Hopping semiconductors

TABLE 7.1 Electrical Conductivity and Mobility of Charge Carriers in Metals, Band-like Semiconductors, and Hopping Semiconductors... 

Metal Band-like Semiconductor Hopping Semiconductor... 

A hopping semiconductor such as an oxide is often best described by Eq. (7.1) rather than by classical band theory. In these materials the conductivity increases with temperature because of the exponential term, which is due to an increase in the successful number of jumps, that is, the mobility, as the temperature rises. Moreover, the concentration of charge carriers increases as the degree of nonstoichiometry increases. 

A comparison of the relevant equations for metals, band theory semiconductors, and hopping semiconductors is given in Table 7.1. These equations can be used in a diagnostic fashion to separate one material type from another. In practice, it is not quite so easy to distinguish between the different conductivity mechanisms. 

Although these materials are often best described as hopping semiconductors in which the mobility is proportional to the exponential of the energy required to liberate the charge carriers, the conduction mechanism in devices is often complex. For application, mechanisms ate sometimes neglected in favor of empirical relationships. 

Insulators or hopping semiconductors. Paramagnetism characteristic off configuration. EuO shows nonmetal-metal transition. 

As is to be expected, inherent disorder has an effect on electronic and optical properties of amorphous semiconductors providing for distinct differences between them and the crystalline semiconductors. The inherent disorder provides for localized as well as nonlocalized states within the same band such that a critical energy, can be defined by distinguishing the two types of states (4). At E = E, the mean free path of the electron is on the order of the interatomic distance and the wave function fluctuates randomly such that the quantum number, k, is no longer vaHd. For E < E the wave functions are localized and for E > E they are nonlocalized. For E > E the motion of the electron is diffusive and the extended state mobiHty is approximately 10 cm /sV. For U <, conduction takes place by hopping from one localized site to the next. Hence, at U =, )J. goes through a... 

A celebrated derivation of the temperature dependence of the mobility within the hopping model was made by Miller and Abrahams 22. They first evaluated the hopping rate y,y, that is the probability that an electron at site i jumps to site j. Their evaluation was made in the case of a lightly doped semiconductor at a very low temperature. The localized states are shallow impurity levels their energy stands in a narrow range, so that even at low temperatures, an electron at one site can easily find a phonon to jump to the nearest site. The hopping rate is given by... 

We note a temperature dependence of the zero field mobility as exp[—( F()/F)2], a behavior which is indeed encountered in real organic semiconductors, and differs from both Millers-Abrahams fixed range and Moll s variable range hopping models. 

The electronic band structure of a neutral polyacetylene is characterized by an empty band gap, like in other intrinsic semiconductors. Defect sites (solitons, polarons, bipolarons) can be regarded as electronic states within the band gap. The conduction in low-doped poly acetylene is attributed mainly to the transport of solitons within and between chains, as described by the intersoliton-hopping model (IHM) . Polarons and bipolarons are important charge carriers at higher doping levels and with polymers other than polyacetylene. 

The above mechanistic aspect of electron transport in electroactive polymer films has been an active and chemically rich research topic (13-18) in polymer coated electrodes. We have called (19) the process "redox conduction", since it is a non-ohmic form of electrical conductivity that is intrinsically different from that in metals or semiconductors. Some of the special characteristics of redox conductivity are non-linear current-voltage relations and a narrow band of conductivity centered around electrode potentials that yield the necessary mixture of oxidized and reduced states of the redox sites in the polymer (mixed valent form). Electron hopping in redox conductivity is obviously also peculiar to polymers whose sites comprise spatially localized electronic states. 

Yu, D. Wang, C. Wehrenberg, B. L. Guyot-Sionnest, P. 2004. Variable range hopping conduction in semiconductor nanocrystal solids. Phys. Rev. Lett. 92 216802-216806. 

Author has introduced the discovery of mechanolysis, a novel phenomenon of water splitting [9,10], which has been understood as a result of frictional electricity between the Teflon stirring rod and the Pyrex glass of the beaker, where pure water containing semiconductor powder is filled. Author [9] has pointed opt that the semiconductor must have the property of the hopping conductivity, and called tribolysis. There exists another type of mechanolysis, which may be due to the piezo electrolysis. This type is called piezolysis, but not discovered yet. 

There are a number of ways in which hopping conductivity can be treated theoretically. In semiconductor... 

Unlike a geometrical factor, the value of the factor

with composition in a predictable way. To illustrate this, suppose that stoichiometric MO2 is heated in a vacuum so that it loses oxygen. Initially, all cations are in the M4+ state, and we expect the material to be an insulator. Removal of O2- to the gas phase as oxygen causes electrons to be left in the crystal, which will be localized on cation sites to produce some M3+ cations. The oxide now has a few M3+ cations in the M4+ matrix, and thermal energy should allow electrons to hop from M3+ to M4+. Thus, the oxide should be an n-type semiconductor. The conductivity increases until

reduction continues, eventually almost all the ions will be in the M3+ state and only a few M4+ cations will remain. In this condition it is convenient to imagine holes hopping from site to site and the material will be a p-typc semiconductor. Eventually at x = 1.5, all cations will be in the M3+ state and M2C>3 is an insulator (Fig. 7.3). 

Ion hopping is a familiar concept in the chemistry of solid-state conductors, e.g. in the semiconductor industry. In the fluoride electrode, fluoride vacancies in.side the solid LaF lattice allow for conduction of charge (see Figure 3.11), in turn registered by the electrode as a potential. The emf is zero if the internal and external solutions are the same because the same numbers of fluoride ion enter the crystal from either face. 

This same equation is, of course, also used to rationalise the general electronic behaviour of metals, semiconductors and insulators. The quantitative application of Eqn (2.1) is handicapped for ionic conductors by the great difficulty in obtaining independent estimates of c,- and u,-. Hall effect measurements can be used with electronic conductors to provide a means of separating c, and u,- but the Hall voltages associated with ionic conduction are at the nanovolt level and are generally too small to measure with any confidence. Furthermore, the validity of Hall measurements on hopping conductors is in doubt. 

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