doped region. Note that the word 'depletion' throughout this work denotes both dopant and majority carrier depletion (compared to the n++ region) as in our previous works in Ref. [48,49]. In principle, the middle-region barrier can be optimized by varying the dopant distribution accordingly, but in this work we assume intrinsic regions and provide the foundations of the design principle. We, thus, refer to these regions interchangeably as 'dopant-depleted', 'clean', or 'intrinsic'.
The TE coefficients in the case of the three-region structure can be computed by combining the individual coefficients of the three regions (well-W, intrinsic-i, barrier-B) as: In a similar manner to the wells and barrier regions, the conductivity and Seebeck coefficient of the dopant-depleted regions (labeled 'i' for intrinsic from here on), can be extracted by: In general, these quantities have a spatial dependence as the band edge changes in the 'i' region, but in most of the analysis below we consider an average band contribution, with that band edge located at mid-energy VB/2 unless specified otherwise.
(We have investigated various cases in our model, i.e. several band edge positions, each providing slightly different outcomes that do not change the foundation and advantages of our design). We also note that the models described by Eqs. 1, 3 and 8 are strictly valid for 1D periodic systems. Nanocomposites, on the other hand, are described by a 3D aperiodic geometry, and the complexity of the transport paths is such that would not allow us to map the 3D onto 1D paths beyond a first order estimation. The design we propose requires the current to flow normal to the wells and the barriers, rather than in parallel to themi.e. the 2D geometry in Fig. 3a consists ideally of columnar grains extending into the page, as in Ref. [48]. Nanocomposites are also subject to geometry variations, and superlattice 1D geometries can also be considered as a limiting case for a nanocomposite system. Thus, we argue that Eqs. 1, 3 and 8 (with the volume fraction included rather than the length of the regions), are at first order applicable to nanocomposite/nanocrystalline materials as well.
Thermoelectric coefficients in the structure with dopant-depleted regions: The TE coefficients for the '3-region' structure are shown by the blue lines in the third column of  Fig. 4h), but the dopant-free region strongly increases the conductivity of the overall domain (Fig. 4g). The carriers can now flow more easily under the weaker phonon-limited scattering conditions that prevail in the dopant-depleted regions. These regions geometrically occupy a significant volume of the structure even when having a narrow Wi = 5nm width. Thus, the overall conductivity acquires a significant phonon-limited part with higher mean-free-paths, rather than an ionized impurity scattering limited part with much lower mean-free-paths. A significant power factor improvement is then achieved in this case as shown by the blue line in Fig.   4i.
Designing optimal attributes for the 3-region structureallow thermal conductivity variations: Finally, we add another component to the design of the material, which brings an independent improvement in the Seebeck coefficient without first order changes in the conductivity. When different thermal conductivities ( ) exist in the different regions, the overall Seebeck coefficient can be generalized to (see derivation in the Appendix) [27,37,64]: Here, we assume a ratio of κW/κB = 5 (and assume κi = κW), which is a reasonable ratio between the conductivities of grains and grain boundaries, for example. An additional increase in the Seebeck coefficient is achieved as shown in Fig. 4k (green lines vs bluedashed lines). This new component leads to a larger power factor as shown in Fig. 4l.
Here we included the TE coefficients from the second and third columns of Fig. 4 Fig. 4l, which indicate the effect on the TE coefficients for each design ingredient (dashed lines). In the first column, by increasing Wi from 5nm to Wi = 10 nm (solid-blue lines), i.e. the length of the intrinsic regions is now longer, the conductivity increases significantly (Fig. 5a). Now a larger area of the material is composed of dopant clean regions, in which transport is phonon-limited with longer MFPs. The Seebeck coefficient is essentially not changed (Fig. 5b), despite the increase in the conductivity, because the elongated intrinsic regions raise the overall well band edge EC compared to the shorter Wi case (see Fig. 3b-3d). This means that the ηF = EC-EF is larger overall (in absolute terms), which tends to increase the Seebeck coefficient, thus mitigating the natural drop in the Seebeck when the conductivity is increased. Overall, therefore, the power factor largely increases to values ~13 mW/mK 2 , dominated by the increase in the conductivity. Importantly, the power factor maximum is achieved for slightly smaller densities, following the shift of the conductivity to lower densities, which is easier to achieve experimentally.
Next, we examine the increase in the barrier height VB from VB = 0.15 eV to VB = 0.25 eV. The results are shown in the second column of Fig. 5, (Fig. 5d-5f) (Fig. 5e), which makes the power factor to also largely increase (Fig. 5f). In this case, however, the power factor peak is shifted towards the higher densities, again following the shift in the conductivity.
The next step is to examine the increase in the ratio κW/κB from 5 to 10 in the third column of Fig. 5, given by the green solid lines in Fig. 5g-5i. Independently of the conductivity, the Seebeck coefficient is improved, which reflects on PF improvements in (red line). Carriers still need high EF levels to be able to overpass the increased VB, which allows for large Seebeck improvements as seen in Fig. 5k. Putting it all together, the increase in the power factor is quite substantial, reaching incredibly high values of >20 mW/mK 2 (magenta line in Fig. 5l), a factor of ~15× over the bulk value where we started from (dashed black line).
These are some very high values predicted by our model and simulations. Our model is simple in considering transport, and has also considered some idealized assumptions, such as energy non-relaxing transport and control on doping regions.
However, we need to stress that even if those idealized conditions are not met in reality, there is still a lot of room for the power factor to be improved substantially over the bulk values. In any case, an efficient TE material can still be realized with even 5× improvement in the power factor. It is also quite remarkable that we consider such high for comparison). It is quite instructive to show the mobility of these structures in Fig. 6d.
The red-dashed line shows for reference the bulk phonon-limited mobility and the blackdashed line the bulk phonon plus ionized impurity scattering mobility, as in Fig. 1a Thermionic emission benefits: In the second part of Fig. 6, (Fig. 6e-h), we repeat the same calculations as in Fig. 6a-d, but in this case we consider thermionic emission over the barrier, as shown in the left panel of Fig. 2e, and described by Eq. 7. The simple assumption of thermionic emission through a thin barrier allows the charge with energies higher than VB to flow 'freely' over the barriers and it is quite advantageous at higher VB, which otherwise introduce strong resistance to the current. Thus, in this case the conductivities are much higher compared to the previous scenario ( Fig. 6a vs Fig. 6e), the mobilities are higher (Fig. 6d vs. Fig. 6h), which doubles the power factor as well ( Fig.   6c vs Fig. 6g) to extremely high values of beyond 60 mW/mK 2 . Of course this is an overestimated value, which will drop once we consider interface resistance or the resistance that arises through quantum mechanical well/barrier momentum state mismatch. This mismatch will be is stronger with the barrier height as well. However, we examine the validity of this further below and point out that thin barriers where thermionic emission prevails can in general provide higher power factors.
We now devote the next part of the work to examining how realistic the conditions that we impose for obtaining such power factors are, using more advanced course the EF will be below the band profile. In the dopant-depleted regions, the band will cross the EF. The question we essentially want to answer here is: how high can we dope the middle region, or how high will the EF be if a certain mobile carrier density needs to be achieved in the entire material region, given that it will only be supplied from the central/core region (blue-colored regions in Fig. 3a)? Ultimately, as shown in Fig. 3c and   3d, the built-in barriers are formed in the undoped region, which is largely extended since the depletion region is preferentially placed in the intrinsic region of the n++/i junction.
On the other hand, the depletion region in the highly doped core is much narrower due to the very high doping (even just a few nanometerssee  (Fig. 3d).
In Fig. 7a we show the PF computations for the case of the different depletion region sizes considered, i.e. Wi = 5 nm (blue line same as in Fig. 4i), 10 nm (red line), and 15 nm (entirely dopant depleted wells, with delta-function shaped core distributed doping, green line). Interestingly, the larger the dopant-depleted region, the higher the power factor, but the smaller the required doping density in the middle of the well to achieve maximum PF, which would be easier to achieve experimentally anyway. Figure   7b shows  [26,27,64]. Energy relaxation in semiconductors is dominated by inelastic scattering processes, primarily electron-optical phonon scattering. In Si, for example, the electron-optical phonon meanfree-path is around λE ~15 nm, which results in wells sizes of LW ~ 50 nm to exhibit semirelaxation of the current energy [27,64]. Well sizes of the order of LW = 30 nm will only exhibit some degree of relaxation as we show below. However, other than the well size, there are other parameters that can contribute to reduced carrier energy relaxation in the design proposed, and they are discussed below. Evidently, whether  or DO is responsible in altering λE, the actual relaxation and overall Seebeck coefficients are different, even at the same λE (simply, the red/blue lines in Fig. 8b   conditions are found when some relaxation is present, such that the barriers are spaced as far as possible to reduce the density of interface resistances, but as close as possible to prevent relaxationwe discuss interface resistance reduction directions further below).
Interestingly, the magenta-dashed line shows an analytical calculation of the SL Seebeck coefficient if the well/barrier are considered independently (as in Fig. 2b). In this case much lower Seebeck coefficients are achieved for the SL.

Reduced relaxation and interface resistance in the 3-region structures: It is also
worth discussing a few other things that point towards reduced energy relaxation in the wells of the proposed design. For example, the shape of the band edge EC in the dopantfree regions, as shown in Fig. 3c and 3d, with oblique band edges: i) provides higher Seebeck coefficient due to the higher average EC, and ii) further reduces the availability of empty states at lower energies for electrons to relax their energy into. This reduces the energy window for optical phonon emission to happen and thus, reduces the energy relaxation. This is clearly indicated in Fig. 9a, where we have performed NEGF simulations using electron-optical phonon scattering alone and altered the barrier potential to the oblique shape as a first order approximation of what shown in Fig. 3b. As the sidewalls of the barrier become more and more oblique, the energy in the wells begins to raise. The oblique sidewalls, however, tend to reduce the energy of the current flow above the barriers, but finally modest improvements to the overall Seebeck coefficient are observed (Fig. 9c). Furthermore, in general, highly doped regions (in the well core), also push the energy of the current upwards as lower energies scatter more effectively off dopants (evident from the anisotropic Brooks-Herring scattering model [56]). Thus, in the structures proposed, the combination of: i) the chosen length for the well to be in the order of λE, ii) the 'oblique' band shape in the dopant-free regions, and iii) the doped core, allows for a significant degree of energy non-relaxing transport in the wells. The important point, here, is that the reduced energy relaxation in the wells is justified also by consideration of quantum transport simulations. This justifies the choice of the beneficial non-relaxing consideration in the BTE simulations earlier.
As a side note, we have shown in the past that nanoinclusions (NI) in the well regions, having barrier heights up to the VB of the superlattice, can push the energy of the current flow upwards and further increase the Seebeck coefficient, also allowing for small, but noticeable power factor improvements [26,66]. That would be something to also provide lower thermal conductivities, with additional benefits for ZT.
Oblique sidewalls reduce interface resistance: It is also quite interesting to observe the conductance of the channels in Fig. 9a, plotted versus the sidewall distance d.
The reduction in the 'local' Seebeck in the barriers in Fig. 9c at first when using oblique profiles is a signal of reduction in well/barrier interface resistance, a consequence of better well/barrier state matching. Indeed, our simulations show that the quantum mechanical transmission over the barriers increases in the oblique cases compared to the sharp barriers, and quantum reflections/oscillations are smoothened out. Due to this, the conductance is improved at first instance by ~20% (Fig. 9b). For larger d, the conductance remains almost constant, whereas the Seebeck coefficient increases from contributions in the wells. Overall, the introduction of the sidewalls increases the power factor monotonically, up to values of ~30% (Fig. 9d).
The validity of thermionic emission (Fig. 10): When it comes to the behavior of carriers over the barrier, we have shown up to now that 'dopant-clean' barriers, with oblique sidewalls for reduced interface resistance, and/or thermionic emission from the wells over the barriers are important ingredients for the design. Here, we examine the validity of the thermionic emission, again using NEGF simulations, this time including electron-acoustic phonon scattering only (elastic scattering to isolate the effect of carrier relaxation on the barrier from inelastic relaxation processes into the wells). We simulate a channel with a single potential barrier in the middle, and vary the length of the barrier LB from LB = 100 nm (taking over the entire channel) to LB = 5 nm and then to zero, i.e. the pristine channel case (as illustrated in the inset of Fig. 10). In Fig. 10 we plot the energy resolved transmission function, defined as Tr = (h/q0 2 )Ich/(f1-f2), where Ich is the NEGF extracted current, h is Plank's constant, and f1 and f2 are the Fermi-Dirac distributions of the left/right contacts, respectively [26]. All these quantities are energy dependent. The Tr is directly related to the transport distribution function in the BTE, and has a linear dependence in energy in the case of acoustic phonon scattering for a single subband [66,67]. That linear dependence is captured in the (multi-band) NEGF simulations for the pristine channel (brown line), as well as the long barrier channel (purple-dashed line), with the initial point being the band edge, i.e. 0 eV in the pristine case, and VB = 0.05 eV in the long barrier case.
As the barrier length LB is scaled, however, there is a clear shift at energies after VB towards the Tr of the pristine channel. The 'jump' in the Tr after VB in the shorter LB = 5 nm barrier channel, clearly indicates that carriers 'see' the barrier, but for energies above the barrier they have a Tr more similar to that of the well. This would be an indication of ballistic thermionic emission, in which case the carriers do not relax (at a large degree) on the bands of the barrier, i.e. overall they do not acquire the low velocities at the top of the barrier, but propagate with the well attributes. The gradual change of the black line Tr in Fig. 10 towards that of the well (reaching at ~80% of that value within 0.01 eV after the barrier at VB = 0.05 eV), could signal that some well/barrier state mismatch and quantum reflections are still present, adding to the interface resistance. We also note that the aforementioned energy window deviations from the pristine Tr would increase as VB increases due to larger mismatch. However, any acquired slope in Tr being larger compared to the one of the large non-thermionic well (brown line in Fig. 10), would be beneficial to the conductivity and the power factor. This effect would be reduced, however, when: i) the barrier sidewalls acquire a slope as explained above in the discussion of Fig. 9a, and ii) if the barrier material has much more transport modes compared to the well material, such that more momentum state matching is achieved [68]. This can be the case of a barrier material with much higher effective mass, for example. Thus, the optimal power factor of our design could be somewhere in between the ~30 mW/mK 2 and ~60 mW/mK 2 indicated in the two examples of Fig. 6.
High electronic conductivity in n++/i SL structures (Fig. 11 With regards to the volume fraction that the boundaries occupy, within the assumption that they remain undoped, our simulations show that moderate PF benefits are possible (even up to 2×) compared to the uniformly doped, pristine material, even if the boundaries occupy a larger volume fraction compared to the wells/grains. To obtain the very large PF improvements we present, however, the boundary regions need to be smaller than 5 nm in length, in order to mitigate electrical resistance and allow for a degree of thermionic emission (as in Fig. 10). On the other hand, boundary regions smaller than 3 nm will allow quantum mechanical tunnelling, and result in smaller temperature drops across them, both which reduce the Seebeck coefficient [24]. Thus, we suggest that the boundaries are optimally of the order of 5 nm thick. In the case of Si, for example, to achieve semi-relaxation of the current flow in the channel, the grain/well size should be in the order of LW = 30 nm -50 nm. This leads to an optimal volume fraction ratio of at most ~25% for the boundary regions.
Example of possible practical realization: The simplest experiment to design and evaluate the potential of the 'clean-filtering' approach is to begin with two regions, and fabricate 2D superlattices formed of n++/i, or n++/n-junctions. In that case the barriers are formed in the intrinsic or lightly doped regions, which will be regions 'cleaner' of dopants, having phonon-limited mobility. Lithography can be used for the definition of windows through an oxide layer (grown by thermal oxidation, for example) on an SOI wafer, and shaped by lithography and etching to act as a mask for the doping process.
Oxide windows, and hence the final doping concentration, can be arranged to form a 2D array using ion implantation (for example). As a next step, one can go even further by lithographically defining lines in the x-and y-directions to form a square 'net', and then dopant diffusion can create highly doped islands in the regions between the 'square net' lines as in the blue-colored islands of Fig. 3a.

V. Conclusions
In conclusion, this work proposes a novel design direction which will allow nanostructured materials to deliver exceptionally high thermoelectric power factors, even more than an order of magnitude compared to the original material's corresponding values. The design is based on an extension and generalization of previously presented strategy that realized experimentally very large power factors (5× compared to optimized pristine material values) [48,49]. In this work, it is shown that much higher power factors can be achieved once the grain/grain-boundary (well/barrier) design is properly optimized. Specifically, the proposed design utilizes energy filtering where carriers flow from heavily doped potential wells into undoped barriers (with a degree of thermionic emission) for reducing the barrier resistance that ionized impurities would have caused.
Importantly, though, it introduces an undoped region, 'clean' of dopant impurities, that separates the core of the wells from the barriers. This essentially allows for higher carrier mean-free-paths and mobility in the wells (approaching phonon-limited), compensating by far the conductivity reduction caused by the barriers. The work also points out that other than the 3-regions (1-heavily doped well core, 2-intrinsic carrier path spacer, 3potential barrier), an essential ingredient is that the energy of the carriers does not relax significantly in the well regions. It is shown, however, that this is the most probable and realistic case in the design we propose, as: i) the potential well length can be chosen such as to be in the same order as the energy relaxation mean-free-path, ii) the band edge shape of the intrinsic regions favors reduced relaxation, and iii) degenerate doping conditions also favor reduced energy relaxation. Thus, the design can provide exceptionally high power factors because it: i) reduces the resistance of the barriers, ii) transfers the high Seebeck coefficient of the barriers into the wells, and iii) allows very high conductivity in the wells. The latter is achieved by utilizing dopant-free intrinsic regions for transport, but with high energy carriers that provide high carrier mean-freepaths and phonon-limited, rather than ionized impurity scattering dominated mobilities.
Although some of the parameters the simulations employ are relevant to Si, the design approach can be applied in general to other materials as well.

Appendix
Here we provide the theoretical proof of the equation for the Seebeck coefficient of a composite system, which leads to Eq. 3b in the main paper. The Seebeck coefficient of an arbitrary irregular system is defined as the weighted average of the local Seebeck coefficients along the length of the material (L), with the weighting factor being the lattice temperature gradient as: where for the applied temperature difference ΔT, it holds Here we assume an 1D channel material. We assume that the lattice temperature (TL) varies according to a simple thermal circuit model. In this case, each of the two materials forming the composite system (barriers and wells) has a different temperature gradient across it depending on its thermal conductivity, as L / B dT dx for the barriers and L / W dT dx for the wells. The entire temperature drop across the material is then decomposed as: where LB and LW are the total lengths of the wells and barrier regions. At an interface between different materials, the heat flux is conserved, thus using Fourier's law we have: Substituting (A3) into (A1), we reach: In the case of a 3D material with irregularities in the distribution of the wells/barriers, for example as in a polycrystalline material of grains/grain boundaries, the lengths are replaced by the volume fractions of the different regions [37], i.e., the overall Seebeck coefficient of a 3D nanocomposite material is approximately the volume weighted average of the Seebeck coefficients of the constituent material phases as: Figure 1: