Exploring Packaging Strategies of Nano-embedded Thermoelectric Generators
EExploring Packaging Strategies of Nano-embedded Thermoelectric Generators
Aniket Singha, Subhendra D. Mahanti, and Bhaskaran Muralidharan ∗ Department of Electrical Engineering,Indian Institute of Technology Bombay, Powai, Mumbai-400076, India Department of Physics and Astronomy,Michigan State University, East Lansing, MI-48824, USA (Dated: September 20, 2018)Embedding nanostructures within a bulk matrix is an important practical approach towards theelectronic engineering of high performance thermoelectric systems. For power generation applica-tions, it ideally combines the efficiency benefit offered by low dimensional systems along with thehigh power output advantage offered by bulk systems. In this work, we uncover a few crucial detailsabout how to embed nanowires and nanoflakes in a bulk matrix so that an overall advantage overpure bulk may be achieved. First and foremost, we point out that a performance degradation withrespect to bulk is inevitable as the nanostructure transitions to being multi moded. It is then shownthat a nano embedded system of suitable cross-section offers a power density advantage over a widerange of efficiencies at higher packing fractions, and this range gradually narrows down to the highefficiency regime, as the packing fraction is reduced. Finally, we introduce a metric - the advantagefactor , to elucidate quantitatively, the enhancement in the power density offered via nano-embeddingat a given efficiency. In the end, we explore the maximum effective width of nano-embedding whichserves as a reference in designing generators in the efficiency range of interest.
I. INTRODUCTION
Low dimensional systems such as nanowires ornanoflakes are potential candidates for high performancethermoelectric systems with the likely enhancement intheir electronic figure of merit [1–4] and hence the maxi-mum thermodynamic efficiency, which is attributed to adistortion in their electronic density of states [5]. How-ever low-dimensional systems operate at much lowerpower outputs in comparison to their bulk counterparts.Bulk systems, on the other hand, offer much higher powerdensities for practical considerations, but at much lowerefficiencies. Therefore, it is essential to beat this so calledpower-efficiency trade-off [6, 7] by intelligently designingnano-structured bulk systems [5, 8, 9] aimed at combin-ing the efficiency benefit of nano structuring with thepower density advantage of bulk.Approaches toward nano-structuring so far haveproven successful in the suppression of phonon mediatedheat flow [10–16], while electronic engineering based onthe DOS [17–19] of low-dimensional systems [20–23] isstill a topic of current and active research. One pos-sible approach for electronic engineering is to pack low-dimensional systems in a bulk matrix [7, 24, 25] as shownin the schematic in Fig. 1(a) and (b), such that the ef-ficiency benefit of nano structuring is tapped along thetransport direction and the power advantage of bulk isharnessed via the areal packaging strategy [9]. With ad-vancements in fabrication and lithographic procedures,atomic layers can be placed with the precision of the or-der of atomic dimensions [26], and hence it is envisionablethat such packaging strategies are quite possible within ∗ [email protected] the current experimental capabilities. Also, with tremen-dous progress in oxide based nanoelectronics, it may alsobe possible to engineer the desired barrier profile on de-mand [27].The key object of this paper is to analyze the pack-ing strategies of such nano-embedded bulk systems andexplore under what operating conditions do they offeradvantages over traditional bulk thermoelectric genera-tors. Specifically, we consider nanowire and nanoflake-embedded bulk systems, as depicted in Fig. 1(a) and (b),where the nano structure concerned is packed in betweensufficiently thick barriers, as shown in the sample cross-sectional band profile depicted in Fig. 1(c). An analysisof packing strategies concerns a judicious choice of thebarriers between the nano-structures and its relationshipwith the effective power density at a given operating ef-ficiency to be discussed in depth here.Traditionally, electronic engineering of low dimensionalthermoelectrics concerns the optimization of the elec-tronic figure of merit z el T defined by z el T = S σκ el T, (1)with S , σ and κ el being the Seebeck coefficient, electricalconductivity and electronic thermal conductivity respec-tively, and T is the operating temperature.While low dimensional systems definitely offer the ad-vantage in terms of an enhanced figure of merit, it is nowincreasingly clear that one has to consider the power-efficiency trade-off [6, 7, 28] for a more complete de-sign strategy. The most glaring example of the power-efficiency trade-off is the Sofo-Mahan thermoelectric [29],which features an infinite electronic figure of merit, butzero power output [6, 28, 30]. That being said, z el T isnot an accurate measure of efficiency when power out-put is of the order of the quantum bound which is the a r X i v : . [ c ond - m a t . m e s - h a ll ] M a y (a) (b)(c) FIG. 1. Device Schematics. (a) Schematic of 1-D quantumwires embedded in a 3-D matrix. (b) Schematic of a 2-Dquantum well embedded in a 3-D matrix (c) Band diagramalong the y direction depicting a typical GaAs/AlAs structurefor the case where GaAs and AlAs have been doped to obtainthe same work-function. Band diagrams are shown along thedotted line of (a) and (b) maximum power any thermoelectric generator can pro-duce [28], [31], [32], [33]. So, in this paper we primarilyfocus on the power density versus efficiency trade-off inembedded nano-systems.A few crucial and fundamental aspects of electron engi-neering via nano-embedding are pointed out in this work.First and foremost, we show that a performance degra-dation is inevitable as the nano structure transitions tobeing multi moded, a trend which has been noted ina few experiments [15]. Turning to packing strategies,we point out that at higher packing fractions, judiciousnano-embedding offers a significant power density ad-vantage when operated over a large range of efficiencies.However, at lower packing fractions, this range graduallynarrows down to the high efficiency regime. Finally, weintroduce a metric - the advantage factor , to elucidatequantitatively, the enhancement in the power density of-fered via nano-embedding at a given efficiency. At theend, we explore the maximum effective width of nano-embedding which serves as a reference in designing nano-embedded generators in the efficiency range of interest. (a) (b) FIG. 2. Engineering optimum packing separation betweennano-structures. Transmission coefficient along the y direc-tion through (a) a 5 nm AlAs energy barrier (b) 10 nm AlAsbarrier of barrier height 1.45eV as shown in Fig. 1(c). It canbe noted from (a) and (b) that the 10 nm barrier offers notmore significant advantage in comparison with the 5 nm bar-rier. II. MODEL DETAILS
We model nano-embedded systems as quantumwires/flakes separated by energy barriers as shown inFig. 1(a) and (b). For the purpose of our calculation,we use the parameters of lightly doped GaAs conduct-ing channels and AlAs potential barriers to separate twoGaAs conducting channels. AlAs serves as a potentialcandidate for energy barrier fabrication because GaAsand AlAs have approximately same lattice constant re-sulting in minimization of lattice strain due to latticeconstant mismatch [34]. We start our calculations withnano devices of cross-sectional dimension 2 . nm embed-ded in a 3-D bulk matrix. The electron affinity of GaAsand AlAs are 4 . eV and 2 . eV respectively. Accord-ing to Anderson’s rule [34], an energy barrier of 1 . eV in a GaAs/AlAs system is feasible if the two materialsare doped in a way such that their work-functions areequal. Further manipulation of the doping concentra-tion of these materials results in higher energy barriers.The conduction band minima of the GaAs/AlAs system,given equal work-functions, are schematically depicted inFig. 1(c). For our discussions we will simply assume thatAnderson’s rule is valid and neglect complicated realisticvariations in band offsets. A more practical DFT basedcalculation of band offsets in GaAs/AlAs junctions is de-picted in [35, 36].The ability of a 5nm AlAs barrier to suppress elec-tronic transmission between two adjacent GaAs conduct-ing channels is depicted in Fig. 2(a). The vanishing val-ues of the transmission coefficient along the y directionover a specific energy range implies an absence of elec-tronic coupling between adjacent GaAs channels in thatenergy range. The transmission coefficient is calculatedin ballistic limit using the non-equilibrium Green’s func-tion (NEGF) formalism with an atomistic tight-bindingHamiltonian [9, 37]. It is clearly seen from Fig. 2(a)that a 5 nm AlAs energy barrier of 1 . eV can suppresselectronic transmission between adjacent GaAs channelsupto E k = 50 k B T at T = 330 K where E k is the kineticenergy of electrons. Fig. 2(b) demonstrates that a 10 nm AlAs energy barrier is only marginally advantageous overa 5 nm AlAs barrier in inhibiting electronic coupling be-low a kinetic energy of E k = 50 k B T . We therefore con-clude that 5 nm is an optimum separation between twoadjacent GaAs conducting channels in a GaAs/AlAs sys-tem.We consider ballistic thermoelectric generators withelectron transport in the z direction such that T ( E z ) ≈ I C = 2 qh (cid:88) s (cid:90) T ( E z )[ f H ( E z + E s ) − f C ( E z + E s )] dE Z (2) I Q = 2 h (cid:88) s (cid:90) ( E z + E s − µ H ) T ( E z )[ f H ( E z + E s ) − f C ( E z + E s )] dE z , (3)where I C is the charge current, I Q is the heat current, E z is the kinetic energy along the transport direction, E s isthe minimum kinetic energy of s th sub-band, f H/C is theelectron occupancy probability at the hot/cold contactsgiven by the respective Fermi-Dirac distributions kept atelectrochemical potentials µ H/C . The summation in (2)and (3) extends over all conducting sub-bands.In the subsequent calculations, the temperatures of thehot and cold contacts, labeled H and C , are assumed tobe 330 K and 300 K respectively. For simplicity, we as-sume parabolic dispersion E − k relationship with a con-stant effective mass m ∗ = 0 . m e . To make calculationssimple, we assume that both the contacts are perfectlysymmetric and the doping of the GaAs conducting chan-nel is uniform and that the voltage drops linearly acrossthe GaAs channels. Assuming that the contacts are inlocal equilibrium, the quasi-Fermi levels in the contacts isgiven by µ H/C = µ ± V / V is the voltage droppedacross the thermoelectric generator due to current flowthrough an external circuit and µ is the equilibrium elec-trochemical potential across the entire device.To generate the power density vs efficiency plots in thenext section, we use a voltage controlled model [6, 9] inorder to vary the bias voltage across the thermoelectricgenerator. The power density and efficiency ( η ) are cal-culated for each value of the bias voltage ( V ). The set ofpoints comprising the power density and efficiency thenconstitute the curves in the η − P plane for a particular µ .With the above description in mind, we now turnour attention to the power density-efficiency trade-off inthree types of embedded nano systems in the absence of electron-phonon interaction (ballistic limit) and phononmediated heat flow- (1) A single moded nanowire of cross-section 2 . nm × . nm (2) A 2-D nanoflake of width2 . nm (3) A multi moded nanowire (28 . nm × . nm )packed in a matrix. III. NANOSTRUCTURES VERSUS BULK
Single moded nanowires:
Our calculations use a singlemoded nanowire of length less than the mean free pathof electrons so that electronic transport occurs predomi-nantly in the ballistic regime. For simplicity, we assumethat the electron wavefunction is confined only withinthe GaAs conducting channels. The kinetic energy of theelectrons with respect to the conduction band minimumis given by E = E s ( x, y ) + (cid:125) k z m ∗ , (4)where E s ( x, y ) is the minimum energy of the s th sub-band and m ∗ = 0 . m e is the conduction band effectivemass of the electrons in GaAs.A nanowire exhibits single moded property when thethermal broadening of the Fermi-Dirac distribution func-tion k B T is much less than the minimum energy separa-tion between two sub-bands. Assuming that the separa-tion between two sub-bands must be greater than 5 k B T ,we deduce that the maximum cross-sectional dimensionsthat preserves the single moded property of a GaAsnanowire is approximately 11 . nm . We also note that themaximum cross-sectional dimension depends on the effec-tive mass of the electrons. For example, for m ∗ = m e , re-doing the above calculation gives max { w, h } ≤ . nm .Although our calculations show that a nanowire re-mains single moded for max ( h, w ) < . nm , thetrue potential of nanowire generators is harnessed at max ( h, w ) << . nm . This is because the total powergenerated by the single moded nanowire does not changewith area making the power density inversely propor-tional to the cross-sectional area. The power efficiencytrade-off for a 2 . nm × . nm nanowire is shown inFig. 3(a). For comparison, power-efficiency plot of abulk thermoelectric generator is also shown in Fig. 3(d).We note that the maximum power density of a sin-gle moded quantum wire of cross-sectional dimensions2 . nm × . nm is approximately 20 M W/m , whilethat of a bulk generator is of the order of 2 M W/m .Such high power density is expected given that the max-imum power produced by a single moded nanowire is ofthe order of 0 . nW [24]. Nanoflakes:
As in the previous case, we consider ananoflake of length less than the mean free path of theelectrons such that electronic transport is ballistic. As-suming that the transport is in the z direction and theelectronic confinement is in the y direction, the kinetic (a) (b)(c) (d) FIG. 3. Power-efficiency trade-off. Power Density vs effi-ciency of a (a) 2 . nm x2 . nm single moded quantum wire(b) 2 . nm quantum flake (c) 28 . nm x28 . nm multi modednano-scale wire and (d) 3-D bulk structure. Plots are shownfor values of η F ranging from − η F = E s − µ k B T ,where, E s is the lowest energy of the first sub-band. energy of the itinerant electrons is given by: E = E s ( y ) + (cid:125) ( k z + k x )2 m ∗ , (5)where E s ( y ) is the minimum energy of the s th sub-banddue to electronic confinement and m ∗ = 0 . m e is theeffective mass of the conduction band electrons in GaAs.Purely two-dimensional electronic band properties areimposed on the nanoflakes when the thermal broaden-ing of the Fermi function ( k B T ) is much less than theminimum energy separation between two sub-bands. As-suming that the separation between two sub-bands mustbe greater than 5 k B T at, we get w ≤ . nm .Similar to a single moded nanowire, the power outputof a nanoflake with only one conducting sub-band doesnot change with the width of the nanoflake. In termsof power density, a nanoflake with the smallest possi-ble width is the most advantageous. Fig. 3(b) showsthe power-density versus efficiency plot of a 2 . nm widenanoflake. The maximum power density of a nanoflake ofwidth 2 . nm is around 6 . M W/m , while a bulk gen-erator provides a maximum power density of 2 M W/m . Multi moded nanowires:
Next, we consider a nanowirewith discrete sub-bands such that energy separation be-tween the lower modes is of the order of thermal broad-ening of the Fermi-Dirac distribution function ( k B T ).Fig. 3(c) shows the power-efficiency trade-off in this case.It can be seen in comparing Figs. 3(a), (b), (c) and (d)that a nanowire with multiple modes provides a lowerperformance not only in power density but also in effi- ciency compared to the single moded nanowire, nanoflakeand bulk thermoelectric generator.A single moded nanowire operates at a higher efficiencyat maximum power density compared to a multi modednanowire. Multiple conducting sub-bands provide a num-ber of higher energy levels for electron transport. Elec-tronic flow through these higher energy levels from thehot contact to the cold contact results in higher elec-tronic heat conductivity thereby reducing the efficiency ofthe thermoelectric generator. Also positioning the Fermilevel near the higher energy sub-bands may result in sig-nificant back flow of electrons from the cold contact tothe hot contact through the lower energy sub-band (holelike conduction) thereby reducing the power output andthe voltage obtained at a given efficiency.By comparing Figs. 3(a),(b),(c) and (d), we can in-fer that single moded nanowires and nanoflakes offer thepossibility for a power density advantage, and hence maybe engineered for an enhancement in the performance vianano-embedding as opposed to multi moded nanowires.So, we conclude that there is a maximum cross-sectionaldimension beyond which the performance of a nanowiremay degrade below the bulk performance, which is whatwe will consider next. Single moded nanowire to bulk transition:
The max-imum cross-sectional width for which nanowires retainsingle moded electronic properties from the perspectiveof power generation is of particular interest in designingthermoelectric generators. For large cross-sectional di-mension of the nanowire in comparison with the latticeconstant, the separation between two sub-bands ∆ E be-comes much less than k B T , and the nanowire begins toshow bulk properties. Two quantities of interest whichcan be used to distinguish between lower dimensional andbulk thermoelectric generators are maximum power den-sity ( P MAX ) and the efficiency at maximum power den-sity ( η PMAX ).In Figs. 4(a) and (b), we show that the advan-tages of single moded nanowires may be harnessed for max ( h, w ) ≤ nm , where the logarithm of the powerdensity decreases linearly and the efficiency at maximumpower ( η P MAX ) remains almost constant with the loga-rithm of the cross-sectional dimensions. Between 8 nm ≤ ( h, w ) ≤ nm , it behaves as a multi moded nanowirewhere the difference ∆ E between two sub-bands is of theorder of k B T . For l ≥ nm the nanowire behaves asa bulk material depicting the expected trend of constantpower-density. As stated previously, the performance ofa nanowire in the multi moded regime degrades in termsof both the maximum power density as seen in Fig. 4(a)as well as the efficiency as seen in Fig. 4(b) in comparisonto single moded nanowire and bulk thermoelectric gener-ators. A multi moded nanowire hence shows no promisein applications to nano-embedded systems.The reason for the degradation of power density inmulti moded nanowires compared to bulk is evident fromFig. 5, which shows the normalized mode density per unitarea of a 28.2nm and a 282nm wide square nanowire. (a)(b) FIG. 4. Transition from nano-scale to bulk (a) in the case ofa nanowire of square cross-section in terms of the maximumpower generated. (b) Change in efficiency at maximum poweras a quantum wire makes transition from being single modedto bulk. The region marked in pink depicts the region wherethere is no utility in using nanowires for power generationapplications.
Above the Fermi energy, the 282nm nanowire containsa higher number of modes per unit area which results inhigher output power density. The degradation of the effi-ciency at maximum power is not so evident via a similarline of argument, since the efficiency is a ratio betweenthe power and the heat current [6, 9]. It is therefore ofsome use to make a back of the envelope calculation byconsidering z el T for the two structures. In the linear re-sponse regime, the efficiency at maximum power is given FIG. 5. Normalized modes per unit area profile as a functionof energy from the minimum of the first sub-band. Above theFermi energy, the normalized number of modes of a 282nmsquare nanowire is either greater than or equal to that forthe 28.2nm nanowire. Greater number of modes facilitatehigher current density against the applied bias resulting in anincreased power density in the case of the 282nm nanowirecompared to that of the 28.2nm nanowire. by [38] η ≈ η C z el Tz el T + 2 , (6)which is a monotonically increasing function of z el T .The calculated value of z el T for the 28 . nm and 282 nm wide nanowires are 1 . . nm would operate athigher η P MAX compared to a nanowire of width 28 . nm .The z el T calculations performed above are through cir-cuit arguments [6, 9], which match the calculations per-formed by evaluating all the linear response coefficientsin (1).The domain of each regime in Figs. 4(a) and (b), how-ever, would scale inversely as the effective mass ( m ∗ ) ofthe material and the temperature of operation T . Thus,nanowires made of materials with higher effective masswould require smaller cross-section to harness the advan-tages of being single moded. IV. OPERATING REGIMES OFNANO-EMBEDDED GENERATORS
We will now consider the effect of packing nanostruc-tures and the details involved in the process of nano-embedding the structures considered so far. Embeddednanowires of cross-sectional dimensions h and w , witha separation t between adjacent nanowires, provide apacking fraction of whwh +( w + h ) t + t . Similarly embeddednanoflakes of width w separated by energy barriers ofwidth t provide a packing fraction of ww + t . The detailsof the calculations are carried out in the appendix. It isof foremost importance to note that despite the higherpower density provided by a single moded nanowire ofcross-section 2 . nm × . nm , its effective power den-sity in a matrix is limited due to a very low packingfraction.A matrix packed with nanowires of square cross-section(2 . nm × . nm ) separated by a 5 nm energy barrierprovides a packing fraction of 0.13. Similarly, 2 . nm nanoflakes packed in a matrix provide a packing fractionof 0.361, when adjacent flakes are separated by a 5 nm en-ergy barrier. These numbers are reduced to 0 . . nm . We now turn ourattention to the operating points of the respective ther-moelectric generators taking their packing fraction intoaccount. We use the term effective power density to de-note the power per unit area of a matrix packed withnanowires/flakes.In the context of our discussion, operating points aredefined as points in the effective power density-efficiencyplane ( η − P ) where the effective power-density is maxi-mum for a given efficiency or the efficiency is maximumfor a given effective power-density. For power genera-tion applications, these points offer the minimum effec-tive power density-efficiency tradeoff. For bulk or nano-embedded generators, depending on the cross-sectionaldimensions, material parameters and packing fractions,there is a set of points in the η − P plane where the devicecan operate [28], which we term as allowed points . Thesubset of all these points constitute the allowed regionof operation in the η − P plane. All other points in the η − P plane constitute the forbidden region [28]. Operat-ing points lie on the boundary of the allowed region andthe forbidden region of operation. These are the pointsalong which the maximum effective power density at agiven efficiency, or equivalently, maximum efficiency ata given effective power density, can be obtained. Fig. 6show the operating points of embedded nano-systems fora separation of 5 nm and 10 nm respectively. Each figurecan be seen to be divided into three regimes: Regime 1:
In this regime, there is no trade-off betweenpower and efficiency for the bulk thermoelectric genera-tors, that is, efficiency increases monotonically with in-creasing power. Thermoelectric generators are not oper-ated in this regime. This regime is marked using the lightblue colour in Figs. 6(a) and (b).
Regime 2 : In this regime, there is a trade-off betweenpower and efficiency for the bulk generators and bulk gen-erators perform better in terms of the power density incomparison to nano-embedded systems. Using the bulkgenerator is advantageous in this regime. This regime ismarked using the light yellow color in Figs. 6(a) and (b).
Regime 3 : In this regime, there is a trade-off betweenpower and efficiency for the bulk generator, but nanoembedded systems perform better than the bulk in termsof power density at a given efficiency. In this region ofoperation, nano-embedding is advantageous. This regime (a)(b)
FIG. 6. Operating Regions. Efficiency vs. effective powerdensity plots for a single moded quantum wire of cross-sectional dimension 2 . nm × . nm , a 2-D quantum flakeof width 2 . nm , and 3-D bulk taking packing fraction intoaccount for (a) 5 nm separation barrier between two nano-structures (b) 10 nm separation barrier between two nanostructure. It can be seen in comparing (a) and (b) thatnano-embedding offers a significant advantage at lower pack-ing fractions over a wide range of efficiencies. is marked using the light purple color in Fig. 6(a) and (b).It is shown in Fig. 6(a) that for nanowires/flake embed-ded systems of cross-sectional dimensions 2 . nm sepa-rated by 5 nm energy barrier, the second regime does notexist. That is, for all relevant points of operation, nano-embedding is advantageous. However when the separa-tion between adjacent nanowires/flakes is increased to10 nm , as noted in Fig. 6(b), nano-embedding is advan-tageous only in the regime of higher efficiency. This couldbe understood by the fact that increasing separation be- FIG. 7. Maximum width of the energy barrier that canbe used for a single moded nanowire of cross sectional di-mensions 2 . nm × . nm and 2-D quantum flake of widthof 2 . nm . For a separation width greater than the giventhe maximum values, bulk provides a better effective powerdensity-efficiency tradeoff. tween adjacent nanowires/flakes decreases the packingdensity resulting in a decrease in the effective power out-put per unit area.In this context, it may be of particular interest toknow what is the maximum width by which two nanodevices may be separated while still retaining the power-efficiency advantage out of nano-embedding. This isshown in Fig. 7, where the maximum possible separationbetween two nano-scale devices increases when the de-vice is operated at a higher efficiency. In other words, forthe same separation between two nano-generators, nano-scaling offers a higher power advantage when they areoperated in higher efficiency regime. V. ADVANTAGE FACTOR DUE TONANO-SCALING
A metric to quantify the power advantage offeredby nano-embedded systems over their bulk counterpartserves as a deciding factor for nano-scaling thermoelec-tric generators. The parameter of interest, is indeed, thefactor by which the power density is increased in nanoembedded systems over bulk generators.In the context of our discussion, we define an advantagefactor at a given efficiency η as χ ( η ) = P nano ( η ) P bulk ( η ) , where P nano is the effective power density of a nano-embedded thermoelectric generator and P bulk is thepower density of the bulk thermoelectric generator. The advantage factor is a basic measure of the advantage weget from nano-embedding. For a given set of materialparameters, the advantage factor depends on (a) the effi-ciency of operation of the thermoelectric generators and(b) the effective area per nano-generator. We define ef-fective area of a nano-generator as A eff = 1 N , where N is the number of nano generators embedded ina square matrix of unit cross-sectional area. Fig. 8(b)shows the logarithm of the advantage factor versus theefficiency for nano-embedded GaAs generators of cross-sectional dimension 2 . nm separated by a 5 nm AlAsenergy barrier. Doubling the effective area per unit nano-scale generator decreases log ( χ ) at each point on Fig. 8 byapproximately 0.3dB. For square embedded nanowires ofcross-sectional dimensions w separated by energy barriersof width t , the effective area is given by A eff = ( w + t ) .For embedded nanoflakes of width w and separatedby energy barriers of width t , the effective area is givenby A eff = w + t . The quantity ( w + t ) is thereforea parameter of importance and we call it the effectivewidth ( W eff ) of the nano-embedded generators. A lower W eff results in a higher advantage factor. Fig. 8(a)shows the maximum effective width that can be allocatedto a unit nano embedded generator with the effectivewidth maximum when χ = 1. In terms of the maxi-mum effective width, the advantage factor may be writ-ten as: χ = ( ( W eff ) MAX W eff ) , for a single moded nanowireand χ = ( W eff ) MAX W eff for a nanoflake with a single modealong confined direction. For the same effective width,nanowires perform better than nanoflakes, the differencein their advantage factors being an increasing function ofthe efficiency. It should be noted that advantage factor iscalculated along the operating points at a given efficiencyfor single moded nanowires and flakes. VI. CONCLUSION
In this work, we have pointed out a few crucial as-pects to be considered for the electronic engineering ofnano-embedded bulk thermoelectric generators. Firstand foremost, we pointed out that a performance degra-dation is inevitable as the nanostructure transitions tobeing multi moded, a trend which has been noted in afew experiments [39, 40]. Turning to packing strategies,we pointed out that at higher packing fractions, nano-embedding offers a significant power density advantagewhen operated over a large range of efficiencies. How-ever, at lower packing fractions, nano-embedding per-forms poorly, displaying only a marginal advantage andthat too, only when operated at much higher efficiencies.Finally, we introduced a metric - the advantage factor , toelucidate quantitatively, the enhancement in the powerdensity offered via nano-embedding at a given efficiency. (a)(b)
FIG. 8. (a)Maximum effective width ( W eff ) of square quan-tum wires and quantum flakes embedded in a bulk matrix.The maximum value W eff depends on the effective mass ofthe material which is used to describe nano-scale thermoelec-tric generators. (b)Advantage factor for square cross-sectionalquantum wires and quantum flakes of cross-sectional dimen-sion 2 . nm separated by 5 nm energy barriers. At the end, we explored the maximum effective width ofnano-embedding which serves as a reference in design-ing nano-embedded generators in the efficiency range ofinterest. From our detailed analysis, we conclude thatalthough a nanowire/flake is capable of demonstratingenhanced nano-scale thermoelectric properties [1, 2] uptoa maximum cross-sectional dimensions of 8 nm , it is ad-vantageous to fix the width of nanowires/flakes to thesmallest possible value permitted by current fabricationcapabilities. In all our calculations, we have neglecteddecrease in efficiency due to lattice heat conductivity. Al-though the performance of both nano-embedded genera-tors and traditional generators are degraded due to heatconduction via phonons, nano-structuring offers a pre-dominant advantage over traditional generators in terms of reduced lattice conductivity. The high density of in-terfaces in nano embedded generators on the length scaleof phonon mean free path effectively scatters long wave-length phonons thereby reducing the lattice conductivity[10–16, 41] and counteracting the reduction in efficiencyto a large extent. Analysis and results obtained in thiswork should provide general design guidelines to embednanostructures within a bulk matrix to offer a greateradvantage over traditional bulk thermoelectrics, withinthe limits of current fabrication technologies. APPENDIXA. Evaluation of Packing Fraction
Let us assume rectangular nano-structures embeddedin a bulk matrix. Let the height and width of each nano-scale device be h and w respectively. Assuming the cross-sectional dimensions of the bulk material (say H and W )to be much greater than the cross-sectional dimensions ofthe nano-structures and the minimum separation widthbetween two nano-scale structures to be t, the total num-ber of nano structures that can be embedded in the bulkmatrix is N = HW ( h + t )( w + t ) . In the context of our discussion, the packing fraction isdefined as the fraction of the 3-D matrix which is occu-pied by the embedded nano structures. From this defini-tion, the packing fraction for embedded nano structuresin the matrix is