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Self-Introduction of Disordered Lattice Distortion by a Polymeric Nanofiber Laser Sheng Li*,†,‡ and Thomas F. George*,‡ Department of Physics, Zhejiang Normal UniVersity, Jinhua, Zhejiang 321004, China, and Office of the Chancellor and Center for Nanoscience, Departments of Chemistry & Biochemistry and Physics & Astronomy, UniVersity of Missouri-St. Louis, St. Louis, Missouri 63121 ReceiVed: March 22, 2010; ReVised Manuscript ReceiVed: June 1, 2010
The developed molecular dynamics shows that upon photoexcitation of a conjugated polymer nanofiber, such as poly(phenylene vinylene)s or the polyfluorene family, singlet excitons initially are formed. Through continuous optical pumping, the electron populations of the excitons are reversed. Different from inorganic materials, the electron population reversion not only generates new localized electron states but also destroys the periodic structure of the polymer chain, inducing localized lattice distortion. These localized modes provide one of the channels to form localized lasing emission in a single conjugated nanofiber laser, which is consistent with recent experimental observations. I. Introduction Solid-state lasers with organic materials date back to the 1960s when Soffer and McFarland fabricated a laser based on dye-doped polymers.1 In 1974, Avanesjan et al. demonstrated the first laser based on a single crystal of fluorine-doped anthracene.2 Despite considerable research and effort during the 1960s and 1970s,1-3 the shortage of high-quality organic single crystals limited the development of lasers based on organic materials. By the end of the 1980s, the inventions of organic transistors4,5 and polymer light-emitting diodes (PLEDs)6 finally broke this limitation. In 1992, the first organic semiconductor laser was developed in a solution consisting of a conjugated polymer.7 In 1996, this was extended to conjugated polymers,8-11 where, in particular, Tessler et al. demonstrated an optically pumped polymer microcavity laser.8 Since then, it has been a topic of vigorous research. Conjugated polymer lasers with a variety of resonators, such as distributed feedback12 and photonic bandgap structures,13 have been extensively fabricated by using a range of coating and imprinting techniques and launching a new field known as “plastic lasers.” The 1D nature of a conjugated polymer gives it special flexibility,14 and by taking advantage of this, the first single conjugated polymer nanowire laser was realized.15 Furthermore, the polymeric laser is also a bit dependent on temperature.16 These prominent virtues of conjugated polymers thereby pave the way to fabricate the nanolaser. Quochi et al. observed laser action in single p-sexiphenyl nanofibers.17-19 The localized lasing emission is attributed to the coherent light propagation in 1D random nanofibers, where a disordered structure is introduced to the polymer chain by air gaps causing breaks in the single nanofibers. Here a question is raised whether there exists a new channel to self-introduce the disorder to a 1D polymer chain besides air-gap breaking. To answer this, we have to focus on the “flexibility” of the conjugated polymer, which reflects that the lattice of the conjugated polymer sensitively depends on the electronic state, namely, self-trapping. The laser, as a typical excitation of * To whom correspondence should be addressed. E-mail:
[email protected] (S.L.);
[email protected] (T.F.G.). † Zhejiang Normal University. ‡ University of Missouri-St. Louis.
electrons, is triggered by electron population inversion. Considering this, an assumption can be proposed that it is highly possible for the polymeric laser to destroy the periodic bond structure of the polymer, inducing the occurrence of disorder structure. To clarify this assumption, descriptions of both electronic states and the lattice structure become two key steps. For the electronic behavior of a polymer laser, by comparing it with the contribution of an exciton to the luminescence of a PLED, scientists generally regard organic lasing to be an excitonic behavior.14 Actually, an exciton (electron-hole pair) has no electron population inversion, which means that it is ill-suited to describe the lasing effect. Therefore, the first challenge is to describe the process of electron transition of the polymeric laser and then to search out the new states contributing to the polymeric lasing effect. Furthermore, because of the strong electron-lattice coupling as mentioned before, the lattice structure is sensitive to the change of electronic state. Therefore, it is necessary to develop a viable method that not only is associated with the evolution of electron transitions but also links lattice structure with the electronic behavior. In this article, we develop a method that combines the electron transition process with the self-trapping effect and helps to extract the new state in the conjugated polymer that contributes to the lasing effect. Because of the self-trapping effect, the new excitation contributing to the polymer laser is also a localized state, and its lattice structure even looks like exciton. However, different from the exciton, this new excitation not only generates the electron population inversion but also strengthens the localized distortion, thus inducing the occurrence of localized lattice disorder that spans many unit clusters/groups along the polymer chain.
II. Model For the prominent self-trapping of these conjugated polymers, the extended Hubbard-Su-Schreiffer-Heeger Hamiltonian provides a convenient and accurate description as follows21
10.1021/jp102582h 2010 American Chemical Society Published on Web 06/24/2010
Self-Introduction of Disordered Lattice Distortion H)-
∑ {t
J. Phys. Chem. B, Vol. 114, No. 27, 2010 8895
+ R(ul+1 - ul) + (-1)lte} ×
0
l,s
K + cl,s + H.c.] + ∑ (ul+1 - ul)2 + H' + HE [cl+1,s 2
(1)
l
Here t0 is a hopping constant, 2.5 to 5.0 eV; R is an electron-lattice coupling constant, 41 eV/nm; c+l, s (cl, s) denotes the electron creation (annihilation) operator with spin s at unit cluster/group land corresponding occupation number nl,s ) c+l,scl,s; ul is the displacement of unit cluster/group l; K is an elastic constant, 2100 ev/nm2; te is the Brazovskii-Kirova term, 0.05 to 0.07 eV; and U (3.0 eV) and V (0.5 eV) are the on-site and nearest-neighbor Coulomb repulsion strengths, respectively. To describe the electron’s behavior, we have to know its energy spectrum εµ and wave function Φµ, which are functionals of the lattice displacement ul, as determined by the eigenequation
HΦµ ) εµΦµ
(3)
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
d2ul 2
occ
)-
dt
∂ε
∑ ∂uµl + K(2ul - ul+1 - ul-1)
(4)
µ
Furthermore, because the polymer is not a strongly correlated system, the electron-electron interaction term can be treated by the Hartree-Fock approximation.20-22 Therefore, assuming an electronic wave function Φµ ) {Zn,s µ}, electron-electron interaction term H′ takes the form
H' )
∑ l,s
{( U
occ
∑ |Z
-s 2 l,µ |
µ
occ
V[
∑ ( ∑ |Z µ occ
∑ (V ∑ Z
+
∑ |Z
-s 2 l+1,µ |
}
† - 2)] cl,s cl,s -
µ
s s † l,µZl+1,µ)(cl,scl+1,s
l,s
)
1 + 2
occ
-s 2 l-1,µ |
s′
-
(5)
+ H.c.)
µ
where occ stands for the occupation or population of electrons. A conventional molecular dynamics approach, which combines all previous equations, fixes the electron occupations. However, this makes it difficult to describe the electronic transitions for the relaxation process associated with the lasing effect. To resolve this dilemma, we introduce electron population rate equations to the molecular dynamics treatment. To begin, if there are three electron occupied energy levels marked by a, b, and c and the gain is designated by g that pumps an electron in c to a, then the evolutions of their related electron populations, Pa, Pb, and Pc are given as
dPa ) gPc - γabPa, dt dPb ) γabPa - γbcPb, dt Pc ) n - Pa - Pb
(6)
where γab (γbc) is the transition rate between energy levels a and b (b and c) and n is the total electron number. If |a〉 and |b〉 are the wave functions for the energy levels a and b with energies Ea and Eb, respectively, then the transition rate between the two levels is γab ) (4e2(Ea - Eb)3)/(3p4c3)|〈a|r|b〉|2. Using these coupled rate 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. III. Results and Discussion As mentioned in Section II, ul is defined as the displacement of the unit cluster/group l along the polymer chain. For convenience of presentation in Figure 1, a sphere is used to depict a unit cluster/group of a polymer, such as one of the poly(phenylene vinylene)s or polyfluorenes. Without electronlattice and electron-electron interactions, it can be expected that there is no displacement for any unit cluster of the polymer, and thus the clusters periodically distribute along the polymeric chain with lattice constant a, as shown in Figure 1a. Actually, the electron-lattice coupling of the conjugated polymer makes the state unstable, and the original symmetry of lattice structure will be broken. The symmetry-breaking drives unit clusters/ groups to move with different directions. Finally, the unit cluster/ group of the conjugated polymer alternates between single and double bonds, giving lattice dimerization. As illustrated in Figure 1b, if the displacement of the lth unit cluster/group is ul ) |ul|, then the displacement of the l + 1-th unit cluster/group is ul+1 ) -|ul+1|. For a clear demonstration, we choose a conjugated polymer consisting of 200 unit clusters/groups as a research sample and use the lattice configuration (-1)lul to describe the lattice structure. Therefore, the lattice dimerization not only makes the lattice configuration constant, as depicted in Figure 2a, but also produces an energy gap, where the valence band is fully occupied while the conduction band is empty, as shown in Figure 2b. When an external laser beam or pulse is applied to excite the polymeric luminescent materials, an electron is excited from the highest-occupied molecular orbital (HOMO) to the lowestunoccupied molecular orbital (LUMO). Given the prominent self-trapping effect of the polymer, the lattice is locally distorted over the background of the homogeneous dimerization of the lattice configuration, as illustrated in Figure 3a. Self-trapping, coupled to the distortion of lattice, moves the HOMO and LUMO states to the center of gap, and the localization of the lattice distortion also localizes Ψu and Ψd in the middle of gap, as shown in Figure 3b. On the basis of the electron spectrum of the exciton depicted in Figure 3b, it is found that each of the localized states Ψu and Ψd is occupied by one electron. There is no electron population inversion, and the singlet exciton contributes only to light emission, not to the lasing effect. As is well-known, the necessary condition for lasing in semiconductors is the existence of three or four energy levels. Fortunately, as shown in Figure 3b, two localized states of the singlet exciton and the HOMO provide the appropriate energy structure of three discrete energy levels. Through the continuous pumping provided by an external optical pulse/beam whose energy matches the difference between Ψu and HOMO, it is possible to realize the lasing effect. The details of the dynamical process also can be theoretically described. The excitation process for the singlet exciton (after the gain exceeds a threshold value) is depicted in Figures 4 and 5. At the beginning, the singlet exciton is composed of an electron-hole
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Figure 1. Periodic lattice structure where a sphere represents each unit cluster of the polymer: (a) periodic distribution of the unit clusters of the polymer chain with lattice constant a and (b) lattice dimerization of the polymer chain with lattice constant a.
Figure 2. Lattice configuration of lattice dimerization: (a) constant configuration and (b) energy gap.
Figure 3. Profile of the singlet exciton: (a) lattice configuration and (b) electronic spectrum. The value of gap is 5.12 eV, and the energy values of Ψu and Ψd are 1.60 and -1.48 eV, respectively.
pair where each of localized states is occupied by one electron (Figure 3b). When the time reaches 100 ps, the electron population of Ψu is 1.48, wherreas that of Ψd becomes 0.52, and the electron population of HOMO is 2, as depicted in Figure
4. At 300 ps, the electron population of Ψu with higher energy is larger than that of Ψd. This means that the electron population undergoes inversion, which also indicates lasing of the polymeric molecule. This is occurring still within the relaxation process
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Figure 4. Time-dependent electron populations in the following three states of a conjugated polymer semiconductor after it undergoes external optical pumping: (a) HOMO, (b) Ψd, and (c) Ψu.
Figure 5. Three-dimensional picture of the time-dependent lattice configuration induced by external optical pumping over the entire time range.
of the excitation. Up to 500 ps, the electron populations of Ψu and Ψd become 0.09 and 1.91, respectively, whereas the electron population of HOMO remains at 2. At 500 ps, the electron population becomes stable compared with that seen before, which is the end of the excitation of the singlet exciton. In other words, the relaxation time of the excitation is ∼500 ps. After that, the electron population inversion of the two localized states, as the signature of lasing, sustains laser emission of the conjugated polymeric molecule. The lattice configuration of the singlet exciton before excitation is shown in Figure 3a. After the optical pumping, the whole relaxation of the lattice configuration is described in detail in Figure 5, and the associated lattice distortion of the conjugated polymer chain becomes more severe than that of the singlet exciton.
As mentioned previously, when an external optical pulse pumps an exciton of the conjugated polymer, the lattice undergoes relaxation, thus giving electron population inversion. At the end of the relaxation of the excitation, that is, 500 ps, the associated lattice becomes stable and also induces localized lattice distortion, which not only totally destroys the initial periodic structure of dimerization but also forms a new lattice structure. As in Figure 1, we use one sphere to represent each unit cluster/group of the conjugated polymer in Figure 6, which shows three series of spheres demonstrating three different lattice structures. The first series of spheres (Figure 6a) describes the periodic distribution of the unit clusters/groups of the conjugated polymer with constant lattice length, a, and Figure 6b illustrates the lattice dimerization whose lattice length also becomes 2a.
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Figure 6. Lattice structure where a sphere represents each unit cluster of the polymer. (a) Periodic lattice structure of the polymer chain with lattice constant a, (b) lattice dimerization of the polymer chain with lattice constant a, and (c) final lattice structure from the 95th to the 100th cluster lattice site along the polymer chain following external pumping.
Figure 7. Final lattice configuration of the polymer after external pumping.
For convenience of presentation, six unit clusters/groups of the polymer from the 95th to the 100th cluster lattice site are chosen to show the final lattice structure of the polymer after external pumping (Figure 6c). Comparing Figure 6c with a and b, it is found that the external pumping finally not only destroys the initial periodic structure of the polymer chain but also induces the occurrence of the disordered lattice structure. Considering the symmetry of the polymer chain, it is appropriate to utilize the lattice configuration (-1)lul to present the size of the newly-formed lattice distortion. Figure 7 shows the final lattice configuration, which is locally distorted and involves ∼60 unit clusters/groups from the 70th to the 130th lattice site. If the size of each unit cluster/group is ∼5 Å, then the width of the self-introduced lattice distortion will be 300 Å. Furthermore, because of the self-trapping and electron population inversion, the newly localized lattice distortion moves the localized states in the middle of the gap, Ψu and Ψd, to 1.452 and -1.260 eV, thus resulting in lasing emission at the wavelength 457.227 nm (2.712 eV). This is well-consistent with the experimental result indicating lasing emission in the range 450-460 nm.16
Therefore, the whole dynamics of the polymeric laser can be described as follows: Once an external optical pulse continuously pumps the conjugated polymer fibers, the induced electron population inversion not only causes laser emission but also triggers lattice relaxation, finally forming local disordered lattice distortion that involves 60 unit clusters/groups along the polymer chain. Then, during the coherent light propagation along the polymeric fiber, the self-introduced local lattice distortion strengthens multiple scattering of coherent light besides the airgap breaking in the polymer fiber, which in turn induces localization of the laser emission labeled as a “random laser” based on isolated individual p-sexiphenyl nanofibers.17-19,23,24 In conclusion, on the basis of the newly developed molecule dynamics approach that includes electronic transitions, it is found that after a conjugated polymer, such as a PLED, undergoes photoexcitation, a singlet exciton is initially formed. Then, because of continuous optical pumping, the electron populations of two localized states of the singlet exciton undergo inversion as well as generate localized lattice distortion. The localized lattice structure opens a channel to form localized
Self-Introduction of Disordered Lattice Distortion lasing emission in a conjugated nanowire, which is consistent with experimental observations. Acknowledgment. We thank S. Y. Liu and X. Sun for helpful discussions. This work was supported by the National Science Foundation of China under grants 20804039 and 10932010 and the Zhejiang Provincial Natural Science Foundation of China under grant Y4080300. References and Notes (1) Soffer, B. H.; McFarland, B. B. Appl. Phys. Lett. 1967, 10, 266. (2) Avanesjan, O. S.; Benderskii, V. A.; Brikenstein, V. K.; Broude, V. L.; Korshunov, L. I.; Lavrushko, A. G.; Tartakovskii, I. I. Mol. Cryst. Liq. Cryst. 1974, 29, 165. (3) Karl, N. Phys. Status Solidi A 1972, 13, 651. (4) Tsumura, A.; Koezuka, H.; Ando, T. Appl. Phys. Lett. 1986, 49, 1210. (5) Tang, C. W.; Vanslyke, S. A. Appl. Phys. Lett. 1987, 51, 913. (6) Burroughes, J. H.; Bradley, D. D. C.; Brown, A. R.; Marks, R. N.; Mackay, K.; Friend, R. H.; Burns, P. L.; Holmes, A. B. Nature 1990, 347, 539. (7) Moses, D. Appl. Phys. Lett. 1992, 60, 3215. (8) Tessler, N.; Denton, G. J.; Friend, R. H. Nature 1996, 382, 695. (9) Hide, F.; DiazGarcia, M. A.; Schwartz, B. J.; Andersson, M. R.; Pei, Q. B.; Heeger, A. J. Science 1996, 273, 1833. (10) Holzer, W.; Penzkofer, A.; Gong, S. H.; Bleyer, A.; Bradley, D. D. C. AdV. Mater. 1996, 8, 974.
J. Phys. Chem. B, Vol. 114, No. 27, 2010 8899 (11) Frolov, S. V.; Ozaki, M.; Gellermann, W.; Vardeny, Z. V.; Yoshino, K. Jpn. J. Appl. Phys., Part 2 1996, 35, L1371. (12) Heliotis, G.; Xia, R. D.; Turnbull, G. A.; Andrew, P.; Barnes, W. L.; Samuel, I. D. W.; Bradley, D. D. C. AdV. Funct. Mater. 2004, 14, 91. (13) Riechel, S.; Kallinger, C.; Lemmer, U.; Feldmann, J.; Gombert, A.; Wittwer, V.; Scherf, U. Appl. Phys. Lett. 2000, 77, 2310. (14) Samuel, I. D. W.; Turnbull, G. A. Chem. ReV. 2007, 107, 1272. (15) O’Carroll, D.; Lieberwirth, I.; Redmond, G. Nat. Nanotechnol. 2007, 2, 180. (16) Kozlov, V. G.; Bulovic, V.; Forrest, S. R. Appl. Phys. Lett. 1997, 71, 2575. (17) Quochi, F.; Cordella, F.; Mura, A.; Bongiovanni, G.; Balzer, F.; Rubahn, H. G. J. Phys. Chem. B 2005, 109, 21690. (18) Quochi, F.; Cordella, F.; Mura, A.; Bongiovanni, G.; Balzer, F.; Rubahn, H. G. Appl. Phys. Lett. 2006, 88, 041106. (19) Andreev, A.; Quochi, F.; Cordella, F.; Mura, A.; Bongiovanni, G.; Sitter, H. J. Appl. Phys. 2006, 99, 034305. (20) (a) Quochi, F.; Cordella, F.; Mura, A.; Bongiovanni, G; Balzer, F; Rubahn, H. G. J. Phys. Chem. B 2005, 109, 21690. (b) Li, S.; George, T. F.; Sun, X. J. Phys. Chem. B 2007, 111, 6097. (21) Li, S.; Chen, L. S.; George, T. F.; Sun, X. Phys. ReV. B 2004, 70, 075201. (22) Li, S.; He, X. L.; George, T. F.; Xie, B. P.; Sun, X. J. Phys. Chem. B 2009, 113, 400. (23) Cao, H. J. Phys. A: Math. Gen. 2005, 38, 10497. (24) Li, S.; Wang, Z. J.; Chen, L. S.; George, T. F.; Sun, X. Appl. Phys. Lett. 2005, 86, 171109. Li, S.; Wang, S.; Wang, Z. J.; George, T. F.; Sun, X. Appl. Phys. Lett. 2006, 88, 111103.
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