Vibronic Coupling in Quantum Wires: Applications to Polydiacetylene

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1 Vibronic Coupling in Quantum Wires: Applications to Polydiacetylene An Exhaustively Researched Report by Will Bassett and Cole Johnson Overall Goal In order to elucidate the absorbance spectra of different polymer semiconductors such as Polydiacetylene (PDA), Yamagata and Spano have put forth a theoretical study including not only the electronic transitions, but also their coupling to vibrational modes to yield vibronic excitons 1. This requires determination of the one dimensional direct band gap energy and structure. These polymers are long linear chains that are treated as quantum wires, simplifying calculations by reducing them to one dimensional problem analogous to the one dimensional particle in a box. Using the 2-MO approximation, to be described later, simplifies the calculations more as well as improves cost. The main focus of this study is to model the vibronic excitons and determine the energy loss from coupling a vibrational mode to an electronic mode for energy dissipation. In this paper the model for Wannier-Mott excitons was improved using the multi-particle basis set for Frenkel excitons, including the charge transfer states, and incorporating linear vibrational modes. From an emission model, the ratio of intensities of the first two vibronic bands formed from an exciton collapsing back into its hole and producing a photon provides information about exciton coherence as well as the effective mass. To further understand the quantum chemistry involved in the described paper, we will detail many quantum phenomena including: excitons, the 2-MO approximation, the Hamiltonian constructed for this model, the basis sets used, the absorption and emission models, the line strength ratio, phonons, k space, and aspects of vibronic coupling in general. Exciton In the simplest description, an exciton is the particle created upon exciting an electron to form an electron-hole pair. The Frenkel model dictates that the electron-hole pair is constrained to the same unit cell where the HOMO/LUMO of each unit cell is weakly coupled to neighboring HOMO/LUMOs. Another way to describe this is to define the Bohr radius of a Frenkel exciton as smaller than the lattice spacing in a crystal where the Bohr radius of an exciton is the average distance between the electron and hole. The coupling of the exciton between two adjacent unit cells is defined as the off-diagonal Hamiltonian elements of each HOMO/LUMO to adjacent HOMO/LUMOs and so a weak coupling ensures the orbitals are far enough apart spatially to prevent the electron from hopping to another unit cell. Conversely, the Wannier-Mott model for an exciton allows for electron-hole hopping to other unit cells, further and further away depending on the magnitude of the transfer integral to be described later. The Bohr radius of Wannier-Mott excitons is therefore on the same order or larger than the lattice spacing, which allows coupling. The two models for excitons are shown in figure 1. In the present study an intermediate coupling Figure 1 Left: An electron-hole pair confined to a single unit-cell. Middle and Right: An electron-hole pair in a charge seperated state with the hole and electron at middle and right respectively.

2 strength is used for the Wannier-Mott exciton and weak coupling for the Frenkel exciton with these strengths to be described later. The simplest exciton is purely electronic in that the formation of an electron-hole pair is simply the excitation of electron from a ground state to an excited electronic state. More correctly, excitons can couple to nuclear vibrations in vibronic transitions, thus removing the applicability of the Born-Oppenheimer approximation. The vibronic transitions allow for non-adiabatic relaxations from excited states to both the ground vibrational and electronic state. Vibronic Transitions The reason we must take into account the coupling of the vibrational and electronic modes of these molecules is that for conjugated polymers, electronic transitions can result in large molecular shifts. Consider the example given in figure 2 for polyacetylene, where the electronic transition from the ground state to the first excited state effects the pi-electron density, and thus shifts the molecule 2. These shifts alter the overlap of the nuclear potential energy surfaces between ground and excited states, which in turn changes the way electronic transitions occur. When this potential shifts, the probability for each electronic transition shifts due to the ability for vertical transitions between ground state vibrational modes of excited states and the ground state to occur, as will be described later in the absorption and emission model section. Thus the vibrational modes of the polymer crystal strongly affect the electronic transitions. For electrons to relax from the excited electronic state, the entire crystal must have a collective vibration to return to ground state geometry, known as a phonon. This coupling of the relaxation to a phonon is what makes it a vibronic transition. 2-Molecular Orbital Approximation Figure 2 Electronic effects on the geometry of polyacetylene. In the model used, a polymer is treated as a one-dimensional semiconductor in which each unit cell is represented by only it s HOMO and LUMO as in figure 3. This approximation is known as the two-molecular-orbital approximation 3 and is used here to greatly simplify calculations. The dominant excitonic transition will be long axis polarized therefore the two-state model will include all necessary components. In principal what this means is that out of the two excitons created, the 1 1 B u - and the 1 A g +, the 1 1 B u - exciton is the only significant contributor due to being created along the axis of the molecule. In solids the most reasonable HOMO and LUMO to use Figure 3 HOMO and LUMO for each unit cell.

3 for each unit cell are not the typical MOs, but rather Wannier functions, which are crystal orbitals. This function includes an extra phase factor to account for the periodicity of the crystal as described in the section on k-space. Using only the HOMO and LUMO makes intuitive sense because the excitation from HOMO to LUMO is the lowest energy and most probable transition. Model Frenkel Hamiltonian The simple Hamiltonian for describing a Frenkel exciton is constructed by simplifying the more complex Hamiltonian describing a Wannier-Mott exciton. The Wannier-Mott Hamiltonian is made up of the summation of an electronic and vibrational component. The electronic Hamiltonian is given in equation 1: (1) where c and d represent the creation and annihilation operators for the hole electron and hole respectively, where the double dagger notates a creation operator and the lack of double dagger is the annihilation operator and t represents the transfer integral. The h.c. in the first summation stands for higher corrections such as the component given by separating the electron and hole more than a single unit cell. The transfer integrals give the overlap between each HOMO and LUMO and its adjacent orbital and give a measure of the energy required for an exciton to jump from one orbital to another and is given by equation 2: (2) where Ĥ is the full Hamiltonian. The values of the transfer integral divide the interactions between unit cells into one of two classifications: weak and intermediate coupling. The weak coupling is defined as when the transfer integral is much less than the potential difference between the unit cell where the exciton resides and neighboring unit cells, and the intermediate coupling is when the transfer integral is roughly equal to the same potential difference as elucidated in equation 3. { (3) The potential interaction in the polymer crystal is given in equation 4: { (4) where U > 0, V 1 > 0, and U > V 1, U is the potential within the unit cell, and V 1 is related to the dielectric constant of the material. Thus e e + e h + V(0) is the energy required to create the electron/hole pair in a unit cell. At V(0) the terms in the second summation simply become the potential within unit cell n times the number operator used to count the electrons and holes in that unit cell.

4 To describe the number operator which is defined as the product of creation and annihilation operators,, we use the harmonic oscillator for an example in equations 5-7: (5) (6) (7) where clearly the result is counting the number of vibrational quanta in this case, or in our case counting the electrons and holes. The basis sets used herein only ever involve the creation of one electron/hole pair and so the number operator always equals one. The vibrational component of the Hamiltonian is given in equation 8 (note: ħ=1). { ( ) } { ( ) ( ) } (8) The symmetric vinyl stretching mode is the only considered mode and has a frequency of ω 0 = 1390 cm -1. The new operators represented with b are associated with the creating and destroying a vibrational quanta in the ground state nuclear potential well corresponding to the absence of electrons and holes in unit cell n. The second term in the summation corresponds to the nuclear stretch when an electron/hole pair is created in nth unit cell (The number operators on the end ensure this, acting as Dirac delta functions) and the third term summation accounts for the same stretching mode when in a charge separated state. The stretches are characterized by Huang-Rhys (HR) factors associated with the neutral unit cell, cationic unit cell, and anionic unit cell given by λ 0, λ +, λ - respectively as shown in figure 4. All nuclear potentials are defined to be harmonic wells with symmetric and equal curvature. The shifts are parameterized to account for nuclear shifts and correspond to experimental values calculated from the vinyl stretching mode. The parenthetical term (b ǂ n + b n ) is known as the position operator that when multiplied by the nuclear shift parameter defines shift of the nuclei in unit cell n during the excitonic transition. Figure 4 The Huang-Rhys factors and corresponding nuclear states. The much simpler Hamiltonian used to describe Frenkel excitons eliminates all components from the charge separated states and combines the electronic and vibrational Hamiltonians into a single Hamiltonian as seen in equation 9. ( ) (9) J mn is a matrix representing couplings between excitons in unit cells m and n, and the creation and annihilation operators B ǂ and B are for Frenkel excitons. The diagonal components of the J mn matrix are the energies of the excitons, and the off-diagonal elements correspond to the transfer integrals between unit cells m and n. The energies for the exciton, as stated before, are just the summation of the energies of the electron, hole, and potential V(0). The number operator again

5 makes an appearance counting both the number of vibrational quanta (b) and the number of excitons (B). The latter ensuring there is only a nuclear shift when there is actually an excitonic transition in unit cell n. Basis Set The set of basis functions used to determine band structure are a combination of five different types of basis functions as seen in Figure 5. The five types are, in order of increasing complexity: the One-Particle Frenkel- State (F-state), Two-Particle F-state, Three- Particle F-state, Charge-Separated Exciton, and Charge-Separated/ Vibrational Exciton. A particle is described as a vibronic or purely vibrational excitation. As you can see in the Figure 5 The five types of basis functions. Three-Particle F-state in figure 5, the n-1 orbital contains a vibronic exciton where the n and n+1 orbitals contain purely vibrational excitations. Similarly the charge-separated exciton contains an orbital with the electron in the ground vibrational state and the hole in a vibronic state. The F-state basis functions represent the Frenkel excitons in that the electron/hole pair is confined to a single unit cell, and the charge separated states compensate for possible Wannier-Mott behavior of the exciton where the electron can migrate from the hole. In order to use the Hamiltonian to completely describe the system, we must rigorously define these basis functions in equations (10) In the equation above we have n as the unit cell, as the number of vibrational quanta in the excited state potential (S 1 ), is the ground electronic state being operated on by creation operators to form an exciton in the same unit cell as the vibrational quanta as shown by the direct product between the electronic state and the vibrational ket. This ensures electronic and vibrational coupling since our basis functions are now a product of electronic and vibrational states. When we extend this basis set to include the vibrationally coupled exciton and one or two pure vibrational excitations, we define other unit cells p and r as well as the ground state vibrational modes ν p and ν r to generate the two and three particle F-state basis functions. (11) (12) In addition to basis functions to describe Frenkel excitons, we need to include basis functions for charge transfer states, with and without additional vibrational quanta. The vibrational quanta in charged states are in the potential states S - and S + for the electron and hole respectively. These potential states are then associated with the λ - and λ + Huang-Rhys factors respectively.

6 (13) (14) The basis sets provided here only go up to two additional vibrational quanta for F-states and one for charge separated states and so is not a complete set. The complete set would require additional basis sets containing more vibrational quanta until you have a vibrational quanta on every unit cell. The number of basis functions increases dramatically as you increase vibrational quanta by N!/(N-ν)!ν! where ν is the number of quanta and N is the number of unit cells. So even with a relatively low number of unit cells such as 100, to account for two additional vibrational quanta requires roughly a million basis functions. The use of a limited number of basis functions is validated by experimental verification that optical responses in weak to intermediately coupled systems are reproducible with high accuracy. This is due to the oscillator strength, and thus the majority of optical responses, primarily being a function of the one-particle F-state. The Hamiltonian is factorable into blocks representing each unit cell and energies of the eigenstate, k and α respectively. k represents the momentum of the crystal and α represents the electronic state. Thus the lowest energy exciton occurs in the k=0 block when α=1. Thus the eigenstate in the k th block is denoted and has energy ε kα for harmonic oscillators where the again the frequency is ω 0 = 1390 cm -1 corresponding the symmetric vinyl stretch. This exciton, as described above, is the 1 1 B u exciton. A non-zero transfer integral is associated mostly with the Wannier-Mott exciton in that the Frenkel exciton is essentially confined to a unit cell by having such a low energy transfer integral relative to the potential between wells, but this does not completely bar a Frenkel exciton from migrating. As shown in figure 6, even with a very small value transfer integral relative to the potential between unit cells, a migration has a small possibility due to quantum effects. As is to be expected, a larger energy for the transfer integral allows greater migration of the hole due to the transfer energy being comparable to the energy required for that transfer. Absorption and Emission Model Figure 6 Probability of an exciton to move to a charge separated state based on transfer integral energy. U = 2eV, V 1 = 1eV. Once the basis sets and Hamiltonian have been described, the authors move to calculate the eigenstates and energies so that absorption and emission can be modeled. These models are used to calculate relative line strengths between peaks in the spectra, which are utilized for determining exciton coherence and effective mass. In the absorption model we neglect the possibility that a unit cell is not in its ground vibrational state because the energy required to be in an excited state is generally higher than thermal energy. The absorption model is described first in equation 15.

7 ( ) ( ) (15) This equation is described by the long-axis polarized dipole moment associated with the S 0 S 1 transition along the vinyl group, μ, the dipole moment operator shown in equation 16, (16) the vibrationless ground state of the polymer shown in equation 17, (17) and the line shape function shown in equation 18, where the gamma function is chosen to mediate line width and is taken as Γ=0.3ω 0 for all measurements herein. In order to define the emission model, we must first define the line strength which depends on the number of vibrational quanta in all unit cells, which are accessed after an electron has been promoted to an excited state seen in equation 19. (19) As you can see the line strength looks similar to the absorption model with the exception that you now have access to the vibrational quanta. As before, the sum of all ν n must equal ν t. With this line strength we can now define the emission spectra in equation 20: (20) where the total emission model differs from the absorption model in three ways. The line strength depends on the vibrational quanta rather than the vibrationless ground state, the third term in the line strength function accounts for a stokes shift, and the emission must be thermally averaged to account for accessibility of all thermally excited states. This thermal average can be seen in equation 21 (18) (21) where Z is the standard partition function Z Σ k,α and beta is 1/kT. The line strength ratio represents the sum of all possible transitions from the state k,α decaying to the ground electronic state with ν t vibrational quanta. Hence the line-strength for the decay of any state to the vibrationless ground state of the ground electronic state as seen in figure 7 can only occur for the case when k=0 creating the 1 1 B u exciton. This is important because it defines the 0-0 transition as uniquely dependent on a single exciton, allowing studies on that exciton with confidence. As shown in figure 7, the 0-2 transition would be the transition from any excited state to the ground electronic state requiring relaxation of two vibrational modes to arrive at the ground vibrational state. As the nuclear potentials shift and increase the overlap of the ground

8 vibrational state of the excited and ground state, other transitions become less likely. The probabilities of each transition, and thus their intensity, are due to the nature of the vibronic transitions which is why we must include the coupling between vibrational modes and electronic when dealing with conjugated polymers as described in the above section on vibronic transitions. Plotting the absorption and emission for the case of weak and intermediate coupling as a function of unit cells reveals the dependence of the emission on the number of unit cells as seen in figure 8. The absorption in the Frenkel model (weak coupling) is relatively unaffected and the Wannier- Mott (intermediate coupling) shows a dependence due to charge transfer states being allowed. In the calculations used to generate the intensity plots in figure 8, several parameters are used: U=2eV, V 1 =1eV,,. The x-axis in the figure is a function of the frequency minus the energy required to create the exciton and normalized to the frequency of the symmetric vinyl stretch. It is worthy to note the large red shift in the intermediate coupling case as a function of the number of unit cells whereas the red shift in the weak coupling case is miniscule due to the ability of the exciton to be more diffuse as the coupling between unit cells increases. Figure 8 The 0-0 emission as noted by the dottod lines. The 0-2 transition is shown by the green line. k-space In order to fully understand the Figure 7 Absoprtion spectra (black) and emission spectra (red) at T=0K. The spectra exactly overlap, but the absorption has been manually blue shifted for clarity.

9 intensity plots and their implications, we must investigate the use of k-space, commonly used in crystallographic studies. In k-space, k is the momentum of the crystal and defines the phase between the Wannier orbitals of each unit cell according to equation 22. (22) The phase factor accounts for rotations in k-space of each unit cell relative to the adjacent unit cells, where the phase in each unit cell is dependent on the crystal momentum. For a vibrational wave function where k=0 all vibrations are in phase. The result of this is that only optical phonons exist, as opposed to acoustic phonons 4. k-space requires boundary conditions such that when you move a distance equal to the lattice spacing, a through a crystal you arrive at the same spatial coordinate. Line Strength Ratio The line strength ratio of the 0-0 to 0-1 transitions is shown at T=0K to be approximately linear with the number of unit cells and is corrected by kappa, where kappa can be calculated from equation 23 which was demonstrated in a previous publication 5 and values the line strength ratio calculated in figure 8. Calculating thermodynamic parameters as a function of the number of unit cells, N, has been shown in a previous paper through the Boltzmann thermal average and integrating in k-space from -π to π to give equation 24 at the thermodynamic limit of high temperature and unit cell count, (23) (24) where ω c is the curvature of the 1B u exciton band which is given in equation 25 and is outside the scope of this paper. Theoretical calculations of the line strength ratio based on the plotted emission at T=0K and the thermodynamic change of the line strength, experimental values have been accurately reproduced. This line strength ratio is instrumental in calculating the coherence of the 1B u exciton and its effective mass, both of which are very important properties of these sorts of systems but will not be covered in this discussion. Conclusions The theoretical study put forth by Yamagata and Spano has developed an improved Hamiltonian and basis set used to study vibronic transitions in long conjugated polymers, referred to as quantum wires. Extending the Frenkel exciton model to include charge separation as in the (25)

10 Wannier-Mott model, a more accurate system is used to simulate absorption and emission spectra. From these simulated spectra and previous knowledge on the temperature dependence of the system, line strength ratios are calculated at a variety of temperatures, which are used to calculate the effective mass and coherence of the lowest energy exciton which is responsible for these transitions. References (1) Yamagata, H.; Spano, F. C. The Journal of Chemical Physics 2011, 135. (2) Research, C. f. M. a. D. f. I. T. In Photonics Wiki; University of Washington: (3) Barford, W.; Bursill, R. J.; Smith, R. W. Physical Review B 2002, 66, (4) Kittel, C. Introduction to Solid State Physics; 7th ed.; John Wiley & Sons, (5) Spano, F. C.; Yamagata, H. The Journal of Physical Chemistry B 2010, 115, 5133.