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Photoexcitation-induced local phonon spectra and local hot excitons in polymer solar cells

Jiahao Chen, Yusong Zhang, Zhe Lin, Jianguo Shen, Thomas F. George, and Sheng Li

Open Access Open Access

Abstract

In this article, based on nonadiabatic molecular dynamics with electronic transitions, the elaborate ultrafast process of hot excitons in conjugated polymer solar cells is revealed. When an external optical beam/pulse with the intensity of 30 µJ/cm−2 is utilized to excite a conjugated polymer, just within only 50 fs, the electronic transition not only redistributes the electron population in the original molecular orbital, but also starts to localize the electron cloud of excited states and to distort the alternating bonds in the polymer chain. Up to 300 fs, the lattice distortion has been stabilized. During the formation of hot excitons, the prominent self-trapping effect of conjugated polymer triggers the occurrence of local infrared active phonon modes, with five peaks in the phonon spectrum as the hot excitons relax. The characteristic phonon spectrum and infrared modes hence form the fingerprint of the hot excitons of a conjugated polymer, which are readily distinguished from other excitation states in the polymer.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

The physical picture of photoinduced hot excitons in conjugated polymer solar cells can be described as follows: With continuous excitation of the optical field, the electron population in the highest-occupied molecular orbital (HOMO) of the conjugated polymer gradually declines, while an electron is excited to the lowest-occupied molecular orbital (LUMO) [1]. Then, each of two electronic orbitals is occupied by an electron, resulting in the so-called photoinduced exciton. Thanks to the prominent self-trapping effect of conjugated polymers, the dynamical process of hot exciton formation involves lattice relaxation and an electronic transition [2–12].

Looking back to 2006, Müller et al, observed that, in a conjugated polymer, the highly-efficient energy transfer of excited states is intrachain rather than interchain [13]. Furthermore, experimental research has uncovered that interchain excitons in a conjugated polymer are unfavorable on the basis of photoexcitation [14]. Especially, the observation of photoexcitation of 2-ethylhexyloxy showed that, once the interchain migration of excitons is inhibited, the quantum efficiency can be highly enhanced and exciton localization strengthened along the polymer chain as well [15]. The intramolecular physical effect is key to discover the mechanism of the formation of hot excitons. Thus, how to uncover the microscopic aspects of hot excitons along a single polymer chain becomes the key step to further understand the mechanism of formation/dissociation with respect to the excited states in polymeric solar cells.

To dig into the underlying dynamical process of excited states of a conjugated polymer, given the ultrafast dynamics, a comparative simulation has been conducted when the electron and hole are injected into a nondegenerate π–conjugated polymer. It is found that the ensuing dynamics of a hot exciton is attributed to its relaxation [16]. The relaxation, as the main part of the nonadiabatic ultrafast process [17], is found out to be tightly related with the intramolecular vibrational relaxation in conjugated polymers, such as PPV/MH-PPV [18]. Especially, the dynamical properties of a hot exciton are largely attributed to lattice relaxation [19,20]. Considering the intrinsic property of polymer self-trapping, a possible mechanism regarding hot exciton is assumed as follows: The external photoexcitation, combined with electron-lattice coupling, not only induces the electronic transition, but also causes vibrations in the lattice of the conjugated polymer, leading to the characteristic phonon spectrum ultimately localizing the electron cloud. To prove this, it is necessary to investigate the nonadiabatic dynamics of a hot exciton under photoexcitation, such as the characteristic phonon spectrum and related phonon modes

2. Methodology

On the basis of previous research with respect to carrier photoexcitation in semiconducting conjugated polymers [21,22], we select a typical conjugated polymer, poly(p-phenylene vinylene) (PPV), as a model to explore the related physical mechanism. The employed Hamiltonian not only accommodates the quasi-one-dimensional polymeric structure, but also incorporates the strong electron-phonon and electron interactions and the self-trapping effect of carriers along with the elastic potential energy of the polymer lattice.

We begin with the extended-Su-Schreiffer-Heeger-Hubbard Hamiltonian:

$$H = {H_e} + {H_l} + H^{\prime}.$$
Here, H is the total energy of the conjugated polymer, He is the electronic component, Hl is for the lattice, and H’ represents the electron-electron interaction:
$${H_e} ={-} \sum\limits_{l,s} {[{t_0} - \alpha ({u_{l + 1}} - {u_l}) + {{( - 1)}^l}{t_e})](a_{l + 1,s}^\dagger {a_{l,s}} + H.c.),}$$
$${H_l} = \frac{M}{2}\sum\limits_l {{{(\mathop {{u_l}}\limits^\cdot )}^2}} + \frac{K}{2}\sum\limits_l {{{({u_{l + 1}} - {u_l})}^2}} ,$$
$$H^{\prime} = U\sum\limits_l {{n_{l, \uparrow }}{n_{l, \downarrow }}} + V\sum\limits_{l,s,s^{\prime}} {{n_{l,s}}{n_{l + 1,s^{\prime}}}} .$$
The specific parameters are as follows: ${t_0}$ is a hopping constant (2.5–3.0 eV); ${t_e}$ is the Brazovskii-Kirova term (0.05–0.10 eV); $\alpha$ is an electron-lattice coupling constant (4.0–6.0 eV/Å); $a_{l,s}^\dagger ({a_{l,s}})$ is the electron creation (annihilation) operator at cluster l with spin s and displacement ${u_l}$; K is an elastic constant (20–30 eV/Å2); and U (2.0–5.0 eV) and V (0.5–2.0 eV) are the on-site and nearest-neighbor Coulomb interactions, respectively. The electron-electron interaction is treated in the Hartree-Fock approximation as
$$\begin{aligned}{H_e} & = \sum\limits_{l,s} {\left\{ {\left. {U\left( {\sum\limits_\mu^{occ} {{{|{Z_{l,\mu }^{ - s}} |}^2}} - \frac{1}{2}} \right) + V\left[ {\sum\limits_{s^{\prime}} {\left( {\sum\limits_\mu^{occ} {{{|{Z_{l + 1,\mu }^{ - s^{\prime}}} |}^2}} + \sum\limits_\mu^{occ} {{{|{Z_{l + 1,\mu }^{ - s^{\prime}}} |}^2} - 2} } \right)} } \right]} \right\}c_{l,s}^\dagger {c_{l,s}}} \right.}\\ & - \sum\limits_{l,s} {\left( {V\sum\limits_\mu^{occ} {Z_{l,\mu }^sZ_{l + 1,\mu }^s} } \right)c_{l,s}^\dagger {c_{l,s}} + H.c.}\end{aligned}$$
where occ stands for the electron occupation or population. Then, the wavefunction for spin s takes the form
$${\varepsilon _\mu }Z_{l,\mu }^s = \left[ {U\left( {\rho_l^{ - s} - \frac{1}{2}} \right) + V\left( {\sum\limits_{s^{\prime}} {\rho_{l - 1}^{s^{\prime}}} + \sum\limits_{s^{\prime}} {\rho_{l + 1}^{s^{\prime}}} - 2} \right) + Ee\left( {l - \frac{{N + 1}}{2}} \right)a} \right]Z_{l,\mu }^s$$
$$- [V\sum\limits_\mu ^{occ} {Z_{l,\mu }^sZ_{l - 1,\mu }^s} + {t_0} + \alpha ({{u_{l - 1}} - {u_l}} )+ {( - 1)^{l - 1}}{t_e}]Z_{l - 1,\mu }^s$$
$$- [V\sum\limits_\mu ^{occ} {Z_{l,\mu }^sZ_{l + 1,\mu }^s + {t_0} + \alpha ({{u_{l + 1}} - {u_l}} )+ {{( - 1)}^{l + 1}}{t_e}} ]Z_{l + 1,\mu }^s.$$
where the charge distribution is defined as ${\rho _l} = \sum\limits_\mu ^{occ} {{{|{{\Psi _{l,\mu }}} |}^2}} - {n_0}$, and ${n_0}$ is the density of the positively-charged background.

At the point where an emissive transition occurs, we need the transition rate equation

$${\gamma _{ab}} = \frac{{4{{({E_a} - {E_b})}^3}}}{{3{\hbar ^4}{c^3}}}{p^2},$$
where p is the transition dipole moment. These formulas together provide a complete description of transport, coupling and transition processes of quasiparticles (including polarons and excitons) in a conjugated polymer system.

As for the lattice vibrations, the vibrational mode matrix can be introduced to explain the localization on the polymer chain. A standard eigenvalue/eigenvector perturbation technique is used to find approximations for the vibrational modes. Suppose the static lattice configuration at site n is $\phi _n^0$, and its perturbation can be written as

$${\phi _n}(t) = \phi _n^0 + \phi _n^{^\prime}(t).$$
The localized vibrational mode can be determined by introducing a second-order perturbation in the calculation of the energy:
$$H({\{{{\phi_n}} \}} )= {E_0} + {E_s}\sum\limits_m^N {{A_m}({\{{{\phi_n}} \}} )\phi _m^{^\prime}} + \frac{1}{2}\sum\limits_{m,n}^N {{B_{m,n}}({\{{\phi_n^0} \}} )\phi _n^{^\prime}\phi _m^{^\prime}}$$
$${B_{m,n}} = k[{({{\delta_{m,n}} + {\delta_{m,n + 1}}} )({1 - {\delta_{m,N}}} )+ ({{\delta_{m,n}} + {\delta_{m,n - 1}}} )({1 - {\delta_{n,1}}} )} ]+ 2{\alpha ^2}{( - 1)^{m + n}}\sum\limits_{\mu ,\nu ( \ne \mu )} {\frac{{C_{\mu ,\nu }^mC_{\mu ,\nu }^n}}{{\varepsilon _\mu ^0 - \varepsilon _\nu ^0}}} .$$
Here, N is the total number of lattice sites; ɛ is the static eigenenergy of the electron; and the parameter $C_{\mu ,\nu }^m$ is
$$C_{\mu ,\nu }^m = (1 - {\delta _{m,N}})(Z_{\mu ,m + 1,s}^0Z_{\nu ,m,s}^0 + Z_{\mu ,m,s}^0Z_{\nu ,m + 1,s}^0) - (1 - {\delta _{m,1}})(Z_{\mu ,m,s}^0Z_{\nu ,m - 1,s}^0 + Z_{\mu ,m - 1,s}^0Z_{\nu ,m,s}^0),$$
where $Z_{\mu ,m,s}^0$ is the corresponding eigenstate of the electron in energy level $\mu$ at lattice site m with spin value s, and ${\delta _{m,n}}$ is the Kronecker delta.

By simply diagonalizing the B matrix the eigenfrequency and eigenvector of the vibrational modes can be found. Consequently, the excitation of phonon modes during the formation process of the hot exciton can be depicted in detail.

3. Results and discussion

To conduct the simulation of the excitation process of the conjugated polymer, we first introduce the lattice configuration ϕn to describe the profiles of the lattice. If the displacement of every group in the PPV chain is un, the lattice configuration is ϕn = (-1)nun. Then, due to the Peierls instability of the one-dimensional structure of conjugated polymer chain, the ground state of the polymer is in a state of dimerization where lattice configuration ϕn is a constant. In the dynamical simulation, if the external laser pulse is applied to stimulate the conjugated polymer, and the energy of the photon just matches the gap of the conjugated polymer, not only does the electron in HOMO (highest occupied molecular orbital) transit into LUMO (lowest unoccupied molecular orbital), but the original LUMO and HOMO move towards the center of the energy gap. A schematic diagram of the process is shown in Fig. 1. Combined with the new electron occupation, the original HOMO and LUMO gradually evolve to two new states, Γu and Γd. Both of these, depicting the electron state and the hole state, compose a new exciton.

 figure: Fig. 1.

Fig. 1. Electronic structure of a conjugated polymer before and after optical excitation.

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Specifically, we take external laser pulses with light intensity of 15, 30 and 40 µJ/cm2 with wavelength of 1000 nm−1 to excite the conjugated polymer, where the photon energy just matches the gap of the PPV. During the photoexcitation, the electron in the HOMO transits into the LUMO after absorbing the photon’s energy. The time-dependent electron occupation of the LUMO is depicted in Figs. 2(A), 3(A) and 4(A), respectively. Under the excitation of laser pulse with light intensity of 15, 30 and 40 µJ/cm2 with wavelength of 1000 nm−1, the electron population increases until the electron occupation of the LUMO approaches 1 just within the first 60, 40 and 30 fs. After 50 fs, the electron occupation of each energy level tends to be 1, which indicates formation of the hot exciton. Referring to the time-dependent spatial distribution of the electron cloud in Figs. 2(B), 3(B) and 4(B), it is found that the electron cloud of the excited state has begun to localize in the conjugated polymer chain following photoexcitation. Here, for convenience, we choose the case of excitation by a laser pulse with light intensity of 30 µJ/cm2. When the conjugated polymer macromolecule is excited by the external laser pulse with 30 µJ/cm2, the localization of the electron occurs within 80 fs, as shown in Fig. 3. Previous experimental research found that the localization appears just within 100 fs after the excitation by the external light, once the hot exciton is formed in the polymer molecule [23,24]. Based on the developed dynamics, we demonstrate that when the conjugated polymer macromolecule is excited by an external laser pulse of 30 µJ/cm2, localization of the electron occurs within 80 fs, as shown in Fig. 3(B), which is in agreement with previous experimental research. Thus, from the evolution of the electron cloud, Fig. 3(B) demonstrates the spatial oscillation of charge distribution within the first 310 fs, which is the result of relaxation, and the localization of electron cloud becomes stable by 670 fs.

 figure: Fig. 2.

Fig. 2. Excitation by a laser pulse with light intensity of 15 µJ/cm2. (A) Time-dependent electron populations of the energy level Γu. (B) Time-dependent spatial distribution of the electron cloud of the excited state in PPV.

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 figure: Fig. 3.

Fig. 3. Excitation by a laser pulse with light intensity of 30 µJ/cm2. (A) Time-dependent electron populations of the energy level Γu. (B) Time-dependent spatial distribution of the electron cloud of the excited state in PPV.

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 figure: Fig. 4.

Fig. 4. Excitation by a laser pulse with light intensity of 45 µJ/cm2. (A) Time-dependent electron populations of the energy level Γu. (B) Time-dependent spatial distribution of the electron cloud of the excited state in PPV.

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The evolution of the lattice configuration is then utilized to depict to the lattice relaxation of the conjugated polymer, and the dynamical process of 140 fs of lattice configuration is exhibited in Fig. 5(A). Within the initial 20 fs, the evolution of the lattice configuration shows that a small valley first appears at the center of the conjugated polymer chain. Later, the photoexcitation continuously drives the dent to become deeper and to oscillate accordingly. Up to 140 fs, an obvious localized distortion has been formed in the middle of the conjugated polymer chain. Figure 5(B) gives a three-dimensional depiction of the time-dependent lattice configuration over 1 ps. At about 300 fs, the lattice distortion has been stabilized. Integrating it with the evolution of the lattice configuration in Fig. 5, both of them constitute the dynamical relaxation process of the hot exciton. Accompanying the appearance of the locally distorted lattice and the localization of the electron cloud, the self-trapping effect, originating from the electron-lattice coupling in the conjugated polymer, will trigger the dynamics of the phonon modes.

 figure: Fig. 5.

Fig. 5. (A) Two-dimensional depiction of the time-dependent lattice configuration over140 fs. (B) Three-dimensional depiction of the time-dependent lattice configuration over 1000 fs.

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Indeed, the previous quantum-dynamical analysis shows that the dynamical process with respect to excited states generally involves multiple vibrational phonons [25,26]. Due to the external photoexcitation, multiple phonons of the conjugated polymer participate in the dynamical process, as discovered based on the polymer-carbon nanotube heterojunction [27]. Thus, our attention focuses on the phonon modes as part of the dynamical process. Undergoing continuous excitation from optical pumping with 30 µJ/cm-2, it is found that five peaks (g1-g5) in the phonon spectrum of Fig. 6 correspond to five vibrational modes as exhibited in the Fig. 7(A-E). Among those five distinct peaks, the highest peak in Fig. 6 is related to the vibrational mode g4 (Fig. 5(D) whose wavenumber is 1490 cm−1. As for the other four vibrational modes (g1,g2,g3,g5), their wavenumbers are 1098 cm−1, 1218 cm−1, 1434 cm−1 and 1501 cm−1, respectively. As shown in Fig. 4, the lattice vibrations lead to five phonon modes that are seen in the phonon spectrum.

 figure: Fig. 6.

Fig. 6. Phonon spectrum of the hot exciton in the conjugated polymer PPV.

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 figure: Fig. 7.

Fig. 7. (A-E) Localized phonon vibration modes of the hot exciton in the conjugated polymer PPV.

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As illustrated in Fig. 7, those five phonon vibrational modes possess the common characteristic property that all of them are of even parity and localized near the lattice center. The five phonon modes in Fig. 7 lie in the far infrared region (209 − 2513 cm−1) in the phonon spectrum. Given these characteristics, all of these are infrared-active vibrational modes. Since the phonon spectrum is obtained from the expansion of the lattice configuration of the hot exciton, those vibration modes can be regarded as the fingerprint information for the hot exciton in the conjugated polymer chain.

4. Summary

In summary, through combining nonadiabatic molecular dynamics of a conjugated polymer with the phonon vibrational modes, the developed molecular dynamics exhibits the elaborate ultrafast process and related profiles with regard to the formation of hot excitons in a conjugated polymer, like PPV. The panoramic landscape of photoexcitation process has been demonstrated in detail: When an external optical beam/pulse with the intensity of 30 µJ/cm−2 is utilized to excite a conjugated polymer, within only 50 fs the electronic transition not only redistributes the electron population in the original molecular orbital, but also begins to localize the electron cloud of the excited state and to distort the alternating bonds in the polymer chain. Up to 300 fs, the lattice distortion has been stabilized. Both the evolution of the lattice and electronic states constitute the dynamical relaxation process of the hot exciton.

With the appearance of the locally distorted lattice and localization of the electron cloud, the self-trapping effect, originating from electron-lattice coupling in the conjugated polymer, will trigger the occurrence of the local infrared-active phonon modes. This is revealed in the five peaks in the phonon spectrum as relaxation of the hot exciton. Mostly, the phonon spectra during the formation of hot exciton in conjugated polymer, which is discovered in this article, can provide a feasible approach to investigate and even distinguish the different excited states in semiconducting conjugated polymer.

Funding

Natural Science Foundation of Zhejiang Province (LY17F050003).

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Figures (7)
Fig. 1.
Fig. 1. Electronic structure of a conjugated polymer before and after optical excitation.

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Fig. 2.
Fig. 2. Excitation by a laser pulse with light intensity of 15 µJ/cm2. (A) Time-dependent electron populations of the energy level Γu. (B) Time-dependent spatial distribution of the electron cloud of the excited state in PPV.

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Fig. 3.
Fig. 3. Excitation by a laser pulse with light intensity of 30 µJ/cm2. (A) Time-dependent electron populations of the energy level Γu. (B) Time-dependent spatial distribution of the electron cloud of the excited state in PPV.

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Fig. 4.
Fig. 4. Excitation by a laser pulse with light intensity of 45 µJ/cm2. (A) Time-dependent electron populations of the energy level Γu. (B) Time-dependent spatial distribution of the electron cloud of the excited state in PPV.

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Fig. 5.
Fig. 5. (A) Two-dimensional depiction of the time-dependent lattice configuration over140 fs. (B) Three-dimensional depiction of the time-dependent lattice configuration over 1000 fs.

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Fig. 6.
Fig. 6. Phonon spectrum of the hot exciton in the conjugated polymer PPV.

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Fig. 7.
Fig. 7. (A-E) Localized phonon vibration modes of the hot exciton in the conjugated polymer PPV.

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Equations (13)

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H = H e + H l + H ′ .
H e = − ∑ l , s [ t 0 − α ( u l + 1 − u l ) + ( − 1 ) l t e ) ] ( a l + 1 , s † a l , s + H . c . ) ,
H l = M 2 ∑ l ( u l ⋅ ) 2 + K 2 ∑ l ( u l + 1 − u l ) 2 ,
H ′ = U ∑ l n l , ↑ n l , ↓ + V ∑ l , s , s ′ n l , s n l + 1 , s ′ .
H e = ∑ l , s { U ( ∑ μ o c c | Z l , μ − s | 2 − 1 2 ) + V [ ∑ s ′ ( ∑ μ o c c | Z l + 1 , μ − s ′ | 2 + ∑ μ o c c | Z l + 1 , μ − s ′ | 2 − 2 ) ] } c l , s † c l , s − ∑ l , s ( V ∑ μ o c c Z l , μ s Z l + 1 , μ s ) c l , s † c l , s + H . c .
ε μ Z l , μ s = [ U ( ρ l − s − 1 2 ) + V ( ∑ s ′ ρ l − 1 s ′ + ∑ s ′ ρ l + 1 s ′ − 2 ) + E e ( l − N + 1 2 ) a ] Z l , μ s
− [ V ∑ μ o c c Z l , μ s Z l − 1 , μ s + t 0 + α ( u l − 1 − u l ) + ( − 1 ) l − 1 t e ] Z l − 1 , μ s
− [ V ∑ μ o c c Z l , μ s Z l + 1 , μ s + t 0 + α ( u l + 1 − u l ) + ( − 1 ) l + 1 t e ] Z l + 1 , μ s .
γ a b = 4 ( E a − E b ) 3 3 ℏ 4 c 3 p 2 ,
ϕ n ( t ) = ϕ n 0 + ϕ n ′ ( t ) .
H ( { ϕ n } ) = E 0 + E s ∑ m N A m ( { ϕ n } ) ϕ m ′ + 1 2 ∑ m , n N B m , n ( { ϕ n 0 } ) ϕ n ′ ϕ m ′
B m , n = k [ ( δ m , n + δ m , n + 1 ) ( 1 − δ m , N ) + ( δ m , n + δ m , n − 1 ) ( 1 − δ n , 1 ) ] + 2 α 2 ( − 1 ) m + n ∑ μ , ν ( ≠ μ ) C μ , ν m C μ , ν n ε μ 0 − ε ν 0 .
C μ , ν m = ( 1 − δ m , N ) ( Z μ , m + 1 , s 0 Z ν , m , s 0 + Z μ , m , s 0 Z ν , m + 1 , s 0 ) − ( 1 − δ m , 1 ) ( Z μ , m , s 0 Z ν , m − 1 , s 0 + Z μ , m − 1 , s 0 Z ν , m , s 0 ) ,
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