Disorder potential

In molecular doped polymers the variance of the disorder potential that follows from a plot of In p versus T 2 is typically 0.1 eV, comprising contributions from the interaction of a charge carrier with induced as well as with permanent dipoles [64-66]. In molecules that suffer a major structural relaxation after removal or addition of an electron, the polaron contribution to the activation energy has to be taken into account in addition to the (temperature-dependent) disorder effect. In the weak-field limit it gives rise to an extra Boltzmann factor in the expression for p(T). More generally, Marcus-type rates may have to be invoked for the elementary jump process [67]. 

Fig. 17 Temperature dependence of the hole mobility measured in an FET with (a) pentacene and (b) P3HT as active layers. Parameter Is the gate voltage. Data fitting using the Fishchuk et al. theory in [102] yields values for the mobility and the disorder potential extrapolated to zero electric field and zero carrier concentration. To is the Meyer-Nedel temperature (see text). From [102] with permission. Copyright (2010) by the American Institute of Physics...

The anion structure of the Mott insulator k-(ET)2Cu2(CN)3 34 in Fig. 21 revealed the disorder in the position of C and N atoms of the C=N groups (L2 part Table 6, Fig. 21a) [205-207, 370], due to the existence of an inversion center. However, NMR experiments observed very sharp resonance lines due to the homogeneous local field in the metallic state [371], which suggests that the C/N disorder, if any, does not work as the disorder potential in the conduction layer. 

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In 1958, Anderson [9] showed that localization of electronic wavefunctions occurs if the random component of the disorder potential is large with respect to the bandwidth of the system, as shown in the schematic diagram in Fig. 3.1. The mean free path ( ) in a system with bandwidth B, random potential Vo, and interatomic distance a is given by... 

The ratio Vo/B determines the transition from coherent diffusive propagation of wavefunctions (delocalized states) to the trapping of wavefunctions in random potential fluctuations (localized states). If I > Vo, then the electronic states are extended with large mean free path. By tuning the ratio Vq/B, it is possible to have a continuous transition from extended to localized states in 3D systems, with a critical value for Vq/B. Above this critical value, wave-functions fall off exponentially from site to site and the delocalized states cannot exist any more in the system. The states in band tails are the first to get localized, since these rapidly lose the ability for resonant tunnel transport as the randomness of the disorder potential increases. If Vq/B is just below the critical value, then delocalized states at the band center and localized states in the band tails could coexist. 

Even in typical disordered metals, the classical model for MR breaks down due to quantum corrections to conductivity, especially at low temperatures [13]. In the presence of weak disorder, carriers get localized by repeated back-scattering due to constructive quantum interference, and this is called weak localization (WL). A weak magnetic field can destroy this interference process and delocalize the carrier. As a result, a negative MR (resistivity decreases with field, usually less than 3%) can be observed at temperatures around 4 K. Another quantum correction to low temperature conductivity is due to e-e interaction contributions. This is mainly due to the fact that carriers interact more often when they diffuse slowly in random disorder potentials. The resistivity increases (usually less than 3%) with field due to e-e interaction contributions. Hence, the total low-field magnetoconductance (MC, Act) due to additive contributions from WL and e-e interactions is given by... 

The M-I transition in doped CPs is mainly governed by the extent of disorder, inter-chain interaction and doping level.88 89 It is well known that disorder potentials can localize the electronic states. If the random component of the disorder potential is large with respect to the bandwidth, then the localization of electronic wave functions can occur. In the presence of strong disorder, the overlap of the wave functions drops off exponentially and the system moves towards the insulating regime. 

The intrinsic disorder of the continuous network is less easily classified in terms of defects. The network has many different configurations, but provided the atomic coordination is the same, all these structures are equivalent and represent the natural variability of the material. Since there is no correct position of an atom, one cannot say whether a specific structure is a defect or not. Instead the long range disorder is intrinsic to the amorphous material and is described by a randomly varying disorder potential, whose effect on the electronic structure is summarized in Section 1.2.5. 

Fig. 1.8. The Anderson model of the potential wells for (a) a crystalline lattice and (b) an amorphous network. is the disorder potential.

An increasing disorder potential causes first strong electron scattering and eventually electron localization, in which the wave-function is confined to a smaU volume of material rather than being extended. The form of the localized wavefunction is illustrated in Fig. 

The critical value of VJB for complete localization is about three. Since the band widths are of order 5 eV, a very large disorder potential is needed to localize all the electronic states. It was apparent from early studies of amorphous semiconductors that the Anderson criterion for localization is not met. Amorphous semiconductors have a smaller disorder potential because the short range order restricts the distortions of the bonds. However, even when the disorder of an amorphous semiconductor is insufficient to meet the Anderson criterion, some of the states are localized and these lie at the band edges. The center of the band comprises extended states at which there is strong scattering and... 

The central issue relates to the strength of the electron-electron interactions relative to the bandwidth, relative to the electron-phonon interaction and relative to the strength of the mean disorder potential. Strong electron-electron... 

There is one localized,unpaired spin per TCNQ molecule. This presumably follows from 1. if the disorder is sufficiently great as to give complete localization of the one-electron states to a single site or if one has a Mott-Hubbard metal to insulator transition and is in the strong-coup ling limit. However, as we shall see, one does not necessarily have one unpaired spin per site when the disorder potential and interaction are comparable. 

In this section we shall discuss different models proposed for the transport mechanism in amorphous silicon, concentrating on the implications for tTo(T), A T), and Q(7 ). The framework of our discussion is based on the picture of localized and delocalized states. This goes back to Anderson s pioneering work on localization in a single band. If the disorder potential Vq is comparable to the bandwidth B, aU states of the band will be localized. If Fq < B, localization will set in at the band edges, leaving the states in the center of the band delocalized. 

Mott has shown in 1967 that in an Anderson model for a disordered metal the conductivity cannot be arbitrarily small if E is above E. The reason is that the position of E, the mobility edge, is given by a balance between the electronic overlap / of wave functions centered at adjacent atomic sites and the disorder potential Fq. For a mobility edge the former energy is a definite fraction of the latter and hence the lower limit of the conductivity is always the minimum metallic conductivity Onjin — 200 D cm" , the value depending somewhat on assumptions about unknown constants. This concept... 

P. Kruger, S. Wildermuth, S. Hofi erberth, L.M. Andersson, S. Groth, I. Bar-Joseph, et al.. Cold atoms close to surfaces Measuring magnetic field roughness and disorder potentials, J. Phys. Conf. Ser. 19 (2005) 56. 

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The recent experimental confirmation of the existence of one-dimensional metallic systems has led to a rapid increase in the experimental and theoretical study of these conducting systems. The objective of this section is to acquaint the reader with the physical basis of the concepts currently being used to explain the experimental results. Emphasis is given to the development of one electron band theory because of its central importance in the description of metals and understanding the effects of lattice distortion (Peierls transition), electron correlation, disorder potentials, and interruptions in the strands. It... 

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