QUE HACE SI SE ROMPE UN TUBO O AHORRADOR
VIDA ÚTIL DE CADA UNA DE LAS ILUMINARIAS.
3.4.2. Sistema de iluminación LED:
where F is the electric field, n is the number of charges, pi are the steady state
populations and Rij is the distance between the centroids of the states i and j. The
contribution of each pair of states to the total mobility is represented by the particle current k pij i(1−pj).
5.2
Results and discussion
5.2.1
Qualitative features of the model
Before considering the mobility results it is convenient to analyze the distribution and characteristics of the electronic states generated by the Hamiltonian in (5.1) with the choice of parameters we considered. Figure 5.1 reports the density of states (DOS), the localization length (LL) and the inverse participation ratio (IPR), all scaled by the number of monomers, for different amounts of off-diagonal disorder. The DOS is defined as usual in terms of the eigenvalues Ej of
el
H and the Dirac delta function as:
DOS( ) ( j)
j
E =
E−E (5.21)The localization length of an individual state is computed as
(
)
1/ 2 2 2 1/ 2 2 2 LL 2 2 j j j j nj j n j n r r r r c = − = −
(5.22)and the energy-dependent localization length is defined as
LL( ) LLj ( j) ( j)
j j
103 The inverse participation ratio for an individual state is an alternative measure of localization and indicates the average number of monomers that share the orbital. It is defined as 1 4 IPRj nj n c − =
(5.24)and, for a convenient representation, it is useful to consider its energy dependent counterpart:
IPR( ) IPRj ( j) ( j)
j j
E =
E−E
E−E (5.25) The data in Figure 5.1 derive from 10 repeated diagonalizations of the Hamiltonian with different realizations of random variables and with the Dirac delta function approximated with a Gaussian of width 0.10.In the absence of disorder (upper panel of Figure 5.1) all states are naturally delocalized and the DOS has the shape familiar for 1D chains (in this limit the hopping mechanism cannot be assumed). The introduction of disorder causes the localization of all states according to Anderson theory[95] and a broadening of the DOS. The localization is not uniform at all energies as the states at the DOS edge are more localized.[83,254] For intermediate disorder, states with a broad range of localization become accessible thermally and the increase in charge carrier density is accompanied by an increase of the localization length at the Fermi level. The presence of this energy dependent localization, together with the inclusion of a microscopic hopping rate expression, is the main innovation of this model with respect to analytical models.
104
Figure 5.1. Density of states, localization length and inverse participation ratio (defined in eqs. (5.21)-(5.25)) of the eigenstates of the electronic Hamiltonian for different levels of off- diagonal disorder .
We consider next a qualitative visualization of the results that will help the following quantitative analysis. In Figure 5.2 we represent, for three levels of disorder
= 0.08, 0.10 and 0.12 0, the spatial distribution of the eigenstates of Hel, their energy and their localization. We have represented only the states observed on a small portion of the polymer chain. One can observe the increase in the density of states with increasing energy and the existence of more delocalized states at higher energy. In Figure 5.2, the particle currents k pij i(1−pj) that are not negligible in magnitude are represented by coloured lines connecting states i and j and, as expected, one observes that particle current (hopping) is appreciable only between states that are relatively close in space. The picture at high disorder is not too different from the
105 assumptions made in most variable range hopping models (all states are localized). However, at lower disorder more long-range hopping events are observed because the system contains relatively delocalized states that can be accessed from many initial states. These delocalized states act as intermediates for long range displacements of the charge and they are clearly not considered in standard variable range hopping models, while they consistently appear in atomistic calculations of the orbitals in polymers.[110,254]
Figure 5.2. Diagram of the spatial distribution, energy and localization of the electronic eigenstates of the system. Only the energy states in the band tail are shown and only a portion of the chain is considered. The length of the chain is L = 1024∙d. Each state is represented by a horizontal segment line whose width is the localization length, and whose central coordinate represents the energy of this state and the position along the chain on the vertical and horizontal axes respectively. The coloured arrows connecting the horizontal segments represent the particle currents k pij i(1−pj) between states, obtained from the steady state
solution of the master equation in the limit of weak electrical field (F = 100 V/cm) and low charge density
(
n =5)
. For clarity, only the particle currents which make up the 99.99% of the total particle current, calculated as ij i (1 j) j j(1 i)ijk p −p +k pi −p
, are shown. The red arrows represent jumps shorter than 0.025 L.To better appreciate the timescale for the hopping events depicted in Figure 5.2 and their dependence on some of the system parameters, we have reported in Figure 5.3 the distribution of the particle currents at a fixed disorder for different temperatures and reorganization energies 1 of the monomer. Particle currents are
106
and, on a logarithmic scale, they are only slightly smaller (up to ~10 times) than the corresponding hopping rate. The fastest particle currents for any reasonable polymer described by our model are in the 109-1011 s-1 range, i.e. the description in terms of incoherent hopping is acceptable as well as the separability between static (slower than 109 s-1) and dynamic (faster than 1011 s-1) disorder.
Figure 5.3. Distribution of the particle currents (k pij i (1−pj)) (for a system of 4000 monomers and /0=0.12) for various system parameters. In the top panel 1 = 0.45 eV and in the bottom panel T = 300 K. Note the logarithmic scale on both axes.