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Energy 93 (2015) 2006e2017

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Energy

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Power density optimization for micro thermoelectric generators

Marc T. Dunham a, Michael T. Barako a, Saniya LeBlanc b, Mehdi Asheghi a, Baoxing Chen c,Kenneth E. Goodson a, *

a Department of Mechanical Engineering, Stanford University, Stanford, CA, 94305, United Statesb Department of Mechanical Engineering, The George Washington University, Washington, DC, 20052, United Statesc Analog Devices, Inc., Norwood, MA, 02062, United States

a r t i c l e i n f o

Article history:Received 30 May 2015Received in revised form30 September 2015Accepted 11 October 2015Available online xxx

Keywords:Micro thermoelectric generatorsPower density optimizationWaste heat recoveryEnergy scavengingThermoelectric figure of merit

* Corresponding author.E-mail address: goodson@stanford.edu (K.E. Good

http://dx.doi.org/10.1016/j.energy.2015.10.0320360-5442/© 2015 Elsevier Ltd. All rights reserved.

a b s t r a c t

Microfabricated thermoelectric generators (mTEGs) can harvest modest temperature differences to pro-vide reliable solid-state electricity for low-power electronics, sensors in distributed networks, andbiomedical devices. While past work on mTEGs has focused on fabrication and demonstration, here wederive and explore comprehensive design guidelines for optimizing power output. A new closed-formthermoelectric device model agrees well with the traditional iterative approach. When thermoelectricleg length is limited by thin-film fabrication techniques, a very low (<10%) active thermoelectric fillfraction is required to optimize device power output, requiring careful selection of filler material.Parasitic resistance due to electrical interconnects is significant when a small number of thermocouplesis used, and this loss can be reduced by increasing the number of thermocouples while decreasing thecross-sectional area of the legs to maintain the same fill fraction. Finally, a discussion of the “incom-pleteness of ZT” shows that different combinations of thermal conductivity, electrical conductivity, andSeebeck coefficient resulting in the same ZT will result in different device performance and optimizationdecisions. For mTEGs, we show it is best to increase Seebeck coefficient, followed by decreasing thermalconductivity for short leg lengths and increasing electrical conductivity for long leg lengths.

© 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Microfabricated thermoelectric generators (mTEGs) are used toproduce electrical power for devices, such as wireless sensors,requiring micro-Watts to milli-Watts of power per device. Thesegenerators scavenge thermal energy from waste heat sources thathave temperature differences or spatial dimensions that are toosmall for conventional thermodynamic heat engines to effectivelyutilize. In this domain, only a small fraction of available thermalenergy needs to be extracted from the thermal reservoir to achievea target power output, oftenwithout concern for thermal efficiency.In the limit where the heat capacities of heat source and sink arevery large, the temperatures of the source and sink are not appre-ciably perturbed by the small amount of heat drawn through themTEG system, which is dependent on device configuration (i.e. thegeometry, interfaces, boundary conditions, etc.). This fixed tem-perature assumption is an ideality that benefits from the relatively

son).

high thermal resistance of a mTEG, compared to larger devices forwhich local temperature perturbations may be a more presentconcern [1]. Under a fixed temperature assumption, optimizationfor either maximum thermal efficiency or maximum power outputwill lead to two different devices [2,3]. Since many microfabricateddevices, sensors, and actuators have well-defined power and foot-print requirements, this work defines and optimizes performancebased on electrical power generated by the device rather thanthermal efficiency and presents a comprehensive design method-ology for microfabricated waste heat scavenging devices.

The field of mTEGs emerged in the early/mid-1990s, reducingfeature sizes to mm and mm following decades of successful use ofhigh-temperature, bulk-scale devices for applications such asspacecraft power generation [4]. Microfabricated devices offer anadvantage over traditional bulk-processed devices by allowing forstreamlined assembly in semiconductor process lines and directon-chip integration. One of the first discussions of the use ofthermoelectric generators for the recovery of “low-grade” wasteheat examined mm-scale devices exposed to temperature sourcesup to 240 �C [5]. In 1997, Fleurial et al. [6] suggested the use of thin-film fabrication methods for “micropower sources” to

Fig. 1. Diagram of vertically-aligned thermoelectric generator, showing (a) side and (b)top views.

M.T. Dunham et al. / Energy 93 (2015) 2006e2017 2007

accommodate the advance in miniaturized systems, particularly forspace applications. In the same year, Stordeur and Stark [7] re-ported on the successful fabrication and testing of a mTEG, producedby thin-film sputtering in a planar pattern followed by dicing andvertical assembly. Fleurial et al. [8,9] achieved thicker (~10e50 mm)thermoelectric films deposited by electrodeposition for low-power,high-voltage harvesting of small temperature gradients. In 2004,B€ottner et al. [10] discussed a novel sandwich-type wafer assemblywith interlocking substrates prepared by co-sputtering. Strasseret al. [11] prepared and analyzed a CMOS-compatible generatordesign based on Si and SiGe semiconductors in a designwhere heatflows laterally in the device. Electrodeposition of bismuth-tellurideand nickel-copper devices using polymer molds was reported byGlatz et al. [12] to extend the regime of thin film thermoelectric leglengths, which is a limiting factor in microfabricated devices.

While significant progress has been made in fabrication andpreliminary demonstration of mTEGs, there has been little attentionon the challenge of optimizing these devices for power output.Optimizing ZT at the materials level has been the focus of much ofthe thermoelectrics community, though these reports may not becomplete as the mTEG community strives to improve power gen-eration per unit footprint area for practical applications, a partic-ularly important metric for small thermoelectric modulesconverting waste heat to electricity. The use of the metric ZT as anindicator of the quality of thermoelectric generators [4,13e15] maynot be appropriate for mTEGs. This quantity encapsulates theoperating temperature and three primary material propertiescontributing to the thermoelectric rate equations into a dimen-sionless parameter, the so-called thermoelectric figure-of-merit:

ZT ¼ S2

rk

�Th þ Tc

2

�(1)

In Eq. (1), S is the Seebeck coefficient (units of mV K�1), r is theelectrical resistivity (U m), k is the thermal conductivity(W m�1 K�1), and Th and Tc are the hot and cold temperaturesacross the thermoelectric. In reality, an “incompleteness of ZT”exists where a single value of ZT does not uniquely predict poweroutput or provide an absolute roadmap to device optimization dueto the many combinations of S, k, and s that yield a single ZT value.

The impact of filler material, which surrounds the active ther-moelectric material (see Fig. 1) and may exist as a byproduct ofmanufacturing or intentionally for mechanical stability, is oftenoverlooked despite observed effects on performance [16]. Externalthermal resistances (i.e. the pathways to heat sources/sinks) play asignificant role in the performance of mTEGs, and some attentionhas been given to these effects [17e19]. However, without the in-clusion of realistic filler material effects and device design consid-erations, these analyses are incomplete. Additional factors becomeimportant in the realization of a practical generator device,including adjustment of device fill fraction to compensate forfabrication limitations to thermoelectric element length andappropriate selection of number of junctions when accounting forthe electrical resistance of interconnects.

The present work develops a comprehensive design method-ology for mTEG devices, particularly those harvesting energy fromsmall temperature differences, paying close attention to severalparameters that are commonly neglected in the existing literature,such as the effects of fill fraction in combinationwith different fillermaterials, limitations and workarounds when restricted to shortleg lengths, the impact of the number of thermocouples, and thedifferent contributions of the parameters in the figure of merit ZT todevice performance. Further, a robust closed-form model for thepower output of a mTEG is derived, which includes the Peltier effect,

external thermal resistances, parasitic resistive losses, fill fraction/filler material, and accounts for variable electric loading.

2. Model

The analysis presented in this paper considers a thermoelectricgenerator with heat flowing through the individual thermoelectricelements in parallel and electric current flowing in a serpentineseries pattern. Simulations use a thermocouple unit cell that can berepeated to scale up to a full device. The unit cell area and relevantsegments of the generator for simulation are illustrated in Fig. 1.

The illustration in Fig.1 shows both a side viewwith heat flowingfrom one substrate to the other and a top-down view with heatflowing into/out of the page. A one-dimensional heat transfer modelwith parallel heat flows is assumed for the vertical structure shownin Fig. 1, and the thermal and electrical resistance networks areprovided in Fig. 2. For simplicity, we assume that the thin passiv-ation layers and high thermal conductivity metal interconnects in

Fig. 2. (a) One-dimensional thermal and (b) electrical resistance networks used to model thermoelectric generators.

M.T. Dunham et al. / Energy 93 (2015) 2006e20172008

series with relatively long and low-conductivity thermoelectric legsdo not contribute significantly to the overall thermal resistance. Thisassumption is confirmed with 3D finite element analysis.

The model accounts for thermal conduction through externalthermal resistances Rt,h and Rt,c (e.g. heat sinks/exchangers), sub-strate resistances, and parallel conduction through the activethermoelectric material and filler material. This model also in-cludes the effects of Joule and Peltier heating. Joule heating ( _Q Joule)is volumetric and well-modeled by introducing half of the totalheat at each of the hot and cold junctions of the device [20]. Thetransfer of heat from the hot junction to the cold junction by thePeltier effect is captured by a removal of heat at the hot end and anaddition of heat at the cold end and degrades the performance ofthermoelectric devices operating as electric generators. Thomsonheat is neglected due to the small temperature gradients encoun-tered in micro-harvesting applications. A nodal energy balance atthe hot junction including heat flow from the source ( _Qh), heat flowthrough the thermoelectric material ( _QTE), heat flow through thefiller material ( _Q filler), heat removed at the hot junction due to thePeltier effect ( _QPeltier;h), and heat added due to Joule heating ( _Q Joule)is given by

0 ¼ � _Qh þ _QTE þ _Q filler þ _QPeltier;h �_Q Joule

2(2a)

where:

_Qh ¼ Th � TTEG;hRt;h

(2b)

_QTE ¼ TTEG;h � TTEG;cRt;TE

(2c)

_Q filler ¼TTEG;h � TTEG;c

Rt;filler(2d)

_QPeltier;h ¼ nTCSnetTTEG;hI (2e)

_Q Joule ¼ I2Re;TEG (2f)

The temperatures of the hot and cold junctions are given byTTEG,h and TTEG,c, respectively. The number of thermocouple junc-tions is expressed by nTC, and Snet is the net Seebeck coefficient(defined as Snet ¼ Sp e Sn). The electrical current through andelectrical resistance of the generator are I and Re,TEG, respectively.The parallel thermal resistances of the thermoelectric material andfiller material are given by:

Rt;TE ¼ lTEkTEAtotalFF

(2g)

Rt;filler ¼lTE

kfillerAtotalð1� FFÞ (2h)

where lTE is the length (or height) of the each thermoelectric leg, kTEand kfiller are the respective thermal conductivities of the thermo-electric and filler material, FF is the filling fraction of active ther-moelectric material, and Atotal is the device footprint area.

Similarly, we construct an energy balance at the cold junction:

0 ¼ _Qc � _QTE � _Q filler � _QPeltier;c �_Q Joule

2(3a)

where:

_Qc ¼TTEG;c � Tc

Rt;c(3b)

_QPeltier;c ¼ nTCSnetTTEG;cI (3c)

The validity of the 1D thermal model (excluding Joule andPeltier contributions) is checked with a 3D finite element simula-tion in COMSOL Multiphysics. The 3D simulation includes two100 nm Si3N4 passivation layers (implemented first as narrowdomain elements and then using the Thin Layer resistance featurewith negligible discrepancy) and 1 mm tall Au interconnectsmatched flush to the sides of the thermoelectric legs. The average

M.T. Dunham et al. / Energy 93 (2015) 2006e2017 2009

temperature of the two TE (thermoelectric) legs is calculated at thehot and cold junctions, and the temperature difference is comparedto that calculated using the 1D heat transfer approximation. Theresults are compared in Fig. 3 for FF ¼ 0.01 and 0.25, and 4 differentfiller materials (vacuum, air, polyimide, and SiO2).

The 1D thermal model neglecting the passivation layers andinterconnects shows good agreement with the 3D finite elementmodel. The 1D model begins to deviate from the 3D model for lowfill fractions and thermoelectric leg lengths on the order of theinterconnect height.

The electric circuit equations are coupled to the thermal modelthrough the junction temperatures TTEG,h and TTEG,c, which deter-mine the generated Seebeck voltage Vgen:

Fig. 3. Comparison of generator temperature difference calculated by the 1D (lines)and 3D (markers) heat transfer models for (a) FF ¼ 0.01 and (b) FF ¼ 0.25 with differentfiller materials.

Vgen ¼ nTCSnet�TTEG;h � TTEG;c

�(4)

The voltage drops across the generator resistance Re,TEG, whichincludes contact and IC (interconnect) resistances, and the con-nected load resistance Re,load are then found using voltage dividerexpressions:

VTEG ¼ Re;TEGRe;TEG þ Re;load

Vgen (5)

Vload ¼ Re;loadRe;TEG þ Re;load

Vgen (6)

The electric current I is calculated as the generated Seebeckvoltage divided by the electrical resistance of the total pathwaythrough generator and load:

I ¼ Vgen

Re;TEG þ Re;load(7)

The electric power delivered to the load is then calculated usingthe voltage across the load resistance given by Eq. (6) to write:

Pload ¼ V2load

Re;load¼ Re:loadn

2TCS

2net�TTEG;h � TTEG;c

�2�Re;TEG þ Re;load

�2 (8)

The electric power and the voltage and current produced bythe generator are of primary interest, and the coupled systemof Eqs. (2)e(8) requires an iterative solution to obtain thetemperatures on the hot and cold active regions of thegenerator, TTEG,h and TTEG,c, respectively. The fraction of tem-perature drop across the thermoelectric versus the total avail-able temperature difference from the reservoirs is less thanunity for nonzero external thermal resistances and can bequantified as a temperature efficiency that is useful for deviceanalysis:

hT ¼ TTEG;h � TTEG;cTh � Tc

(9)

3. Closed-form solution for temperature

The coupled system described by Eqs. (2e8) requires an iter-ative numerical approach to converge to a solution. However,simplifications can be used to obtain a closed-form solution. Onefirst-order approach [17,21] neglects both the Peltier and Jouleheating contributions and calculates the relevant temperaturedifference from a ratio of thermal resistances analogous to avoltage divider:

�TTEG;h � TTEG;c

�1st�order ¼

1

Rt;TEþ 1

Rt;filler

!�1

Rt;h þ

1Rt;TE

þ 1Rt;filler

!�1

þ Rt;c

ðTh � TcÞ

(10)

The resulting relation requires no numerical iteration, but leadsto a non-negligible error, on the order of 10%, primarily due toabsence of the Peltier heats.We obtain amore accurate closed-formexpression by retaining a form of the Peltier heating term. In thedomain of small temperature differences the conversion efficiencyis very small due to the theoretical limits of heat engine thermalefficiency. This allows us to neglect the effect of Joule heating in Eqs.

M.T. Dunham et al. / Energy 93 (2015) 2006e20172010

(2) and (3), which is negligible compared to the total heat flow inthe device (~3% for a Carnot engine operating between 20 and 30 C).If Eqs. (2) and (3) are then added and combined with Eqs. (4) and(7), the following equation is produced:

Th � TTEG;hRt;h

þ TTEG;c � TcRt;c

¼ 2TTEG;h � TTEG;c

Rt;TEGþ2

TTEG;h� TTEG;cRt;filler

þ…

n2TCS2net�TTEG;h � TTEG;c

�Re;loadþRe;TEG

�TTEG;h þ TTEG;c

�(11)

For small temperature difference near room temperature, suchas seen by microscale generators in waste heat recovery applica-tions, it can be assumed with reasonable accuracy that(TTEG,h þ TTEG,c) z (Th þ Tc), which approximates the Peltier heatcontribution in terms of the known reservoir temperatures. Thesetemperatures are assumed constant due to the small amount ofheat extracted by a mTEG compared to the heat capacity of a hotwall or pipe, for example. In a final step, it is assumed that theexternal thermal resistances Rt,h and Rt,c are expressible as anaverage Rt,ext,avg. With these steps, a closed-form solution for thetemperature difference across the active region of the generator isderived:

TTEG;h � TTEG;c� �

closed ¼ Th � Tc

1þ Rt;ext;avg 2Rt;TEG

þ 2Rt;filler

þ n2TCS

2net

Re;loadþRe;TEGTh þ Tcð Þ

h i (12)

The temperature difference calculated in Eq. (12) can then beused directly in Eq. (8) to determine electric power delivered to aresistive load. Two limiting cases for Eq. (12) can be considered toverify physical intuition. First, as Rt,ext,avg / 0 KW�1, the case of nothermal resistance between the generator junctions and the ther-mal reservoirs, the temperature efficiency hT / 1, and the junctiontemperature difference (TTEG,h e TTEG,c) simplifies to the reservoirtemperature difference (Th e Tc). Second, as Rt,ext,avg / ∞ K W�1,the case of external thermal resistances much greater than thegenerator and filler thermal resistance, the temperature efficiencyhT / 0 and the junction temperature difference (TTEG,h e TTEG,c)approaches 0 K because effectively all of the available temperaturedifference is dropped across the external resistances.

The closed-formmodel, Eq. (12), can be compared quantitativelywith the iterative, Eqs. (2)e(3), and 1st order, Eq. (10), models byconsidering a device with nominal thermophysical properties ofthermoelectric material comparable to bismuth telluride com-pounds, given in Table 1. Thermoelectric legs for the model aresquare in cross-section (side length 15 mm) with a fill fraction ofFF ¼ 0.01. The fill fraction of vertically-aligned thermoelectric de-vices is defined to be the fraction of Atotal occupied by active ther-moelectric material. For electrical resistance calculations,interconnects are modeled as gold with width equal to the width ofthe adjacent thermoelectric leg, and thickness of 1 mm. Filler

Table 1Input thermoelectric material parameters for one possible device configuration.

S (mV K�1) r (U m) k (W m�1 K�1) ZT@300K

p-type 200 1.0e�5 2.0 0.6n-type �200 1.0e�5 2.0 0.6

material is air with thermal conductivity of 0.024 W m�1 K�1 andnegligible convective thermal transport inside the module.

The values for Rt,h and Rt,c include thermal resistance through a525 mm-thick monosilicon substrate as well as an external thermalresistance (i.e. heat exchanger) between the substrate and thermalreservoir. The nominal values for these external thermal resistancesare 1 � 10�5 m2 K W�1 to represent conduction from a solid hot-side heat reservoir (e.g. hot water pipe) and 1 � 10�4 m2 K W�1

to represent fin-assisted natural convection to a cold-side reservoir(e.g. static air) at ambient conditions. These values are characteristicof a solid bond contact with spreading into a metal surface for thehot side and small commercially-available machined-aluminum finarrays (approximately 10 K W�1 total thermal resistance), madearea-specific using a nominal thermoelectric device footprint of10 mm2. The results from the 1D iterative, closed-form, and firstorder approaches are compared in Fig. 4, using matched electricalloading conditions, where the load electrical resistance in Fig. 2b isequal to the internal electrical resistance of the thermoelectricgenerator.

The proposed closed-form solution shows excellent agreementwith the iterativemodel and a noticeable improvement over the 1storder model, particularly near maximum power conditions, wherethese devices are intended to operate. Nominal parameters used toproduce simulation data are given in Table 2. We note that in the

domain of microscale generators, electrical contact resistance in theseries electrical pathway is significant and included, while thermalcontact resistances in the parallel heat pathway are not andignored.

Fig. 4. Comparison of iterative 1D model (black circles calculated using Eqs. (2)e(8)),simplified first-order model (dashed red line calculated using Eqs. (10) and (8)), andthe proposed closed-form model (solid black line calculated using Eqs. (12) and (8)).(For interpretation of the references to colour in this figure legend, the reader isreferred to the web version of this article.)

Table 2Nominal simulation parameters.

Parameter Nominal value

Net Seebeck coefficient, Snet 400 mV K�1

Thermoelectric material thermal conductivity, kTE 2 W m�1 K�1

Thermoelectric material electrical resistivity, rTE 1.0 � 10�5 U mSemiconductor/metal electrical contact resistance, rcontact 5.0 � 10�11 U m2 [12,22]Cold reservoir temperature, Tc 293.15 KHot reservoir temperature, Th 303.15 KTotal device area, Atotal 1 � 10�5 m2

Individual leg cross-sectional area, Aleg 2.25 � 10�10 m2

Au interconnect thickness, tIC 1 mmArea-specific thermal resistance to heat source, Rt,h � Atotal 1.35 � 10�5 m2 K W�1

Area-specific thermal resistance to heat sink, Rt,c � Atotal 1.035 � 10�4 m2 K W�1

Silicon substrate thickness 525 mmSi3N4 passivation thickness 100 nm

M.T. Dunham et al. / Energy 93 (2015) 2006e2017 2011

4. Device design considerations

Proper design of a thermoelectric generator for maximum po-wer output relies on optimization of thermal resistance versuselectrical resistance. To maximize electrical power from thegenerator, the thermal resistance of the active thermoelectric layershould be comparable to that of any inactive layers such as sub-strates and heat sinks. Increasing the ratio Rt,TEG/(Rt,hþ Rt,c) leads toa higher temperature efficiency, initially increasing the powerthrough the (TTEG,h e TTEG,c) term in Eq. (8). However, the means ofincreasing this thermal resistance geometrically (i.e. by increasingleg length or decreasing leg cross-section) also increase the elec-trical resistance, which increases Re,TEG. With all other parametersremaining constant, power output is maximized for a specific valueof leg length or a specific value of fill fraction.

4.1. Thermoelectric element length

Changing the length of each thermoelectric element is astraightforward way to adjust the thermal and electrical resistance.Under fixed reservoir temperature conditions, an increase in leglength at first rapidly increases the temperature drop across thethermoelectric layer before the rate of change slows and asymp-totically approaches the reservoir temperature difference (hT / 1).In contrast, the electrical resistance continues to increase linearlywith increasing leg length. These competing effects lead to amaximum, where we find an initial rise in power output due torapidly increasing thermal resistance for short leg lengths followedby a decrease in power output due to the continuing increase inelectrical resistance with diminishing benefits in thermal resis-tance. Key results from this work plotted vs. thermoelectric leglength, lTE, are given in Fig. 5. While the present analysis focuses ona small temperature differential, raising the source temperature Thdoes not appreciably alter the optimal leg length.

The competing temperature and electrical resistance effects areshown in Fig. 5a. In an ideal system, themaximummatched power isreached when half of the available reservoir temperature differenceoccurs across the thermoelectric layer (hT ¼ 0.5). However, whenparasitic losses such as interconnect electrical resistance are includedas they are here, the target fractional temperature difference hTchanges, and the maximum power condition does in fact occur at alonger leg length than the hT ¼ 0.5 condition resulting in hT,max po-

wer > 0.5. This observation is discussed in a later section of this work.

4.2. Fill fraction

The fabrication method imposes limits on maximum attainableleg length, particularly in the domain of microdevices constructed

using bottomeup processes like sputtering or electrodeposition.This poses a constraint on the maximum thermal resistance thatcan be attained by simply extending the thermoelectric element.The fill fraction can be used to compensate for this limitation.Whilea very high fill fraction may seem desirable for efficient use of de-vice real estate, it will be shown here that it is not desirable forpower output except for very long elements. In fact, when elementlengths are limited to ~1e10 mm as is common with current thinfilm technology, optimal fill fractions are very low (<10%).

The impact of thermoelectric leg length and device fill fraction(varied by changing nTC with constant element cross section) onpower output is illustrated in Fig. 6. The data show that amaximumpower output exists at some fill fraction for a fixed leg length, andthat for very short leg lengths, this optimum fill fraction approacheszero. At very long leg lengths, the maximumwould appear to occurfor FF > 1, which is non-physical. The thermal resistance due to thethermoelectric under these conditions is too large to reach themaximumpower conditionwith any fill fraction. This motivates thethin-film fabrication approach, as optimal leg lengths for achiev-able fill fractions may be much shorter than what is possible withbulk machining processes. As seen in Fig. 6, the maximum attain-able power output occurs when higher FF is used in combinationwith longer leg length, as long as their combined effect on thermalresistance remains optimal. It is important to note here that themaximum power output for each leg length would be identical,though still occurring for different fill fractions, if electrical lossesdue to the interconnects and thermal shunting due to the fillermaterial were neglected. It will be shown that increasing fill frac-tion by increasing the number of thermocouples reduces theparasitic effect of non-ideal interconnects.

Fill fraction also has an effect on the sensitivity of the optimal leglength to the reservoir temperature gradient. Increasing the fillfraction leads to a longer optimal leg length, as shown in Fig. 6, andat this leg length, a larger electric power generation. The resultinglarger Peltier heats diminish the steady state junction temperaturedifference, and a longer leg length must be reached to compensate.This effect is shown for two fill fractions, 0.01 and 0.50, in Fig. 7. Theplots also show an increasing error with increased temperaturegradient at higher fill fractions due to the approximation made forthe Peltier heat contribution in the closed-form equation, but re-sults in only a 2.3% difference in calculated optimal leg length for airfiller, 50% fill fraction, and 100 K available reservoir temperaturedifference.

If the Peltier heat contribution is removed from the equations,the optimal leg length is independent of temperature difference atany fill fraction. For significant temperature differences across thethermoelectric material, temperature-dependent properties willbecome important and further impact this relationship.

Fig. 5. Plot of (a) matched load voltage, load power, and device electrical resistance trends vs. thermoelectric leg length; matched power vs. leg length for (b) FF ¼ 0.01 and (c)FF ¼ 0.50 assuming different filler materials; (d) power vs. leg length curves calculated using closed-form solution for three different parameter combinations giving the same valueof ZT (0.9) at room temperature.

M.T. Dunham et al. / Energy 93 (2015) 2006e20172012

4.3. Filler material

When optimization leads to a very low fill fraction, the thermalproperties of the filler material become critically important as therelatively large cross-sectional area may create a significant ther-mal shunt and reduce the temperature drop across the thermo-electric elements. The ideal filler material should have low thermalconductivity, considered here with a limiting case of vacuum(k¼ 0.0001Wm�1 K from first-order kinetic theory calculations forair at 1 Pa, 300 K, and a mean free path size-limiting dimension of~100 mm). In a real device this is not always a practical option aspreservation of vacuum conditions means carefully sealing a mm-scale structure and adding a competing thermal shunt through theseal. Air (k¼ 0.024Wm�1 K�1) is a simple option, but oxidation can

be a concern and it does not provide structural support duringfabrication or operation. Polymers with low thermal conductivitysuch as polyimide (k ¼ 0.14 W m�1 K�1) and SU-8(k ¼ 0.208 W m�1 K�1) [12,21] have been used in an attempt toadd structural integrity while minimizing thermal shunting, asopposed to the use of SiO2 (k ¼ 1.38 W m�1 K�1) [16] which has athermal conductivity comparable to bismuth telluride thermo-electric material.

Matched power per unit area vs. thermoelectric leg length isshown in Fig. 5b and c for FF ¼ 0.01 and FF ¼ 0.50, respectively, anda variety of candidate filler materials. Each curve shows the samegeneral trend: maximum matched power decreases and occurs atlonger leg lengths with increasing filler thermal conductivity. Witha more prominent thermal shunt (or large thermal gradients

Fig. 6. Matched power per-unit-area vs. fill fraction for different thermoelectricelement lengths, using the results of the closed-form solution from Eq. (12).

M.T. Dunham et al. / Energy 93 (2015) 2006e2017 2013

combined with moderate-to-high fill fractions), a longer leg lengthmust be attained to reach the optimal temperature drop across thethermoelectric. This increased leg length increases electricalresistance and decreases power output. When the fill fraction isincreased, two primary effects are seen. First, the impact of thespecific filler material becomes less significant, and in fact, vacuumand air show nearly identical power curves at 0.50 fill fraction. Thisis simply due to the fact that there is less filler material, and itspresence becomes negligible as FF/1. Second, higher maximummatched powers are reached, but at longer leg lengths. The reasonthat different maximum power outputs are reached is due to thereduction in parasitic losses, both thermal losses from a smaller

Fig. 7. Dependence of optimal leg length on reservoir temperature gradient for fill fractionslines calculated using the full iterative 1D model are given for accuracy comparison.

thermal shunt and electrical losses from the interconnects. For lowfill fractions, the thermal resistance is very high and the fraction ofreservoir temperature difference dropped across the thermoelec-tric is large at short leg lengths. Hence, the electrical resistance dueto the legs is relatively small, and parasitic losses through the in-terconnects contribute significantly. When fill fraction increases,the thermal resistance decreases, and a longer leg length must bereached to force the optimal temperature drop. The presence of afiller material with nonzero thermal conductivity will reduce theparallel thermal resistance. The following relationship appropri-ately matches the filler and thermoelectric thermal conductivitiesbased on fill fraction FF and maximum desired thermal resistancereduction factor FR,max:

kfillerkTE

� FR;maxFF

1� FF(13)

For example, with a fill fraction of 0.1 and a desired maximumthermal resistance reduction (from perfect vacuum filler condi-tions) of 0.1 (or 10%), materials should be selected such thatkfiller � 0.011kTE.

4.4. Number of thermocouples

Another design consideration is the selection of the number ofthermocouples. The fill fraction is non-unique in that the total areaoccupied by thermoelectric material depends on both the numberof legs and the cross-sectional area of each leg. Therefore multiplecombinations of thermocouple number density and individual legcross-section will result in the same fill fraction, as illustrated inFig. 8.

Under ideal conditions (i.e. no parasitic thermal or electricallosses), matched power output will be identical for any combina-tion of thermocouple count and leg cross-section which results inthe same fill fraction for the same total area. Beginning with thedefinition of fill fraction as the ratio of thermoelectric material crosssectional area to total footprint:

of (a) 0.01 and (b) 0.50. Solid lines utilize the derived closed-form Eq. (12), while dotted

Fig. 8. Configurations with (a) nTC ¼ 2 and (b) nTC ¼ 8 each resulting in the same fillfraction, with different numbers of thermocouples and leg cross-section.

M.T. Dunham et al. / Energy 93 (2015) 2006e20172014

FF ¼ 2nTCAleg

Atotal/Aleg ¼ AtotalFF

2nTC(14a)

Under matched loading conditions, the voltages dropped acrossthe generator and the load resistance are equivalent:

Vmatched ¼ nTCSnet�TTEG;h � TTEG;c

�2

(14b)

The electrical resistance of the series of thermoelectric elementsnot accounting for parasitic interconnect or contact resistance, andtherefore the load for matched loading conditions, can be related tothe fill fraction FF:

Re ¼ 2nTCrTElTEAleg

¼ 4n2TCrTElTEAtotalFF

(14c)

We then calculate the power delivered to the load using Eqs.(14b) and (14c):

Pmatched ¼ V2matchedRe

¼ AtotalS2netFF

�TTEG;h � TTEG;c

�216rTElTE

(14d)

With fill fraction constant, the thermal resistance is unchanged.The Peltier and Joule heating terms are the only remaining pa-rameters in the heat balance which could affect the matched powercalculation. The electrical current in terms of the number of ther-mocouples nTC is:

I ¼ VmatchedRe

¼ AtotalSnetFF�TTEG;h � TTEG;c

�8nTCrTElTE

(15a)

The Peltier heating contributions at the hot and cold junctionsare dependent on fill fraction FF, but independent of the number ofthermocouples nTC:

_QPeltier;h ¼ �nTCSnetITTEG;h ¼ �TTEG;hAtotalS2netFF

�TTEG;h � TTEG;c

�8rTElTE

(15b)

_QPeltier;c ¼TTEG;cAtotalS2netFF

�TTEG;h � TTEG;c

�8rTElTE

(15c)

Under matched electrical loading, the power delivered to theload is equivalent to the power dissipated in the generator by Jouleheating.

For matched load conditions and constant total area, the heatbalance equations will not change for any combination of ther-mocouple number and leg cross-section that gives the same fillfraction. In the absence of parasitic losses, this property can be usedto design the generator for a desired voltage output or electricalresistance. Increasing the number of thermocouples should be thepreferred approach to reach a target load voltage when expectedsource/sink temperatures are known, as voltage boost circuits mayresult in significant electrical conversion penalties.

nTC ¼ Vgen;desired

Snet�TTEG;h � TTEG;c

� (16)

If parasitic losses are considered, particularly due to electricalresistance over the interconnect length between each thermo-electric element and contact resistance at the semi-conductoremetal junctions, the power output may vary withdifferent combinations of element area and number of thermo-couples. The electrical resistance along total length of all in-terconnects within the device is derived geometrically as:

Re;total;IC ¼ 2nTCrIC

ffiffiffiffiffiffiATEFF

qþ

ffiffiffiffiffiffiffiffiATE

ptIC

ffiffiffiffiffiffiffiffiATE

p (17a)

Utilizing the generator electrical resistance from Eq. (14c) andcalculating the ratio of parasitic to generator electrical resistance:

Re;total;IC þ Re;total;contactRe;total;TE

¼rIC

ffiffiffi1FF

pþ1

tICþ 2 rcontact

Aleg

rTElTEAleg

¼AtotalFFrIC

ffiffiffiffi1FF

qþ 1

!

2nTCrTEtIClTEþ 2

rcontactrTElTE

(17b)

Eq. (17b) shows that when keeping fill fraction constant theparasitic electrical resistance of the interconnects and semi-conductor/metal junctions relative to the resistance of active

M.T. Dunham et al. / Energy 93 (2015) 2006e2017 2015

thermoelectric material is related to the inverse of the number ofthermocouples. Once the fill fraction has been chosen for optimalthermal resistance, it is beneficial to utilize many thermocoupleswith smaller cross-sectional area. This is typically convenient asincreasing the number of thermocouples will increase the outputSeebeck voltage from the device. The legs should also be as long aspossible to minimize the relative impact of contact resistance.

The required temperature efficiency for maximum matchedpower versus fill fraction is shown in Fig. 9 for different numbers ofthermoelectric leg pairs and filler materials in a 10 mm2 footprint.In the idealized case of interconnects with infinite electricalconductance, no electrical contact resistance, and negligible para-sitic thermal losses, the optimal temperature drop is always half ofthe available temperature difference from the thermal reservoirs,as discussed in the literature [18]. When parasitic losses areincluded, a larger thermal resistance (i.e. longer leg length) must bereached to compensate. Furthermore, the required temperatureefficiency for maximummatched power is significantly higher thanthe ideal temperature efficiency when the thermal conductivity ofthe filler material is very low. This demonstrates a weakness of thevast majority of previous system analyses, which not only neglectthe impact of interconnect and contact resistance but assume nocontribution from the filler material, (i.e. assuming no parasiticinterconnect resistance and the case for which this resistance ismost significant). The concavity of the curve depends on the ratio ofthermal resistance of the filler region to that of the active ther-moelectric elements. If the filler material has very low conductivity,then the thermal shunt through filler material is very small even atlow fill fractions, and it is beneficial to have a higher fraction of theavailable temperature difference across the active thermoelectricmaterials. Conversely, for high filler material conductivities there isa significant thermal shunt through the filler region, particularly atlow fill fractions. In this case it is less desirable to have a largefraction of the available temperature difference dropped across thefiller/thermoelectric.

The varying optimal temperature efficiency is a factor of theparasitic loss introduced by the interconnect electrical resistancesince the geometry (e.g. leg length, fill fraction, etc.) must be alteredto give a larger temperature difference, and the resulting increasedSeebeck voltage compensates for the additional voltage dropsthrough the series circuit. Inclusion of the Peltier heat contributionalso affects the optimal temperature efficiency, although lesssignificantly. If it is neglected as in the first order approximation

Fig. 9. Temperature efficiency at maximum matched power output vs. fill fraction,where fill fraction is adjusted by changing the area of thermoelectric elements withconstant hTC and tIC. At each fill fraction value, the thermoelectric leg length lTE isadjusted to reach maximum matched power.

[17,21], the optimal temperature efficiency is even higher, as thattemperature-dependent loss is no longer contributing to the ther-mal equation.

4.5. Incompleteness of ZT

Many analyses suggest that the magnitude of the parameter ZT,regardless of its composition, is the ultimate rule to device per-formance. However, due to the optimization balance of thermal andelectrical resistance and the concurrent presence of thermal andelectrical conductivity in Z, the optimal device design will bedifferent depending on the ratio of these parameters even foridentical values of Z.

The data in Fig. 5d and Table 3 reveal key points to consider forthermoelectric device design and materials selection. First, thetuning of individual material parameters does not give equivalentdevice performance, even if they result in the same ZT. This is due tothe competing interaction between thermal and electrical resis-tance and how they differ according to device design. Second,depending on the attainable film thickness, it may be beneficial tooptimize one parameter over another, if possible. While increasingthe Seebeck coefficient consistently increases the performance inthis example more than changes in k or r for a given design, if theleg lengths are very short (e.g. less than about 5 mm in Fig. 5d) it isbetter to reduce thermal conductivity than to increase electricalconductivity. If longer leg lengths are attainable, the opposite istrue. Finally, the optimization of the different parameters can leadto a shift in the optimal thermoelectric leg length. While increasingS does not alter the condition of maximum matched power, adecrease in k results in a decreased optimal leg length. This isbecause the thermal resistance of the device is increased and theideal temperature drop across the thermoelectric elements can beachieved without adding increased electrical resistance.Conversely, when the device is optimized by increasing electricalconductivity the leg length at maximum matched power isincreased. This is because a longer leg length can be reached torealize an optimal temperature drop before added electrical resis-tance from increased leg length becomes dominant.

These trends become even more impactful when examined inmulti-parameter space. A contour plot of maximum matched po-wer for varying thermal conductivity and net Seebeck coefficient isgiven in Fig. 10. Electrical resistivity and fill fraction each remainfixed to nominal values, and the leg length is adjusted at each datapoint to maximize the matched power output. Iso-ZT lines areshown in red, while two isopower lines are provided in blackindicating 1.0 and 2.0 mW cm�2. As one example from this plot, thesame maximummatched power of 2.0 mW cm�2 can be reached ina device with ZT values of 0.6 and 1.4 within this parametric space.The variable system parameters for each point (indicated with ablack box marker) are Snet ¼ 435.4 mV K�1, kTE ¼ 2.36 W m�1 K�1,lTE,max ¼ 16.4 mm (ZT ¼ 0.5987), and Snet ¼ 396.0 mV K�1,kTE ¼ 0.822 W m�1 K�1, lTE,max ¼ 13.0 mm (ZT ¼ 1.422).

These findings are critical for device design decisions since areview of thermoelectric literature reveals an abundance of focuson tuning only ZT values for material and device performance.

5. Summary and concluding remarks

This work derives a closed-form solution that closely matchesthe coupled iterative solution for use in thermoelectric micro-generator modeling, targeting low-temperature waste heat recov-ery applications. The proposed formula yields a noticeableimprovement in accuracy over simplified first-order models foundin literature, while eliminating the need for numerical iteration inthe coupled 1D heat transfer model. Additionally, the importance of

Table 3Maximum matched power and corresponding TE element leg length for a 50% increase in ZT by modifying each parameter of Z individually.

kTE (W/m-K) rTE (U-m) Snet (mV/K) ZT @ 300 K Pmatched,max (mW/cm2) lTE,max (mm)

1.325 1e�5 400 0.9 1.91 14.12 6.63e�6 400 0.9 2.13 18.52 1e�5 491 0.9 2.62 15.7

Fig. 10. Contour plot of maximum matched power per area for ranges of thermalconductivity and net Seebeck coefficient and 0.01 fill fraction. ZT values are included asred isolines for reference, and two black isopower lines are included at 1.0 and2.0 mW cm�2. (For interpretation of the references to colour in this figure legend, thereader is referred to the web version of this article.)

M.T. Dunham et al. / Energy 93 (2015) 2006e20172016

often-neglected device design considerations has been illustrated,including fill fraction, filler material, number of thermocouples, andthe inequality of the different parameters of the thermoelectricfigure of merit ZT, focusing on matched power output under near-ambient fixed-temperature thermal reservoir conditions.

Key conclusions include the fact that a low fill fraction may beused to compensate for inadequate thermal resistance fromfabrication-limited leg lengths. Furthermore, the transport prop-erties of filler material may seriously impact the performance ofdevices fabricated with thin-film processing techniques. Increasingthe number of thermocouples as much as possible for a given fillfraction is desirable to minimize parasitic losses due to the elec-trical interconnects. Finally, when attempting to increase ZT forimproved device performance, it is best to increase Seebeck coef-ficient, followed by decreasing thermal conductivity for short leglengths and increasing electrical conductivity for long leg lengths.The reported model and observed trends can be used to guidedesign decisions for thermoelectric generators, particularly in themicroscale thermal energy harvesting domain.

Acknowledgments

This work was supported by Analog Devices, Inc., as well as bythe United States Department of Defense through the NationalDefense Science and Engineering Graduate Fellowship program.

Nomenclature

A area (m2)FR,max thermal resistance reduction factorFF fill fraction (�)I electric current (A)

k thermal conductivity (W m�1 K�1)lTE thermoelectric leg length (mm)nTC number of thermocouple pairs_Q heat rate (W)Re electrical resistance (U)Rt thermal resistance (K W�1)S Seebeck coefficient (mV K�1)t thickness (mm)T temperature (K)V electric voltage (V)ZT thermoelectric figure of merit (�)

GreekhT temperature efficiencyr electrical resistivity (U m)

Subscriptc cold sidefiller related to material surrounding the thermoelectric

materialgen generated by Seebeck effecth hot sideIC interconnectJoule related to Joule heating effectsleg related to a single thermoelectric legload related to the connected electrical loadmatched under matched electrical loading conditionsmax corresponding to maximum power conditionnet difference of n- and p-type material propertiesPeltier related to Peltier heating/cooling effectsTE related to the thermoelectric materialTEG related to the thermoelectric generator moduletotal related to the total device

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