Band Regime versus Hopping Regime

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The band regime and hopping regime are the two limiting systems for transport although there are variations of each.

In band regime the charge carrier (wavefunction) is delocalized over the entire molecule or system.

In hopping regime there is a localized charge carrier on a single molecule and you can follow the movement of the charge from molecule to molecule or chain segment to chain segment. Depending on the geometry and orientation of crystals hopping may result in differing mobilities in different directions or when hopping between layers. The band regime has the highest charge mobility and it is critical to understand the circumstances that encourage it.

Band regime vs Hopping regime

In a band regime the delocalization over the whole system means there is only a probability of finding the charge carrier in any location. This results in coherent transport. In a sense the charge carrier has a “memory” of where it was before.

In the case of hopping the transport is incoherent because each hop is random and has no correlation to the immediately preceding hop.

Electronic coupling is large in the band regime and is the dominant factor. Whereas in the hopping regime coupling of charge carrier with vibrations, caused by geometry relaxations, is the dominant factor. In π conjugated systems with a charge localized somewhere on a chain or a molecule there will be a geometry relaxation around that charge leading to the formation of a phonon. A phonon is a mode of vibration in an ordered solid material or molecular substance that involve then entire lattice.

Geometry relaxations can lead to intramolecular vibrations as well as intermolecular vibrations in which adjacent molecules move with respect to one another.

band regime	hopping regime
delocalization of the charge carrier wavefunctions	localization of the charge carrier wavefunctions
coherent transport	incoherent hops
electronic coupling	electron-phonon coupling- leading to geometry relaxations intramolecular modes intermolecular modes.

See Chemical Reviews 2004

See Chemical Reviews 2007

Requirements for Band Regime

In order to achieve a band regime (which has the largest carrier mobility) you must have a highly ordered structure or single crystal with few impurities. Low temperature is required because vibrations in adjacent coupled molecules due to temperature will decrease the electronic coupling and therefore decrease the mobility. For this reason mobility decreases with increasing temperature. In the case of methyl, which has a high charge carrier density, the decrease in mobility means the conductivity also decreases with temperature. The increase in intermolecular and intramolecular vibrations has generally a negative impact on the mobility of charge carriers.

The second requirement for a band regime is the well ordered structure provides a large electronic coupling between adjacent sites on adjacent molecules or polymer repeat units. In a perfectly ordered polyacetylene the wavefunctions for the charge carriers are delocalized over the whole system. If there are polarons and solitons this means there must not be a perfect crystalline structure because these polaronic structures are associated with a geometry relaxation of a the system. Charge polarons and solitons dissappear when there is complete delocalization. Polysulfur nitride ( (SN)_x ) has crystals with strong interactions between the chains and no distortions. As a result there are no solitons. Normally samples of polyacetylene and transpolyacetylene are somewhat disordered and soliton and polaron behavior is observed.

Evidence for band regime vs hopping regime

Time domain

In the time domain the typical residence time t of a charge carrier on a site in a molecular materials or polymer repeat chain is described by an Heisenberg-like equation:

<math>\tau \approx \frac {\hbar} {W} \rightarrow \tau \approx \frac {2}{3} \frac {10^{-15}} {W_{(eV)}} S\,\!</math>

Where:

<math>\tau\,\!</math> is the residence time on a unit

<math>\hbar\,\!</math> is the period in the Heisenberg uncertainty principle equation

<math>W\,\!</math> is the bandwidth

The wider the bandwidth the larger the electronic coupling between adjacent units.

If W, the conduction bandwidth (for electrons) (and, or) the valance bandwidth (for holes) is large enough (.2-.3 eV) then τ is shorter than 10^-13 s and this makes band transport is possible.

This is because the carrier will not “sit” long enough on any single molecule for that molecule to have time to geometrically relax (relaxaton for C-C requires 20 ms) making band transport possible.

In a simple tight-binding approximation:

<math>W = 4 \cdot t\,\!</math>

where:

<math>t\,\!</math> is the electronic coupling between adjacent sites or adjacent repeating units. T is can also be called the transfer integral or the resonance integral Β(in Hückel terminology), or tunneling matrix element (in physics).

Energy Domain

There are two possibilities from an energy perspective. The charge carrier can fully delocalize along the system in which case there is no geometry relaxation and the energy gained by electronic coupling and charge carrier delocalization is larger than the energy gained by electron-phonon coupling (geometry relaxations around the charge) leading to charge carrier localization and polaron formation. In delocalization π electrons from several isolated orbitals contribute to form a filled lower energy valence band and an empty higher energy conduction band. Thus the energy was lowered by delocalization. If the energy gained the electronic coupling and charge carrier delocalization is larger than the energy gained by electron-phonon coupling (geometry relaxations around the charge) leading to charge carrier localization and polaron formation, then the band regime is favored. In Marcus theory this is called minimizing the reorganization of the energy of the system.

However if more energy is gained by geometry relaxation then the hopping regime is favored.

In molecular materials we distinguish between intramolecular vibrations for instance the stretching of a C-C in a pentacene molecule (which is not not affected by adjacent molecules) and intermolecular vibrations (phonons) which propagate across an entire lattice.

Lattice modes have frequency of 50-200 wavenumbers whereas intramolecular C-C stretch can have wavenumbers of 1500-1600. Phonon modes have 20-50 times less energy than intramolecular modes. As temperature in the system increases the phonon mode of vibration are the first to be excited. The frequency is lower so they are 20-50 times slower than a C-C bond stretch. In addition the higher temperature decreases the electronic coupling and increases localization.

References

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