Photoinduced Localized Phonons and Instantaneous Structure

Jan 4, 2017 - 5. Hide , F.; Díaz-García , M. A.; Schwartz , B. J.; Andersson , M. R.; Pei , Q. B.; Heeger , A. J. Semiconducting Polymers: A New Cla...
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Photoinduced Localized Phonons and Instantaneous Structure Contributing to Amplified Spontaneous Emission of Conjugated Polymers Deyao Jiang,†,‡ Weikang Chen,† Yusong Zhang,†,‡ Sheng Li,*,†,‡,§ and Thomas F. George*,§ †

Department of Physics, Zhejiang Normal University, Jinhua, Zhejiang 321004, China Department of Physics and Key State Laboratory of Surface Physics, Fudan University, Shanghai 200433, China § Office of the Chancellor and Center for Nanoscience, Departments of Chemistry & Biochemistry and Physics & Astronomy, University of MissouriSt. Louis, St. Louis, Missouri 63121, United States ‡

ABSTRACT: Amplified spontaneous emission (ASE), as a microscopic dynamical process, significantly influences the quality of optically pumped conjugated polymeric lasers. Based on the continuous optical pumping that couples with the prominent selftrapping process of conjugated polymer, four localized phonon modes are distinguished in conjugated polymers, all of which greatly contribute to localization of the excited state as well as local distortion along alternating single and double bonds. Consequently, the ultrafast localized distortion of alternating bonds constructs “instantaneous” structuresan effective four-level electronic structure and population inversionfor ASE. It is shown that all the evolving localized vibrational modes possess even parity, which not only makes it highly possible to be probed through the infrared phonon spectrum, but also opens up an opportunity to improve the quantum efficiency of a polymer ASE/laser by modification of localized phonon modes.

1. INTRODUCTION The first polymer laser was based on a conjugated polymer in solution that dates back to 1992.1 Since then, progress on polymeric light-emitting diodes (PLEDs)2,3 has extended the range of candidates for solid conjugated polymeric lasers, and the first solid polymeric laser with a microcavity was fabricated in 1996, which was triggered by optical pumping.4−7 Since then, conjugated polymer lasers with various types of resonators have been designed and reported, most of which are composed of the resonators of distributed Bragg reflectors,8−12 distributed feedback structures,13−15 whispering galleries,16−18 photonic band gap fibers,19 two-dimensional (2D) photonic crystals,20−23 and even flexible polymer fibers.24 The general mechanism for optical pumped lasing can be briefly presented as follows: Given a three- or four-level energy structure, continuous pumping results in electron population inversion, leading to amplified spontaneous emission (ASE). The resonator confines the emitted light as well as tunes the phase of the light, resulting in laser emission. For conventional inorganic lasers, the research generally has concentrated on the modification of the resonator and omitted the microscopic © XXXX American Chemical Society

dynamical process. In the same vein, researchers accordingly have modified the localized modes of the resonator to lower the threshold of the polymer organic lasers, thus enhancing the light absorption,25 which seems to open a pathway to improve the quality of organic lasers. Yet, it has been revealed that an optically pumped organic semiconductor laser largely depends on the microscopic ultrafast dynamics,26 particularly the light absorption and excited state of the organic laser.27−29 At the beginning of the external pumping, the polymer laser emission is easily affected by a multiple quantum process which involves phonons.30 Therefore, ASE, as the microscopic state of the lasing, naturally becomes the key stage for elucidating the underlying mechanism of the polymer laser, which is the topic of this article. Considering the contribution of an exciton to the luminescence of a PLED, researchers usually take it for granted that polymer lasing/ASE has excitonic behavior. An exciton Received: November 15, 2016 Revised: January 1, 2017 Published: January 4, 2017 A

DOI: 10.1021/acs.jpcc.6b11496 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C (electron−hole pair), however, does not possess electron population inversion, which makes it ill-suited to describe the lasing/ASE. So what is the excited state that is responsible for ASE/lasing with the electron population inversion? To answer this question, we propose the following approach that incorporates the electron transition process with conventional molecular dynamics, thus making it possible to clarify the dynamical evolution with regard to the excited state and electronic structure of the conjugated polymer. It has been observed for organic semiconductor materials that an effective four-level structure for TPD and DPABP ASE/ lasing is associated with a relatively high-frequency phonon mode.31 Bearing in mind that organic molecules can have complicated structures, this raises the question as to whether the simple chain-like conjugated polymer is able to form an analogous structure for ASE/lasing due to the similar highfrequency phonon mode. In 2005, Y. R. Shen et al. uncovered a vibrational−electronic double resonance resulting in the selftrapping effect in conjugated PPV.32 This intramolecular selftrapping enables the optical phonon to spatially track the excited-state relaxation along the polymer chain.33 Given π-electrons in low-dimensional materials, such as graphene, the energy levels and band gap have a close relationship with the excitation of phonon that drives the structural changes.34 Once the low-dimensional π-electrons undergo excitation, the ultrafast energy relaxation is increasingly dominated by phonon emission, as recently reported.35 During the dynamical process of the excited state, a quantumdynamical analysis shows multiple phonons taking part in exciton dissociation at polymer heterojunctions.36,37 Research on polymer−carbon nanotube heterojunctions has furthermore clarified that the excitation of multiple phonons on the conjugated polymer involves the whole dynamical process under the external photoexcitation.38 During the ultrafast excited-state dynamics, evolution of the wave function is driven by the efficient coupling to high-frequency vibrational modes, which moreover links with the ultrafast localization of the photoinduced wave function and nonadiabatic electronic transitions.39 Thus, on the basis of the above analysis, the assumption is made that multiple optical phonon vibrations, coupled with the prominent self-trapping process of the one-dimensional conjugated polymer, change the original energy structure to produce the special energy level for ASE. In this case, with a detailed depiction of the various aspects regarding the phonons and electronic states, we open up a path to understand the underlying mechanism affecting the quality of polymeric lasers, illustrating how the phonon vibrations influence the evolution of the excited state for ASE.

He = −∑ [t0 − α(ul + 1 − ul) + ( −1)l te](cl†+ 1, scl , s + Hc) l ,s

H′ = U ∑ nl , ↑nl , ↓ + V ∑ nl , snl + 1, s ′ l

HL =

K 2

(3)

l ,s,s′

∑ (ul+ 1 − ul)2 + l

M 2

∑ ul̇ 2

(4)

l

Here, PPV is selected to be a model to exhibit the related dynamical properties, where the parameters here are accordingly specified as follows: t0 is a hopping constant (2.5−5.0 eV); te is the Brazovskii−Kirova term (0.05−0.10 eV); α is an electron−lattice coupling constant (4.3−5.6 eV/Å); c†l,s (cl,s) denotes the electron creation (annihilation) operator at cluster l with spin s; ul is the displacement of cluster l; K is an elastic constant (eV/Å2); U (2.0−5.0 eV) and V (0.5−2.0 eV) are the on-site and nearest-neighbor Coulumb interactions, respectively; the lattice constant is 1.2−3.8 Å. The Hamiltonian for the electron−electron interaction can be treated with the Hartree−Fock approximation. The electronic energy spectrum and its quantum states are functionals of the lattice displacement. We can then depict the lattice displacement as describing the electronic behavior by means of the conventional dynamical equation combined with the Feynman−Hellmann theorem: Fl = − Ψ

M

d2ul dt

2

∂H Ψ ∂ul

(5)

⎫ ⎧ occ ∂E = −⎨∑ ν + K (2ul − ul + 1 − ul − 1)⎬ ⎩ ν ∂ul ⎭ ⎪







(6)

Given four energy levels (molecular orbitals) |A⟩, |B⟩, |C⟩, and |D⟩, the rate of the spontaneous emission between two levels (taking |A⟩ and |B⟩, for example) can be expressed as

γAB =

4(EA − EB)3 4 3

3ℏ c

p2

(7)

and the rate of stimulated emission as λAB =

π 2 p ρ(ω0) 3ε0ℏ2

(8)

Here, EA and EB are the energies of |A⟩ and |B⟩, ℏ is Planck’s constant, c is the speed of light in a vacuum, ε0 is the permittivity of a vacuum, p = e⟨A|r|B⟩ denotes the dipole moment of the two energy levels; and ρ(ω0)is the energy density of the external photoelectric field, per unit frequency, E −E evaluated at ω0 = A ℏ B . Within the four energy levels of the conjugated polymer, the electron transition will be accompanied by a nonradiative transition. Actually, this transition is of great significance in the whole transition process, and J. S. Wilson et al. experimentally find its transition rate in a conjugated polymer to be about knr = 105 s−1.40 Thus, once the optical pumping has begun, electrons will be pumped from energy level |D⟩ to |A⟩, where the transition equations for the four energy levels can be expressed as

2. METHODS Due to the quasi-one-dimensional structure of conjugated polymers, it is necessary to take the electron−photon coupling and electron−electron interaction into account. For the construction of the Hamiltonian describing a conjugated polymer, we start with the typical one-dimension model of the extended Su−Schreiffer−Heeger−Hubbard Hamiltonian: H = He + H′ + HL

(2)

+ H′

(1)

where He is the electronic component, H′ is the electron− electron interaction, and HL is for the lattice: B

DOI: 10.1021/acs.jpcc.6b11496 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C dPA = gPD − (γAD + k nr)PA dt

the configuration of the alternating single and double bonds in the PPV chain. Thanks to the self-trapping effect of conjugated polymers, once the chain is excited, the excitation drives the configuration of alternating bonds to be locally distorted in the background of the homogeneous alternating bonds, as shown in Figure 1A, to form an exciton, i.e., relaxed “electron−hole pair”.

(9)

dPB = k nrPA + λCBPC − (λBC + γBC)PB dt

(10)

dPC = (λBC + γBC)PB − (k nr + λCB)PC dt

(11)

PD = n − PA − PB − PC

(12)

Here, g is the gain, n is the total electron number, PA is the electron population in level |A⟩, and so forth. We thus have combined classical molecular dynamics with the electron transition equations to show completely the dynamical properties of the polymer and electrons, and hence the process of amplified spontaneous emission in a conjugated polymer. 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 ϕn0, and its perturbation can be written as ϕn(t ) = ϕn0 + ϕn′(t )

Figure 1. (A) Locally distorted alternating bonds. (B) Electronic energy spectrum with HOMO, Γd, Γu, and LUMO.

(13)

Simultaneously, the locally distorted lattice of the polymer further induces changes in the electronic structure, where the original HOMO and LUMO in the electronic spectrum are pulled into the energy gap. Along with local distortion of the alternating bonds in Figure 1A, two energy levels, Γu and Γd, relative to the two localized molecular orbitals, are formed, which are depicted as Figure 1B. As previously mentioned, the quantum-dynamical analysis shows that the dynamical process with respect to excited states generally involves multiple vibrational phonons.36,37 It has been further clarified that multiple phonons of the conjugated polymer are favorable to the whole dynamical process at a polymer−carbon nanotube heterojunction due to the external photoexcitation.38 Thus, our attention focuses on the phonon modes along with the dynamical process. Based on the formed exciton, with continuous excitation of the optical pumping with 60 μJ/cm−2, it is found that, four vibrational modes, as shown in Figure 2A−D, are selected, being identified as the localized phonon modes, which also can serve as a special fingerprint for the stable local distortion of the PPV chain. Interestingly, from Figure 2, those four localized vibrational modes have even parity, leading to the resultant infrared phonon spectrum (Figure 2E), where the highest absorption peak located at about 780 cm−1 is due to the phonon mode m1 (Figure 2A), while the other three salient absorption peaks correspond to the localized vibrational modes m2, m3, and m4 (Figure 2B−D). Due to the prominent self-trapping effect of conjugated polymer, it is further discovered that the occurrence of these four local phonon modes certainly induces the change of the electronic structure of PPV. During the excitation of the local phonon modes, the optical pumping by the 60 μJ/cm−2 ultrafast laser changes the electronic structure as well, where the HOMO and LUMO (Figure 1 B) of the exciton in the electronic spectrum are pulled into the energy gap within the first 50 fs besides the Γu and Γd, and then the four energy levels are formed by 100 fs, as depicted in Figure 3A. At the beginning of the excitation, the configuration of the alternating bonds of PPV is locally distorted as shown in Figure 1A. After the exciton undergoes optical pumping by the 60 μJ/ cm−2 laser, referring to the excitation of the local phonon

The localized vibrational mode can be determined by introducing a second-order perturbation in the calculation of energy: N

H({ϕn}) = E0 + Es ∑ A m({ϕn})ϕm′ m

+

1 2

N

∑ Bm, n({ϕn0})ϕn′ϕm′

(14)

m,n

Bm , n = k[(δm , n + δm , n + 1)(1 − δm , N ) + (δm , n + δm , n − 1) (1 − δn ,1)] + 2α 2( −1)m + n

∑ μ , ν(≠ μ)

Cμm, νCμn, ν εμ0 − εν0

(15)

Here, N is the total number of lattice sites, ε is the static eigenenergy of the electron, and the parameter Cmμ,ν is Cμm, ν = (1 − δm , N )(Zμ0, m + 1, sZν0, m , s + Zμ0, m , sZν0, m + 1, s) − (1 − δm ,1)(Zμ0, m , sZν0, m − 1, s + Zμ0, m − 1, sZν0, m , s)

(16)

0 Zμ,m,s

where is the corresponding eigenstate of the electron in energy level μ at lattice site m with spin value s, and δm,n is the Kronecker delta. By simply diagonalizing the matrix B, the eigenfrequency and eigenvector of the vibrational modes can be found. Consequently, the excitation of phonon modes during the formation process of the ASE or laser effect can be depicted in detail.

3. RESULTS AND DISCUSSION For the convenience of our calculation, let us choose a single PPV chain consisting of 200 unit clusters. When an external laser beam of 60 μJ/cm−2, whose energy matches the gap, is utilized to excite the chain, the electron is stimulated from the highest occupied molecular orbital (HOMO) to the lowest unoccupied molecular orbital (LUMO). If ul is the displacement of cluster l along the polymer chain, (−1)lul is regarded as C

DOI: 10.1021/acs.jpcc.6b11496 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C

Figure 2. (A−D) Localized phonon vibrational modes. (E) Infrared phonon spectrum for ASE of the polymer.

modes as in Figure 2, the configuration of the alternating bonds evolves into a new situation up to 5 ps (or 5000 fs), reflected by the evolution of the configuration of the alternating bonds in Figure 3B. For the configuration up to 200 fs, it is demonstrated in Figure 3C that the original localized distortion of the alternating bonds of exciton splits into a double-valleylike distortion. More importantly, these localized vibrational modes contribute almost 99.98% to terminal distortion of the alternating bonds, while the extended vibrational modes are greatly suppressed. It also means that the localized phonon mode, especially, plays an increasingly important role in the photoexcitation. With the excitation of the four localized phonon modes, the evolution of the electronic energy levels, as displayed in Figure 3A up to 200 fs, and the evolving four discrete energy levels in the gap as depicted in Figure 4A, provide an instantaneous (not fixed) energy level structure for ASE or lasing. Being different from the conventional fixed energy levels for ASE, it therefore poses a question as to whether the instantaneous four discrete

energy levels favor ASE in a conjugated polymer. In another words, does there exist the possibility that the population inversion for ASE in PPV appears due to the formation of the instantaneous four discrete energy levels? To clarify this, both parts B and C of Figure 4 describe the time-dependent evolution of the electron populations of the energy levels Γu and Γd within the time span of 5 ps, where the extrinsic photoexcitation drives the electron population of Γu to grow from 1.0 to about 1.38, while that of Γd decreases from 1.0 to about 0.68. At the end, the population in the high energy level Γu is more than Γd, accordingly creating the electron population inversion, which just paves the way to realize ASE in PPV, or even the lasing effect.

4. CONCLUSION In summary, a microscopic dynamical process with respect to ASE for a conjugated polymer is revealed. Given the selftrapping effect of the quasi-one-dimensional structure of the polymer, continuous external optical pumping changes the D

DOI: 10.1021/acs.jpcc.6b11496 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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Figure 3. (A) Evolution of the energy levels within 200 fs. Three-dimensional depiction of the time-dependent configuration of the alternating bonds over (B) 5000 and (C) 200 fs.

Figure 4. (A) Electron energy spectrum. (B, C) Time-dependent electron populations of the energy levels Γu and Γd in the band gap after undergoing external optical pumping.

structure through local distortion of the alternating bonds and electronic energy structure, with ultrafast construction of an instantaneous (not fixed) four-level electronic structure and population inversion for ASE. It is seen that, during the ultrafast dynamical process, four distinguished localized phonon modes greatly contribute to the localization of the excited state and local distortion along the alternating bonds, leading to the

emergence of the four-level structure just within 100 fs. It is furthermore revealed that the four localized vibrational modes have even parity that can be detected through the infrared phonon spectrum. Above all, this opens up an opportunity to improve the quality of polymeric ASE and lasing through modifying the appropriate localized phonon modes in a conjugated polymer. E

DOI: 10.1021/acs.jpcc.6b11496 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. Tel.: 314-516-5252. ORCID

Thomas F. George: 0000-0003-1225-6778 Notes

The authors declare no competing financial interest.



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



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