How do semi-conducting polymers work?

Overview

Conducting polymers (a.k.a. conjugated polymers) are a unique class of materials because they possess the electronic properties of metals or semi-conductors, with the advantageous processing and mechanical characteristics of polymers.  Either of these properties by itself is not very special.  Metals and semi-conductors exhibit exceptional electronic properties that would be difficult to improve upon.  Similarly, there are a huge variety of polymers with better (much better) processability and mechanical properties than conducting polymers.  It is the unique duality of metal-like conductivity and polymer-like processability that conductive polymers become an exceptional class of materials.

The most important potential applications of these materials is in the production of large-scale electronic materials at low temperatures.  This offers the promise of low-cost manufacturing with electronic inks containing conducting and metallic polymers.  In this case, we can utilize existing advanced printing technologies (i.e. ink-jet printing) that, when combined with continuing developments in conducting polymers, have potential to revolutionize the electronic device manufacturing industry.

Current commercial realizations of conductive polymers include organic light-emitting diodes, organic solar cells, organic transistors, organic photo-detectors, and organic biosensors.

 

Electronic Structure

Consider a linear chain of chain atoms, where each carbon atom has a total of 4 valence electrons.  These electrons are distributed throughout the s- and p-orbitals.  If the potential is spherically symmetric, then the 1s and 2s orbitals of the carbon will be filled, which leaves the 2p orbital with only 2 electrons.  In order to minimize the free energy during bonding, this will lead to either (1) tetrahedrally directed covalent bonds, such as in insulating materials like diamond and saturated polymers, or (2) hexagonally directed covalent bonds, which are conductive and characteristic of graphite and conjugated polymers.

Covalent bonds with directed orbitals form by borrowing an electron from the 2s orbital of a carbon and subsequently combining the s- and p- wave functions to form a linearly combined hybridized orbital.  The important point here is that the s- orbital is spherically symmetric, whereas the p-orbital is directed and has three orthogonal components, px, py, and pz, which represent its particular coordinate axis.

Now let's start wtih the tetrahedral configuration.  This consists of sp3 hybrdized carbon atoms in which all valence electrons are bound in covalent bonds and, therefore, are unable to participate in electronic transport.  Polymers with this backbone structure will be referred to as saturated polymers, and are characteristically flexible and easily processed by extrusion or melt-processing.  This is the structure of most polymers you are familiar with, and may exhibit remarkable strength if sufficiently well-aligned.

Structure of polyethylene consists of directed tetrahedral configuration of sp3 hybridized carbon bonds along repeat structure with no free valence electrons.  The four valence electrons are saturated: each carbon is bonded with two other carbon atoms and two hydrogen atoms.

Structure of polyethylene consists of directed tetrahedral configuration of sp3 hybridized carbon bonds along repeat structure with no free valence electrons.  The four valence electrons are saturated: each carbon is bonded with two other carbon atoms and two hydrogen atoms.

Conductive polymers are formed from hexagonally directed orbitals which are sp2pz hybridized.  In other words, the pz orbital is available to form a π-bond containing delocalized electrons.  Polyacetylene is the simplest example of a conjugated polymer.  It has 3 in-plane, σ-orbitals that form its backbone, with two electrons bonded to neighboring carbons and a third bonded to a hydrogen atom.  The fourth electron resides in the pz orbital, and is essentially independent of the three σ-bonds.  This "independence" results from its orthogonality to the plane formed by the 3 σ-bonds.  The pz-electron can therefore be referred to as a delocalized electron, and is generally referred to as the π-electron, which resides in the π-orbital (pz orbital).

 

Structure of polyacetylene consists of planar backbone of sp2 hybridized sigma bonds, where three sigma orbitals are in-plane, and the fourth pz-orbital is orthogonal to the backbone plane and delocalized.

Structure of polyacetylene consists of planar backbone of sp2 hybridized sigma bonds, where three sigma orbitals are in-plane, and the fourth pz-orbital is orthogonal to the backbone plane and delocalized.

In other words, the pz-electron is decoupled  from the backbone of σ-orbitals, which gives the polymer its special electronic properties.  In particular, the σ-electrons define the covalent bonds along the backbone of the polymer, and are at a lower energy state than the π-electrons.  Therefore, the π-electron in a particular carbon essentially thinks it is part of a lithium atom, which means that it is able to freely move.  This π-electron is attracted to the nuclei of neighboring carbon atoms, which results in the same delocalization phenomena observed in Li-metal and, hence, polyacetylene should essentially behave as a metal.  In reality, it is a semi-conductor and can become conducting, and even metallic, with doping.

The electronic structure of π-conjugated polymers contains 2 main types of energy structures: 1) energy bands that originate from the bonding and anti-bonding energy levels of the σ-orbitals, and 2) a π-band that originates from the delocalized pz wave function.

The polymer chains themselves are held together by the covalent σ-bonds along their backbone.  In traditional polymers, any excitations from this band cause structural instability by promoting electrons into the anti-bonding σ*-band orbital.  In this case, degradation will occur because the electrons in the σ-bonding orbitals are what hold the polymer chain together!  This is a useful property for photoresists, but is responsible for the rather bland electronic and optical properties of saturated polymers.  In π-conjugated polymers, excitations within the π-band do not typically lead to bond breaking, and account for the useful photo-conductive properties of π-conjugated polymers as opposed to the photo-resistive properties of saturated polymers.

 

Coupling Bond Length to Electronic Structure

If we assume that polyacetylene has a uniform bond length, then there will be a single π-band which accomodates 2N electrons, where N is the number of atoms in the chain.  The factor of 2 accounts for electron spin.  Since each carbon atom contributes one pz-electron, the π-band will be half-filled and polyacetylene with a uniform bond-length should behave as a metal!

This, of course, is not the case.  The first model of the electronic structure of π-conjugated polymers was the Su-Schrieffer-Heeger Hamiltonian, which assumed that 1) the π-electronic structure can be treated with a tight-binding approximation and transfer (hopping) interactions, and 2) the chain of carbon atoms is coupled to the local electron density  through the length of the chemical bonds.  The first assumption defines the lowest order transfer energy and accounts for the delocalization of the π-electrons along the chain.  The second assumption is essentially a first-order correction factor and provides terms related to the lattice degrees of freedom; namely, a harmonic spring constant representing the increase in potential energy due to the deviation from the uniform bond length assumption, and a kinetic energy term for nuclear motion.  The two lattice terms are the origin of the non-linear excitations of the polymer chain : solitons, polarons, and bipolarons.  They are coupled to the electronic structure of the polymer via the bond length dependent transfer term.

The transfer energy arises from the attractive interaction between an electron on a carbon atom at a site i along the chain, and the nucleus of the carbon atom at site i +/- 1.  The magnitude of the transfer energy is roughly proportional to the negative ionization potential of the carbon atom, which gives a relatively large implied overlap of the wave functions and reveals the strong tendency of the electrons to delocalize along the conjugated chain of carbon atoms.

The magnitude of the transfer interaction is strongly dependent on the distance between successive carbon atoms.  The SSH Hamiltonian model accounts for the coupling effect of the electronic structure to the molecular structure (i.e. it states that the magnitude of the transfer interaction depends on bond length), which is given by the bond-length dependent hopping interaction between sites.

Bond-length dependence of transfer energy is an essential feature of the description of conjugated polymers since the ground state bond alternation makes the hopping interaction across double bonds larger than the corresponding hopping interaction across single bonds.

Even though the π-electrons in polyacetylene are delocalized along the chain, polyacetylene is not a metal.  The polymerization of polyacetylene from the monomer acetylene yields a dimerized structure, which means that it has an alternating single and double bonded backbone.  The resulting polymer is insoluble and intractable, hence its molecular weight cannot be directly determined!  Since its degree of polymerization is characteristically unknown, polyacetylene is typically designated as (CH)x.

The molecular structure of real polyacetylene consists of alternating single and double bonds which are, respectively, longer and shorter than the equilibrium value of bond length in uniform (metallic) (CH)x.  In this structure, π-electrons on neighboring carbon atoms form weaker and stronger π-π bonds, which results in the bond alternating structure.

The bond alternating doubles the size of the unit cell, which thereby introduces a gap in the electronic structure.  Therefore, the semi-conducting nature of polyacetylene is a direct consequence of the bond-alternating structure!  In other words, all the states in the π-band (valence band) are filled, and all the states in the π*-band (conduction band) are empty.

High enough carrier densities can be reached via doping to enable semi-conducting polymers to reach a metallic state.  The metallic state of doped conjugated polymers is stabilized by interchain interactions sufficiently strong to result in a system which is effectively an anisotropic three-dimensional metal (quasi-1D metal).

In general, conjugated polymers are defined as bond-alternating linear chains of carbon atoms in the sp2pz configuration.  The number of π-bands in the electronic structure of a conjugated polymer is determined by the number of atoms in the repeat unit.  If there are n atoms in the main chain of the repeat unit, the π-band is divided into n subbands.

Chemical structure of PPV - Poly(phenylene vinylene)Poly(phenylene vinylene) (PPV) is an alternating copolymer with repeat units of polyacetylene and poly(paraphenylene).  PPV and its soluble derivatives are the prototypical luminescent semi-conducting polymers.  There are eight carbon atoms in the repeat unit, hence the pi-band is split into eight subbands.  Each subband can hold two electrons per atom.  Thus, four lowest subbands are filled (π-subbands) and the four highest subbands are empty (π*-subbands).  Consequently, PPV is not metallic, as there are no partially filled bands.

 

The energy difference between the highest occupied pi-subband and the lowest unoccupied pi*-subband is the π-π* energy gap of the polymer.


Problems with energy band theory for conjugated polymers:

  1. In the solid state, all macromolecules have some degree of disorder.  This causes localization of electronic states, band tailing, and general broadening of the optical transitions.  Due to this disorder, even heavily doped polymers may behave as insulators rather than metals.  This can be mitigated by using chain extended and chain aligned polymers with structures which approach those of idealized ordered materials.  For example, by gel-processing blends of cunjugated polymers in polyethylene, a high degree of structural order may be obtained.
  2. Two-fold coordination of linear conjugated polymers and the bond-length proportionality to bond-strength in conjugated systems results in a fundamental coupling between electronic and chemical structures.  As a result, the elementary excitations in semi-conducting polymers are not the same as in classic inorganic semi-conductors.  In particular, the electrons and holes that we are familiar with as charge carriers must also involve lattice relaxation terms.  This lattice relaxation in the excited state is a geometric effects which results in self-localization and is responsible for the formation of solitons, polarons, and bipolarons.
  3. The basic band description ignores correlations between electrons which originate from electron-electron repulsive interactions.

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