\documentstyle[12pt,fleqn]{article} \setlength{\textwidth}{5.5in} \setlength{\textheight}{9in} \setlength{\topmargin}{-0.25in} \renewcommand{\baselinestretch}{1.8} \def\nel{N_{\rm el}} \def\nc{N_{\rm c}} \def\nb{N_{\rm b}} \def\eg{E_{\rm g}} \def\tc{T_{\rm c}} \def\eps{\epsilon} \def\la{\langle} \def\ra{\rangle} \def\beeq{\begin{equation}} \def\eneq{\end{equation}} \def\beeqa{\begin{eqnarray}} \def\eneqa{\end{eqnarray}} \def\theequation{\arabic{equation}} \def\tmptheequation{\arabic{tmpequation}} \def\thefootnote{\fnsymbol{footnote}} \def\thesection{\Roman{section}.} \def\thepage{-- \arabic{page} --} \setcounter{section}{1} \addtocounter{section}{-1} \setcounter{page}{1} \begin{document} \begin{center} \vspace{2in} {\large {\bf{Nonlinear optical response from excitons\\ in soliton lattice systems: II.\\ Effects of boundary conditions\\ } } }\\ \vspace{1cm} (Running head: {\sl Nonlinear optical response in soliton lattice systems}) \vspace{1cm} {\rm Kikuo Harigaya\footnote[1]{E-mail address: harigaya@etl.go.jp; URL: http://www.etl.go.jp/~harigaya/.}}\\ \vspace{1cm} {\sl Physical Science Division,\\ Electrotechnical Laboratory,\\ Umezono 1-1-4, Tsukuba, Ibaraki 305, Japan} \vspace{1cm} (Received ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~) \end{center} \vspace{1cm} \noindent {\bf Abstract}\\ Exciton effects on conjugated polymers and the resulting optical nonlinearities are investigated in soliton lattice states. We use the Su-Schrieffer-Heeger model with long-range Coulomb interactions. The third-harmonic generation (THG) at off-resonant frequencies is calculated as functions of the soliton concentration and the chain length of the polymer. The magnitude of the THG at the 10 percent doping increases by the factor about 10$^2$ from that of the neutral system. The huge increase by the order two is common for the several choices of Coulomb interaction strengths, and is seen in open systems as well as in periodic systems. We also find that the boundary condition effects are more remarkable when the number of solitons is two in the open boundaries. We obtain much huge enhancement of THG here. This new fact should be attractive to the experimentalists. \mbox{} \noindent PACS numbers: 7135, 7866, 7138 \pagebreak The doping effects in conjugated polymers and their linear and nonlinear optical responses are fascinating research topics because of their importance in the scientific interests as well as technological developments. In the previous papers [1,2], we have theoretically considered exciton effects in the soliton lattice states of doped systems, and have investigated the nonlinear optical response properties. There is one kind of exciton in the undoped system with half-filled electronic states [3], where the excited electron (hole) sits at the bottom of the conduction band (top of the valence band). We have called this exciton as the intercontinuum exciton [1]. In the soliton lattice states of the doped Su-Schrieffer-Heeger (SSH) model for degenerate conjugated polymers [4], there are small energy gaps between the soliton band and the continuum states, i.e., valence and conduction bands. The number of the kind of excitons increases, and their presence effects on structures of the optical spectra. A new exciton, which we have named the soliton-continuum exciton [1], appears when the electron-hole excitation is considered between the soliton band and one of the continuum bands. In the next study [2], we have investigated the nonlinear optical properties in order to look at how characters of optical excitations effect on the nonlinear optical properties. The central issue has been how large optical nonlinearities would be obtained when conjugated polymers are doped with electrons or holes up to as much as 10 percent. The SSH model [4] with the long range Coulomb interactions of the Ohno expression [5] has been solved with the Hartree-Fock (HF) approximation, and the excitation wavefunctions of electron-hole pairs have been calculated by the single-excitation configuration-interaction (single-CI) method. In [2], the systems with periodic boundary conditions have been considered. We have calculated the off-resonant nonlinear susceptibility as a guideline of the magnitude of the nonlinearity. We might assume that multi excitations, such as double (triple) excitations [6], do not contribute significantly. Then, we have looked at the characters of excitations at off-resonances resulted from the single-CI calculations. The third harmonic generation (THG) at the zero frequency, $\chi_{\rm THG}^{(3)} (0) = \chi^{(3)} (3\omega;\omega,\omega,\omega)|_{\omega = 0}$, has been considered with changing the chain length and the soliton concentration. We have shown that the magnitude of the THG at the 10 percent doping increases by the factor about 10$^2$ from that of the neutral system. In the above works, the periodic systems have been considered. This simulates the reasonably long systems where chain end effects are small. However, all of the polymers are not regarded as long enough to neglect end effects, and shorter chains would be present. The chain ends might effect on their electronic structures, excitation properties, and also on the resulting nonlinear optical responses. In this paper, we shall study how the chain ends will effect on the huge increase of the optical nonlinearity which has been found in the periodic systems. We will show that the huge increase of the THG upon doping is seen in open systems, too. The effects of the chain ends are stronger when the number of the solitons is two. This is due to the trapping of the solitons near to the chain ends. It may seem that this paper is an addendum to the previous paper [2]. However, we find that the boundary condition effects are more dramatic when the number of solitons is two, and we obtain much huge enhancement of THG. This is a new fact of this paper. We use the SSH hamiltonian [4] with the Coulomb interactions: \beeq % eq 1 H = H_{\rm SSH} + H_{\rm int}. \eneq The first term of eq. (1) is: \beeqa % eq 2 H_{\rm SSH} &=& - \sum_{i,\sigma} ( t - \alpha y_i ) ( c_{i,\sigma}^\dagger c_{i+1,\sigma} + {\rm h.c.} ) \nonumber \\ &+& \frac{K}{2} \sum_i y_i^2, \eneqa where $t$ is the hopping integral of the system without the bond alternation; $\alpha$ is the electron-phonon coupling constant which changes the hopping integral linearly with respect to the bond variable $y_i$; the bond variable is assigned to the bond between the $i$th and $(i+1)$th sites ($1 \leq i \leq N -1$, $N$ being the system size); $c_{i,\sigma}$ is an annihilation operator of the $\pi$-electron at the site $i$ with spin $\sigma$; the sum is taken over $i$ within $1 \leq i \leq N -1$ due to the open open boundary conditions; and the last term with the spring constant $K$ is the harmonic energy of the classical spring simulating the $\sigma$-bond effects. The second term of eq. (1) is the long-range Coulomb interaction in the form of the Ohno potential [5]: \beeqa % eq 3 H_{\rm int} &=& U \sum_i (c_{i,\uparrow}^\dagger c_{i,\uparrow} - \frac{n_{\rm el}}{2}) (c_{i,\downarrow}^\dagger c_{i,\downarrow} - \frac{n_{\rm el}}{2}) \nonumber \\ &+& \sum_{i \neq j} W(r_{i,j}) (\sum_\sigma c_{i,\sigma}^\dagger c_{i,\sigma} - n_{\rm el}) (\sum_\tau c_{j,\tau}^\dagger c_{j,\tau} - n_{\rm el}), \eneqa where $n_{\rm el}$ is the number of $\pi$-electrons per site, $r_{i,j}$ is the distance between the $i$th and $j$th sites, and \beeq % eq 4 W(r) = \frac{1}{\sqrt{(1/U)^2 + (r/a V)^2}} \eneq is the Ohno potential. The quantity $W(0) = U$ is the strength of the onsite interaction, $V$ means the strength of the long range part, and $a$ is the mean bond length. The model is treated by the HF approximation and the single-CI for the Coulomb potential. The bond variables are calculated by the adiabatic approximation. The selfconsistent formalism has been explained in the previous paper [1]. The electric field of light is parallel to the chain which is along with the $x$-axis. The optical absorption spectra are calculated by the formula: \beeq % eq 5 \sum_\kappa E_{\kappa} P (\omega - E_{\kappa}) \langle g | x |\kappa \rangle \langle \kappa | x | g \rangle, \eneq where $P (\omega) = \gamma/[ \pi (\omega^2 + \gamma^2)]$ is the Lorentzian distribution ($\gamma$ is the width), $E_{\kappa}$ is the electron-hole excitation energy, $| \kappa \rangle$ is the $\kappa$th excitation, and $| g \rangle$ means the ground state. The THG is calculated with the conventional formula [7-9], which is sometimes called the sum-over-states method. The expression has been shown as eq. (7) of the previous paper [2]. In order to demonstrate the magnitude of the THG, we use the number density of the CH unit, which is taken from {\sl trans}-polyacetylene: $5.24\times 10^{22} {\rm cm}^{-3}$ [10]. The same value has been used in ref. [2]. We also use $t=1.8$eV in order to look at numerical data in the esu unit. We include a small imaginary part $\eta$ in the denominator, which assumes a lifetime broadening and suppresses the height of the $\delta$-function peaks. The THG at $\omega = 0$ does not sensitively depend on the choice of $\eta$. This can be checked by varying the broadening. Here, we report the results with the value $\eta = 0.02 t$. The system size is chosen as $N= 80$, 100, 120 when the electron number is even (it is varied from $\nel = N, N+2, N+4, N+6, N+8, N+10$ to $N+12$), in order to compare with the results of the previous paper [2]. We change Coulomb interaction parameters arbitrary in a reasonable range in order to look at general properties of the optical nonlinearities of the soliton lattice systems. We take two combinations of the Coulomb parameters $(U,V) = (2t,1t)$ and $(4t,2t)$ as the representative cases. The other parameters, $t = 1.8$eV, $K = 21$eV/\AA$^2$, and $\alpha = 4.1$eV/\AA, are fixed in view of the general interests of this paper. All the quantities of energy dimension are shown in the units of $t$. As experimental developments of the doped conjugated polymers have not been reported so much, we will not compare with experiments. This paper is a pure theoretical work. However, the calculated THG data should be attractive to experimentalists, too. Figure 1 shows the typical lattice configuration and excess-electron density distribution for $N=100$, $\nel = 102$, $U=4t$, and $V=2t$. Both quantities are smoothed by removing small oscillations between even- and odd-number sites. There are two charged solitons due to the excess-electron number $\nel - N = 2$. The two solitons are centered around the 20th and 80th lattice sites. They would be around the 25th and 75th sites in the periodic boundary system. The solitons are moved near to the chain ends slightly. This is the end effects, due to the enhanced bond-alternation strengths near the 1st and 100th sites. The enhanced bond variables pull the two solitons in the direction of the chain ends. The effects have been seen in the earlier paper already [11]. The excess-electron density shows that the doped charges accumulate with the maxima at the soliton centers. In the following, we calculate the optical spectra -- linear absorption and THG -- and consider exciton effects. Figure 2(a) shows the typical optical absorption spectrum at the 2\% soliton concentration for $(U,V) = (4t,2t)$. The broadening $\gamma = 0.05t$ is used. There are two main features around the energies 0.7$t$ and 1.4$t$. The former originates from the soliton-continuum exciton, and the latter is from the intercontinuum exciton. The energy positions of the two main features do not change from those of the periodic system which have been reported in Fig. 1 of ref. [2]. The presence of the chain ends does not effect on the excitation energies so much. However, the oscillator strengths of the soliton-continuum exciton become relatively larger than in the periodic system. This is a new property which is now found in the system with boundaries. Figure 2(b) displays the absolute value of the THG against the excitation energy $\omega$. The abscissa is scaled by the factor 3 so that the features in the THG locate at the similar points in the abscissa of Fig. 2(a). The large feature at about $\omega=0.22t$ comes from the lowest excitation of the soliton-continuum exciton and the other feature at about $\omega=0.26t$ comes from the higher excitations. The features from the intercontinuum exciton extend from $\omega = 0.48t$ to the higher energies. The soliton-continuum exciton gives rise to the large optical nonlinearity, as we looked in the linear absorption. The THG spectra like in Fig. 2(b) are calculated for the three system sizes, $N=80$, 100, 120, and for the soliton concentrations up to 10\%, in order to compare with the calculations with periodic boundaries. As in Fig. 2(b), the off-resonant THG at $\omega = 0$ is quite far from features coming from soliton-continuum and intercontinuum excitons. The contributions from multi excitations [6] would be small. Therefore, the single-CI calculations could be used as a measure of the optical nonlinearities of soliton lattice systems. Figures 3(a) and (b) display the variations of the absolute value of $\chi_{\rm THG}^{(3)} (0)$ for $(U,V) = (2t,1t)$ and $(4t,2t)$, respectively. The plots are the numerical data: the open symbols are for the periodic systems (already reported in ref. [2]), and the closed symbols are for the present calculations of the systems with chain ends. The dashed lines are the guide for eyes for the plots of the periodic systems, showing the overall behavior for each system size. The finite system size effects [12] appear here. It is known that the THG is not size consistent, and spectral shapes depend on the system size when $N$ is as large as 100 [12]. This fact reflects in the apparent separation of the plots of the three system sizes with periodic and open boundaries. The off-resonant THG near zero concentration increases very rapidly, but the THG is still increasing for a few percent to 10\% soliton concentration. The increase between the zero concentration and the 10\% concentration is by the factor about 100. The behavior is common for the two boundary conditions and the Coulomb interaction parameters. The main difference between the two boundary conditions is that the THG with open boundaries is larger than that of the periodic system, when the concentration, system size, and the Coulomb parameters are the same. The similar property has been discussed in the calculations of the half-filled systems [3]. When we look at the enhancement factor in detail, it is found that the enhancement is particularly large for the systems with two solitons. This is seen for the three closed symbols near the 2\% concentration in Figs. 3(a) and (b). These are the cases of the concentrations, 2.5\%, 2.0\%, and 1.67\%, for $N=80$, 100, and 120, respectively. Therefore, if the limit of the long chain is taken, there is a jump of the THG between the zero concentration and the infinitesimal concentration. This is different from the expected smooth behaviors of the periodic chains. The presence of the chain ends results in the trapping of solitons near the ends, and thus the THG jumps suddenly at the zero concentration limit. The dramatic enhancement of the THG in the case of two solitons is clearly seen when we take the ratio of the THG value of the open system with respect to that of the periodic system. The calculations have been done for $(U,V)=(4t,2t)$, and the ratio is shown in Fig. 4. The numerical data are shown by the triangles ($N=80$), circles ($N=100$), and squares ($N=120$), respectively. Except for the three plots near the concentration 2\%, the ratio is almost constant and about 3. It becomes more than 10 for the two soliton systems. Thus, we have found that the case with two solitons is special mainly owing to the chain end effects. It should be stressed that the large THG due to the presence of chain ends could be used as a tool for increasing nonlinear optical responses experimentally. The reason of the large enhancement can be understood as follows. In two soliton solutions with the open boundary condition, the inter-soliton distance becomes longer than in the system with periodic boundaries. This gives rise to the larger expectation values of the dipole moment operators, and thus we obtain the hugely enhanced THG in the system with two solitons. In summary, we have considered the off-resonant nonlinear susceptibility as a guideline of the strength of the nonlinearity in the doped conjugated polymers. We have calculated the off-resonant THG with changing the system size and the soliton concentration for the chains with open boundaries. We have shown that the magnitude of the THG at the 10 percent doping increases by the factor about 10$^2$ from that of the neutral system. The huge increase by the order two is common for the several choices of Coulomb interaction strengths, and is seen in the open systems as well as in the periodic systems. There would be a problem whether the single-CI approximation is correct in the large values of the interaction parameters used in this paper. However, the single-CI has been used by our colleagues. Theory and experiments agree well in several situations [12,13,14]. Therefore, the present calculations can be used as one of tools for THG enhancement. A comparison with more elaborated methods, for example, exact diagonalization calculations of short chains, can be made in future. \mbox{} \noindent {\bf Acknowledgments}\\ The author acknowledges useful discussion with Prof. T. Kobayashi, Prof. S. Stafstr\"{o}m, Dr. S. Abe, Dr. Y. Shimoi, and Dr. A. Takahashi. \pagebreak \begin{flushleft} {\bf References} \end{flushleft} \noindent $[1]$ K. Harigaya, Y. Shimoi, and S. Abe, J. Phys.: Condens. Matter {\bf 7}, 4061 (1995).\\ $[2]$ K. Harigaya, J. Phys.: Condens. Matter {\bf 7}, 7529 (1995).\\ $[3]$ S. Abe, J. Yu, and W. P. Su, Phys. Rev. B {\bf 45}, 8264 (1992).\\ $[4]$ W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. B {\bf 22}, 2099 (1980).\\ $[5]$ K. Ohno, Theor. Chem. Acta {\bf 2}, 219 (1964).\\ $[6]$ V. A. Shakin, and S. Abe, Phys. Rev. B {\bf 50}, 4306 (1994).\\ $[7]$ N. Bloembergen, {\sl Nonlinear Optics} (Benjamin, New York, 1965).\\ $[8]$ B. J. Orr and J. F. Ward, Mol. Phys. {\bf 20}, 513 (1971).\\ $[9]$ J. Yu, B. Friedman, P. R. Baldwin, and W. P. Su, Phys. Rev. B {\bf 39}, 12814 (1989).\\ $[10]$ C. R. Fincher, C. E. Chen, A. J. Heeger, A. G. MacDiarmid, and J. B. Hastings, Phys. Rev. Lett. {\bf 48}, 100 (1982).\\ $[11]$ S. Stafstr\"{o}m and K. A. Chao, Phys. Rev. B {\bf 30}, 2098 (1984).\\ $[12]$ S. Abe, M. Schreiber, W. P. Su, and J. Yu, J. Lumin. {\bf 53}, 519 (1992).\\ $[13]$ Y. Shimoi and S. Abe, Phys. Rev. B {\bf 49}, 14113 (1994).\\ $[14]$ Y. Shimoi, S. Abe, S. Kuroda, and K. Murata, Soild State Commun. {\bf 95}, 137 (1995).\\ \pagebreak \begin{flushleft} {\bf Figure Captions} \end{flushleft} \mbox{} \noindent Fig. 1. (a) The bond alternation order parameter, $(-1)^n (y_{n+1} - y_n)/2$, and (b) the excess-electron density, $(\rho_{n-1} + 2 \rho_n + \rho_{n+1})/4$. The parameters are $U=4t$, $V=2t$, $N=100$, and $\nel = 102$. See the text for the other parameters. \mbox{} \noindent Fig. 2. (a) The optical absorption spectrum and (b) the absolute value of the THG, for the system size $N=100$, the electron number $\nel = 102$, and $(U,V) = (4t,2t)$. The broadening $\gamma = 0.05t$ is used in (a), and $\eta = 0.02t$ is used in (b). The absorption is shown in the arbitrary units, and the nonlinear optical response is in the esu unit. \mbox{} \noindent Fig. 3. The absolute value of the THG at $\omega = 0$ v.s. the soliton concentration for (a) $(U,V) = (2t,1t)$ and (b) $(4t,2t)$. The numerical data are shown by the triangles ($N=80$), circles ($N=100$), and squares ($N=120$), respectively. The data of the system with the periodic boundary condition are shown by the open symbols. And, the data with the open boundaries are shown by the closed symbols. The dashed lines are the guide for eyes. \mbox{} \noindent Fig. 4. The ratio of the absolute values of the THG between the open and periodic boundary conditions, shown against the soliton concentration. The Coulomb parameters are $(U,V) = (4t,2t)$. The numerical data are shown by the triangles ($N=80$), circles ($N=100$), and squares ($N=120$), respectively. \end{document}