Article pubs.acs.org/JPCC
Thermoelectric Transport Across Nanoscale Polymer− Semiconductor−Polymer Junctions Yuanyuan Wang, Jun Liu, Jun Zhou, and Ronggui Yang* Department of Mechanical Engineering, University of Colorado, Boulder, Colorado 80309, United States ABSTRACT: There is an increasing interest in the thermoelectric (TE) properties of hybrid organic−inorganic structures, such as molecular junctions, organic−inorganic multilayers, and nanocomposites, owing to the recent success in nanostructuring of inorganic materials for high-efficiency thermoelectrics and the ability to synthesize hybrid materials in close analog to inorganic counterparts with much lower cost and greater flexibility. Compared to inorganic counterparts, the development of hybrid inorganic-polymer structures for TE applications are in a nascent stage, where theoretical understanding is very much needed and many potential nanoscale structures are yet to be explored. In this work, we study quasi-one-dimensional TE transport in a nanoscale polymer−semiconductor−polymer (PSP) junction, where a semiconductor quantum dot (QD) is trapped between two bulk polymers with aligned polymer chains. The Holstein small polaron model, which can be used for strong electron−phonon interaction in polymers beyond the perturbation theory, is used to model the transport in such a nanoscale PSP junction. We then use the Green’s function method along with the Landauer formula to calculate the TE properties of a nanoscale PSP junction, including the electrical conductance G, the Seebeck coefficient S, and the power factor GS2. Due to the sharp distribution of electron density of states in the polymer leads and the discrete energy levels in a QD, simultaneous enhancement of the Seebeck coefficient and the electrical conductance in nanoscale PSP junctions can be achieved when the energy levels are appropriately aligned, compared to metal−molecule−metal junctions. The theoretical approach to study nanoscale PSP junctions can be readily extended to the study of QD−polymer nanocomposites. The quantitative results obtained in this work can shed some light in material selection for the synthesis of hybrid inorganic−polymer nanocomposites, where theoretical guidance is much in need.
I. INTRODUCTION The thermoelectric (TE) effect, which directly converts heat to electricity, and vice versa, is attracting great interests for its applications in waste heat recovery, solar thermal utilization, and thermal management.1−3 The energy conversion efficiency of TE devices is characterized by the dimensionless TE figureof-merit ZT, which is defined as ZT =
GS2 T κc + κ p
For thermoelectricity to play a significant role in energy sectors, TE materials should not only be highly efficient, but also be low-cost, manufacturing-scalable, and environmentally friendly. Conducting polymer-inorganic hybrid materials, which might meet such challenging demands, have raised great interests in the materials community for potential applications as TE materials.5−8 Due to the recent success in developing highly efficient inorganic TE nanocomposites,9−12 many efforts have thus been directed toward the synthesis of polymer− inorganic nanocomposites for TE applications where semiconductor nanoparticles, nanowires, or carbon nanostructures are embedded in polymer matrix.13−19 For example, See et al.13 reported a promising ZT ∼0.1 when tellurium nanorods are embedded into PEDOT:PSS matrix. He et al.15 reported that the P3HT-Bi2Te3 nanocomposites exhibit a high power factor of 13.6 μW/mK2 due to the energy-filtering effect at the organic−inorganic interfaces. Though good experimental progress has been made, many of these studies lack a wellgrounded theoretical guidance. On the other hand, significant amount of theoretical and experimental works have been
(1)
where T is the absolute temperature, G and S are the electrical conductance and the Seebeck coefficient, and κc and κp are the thermal conductance of electronic and phononic (lattice) contributions, respectively. It is a grand challenge to increase ZT in conventional bulk materials beyond a value of unity since G, S, and κc are interdependent.4 Significant progress has been made over the past two decades in the improvement of ZT of inorganic TE materials, especially with the nanostructuring approach, in which the power factor is increased due to the quantum confinement effect and/or low-energy filtering effect of electrons while the lattice thermal conductivity can be reduced due to the phonon-interface/boundary scattering.2 © 2013 American Chemical Society
Received: August 22, 2013 Revised: October 19, 2013 Published: October 28, 2013 24716
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devoted to the understanding of molecular junctions,20 such as the metal−molecule−metal (MMM) and self-assembly monolayer (SAM) junctions. In a MMM junction, a single molecule is (or several molecules are) trapped between two metal electrodes.20−28 Due to the sharp distribution of electron density of states (DOS) in the molecule, the Seebeck coefficient of the MMM junction is theoretically predicted22,24 to be larger than 100 μV/K whereas a value of 15 μV/K has been demonstrated experimentally.21,25,27 However, due to the limitation of small electrical conductance (0.22 eV) to cover both the lowest and second lowest QD energy levels at 1.33 and 1.55 eV. As a result, there appear two peaks in the transmission coefficient. The transmission coefficient at certain carrier energy ranges could thus be larger than unity due to the superposition of the transmission coefficient at εQD and εQD 1 2 . Figure 4 also shows that a smaller energy range is covered by nanozero transmission coefficient when tz is increased, because the half width of the transmission coefficient through the nth energy level in QD is proportional to the DOS (ΓLnn(ε) ∝ ρ(ε)V02) and the DOS decreases with the increase of tz. Figure 5 shows the dependence of TE properties on the small polaron bandwidth by varying tz, assuming that the chemical potential is located at εQD 1 . When the chemical potential is set to be equal to εQD 1 , only the carriers close to the lowest energy level in the QD (εQD 1 ), within ∼3kBT, contribute to the electrical conductance. The increase of tz results in a smaller energy range of carriers with nonzero transmission coefficient as shown in Figure 4, which in turn reduces the electrical conductance, as shown in Figure 5a. Figure 5b shows that the absolute Seebeck coefficient abruptly increases at tz = 0.1 eV. As shown in Figure 4, when tz > 0.09 eV, only the carriers around εQD has nonzero transmission coefficient and 1 they are distributed almost symmetrical to the chemical potential (μ = ε1QD), which leads to negligible Seebeck coefficient. When tz > 0.09 eV, the carriers with the energy close to the second lowest energy levels (εQD 2 ) start to play a role, which causes asymmetrical distribution around the chemical potential (μ = εQD 1 ), and as a result the absolute Seebeck coefficient abruptly increases. Figure 5c shows that the power factor has a maximum value (0.016 pW/K2) when the intrachain coupling tz is 32 meV. This intrachain coupling value is indeed very close to the intrachain coupling constant of P3HT that is reported to be 34−42 meV.17 As a result, 24721
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Figure 6. Transmission coefficient Tr(ε) as a function of carrier energy (y axis) and the size l of the cubic QD (x axis) for a PSP junction made of P3HT (ε0 = 1.25 eV) and PbTe QD, assuming the coupling strength between leads and QD is 15 meV.
QD size. Figure 6 also shows that the transmission coefficient has two peaks at both the upper and lower edges of the small polaron band. This is due to the larger DOS at the edges of the small polaron band. The width function ΓLnn′(ε) is proportional to the DOS, ΓLnn′(ε) = 2πρ(ε)V̅ 0V̅ 0*. Therefore the transmission coefficient is larger at the edges. Figure 7 shows the dependence of TE properties on the size of QD. As the QD size increases, the energy levels in QD
Figure 5. (a) Electrical conductance G, (b) absolute Seebeck coefficient |S|, and (c) power factor GS2 as a function of intrachain coupling tz in polymer leads (x axis) and coupling strength (y axis) between polymer leads and QD assuming the size of cubic QD l = 5 nm.
nanoscale PSP junctions with P3HT leads could have a large power factor. The maximum value of the power factor is 0.005 pW/K2 when P3HT is coupled to a 5 nm PbTe cubic QD, assuming the coupling strength V0 = 15 meV and tz = 38 meV. C. Dependence on the QD Size. In this subsection, we study the effect of QD size on the TE properties of nanoscale PSP junctions by choosing P3HT as the lead material (ε0 = 1.25 eV) and changing the size of the PbTe QD trapped in between, l = 3−23 nm. The parameters of P3HT are listed in Table 3. Figure 6 shows that the transmission coefficient Table 3. Physical Parameters of P3HT Used in This Work affinity (eV)
intermonomer coupling tz (meV)
interchain coupling t∥ (meV)
optical phonon energy ℏω0 (meV)
dielectric constant ε (ε0)a
3.2−3.560
3454
1−2552
17066
3.467
ε is the vacuum dielectric constant.
a 0
increases with the QD size. In the P3HT leads, the center of the small polaron band lies at ε0 = 1.25 eV with a bandwidth of 0.12 eV. When the QD size is increased, more QD energy levels fall into the range of the small polaron band (1.19−1.31 eV), which leads to more resonant tunneling channels. Consequently, the total transmission coefficient increases with the increase of the
Figure 7. (a) Electrical conductance G, (b) Seebeck coefficient S, and (c) power factor GS2 as a function of the coupling strength (y axis) and the size of cubic QD l (x axis) for a PSP junction assuming the coupling strength V0 is 15 meV and μ = ε0 = 1.25 eV. 24722
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become denser, which leads to more carrier transport channels and consequently the increase of the electrical conductance, as shown in Figure 7a. Figure 7b shows that there are many peaks in the absolute Seebeck coefficient, which is because more energy levels in QD are involved in TE transport when the QD size becomes larger. However, the maximum value of these peaks decreases with the increase of the QD size. For a cubic QD with a small size, the number of energy levels is small and a newly involved energy level strongly affects the transport. For a cubic QD with a larger size, there are already many energy levels involved in the transport and a newly involved energy level would only slightly affects the transport. Similarly, there exist multiple peaks in the power factor with the QD size, as shown in Figure 7c. The maximum power factor is 0.43 pW/K2 with a QD size l = 14.5 nm. The power factor decreases when the QD size diverges from 14.5 nm. For example, when l = 9.5 nm and 17.5 nm, the maximum values are both decreased to 0.34 pW/K2. When l = 5 nm, the maximum value of the peak is even smaller (0.18 pW/K2). D. Dependence on Chemical Potential. The doping concentration, which controls the location of the chemical potential in the band diagram shown in Figure 1, can play a critical role in the TE transport. The doping concentration (the chemical potential) needs to be optimized for optimal TE transport properties. Figure 8 shows the dependence of TE properties on the chemical potential for a nanoscale PSP junction when the small polaron band is aligned with the lowest energy level of a 5 nm PbTe QD, i.e., εQD 1 = ε0 = 1.33 eV. As expected, when the chemical potential is aligned with the lowest QD energy level, μ = εQD = ε0, both the transmission 1 coefficient and the electrical conductance reach their maximum values, which decreases rapidly when the chemical potential deviates from the small polaron band. However, when the QD chemical potential aligns with ε0 and εQD 1 (μ = ε1 = ε0), the carrier energy distribution is symmetrical to the chemical potential, which results in a zero Seebeck coefficient since the transmission coefficient is symmetrical to ε0 and εQD 1 , as shown in Figure 8b. When the chemical potential shifts away from ε0 and εQD 1 , the transmission coefficient becomes asymmetrical to the chemical potential, which leads to an increase in the Seebeck coefficient. As a result, the power factor GS2 reaches the maximum when μ is about 3kBT away from the small polaron band, as shown in Figure 8c. E. Applications to P3HT-PbTe-P3HT Nanoscale Junction. In the previous subsections, the dependence of different parameters of both polymer leads and semiconductor QD on the TE properties are studied and the optimized parameters to enhance the TE performance are discussed. We can now apply our model to study the TE properties of a P3HT-PbTe QDP3HT junction. The affinity mismatch is 1.25 eV (ε0 = 1.25 eV). According to the previous discussion, the coupling strength is chosen to be the optimized value (15 meV), the size of the QD is set to be 5 nm (one of the maximum values), and the chemical potential is taken to be 3kBT lower than the small polaron band (μ = ε0 − 3kBT). Then the calculated power factor of the P3HT-PbTe QD-P3HT junction is 1.45 pW/K2, which is much larger than the reported measured value of the MMM molecular junction (8.25 × 10−6 pW/K2).27,50
Figure 8. (a) Electrical conductance G, (b) Seebeck coefficient S, and (c) power factor GS2 as a function of chemical potential μ for a PSP junction with the size of cubic QD l = 5 nm assuming that the small polaron band in the polymer leads is aligned with the lowest energy level in QD. The coupling strength between leads and QD is assumed to be 15 meV.
trapped between two polymer leads with aligned polymer chains. The Holstein small polaron model is used to describe such a nansocale PSP junction. We then used Green’s function method along with the Landauer formula to calculate the TE properties. Due to the sharp distribution of electron density of states in the polymer leads and the discrete energy levels in a QD, the Seebeck coefficient and electrical conductance can be greatly tuned by appropriately aligning the energy levels. Such alignment can be physically achieved through appropriate selection of polymer materials, the size and the material of QD, and doping control. The calculated power factor of a P3HTPbTe QD-P3HT junction with 5 nm PbTe QD is as high as 1.45 pW/K2 as shown in section III.E, which is many orders of magnitude higher than the reported experimental value of the Au−molecule−Au molecular junction (8.25 × 10−6 pW/ K2).27,50 The theoretical approach to study a QD trapped between two polymer leads can be extended to the study of QD−polymer nanocomposites. The quantitative results obtained in this work can shed some light in material selection
IV. SUMMARY We studied the TE properties, including the electrical conductance G, the Seebeck coefficient S, and the power factor GS2, in a nanoscale PSP junction where a semiconductor QD is 24723
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for high-efficiency hybrid inorganic-polymer nanocomposites where a theoretical guidance is much in need.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work is supported by AFOSR (Grant Nos. FA9550-11-10109 and FA9550-11-C-0034) and NSF (Grant No. CBET 0846561).
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