Low-Temperature Seebeck Coefficients for Polaron-Driven

Article pubs.acs.org/JPCA

Low-Temperature Seebeck Coefficients for Polaron-Driven Thermoelectric Effect in Organic Polymers Pedro Henrique de Oliveira Neto,*,†,‡ Demétrio A. da Silva Filho,† Luiz F. Roncaratti,† Paulo H. Acioli,§,† and Geraldo Magela e Silva† †

Institute of Physics, University of Brasilia, Brasilia 70.919-970, Brazil Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307, United States § Department of Physics, Northeastern Illinois University, Chicago, Illinois 60625, United States ‡

ABSTRACT: We report the results of electronic structure coupled to molecular dynamics simulations on organic polymers subject to a temperature gradient at lowtemperature regimes. The temperature gradient is introduced using a Langevin-type dynamics corrected for quantum effects, which are very important in these systems. Under this condition we were able to determine that in these no-impurity systems the Seebeck coefficient is in the range of 1−3 μV/K. These results are in good agreement with reported experimental results under the same low-temperature conditions.

1. INTRODUCTION There has been renewed interest in thermoelectric (TE) devices in recent years due to the improvement of the figure of merit and the growth of the green energy industry. Traditionally, the most efficient TE materials are based in inorganic materials; however, the recent efforts in the development of organic-based TE materials are bringing their efficiency to levels comparable to their inorganic counterparts. There are two main phenomena associated with thermoelectricity, the Peltier effect, the appearance of a temperature gradient as a response to a difference of potential, and the Seebeck effect that is the appearance of a difference of potential as a response to a temperature gradient. Peltier effect TE devices are used in compact cooling applications, where efficiency is not the most important aspect, while Seebeck-based devices may be used in conjunction with solar panels to generate electricity when sunlight is not available. Although inorganic-based TE devices are more efficient, organic TE devices are gaining in popularity as being a cheaper alternative and because they can be flexible, bendable, and oxidation-resistant. The efficiency of a TE device is defined as the ratio of the power output and the amount of heat put into the system. In terms of the Carnot efficiency it can be expressed as ϕmax = ϕC

engineering new TE devices has been devoted to increasing its value. It can be shown1 that the figure of merit can be expressed as ZT =

(1)

Special Issue: Piergiorgio Casavecchia and Antonio Lagana Festschrift

where ϕC = 1 − TC/TH is the Carnot efficiency, TH and TC are the temperatures of the hot and cold reservoirs, and ZT is called the figure of merit. The figure of merit can be viewed as another measure of efficiency, and much of the effort in © XXXX American Chemical Society

(2)

where S is the Seebeck coefficient, σ is the isothermal electric conductivity, and k is the coefficient of thermal conductivity. From eq 2 it is clear that one can increase the figure of merit and therefore the efficiency by increasing the electric conductivity and Seebeck coefficient or by decreasing the thermal conductivity. The current state of the art for inorganic materials is an efficiency of ∼28% and Z ≈ 2.2 Some of the efforts in increasing the figure of merit in inorganic materials is to decrease its dimensionality. The idea is to locate the Fermi energy at the maximum of the density of states, thus increasing the Seebeck coefficient. The problem with this approach is the potential to increase the thermal conductivity that in turn will reduce the figure of merit. One of the appeals of organic TE is that, in general, they have small thermal conductivity and a potential to increase the figure of merit by increasing the other factors. In the past decade, the figure of merit in organic materials has increased from 10−4 to ∼0.5. Some authors2 believe that it is just a matter of time until they reach the same

1 + ZTav − 1 1 + ZTav + TH/TC

S2σT k

Received: December 22, 2015 Revised: February 11, 2016

A

DOI: 10.1021/acs.jpca.5b12524 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A

prepared. Classical molecular dynamics is performed to simulate the nuclear degrees of freedom with the force on site n given by

values as those of inorganic-based devices. It is very important to understand how one can improve the Seebeck coefficient, the thermal conductivity, and consequently the figure of merit in organic-based materials. In this work we used a model Hamiltonian coupled to molecular dynamics to simulate long polymeric chains subject to a temperature gradient. We compute the thermal and electric currents and the Seebeck coefficient for temperatures ranging from 10 to 40 K.

Fn(t ) = −K[2un(t ) − un + 1(t ) − un − 1(t )] + α[Bn , n + 1(t ) − Bn − 1, n(t ) + Bn + 1, n(t ) − Bn , n − 1(t )]

where Bn,n′(t) = ∑′k,sψ*k,s(n,t) ψk,s (n′,t) couples the electronic and lattice parts of the model Hamiltonian. The primed summation represents a sum over occupied states. With eqs 5 and 6 one can represent the electronic ground state and excited states as changing the occupation of the electronic levels. The resulting equations of motion for the lattice

2. METHODOLOGY Thermoelectric effects can be explained in terms of the following simplified transport equations ΔV = RIel + SΔT Q̇ = ΠIel − k ΔT

un(t j + 1) = un(t j) + uṅ (t j)Δt

(3)

where R is the electric resistance, Iel is the electric current, ΔV is the electric potential difference, Q̇ is the rate of heat transfer, and Π is the Peltier coefficient. This set of equations describes the Peltier−Seebeck effect. In the case where there is no applied electric field we can write the potential difference as the Ohmic resistance times the thermoelectric current, and we can calculate the Seebeck coefficient as RI S = th ΔT

uṅ (t j + 1) = uṅ (t j) +

ψk , s(n , t j + 1) =

n,s

n

K 2 y + 2 n

∑ n

M

Δt

(8)

∑ [∑ ϕl*,s(m , t j)ψk ,s(m , t j)] l

(4)

∑

Fn(t j)

(7)

were numerically integrated using the Euler method. The solution to the electronic part of eq 5 can be expressed as a linear combination of instantaneous eigenstates m (−iεl Δt / ℏ)

×e

and Ith = nev, where n is the linear density of electrons, e is the electron charge, and v is the drift velocity. To evaluate the Seebeck coefficient, we need to determine the resistance of the device. To evaluate the resistance, we study the system in the case the gradient of temperature is zero and apply a small electric field. Considering a polymeric chain of length L and an applied electric field of magnitude E, we evaluate the resistance as R = LE/nev. We determine the resistance as a function of the applied electric field and the absolute temperature of the chain. To calculate the drift velocity, the resistance, and the diffusion of polarons in organic polymers, we use a model Hamiltonian3,4 coupled to classical molecular dynamics to simulate the motion of the nuclei. The model Hamiltonian is written as H = −∑ (tn , n + 1Cn†+ 1, sCn , s + h. c . ) +

(6)

ϕl , s(n , t j)

(9)

where ϕl(n) and εl are the eigenfunctions and the eigenvalues of the electronic part for the Hamiltonian at a given time tj. At time tj the wave functions ψk,s(n,tj) are expressed as an expansion of the eigenfunctions ϕl,s, that is, ψk,s(n,tj) = s s ∑Nl=1Cl,k ϕl,s(n), in which Cl,k are the expansion coefficients. For simulations at a fixed temperature we use a canonical Langevin dynamics that was implemented in previous works12,13 and widely applied in the literature. The temperature is introduced in the nuclear dynamics, and its influence on the electronic part of the system is through the electron−phonon coupling terms. The temperature is introduced in the Langevin equation as a white stochastic signal ζ(t), and a Stokes-like dissipation term is used to keep the temperature (and energy) constant. This terms are introduced in the site equation that becomes Mun̈ = −γuṅ + ζ(t ) + Fn(t ). The white noise fluctuations are characterized by ⟨ζ(t)⟩ = 0 and ⟨ζ(t) ζ(t′)⟩ = Bδ(t − t′). The relationship between ζ, the damping term γ, and the temperature T of the system is given by the fluctuation− dissipation theorem B = 2γMkBT, where kB is the Bolztmann constant. A value of γ = 0.06 fs−1 is used in this work for constant temperature simulations. This value was chosen, taking into account the typical frequency of a C−C bond oscillation on π-conjugated systems (∼1500 cm−1). We have shown that this methodology was very successful in the study of exciton diffusion in pi-conjugated systems.14 The methodology previously described does not account for quantum effects that are important at low temperatures, below the Debye temperature. This means, in principle, that we cannot treat the nuclear degrees of freedom classically at that regime; however, at low temperatures only the phonon normal modes with low frequencies will be excited. Only when we are close to the classical limit will the contributions from the very high optical modes be important. It is therefore reasonable that we can study the behavior of these systems using classical molecular dynamics and correct the final results for quantum effects.

pn2 2M (5)

where n labels each CH site. To simplify the calculations we use the relative displacement yn ≡ un+1 − un, where un is the lattice displacement of a particular site n. pn is the conjugated momentum and K is the harmonic constant for a σ bond and M is the mass of a CH monomer. The operator C†n,s (Cn,s) creates (annihilates) a π-electron with spin s at the nth site. The interaction between each neighboring site is represented by tn,n+1 = [(1 + (−1)nδ0) t0 − αyn], where t0 is the hopping integral of a π electron in a fully undimerized chain and α is the electron−phonon coupling. A Brazovskii−Kirova symmetrybreaking term δ0 is added to simulate armchair π-conjugated polymers. We used the standard values of these parameters: t0 = 2.5 eV, M = 1349.14 eV × fs2/Å2, K = 21 eV/Å2, δ0 = 0.05, α = 4.1 eV/Å, and a = 1.22 Å. The validation of these parameters was performed in previous works.5−11 Before the simulation starts, a fully self-consistent stationary state in all degrees of freedom of electrons and phonons is B


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L for different values of the temperature TMD. One can clearly see that the behavior is typical of a non-Ohmic device. The resistance ranges from 1050 Ω at 10 K to 1225 Ω at 50 K. Finally, to study the thermoelectric effect in the polymeric chain we need to include a temperature gradient. The temperature gradient is added to the chain using the Langevin dynamics with a different T for each site. The polaron is initially kept stationary by means of a small impurity that is removed once the desired gradient profile in the chain is reached. For each temperature gradient, the propagation of the polaron was simulated 200 times, every time with a different random number seed. The polaron propagation is then averaged in the same manner as described in ref 14. From the position−time graph we extract the TE current needed to compute the Seebeck coefficient as a function of the temperature. In this work we considered the low-temperature regime with temperatures of 10, 20, and 40 K and a temperature gradient of 0.1 K/ Å.

We included quantum corrections using the method described in ref 15 and used in many classical molecular dynamics simulations in the low-temperature regime.14,16−19 The main idea is to equate the molecular dynamics kinetic energy, using the equipartition theorem, with the energy at a temperature T given by NkBTMD =

⎡

∫ ℏωD(ω)⎢⎣ 12

+

1 e

ℏω / kBT

⎤ ⎥ dω − 1⎦

(10)

where D(ω) is the phonon density of states. Given the desired temperature T we obtain the corresponding simulation temperature TMD by use of the above expression. Because we are mostly interested in phonon excitations we did not include the zero point energy correction (first term in the previous equation). In Figure 1 one can see the correspondence between

3. RESULTS To study the thermoelectric effect we added a charged polaron to the polymeric chain with 160 sites and subjected it to different temperature conditions. Figure 3 illustrates the results

Figure 1. Quantum-corrected temperature as a function of the molecular dynamics temperature. Figure adapted from Ref 14.

the molecular dynamics temperature TMD and the quantum temperature. The main difference occurs in the low-temperature regime, where the real temperature is much higher than the one used in the molecular dynamics simulation; however, as the temperature increases the two values approach each other. All temperatures reported in this work included the quantum corrections previously described. To determine the electrical resistance of a single polymeric chain, we need to include an electric field and determine the charge propagation (drift) velocity. The electric field E is included in the force on site n as described in ref 20. In Figure 2 we display the applied potential difference ΔV = EL as a function of the electric current I in a polymeric chain of length

Figure 3. Position of the polaron as a function of time. In the left panel, the average temperature TQ = 20 K, temperature gradient ΔT/ Δr = 0.1 K/Å, and external electric field E = 0 mV/Å. In the right panel, TQ = 20 K, ΔT/Δr = 0.1 K/Å, and E = 0.13 mV/Å.

of two simulations that show the typical propagation of the polaron under (a) TQ = 20 K, ΔT/Δr = 0.1K/Å, E = 0 mV/Å and (b) TQ = 20 K, ΔT/Δr = 0.1 K/Å, E = 0.13 mV/Å. A constant temperature yields a typical random walk behavior as previously studied in ref 21. For a small temperature gradient and zero electric field, the polaron moves with a small drift velocity. For the nonzero field case and a nonzero temperature gradient the polaron propagates with an increased drift velocity, as indicated in right panel of Figure 3, as expected. We must remind the reader that these are individual simulations. The results used to compute the Seebeck coefficients are from ensemble averages over 200 simulations with the same temperature gradient conditions. Figure 4 illustrates the displacement of the polaron position for each of the 200 simulations for an average temperature of 20 K and a gradient of 0.1 K/Å applied on the polymeric chain. In this case the temperature of the system is kept constant throughout the polymeric chain, and the polaron moves as random walker with average position equal to zero. When a gradient of temperature is applied, the distribution is now biased toward positive displacements. For instance, after 1.4 ps it is released, and the polaron has moved over an average distance of 1.2 Å as a response to the temperature gradient. In Figure 5 we plot the average position of the polaron as a function of time for different values of the average temperature

Figure 2. Electron current due to a polaron in the presence of an external difference of potential for different temperatures and no temperature gradient. C


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tendency to saturate for high enough values of the temperature. This saturation is a result of the competition of the high diffusivity of the polaron and an increase in its resistance as the temperature increases. A comparison of the electric currents due to an electric field (Figure 2) with the thermoelectric currents shows that the thermoelectric currents are 1 order of magnitude smaller. This is an expected result as the electric current is a direct response of a charged quasi-particle to an applied electric field, while the thermoelectric current is a response due to the electron−phonon coupling to a temperature gradient. This interaction of the polarons with the lattice increases when more phonons in the lattice are excited by the increased electric field. A similar effect takes place when there is an increase in temperature. Although the interaction of the lattice with the quasi-particle is responsible for its motion and the creation of the thermoelectric current, the same interactions also work as an electric resistance. We use the currents in the left panel of Figure 6 and the resistances extracted from Figure 2 to calculate the Seebeck coefficients at the low-temperature regime. These results are plotted in the right panel of Figure 6. Because the resistance is nearly constant at this range of temperatures the Seeback coefficient follows the same trend as the TE current: S grows as a function of temperature, with a tendency of leveling of at higher temperatures. The behavior and values of the Seeback coefficient are in line with what is observed for conjugated polymers at the low-temperature regime.22

Figure 4. (a) Positions (Å) of the polaron as a function of time (ps) for 200 realizations at the average temperature TQ = 20 K and temperature gradient ΔT/Δr = 0.1 K/Å.

4. CONCLUSIONS We report low-temperature Seebeck coefficients of polaron driven thermoelectric effects in organic polymers. We use a model Hamiltonian coupled to classical molecular dynamics and Langevin dynamics to include the temperature effects. All results report a quantum correction as described in our recent work on exciton diffusion in highly ordered π-conjugated system. Our computed Seebeck coefficients are in the range of 1−3 μV/K, which is in line with experimental results in organic polymers in this temperature range. We plan to extend this work to more complex organic polymers in an effort to predict candidates with increased Seebeck coefficient to improve the efficiency of organic thermoelectric devices.

Figure 5. Average position of the polaron as a function of time at: T = 10 (red), 20 (green), and 40 (blue). In all cases, the temperature gradient ΔT/Δr = 0.1 K/Å and the electric field E = 0.13 mV/Å.

after 1 ps because the polaron remains stationary during that time due to an impurity pinning it at the center of the polymeric chain. Once the temperature gradient in the chain reaches the desired profile, that impurity is removed and the polaron is free to move. In all of the cases the average position of the polaron is toward the colder side of the chain. In addition, the slope of the average position is an increasing function of the temperature. We obtain the average velocity of the polaron and consequently the thermoelectric current by fitting the average position to a linear function of time. In the left panel of Figure 6 we plot the thermoelectric current as a function of the temperature. As one expects, the thermoelectric current increases with the temperature with a

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]; [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We gratefully acknowledge the financial support from the Brazilian Research Councils CNPq, CAPES, and FINATEC. D.A.S.F. gratefully acknowledges the financial support from the Brazilian Research Council CNPq, grant 306968/2013-4.

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REFERENCES

(1) Goupil, C.; Seifert, W.; Zabrocki, K.; Muller, E.; Snyder, G. J. Entropy 2011, 13, 1481−1517. (2) Culebras, M.; Gomez, C. M.; Cantarero, A. Materials 2014, 7, 6701−6732. (3) Su, W.; Schrieffer, J.; Heeger, A. Phys. Rev. Lett. 1979, 42, 1698− 1701. (4) Su, W.; Schrieffer, J.; Heeger, A. Phys. Rev. B: Condens. Matter Mater. Phys. 1980, 22, 2099−2111.

Figure 6. Thermoelectric current (left) and Seebeck coefficient (right) as a function of the temperature. D


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The Journal of Physical Chemistry A (5) de Oliveira Neto, P. H.; da Cunha, W.; Teixeira, J.; Gargano, R.; e Silva, G. J. Phys. Chem. Lett. 2012, 3, 3039−3042. (6) Ribeiro, L. A.; de Oliveira Neto, P. H.; da Cunha, W. F.; Gargano, R.; e Silva, G. Chem. Phys. Lett. 2013, 555, 168−172. (7) Junior, L. A. R.; Neto, P. H. O.; da Cunha, W. F.; Silva, G. M. e. Phys. Procedia 2012, 28, 112−116. (8) Johansson, A.; Stafstrom, S. Phys. Rev. Lett. 2001, 86, 3602−3685. (9) Johansson, A.; Stafstrom, S. Phys. Rev. B: Condens. Matter Mater. Phys. 2004, 69, 235205. (10) An, Z.; Wu, C. Q.; Sun, X. Phys. Rev. Lett. 2004, 93, 216407. (11) Magela e Silva, G.; Terai, A. Phys. Rev. B: Condens. Matter Mater. Phys. 1993, 47, 12568−12577. (12) da Cunha, W. F.; Junior, L. A. R.; Gargano, R.; e Silva, G. M. Phys. Chem. Chem. Phys. 2014, 16, 17072−17080. (13) Junior, L. A. R.; da Cunha, W. F.; de Almeida Fonseca, A. L.; Gargano, R.; e Silva, G. M. Chem. Phys. Lett. 2014, 614, 151−155. (14) de Oliveira Neto, P. H.; da Silva Filho, D. A.; da Cunha, W. F.; Acioli, P. H.; e Silva, G. M. J. Phys. Chem. C 2015, 119, 19654−19659. (15) Lee, Y. H.; Biswas, R.; Soukoulis, C. M.; Wang, C. Z.; Chan, C. T.; Ho, K. M. Phys. Rev. B: Condens. Matter Mater. Phys. 1991, 43, 6573−6580. (16) Kopidakis, G.; Wang, C. Z.; Soukoulis, C. M.; Ho, K. M. J. Phys.: Condens. Matter 1997, 9, 7071−7080. (17) Wu, M. C. H.; Hsu, J.-Y. Nanotechnology 2009, 20, 145401. (18) Hu, J.; Ruan, X.; Jiang, Z.; Chen, Y. P. AIP Conf. Proc. 2009, 1173, 135−138. (19) Turney, J. E.; McGaughey, A. J. H.; Amon, C. H. Phys. Rev. B: Condens. Matter Mater. Phys. 2009, 79, 224305. (20) Roncaratti, L. F.; Gargano, R.; e Silva, G. M. J. Phys. Chem. A 2009, 113, 14591−14594. (21) da Cunha, W. F.; de Oliveira Neto, P. H.; Gargano, R.; e Silva, G. M. Int. J. Quantum Chem. 2008, 108, 2448−2453. (22) Shakouri, A.; Li, S. In Proceedings of International Conference on Thermoelectrics, Baltimore, MD, 1999.

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Low-Temperature Seebeck Coefficients for Polaron-Driven

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