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Efficiency assessment of novel materials based flexible thermoelectric devices by a multiscale modeling approach
Computational Materials Science xxx (2015) xxx–xxx
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Efficiency assessment of novel materials based flexible thermoelectric devices by a multiscale modeling approach Malika Bella a,b,c,⇑, Sylvain Blayac b, Christian Rivero a, Valérie Serradeil a, Pascal Boulet c,⇑ a
STMicroelectronics, ZI Rousset-Peynier, 13106 Rousset, France Ecole Nationale Supérieure des Mines de Saint-Etienne, Centre Microélectronique de Provence Georges Charpak, 13541 Gardanne, France c Aix Marseille Université, CNRS, MADIREL UMR 7246, 13397 Marseille, France b
a r t i c l e
i n f o
Article history: Received 16 February 2015 Received in revised form 19 June 2015 Accepted 25 June 2015 Available online xxxx Keywords: Tetrahedrite First-principle calculations Thermoelectric properties Finite-element method Module Output power
a b s t r a c t The presented work demonstrates a multiscale approach for evaluating novel materials for room temperature thermoelectric applications and provides some insights into the development of flexible devices composed of those materials. Tetrahedrite is studied as it is a promising p-type thermoelectric material that exhibits good thermoelectric properties at room temperature. Considering our target application, analysis of the theoretical results reveals that tetrahedrite is an interesting surrogate material to bismuth telluride for room temperature applications with a power factor ranging from 4.16 lW/cm K2, for the pristine tetrahedrite compound, to around 9 lW/cm K2, for a doped tetrahedrite. A single thermocouple made of p-type pristine tetrahedrite and n-type natural chalcopyrite has an optimum output power of 5.53 nW/K. This output power can reach 7.47 nW/K when optimally doping tetrahedrite. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction The complete understanding of thermoelectric phenomena has concentrated years of intensive work from materials study to device integration in self-powered systems [1,2]. Thermoelectricity, capitalizing on waste heat recuperation, offers good prospects for the development of autonomous systems in the case of wearable electronics. The most critical obstacle for technology development is to obtain thermoelectric materials that exhibit high efficiencies under low thermal gradient. These target materials should be made of abundant, non-toxic chemical elements. The main issue regarding materials currently used for room temperature wearable thermoelectric applications, bismuth telluride alloys, is their scarcity and reported health hazardousness [3]. The aim of this study is thus to propose a solution for flexible thermoelectric generators based on novel abundant, cheap and non-health hazardous materials. In this perspective, tetrahedrite material is studied as it is promising p-type thermoelectric material for room temperature applications which transport properties can be significantly improved by chemical substitutions [4]. Targeting the development
⇑ Corresponding authors at: MADIREL UMR 7246, Aix-Marseille Université, CNRS, Campus de St Jérôme, 13397 Marseille Cedex 20, France (M. Bella). E-mail addresses: [email protected] (M. Bella), [email protected] (P. Boulet).
of flexible printed thermoelectric devices, we propose an innovative approach for device optimization based on the multiscale modeling of a complete thermoelectric system. In this scope, two levels of modeling are addressed, from nano to macroscale. At the nanoscopic level, quantum density-functional theory is used in conjunction with semi-classical approach using Boltzmann transport theory to calculate electronic properties such as Seebeck coefficient and electrical conductivity [5] of tetrahedrite. Simulation results are then compared with experimental data. The impact of doping on thermoelectric properties is also investigated. An interesting way of fabricating low cost flexible devices is by using printing techniques as it enables the processing of flexible substrates under standard temperature and pressure conditions. Beside the technological challenges (ink preparation, printing parameters control, post treatment), it is necessary to evaluate the efficiency of an ‘‘ideal’’ printed device. In this context, a virtual prototype of a flexible thermoelectric device with an innovative design is proposed and evaluated with finite-elements simulation. 2. Computational methods and models 2.1. Atomistic simulation of the Cu12Sb4S13 compound Tetrahedrite with classical formula Cu12Sb4S13 is used as the starting material. The general nature of the tetrahedrite structure
http://dx.doi.org/10.1016/j.commatsci.2015.06.038 0927-0256/Ó 2015 Elsevier B.V. All rights reserved.
Please cite this article in press as: M. Bella et al., Comput. Mater. Sci. (2015), http://dx.doi.org/10.1016/j.commatsci.2015.06.038
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Fig. 2. Schematic representation of the device.
2.3. Finite-element analysis of an innovative flexible TEG architecture
Fig. 1. Crystal structure of Cu12Sb4S13.
was first described by Machatschki (1928). The ideal structure was determined to be Cu3SbS3. The correct formula Cu12Sb4S13 has been first found and described by Pauling and Neuman (1934) from experimental analysis of the composition. However the crystallographic structure proposed was found to be slightly inaccurate. A correction, for atomic positions in particular, has been provided by Wuensch (1963) [6]. Tetrahedrite crystallizes in the cubic system with a = 10.33 Å (space group I43m) at room temperature. Atoms form SbS3 pyramids, CuS4 tetrahedra, and unique CuS3 triangles. The tetrahedrite has 58 atoms in the unit cell (Fig. 1). The electronic band structure is calculated using ab initio Density Functional Theory (DFT) [7,8] and GGA-PW91 exchange correlation functionals [9]. Calculations are performed using the Quantum Espresso software (v4.3.2) [10]. The kinetic energy cutoff is 460 eV. The structure is optimized (atomic positions and cell parameters) at hydrostatic pressures equal to 0 GPa. The k-point selection is based on the Monkhorst–Pack scheme. The k-point mesh used to sample the Brillouin zone is set to 30 30 30. 2.2. Electronic transport simulation Electronic transport properties were calculated by solving the Boltzmann transport equation within the constant relaxation time approximation (CRTA) as implemented in the BoltzTraP code [11]. From the solution of the Boltzmann equation, in the relaxation time approximation, the transport coefficients, namely the electrical conductivity r and the Seebeck coefficient S, can be written as [12]:
r ¼ e2 S¼
Z
ekB
r
Z
@f de 0 NðeÞ; @e @f el ; de 0 NðeÞ @e kB T
ð1Þ
ð2Þ
where s is the relaxation time, f0 is the Fermi distribution, l the chemical potential, e and kB the Boltzmann’s constant. N is the transport distribution and is expressed as follows:
N¼
X ~ v~k~ v~k s~k
ð3Þ
k
~ v is the group velocity and s is the relaxation time. A dense k-mesh of over 3000 k-points in the IBZ was used for transport calculations to minimize possible band-crossing effects [11].
The main issue preventing the widespread use of TE modules as human body heat energy scavengers is their low efficiency under low thermal gradient consequence of their poor ability to evacuate heat. A solution for keeping the thermal gradient naturally present between skin and air across the device can be achieved by increasing the device surface at the cold side in contact with air. We thus propose the following device architecture: The device architecture is represented in Fig. 2. The plastic substrate is used as a thermal insulator for the hot part (bottom side Fig. 2) and the metal contacts as heat sinks for better heat removal (top side Fig. 2). A thermoelectric generator is made of multiple thermocouples, and each thermocouple is composed of an n-type and a p-type material. In our model, tetrahedrite is used for the p-type part. Chalcopyrite CuFeS2 can be considered as an interesting n-type counterpart to Cu12Sb4S13 as it is also a natural occurring and abundant mineral. Compared with tetrahedrite, chalcopyrite shows a higher power factor that can be increased by adjusting the Cu/Fe ratio. The thermoelectric properties of the CuFeS2 compound considered in our model for the device simulation are taken from natural samples characterization [13]. Chalcopyrite can be either n-type or p-type depending on the exact stoichiometry. The numerical analysis of the device is carried out using COMSOL Multiphysics, a finite-element method based code. The equations governing the temperature and electrical potential distributions and implemented in the software are [14]:
~ q ¼ krT þ P~ J
ð4Þ
~ J ¼ rrV rSrT
ð5Þ
q represents the heat flux and ~ where k is the thermal conductivity, ~ J the electric current ‘‘flux’’ and:
~ E ¼ rV
ð6Þ
Q ¼~ J ~ E
ð7Þ
~ E is the electric field and Q joule heating. Before applying this model to our specific configuration, it has been tested on a commercially available module model and validated by comparison between simulated and experimental characterization data. Cooling of the device by natural convection in air is modeled by adding an air channel around the device and defining the following boundary conditions: air at 297 K flows through the channel and the bottom of the device is kept at 307 K. 3. Results and discussion 3.1. Electronic structure of Cu12Sb4S13 The band structure of Cu12Sb4S13 is represented in Fig. 3. The valence band and conduction band edges are separated by a
Please cite this article in press as: M. Bella et al., Comput. Mater. Sci. (2015), http://dx.doi.org/10.1016/j.commatsci.2015.06.038
M. Bella et al. / Computational Materials Science xxx (2015) xxx–xxx
3
Fig. 3. Cu12Sb4S13 energy-band diagram (on the left), density of states (DOS) and projected density of states (pDOS) (on the right).
Fig. 4. Cu12Sb4S13 Seebeck coefficient evolution with temperature.
Fig. 5. Cu12Sb4S13 electrical conductivity evolution with temperature.
semiconducting gap of 1.25 eV, which is relatively lower than the experimental value of 1.7 (±0.2) eV from [15]. Cu12Sb4S13 is a metal in which the Fermi level lies in the valence band, near the edge. We thus expect the material to behave electrically as a metal. From our calculation, the compound has an indirect band gap which is in accordance with the experimental observation [16].
the band structure does not change with temperature and doping, in order to suppress the chemical potential and temperature dependence of the electrical conductivity. The material being intrinsically heavily doped, the doping concentration does not strongly changes with temperature. From the results (Fig. 5) we can clearly state that the adopted approximation do not appear to be valid above 450 K. Time relaxation is indeed strongly temperature dependent, especially at higher temperatures (above 500 K). However, there is a fair accordance between calculated and measured values in the temperature region of interest (below 450 K). A way to tune the thermoelectric properties of a semiconducting material can be done by doping. The power factor represents the capability of a material to produce useful electrical power and is defined as PF = S2r. A second step has thus been the evaluation of the thermoelectric properties as a function of free carrier concentration in order to find out the optimum doping concentration for power factor (PF) maximization. The free carrier concentration of pure tetrahedrite calculated using BoltzTraP is 1.02 1021 cm3 at 300 K. The Hall coefficient calculated in BoltzTraP can be used for carrier concentration determination. As RH = rh/q p, in order to evaluate the free carrier concentration, we take the following approximation: rh = 1, the Hall factor being close to unity for most semiconductors assuming single carrier model. With a calculated Hall coefficient of 3 109 m3/C at
3.2. Electronic transport properties of Cu12Sb4S13 The calculated results have been compared with experimental data. Concerning the Seebeck coefficient, there is a good agreement between calculated and experimental values for the range of temperature from 300 to 600 K [17] (Fig. 4). As there is no available data on the carrier lifetime for this compound, it is difficult to correctly determine the electrical conductivity since only r/s is calculated (Eq. (1)). In order to have a better idea on the electrical behavior of the compound as a function of temperature (Fig. 5), we add the following strong approximation: s is temperature independent. A way to approximate the time relaxation can be done by using the resistivity measurement on Cu12Sb4S13 done by Heo et al. [17]. We thus fit our experimental data at a given temperature and carrier concentration. We consider the pure crystal at 320 K and we obtain s = 5.9 1015 s by fitting the conductivity r (320 K) = 72,100 S/m of bulk Cu12Sb4S13 (Heo et al.). We also apply the ‘‘rigid band approach’’, assuming that
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M. Bella et al. / Computational Materials Science xxx (2015) xxx–xxx
Fig. 6. PF/s, seebeck coefficient and r/s evolution with carrier concentration.
[18]: 3.44 1021 cm3. The thermoelectric parameters evolution with carrier concentration have been evaluated at 300 K and are represented in Fig. 6. The optimum PF/s of 1.4 1011 W/m K2 s is achieved for a free carrier concentration of 5.85 1020 cm3 at 300 K. In order to evaluate the electrical conductivity of the doped compound, we fixe s = 5.9 1015 s. The power factor for such the optimally doped compound is thus estimated to be around 9 lW/cm K2. The doping concentration needed to achieve this concentration is high, in the order of 1020 cm3 electron, and thus achievable by chemical doping only. A way to achieve quite efficiently and easily such a concentration would be by substitution of some copper atoms (Fig. 3). Indeed, in order to lower the free hole concentration, the valence band has to be slightly filled and copper being the main element responsible for the compound ‘‘intrinsic’’ doping, the adjustment can be done by replacing some copper atoms by elements providing more electrons, such as zinc. Fig. 7. Device configuration.
3.3. Device performances
Fig. 8. Temperature gradient as a function of the thermal conductivity of the thermoelectric elements.
300 K, the free carrier concentration of pure tetrahedrite is estimated at 2.08 1021 cm3 .Those results are in fair accordance with the formal charge counting accounting for 2 holes per formula cell
In the following section, we evaluate the cooling efficiency of the device presented in Fig. 2 as well as its electrical performances. The performances of the device containing optimally doped tetrahedrite will be evaluated and compared to the one containing pristine tetrahedrite. First, the temperature gradient across the device heated in one side by the human body and cooled by natural convection in air (Fig. 7) is evaluated. We evaluate the temperature gradient difference and note that for an external temperature gradient of 10 °C: temperature gradient naturally present between a wrist at 34 °C and air at 24 °C, the gradient across the device stabilizes at 3 °C for our steady state simulation. In our particular device configuration, we observed that there is no significant impact of the increase in thermal conductivity of the thermoelectric elements on the temperature gradient kept across the device (Fig. 8). For instance, an increase of thermal conductivity from 1 W/m K to 7 W/m K induces a temperature gradient loss of 0.33 °C across the device. Table 1 gives the material parameters used for the finite-element analysis. We consider the same value of thermal conductivity for pristine and doped tetrahedrite as we expect no higher value, the optimally doped material having a lower carrier
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M. Bella et al. / Computational Materials Science xxx (2015) xxx–xxx
Table 1 Materials parameters at 300 K. Materials
S (lV/K)
r (S/m)
j (W/m K)
Cu12Sb4S13 (p-type) Optimally doped Cu12Sb4S13 (p-type) CuFeS2 (n-type) Sb/Te (p-type) Bi/Te (n-type)
76 (this work) 177(this work) 405 [13] 209 209
7.38 104 (this work) 2.65 104 (this work) 3.85 103 [13] 1.16 105 1.16 105
1.1 [17] 1.1 [17] 5.9 [19] 1.72 1.72
Fig. 9. Power output as a function of the resistance load for pristine tetrahedrite (circles), optimally doped tetrahedrite p = 5.85 1020 cm3 (squares) and Bi2Te3 (triangles) based devices.
concentration [17]. The thermoelectric properties of Sb/Te and Bi/Te alloys have been extracted from the characterization of a commercial module purchased from Marlow (RC12–6LS). The second step is the evaluation of the thermoelectric generator performance. The contact resistance has to be taken into account in the model as in practice the contact between the thermoelectric material and the metal is not ideal. For evaluation purpose and as there is no available data, we fix RC = 106 X cm2, value measured in a bulk Bi2Te3 device [20]. Only one thermocouple is modeled. In the presented configuration (one thermocouple) and for a matched resistance load, the maximum output power is 16.6 nW for a temperature gradient of 3 °C in the case of pristine tetrahedrite (Fig. 9). We can note that the internal electrical resistance of 31 X is quite high, which is mainly due to the relatively poor electrical conductivity of CuFeS2. In the literature [19], a strategy for this compound thermoelectric properties improvement has been discussed. An electrical conductivity of 2.11 104 S/m for the Cu0.9Fe1.1S2 compound has been achieved which is nearly 10 times higher than for the classical formula CuFeS2. The device power output can be increased from 16.6 nW to 22.4 nW by correctly doping tetrahedrite (Fig. 9). However, as the optimally doped material has a lower electrical conductivity, the device electrical resistance increases slightly from 31 X to 34 X. For comparative purpose, a simulation with the same device architecture but with Sb/Te and Bi/Te alloys instead of tetrahedrite is done. The comparative results (Fig. 9) demonstrate that the tetrahedrite based modules are less efficient than conventional materials based modules with a power output divided by six. In order to evaluate a device that would be composed of the evaluated thermocouple, we fix our target output power at 40 lW, which represents the minimum output power needed to trigger a microcontroller unit. An output power of 40 lW is reached for a device composed of 2411 thermocouples which
represents a surface of around 482 cm2. A wristwatch covering approximately 100 cm2 of skin, the thermoelectric generator performances can be further improved either by improving the design, by working on the heat sink efficiency, and increase the thermal gradient kept across the device and/or by improving the thermoelectric properties of the embedded materials, which can be achieved by appropriate carrier concentration adjustment.
4. Conclusions First principle calculations coupled with the Boltzmann transport equation resolution under the constant relaxation time approximation (CRTA) were used to investigate the electronic structure together with the thermoelectric properties of the tetrahedrite compound. The impact of doping on thermoelectric properties of pure tetrahedrite was investigated. In order to achieve a higher power factor, optimized carrier concentration should be around 5.8 1020 cm3. We have proposed a new method for evaluating novel materials potential for room temperature thermoelectric applications with a multiscale modelling approach by the mean of a macroscale prototype compatible with our target application. This method enlightens some key parameters: the temperature gradient across the device and the device internal electrical resistance. In order to achieve higher temperature gradient, an additional work on the heat sink has to be done. The electrical conductivity of the thermoelectric materials can be adjusted by appropriate atom substitutions as suggested by the computational results. Decreasing the device internal electrical resistance can also be done by improving the contact resistance between the thermoelectric material and the metal contact. We have also shown that thermal conductivity increase (with electrical conductivity) is not a hindrance in this specific configuration considering thermal conductivities of the studied materials do not exceed a few W/m K.
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References [1] T.M. Tritt, M.G. Kanatzidis, H.B. Lyon, G.D. Mahan, MRS Proc. 478 (1997). [2] B.C. Sales, Int. J. Appl. Ceram. Technol. 4 (4) (2007) 291. [3] Occupational Safety And Health Guideline For Bismuth Telluride, Centers for Disease Control and Prevention, Atlanta. http://www.cdc.gov/, 1992. [4] X. Lu, D.T. Morelli, Y. Xia, F. Zhou, V. Ozolins, H. Chi, X. Zhou, C. Uher, Adv. Energy Mater. 3 (2013) 342–348. [5] B.-L. Huang, M. Kaviany, Phys. Rev. B 77 (2008) 125209. [6] Bernhardt J. Wuensch, Zeitschrift Fur Kristallographie 119 (5-6) (1964) 437– 453. [7] P. Hohenberg, W. Kohn, Phys. Rev. 136 (1964) B864–B871. [8] W. Kohn, L.J. Sham, Phys. Rev. 140 (1965) A1133–A1138. [9] J.P. Perdew, in: P. Ziesche, H. Eschrig (Eds.), Electronic Structure of Solids ’91, vol. 11, Akademie Verlag, Berlin, 1991. [10] P. Giannozzi et al., J. Phys.: Condens. Matter 21 (2009) 395502.
[11] G.K.H. Madsen, D.J. Singh, Comput. Phys. Commun. 175 (2006) 67–71. [12] T.J. Scheidemantel, D. Ambrosch, T. Thounhauser, J.V. Bading, J.O. Sofo, Phys. Rev. B 68 (2003) 125210. [13] M. Telkes, Amer. Miner. 35 (1950) 536–555. [14] M. Jaegle, Multiphysics simulation of thermoelectric systems – modeling of peltier-cooling and thermoelectric generation. in: Proceedings of the COMSOL Conference, Hannover, 2008. [15] R. Jeanloz, M.L. Johnson, Phys. Chem. Miner. 11 (1) (1984) 52–54. [16] J. Van Embden, K. Latham, N.W. Duffy, Y. Tachibana, J. Am. Chem. Soc. 135 (2013) 11562. [17] J. Heo, G. Laurita, S. Muir, M.A. Subramanian, D.A. Keszler, Chem. Mater. 26 (2014) 2047. [18] A. Pfitzner, M. Evain, V. Petricek, Acta Crystallogr. B 53 (3) (1997) 337–345. [19] Y.L. Li, T.S. Zhang, Y.T. Qin, T. Day, G.J. Snyder, X. Shi, L.D. Chen, J. Appl. Pys. 116 (20) (2014) 203705. [20] R.P. Gupta, R. McCarty, J. Sharp, J. Elec. Mat. 43 (6) (2014) 1608–1612.
Please cite this article in press as: M. Bella et al., Comput. Mater. Sci. (2015), http://dx.doi.org/10.1016/j.commatsci.2015.06.038
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Efficiency assessment of novel materials based flexible thermoelectric devices by a multiscale modeling approach Malika Bella a,b,c,⇑, Sylvain Blayac b, Christian Rivero a, Valérie Serradeil a, Pascal Boulet c,⇑ a
STMicroelectronics, ZI Rousset-Peynier, 13106 Rousset, France Ecole Nationale Supérieure des Mines de Saint-Etienne, Centre Microélectronique de Provence Georges Charpak, 13541 Gardanne, France c Aix Marseille Université, CNRS, MADIREL UMR 7246, 13397 Marseille, France b
a r t i c l e
i n f o
Article history: Received 16 February 2015 Received in revised form 19 June 2015 Accepted 25 June 2015 Available online xxxx Keywords: Tetrahedrite First-principle calculations Thermoelectric properties Finite-element method Module Output power
a b s t r a c t The presented work demonstrates a multiscale approach for evaluating novel materials for room temperature thermoelectric applications and provides some insights into the development of flexible devices composed of those materials. Tetrahedrite is studied as it is a promising p-type thermoelectric material that exhibits good thermoelectric properties at room temperature. Considering our target application, analysis of the theoretical results reveals that tetrahedrite is an interesting surrogate material to bismuth telluride for room temperature applications with a power factor ranging from 4.16 lW/cm K2, for the pristine tetrahedrite compound, to around 9 lW/cm K2, for a doped tetrahedrite. A single thermocouple made of p-type pristine tetrahedrite and n-type natural chalcopyrite has an optimum output power of 5.53 nW/K. This output power can reach 7.47 nW/K when optimally doping tetrahedrite. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction The complete understanding of thermoelectric phenomena has concentrated years of intensive work from materials study to device integration in self-powered systems [1,2]. Thermoelectricity, capitalizing on waste heat recuperation, offers good prospects for the development of autonomous systems in the case of wearable electronics. The most critical obstacle for technology development is to obtain thermoelectric materials that exhibit high efficiencies under low thermal gradient. These target materials should be made of abundant, non-toxic chemical elements. The main issue regarding materials currently used for room temperature wearable thermoelectric applications, bismuth telluride alloys, is their scarcity and reported health hazardousness [3]. The aim of this study is thus to propose a solution for flexible thermoelectric generators based on novel abundant, cheap and non-health hazardous materials. In this perspective, tetrahedrite material is studied as it is promising p-type thermoelectric material for room temperature applications which transport properties can be significantly improved by chemical substitutions [4]. Targeting the development
⇑ Corresponding authors at: MADIREL UMR 7246, Aix-Marseille Université, CNRS, Campus de St Jérôme, 13397 Marseille Cedex 20, France (M. Bella). E-mail addresses: [email protected] (M. Bella), [email protected] (P. Boulet).
of flexible printed thermoelectric devices, we propose an innovative approach for device optimization based on the multiscale modeling of a complete thermoelectric system. In this scope, two levels of modeling are addressed, from nano to macroscale. At the nanoscopic level, quantum density-functional theory is used in conjunction with semi-classical approach using Boltzmann transport theory to calculate electronic properties such as Seebeck coefficient and electrical conductivity [5] of tetrahedrite. Simulation results are then compared with experimental data. The impact of doping on thermoelectric properties is also investigated. An interesting way of fabricating low cost flexible devices is by using printing techniques as it enables the processing of flexible substrates under standard temperature and pressure conditions. Beside the technological challenges (ink preparation, printing parameters control, post treatment), it is necessary to evaluate the efficiency of an ‘‘ideal’’ printed device. In this context, a virtual prototype of a flexible thermoelectric device with an innovative design is proposed and evaluated with finite-elements simulation. 2. Computational methods and models 2.1. Atomistic simulation of the Cu12Sb4S13 compound Tetrahedrite with classical formula Cu12Sb4S13 is used as the starting material. The general nature of the tetrahedrite structure
http://dx.doi.org/10.1016/j.commatsci.2015.06.038 0927-0256/Ó 2015 Elsevier B.V. All rights reserved.
Please cite this article in press as: M. Bella et al., Comput. Mater. Sci. (2015), http://dx.doi.org/10.1016/j.commatsci.2015.06.038
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Fig. 2. Schematic representation of the device.
2.3. Finite-element analysis of an innovative flexible TEG architecture
Fig. 1. Crystal structure of Cu12Sb4S13.
was first described by Machatschki (1928). The ideal structure was determined to be Cu3SbS3. The correct formula Cu12Sb4S13 has been first found and described by Pauling and Neuman (1934) from experimental analysis of the composition. However the crystallographic structure proposed was found to be slightly inaccurate. A correction, for atomic positions in particular, has been provided by Wuensch (1963) [6]. Tetrahedrite crystallizes in the cubic system with a = 10.33 Å (space group I43m) at room temperature. Atoms form SbS3 pyramids, CuS4 tetrahedra, and unique CuS3 triangles. The tetrahedrite has 58 atoms in the unit cell (Fig. 1). The electronic band structure is calculated using ab initio Density Functional Theory (DFT) [7,8] and GGA-PW91 exchange correlation functionals [9]. Calculations are performed using the Quantum Espresso software (v4.3.2) [10]. The kinetic energy cutoff is 460 eV. The structure is optimized (atomic positions and cell parameters) at hydrostatic pressures equal to 0 GPa. The k-point selection is based on the Monkhorst–Pack scheme. The k-point mesh used to sample the Brillouin zone is set to 30 30 30. 2.2. Electronic transport simulation Electronic transport properties were calculated by solving the Boltzmann transport equation within the constant relaxation time approximation (CRTA) as implemented in the BoltzTraP code [11]. From the solution of the Boltzmann equation, in the relaxation time approximation, the transport coefficients, namely the electrical conductivity r and the Seebeck coefficient S, can be written as [12]:
r ¼ e2 S¼
Z
ekB
r
Z
@f de 0 NðeÞ; @e @f el ; de 0 NðeÞ @e kB T
ð1Þ
ð2Þ
where s is the relaxation time, f0 is the Fermi distribution, l the chemical potential, e and kB the Boltzmann’s constant. N is the transport distribution and is expressed as follows:
N¼
X ~ v~k~ v~k s~k
ð3Þ
k
~ v is the group velocity and s is the relaxation time. A dense k-mesh of over 3000 k-points in the IBZ was used for transport calculations to minimize possible band-crossing effects [11].
The main issue preventing the widespread use of TE modules as human body heat energy scavengers is their low efficiency under low thermal gradient consequence of their poor ability to evacuate heat. A solution for keeping the thermal gradient naturally present between skin and air across the device can be achieved by increasing the device surface at the cold side in contact with air. We thus propose the following device architecture: The device architecture is represented in Fig. 2. The plastic substrate is used as a thermal insulator for the hot part (bottom side Fig. 2) and the metal contacts as heat sinks for better heat removal (top side Fig. 2). A thermoelectric generator is made of multiple thermocouples, and each thermocouple is composed of an n-type and a p-type material. In our model, tetrahedrite is used for the p-type part. Chalcopyrite CuFeS2 can be considered as an interesting n-type counterpart to Cu12Sb4S13 as it is also a natural occurring and abundant mineral. Compared with tetrahedrite, chalcopyrite shows a higher power factor that can be increased by adjusting the Cu/Fe ratio. The thermoelectric properties of the CuFeS2 compound considered in our model for the device simulation are taken from natural samples characterization [13]. Chalcopyrite can be either n-type or p-type depending on the exact stoichiometry. The numerical analysis of the device is carried out using COMSOL Multiphysics, a finite-element method based code. The equations governing the temperature and electrical potential distributions and implemented in the software are [14]:
~ q ¼ krT þ P~ J
ð4Þ
~ J ¼ rrV rSrT
ð5Þ
q represents the heat flux and ~ where k is the thermal conductivity, ~ J the electric current ‘‘flux’’ and:
~ E ¼ rV
ð6Þ
Q ¼~ J ~ E
ð7Þ
~ E is the electric field and Q joule heating. Before applying this model to our specific configuration, it has been tested on a commercially available module model and validated by comparison between simulated and experimental characterization data. Cooling of the device by natural convection in air is modeled by adding an air channel around the device and defining the following boundary conditions: air at 297 K flows through the channel and the bottom of the device is kept at 307 K. 3. Results and discussion 3.1. Electronic structure of Cu12Sb4S13 The band structure of Cu12Sb4S13 is represented in Fig. 3. The valence band and conduction band edges are separated by a
Please cite this article in press as: M. Bella et al., Comput. Mater. Sci. (2015), http://dx.doi.org/10.1016/j.commatsci.2015.06.038
M. Bella et al. / Computational Materials Science xxx (2015) xxx–xxx
3
Fig. 3. Cu12Sb4S13 energy-band diagram (on the left), density of states (DOS) and projected density of states (pDOS) (on the right).
Fig. 4. Cu12Sb4S13 Seebeck coefficient evolution with temperature.
Fig. 5. Cu12Sb4S13 electrical conductivity evolution with temperature.
semiconducting gap of 1.25 eV, which is relatively lower than the experimental value of 1.7 (±0.2) eV from [15]. Cu12Sb4S13 is a metal in which the Fermi level lies in the valence band, near the edge. We thus expect the material to behave electrically as a metal. From our calculation, the compound has an indirect band gap which is in accordance with the experimental observation [16].
the band structure does not change with temperature and doping, in order to suppress the chemical potential and temperature dependence of the electrical conductivity. The material being intrinsically heavily doped, the doping concentration does not strongly changes with temperature. From the results (Fig. 5) we can clearly state that the adopted approximation do not appear to be valid above 450 K. Time relaxation is indeed strongly temperature dependent, especially at higher temperatures (above 500 K). However, there is a fair accordance between calculated and measured values in the temperature region of interest (below 450 K). A way to tune the thermoelectric properties of a semiconducting material can be done by doping. The power factor represents the capability of a material to produce useful electrical power and is defined as PF = S2r. A second step has thus been the evaluation of the thermoelectric properties as a function of free carrier concentration in order to find out the optimum doping concentration for power factor (PF) maximization. The free carrier concentration of pure tetrahedrite calculated using BoltzTraP is 1.02 1021 cm3 at 300 K. The Hall coefficient calculated in BoltzTraP can be used for carrier concentration determination. As RH = rh/q p, in order to evaluate the free carrier concentration, we take the following approximation: rh = 1, the Hall factor being close to unity for most semiconductors assuming single carrier model. With a calculated Hall coefficient of 3 109 m3/C at
3.2. Electronic transport properties of Cu12Sb4S13 The calculated results have been compared with experimental data. Concerning the Seebeck coefficient, there is a good agreement between calculated and experimental values for the range of temperature from 300 to 600 K [17] (Fig. 4). As there is no available data on the carrier lifetime for this compound, it is difficult to correctly determine the electrical conductivity since only r/s is calculated (Eq. (1)). In order to have a better idea on the electrical behavior of the compound as a function of temperature (Fig. 5), we add the following strong approximation: s is temperature independent. A way to approximate the time relaxation can be done by using the resistivity measurement on Cu12Sb4S13 done by Heo et al. [17]. We thus fit our experimental data at a given temperature and carrier concentration. We consider the pure crystal at 320 K and we obtain s = 5.9 1015 s by fitting the conductivity r (320 K) = 72,100 S/m of bulk Cu12Sb4S13 (Heo et al.). We also apply the ‘‘rigid band approach’’, assuming that
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Fig. 6. PF/s, seebeck coefficient and r/s evolution with carrier concentration.
[18]: 3.44 1021 cm3. The thermoelectric parameters evolution with carrier concentration have been evaluated at 300 K and are represented in Fig. 6. The optimum PF/s of 1.4 1011 W/m K2 s is achieved for a free carrier concentration of 5.85 1020 cm3 at 300 K. In order to evaluate the electrical conductivity of the doped compound, we fixe s = 5.9 1015 s. The power factor for such the optimally doped compound is thus estimated to be around 9 lW/cm K2. The doping concentration needed to achieve this concentration is high, in the order of 1020 cm3 electron, and thus achievable by chemical doping only. A way to achieve quite efficiently and easily such a concentration would be by substitution of some copper atoms (Fig. 3). Indeed, in order to lower the free hole concentration, the valence band has to be slightly filled and copper being the main element responsible for the compound ‘‘intrinsic’’ doping, the adjustment can be done by replacing some copper atoms by elements providing more electrons, such as zinc. Fig. 7. Device configuration.
3.3. Device performances
Fig. 8. Temperature gradient as a function of the thermal conductivity of the thermoelectric elements.
300 K, the free carrier concentration of pure tetrahedrite is estimated at 2.08 1021 cm3 .Those results are in fair accordance with the formal charge counting accounting for 2 holes per formula cell
In the following section, we evaluate the cooling efficiency of the device presented in Fig. 2 as well as its electrical performances. The performances of the device containing optimally doped tetrahedrite will be evaluated and compared to the one containing pristine tetrahedrite. First, the temperature gradient across the device heated in one side by the human body and cooled by natural convection in air (Fig. 7) is evaluated. We evaluate the temperature gradient difference and note that for an external temperature gradient of 10 °C: temperature gradient naturally present between a wrist at 34 °C and air at 24 °C, the gradient across the device stabilizes at 3 °C for our steady state simulation. In our particular device configuration, we observed that there is no significant impact of the increase in thermal conductivity of the thermoelectric elements on the temperature gradient kept across the device (Fig. 8). For instance, an increase of thermal conductivity from 1 W/m K to 7 W/m K induces a temperature gradient loss of 0.33 °C across the device. Table 1 gives the material parameters used for the finite-element analysis. We consider the same value of thermal conductivity for pristine and doped tetrahedrite as we expect no higher value, the optimally doped material having a lower carrier
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Table 1 Materials parameters at 300 K. Materials
S (lV/K)
r (S/m)
j (W/m K)
Cu12Sb4S13 (p-type) Optimally doped Cu12Sb4S13 (p-type) CuFeS2 (n-type) Sb/Te (p-type) Bi/Te (n-type)
76 (this work) 177(this work) 405 [13] 209 209
7.38 104 (this work) 2.65 104 (this work) 3.85 103 [13] 1.16 105 1.16 105
1.1 [17] 1.1 [17] 5.9 [19] 1.72 1.72
Fig. 9. Power output as a function of the resistance load for pristine tetrahedrite (circles), optimally doped tetrahedrite p = 5.85 1020 cm3 (squares) and Bi2Te3 (triangles) based devices.
concentration [17]. The thermoelectric properties of Sb/Te and Bi/Te alloys have been extracted from the characterization of a commercial module purchased from Marlow (RC12–6LS). The second step is the evaluation of the thermoelectric generator performance. The contact resistance has to be taken into account in the model as in practice the contact between the thermoelectric material and the metal is not ideal. For evaluation purpose and as there is no available data, we fix RC = 106 X cm2, value measured in a bulk Bi2Te3 device [20]. Only one thermocouple is modeled. In the presented configuration (one thermocouple) and for a matched resistance load, the maximum output power is 16.6 nW for a temperature gradient of 3 °C in the case of pristine tetrahedrite (Fig. 9). We can note that the internal electrical resistance of 31 X is quite high, which is mainly due to the relatively poor electrical conductivity of CuFeS2. In the literature [19], a strategy for this compound thermoelectric properties improvement has been discussed. An electrical conductivity of 2.11 104 S/m for the Cu0.9Fe1.1S2 compound has been achieved which is nearly 10 times higher than for the classical formula CuFeS2. The device power output can be increased from 16.6 nW to 22.4 nW by correctly doping tetrahedrite (Fig. 9). However, as the optimally doped material has a lower electrical conductivity, the device electrical resistance increases slightly from 31 X to 34 X. For comparative purpose, a simulation with the same device architecture but with Sb/Te and Bi/Te alloys instead of tetrahedrite is done. The comparative results (Fig. 9) demonstrate that the tetrahedrite based modules are less efficient than conventional materials based modules with a power output divided by six. In order to evaluate a device that would be composed of the evaluated thermocouple, we fix our target output power at 40 lW, which represents the minimum output power needed to trigger a microcontroller unit. An output power of 40 lW is reached for a device composed of 2411 thermocouples which
represents a surface of around 482 cm2. A wristwatch covering approximately 100 cm2 of skin, the thermoelectric generator performances can be further improved either by improving the design, by working on the heat sink efficiency, and increase the thermal gradient kept across the device and/or by improving the thermoelectric properties of the embedded materials, which can be achieved by appropriate carrier concentration adjustment.
4. Conclusions First principle calculations coupled with the Boltzmann transport equation resolution under the constant relaxation time approximation (CRTA) were used to investigate the electronic structure together with the thermoelectric properties of the tetrahedrite compound. The impact of doping on thermoelectric properties of pure tetrahedrite was investigated. In order to achieve a higher power factor, optimized carrier concentration should be around 5.8 1020 cm3. We have proposed a new method for evaluating novel materials potential for room temperature thermoelectric applications with a multiscale modelling approach by the mean of a macroscale prototype compatible with our target application. This method enlightens some key parameters: the temperature gradient across the device and the device internal electrical resistance. In order to achieve higher temperature gradient, an additional work on the heat sink has to be done. The electrical conductivity of the thermoelectric materials can be adjusted by appropriate atom substitutions as suggested by the computational results. Decreasing the device internal electrical resistance can also be done by improving the contact resistance between the thermoelectric material and the metal contact. We have also shown that thermal conductivity increase (with electrical conductivity) is not a hindrance in this specific configuration considering thermal conductivities of the studied materials do not exceed a few W/m K.
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Please cite this article in press as: M. Bella et al., Comput. Mater. Sci. (2015), http://dx.doi.org/10.1016/j.commatsci.2015.06.038