Thermoelectric Properties of Molecular Nanowires - The Journal of

ARTICLE pubs.acs.org/JPCC

Thermoelectric Properties of Molecular Nanowires Yuanyuan Wang, Jun Zhou, and Ronggui Yang* Department of Mechanical Engineering, University of Colorado, Boulder, Colorado 80309, United States ABSTRACT: Recently there has been an increasing interest in using organic materials for thermoelectric applications due to the noteworthy advantages of mechanical flexibility, low-cost synthesis, and solution processability. While the thermal conductivity of organic semiconductors is known to be small, very few theoretical works are available to predict thermoelectric properties of organic semiconductors and guide experimental efforts. In this work, we studied the thermoelectric properties (electrical conductivity, Seebeck coefficient, and power factor) of quasi-one-dimensional (Q1D) self-assembled molecular nanowires (NWs) based on a rigorous evaluation of the Kubo formula using the Holstein model. The molecular NWs are composed of either one-dimensional conducting polymer chains, coupled by covalent bonds, or linear stacks of planar building blocks, held together by van der Waals or ππ interactions; these are also known as molecule metals (or semiconductors), charge transfer salts, or π-stacked organic semiconductors. The dependence of thermoelectric properties on a variety of physical parameters, including intersite coupling, electronphonon interaction, chemical potential (doping concentration), and temperature, are systematically studied. We found that the thermoelectric properties are strongly affected by the intersite coupling and the dielectric constant. If the phonon thermal conductivity is assumed to be 0.2 W/mK, the thermoelectric figure of merit (ZT) of a molecular NW can reach 15.2, which is much larger than the best known inorganic thermoelectric materials obtained so far. Finally, we applied our model to study the thermoelectric properties of 1D polymer chains of P3HT and PEDOT:PSS. The power factor reaches 500 mW/cmK for a PEDOT:PSS chain. This study indicates that low-dimensional conducting polymers could be promising high-ZT materials. The theoretical study presented in this work could be useful for guiding the searchfor high-efficiency thermoelectric materials that are potentially low-cost and compatible with environmental-friendly processing.

I. INTRODUCTION Thermoelectric (TE) devices are becoming increasingly important for solarthermal conversion, waste heat recovery, and electronics cooling.13 The conversion efficiency between thermal energy and electricity for TE devices is determined by the dimensionless thermoelectric figure of merit (ZT) of the material: ZT ¼

σS2 T kc þ kp

ð1Þ

where S is the Seebeck coefficient, σ is electrical conductivity, T is the absolute temperature, kc is thermal conductivity due to the contribution of charge carriers, and kp is lattice (phonon) thermal conductivity. There have been significant efforts and progress in enhancing ZT by increasing the electronic power factor (σS2) or by reducing the lattice thermal conductivity over the past two decades, mostly for known inorganic materials.413 More recently, there is an increasing interest in using organic materials for TE applications.1422 Compared to inorganic materials, organic materials have the noteworthy advantages of mechanical flexibility, low-cost synthesis, and solution processability.22 The thermal conductivity of polymer is quite small compared to inorganic material, which usefully falls in the range of 0.11 W/m 3 K. r 2011 American Chemical Society

Moreover, thanks to advances in organic electronics over the past two decades, the physical and chemical properties of conducting polymers can be tuned in a fairly large range, providing great material flexibility to potentially meet the stringent requirements of TE applications. Many experimental works reported that large electrical conductivity values and large Seebeck coefficients can be found in some types of conducting polymers when the materials are appropriately doped. Combined with the low thermal conductivity, large ZT values have been speculated. For example, a ZT value of ∼0.38 has been reported for iodine-doped polyacetylene19 and the value is 0.25 in poly(3,4-ethylenedioxythiophene) PEDOT.22 Similarly, the nanostructuring approach1 that was proposed to enhance TE properties in inorganic materials has also been applied for conducting polymers. The TE properties of organic films,2325 chainlike polymer structures,20,26,27 and organic hybrid materials28 have been studied experimentally. Among these works, a power factor of 0.02 mW/mK2 has been obtained in the PEDOT films.23 More interestingly, advances in organic materials synthesis render the possibility of synthesizing self-assembled molecular Received: September 2, 2011 Revised: October 20, 2011 Published: October 22, 2011 24418

dx.doi.org/10.1021/jp208490q | J. Phys. Chem. C 2011, 115, 24418–24428

The Journal of Physical Chemistry C nanowires (NWs) that are composed of 1D stacks of planar building blocks loosely held together by van der Waals or ππ interactions.29 These 1D molecular NWs, also known as molecule metals (semiconductors), charge transfer salts, or π-stacked organic semiconductors, are expected to be excellent TE materials because the thermal conductivity along the stacking direction could be very small due to the interfacephonon scattering and the Seebeck coefficient could be very large due to the narrow and sharp density of states.29 In comparison to the ever-increasing experimental efforts, there are very limited theoretical works that can be used to predict TE properties and to guide the search for potentially high-efficiency organic TE materials. TE transport mechanisms in conducting polymers are not well understood. Casian et al.30,31 predicted that ZT ≈ 20 at room temperature is possible for quasi-one-dimensional (Q1D) organic crystals by using the Boltzmann transport equations of conducting electrons. Wang et al.32 coupled Boltzmann transport theory with first-principles band-structure calculations to study the thermoelectric properties in pentacene and rubrene crystals and reported ZT values ranging from 0.8 to 1.1 for these materials. In these works, electrons are considered as the electrical conducting carriers in the organic materials, which is appropriate only when the electronphonon interaction is much weaker than the intermolecular coupling. In organic materials, polaron transport should be considered since the electron phonon interaction is much stronger compared to the inorganic materials. When the electronphonon interaction is sufficiently small in comparison to the intermolecular coupling, the Fr€ohlich-type large polaron transport33 should be considered, while the Holstein small-polaron model should be applied when the electronphonon interaction is comparable to or stronger than the intermolecular coupling.3338 Very recently, Ortmann et al.38 presented a theory beyond the Holstein model to describe the transport of both small and large polarons. In this work, we limit our study to organic materials when the electronphonon interaction is comparable to or even stronger than the intermolecular coupling, where the Holstein small-polaron model is adequate to describe the carrier transport. We need to note that Schotte39 considered the TE properties of small polarons in bulk conducting polymers in 1966. In this work, we applied the Holstein small-polaron model40 to study TE properties of Q1D molecular NWs. The Holstein small-polaron model can describe carrier transport not only in the Q1D chainlike conjugated conducting polymers but also in the emerging Q1D planar molecular stacking structures when the intermolecular coupling is weaker than the electronphonon interaction, depending on whether each repeating monomer or each planar stacking unit is considered to be one site in this model. The rest of this paper is organized as follows. In section II, we present the Holstein Hamiltonian for small polarons and show the derivations of TE properties using the Kubo formula. In section III, we present detailed numerical results about the dependence of TE properties on various physical parameters, including intersite coupling, electronphonon interaction, chemical potential (doping concentration), and temperature. In section IIIF, we apply our model to study TE properties of 1D molecular chains of poly(3-hexylthiophene) (P3HT) and poly(3,4-ethylenedioxythiophene)poly(styrenesulfonate) (PEDOT:PSS). Finally, we conclude this study in section IV.

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Figure 1. (a) Q1D self-assembled molecular NWs that are composed of 1D stacks of planar building blocks loosely held together by van der Waals or ππ interactions. (b) 1D single molecular chains of conducting polymers such as P3HT and PEDOT:PSS. (c) Schematic of small-polaron transport model to describe the thermoelectric transport in organic materials shown in panels a and b. The strong electronphonon interaction in these organic materials causes lattice distortions around the electron. The electron is trapped by the polarized field formed itself.

II. THEORETICAL MODEL We consider Q1D molecular NWs as shown in Figure 1. Figure 1a depicts a Q1D molecular chain consisting of planar macrocycles held together by van der Waals or ππ interactions. Figure 1b depicts a linear molecular chain made of simple conducting monomers such as those in P3HT and PEDOT:PSS. In both organic materials, the strong electronphonon interaction causes lattice distortion around the electron, which moves along the chain but is also trapped by the polarized field formed itself. A small-polaron model should be used to describe such strong interaction of electron and phonon in organic materials.33,35 To study TE transport along the chain direction in Figure 1a,b, we set up a Q1D polaron transport model composed of N sites with intersite distance d as shown in Figure 1c. Each planar macrocycle in Figure 1a can be viewed as a site in Figure 1c. Similarly, each monomer unit of the conducting polymer in Figure 1b can be viewed as a site. Electrical transport comes from the polarons hopping from one site to another caused by overlap of the electron wave functions on adjacent sites, where the intersite coupling is essential for TE properties. In organic materials, the type and size of polymer molecules, the carrier concentration, and the type of doped atoms can be used to tune the physical parameters that are important for TE properties, including intermolecular coupling, electronphonon interaction strength, mass of molecules, and chemical potential, and thus the TE properties.41,42 24419

dx.doi.org/10.1021/jp208490q |J. Phys. Chem. C 2011, 115, 24418–24428

The Journal of Physical Chemistry C

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A. Hamiltonian and Current Operators. The Holstein Hamiltonian for the small-polaron system can be written as33

H ¼t

∑jδ cþjþδ cj þ ∑q pωq aþq aq

þ

∑j Mq eiq 3 R cþj cjðaq þ aþq Þ j

ð2Þ

where the first term on the right describes the electrons hopping between adjacent sites, the second term is the phonon spectrum, and the third term describes the local interaction between electrons and phonons. We neglect the electron electron interaction because of the low carrier densities we studied. In eq 2, cj(c+j ) is the annihilation (creation) operator of an electron at site Rj with j = 1, ..., N standing for the site index, t is intersite coupling between the nearest neighboring sites coupling (δ = ( 1), and the on-site electron energy is set to 0 as reference. aq (a+q ) is the annihilation (creation) operator of a phonon with wave vector q and frequencies ωq. For the electronacousticphonon interaction, the scattering matrix can be written as Mq = iqΞd(p/2FωqV)1/2, and for the electronopticalphonon interaction, the scattering matrix is Mq = (ie/q)[pω0/2εV]1/2, where Ξd is the deformation potential, F is the mass density, V is the volume of the material, pω0 stands for the energy of the optical phonon, and ε is the dielectric constant of the material. In the Holstein Hamiltonian, the electronphonon interaction term (nondiagonal term) is considered to be more significant than the electron term (diagonal term), which is usually considered as a small perturbation.33 We apply a canonical transformation H = eTHeT with T = ∑ jq c+j cjeiq 3 RjMq(aq a+q) to diagonalize the electronphonon interaction term in the small-polaron Hamiltonian.33 By neglecting the higher-order polaronpolaron interaction,38 we can obtain the transformed Hamiltonian H as H̅ ¼ Δ

þ Xj ∑j cþj cj þ ∑q pωq aþq aq þ t ∑jδ cþjþδ cjXjþδ

ð3Þ Δ = ∑q M2q/pωq is the polaron self-energy, and exp{∑q eiq 3 Rj(Mq/pωq)(aq a+q)} arises

at Δ and width is 4tΓ(T). When i 6¼ f, the hopping process is inelastic. The small polarons lose their phase coherence by emitting or absorbing phonons. The transition rate w for these nondiagonal transitions can be obtained via Fermi’s golden rule:33

the factor where from the Xj = canonical transformation of the electron operators eTcjeT = cjXj. Here we have introduced a new basis of polaron wave functions c+j X+j |0æ, where |0æ is the vacuum state. The first term on the right-hand side of eq 3 gives the polaron self-energy, where c+j cj can actually be written as c+j cjX +j Xj since X +j Xj = 1. The second term remains the same as that in eq 2 for phonons, since this term has already been diagonal in eq 2. The third term describes the kinetic energy of the polarons, which explicitly expresses the motion of lattice deformation described by the operators Xj and indicates the hopping of polarons from site j to j + δ. The hopping energy amplitude is tÆf|X +j+δXj|iæ, where i and f describe the initial and final states. When the transition is diagonal with i = f, the polarons hop from one site to another coherently. We can derive the diagonal part of the hopping possibility as Γ(T) Æi|X +j+δXj|iæ = exp[ξ(T)/2] with the thermal factor ξ(T) = ∑q |uq|2(1 + 2Nq), where uq (Mq/ pωq)(eq 3 δ 1) and Nq = 1/(epωq/kBT 1) which is the BoseEinstein distribution function.33 The thermal factor ξ(T) describes the effect of electronphonon interaction on the small polaron band. Now the renormalized polaron band is ~ε = tΓ(T)∑δ eik 3 δ Δ = 2tΓ(T) cos (kδ) Δ,33 whose center is

w¼

1Z p2

∞ ∞

WðtÞ dt

ð4Þ

where WðtÞ ¼ t 2

∑ ÆijXjþðtÞXjþδ ðtÞjf æÆf jXjþþ δ Xjjiæ

f¼ 6 i

ð5Þ

By applying the saddle-point integration,33 the transition rate can be rewritten as rffiffiffiffiffiffiffiffiffiffi t2 π 2ξðTÞ þ ηðTÞ e w¼ 2 ð6Þ p ζðTÞ where ξ(T) and η(T) introduce the activation energy Ea = [2ξ(T) η(T)]kBT. The expressions for η(T) and ξ(T) are given in the Appendix. On the basis of the canonical transformation, we can now write down the operators for electrical current J, energy current JE, and heat current U as33,39 te J̅ ¼ i δcþ cj X þ Xj ð7Þ p jδ jþδ jþδ

∑

J̅ E ¼ iΔ

þ Xj ∑jδ δcþjþδcj Xjþδ

t2 i ðδ þ δ0 Þcþ c Xþ X j þ δ þ δ0 j j þ δ þ δ0 j 2p jδδ0

∑

U ̅ ¼ J̅ E

μ J̅ e

ð8Þ

ð9Þ

The electrical current J is induced by the hopping of small polarons. The first term of the energy current JE describes the flow of the electronlattice interaction energy, while the second term is associated with the kinetic energy of the polarons. B. Kubo Formula for Transport Properties. The electrical conductivity σ, the Seebeck coefficient S, and the electronic thermal conductivity kc can be found from the currentcurrent correlation functions using the Kubo formula as:33 σ ¼ βL11

ð10Þ

1 L12 T L11 " # 1 22 ðL12 Þ2 kc ¼ 2 L 11 T L S¼

ð11Þ ð12Þ

where β = 1/kBT, kB is the Boltzmann constant, and

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L11 ¼

1 Z 2V

þ∞

L12 ¼

1 Z 2V

þ∞

L22 ¼

1 Z 2V

þ∞

∞

∞

∞

ÆJαþ ðτÞJα ð0Þæ dτ

ð13Þ

ÆUαþ ðτÞJα ð0Þæ dτ

ð14Þ

ÆUαþ ðτÞUα ð0Þæ dτ

ð15Þ



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where the subscript α = x, y, or z describes the transport direction studied and V = S0l is the volume of the sample, with S0 being the cross-sectional area perpendicular to the transport direction and l being the length along the transport direction. By introducing the cross-sectional area, one can apply the above equations for the TE properties in a Q1D system. However, for a rigorous 1D system, the electrical and thermal conductivity cannot be defined and the electrical and thermal conductance must be used.43 The Kubo formula still holds when electrical conductivity σ, thermal conductivity kc and volume V are respectively replaced with electrical conductance G, thermal conductance U, and length l in eqs 10 and 1215. By substituting eqs 7 and 9 into eqs 13, 14 and 15, we obtain the expressions for L11, L12, and L22 as follows: L11 ¼

te2 d Nz ηðTÞξðTÞ π 1=2 e ζðTÞ 2p S0 N 1

fk ð1 fk Þeðε~ ∑ k k 1

k1

2

ε~k2 Þ=2kB T ðε~k1 ε~k2 Þ2 =4ζ2

e

ð16Þ

1 2

L12 ¼ P

Δ þ μ 11 L e

ð17Þ

" 2 # Δ þ μ 2 J Δ þ μ P ¼ þ L11 þ 2 e 16e e

22

L

ð18Þ where P¼

t2e d 1 4p S0 N 1

∑jj ∑ ðδ þ δ0 Þδ1 eηðj, j , δ, δ , TÞ ξðTÞ 0

0

0

0

δδ δ1

!1=2

π ζðj, j0 , δ, δ0 , TÞ

∑ fk ð1 fk ÞeiðR R Þ 3 ðk k k j

1

2

j0

1

k2 Þ ik2 3 ðδ þ δ0 δ1 Þ ðε~k1 ε~k2 Þ=2kB T

e

e

1 2

eðε~k1

ε~k1 Þ2 =4ζ

ð19Þ

fk = 1/[e(εkμ)/kBT 1] is the FermiDirac distribution function, μ is the chemical potential, and z = ∑δ=(1 δ2 = 2. Explicit expressions for η(j,j0 ,δ,δ0 ,T), ζ(j,j0 ,δ,δ0 ,T), η(T), ζ(T), and ξ(T) are given in the Appendix. C. Physical Relationships. Let us now discuss the dependence of band structure [including the center of the band Δ and the bandwidth 4tΓ(t)] and the nondiagonal transition rate w of small polarons on different physical parameters. The center of the polaron band Δ is determined by the electronphonon interaction with Δ = ∑q (M2q/pωq) = ∑q (qΞd)2p/(2Fω2qV) v2Ξ2d for the acoustic-phonon-induced small polarons and Δ = ∑q (M2q/ pωq) = ∑q (e/q)2p/2εV ε1 for the optical-phonon-induced small polarons. Both the intersite coupling t and the electron phonon interaction affect the width of small polaron band 4tΓ(T) = 4t exp[ξ(T)/2]: (i) The bandwidth is proportional to the intersite coupling t. (ii) Since the average phonon number Nq increases with temperature, the bandwidth decreases with temperature through its relationship with the thermal factor ξ(T). (iii) For optical-phonon-induced small polarons, the thermal factor ξ(T) = ∑q |Mq/pωq|2(1 + 2Nq) = ∑q (e/q)2(p/2εω0V)(1 + 2Nq) (εω0)1 decreases with dielectric constant ε and optical phonon

frequency ω0. Therefore, the bandwidth increases with ε and phonon energy E0 = pω0. (iv) For acoustic-phonon-induced small polarons, ξ(T) = ∑q |Mq/pωq|2(1 + 2Nq) = ∑q (qΞd)2(p/2Fω3qV)(1 + 2Nq) Ξ2dv3 increases with deformation potential Ξd and decreases with sound velocity v. Therefore, the bandwidth decreases with Ξd and increases with v. The nondiagonal transition rate w increases with temperature, which indicates that the smallpolaron hopping rate is thermally activated. The transition rate w increases with ω0 and ε for optical-phonon-induced small polarons since ξ(T) + η(T) ≈ ∑q |Mq/pωq|2 (ω0ε)1, while the transition rate w increases with v and decreases with Ξd for acoustic-phonon-induced small polarons since ξ(T) + η(T) ≈ ∑q |Mq/pωq|2 Ξ2dv3(Nq 1). In the limit of narrow bandwidth, electrical conductivity is proportional to the nondiagonal transition rate w. For opticalphonon-induced polarons, electrical conductivity increases with temperature T, phonon energy pω0, and dielectric constant ε. For acoustic-phonon-induced polarons, electrical conductivity increases with sound velocity v and decreases with deformation potential Ξd. From eqs 16 and 17, the Seebeck coefficient S can be written as 1 P Δ μ ð20Þ S¼ T L11 e e In eq 20, L11 is related to the nondiagonal transition rate w and P is proportional to the second-order transition rate W(2) = t2∑f6¼i Æi|X +j (t)Xj+δ(t)|fæ Æf|X+j+δ0 Xj|iæ, where δ 6¼ δ0 . Therefore, the Seebeck coefficient is affected by the optical phonon energy and is temperature-dependent. Since the decrease of the secondorder transition rate with optical phonon energy is faster than that of the first-order transition rate, the first term on the righthand side of eq 20, P/TL11, decreases with optical phonon energy. Therefore, the Seebeck coefficient decreases with the optical phonon energy. P/TL11 also decreases with temperature since the increase of the second-order transition rate is slower than that of the first-order transition rate. However, the second term on the right-hand side of eq 20 decreases with T. The combined effects determine a complicated temperature dependence of the Seebeck coefficient. D. Physical Parameters. As discussed in previous sections, the band structure and nondiagonal transition rate w of small polarons, and thus the TE properties of organic molecules, are determined by a few physical parameters, including mass M of the organic molecules, distance d between two neighboring sites, intersite coupling strength t, optical phonon energy pω0, dielectric constant ε, sound velocity v, and deformation potential Ξd. In Table 1, we have summarized the typical values of these physical parameters for a few widely investigated conducting polymers that might be of interest to the TE community. The mass of polymer molecules is in the range (105106)me,31,4447 where me is the electron mass. The distance between the nearest neighboring sites is on the order of the bond length, which is about several angstroms.31,4851 Usually, the larger the intersite distance, the smaller the intersite coupling strength. For example, the intersite coupling strength for P3HT52 (3442 meV) is much smaller than that for other materials such as PEDOT:PSS53 (∼280 meV) since the bond length of P3HT is about 23 times longer than that of PEDOT:PSS. The optical phonon energy varies in a quite large range, from 102 meV (PEDOT:PSS54) to 102 meV (P3HT,52 trans-polyacetylene47). The dielectric constants for most polymer materials are less than 10ε0,5558 where 24421



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Table 1. Physical Parameters in the Conducting Polymers That Might Be Good TE Materials pω0 (meV)

v (m/s)

280

0.17

1.58 103

48

53

54

59

(40.2 105)(346 105)

3.7

3442

170

2.87 103

2280

46

50

52

68

60

68

M (me)

d (Å)

PEDOT:PSS

(14.5 105)(72.0 105)

1.3451.453

ref

44

P3HT ref C60 polymers51 tetrathiotetracene iodide31

t (meV)

1.534

Ξd (meV)

3.9 55 3.4 57

1160

6.5 105

1.5 103

160

electroactive polymers58

0.6 eV. |R| is always larger than 1 for the acoustic-phonon-induced small polarons. In contrast to Schotte,39 Figure 2c indicates that the contribution of the small-polaron kinetic energy is not negligible. Figure 2d shows that the power factor σS2 increases with t for the optical-phonon-induced small polarons, while σS2 has a maximum value when t = 0.1 eV for acoustic-phonon-induced small polarons. The acoustic-phonon-induced small polarons play the leading role when t < 0.1 eV, while the optical-phonon-induced small polarons dominate when t g 0.1 eV. B. Dependence on ElectronPhonon Coupling. The TE transport properties of molecular NWs depend strongly on electronphonon coupling. The type and structure of the molecular material determines the electronphonon coupling61 and thus greatly affects the TE transport properties.41 Figure 3 shows the dependence of TE properties on the optical phonon energy E0 for optical-phonon-induced small polarons. Four cases are shown with different dielectric constants, ε = 2ε0, 4ε0, 7ε0, and 10ε0. In the calculation, t is set to be 0.8 eV. Figure 3a shows that the electrical conductivity remains almost unchanged with E0, no matter how large the dielectric constant ε is. As mentioned in section IIIA, the bandwidth for optical-phonon-induced small polarons is negligible. The electrical conductivity is thus determined only by the nondiagonal transition rate w. In the range of E0 we applied, w almost does not change, which results in a constant electrical conductivity. Comparing the curves for different dielectric constants, we find that the electrical conductivity increases dramatically with ε since the nondiagonal transition rate w decreases with ε. For example, the electrical conductivity for ε = 10ε0 is more than 2 orders of magnitude larger than that for ε = 2ε0. Figure 3b shows that the Seebeck coefficient almost

does not vary with E when ε is small (= 2ε0, 4ε0, or 7ε0). However, |S| decreases with E0 when ε is larger (= 10ε0). When ε is small, the contribution of the small-polaron self-energy is dominant and |S| Δ/eT. As we analyzed in section IIC, Δ does not depend on E0, and consequently |S| is a constant when E0 changes. With increasing ε, the contribution of the small-polaron self-energy becomes smaller since Δ ε1, while the contribution of electronphonon interaction starts to play a role and S P/ L11, which increases with phonon energy. As a result, |S| decreases with optical phonon energy. Figure 3c shows that the power factor decreases with E0, especially when ε is larger than 10ε0. Figure 4 shows the dependence of TE properties on sound velocity and deformation potentials for acoustic-phonon-induced small polarons. There exists a maximum electrical conductivity at certain sound velocity. For example, the maximum value appears when v = 1.25 m/s for Ξd = 0.75 eV. The phonon velocity value at which the electrical conductivity reaches a maximum becomes larger with increasing deformation potential Ξd. For acoustic-phonon-induced small polarons, both the nondiagonal transition rate w and the small polaron energy dispersion change with the sound velocity, but in the opposite trend. As a result, a maximum electrical conductivity is observed at certain sound velocity value for acoustic-phonon-induced small polarons. Very similar to the results for optical-phononinduced small polarons, Figure 4b shows that the Seebeck coefficient for acoustic-phonon-induced small polarons increases rapidly with E0. The self-energy Δ is proportional to v2; therefore |S| decreases with v, since the Seebeck coefficient is essentially determined by the self-energy of the small polarons. Figure 4c shows that there exists a maximum power factor. The phonon velocity at which the power factor reaches the maximum value increases with deformation potential. From the above analysis, we find that the optical phonon energy should be small and the dielectric constant should be large 24423



Figure 5. (a) Electrical conductivity σ, (b) Seebeck coefficient S, and (c) power factor σS2 as functions of chemical potential μ. T = 300 K was used for the calculation.

to obtain good TE properties for optical-phonon-induced small polarons. Similarly, the acoustic phonon velocity and deformation potential should be small to obtain good TE properties for acoustic-phonon-induced small polarons. We have thus chosen E0 = 10 meV and ε = 10ε0 for the optical-phonon-induced small polarons and Ξd = 0.075 eV and v = 1.25 km/s for acousticphonon-induced small polarons to calculate the dependence of TE properties on chemical potential and temperature in sections IIIC and IIID. C. Dependence on Chemical Potential. The chemical potential could be greatly tuned by the doping concentrations in organic materials.19,24 We now study the dependence of TE properties on chemical potential. The temperature is fixed at 300 K. Figure 5a shows that the electrical conductivity has a maximum value, for small polarons formed by both optical and acoustic phonons. The maximum value appears when μ is 10 meV for optical-phonon-induced small polarons and when μ is 0.1 meV for acoustic-phonon-induced small polarons. When the chemical potential is away from these values, the electrical conductivity gradually decreases. When the chemical potential is in the center of the band, the carriers are most active and thus the electrical conductivity reaches its maximum since the term fk(1 fk) in eq 16 reaches its maximum. As shown in section IIC, the center of the small polaron band is Δ and the bandwidth is 4tΓ. With the parameters taken here, Δ is about 10 meV for optical-phonon-induced small polarons and about 0.1 meV for acoustic-phonon-induced small polarons. Not surprisingly, the maximum values of electrical conductivity for optical- and acoustic-phonon-induced small polarons appear at 10 and 0.1 meV, respectively. Figure 5b shows that the absolute value of the Seebeck coefficient reached its minimum at the center of the small polaron band, 0.1 and 10 meV of acoustic- and optical-phonon-induced small polarons, respectively. The absolute value of the Seebeck coefficient increases gradually when μ moves away from the center of the band. The Seebeck coefficient

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Figure 6. (a) Electrical conductivity σ, (b) Seebeck coefficient S, and (c) power factor σS2 as functions of temperature T for both optical- and acoustic-phonon-induced small polarons. t = 20 eV and Ef = 0.05 eV were used for the calculation. Note the dotted curve with 9 in panel b was obtained from the parameters of PEDOT:PSS listed in Table 1.

is the average entropy of the carriers, which implies the carrier distribution symmetry. The farther the chemical potential is from the center of the band, the more unsymmetrical the carrier distribution is, and thus |S| increases. Figure 5c shows that the power factor σS2 of optical-phonon-induced small polarons reaches the largest value, 160 mW/cm 3 K2, when m is about 50 meV (≈ 2kBT for T = 300 K) away from the center of the band. The power factor of the acoustic-phonon-induced small polarons reaches 460 mW/cm 3 K2 when m is about 70 meV (≈ 2.7kBT for T = 300 K) away from the center of the band. Since σ decreases and |S| increases when the chemical potential is away from the center of the band, there exists a maximum power factor when the doping level (chemical potential) changes. D. Temperature Dependence. Figure 6a shows that electrical conductivity σ increases with temperature for both the opticaland acoustic-phonon-induced small polarons due to the increase of hopping probability with temperature. Figure 6b shows that the absolute Seebeck coefficient decreases with temperature for both optical- and acoustic-phonon-induced small polarons. Note that the parameters for the calculation in this section are taken on the basis of optimized values obtained in the previous three subsections with intersite coupling t = 0.8 eV, optical phonon energy E0 = 10 meV, dielectric constant ε = 10ε0, sound velocity v = 1.5 km/s, and deformation potential Ξd = 75 meV. However, if we use the physical parameters based on PEDOT:PSS (see Table 1) for the calculation, the absolute Seebeck coefficient for opticalphonon-induced small polarons increases first and then decreases with T. This is because when the optimized parameters are taken, the Seebeck coefficient is dominant by the contribution of the small-polaron self-energy |S| Δ/eT, and thus |S| decreases with increasing T. These results are consistent with the experimental results of P3HT.19 If we take the PEDOT:PSS parameters, the contribution of the electronphonon interaction cannot be 24424



Figure 7. (a) Lorentz number L and (b) ZT as functions of chemical potential μ for optical-phonon-induced small polarons. t = 0.2 eV, M = (50 105)me, ω0 = 10 meV, and ε = 10ε0 were used for the calculation.

Figure 8. (a) Electrical conductance G, (b) Seebeck coefficient S, (c) GS2 and (d) power factor σS2 as functions of chemical potential μ for single polymer chain made of P3HT and PEDOT:PSS at room temperature.

neglected compared to that of the small-polaron self-energy, and thus |S| (P/L11 Δ)/T. When both P/L11 and Δ decrease with T, P/L11 Δ does not necessarily decrease with T. Figure 6c shows that the maximum values of the power factor are 700 μW/ cmK2 at T = 250 K for optical-phonon-induced small polarons and 180 μW/cmK2 at T = 190 K for acoustic-phonon-induced small polarons. E. ZT and Lorentz Number. We now calculate the Lorentz number L and figure of merit ZT of the Q1D molecular NWs. We have used the optimal parameters derived from the previous sections: intermolecular coupling t = 10 meV and optical phonon energy E0 = 10 meV. The temperature is fixed at 300 K. Figure 7a shows that the Lorentz numbers for all doping levels are smaller than (π2/3)(kB2/e2). Clearly a violation of the WiedemannFranz law in the Q1D molecular NW is observed as expected. For most polymer materials, the thermal conductivity due to phonon contribution is about 0.11.0 W/mK for bulk materials.62,63 Although some recent works show that phonon thermal conductivity can increase significantly in low-dimensional polymers,64,65 we assume

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the phonon thermal conductivity kp = 0.2 W/mK to plot the ZT of Q1D molecular NW in Figure 7b. It is exciting to find that the largest ZT can reach 15.2 in Q1D molecular NWs, which is much better than the known best inorganic TE material.4,5,7,8 We also note that the ZT value could be lower if kp of the molecular NW is larger, which remains to be investigated. F. Applications to Molecular Chains of P3HT and PEDOT: PSS. Here we apply our small-polaron transport model to the 1D single molecular chains of P3HT and PEDOT:PSS to study their TE properties. Only TE properties for the optical-phononinduced small polarons are considered here, since they play dominant roles when the intersite coupling is large. The calculation is conducted for T = 300 K with the physical parameters of these conducting polymers shown in Table 1. For molecular chains, the electrical conductivity cannot be defined and the electrical conductance must be used since a molecular chain is a rigorous 1D system. Figure 8a shows that the electrical conductance for PEDOT: PSS has maximum values at μ = 10 meV, while the maximum of the electrical conductance for P3HT occurs at μ = 20 meV. For different materials, the strength of the electronphonon interaction varies and then the central positions of the small-polaron band are different, which affects the position where the maximum value of electrical conductance occurs. Figure 8b shows that the Seebeck coefficient becomes smaller when μ moves gradually away from 10 meV for PEDOT:PSS, while it moves away from 20 meV for P3HT. The electrical conductance of PEDOT:PSS are at least 2 orders of magnitude larger than that of P3HT because the intersite coupling t for P3HT is much smaller than that of PEDOT:PSS as shown in Table 1. The Seebeck coefficient of P3HT is also smaller than that of PEDOT:PSS when μ is smaller than 25 meV, which is also due to the relatively smaller intersite coupling for P3HT. Figure 8c shows that GS2 has two peak values due to the different dependences of G and S on chemical potential. The largest GS2 of PEDOT:PSS is about 102 mW/K2, which is 2 orders of magnitude larger than that for P3HT. This indicates that intersite coupling is the most important parameter affecting TE properties in the molecular chains. As we have known, for the rigorous 1D molecular chain, one cannot define the electrical conductivity. However, to compare the TE properties of the single molecular chain with other inorganic systems, we can introduce the crystal lattice parameters to approximately define the cross-sectional areas, which are S0 = 0.296 nm2 and 0.540 nm2 for P3HT50 and PEDOT:PSS,66 respectively, to obtain the power factor σS2. Figure 8d shows that the largest power factor can reach 500 mW/cmK2 for PEDOT:PSS molecular chain, which is much larger than that for inorganic NWs.10,11

IV. CONCLUSIONS We have investigated the TE properties, including electrical conductivity σ, Seebeck coefficient S, and power factor σS2, in molecular NWs, based on the Holstein small-polaron model. We gave detailed expressions for electrical conductivity and Seebeck coefficient for both acoustic- and optical-phonon-induced small polarons. Dependence of TE properties on intersite coupling, optical phonon energy, dielectric constant, deformation potential, sound velocity, chemical potential (doping concentration), and temperature are studied systematically. Intersite coupling plays a key role in the TE properties of organic materials. The larger the intersite coupling t, the better the TE properties. When t is small (< 0.11 eV), the acoustic-phonon-induced small polarons have larger contribution to the electrical conductivity. Small sound velocity and small deformation potential are desirable for 24425



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enhancing the power factor. When t is larger (> 0.11 eV), the optical-phonon-induced small polarons have more significant contributions to TE properties. Large dielectric constant and small optical phonon energy are beneficial to achieve large power factor. If the phonon thermal conductivity is assumed to be 0.2 W/mK, the largest ZT can reach 15.2 for molecular NWs, which is much larger than the known best inorganic TE materials so far. At last, we applied our model to 1D molecular chains of two promising conducting polymer TE materials, P3HT and PEDOT:PSS. The power factor σS2 reaches 500 μW/cmK2 for PEDOT:PSS molecular chain. This study indicates that low-dimensional structures of conducting polymers could be promising directions for obtaining high-ZT materials with green processing. The theoretical study presented in this work provides good theoretical guidance for searching high-efficiency TE materials that are low-cost and compatible with environmental-friendly processing.

’ APPENDIX: CALCULATION OF CORRELATION FUNCTIONS We calculate the correlation functions of phonons and electrons in eqs 1315 based on the Hamiltonian H in eq 3. We derive first the correlation function of phonons. The exponents in the factor X +j Xj+δ can be combined since they commute and Mq is symmetric: " # þ iq 3 δ iq 3 δ Mq þ e ð1 e Þ ða aq Þ Xj Xjþδ ¼ exp pωq q q

∑

ðA1Þ Then we have Xjþ ðτÞXjþδ ðτÞXjþ0 þ δ0 Xj0 ( ) h i 1 2 0 2 0 iωq τ þ ¼ exp =2 ðjΛj þ jΛ j Þ þ ΛΛ e þ λaq λaq

∑q

in which u q (M q /pω q )(e q 3 δ 1) and v q (M q /pω q )[e q 3 R j (e q 3 δ 1). In the vicinity of the saddle point, iβp/2, Φ(Rj Rj0 ,δ,δ0 ,τ) (eq A7) is expanded as a power series about the saddle point iβp/2 and the terms decrease gradually with increasing order since βp/2 , 1: ΦðRj Rj0 , δ, δ0 , τÞ ¼ ξðTÞ ηðj, j0 , δ, δ0 Þ þ ζðj, j0 , δ, δ0 Þz2 þ Oðz4 Þ

where z = τ + iβp/2 with β = 1/kBT, and ξ(T), η(j,j0 ,δ,δ0 ,T), and ζ(j,j0 ,δ,δ0 ,T) are given by ξðTÞ ¼

ðA11Þ In eq A8, we keep only the first three terms since the higherorder terms are negligible. We can introduce the ratio between z4 term (the largest term among those we neglected) and z2 term and show the applicability of this approximation: 1/12∑q (pωq)4vq(j,δ)v/q(j,δ)[Nq(Nq + 1)]1/2z4/ζ(j,j0 ,δ,δ0 )z2 ≈ 1/12 [pωq/(βp/2)]2. The approximation is valid only when the ratio is much smaller than 1. When the electronphonon interaction is strong, the ratio is indeed much smaller than 1, which indicates that the saddle-point approximation is rather accurate. For example, the ratio is about 1 106 for PEDOT:PSS at 300 K. When j = j0 and σ = σ0 , we can define ζðTÞ ζðj, j0 , δ, δ0 , TÞjj ¼ j0 , δ ¼ δ0

ðA3Þ

Mq pωq

Æeλ

aþ λaq q

2

æ ¼ ejλj Nq

ðA5Þ

¼2

ÆXjþ ðτÞXjþδ ðτÞXjþ0 þδ0 Xj0 æ ¼ exp½ ΦðRj Rj0 , δ, δ0 , τÞ

ΦðRj Rj0 , δ, δ0 , τÞ ¼ 2

∑q juq j2 ð1 þ 2Nq Þ

∑q vq ðj, δÞvq ðj, δ0 Þ½Nq ðNq

βp þ 1Þ1=2 cos ωq τ þ i 2 ðA7Þ

∑q juq j2½Nq ð1 þ Nq Þ1=2

ðA13Þ

Æcj þ ðτÞcjþδ ðτÞcþ c 0æ j0 þδ0 j ¼

where

ðA12Þ

After derivation of the phonon correlation function, we now derive the electron correlation function as

Then we can derive ÆX+j (τ)Xj+δ(τ)X+j0 +δ0 Xj0 æ as follows:67 ðA6Þ

∑q pωq juq j2½Nq ð1 þ Nq Þ1=2

ηðTÞ ηðj, j0 , δ, δ0 , TÞjj ¼ j0 , δ ¼ δ0

ðA4Þ

phonon Hamiland λ = Λeiωqτ Λ0 . For the noninteracting / + tonian, the thermodynamic average of Æeλ aqλaqæ is given by33,67

∑q vq ðj, δÞvq ðj, δÞ½Nq ðNq þ 1Þ1=2

∑q ðpωq Þ2vq ðj, δÞvq ðj, δÞ½Nq ðNq þ 1Þ1=2

ζðj, j0 , δ, δ0 , TÞ ¼

¼

0

ðA9Þ

ðA10Þ

where

Λ0 ¼ eiq 3 R j0 ðeiq 3 δ 1Þ

∑q juq j2ð1 þ 2Nq Þ

ηðj, j0 , δ, δ0 , TÞ ¼ 2

ðA2Þ Mq Λ ¼ eiq 3 R j ðeiq 3 δ 1Þ pωq

ðA8Þ

∑ fk ð1 fk Þeiðε~ k k 1

k1

2

ε~k2 Þτ iðR j R j0 Þ 3 ðK 1 K 2 Þ ik2 3 ðδ δ0 Þ e e

1 2

ðA14Þ ik 3 δ

where ~ε = tΓ(T)∑δ e Δ. After the correlation functions of phonons and electrons (eqs A6 and A14) are obtained, we can treat the τ integrations in eqs 1315 as contour integrals in a complex space.

’ AUTHOR INFORMATION Corresponding Author

*E-mail [email protected]. 24426



’ ACKNOWLEDGMENT This work is partly supported by DARPA Active Cooling Modules Program (Contract N66001-10-C-4002) and AFOSR STTR Grant FA9550-11-C-0034 (ADA Tech Inc., PI Dr. Sayan Naha, managed by John Hottle). ’ REFERENCES (1) Dresselhaus, M. S.; Chen, G.; Tang, M. Y.; Yang, R. G.; Lee, H.; Wang, D. Z.; Ren, Z. F.; Fleurial, J. P.; Gogna, P. Adv. Mater. 2007, 19, 1043–1053. (2) Snyder, G. J.; Toberer, E. S. Nat. Mater. 2008, 7, 105–114. (3) Tritt, T. M. In Recent Trends in Thermoelectric Materials Research IIII; Tritt, T. M., Ed.; Semiconductors and Semimetals, Vol. 6971; Academic Press: San Diego, CA, 2001. (4) Chung, D. Y.; Hogan, T.; Brazis, P.; Rocci-Lane, M.; Kannewurf, C.; Bastea, M.; Uher, C.; Kanatzidis, M. G. Science 2000, 287, 1024–1027. (5) Venkatasubramanian, R.; Siivola, E.; Colpitts, T.; O’Quinn, B. Nature 2001, 413, 597–602. (6) Harman, T. C.; Taylor, P. J.; Walsh, M. P.; LaForge, B. E. Science 2002, 297, 2229–2232. (7) Hsu, K. F.; Loo, S.; Guo, F.; Chen, W.; Dyck, J. S.; Uher, C.; Hogan, T.; Polychroniadis, E. K.; Kanatzidis, M. G. Science 2004, 303, 818–821. (8) Poudel, B.; Hao, Q.; Ma, Y.; Lan, Y. C.; Minnich, A.; Yu, B.; Yan, X. A.; Wang, D. Z.; Muto, A.; Vashaee, D.; Chen, X. Y.; Liu, J. M.; Dresselhaus, M. S.; Chen, G.; Ren, Z. F. Science 2008, 320, 634–638. (9) Heremans, J. P.; Jovovic, V.; Toberer, E. S.; Saramat, A.; Kurosaki, K.; Charoenphakdee, A.; Yamanaka, S.; Snyder, G. J. Science 2008, 321, 554–557. (10) Boukai, A. I.; Bunimovich, Y.; Tahir-Kheli, J.; Yu, J. K.; Goddard, W. A.; Heath, J. R. Nature 2008, 451, 168–171. (11) Hochbaum, A. I.; Chen, R. K.; Delgado, R. D.; Liang, W. J.; Garnett, E. C.; Najarian, M.; Majumdar, A.; Yang, P. D. Nature 2008, 451, 163–167. (12) Rhyee, J. S.; Lee, K. H.; Lee, S. M.; Cho, E.; Il Kim, S.; Lee, E.; Kwon, Y. S.; Shim, J. H.; Kotliar, G. Nature 2009, 459, 965–968. (13) Biswas, K.; He, J. Q.; Zhang, Q. C.; Wang, G. Y.; Uher, C.; Dravid, V. P.; Kanatzidis, M. G. Nat. Chem. 2011, 3, 160–166. (14) Reddy, P.; Jang, S.-Y.; Segalman, R. A.; Majumdar, A. Science 2007, 315, 1568–1571. (15) Kaiser, A. B. Adv. Mater. 2001, 13, 927. (16) Kaiser, A. B. Phys. Rev. B 1989, 40, 2806–2813. (17) Shakouri, A.; Li, S. In Proceedings of the IEEE 18th International Conference on Themoelectric, 1999; p 402. (18) Gao, X.; Uehara, K.; Klug, D. D.; Patchkovskii, S.; Tse, J. S.; Tritt, T. M. Phys. Rev. B 2005, 72, No. 125202. (19) Xuan, Y.; Liu, X.; Desbief, S.; Leclere, P.; Fahlman, M.; Lazzaroni, R.; Berggren, M.; Cornil, J.; Emin, D.; Crispin, X. Phys. Rev. B 2010, 82, 115454. (20) Nogami, Y.; Kaneko, H.; Ishiguro, T.; Takahashi, A.; Tsukamoto, J.; Hosoito, N. Solid State Commun. 1990, 76, 583–586. (21) Dubey, N.; Leclerc, M. J. Polym. Sci., Part B: Polym. Phys. 2011, 49, 467–475. (22) Bubnova, O.; Khan, Z. U.; Malti, A.; Braun, S.; Fahlman, M.; Berggren, M.; Crispin, X. Nat. Mater. 2011, 10, 429–433. (23) Rusu, G. I.; Rusu, G. G.; Popa, M. E. Mater. Res. Innovations 2003, 7, 372–380. (24) Taggart, D. K.; Yang, Y. G.; Kung, S. C.; McIntire, T. M.; Penner, R. M. Nano Lett. 2011, 11, 2192–2193. (25) Maddison, D. S.; Unsworth, J.; Roberts, R. B. Synth. Met. 1988, 26, 99–108. (26) Yoon, C. O.; Na, B. C.; Park, Y. W.; Shirakawa, H.; Akagi, K. Synth. Met. 1991, 41, 125–128. (27) Pukacki, W.; Plocharski, J.; Roth, S. Synth. Met. 1994, 62, 253–256.

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