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Modeling of on-membrane thermoelectric power supplies
Sensors and Actuators A 116 (2004) 501–508
Modeling of on-membrane thermoelectric power supplies A. Jacquot a,∗ , G. Chen b , H. Scherrer a , A. Dauscher a , B. Lenoir a a
Laboratoire de Physique des Matériaux, UMR CNRS-INPL-UHP 7556, Ecole Nationale Supérieure des Mines de Nancy, Parc de Saurupt, F-54042 Nancy, France b Mechanical Engineering Department, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Received 12 May 2004; accepted 25 May 2004 Available online 27 July 2004
Abstract The geometry of on-membrane thermoelectric power supplies has been optimized to obtain the highest useful electric power for a given heating power. Assuming the use of materials with moderate thermoelectric performances, an analytical one-dimensional model has been developed to find the optimum thermoelectric film thickness and leg length together with the best heat source configuration. The optimal geometry of these thermoelectric devices is discussed according to the thermal properties and emissivity of the materials as well as based on their working environment, i.e. air or vacuum, and on their operational mode. These investigations have been extended to more realistic micro-device geometries and to more efficient thermoelectric materials, using a numerical model. The performance of on-membrane thermopile is predicted for various materials and geometry combinations. © 2004 Elsevier B.V. All rights reserved. Keywords: Power supplies; Thermoelectricity; Thermopile; Modeling; Microgenerator
1. Introduction There is a growing demand in consumer electronics for small, inexpensive and environmentally friendly power sources. This demand is driven by progresses made in electronic components miniaturization and smart power management. Thermoelectric technology can readily be miniaturized in principle without loss in efficiency using micro-fabrication technology. No moving part is required and various heat sources can be used, thus making micro-thermoelectric devices good candidates to power wearable consumer electronics. Miniaturization decreases both material-cost and device weight and increases the output voltage of thermoelectric generators in addition. On the other hand, in practice, it has been proven difficult to make both cost-competitive and efficient small thermoelectric devices with the geometry of commercially available thermoelectric modules mainly because the micro-fabrication technologies are planar by nature and because thick films technology of thermolectrics is in its infancy [1–3]. The impact of electrical contact resistance on micromachined device performance is an additional ∗ Corresponding author. Tel.: +33 3 83 58 41 70; fax: +33 3 83 57 97 94. E-mail address: [email protected] (A. Jacquot).
0924-4247/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.sna.2004.05.023
source of concern since it becomes much more severe when downscaling thermoelectric devices [4]. Planar devices using thermoelectric effects have been successfully miniaturized very early to make radiation sensors and, quite recently, to make single-chip micro-thermostats for both active heating and active cooling applications [4–6]. The thermoelements used to make these micro-devices are supported by a thin film free of its substrate called a membrane. These microsystems take advantage either of the large surface-to-volume ratio of thermoelectric thin films to increase the sensitivity and decrease the response time of radiation sensors, or of the active heating and cooling enabled by thermoelectric technology to potentially extend the thermal operating range of micro-thermostats to well-below ambient temperature [7]. Although works have already been done on small-scale thermoelectric generators [8,9], there is still some need for a theoretical framework to optimize their geometry since the actual performances of such devices are low. One reason is that thin films technology is not suitable to get high heat flux and high output power. Furthermore, large forecasted heat losses by convection and radiation have weakened the trust people put in thermoelectric devices with this geometry for power supplies. In the meantime, however, the large thermal resistance of on-membrane thermopiles could be used to get large temperature differences along the thermoelements with a very
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small heat source and the large surface-to-volume ratio could be used to efficiently cool thermoelectric devices by air. These intrinsic characteristics of on-membrane thermopiles may improve the Carnot efficiency of thermoelectric generators in a given working environment. Recently the fabrication of such in-plane thermoelectric microgenerators has been demonstrated with polycrystalline silicon as the thermoelectric material and proof of this concept was obtained experimentally [10]. In this paper, the modeling of an on-membrane thermopile is presented. When the membrane length to width ratio is high, analytical formulas are derived to calculate the best heat source configuration and both the optimum thermoelectric leg length and thickness. A numerical model valid for any membrane length to width ratio has been developed to gain further insight into the performance of these thermoelectric devices. Three potential applications have received our attention: a solar-powered micro-thermoelectric generator (SmTG), a radioisotope-powered micro-thermoelectric generator (RmTG) and a body-heat powered micro-thermoelectric generator (BHPmTG).
2. Analytical modeling The building-block of thermopile-like microgenerators is shown in Fig. 1. It consists of a frame having a high thermal conductivity supporting a thin film being the substrate for thermoelectric legs connected in serial. Two working modes can be considered for this building-block. The first working mode is when the heat source is on the thin film substrate and the supporting frame is the cold side. The potential applications of this working mode are the SmTG and RmTG (Fig. 1a). The second working mode is when the heat source is attached to the supporting frame and the membrane is cooled by air (Fig. 1b). The large surface-to-volume ratio and high thermal resistance of the membrane are used to efficiently dissipate the heat in air and to produce a large temperature gradient along the thermoelements. A BHPmTG
could advantageously make use of this configuration, for instance. 2.1. Simplified model when neglecting the heat losses by radiation and convection In this model, it is assumed that the electrical current in the thermoelectric legs does not change the temperature distribution on the membrane. This assumption is valid if the thermoelectric materials used to make the thermoelectric legs do possess low dimensionless figures of merit ZT (ZT = σS 2 T/λ, where T, σ, S and λ are the absolute temperature, the electrical conductivity, the thermopower and the thermal conductivity, respectively). Under this binding hypothesis, the temperature distribution on the membrane can be simply calculated by solving the heat transfer equation. If a one-dimensional heat transfer along the thermoelectric legs is assumed and if the heat losses by radiation and convection are neglected, it can be easily demonstrated that the temperature drop along the thermoelements T is given by α (1 − α) α2 β 2 α2 β (1 − β) T = q0 l2 (1) + + λ3 d 3 2λ2 d2 λ2 d 2 In this equation, q0 is the heat generated by the heat source per surface unit, l = l1 + l2 + l3 is the half width of the thin film substrate (l1 , l2 and l3 are defined in Fig. 2), α = (l1 + l2 )/ l is the coverage of the thin film substrate by the heat source and β = l2 /(l1 + l2 ) is the overlapping of the heat source by the thermoelements. λi and di are the average thermal conductivity and the thickness of part i, respectively. The thermal conductivity of the part 3 is defined by λ3 =
λ S d S + λ T dT dS + d T
(2)
where λS dS and λT dT are the thermal conductivities and thicknesses of the thin film substrate and of the thermoelectric material averaged over the values of the n and p type materials, respectively.
Fig. 1. Overview of the building-block of the on-membrane thermopile taken into consideration. Applications of the on-membrane thermopile building-block to make (a) solar or radioactive powered micro-thermoelectric generators and (b) body-heat powered micro-thermoelectric generators. In the first case, the supporting frame is the cold side and in the second case it is attached to the heat source while the membrane is being cooled by air.
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Fig. 2. Cross-sectional view of an on-membrane thermopile. (1) Thermoelectric legs; (2) absorbing layer for the SmTG, radioactive heat source for the RmTG or metallic heat spreader for the BHPmTG; (3) thin film substrate; (4) heat sink for the SmTG and RmTG or frame in contact with the heat source for a BHPmTG. The calculations are done on half of the membrane according to the symmetry plan.
result of a competition between the heating power that rises when the heater surface increases, the increase of the thermal conductance (i.e. the decrease of the temperature rise along the thermoelectric legs) and the reduction of the electrical resistance when the thermoelectric leg length decreases. Note that these results are strictly speaking only valid under the binding hypotheses of the model and in the particular working mode of the RmTG. Nevertheless, we will see that this crude model is very useful to understand the results developed in the next paragraphs of both the analytical model, that takes into account the heat losses by radiation and convection, and the numerical simulations of more realistic devices in RmTG, SmTG and BHPmTG working modes.
The maximum output electrical power WL per unit of length of on-membrane thermopiles is given by
2.2. Simplified model when accounting for the heat losses by convection and radiation
WL =
N 2 S 2 T 2 4(RL = RG )
(3)
where N and RG are the number of thermoelectric legs and the internal resistance per unit of length of the thermopile, respectively and S, the Seebeck coefficient averaged over the values of the n and p type materials. The load resistance RL is adjusted to match the internal resistance of the thermopile to calculate the maximum useful electric power on the load. Interesting information can be obtained when l2 = 0 (β = 0). The maximum electrical power produced at a given heating power is obtained for an optimum thermoelectric leg thickness derived from ∂WL /∂dT |β=0 = 0 and is given by λT dT = λS dS
2 l1 = l 3
T1 =
T2 =
(4)
The associated optimum leg length is derived from ∂WL /∂α|β=0 = 0 and is given by α=
If the heat losses by radiation and convection are taken into account, the temperature distributions Ti in part i (Fig. 2) are derived by solving the heat transfer equation and are given by
(5)
Eq. (4) shows that optimum thermoelectric film thickness is reached when the thermal conductance of the thermoelectric film is equal to the thermal conductance of the d1 exp(ρ2 αβl) ch (ρ2 αβl) 1 − λ2 ρ 2 d 2 sh (ρ2 αβl) −1 ch (ρ1 α(1 − β)l) d1 ch (ρ2 αβl) − λ1 ρ1 sh (ρ1 α(1 − β)l) λ2 ρ2 d2 sh (ρ2 αβl) q0 /A3 = q0 /A2 − q0 /A1 substrate. The origin of this optimum arises from a competition between the temperature difference along the thermoelectric legs, that will be higher with thinner legs, and the electrical resistance of the device that increases when the thermoelements’ thickness decreases. If we assume the heating power to be proportional to the area covered by the heater, the optimum leg length calculated from Eq. (5) is the
T3 =
q1 ch(ρ1 x1 ) q0 − + T0 A1 λ1 ρ1 sh(ρ1 α(1 − β)l)
(6)
q d1 2 − 1 exp(ρ2 x2 ) + λ 2 ρ2 d2 1 − exp(−2ρ2 αβl) q1 q q0 × − 2 exp(−ρ2 αβl) ch(ρ2 x2 ) + + T0 d2 d1 A2 (7) 1 d2 sh (ρ3 ((1 − α)l − x3 )) q + T0 λ3 ρ 3 d 3 2 ch (ρ3 (1 − α)l)
(8)
where xi ∈ [0, li ]
(i = 1, 2, 3)
(9)
In Eqs. (6–8), q1 and q2 are the solutions of the following linear system of equations: 1 ch (ρ2 αβl) d2 sh (ρ3 (1 − α)l) + λ3 ρ3 d3 ch (ρ3 (1 − α)l) λ2 ρ2 sh (ρ2 αβl) q1 q2 1 1 λ2 ρ2 sh (ρ2 αβl) (10) where
Ai λi d i
(11)
Ai = 2hi + 8σB εi T03
(12)
ρi =
The coefficients hi , εi , σ B and T0 are the heat transfer coefficient of convection and the emissivity of part i, the
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Stefan–Boltzmann constant and the room temperature, respectively. By setting β equal to zero in Eq. (10) and by equaling T1 |x1 =l1 and T3 |x3 =0 , we can easily verify that Eqs. (6), (8) and (10) are reduced to the equation obtained by Kozlov [11]. By doing similar calculations, it is easy to show that Eqs. (7) and (10) are reduced to the equation obtained by the same author when β is equal to one [12]. The optimum geometry of on-membrane thermopiles is found numerically by maximizing the electrical power calculated by Eq. (3) by adjusting the value of the three independent variables α, β and dT . The initial values for α and β are first obtained with dT = λS dS /λT . These values are then used to calculate a new value for dT . This procedure is repeated until the electrical power does not change significantly. The temperature drop along the thermoelements in Eq. (3) is given by T2 |x2 =0 − T0 .
The results of the geometry optimizations are presented in Fig. 3. They are obtained with an overall emissivity of 0 and 1 (εi = 0 or 1) with no heat transfer by convection (hi = 0) and as a function of the heat source thermal conductivity λQ , which can either be an absorbing layer (SmTG) or a radioactive source (RmTG). The calculations were performed for thermoelements made of silicon, silicon germanium and bismuth telluride. The material properties and film thicknesses used to make these calculations are given in Table 1. The thermoelectric parameters are considered to be an average of the n and p type material properties. It can be seen that when the overall emissivity is zero and λQ is larger than 2.8 W m−1 K−1 , the overlapping of the heat source by the thermoelements β is zero. Moreover, it does not depend on the thermoelectric materials and the coverage α, of the thin film substrate by the heat source, which is 2/3. This last result is consistent with the result that could have been obtained straightforwardly from Eq. (5).
Fig. 3. Optimum geometry of the microgenerator to maximize its electrical power. Dotted line: optimal coverage α of the thin film substrate by the absorbing layer (SmTG) or the radioactive source (RmTG). Solid line: overlapping β of the absorbing or the radioactive source by the thermoelement. Dashed line: thickness of the thermoelectric elements dT made respectively of polysilicon, silicon germanium and bismuth telluride as a function of the absorbing layer or the radioactive heat source thermal conductivity λQ for an overall emissivity of 0 and 1.
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Table 1 Physical propertiesa of the materials selected for the simulations according to the literature
substrateb
Thin film Heat source Polysiliconb SiGec Bi2 Te3 d
d (nm)
σ (−1 m−1 )
S (V/K)
λ (W m−1 K−1 )
ZT300 K
900 800 Variable Variable Variable
– – 50000 102850 100000
– – 165 114 195
2.24 Variablee 14.5 4.7 1.9
– 0.028 0.085 0.60
a
The data are an average of the thermoelectric properties of n and p type materials. Data taken from [13,14]. c Data taken from [15]. d Data taken from [16]. e Ranging from 0.02 to 200 W m−1 K−1 when used in the analytical models and equal to 140 W m−1 K−1 when used in the numerical simulations. b
When λQ is lower than 2.8 W m−1 K−1 , β increases and the heater completely overlaps the thermoelements at λQ = 0.02 W m−1 K−1 . An overlapping of the thermoelements by the heater is advantageous when the thermal conductivity of the heater is low because the surface of the heater as well as the heating power are increased while the thermal by-pass through the heater at the thermoelements level is low. At intermediary values of the heater, the optimum thermoelement thicknesses raise sharply when λQ is reduced and then decrease slowly. This increase of the optimum thermoelectric film thickness can also be understood by the thermal by-pass through the heater at the thermoelements level and can be interpreted as an increase of λS dS in Eq. (4). Consequently, it is not surprising that the thickness of the thermoelements tends to their values calculated from Eq. (4) when λQ tends to zero even if the overlapping is complete. When the overall emissivity is one, the value of the heater thermal conductivity, at which β is no more zero, is 9.5 W m−1 K−1 . This value of the thermal conductivity does not depend on the materials the thermoelements are made of as previously noticed when the emissivity was zero. The heater covers a little more than 2/3 of the thin film substrate, whatever the thermal conductivity of the heat source is. The raise in α enhances the heating power, because the heat source surface is increased, while it reduces the effect of the heat losses by radiation, because the optimal thermal conductance of the thermoelements is increased. The optimum thermoelement film thicknesses are higher than when the emissivity is zero, because the heat losses by radiation can be interpreted as an increase of λS dS in Eq. (4). It is worth noticing that the better the thermoelectric materials (the larger the ZT values), the larger are the optimum thermoelements film thicknesses. This result can easily be understood by Eq. (4), because good thermoelectric materials usually have a very low thermal conductivity.
3. Numerical simulation To overcome the limitation of the analytical models, we have solved the coupled electrical and heat transfer problems of on-membrane thermopiles by the finite volume method.
This numerical model was used to evaluate the change in the optimum heater coverage of the thin film substrate and the change of the thermoelements thicknesses when the coupled electric charges and heat transfer as well as the finite thin film substrate size are taken into account. The thermal conductivity of both the heat source and the heat spreader is set to be 140 W m−1 K−1 . Consequently, we are assuming that there is no overlapping of the thermoelements by the heat source in the RmTG and SmTG as well as in the BHPmTG working mode. The calculations are done with the data reported in Table 1. The general problem should be solved using a three-dimensional grid. However, it can be reduced to a 2D problem by assuming that the heat flows in the plane of the membrane and that the temperature of the supporting frame is constant. The Peltier effect is taken into account through an additional heat flux Q at the junctions between the thermoelements and the interconnections not in contact with the silicon frame. Assuming that the Seebeck coefficient of the material used to make the interconnection is negligible compared to the thermoelectric material, the expression for Q is Q = SIT
(13)
where I is the electrical current in the thermoelements. The material properties were assumed to not depend on temperature. Consequently, the Thompson effect was neglected. The heat losses by radiation and the convective heat transfer were taken into account through the emissivity (ε) and the heat transfer coefficient for convection (h). We used linearized approximations for the radiative and the convective heat transfers. Numerical simulations have been performed to predict the optimum geometry and the useful electrical power of in-plane thermoelectric devices of various membrane sizes in the RmTG and SmTG (Table 2) and BHPmTG working modes (Table 3). The membrane areas are 1.6 × 1.6 mm2 for the calculation results presented in Tables 2 and 4 and 1.6 × 16 mm2 for those presented in Tables 3, 5 and 6. The results obtained by using the numerical simulation and the analytical formulas are compared in the case of the larger membrane length to width ratio, i.e. when the heat flow is almost one-dimensional.
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Table 2 RmTG and SmTG working modes, membranes 1.6 × 1.6 mm2 in size, 50 thermoelements, the heating power is 1 mW, V and W are the output voltage and the useful electrical power, respectively
Polysilicon SiGe Bi2 Te3
ε
lT ()
dT (nm)
ZTm
T (K)
V (V)
W (W)
0 1 0 1 0 1
180 150 180 150 180 150
150 190 460 570 900 1150
0.014 0.016 0.043 0.048 0.27 0.30
8.0 5.1 7.9 5.0 8.1 5.0
0.13 0.084 0.091 0.057 0.16 0.098
0.090 0.058 0.27 0.16 1.5 0.94
ZTm is the dimensionless figure of merit at the average temperature along the thermoelements, when the thermal bypass through the substrate is taken into account. Table 3 RmTG and SmTG working modes, membranes 1.6 × 16 mm2 in size, 500 thermoelements, the heating power is 5 mW
Polysilicium SiGe Bi2 Te3
ε
lT ()
dT (nm)
ZTm
T (K)
V (V)
W (W)
0 1 0 1 0 1
270 200 270 200 270 200
140 240 410 750 840 1450
0.014 0.018 0.041 0.053 0.26 0.34
9.9 4.2 9.9 4.1 9.8 4.2
1.6 0.70 1.1 0.47 1.9 0.83
0.58 0.24 1.7 0.7 9.4 4.0
3.1. Results and discussion The optimum legs thicknesses obtained by numerical simulations in the RmTG and SmTG modes for the largest thin film substrate (Table 3), when neglecting the heat losses by radiation, are close but a little lower than the values predicted by Eq. (4) (140, 425 and 1063 nm for polysilicon, SiGe and Bi2 Te3 , respectively). The agreement is fairly good for the materials with the lowest ZT, but is less good for the best thermoelectric material. These lower values can be understood when considering the assumptions made to derive Eq. (4). It was assumed that the electrical current passTable 4 BHPmTG working mode, membranes 1.6 × 1.6 mm2 in size, 50 thermoelements, h = 52.6 W m−2 K−1
Polysilicon SiGe Bi2 Te3
lT ()
dT (nm)
ZTm
T (K)
V (V)
W (W)
220 230 220
750 2300 4400
0.024 0.073 0.49
−7.4 −7.5 −7.4
0.12 0.086 0.14
0.29 0.86 4.87
Table 5 BHPmTG working mode, membranes 1.6 × 16 mm2 in size, 500 thermoelements, h = 52.6 W m−2 K−1
Polysilicon SiGe Bi2 Te3
lT ()
dT (nm)
ZTm
T (K)
V (V)
W (W)
290 300 290
1300 4000 7800
0.026 0.078 0.54
−8.3 −8.3 −8.2
1.37 0.95 1.6
3.5 10 57
Table 6 BHPmTG working mode, membranes 1.6 × 16 mm2 in size, 500 thermoelements, h = 26.3 W m−2 K−1
Polysilicon SiGe Bi2 Te3
lT ()
dT (nm)
ZTm
T (K)
V (V)
W (W)
330 340 330
800 2500 4900
0.025 0.074 0.50
−7.9 −7.8 −7.7
1.30 0.89 1.5
1.7 4.9 27
ing through the thermoelectric elements does not change the temperature distribution on the device (low ZT). In reality, it reduces the temperature difference along the thermoelectric legs because of the Peltier effect. It is therefore advantageous to decrease the thermoelectric legs thicknesses in the numerical simulation to get a higher temperature rise along the thermoelements, because the coupled electric charges and heat transfer are fully taken into account. The leg length obtained by numerical simulation when neglecting the heat losses by radiation is very close to the length calculated by Eq. (5) (270 m versus 266 m). When the membrane changes in area from 1.6×16 to 1.6× 1.6 mm2 , the optimum thermoelectric leg lengths decrease and the thermoelements’ thicknesses increase (Table 3). The thermal conductance of the smallest thin film substrate is larger than the thermal conductance of the largest thin film substrate for the same membrane thicknesses. Eq. (2) predicts in fact larger thermoelectric leg thicknesses when the thermal conductance of the thin film substrate increases. The decreasing leg length can be understood considering the thin film substrate perimeter-to-surface ratio. This ratio is lower when the membrane is a square than when the membrane is a rectangle, if the membranes are of equal surfaces. It is therefore advantageous to increase the heater surface (i.e. to decrease the leg length) of the square-shaped membrane, because the benefit of the increased heating power will be greater than the corresponding increase of the membrane’s thermal conductance. The optimum thermoelectric leg length decreases and the optimum thermoelectric thickness increases when the overall emissivity of the device is increased. The effect of the heat losses by radiation is similar to the thermal by-pass through the substrate in Eq. (4). As a consequence, the optimum film thickness will be greater. The optimum thermoelement lengths decrease, since this increases the device conductance and therefore the heat flowing by conduction versus the heat losses by radiation. The optimal thermoelectric legs thickness dT and leg length lT calculated by numerical simulations for the largest length-to-width ratio of the thin film substrate, are very close to those calculated by Eqs. (3), (7) and (10) when the overall emissivity is equal to one (polysilicon: lT = 200 m, dT = 240 nm versus 195 m and 250 nm for the numerical simulation results; SiGe: lT = 200 m, dT = 750 nm versus 195 m and 765 nm; Bi2 Te3 : lT = 200 m, dT = 1450 nm versus 195 m and 1900 nm). The deviation is again larger when ZT increases, because the Peltier effect is taken into account in the numerical simulation, whereas it is not in the analytical models.
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3.2. Optimization of body-heat powered micro-thermoelectric generators The useful electrical powers and the optimum geometry have also been calculated in the BHPmTG working mode by numerical simulations (Table 3). The air temperature is set to 293 K. The temperature of the supporting frame is set to 310.2 K, even if the skin temperature is probably lower than the internal body temperature. The membrane areas are 1.6 × 1.6 mm2 in Table 4 and 1.6 × 16 mm2 in Tables 5 and 6. The heat transfer coefficients of convection are 52.5 in Tables 4 and 5 and 23.6 W m−2 K−1 in Table 6. The optimum thermoelectric film thicknesses are far thicker in the BHPmTG working mode than in the RmTG and SmTG working modes. They are typically in the 5–10 m range, because heat is more effectively dissipated in air than in vacuum. The conclusions are the same for the leg lengths which are longer than in the RmTG working mode, because the heat source is at the supporting frame, the membrane being the cold side. It is found that the larger the heat transfer coefficient of convection is, the thicker the optimum thermoelectric film thickness (Tables 5 and 6). The optimum thermoelectric film thicknesses are two times thinner with a squared membrane than with a long rectangular membrane of the same width, but 10 times longer. A power of about 57 W, at a voltage of 1.5 V, can be produced from heat wasted by the body in the case of the largest membrane, by a light-weighted and compact thermoelectric generator, based on Bi2 Te3 thick-film technologies. This is potentially far larger than the thermoelectric microgenerator developed by Seiko to power their wristwatches [1].
4. Conclusions Analytical models and numerical simulations demonstrate that an optimum thermoelectric leg length and thickness exist to get the maximum useful electrical power from an on-membrane thermopile. The lower the thermal conductivity of the thermoelectric materials or the higher the heat losses by radiation and convection are, the thicker are the optimum thermoelectric film thicknesses. These modelings show that heat losses by radiation are detrimental to the radioisotope micro-thermoelectric generator working mode, but improve the efficiency of microgenerators when the heat source is attached to the frame of the thermopile. The large surface-to-volume ratio of the thin film substrate is used to improve the coupling between the heat reservoirs and the thermoelements in this case. Due to the large thermal resistance of these devices, small heat sources can be efficiently used by on-membrane thermopiles, because large temperature differences along the thermoelements can be obtained, resulting in an increase in Carnot efficiency. On-membrane thermopiles are all the more promising since they should easily be fabricated [9].
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Acknowledgements This work was supported by the Jet Propulsion Laboratory (contract 1217092) and DoD/ONR MURI (N00014-97-1-0576).
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Biographies A. Jacquot was born in Saint-Dié, France, in 1974. From 1998 to 2003 he was a doctoral student under the joint guardianship of the Polytechnical Institute of Lorraine, France, and the Martin-Luther-University of Halle-Wittenberg, Germany. From 2001 to 2002 he was an assistant researcher at the University of California in Los Angeles in the framework of his Military Service. He obtained his PhD degree in material science and engineering from the Polytechnical Institute of Lorraine in 2003. His current research interests are the measurement and improvement of thermoelectric properties of nanocomposites and superlattices. He is an author or co-author of more than 20 publications and proceedings. G. Chen received his BS and MS from Huazhong University of Science and Technology (China) in 1984 and 1987, respectively, and his PhD from UC Berkeley in 1993. He taught at Duke University (1993–1997) and UCLA (1997–2001) and is currently a professor at MIT. His research interests are focused on nanoscale transport phenomena, particularly thermal energy transport, and their applications in energy and information technologies. He is a recipient of the NSF Young Investigator Award and a Guggenheim Fellowship, and serves on the editorial board four journals. H. Scherrer was born in 1941 in France. He is Professor at the Henri Poincaré University in Nancy (France). Since 1985, his research activities have mainly focused on the field of thermoelectricity. He is the head of the thermoelectric team in the Laboratoire de Physique des
Materiaux. He is the President of the European Thermoelectric Society and a member of the board of the International Thermoelectric Society, of the Thermoelectric Academy of the Ukraine and of the Refrigeration Academy of Saint Petersburg. He is an author or co-author of more than 100 publications and proceedings and of seven book chapters. A. Dauscher was born in Strasbourg, France in 1958. She received her PhD degree in sciences from Louis Pasteur University in Strasbourg (France) in 1987. She worked as student during 10 years and then as a CNRS researcher in the field of heterogeneous catalysis in Strasbourg. In 1993, she joined the research team on thermoelectric materials in the Laboratoire de Physique des Materiaux in Nancy. Her current research interest is focusing on the preparation of both thermoelectric thin films, by pulsed laser deposition, and new thermoelectric bulk materials, besides the microstructural and physical characterizations of the samples. B. Lenoir was born in Rambervillers, France in 1965. He received the ESSTIN (Graduate School at Nancy, France) engineering degree in physics in 1990 and the PhD degree in material science and engineering from the National Polytechnics Institute of Lorraine in 1994. He is now an Assistant Professor at the École Nationale Supérieure des Mines de Nancy in the Laboratoire de Physique des Materiaux. His research activities are mainly concerned with the synthesis, the structural and physical characterization of thermoelectric materials prepared in the bulk or thin film form. He is an author or co-author of more than 70 papers and proceedings.
Modeling of on-membrane thermoelectric power supplies A. Jacquot a,∗ , G. Chen b , H. Scherrer a , A. Dauscher a , B. Lenoir a a
Laboratoire de Physique des Matériaux, UMR CNRS-INPL-UHP 7556, Ecole Nationale Supérieure des Mines de Nancy, Parc de Saurupt, F-54042 Nancy, France b Mechanical Engineering Department, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Received 12 May 2004; accepted 25 May 2004 Available online 27 July 2004
Abstract The geometry of on-membrane thermoelectric power supplies has been optimized to obtain the highest useful electric power for a given heating power. Assuming the use of materials with moderate thermoelectric performances, an analytical one-dimensional model has been developed to find the optimum thermoelectric film thickness and leg length together with the best heat source configuration. The optimal geometry of these thermoelectric devices is discussed according to the thermal properties and emissivity of the materials as well as based on their working environment, i.e. air or vacuum, and on their operational mode. These investigations have been extended to more realistic micro-device geometries and to more efficient thermoelectric materials, using a numerical model. The performance of on-membrane thermopile is predicted for various materials and geometry combinations. © 2004 Elsevier B.V. All rights reserved. Keywords: Power supplies; Thermoelectricity; Thermopile; Modeling; Microgenerator
1. Introduction There is a growing demand in consumer electronics for small, inexpensive and environmentally friendly power sources. This demand is driven by progresses made in electronic components miniaturization and smart power management. Thermoelectric technology can readily be miniaturized in principle without loss in efficiency using micro-fabrication technology. No moving part is required and various heat sources can be used, thus making micro-thermoelectric devices good candidates to power wearable consumer electronics. Miniaturization decreases both material-cost and device weight and increases the output voltage of thermoelectric generators in addition. On the other hand, in practice, it has been proven difficult to make both cost-competitive and efficient small thermoelectric devices with the geometry of commercially available thermoelectric modules mainly because the micro-fabrication technologies are planar by nature and because thick films technology of thermolectrics is in its infancy [1–3]. The impact of electrical contact resistance on micromachined device performance is an additional ∗ Corresponding author. Tel.: +33 3 83 58 41 70; fax: +33 3 83 57 97 94. E-mail address: [email protected] (A. Jacquot).
0924-4247/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.sna.2004.05.023
source of concern since it becomes much more severe when downscaling thermoelectric devices [4]. Planar devices using thermoelectric effects have been successfully miniaturized very early to make radiation sensors and, quite recently, to make single-chip micro-thermostats for both active heating and active cooling applications [4–6]. The thermoelements used to make these micro-devices are supported by a thin film free of its substrate called a membrane. These microsystems take advantage either of the large surface-to-volume ratio of thermoelectric thin films to increase the sensitivity and decrease the response time of radiation sensors, or of the active heating and cooling enabled by thermoelectric technology to potentially extend the thermal operating range of micro-thermostats to well-below ambient temperature [7]. Although works have already been done on small-scale thermoelectric generators [8,9], there is still some need for a theoretical framework to optimize their geometry since the actual performances of such devices are low. One reason is that thin films technology is not suitable to get high heat flux and high output power. Furthermore, large forecasted heat losses by convection and radiation have weakened the trust people put in thermoelectric devices with this geometry for power supplies. In the meantime, however, the large thermal resistance of on-membrane thermopiles could be used to get large temperature differences along the thermoelements with a very
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small heat source and the large surface-to-volume ratio could be used to efficiently cool thermoelectric devices by air. These intrinsic characteristics of on-membrane thermopiles may improve the Carnot efficiency of thermoelectric generators in a given working environment. Recently the fabrication of such in-plane thermoelectric microgenerators has been demonstrated with polycrystalline silicon as the thermoelectric material and proof of this concept was obtained experimentally [10]. In this paper, the modeling of an on-membrane thermopile is presented. When the membrane length to width ratio is high, analytical formulas are derived to calculate the best heat source configuration and both the optimum thermoelectric leg length and thickness. A numerical model valid for any membrane length to width ratio has been developed to gain further insight into the performance of these thermoelectric devices. Three potential applications have received our attention: a solar-powered micro-thermoelectric generator (SmTG), a radioisotope-powered micro-thermoelectric generator (RmTG) and a body-heat powered micro-thermoelectric generator (BHPmTG).
2. Analytical modeling The building-block of thermopile-like microgenerators is shown in Fig. 1. It consists of a frame having a high thermal conductivity supporting a thin film being the substrate for thermoelectric legs connected in serial. Two working modes can be considered for this building-block. The first working mode is when the heat source is on the thin film substrate and the supporting frame is the cold side. The potential applications of this working mode are the SmTG and RmTG (Fig. 1a). The second working mode is when the heat source is attached to the supporting frame and the membrane is cooled by air (Fig. 1b). The large surface-to-volume ratio and high thermal resistance of the membrane are used to efficiently dissipate the heat in air and to produce a large temperature gradient along the thermoelements. A BHPmTG
could advantageously make use of this configuration, for instance. 2.1. Simplified model when neglecting the heat losses by radiation and convection In this model, it is assumed that the electrical current in the thermoelectric legs does not change the temperature distribution on the membrane. This assumption is valid if the thermoelectric materials used to make the thermoelectric legs do possess low dimensionless figures of merit ZT (ZT = σS 2 T/λ, where T, σ, S and λ are the absolute temperature, the electrical conductivity, the thermopower and the thermal conductivity, respectively). Under this binding hypothesis, the temperature distribution on the membrane can be simply calculated by solving the heat transfer equation. If a one-dimensional heat transfer along the thermoelectric legs is assumed and if the heat losses by radiation and convection are neglected, it can be easily demonstrated that the temperature drop along the thermoelements T is given by α (1 − α) α2 β 2 α2 β (1 − β) T = q0 l2 (1) + + λ3 d 3 2λ2 d2 λ2 d 2 In this equation, q0 is the heat generated by the heat source per surface unit, l = l1 + l2 + l3 is the half width of the thin film substrate (l1 , l2 and l3 are defined in Fig. 2), α = (l1 + l2 )/ l is the coverage of the thin film substrate by the heat source and β = l2 /(l1 + l2 ) is the overlapping of the heat source by the thermoelements. λi and di are the average thermal conductivity and the thickness of part i, respectively. The thermal conductivity of the part 3 is defined by λ3 =
λ S d S + λ T dT dS + d T
(2)
where λS dS and λT dT are the thermal conductivities and thicknesses of the thin film substrate and of the thermoelectric material averaged over the values of the n and p type materials, respectively.
Fig. 1. Overview of the building-block of the on-membrane thermopile taken into consideration. Applications of the on-membrane thermopile building-block to make (a) solar or radioactive powered micro-thermoelectric generators and (b) body-heat powered micro-thermoelectric generators. In the first case, the supporting frame is the cold side and in the second case it is attached to the heat source while the membrane is being cooled by air.
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Fig. 2. Cross-sectional view of an on-membrane thermopile. (1) Thermoelectric legs; (2) absorbing layer for the SmTG, radioactive heat source for the RmTG or metallic heat spreader for the BHPmTG; (3) thin film substrate; (4) heat sink for the SmTG and RmTG or frame in contact with the heat source for a BHPmTG. The calculations are done on half of the membrane according to the symmetry plan.
result of a competition between the heating power that rises when the heater surface increases, the increase of the thermal conductance (i.e. the decrease of the temperature rise along the thermoelectric legs) and the reduction of the electrical resistance when the thermoelectric leg length decreases. Note that these results are strictly speaking only valid under the binding hypotheses of the model and in the particular working mode of the RmTG. Nevertheless, we will see that this crude model is very useful to understand the results developed in the next paragraphs of both the analytical model, that takes into account the heat losses by radiation and convection, and the numerical simulations of more realistic devices in RmTG, SmTG and BHPmTG working modes.
The maximum output electrical power WL per unit of length of on-membrane thermopiles is given by
2.2. Simplified model when accounting for the heat losses by convection and radiation
WL =
N 2 S 2 T 2 4(RL = RG )
(3)
where N and RG are the number of thermoelectric legs and the internal resistance per unit of length of the thermopile, respectively and S, the Seebeck coefficient averaged over the values of the n and p type materials. The load resistance RL is adjusted to match the internal resistance of the thermopile to calculate the maximum useful electric power on the load. Interesting information can be obtained when l2 = 0 (β = 0). The maximum electrical power produced at a given heating power is obtained for an optimum thermoelectric leg thickness derived from ∂WL /∂dT |β=0 = 0 and is given by λT dT = λS dS
2 l1 = l 3
T1 =
T2 =
(4)
The associated optimum leg length is derived from ∂WL /∂α|β=0 = 0 and is given by α=
If the heat losses by radiation and convection are taken into account, the temperature distributions Ti in part i (Fig. 2) are derived by solving the heat transfer equation and are given by
(5)
Eq. (4) shows that optimum thermoelectric film thickness is reached when the thermal conductance of the thermoelectric film is equal to the thermal conductance of the d1 exp(ρ2 αβl) ch (ρ2 αβl) 1 − λ2 ρ 2 d 2 sh (ρ2 αβl) −1 ch (ρ1 α(1 − β)l) d1 ch (ρ2 αβl) − λ1 ρ1 sh (ρ1 α(1 − β)l) λ2 ρ2 d2 sh (ρ2 αβl) q0 /A3 = q0 /A2 − q0 /A1 substrate. The origin of this optimum arises from a competition between the temperature difference along the thermoelectric legs, that will be higher with thinner legs, and the electrical resistance of the device that increases when the thermoelements’ thickness decreases. If we assume the heating power to be proportional to the area covered by the heater, the optimum leg length calculated from Eq. (5) is the
T3 =
q1 ch(ρ1 x1 ) q0 − + T0 A1 λ1 ρ1 sh(ρ1 α(1 − β)l)
(6)
q d1 2 − 1 exp(ρ2 x2 ) + λ 2 ρ2 d2 1 − exp(−2ρ2 αβl) q1 q q0 × − 2 exp(−ρ2 αβl) ch(ρ2 x2 ) + + T0 d2 d1 A2 (7) 1 d2 sh (ρ3 ((1 − α)l − x3 )) q + T0 λ3 ρ 3 d 3 2 ch (ρ3 (1 − α)l)
(8)
where xi ∈ [0, li ]
(i = 1, 2, 3)
(9)
In Eqs. (6–8), q1 and q2 are the solutions of the following linear system of equations: 1 ch (ρ2 αβl) d2 sh (ρ3 (1 − α)l) + λ3 ρ3 d3 ch (ρ3 (1 − α)l) λ2 ρ2 sh (ρ2 αβl) q1 q2 1 1 λ2 ρ2 sh (ρ2 αβl) (10) where
Ai λi d i
(11)
Ai = 2hi + 8σB εi T03
(12)
ρi =
The coefficients hi , εi , σ B and T0 are the heat transfer coefficient of convection and the emissivity of part i, the
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Stefan–Boltzmann constant and the room temperature, respectively. By setting β equal to zero in Eq. (10) and by equaling T1 |x1 =l1 and T3 |x3 =0 , we can easily verify that Eqs. (6), (8) and (10) are reduced to the equation obtained by Kozlov [11]. By doing similar calculations, it is easy to show that Eqs. (7) and (10) are reduced to the equation obtained by the same author when β is equal to one [12]. The optimum geometry of on-membrane thermopiles is found numerically by maximizing the electrical power calculated by Eq. (3) by adjusting the value of the three independent variables α, β and dT . The initial values for α and β are first obtained with dT = λS dS /λT . These values are then used to calculate a new value for dT . This procedure is repeated until the electrical power does not change significantly. The temperature drop along the thermoelements in Eq. (3) is given by T2 |x2 =0 − T0 .
The results of the geometry optimizations are presented in Fig. 3. They are obtained with an overall emissivity of 0 and 1 (εi = 0 or 1) with no heat transfer by convection (hi = 0) and as a function of the heat source thermal conductivity λQ , which can either be an absorbing layer (SmTG) or a radioactive source (RmTG). The calculations were performed for thermoelements made of silicon, silicon germanium and bismuth telluride. The material properties and film thicknesses used to make these calculations are given in Table 1. The thermoelectric parameters are considered to be an average of the n and p type material properties. It can be seen that when the overall emissivity is zero and λQ is larger than 2.8 W m−1 K−1 , the overlapping of the heat source by the thermoelements β is zero. Moreover, it does not depend on the thermoelectric materials and the coverage α, of the thin film substrate by the heat source, which is 2/3. This last result is consistent with the result that could have been obtained straightforwardly from Eq. (5).
Fig. 3. Optimum geometry of the microgenerator to maximize its electrical power. Dotted line: optimal coverage α of the thin film substrate by the absorbing layer (SmTG) or the radioactive source (RmTG). Solid line: overlapping β of the absorbing or the radioactive source by the thermoelement. Dashed line: thickness of the thermoelectric elements dT made respectively of polysilicon, silicon germanium and bismuth telluride as a function of the absorbing layer or the radioactive heat source thermal conductivity λQ for an overall emissivity of 0 and 1.
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Table 1 Physical propertiesa of the materials selected for the simulations according to the literature
substrateb
Thin film Heat source Polysiliconb SiGec Bi2 Te3 d
d (nm)
σ (−1 m−1 )
S (V/K)
λ (W m−1 K−1 )
ZT300 K
900 800 Variable Variable Variable
– – 50000 102850 100000
– – 165 114 195
2.24 Variablee 14.5 4.7 1.9
– 0.028 0.085 0.60
a
The data are an average of the thermoelectric properties of n and p type materials. Data taken from [13,14]. c Data taken from [15]. d Data taken from [16]. e Ranging from 0.02 to 200 W m−1 K−1 when used in the analytical models and equal to 140 W m−1 K−1 when used in the numerical simulations. b
When λQ is lower than 2.8 W m−1 K−1 , β increases and the heater completely overlaps the thermoelements at λQ = 0.02 W m−1 K−1 . An overlapping of the thermoelements by the heater is advantageous when the thermal conductivity of the heater is low because the surface of the heater as well as the heating power are increased while the thermal by-pass through the heater at the thermoelements level is low. At intermediary values of the heater, the optimum thermoelement thicknesses raise sharply when λQ is reduced and then decrease slowly. This increase of the optimum thermoelectric film thickness can also be understood by the thermal by-pass through the heater at the thermoelements level and can be interpreted as an increase of λS dS in Eq. (4). Consequently, it is not surprising that the thickness of the thermoelements tends to their values calculated from Eq. (4) when λQ tends to zero even if the overlapping is complete. When the overall emissivity is one, the value of the heater thermal conductivity, at which β is no more zero, is 9.5 W m−1 K−1 . This value of the thermal conductivity does not depend on the materials the thermoelements are made of as previously noticed when the emissivity was zero. The heater covers a little more than 2/3 of the thin film substrate, whatever the thermal conductivity of the heat source is. The raise in α enhances the heating power, because the heat source surface is increased, while it reduces the effect of the heat losses by radiation, because the optimal thermal conductance of the thermoelements is increased. The optimum thermoelement film thicknesses are higher than when the emissivity is zero, because the heat losses by radiation can be interpreted as an increase of λS dS in Eq. (4). It is worth noticing that the better the thermoelectric materials (the larger the ZT values), the larger are the optimum thermoelements film thicknesses. This result can easily be understood by Eq. (4), because good thermoelectric materials usually have a very low thermal conductivity.
3. Numerical simulation To overcome the limitation of the analytical models, we have solved the coupled electrical and heat transfer problems of on-membrane thermopiles by the finite volume method.
This numerical model was used to evaluate the change in the optimum heater coverage of the thin film substrate and the change of the thermoelements thicknesses when the coupled electric charges and heat transfer as well as the finite thin film substrate size are taken into account. The thermal conductivity of both the heat source and the heat spreader is set to be 140 W m−1 K−1 . Consequently, we are assuming that there is no overlapping of the thermoelements by the heat source in the RmTG and SmTG as well as in the BHPmTG working mode. The calculations are done with the data reported in Table 1. The general problem should be solved using a three-dimensional grid. However, it can be reduced to a 2D problem by assuming that the heat flows in the plane of the membrane and that the temperature of the supporting frame is constant. The Peltier effect is taken into account through an additional heat flux Q at the junctions between the thermoelements and the interconnections not in contact with the silicon frame. Assuming that the Seebeck coefficient of the material used to make the interconnection is negligible compared to the thermoelectric material, the expression for Q is Q = SIT
(13)
where I is the electrical current in the thermoelements. The material properties were assumed to not depend on temperature. Consequently, the Thompson effect was neglected. The heat losses by radiation and the convective heat transfer were taken into account through the emissivity (ε) and the heat transfer coefficient for convection (h). We used linearized approximations for the radiative and the convective heat transfers. Numerical simulations have been performed to predict the optimum geometry and the useful electrical power of in-plane thermoelectric devices of various membrane sizes in the RmTG and SmTG (Table 2) and BHPmTG working modes (Table 3). The membrane areas are 1.6 × 1.6 mm2 for the calculation results presented in Tables 2 and 4 and 1.6 × 16 mm2 for those presented in Tables 3, 5 and 6. The results obtained by using the numerical simulation and the analytical formulas are compared in the case of the larger membrane length to width ratio, i.e. when the heat flow is almost one-dimensional.
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Table 2 RmTG and SmTG working modes, membranes 1.6 × 1.6 mm2 in size, 50 thermoelements, the heating power is 1 mW, V and W are the output voltage and the useful electrical power, respectively
Polysilicon SiGe Bi2 Te3
ε
lT ()
dT (nm)
ZTm
T (K)
V (V)
W (W)
0 1 0 1 0 1
180 150 180 150 180 150
150 190 460 570 900 1150
0.014 0.016 0.043 0.048 0.27 0.30
8.0 5.1 7.9 5.0 8.1 5.0
0.13 0.084 0.091 0.057 0.16 0.098
0.090 0.058 0.27 0.16 1.5 0.94
ZTm is the dimensionless figure of merit at the average temperature along the thermoelements, when the thermal bypass through the substrate is taken into account. Table 3 RmTG and SmTG working modes, membranes 1.6 × 16 mm2 in size, 500 thermoelements, the heating power is 5 mW
Polysilicium SiGe Bi2 Te3
ε
lT ()
dT (nm)
ZTm
T (K)
V (V)
W (W)
0 1 0 1 0 1
270 200 270 200 270 200
140 240 410 750 840 1450
0.014 0.018 0.041 0.053 0.26 0.34
9.9 4.2 9.9 4.1 9.8 4.2
1.6 0.70 1.1 0.47 1.9 0.83
0.58 0.24 1.7 0.7 9.4 4.0
3.1. Results and discussion The optimum legs thicknesses obtained by numerical simulations in the RmTG and SmTG modes for the largest thin film substrate (Table 3), when neglecting the heat losses by radiation, are close but a little lower than the values predicted by Eq. (4) (140, 425 and 1063 nm for polysilicon, SiGe and Bi2 Te3 , respectively). The agreement is fairly good for the materials with the lowest ZT, but is less good for the best thermoelectric material. These lower values can be understood when considering the assumptions made to derive Eq. (4). It was assumed that the electrical current passTable 4 BHPmTG working mode, membranes 1.6 × 1.6 mm2 in size, 50 thermoelements, h = 52.6 W m−2 K−1
Polysilicon SiGe Bi2 Te3
lT ()
dT (nm)
ZTm
T (K)
V (V)
W (W)
220 230 220
750 2300 4400
0.024 0.073 0.49
−7.4 −7.5 −7.4
0.12 0.086 0.14
0.29 0.86 4.87
Table 5 BHPmTG working mode, membranes 1.6 × 16 mm2 in size, 500 thermoelements, h = 52.6 W m−2 K−1
Polysilicon SiGe Bi2 Te3
lT ()
dT (nm)
ZTm
T (K)
V (V)
W (W)
290 300 290
1300 4000 7800
0.026 0.078 0.54
−8.3 −8.3 −8.2
1.37 0.95 1.6
3.5 10 57
Table 6 BHPmTG working mode, membranes 1.6 × 16 mm2 in size, 500 thermoelements, h = 26.3 W m−2 K−1
Polysilicon SiGe Bi2 Te3
lT ()
dT (nm)
ZTm
T (K)
V (V)
W (W)
330 340 330
800 2500 4900
0.025 0.074 0.50
−7.9 −7.8 −7.7
1.30 0.89 1.5
1.7 4.9 27
ing through the thermoelectric elements does not change the temperature distribution on the device (low ZT). In reality, it reduces the temperature difference along the thermoelectric legs because of the Peltier effect. It is therefore advantageous to decrease the thermoelectric legs thicknesses in the numerical simulation to get a higher temperature rise along the thermoelements, because the coupled electric charges and heat transfer are fully taken into account. The leg length obtained by numerical simulation when neglecting the heat losses by radiation is very close to the length calculated by Eq. (5) (270 m versus 266 m). When the membrane changes in area from 1.6×16 to 1.6× 1.6 mm2 , the optimum thermoelectric leg lengths decrease and the thermoelements’ thicknesses increase (Table 3). The thermal conductance of the smallest thin film substrate is larger than the thermal conductance of the largest thin film substrate for the same membrane thicknesses. Eq. (2) predicts in fact larger thermoelectric leg thicknesses when the thermal conductance of the thin film substrate increases. The decreasing leg length can be understood considering the thin film substrate perimeter-to-surface ratio. This ratio is lower when the membrane is a square than when the membrane is a rectangle, if the membranes are of equal surfaces. It is therefore advantageous to increase the heater surface (i.e. to decrease the leg length) of the square-shaped membrane, because the benefit of the increased heating power will be greater than the corresponding increase of the membrane’s thermal conductance. The optimum thermoelectric leg length decreases and the optimum thermoelectric thickness increases when the overall emissivity of the device is increased. The effect of the heat losses by radiation is similar to the thermal by-pass through the substrate in Eq. (4). As a consequence, the optimum film thickness will be greater. The optimum thermoelement lengths decrease, since this increases the device conductance and therefore the heat flowing by conduction versus the heat losses by radiation. The optimal thermoelectric legs thickness dT and leg length lT calculated by numerical simulations for the largest length-to-width ratio of the thin film substrate, are very close to those calculated by Eqs. (3), (7) and (10) when the overall emissivity is equal to one (polysilicon: lT = 200 m, dT = 240 nm versus 195 m and 250 nm for the numerical simulation results; SiGe: lT = 200 m, dT = 750 nm versus 195 m and 765 nm; Bi2 Te3 : lT = 200 m, dT = 1450 nm versus 195 m and 1900 nm). The deviation is again larger when ZT increases, because the Peltier effect is taken into account in the numerical simulation, whereas it is not in the analytical models.
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3.2. Optimization of body-heat powered micro-thermoelectric generators The useful electrical powers and the optimum geometry have also been calculated in the BHPmTG working mode by numerical simulations (Table 3). The air temperature is set to 293 K. The temperature of the supporting frame is set to 310.2 K, even if the skin temperature is probably lower than the internal body temperature. The membrane areas are 1.6 × 1.6 mm2 in Table 4 and 1.6 × 16 mm2 in Tables 5 and 6. The heat transfer coefficients of convection are 52.5 in Tables 4 and 5 and 23.6 W m−2 K−1 in Table 6. The optimum thermoelectric film thicknesses are far thicker in the BHPmTG working mode than in the RmTG and SmTG working modes. They are typically in the 5–10 m range, because heat is more effectively dissipated in air than in vacuum. The conclusions are the same for the leg lengths which are longer than in the RmTG working mode, because the heat source is at the supporting frame, the membrane being the cold side. It is found that the larger the heat transfer coefficient of convection is, the thicker the optimum thermoelectric film thickness (Tables 5 and 6). The optimum thermoelectric film thicknesses are two times thinner with a squared membrane than with a long rectangular membrane of the same width, but 10 times longer. A power of about 57 W, at a voltage of 1.5 V, can be produced from heat wasted by the body in the case of the largest membrane, by a light-weighted and compact thermoelectric generator, based on Bi2 Te3 thick-film technologies. This is potentially far larger than the thermoelectric microgenerator developed by Seiko to power their wristwatches [1].
4. Conclusions Analytical models and numerical simulations demonstrate that an optimum thermoelectric leg length and thickness exist to get the maximum useful electrical power from an on-membrane thermopile. The lower the thermal conductivity of the thermoelectric materials or the higher the heat losses by radiation and convection are, the thicker are the optimum thermoelectric film thicknesses. These modelings show that heat losses by radiation are detrimental to the radioisotope micro-thermoelectric generator working mode, but improve the efficiency of microgenerators when the heat source is attached to the frame of the thermopile. The large surface-to-volume ratio of the thin film substrate is used to improve the coupling between the heat reservoirs and the thermoelements in this case. Due to the large thermal resistance of these devices, small heat sources can be efficiently used by on-membrane thermopiles, because large temperature differences along the thermoelements can be obtained, resulting in an increase in Carnot efficiency. On-membrane thermopiles are all the more promising since they should easily be fabricated [9].
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Acknowledgements This work was supported by the Jet Propulsion Laboratory (contract 1217092) and DoD/ONR MURI (N00014-97-1-0576).
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Biographies A. Jacquot was born in Saint-Dié, France, in 1974. From 1998 to 2003 he was a doctoral student under the joint guardianship of the Polytechnical Institute of Lorraine, France, and the Martin-Luther-University of Halle-Wittenberg, Germany. From 2001 to 2002 he was an assistant researcher at the University of California in Los Angeles in the framework of his Military Service. He obtained his PhD degree in material science and engineering from the Polytechnical Institute of Lorraine in 2003. His current research interests are the measurement and improvement of thermoelectric properties of nanocomposites and superlattices. He is an author or co-author of more than 20 publications and proceedings. G. Chen received his BS and MS from Huazhong University of Science and Technology (China) in 1984 and 1987, respectively, and his PhD from UC Berkeley in 1993. He taught at Duke University (1993–1997) and UCLA (1997–2001) and is currently a professor at MIT. His research interests are focused on nanoscale transport phenomena, particularly thermal energy transport, and their applications in energy and information technologies. He is a recipient of the NSF Young Investigator Award and a Guggenheim Fellowship, and serves on the editorial board four journals. H. Scherrer was born in 1941 in France. He is Professor at the Henri Poincaré University in Nancy (France). Since 1985, his research activities have mainly focused on the field of thermoelectricity. He is the head of the thermoelectric team in the Laboratoire de Physique des
Materiaux. He is the President of the European Thermoelectric Society and a member of the board of the International Thermoelectric Society, of the Thermoelectric Academy of the Ukraine and of the Refrigeration Academy of Saint Petersburg. He is an author or co-author of more than 100 publications and proceedings and of seven book chapters. A. Dauscher was born in Strasbourg, France in 1958. She received her PhD degree in sciences from Louis Pasteur University in Strasbourg (France) in 1987. She worked as student during 10 years and then as a CNRS researcher in the field of heterogeneous catalysis in Strasbourg. In 1993, she joined the research team on thermoelectric materials in the Laboratoire de Physique des Materiaux in Nancy. Her current research interest is focusing on the preparation of both thermoelectric thin films, by pulsed laser deposition, and new thermoelectric bulk materials, besides the microstructural and physical characterizations of the samples. B. Lenoir was born in Rambervillers, France in 1965. He received the ESSTIN (Graduate School at Nancy, France) engineering degree in physics in 1990 and the PhD degree in material science and engineering from the National Polytechnics Institute of Lorraine in 1994. He is now an Assistant Professor at the École Nationale Supérieure des Mines de Nancy in the Laboratoire de Physique des Materiaux. His research activities are mainly concerned with the synthesis, the structural and physical characterization of thermoelectric materials prepared in the bulk or thin film form. He is an author or co-author of more than 70 papers and proceedings.