Conducting polymers

Conventionally, polymers are considered as good insulators, which are usually used to insulate good electrical conductors such as metals. As research moves along, it was found that the electrical conductivity of polymers can be possibly tuned over a range from insulating to metallic. Conducting polymers have received substantial attention in tech-nological applications. The significant breakthrough in the development of conducting polymers occurs in 1977 when a doping process was applied to an intrinsically insula-tor, polyacetylene (PA). This polymer has an intrinsic electrical conductivity lower than 10⁻⁵ Ω⁻¹ cm⁻¹. After doping with oxidizing or reducing agents the electrical conductivity

can reach values up to 10³ Ω⁻¹cm⁻¹[7]. Due to this important discovery, the Nobel Prize was awarded to Shirakawa, MacDiarmid and Heeger in 2000. For conducting polymers, a conjugated π-electron system with alternating single and multiple bonds is required. A region of overlapping p-orbitals allows the delocalized π electrons cross all the adjacent aligned p-orbitals. In this way, the π-electrons belong to a group of atoms rather than a single atom. From an energetic perspective, bonding π- and antibonding π^∗- bands form as a consequence of the overlap of π orbitals. The π- and π^∗- bands are referred as the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), respectively. If the system is in the ground state, all of the π-electrons are located in the HUMO and leaving the LUMO empty. However, electrons can be excited into higher energetic states and then occupy states in the LUMO.

Band structure and charge carriers

Figure 2.5: Schematic illustration of Peierl’s instability theorem: band structures and sketches of a) an undistorted 1D lattice with lattice spacing a, and b) a distorted 1D lattice with distortion of δ and periodicity of 2a.

The origin of the band structure of conjugated polymers can be described by Peierl’s instability. In this theorem, a one dimensional (1D) crystal with constant lattice spacing a is assumed. If each atom contributes one electron, the band is half-filled up to the Fermi level E_f as shown in figure 2.5a. However, in case of conducting polymers such as polyacetylene, the backbone is conjugated, the alternation of sigma and double bonds leads to a new periodicity of the 1D crystal since they differ in length (figure 2.5b). In order to lower the total energy of the system, the Brioullin zone is broken at the position

of k = ±π/2ain case of the scheme described in figure 2.5b and a new band gap ∆E_gap is formed. In polymers, the values of ∆E_gap matters strongly with structure, degree of polymerization and doping. Typically, the Peierl’s instability for polymers gives rise to band gaps ranging from 1.5 eV to 3 eV [29]. For example, P3HT is a widely used conducting polymer with a band gap of 1.9 eV - 2 eV.

In conducting polymers, charge carriers are generally generated by doping or by light excitation. Unlike electrons or holes in inorganic systems, charge carriers are transported within polymers by quasiparticles (QPs), which combine the charges and lattice distortion.

Three primary quasiparticles, termed as solitons, polarons and bipolarons, are present in systems consisting of degenerate ground states. Taking PA as an example, an overview of these three quasiparticles is schematically described in figure 2.6. Solitons have three

Figure 2.6: Schematic illustrations of a) solitons, b) polarons, and c) bipolarons in a con-ducting polymer PA. Solitons exist in the form of S⁰, S⁺ and S⁻. Positive and negative polarons are represented as P⁺ and P⁻. Bipolarons have two positive B⁺ or two negative charges B⁻. The three red short lines in a), four blue short lines in b) and four green short lined in c) represent the energetic band states of all quasiparticles which stay within the band gap E_g.

variation (figure 2.6a). A neutral soliton (S⁰) with no charge and a spin of 1/2 is created when two degenerate ground states meet. When an electron is taken away from the polymer chain, a positive charge is left behind, a positive soliton (S⁺) with a spin of 0 is generated. Similarly a negative soliton (S⁻) is produced by donating an electron to the polymer chain. Since no unpaired spins exist, the spin of S⁻ is 0. A polaron can be thought as a bound state of a neutral soliton and a charged soliton (figure 2.6b).

Therefore, polarons are charged and have a spin of 1/2. While bipolarons are charged and spinless (figure 2.6c). Bipolarons are formed out of 2 polarons.

Charge transport

Figure 2.7: Schematic illustration of hopping transport of a charge carrier in a conducting polymer. In left, black and blue curves indicate density of states distribution and charge density distribution, respectively. Right, dashed lines represent localized energetic states.

A generated charge hops downwards accompanied with relaxation, and hops upwards with prerequisite of obtaining external energy. The middle dotted line indicates the transport energy. Adapted from reference [30].

The transport of charge carriers in polymer systems is distinct from band transport found in inorganic crystalline semiconductors. The aforementioned QPs are able to move freely along the chain backbone due to the existence of a conjugated system of overlap-pingπ-orbitals. However, the so-called hopping transport occurs when the charge carriers transfer to neighboring chains. This is because of spatial and energetic disorders in con-ducting polymers. The Gaussian disorder model has been developed to illustrate the hopping mechanism, as schematically shown in figure 2.7. A Gaussian density of states with width σ is used to describe the energetic disorder of hopping sites. Under steady-state conditions, the charge density lies below the center of density of steady-states by a thermal activation energy –σ²/kBT. A charge is generated at a high energetic site (figure 2.7) and then hops to a neighboring state via tunneling. The neighboring state has a lower energy, thus the hopping process is accompanied with a relaxation. When the charge reaches the area between the centers of the charge density and the density of states, it can potentially be thermalized. If the charge gains the thermal energy from the system, it is able to hop towards energetically higher states. If not, it will hop down until it reaches a trap state.

In the former process, there will be more accessible states for charge carriers if a higher temperature is applied to the system. For the latter case, the trap state makes charge carriers immobile, thereby they do not contribute to the electrical conductivity. It is noteworthy that, the mobility of charge carriers is much higher in crystalline regions than in amorphous regions. For example, the mobility of positive charge carriers in the P3HT crystalline regions can reach 0.1 cm²/Vs, whereas lower than 10⁻⁵ cm²/Vs in amorphous regions [16,17]. But in general, charge carriers travel slower in polymers than in metals.