Ultrafast optical pulse convertor caused by oscillations of the energy level structure in the conjugated polymer poly(p-phenylenevinylene)

Yusong Zhang; Weikang Chen; Zhe Lin; Sheng Li; Thomas F. George

doi:10.1364/OE.25.020233

1. Introduction

The development of high-performance polymeric optoelectronics has been studied with respect to conjugated polymers [1–14]. Understanding the underlying mechanism regarding the gap of the excited state of a polymeric semiconductor is a key step to controlling its optoelectronic performance.

For a quasi-one-dimensional π-electron system of carbon nanotubes, researchers have calculated the energy gaps of three types of short armchair carbon nanotubes with various lengths and found that each gap oscillates with a certain period [15]. Especially for finite single-wall carbon nanotubes, the maxima of the energy gap oscillations not only depend on the lengths of the nanotubes, but also are modified by the periodicities of the oscillations [16].

Extending the studies to two-dimensional π-electron systems − carbon nanoribbons – it is revealed that the band gap of a semiconducting graphene nanoribbon is sensitive to its width. Just as for chiral semiconducting graphene nanoribbons, the band gap oscillations rapidly change as a function of the chiral angle [17,18]. The band gap of a graphene nanoribbon can be changed by both uniaxial and shear strains. For an armchair/zigzag graphene nanoribbon, a large strain results in periodic oscillations of the gap [19]. When a magnetic flux penetrates into a carbon nanotube, gap oscillations also can be induced by the anomalous Aharonov-Bohm effect [20].

Once small molecules are adsorbed on nanoscale graphene, it is found that the adsorbed molecules are able to break the original structure and molecular symmetry and control the magnetic ordering of localized states at the edges, thus facilitating the tunability of the band gap [21]. Besides the structural factors mentioned above, this raises a question as to whether an external field is able to modify the gap.

In small Fe₈molecular nanomagnets, the applied static transverse magnetic field directly causes quantum tunneling gap oscillations [22].First-principles calculations show that due to the surface interactions, the competition between the surface-bonding effect and the quantum-confinement effect leads to the gap oscillatory behavior in a TiO₂ nanowire and nanotube [23]. Moving to a one-dimensional π-electron system (conjugated polymer), it is found out that for carbon-based ladder polymers and polynuclear aromatic hydrocarbons, there also exists similar oscillating behavior in the band gap that depends on the number of the subgroups [24]. Recent research on iron pyrite has uncovered the underlying mechanism that an ultrafast (≈100 fs) oscillation in the band gap is strongly coupled to a phonon mode, where the coherency of the phonons is able to cause changes of up to ± 0.3 eV in the band gap from its average value [25].

Considering the one-dimensional structure of the conjugated polymer, the strong electron-phonon interaction causes the change of electronic structure readily with the occurrence of phonons and the evolution of lattice structure. Here we propose an assumption that, if the appropriate external factors are utilized to excite the conjugated polymer, the newly excited state is able to not only induce vibrations of the lattice but to also change the electronic structure, thus realizing the gap oscillation. Meanwhile, due to the conductivity of the conjugated polymer, the external optical field becomes an appropriate candidate to induce new excitations and naturally change the electronic structure. It has to be emphasized that, during the oscillation of the gap, it is just like a convertor for the external optical field to reform as an effective light field. Hence, to validate our proposal becomes the main mission of the article. This should not only give new insight into the fundamental understanding of conjugated polymers, but also provide valid information as to how to design polymeric nanostructures for different applications which can be designed as an ultrafast response optical convertor that is able to convert the external optical pulse to a new effective light field with certain oscillation period.

2. Methodology

Recently, it has been demonstrated that once external light excites a conjugated polymer, such as PBDTTT or poly(3-hexylthiophene), to yield an exciton, the evolution of the fluorescence spectrum within 1 ns reveals the details of the evolution for lasers with different wavelengths, and provides the time scale of the exciton formation, relaxation and radiative decay [26,27]. Furthermore, researchers have combined experimental and theoretical studies of excitation relaxation in poly[2-methoxy-5-(2-ethylhexyloxy)-p-phenylenevinylene], where the paradigm is based on the basic characteristics of conjugated polymers that are decided by conformational subunits [28]. To clarify the mechanism regarding the excited polymer, it becomes necessary to consider the quasi-one-dimensional structure of conjugated polymers for the modeling of the Hamiltonian that is used to describe the polymeric properties. Thus, the Hamiltonian has to include both the electron-lattice and electron-electron interactions. We should point out that we are using a quantum model as opposed to a thermal bath model. Based on previous research with respect to the semiconducting conjugated polymer poly(p-phenylenevinylene) (PPV) [29,30], we select PPV to be a model for the calculation, where the related Hamiltonian, starting with the typical one-dimension model, is constructed.

The strong electron-phonon interaction in a conjugated polymer reflects the prominent one-dimensional aspects, which can lead to the self-trapping effect of carriers in the polymer. Considering electron-phonon and electron-electron interactions along with the elastic potential energy of the lattice, the Hamiltonian for the conjugated polymer can be written as

\begin{array}{l} \begin{matrix} H = H_{e - p} + H_{e - e} + H_{l}, \\ H_{e - p} = - \sum_{l, s} [t_{0} + α (u_{l + 1} - u_{l}) + {(- 1)}^{l} t_{e}] \times [c_{l + 1, s}^{+} c_{l, s} + H . c .], \\ H_{e - e} = U \sum_{l} n_{l, ↑} n_{l, ↓} + V \sum_{l, s, s^{'}} n_{l, s} n_{l + 1, s^{'}}, \\ H_{l} = \frac{K}{2} \sum_{l} {(u_{l + 1} - u_{l})}^{2} . \end{matrix} \end{array}

Here,

t_{0}

is a hopping constant (2.5 eV);

α

is an electron-lattice coupling constant (4.78 eV/nm);

c_{l, s}^{+} (c_{l, s})

denotes the electron creation (annihilation) operator with spin

s

at unit cluster/group

l

with displacement

u_{l}

and corresponding electron occupation number

n_{l, s} = c_{l, s}^{+} c_{l, s}

;

t_{e}

is the Brazovskii-Kirova term (0.12 eV);

K

is an elastic constant (176.86 eV/nm²); 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 total Hamiltonian describing the polymer chain is called the extended Su-Schreiffer-Heeger-Hubbard Hamiltonian.

In order to describe the electron’s behavior, we have to know its energy spectrum $ε_{μ}$ and wavefunction $Φ_{μ} = {Z_{l, μ}^{s}}$ , which are functionals of the lattice displacement $u_{l}$ , as determined by the eigenequation

H Φ_{μ} = ε_{μ} Φ_{μ} .

The electron-electron interaction is treated in the Hartree-Fock approximation, whereby the eigenequation above can be written as follows:

\begin{array}{l} ε_{μ} Z_{l, μ}^{s} = [U (ρ_{l}^{- s} - \frac{1}{2}) + V (\sum_{s'} ρ_{l - 1}^{s'} + \sum_{s'} ρ_{l + 1}^{s'} - 2)] Z_{l, μ}^{s} \\ - [V \sum_{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 \sum_{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} . \end{array}

The charge distribution is defined as

ρ_{l}^{s} = \sum_{μ}^{o c c} | Z_{l, μ}^{s} |^{2} - n_{0}

, where

n_{0}

is the density of the positively-charged background.

Realizing that atoms are much heavier than electrons and using the Feynman-Hellmann theorem, we can describe the atomic movement of the lattice through classical dynamics by the equation

M \frac{d^{2} u_{l}}{d t^{2}} = - \frac{\partial E_{t o t} ({u_{l}})}{\partial u_{l}} u_{l} + K (2 u_{l} - u_{l + 1} - u_{l - 1}) .

Here,

E_{t o t} = \sum_{μ}^{o c c} ε_{μ} + \frac{K}{2} \sum_{l} {(u_{l + 1} - u_{l})}^{2}

, where the first term is the total electron energy reflecting the quantum microscopic properties of the excited polymer, and the second term is the elastic potential energy of polymer lattice. Thus, the Eq. (4) becomes

M \frac{d^{2} u_{l}}{d t^{2}} = - \sum_{μ}^{o c c} \frac{\partial ε_{μ}}{\partial u_{l}} + K (2 u_{l} - u_{l + 1} - u_{l - 1}),

where occ stands for the occupation or population of electrons. Since the polymer PPV is not a strongly correlated system, by using the above coupled equations and conventional molecular dynamics, we can quantitatively describe the dynamical evolution of not only the electronic states but also the lattice structure in a conjugated polymer chain.

When an external optical pulse/beam focuses on a polymer light-emitting diode or polymer solar cell and its photon energy matches the band gap of the polymer, there are two energy levels − $Γ_{u}$ and $Γ_{d}$ − that compose a singlet exciton. $| u 〉$ and $| d 〉$ are taken as the wavefunctions of the energy levels, whose energies are $E_{u}$ and $E_{d}$ ( $E_{u} > E_{d}$ ), and the electron populations are $P_{u}$ and $P_{d}$ ,respectively. Without the restriction of Pauli repulsion, the dipole moment of the two localized states can be expressed as $p = e 〈 u | r | d 〉$ , where $r$ is the dipole operator. When an appropriate external optical field with frequency $ω$ and energy density $ρ (ω)$ is applied to these two energy states of the energy spectrum of the conjugated polymer molecule, photon absorption occurs, and the electron in energy level $Γ_{d}$ is pumped to level $Γ_{u}$ . The stimulated transition rate between the two levels can be written as

W_{d \to u} = \frac{4 π^{2}}{3 ℏ^{2}} p^{2} ρ (ω) δ (ω - \frac{E_{u} - E_{d}}{ℏ}) .

It is obvious that the frequency/photon energy affects the possibility of the transition, and the energy density $ρ$ of the optical field with a certain frequency $ω$ can affect the rate of the transition between $Γ_{u}$ and $Γ_{d}$ . Also, an important factor along with a stimulated transition is a spontaneous transition, whose rate $γ_{u d}$ between these states is

γ_{u \to d} = \frac{4 {(E_{u} - E_{d})}^{3}}{3 ℏ^{4} c^{3}} p^{2} .

While the external optical pulse field with certain bandwidth determines the transition process, the inclusion of the spontaneous transition rate modifies and ensures the accuracy of the transition rate equation for the electron population. Thus, the evolution of the electron populations $P_{u}$ and $P_{d}$ during the electron transition are expressed by the decay equation

\begin{array}{l} \frac{d P_{u}}{d t} = (W_{d \to u} - γ_{u \to d}) P_{u}, \\ P_{d} = n - P_{u}, \end{array}

Where

n

is the total electron number. Combining the above equations, we can quantitatively and dynamically illustrate the whole process of the self-localization of photoexcitation and exciton formation.

3. Results and Discussion

Recently, related experimental research has illustrated the whole evolution of the dynamical fluorescence spectrum after a conjugated polymer undergoes excitation by a pulsed laser with different wavelengths, which provides information on hot exciton formation, relaxation and radiative decay [26,27]. Once the external optical field is introduced to excite the conjugated polymer, the polymer absorbs a photon, causing an electron in the highest occupied molecular orbital (HOMO) to jump to the lowest unoccupied molecular orbital (LUMO). This forms the hot exciton, which is depicted schematically in Fig. 1.

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

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In this article, we choose two laser beams with pulse widths of 13 fs (pulse 1) and24 fs(pulse 2) as the external optical sources. For pulse 1, its relationships between the light intensity and the time are presented in Fig. 2(a).

Fig. 2 Light intensity vs time forpulse 1 (A) and pulse 2(C), and light intensity vs wavelength forpulse 1 (B) and pulse 2 (D).

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After the discrete Fourier transformation is applied to pulse 1 as shown in Fig. 2(a), the relationship between the light intensity and the wavelength can be obtained. As exhibited in Fig. 2(b), we see that, the light intensity of pulse 1 is distributed within a certain range of wavelengths, reaching the maximum value at the wavelength of 1027 nm. For pulse 2, the relationship between the light intensity and time is depicted in Fig. 2(c). Based on the Fourier transform of pulse 2, the function between the light intensity and wavelength is illustrated in Fig. 2(d), where the pulse 2 distribution is seen to be similar to pulse 1, with the same wavelength maximum at 1027 nm.

When the conjugated polymer undergoes excitation, recent experiments have shown the electron localization to occur in the first 100 fs [26,27]. In our simulation here, given the excitation of the external laser pulse 1 as described in Figs. 2(a) and 2(b), it is found out that the electron begins to be localized around30 fs. And by 80 fs, the electron of the electron-hole pair, due to the photoexcitation by pulse 1, is well localized and becomes stable, as illustrated in Fig. 3(a). Under excitation by the external laser pulse 2, the electron starts to be localized at 25 fs. Following the continuous excitation by pulse 2 driving the electron cloud of the resultant hot exciton to be localized at the center of the polymer chain, it becomes stable by 70 fs. These results are in accord with the time scale of localization seen in experiments [26,27].

Fig. 3 Localization of the electron in the HOMO induced by the external laser pulse 1 (A) and pulse 2 (B).

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Different from inorganic semiconductors, the self-trapping effect of the conjugated polymer makes it highly possible that the evolution of the electron state is strongly correlated to the movement of the lattice. So, when the electron cloud has been localized during the formation of the electron-hole pair, i.e., hot exciton, the concentrate has moved to the evolution of the lattice structure of polymer. Actually, the photoexcitation at the beginning has already triggered the oscillation of the lattice energy that involves the kinetic energy and the potential energy, which is described in Fig. 4. Here, the period of oscillation can be estimated as 50 fs. In particular, the oscillation of the lattice energy at the first period reaches the largest amplitude of 2.5 eV with pulse 1 in Fig. 4(a) and 3 eV with pulse 2 in Fig. 4(b). This not only directly induces the lattice to seriously vibrate, but also destroys and breaks the original lattice structure and binding energy, finally resulting in the new lattice structure. It also poses a new question as to how the lattice configuration changes at the same time.

Fig. 4 Lattice energy over 600 fs with pulse 1 (A) and pulse 2 (B).

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If $u_{l}$ is the displacement of cluster l along the polymer chain, we construct (−1)^{l $u_{l}$} and regard it as a new parameter in the chain – the configuration of alternating single and double bonds. Figure 4 depicts the evolution of the lattice configuration after the conjugated polymer undergoes the excitation. As shown in Figs. 5(a) and 5(c), once the photoexcitation of pulse 1/pulse2 is triggered, it is observed that the alternating single and double bonds of the polymer immediately began to vibrate. After undergoing the vibration within 120 fs, the configuration of the alternating single and double bonds has been locally distorted along the polymer chain. The stable distortion of the configuration is finally established by 960 fs, as described in Figs. 5(b) and 5(d). Comparing Figs. 4 and 5, we see that, during the formation of the electron-hole pair in the conjugated polymer, the whole process of local distortion of the configuration accompanies the localization of the electron cloud (Fig. 3).

Fig. 5 Lattice configuration over 140 fs with pulse 1 (A) and pulse 2 (C); lattice configuration over 1200 fs with pulse 1 (B) and pulse 2 (D).

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Thanks to the prominent electron-lattice coupling in the conjugated polymer, both the appearance of the local distortion of the lattice configuration and the electron localization are able to change the electronic structure in the conjugated polymer. Especially, considering the oscillation of the lattice energy as depicted in Fig. 4, it can be assumed that the electronic structure will be driven to oscillate with the same period. As described in Fig. 6, once the photoexcitation of pulse 1 is turned on, the band gap width and the difference between the HOMO and LUMO of the conjugated polymer has been triggered to oscillate. Within 174 fs, the difference between the HOMO and LUMO becomes narrowest, changing from 1.83 eV to 0.87 eV. The time-dependent oscillation is most sharp in the first 200 fs of photoexcitation. After the relaxation of 1 ps, based on Fig. 6, the difference between the HOMO and LUMO gradually reaches the value of 0.92 eV.

Fig. 6 Time-dependent evolution of the difference between the HOMO and LUMO due to excitation from pulse 1 (A) and pulse 2 (B).

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On the basis of the profiles with respect to light intensity vs frequency of the external laser pulse1/pulse 2, when the external optical pulse is introduced into the conjugated polymer films, only the photon whose energy just matches the difference between the HOMO and LUMO can tunnel through the polymer film. So, the oscillation of the band gap acts as a filter to convert the intensity of the original optical pulse to a new effective one. The original pulse 1 with an average intensity of 10 µJ/(s·cm²) (Figs. 2(a) and 2(b)) has been changed to an effective laser pulse as illustrated in Fig. 7(a). Pulse 2 with an average intensity of 30 µJ/(s·cm²) (Figs. 2(c) and 2(d)) is converted to a new pulse as illustrated in Fig. 7(b).

Fig. 7 Time variation of the effective light intensity of pulse 1 (A) and pulse 2 (B).

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For a clearer presentation, the Fourier transforms of the related time-dependent evolutions are plotted in Fig. 8. It is easily seen that the period of the oscillation of the new effective optical field has a similar trend to the oscillation of the band gap and lattice energy, which is determined by the resonance aspects of the lattice.

Fig. 8 Fourier transform (FT) of the time variation of the effective light intensity of pulse 1(A)/pulse 2(B). FT of the time-dependent evolution of the energy gap due to the excitation by pulse 1(C)/pulse 2(D). FT of the time-dependent evolution of the lattice energy with pulse 1(E)/pulse 2(F).

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The above mechanism provides a possible approach to design an ultrafast optical field convertor that is composed of a polymer film. When an optical pulse is introduced into the polymer, it undergoes photoexcitation and induces oscillations of the energy level structure. Following this, the oscillations, acting as a filter, convert the intensity of the original optical pulse to a new effective one. The resonance aspects of the polymer chain determine the period of the new resultant optical pulse. The whole process finishes within 1 ps, which qualifies as an ultrafast response optical field convertor.

4. Summary

In summary, this article investigates the whole process of excitation after a conjugated polymer undergoes photoexcitation. Two pulses are used to stimulate the conjugated polymer, where the localization of the electron cloud in the first 100 fs not only induces oscillations of the lattice of the polymer chain, but also triggers oscillations of the energy level structure. Following this, the oscillations naturally change the absorption of the external optical field inversely. The mechanism sheds new insight into designing polymeric nanostructures for different applications. Especially, the oscillation of the band gap is able to act as an ultrafast response optical convertor for the external optical field.

Funding

This work was supported by the National Natural Science Foundation of China under Grant 21374105 and the Zhejiang Provincial Science Foundation of China under Grant R12B040001.

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Ultrafast optical pulse convertor caused by oscillations of the energy level structure in the conjugated polymer poly(p-phenylenevinylene)

Abstract

1. Introduction

2. Methodology

3. Results and Discussion

4. Summary

Funding

References and links

Cited By

Figures (8)

Equations (8)

Optics Express