5.1 Molecular Orbital Method

In order to design EO polymer wires, the molecular orbital (MO) method is used. In the present section, the MO method is briefly reviewed.

Figure 5.1 shows models for Hamiltonian in the MO calculations. In this example, the system consists of N electrons. The Hamiltonian of the whole system is written as follows.

Here, i(i = 1,2, … , N) and j(j = 1,2, … , N) represent i^th electron and j^th electron, respectively. q_i is the position of the i^th electron, q_j is the position of the j^th electron, Δ_i is the Laplacian for the i^th electron, V(q_i) is the interaction energy between the i^th electron and nuclei, m is the mass of an electron, and e is the charge of an electron. The second term represents interaction energy arising from the Coulomb force between electrons.

A one-electron Hamiltonian for the i^th electron is expressed as follows:

Here, ϕ_j(q_j) is the wavefunction of the j^th electron. ϕ_j(q_j) is called the molecular orbital. The second term represents the interaction energy between the i^th electron and the j^th electron, where the j^th electron is regarded as an electron cloud. The interaction energy is calculated as an expected value determined by using the charge density distribution of the j^th electron, eϕ_j^*(q_j) ϕ_j (q_j).

Schrödinger equations for the whole system Hamiltonian and the one-electron Hamiltonian are respectively given as follows:

Here, Ψ is a wavefunction of the whole system.

The molecular orbitals can be expressed as linear combinations of wavefunctions in atoms (atomic orbitals) χ_r.

An example of the relationship between ϕ_i and χ_r is schematically shown in Figure 5.2 for a case of C₂H₂. ϕ_i is built up by χ_r with amplitude of c_ir. c_irs are determined by the variation principle, namely, by finding a set of c_irs that minimize ε_i in the following formula.

The minimized ε_i gives the energy eigen value for the molecular orbital of ϕ_i = Σ_rC_irχ_r. In a system with N electrons, N molecular orbitals are derived. However, in real systems, two spin states (α, β) are involved for one molecular orbital, resulting in 2N molecular orbitals. So, we should renumber the molecular orbitals including spin states as ϕ_i(i = 1,2,…,2N).

Figure 5.3 shows schematic illustrations for energy levels of molecular orbitals and electron occupation in many-electron wavefunctions built up from the molecular orbitals. The many-electron wavefunctions represent the actual electronic state of the whole system. The wavefunction for the ground state is expressed as follows,

where, P^ is a permutation operator and p is the number of permutations. Electrons occupy molecular orbitals ϕ₁, ϕ₂,..., and ϕ_N.

In the 0^th approximation, many-electron wavefunctions for the excited states are written as follows

where, i→j represents that an excited state, where an electron is excited from ϕ_i to ϕ_j. Φ_i→j is called the configuration function.

In order to improve the accuracy of the molecular orbitals obtained previously, the variation principle is again applied to the following equation.

Then, a set of improved c_irs that minimize E are obtained. The minimized E gives the ground state energy E_g for Ψ_g.

In general, the excited states are expressed as linear combinations of Φ_i→j as follows.

C_{n, i→j} s are determined by the variation principle, namely, by finding a set of C_{n, i→j} s that minimize E_n in the following formula.

The minimized E_n gives the energy eigen values for the excited states of Ψ_n = Σ_i,jC_n,i→jΦ_i→j.

Thus, wavefunctions and energy eigen states are obtained.

5.2 Nonlinear Optical Effects

Nonlinear optical phenomena can be understood using the spring analogy shown in Figure 5.4. In a spring, displacement x is proportional to external force F for small F. With increasing F, the spring stretches rapidly with a nonlinear relationship between x and F. Nonlinear optical phenomena can be similarly described by replacing x with polarization P and F with electric field E as follows.

Here, ε₀ is the dielectric constant in vacuum, χ⁽¹⁾ is linear susceptibility, and χ⁽²⁾ and χ⁽³⁾ are nonlinear susceptibilities. When E is small, P is proportional to E.

With increasing E, the contribution of the E² or E³ term becomes dominant. The E2 and E³ terms induce second-order and third-order nonlinear optical effects, respectively.

E can be expressed by a sum of electric fields of light with an angular frequency of ω, E_opt = Ee^–jωt, and an externally applied electric field, E_DC, as follows:

The term E² gives E_opt²,E_DC,E_opt,E_DC² which are origins of the second-order nonlinear optical effects, and the term E³ gives E_opt³,E_opt²,E_DC,E_optE_DC²,E_DC³ which are origins of the third-order nonlinear optical effects. When we focus on the term E_opt², Equation (5.12) can be written as

Polarization oscillating at 2ω generates light with angular frequency of 2ω, that is, the second harmonic generation (SHG) arises. When we focus on the term E_opt³, Equation (5.12) can be written as

Polarization contains a 3ω component, generating light with angular frequency of 3ω, that is, the third harmonic generation (THG) arises.

When we focus on the term E_DCE_opt, Equation (5.12) can be written as

Since the refractive index, n, is a function of ε₀χ⁽¹⁾ + ε₀χ⁽²⁾ E_DC, n can be expressed as follows.

Considering the relationship of ε₀χ⁽¹⁾ ≫ ε₀χ⁽²⁾ E_DC, n is rewritten using a Taylor expansion as

By putting n₀ = f(ε₀χ⁽¹⁾) and r ∝ f′(ε₀χ⁽¹⁾)ε₀χ⁽²⁾

is obtained. This formula implies that the refractive index changes linearly with an applied electric field, which is the Pockels effect. r is the Pockels coefficient.

When we focus on the term E_DC²E_opt, Equation (5.12) can be written as

As in the Pockels effect,

is derived. This implies that the refractive index changes in proportion with E², which is the Kerr effect. R is the Kerr coefficient. The Pockels effect and the Kerr effect are used for various kinds of EO devices such as light modulators, optical switches, tunable filters, and so on.

The relationship between E and P can be translated into the electronic potential energy vs. polarization curve as Figures 5.4(b) and (c) show. When E and P are linearly related, the potential energy curve is parabolic. This corresponds to the fact that the potential energy of a spring is proportional to x². When nonlinear terms predominate, the potential energy curve deviates from the parabola.

In Figure 5.5, various kinds of nonlinear optical effects are drawn visually. SHG can be understood by the illustration shown in Figure 5.5(a). Due to the term E² in Equation (5.12), the potential energy vs. polarization curve becomes asymmetric. Therefore, when light with angular frequency of ω is introduced, asymmetric responses of P are induced, resulting in a P component with 2ω. The P oscillating at 2ω emits light with angular frequency of 2ω. For THG, due to the term E³ in Equation (5.12), the potential energy vs. polarization curve deviates from the parabola symmetrically as shown in Figure 5.5(b). Then, when light with an angular frequency of ω is introduced, the P component with 3ω is induced, emitting light with an angular frequency of 3ω.

For the Pockels effect, in which the refractive index changes linearly with an applied electric field E_DC, the potential energy vs. polarization curve deviates from the parabola asymmetrically as shown in Figure 5.5(c). When light is introduced, in the model shown in Figure 5.5(c), a larger P disturbance, ΔP, is induced for E_DC > 0 than for E_DC < 0 since the slope of the curve is smaller for E_DC > 0 than for E_DC < 0. Large ΔP tends to give rise to a large refractive index. Thus, with an increase of the applied electric field, the refractive index increases.

Measures of second-order and third-order optical nonlinearities for a molecule are respectively given by molecular second-order and third-order nonlinear susceptibilities β and γ. For aggregates of many molecules, macroscopic measures of second-order and third-order optical nonlinearities can be given by χ⁽²⁾ originating from βs and χ⁽³⁾ originating from γs. EO coefficient r, which is proportional to χ⁽²⁾, is used as the measure for the Pockels effect.

In general, the potential curve deviation becomes large in highly polarizable systems. Figure 5.6 schematically illustrates electron-cloud polarizations induced by an electric field. In dielectric materials such as lithium niobate (LN), electronic polarization is not large because electrons are localized near atoms, regarded as a strong spring. In semiconductors, electron clouds are widely spread, regarded as a weak spring. Therefore, compared to dielectric materials, large electronic polarization is attainable in semiconductors. Organic materials, especially polymers with long conjugated wires, have electrons delocalized one dimensionally, that is, they are regarded as natural quantum wires. The one-dimensional characteristics are favorable for efficient electronic polarization induced by an electric field in the wire direction. This is why organic materials seem promising for nonlinear optical devices.

5.3 Procedure for Evaluation of the eo Effects by the Molecular Orbital Method

Molecular second-order nonlinear susceptibility, or hyperpolarizability in the wire direction, is shown in Equation (5.22) based on Ward’s expression [4,5]

for the Pockels effect, and

for SHG.

In Equations (5.22) and (5.23), r_gn is the transition dipole moment between the ground and excited states and r_n’n is that between two excited states. Δr_n is the difference in the dipole moment between the excited and ground states. ħω_ng is the excitation energy from the ground state to the excited state. ħω is the energy of incident photons.

r_gn, r_n’n, and Δr_n are expressed by the following equations,

where, Ψ_g is the wavefunction of the ground state and Ψ_n is the wavefunction of the excited state. These wavefunctions are expressed by Equations (5.7) and (5.10), and are constructed by molecular orbitals ϕ_i. Then, r_gn, r_n’n, and Δr_n are expressed as follows [5].

Here, the primed summations indicate to exclude terms, in which i = k and j = l simultaneously. r_gn, r_n’n, and Δr_n can be calculated from Equations (5.27) through (5.30) once a set of molecular orbitals are obtained by the MO method. Using the r_gn, r_n’n, and Δr_n, β can be calculated from Equations (5.22) and (5.23).

In the present simulation [1–3], the Austin Model 1 (AM1) method, a semi-empirical molecular orbital method was used for calculations of molecular orbitals. AM1 was developed by modifying the core repulsion function in the modified neglect of the diatomic overlap (MNDO) method. Using these molecular orbitals, the ground state wavefunction Ψ_g, a product of the occupied orbitals, and configuration functions, Φ_i→j, a product of the orbitals with one-electron excitation from occupied orbital ϕ_i to unoccupied orbital ϕ_j were made. Excited states Ψ_n, expressed by linear combinations of configuration functions as Equation (5.10), were determined by single-excitation configuration interaction (CIS). CI calculation involved eight unoccupied orbitals above the lowest unoccupied molecular orbital (LUMO) and six occupied orbitals below the highest occupied molecular orbital (HOMO). 2-methyl-4-nitroaniline (MNA) was used as the standard molecule. From the calculation, β of 13 × 10⁻³⁰ esu for SHG with a fundamental wavelength of 1.06 µm was obtained. This agrees fairly well with the 12 × 10⁻³⁰ esu reported by Garito et al. [6].

5.4 Qualitative Guidelines for Improving Optical Nonlinearities

Figure 5.7 shows the microscopic origins of second-order and third-order optical nonlinearities. As mentioned in Section 5.2, the potential energy curve in linear systems is parabolic. In nonlinear systems, however, it is not. This potential curve deviation from the parabola induces a variety of nonlinear phenomena. Second-order optical nonlinearity arises from the asymmetric potential curve deviation (see Figure 5.7(a)). Third-order optical nonlinearity arises from the symmetric potential curve deviation (see Figure 5.7(b)). The larger the potential curve deviation becomes, the larger the induced nonlinear optical effect becomes.

The driving force for this potential curve deformation is explained in terms of the shape of the wavefunction as shown in Figures 5.7(a) and (b). Consider the ground state g and one excited state e, that is, a two-level model, for simplicity. When perturbation is induced in a molecule by incident light or electric field, mixing of the excited state wavefunction into the ground state wavefunction occurs, deforming the potential curve. Therefore, the first approach to improving optical nonlinearity lies in promoting the mixing rate of ground and excited states, that is, promoting the oscillator strength, f, between the ground and excited states. Here, f ∝ r_ge², which can be increased by promoting the wavefunction overlap between the ground and excited states.

However, increasing only the wavefunction mixing rate is not in itself sufficient to increase the potential curve deformation. For example, when the ground and excited state wavefunctions have a similar shape, mixing the wavefunctions results in only a small change in electron distribution. Consequently, there is only a small potential curve deformation. Thus, another approach to improving optical nonlinearity is promoting the wavefunction difference between the ground and excited states. From the arguments mentioned above, the qualitative guidelines described in subsections 5.4.1 and 5.4.2 are derived.

5.5 Enhancement of Second-Order Optical Nonlinearity by Controlling Wavefunctions

The effect of wavefunction shapes of the excited state (n) and the ground state (g) on r_gn , Δr= r_n−r_g, and β is schematically illustrated in Figure 5.8. From left to right, wavefunction separation between the excited and the ground states increases while wavefunction overlap between the two states decreases, that is, Δr increases and r_gn decreases in a trade-off relationship. According to Equation (5.31), in the two-level model, β is expressed by β ∝ r_gn² Δr . So, β becomes the maximum in the middle wavefunction shape. Both the wavefunction separation and wavefunction overlap must simultaneously be considered to optimize β.

Furthermore, as described in Section 5.5.2, Δr and r_gn depend on the dimensionality of the conjugated systems. Therefore, it is concluded that β is expected to have its maximum in a wavefunction shape of intermediate wavefunction separation with an appropriate conjugated system dimensionality existing somewhere between zero and one, that is, between the quantum dot condition and the quantum wire condition.

5.6 ENHANCEMENT OF THIRD-ORDER OPTICAL NONLINEARITY BY CONTROLLING WAVEFUNCTIONS

To improve the Kerr effect, which is one of the third-order nonlinear optical effects, the shape of the wavefunction was controlled by constructing quantum dot structures in one-dimensional conjugated systems [3] according to the qualitative guidelines mentioned in Section 5.4. Figures 5.20(a) and (b) give an example of a quantum dot structure and corresponding wavefunctions. Varying the dot length in the conduction band (ML) and that in the valence band (W) adjusts the wavefunction spread difference between the ground and excited states.

D is the dot width. Reducing D changes the system from two-dimensional to one-dimensional. γ per unit length for the light polarization with the well length direction was calculated from Equation (5.33). For simplicity, it was assumed that the ground state wavefunction has even parity, the excited state wavefunction odd parity, and wavefunctions are square waves. The results are shown in Figure 5.20(c). With reduced D, both (r²)_e-(r²)_g, and r_ge² increase, then γ increases rapidly. This clearly indicates that the one-dimensional system is better than a two-dimensional system for inducing large optical nonlinearity, as mentioned in Section 5.2. That is, a wavefunction extending perpendicular to the direction of light polarization contributes little to optical nonlinearity, simply diluting the wavefunction density and reducing optical nonlinearity.

With increasing W, that is, increasing the overlap and decreasing the difference in wavefunction spread between the ground and excited states, r_ge² increases and (r²)_e-(r²)_g decreases, with γ peaking at an intermediate region in W.

Figures 5.21(a) and (b) show a structure and wavefunctions for a double quantum dot (DQD) with subdots at both sides. The wavefunction spread difference is controlled by the difference in dot depth between the conduction band and the valence band. With increasing dot depth, the wavefunction tends to localize on both sides. The calculated values of γ, (r²)_e-(r²)_g, and r_ge² are shown in Figure 5.21(c) as a function of R = C/D. R is a measure of the wavefunction extent in the valence band as defined in Figure 5.21(b). With increasing R, (r²)_e-(r²)_g increases and r_ge² decreases. γ peaks at an intermediate region in R of about 1 when the balance between the wavefunction overlap and the wavefunction spread difference is optimum. In this case, γ is improved about five times over that for a single quantum dot (SQD). It was also found that further confinement of electrons in three or more subdots would make it possible to enhance γ more than ten times.

5.7 Multiple Quantum Dots (MQDS) in Conjugated Polymer Wires

To apply the tailored materials described in Sections 5.5 and 5.6 to practical devices, it is necessary to fabricate the materials in ordered structures. This involves constructing MQD structures in one-dimensional conjugated polymer wires. One way to do this is to insert molecules for cutting the π-conjugation in polymer wires as described in Chapter 4.

The author found another way to form quantum dots [3]. It was simulated by the MO calculation that the energy gap of conjugated polymer wires with PDA backbones is reduced when donors and acceptors are substituted in the DAAD configuration. Figure 5.22 shows one of the examples. The energy gap is decreased from 3.3 eV of the intrinsic PDA wires to 1.6 eV by donor–acceptor substitutions. The phenomena can be used for MQD construction.

The mechanism of the energy gap narrowing in conjugated polymer wires by donor and acceptor substitution is as follows. From the viewpoint of charge distributions, it is found from Figure 5.23 that electrons are localized around donor sites in HOMO while they are localized around acceptor sites in LUMO. The electron distribution in HOMO corresponds to hole distributions in HOMO when the electron is excited. So, with decreasing the distance between donors and acceptors, electron–hole distance becomes short, increasing electric fields in the area, resulting in a large potential energy slope as shown in Figure 5.24. Then, energy gaps become narrow.

In Figure 5.25, DAAD18 and PDA with an inserted DAAD18 structure are compared. Although wavefunctions slightly penetrate into the barrier regions by tunneling in the conjugated polymer wire with an inserted DAAD18 structure, the molecular orbitals and β are almost the same for both. Therefore, using this effect, it will be possible to insert many DAAD structures into the conjugated polymer wire, constructing a polymer MQD of a polymer superlattice, as shown in Figure 5.26. This structure enables us to align wavefunctions of DAAD structures in the wire direction perfectly, which is favorable to attaining a large Pockels effect.

As mentioned above, using energy gap narrowing, it should be possible to form quantum dots in one-dimensional conjugated wires by modulating the energy gap with selective substitutions of donors and acceptors into the wires. A DQD structure with two 1.4-nm long dots is shown in Figure 5.27 together with an SQD structure. Quantum dots are formed in a PDA wire with 38 carbon sites. Areas with donor and acceptor substitution are quantum dots, and areas with hydrogen substitution are barriers. The energy gap in the quantum dot region is about 1.6 eV, about one half of that in the barrier region. Molecular orbitals show that electrons are spread throughout the conjugated wires in the SQD. In the DQD, electrons tend to be confined to both sides of the wire. Here, note that although electrons are localized on both sides in all E1 , E2, and E3 unoccupied orbitals, they are in the barrier region in the G3 occupied orbital. This suggests that the wavefunction tends to be delocalized throughout the wire more in the valence band than in the conduction band, reproducing the condition in Figure 5.21.

γ of DQD and SQD were calculated by the AM1 MO calculation. As expected, γ in the DQD is more than twice that in the SQD. This confirms that it may be possible to improve third-order optical nonlinearity by adjusting the well structures.

Research on artificial materials, like PDA, poly-AM or other conjugated systems with MQDs, has far-reaching consequences. If a way is developed to connect a molecule on a molecule one by one with direction control, the donor and acceptor substitution locations and the conjugated length could be controlled. MLD is the most promising method for making artificial materials. Figure 5.28 shows possible configurations of EO waveguides consisting of artificial materials fabricated by MLD.

References

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